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[ [ "Dynamical generation of wormholes with charged fluids in quadratic\n Palatini gravity" ], [ "Abstract The dynamical generation of wormholes within an extension of General Relativity (GR) containing (Planck's scale-suppressed) Ricci-squared terms is considered.", "The theory is formulated assuming the metric and connection to be independent (Palatini formalism) and is probed using a charged null fluid as a matter source.", "This has the following effect: starting from Minkowski space, when the flux is active the metric becomes a charged Vaidya-type one, and once the flux is switched off the metric settles down into a static configuration such that far from the Planck scale the geometry is virtually indistinguishable from that of the standard Reissner-Nordstr\\\"om solution of GR.", "However, the innermost region undergoes significant changes, as the GR singularity is generically replaced by a wormhole structure.", "Such a structure becomes completely regular for a certain charge-to-mass ratio.", "Moreover, the nontrivial topology of the wormhole allows to define a charge in terms of lines of force trapped in the topology such that the density of lines flowing across the wormhole throat becomes a universal constant.", "To the light of our results we comment on the physical significance of curvature divergences in this theory and the topology change issue, which support the view that space-time could have a foam-like microstructure pervaded by wormholes generated by quantum gravitational effects." ], [ "Introduction", "The Vaidya metric [1] $ds^2=-\\left[ 1-\\frac{2m(v)}{r} \\right] dv^2 +2\\epsilon \\,dv dr +r^2 d \\Omega ^2,$ is a nonstatic spherically symmetric solution of the Einstein equations generated by a null stream of radiation.", "Depending on $\\epsilon =+1(-1)$ it corresponds to an ingoing (outgoing) radial flow and $m(v)$ is a monotonically increasing (decreasing) function in the advanced (retarded) time coordinate $-\\infty <v<+\\infty $ .", "Both the Vaidya solution and its extension to the charged case, the Bonnor-Vaidya solution [2], have been widely employed in a variety of physical situations, including the spherically symmetric collapse and the formation of singularities [3], the study of Hawking radiation and black hole evaporation [4], the gravitational collapse of charged fluids (plasma) [5] or as a testing tool for various formulations of the cosmic censorship conjecture.", "In addition to this, several theorems on the existence of exact spherically symmetric dynamical black hole solutions have been established [6].", "In the context of modified gravity, Vaidya-type solutions have been found in metric $f(R)$ gravity coupled to both Maxwell and non-abelian Yang-Mills fields [7] and in Lovelock gravity [8].", "The Vaidya metric has also been used to consider whether a wormhole could be generated out of null fluids.", "More specifically, in [9] a crossflow of a two-component radiation was considered, and the resulting solution was interpreted as a wormhole (this analysis extended the results presented in [10]).", "It was indeed shown that a black hole could be converted into a wormhole by irradiating the black-hole horizon with pure phantom radiation, which may cause a black hole with two horizons to merge and consequently form a wormhole.", "Conversely, switching off the radiation causes the wormhole to collapse to a Schwarzschild black hole [11].", "These results were further extended in [12] showing that two opposite streams of radiation may support a static traversable wormhole.", "Furthermore, analytic solutions describing wormhole enlargement were presented, where the amount of enlargement was shown to be controlled by the beaming in and the timing of negative-energy and positive-energy impulses.", "It was also argued that the wormhole enlargement is not a runaway inflation, but an apparently stable process.", "The latter issue addressed the important point that though wormholes were possible, and even expected at the Planck scale, macroscopic wormholes were unlikely.", "In fact, the generation/construction of wormholes has also been extensively explored in the literature, in different contexts.", "The late-time cosmic accelerated expansion implies that its large-scale evolution involves a mysterious cosmological dark energy, which may possibly lie in the phantom regime, i.e., the dark energy parameter satisfies $w<-1$ [13].", "Now, phantom energy violates the null energy condition, and as this is the fundamental ingredient to sustain traversable wormholes [14], this cosmic fluid presents us with a natural scenario for the existence of these exotic geometries [15].", "Indeed, due to the fact of the accelerating Universe, one may argue that macroscopic wormholes could naturally be grown from the submicroscopic constructions, which envisage transient wormholes at the Planck scale that originally pervaded the quantum foam [16], much in the spirit of the inflationary scenario [17].", "It is also interesting to note that self-inflating wormholes were also discovered numerically [12].", "In the context of dark energy, and in a rather speculative scenario, one may also consider the existence of compact time-dependent dark energy stars/spheres [18], with an evolving dark energy parameter crossing the phantom divide [19].", "Once in the phantom regime, the null energy condition is violated, which physically implies that the negative radial pressure exceeds the energy density.", "Therefore, an enormous negative pressure in the center may, in principle, imply a topology change, consequently opening up a tunnel and converting the dark energy star into a wormhole.", "The criteria for this topology change were also discussed, in particular, a Casimir energy approach involving quasi-local energy difference calculations that may reflect or measure the occurrence of a topology change.", "As the Planck scale plays a fundamental importance in quantum gravitational physics, an outstanding question is whether large metric fluctuations may induce a change in topology.", "Wheeler suggested that at distances below the Planck length, the metric fluctuations become highly nonlinear and strongly interacting, and thus endow space-time with a foamlike structure [16].", "This behaviour implies that the geometry, and the topology, may be constantly fluctuating, and thus space-time may take on all manners of nontrivial topological structures, such as wormholes.", "However, paging through the literature, one does encounter a certain amount of criticism to Wheeler's notion of space-time foam, for instance, in that stability considerations may place constraints on the nature or even existence of Planck-scale foamlike structures [20].", "Indeed, the change in topology of spacelike sections is an extremely problematic issue, and a number of interesting theorems may be found in the literature on the classical evolution of general relativistic space-times [21], namely, citing Visser [22]: (i) In causally well-behaved classical space-times the topology of space does not change as a function of time; (ii) In causally ill-behaved classical space-times the topology of space can sometimes change.", "Nevertheless, researchers in quantum gravity have come to accept the notion of space-time foam, in that this picture leads to topology-changing quantum amplitudes and to interference effects between different space-time topologies [22], although these possibilities have met with some disagreement [23].", "Despite the fact that topology-changing processes, such as the creation of wormholes and baby universes, are tightly constrained [24], this still allows very interesting geometrical (rather than topological) effects, such as the shrinking of certain regions of space-time to umbilical cords of sufficiently small sizes to effectively mimic a change in topology.", "Recently, the possibility that quantum fluctuations induce a topology change, was also explored in the context of Gravity's Rainbow [25].", "A semi-classical approach was adopted, where the graviton one-loop contribution to a classical energy in a background space-time was computed through a variational approach with Gaussian trial wave functionals [26] (note that the latter approach is very close to the gravitational geon considered by Anderson and Brill [27], where the relevant difference lies in the averaging procedure).", "The energy density of the graviton one-loop contribution, or equivalently the background space-time, was then let to evolve, and consequently the classical energy was determined.", "More specifically, the background metric was fixed to be Minkowskian in the equation governing the quantum fluctuations, which behaves essentially as a backreaction equation, and the quantum fluctuations were let to evolve; the classical energy, which depends on the evolved metric functions, is then evaluated.", "Analyzing this procedure, a natural ultraviolet (UV) cutoff was obtained, which forbids the presence of an interior space-time region, and may result in a multiply-connected space-time.", "Thus, in the context of Gravity's Rainbow, this process may be interpreted as a change in topology, and in principle results in the presence of a Planckian wormhole.", "In this work, we consider the dynamical generation of wormholes in a quadratic gravity theory depending on the invariants $R=g_{\\mu \\nu }R^{\\mu \\nu }$ and $Q=R_{\\mu \\nu }R^{\\mu \\nu }$ , which are Planck scale-suppressed [see Eq.", "(REF ) below for details].", "This theory is formulated a la Palatini, which means that the metric and connection are regarded as independent entities.", "Though in the case of General Relativity (GR) this formulation is equivalent to the standard metric approach (where the connection is imposed a priori to be given by the Christoffel symbols of the metric) this is not so for modified gravity.", "Interestingly, the Palatini formulation yields second-order field equations that in vacuum boil down to those of GR and, consequently, are ghost-free, as opposed to the usual shortcomings that plague the metric formulation.", "To probe the dynamics of our theory, in a series of papers [28], [29] we have studied spherically symmetric black holes with electric charge.", "As a result we have found electrovacuum solutions that macroscopically are in excellent qualitative agreement with the standard Reissner-Nordström solution of GR, but undergo important modifications in their innermost structure.", "Indeed, the GR singularity is generically replaced by a wormhole structure with a throat radius of order $r_c \\sim l_P$ .", "The behaviour of the curvature invariants at $r_c$ shows that for a particular charge-to-mass ratio the space-time is completely regular.", "The topologically non-trivial character of the wormhole allows us to define the electric charge in terms of lines of electric force trapped in the topology, such that the density of lines of force is given by a universal quantity (independent of the specific amounts of mass and charge).", "These facts allow to consistently interpret these solutions as geons in Wheeler's sense [16] and raise the question on the true meaning of curvature divergences in our theory since their existence seems to pose no obstacle for the wormhole extension.", "Let us note that these wormhole solutions correspond to static solutions of the field equations.", "Here we shall see that such solutions can be dynamically generated by probing the Minkowski space with a charged null fluid.", "In this way we obtain a charged Vaidya-type metric such that when the flux is switched off, the space-time settles down into a Reissner-Nordström-like configuration containing a wormhole structure and thus a multiply-connected topology in its interior.", "As we shall see, these results have important consequences for the issue of the foam-like structure of space-time.", "This work largely extends the results and discussion of [30].", "This paper is organized in the following manner: In Sec.", ", we present the Palatini formalism for Ricci-squared theories that are used throughout the paper.", "In Sec.", ", we consider general electrovacuum scenarios with a charged null fluid, and in Sec.", "we solve the gravitational field equations.", "In Sec.", ", we analyze the different contributions to the metric and discuss some particular scenarios.", "A discussion on the physical implications of these results follows in Sec.", ", where we conclude with a brief summary and some future perspectives." ], [ "Palatini formalism for Ricci-squared theories", "Our initial setup corresponds to that of a generic Palatini Lagrangian coupled to matter, defined by the following action $S[g,\\Gamma ,\\psi _m]=\\frac{1}{2\\kappa ^2}\\int d^4x \\sqrt{-g}f(R,Q) +S_m[g,\\psi _m] \\ ,$ where $\\mathcal {L}_G=f(R,Q)/(2\\kappa ^2)$ represents the gravity Lagrangian, $\\kappa ^2$ is a constant with suitable dimensions (in GR, $\\kappa ^2 \\equiv 8\\pi G$ ), $g_{\\mu \\nu }$ is the space-time metric, $R=g^{\\mu \\nu }R_{\\mu \\nu }$ , $Q=g^{\\mu \\alpha }g^{\\nu \\beta }R_{\\mu \\nu }R_{\\alpha \\beta }$ , $R_{\\mu \\nu }={R^\\rho }_{\\mu \\rho \\nu }$ and ${R^\\alpha }_{\\beta \\mu \\nu }=\\partial _{\\mu }\\Gamma ^{\\alpha }_{\\nu \\beta }-\\partial _{\\nu }\\Gamma ^{\\alpha }_{\\mu \\beta }+\\Gamma ^{\\alpha }_{\\mu \\lambda }\\Gamma ^{\\lambda }_{\\nu \\beta }-\\Gamma ^{\\alpha }_{\\nu \\lambda }\\Gamma ^{\\lambda }_{\\mu \\beta } \\,,$ is the Riemann tensor constructed by the connection $\\Gamma \\equiv \\Gamma ^{\\lambda }_{\\mu \\nu }$ .", "The term $S_m[g,\\psi _m]$ represents the matter action, where $\\psi _m$ are the matter fields, to be specified later.", "To obtain the field equations from the action (REF ), in the Palatini approach one assumes that the connection $\\Gamma _{\\mu \\nu }^{\\lambda }$ , which defines the affine structure, is a priori independent of the metric, which defines the chrono-geometric structure (see [31] for a pedagogical discussion).", "This approaches reduces the number of assumptions on the structure of spacetime beyond GR, and has important consequences for the dynamics of the theory, as we shall see later.", "The variational principle thus leads to two sets of field equations resulting from the variation of (REF ) with respect to metric and connection as $f_R R_{\\mu \\nu }-\\frac{f}{2}g_{\\mu \\nu }+2f_QR_{\\mu \\alpha }{R^\\alpha }_\\nu &=& \\kappa ^2 T_{\\mu \\nu }\\\\\\nabla _{\\beta }^\\Gamma \\left[\\sqrt{-g}\\left(f_R g^{\\mu \\nu }+2f_Q R^{\\mu \\nu }\\right)\\right]&=&0 \\ ,$ respectively.", "In deriving these field equations, for simplicity, we have set the torsion to zero and assumed $R_{[\\mu \\nu ]}=0$ , which guarantees the existence of invariant volumes in our theory [32].", "The connection equation () can be solved by means of algebraic manipulations, which are described in a number of previous works [33], [28], [29].", "One thus finds that Eq.", "() can be written as $ \\nabla _{\\beta }^\\Gamma [\\sqrt{-h} h^{\\mu \\nu }]=0 \\ ,$ with $h_{\\mu \\nu }$ defined as $ h^{\\mu \\nu }=\\frac{g^{\\mu \\alpha }{\\Sigma _{\\alpha }}^\\nu }{\\sqrt{\\det \\hat{\\Sigma }}} \\ , \\quad h_{\\mu \\nu }=\\left(\\sqrt{\\det \\hat{\\Sigma }}\\right){{\\Sigma ^{-1}}_{\\mu }}^{\\alpha }g_{\\alpha \\nu } \\ ,$ where $ {\\Sigma _\\alpha }^{\\nu }=\\left(f_R \\delta _{\\alpha }^{\\nu } +2f_Q {P_\\alpha }^{\\nu }\\right) \\ ,$ and ${P_\\mu }^\\nu \\equiv R_{\\mu \\alpha }g^{\\alpha \\nu }$ .", "It is easy to verify from Eq.", "(REF ) that $\\Gamma _{\\mu \\nu }^{\\lambda }$ can be written as the Levi-Civita connection of the (auxiliary) metric $h_{\\mu \\nu }$ .", "It can be shown that $h_{\\mu \\nu }$ is algebraically related to $g_{\\mu \\nu }$ and the stress-energy tensor of matter.", "In fact, in terms of the object ${P_\\mu }^\\nu $ , we can write Eq.", "(REF ) as $f_R {P_\\mu }^\\nu -\\frac{f}{2}{\\delta _\\mu }^\\nu +2f_Q{P_\\mu }^\\alpha {P_\\alpha }^\\nu = \\kappa ^2 {T_\\mu }^\\nu \\,,$ or, in matrix form, as (here a hat denotes a matrix) $2f_Q\\hat{P}^2+f_R \\hat{P}-\\frac{f}{2}\\hat{I} = \\kappa ^2 \\hat{T} \\ ,$ which represents a quadratic algebraic equation for ${P_\\mu }^\\nu $ as a function of ${T_\\mu }^\\nu $ .", "This implies that $R={[\\hat{P}]_\\mu }^\\mu $ , $Q={[\\hat{P}^2]_\\mu }^\\mu $ , and ${\\Sigma _\\alpha }^{\\nu }$ are just functions of the matter sources.", "Using the definition of ${\\Sigma _\\mu }^\\nu $ and the relations (REF ), we can write Eq.", "(REF ) [or, alternatively, Eq.", "(REF )] as ${P_\\mu }^\\alpha {\\Sigma _\\alpha }^\\nu =R_{\\mu \\alpha }h^{\\alpha \\nu } \\sqrt{\\det \\hat{\\Sigma }}=\\frac{f}{2}{\\delta _\\mu ^\\nu }+\\kappa ^2{T_\\mu }^\\nu \\,,$ which allows to express the metric field equations using $h_{\\mu \\nu }$ as follows $ {R_{\\mu }}^{\\nu }(h)=\\frac{\\kappa ^2}{\\sqrt{\\det \\hat{\\Sigma }}}\\left(\\mathcal {L}_G\\delta _{\\mu }^{\\nu }+ {T_\\mu }^{\\nu } \\right) \\ .$ This representation of the metric field equations puts forward that $h_{\\mu \\nu }$ satisfies a set of GR-like second-order field equations.", "Since $h_{\\mu \\nu }$ and $g_{\\mu \\nu }$ are algebraically related, it follows that $g_{\\mu \\nu }$ also verifies second-order equations.", "Additionally, we note that in vacuum, $\\hat{T}=0$ , implies that $\\hat{P}$ can be written as $\\hat{P}=\\Lambda (R^{vac},Q^{vac}_S)\\hat{I}$ , where the explicit form of $\\Lambda (R^{vac},Q^{vac}_S)$ can be found straightforwardly from Eq.", "(REF ).", "However, this is not essential for the current discussion.", "We note that the relations $R^{vac}={P_{\\mu }}^{\\mu }=4\\Lambda (R^{vac},Q^{vac})$ , and $Q^{vac}={[P^2]_{\\mu }}^{\\mu }=4\\Lambda ^2(R^{vac},Q^{vac})$ imply that the values of $R^{vac}$ and $Q^{vac}$ that simultaneously solve Eq.", "(REF ) are constant and are related by $Q^{vac}=(R^{vac})^2/4$ .", "In addition, $\\hat{P}=\\Lambda (R^{vac},Q^{vac})\\hat{I}$ also implies that $h_{\\mu \\nu }$ and $g_{\\mu \\nu }$ are related by a constant conformal factor [see Eqs.", "(REF ) and (REF )].", "As a result, Eq.", "(REF ) tells us that $R_{\\mu \\nu }(h)=C^{vac}h_{\\mu \\nu } \\ \\leftrightarrow \\ R_{\\mu \\nu }(g)=\\tilde{C}^{vac}g_{\\mu \\nu } $ , with $C^{vac}$ and $\\tilde{C}^{vac}$ constant (and identical in an appropriate system of units).", "This shows that the vacuum field equations of Palatini theories of the form (REF ) coincide with the vacuum Einstein equations with a cosmological constant (whose magnitude depends on the particular gravity Lagrangian $\\mathcal {L}_G$ ), which is a manifestation of the observed universality of the Einstein equations in the Palatini formalism [34].", "These theories, therefore, do not introduce any new propagating degrees of freedom besides the standard massless spin-2 gravitons, and are free from the ghost-like instabilities present in the (higher-derivative) metric formulation of four-dimensional theories containing Ricci-squared terms." ], [ "Electrovacuum scenarios with a charged null fluid", "In this section, we will consider the problem of a spherically symmetric charged space-time perturbed by an ingoing null flux of energy and charge.", "The electromagnetic field is described by the free Maxwell action plus a coupling to an external current $J^\\mu $ , $S_{em}=-\\frac{1}{16\\pi } \\int d^4x \\sqrt{-g} F_{\\mu \\nu }F^{\\mu \\nu }-\\int d^4x \\sqrt{-g}A_\\mu J^\\mu ,$ where $F_{\\mu \\nu }=\\partial _{\\mu }A_{\\nu }-\\partial _{\\nu }A_{\\mu }$ is the field strength tensor of the vector potential $A_{\\mu }$ .", "The Maxwell stress-energy tensor is obtained as $T_{\\mu \\nu }^{em}=\\frac{1}{4\\pi }\\left[F_{\\mu \\alpha }{F_{\\nu }}^\\alpha -\\frac{1}{4}F_{\\alpha \\beta }F^{\\alpha \\beta } g_{\\mu \\nu }\\right] \\ .$ On the other hand, the pressureless flux of ingoing charged matter has a stress-energy tensor $T_{\\mu \\nu }^{flux}= \\rho _{in} k_\\mu k_\\nu \\,,$ where $k_{\\mu }$ is a null vector, satisfying $k_{\\mu }k^{\\mu }=0$ , and $\\rho _{in}$ is the energy density of the flux.", "In order to write the field equations (REF ) in combination with the matter source given by Eqs.", "(REF ) and (REF ) in a form amenable to calculations, we need first to obtain the explicit expression of $Q$ .", "To do this we note that Eq.", "(REF ) can also be written as $ 2f_Q\\left(\\hat{P}+\\frac{f_R}{4f_Q}\\hat{I}\\right)^2=\\left(\\frac{f}{2}+\\frac{f_R^2}{8f_Q}\\right)\\hat{I}+\\kappa ^2 \\hat{T} \\ .$ In order to obtain an explicit expression for ${P_\\mu }^{\\nu }$ , we need to compute the square root of the right-hand side of this equation.", "To this effect, let us assume a line element of the form $ds^2=-A(x,v) e^{2\\psi (x,v)}dv^2\\pm 2e^{\\psi (x,v)}dv dx+r^2(v,x)d\\Omega ^2 \\ ,$ where $-\\infty <v< +\\infty $ is an Eddington-Finkelstein-like null ingoing coordinate (outgoing if the minus sign is chosen) and $x$ a radial coordinate.", "Note that $r^2$ is not a coordinate but a function, in general.", "For this line element we find that a suitable null tetrad is given by $k_\\mu &=&(-1,0,0,0), \\\\l_\\mu &=&\\left(-\\frac{A}{2} e^{2\\psi (x,v)},\\pm e^{\\psi (x,v)},0,0\\right) \\,, $ $m_\\mu &=&\\left(0,0,\\frac{r}{\\sqrt{2}},\\frac{ir\\sin \\theta }{\\sqrt{2}}\\right)\\,, \\\\\\bar{m}_\\mu &=&\\left(0,0,\\frac{r}{\\sqrt{2}},-\\frac{i r\\sin \\theta }{\\sqrt{2}}\\right) $ and its dual yields $k^\\mu &=&\\left(0,\\mp e^{-\\psi (x,v)},0,0\\right) \\,, \\\\l^\\mu &=&\\left(1,\\pm \\frac{A e^{\\psi (x,v)}}{2},0,0 \\right) \\,, $ $m^\\mu &=&\\left(0,0,\\frac{1}{r\\sqrt{2}},\\frac{i}{\\sqrt{2}r\\sin \\theta }\\right) \\,, \\\\\\bar{m}^\\mu &=&\\left(0,0,\\frac{1}{r\\sqrt{2}},-\\frac{i}{\\sqrt{2}r\\sin \\theta } \\right) \\,, $ respectively.", "Thus, in this representation the only non-vanishing products are $k^\\mu l_\\mu =-1$ and $m^\\mu \\bar{m}_\\mu =1$ ." ], [ "The matter field equations", "With the above null tetrad, the stress-energy tensor (REF ) for a spherically symmetric non-null electromagnetic field can be expressed as [35] $ T_{\\mu \\nu }^{em}=\\chi \\left(m_\\mu \\bar{m}_\\nu +m_\\nu \\bar{m}_\\mu +k_\\mu l_\\nu +k_\\nu l_\\mu \\right) \\ ,$ where the form of $\\chi (v)$ can be obtained by solving explicitly the field equations for the Maxwell field and comparing with Eq.", "(REF ) written in matrix representation [see Eq.", "(REF ) below].", "From the action (REF ) and taking into account the presence of the null fluid (REF ) the Maxwell equations read $\\nabla _\\mu F^{\\mu \\nu }=4\\pi J^\\nu \\,,$ where $J^\\nu \\equiv \\Omega (v) k^\\nu $ is the current of the null ingoing flux, with $\\Omega (v)$ a function to be determined.", "With this expression and knowing that $\\nabla _\\mu F^{\\mu \\nu }\\equiv \\frac{1}{\\sqrt{-g}}\\partial _\\mu \\left(\\sqrt{-g} F^{\\mu \\nu }\\right)$ , the only non-trivial equations are $\\partial _x\\left(\\sqrt{-g} F^{xv}\\right)&=&0 \\,, \\\\\\partial _v\\left(\\sqrt{-g} F^{vx}\\right)&=&-4\\pi \\sqrt{-g} e^{-\\psi (x,v)} \\Omega \\,, $ From Eq.", "(REF ) we find that $r^2 e^{\\psi (x,v)}F^{xv}=q(v)$ , where $q(v)$ is an integration function.", "Inserting this back in Eq.", "(), it follows that $\\Omega (v)=q_v/4\\pi r^2$ .", "Note that the function $q_v\\equiv \\partial _v q(v)$ is our input and, therefore, can be freely specified.", "Having defined the current that gives consistency to the Maxwell field equations, we can compute explicitly the form of $T_{\\mu \\nu }^{em}$ in Eq.", "(REF ) to obtain $\\chi $ , which yields $\\chi =\\frac{q^2(v)}{8\\pi r^4}\\,.", "$" ], [ "Energy-momentum conservation", "To verify that charge and momentum are conserved in our model we consider the following equation $\\nabla _\\mu {T^\\mu }_{\\nu , em}= J^\\alpha F_{\\nu \\alpha }+\\frac{1}{4\\pi }\\left[F^{\\mu \\alpha } \\nabla _\\mu F_{\\nu \\alpha }-\\frac{1}{2}F^{\\alpha \\beta }\\nabla _\\nu F_{\\alpha \\beta }\\right] \\ ,$ where $\\nabla _\\mu $ is the derivative operator of the metric $g_{\\alpha \\beta }$ .", "Now we use the Bianchi identities $\\nabla _{[\\nu }F_{\\alpha \\beta ]}=0$ to express $F^{\\alpha \\beta }\\nabla _\\nu F_{\\alpha \\beta }=-F^{\\alpha \\beta }(\\nabla _\\beta F_{\\nu \\alpha }+\\nabla _\\alpha F_{\\beta \\nu })=2F^{\\beta \\alpha }\\nabla _\\beta F_{\\nu \\alpha } \\ .$ Inserting this result in Eq.", "(REF ) we find $\\nabla _\\mu {T^\\mu }_{\\nu , em}=J^{\\alpha } F_{\\nu \\alpha }$ .On the other hand, the null fluid yields $\\nabla _\\mu {T^{\\mu }}_{\\nu , \\ flux}=k_\\nu \\nabla _\\mu \\left(\\rho _{in}k^\\mu \\right)+\\rho _{in}k^\\mu \\nabla _\\mu k_\\nu \\ .$ One can see by direct calculation that $k^\\mu \\nabla _\\mu k_\\nu =0$ , which verifies that $k_\\mu $ is a geodesic vector.", "Since $\\nabla _\\mu {T^{\\mu }}_{\\nu , \\ Total}=\\nabla _\\mu {T^{\\mu }}_{\\nu ,em} + \\nabla _\\mu {T^{\\mu }}_{\\nu ,flux}=0$ , contracting with $l_\\nu $ we find $\\partial _\\mu \\left(\\sqrt{-g}\\rho _{in}k^\\mu \\right)+\\sqrt{-g}F^{\\mu \\nu } l_\\mu J_\\nu =0 \\ ,$ which becomes $\\partial _x\\left(\\rho _{in}r^2\\right)=e^{\\psi (x,v)} \\frac{q q_v}{4\\pi r^2} \\ .$ This condition should be satisfied by the solution of the problem on consistency grounds.", "We note that an analogous procedure for the derivation of these expressions can be carried out when a magnetic field, and consequently, a magnetic flux, is present.", "Thereby, the expressions in the previous subsections can be trivially extended to those of the magnetic case by just swapping the electric charge $q$ with a magnetic charge $g$ ." ], [ "The ${\\Sigma _{\\mu }}^{\\nu }$ matrix.", "With the representation of the stress-energy tensor given by Eq.", "(REF ), we can proceed to obtain the square root of Eq.", "(REF ) for the ingoing flux of charged null matter represented by the stress-energy tensor in Eq.", "(REF ).", "To do this, we identify the right-hand side of Eq.", "(REF ) with the squared matrix $ {M_a}^c M{_c}^b &=& \\lambda \\delta _a^b + \\kappa ^2\\chi (m_a\\bar{m}^b + m^b \\bar{m}_a +k_a l^b+k^b l_a) \\nonumber \\\\ &&+ \\kappa ^2 \\rho _{in} k_a k^b \\ ,$ where $\\lambda \\equiv \\frac{f}{2} + \\frac{f_R^2}{8f_Q}$ .", "We now propose the ansatz $ {M_a}^b&=&\\alpha \\delta _a^b + \\beta (m_a \\bar{m}_b + m^b \\bar{m}_a)+\\gamma (k_al^b+k^bl_a) \\nonumber \\\\ && + \\delta k_a k^b + \\epsilon l_al^b,$ where $\\alpha , \\beta , \\gamma , \\delta , \\epsilon $ are functions to be determined by matching the right-hand side of Eq.", "(REF ) with the square ${M_a}^c M{_c}^b$ using the ansatz (REF ).", "This leads to the set of equations $\\alpha ^2 = \\lambda ; \\quad \\beta (\\beta +2\\alpha )&=&\\kappa ^2 \\chi ; \\quad \\gamma (2\\alpha -\\gamma )-\\delta \\epsilon = \\kappa ^2 \\chi \\nonumber \\\\2\\delta (\\alpha -\\gamma )= \\kappa ^2 \\rho _{in} ; && \\quad 2\\epsilon (\\alpha -\\gamma )=0$ whose unique consistent solution is $&&\\alpha = \\lambda ^{1/2} \\,;\\qquad \\beta =-\\lambda ^{1/2} + \\sqrt{\\lambda + \\kappa ^2 \\chi } \\nonumber \\\\&& \\hspace{-11.38092pt} \\gamma =\\lambda ^{1/2}-\\sqrt{\\lambda - \\kappa ^2 \\chi } \\,;\\quad \\delta =\\frac{\\kappa ^2 \\rho _{in}}{2\\sqrt{\\lambda - \\kappa ^2 \\chi }} \\,;\\quad \\epsilon =0 \\,.", "$ With these results, we can use the expression $\\hat{M}=\\sqrt{2f_Q} \\left(\\hat{P} + \\frac{f_R}{4f_Q} \\hat{I} \\right) \\,,$ to write the matrix ${\\Sigma _{\\mu }}^{\\nu }$ defined in Eq.", "(REF ) as ${\\Sigma _{\\mu }}^{\\nu }=\\frac{f_R}{2}{\\delta _{\\mu }}^{\\nu } + \\sqrt{2f_Q}{M_{\\mu }}^{\\nu } \\,,$ which finally becomes ${\\Sigma _{\\mu }}^{\\nu }&=& \\left(\\frac{f_R}{2}+ \\sqrt{2f_Q} \\lambda ^{1/2} \\right)\\delta _{\\mu }^{\\nu }\\nonumber \\\\&&+ \\sqrt{2f_Q} \\Big [\\beta (m_{\\mu } \\bar{m}^{\\nu } +m^{\\nu } \\bar{m}_{\\mu })\\nonumber \\\\&& + \\gamma (k_{\\mu }l^{\\nu }+ k^{\\nu } l_{\\mu }) + \\delta k_{\\mu }k^{\\nu }\\Big ] \\,.$ Note that this expression only depends on the electromagnetic, $\\chi $ , and fluid, $\\rho _{in}$ , parameters and on the Lagrangian density $f(R,Q)$ ." ], [ "The $f(R,Q)$ Lagrangian", "In what follows we shall only be concerned with the quadratic Lagrangian $ f(R,Q)=R+l_P^2 (a R^2+bQ) \\ ,$ where $l_P^2 \\equiv \\hbar G/c^3$ represents Planck's length squared; $a$ and $b$ are dimensionless constants.", "The physical reason underlying this models is the fact that quadratic corrections of the form above arise in the quantization of fields in curved space-time [36] and also in the low-energy limits of string theories [37], [38].", "It has also been argued that this kind of models are natural in an effective field theory approach to quantum gravity [39].", "Palatini Lagrangians with quadratic and higher-order curvature corrections also arise in effective descriptions [40] of the dynamics of loop quantum cosmology [41], a scenario in which the big bang singularity is replaced by a cosmic bounce.", "Alternatively, one could use other models of Palatini gravity, such as the proposal of Deser and Gibbons [42], dubbed Eddington-inspired Born-Infeld gravity [43], which in the static electrovacuum case is exactly equivalent (not only perturbatively) to the quadratic Lagrangian (REF ), as shown in [44].", "As a working hypothesis, we shall assume that neither the quadratic Lagrangian (REF ), nor the perturbation induced by the flux will spoil the geometrical nature of gravitation as we approach the scale where the $l_P^2$ effects in (REF ) begin to play an important role.", "Tracing in Eq.", "(REF ) with the metric $g^{\\mu \\nu }$ it follows that $R=-k^2 T$ , where $T$ is the trace of the stress-energy tensor.", "For the electric field we are considering, one has $T=0$ , which implies $R=0$ .", "As a result, the dependence of the Lagrangian on the parameter $a$ becomes irrelevant.", "Note, however, that for nonlinear theories of electrodynamics (for which $T \\ne 0$ ), the parameter $a$ does play a role [45], [46].", "From now on we consider the case $b>0$ and, for simplicity, set $b=1$ .", "To obtain the expression for $Q$ we take the trace of ${M_a}^{b}$ and use the tetrad relations to write $ \\frac{1}{\\sqrt{2}l_P}=\\sqrt{\\lambda +\\kappa ^2 \\chi } - \\sqrt{\\lambda -\\kappa ^2 \\chi } \\,.$ For our theory given by Eq.", "(REF ), we have $f=l_P^2 Q$ , $f_R=1, f_Q=l_P^2$ and then from Eq.", "(REF ) we obtain $\\lambda =\\frac{1}{8l_P^2}(1+4l_P^2 Q)$ .", "Inserting this in Eq.", "(REF ), we obtain the solution $Q=4\\kappa ^4 \\chi ^{2}=\\frac{\\tilde{\\kappa }^4q^4}{r^8} \\,,$ where $\\tilde{\\kappa }^2=\\kappa ^2/(4\\pi )$ .", "We note that this expression remains unchanged if the null fluid is absent [47].", "To obtain the field equations (REF ) for the theory given by Eq.", "(REF ) we need both the explicit expression of ${\\Sigma _{\\mu }}^{\\nu }$ and of $(\\mathcal {L}_G\\delta _{\\mu }^{\\nu }+ {T_{\\mu }}^{\\nu })$ appearing on the right-hand side of Eq.", "(REF ).", "From the tetrad definitions (REF )-() it is easily seen that $ m_{\\mu }\\bar{m}^{\\nu }+m^{\\nu }\\bar{m}_{\\mu }=\\left(\\begin{array}{cc}\\hat{0}& \\hat{0} \\\\\\hat{0} & \\hat{I} \\\\\\end{array}\\right) \\,,\\\\\\hspace{2.84544pt}k_{\\mu }l^{\\nu }+l^{\\nu }k_{\\mu }=\\left(\\begin{array}{cc}-\\hat{I}& \\hat{0} \\\\\\hat{0} & \\hat{0} \\\\\\end{array}\\right)\\,,$ where $\\hat{I}$ and $\\hat{0}$ are the $2 \\times 2$ identity and zero matrices, respectively.", "In this formalism, we immediately recover the usual expression for a spherically symmetric electromagnetic field, namely, $T_{\\mu \\nu }^{em}=\\chi \\, \\text{diag}(-1,-1,1,1)$ .", "On the other hand, for the null fluid contribution we have $ k_{\\mu }k^{\\nu }=\\left(\\begin{array}{cccc}0 & e^{-\\psi } & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\\\end{array}\\right) \\ .$ Taking into account all these elements, one readily finds that $ {\\Sigma _{\\mu }}^{\\nu }=\\left(\\begin{array}{cccc}\\sigma _- & \\sigma _{in} & 0 & 0 \\\\0 & \\sigma _- & 0 & 0 \\\\0 & 0 & \\sigma _+ & 0 \\\\0 & 0 & 0 & \\sigma _+ \\\\\\end{array}\\right) ,$ where $\\sigma _{\\pm }&=&1\\pm \\frac{\\tilde{\\kappa }^2 l_P^2 q^2(v)}{r^4} \\,, \\\\\\sigma _{in}&=&\\frac{2\\kappa ^2 l_P^2 \\rho _{in}}{1-2\\tilde{\\kappa }^2 l_P^2 q^2(v)/r^4} \\,.", "$ With all these results, we can finally write the field equations (REF ) for our problem as ${R_{\\mu }}^{\\nu }(h)=\\left(\\begin{array}{cccc}-\\frac{\\tilde{\\kappa }^2q^2(v)}{2r^4\\sigma _+} & \\frac{e^{-\\psi }\\kappa ^2\\rho _{in}}{\\sigma _+\\sigma _-} & 0 & 0 \\\\0 & -\\frac{\\tilde{\\kappa }^2q^2(v)}{2r^4\\sigma _+} & 0 & 0 \\\\0& 0& \\frac{\\tilde{\\kappa }^2q^2(v)}{2r^4\\sigma _-} & 0 \\\\0& 0& 0 & \\frac{\\tilde{\\kappa }^2q^2(v)}{2r^4\\sigma _-}\\end{array}\\right) .$ Note that in the limit $l_P \\rightarrow 0$ , we have $\\sigma _{\\pm } \\rightarrow 1$ and $\\sigma _{in}\\rightarrow 0$ , which entails $h_{\\mu \\nu }=g_{\\mu \\nu }$ and Eq.", "(REF ) recovers the equations of GR.", "Having obtained the field equations in the form of Eq.", "(REF ), we now proceed to solve them as follows.", "Firstly, we propose a spherically symmetric line element associated to the metric $h_{\\mu \\nu }$ following the structure given in Eq.", "(REF ), namely $d\\tilde{s}^2&=& -F(v,x)e^{2\\xi (v,x)}dv^2+2e^{\\xi (v,x)}dv dx \\nonumber \\\\&&+ \\tilde{r}^2(v,x)d\\Omega ^2 \\,.", "$ Direct comparison of $g_{\\mu \\nu }$ and $h_{\\mu \\nu }$ using $\\hat{\\Sigma }$ implies that $g_{vv}&=&\\frac{h_{vv}}{\\sigma _+}+\\frac{\\sigma _{in} h_{vx}}{\\sigma _+\\sigma _-} \\,, \\\\g_{vx}&=& \\frac{h_{vx}}{\\sigma _+} \\,,$ which leads to $e^\\psi =\\frac{e^\\xi }{\\sigma _+}$ and $\\tilde{r}^2=r^2\\sigma _-$ .", "Note that given the $v$ -dependence of $q(v)$ and the $(x,v)$ -dependence of $\\sigma _\\pm $ in Eq.", "(REF ), it is reasonable to expect a priori some $v$ -dependence on $\\tilde{r}$ [as we have assumed in Eq.", "(REF )].", "In fact, from the relation $\\tilde{r}^2=r^2\\sigma _-$ , one finds $r^2=\\frac{\\tilde{r}^2+\\sqrt{\\tilde{r}^4+4l_P^2 \\tilde{\\kappa }^2q^2(v)}}{2} \\ ,$ which establishes a non-trivial relation between $r,\\tilde{r}$ , and $q(v)$ .", "For this reason, we have not used $\\tilde{r}$ as a variable and have kept the independent coordinate $x$ in the non-spherical sector of the line element (REF ).", "The explicit relation between $\\tilde{r}, x$ and $v$ must thus follow from the field equations.", "From the line element (REF ) we obtain, using the algebraic manipulation package xAct [48], ${R_x}^v\\equiv \\frac{2e^{-\\xi }(\\tilde{r}_x \\xi _x-\\tilde{r}_{xx})}{\\tilde{r}}=0 \\ .$ This implies that $e^{\\xi (x,v)}=C(v)\\tilde{r}_x$ , and inserting the latter in Eq.", "(REF ), yields $ d\\tilde{s}^2= -F(v,x)\\tilde{r}_x^2dv^2+2\\tilde{r}_xdv dx+\\tilde{r}^2(v,x)d\\Omega ^2 \\ .$ where the function $C(v)$ has been reabsorbed into a redefinition of $v$ .", "Working now with the ansatz (REF ), we get ${R_x}^v\\equiv -\\frac{2\\tilde{r}_{xx}}{\\tilde{r}}=0 \\ ,$ which implies that $\\tilde{r}=\\alpha (v)x+\\beta (v) \\ ,$ where $\\alpha (v)$ and $\\beta (v)$ are, so far, two arbitrary functions.", "Assuming that $F(x,v)=1-2M(x,v)/\\tilde{r}$ , we obtain $R_{vx}&\\equiv &\\frac{1}{\\tilde{r}}\\left[M_{xx}-2\\alpha _v\\right]={R_v}^v={R_x}^x \\,,\\\\{R_\\theta }^\\theta &\\equiv & \\frac{1}{\\tilde{r}^2}\\left[1-\\alpha ^2-2\\beta \\alpha _v-2\\alpha [\\beta _v+(2x\\alpha _v-M_x)]\\right] \\,,\\\\R_{vv}&\\equiv &-\\frac{1}{\\tilde{r}^2}\\left[(\\tilde{r}-2M)M_{xx}+2\\tilde{r}\\tilde{r}_{vv}+2\\tilde{r}_vM_x-2\\alpha M_v\\right] \\,.$ From the first of these equations, we find $M_x=\\frac{\\tilde{\\kappa }^2q^2}{4\\alpha r^2}+2\\alpha _v x \\ .$ Inserting this result in ${R_\\theta }^\\theta $ and performing some manipulations, one obtains $1-\\alpha ^2=\\partial _v(\\alpha \\beta ) \\ .$ A consistent solution of this equation is $\\beta =0$ , and $\\alpha =1$ , which implies that $\\tilde{r}=x$ is independent of $v$ .", "Assuming this solution from now on, we find that ${R_v}^x\\equiv R_{vv}+FR_{vx}=\\frac{2M_v}{\\tilde{r}^2} \\ .$ Since ${R_v}^x=\\frac{\\kappa ^2\\rho _{in}}{\\sigma _-}$ , the above relation implies $M_v=\\frac{\\kappa ^2\\rho _{in} r^2}{2} \\,.$ With the above results, and using the relations $\\partial _v(q^2/r^2)\\big \|_{x}=2q q_v/r^2\\sigma _+$ (at constant $x$ ) and $dr\\big \|_v=\\frac{\\sigma _{-}^{1/2}}{\\sigma _{+}}dx$ (at constant $v$ ), one can show that the integrability condition $\\partial _v M_x=\\partial _x M_v$ implies the conservation equation (REF ) (where the relation (REF ), which leads to $\\sqrt{x^4+4l_P^2 \\tilde{\\kappa }^2q^2}=r^2\\sigma _+$ , must be used).", "From Eq.", "(REF ), with $\\alpha =1$ , by direct integration we find $M(x,v)=\\frac{\\tilde{\\kappa }^2q^2(v)}{4}\\int \\frac{dx}{r^2}+\\gamma (v) \\ .$ Computing $M_v$ from this expression and comparing with Eq.", "(REF ), we find (recall that $\\partial _v(q^2/r^2)\\big \|_{x}=2q q_v/r^2\\sigma _+$ ) $\\gamma _v=\\frac{{\\kappa }^2}{2}\\left(\\rho _{in} r^2-\\frac{q q_v}{4\\pi }\\int \\frac{dx}{r^2\\sigma _+}\\right) \\ .$ Since $\\gamma =\\gamma (v)$ , defining $L(v) \\equiv \\gamma _v$ as the luminosity function, it follows that $\\rho _{in} r^2=\\frac{2}{\\kappa ^2} \\left[L(v) + \\frac{\\kappa ^2 q q_v}{8\\pi }\\int \\frac{dx}{r^2\\sigma _+} \\right] \\,, $ which is fully consistent with the conservation equation (REF ) because Eq.", "() and the subsequent manipulations imply $e^\\psi =1/\\sigma _+$ .", "In summary, we conclude that $\\tilde{r}(x,v)&=&x \\ , \\\\r^2(x,v)&=&\\frac{x^2+\\sqrt{x^4+4l_P^2 \\tilde{\\kappa }^2q^2(v)}}{2} \\ , \\\\F(x,v)&=&1-\\frac{2M(x,v)}{x} \\,, \\\\M(x,v)&=& M_0+\\gamma (v)+\\frac{\\tilde{\\kappa }^2q^2(v)}{4}\\int \\frac{dx}{r^2} \\,, \\\\\\gamma (v)&=& \\int dv L(v) \\,,\\\\\\rho _{in}&=&\\frac{2}{\\kappa ^2 r^2} \\left[L(v) + \\frac{\\kappa ^2 q q_v}{8\\pi }\\int \\frac{dx}{r^2\\sigma _+} \\right] \\,.$ This set of equations provides a consistent solution to the Palatini $f(R,Q)$ Lagrangian (REF ) with null and non-null electromagnetic fields satisfying $\\nabla _\\mu F^{\\mu \\nu }=\\frac{q_v}{4\\pi r^2}k^\\nu $ , where $q_v\\equiv \\partial _v q(v)$ and $\\gamma _v\\equiv L(v)$ are free functions.", "Their dependence on $v$ reflects the presence of the charged stream of null particles.", "Given the structure of the mass function in Eq.", "() and to make contact with previous results on static configurations, we find it useful to write it as $M(x,v)= M_0+\\gamma (v) +\\frac{r_q(v)^2}{4r_c(v)}\\left(\\int dz G_z\\right)\\Big \|_{z=r/r_c} \\ ,$ with $r_c(v)=\\sqrt{r_q(v)l_P}$ , $z(x,v)=r(x,v)/r_c(v)$ and $G_z=\\frac{z^4+1}{z^4 \\sqrt{z^4-1}} \\,,$ where we have used the relation $dr/dx=\\sigma _{-}^{1/2}/\\sigma _{+}$ (at constant $v$ ).", "This can be expressed in a more compact form as $M(x,v)=M(v)\\left[1+\\delta _1(v) G\\left(z\\right)\\right]\\big \|_{z=\\frac{r}{r_c}}\\ ,$ where $M(v)=M_0+\\gamma (v)\\equiv r_S(v)/2 $ and $ \\delta _1(v)=\\frac{1}{2r_S(v)} \\sqrt{\\frac{r_q^3(v)}{l_P}} \\ .$ The function $G(z)$ can be written as an infinite power series and its form was given in [28].", "Using these results, we can write $g_{vv}$ in Eq.", "(REF ) as $ g_{vv}=-\\frac{F(x,v)}{\\sigma _{+}} + \\frac{2l_P^2 \\kappa ^2 \\rho _{in}}{\\sigma _{-}(1-\\frac{2r_c^4}{r^4})} \\,,$ where (recall that $z=r(x,v)/r_c(v)$ ) $F(x,v)=1-\\frac{1+\\delta _1 (v) G(z)}{\\delta _2(v) z \\sigma _{-}^{1/2}} \\,,$ and we have introduced the parameter $ \\delta _2(v)=\\frac{r_c(v)}{r_S(v)} \\,.$ Using the above results, the line element (REF ) becomes $ds^2&=&-\\left[\\frac{1}{\\sigma _+}\\left(1-\\frac{1+\\delta _1 (v) G(z)}{\\delta _2(v) z \\sigma _{-}^{1/2}}\\right)- \\frac{2l_P^2 \\kappa ^2 \\rho _{in}}{\\sigma _{-}(1-\\frac{2r_c^4}{r^4})}\\right]dv^2\\nonumber \\\\&&+ \\frac{2}{\\sigma _+}dvdx+r^2(x,v) d\\Omega ^2 \\ , $ Equation (REF ) with the definitions given by Eqs.", "(REF ), (REF ) and (REF ), and the function $G_z$ in Eq.", "(REF ), which contains the contribution of the non-null electromagnetic field, constitutes the main result of this paper." ], [ "Physical properties", "In the previous section, we found the exact analytical solution to the problem of a spherically symmetric ingoing null fluid carrying electric charge and energy in a space-time whose dynamics is governed by the Palatini theory given by Eq.", "(REF ).", "In this section, we discuss the different contributions appearing in the line element (REF ) and their properties." ], [ "The GR limit", "Let us first note that when $l_P \\rightarrow 0$ in the Lagrangian density (REF ) we recover the GR limit, since it implies $\\sigma _{\\pm } \\rightarrow 1$ and allows to perform the integration in $dx=dr \\sigma _{+}/\\sigma _{-}^{1/2}=dr$ , finding that $\\int dx/r^2=-1/r$ , which leads to $G(z) = -1/z$ .", "With these elements the metric component $g_{vv}$ in Eq.", "(REF ) boils down to $g_{vv}=-\\left(1 - \\frac{r_s(v)}{r} + \\frac{r_q(v)^2}{2r^2} \\right) \\,,$ while the relation (REF ) becomes $\\rho _{in}r^2=\\frac{2}{\\kappa ^2} \\left( L(v) - \\frac{\\kappa ^2 q q_v}{8\\pi r} \\right) \\,.$ Thus, these expressions reproduce the well known Bonnor-Vaidya solution of GR [2]." ], [ "Uncharged solutions with null fluid", "This dynamical scenario was considered in [47], where more details and examples can be found.", "Here we summarize the main features of this case.", "When the electrovacuum field is not present ($q=0$ , $\\sigma _{\\pm } \\rightarrow 1$ ), the line element (REF ) becomes [47] $ ds^2=-B(v,r)dv^2+2dvdr + r^2 d \\Omega ^2 \\,,$ with $ B(v,r)=1-\\frac{2 M(v)}{r} - \\frac{Q^2(v)}{r^2} \\,,$ where $M(v)\\equiv \\int _{v_0}^v L(v^{\\prime })dv^{\\prime }$ represents the mass term, $Q^2(v)\\equiv 4L(v)/\\rho _P$ represents a charge-like term, and we have defined $\\rho _P=\\frac{c^2}{l_P^2 G} \\sim 10^{96}kg/m^3$ as the Planck density.", "The luminosity function $L(v)=\\kappa ^2 r^2 \\rho _{in}/2$ follows from Eq.", "(REF ).", "The metric (REF ) is formally that of a (nonrotating) Reissner-Nordström black hole but with the wrong sign in front of the charge term.", "The single horizon of this solution is located at $r_{+}(v)=M(v) + \\sqrt{M(v)+Q^2(v)}$ and is larger than in the dynamical Schwarzschild solution of GR, $r_S(v)=2M(v)$ .", "When the flux of radiation ceases, $Q^2(v)$ vanishes and $r_+$ retracts to its GR value $r_S$ .", "The metric function (REF ) puts forward that the null fluid is leaving its imprint on the structure of the space-time not only through its integrated luminosity (the mass term $M(v)$ ), but also directly through the luminosity function $L(v)$ (suppressed by the Planck density), which contains full details about the distribution of the incoming fluid.", "It should be noted that if a scalar field is quantized in such a background, the field mode functions will be sensitive to $L(v)$ , thus having access to all the information contained in the source that forms the black hole.", "As a result, the emitted Hawking quanta will contain crucial information not only about the integrated energy profile $M(v)$ , but also about the most minute details of its time distribution $L(v)$ ." ], [ "Static charged configurations", "When there is no incoming flux of charge and energy, $q(v)$ and $M(v)$ remain constant.", "In this case, the metric gets simplified in a number of ways.", "Firstly, the term $\\rho _{in}$ disappears, and $\\delta _1(v)$ , $\\delta _2(v)$ , and $r_c(v)$ take the constant values $\\delta _1^{(0)}$ , $\\delta _2^{(0)}$ , and $r_c^{(0)}$ , respectively, where the superindex denotes the amounts of mass and charge, $M=M_0$ and $q=q_0$ , characterizing the solution.", "The line element (REF ) can then be written as $ds^2&=&-\\frac{\\left(1-\\frac{1+\\delta _1^{(0)} G(z)}{\\delta _2^{(0)} z \\sigma _{-}^{1/2}}\\right)}{\\sigma _+(x)}dv^2+\\frac{2dvdx}{\\sigma _+(x)}+r^2(x)d\\Omega ^2 \\ , $ where here $r=r_c^{(0)}z(x)$ is just a function of $x$ , i.e., there is no time-dependence on $v$ .", "Accordingly, $\\sigma _\\pm =\\sigma _\\pm (z)$ and $G(z)$ are $v$ -independent functions.", "For $\|x\|\\gg r_c^{(0)}$ , one finds that $r^2(x)\\approx x^2$ , $\\sigma _\\pm \\approx 1$ , and $g_{vv}\\approx -\\left(1-\\frac{r_S^{(0)}}{r}+\\frac{{r_q^{(0)}}^2}{2r^2}\\right)$ , which turns (REF ) into the expected GR limit.", "On the other hand, from the relation (REF ), it is easy to see that $r(x)$ reaches a minimum $r_{min}=r_c^{(0)}$ at $x=0$ .", "At that point, one can verify [28] that curvature scalars generically diverge except if the charge-to-mass ratio $\\delta _1^{(0)}$ takes the value $\\delta _1^{(0)}=\\delta _1^$ In terms of horizons, configurations with $\\delta _1>\\delta _1^$ are similar to the standard Reissner-Norström solution of GR, having two horizons, a single (extreme) one or none, while those with $\\delta _1<\\delta _1^$ have always a single (non-degenerate) horizon, resembling the Schwarzschild solution.", "For more details see the first of Refs.", "[28]., where $\\delta _1^ \\simeq 0.572$ is a constant that appears in the series expansion of $G(z)=-1/\\delta _1^+2\\sqrt{z-1}+\\ldots $ as $z\\rightarrow 1$ .", "The smoothness of the geometry when $\\delta _1^{(0)}=\\delta _1^$ together with the fact that $r(x)$ reaches a minimum at $x=0$ allow to naturally extend the coordinate $x$ to the negative real axis, thus showing that the radial function $r^2(x)$ bounces off to infinity as $x\\rightarrow -\\infty $ (see Fig.REF ).", "Figure: The minimum of the radial function z(x)z(x) implies the existence of a wormhole extension of the geometry, with xx covering the whole real axis -∞<x<+∞-\\infty <x<+\\infty .", "Note the smoothness of the function dG/dxdG/dx and the bounce of z(x)z(x) at x=0x=0.", "In this plot, r c =1r_c=1.This implies the existence of a wormhole structure with its throat located at $r=r_c^{(0)}$ , where $dr/dx=0$ .", "This interpretation is further supported by the existence of an electric flux coming out from the wormhole mouth and responsible for the spherically symmetric electric field.", "An electric field of this kind does not require the existence of point-like sources for its generation, as first shown by Wheeler and Misner in [49].", "The non-trivial wormhole topology implies that the flux $\\Phi = \\int _S F$ , where $F$ is the 2-form dual to the Faraday tensor, through any closed 2-surface $S$ enclosing one of the wormhole mouths is non-zero and can be used to define a charge $\\Phi =4\\pi q$ .", "On practical grounds, there is no difference between this kind of charge, arising from a pure electric field trapped in the topology (going through a wormhole), and a standard point-like charge.", "Remarkably, one can easily verify [28] that this flux is independent of the particular value of $\\delta _1^$ , which entails that the wormhole structure exists even when the curvature scalars diverge at $r=r_c^{(0)}$ .", "This result gives consistencyNote, in this sense, that the Reissner-Nordström solution of GR represents an incomplete problem because the source term is generally not considered, restricting the discussion to the region external to the sources [38].", "to the field equations of the static problem, in which the electric field is assumed sourceless, and demands a debate on the physical meaning and implications of curvature divergences since, as we have shown, they pose no obstacle to the existence of a well-defined (topological) electric flux through them.", "Since the line element (REF ) recovers the Reissner-Nordström geometry when $\|x\|\\gg r_c^{(0)}$ , one can verify that for $\\delta _1 = \\delta _1^$ an external horizon exists in general.", "One thus expects that the existence of the horizon forces the regular configurations to decay into those with $\\delta _1 \\ne \\delta _1^$ via Hawking radiation.", "However, as shown in [28], [29], when the number of charges $N_q\\equiv q/\|e\|$ drops below the critical value $N_q^c=\\sqrt{2/\\alpha _{em}} \\approx 16.55$ (where $\\alpha _{em}$ is the fine structure constant) the event horizon disappears, yielding an object which is stable against Hawking decay and whose charge is conserved and protected on topological grounds.", "Such everywhere regular and horizonless objects can be connected with black hole states, which posses an event horizon, in a continuous way, thus suggesting that they can be interpreted as black hole remnants.", "As a final remark, we point out that for arbitrary $\\delta _1$ the spatial integration of the action, representing the addition of electromagnetic plus gravitational energies, yields a finite result, which implies that the total energy is finite regardless of the existence or not of curvature divergences at the wormhole throat.", "In the particular case of the regular solutions $\\delta _1 = \\delta _1^$ , the action defined by Eqs.", "(REF ) and (REF ) evaluated on the solutions coincides with the action of a point-like massive particle at rest.", "Additionally, the surface $r=r_c^{(0)}$ becomes timelike when $N_q<N_q^c$ , which further supports the idea that such regular solutions possess particle-like properties [28], [29], representing a specific realization of Wheeler's geon [16]." ], [ "Dynamical charged configurations", "When the incoming null flux of radiation carries electric charge, the geometry changes in a highly non-trivial way.", "This setup should provide a good description of highly relativistic charged particles collapsing in a spherically symmetric way.", "To illustrate this complexity, consider first that the initial state is flat Minkowski space.", "Assume that a charged perturbation of compact support propagates within the interval $[v_i,v_f]$ .", "Given the relation (REF ), which for future reference we write as $r^2(x,v)=\\frac{x^2+\\sqrt{x^4+4r_c^4(v)}}{2} \\ ,$ where $r_c^4 (v)\\equiv l_P^2 \\tilde{\\kappa }^2q^2(v)$ , it follows that for $v<v_i$ the radial function $r^2(x,v)=x^2$ extends from zero to infinity [30].", "As we get into the $v\\ge v_i$ region, this radial function, which measures the area of the 2-spheres of constant $x$ and $v$ , never becomes smaller than $r_c^2 (v)$ .", "In the region $v>v_f$ , in which the ingoing flux of charge and radiation is again zero, the result is a static geometry identical to that described above in Sec.", "REF .", "This change in the geometry can be interpreted as the formation of a wormhole whose throat has an area $A_{WH}=4\\pi r_c^2(v_f)$ .", "Depending on the total amounts of charge and energy conveyed by the incoming flux, the space-time may have developed event horizons (see Sec.", "REF , and [28] for full details on the different configurations).", "The existence or not of curvature divergences at $r=r_c(v_f)$ depends on the (integrated) charge-to-mass ratio of the flux.", "For simplicity, one can assume situations where $\\delta _1(v_f)=\\delta _1^$ , for which the final configuration has no curvature divergences, and $\\delta _2(v_f)>\\delta _1^$ , for which there are no event horizons.", "Related to this, we emphasize that, as shown in Sec.", "REF , the electric flux $\\Phi $ through any 2-surface enclosing the region $r=r_c(v)$ is always well-defined regardless of the value of $\\delta _1(v_f)$ .", "Figure: Penrose diagram for the nonsingular case δ 1 =δ 1 * \\delta _1=\\delta _1^* without event horizon, N q <N q c N_q<N_q^c.Consider first that the initial state is flat Minkowski space.", "Next, assume that a charged perturbation of compact support propagates within the interval [v i ,v f ][v_i,v_f].", "We verify that the area of the 2-spheres of constant xx and vv increases and never becomes smaller than r c 2 (v)r_c^2 (v).", "If we consider now the region v>v f v>v_f, where the ingoing flux of charge and radiation is switched off, the result is a static geometry identical to that described in Section .", "This change in the geometry can be interpreted as the formation of a wormhole whose throat has an area A WH =4πr c 2 (v f )A_{WH}=4\\pi r_c^2(v_f).", "See the text for more details.In a first approximation, this process of wormhole formation could be visualized as depicted in Fig.", "REF .", "This diagram suggests that the ingoing charged flux of radiation generates a whole new region of space-time as it propagates.", "This view would imply a change in the global properties of space-time and, therefore, in its topology.", "A more careful examination of this process is necessary to understand how the other side of the wormhole arises and how this affects the topology of the problem.", "In fact, from a mathematical point of view, the exactly Minkowskian case $q=0$ can be seen as an exceptional situation in which the derivative of $r^2=x^2$ takes the values $\\pm 1$ and has a discontinuity at $x=0$ .", "However, in a physical context with continuous virtual pair creation/annihilation out of the quantum vacuum, it seems reasonable to expect that the exact case $q=0$ is never physically realized and that only the limiting case $q\\rightarrow 0$ makes senseIn such a scenario, the vanishing of (the quantum average) $<q>$ in a given region can still be compatible with $<q^2>\\ne 0$ .", "In this sense, we understand that it is $<q^2>$ which should enter in the definition of $r_q^4(v)=l_P^2\\tilde{\\kappa }^2 <q^2(v)>$ .. One can thus assume that in the physical branch of the theory, $q$ can be arbitrarily small but non-zero, with the limit $q\\rightarrow 0$ leading to vanishing derivative $dr/dx$ at $x=0$ and quickly converging to $\\pm 1$ away from $x=0$ .", "In this scenario, the initial $q\\rightarrow 0$ configuration could be seen as consisting of two identical pieces of Minkowski space-time connected along the line $x=0$ (see Fig.", "REF ) through a wormhole of area $A_{WH}\\propto q \\rightarrow 0$ , which can be as small as one wishes but never zero due to the vacuum fluctuations.", "We can now consider again the collapse of a spherical shell of charged radiation.", "As shown in Fig.", "REF , the geometry inside the collapsing shell is essentially Minkowskian, up to the existence of infinitesimally small wormholes generated by quantum fluctuations (which realize the idea of a space-time foam).", "Though the details of the transient are complex and require a case-by-case numerical analysis because of the $\\rho _{in}$ term appearing in the line element (REF ), the result of the collapse is the stretching of an initially infinitesimal wormhole to yield a finite size hole of area $A_{WH}(v)=4\\pi r_c^2(v)$ .", "This occurs in such a way that the density of lines of force at the wormhole throat is kept constant, $\\Phi (v)/A_{WH}(v)=\\sqrt{c^7/(2\\hbar G^2)}$ .", "Note that this constraint between the flux and the area of the wormhole is valid for arbitrary charge and, in particular, in the limit $q\\rightarrow 0$ .", "Only if $q=0$ exactly, this ratio becomes indefinite.", "In GR, where an electric flux is assumed to be generated by a point-like particle (of zero area), one finds a divergent result.", "This divergence corresponds to taking the limit $\\hbar \\rightarrow 0$ in the above ratio and indicates that the wormhole closes in the limit in which classical GR is recovered, which is fully consistent with the fact that wormholes supported by electromagnetic fields do not exist in the case of GR [50].", "Figure: Penrose diagram for the formation of a wormhole (with δ 1 =δ 1 * \\delta _1=\\delta _1^* and N q <N q c N_q<N_q^c) out of Minkowski space resulting from the perturbation of a charged null fluid of integrated charge +q+q, and its subsequent removal due to a second flux of integrated charge -q-q.", "Note that we have chosen positive energy fluxes in both cases, which implies that the final state is a Schwarzschild black hole instead of Minkowski space.The sudden generation of a new space-time region depicted in Fig.", "(REF ) can thus be avoided by assuming from the very beginning that the space-time admits a foam-like microstructure in which electric field lines may sustain wormholes that connect two different regions (the two sides of each wormhole).", "The spontaneous generation of virtual pairs of electrically charged particles could be seen as the spontaneous formation of two nearby wormholes with identical but opposite charges [51].", "The energy deficit resulting from the generation of this pair would be released when the pair meets and the wormholes disappear.", "In this picture, the universe that we perceive would thus be a copy of another universe containing the same particles but with opposite charges (due to the different orientation of the fluxes on each side of the wormholes).", "If a second flux of charged radiation is considered, the wormhole can be reduced again to an infinitesimal structure (associated to quantum fluctuations) if at $v\\ge v_{f_2}$ we have $q\\rightarrow 0$ (see Fig.", "REF ).", "The geometry then becomes essentially identical to that of a Schwarzschild black hole on both sides, with curvature scalars diverging at $x=0$ .", "If a new flux of charged matter reaches $x=0$ , then the wormhole throat should grow again to give consistency to the electromagnetic field equations and conservation laws." ], [ "Summary and Discussion", "We have worked out a simplified scenario of gravitational collapse in which new gravitational physics at high energies is introduced by means of quadratic curvature corrections in the gravitational Lagrangian.", "We have made use of two elements that simplify the mathematical analysis, namely, 1) spherical symmetry, and 2) a pressureless fluid.", "These simplifications have been traditionally used in theoretical discussions about gravitational collapse and the study of the properties of singularities.", "Obviously, neither 1) nor 2) can be exactly realized in nature but, nonetheless, they are very useful for theoretical analysis of the type considered here.", "Note, in this sense, that already in the first models of gravitational collapse worked out by Oppenheimer and Snyder [56], the internal pressures of the collapsing fluids were neglected as it was understood that, above a certain threshold, rather than helping to prevent the collapse they contribute to increase the energy density, which further accelerates the process.", "Similarly, in the case of electrically charged black holes (the well-known Reissner-Nordström solution), for instance, the repulsive electric force of the collapsed matter does not help to alleviate the strength of the central singularity.", "Rather, the squared of the Riemann tensor increases its degree of divergence, going from ${R^\\alpha }_{\\beta \\mu \\nu }{R_\\alpha }^{\\beta \\mu \\nu }\\sim 1/r^4$ in the Schwarzschild case to ${R^\\alpha }_{\\beta \\mu \\nu }{R_\\alpha }^{\\beta \\mu \\nu }\\sim 1/r^8$ in the charged case.", "The energy of the electric field, therefore, worsens the degree of divergence of the curvature scalars.", "In our model, we have considered a radiation fluid carrying a certain amount of energy and also electric charge.", "The repulsive forces or pressures that the particles making up the fluid could feel have been neglected as they are not essential for the study of the end state of the collapse.", "As a result, the fluid follows geodesics of the metric, which have been determined dynamically by taking into account the energy and charge conveyed by the fluid.", "The fluid motion, therefore, is not given a priori, but follows from the consistent resolution of the coupled system of radiation, electric field, and gravity.", "The key point of this work has been the study of the end state of the collapse of this idealized system.", "In general relativity, this configuration unavoidably leads to the formation of a point-like singularity.", "In our model, however, the geometry and the topology undergo important changes.", "When the energy density of the collapsing fluid reaches a certain scale (of order the Planck scale), gravity is no longer attractive and becomes repulsive.", "This has a dramatic effect on the geodesics followed by the fluid which, rather than focusing into a point-like singularity, expand into a growing sphere.", "The wormhole is thus somehow produced by the repulsive character of gravitation at high energy-densities and the need to conserve the electric flux.", "In principle, in our model wormholes of arbitrary charge and mass can be formed.", "However, this cannot be completely true since our results are valid as long as the approximations involved hold with sufficient accuracy.", "Therefore, one should note that in low-energy scenarios pressure and other dispersion effects should act so as to prevent the effective concentration of charge and energy way before it can concentrate at Planckian scales, thus suppressing wormhole production.", "However, for adequate concentrations of charge and energy, gravitational collapse cannot be halted and our analysis should be regarded as a good approximation.", "In this sense, we note that Hawking already analyzed the process of classical collapse in the early universe, finding that (primordial) black holes with a Planck mass or higher and up to 30 units of charge could be formed out of a charged plasma [57].", "Stellar collapse offers another robust mechanism to generate the conditions under which our approximations are valid.", "In fact, in order to build a completely regular configuration with a solar mass, about $\\sim 10^{57}$ protons, one needs $\\sim 2.91 \\times 10^{21}$ electrons [28], which is a tiny fraction of the total available charge ($10^{-31}$ ) and mass.", "Therefore, the generation of wormholes under realistic situations is possible.", "The dynamical generation of wormholes outlined above, in the context of charged fluids in quadratic Palatini gravity, differs radically in nature to the construction of general relativistic traversable wormholes, with the idealization of impulsive phantom radiation considered extensively in the literature [10], [9], [11], [12], [52], [53], [54], [55].", "In the latter, it was shown that two opposing streams of phantom radiation, which form an infinitely thin null shell, may support a static traversable wormhole [9].", "Essentially, one begins with a Schwarzschild black hole region, and triggers off beams of impulsive phantom radiation, with constant energy density profiles, from both sides symmetrically, consequently forming Vaidya regions.", "Now, in principle, if the energies and the emission timing are adequately synchronized, the regions left behind the receding impulses after the collision results in a static traversable wormhole geometry.", "Furthermore, it is interesting to note that it was shown that with a manipulation of the impulsive beams, it is possible to enlarge the traversable wormhole (see [12] for more details).", "These solutions differ radically from the self-inflating wormholes discovered numerically [54] and the possibility that inflation might provide a natural mechanism for the enlargement of Planck-size wormholes to macroscopic size [17].", "The difference lies in the fact that the amount of enlargement can be controlled by the amount of energy or the timing of the impulses, so that a reduction of the wormhole size is also possible by reversing the process of positive-energy and negative-energy impulses outlined in [12].", "The theory presented here allows to generate static wormholes by means of a finite pulse of charged radiation, without the need to keep two energy streams active continuously or to synchronize them in any way across the wormhole.", "Regarding the size of the wormholes, we note that if instead of using $l_P^2$ to characterize the curvature corrections one considers a different length scale, say $l_\\epsilon ^2$ , then their area would be given by $A_{WH}=\\left(\\frac{l_\\epsilon }{l_P}\\right)\\frac{2N_q}{N_q^c}A_P$ , where $A_P=4\\pi l_P^2$ , $N_q=\|q/e\|$ is the number of charges, and $N_q^c\\approx 16.55$ .", "Though this could allow to reach sizes orders of magnitude larger than the Planck scale, it does not seem very likely that macroscopic wormholes could arise from any viable theory of this form, though the role that other matter/energy sources could produce might be nontrivial.", "Relative to the issue of classical singularities, the meaning and implications of the latter has been a subject of intense debate in the literature for years.", "Their existence in GR is generally interpreted as a signal of the limits of the theory, where quantum effects should become relevant and an improved theory would be necessary.", "This is, in fact, the reason that motivates our heuristic study of quadratic corrections beyond GR.", "As pointed out above and shown in detail in [28], the curvature divergences for the static wormhole solutions arising in quadratic Palatini gravity with electrovacuum fields (and also in the Palatini version of the Eddington-inspired Born-Infeld theory of gravity, see [44]) are much weaker than their counterparts in GR (from $\\sim 1/r^8$ in GR to $\\sim 1/(r-r_c)^3$ in our model).", "Additionally, the existence of a wormhole structure that prevents the function $r^2$ from dropping below the scale $r_c^2$ implies that the total energy stored in the electric field is finite (see [44], [29] for details), which clearly contrasts with the infinite result that GR yields.", "Therefore, even though curvature scalars may diverge, physical magnitudes such as total mass-energy, electric charge, and density of lines of force are insensitive to those divergences, which demands for an in-depth analysis of their meaning and implications.", "In this sense, we note that topology is a more primitive concept than geometry, in the sense that the former can exist without the latter.", "Comparison between a sphere and a cube is thus pertinent and enlightening in this context to better understand the physical significance of curvature divergences.", "It turns out that a cube and a sphere are topological equivalent.", "However, the geometry of the former is ill-defined along its edges and vertices.", "The divergent behavior of curvature scalars for certain values of $\\delta _1$ , therefore, simply indicates that for those cases the geometry is not smooth enough at the wormhole throat, but that does not have any impact on the physical existence of the wormhole.", "Regarding the existence of curvature divergences at $x=0$ in the Schwarzschild case ($q\\rightarrow 0$ ), our view is that there exist reasons to believe that such divergences could be an artifact of the approximations and symmetries involved in our analysis.", "These suspects are supported by the fact that radiation fluids (with equation of state $P/\\rho =1/3$ ) in cosmological scenarios governed by the dynamics of the theory under study are able to avoid the Big Bang singularity, which is replaced by a cosmic bounce [33].", "For the radiation fluid, the cosmic bounce occurs in both isotropic and anisotropic homogeneous scenarios when the energy density approaches the Planck scale.", "One would thus expect that a process of collapse mimicking the Oppenheimer-Snyder model with a radiation fluid should avoid the development of curvature divergences.", "This, in fact, occurs in Eddington-inspired Born-Infeld gravity [44], studied recently in [58].", "The generic existence of curvature divergences in the uncharged case involving a Vaidya-type scenario with null fluids is thus likely to be due to the impossibility of normalizing the null fluid, which is therefore insensitive to the existence of a limiting density scale.", "The consideration of more realistic non-null charged fluids could thus help to improve the current picture and avoid the shortcomings of the uncharged ($q\\rightarrow 0$ ) Schwarzschild configurations.", "As a final comment, we note that since in our theory the field equations outside the matter sources recover those of vacuum GR, Birkhoff's theorem must hold in those regions.", "This means that for $v<v_i$ we have Minkowski space, whereas for $v>v_f$ we have a Reissner-Nordström-like geometry of the form (REF ).", "The departure from Reissner-Nordström is due to the Planck scale corrections of the Lagrangian, which are excited by the presence of an electric field, and only affect the microscopic structure, which is of order $\\sim r_c(v)$ (see Sec.", "REF and [28]).", "Due to the spherical symmetry and the second-order character of the field equations, Birkhoff's theorem guarantees the staticity of the solutions for $v>v_f$ .", "To conclude, in this work an exact analytical solution for the dynamical process of collapse of a null fluid carrying energy and electric charge has been found in a quadratic extension of GR formulated à la Palatini.", "This scenario extends the well-known Vaidya-Bonnor solution of GR [2], thus allowing to explore in detail new physics at the Planck scale.", "In the context of the static configurations, we have shown that wormholes can be formed out of Minkowski space by means of a pulse of charged radiation, which contrasts with previous approaches in the literature requiring artificial configurations and synchronization of two streams of phantom energy.", "Our results support the view that space-time could have a foam-like microstructure with wormholes generated by quantum fluctuations.", "Though such geometric structures develop, in general, curvature divergences, they are characterized by well-defined and finite electric charge and total energy.", "The physical role that such divergences could have is thus uncertain and requires an in-depth analysis, though from a topological perspective they seem not to play a relevant role.", "To fully understand these issues our model should be improved to address several important aspects including, for instance, the presence of gauge field degrees of freedom, to take into account the dynamics of counter-streaming effects due to the presence of simultaneous ingoing and outgoing fluxes, or to consider other theories of gravity beyond the quadratic Lagrangian (REF ).", "These and related research issues are currently underway." ], [ "Acknowledgments", "F.S.N.L.", "acknowledges financial support of the Fundação para a Ciência e Tecnologia through an Investigador FCT Research contract, with reference IF/00859/2012, funded by FCT/MCTES (Portugal), and grants CERN/FP/123615/2011 and CERN/FP/123618/2011.", "G.J.O.", "is supported by the Spanish grant FIS2011-29813-C02-02, the Consolider Program CPANPHY- 1205388, the JAE-doc program of the Spanish Research Council (CSIC), and the i-LINK0780 grant of CSIC.", "D.R.-G. is supported by CNPq (Brazilian agency) through project No.", "561069/2010-7 and acknowledges hospitality and partial support from the Department of Physics of the University of Valencia, where this work initiated.", "This work has also been supported by CNPq project No.", "301137/2014-5." ] ]	1403.0105
[ [ "Tunable plasmons in atomically thin gold nanodisks" ], [ "Abstract The ability to modulate light at high speeds is of paramount importance for telecommunications, information processing, and medical imaging technologies.", "This has stimulated intense efforts to master optoelectronic switching at visible and near-infrared frequencies, although coping with current computer speeds in integrated architectures still remains a major challenge.", "As a partial success, midinfrared light modulation has been recently achieved through gating patterned graphene.", "Here we show that atomically thin noble metal nanoislands can extend optical modulation to the visible and near-infrared spectral range.", "We find plasmons in thin metal nanodisks to produce similar absorption cross-sections as spherical particles of the same diameter.", "Using realistic levels of electrical doping, plasmons are shifted by about half their width, thus leading to a factor-of-two change in light absorption.", "These results, which we substantiate on microscopic quantum theory of the optical response, hold great potential for the development of electrical visible and near-infrared light modulation in integrable, nanoscale devices." ], [ "Introduction", "Surface plasmons, the collective oscillations of conduction electrons in metallic structures, allow us to confine light down to deep subwavelength volumes [1].", "Additionally, they couple strongly to electromagnetic fields [2].", "Because of these properties, plasmons are excellent tools to engineer nanoscale devices for manipulating optical signals, without the limitation imposed by diffraction in far-field setups.", "This has triggered a number of applications in areas as diverse as ultrasensitive biosensing [3], improved photovoltaics [4], plasmon enhanced photodetection [5], and photothermal cancer therapy [6].", "The design of plasmonic structures with suitable spectral characteristics involves a careful choice of geometry and composition.", "In recent years, a vast amount of work has been devoted to producing nanostructures made of noble metals with controlled size and morphology, using in particular colloidal methods [7] and lithography [8].", "Despite these advances in the control over the static characteristics of plasmons, the dynamical modulation of their frequencies and spatial profiles remains ellusive, particularly in the visible and near-infrared (vis-NIR) parts of the spectrum.", "In this context, slow mild changes of the plasmon frequency have been produced by electrochemically injecting electrons in metal nanoparticles [9], by electrically driving liquid crystals containing plasmonic particles [10], and through controllable metamaterial designs [11], [12].", "Magneto-optical modulation has also been explored to control plasmons in noble metal structures [13].", "Hybrids of plasmonic and conductive oxides have been proposed [14], [15], as well as colloids based on different materials [16].", "However, we still need to devise new methods to produce larger and faster control over plasmons, as required for nanoscale optical commutation and light modulation at high speeds.", "Recently, the emergence of graphene [17] as a novel plasmonic material [18], [19], [20] has opened up new paths towards the design of dynamically tunable plasmonic devices.", "Electrically doped graphene supports surface plasmons whose frequency can be efficiently varied by changing the level of doping [21], [22], [23].", "Consequently, the resulting modulation is intrinsically fast because it can be driven by charge-carrier injection using conventional electric gating technology.", "This promising material has been so far shown to support mid-infrared and lower-frequency plasmons [24], [21], [22], [23], [25], while vis-NIR modes are being pursued by reducing the size of the structures [26], [27] and increasing the level of doping [23].", "The search for plasmon modulation in the vis-NIR is thus still ongoing, as these are spectral regions of utmost importance for sensing and optical signal processing technologies.", "The origins of the excellent tunability of plasmons in graphene can be found in both the atomic thickness and the peculiar electronic structure of this material.", "The latter is characterized by a linear dispersion relation, which leads to a vanishing of the density of states at the Fermi level, so that a relatively small density of injected charge carriers produces substantial optical gaps in which collective plasmon modes emerge [17], [19].", "Although this unique feature cannot be easily transported to conventional plasmonic materials, such as gold, we can still mimic graphene plasmonics by going to atomically thin noble metals, whose optical response should be more susceptible to doping than traditional thicker layers.", "In particular, monolayer gold, the synthesis of which has been mastered for a long time in the context of surface science [28], presents the advantage of having a plasma frequency compatible with the existence of plasmons in the vis-NIR [29].", "Here, we show that single-monolayer gold disks (SMGDs) with diameters of the order of $10\\,$ nm support surface plasmons with large cross-sections comparable to their geometrical areas.", "The frequencies of these excitations lie in the vis-NIR and can be efficiently modulated using attainable concentrations of doping charge carriers, which can be provided via electrical doping using for example backgating technology.", "We also analyze the optical response of periodic arrays of SMGDs, for which we predict an absorbance $\\sim 25\\%$ for metal layer filling fractions $\\sim 40\\%$ .", "The system under study is depicted in Fig.", "1a.", "We consider a gold nanodisk of diameter $D$ , extracted from a single (111) atomic layer of gold.", "We take the thickness of the gold monolayer to be equal to the separation between (111) atomic planes in bulk gold (i.e., $a_0/\\sqrt{3}$ , where $a_0=0.408\\,$ nm is the atomic lattice constant).", "Incidentally, our results are rather independent on the choice of disk thickness when this is small compared with the diameter, as long as the total valence charge is preserved (see Supplementary Fig.", "6).", "As a first step in our analysis, we describe the optical response of a SMGD classically by modeling it as a thin disk described by a frequency-dependent homogeneous dielectric function $\\epsilon (\\omega )$ .", "More precisely, we calculate the extinction cross-section $\\sigma $ by solving Maxwell's equations using the boundary-element method (BEM) [30].", "Interestingly, for a diameter $D=20\\,$ nm, the cross section is dominated by a NIR plasmon at an energy $\\sim 1\\,$ eV and it exceeds the geometrical area of the disk (see left part of Fig.", "1b).", "It is convenient to separate the contribution from s-band electrons in the dielectric function as a Lorentzian term, $\\epsilon \\left(\\omega \\right)= \\epsilon _{\\rm b} - \\frac{\\omega ^2_{\\rm p}}{\\omega \\left(\\omega +{\\rm i}\\gamma \\right)}, $ where $\\epsilon _{\\rm b}$ accounts for the effect of background screening due to d-band electrons, $\\hbar \\omega _{\\rm p}=9.06\\,$ eV is the classical plasmon energy associated with s valence electrons (see Methods), and $\\hbar \\gamma =71\\,$ meV is an inelastic width (we adopt this value of the damping throughout this work).", "As explained below, we introduce doping in the classical model by changing $\\omega _{\\rm p}$ in Eq.", "(1).", "In general, the description of the response of gold in the vis-NIR region including interband transitions requires to use either experimental data [29] or a sophisticated multi-Lorentzian model [31] for $\\epsilon (\\omega )$ , which yields a $\\omega $ -dependent background $\\epsilon _{\\rm b}=\\epsilon (\\omega )+\\omega ^2_{\\rm p}/[\\omega (\\omega +{\\rm i}\\gamma )]$ .", "However, we show in Fig.", "1b that a simple Drude model for Eq.", "(1) (dashed curves), consisting in fixing $\\epsilon _{\\rm b}=9.5$ for all frequencies, produces a satisfactory level of accuracy at the observed relatively low disk-plasmon energies compared with the results obtained from tabulated optical data [29] (solid curves).", "Additionally, as the Drude model (i.e., constant $\\epsilon _{\\rm b}$ ) provides a natural connection with the quantum-mechanical approach described below, we use for disks it in what follows.", "Figure: Optical response and electrical tunability of single-monolayer gold disks.", "(a) We consider a single-monolayer gold disk (SMGD) of diameter DD carved from a single (111) atomic layer.", "The disk thickness is a 0 /3a_0/\\sqrt{3}, where a 0 =0.408a_0=0.408\\,nm is the atomic lattice constant.", "(b) Extinction cross-section of a D=20D=20\\,nm SMGD for different doping charge-carrier densities (see upper legend).", "A doping density of 13.8×10 13 13.8\\times 10^{13}\\,cm -2 ^{-2} corresponds to total of 440 additional charge carriers in the disk.", "For comparison, we also plot the cross section for a gold nanosphere of the same diameter and total doping charge, clearly showing an almost negligible degree of tunability.", "The particles are assumed to be homogeneously doped and described classically through the local dielectric function tabulated from measured optical data (solid curves).", "Results obtained from a Drude dielectric function (Eq.", "(1)) are shown for comparison (broken curves).", "(c) Plasmon frequency shift relative to the width of the plasmon resonance for the disk (orange) and the sphere (green) of panel (b) as a function of doping density.", "The small scales indicate the potential at the disk/sphere surface for different disk doping densities.We consider next the effect of electrical doping.", "The spatial distribution of additional charge carriers depends on the doping configuration, as it can be for example homogenous for disks connected to a non-absorbing gate (e.g., ITO) or inhomogeneous in self-standing charged disks, although the plasmon energies and spatial profiles are expected to be similar in both cases based upon our experience with graphene disk plasmons [32].", "For simplicity, we assume homogeneously doped disks in what follows.", "The additional doping charge density $n$ adds up to the undoped s band density $n_0=m_{\\rm e}\\omega _{\\rm p}^2/(4\\pi e^2)\\approx 1.4\\times 10^{15}\\,$ cm$^{-2}$ , which is rather close to the s-band areal electron density in neutral monolayer gold, $4/\\sqrt{3}a_0^2$ .", "The doping charge is thus introduced by changing the bulk plasma frequency to $\\omega _{\\rm p}=\\left[\\left(4\\pi e^2/m_{\\rm e}\\right)(n_0+n)\\right]^{1/2}$ in Eq.", "(1).", "Now, the addition of a moderate amount of doping electrons ($\\sim 5-10\\%$ of $n_0$ ) results in significant blue shifts and increase in the strength of the plasmon resonance (cf.", "purple and green curves of Fig.", "1b).", "Obviously, the injection of similar amounts of holes produces the opposite effects (Fig.", "1b, orange curve).", "The small thickness of the disk is a key factor in producing such dramatic modifications in the optical response using realistic doping densities.", "In fact, repeating this operation with a gold nanosphere of the same diameter, we also observe a prominent plasmon (Fig.", "1b, $\\sim 2.5\\,$ eV region), but it remains unchanged when adding similar amounts of doping charges.", "In the sphere, the doping charges pileup in the outermost atomic layer [33], but this produces the same extinction cross-section as if the charges where homogenously distributed over its entire volume, and therefore, the change in bulk charge density is substantially reduced with respect to the disk.", "Figure 1c compares the modulation of the nanodisk and the sphere.", "In particular, we plot the frequency shift normalized to the full width at half maximum (FWHM) for the plasmon resonance as a function of doping charge density.", "In contrast to the negligible tunability of the sphere, the disk allows shifts comparable to the FWHM to be electrically induced.", "Incidentally, the doping densities here considered produce realistic values of the electrostatic potential at the surface of these nanoparticles (Fig.", "1c, small scales), indicating that they are compatible with currently available backgating technology [23].", "Figure: Optical response of individual single-monolayer gold disks.", "We plot the extinction cross-section normalized to the geometrical area for different diameters DD, calculated using the quantum model (solid curves) and a classical description (dashed curves).", "Results obtained with and without inclusion of d-band screening are shown in (b) and (a), respectively." ], [ "Quantum-mechanical effects", "For nanoparticles of only a few nanometers in size, the above classical description fails to account for spatial dispersion and quantum confinement effects [34], [35], which generally require models based on a quantum-mechanical treatment of valence electrons and their interactions.", "Here, we use the random-phase approximation (RPA) to calculate the optical response of SMGDs (see Methods) within the electrostatic approximation, which should be rather accurate given the small sizes of the particles under consideration.", "This allows us to determine the validity of the classical approach and explore the response of nanodisks with smaller diameters.", "We use particle-in-a-box states to describe independent s-band electrons.", "The cylindrical box has the same dimensions as in the classical calculations (see above) and it is surrounded by an infinite potential.", "The RPA susceptibility is then evaluated using these electron states to obtain the induced charge density, which in turn allows us to compute the optical extinction of the disk.", "Additionally, we model screening due to d-band electrons through an array of point dipoles placed at the atomic positions in the (111) layer and with polarizability adjusted to render an effective background permittivity $\\epsilon _{\\rm b}$ in the bulk material (see Methods).", "Apart from the relative position of these dipoles with respect to the disk center, our quantum description only depends on the three same parameters as the classical Drude theory (i.e., $\\epsilon _{\\rm b}$ , $\\omega _{\\rm p}$ , and $\\gamma $ ).", "A major assumption we are making is that $\\gamma $ takes the same value as in the bulk metal.", "We use this as a reasonable estimate because s-band electrons are rather delocalized along directions parallel to the layer (i.e., similar to the bulk), while they are narrowly confined to the ground state across the transversal direction, so that plasmons result from in-plane motion.", "However, the actual value of $\\gamma $ might depend on the detailed coupling of valence electrons to impurities and to the atomic lattice.", "Concerning d-band screening, our effective dipoles approach should provide a more realistic description than a homogeneous polarizable background.", "Although the discreteness of the dipole lattice can have strong effects in small islands, we find converged results for large islands, which are independent of the alignment of the dipole lattice relative to the disk center.", "Figure 2 shows the extinction cross-section normalized to the disk area for SMGDs of different diameters ranging from 3 to $15\\,$ nm.", "The main conclusions from this figure are as follows: (1) the extinction cross-sections are of the order of the disk area; (2) the plasmon energy increases with decreasing diameter $D$ , exhibiting an approximate $\\propto \\sqrt{D}$ dependence, similar to what one finds in graphene nanodisks [23]; (3) quantum calculations produce energies above those predicted by classical theory, as well as broader plasmon peaks, but the discrepancy between the two models decreases with increasing diameter; (4) in the absense of d-band screening (Fig.", "2a, obtained by setting $\\epsilon _{\\rm b}=1$ ), both levels of description give rise to smooth plasmon peaks, in contrast to the quantum results obtained when d-band screening is switched on (Fig.", "2b, with $\\epsilon _{\\rm b}=9.5$ ); (5) d-band screening also leads to a redshift of the plasmons, which is more pronounced at small sizes.", "Incidentally, the induced charge associated with the plasmon exhibits a dipolar profile dressed with radial oscillations mimicking those of Friedel oscillations, which are particularly intense for small diameters (see Supplementary Fig.", "7).", "Similar blue shifts with respect to classical local theory are also found in small noble metal particles [36], the origin of which is a combination of nonlocal and quantum effects, particularly due to the surface spill out of s electrons beyond the polarizable background of d-band electrons.", "In simple metals such as aluminum, the spill out produces smaller electron densities near the surface, and consequently, also smaller surface plasmon frequencies.", "In contrast, in noble metals, the spill out results in a weaker interaction with the localized d electrons, and thus, it leads to an increase in the observed frequency, which overcomes the redshift due to the smaller electron density.", "[37] We incorporate here the finite extension of s electrons across the normal direction, combined with the localization of the effective d-band dipoles, leading to similar blue shifts.", "Interestingly, our quantum model predicts splitting of the plasmon into multiple peaks for small disks when d-band screening is included (see for example the $D=3\\,$ nm spectrum in Fig.", "2b).", "The presence of these peaks, which are rapidly coalescing into a single plasmon resonance for $D>8\\,$ nm, is a manifestation of the discrete character of the interaction with d-band electrons, which is more pronounced for small $D$ 's.", "The jumps observed in the FWHM also shares a similar origin.", "It should be noted that these effects could be sensitive to the exact form of the s-electron transversal wave function in the smallest islands under consideration, which require a more atomistic analysis, based for example upon density-functional theory [38], [39].", "Likewise, the spectra for the smallest disks depend on the alignment of the d-band dipole lattice relative to the edges.", "In practice, 1D faceting of the edges becomes an important source of anisotropy, which can contribute to broaden the spectra for $D<5\\,$ nm.", "Figure: Comparison of quantum and classical plasmon energies and widths.", "Energy (a,b) and FWHM (c,d) of the plasmonic resonance of individual SMGDs as a function of disk diameter, calculated from quantum (green circles) and classical (orange triangles) models.", "Results obtained with and without inclusion of d-band screening are shown in (b,d) and (a,c), respectively.", "The dashed curves in (c,d) indicate the intrinsic broadening ℏγ=71\\hbar \\gamma =71\\,meV introduced in the Drude formula (Eq.", "(1)) and in the RPA susceptibility (Eq.", "(4)).The convergence of the quantum model to the classical description for increasing diameter is clearly observed in Fig.", "3, which summarizes the plasmon energies and widths observed in the spectra of Fig.", "2.", "Here, we define the FWHM as the frequency interval around the peak maximum that contains half of its area; this definition coincides with the standard FWHM for individual Lorentzian resonances, but it can be applied to multiple resonances as well to yield an overall width (in particular to the lower quantum-model spectra of Fig.", "2b).", "Within the electrostatic limit under consideration, the FWHM predicted by the classical model is independent of diameter and equals the damping energy $\\hbar \\gamma =71\\,$ meV (see Eq.", "(1)).", "In contrast, the quantum model leads to a significant increase in the FWHM for small diameters, essentially as a consequence of Landau damping, which involves inelastic decay of plasmons to electron-hole pairs for momentum transfers $\\sim \\omega /v_{\\rm F}$ , where $v_{\\rm F}$ is the Fermi velocity ($\\sim 10^6$ m s$^{-1}$ , see Supplementary Fig.", "8a).", "As the momentum transfer provided by the breaking of translational invariance in a disk is $\\sim 2\\pi /D$ , the onset of Landau damping is expected to occur at $D\\sim 2\\pi v_{\\rm F}/\\omega \\sim 4\\,$ nm, in qualitative agreement with the results shown in Fig.", "3c,d.", "An intuitive estimate can be stablished from the electron mean free path $v_{\\rm F}/\\gamma \\sim 10\\,$ nm, which determines the rate of collisions with the edges (i.e., events that provide the noted momentum), and is also in agreement with the trends observed in Fig.", "3c,d, although the value of $\\gamma $ regarded as a parameter simply produces an additional contribution to the FWHM that is independent of $D$ , and the ultimate origin of broadening for small sizes can be found in Landau damping.", "Figure: Quantum vs classical analysis of the electrically tunable optical response.", "(a,d) Extinction cross-section normalized to the geometrical area for a D=8D=8\\,nm single-monolayer gold disk calculated with different doping charge densities.", "(b,e) Plasmon energy as a function of doping charge density.", "We indicate the FWHM of the plasmon resonances as shadowed regions.", "(c,f) Optical extinction cross-section at the plasmon peak energy (green curves and symbols, left scale) and FWHM (orange, right scale) as a function of doping density.", "Quantum mechanical calculations (a-c) are compared with classical results (d-f), including d-band screening in all cases.As discussed in Fig.", "1, the optical response of SMGDs can be modified through the addition of small amounts of charge carriers to relatively large disks ($D=20\\,$ nm), for which classical theory is rather accurate (see Figs.", "2 and 3).", "Using smaller SMGDs, we obtain qualitatively similar results as for larger disks (see Fig.", "4 for an analysis of a $D=8\\,$ nm doped disk).", "Given the small disk size, we compare classical (Fig.", "4d-f) and quantum (Fig.", "4a-c) results, showing again a blue shift and plasmon broadening in the latter relative to the former.", "In contrast to the nearly linear plasmon shift with doping charge density predicted by classical theory (Fig.", "4e), the quantum model leads to initially smaller modulation at low doping (Fig.", "4b), which increases to a faster pace than the classical results for higher doping.", "This nonlinear dependence of the plasmon energy on the doping density could be exploited for improved light modulation by operating around a highly doped SMGD configuration.", "In particular, the plasmon shift can be as large as the FWHM when the density of s-band electrons is changed by $\\pm (5-10)\\%$ .", "Interestingly, the nonlinear plasmon shifts observed in the quantum model becomes oscillatory when examining the maximum extinction cross-section and the FWHM (Fig.", "4c).", "The oscillations of these two quantities are out of phase, as required to satisfy the $f$ -sum rule [40], and can be traced back to the discreteness of the electronic energies.", "Importantly, in all cases the maximum cross-section is of the order of the disk area (Fig.", "4c,f), thus providing good coupling to light for potential applications to modulation devices.", "Figure: Electrical modulation of the absorbance of an hexagonal periodic arrangement of single-monolayer gold disks.", "(a) Scheme of the system under study.", "(b) Absorbance spectrum of undoped nanodisks (diameter D=8D=8\\,nm) for different values of the array spacing dd.", "(c) Modulation of the absorbance relative to the undoped state as a function of doping charge density for different array spacings." ], [ "Periodic arrangement of single-monolayer gold nanodisks", "The large optical strength and degree of electrical tunability discussed above for SMGDs can be exploited to modulate light that is either transmitted or reflected by a periodic array of such structures.", "We consider an hexagonal array of $D=8\\,$ nm disks in Fig.", "5 with different values of the array spacing $d$ .", "Given the large mismatch between $D$ and the resonant light wavelength ($\\sim 830\\,$ nm), we approximate the disks as point dipoles of polarizability extracted as explained in Methods (see also Supplementary Fig.", "9, where we show that higher-order multipoles play only a small role for the relative distances under consideration).", "Following previous analytical methods [41] to compute the absorbance $A$ , we find remarkably large values (e.g., $A=25\\%$ for $d=1.5\\,D$ , see Fig.", "5b), given the small amount of gold in the structure (sub-monoatomic layer film).", "The fractional change in absorbance driven by electrical doping (Fig.", "5c) is rather independent of lattice spacing and reaches $\\sim 70\\%$ for a $10\\%$ variation in the s-band electron density.", "The potential of patterned gold monolayers for electro-optical modulation in the NIR is thus excellent.", "In summary, we have simulated both classically and quantum-mechanically the plasmonic response and performance in electro-optical modulation of gold nanodisks carved from a single (111) atomic layer.", "Our RPA calculations incorporate the wave functions of free s valence electrons evolving in a circular box, as well as an adjusted distribution of dipoles to account for d-band screening.", "Despite the atomic thickness of the disks, this quantum-mechanical description converges smoothly to the results of classical dielectric theory, based upon the bulk, frequency-dependent dielectric function of gold.", "This is a remarkable result, which can be intuitively understood from the fact that the electron current associated with the plasmons flows along the gold layer, and thus, it is rather insensitive to electron confinement within the small film thickness.", "Nontheless, nonlocality plays a crucial role, leading to strong plasmon blue shifts, as well as splitting due to the complex interaction with the d band.", "We estimate that nonlocal effects become dominant when the disk diameter is below $\\sim 10\\,$ nm.", "Remarkably, the disks interact strongly with light, giving rise to extinction cross-sections exceeding their geometrical areas in the vis-NIR.", "We have also shown that the optical response of SMGDs can be efficiently modulated through the addition or removal of realistic concentrations of doping charge carriers using for example gating technology.", "In particularly, periodic patterns of monolayer gold appear to be a suitable solution for combining strong plasmonic response and high doping, for example using an electrical backgate, because the average charge density of the layer is simply determined by capacitor theory for a fixed distance from the gate, and thus, the actual doping charge density in the metal scales with the inverse of the areal filling fraction occupied by the gold.", "Similar results are expected for films consisting of only a few atomic layers, although the degree of modulation is then reduced because the doping charge has to be shared across the increased thickness.", "Other plasmonic metals such as silver and copper should find similar degree of tunability (see Supplementary Fig.", "10).", "In particular, the small plasmon width of silver compared with gold makes it an attractive candidate to drive plasmon shifts well beyond the FWHM.", "Additionally, the lower d-band screening in this material should result in higher plasmon energies, reaching the visible in small disks, or equivalently, the NIR for larger disk diameters.", "It should be stressed that, while the synthesis of single-layer gold is a mature field [28], the fabrication of laterally confined thin gold nanostructures represents a technical challenge, which could benefit from advances in lithography and self-assembly.", "Alternatively, one could use a continuous gold layer, which also exhibits large electrical tunability of its propagating plasmons (see Supplementary Fig.", "11), coupled to external light by decorating it with dielectric colloids in order to bridge the light-plasmon momentum mismatch (i.e., this is essentially what nanostructuration does in the SMGDs that we study above).", "The resulting planar structures hold great potential for light modulation at vis-NIR frequencies, which could be the basis of a new generation of electrically tunable optical devices with applications ranging from sensing to nanoscale spectroscopy." ], [ "Quantum-mechanical RPA simulations", "We consider small disk sizes compared with the light wavelength, so that we work in the electrostatic limit.", "Within this approximation, assuming an overall monochromatic time dependence ${\\rm e}^{-{\\rm i}\\omega t}$ with frequency $\\omega $ , the induced charge density $\\rho ^{\\rm ind}$ can be expressed in terms of the self-consistent potential $\\phi $ as $\\rho ^{\\rm ind}({\\bf r},\\omega )=\\int d^3{\\bf r}^{\\prime }\\chi ^0({\\bf r},{\\bf r}^{\\prime },\\omega )\\phi ({\\bf r}^{\\prime },\\omega )\\equiv \\chi _0\\cdot \\phi ,$ where $\\chi ^0$ is the noninteracting susceptibility associated with the s valence electrons of gold, and the last identity defines a matrix notation in which matrix multiplication involves integration over space coordinates.", "We obtain $\\chi ^0$ within the RPA[40], in which a one-electron picture is assumed and only individual electron-hole pair excitations are explicitly considered.", "We further approximate the wave functions of valence electrons by the solutions of a cylindrical box with the same diameter as the nanodisk and a thickness corresponding to the separation between (111) atomic planes in bulk gold (i.e., $a_0/\\sqrt{3}\\approx 0.236\\,$ nm, see Fig.", "1a).", "More precisely, $\\psi _{lm}\\left(\\textbf {r}\\right)=N_{lms}J_{m}\\;\\left(Q_{lm}R\\right){\\rm e}^{{\\rm i}m\\varphi }g_1\\left(z\\right),$ where $N_{lms}$ is a normalization constant, $Q_{lm}=2\\zeta _{lm}/D$ , $\\zeta _{lm}$ is the $l^{\\rm th}$ zero of the Bessel function $J_m$ , and $g_1\\left(z\\right)=\\sin \\left(\\pi \\sqrt{3}z/a_0\\right)$ yields the dependence on the coordinate $z$ normal to the disk.", "For simplicity, we are assuming that the $z$ dependence is separable in the wave function, so that electron diffraction effects at the disk edges are not important.", "Moreover, we assume that the electrons remain in the ground state of the vertical cavity of thickness $a_0/\\sqrt{3}$ , which is a reasonable approximation if we consider that the first excited state (i.e., $g_2(z)=\\sin \\left(2\\pi \\sqrt{3}z/a_0\\right)$ ) lies $\\sim 20\\,$ eV above $g_1$ , well beyond the Fermi and vacuum levels.", "With the wave functions of Eq.", "(3), we can write the susceptibility as $\\chi ^0\\left(\\textbf {r},\\textbf {r}^{\\prime },\\omega \\right)=\\frac{2e^2}{\\hbar }\\sum _{l,l^{\\prime },m,m^{\\prime }}\\left(f_{l^{\\prime }m^{\\prime }}-f_{lm}\\right)\\frac{\\psi _{lm}\\left(\\textbf {r}\\right)\\psi ^{\\ast }_{lm}\\left(\\textbf {r}^{\\prime }\\right)\\psi ^{\\ast }_{l^{\\prime }m^{\\prime }}\\left(\\textbf {r}\\right)\\psi _{l^{\\prime }m^{\\prime }}\\left(\\textbf {r}^{\\prime }\\right)}{\\omega -\\varepsilon _{lm}+\\varepsilon _{l^{\\prime }m^{\\prime }}+{\\rm i}\\gamma /2},$ where spin degeneracy is simply included through an overall factor of 2, $\\hbar \\varepsilon _{lm}=\\hbar ^2 Q^2_{lm}/2 m_{\\rm e}$ is the energy of state $\\psi _{lm}$ (notice that the energy associated with $z$ motion cancels out in Eq.", "(4), so we disregard it), $\\gamma $ is an intrinsic relaxation time, which we take from a Drude fit (Eq.", "(1)) to measured optical data [29] ($\\hbar \\gamma =71\\,$ meV), and $f_{lm}=\\left\\lbrace \\exp \\left[\\left(\\varepsilon _{lm}-E_{\\rm F}\\right)/k_{\\rm B}T\\right]+1\\right\\rbrace ^{-1}$ is the Fermi-Dirac distribution function, evaluated here at $T=0$ .", "The method using to fill the energy levels in the disk is discussed in the Supplementary Fig.", "8.", "The total potential $\\phi $ is the sum of the external potential $\\phi ^{\\rm ext}$ and the potential produced by the induced charges $\\phi =\\phi ^{\\rm ext}+v\\cdot \\rho ^{\\rm ind},$ where $v({\\bf r}-{\\bf r}^{\\prime })=1/\|{\\bf r}-{\\bf r}^{\\prime }\|$ is the Coulomb interaction.", "From here and Eq.", "(2), we solve the induced charge density as $\\rho ^{\\rm ind}=\\chi ^0\\cdot \\left(1-v\\cdot \\chi ^0\\right)^{-1}\\cdot \\phi ^{\\rm ext}.$ As the polarization along the direction normal to the disk is expected to be negligible, we focus on parallel components and write $\\phi ^{\\rm ext}=-R{\\rm e}^{{\\rm i}\\varphi }$ (i.e., we focus on solutions with $m=1$ azimuthal symmetry), where $\\varphi $ is the azimuthal angle of $(x,y)$ and $R=\\sqrt{x^2+y^2}$ .", "This allows us to obtain the in-plane polarizability by calculating $\\alpha \\left(\\omega \\right)=\\frac{1}{2}\\int d^3\\textbf {r}\\;R{\\rm e}^{-{\\rm i}\\varphi }\\;\\rho ^{\\rm ind}\\left(\\textbf {r},\\omega \\right).$ Finally, the extinction cross-section is obtained from $\\sigma (\\omega )=\\left(4\\pi \\omega /c\\right)\\mbox{Im}\\left\\lbrace \\alpha \\left(\\omega \\right)\\right\\rbrace .$" ], [ "Inclusion of d-band screening", "Deeper electrons in the d band are relatively localized in the gold atoms, and therefore, we model them by assuming a background of polarizable point particles at the atomic positions in the (111) layer (see lower part of Fig.", "1a).", "The polarizability $\\alpha _{\\rm b}$ of these particles is adjusted to fit the experimentally measured bulk dielectric function of gold $\\epsilon _{\\rm exp}$ .", "That is, if we subtract the Drude s-band contribution from $\\epsilon _{\\rm exp}$ (see Eq.", "(1)), we obtain the background permittivity $\\epsilon _{\\rm b}=\\epsilon _{\\rm exp}+\\omega _{\\rm p}^2/\\omega (\\omega +{\\rm i}\\gamma )$ , where $\\omega _{\\rm p}^2=4\\pi e^2 n_0/m_{\\rm e}$ is determined by the s-band electron density $n_0=4/a_0^3\\approx 5.9\\times 10^{28}\\,$ m$^{-3}$ .", "This yields $\\hbar \\omega _{\\rm p}\\approx 9.01\\,$ eV, which is slightly different from the best fit of Eq.", "(1) to measured data [29] (9.06 eV), from which we also find $\\epsilon _{\\rm b}=9.5$ .", "Now the Clausius-Mossotti relation[42] leads to $\\alpha _{\\rm b}=\\frac{3}{4\\pi n_0}\\;\\frac{\\epsilon _{\\rm b}-1}{\\epsilon _{\\rm b}+2}.\\nonumber $ Using dyadic notation, the susceptibility tensor of the background dipoles reduces to $\\chi _{\\rm b}^0({\\bf r},{\\bf r}^{\\prime })=\\sum _j\\overleftarrow{\\nabla }\\cdot \\alpha _{\\rm b}\\delta ({\\bf r}-{\\bf r}_j)\\delta ({\\bf r}^{\\prime }-{\\bf r}_j)\\cdot \\overrightarrow{\\nabla }^{\\prime }$ , where $j$ runs over the positions of the metal atoms, whereas $\\overleftarrow{\\nabla }$ ($\\overrightarrow{\\nabla }^{\\prime }$ ) acts on ${\\bf r}$ -dependent (${\\bf r}^{\\prime }$ -dependent) functions to the left (right) of operator $\\chi _{\\rm b}^0$ .", "As the charge induced through both s and d bands contribute together to the full potential, we can rewrite Eq.", "(2) as $\\rho ^{\\rm ind}=\\left(\\chi ^0+\\chi ^0_{\\rm b}\\right)\\cdot \\phi \\nonumber $ to take into account the effect of d-band screening.", "Using this expression together with Eq.", "(5), the total induced charge density becomes $\\rho ^{\\rm ind}=\\left(\\chi ^0+\\chi ^0_{\\rm b}\\right)\\cdot \\left[1-v\\cdot \\left(\\chi ^0+\\chi ^0_{\\rm b}\\right)\\right]^{-1}\\cdot \\phi ^{\\rm ext},$ from which we calculate the disk polarizability and the extinction cross-section as explained above." ], [ "Acknowledgments", "This work has been supported in part by the European Commission (Graphene Flagship CNECT-ICT-604391 and FP7-ICT-2013-613024-GRASP).", "A.M. acknowledges financial support from the Spanish MEC through the FPU program and from the Evans Attwell-Welch Postdoctoral Fellowship for Nanoscale Research, administered by the Richard E. Smalley Institute for Nanoscale Science and Technology." ] ]	1403.0084
[ [ "Nodeless superconducting gaps in\n Ca$_{10}$(Pt$_{4-\\delta}$As$_8$)((Fe$_{1-x}$Pt$_{x}$)$_2$As$_2$)$_5$ probed\n by quasiparticle heat transport" ], [ "Abstract The in-plane thermal conductivity of iron-based superconductor Ca$_{10}$(Pt$_{4-\\delta}$As$_8$)((Fe$_{1-x}$Pt$_{x}$)$_2$As$_2$)$_5$ single crystal (``10-4-8\", $T_c$ = 22 K) was measured down to 80 mK.", "In zero field, the residual linear term $\\kappa_0/T$ is negligible, suggesting nodeless superconducting gaps in this multiband compound.", "In magnetic fields, $\\kappa_0/T$ increases rapidly, which mimics those of multiband superconductor NbSe$_2$ and LuNi$_2$B$_2$C with highly anisotropic gap.", "Such a field dependence of $\\kappa_0/T$ is an evidence for multiple superconducting gaps with quite different magnitudes or highly anisotropic gap.", "Comparing with the London penetration depth results of Ca$_{10}$(Pt$_3$As$_8$)((Fe$_{1-x}$Pt$_{x}$)$_2$As$_2$)$_5$ (``10-3-8\") compound, the 10-4-8 and 10-3-8 compounds may have similar superconducting gap structure." ], [ "Introduction", "To understand the electronic pairing mechanism of a superconductor, it is very important to know the symmetry and structure of its superconducting gap.", "For the iron-based high-temperature superconductors, there are many families, such as LaO$_{1-x}$ F$_{x}$ FeAs (“1111”),[1] Ba$_{1-x}$ K$_{x}$ Fe$_{2}$ As$_{2}$ (“122”),[2] NaFe$_{1-x}$ Co$_x$ As (“111”),[3] and FeSe$_{x}$ Te$_{1-x}$ (“11”).", "[4] The most notable character of these families is the multiple Fermi surfaces, which may be the reason why their superconducting gap structure is so complicated.", "[5], [6] Different from other families, the new compounds Ca$_{10}$ (Pt$_3$ As$_8$ )((Fe$_{1-x}$ Pt$_x$ )$_2$ As$_2$ )$_5$ (“10-3-8\") and Ca$_{10}$ (Pt$_{4}$ As$_8$ )((Fe$_{1-x}$ Pt$_x$ )$_2$ As$_2$ )$_5$ (“10-4-8\") consist of semiconducting [Pt$_{3}$ As$_{8}$ ] layers or metallic [Pt$_{4}$ As$_{8}$ ] layers sandwiched between [Fe$_{10}$ As$_{10}$ ] layers, and show superconductivity with maximal $T_c \\sim $ 15 and 38 K, respectively.", "[7], [8], [9] The metallic [Pt$_{4}$ As$_{8}$ ] layers lead to stronger FeAs interlayer coupling in 10-4-8 compound, thus higher $T_c$ as compared to the 10-3-8 compound.", "[8] The upper critical field of both 10-3-8 and 10-4-8 compounds show strong anisotropy.", "[10], [11] For the 10-3-8 compound, the London penetration depth exhibits power-law variation, which suggests nodeless superconducting gap.", "[12] For the 10-4-8 compound, the angle-resolved photoemission spectroscopy (ARPES) experiments have revealed a multiband electronic structure,[13], [14] but so far there is still no any investigation of its superconducting gap structure.", "As the 10-4-8 has a much higher $T_c$ than the 10-3-8 compound, it will be interesting to study its superconducting gap structure and compare with the 10-3-8 compound.", "Low-temperature thermal conductivity measurement is a bulk technique to study the superconducting gap structure.", "[15] According to the magnitude of residual linear term $\\kappa _0/T$ in zero field, one may judge whether there exist gap nodes or not.", "The field dependence of $\\kappa _0/T$ can give further information on nodal gap, gap anisotropy, or multiple gaps.", "[15] In this paper, we measure the thermal conductivity of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ ($T_c$ = 22 K) single crystal down to 80 mk.", "A negligible residual linear term $\\kappa _0/T$ is found in zero magnetic field.", "The field dependence of $\\kappa _0/T$ is very similar to those in multigap $s$ -wave superconductor NbSe$_2$ and LuNi$_2$ B$_2$ C with highly anisotropic gap.", "Our data strongly suggest Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ has nodeless superconducting gaps.", "The magnitudes of these gaps could be quite different, or some gap may be anisotropic." ], [ "Experiment", "Single crystals of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ ($T_c$ = 22 K) were grown by the flux method.", "[13] The composition of the sample was determined as Ca:Fe:Pt:As = 2:1.73:0.79:3.39 by wavelength-dispersive spectroscopy (WDS), utilizing an electron probe microanalyzer (Shimadzu EPMA-1720).", "This doping level is close to the $T_c$ = 26 K sample with Ca:Fe:Pt:As = 2:1.8:0.9:3.5 in Ref.", "8, which has the chemical formula Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{0.97}$ Pt$_{0.03}$ )$_2$ As$_2$ )$_5$ ($\\delta $ = 0.26) determined by single crystal structure refinement.", "The dc magnetization was measured at $H$ = 20 Oe, with zero-field cooling process, using a SQUID (MPMS, Quantum Design).", "The sample was cleaved to a rectangular shape with dimensions of 1.5 $\\times $ 0.74 mm$^2$ in the $ab$ plane and $\\sim $ 25 $\\mu $ m along the $c$ axis.", "Contacts were made directly on the fresh sample surfaces with silver paint, which were used for both resistivity and thermal conductivity measurements.", "In-plane thermal conductivity was measured in a dilution refrigerator using a standard four-wire steady-state method with two RuO$_2$ chip thermometers, calibrated $in$ $situ$ against a reference RuO$_2$ thermometer.", "Magnetic fields are applied along the $c$ axis.", "To ensure a homogeneous field distribution in the samples, all fields are applied at a temperature above $T_c$ ." ], [ "Results and Discussion", "Figure 1(a) shows the in-plane resistivity of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ single crystal in zero field.", "Defined by $\\rho $ = 0, $T_c$ = 22.2 K is obtained.", "The solid line is a fit of the data between 50 and 125 K to $\\rho = \\rho _0 +AT^\\alpha $ , which gives residual resistivity $\\rho _0$ = 82.5 $\\mu \\Omega $ cm and $\\alpha = 1.15$ .", "The dc magnetization is shown in Fig.", "1(b), and a slightly lower $T_c$ = 21.7 K is found.", "Blow we take $T_c$ = 22 K. This value is lower than the $T_c$ = 38 K at optimal doping.", "Since the phase diagram, $T_c$ vs $x$ ($\\delta $ ), of 10-4-8 system has not been well studied, it is not sure that our sample is underdoped or overdoped.", "Figure 2(a) shows the low-temperature resistivity of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ single crystal in magnetic fields up to 14.5 T. The superconducting transition becomes broader and the $T_c$ decreases with increasing field.", "The temperature dependence of upper critical field $H_{c2}(T)$ , defined by $\\rho =0$ in Fig.", "2(a), is plotted in Fig.", "2(b).", "To estimate the zero-temperature limit of $H_{c2}$ , one usually fits the curve according to the Werthamer-Helfand-Hohenberg (WHH) theory.", "[16] However, our $H_{c2}(T)$ with upward curvature apparently can not be fitted well by WHH formula.", "As explained in Ref.", "11, the underlying reason is that the WHH theory is for superconductors with single band, while the iron-based superconductors have multiple bands.", "In Ref.", "11, the $H_{c2}^{\|\|c}(T)$ curve of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{0.97}$ Pt$_{0.03}$ )$_2$ As$_2$ )$_5$ ($T_c$ = 26 K) sample was fitted by the two-band model,[17] giving $H_{c2}^{\|\|c}(0) \\approx $ 90 T. Taking the same process as in Ref.", "11, we also fit the $H_{c2}(T)$ data in Fig.", "2(b) with the two-band model, and get $H_{c2}$ (0) = 52 T for our Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ ($T_c$ = 22 K) sample.", "Note that a slightly different $H_{c2}$ (0) does not affect our discussion on the field dependence of normalized $\\kappa _0/T$ blow.", "Figure: (Color online)Low-temperature thermal conductivity ofCa 10 _{10}(Pt 4-δ _{4-\\delta }As 8 _8)((Fe 1-x _{1-x}Pt x _{x}) 2 _2As 2 _2) 5 _5with magnetic fields applied along the cc axis (HH = 0, 2, 3, 6, 9, and 12 T).The solid lines are κ/T\\kappa /T = a+bT α-1 a + bT^{\\alpha -1} fits, and the dashed line is the normal-stateWiedemann-Franz law expectation L 0 L_0/ρ 0 \\rho _0.In Fig.", "3, the temperature dependence of the in-plane thermal conductivity for Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ in $H$ = 0, 2, 3, 6, 9, and 12 T magnetic fields are plotted as $\\kappa /T$ vs $T$ .", "To get the residual linear term $\\kappa _0/T$ , we fit the curves to $\\kappa /T$ = $a + bT^{\\alpha -1}$ blow 0.4 K, in which the two terms $aT$ and $bT^{\\alpha }$ come from contributions of electrons and phonons, respectively.", "[18], [19] The power $\\alpha $ of the phonon term is typically between 2 and 3, due to the specular reflections of phonons at the sample surfaces.", "[18], [19] In zero field, the fitting gives $\\kappa _0/T$ = 0.005 $\\pm $ 0.013 mW K$^{-2}$ cm$^{-1}$ and $\\alpha $ = 2.57 $\\pm $ 0.03.", "Such a tiny value of $\\kappa _0/T$ is within our experimental error bar $\\pm $ 0.005 mW K$^{-2}$ cm$^{-1}$ .", "[19] Therefor it is negligible, comparing to the normal-state Wiedemann-Franz law expectation $L_0$ /$\\rho _0$ = 0.297 mW K$^{-2}$ cm$^{-1}$ , with $L_0$ = 2.45 $\\times $ 10$^{-8}$ W$\\Omega $ K$^{-2}$ and $\\rho _0$ = 82.5 $\\mu \\Omega $ cm.", "The absence of $\\kappa _0/T$ in zero field means that there are no fermionic quasiparticles to conduct heat as $T\\rightarrow 0$ , which provides bulk evidence for nodeless superconducting gaps in Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ , at least in the $ab$ plane.", "The data in magnetic fields $H$ = 2, 3, 6, 9, and 12 T are also fitted, as seen in Figs.", "3(b)-3(f).", "The $\\kappa _0/T$ increases significantly with increasing field, although the maximum applied field $H$ = 12 T is only about 23% of the $H_{c2}$ (0) = 52 T. Figure: (Color online)Normalized residual linear termκ 0 /T\\kappa _0/T ofCa 10 _{10}(Pt 4-δ _{4-\\delta }As 8 _8)((Fe 1-x _{1-x}Pt x _{x}) 2 _2As 2 _2) 5 _5as a function of H/H c2 H/H_{c2}.", "For comparison, similar data are shown for the cleanss-wave superconductor Nb, the dirty ss-wavesuperconducting alloy InBi, the multiband ss-wavesuperconductor NbSe 2 _2, the ss-wave superconductor LuNi 2 _2B 2 _2C with highly anisotropic gap,an overdoped dd-wave cuprate superconductor Tl-2201, and the iron-based superconductor BaFe 1.73 _{1.73}Co 0.27 _{0.27}As 2 _2.To see the field dependence of $\\kappa _0/T$ more clearly, the normalized $\\kappa _0/T$ of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ as a function of $H/H_{c2}$ is plotted in Fig.", "4.", "Similar data are shown for the clean $s$ -wave superconductor Nb, [20] the dirty $s$ -wave superconducting alloy InBi, [21] the multiband $s$ -wave superconductor NbSe$_2$ , [22] the $s$ -wave superconductor LuNi$_2$ B$_2$ C with highly anisotropic gap,[23] an overdoped $d$ -wave cuprate superconductor Tl$_2$ Ba$_2$ CuO$_{6+\\delta }$ (Tl-2201), [24] and the iron-based superconductor BaFe$_{1.73}$ Co$_{0.27}$ As$_2$ .", "[25] For clean $s$ -wave superconductor with a single isotropic gap, $\\kappa _0/T$ is expected to grow exponentially with increasing $H$ , as found in Nb.", "[20] In a $d$ -wave superconductor, $\\kappa _0/T$ increases roughly proportional to $\\sqrt{H}$ at low field due to the Volovik effect,[26] as found in Tl-2201.", "[24] From Fig.", "4, the normalized $\\kappa _0/T$ of our 10-4-8 compound starts from a negligible value at zero field, then increases very rapidly with increasing field.", "This behavior clearly mimics those of NbSe$_2$ and LuNi$_2$ B$_2$ C. [22], [23] For the multiband $s$ -wave superconductor NbSe$_2$ , the gap on the $\\Gamma $ band is approximately one third of the gap on the other two Fermi surfaces and magnetic field first suppresses the superconductivity on the Fermi surface with smaller gap.", "[22] For the $s$ -wave superconductor LuNi$_2$ B$_2$ C with highly anisotropic gap, the gap minimum $\\Delta _{min}$ is at least 10 times smaller than the gap maximum, $\\Delta _{min} \\le \\Delta _0/10$ .", "[23] The nearly identical field dependence of normalized $\\kappa _0/T$ between NbSe$_2$ and LuNi$_2$ B$_2$ C indicates that bulk thermal conductivity measurement can not distinguish these two kinds of superconducting gap structures.", "Nevertheless, the field dependence of $\\kappa _0/T$ suggests that the nodeless superconducting gaps in multiband 10-4-8 compound may have quite different magnitudes, or some gap could be anisotropic.", "Note that similar field dependence of $\\kappa _0/T$ was also observed in iron-based superconductors BaFe$_{1.73}$ Co$_{0.27}$ As$_2$ and FeSe$_x$ .", "[25], [27] In a theoretical calculation of $\\kappa _0(H)/T$ with unequal size of isotropic $s_{\\pm }$ -wave gaps, the shape of $\\kappa _0(H)/T$ changes systematically with the gap size ratio $\\Delta _{S}/\\Delta _{L}$ .", "[28] In case of isotropic $s$ -wave gaps with unequal size, the ratio $\\Delta _{S}/\\Delta _{L} \\approx 1/4$ is estimated for our 10-4-8 compound, by comparing with the theoretical curves.", "However, we can not rule out that some gap may be anisotropic.", "In fact, the robust power-law variation of London penetration depth observed in 10-3-8 compound was interpreted as a multigap behavior, and the anisotropy of some superconducting gaps may increase towards the edges of the superconducting dome.", "[12] In the sense that both thermal conductivity and London penetration depth measurements are bulk probe of the low-energy quasiparticles, the 10-4-8 and 10-3-8 compounds may have similar superconducting gap structure." ], [ "Summary", "In summary, we have measured the thermal conductivity of Ca$_{10}$ (Pt$_{4-\\delta }$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ single crystal down to 80 mK.", "The absence of $\\kappa _0/T$ in zero field gives strong evidence for nodeless superconducting gaps in such a multiband compound.", "The rapid field dependence of $\\kappa _0/T$ suggests multiple superconducting gaps with quite different magnitudes or highly anisotropic gap, which may be similar to that of Ca$_{10}$ (Pt$_3$ As$_8$ )((Fe$_{1-x}$ Pt$_{x}$ )$_2$ As$_2$ )$_5$ compound.", "ACKNOWLEDGEMENTS This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No.", "2012CB821402), and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.", "$^*$ E-mail: shiyan$\\_$ [email protected]" ] ]	1403.0194
[ [ "A 4-approximation for scheduling on a single machine with general cost\n function" ], [ "Abstract We consider a single machine scheduling problem that seeks to minimize a generalized cost function: given a subset of jobs we must order them so as to minimize $\\sum f_j(C_j)$, where $C_j$ is the completion time of job $j$ and $f_j$ is a job-dependent cost function.", "This problem has received a considerably amount of attention lately, partly because it generalizes a large number of sequencing problems while still allowing constant approximation guarantees.", "In a recent paper, Cheung and Shmoys provided a primal-dual algorithm for the problem and claimed that is a 2-approximation.", "In this paper we show that their analysis cannot yield an approximation guarantee better than $4$.", "We then cast their algorithm as a local ratio algorithm and show that in fact it has an approximation ratio of $4$.", "Additionally, we consider a more general problem where jobs has release dates and can be preempted.", "For this version we give a $4\\kappa$-approximation algorithm where $\\kappa$ is the number of distinct release dates." ], [ "Introduction", "We study the problem of scheduling jobs on a single machine with generalized min-sum cost function.", "Consider a set of jobs $\\mathcal {J}$ each with a given processing time $p_j$ and an arbitrary non-decreasing non-negative cost function $f_j$ .", "We aim to find a single machine schedule that minimizes $\\sum _jf_j(C_j)$ , where $C_j$ denotes the completion time of job $j$ .", "In the 3-field notation by Graham et al.", "[8] this problem is denoted by $1\|\|\\sum f_j$ .", "The problem is strongly NP-hard even in the case where the cost functions are of the form $f_j= w_j \\max \\lbrace C_j-d_j,0\\rbrace $ , which corresponds to minimizing weighted total tardiness [10].", "A natural variant additionally considers a release date $r_j$ for each job $j\\in \\mathcal {J}$ and seeks a preemptive schedule with the same min-sum objective.", "In the case that all release dates are zero, preempting a job cannot help decrease the objective function, and therefore this problem generalizes $1\|\|\\sum f_j$ .", "The more general version is denoted by $1\|r_j,\\text{pmtn}\|\\sum f_j$ .", "Bansal and Pruhs [1] were the first to study these two problems: They gave a 16-approximation for $1\|\|\\sum f_j$ and an $O(\\log \\log (n \\max _j p_j))$ -approximation for $1\|r_j,\\text{pmtn}\|\\sum f_j$ .", "For the case without release dates Cheung and Shmoys [6] give a primal-dual algorithm and claim that has an approximation ratio of 2.", "Their approach is based on a natural time-indexed LP relaxation strengthen with the so called knapsack-cover inequalities, introduced by Carr et al. [5].", "Although their algorithm has pseudo-polynomial running time, with standard techniques they can make the algorithm polynomial by only increasing the approximation ratio in a $1+\\varepsilon $ factor.", "Our first result is an example showing that the analysis of Cheung and Shmoys cannot imply an approximation guarantee better than 4.", "Then we interpret their algorithm in the local ratio framework and show that it is in fact 4-approximate.", "We also give a natural generalization of the techniques to $1\|r_j,\\text{pmtn}\|\\sum f_j$ that yields a $4\\kappa $ -approximation algorithm, where $\\kappa $ is the number of distinct release dates.", "With the same technique of Cheung and Shmoys these algorithms can be made to run in polynomial time by loosing a $1+\\varepsilon $ factor in the approximation guarantee." ], [ "Previous Work", "Besides the previously mentioned results by Bansal and Pruhs [1] and Cheung and Shmoys [6], there is plenty of literature on special cases of the problem.", "Epstein et al.", "[7] study the problem $1\|\|\\sum _j w_j f$ , corresponding to the case where functions $f_j$ are a scaled versions of a common function $f$ .", "This problem is equivalent to minimize $\\sum _j w_j C_j$ on a machine that changes its speed over time.", "They give an algorithm that computes a sequence of jobs which simultaneously yields a $(4+\\varepsilon )$ -approximate schedule for any function $f$ .", "Using randomization this result can be improved to a $(e+\\varepsilon )$ -approximation.", "They also consider the version with release dates and obtain analogous results.", "However, in this setting a schedule is specified by a priority order used in a preemptive list schedule procedure.", "Very recently Megow and Verschae [11] consider the version without release dates, $1\|\|\\sum _j w_j f$ , and obtain a PTAS.", "This result is best possible since the problem is strongly NP-hard.", "Höhn and Jacobs [9] study the problem for a convex or concave function $f$ .", "Their main result is a method to compute the exact approximation ratio of Smith's rule.", "Additionally, they show that the problem is strongly NP-hard even for piece-wise linear functions $f$ .", "In its full generality, $1\|r_j,\\text{pmtn}\|\\sum f_j$ models several problems that are far from being understood from an approximation point of view.", "The most prominent of these problems is minimizing weighted flow time $1\|r_j\|\\sum _j w_j (C_j-r_j)$ .", "Here $C_j-r_j$ is the flow time of job $j$ , i.e., the amount of time that the job is alive until it is served.", "Another related objective is minimizing squared flow time $\\sum _j(C_j-r_j)^2$ , or more generally $\\sum _j (C_j-r_j)^k$ for some $k\\in \\left\\lbrace 2,3,\\ldots \\right\\rbrace $ , which were recently proved to be NP-hard by Moseley et al. [12].", "On the positive side, the best approximation guarantee for all these problems is the $O(\\log \\log (n\\max _j p_j))$ -approximation by Bansal and Pruhs [1].", "In principle there is no theoretical reason to rule out the existence of a PTAS, even for the general problem $1\|r_j,\\text{pmtn}\|\\sum f_j$ .", "This remains one of the most intriguing questions in this area.", "The local ratio technique has been widely used in the design of approximation algorithms [3].", "Local-ratio and primal-dual algorithms are closely related; indeed, the two frameworks are known to be equivalent [4].", "Our local ratio interpretation of the Cheung-Shmoys algorithm is inspired by the local ratio algorithm for the resource allocation problem studied by Bar-Noy et al. [2].", "In Section we review the Cheung-Shmoys algorithm.", "As mentioned before, the algorithm is based on a time-index LP relaxation.", "This LP is used in a primal-dual framework that construct simultaneously a primal and a dual solution.", "Here the primal solution defines a schedule while the dual serves to define a lower bound on the optimal cost.", "In Section REF we give an instance where the algorithm returns a primal solution whose cost is roughly 4 times the cost of the dual solution constructed.", "This implies that the primal-dual analysis by Cheung and Shmoys showing an approximation guarantee of 2 is incorrect; in Appendix we discuss the precise issue with their proof.", "We must remark, however, that our instance does not rule out that the algorithm could be a 2-approximation.", "However, it does show that a different proof strategy would be necessary.", "Our main result is given in Section , where we show that the Cheung-Shmoys is in fact a pseudo-polynomial time 4-approximation.", "We do this by interpreting their approach in a local ratio framework.", "Finally, we generalize this algorithm to the setting with release dates in Section , obtaining a pseudo-polynomial time $4\\kappa $ -approximationThe reason of presenting the algorithm in the local ratio framework is twofold.", "First, we believe that a different interpretation helps shed some light on the techniques of Cheung and Shmoys.", "Second, it greatly simplifies the notation, specially for the case with release dates..", "Both algorithms can be modified so as to run in polynomial time at the expense of increasing the approximation factor by $\\varepsilon $ ." ], [ "The Cheung-Shmoys algorithm", "Before describing the Cheung-Shmoys algorithm we need to introduce some notation.", "We assume the processing times $p_j$ are integral.", "Let $\\mathcal {T} = \\left\\lbrace 1, \\ldots , T \\right\\rbrace $ be the set of time slots our schedule is allowed to use and $\\mathcal {J} = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ be the set of jobs we are to schedule.", "Note that $T = \\sum _{j \\in \\mathcal {J}} p_j$ .", "For any given time $t\\in \\mathcal {T}$ we denote by $D(t)=T-t+1$ the demand at time $t$ .", "The total processing time of jobs finishing at $t$ or later needs to be at least $D(t)$ .", "Moreover, given $t \\in \\mathcal {T}$ and a subset of jobs $A \\subseteq \\mathcal {J}$ we define $D(t,A) = \\max \\left\\lbrace 0, T- t + 1 - p(A) \\right\\rbrace $ , which corresponds to the demand to be cover at $t$ even if jobs in $A$ finish at $t$ or later.", "We refer to $D(t,A)$ as the residual demand at time $t$ given set $A$ .", "Similarly, $p_j(t,A) = \\min \\left\\lbrace p_j, D(t,A) \\right\\rbrace $ is the truncated processing time of job $j$ at time $t$ given set $A$ .", "The interpretation of this value is as follows: if all jobs in $A$ are processed at $t$ or later, $p_j(t,A)$ is the maximum amount of residual demand $D(t,A)$ that job $j$ can cover.", "The algorithm is based on the following pair of primal and dual linear programs.", "The primal uses binary variables $x_{j,t}$ that indicate the time $t$ at which job $j$ completes.", "$\\begin{array}{cr@{\\hspace{3.0pt}}l@{\\hspace{20.0pt}}{l}}\\vspace{8.00003pt} \\mathrm {minimize}& \\displaystyle \\sum _{j \\in \\mathcal {J}} \\sum _{t \\in \\mathcal {T}} f_j(t) \\, x_{j,t} \\\\\\mathrm {subject\\ to}& \\displaystyle \\sum _{j \\notin A} \\, \\sum _{s \\in \\mathcal {T}: s \\ge t} p_j(t,A)\\, x_{j,s} & \\ge D(t,A) & \\text{for all } t \\in \\mathcal {T} \\text{ and } A \\subseteq \\mathcal {J}, \\\\& x_{j,t} & \\ge 0 & \\text{for all } j \\in \\mathcal {J} \\text{ and } t \\in \\mathcal {T},\\end{array}$ $\\begin{array}{cr@{\\hspace{3.0pt}}l@{\\hspace{20.0pt}}{l}}\\vspace{8.00003pt} \\mathrm {maximize}& \\displaystyle \\sum _{t \\in \\mathcal {T}} \\sum _{A \\subseteq \\mathcal {J}} D(t,A) \\, y_{t,A} \\\\\\mathrm {subject\\ to}& \\displaystyle \\sum _{t \\in \\mathcal {T} : t \\le s} \\, \\sum _{A \\subseteq \\mathcal {J} : j \\notin A} p_j(t,A) \\, y_{t,A} & \\le f_j(s) & \\text{for all } j \\in \\mathcal {J} \\text{ and } s \\in \\mathcal {T}, \\\\\\, & y_{t,A} & \\ge 0 & \\text{for all } t \\in \\mathcal {T} \\text{ and } A \\subseteq \\mathcal {J}.", "\\\\\\end{array}$ The pseudo-code of the procedure appears in Algorithm .", "It has a primal growing phase followed by a pruning phase.", "In the growing phase (Lines –) the algorithm builds a tentative primal solution $x$ and a dual solution $y$ .", "In the pruning phase (Lines –) the algorithm resets some of the primal variables that are not needed anymore to maintain feasibility.", "Finally, in the analysis one shows that the cost of the primal solution is at most $\\beta $ times the cost of the dual solution, where $\\beta $ is the targeted approximation factor.", "[H] cheung-shmoys$(f,p)$ linenosize=, linenodelimiter=.", "[1] $x, y = 0$ primal and dual solutions $A_t = \\emptyset $ for each $t \\in \\mathcal {T}$ set of jobs assigned to $t$ or later by $x$ $k = 1$ iteration counter $\\exists t : D(t, A_t) > 0$ $t^k = \\operatorname{argmax}_t D(t, A_t)$ (break ties by choosing largest time index) Increase $y_{t^k, A_{t^k}}$ until a dual constraint with right-hand side $f_j(t)$ becomes tight (break ties by choosing largest time index) $x_{j,t} = 1$ $A_s = A_s \\cup \\left\\lbrace j \\right\\rbrace $ for all $s \\in \\mathcal {T} : s \\le t$ $k = k + 1$ $(j,t) : x_{j,t}= 1$ , in reverse order in which the variables were set $j \\in A_{t+1}$ $x_{j,t}= 0$ $S = \\left\\lbrace s \\le t : \\text{$j$ was added to $A_s$ in the same iteration we set $x_{j,t} = 1$} \\right\\rbrace $ $\\sum _{j^{\\prime } \\in A_s \\setminus \\left\\lbrace j \\right\\rbrace } p_{j^{\\prime }} \\ge D(s)$ for all $s \\in S$ $x_{j,t} = 0$ $A_s = A_s \\setminus \\left\\lbrace j \\right\\rbrace $ for all $s \\in S$ $j \\in \\mathcal {J}$ set due date $d_j$ to $t$ if $x_{j,t} = 1$ Schedule jobs using the EDD rule" ], [ "Counterexample", "We will show that a primal-dual analysis of cheung-shmoys cannot yield an approximation ratio better than 4.", "Appendix discusses the specific issue with the argument provided by Cheung and Shmoys [6].", "Lemma 1 For any $\\varepsilon >0$ there exists an instance where the cheung-shmoys algorithm constructs a pair of primal-dual solutions with a gap of $4-\\varepsilon $ .", "Consider an instance with 4 jobs.", "Let $p\\ge 4$ be an integer.", "For $j \\in \\left\\lbrace 1,2,3,4 \\right\\rbrace $ , we define the processing times as $p_j=p$ and the cost functions as $f_1(t) = f_2(t) & ={\\left\\lbrace \\begin{array}{ll}0 \\qquad \\qquad & \\text{if } 1\\le t \\le p-1,\\\\p \\qquad \\qquad & \\text{if } p\\le t \\le 3p-1,\\\\\\infty \\,\\quad & \\text{otherwise}, \\text{ and}\\end{array}\\right.", "}\\ $ $f_3(t) = f_4(t) & ={\\left\\lbrace \\begin{array}{ll}0 \\qquad \\qquad & \\text{if } 1\\le t \\le 3p-2,\\\\p & \\text{otherwise}.\\end{array}\\right.", "}$ The following table shows the key variables of the algorithm in each iteration of the growing phase and the corresponding updates to the dual and primal solutions.", "It is worth noting that the algorithm breaks ties by favoring the largest time index (Lines and ) and that our example follows this rule.", "Table: NO_CAPTIONNotice that the only non-zero dual variable the algorithm sets is $y_{3p-1,\\emptyset }=1$ .", "Thus the dual value achieved is $y_{3p-1,\\emptyset }D(3p-1,\\emptyset ) = p+2$ .", "It is not hard to see that the pruning phase keeps the largest due date for each job and has cost $4p$ .", "In fact, it is not possible to obtain a primal (integral) solution with cost less than $4p$ : We must pay $p$ for each job 3 and 4 in order to cover the demand at time $3p$ , and we must pay $p$ for each job 1 and 2 since they cannot finish before time $p$ .", "Therefore the pair of primal-dual solutions have a gap of $4p/(p+2)$ , which converges to 4 as $p$ tends to infinity.", "The attentive reader would complain that the cost functions used in the proof Lemma REF are somewhat artificial.", "Indeed, jobs 1 and 2 cost 0 only in $[0,p-1]$ even though it is not possible to finish them before $p$ .", "This is, however, not an issue since given any instance $(f, p)$ of the problem we can obtain a new instance $(f^{\\prime }, p^{\\prime })$ where $f^{\\prime }_j(t) \\ge f^{\\prime }_j(p^{\\prime }_j)$ for all $t$ where we observe essentially the same primal-dual gap in $(f,p)$ and $(f^{\\prime }, p^{\\prime })$ .", "The transformation is as follows: First, we create a dummy job with processing time $T =\\sum _{j} p_j$ that costs 0 up to time $T$ and infinity after that.", "Second, for each of the original jobs $j$ , we keep their old processing times, $p^{\\prime }_j =p_j$ , but modify their cost function: $f^{\\prime }_j(t) = {\\left\\lbrace \\begin{array}{ll}\\delta p_j & \\text{if } t \\ge T, \\\\\\delta p_j + f_j(t - T) & \\text{if } T < t \\le 2T.\\end{array}\\right.", "}$ In other words, to obtain $f^{\\prime }_j$ we shift $f_j$ by $T$ units of time to the right and then add $\\delta p_j$ everywhere, where $\\delta $ is an arbitrarily small value.", "Consider the execution of cheung-shmoys on the modified instance $(f^{\\prime },p^{\\prime })$ .", "In the first iteration, the algorithm sets $y_{1, \\emptyset }$ to 0 and assigns the dummy job to time $T$ .", "In the second iteration, the algorithm chooses to increase the dual variable $y_{T+1, \\emptyset }$ .", "Imagine increasing this variable in a continuous way and consider the moment when it reaches $\\delta $ .", "At this instant, the slack of the dual constraints for times in $[T+1, 2T]$ in the modified instance are identical to the slack for times in $[1, T]$ at the beginning of the execution on the original instance $(f,p)$ .", "From this point in time onwards, the execution on the modified instance will follow the execution on the original instance but shifted $T$ units of time to the right.", "The modified instance gains only an extra $\\delta T$ of dual value, which can be made arbitrarily small, so we observe essentially the same primal-dual gap on $(f^{\\prime }, p^{\\prime })$ as we do on $(f, p)$ ." ], [ "A 4-approximation via local-ratio", "In this section we cast the primal-dual algorithm of Cheung and Shmoys as a local-ratio algorithm and prove that it is a pseudo-polynomial time 4-approximation.", "At the end of the section, we discuss how to turn this algorithm into a polynomial time $(4 + \\varepsilon )$ -approximation.", "We will work with due date assignment vectors $\\mathbf {\\sigma }=(\\sigma _1, \\ldots ,\\sigma _n) \\in (\\mathcal {T}\\cup \\lbrace 0\\rbrace )^n$ , where $\\sigma _j=t$ means that job $j$ has a due date of $t$ .", "We will use the short-hand notation $(\\mathbf {\\sigma }_{-j}, s)$ to denote the assignment where $j$ is given a due date $s$ and all other jobs get their $\\mathbf {\\sigma }$ due date; that is, $ (\\mathbf {\\sigma }_{-j}, s) = (\\sigma _1, \\ldots , {\\sigma }_{j-1}, s, {\\sigma }_{j+1}, \\ldots , {\\sigma }_n).", "$ We call an assignment $\\mathbf {\\sigma }$ feasible, if there is a schedule of the jobs that meets all due dates.", "We say that job $j\\in \\mathcal {J}$ covers time $t$ if $\\sigma _j \\ge t$ .", "The cost of $\\mathbf {\\sigma }$ under the cost function vector $\\mathbf {g}=(g_1, \\ldots , g_n)$ is defined as $\\mathbf {g}(\\mathbf {\\sigma }) = \\sum _{j\\in \\mathcal {J}}g_j(\\sigma _j)$ .", "We denote by $A_t^{\\mathbf {\\sigma }}=\\left\\lbrace j\\in \\mathcal {J}: \\sigma _j\\ge t \\right\\rbrace $ , the set of jobs that cover $t$ .", "We call $ D(t, \\mathbf {\\sigma }) = D(t, A^{\\mathbf {\\sigma }}_{t})= \\max \\left\\lbrace T - t + 1 - p(A_t^{\\mathbf {\\sigma }}), 0 \\right\\rbrace $ the residual demand at time $t$ with respect to assignment $\\mathbf {\\sigma }$ .", "And $ p_j(t, \\mathbf {\\sigma }) = p_j (t, A^{\\mathbf {\\sigma }}_t) = \\min \\left\\lbrace p_j, D(t, \\mathbf {\\sigma }) \\right\\rbrace $ the truncated processing time of $j$ with respect to $t$ and $\\mathbf {\\sigma }$ .", "The following is a well-known fact from Scheduling Theory.", "An assignment $\\mathbf {\\sigma }$ is feasible if there is no residual demand at any time step; namely, if $D(t, \\mathbf {\\sigma })=0$ for all $t \\in \\mathcal {T}$ .", "Furthermore, scheduling the jobs according to earliest due date first (EDD) yields a feasible schedule.", "For a fix job $j$ , let $C_j$ be its completion time in the EDD schedule and $t=\\sigma _j$ its due date.", "We show that $C_j\\le t$ .", "Consider the set $A=\\left\\lbrace i \\in \\mathcal {J} : \\sigma _i \\ge \\sigma _j+1 \\right\\rbrace $ of jobs with due dates at $\\sigma _j+1$ or later.", "Since $\\mathbf {\\sigma }$ leaves no residual demand at $t+1$ , then $\\sum _{i\\in A} p_i \\ge D(t+1) = T-t$ .", "Noticing that all jobs in $A$ are processed after $j$ and that the schedule does not leave any idle time, we obtain that $C_j\\le T - \\sum _{i\\in A} p_i \\le T - (T-t) = t.$ At a very high level, the algorithm, which we call lr-cs, works as follows: We start by assigning a due date of 0 to all jobs; then we iteratively increase the due dates until the assignment is feasible; finally, we try to undo each increase in reverse order as long as it preserves feasibility.", "In the analysis we will argue that the due date assignment that the algorithm ultimately returns is feasible and that the cost of any schedule that meets the due dates is a 4-approximation.", "Together with Lemma this implies the main result in this section.", "Theorem 2 There is a pseudo-polynomial time 4-approximation algorithm for scheduling jobs on a single machine with generalized cost functions.", "Figure: lr-cs(σ,𝐠)(\\mathbf {\\sigma }, \\mathbf {g})We now describe the algorithm in more detail.", "Then we prove that is a 4-approximation.", "For reference, its pseudo-code is given in Algorithm REF ." ], [ "Formal description of the algorithm", "The algorithm is recursive.", "It takes as input an assignment vector $\\mathbf {\\sigma }$ and a cost function vector $\\mathbf {g}$ , and returns a feasible assignment $\\mathbf {\\rho }$ .", "Initially, the algorithm is called on the trivial assignment $(0, \\ldots , 0)$ and the instance cost function vector $(f_1,\\ldots , f_n)$ .", "As the algorithm progresses, both vectors are modified.", "We assume, without loss of generality, that $f_j(0) = 0$ for all $j \\in \\mathcal {J}$ .", "First, the algorithm checks if the input assignment $\\mathbf {\\sigma }$ is feasible.", "If that is the case, it returns $\\mathbf {\\rho } = \\mathbf {\\sigma }$ .", "Otherwise, it decomposes the input vector function $\\mathbf {g}$ into two cost function vectors $\\mathbf {\\widetilde{g}}$ and $\\mathbf {\\widehat{g}}$ as follows $ \\mathbf {g} = \\mathbf {\\widetilde{g}} + \\alpha \\cdot \\mathbf {\\widehat{g}}, $ where $\\alpha $ is largest value such that $\\mathbf {\\widetilde{g}} \\ge \\mathbf {0}$ By $\\mathbf {g} = \\mathbf {\\widetilde{g}} + \\alpha \\cdot \\mathbf {\\widehat{g}}$ , we mean $g_j(t) = \\widetilde{g}_j(t) + \\alpha \\cdot \\widehat{g}_j(t)$ for all $t \\in \\mathcal {T}$ and $j \\in \\mathcal {J}$ .", "By $\\mathbf {\\widetilde{g}} \\ge \\mathbf {0}$ , we mean $\\widetilde{g}_j(t) \\ge 0$ for all $j \\in \\mathcal {J}$ , $t \\in \\mathcal {T}$ .", ", and $\\mathbf {\\widehat{g}}$ will be specified later.", "It selects a job $j$ and a time $s$ such that $\\widehat{g}_j(s) > 0$ and $\\widetilde{g}_j(s) = 0$ , and builds a new assignment $\\mathbf {\\widetilde{\\sigma }}=(\\mathbf {\\sigma }_{-j}, s)$ thus increasing the due date of $j$ to $s$ while keeping the remaining due dates fixed.", "It then makes a recursive call lr-cs$(\\mathbf {\\widetilde{g}},\\mathbf {\\widetilde{\\sigma }})$ , which returns a feasible assignment $\\mathbf {\\widetilde{\\rho }}$ .", "Finally, it tests the feasibility of reducing the deadline of job $j$ in $\\mathbf {\\widetilde{\\rho }}$ back to $\\sigma _j$ .", "If the resulting assignment is still feasible, it returns that; otherwise, it returns $\\mathbf {\\widetilde{\\rho }}$ .", "The only part that remains to be specified is how to decompose the cost function vector.", "Let $t^$ be a time slot with maximum residual unsatisfied demand with respect to $\\mathbf {\\sigma }$ : $t^\\in \\operatorname{argmax}_{t\\in \\mathcal {T}} D(t, \\mathbf {\\sigma }).$ The algorithm creates, for each job $i \\in \\mathcal {J}$ , a model cost function $\\widehat{g}_i(t)={\\left\\lbrace \\begin{array}{ll}p_i(t^,\\mathbf {\\sigma }) & \\text{if } \\sigma _i < t^ \\le t, \\\\0 & \\text{otherwise}.", "\\\\\\end{array}\\right.", "}$ and chooses $\\alpha $ to be the largest value such that $\\widetilde{g}_i(t) = g_i(t) - \\alpha \\widehat{g}_i(t)\\ge 0 \\qquad \\text{for all } i\\in \\mathcal {J} \\text{ and } t \\in \\mathcal {T}.$ In the primal-dual interpretation of the algorithm, $\\alpha $ is the value assigned to the dual variable $y(t^,A_{t^}^{\\sigma })$ .", "Let $(j,s)$ be a job-time pair that prevented us from increasing $\\alpha $ further.", "In other words, let $(j,s)$ be such that $\\widetilde{g}_{j}(s) = 0$ and $\\widehat{g}_j(s) > 0$ .", "Observation 3 If $j \\in \\mathcal {J}$ and $s \\in \\mathcal {T}$ are such that $\\widehat{g}_j(s) > 0$ then $\\sigma _j< t^* \\le s$ .", "Intuitively, assigning a due date of $s$ to job $j$ helps cover some of the residual demand at $t^$ .", "This is precisely what the algorithm does: The assignment used as input for the recursive call is $\\mathbf {\\widetilde{\\sigma }} = (\\mathbf {\\sigma }_{-j}, s)$ ." ], [ "Analysis", "For a given vector $\\mathbf {g}$ of non-negative functions, we denote by $\\operatorname{opt}(\\mathbf {g})$ the cost of an optimal schedule under these cost functions.", "We say an assignment $\\mathbf {\\rho }$ is $\\beta $ -approximate with respect to $\\mathbf {g}$ if $\\sum _{i \\in \\mathcal {J}} g_i(\\rho _i) \\le \\beta \\cdot \\operatorname{opt}(\\mathbf {g})$ .", "The correctness of the algorithm rests on the following Lemmas.", "Let $(\\mathbf {\\sigma ^{(1)}}, \\mathbf {g^{(1)}}), (\\mathbf {\\sigma ^{(2)}}, \\mathbf {g^{(2)}}), \\ldots , (\\mathbf {\\sigma ^{(k)}}, \\mathbf {g^{(k)}})$ be the inputs to the successive recursive calls to lr-cs and let $\\mathbf {\\rho ^{(1)}}, \\mathbf {\\rho ^{(2)}}, \\ldots , \\mathbf {\\rho ^{(k)}}$ be their corresponding outputs.", "The following properties hold: $\\mathbf {\\sigma ^{(1)}} \\le \\mathbf {\\sigma ^{(2)}} \\le \\cdots \\le \\mathbf {\\sigma ^{(k)}}$ , $\\mathbf {\\rho ^{(1)}} \\le \\mathbf {\\rho ^{(2)}} \\le \\cdots \\le \\mathbf {\\rho ^{(k)}}$ , $\\mathbf {\\sigma ^{(i)}} \\le \\mathbf {\\rho ^{(i)}}$ for all $i =1, \\ldots , k$ , $g^{(i)}_j(\\sigma ^{(i)}_j) = 0$ and $g^{(i)}_j$ is non-negative for all $i=1, \\ldots , k$ and $j \\in \\mathcal {J}$ .", "The first property follows from the fact that $\\mathbf {\\sigma ^{(i+1)}}$ is constructed by taking $\\mathbf {\\sigma ^{(i)}}$ and increasing the due date of a single job.", "The second property follows from the fact that $\\mathbf {\\rho ^{(i)}}$ is either $\\mathbf {\\rho ^{(i+1)}}$ or it is constructed by taking $\\mathbf {\\rho ^{(i+1)}}$ and decreasing the due date of a single job.", "The third property follows by an inductive argument.", "The base case is the base case of the recursion, where $\\mathbf {\\sigma ^{(k)}} = \\mathbf {\\rho ^{(k)}}$ .", "For the recursive case, we need to show that $\\mathbf {\\sigma ^{(i)}} \\le \\mathbf {\\rho ^{(i)}}$ , by recursive hypothesis we know that $\\mathbf {\\sigma ^{(i+1)}}\\le \\mathbf {\\rho ^{(i+1)}}$ and by the first property $\\mathbf {\\sigma ^{(i)}}\\le \\mathbf {\\sigma ^{(i+1)}}$ .", "The algorithm either sets $\\mathbf {\\rho ^{(i)}} = \\mathbf {\\rho ^{(i+1)}}$ , or $\\mathbf {\\rho ^{(i)}}$ is constructed by taking $\\mathbf {\\rho ^{(i+1)}}$ and decreasing the due date of some job to its old $\\mathbf {\\sigma ^{(i)}}$ value.", "In both cases the property holds.", "The forth property also follows by induction.", "The base case is the first call we make to lr-cs, which is $\\mathbf {\\sigma ^{(1)}} = (0,\\ldots , 0)$ and $\\mathbf {g^{(1)}} = (f_1, \\ldots , f_n)$ , where it holds by our assumption.", "For the inductive case, we note that $\\mathbf {{g}^{(i+1)}}$ is constructed by taking $\\mathbf {{g}^{(i)}}$ and subtracting a scaled version of the model function vector, so that $\\mathbf {0} \\le \\mathbf {g^{(i+1)}} \\le \\mathbf {g^{(i)}}$ , and $\\mathbf {\\sigma ^{(i+1)}}$ is constructed by taking $\\mathbf {\\sigma ^{(i)}}$ and increasing the due date of a single job $j^{(i)}$ .", "The way this is done guarantees $g^{(i+1)}_{j^{(i)}} ( \\sigma ^{(i+1)}_{j^{(i)}}) = 0$ , which ensures that the property holds.", "Lemma 4 Let lr-cs$(\\mathbf {\\sigma }, \\mathbf {g})$ be a recursive call returning $\\mathbf {\\rho }$ then $\\sum _{i \\in \\mathcal {J}\\, :\\, \\sigma _i < t^ \\le \\rho _i} p_i(t^, \\mathbf {\\sigma })\\le 4 \\cdot D(t^, \\mathbf {\\sigma }).$ where $t^$ is the value used to decompose the input cost function vector $\\mathbf {g}$ .", "Our goal is to bound the $p_i(t^, \\mathbf {\\sigma })$ value of jobs in $X = \\left\\lbrace i \\in \\mathcal {J} : \\sigma _i < t^* \\le \\rho _i \\right\\rbrace .$ Notice that the algorithm increases the due date of these jobs in this or a later recursive call.", "Furthermore, and more important to us, the algorithm decides not to undo the increase.", "For each $i \\in X$ , consider the call lr-cs$(\\mathbf {\\sigma ^{\\prime }}, \\mathbf {g^{\\prime }})$ when we first increased the due date of $i$ beyond $\\sigma _i$ .", "Let $\\mathbf {{\\rho }^{\\prime }}$ be the assignment returned by the call.", "Notice that $\\rho ^{\\prime }_i > \\sigma _i$ and that $(\\mathbf {\\rho ^{\\prime }}\\!_{-i},\\sigma _i)$ is not feasible—otherwise we would have undone the due date increase.", "By Lemma REF , we know that $\\mathbf {\\rho } \\le \\mathbf {\\rho ^{\\prime }}$ , so we conclude that $(\\mathbf {\\rho }_{-i}, \\sigma _i)$ is not feasible either.", "Let $t_i$ be a time with positive residual demand in this unfeasible assignment: $ D(t_i, (\\mathbf {{\\rho }}_{-i}, \\sigma _i)) > 0.", "$ Note that $\\sigma _i < t_i \\le \\rho _i$ , otherwise $\\mathbf {\\rho }$ would not be feasible, contradicting Lemma REF .", "We partition $X$ into two subsets $L = \\left\\lbrace i\\in X: t_i \\le t^* \\right\\rbrace \\text{ and } R = \\left\\lbrace i\\in X: t_i > t^* \\right\\rbrace ,$ and we let $t_L = \\max \\left\\lbrace t_i : i \\in L \\right\\rbrace $ and $i_L$ be a job attaining this value.", "Similarly, we let $t_R = \\min \\left\\lbrace t_i : i \\in R \\right\\rbrace $ and $i_R$ be a job attaining this value.", "We will bound the contribution of each of these sets separately.", "Our goal will be to prove that $\\sum _{i \\in L - i_L} p_i & \\le D(t^, \\mathbf {\\sigma }), \\text{ and }\\\\\\sum _{i \\in R - i_R} p_i & \\le D(t^,\\mathbf {\\sigma }).$ Let us argue (REF ) first.", "Since $D\\left(t_L, (\\mathbf {\\rho }_{-i_L},\\sigma _{i_L})\\right) > 0$ , it follows that $\\sum _{i \\in \\mathcal {J} - i_L : \\rho _i \\ge t_L} p_i & < T-t_L + 1 \\\\\\sum _{i \\in \\mathcal {J} : \\sigma _i \\ge t_L} p_i + \\sum _{i \\in \\mathcal {J} - i_L : \\rho _i \\ge t_L > \\sigma _i} p_i & < T- t_L +1 \\\\\\sum _{i \\in \\mathcal {J} - i_L : \\rho _i \\ge t_L > \\sigma _i} p_i & < D(t_L, \\mathbf {\\sigma })$ Recall that $\\sigma _i < t_i \\le \\rho _i$ for all $i \\in X$ and that $t_i\\le t_L \\le t^$ for all $i \\in L$ .", "It follows that the sum on the left-hand side of the last inequality contains all jobs in $L - i_L$ .", "Finally, we note that $D(t_L, \\mathbf {\\sigma } )\\le D(t^, \\mathbf {\\sigma })$ due to the way lr-cs chooses $t^$ , which gives us (REF ).", "Now let us argue ().", "Since $D\\left(t_R, (\\mathbf {\\rho }_{-i_R},\\sigma _{i_R})\\right) > 0$ , it follows that $\\sum _{i \\in \\mathcal {J} - i_R : \\rho _i \\ge t_R} p_i & < T-t_R + 1 \\\\\\sum _{i \\in \\mathcal {J} : \\sigma _i \\ge t_R} p_i + \\sum _{i \\in \\mathcal {J} - i_R : \\rho _i \\ge t_R > \\sigma _i} p_i & < T- t_R +1 \\\\\\sum _{i \\in \\mathcal {J} - i_R : \\rho _i \\ge t_R > \\sigma _i} p_i & < D(t_R, \\mathbf {\\sigma }).$ Recall that $\\sigma _i < t^$ for all $i \\in X$ and that $t^* < t_R \\le t_i \\le \\rho _i$ for all $i \\in R$ .", "It follows that the sum in the left-hand side of the last inequality contains all jobs in $R- i_R$ .", "Finally, we note that $D(t_R, \\mathbf {\\sigma } )\\le D(t^, \\mathbf {\\sigma })$ due to the way lr-cs chooses $t^$ , which gives us ().", "Finally, we note that $p_i(t^, \\mathbf {\\sigma }) \\le D(t^, \\mathbf {\\sigma })$ for all $i \\in \\mathcal {J}$ .", "Therefore, $\\sum _{i \\in X} p_i(t^, \\mathbf {\\sigma })&\\le \\sum _{i \\in L-i_L} p_i + p_{i_L}(t^, \\mathbf {\\sigma }) +\\sum _{i \\in R-i_R} p_i + p_{i_R}(t^, \\mathbf {\\sigma }) \\\\& \\le 4 \\cdot D(t^, \\mathbf {\\sigma }).$ Lemma 5 Let lr-sc$(\\mathbf {\\sigma }, \\mathbf {g})$ be a recursive call and $\\mathbf {\\rho }$ be its output.", "Then $\\mathbf {\\rho }$ is a feasible 4-approximation w.r.t.", "$\\mathbf {g}$ .", "The proof is by induction.", "The base case corresponds to the base case of the recursion, where we get as input a feasible assignment $\\mathbf {\\sigma }$ , and so $\\mathbf {\\rho } = \\mathbf {\\sigma }$ .", "From Lemma REF we know that $g_i(\\sigma _i) =0$ for all $i \\in \\mathcal {J}$ , and that the cost functions are non-negative.", "Therefore, the cost of $\\mathbf {\\rho }$ is optimal since $ \\sum _{i \\in \\mathcal {J}} g_i(\\rho _i) = 0.", "$ For the inductive case, the cost function vector $\\mathbf {g}$ is decomposed into $\\mathbf {\\widetilde{g}} + \\alpha \\cdot \\mathbf {\\widehat{g}}$ .", "Let $(j,s)$ be the pair used to define $\\mathbf {\\widetilde{\\sigma }} = (\\mathbf {\\sigma }_{-j}, s)$ .", "Let $\\mathbf {\\widetilde{\\rho }}$ be the assignment returned by the recursive call.", "By inductive hypothesis, we know that $\\mathbf {\\widetilde{\\rho }}$ is feasible and 4-approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "After the recursive call returns, we check the feasibility of $(\\mathbf {\\widetilde{\\rho }}_{-j}, \\sigma _j)$ .", "If the vector is feasible, we return the modified assignment; otherwise, we return $\\mathbf {\\widetilde{\\rho }}$ .", "In either case $\\mathbf {\\rho }$ is feasible.", "We claim that $\\mathbf {\\rho }$ is 4-approximate w.r.t.", "$\\mathbf {\\widehat{g}}$ .", "Indeed, $ \\sum _{i \\in \\mathcal {J}} \\widehat{g}_i(\\rho _i) = \\sum _{i \\in \\mathcal {J}: \\sigma _i < t^\\le \\rho _i} p_i(t^, \\mathbf {\\sigma }) \\le 4 \\cdot D(t^,\\mathbf {\\sigma }) \\le 4 \\cdot \\operatorname{opt}(\\mathbf {\\widehat{g}}),$ where the first inequality follows from Lemma REF and the last inequality follows from the fact that the cost of any schedule under $\\mathbf {\\widehat{g}}$ is given by the $p_i(t^, \\mathbf {\\sigma })$ value of jobs $i\\in \\mathcal {J}$ with $\\sigma _i < t^* \\le \\rho _i$ , which must have a combined processing time of at least $D(t^, \\mathbf {\\sigma })$ on any feasible schedule.", "Hence, $\\operatorname{opt}(\\mathbf {\\widehat{g}}) \\ge D(t^, \\mathbf {\\sigma })$ .", "We claim that $\\mathbf {\\rho }$ is 4-approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "Recall that $\\mathbf {\\widetilde{\\rho }}$ is 4-approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ ; therefore, if $\\mathbf {\\rho } =\\mathbf {\\widetilde{\\rho }}$ then $\\mathbf {\\rho }$ is 4-approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "Otherwise, $\\mathbf {\\rho } = (\\mathbf {\\widetilde{\\rho }}_{-j}, \\sigma _j)$ , in which case $\\widetilde{g}_j(\\rho _j) = 0$ , so $\\mathbf {\\rho }$ is also 4-approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "At this point we can invoke the Local Ratio Theorem to get that $\\sum _{j \\in \\mathcal {J}} g_j(\\rho _j) &= \\sum _{j \\in \\mathcal {J}} \\widetilde{g}_j(\\rho _j) +\\sum _{j \\in \\mathcal {J}} \\alpha \\cdot \\widehat{g}_j(\\rho _j), \\\\& \\le 4 \\cdot \\operatorname{opt}(\\mathbf {\\widetilde{g}}) + 4 \\alpha \\cdot \\operatorname{opt}(\\mathbf {\\widehat{g}}), \\\\& = 4 \\cdot \\big ( \\operatorname{opt}(\\mathbf {\\widetilde{g}}) + \\operatorname{opt}(\\alpha \\cdot \\mathbf {\\widehat{g}}) \\big ), \\\\& \\le 4 \\cdot \\operatorname{opt}(\\mathbf {g}),$ which finishes the proof of the lemma.", "Note that the number of recursive calls in Algorithm REF is at most $\|\\mathcal {J}\|\\cdot \|\\mathcal {T}\|$ .", "Indeed, by Observation REF in each call the due date of some job is increased.", "Therefore we can only guarantee a pseudo-polynomial running time.", "To obtain a polynomial time algorithm we can use the exact same technique as Cheung and Shmoys and reduce the number of time slots by loosing a factor of $1+\\varepsilon $ .", "This is done by considering time intervals in which some cost function increases by a $1+\\varepsilon $ factor.", "This yields an algorithm with running time a polynomial in $\|\\mathcal {J}\|$ , $1/\\varepsilon $ , $\\log ( \\max _{j}p_j)$ , and $\\log (\\max _j f_j(T))$ ." ], [ "Release dates", "This section discusses how to generalize the ideas from the previous section to instances with release dates.", "We assume that there are $\\kappa $ different release dates.", "Our main result is a pseudo-polynomial $4\\kappa $ -approximation algorithm.", "The generalization is surprisingly easy: We only need to re-define our residual demand function to take into account release dates.", "For a given due date assignment vector $\\mathbf {\\sigma }$ and an interval $[r,t)$ we denote by $ D(r,t,\\mathbf {\\sigma }) = \\max \\left\\lbrace r + p \\left(\\left\\lbrace j \\in \\mathcal {J} : r \\le r_j \\le \\sigma _j < t \\right\\rbrace \\right)- t+ 1 ,0 \\right\\rbrace $ the residual demand for $[r,t)$ .", "Intuitively, this quantity is the amount of processing time of jobs released in $[r,t)$ that currently have a due date strictly less than $t$ that should be assigned a due date of $t$ or greater if we want feasibility.", "The truncated processing time of $j$ with respect to $r$ , $t$ , and $\\mathbf {\\sigma }$ is $ p_j(r,t, \\mathbf {\\sigma }) = \\min \\left\\lbrace p_j, D(r,t, \\mathbf {\\sigma }) \\right\\rbrace .$ The algorithm for multiple release dates is very similar to Algorithm REF .", "The only difference is in the way we decompose the input cost function vector $\\mathbf {g}$ .", "First, we find values $r^$ and $t^$ maximizing $D(r^, t^, \\mathbf {\\sigma })$ .", "Second, we define the model cost function for job each $i \\in \\mathcal {J}$ as follows $ \\widehat{g}_i(t) ={\\left\\lbrace \\begin{array}{ll}p_i(r^,t^, \\mathbf {\\sigma }) & \\text{if } r^* \\le r_i < t^* \\text{ and } \\sigma _i < t^* \\le t, \\\\0 & \\text{otherwise}.", "\\\\\\end{array}\\right.", "}$ [H] lr-cs-rd$(\\mathbf {\\sigma }, \\mathbf {g})$ linenosize=, linenodelimiter=.", "[1]$\\mathbf {\\sigma }$ is feasible $\\mathbf {\\rho }$ = $\\mathbf {\\sigma }$ $(t^, r^) = \\operatorname{argmax}_{(t, r) \\in \\mathcal {T} \\times R} D(r, t,\\mathbf {\\sigma })$ break ties arbitrarily For each $i \\in \\mathcal {J}$ let $\\widehat{g}_i(t)={\\left\\lbrace \\begin{array}{ll}p_i(r^,t^, \\mathbf {\\sigma }) & \\text{if } r^* \\le r_i < t^* \\text{ and } \\sigma _i < t^* \\le t, \\\\0 & \\text{otherwise}.\\end{array}\\right.", "}$ Set $\\mathbf {\\widetilde{g}} = \\mathbf {g} - \\alpha \\cdot \\mathbf {\\widehat{g}}$ where $\\alpha $ is the largest value such that $\\mathbf {\\widetilde{g}} \\ge 0$ Let $j$ and $s$ be such that $ \\widetilde{g}_j(s) = 0 \\text{ and } \\widehat{g}_j(s) > 0 $ $\\mathbf {\\widetilde{\\sigma }} = (\\mathbf {\\sigma }_{-j}, s)$ $\\mathbf {\\widetilde{\\rho }}$ = lr-cs-rd$(\\widetilde{\\mathbf {\\sigma }}, \\mathbf {\\widetilde{g}})$ $(\\mathbf {\\widetilde{\\rho }}_{-j}, \\sigma _j)$ is feasible $\\mathbf {\\rho } = (\\mathbf {\\widetilde{\\rho }}_{-j}, \\sigma _j)$ $\\mathbf {\\rho } = \\mathbf {\\widetilde{\\rho }}$ $\\mathbf {\\rho }$ The rest of the algorithm is exactly as before.", "We call the new algorithm lr-cs-rd.", "Its pseudocode is given in Algorithm .", "The initial call to the algorithm is done on the assignment vector $(r_1, r_2, \\ldots ,r_n)$ and the function cost vector $(f_1, f_2, \\ldots , f_n)$ .", "Without loss of generality, we assume $f_j(r_j) = 0$ for all $j \\in \\mathcal {J}$ .", "Theorem 6 There is a pseudo-polynomial time $4\\kappa $ -approximation for scheduling jobs with release dates on a single machine with generalized cost function.", "The proof of this theorem rests on a series of Lemmas that mirror Lemmas , REF , REF , and REF .", "An assignment $\\mathbf {\\sigma }$ is feasible if there is no residual demand at any interval $[r,t)$ ; namely, $\\mathbf {\\sigma }$ is feasible if $D(r, t,\\mathbf {\\sigma })=0$ for all $r\\in R$ and $r<t \\in \\mathcal {T}$ .", "Furthermore, scheduling the jobs according to early due date first yields a feasible preemptive schedule.", "We start by noting that one can use a simple exchange argument to show that if there is some schedule that meets the dealines $\\mathbf {\\sigma }$ , then the earliest due date (EDD) schedule must be feasible.", "First, we show that if there is a job $j$ in the EDD schedule that does not meet its deadline, then there is an interval $[r,t)$ such that $D(r,t,\\mathbf {\\sigma }) > 0$ .", "Let $t = \\sigma _j + 1$ and let $r < t$ be latest point in time that the machine was idle in the EDD schedule.", "Let $X = \\left\\lbrace i \\in \\mathcal {J} : r \\le r_i , \\sigma _i < t \\right\\rbrace $ .", "Clearly, $r + p(X) \\ge t$ , otherwise $j$ would have met its due date.", "Therefore, $0 & < r + p(X) - t + 1\\\\& = r + p \\left(\\left\\lbrace i \\in \\mathcal {J} : r \\le r_i \\le \\sigma _i < t \\right\\rbrace \\right) - t + 1 \\\\& \\le D(r, t, \\mathbf {\\sigma }).$ Second, we show that if an interval $[r,t)$ such that $D(r,t,\\mathbf {\\sigma }) > 0$ , then there is a job $j$ in the EDD schedule that does not meet its deadline.", "Let $X = \\left\\lbrace i \\in \\mathcal {J} : r \\le r_i, \\sigma _i<t \\right\\rbrace $ .", "Then, $0 < D(r, t, \\mathbf {\\sigma }) = r + p(X) - t + 1 \\quad \\Longrightarrow \\quad r + p(X) \\ge t.$ Let $j$ be the job in $X$ with the largest completion time in the EDD schedule.", "Notice that the completion time of $j$ is at least $r + p(X) \\ge t$ .", "On the other hand, its due date is $\\sigma _j < t$ .", "Therefore, the EDD schedule missing $j$ 's due date.", "Let $(\\mathbf {\\sigma ^{(1)}}, \\mathbf {g^{(1)}}), (\\mathbf {\\sigma ^{(2)}}, \\mathbf {g^{(2)}}), \\ldots , (\\mathbf {\\sigma ^{(k)}}, \\mathbf {g^{(k)}})$ be the inputs to the successive recursive calls to lr-cs-rd and let $\\mathbf {\\rho ^{(1)}}, \\mathbf {\\rho ^{(2)}}, \\ldots , \\mathbf {\\rho ^{(k)}}$ be their corresponding outputs.", "The following properties hold: $\\mathbf {\\sigma ^{(1)}} \\le \\mathbf {\\sigma ^{(2)}} \\le \\cdots \\le \\mathbf {\\sigma ^{(k)}}$ , $\\mathbf {\\rho ^{(1)}} \\le \\mathbf {\\rho ^{(2)}} \\le \\cdots \\le \\mathbf {\\rho ^{(k)}}$ , $\\mathbf {\\sigma ^{(i)}} \\le \\mathbf {\\rho ^{(i)}}$ for all $i =1, \\ldots , k$ , $g^{(i)}_j(\\sigma ^{(i)}_j) = 0$ and $g^{(i)}_j$ is non-negative for all $i=1, \\ldots , k$ and $j \\in \\mathcal {J}$ .", "Properties (i)-(iii) follows from the exactly same reasoning as in Lemma REF .", "The forth property follows by induction.", "The base case is the first call we make to lr-cs-rd, which is $\\mathbf {\\sigma ^{(1)}} = (r_1,\\ldots , r_n)$ and $\\mathbf {g^{(1)}} = (f_1, \\ldots , f_n)$ , where it holds by our assumption.", "For the inductive case, we note that $\\mathbf {{g}^{(i+1)}}$ is constructed by taking $\\mathbf {{g}^{(i)}}$ and subtracting a scaled version of the model function vector, so that $\\mathbf {0} \\le \\mathbf {g^{(i+1)}} \\le \\mathbf {g^{(i)}}$ , and $\\mathbf {\\sigma ^{(i+1)}}$ is constructed by taking $\\mathbf {\\sigma ^{(i)}}$ and increasing the due date of a single job $j^{(i)}$ .", "The way this is done guarantees $g^{(i+1)}_{j^{(i)}} ( \\sigma ^{(i+1)}_{j^{(i)}}) = 0$ , which ensures that the property holds.", "Let lr-cs-rd$(\\mathbf {\\sigma }, \\mathbf {g})$ be a recursive call returning $\\mathbf {\\rho }$ then $\\sum _{i \\in \\mathcal {J} \\atop r^* \\le r_i \\le \\sigma _i < t^* \\le \\rho _i} p_i(r^, t^, \\mathbf {\\sigma })\\le 4 \\kappa \\cdot D(r^, t^, \\mathbf {\\sigma }).$ where $(r^, t^)$ are the values used to decompose the input cost function vector $\\mathbf {g}$ .", "Our goal is to bound the $p_i(r^,t^, \\mathbf {\\sigma })$ value of jobs $X = \\left\\lbrace i \\in \\mathcal {J} : r \\le r_i \\le \\sigma _i < t^* \\le \\rho _i \\right\\rbrace .$ Notice that the algorithm increases the due date of these jobs in this or a later recursive call.", "Furthermore, and more important to us, the algorithm decides not to undo the increase.", "For each $i \\in X$ , consider the call lr-cs-rd$(\\mathbf {\\sigma ^{\\prime }},\\mathbf {g^{\\prime }})$ when we first increased the due date of $i$ beyond $\\sigma _i$ .", "Let $\\mathbf {{\\rho }^{\\prime }}$ be assignment returned by the call.", "Notice that $\\rho ^{\\prime }_i >\\sigma _i$ and that $(\\mathbf {\\rho ^{\\prime }}\\!_{-i}, \\sigma _i)$ is not feasible—otherwise we would have undone the due date increase.", "By Lemma REF , we know that $\\mathbf {\\rho } \\le \\mathbf {\\rho ^{\\prime }}$ , so we conclude that $(\\mathbf {\\rho }_{-i}, \\sigma _i)$ is not feasible either.", "We define $r(i) \\le r_j$ and $\\sigma _i < t(i) \\le \\rho _i$ such that the interval $[r(i), t(i))$ has a positive residual demand in this unfeasible assignment: $ D(r(i), t(i), (\\mathbf {\\rho }_{-i}, \\sigma _i)) > 0.", "$ Note that such an interval must exist, otherwise $\\mathbf {\\rho }$ would not be feasible.", "We partition $X$ in $2 \\kappa $ subsets.", "For each release date $r$ we define $L(r) = \\left\\lbrace i\\in X: t(i) \\le t^, r(i) = r \\right\\rbrace \\text{ and } R(r) = \\left\\lbrace i\\in X: t(i) > t^, r(i) = r \\right\\rbrace ,$ Let $t_L^r = \\max \\left\\lbrace t(i) : i \\in L(r) \\right\\rbrace $ and $i_L^r$ be a job attaining this value.", "Similarly, consider $t_R^r = \\min \\left\\lbrace t(i) : i \\in R(r) \\right\\rbrace $ and $i_R^r$ be a job attaining this value.", "We will bound the contribution of each of these sets separately.", "Our goal will be to prove that for each release date $r$ we have $\\sum _{i \\in L(r) - i_L^r} p_i & \\le D(r^, t^, \\mathbf {\\sigma }), \\text{ and }\\\\\\sum _{i \\in R(r) - i_R^r} p_i & \\le D(r^, t^,\\mathbf {\\sigma }).$ Let us argue (REF ) first.", "Assume $L(r) \\ne \\emptyset $ , so $t_L^r$ is well defined; otherwise, the claim is trivial.", "Since $D\\left(r, t_L^r,(\\mathbf {\\rho }_{-i_L^r}, \\sigma _{i_L^r})\\right) > 0$ , it follows that $\\sum _{i \\in \\mathcal {J} - i_L^r \\atop r \\le r_i < t_L^r \\le \\rho _i } p_i & < r + \\sum _{i \\in \\mathcal {J} \\atop r \\le r_i < t_L^r } p_i - t_L^r + 1 \\\\\\sum _{i \\in \\mathcal {J} \\atop r \\le r_i < t_L^r \\le \\sigma _i } p_i +\\sum _{i \\in \\mathcal {J} - i_L^r \\atop r \\le r_i \\le \\sigma _i < t_L^r \\le \\rho _i } p_i & < r + \\sum _{i \\in \\mathcal {J} \\atop r \\le r_i < t_L^r } p_i - t_L^r + 1 \\\\\\sum _{i \\in \\mathcal {J} - i_L^r \\atop r \\le r_i \\le \\sigma _i < t_L^r \\le \\rho _i } p_i & < D(r,t_L^r, \\mathbf {\\sigma }).$ Recall that $\\sigma _i < t(i)$ for all $i \\in X$ .", "Furthermore, $t(i) \\le t_L^r$ , and thus $\\sigma _i < t_L^r$ , for all $i \\in L(r)$ .", "Also, $t(i) \\le \\rho _i$ for all $i \\in X$ .", "Therefore the sum in the left-hand side of the last inequality contains all jobs in $L(r) - i_L^r$ .", "Finally, we note that $D(r,t_L, \\mathbf {\\sigma } ) \\le D(r^, t^, \\mathbf {\\sigma })$ due to the way lr-cs-rd chooses $r^$ and $t^$ , which gives us (REF ).", "Let us argue ().", "Assume $R(r) \\ne \\emptyset $ , so $t_R^r$ is well defined; otherwise, the claim is trivial.", "Since $D\\left(r,t_R^r, (\\mathbf {\\rho }_{-i_R^r},\\sigma _{i_R^r})\\right) > 0$ , it follows that $\\sum _{i \\in \\mathcal {J} - i_R^r \\atop r \\le r_i < t_R^r \\le \\rho _i } p_i & < r + \\sum _{i \\in \\mathcal {J} \\atop r \\le r_i < t_R^r } p_i - t_R^r + 1 \\\\\\sum _{i \\in \\mathcal {J} \\atop r \\le r_i < t_R^r \\le \\sigma _i } p_i +\\sum _{i \\in \\mathcal {J} - i_R^r \\atop r \\le r_i \\le \\sigma _i < t_R^r \\le \\rho _i } p_i & < r + \\sum _{i \\in \\mathcal {J} \\atop r \\le r_i < t_R^r } p_i - t_R^r + 1 \\\\\\sum _{i \\in \\mathcal {J} - i_R^r \\atop r \\le r_i \\le \\sigma _i < t_R^r \\le \\rho _i } p_i & < D(r,t_R^r, \\mathbf {\\sigma })$ Recall that $t(i) \\le \\rho _i$ for all $i \\in X$ .", "Furthermore, $t_R^r \\le t(i)$ , and thus $t_R^r \\le \\rho _i$ , for all $i \\in R(r)$ .", "Also, $t_i > \\sigma _i$ for all $i \\in X$ .", "Therefore, the sum in the left-hand side of the last inequality contains all jobs in $R(r)- i_R^r$ .", "Finally, we note that $D(r,t_R^r,\\mathbf {\\sigma } )\\le D(r^, t^, \\mathbf {\\sigma })$ due to the way lr-cs chooses $r^$ and $t^$ , which gives us ().", "Finally, we note that $p_i(r^, t^, \\mathbf {\\sigma }) \\le D(r^,t^, \\mathbf {\\sigma })$ for all $i \\in \\mathcal {J}$ .", "Therefore, $\\sum _{i \\in \\mathcal {J}: \\rho _i \\ge t^} p_i(r^, t^, \\mathbf {\\sigma })&= \\sum _{i \\in X} p_i(r^, t^, \\mathbf {\\sigma }) \\\\& = \\sum _r \\left( \\sum _{i \\in L(r)} p_i(r^, t^, \\mathbf {\\sigma }) + \\sum _{i \\in R(r)} p_i(r^, t^, \\mathbf {\\sigma }) \\right) \\\\& \\le \\sum _r \\Big ( 2 \\cdot D(r^, t^, \\mathbf {\\sigma }) + 2 \\cdot D(r^, t^, \\mathbf {\\sigma }) \\Big ) \\\\& = 4 \\kappa \\cdot D(r^, t^, \\mathbf {\\sigma }).$ Let lr-sc-rd$(\\mathbf {\\sigma }, \\mathbf {g})$ be a recursive call and $\\mathbf {\\rho }$ be its output.", "Then $\\mathbf {\\rho }$ is a feasible $4\\kappa $ -approximation w.r.t.", "$\\mathbf {g}$ .", "The proof is by induction.", "The base case corresponds to the base case of the recurrence where we get as input a feasible assignment $\\mathbf {\\sigma }$ , and so $\\mathbf {\\rho } = \\mathbf {\\sigma }$ .", "From Lemma REF we know that $g_i(\\sigma _i) = 0$ for all $i \\in \\mathcal {J}$ , and that the cost functions are non-negative.", "Therefore, the cost of $\\mathbf {\\rho }$ is optimal since $ \\sum _{i \\in \\mathcal {J}} g_i(\\rho _i) = 0.", "$ For the inductive case, the cost function vector $\\mathbf {g}$ is decomposed into $\\mathbf {\\widetilde{g}} + \\alpha \\cdot \\mathbf {\\widehat{g}}$ .", "Let $(j,s)$ be the pair used to define $\\mathbf {\\widetilde{\\sigma }} = (\\mathbf {\\sigma }_{-j}, s)$ .", "Let $\\mathbf {\\widetilde{\\rho }}$ be the assignment returned by the recursive call.", "By induction hypothesis, we know that $\\mathbf {\\widetilde{\\rho }}$ is feasible and $4\\kappa $ -approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "After the recursive call returns, we check the feasibility of $(\\mathbf {\\widetilde{\\rho }}_{-j}, \\sigma _j)$ .", "If the vector is feasible, then we return the modified assignment; otherwise, we return $\\mathbf {\\widetilde{\\rho }}$ .", "In either case $\\mathbf {\\rho }$ is feasible.", "We claim that $\\mathbf {\\rho }$ is $4\\kappa $ -approximate w.r.t.", "$\\mathbf {\\widehat{g}}$ .", "Indeed, $ \\sum _{i \\in \\mathcal {J}} \\widehat{g}_i(\\rho _i) = \\sum _{i \\in \\mathcal {J} \\atop r^\\le r_i < t^* \\le \\rho _i} p_i(r^,t^, \\mathbf {\\sigma }) \\le 4 \\kappa \\cdot D(r^,t^,\\mathbf {\\sigma }) \\le 4 \\kappa \\cdot \\operatorname{opt}(\\mathbf {\\widehat{g}}),$ where the first inequality follows from Lemma REF and the last inequality follows from the fact that the cost of any schedule under $\\mathbf {\\widehat{g}}$ is given by the $p_i(r^,t^, \\mathbf {\\sigma })$ value of jobs $i \\in \\mathcal {J}$ with $r^\\le r_i < t^$ and $\\sigma _i < t^$ that cover $t^$ , which must have a combined processing time of at least $D(r^,t^, \\mathbf {\\sigma })$ .", "Hence, $\\operatorname{opt}(\\mathbf {\\widehat{g}}) \\ge D(r^, t^, \\mathbf {\\sigma })$ .", "We claim that $\\mathbf {\\rho }$ is $4\\kappa $ -approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "Recall that $\\mathbf {\\widetilde{\\rho }}$ is $4\\kappa $ -approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ ; therefore, if $\\mathbf {\\rho } =\\mathbf {\\widetilde{\\rho }}$ then $\\mathbf {\\rho }$ is $4\\kappa $ -approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "Otherwise, $\\mathbf {\\rho } = (\\mathbf {\\widetilde{\\rho }}_{-j}, \\sigma _j)$ , in which case $\\widetilde{g}_j(\\rho _j) = 0$ , so $\\mathbf {\\rho }$ is also 4-approximate w.r.t.", "$\\mathbf {\\widetilde{g}}$ .", "At this point we can invoke the Local Ratio Theorem to get that $\\sum _{j \\in \\mathcal {J}} g_j(\\rho _j) &= \\sum _{j \\in \\mathcal {J}} \\widetilde{g}_j(\\rho _j) +\\sum _{j \\in \\mathcal {J}} \\alpha \\cdot \\widehat{g}_j(\\rho _j), \\\\& \\le 4 \\kappa \\cdot \\operatorname{opt}(\\mathbf {\\widetilde{g}}) + 4 \\kappa \\cdot \\alpha \\cdot \\operatorname{opt}(\\mathbf {\\widehat{g}}), \\\\& = 4 \\kappa \\cdot \\big ( \\operatorname{opt}(\\mathbf {\\widetilde{g}}) + \\operatorname{opt}(\\alpha \\cdot \\mathbf {\\widehat{g}}) \\big ), \\\\& \\le 4 \\kappa \\cdot \\operatorname{opt}(\\mathbf {g}),$ which finishes the proof of the lemma." ], [ "The issue with the analysis of Cheung-Shmoys", "The analysis of Cheung and Shmoys hinges on the following claim [6]: $\\sum _{j \\in \\bar{A}_t \\setminus A} p_j(t,A) \\le 2 D(t,A) \\qquad \\text{for each } (t,A): y_{t,A} > 0,$ where $\\bar{A}_t = \\left\\lbrace j \\in \\mathcal {J} : d_j \\ge t \\right\\rbrace $ is the set of jobs assigned a due date greater than or equal to $t$ (at the end of the pruning phase).", "Notice that in the example from Section REF this property does not hold for $t = 3p-1$ and $A = \\emptyset $ .", "Indeed, $y_{3p-1, \\emptyset } > 0$ and in the final solution all jobs are assigned to $3p-1$ or later, and thus, $\\sum _{j\\in \\bar{A}_{3p-1}} p_j(3p-1,\\emptyset ) = 4p,$ while on the other hand $D(3p-1,\\emptyset ) = p+2$ .", "Let $A_t^k$ denote the set $A_t$ at the beginning of the $k$ -th iteration of the growing phase.", "Property (REF ) is stated as Lemma 5 in [6].", "Its proof aims to show that each time $t^k$ is critical with respect to $\\bar{x}$ and $A^k_{t_k}$ , meaning there exists a job $\\ell \\in \\bar{A}_{t_k} \\setminus A^k_{t_k}$ such that $\\sum _{j \\in \\bar{A}_{t_k} \\setminus (A^k_{t_k} \\cup \\left\\lbrace \\ell \\right\\rbrace )} p_j < D(t^k, A^k_{t_k})$ .", "Notice that this is not in the case in our example for $t^k = 3p-1$ , where $A^k_{t_k}= \\emptyset $ and $\\bar{A}_{t_k} = \\left\\lbrace 1,2,3,4 \\right\\rbrace $ .", "Looking further in the proof, we see that the critical nature of $t^k$ is shown by contradiction, saying that if $t^k$ is not critical then for each job $\\ell \\in \\bar{A}_{t^k}\\setminus A^k_{t_k}$ , then there must exists a $t_{\\ell }$ such that job $\\ell $ is critical with respect to $t_{\\ell }$ and $A^k_{t_\\ell }$ , meaning $\\sum _{j \\in \\bar{A}_{t_\\ell } \\setminus (A^k_{t_\\ell } \\cup \\left\\lbrace \\ell \\right\\rbrace )} p_j <D(t_\\ell , A^k_{t_\\ell })$ .", "Two cases are considered: In Case 1 there exists $\\ell $ such that $t_\\ell < t^k$ and in Case 2 we have $t_\\ell > t^k$ for all $\\ell $ .", "In our example, we see that for $t^k = 3p-1$ , job $\\ell =1$ has $t_{\\ell }=1$ .", "Since $t_{\\ell } < t^k$ that puts us in Case 1 of the proof.", "For this case the proof claims that $\\bar{A}_{t_\\ell }\\setminus (A^k_{t_{\\ell }} \\cup \\lbrace \\ell \\rbrace ) \\supseteq \\bar{A}_{t^k}\\setminus (A^k_{t^k} \\cup \\lbrace \\ell \\rbrace ).$ However, this is not true in our example, since $\\bar{A}_{t_\\ell }\\setminus (A^k_{t_{\\ell }} \\cup \\lbrace \\ell \\rbrace ) = \\lbrace 1,2,3,4\\rbrace \\setminus (\\lbrace 2,3,4\\rbrace \\cup \\lbrace 1\\rbrace ) = \\emptyset ,$ and $\\bar{A}_{t^k}\\setminus (A^k_{t^k} \\cup \\lbrace \\ell \\rbrace ) = \\lbrace 1,2,3,4\\rbrace \\setminus \\lbrace 1\\rbrace =\\lbrace 2,3,4\\rbrace .$ The issue is that jobs 3 and 4 are in $A_{t_{\\ell }}^k\\setminus A^k_{t^k}$ , and they finally belong to $\\bar{A}_{t^k}\\setminus A^k_{t^k}$ ." ] ]	1403.0298
[ [ "Ergodicity of regime-switching diffusions in Wasserstein distances" ], [ "Abstract Based on the theory of M-matrix and Perron-Frobenius theorem, we provide some criteria to justify the convergence of the regime-switching diffusion processes in Wasserstein distances.", "The cost function we used to define the Wasserstein distance is not necessarily bounded.", "The continuous time Markov chains with finite and countable state space are all studied.", "To deal with the countable state space, we put forward a finite partition method.", "The boundedness for state-dependent regime-switching diffusions in an infinite state space is also studied." ], [ "Introduction", "The regime-switching diffusion processes can be viewed as diffusion processes in random environments, which are characterized by continuous time Markov chains.", "The behavior of the diffusion in each fixed environment may be very different.", "Hence, they can provide more realistic models for many applications, for instance, control problems, air traffic management, biology and mathematical finance.", "We refer the reader to [9], [10], [12], [11], [21] and references therein for more details of regime-switching diffusion processes and their applications.", "In view of the usefulness of regime-switching diffusion processes, the recurrent properties of these processes are rather complicated due to the appearance of the diffusion process and jump process at the same time.", "One can get this viewpoint from the examples constructed in [17].", "In [17], the authors showed that even in every fixed environment the corresponding diffusion process is recurrent (transient), the diffusion process in random environment could be transient (positive recurrent).", "In [16], [17], [18], [21], there are some studies on the transience, null recurrence, ergodicity, strong ergodicity of regime-switching diffusion processes.", "In these works, the convergence of the semigroups to their stationary distributions is in the total variation distance.", "Recently, in [8], besides the total variation distance, the authors also considered the exponential ergodicity in the Wasserstein distance.", "They provided an on-off type criterion.", "In [8], the state-independent and state-dependent regime-switching diffusion processes in a finite state space were studied.", "The cost function used in [8] to define the Wasserstein distance is bounded.", "The work [8] attracts us to studying the ergodicity of regime-switching diffusion processes in Wasserstein distance.", "In this work, we consider the regime-switching diffusion process $(X_t,\\Lambda _t)$ in the following form: $(X_t)_{t\\ge 0}$ satisfies a stochastic differential equation (SDE) $\\text{\\rm {d}}X_t=b(X_t, \\Lambda _t)\\text{\\rm {d}}t +\\sigma (X_t,\\Lambda _t)\\text{\\rm {d}}B_t, \\quad X_0=x\\in \\mathbb {R}^d,$ where $(B_t)$ is a $d$ -dimensional Brownian motion, and $(\\Lambda _t)$ is a continuous time Markov chain with a state space $\\mathcal {S}=\\lbrace 1,2,\\ldots , N\\rbrace $ , $1\\le N \\le \\infty $ , such that $\\mathbb {P}(\\Lambda _{t+\\delta }=l\|\\Lambda _t=k, X_t=x)={\\left\\lbrace \\begin{array}{ll} q_{kl}(x)\\delta +o(\\delta ), &\\text{if}\\ k\\ne l,\\\\1+q_{kk}(x)\\delta +o(\\delta ), & \\text{if}\\ k=l,\\end{array}\\right.", "}$ for $\\delta >0$ .", "The $Q$ -matrix $Q_x=(q_{kl}(x))$ is irreducible and conservative for each $x\\in \\mathbb {R}^d$ .", "If the $Q$ -matrix $(q_{kl}(x))$ does not depend on $x$ , then $(X_t,\\Lambda _t)$ is called a state-independent regime-switching diffusion; otherwise, it is called a state-dependent one.", "The state-independent switching $(\\Lambda _t)$ is also called Markov switching.", "When $N$ is finite, namely, $(\\Lambda _t)$ is a Markov chain in a finite state space, we call $(X_t,\\Lambda _t)$ a regime-switching diffusion process in finite state space.", "And it is easy to see the recurrent property of $(X_t,\\Lambda _t)$ is equivalent to that of $(X_t)$ as pointed out in [16].", "When $N$ is infinite, we call $(X_t,\\Lambda _t)$ a regime-switching diffusion process in infinite state space.", "There is few study on the recurrent property of the regime-switching processes in an infinite state space.", "The infinity of $N$ causes some well studied methods useless.", "In this setting, the effect of recurrent property of $(\\Lambda _t)$ to that of $(X_t,\\Lambda _t)$ is still not clear.", "According to [21], we suppose that the following conditions hold throughout this work, which ensure that there exists a unique, non-explosive solution of (REF ) and (REF ): there exists $\\bar{K}>0$ such that $1^\\circ $ $q_{ij}(x)$ is a bounded continuous function for each pair of $i,j\\in \\mathcal {S}$ ; $2^\\circ $ $\|b(x,i)\|+\\Vert \\sigma (x,i)\\Vert \\le \\bar{K}(1+\|x\|),\\quad x\\in \\mathbb {R}^d,\\ i\\in \\mathcal {S}$ ; $3^\\circ $ $\|b(x,i)-b(y,i)\|+\\Vert \\sigma (x,i)-\\sigma (y,i)\\Vert \\le \\bar{K}\|x-y\|,\\quad x,\\,y\\in \\mathbb {R}^d,\\ i\\in \\mathcal {S}$ , where $\\Vert \\sigma \\Vert $ denotes the operator norm of matrix $\\sigma $ .", "In this work, we establish some new criteria for the ergodicity of regime-switching processes in Wasserstein distance.", "We assume the semigroup $P_t$ of $(X_t,\\Lambda _t)$ converges weakly to some probability measure $\\nu $ , and consider under what condition $P_t$ also converges in Wasserstein distance to $\\nu $ .", "The existence of $\\nu $ can be obtained by the results on the boundedness in moments of $(X_t,\\Lambda _t)$ (cf.", "[5]).", "There are some studies on the asymptotic boundedness in moments for regime-switching processes in a finite state space.", "See, for instance, [13] for state-independent switching and [21] for state-dependent switching.", "In Section 4, we shall discuss the boundedness for state-dependent regime-switching in an infinite state space.", "In Section 3, we first consider the ergodicity in Wasserstein distance for state-independent regime-switching diffusion process in a finite state space.", "We provides a general result in Theorem REF .", "Based on it, we find three kinds of conditions to verify its assumption.", "In Theorem REF , we provide an easily verifiable condition using the theory of M-matrix.", "In Theorem REF , we relate it to the well studied topic on the estimate of lower bound of principal eigenvalue of Dirichlet form when $(\\Lambda _t)$ is reversible.", "In Theorem REF , we give a concise condition using the Perron-Frobenius theorem, which recover the on-off type criterion established in [8].", "But different to [8], the cost function used by us to define the Wasserstein distance is not necessarily bounded.", "In Section 3, we proceed to study the state-independent regime-switching diffusion process in an infinite state space.", "We put forward a finite partition procedure to study the ergodicity in Wasserstein distance.", "By this method, we first divide the infinite state space $\\mathcal {S}$ into finite number of subsets.", "Then we construct a new Markov chain in a finite state space, which induces a new regime-switching diffusion process.", "We show that if the new regime-switching process is ergodic in Wasserstein distance according to our criterion then so is the original one.", "Moreover, this finite projection method owns some kind of consistency, which is explained in Proposition REF below.", "Examples are constructed to show the usefulness of our criteria.", "In Section 4, we study the ergodicity of state-dependent regime-switching process in an infinite state space.", "For the state-dependent switching, it is more difficult to construct coupling process to estimate the Wasserstein distance.", "See [20] for some study on this topic.", "At the present stage, we provide a criterion by M-matrix theory to guarantee the existence of the stationary distribution in weak topology.", "We extend the result in [22] on boundedness of state-independent switching in a finite state space to a state-dependent switching in an infinite state space.", "Moreover, this section also serves as providing method to check our assumption (A2) (A3) used in Section 3.", "At last, note that the convergence in the weak topology is equivalent to the convergence in Wasserstein distance if the Wasserstein distance is induced by a bounded cost function.", "This work is organized as follows.", "In Section 2, we give out some introduction on the Wasserstein distances, M-matrix theory and optimal couplings for diffusion processes.", "In Section 3, we provide a criterion on the exponential ergodicity of state-independent regime-switching process in a finite state space.", "The Section 4 is devoted to studying the state-independent regime-switching process in an infinite state space.", "In Section 5, we study the existence of stationary distribution of state-dependent regime-switching processes in a finite state space." ], [ "Preliminaries", "Let $(X_t,\\Lambda _t)$ be defined by (REF ) and (REF ).", "Letting $x=(x_1,\\ldots ,x_d),\\,y=(y_1,\\ldots ,y_d)\\in \\mathbb {R}^d$ , their Euclidean distance is $\|x-y\|:=\\sqrt{\\sum _{i=1}^d(x_i-y_i)^2}$ .", "Let $\\rho :[0,\\infty )\\rightarrow [0,\\infty )$ satisfying $\\rho (0)=0$ , $\\rho ^{\\prime }>0$ , $\\rho ^{\\prime \\prime }\\le 0$ and $\\lim _{r\\rightarrow \\infty } \\rho (r)=\\infty $ .", "Then $(x,y)\\mapsto \\rho (\|x-y\|)$ is a new metric on $\\mathbb {R}^d$ .", "Replacing the original Euclidean distance to the new distance $\\rho (\|x-y\|)$ is useful in application.", "See, for instance, [3] for application in estimating the spectral gap of Laplacian operator on manifolds.", "Define two distances $\\tilde{\\rho }$ and $\\tilde{\\rho }_b$ on $\\mathbb {R}^d\\times \\mathcal {S}$ by $\\tilde{\\rho }((x,i),(y,j))=\\sqrt{\\mathbf {1}_{i\\ne j}+\\rho (\|x-y\|)},\\quad x,\\,y\\in \\mathbb {R}^d,\\, i,\\,j\\in \\mathcal {S},$ and $\\tilde{\\rho }_p((x,i),(y,j))=\\sqrt{\\mathbf {1}_{i\\ne j}+\\rho ^p(\|x-y\|) },\\quad x,\\,y\\in \\mathbb {R}^d,\\, i,\\,j\\in \\mathcal {S}, \\ p>0.$ Let $\\mathcal {P}(\\mathbb {R}^d\\times \\mathcal {S})$ be the collection of all probability measures on $\\mathbb {R}^d\\times \\mathcal {S}$ .", "Using $\\tilde{\\rho }$ and $\\tilde{\\rho }_p$ to be the cost function separatively, we can define the Wasserstein distance between every two probability measures $\\mu $ and $\\nu $ in $\\mathcal {P}(\\mathbb {R}^d\\times \\mathcal {S})$ by $W_{\\tilde{\\rho }}(\\mu ,\\nu )=\\inf \\big \\lbrace \\mathbb {E}[\\tilde{\\rho }(X_1,X_2)]\\big \\rbrace ,\\quad W_{\\tilde{\\rho }_p}(\\mu ,\\nu )=\\inf \\big \\lbrace \\mathbb {E}[\\tilde{\\rho }_p(X_1,X_2)]\\big \\rbrace ,\\ \\ p>0,$ where the infimum is taken over all pairs of random variables $X_1$ , $X_2$ on $\\mathbb {R}^d\\times \\mathcal {S}$ with respective laws $\\mu $ , $\\nu $ .", "As our criteria on the ergodicity of $(X_t,\\Lambda _t)$ are related to the theory of M-matrix, here we introduce some basic definition and notation of M-matrices, and refer the reader to the book [1] for more discussions on this well studied topic.", "The theory of M-matrix has also been used to study the stability of state-independent regime-switching processes in a finite state space (see [13]).", "Let $B$ be a matrix or vector.", "By $B\\ge 0$ we mean that all elements of $B$ are non-negative.", "By $B>0$ we mean that $B\\ge 0$ and at least one element of $B$ is positive.", "By $B\\gg 0$ , we mean that all elements of $B$ are positive.", "$B\\ll 0$ means that $-B\\gg 0$ .", "Definition 2.1 (M-matrix) A square matrix $A=(a_{ij})_{n\\times n}$ is called an M-Matrix if $A$ can be expressed in the form $A=sI-B$ with some $B\\ge 0$ and $s\\ge \\mathrm {Ria}(B)$ , where $I $ is the $n\\times n$ identity matrix, and $\\mathrm {Ria}(B)$ the spectral radius of $B$ .", "When $s>\\mathrm {Ria}(B)$ , $A$ is called a nonsingular M-matrix.", "In [1], the authors gave out 50 conditions which are equivalent to $A$ is a nonsingular M-matrix.", "We cite some of them below.", "Proposition 2.2 The following statements are equivalent.", "$A$ is a nonsingular $n\\times n$ M-matrix.", "All of the principal minors of $A$ are positive; that is, $\\begin{vmatrix} a_{11}&\\ldots &a_{1k}\\\\ \\vdots & &\\vdots \\\\ a_{1k}&\\ldots &a_{kk}\\end{vmatrix}>0 \\ \\ \\text{for every $k=1,2,\\ldots ,n$}.$ Every real eigenvalue of $A$ is positive.", "$A$ is semipositive; that is, there exists $x\\gg 0$ in $\\mathbb {R}^n$ such that $Ax\\gg 0$ .", "There exists $x\\gg 0$ with $Ax>0$ and $\\sum _{j=1}^ia_{ij}x_j>0$ , $i=1,\\ldots , n$ .", "Now we recall some results on the optimal couplings of Wasserstein distances, which will help us to check the assumption (A1) below.", "Let $(E,\\tilde{d},\\mathcal {E})$ be a complete separable metric space.", "Let $\\mathcal {P}(E)$ denote the set of all probability measures on $E$ .", "For two given probability measures $\\mu $ and $\\nu $ on $E$ , define $W_p(\\mu ,\\nu )=\\inf _{\\pi \\in \\mathcal {(}\\mu ,\\nu )}\\Big \\lbrace \\int _{E\\times E}\\tilde{d}(x,y)^p\\pi (\\text{\\rm {d}}x,\\text{\\rm {d}}y)\\Big \\rbrace ^{1/p},\\quad p\\ge 1,$ where $\\mathcal {C}(\\mu ,\\nu )$ denotes the set of all couplings of $\\mu $ and $\\nu $ .", "It is well known that given $\\mu $ and $\\nu $ , the infimum is attained for some coupling $\\pi $ .", "For a sequence of probability measures $\\mu _n$ in $\\mathcal {P} (E)$ , the statement that $\\mu _n$ converges to some $\\mu \\in \\mathcal {P}(E)$ in the metric $W_p$ is equivalent to that $\\mu _n$ converges weakly to $\\mu $ and for some (or any) $a\\in E$ , $\\lim _{K\\rightarrow \\infty } \\sup _n\\int _{\\lbrace x:\\tilde{d}(x,a)>K\\rbrace }\\tilde{d} (x,a)^p\\mu _n(\\text{\\rm {d}}x)=0.$ Hence, when $\\tilde{d}$ is bounded, the convergence in Wasserstein distance $W_p$ is equivalent to the weak convergence.", "It can be seen from the definition of the Wasserstein distance that the calculation of this distance between two probability measures is not an easy task.", "Usually, one tries to find some suitable estimates on it.", "In [3], some constructions of optimal couplings for Markov chain and diffusion processes are given.", "We recall some results on the optimal couplings of diffusion processes.", "Consider a diffusion process in $\\mathbb {R}^d$ with operator $L=\\frac{1}{2}\\sum _{i,j=1}^d a_{ij}(x)\\frac{\\partial ^2}{\\partial x_i\\partial x_j}+\\sum _{i=1}^db_i(x)\\frac{\\partial }{\\partial x_i}$ .", "For simplicity, we write $L\\sim (a(x),b(x))$ .", "Given two diffusions with operators $L_k\\sim (a_k(x),b_k(x))$ , $k=1,2$ respectively, an operator $\\tilde{L}$ on $\\mathbb {R}^d\\times \\mathbb {R}^d$ is called a coupling operator of $L_1$ and $L_2$ if $\\tilde{L} f(x,y)&=L_1f(x), \\ \\text{if $f\\in C_b^2(\\mathbb {R}^d)$ and independent of $y$},\\\\\\tilde{L} f(x,y)&=L_2f(y), \\ \\text{if $f\\in C_b^2(\\mathbb {R}^d)$ and independent of $x$}.$ Let $\\tilde{d}\\in C^2(\\mathbb {R}^d\\times \\mathbb {R}^d\\backslash \\lbrace (x,x);x\\in \\mathbb {R}^d\\rbrace )$ be a metric on $\\mathbb {R}^d$ .", "A coupling operator $\\bar{L}$ is called $\\tilde{d}$ -optimal if $\\bar{L}\\tilde{d}(x ,y)=\\inf _{\\tilde{L}}\\tilde{L}\\tilde{d}(x,y)$ for all $x\\ne y$ , where $\\tilde{L}$ varies over all coupling operators of $L_1$ and $L_2$ .", "The coefficients of any coupling operator must be of the form $\\tilde{L}\\sim \\big (a(x,y), b(x,y)\\big )$ , $a(x,y)=\\begin{pmatrix} a_1(x) & c(x,y)\\\\c(x,y)^\\ast &a_2(y)\\end{pmatrix},\\ b(x,y)=\\begin{pmatrix} b_1(x)\\\\ b_2(y)\\end{pmatrix},$ where $c(x,y)$ is a matrix such that $a(x,y)$ is non-negative definite and $c(x,y)^\\ast $ denotes the transpose of $c(x,y)$ .", "Next, we recall a result for optimal couplings of one-dimensional diffusion processes due to [3], and refer to [3] for results in multidimensional case and more discussion.", "Theorem 2.3 ([3] Theorem 5.3) Let $\\rho \\in C^2(\\mathbb {R}_+,\\mathbb {R}_+)$ with $\\rho (0)=0$ , $\\rho ^{\\prime }>0$ and $\\rho ^{\\prime \\prime }\\le 0$ .", "Set $\\tilde{d}(x,y)=\\rho (\|x-y\|)$ .", "If $d=1$ , then the $\\tilde{d}$ -optimal solution $c(x,y)$ is given by $c(x,y)=-\\sqrt{a_1(x)a_2(y)}$ ." ], [ " Markovian switching in a finite state space", "Let $(X_t,\\Lambda _t)$ be a regime-switching diffusion process defined by (REF ) and (REF ) with $N<\\infty $ and $Q=(q_{ij})$ state-independent switching.", "For each fixed environment $i\\in \\mathcal {S}$ , the corresponding diffusion $X_t^{(i)}$ is defined by $\\text{\\rm {d}}X_t^{(i)}=b(X_t^{(i)},i)\\text{\\rm {d}}t+\\sigma (X_t^{(i)},i)\\text{\\rm {d}}B_t, \\quad X_0^{(i)}=x\\in \\mathbb {R}^d.$ Let $a^{(i)}(x)=\\sigma (x,i)\\sigma (x,i)^\\ast $ , then the infinitesimal operator $L^{(i)}$ of $(X_t^{(i)})$ is $L^{(i)}=\\frac{1}{2}\\sum _{k,l=1}^da^{(i)}_{kl}(x)\\frac{\\partial ^2}{\\partial x_k\\partial x_l} +\\sum _{k=1}^d b_k(x,i)\\frac{\\partial }{\\partial x_k}.$ Let $\\rho :[0,\\infty )\\rightarrow [0,\\infty )$ satisfying $\\rho (0)=0,\\ \\rho ^{\\prime }>0,\\ \\rho ^{\\prime \\prime }\\le 0\\ \\text{and}\\ \\rho (x)\\rightarrow \\infty \\ \\text{as}\\ x\\rightarrow \\infty .$ It is clear that $\\rho (x)=x$ satisfying previous conditions.", "In the sequel, the function $\\rho $ we used satisfies (REF ).", "We pose the following assumption to study the ergodic property of $(X_t,\\Lambda _t)$ in this section.", "(A1) For each $i\\in \\mathcal {S}$ , there exist a coupling operator $\\tilde{L}^{(i)}$ of $L^{(i)}$ and itself, and a constant $\\beta _i$ such that $\\tilde{L}^{(i)}\\rho (\|x-y\|)\\le \\beta _i\\rho (\|x-y\|), \\quad x,\\, y\\in \\mathbb {R}^d,\\ x\\ne y.$ (A2) There exists constants $C_1, \\gamma >0$ such that $\\mathbb {E}[\\rho (\|X_t\|)]\\le C_1\\big (1+\\mathbb {E}[\\rho (\|X_0\|)]e^{-\\gamma t}\\big ),\\, t>0$ .", "Note that (A2) can not be deduced from (A1) by setting $y=0$ even when $\\beta _i<0$ .", "According to [7], without loss of generality, we assume that once the coupling processes corresponding to $\\tilde{L}^{(i)}$ meet each other, then they will move together.", "This makes us to consider the inequality of (A1) only for $x\\ne y$ in $\\mathbb {R}^d$ .", "This property will be used directly later, rather than mentioning the details.", "Our introduction of $\\rho (\|\\cdot \|)$ -optimal coupling operator could be used here to help us check this assumption.", "Note that the constant $\\beta _i$ could be positive and negative similar to [8].", "If $\\beta _i$ is negative, condition (A1) implies that the process $(X_t^{(i)})$ is exponentially convergent (see [15]).", "When $\\beta _i<0$ is smaller, the process $(X_t^{(i)})$ converges more rapidly.", "These constants $\\beta _i$ are used to characterize the ergodic behavior of (REF ).", "In [17], R. Pinsky and M. Scheutzow constructed examples on $[0,\\infty )\\times \\lbrace 1,2\\rbrace $ to show that even when all $(X^{(k)}_t)$ , $k=1,2$ , are positive recurrent (transient), $(X_t)$ could be transient (positive recurrent).", "These examples reveal the complexity in studying the recurrence properties of regime-switching diffusions.", "One aim of this work is to show how the coaction of jumping process and diffusion process in each fixed environment determines the recurrent property of diffusion process in random environment.", "Besides, in [18], we showed that in one dimensional space, if for each fixed environment $i\\in \\mathcal {S}$ with $N<\\infty $ , $(X_t^{(i)})$ is strongly ergodic, then so is $(X_t)$ .", "Recall the definition of Wasserstein distance $W_{\\tilde{\\rho }}$ , $W_{\\tilde{\\rho }_p}$ given by (REF ).", "In this work, we write $\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)$ to denote the diagonal matrix induced by vector $(\\beta _1,\\ldots , \\beta _N)^\\ast $ as usual.", "We now come to our first main result.", "Theorem 3.1 Assume that (A1) (A2) and (A3) hold.", "If there exists a vector $\\xi \\gg 0$ such that $\\lambda =(\\lambda _1,\\ldots ,\\lambda _N)^\\ast :=\\big (Q+\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)\\big )\\xi \\ll 0$ , then there exists a probability measure $\\nu $ on $\\mathbb {R}^d\\times \\mathcal {S}$ such that $W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\nu )\\le 2\\tilde{C}(\\sqrt{3+\\rho (\|x\|)}+\\tilde{C}) e^{-\\tilde{\\alpha }t},$ where $P_t$ is the Markovian semigroup associated with $(X_t,\\Lambda _t)$ , $\\delta _{(x,i)}$ denotes the Dirac measure at $(x,i)$ , $\\tilde{\\alpha }$ and $\\tilde{C}$ are positive constants, defined by $\\tilde{\\alpha }=\\min \\lbrace \\alpha ,\\theta \\rbrace /4$ , $\\tilde{C}=\\max \\lbrace C_1, C_2, 1\\rbrace $ with $\\alpha ,\\,\\theta , C_2$ given by (REF ) and Lemma REF below.", "The existence of $\\xi \\gg 0$ such that $\\big (Q+\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)\\big )\\xi \\ll 0$ is not an easily checked condition in general.", "Therefore, in Theorem REF below, we provide a sufficient condition by using the theory of M-matrix.", "When $(\\Lambda _t)$ is reversible, we relate the existence of such $\\xi $ with the positiveness of principal eigenvalue in Theorem REF below.", "In Theorem REF , we modify the definition of Wasserstein distance and provide another sufficient condition by using Perron-Frobenius theorem.", "Before proving Theorem REF , we make some necessary preparation.", "First we construct the coupling process $(Y_t,\\Lambda ^{\\prime }_t)$ of $(X_t,\\Lambda _t)$ to be used in the arguments of this Theorem.", "Let $(\\Lambda _t, \\Lambda _t^{\\prime })$ be the classical coupling for $(\\Lambda _t)$ , whose infinitesimal operator is defined by $\\begin{split}\\tilde{Q}f(k,l)&:= \\sum _{m,n\\in \\mathcal {S}} q_{(k,l)(m,n)}(f(m,n)-f(k,l))\\\\&=\\mathbf {1}_{\\lbrace k=l\\rbrace }\\!\\sum _{m\\in \\mathcal {S}} q_{km}(f(m,m)-f(k,l))+\\mathbf {1}_{\\lbrace k\\ne l\\rbrace }\\!\\!\\sum _{m\\in \\mathcal {S},m\\ne k}\\!\\!q_{km}(f(m,l)-f(k,l))\\\\&\\quad +\\mathbf {1}_{\\lbrace k\\ne l\\rbrace }\\!\\sum _{m\\in \\mathcal {S},m\\ne l}\\!\\!q_{lm}(f(k,m)-f(k,l)),\\end{split}$ for every measurable function $f$ on $\\mathcal {S}\\times \\mathcal {S}$ .", "This implies that once $\\Lambda _t=\\Lambda ^{\\prime }_t$ , then $\\Lambda _s=\\Lambda _s^{\\prime }$ for all $s>t$ .", "For $(x,k,y,l)\\in \\mathbb {R}^d\\times \\mathcal {S}\\times \\mathbb {R}^d\\times \\mathcal {S}$ , set $a(x,k,y,l)=\\mathbf {1}_{\\Delta }(k,l)a_k(x,y)+\\mathbf {1}_{\\Delta ^c}(k,l)\\begin{pmatrix} a(x,k)&0\\\\0&a(y,l)\\end{pmatrix},$ where $\\Delta =\\lbrace (k,k);\\ k\\in \\mathcal {S}\\rbrace $ , and $a_k(x,y)$ is determined by the coupling operator $\\tilde{L}^{(k)}$ provided $\\tilde{L}^{(k)}\\sim (a_k(x,y), (b_k(x),b_k(y))^\\ast )$ .", "Let $(X_t,Y_t)$ satisfy the following SDE, $\\text{\\rm {d}}\\begin{pmatrix} X_t\\\\ Y_t\\end{pmatrix}=\\Psi (X_t,\\Lambda _t,Y_t,\\Lambda ^{\\prime }_t)\\text{\\rm {d}}W_t+\\begin{pmatrix} b(X_t,\\Lambda _t)\\\\b(Y_t,\\Lambda ^{\\prime }_t)\\end{pmatrix}\\text{\\rm {d}}t,$ where $\\Psi (x,k,y,l)$ is a $2d\\times 2d$ matrix such that $\\Psi (x,k,y,l)\\Psi (x,k,y,l)^\\ast =a(x,k,y,l)$ , and $(W_t)$ is a Brownian motion on $\\mathbb {R}^{2d}$ .", "Let $\\tau =\\inf \\lbrace t\\ge 0;\\ \\Lambda _t=\\Lambda _t^{\\prime }\\rbrace $ be the coupling time of $(\\Lambda _t,\\Lambda ^{\\prime }_t)$ .", "Since $\\mathcal {S}$ is a finite set, and $\\tilde{Q}$ defined by (REF ) is irreducible, it is well known that there exists a constant $\\theta >0$ such that $\\mathbb {P}(\\tau >t)\\le e^{-\\theta t},\\quad t>0.$ The processes $(X_t)$ and $(Y_t)$ defined by (REF ) will evolve independently until time $\\tau $ , then they evolve as the diffusion process $(X_t^{(k)}, Y_t^{(k)})$ corresponding to $\\tilde{L}^{(k)}$ when $\\Lambda _t=\\Lambda _t^{\\prime }=k$ .", "So once $X_t=Y_t$ at some $t>\\tau $ , they will move together after $t$ .", "This kind of coupling processes has appeared in [20] and [8].", "In [20], together with F. Xi, we discussed the question what conditions could ensure this kind of coupling to be successful when $(X_t,\\Lambda _t)$ is a state-dependent regime-switching process in a finite state space.", "The following lemma is the key point in the argument of Theorem REF .", "Lemma 3.2 Let the assumption of Theorem REF be satisfied, and assume further $\\Lambda _0=\\Lambda _0^{\\prime }$ and $X_0=x$ , $Y_0=y$ .", "Then for $C_2=\\xi _{\\mathrm {max}}/\\xi _{\\mathrm {min}}>0$ , $\\alpha =-\\lambda _{\\mathrm {max}}/\\xi _{\\mathrm {max}}>0$ , where $\\xi _{\\mathrm {max}}=\\max _{1\\le i\\le N}\\xi _i$ , $\\xi _{\\mathrm {min}}=\\min _{1\\le i\\le N}\\xi _i$ , and $\\lambda _{\\mathrm {max}}=\\max _{1\\le i\\le N}\\lambda _i<0$ , it holds, for every $t>s\\ge 0$ , $\\mathbb {E}[\\rho (\|X_t-Y_t\|)]\\le C_2\\mathbb {E}[\\rho (\|X_s-Y_s\|)]e^{-\\alpha (t-s)}.$ Proof.", "Since $\\Lambda _0=\\Lambda ^{\\prime }_0$ , by our construction of coupling process $(\\Lambda _t,\\Lambda ^{\\prime }_t)$ , $\\Lambda _t=\\Lambda _t^{\\prime }$ for all $t>0$ .", "As $N<\\infty $ and $\\lambda \\ll 0$ , we have $ \\lambda _{\\mathrm {max}}=\\max _{1\\le i\\le N} \\lambda _i<0$ .", "Applying Itô's formula (cf.", "[19]), we get $&\\mathbb {E}[\\rho (\|X_t-Y_t\|)\\xi _{\\Lambda _t}]\\\\&=\\mathbb {E}[\\rho (\|X_s-Y_s\|)\\xi _{\\Lambda _s}]+\\mathbb {E}\\Big [\\int _s^t(Q\\xi )(\\Lambda _r)\\rho (\|X_r-Y_r\|)+\\tilde{L}^{(\\Lambda _r)} \\rho (\|X_r-Y_r\|)\\xi _{\\Lambda _r}\\,\\text{\\rm {d}}r\\Big ]\\\\&\\le \\mathbb {E}[\\rho (\|X_s-Y_s\|)\\xi _{\\Lambda _s}]+\\mathbb {E}\\Big [\\int _s^t\\big ((Q\\xi )(\\Lambda _r)+\\beta _{\\Lambda _r}\\xi _{\\Lambda _r}\\big )\\rho (\|X_r-Y_r\|)\\,\\text{\\rm {d}}r\\Big ]\\\\&\\le \\mathbb {E}[\\rho (\|X_s-Y_s\|)\\xi _{\\Lambda _s}]+ \\lambda _{\\mathrm {max}} \\mathbb {E}\\Big [\\int _s^t \\rho (\|X_r-Y_r\|)\\text{\\rm {d}}r\\Big ].$ This implies that $&\\mathbb {E}[\\rho (\|X_t-Y_t\|)\\xi _{\\Lambda _t}]\\le \\mathbb {E}[\\rho (\|X_s-Y_s\|)\\xi _{\\Lambda _s}]+\\frac{ \\lambda _{\\mathrm {max}}}{\\xi _{\\mathrm {max}}}\\mathbb {E}\\Big [\\int _s^t\\rho (\|X_r-Y_r\|)\\xi _{\\Lambda _r}\\text{\\rm {d}}r\\Big ].$ The previous inequality still holds if we replace $s$ with $u$ satisfying $s<u<t$ .", "So we can apply Gronwall's inequality in the differential form to obtain $\\mathbb {E}[\\rho (\|X_t-Y_t\|)\\xi _{\\Lambda _t}]\\le \\mathbb {E}[\\rho (\|X_s-Y_s\|)\\xi _{\\Lambda _s}]e^{\\lambda _{\\mathrm {max}}(t-s)/\\xi _{\\mathrm {max}}},$ and further $\\mathbb {E}[\\rho (\|X_t-Y_t\|)]\\le \\frac{\\xi _{\\mathrm {max}}}{\\xi _{\\mathrm {min}}}\\mathbb {E}[\\rho (\|X_s-Y_s\|)]e^{ \\lambda _{\\mathrm {max}}(t-s)/\\xi _{\\mathrm {max}}},$ which yields the inequality (REF ).", "Proof of Theorem REF .", "Now we fix the initial point of the process $(X_t,\\Lambda _t,Y_t,\\Lambda _t^{\\prime })$ to be $(x,i,y,j)$ with $i\\ne j$ , and go to estimate the Wasserstein distance between the distributions of $(X_t,\\Lambda _t)$ and $(Y_t,\\Lambda _t^{\\prime })$ .", "By the inequality (REF ) and Lemma REF , we obtain $&\\mathbb {E}[\\tilde{\\rho }((X_t,\\Lambda _t), (Y_t,\\Lambda _t^{\\prime }))]\\\\ &=\\mathbb {E}\\big [\\sqrt{\\mathbf {1}_{\\Lambda _t\\ne \\Lambda ^{\\prime }_t}+\\rho (\|X_t-Y_t\|)}\\mathbf {1}_{\\tau >t/2}\\big ]+\\mathbb {E}\\big [\\sqrt{\\rho (\|X_t-Y_t\|)}\\mathbf {1}_{\\tau \\le t/2}\\big ]\\\\&\\le \\sqrt{\\mathbb {P}(\\tau >t/2)}\\sqrt{\\mathbb {E}[1+\\rho (\|X_t-Y_t\|)]}+\\sqrt{\\mathbb {E}[\\rho (\|X_t-Y_t\|)\\mathbf {1}_{\\tau \\le t/2}]}\\\\&\\le \\sqrt{1+\\mathbb {E}[\\rho (\|X_t\|)+\\rho (\|Y_t\|)]} e^{-\\frac{\\theta t}{4}}+\\sqrt{\\mathbb {E}[\\mathbb {E}[\\rho (\|X_t-Y_t\|)\\big \|F_\\tau ]\\mathbf {1}_{\\tau \\le t/2}]}\\\\&\\le \\sqrt{1+C_1(2+\\rho (\|x\|)+\\rho (\|y\|)) } e^{-\\frac{\\theta t}{4}}+\\sqrt{C_2\\mathbb {E}[\\rho (\|X_\\tau -Y_\\tau \|)e^{\\frac{-\\alpha t}{2}}]}\\\\&\\le \\sqrt{1+C_1(2+\\rho (\|x\|)+\\rho (\|y\|)) }e^{-\\frac{\\theta t}{4}}+\\sqrt{C_2C_1(2+\\rho (\|x\|)+\\rho (\|y\|)) }e^{\\frac{-\\alpha t}{4}}\\\\&\\le 2\\tilde{C}\\sqrt{3+\\rho (\|x\|)+\\rho (\|y\|)}e^{ -\\tilde{\\alpha }t },$ where $\\tilde{C}=\\max \\lbrace C_1, C_2,1\\rbrace $ independent of $(x,i,y,j)$ and $\\tilde{\\alpha }=\\min \\lbrace \\alpha ,\\theta \\rbrace /4>0$ .", "This implies $W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\delta _{(y,j)}P_t)\\le 2\\tilde{C}\\sqrt{3+\\rho (\|x\|)+\\rho (\|y\|)} e^{-\\tilde{\\alpha }t}.$ By (A2), we know that $\\mathbb {E}[\\rho (\|X_t\|)]$ is bounded for all $t>0$ .", "This yields that the family of probability measures $(\\delta _{(x,i)}P_t)_{t>0}$ is weakly compact since for each $c>0$ , $\\lbrace x\\in \\mathbb {R}^d;\\ \\rho (\|x\|)\\le c\\rbrace $ is a compact set.", "Moreover, $\\lim _{K\\rightarrow \\infty }\\sup _{t}\\!\\sum _{j\\in \\mathcal {S}}\\int _{\\tilde{\\rho }((y,j),(x,i))\\ge \\!", "K} \\!\\!\\!\\tilde{\\rho }((y,j),(x,i))\\delta _{(x,i)}P_t(\\text{\\rm {d}}x,j)\\le \\lim _{K\\rightarrow \\infty }\\sup _t\\frac{\\mathbb {E}[1+\\rho (\|x\|)+\\rho (\|X_t\|)]}{K}=0.$ Hence, $(\\delta _{(x,i)}P_t)_{t>0}$ is also compact in the Wassestein distance $W_{\\tilde{\\rho }}$ .", "There exists a subsequence $\\delta _{(x,i)}P_{t_k}$ , $t_k\\rightarrow \\infty $ as $k\\rightarrow \\infty $ , converging in $W_{\\tilde{\\rho }}$ -metric to some probability measure $\\nu $ on $\\mathbb {R}^d\\times \\mathcal {S}$ .", "Inequality (REF ) implies that for all $(y,j)\\in \\mathbb {R}^d\\times \\mathcal {S}$ , $\\delta _{(y,j)}P_{t_k}$ converges in $W_{\\tilde{\\rho }}$ -metric to $\\nu $ , and further that $\\nu _0P_{t_k}$ converges in $W_{\\tilde{\\rho }}$ -metric to $\\nu $ for every probability measure $\\nu _0$ on $\\mathbb {R}^d\\times \\mathcal {S}$ with $\\displaystyle \\int _{\\mathbb {R}^d}\\sqrt{\\rho (\|x\|)}\\nu _0(\\text{\\rm {d}}x,\\mathcal {S})<\\infty $ .", "This yields that for each $s>0$ , $\\delta _{(x,i)}P_sP_{t_k}$ converges in $W_{\\tilde{\\rho }}$ to $\\nu $ .", "It is easy to see $\\delta _{(x,i)}P_{t_k}P_s$ converges weakly to $\\nu P_s$ , hence converges in $W_{\\tilde{\\rho }}$ -metric to $\\nu P_s$ .", "In all, we get for each $s>0$ , $\\nu P_s=\\nu $ , then $\\nu $ is a stationary distribution of $P_t$ .", "As $(\\delta _{(x,i)}P_{t_k})$ converges weakly to $\\nu $ , by (A2), we get by Fatou's lemma $\\int _{\\mathbb {R}^d\\times \\mathcal {S}}\\rho (\|y\|)\\text{\\rm {d}}\\nu \\le \\liminf _{k\\rightarrow \\infty }\\int _{\\mathbb {R}^d\\times \\mathcal {S}}\\rho (\|y\|)\\text{\\rm {d}}\\big (\\delta _{(x,i)}P_{t_k}\\big )=\\liminf _{k\\rightarrow \\infty }\\mathbb {E}[\\rho (\|X_t\|)]\\le C_1.$ By (REF ), $&W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\nu )=W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\nu P_t)\\\\&=\\sup _{\\varphi :\\mathrm {Lip}(\\varphi )\\le 1}\\Big \\lbrace \\int _{\\mathbb {R}^d\\times \\mathcal {S}} \\varphi (y,j)\\text{\\rm {d}}\\big (\\delta _{(x,i)}P_t\\big )-\\int _{\\mathbb {R}^d\\times \\mathcal {S}}\\varphi (y,j)\\text{\\rm {d}}\\big (\\nu P_t\\big )\\Big \\rbrace \\\\&\\le \\int _{\\mathbb {R}^d\\times \\mathcal {S}}\\nu (\\text{\\rm {d}}y, j) W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\delta _{(y,j)}P_t)\\\\&\\le 2\\tilde{C}(\\sqrt{3+\\rho (\|x\|)}+ \\sqrt{C_1}) e^{-\\tilde{\\alpha }t}.$ Here we have used the duality formula for the Wasserstein distance, and $\\mathrm {Lip}(\\varphi ):=\\sup \\Big \\lbrace \\frac{\\varphi (y,j)-\\varphi (z,k)}{\\tilde{\\rho }((y,j),(k,l))};\\ (y,j)\\ne (z,k) \\Big \\rbrace .$ Till now, we have completed the proof of this theorem.", "Theorem 3.3 Assume that (A1) and (A2) hold.", "If the matrix $-\\big (Q+\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)\\big )$ is a nonsingular M-matrix, then there exists a probability measure $\\nu $ on $\\mathbb {R}^d\\times \\mathcal {S}$ such that $W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\nu )\\le 2\\tilde{C}(\\sqrt{3+\\rho (\|x\|)}+\\tilde{C}) e^{-\\tilde{\\alpha }t },$ The constants $\\tilde{\\alpha }$ and $\\tilde{C}$ are defined in Theorem REF .", "Proof.", "By Proposition REF , since $-(\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)+Q)$ is a nonsingular M-matrix, there exists a vector $\\xi =(\\xi _1,\\ldots , \\xi _N)^\\ast \\gg 0$ such that $\\lambda :=(\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)+Q)\\xi \\ll 0.$ According to Theorem REF , the desired results hold.", "Next, we assume that $(\\Lambda _t)$ is reversible with $\\pi =(\\pi _i)$ being its reversible probability measure.", "So it holds $\\pi _iq_{ij}=\\pi _j q_{ji}$ , $i,\\,j\\in \\mathcal {S}$ .", "Let $L^2(\\pi )=\\lbrace f\\in B(\\mathcal {S});\\ \\sum _{i=1}^N \\pi _if_i^2<\\infty \\rbrace $ , and denote by $\\Vert \\cdot \\Vert $ and $\\langle \\cdot , \\cdot \\rangle $ respectively the norm and inner product in $L^2(\\pi )$ .", "Let $D(f)=\\frac{1}{2}\\sum _{i,j=1}^N\\pi _iq_{ij}(f_j-f_i)^2-\\sum _{i=1}^N\\pi _i\\beta _if_i^2,\\quad f\\in L^2(\\pi ),$ where $(\\beta _i)$ is given by condition (A1).", "We borrow the notation $D(f)$ from the Dirichlet theory for continuous time Markov chain, but we should note that in our case $\\beta _i$ could be positive, so $D(f)$ may be negative which is different to the standard Dirichlet theory.", "Define the principal eigenvalue by $\\lambda _0=\\inf \\big \\lbrace D(f);\\ f\\in L^2(\\pi ),\\ \\Vert f\\Vert =1\\big \\rbrace .$ Theorem 3.4 Let (A1) and (A2) be satisfied and assume that $(\\Lambda _t)$ is reversible with respect to the probability measure $(\\pi _i)$ .", "Assume the principal eigenvalue $\\lambda _0>0$ .", "Then there are positive constants $\\tilde{C}$ , $\\tilde{\\alpha }$ and a probability measure $\\nu $ on $\\mathbb {R}^d\\times \\mathcal {S}$ so that $W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\nu )\\le 2\\tilde{C}(\\sqrt{3+\\rho (\|x\|)}+\\tilde{C}) e^{-\\tilde{\\alpha }t }.$ Proof.", "As $N<\\infty $ and $\\lambda _0>0$ , there exists a $g\\in L^2(\\pi )$ such that $g\\lnot \\equiv 0$ , $D(g)=\\lambda _0\\Vert g\\Vert ^2$ .", "We shall show that $g\\gg 0$ and $Qg(i)+\\beta _ig_i=-\\lambda _0 g_i$ , $i\\in \\mathcal {S}$ , then this theorem follows immediately from Theorem REF by taking $\\xi =g$ .", "We use the variational method in [4].", "It is easy to check $D(f)\\ge D(\|f\|)$ , so it must hold $g\\ge 0$ .", "For a fixed $k\\in \\mathcal {S}$ , let $\\tilde{g}_i=g_i$ for $i\\ne k$ and $\\tilde{g}_k=g_k+\\varepsilon $ .", "It holds $Q\\tilde{g}(i)=Qg(i)+\\varepsilon q_{ik}$ for $i\\ne k$ and $Q\\tilde{g}(k)=Qg(k)-\\varepsilon q_k$ .", "We have $D(\\tilde{g})&=\\langle \\tilde{g},-Q\\tilde{g}\\rangle -\\sum _{i=1}^N\\pi _i\\beta _i\\tilde{g}_i^2\\\\&=\\langle g, -Qg\\rangle -\\sum _{i=1}^N\\pi _i \\beta _i g_i^2 +2\\varepsilon \\pi _k(-Qg)(k) -2\\varepsilon \\pi _k\\beta _kg_k-\\varepsilon ^2\\pi _k(q_k-\\beta _k),$ where we have used $\\pi _iq_{ik}=\\pi _kq_{ki}$ .", "Because $D(\\tilde{g})\\ge \\lambda _0\\Vert \\tilde{g}\\Vert ^2$ and $D(g)=\\lambda _0\\Vert g\\Vert ^2$ , we get $-2\\varepsilon \\pi _k\\big (\\lambda _0g_k+Qg(k)+\\beta _kg_k\\big )+\\varepsilon ^2\\pi _k(g_k-\\beta _k)-2\\lambda _0\\varepsilon ^2\\pi _k\\ge 0.$ This yields $Qg(k)+\\beta _kg_k=-\\lambda _0g_k$ since $\\varepsilon $ is arbitrary, and then $Qg(i)+\\beta _ig_i=-\\lambda _0 g_i$ for each $i\\in \\mathcal {S}$ since $k$ is arbitrary.", "Since $g\\lnot \\equiv 0$ and $g\\ge 0$ , there exists $k$ such that $g_k>0$ .", "If $q_{ik}>0$ , then $0<q_{ik}g_k\\le \\sum _{j\\ne i}q_{ij}g_j=(q_i-\\beta _i-\\lambda _0)g_i,$ so $g_i>0$ and $q_i-\\beta _i-\\lambda _0>0$ .", "As $Q$ is irreducible, by an inductive procedure, we can prove that $g_i>0$ for every $i\\in \\mathcal {S}$ .", "Remark 3.5 According to the argument of Theorem REF and the statement 3 of Proposition REF , we obtain that $\\lambda _0>0$ is equivalent to the statement $-(Q+\\mathrm {diag}(\\beta ))$ is a nonsingular M-matrix when $(\\Lambda _t)$ is reversible process in a finite state space, i.e.", "$N<\\infty $ .", "However, the criterion expressed by the principal eigenvalue $\\lambda _0$ of a bilinear form can be extended directly to deal with the situation $N=\\infty $ and the criterion expressed by nonsingular M-matrix can not.", "To apply the criterion expressed by the principal eigenvalue, one has to justify the positiveness of $\\lambda _0$ which is not easy when $N=\\infty $ .", "But this is not the main topic of present work, and we are satisfied with this connection at present stage and leave further study of $\\lambda _0$ to another work.", "If we use the metric $\\tilde{\\rho }_p$ on $\\mathbb {R}^d\\times \\mathcal {S}$ , we can recover the condition given by [8] to justify the exponential ergodicity of $(X_t,\\Lambda _t)$ .", "The advantage of this criterion (see (REF ) below) is that it has very concise expression, and the disadvantage is that we can not fix explicitly the power $p$ .", "Theorem 3.6 Assume that (A1) and (A2) hold.", "Let $\\mu =(\\mu _i)_{i\\in \\mathcal {S}}$ be the invariant probability measure of $(q_{ij})$ .", "If $\\sum _{i=1}^N\\mu _i\\beta _i<0,$ then there exist positive constants $p$ , $\\alpha _p$ , $\\tilde{C}_1$ , such that $W_{\\tilde{\\rho }_p}(\\delta _{(x,i)}P_t,\\nu )\\le 2\\tilde{C}_1(\\sqrt{3+\\rho (\|x\|)}+\\tilde{C}_1) e^{-\\alpha _p t},$ where $\\tilde{C}_1$ is independent of $(x,i)$ .", "Proof.", "Let $Q_p=Q+p\\,\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)$ , and $\\eta _p=-\\max _{\\gamma \\in \\mathrm {spec}(Q_p)}\\mathrm {Re}\\,\\gamma , \\quad \\text{where $\\mathrm {spec}(Q_p)$ denotes the spectrum of $Q_p$}.$ Let $Q_{(p,t)}=e^{tQ_p}$ , then the spectral radius $\\mathrm {Ria}(Q_{(p,t)})$ of $Q_{(p,t)}$ equals to $e^{-\\eta _p t}$ .", "Since all coefficients of $Q_{(p,t)}$ are positive, Perron-Frobenius theorem (see [1]) yields $-\\eta _p$ is a simple eigenvalue of $Q_p$ .", "Moreover, note that the eigenvector of $Q_{(p,t)}$ corresponding to $e^{-\\eta _p t}$ is also an eigenvector of $Q_p$ corresponding to $-\\eta _p$ .", "Then Perron-Frobenius theorem ensures that there exists an eigenvector $\\xi \\gg 0$ of $Q_p$ corresponding to $-\\eta _p$ .", "Now applying Proposition 4.2 of [2] (by replacing $A_p$ there with $Q_p$ ), if $\\sum _{i=1}^N\\mu _i\\beta _i<0$ , then there exists some $p_0>0$ such that $\\eta _p>0$ for any $0<p<p_0$ .", "Fix a $p$ with $0<p<\\min \\lbrace 1,p_0\\rbrace $ and an eigenvector $\\xi \\gg 0$ , then we obtain $Q_p\\,\\xi =(Q+p\\,\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N))\\xi =-\\eta _p\\,\\xi \\ll 0.$ For the coupling operator $\\tilde{L}^{(i)}\\sim (a^{(i)}(x,y),b^{(i)}(x,y))$ , due to the nonnegative definiteness of $a^{(i)}(x,y)$ and $ 0<p<1$ , it can be checked by direct calculus that $\\tilde{L}^{(i)} \\rho ^p(\|x-y\|)\\le p \\rho ^{p-1}(\|x-y\|)\\tilde{L}^{(i)}\\rho (\|x-y\|),\\quad x,\\,y\\in \\mathbb {R}^d,\\ x\\ne y.$ Combining with Assumption (A1), we get $\\tilde{L}^{(i)}\\rho ^p(\|x-y\|)\\le p\\beta _i\\rho ^p(\|x-y\|),\\quad x,\\,y\\in \\mathbb {R}^d,\\ x\\ne y.$ Similar to the argument of Lemma REF , if $\\Lambda _s=\\Lambda _s^{\\prime }$ for some $0\\le s<t$ , by Itô's formula, we obtain $&\\mathbb {E}[\\rho ^p(\|X_t-Y_t\|)\\xi _{\\Lambda _t}]\\\\&\\le \\mathbb {E}[\\rho ^p(\|X_s-Y_s\|)\\xi _{\\Lambda _s}]+\\mathbb {E}\\Big [\\int _s^t\\big ((Q+p\\,\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N))\\xi \\big )(\\Lambda _r)\\rho ^p(\|X_r-Y_r\|)\\text{\\rm {d}}r\\Big ]\\\\&\\le \\mathbb {E}[\\rho ^p(\|X_s-Y_s\|)\\xi _{\\Lambda _s}]-\\eta _p\\mathbb {E}\\Big [\\int _s^t\\rho ^p(\|X_r-Y_r\|)\\xi _{\\Lambda _r}\\text{\\rm {d}}r\\Big ].$ Due to the arbitrariness of $s$ , $0\\le s<t$ , we can apply Gronwall's inequality in differential form to get $\\mathbb {E}[\\rho ^p(\|X_t-Y_t\|)\\xi _{\\Lambda _t}]\\le \\mathbb {E}[\\rho ^p(\|X_s-Y_s\|)\\xi _{\\Lambda _s}]e^{-\\eta _p(t-s)}.$ Consequently, $\\mathbb {E}[\\rho ^p(\|X_t-Y_t\|)]\\le C_3\\mathbb {E}[\\rho ^p(\|X_s-Y_s\|)]e^{-\\eta _p(t-s)},$ where $C_3=\\max _{k,l\\in \\mathcal {S}} \\big (\\xi _k/ \\xi _l\\big )\\ge 1$ .", "Now we go to estimate the Wasserstein distance between the distributions of $(X_t,\\Lambda _t)$ and $(Y_t,\\Lambda _t^{\\prime })$ with $(X_0,\\Lambda _0,Y_0,\\Lambda _0^{\\prime })=(x,i,y,j)$ and $i\\ne j$ .", "$&\\mathbb {E}\\big [\\tilde{\\rho }_p((X_t,\\Lambda _t),(Y_t,\\Lambda _t^{\\prime }))\\big ]\\\\&=\\mathbb {E}\\big [\\sqrt{\\mathbf {1}_{\\Lambda _t\\ne \\Lambda _t^{\\prime }}+\\rho ^p(\|X_t-Y_t\|)}\\mathbf {1}_{\\tau >t/2}\\big ]+\\mathbb {E}\\big [\\sqrt{\\rho ^p(\|X_t-Y_t\|)}\\mathbf {1}_{\\tau \\le t/2}\\big ]\\\\&\\le \\sqrt{\\mathbb {P}(\\tau >t/2)}\\sqrt{1+\\big (\\mathbb {E}[\\rho (\|X_t-Y_t\|)])^{p}}+\\sqrt{\\mathbb {E}[\\rho ^p(\|X_t-Y_t\|)\\mathbf {1}_{\\tau \\le t/2}]}\\\\&\\le 2\\tilde{C}_1\\sqrt{3+\\rho (\|x\|)+\\rho (\|y\|)} e^{- \\alpha _p t },$ where $\\tilde{C}_1=\\max \\lbrace C_1, C_3\\rbrace $ , $ \\alpha _p=\\min \\lbrace \\theta ,\\eta _p\\rbrace /4>0$ .", "This yields that $W_{\\tilde{\\rho }_p}(\\delta _{(x,i)}P_t,\\delta _{(y,i)}P_t)\\le 2\\tilde{C}_1\\sqrt{3+\\rho (\|x\|)+\\rho (\|y\|)} e^{- \\alpha _p t }.$ As in the late part of the argument of Theorem REF , (REF ) can yield the desired result.", "The proof is completed.", "Remark 3.7 From the argument of Theorem REF , we essentially use the Perron-Frobenius theorem to ensure the existence of a vector $\\xi \\gg 0$ such that $\\big (Q+p\\,\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)\\big )\\xi \\ll 0$ .", "Moreover, in the proof of [2] the fact $-\\eta _p=\\pi _pQ_p\\mathbf {1}=\\pi _p\\,\\mathrm {diag}(\\beta _1,\\ldots ,\\beta _N)\\mathbf {1}$ has been used for the left eigenvector $\\pi _p$ of $Q_p$ associated to $-\\eta _p$ with $\\pi _p\\mathbf {1}=1$ .", "This prevents us from applying this method to regime-switching processes with a countable state space $\\mathcal {S}$ ." ], [ "Markovian Switching in a countable state space", "In this section, we consider the regime-switching diffusion $(X_t,\\Lambda _t)$ given by (REF ) and (REF ) with state-independent switching in a countable set, i.e.", "$\\mathcal {S}=\\lbrace 1,2,\\ldots ,N\\rbrace $ and $N=\\infty $ .", "There few result on the ergodicity of regime-switching diffusion process when the state space $\\mathcal {S}$ of $(\\Lambda _t)$ is a countable set.", "In this case, the criteria expressed by Lyapunov function or drift condition for general Markov processes still work.", "But it is well known that constructing Lyapunov functions is a difficult job even for diffusion processes.", "To construct a Lyapunov function for a regime-switching diffusion becomes more difficult due to the appearance of diffusion operator and jump operator in its infinitesimal generator at the same time.", "In this part, we put forward a method to transform the switching process $(\\Lambda _t)$ in a countable state space into a new one in a finite state space.", "By using the criterion established in previous section by M-matrix theory, we can guarantee that if the new regime-switching diffusion process is ergodic in Wasserstein distance, then so is the original one.", "In this section, we assume that (A1) holds, and further $M:=\\sup _{i\\in \\mathcal {S}}\\beta _i<\\infty $ .", "As $\\mathcal {S}$ is a countable set, we need more assumption on $(\\Lambda _t)$ so that the coupling method could be applied.", "(A3) The $Q$ -matrix of $(\\Lambda _t)$ is conservative irreducible and $\\sup _{i\\in \\mathcal {S}} q_i<\\infty $ .", "There is a coupling process $(\\Lambda _t,\\Lambda _t^{\\prime })$ with operator $\\tilde{Q}$ on $\\mathcal {S}\\times \\mathcal {S}$ .", "Suppose there is a bounded function $g\\ge 0$ in the domain of $\\tilde{Q}$ such that $g(i,i)=0$ and $\\tilde{Q}g(i,j)\\le -1,\\quad i\\ne j.$ According to [6], (A2) implies that for $0<\\theta <\\Vert g\\Vert _{\\infty }$ , $\\tilde{E}[ e^{\\theta \\, \\tau }]\\le \\frac{1}{1-\\theta \\, \\Vert g\\Vert _{\\infty }},$ where $\\tau =\\inf \\lbrace t\\ge 0,\\Lambda _t=\\Lambda _t^{\\prime }\\rbrace $ .", "We choose and fix a $\\theta $ with $0<\\theta <\\Vert g\\Vert _{\\infty }$ in the rest of this section.", "The inequality (REF ) is what we need to estimate the Wasserstein distance directly.", "Assumption (A2) provides a sufficient condition to guarantee (REF ) hold.", "The coupling process for a continuous time Markovian chain is a well studied topic.", "There are lots of work on this topic.", "For example, due to [23], [14], there are explicit conditions in terms of birth rate and death rate to check condition (A2) for birth-death process.", "We refer the reader to [6] for more discussion on this condition.", "First, we divide $\\mathcal {S}$ into finite subsets according to $\\beta _i$ .", "Precisely, choose a finite partition $\\Gamma $ of $(-\\infty ,M]$ , that is, $\\Gamma :=\\big \\lbrace -\\!\\infty =:k_0<k_1<\\cdots <k_m<k_{m+1}:=M\\big \\rbrace .$ Corresponding to $\\Gamma $ , there is a finite partition of $\\mathcal {S}$ , denoted by $ F:= \\lbrace F_1,\\ldots , F_{m+1}\\rbrace $ , where $F_i=\\big \\lbrace j\\in \\mathcal {S};\\ \\beta _j\\in (k_{i-1},k_{i}]\\big \\rbrace ,\\quad i=1,\\ldots , m+1.$ We assume that each $F_i$ is not empty, otherwise, we can delete some points in the partition $\\Gamma $ to ensure it.", "Let $\\phi :\\mathcal {S}\\rightarrow \\lbrace 1,\\ldots ,m+1\\rbrace $ be a map defined by $\\phi (j)=i$ if $j\\in F_i$ .", "Let $\\beta ^F_i=\\sup _{j\\in F_i}\\beta _{j} \\ \\text{for}\\ i=1,\\ldots ,m+1, \\ \\text{so}\\ \\beta _j\\le \\beta _{\\phi (j)}^F \\ \\text{for every $j\\in \\mathcal {S}$}, \\ \\text{and}\\ \\beta _i^F<\\beta _{i+1}^F.$ Set $Q^F=(q_{ij}^F)$ be a new $Q$ -matrix on state space $\\lbrace 1,2,\\ldots ,m+1\\rbrace $ corresponding to $F$ defined by $q_{ik}^F=\\inf _{r\\in F_i}\\sum _{j\\in F_k} q_{rj},\\ k>i;\\quad q_{ik}^F=\\sup _{r\\in F_i}\\sum _{j\\in F_k} q_{rj},\\ k<i,\\ \\text{and}\\ q_{ii}^F=-\\sum _{k\\ne i} q_{ik}^F.$ As each $F_i$ is nonempty and $(q_i)_{i\\in \\mathcal {S}}$ is bounded, we get $0\\le q_{ik}^F\\le \\sup _{i\\in \\mathcal {S}} q_i<\\infty ,\\, k\\ne i$ .", "It is not easy to check whether $Q^F$ is irreducible.", "But this does not impact the criterion provided by the theory of M-matrix.", "It is an advantage that there is no demand on irreducibility in checking a matrix to be nonsingular M-matrix.", "However, in the study of Perron-Frobenius theorem, irreducibility of a matrix plays important role.", "Theorem 4.1 Assume that (A1) (A2) and (A3) hold.", "For the partition $F$ given above, if the $(m+1)\\times (m+1)$ -matrix $-\\big (Q^F+\\mathrm {diag}(\\beta _1^F,\\ldots ,\\beta _{m+1}^F)\\big )H_{m+1}$ is a nonsingular M-matrix, where $H_{m+1}=\\begin{pmatrix}1&1&1&\\cdots &1\\\\ 0&1&1&\\cdots &1\\\\ \\vdots &\\vdots &\\vdots &\\cdots &\\vdots \\\\ 0&0&0&\\cdots &1\\end{pmatrix}_{(m+1)\\times (m+1)},$ then there exist constants $\\tilde{C},\\,\\tilde{\\alpha }>0$ and a probability measure $\\nu $ on $\\mathbb {R}^d\\times \\mathcal {S}$ such that $W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\nu )\\le 2\\tilde{C}(\\sqrt{3+\\rho (\|x\|)}+\\tilde{C}) e^{-\\tilde{\\alpha }t},\\quad (x,i)\\in \\mathbb {R}^d\\times \\mathcal {S},$ where $\\tilde{C}>0$ is independent of $(x,i)$ .", "Proof.", "Let $(\\Lambda _t,\\Lambda ^{\\prime }_t)$ be the coupling given by (A2).", "Let $(X_t,Y_t)$ be defined by (REF ).", "Similar to the proof of Theorem REF , the key point is also the estimates given by Lemma REF .", "As $-\\big (Q^F+\\mathrm {diag}(\\beta _1^F,\\ldots ,\\beta _{m+1}^F)\\big )H_{m+1}$ is a nonsingular M-matrix, there exists a vector $\\eta ^F=(\\eta _1^F,\\ldots ,\\eta _{m+1}^F)^\\ast \\gg 0$ such that $\\lambda ^F=(\\lambda _1^F,\\ldots ,\\lambda _{m+1}^F)^\\ast =\\big (Q^F+\\mathrm {diag}(\\beta _1^F,\\ldots ,\\beta _{m+1}^F)\\big )H_{m+1}\\eta ^F\\ll 0.$ Then $\\bar{\\lambda }:=\\max _{1\\le i\\le m+1} \\lambda _i^F<0$ .", "Set $\\xi ^F=H_{m+1}\\eta ^F$ .", "Then $\\xi _i^F=\\eta _{m+1}^F+\\cdots +\\eta _i^F,\\ i=1,\\ldots m+1.", "$ Hence, $\\xi _{i+1}^F<\\xi _i^F$ , $i=1,\\ldots ,m$ , and $\\xi ^F\\gg 0$ .", "We extend the vector $\\xi ^F$ to a vector on $\\mathcal {S}$ by setting $\\xi _r=\\xi _i^F$ , if $r\\in F_i$ .", "For $r\\in F_i$ , we obtain $Q\\xi (r)&=\\sum _{j\\in \\mathcal {S},j\\ne r} q_{rj}(\\xi _j-\\xi _r)=\\sum _{j\\notin F_i,j\\in \\mathcal {S}}q_{rj}(\\xi _j-\\xi _r)\\\\&=\\sum _{k<i}\\big (\\sum _{j\\in F_k} q_{rj}\\big )(\\xi _k^F-\\xi _i^F)+\\sum _{k>i}\\big (\\sum _{j\\in F_k}q_{rj}\\big )(\\xi _k^F-\\xi _i^F)\\\\&\\le \\sum _{k<i} q_{ik}^F(\\xi _k^F-\\xi _i^F)+\\sum _{k>i}q_{ik}^F(\\xi _k^F-\\xi _i^F)=\\big (Q^F\\xi ^F\\big )(i),$ where we have used (REF ).", "Applying Itô's formula to $(X_t,\\Lambda _t)$ , $(Y_t,\\Lambda _t^{\\prime })$ with $X_0=x$ , $Y_0=y$ and $\\Lambda _0=\\Lambda _0^{\\prime }=r$ , we have, for every $0<u<t$ , $&\\mathbb {E}[\\rho (\|X_t-Y_t\|)\\xi _{\\Lambda _t}]\\\\&\\le \\mathbb {E}[\\rho (\|X_u-Y_u\|)\\xi _{\\Lambda _u}]+\\mathbb {E}\\Big [\\int _u^t\\big ((Q \\xi )(\\Lambda _s)+\\beta _{\\Lambda _s}\\xi _{\\Lambda _s}\\big )\\rho (\|X_s-Y_s\|)\\text{\\rm {d}}s\\Big ]\\\\&\\le \\mathbb {E}[\\rho (\|X_u-Y_u\|)\\xi _{\\Lambda _u}]+\\mathbb {E}\\Big [\\int _u^t\\big ((Q^F\\xi ^F)(\\phi (\\Lambda _s))+\\beta ^F_{\\phi (\\Lambda _s)}\\xi _{\\phi (\\Lambda _s)}^F\\big )\\rho (\|X_s-Y_s\|)\\text{\\rm {d}}s\\Big ]\\\\&\\le \\mathbb {E}[\\rho (\|X_u-Y_u\|)\\xi _{\\Lambda _u}]+\\frac{\\bar{\\lambda }}{\\xi _{\\mathrm {min}}^F}\\mathbb {E}\\Big [\\int _u^t\\rho (\|X_s-Y_s\|)\\xi _{\\Lambda _s}\\text{\\rm {d}}s\\Big ],$ where $\\phi :\\mathcal {S}\\rightarrow F$ denotes the projection map, $\\xi _{\\mathrm {max}}^F=\\max _{1\\le i\\le m+1}\\xi ^F_i>0$ .", "Set $\\tilde{\\alpha }=-\\bar{\\lambda }/\\xi _{\\mathrm {max}}^F>0$ .", "Due to the arbitrariness of $0<u<t$ , we can apply Gronwall's inequality in differential form to get $\\mathbb {E}\\big [\\rho (\|X_t-Y_t\|)\\xi _{\\Lambda _t}\\big ]\\le \\mathbb {E}\\big [\\rho (\|X_u-Y_u\|)\\xi _{\\Lambda _u}\\big ]e^{-\\tilde{\\alpha }(t-u)},\\quad 0<u<t.$ Thanks to (REF ) and (A2), we can prove that $W_{\\tilde{\\rho }}(\\delta _{(x,i)}P_t,\\delta _{(y,j)}P_t)\\le 2\\tilde{C}\\sqrt{3+\\rho (\|x\|)+\\rho (\|y\|)} e^{-\\tilde{\\alpha }t}$ for some $\\tilde{C}>0$ and $\\tilde{\\alpha }>0$ .", "According to (A3), we know that $(\\Lambda _t)$ is exponential ergodic, hence there exists a compact function $h$ on $\\mathcal {S}$ such that $\\sup _{t>0}\\mathbb {E}[h(\\Lambda _t)]\\le C_4$ , where $C_4$ is a positive constant (see [5]).", "Combining with (A2), there is a constant $C_5>0$ so that $\\sup _{t>0}\\mathbb {E}[\\rho (\|X_t\|)+h(\\Lambda _t)]\\le C_5.$ Therefore, the family of probability measures $(\\delta _{(x,i)}P_t)_{t>0}$ is weakly compact.", "Then, following the similar argument as in Theorem REF , we can conclude the proof.", "Now we consider the consistency of our method on the finite partitions.", "Under the assumption (A1), consider two finite partitions $\\tilde{\\Gamma }$ and $\\Gamma $ of $(-\\infty ,M]$ such that $\\tilde{\\Gamma }$ is a refinement of $\\Gamma $ .", "Associated with $\\tilde{\\Gamma }$ and $\\Gamma $ , there are respectively two finite partitions $\\tilde{F}$ and $F$ of $\\mathcal {S}$ given by $\\tilde{F}=\\lbrace \\tilde{F}_1,\\ldots ,\\tilde{F}_{n+1}\\rbrace ,\\ \\text{and}\\ F=\\lbrace F_1,\\ldots ,F_{m+1}\\rbrace .$ Therefore, each $\\tilde{F}_k$ is a subset of some $F_i$ .", "Without loss of generality, assume $\\tilde{F}_k$ is nonempty for each $k$ .", "Let $\\beta ^F=(\\beta _1^F,\\ldots ,\\beta _{m+1}^F)^\\ast $ , $\\beta ^{\\tilde{F}}=(\\beta _1^{\\tilde{F}},\\ldots ,\\beta _{n+1}^{\\tilde{F}})^\\ast $ , $(q_{ij}^F)$ and $(q_{kl}^{\\tilde{F}})$ be defined similarly by (REF ) and (REF ).", "Proposition 4.2 Suppose that for each $k$ , $1\\le k\\le n+1$ , $\\begin{split}q_{ij}^F&\\ge \\sum _{l:\\tilde{F}_l\\subseteq F_j} q_{kl}^{\\tilde{F}}, \\ \\text{if}\\ i<j;\\quad q_{ij}^F\\le \\sum _{l:\\tilde{F}_l\\subseteq F_j} q_{kl}^{\\tilde{F}},\\ \\text{if}\\ i>j.\\end{split}$ Then the fact $-(Q^F+\\mathrm {diag}(\\beta ^F))H_{m+1}$ is a nonsingular M-matrix yields that so is the matrix $-(Q^{\\tilde{F}}+\\mathrm {diag}(\\beta ^{\\tilde{F}}))H_{n+1}$ .", "Proof.", "According to Proposition REF , as $-(Q^F+\\mathrm {diag}(\\beta ^F))H_{m+1}$ is a nonsingular M-matrix, there exists a vector $\\eta ^F\\gg 0$ such that $\\big (Q^F+\\mathrm {diag}(\\beta ^F))H_{m+1}\\eta ^F\\ll 0$ .", "Let $\\xi ^F=H_{m+1}\\eta ^F$ , then $\\xi ^F\\gg 0$ and $\\xi _i^F\\le \\xi _{i+1}^F$ for $i=1,\\ldots ,m$ .", "Let $\\xi _k^{\\tilde{F}}=\\xi _i^F$ if $\\tilde{F}_k\\subseteq F_i$ , $k=1,\\ldots , n+1$ , and $\\xi ^{\\tilde{F}}=\\lbrace \\xi _1^{\\tilde{F}},\\ldots , \\xi _{n+1}^{\\tilde{F}}\\rbrace $ .", "Then by (REF ) and the fact $\\beta _k^{\\tilde{F}}\\le \\beta _i^F$ if $\\tilde{F}_k\\subseteq F_i$ , we have, $&\\big (Q^{\\tilde{F}}+\\mathrm {diag}(\\beta ^{\\tilde{F}})\\xi ^{\\tilde{F}}=\\sum _{l=1}^{n+1}q_{kl}^{\\tilde{F}}\\xi _l^{\\tilde{F}}+\\beta _k^{\\tilde{F}}\\xi _k^{\\tilde{F}}\\\\&=\\sum _{j<i}\\big (\\sum _{l:\\tilde{F}_l\\subseteq F_j}q_{kl}^{\\tilde{F}}\\big )(\\xi _j^F-\\xi _i^F)+\\sum _{j>i}\\big (\\sum _{l:\\tilde{F}_l\\subseteq F_j}q_{kl}^{\\tilde{F}}\\big )(\\xi _j^F-\\xi _i^F)+\\beta _k^{\\tilde{F}}\\xi _i^F\\\\&\\le \\sum _{j\\ne i}q_{ij}^F(\\xi _j^F-\\xi _i^F)+\\beta _i^F\\xi _i^F\\\\&=\\big (Q^F+\\mathrm {diag}(\\beta ^F)\\big )\\xi ^F(i)\\ll 0.$ Therefore, $-\\big (Q^{\\tilde{F}}+\\mathrm {diag}(\\beta ^{\\tilde{F}})H_{n+1}$ is also a nonsingular M-matrix due to Proposition REF .", "By Theorem REF , we can provide some examples of regime-switching processes in an infinite state space, which are exponentially ergodic in the Wasserstein distance $W_{\\tilde{\\rho }}$ .", "Example 4.1 Let $\\mathcal {S}=\\lbrace 1,2,\\ldots \\rbrace $ be a countable set.", "Let $(X_t,\\Lambda _t)$ be a state-independent regime-switching diffusion process given by (REF ) and (REF ).", "Assume (A1-A4) hold.", "Let $F_1=\\lbrace j\\in \\mathcal {S}; \\beta _j<0\\rbrace $ and $F_2=\\lbrace j\\in \\mathcal {S};\\beta _j>0\\rbrace $ .", "Set $\\beta _1^F=\\sup _{j\\in F_1} \\beta _j$ and $\\beta _2^F=\\sup _{j\\in F_2}\\beta _j$ .", "$Q^F=(q_{ij}^F)$ is induced from $Q$ as above.", "We now check the condition that $-\\big (Q^F+\\mathrm {diag}(\\beta _1^F,\\beta _2^F)\\big )H_2=\\begin{pmatrix}-q_{11}^F-\\beta _1^F& -\\beta _1^F\\\\ q_{22}^F& -\\beta _2^F\\end{pmatrix}$ is a nonsingular M-matrix.", "By Proposition REF , it is equivalent to $\\beta _1^F<-q_{11}^F\\ \\text{and}\\ \\beta _1^F<\\beta _2^F<\\frac{q_{22}^F\\beta _1^F}{-q_{11}^F-\\beta _1^F}.$ This ensures that there are many regime-switching diffusion processes $(X_t,\\Lambda _t)$ with infinite state space $\\mathcal {S}$ such that the conditions of Theorem REF hold.", "Next, we provide a more concrete example.", "Example 4.2 Let $(\\Lambda _t)$ be a birth-death process on countable set $\\mathcal {S}=\\lbrace 1,2,\\ldots \\rbrace $ .", "For each $i>1$ , set $q_{i i+1}=b_i>0$ and $q_{i i-1}=a_i>0$ , and $q_{ij}=0$ for $j\\ne i+1$ or $i-1$ .", "Let $q_{12}=b_1>0$ .", "Set $\\mu _1=1$ and $\\mu _n=b_1b_2\\cdots b_{n-1}/a_2a_3\\cdots a_n$ for $n\\ge 2$ .", "Assume $\\sum _{i=1}^\\infty \\frac{1}{\\mu _ib_i}\\sum _{j=i+1}^\\infty \\mu _j<\\infty .$ Let $(\\Lambda _t,\\Lambda _t^{\\prime })$ be the classical coupling whose generator is given by $\\tilde{Q}h(i,j)={\\left\\lbrace \\begin{array}{ll} [a_i(h(i-1,j)-h(i,j))+b_i(h(i+1,j)-h(i,j))]\\\\\\quad +[a_j(h(i,j-1)-f(i,j))+b_j(h(i,j+1)-h(i,j))], \\quad i\\ne j,\\\\a_i(h(i-1,j-1)-h(i,j))+b_i(h(i+1,j+1)-h(i,j)), \\quad i=j.\\end{array}\\right.", "}$ Let $g(i,j)=\\sum _{k=1}^{j-1}\\frac{1}{\\mu _k b_k}\\sum _{l=k+1}\\mu _l.$ Then $g$ satisfies $\\displaystyle \\Vert g\\Vert _\\infty :=\\sup _{(i,j)\\in \\mathcal {S}^2} g(i,j)\\le \\sum _{k=1}^\\infty \\frac{1}{\\mu _k b_k}\\sum _{l=k+1}^\\infty \\mu _l<\\infty $ by assumption.", "It is easy to check that $\\tilde{Q} g(i,j)=-1$ by direct calculation.", "Therefore, assumption (A4) is satisfied.", "For each $i\\ge 1$ , let $(X_t^{(i)})$ be a diffusion process on $[0,\\infty )$ with reflecting boundary at 0 satisfying following SDE: $\\text{\\rm {d}}X_t^{(i)}=\\beta _i X_t^{(i)}\\text{\\rm {d}}t+\\sqrt{2}\\text{\\rm {d}}B_t,$ where $\\beta _i$ is a constant and $(B_t)$ is a Brownian motion.", "When $\\beta _i<0$ , $(X_t^{(i)})$ is an Ornstein-Uhlenbeck process, which is exponential ergodic.", "But when $\\beta _i>0$ , $(X_t^{(i)})$ is not recurrent.", "For each $i\\ge 1$ , define a reflecting coupling for $(X_t^{(i)})$ with infinitesimal generator $\\tilde{L}^{(i)}\\sim (a^{(i)}(x,y),b^{(i)}(x,y))$ , where $a^{(i)}(x,y)=\\begin{pmatrix}1& -1\\\\ -1& 1\\end{pmatrix},\\quad b^{(i)}(x,y)=\\begin{pmatrix}\\beta _i\\, x\\\\ \\beta _i\\, y\\end{pmatrix}.$ Let $\\rho (\|x-y\|)=\|x-y\|$ , then it is easy to see $\\tilde{L}^{(i)}\\rho (\|x-y\|)=\\beta _i\\rho (\|x-y\|),\\quad x\\ne y.$ Therefore, Assumption (A1) holds.", "Let $\\beta _1=-\\kappa _1$ and $\\beta _i=\\kappa _2-i^{-1}$ for $i\\ge 2$ , where $\\kappa _1$ and $\\kappa _2$ are two positive constants.", "Let $(X_t)$ be a solution of the following SDE: $\\text{\\rm {d}}X_t=\\beta _{\\Lambda _t}X_t\\text{\\rm {d}}t+\\sqrt{2}\\text{\\rm {d}}B_t, X_0=x>0.$ Then $(X_t,\\Lambda _t)$ is a state-independent regime-switching diffusion process satisfying assumptions (A1-A4).", "Take $F_1=\\lbrace 1\\rbrace $ and $F_2=\\lbrace 2,3,\\ldots \\rbrace $ , which is a finite partition of $\\mathcal {S}=\\lbrace 1,2,\\ldots \\rbrace $ .", "Then $\\beta _1^F=\\beta _1=-\\kappa _1$ , $\\beta _2^F =\\kappa _2$ , $q^F_{12}=\\sum _{j\\in F_2}q_{1j}=b_1$ and $q^F_{21}=\\sup _{i\\in F_2} q_{i1}=a_2$ .", "When $\\displaystyle \\kappa _2<\\frac{a_2\\kappa _1}{b_1+\\kappa _1}$ , the matrix $-(Q^F+\\mathrm {diag}(\\beta _1^F,\\beta _2^F)H_2$ is a nonsingular M-matrix.", "Hence, the regime-switching process $(X_t,\\Lambda _t)$ is exponentially ergodic in the Wasserstein distance $W_{\\tilde{\\rho }}$ with $\\tilde{\\rho }((x,i),(y,j))=\\sqrt{\\mathbf {1}_{i\\ne j}+\|x-y\|}$ , if $\\kappa _2<\\frac{a_2\\kappa _1}{b_1+\\kappa _1}.$ This example shows that although the diffusion process $(X_t)$ in a random environment characterized by $(\\Lambda _t)$ is transient in infinitely many environments ($i\\ge 2$ ), and is recurrent only in a environment ($i=1$ ), the process $(X_t)$ could be recurrent." ], [ "State-dependent switching in an infinite state space", "In this section, we study state-dependent regime-switching diffusion processes $(X_t,\\Lambda _t)$ defined by (REF ) and (REF ), that is, the Q-matrix of $(\\Lambda _t)$ depends on $(X_t)$ .", "This makes the coupling process used in previous two sections useless because we can not make the coupling process $(\\Lambda _t,\\Lambda _t^{\\prime })$ moves together after their first meeting.", "So it is difficult in this case to construct successful coupling $(Y_t,\\Lambda _t^{\\prime })$ of $(X_t,\\Lambda _t)$ to estimate the Wasserstein distance between them.", "In [20], we discussed how to construct successful couplings for state-dependent regime-switching process with $\\mathcal {S}$ being finite.", "In this section, we shall study the asymptotic boundedness of $(X_t,\\Lambda _t)$ .", "We extend the known results to state-dependent regime-switching diffusion processes in an infinite state space.", "In this section, $\\mathcal {S}$ is an infinite set, i.e.", "$N=\\infty $ .", "Let $\\rho :[0,\\infty )\\rightarrow [0,\\infty )$ satisfying $\\rho (0)=0$ , $\\rho ^{\\prime }>0$ , $\\lim _{r\\rightarrow \\infty } \\rho (r)=\\infty $ .", "We assume that (H) For each $i\\in \\mathcal {S}$ , there exists constants $\\theta _i\\in \\mathbb {R}$ , $K_i\\in [0,\\infty )$ such that $L^{(i)}\\rho (\|x\|)\\le \\theta _i\\rho (\|x\|)+K_i,\\quad x\\in \\mathbb {R}^d,$ and $M_1:=\\sup _{i\\ge 1} \\theta _i<\\infty $ , $M_2:=\\sup _{i\\ge 1} K_i<\\infty $ .", "Divide $\\mathcal {S}$ into finite nonempty subsets according to $\\theta _i$ .", "Let $\\Gamma :=\\lbrace -\\infty =:k_0<k_1<\\cdots <k_m<k_{m+1}=M_1\\rbrace .$ Corresponding to $\\Gamma $ , there is finite partition of $\\mathcal {S}$ , denoted by $F=\\lbrace F_1,\\ldots ,F_{m+1}\\rbrace $ , where $F_i=\\lbrace j\\in \\mathcal {S};\\ \\theta _j\\in (k_{i-1},k_i]\\rbrace .$ Let $\\phi :\\mathcal {S}\\rightarrow \\lbrace 1,\\ldots ,m+1\\rbrace $ be defined by $\\phi (j)=i$ if $j\\in F_i$ .", "Set $\\theta _i^F=\\sup _{j\\in F_i}\\theta _j,\\quad \\text{for $i=1,\\ldots ,m+1$}, \\ \\text{so}\\ \\theta _j\\le \\theta _{\\phi (j)}^F\\ \\ \\text{and $\\theta _i^F<\\theta _{i+1}^F$}.$ Define a new $Q$ -matrix $Q^F=(q_{ij}^F)$ on the space $\\lbrace 1,\\ldots ,m+1\\rbrace $ corresponding to partition $F$ by $q_{ik}^F=\\inf _{r\\in F_i}\\inf _{x\\in \\mathbb {R}^d} \\sum _{j\\in F_k} q_{rj}(x), \\ k>i;\\quad q_{ik}^F=\\sup _{r\\in F_i}\\sup _{x\\in \\mathbb {R}^d}\\sum _{j\\in F_k} q_{rj}(x),\\ k<i,\\ \\ q_{ii}^F=-\\sum _{k\\ne i} q_{ik}^F.$ Theorem 5.1 Assume that (H) holds.", "Use the notation defined above.", "If the $(m+1)\\times (m+1)$ matrix $-\\big (Q^F+\\mathrm {diag}(\\theta _1^F,\\ldots ,\\theta _{m+1}^F)\\big )H_{m+1}$ is a nonsingular M-matrix, where $H_{m+1}$ is defined by (REF ), then there are constants $\\alpha ,\\,c_1,\\,c_2>0$ such that $\\mathbb {E}[\\rho (\|X_t\|)]\\le c_1\\mathbb {E}[\\rho (\|X_0\|)]e^{-\\alpha t}+c_2,\\quad t>0.$ Proof.", "As $-\\big (Q^F+\\mathrm {diag}(\\theta _1^F,\\ldots ,\\theta _{m+1}^F)\\big )H_{m+1}$ is a nonsingular M-matrix, there is a vector $\\eta ^F=(\\eta _1^F,\\ldots ,\\eta _{m+1}^F)^\\ast \\gg 0$ such that $\\lambda ^F=(\\lambda _1^F,\\ldots ,\\lambda _{m+1}^F)^\\ast :=(Q^F+\\mathrm {diag}(\\theta _1^F,\\ldots ,\\theta _{m+1}^F))H_{m+1}\\eta ^F\\ll 0.$ Then $\\lambda _{\\mathrm {max}}:=\\max _{1\\le i\\le m+1} \\lambda _i^F<0$ .", "Set $\\xi ^F=H_{m+1}\\eta ^F$ .", "It is easy to see $\\xi _i^F=\\eta _{m+1}^F+\\cdots +\\eta _i^F,\\ i=1,\\ldots ,m+1.$ Hence, $\\xi _{i+1}^F<\\xi _i^F$ , $i=1,\\ldots , m$ , and $\\xi ^F\\gg 0$ .", "To proceed, we extend $\\xi ^F$ to a vector on $\\mathcal {S}$ by setting $\\xi _j=\\xi _i^F$ if $j\\in F_i$ .", "Then we have, for $r\\in F_i$ , $x\\in \\mathbb {R}^d$ , $Q_x\\xi (r)&=\\sum _{j\\in \\mathcal {S},j\\ne r} q_{rj}(x)(\\xi _j-\\xi _r)=\\sum _{j\\in F_i,j\\in \\mathcal {S}} q_{rj}(x)(\\xi _j-\\xi _r)\\\\&=\\sum _{k< i}\\big (\\sum _{j\\in F_k} q_{rj}(x)\\big )(\\xi _k^F-\\xi _i^F)+\\sum _{k> i}\\big (\\sum _{j\\in F_k} q_{rj}(x)\\big )(\\xi _k^F-\\xi _i^F)\\\\&\\le \\sum _{k\\ne i} q_{ik}^F(\\xi _k^F-\\xi _i^F)=\\big (Q^F\\xi ^F\\big )(i).$ By Itô's formula, we obtain $&\\mathbb {E}[\\rho (\|X_t\|)\\xi _{\\Lambda _t}]\\\\&\\le \\mathbb {E}[\\rho (\|X_0\|)\\xi _{\\Lambda _0}]+\\mathbb {E}\\Big [\\int _0^t\\big ((Q_{X_s}\\xi )(\\Lambda _s)+\\theta _{\\Lambda _s}\\xi _{\\Lambda _s}\\big )\\rho (\|X_s\|)+K_{\\Lambda _s}\\xi _{\\Lambda _s}\\text{\\rm {d}}s\\Big ]\\\\&\\le \\mathbb {E}[\\rho (\|X_0\|)\\xi _{\\Lambda _0}]+\\mathbb {E}\\Big [\\int _0^t\\big ((Q^F\\xi ^F)(\\phi (\\Lambda _s))+\\theta _{\\phi (\\Lambda _s)}^F\\xi _{\\phi (\\Lambda _s)}^F\\big )\\rho (\|X_s\|)+K_{\\Lambda _s}\\xi _{\\phi (\\Lambda _s)}^F\\text{\\rm {d}}s\\Big ]\\\\&\\le \\mathbb {E}[\\rho (\|X_0\|)\\xi _{\\Lambda _0}]+\\lambda _{\\mathrm {max}}^F\\mathbb {E}\\Big [\\int _0^t\\rho (\|X_s\|)\\text{\\rm {d}}s\\Big ]+M_2\\xi _{\\mathrm {max}}^Ft.$ This yields that $\\mathbb {E}[\\rho (\|X_t\|)\\xi _{\\Lambda _t}]\\le \\mathbb {E}[\\rho (\|X_0\|)\\xi _{\\Lambda _0}]e^{\\frac{\\lambda _{\\mathrm {max}}^F}{\\xi _{\\mathrm {max}}^F} t}-\\frac{M_2(\\xi _{\\mathrm {max}}^F)^2}{\\lambda _{\\mathrm {max}}}.$ Therefore, $\\mathbb {E}[\\rho (\|X_t\|)]\\le \\mathbb {E}[\\rho (\|X_0\|)]\\frac{\\xi _{\\mathrm {max}}^F}{\\xi _{\\mathrm {min}}^F}e^{\\frac{\\lambda _{\\mathrm {max}}^F}{\\xi _{\\mathrm {mx}}^F} t}-\\frac{M_2(\\xi _{\\mathrm {max}}^F)^2}{\\lambda _{\\mathrm {max}}\\xi _{\\mathrm {min}}^F},\\ \\ t>0.$ We conclude the proof by taking $c_1=\\frac{\\xi _{\\mathrm {max}}^F}{\\xi _{\\mathrm {min}}^F}$ , $\\alpha =-\\frac{\\lambda _{\\mathrm {max}}^F}{\\xi _{\\mathrm {mx}}^F}$ and $c_2=-\\frac{M_2(\\xi _{\\mathrm {max}}^F)^2}{\\lambda _{\\mathrm {max}}\\xi _{\\mathrm {min}}^F}$ ." ] ]	1403.0291
[ [ "Critical coupling and coherent perfect absorption for ranges of energies\n due to a complex gain and loss symmetric system" ], [ "Abstract We consider a non-Hermitian medium with a gain and loss symmetric, exponentially damped potential distribution to demonstrate different scattering features analytically.", "The condition for critical coupling (CC) for unidirectional wave and coherent perfect absorption (CPA) for bidirectional waves are obtained analytically for this system.", "The energy points at which total absorption occurs are shown to be the spectral singular points for the time reversed system.", "The possible energies at which CC occurs for left and right incidence are different.", "We further obtain periodic intervals with increasing periodicity of energy for CC and CPA to occur in this system." ], [ "Introduction", "The recent ideas of $PT$ -symmetric non-Hermitian quantum mechanics [1], [2], [3] have been fruitfully extended to optics due to formal equivalence between Schroedinger equation and certain wave equations in optics [4], [5], [6].", "The parity operator $P$ stands for spatial reflections $(x\\rightarrow -x, p\\rightarrow -p)$ , while the anti-liner time reversal operator $T$ leads to $(i\\rightarrow -i, p\\rightarrow -p, x\\rightarrow x)$ .", "The equivalence between quantum mechanics and optics becomes possible due to the judicious inclusion of complex refractive index distribution $V(x)=\\eta _R(x)+i\\eta _I(x)$ , in the electromagnetic wave equation [5], [7].", "To realize this consider a one dimensional optical system with effective refractive index $\\eta _R(x)+i\\eta _I(x)$ in the background of constant refractive index $\\eta _0$ , $\\eta _I$ stands for gain and loss component.", "The electric field $E=E_0(x,z)e^{i(wt-kz)}$ of a light wave propagating in this medium (with $\\eta _0>>\\eta _I,\\eta _R$ ) satisfies the Schroedinger like equation, $i\\frac{\\partial }{\\partial z}E_0(x,z)&=&\\Big [\\frac{1}{2k}\\frac{\\partial ^2}{\\partial x^2}+k_0[\\eta _R(x)+i\\eta _I(x)]\\Big ]E_0(x,z) \\nonumber \\\\&=& HE_0(x,z)$ $k=k_0\\eta _0$ , $k_0$ being the wave number in vacuum.", "This Hamiltonian $H$ has gain and loss symmetry for $\\eta _R(x)=\\eta _R(-x)$ and $\\eta _I(x)=-\\eta _I(-x)$ .", "Thus complex optical gain and loss potential can be realized by judiciously incorporating gain and loss profile on an even index distribution.", "This important realization opens a wide window to study optical systems with gain and loss medium and leads to the experimental observation of PT-invariance and its breaking [8]-[17] in various optical systems [4], [5], [6], [7], [18], [19].", "Various features of quantum scattering due to non-Hermitian potentials like exceptional points [20]-[22] and spectral singularity (SS) [23]-[26], reflectionlessness and invisibility [25], [27], [28], [29], reciprocity [25], [26], [30] etc have generated huge interest due to their applicability and usefulness in the study of optics.", "Recently the observation of perfect absorption [31]-[44] of incident electromagnetic wave by an optical media with complex refractive index distribution is counted as a big achievement in optics.", "The coherent perfect absorber (CPA) which is actually the time-reversed counterpart of a laser has become the center to all such studies in optics due to the discovery of anti-laser [31], [32], [33] in which incoming beams of light interfere with one another in such a way as to perfectly cancel each other out.", "This phenomena of perfect absorption in optics can also be observed in quantum scattering when particles (with a mass $m$ and total energy $E$ ) interact with the surrounding medium through a complex potential distribution $V(x)$ .", "Scattering due to complex potential can be described in a simple mathematical language as follows.", "If A and B are the incident wave amplitudes from left and right directions and C and D are the outgoing wave amplitudes to left and right respectively, then the scattering amplitudes are related through scattering matrix as, $\\left(\\begin{array}{clcr}C \\\\D \\\\\\end{array} \\right) &=&S\\left(\\begin{array}{clcr}A \\\\B \\\\\\end{array} \\right) , \\ \\ \\mbox{where} \\ \\ S=\\left(\\begin{array}{clcr}t_l & r_r \\\\r_l & t_r \\\\\\end{array} \\right) .$ For the perfect absorption the outgoing amplitudes C and D vanish leading to, $t_lA +r_rB=0 \\ ; \\ \\ \\ r_lA+t_rB=0 \\ ;$ The condition for perfect absorption for unidirectional incident waves can be written as, $t_l=0 \\ ; r_l&=&0 , \\ \\ \\mbox{for left incidence} \\ (B=0) \\nonumber \\\\t_r=0 \\ ; r_r&=&0 , \\ \\ \\mbox{for right incidence} \\ (A=0)$ These situations are known in literature as critical coupling (CC) [40]-[44].", "On the other hand for bidirectional incident wave the condition for perfect absorption is, $\\Big \|det[S]\\Big \|=\\Big \| t_lt_r -r_lr_r\\Big \|=0$ This condition refers as coherent perfect absorption which has recently created lots of excitements [31]-[39] due to the discovery of anti-laser.", "This could pave the way for a number of novel technologies with various applications from optical computers to radiology [31], [33].", "Thus it is extremely important to investigate different aspects of CPA using different non-Hermitian systems.", "The purpose of this work is to investigate various prospectives of scattering of particles due to a complex potential.", "In particular we would like to study different aspects of null scattering (CC and CPA) and super scattering in the context of a particular non-Hermitian gain and loss symmetric system which shows rich scattering properties.", "We consider non-Hermitian PT-symmetric version of an exponentially damped optical potential to derive the condition for CC and CPA analytically.", "This specific potential is rather known as Wood-Saxon (WS) [45], [46] potential, plays an important role in describing microscopic particle interactions.", "CPA never happens for Hermitian or PT-symmetric non-Hermitian systems as $\\Big \|det[S]\\Big \|$ is always unity in these cases.", "Here the complex PT-symmetric Wood-Saxon potential with an additive imaginary shift to the real axis shows scattering spectrum with total absorption for discrete as well as continuous ranges of energies.", "We further explicitly show the energy points at which CC or CPA occur correspond to SS points of the time reversed system.", "So far discrete energy points for CC and CPA are obtained for different optical media.", "We are able to obtain narrow but finite ranges of energies for CC and CPA to occur in this system.", "The nature of CC depends on the direction of incident wave.", "No energy range exists for right incidence.", "This happens due to the asymmetric left-right asymptotic behavior of this particular potential.", "More interestingly these ranges occur periodically with increasing periodicity for this particular system.", "This demonstration can give ideas to build perfect absorbers of matter waves which would be flexible to work for different ranges of energies of the incident particles.", "As the potential distribution is analogous to the medium's refractive index profile this work can be extended to quantum optics for the absorption of electromagnetic waves.", "Now we present the plan of this paper.", "In section II we discuss the scattering of non-Hermitian PT-symmetric WS potential and its time reversal partner potential.", "The CC for this system at discrete positive energies is reported in Sec.", "III.", "In Sec.", "IV we calculate the periodic ranges for CC in this potential.", "Sec.", "V is devoted to obtain the condition for CPA and its ranges while Sec.", "VI is left for conclusions and discussions." ], [ "Scattering from gain and loss symmetric WS potential", "In this section we explore the nature of scattering when the single-particle space in expanded in a Wood-Saxon basis [47].", "If particle scatters through a medium consists of semi-infinite nuclear matter it is more reasonable to take the nucleon distribution of a Wood-Saxon type rather than an uniform distribution [48], [49].", "The Hermitian WS potential in a simplified form [50] can be written as , $V(x)=-\\frac{V_{0}}{1+e^{\\frac{x-r_0}{a}}} \\equiv -\\frac{V_{0}}{1+qe^{\\delta x}} \\ ,$ where $V_0$ is the WS potential depth, $r_0$ is the width of the potential i.e the nuclear radius and $a$ is the diffuseness of semi-infinite nuclear matter [48].", "The WS-potential is rewritten in a more simplified form in Eq.", "(REF ) with $\\delta =1/a$ and $q=e^{-r_0\\delta }$ , for its quantum mechanical study.", "In this simplified form $V_0$ and $\\delta $ determines the height and shape of the potential respectively.", "For positive $\\delta $ and $V_0$ this potential asymptotically becomes $-V_0$ for $x\\rightarrow -\\infty $ and vanishes for $x\\rightarrow +\\infty $ .", "The solutions of Schroedinger equation for this potential correspond to both bound states and scattering states and are useful in studying different problems.", "Two independent scattering state solutions of the particle wave equation (with $q=1$ and $\\hbar =1$ ) for WS potential are written as [50], $\\psi _1(x)&=& M(1+e^{\\delta x})^{-\\alpha _2-\\alpha _3} \\ e^{ \\alpha _3\\delta x}\\nonumber \\\\&\\times & _2F_1(1+\\alpha _2+\\alpha _3,\\alpha _2+\\alpha _3;1+2\\alpha _2; \\frac{1}{1+e^{\\delta x}}) \\ ;$ $\\mbox{and} \\ \\ \\ \\ \\psi _{2} (x)&=& M \\Big (\\psi _1(x)\\mid _{\\alpha _2\\rightarrow -\\alpha _2}\\Big ) \\ ,\\ M=(-1)^{2\\alpha _3 }\\frac{\\Gamma (\\alpha _2-\\alpha _3)\\Gamma (1+\\alpha _2-\\alpha _3)}{\\Gamma (1+2\\alpha _2))\\Gamma (-2\\alpha _3)} \\ ,$ $\\mbox{with} \\ \\ \\ \\ \\ \\alpha _{2}=\\frac{2i}{ \\delta } \\sqrt{m E} \\equiv \\frac{2i}{ \\delta } k_1; \\ \\ \\ \\alpha _{3}=\\frac{2i}{ \\delta } \\sqrt{m (E+V_{0})} \\equiv \\frac{2i}{ \\delta } k_2 \\ .$ One can calculate the different scattering amplitudes by considering the superposition of these two independent scattering state solutions and looking at the asymptotic behaviors of these.", "We complexify the WS potential by taking the shape parameter imaginary (i.e.", "$\\delta \\rightarrow i\\rho $ ) such that it becomes PT-symmetric.", "Moreover for latter convenience we would like to consider this complex potential with an imaginary shift to the real axis as [51], [52], $\\tilde{V}(\\bar{x})=-\\frac{V_{0}}{1+\\bar{q}e^{i \\rho \\bar{x}}}, \\ \\mbox{where} \\ \\bar{x}=x-i \\zeta \\ , \\ \\zeta \\ \\ \\mbox{is real,}$ where $\\bar{q}=e^{-ir_0\\rho }=1$ and the complex part of the potential $\\tilde{V}(\\bar{x})$ working as the gain and loss component for this complex system.", "Figure: NO_CAPTIONFigure: NO_CAPTION Fig.", "1: Distributions of the real and imaginary components of PT-symmetric complex potential $\\tilde{V}(\\bar{x})$ are shown against $\\zeta $ (for $V_0 = 1.2$ and $\\rho = 1.8$ ) with the values $x=2$ (continuous lines) and $x=4$ (dashed lines).", "The Schroedinger equation with respect to $\\bar{x}$ will have the same form as with respect to real $x$ .", "In this case $x$ bears the periodic nature of $\\tilde{V}(\\bar{x})$ where $\\zeta $ decides the asymptotic behavior of the potential.", "The complexified WS potential in Eq.", "(REF ) has the same asymptotic behaviors as $\\lbrace x, \\zeta \\rbrace \\rightarrow +\\infty $ and $\\lbrace x, \\zeta \\rbrace \\rightarrow -\\infty $ shown in Fig.", "1.", "This observation leads us to obtain the scattering state solutions of PT-symmetric non-Hermitian WS potential in the same form as in the Eq.", "(REF ) and (REF ) subjected to the following modifications, $x\\rightarrow \\bar{x} \\ ; \\ \\ \\alpha _2\\rightarrow a_2=\\frac{2}{\\rho } \\sqrt{m E}\\equiv \\frac{2}{\\rho } k_1 \\ ; \\nonumber \\\\\\alpha _3\\rightarrow a_3=\\frac{2}{\\rho } \\sqrt{m (E+V_0)}\\equiv \\frac{2}{\\rho } k_2 \\ .$ The above solutions for the non-Hermitian case have the following asymptotic behaviors, $\\psi _1(\\bar{x}\\rightarrow +\\infty )&=& Me^{-ik_1x}.e^{-k_1\\zeta }=Me^{-ik_1\\bar{x}}; \\nonumber \\\\ \\ \\psi _1(\\bar{x}\\rightarrow -\\infty )&=& M\\Big [G1 \\ e^{ik_2x}.e^{ k_2\\zeta }+G2 \\ e^{-ik_2x}.e^{-k_2\\zeta }\\Big ]=M\\Big [G1 \\ e^{ik_2\\bar{x}}+G2 \\ e^{-ik_2\\bar{x}}\\Big ]; \\nonumber \\\\\\psi _{2}(\\bar{x}\\rightarrow +\\infty )&=& Me^{ik_1x}.e^{k_1\\zeta }=Me^{ik_1\\bar{x}} ; \\nonumber \\\\\\psi _{2}(\\bar{x}\\rightarrow -\\infty )&=& M\\Big [G3 \\ e^{ik_2x}.e^{ k_2\\zeta }+G4 \\ e^{-ik_2x}.e^{- k_2\\zeta }\\Big ]=M\\Big [G3 \\ e^{ik_2\\bar{x}}+G4 \\ e^{-ik_2\\bar{x}}\\Big ] \\ ,\\nonumber \\\\$ where G1, G2, G3, and G4 are given in terms of Gamma functions as, $G1 &=& \\frac{\\Gamma (1 + 2 a_2) \\Gamma (-2 a_3)}{\\Gamma (a_2 - a_3) \\Gamma (1 + a_2 - a_3)};\\nonumber \\\\G2 &=& \\frac{\\Gamma (1 + 2 a_2) \\Gamma (2 a_3)}{\\Gamma (a_2 + a_3) \\Gamma (1 + a_2 + a_3)} ;\\nonumber \\\\G3 &=& \\frac{\\Gamma (1 - 2 a_2) \\Gamma (-2 a_3)}{\\Gamma (-a_2 - a_3) \\Gamma (1 - a_2 - a_3)};\\nonumber \\\\G4 &=& \\frac{\\Gamma (1 - 2 a_2)\\Gamma (2 a_3)}{\\Gamma (-a_2 + a_3) \\Gamma (1 - a_2 + a_3)} \\ .$ The scattering wavefunctions from the solutions in Eq.", "(REF ) asymptotically diverges as the potential becomes 0 and $-V_0$ for $\\zeta \\rightarrow \\pm \\infty $ in the Fig.", "1.", "Different scattering amplitudes for this gain and loss symmetric complex WS potential thus can be readily read out as, $r_{l} = \\frac{G4}{G3}=\\frac{\\Gamma (2 a_3)\\Gamma (-a_2 - a_3) \\Gamma (1 - a_2 - a_3)}{\\Gamma (-2 a_3)\\Gamma (-a_2 + a_3) \\Gamma (1 - a_2 + a_3)} ;$ $t_{l} = \\sqrt{\\frac{k_1}{k_2}} \\ \\frac{1}{G3}= \\sqrt{\\frac{k_1}{k_2}} \\ \\frac{\\Gamma (-a_2 - a_3) \\Gamma (1 - a_2 - a_3)}{\\Gamma (1-2 a_2)\\Gamma (-2 a_3)} =t_r;$ $r_{r} = -\\frac{G1}{G3}=-\\frac{\\Gamma (1+2 a_2)\\Gamma (-a_2 - a_3) \\Gamma (1 - a_2 - a_3)}{\\Gamma (1-2 a_2)\\Gamma (a_2 - a_3) \\Gamma (1 + a_2 - a_3)} \\ .$ It is easy to see that $\|r_l\|^2\\equiv R_{l}\\ne R_{r}\\equiv \|r_r\|^2$ as $ a _2$ $a_3$ are real, as expected for PT-symmetric non-Hermitian systems.", "However we have $\|t_l\|^2\\equiv T_{l}=T_{r}\\equiv \|t_r\|^2$ in this case unitarity is violated, $R+T\\ne 1$ .", "On the other hand reciprocity and unitarity are restored for Hermitian case as $G1^=G4$ and $G2^=G3$ , since $ \\alpha _2$ and $\\alpha _3$ are purely imaginary.", "Under time reversal transformation the WS potential becomes, $\\tilde{V}^(\\bar{x})=-\\frac{V_{0}}{1+e^{-i \\rho \\bar{x}^}} \\ .$ The scattering amplitudes for $\\tilde{V}^(\\bar{x})$ (denoted with prime) are obtained from that of $\\tilde{V}(\\bar{x})$ by changing the parameter $\\rho \\rightarrow -\\rho $ i.e.", "$a_2\\rightarrow -a_2$ and $a_3\\rightarrow -a_3$ as, $r^{\\prime }_{l} = r_{l}\\Big \|_{\\stackrel{a_2\\rightarrow -a_2}{a_3\\rightarrow -a_3}}=\\frac{\\Gamma (-2 a_3)\\Gamma (a_2 + a_3) \\Gamma (1 + a_2 + a_3)}{\\Gamma (2 a_3)\\Gamma (a_2 - a_3) \\Gamma (1 + a_2 - a_3)} \\ ;$ $t^{\\prime }_{l} = t_{l}\\Big \|_{\\stackrel{a_2\\rightarrow -a_2}{a_3\\rightarrow -a_3}}= \\sqrt{\\frac{k_1}{k_2}} \\ \\frac{\\Gamma (a_2 + a_3) \\Gamma (1 + a_2 + a_3)}{\\Gamma (1+2 a_2)\\Gamma (2 a_3)} =t^{\\prime }_r \\ ;$ $r^{\\prime }_{r} = r_{r}\\Big \|_{\\stackrel{a_2\\rightarrow -a_2}{a_3\\rightarrow -a_3}}=-\\frac{\\Gamma (1-2 a_2)\\Gamma (a_2 + a_3) \\Gamma (1 +a_2 + a_3)}{\\Gamma (1+2 a_2)\\Gamma (a_2 + a_3) \\Gamma (1 - a_2 + a_3)} \\ .$ Different properties of null scattering and super scattering will be discussed using these scattering amplitudes in the following sections." ], [ "Total absorption of unidirectional wave: Critical Coupling", "Particle waves of certain specific frequencies when incident from one direction on a potential are completely absorbed by the potential due to critical coupling (CC).", "In this section we discuss the CC due to non-Hermitian gain and loss symmetric WS potential and calculate the frequencies which are absorbed by the system.", "The transmission and reflection coefficients become identically zero at these energies.", "We observe that the frequencies of the waves for CC for left incidence are different from that of for right incidence." ], [ "CC for left incident wave", "The transmission and reflection coefficients of $\\tilde{V}(\\bar{x})$ are written from Eqs.", "( REF ) and (REF ) as, $R_l=\\mid r_l\\mid ^2=\\mid \\frac{G4}{G3}\\mid ^2 \\ ;T_l=\\mid t_l\\mid ^2=\\mid \\frac{1}{G3}\\mid ^2 \\ .$ $R_l$ and $T_l$ become simultaneously zero if $2a_3=n$ , a positive integer.", "This happens for the positive discrete energy, $E_{n}^l= \\frac{\\rho ^{2}}{16m} n^{2}-V_{0}, \\ \\mbox{with} \\ \\ n> \\frac{4}{\\rho }\\sqrt{mV_0} \\ .$ This physically means when matter wave with energy $E_n^l$ is incident on a medium with potential distribution $\\tilde{V}(\\bar{x})$ then the wave will be completely absorbed as $R_l=0$ and $T_l=0$ .", "The successive energy separations for this null scattering, $\\Delta E_n^l\\equiv E_{n+1}^l-E_{n}^l=\\frac{\\rho ^{2}}{16m} (2n+1)$ are independent of depth of the potential.", "Now we consider the time reversed case of $\\tilde{V}(\\bar{x})$ with reflection and transmission amplitudes $r^{\\prime }_{l,r}$ and $t^{\\prime }_{l,r}$ as given in Eqs.", "(REF ),(REF ) and (REF ).", "In this case $R^{\\prime }_l$ diverges when $2a_3=n$ , leading to the SS at the same energy point $E_n^l$ .", "Thus we analytically see CC of a gain and loss symmetric non-Hermitian WS potential are the SS points of the time reversed of the same potential.", "This is demonstrated graphically in Fig.", "2.", "Figure: NO_CAPTIONFigure: NO_CAPTION Fig.", "2: Critical couplings for left incident waves against incident energies are shown for the potential $\\tilde{V}(\\bar{x})$ in Fig.", "2(a) and its time reversal partner $\\tilde{V}^(\\bar{x})$ in Fig.", "2(b).", "In Fig.", "2(a) both $R_l$ and $T_l$ vanish at the energies $E_n^l$ with $n_{min}=3$ (for $V_0 = 1.2, \\rho = 1.8$ and $m = 1$ ).", "On the other hand Fig.", "2(b) shows divergence of total scattering coefficient (as $R^{\\prime }_l\\rightarrow \\infty $ ) at the same incident energies for the time reversed potential." ], [ "CC for right incident wave", "In this subsection we would like to emphasis that condition for CC depends on the direction of incidence.", "In particular we show that condition of CC for left incidence is different from that of for right incidence for gain and loss symmetric non-Hermitian WS potential.", "From Eqs.", "(REF ) and (REF ) the scattering coefficients $R_r=\\mid r_r\\mid ^2=\\mid \\-\\frac{G1}{G3}\\mid ^2 $ and $T_r=\\mid t_r\\mid ^2=\\mid \\frac{1}{G3}\\mid ^2$ vanish simultaneously when $2a_2$ is a positive integer ($n^{\\prime }$ ).", "This leads to CC at the energy, $E_{n^{\\prime }}^r=\\frac{\\rho ^{2}}{16m} n^{\\prime 2} $ , which is different from $E_n^l$ at which CC occurs for left incidence.", "This happens due to the asymmetry in the left and right asymptotic behavior of the potential.", "However the energy separation between two consecutive CC is independent of direction of incident wave for same values of $n$ and $n^{\\prime }$ as, $\\Delta E_{n^{\\prime }}^r\\equiv E_{n^{\\prime }+1}^r-E_{n^{\\prime }}^r=\\frac{\\rho ^{2}}{16m} (2n^{\\prime }+1)= \\Delta E_{n=n^{\\prime }}^l \\ .$ Following the discussion of the previous subsection it can be shown easily that null scattering due to $\\tilde{V}(\\bar{x})$ is same as the super scattering due to $\\tilde{V}^(\\bar{x})$ at the same energy even for right incidence.", "This situation is nicely demonstrated in Fig.", "3.", "Figure: NO_CAPTIONFigure: NO_CAPTION Fig.", "3: Critical couplings for right incidence shown for gain and loss symmetric WS potential.", "In Fig.", "3(a) both $R_r$ and $T_r$ for $\\tilde{V}(\\bar{x})$ vanish at the energies $E_n^r$ with $n=1,2,3..$ (for $V_0 = 1.2, \\rho = 1.8$ and $ m = 1$ ).", "On the other hand Fig.", "3(b) shows the divergences of $R^{\\prime }_r$ at the same incident energies for $\\tilde{V}^(\\bar{x})$ .", "Thus the gain and loss symmetric WS potential potential can work as a critical coupler for both left and right incidences but for different frequencies.", "The waves with these frequencies when incident from left or right on the time reversed gain and loss symmetric WS potential result in super reflectivity ($R^{\\prime }_l\\rightarrow \\infty $ or $R^{\\prime }_r\\rightarrow \\infty $ , with finite $T^{\\prime }$ )." ], [ "Critical coupling for ranges of incident energies", "In this section we find critical coupling for different energy ranges.", "To demonstrate this we would like to consider time reversed potential $\\tilde{V}^(\\bar{x})$ for which the transmission and reflection amplitudes are written in Eqs.", "(REF )-(REF ).", "For this time reversed case left handed transmission amplitude neither vanishes nor diverges as $a_2, a_3$ are real positive numbers.", "However we can adjust the values of the parameter $\\rho $ and $V_0$ in such a manner that $T^{\\prime }\\equiv \\mid t^{\\prime }_{l,r}\\mid ^2$ becomes negligibly small over an interval of energy.", "This is achieved by considering $\\rho $ very small such that $a_2$ and $ a_3$ are very high and by adjusting $V_0$ in such a manner that $a_3\\gg a_2$ .", "In that case the dominating term $\\Gamma (2a_3)$ in the denumerator of the transmission amplitude will be very high leading to very small or negligible transmission.", "At the same time left handed reflection amplitude also becomes negligible for a certain interval of energy due to the presence of $\\Gamma (2a_3)$ term in denumerator.", "Simultaneously $R^{\\prime }_l\\equiv \\mid r^{\\prime }_{l}\\mid ^2$ also vanishes for discrete positive energies at which $a_2-a_3=-n$ condition is satisfied.", "But this interval is interrupted by certain singularities when $2a_3=N$ occur.", "Both n and N are positive integers.", "The positive discrete energies at which $R^{\\prime }_l$ diverges are written as, $E_{ss}^N=\\frac{N^2\\rho ^2}{16 m}-V_0, \\ \\ \\ \\mbox{for} \\ \\ N>\\frac{4}{\\rho }\\sqrt{mV_0} \\ .$ Energy gap between two such successive SS is, $\\bigtriangleup E_{ss} =E_{ss}^{N+1}-E_{ss}^{N}=(2N+1)\\frac{\\rho ^2}{16m} \\ .$ Therefore between two consecutive spectral singularities we have a certain energy range where $R^{\\prime }_l\\approx 0, T^{\\prime }_l\\approx 0$ .", "This range is linearly increasing with the integer values of N, and also can be controlled by the parameters $\\rho $ and $V_0$ .", "In such intervals of energy $R^{\\prime }_l$ becomes exactly zero at certain discrete positive energies (for $a_2-a_3=-n$ ) which are calculated as, $E- \\sqrt{E^2+EV_{0}}&=&\\frac{1}{2} \\Big ( \\frac{n \\rho }{2 \\sqrt{m}}\\Big )^{2} -\\frac{V_{0}}{2}\\equiv p_n \\ , \\nonumber \\\\\\mbox{i.e.}", "\\ \\ E_n&=&\\frac{p_n^{2}}{V_{0}+2p_n} \\ .$ These discrete energies $E_n$ is real and positive for $n>\\sqrt{\\frac{4mV_0}{\\rho ^2}}$ .", "The separation between two such consecutive zeros of $R^{\\prime }_l$ is computed as, $E_{n+1}-E_{n}=(2n+1) \\Big (\\frac{\\rho ^{2}}{18 m}- \\frac{mV_{0}^{2}}{n^{2} (n+1)^{2}\\rho ^2} \\Big ) \\ .$ Thus in case of $\\tilde{V}^(\\bar{x})$ we obtain different range of CC separated by SS for left incidence.", "On the other hand for $\\tilde{V}(\\bar{x})$ we obtain ranges of SS separated by CC at the same energy values.", "All these features of WS potential are well illustrated in Fig.", "4.", "However we would like to point out no such ranges of energy exist for $\\tilde{V}^(\\bar{x})$ in the case of right incidence.", "Figure: NO_CAPTIONFigure: NO_CAPTION Figure: NO_CAPTIONFigure: NO_CAPTION Fig.", "4: $log_{10}(R^{\\prime }_l)$ and $log_{10}(T^{\\prime }_l)$ are plotted to demonstrate different energy ranges of CC due to the potential $\\tilde{V}^(\\bar{x})$ .", "In Figs.", "4(a) and (b) one particular range of energy is shown between two successive SS ($\\rho =.0006, V_0 = 1, m = 1$ ) where $R^{\\prime }_l\\approx 0$ and $T^{\\prime }_l\\approx 0$ .", "Different energy ranges for $R^{\\prime }_l\\approx 0$ are shown in Fig.", "4(c) (for $\\rho =60; V_0 = 5.5\\times 10^{6}, m = 1$ ), whereas Fig.", "4(d) shows that $T^{\\prime }_l$ is vanishingly small in those intervals.", "Any desired ranges of incident energy for CC can be achieved with the potential $\\tilde{V}^(\\bar{x})$ by adjusting the parameters in the potential." ], [ "Perfect absorption of bidirectional waves", "In this section we investigate the absorption through this particular potential distribution when waves are coming from both directions.", "As mentioned in the introduction perfect absorption for bidirectional waves will occur if $\\Big \| t_lt_r -r_lr_r=0\\Big \|$ .", "From Eqs.", "(REF )-(REF ) we obtain, $\\frac{1}{(G3)^2}\\Big [\\frac{k_1}{k_2}+G1 G4\\Big ]=0 \\ .$ Using the properties of $\\Gamma $ function we find the relation $G4 G1+\\frac{k_{1}}{k_{2}}=G2 G3$ .", "Thus perfect absorption occurs for the bidirectional waves if $\\frac{G2}{G3}=0 \\ (G3\\ne 0)$ .", "Since $a_2, a_3$ are real positive numbers, CPA will occur for the potential $\\tilde{V}(\\bar{x})$ when $G3$ is infinity ($G2$ can not be zero for $\\tilde{V}(\\bar{x})$ ).", "From Eq.", "(REF ) we see that this situation occurs in two possible ways either $2a_2=n_1+1$ or $2a_3=n_2$ .", "Thus we obtain CPA for $\\tilde{V}(\\bar{x})$ at two different energies $E_{n_1}=\\frac{(n_1+1)^2\\rho ^2}{16 m} \\ \\ \\mbox{and} \\ \\ E_{n_2}=\\frac{n_2^2\\rho ^2}{16 m}-V_0 \\ ,$ where $n_1, n_2$ are positive integers.", "At energies $E_{n_1}$ , $T=0, R_l=0$ and at energies $E_{n_2}$ , $T=0, R_r=0$ [as shown in Fig.", "5].", "Alternatively for the time reversed case $a_2$ and $a_3$ changes sign and CPA can only occur for $G2=0$ ($G3$ is always finite in the case of $\\tilde{V}^(\\bar{x})$ ).", "The discrete positive energies for CPA are calculated as, $E+\\sqrt{E^2+EV_0}&=&\\frac{M^2\\rho ^2}{8}-\\frac{V_0}{2}\\equiv q_M \\ , \\nonumber \\\\\\ \\mbox{i.e.}", "\\ \\ E&=&E^_M=\\frac{q_M^2}{V_0+2q_M} \\ (\\mbox{where M is positive integer}).$ To ensure real positive energy $M$ must be greater than $\\sqrt{\\frac{4mV_0}{\\rho ^2}}$ .", "This depicts that at least one such discrete positive energy for CPA will exist for the potential $\\tilde{V}^(\\bar{x})$ if $\\rho ^2>\\frac{\|V_0\|}{4}$ .", "At these energies left incident and right incident waves interfere destructively and absorbed by the potential completely.", "Waves with energies $E_M^$ when incident on the potential $\\tilde{V}(\\bar{x})$ produce lasing behavior where all the scattering amplitudes diverge (as $a_2+a_3$ is equal to a positive integer).", "Similarly for $\\tilde{V}^(\\bar{x})$ , $R^{\\prime }_l\\rightarrow \\infty $ and $R^{\\prime }_r\\rightarrow \\infty $ at energies $E_{n_1}$ and $E_{n_2}$ respectively where CPA occur for $\\tilde{V}(\\bar{x})$ .", "Thus spectral singularity of $\\tilde{V}^(\\bar{x})$ and CPA of $\\tilde{V}(\\bar{x})$ occur at the same energy points.", "Figure: NO_CAPTIONFigure: NO_CAPTION Figure: NO_CAPTIONFigure: NO_CAPTION Fig.", "5: Figs (a) and (c) are for the potentials $\\tilde{V}^(\\bar{x})$ whereas (b) and (d) are for $\\tilde{V}(\\bar{x})$ .", "Behavior of R and T are shown in (a) and (b) for $V_0 = 2, \\rho = 2, m = 1$ .", "On the other hand CPA and its time reversed situation are shown in (c) and (d).", "To obtain the ranges of energy for CPA for the potential $\\tilde{V}^(\\bar{x})$ we choose a small $\\rho $ such that $a_2, a_3 $ are large even for lower energies.", "Then we adjust $V_0$ in such a manner that $a_3$ is large enough compare to $a_2$ .", "In such situation we obtain a range of energy for which $t^{\\prime }_{r,l}\\approx 0$ and $r^{\\prime }_l\\approx 0$ and hence $\\Big \|det[S]\\Big \|\\approx 0$ , leading to CPA.", "This range is interrupted by the singular points of $r^{\\prime }_l$ and $r^{\\prime }_r$ which occurs due to the presence of $\\Gamma (-2 a_3)$ and $\\Gamma (1-2 a_2)$ in the numerator of $r^{\\prime }_l$ and $r^{\\prime }_r$ respectively.", "Thus range of CPA is separated by these SS as shown in Fig.", "6.", "The $n^{th}$ spectral singular point for $r^{\\prime }_l$ is at the positive discrete energies $E_n=\\frac{n^2\\rho ^2}{16m}-V_0$ for which $2a_3=n$ , with $n_{min}=Int(\\sqrt{\\frac{16m V_0}{\\rho ^2}})+1$ .", "The energy interval for any two consecutive singularities of $r^{\\prime }_l$ is, $\\bigtriangleup E_{ss}^{rl}=E_{n+1}-E_{n}=\\frac{\\rho ^2}{16m}(1+2n).$ On the other hand $r^{\\prime }_r$ has singularities for $2a_2=n^{\\prime }$ (i.e.", "at $E_n^{\\prime }=\\frac{n^{\\prime 2}\\rho ^2}{16m}$ , where $n^{\\prime }$ is another positive integer) which occur at the energy intervals of, $\\bigtriangleup E_{ss}^{rr}=E_{n^{\\prime }+1}-E_n^{\\prime }=\\frac{\\rho ^2}{16m}(1+2n^{\\prime }).$ Both the energy intervals occurs periodically and $\\bigtriangleup E_{ss}^{rl}>\\bigtriangleup E_{ss}^{rr}$ as $n>n^{\\prime }$ .", "The energy span of these ranges increase with increasing $n^{\\prime }$ for a fixed $\\rho $ .", "Fig.", "6 shows the ranges of energies of perfect absorption for the time reversed potential with different parametric regimes.", "Figure: NO_CAPTIONFigure: NO_CAPTION Fig.", "6: Energy ranges for CPA for the potential $\\tilde{V}^(\\bar{x}))$ are demonstrated.", "Fig.", "6(a) shows a range of energy with $\\rho = .001, V_0 = 15, m=1$ .", "In Fig.", "6(b) we have shown periodic ranges of energies separated by spectral singular points for a different parametric regime ($\\rho =60; V_0 = 5.5\\times 10^{6}, m=1)$ ." ], [ "Conclusions and discussion", "It worths studying CC and CPA and its properties for interaction of particle waves with various non-Hermitian models to search for new features and possibilities for perfect absorption.", "In this work we have shown that for a particular gain and loss symmetric non-Hermitian optical potential (WS potential) it is possible to achieve CC and CPA for a range of frequencies due to quantum scattering.", "We have obtained that the conditions of CC depend on the direction of incident waves and no range exists for right incident case for this non-Hermitian potential.", "More interestingly by adjusting the parameters in the potential we can have these total absorptions in any desired ranges of frequencies.", "For a PT-symmetric non-Hermitian optical potential we have derived the analytical conditions of null scattering (CC and CPA) to occur.", "The energy points at which the null scattering occurs are shown to be the SS points for the time reversed system.", "An estimation of the values of parameters are provided in in Table.", "1.", "The CC and CPA energies and energy ranges increases with increasing shape parameter, i.e.", "decreasing diffuseness of the potential.", "As this specific potential plays an important role in describing interactions of nucleon with heavy nucleus [45], [46], our theoretical demonstration will open the possibility of future work on absorption of interacting microscopic particles of different masses and energies.", "Table: Energy and energy ranges for CC and CPA for different values ofparameters (with incident particle mass m=1m=1, in atomic unit) ofnon-Hermitian PT-symmetric WS potential.Acknowledgments: Two of the authors (A. Ghatak and B. P. Mandal) acknowledge the hospitality of TIFR, Mumbai, where a part of revision of this manuscript is done." ] ]	1403.0539
[ [ "Dynamics-Adapted Cone Kernels" ], [ "Abstract We present a family of kernels for analysis of data generated by dynamical systems.", "These so-called cone kernels feature an explicit dependence on the dynamical vector field operating in the phase-space manifold, estimated empirically through finite-differences of time-ordered data samples.", "In particular, cone kernels assign strong affinity to pairs of samples whose relative displacement vector lies within a narrow cone aligned with the dynamical vector field.", "As a result, in a suitable asymptotic limit, the associated diffusion operator generates diffusions along the dynamical flow, and is invariant under a weakly restrictive class of transformations of the data, which includes conformal transformations.", "Moreover, the corresponding Dirichlet form is governed by the directional derivative of functions along the dynamical vector field.", "The latter feature is metric-independent.", "The diffusion eigenfunctions obtained via cone kernels are therefore adapted to the dynamics in that they vary predominantly in directions transverse to the flow.", "We demonstrate the utility of cone kernels in nonlinear flows on the 2-torus and North Pacific sea surface temperature data generated by a comprehensive climate model." ], [ "Introduction", "Large-scale datasets generated by dynamical systems arise in a diverse range of disciplines in science and engineering, including fluid dynamics [32], [50], materials science [36], [35], molecular dynamics [20], [43], and geophysics [40], [22].", "A major challenge in these domains is to utilize the vast amount of data that is being collected by observational networks or output by large-scale numerical models to understand the operating physical processes, and make inferences about aspects of the system which are not accessible to observation.", "For instance, in climate atmosphere ocean science (CAOS) the dynamics takes place in an infinite-dimensional phase space where the coupled nonlinear partial differential equations for fluid flow and thermodynamics are defined, and the observed data correspond to functions of that phase space, such as temperature or circulation measured over a geographical region of interest.", "There exists a strong need for data analysis algorithms to extract and create reduced representations of the large-scale coherent patterns which are an outcome of these dynamics, including the El Niño Southern Oscillation (ENSO) in the ocean [53] and the Madden-Julian Oscillation in the atmosphere [39].", "Advances in the scientific understanding and forecasting capability of these phenomena have potentially high socioeconomic impact.", "Despite the high phase space dimension of many systems of interest, their dynamics evolve asymptotically (at long times) on low-dimensional submanifolds of phase space (attractors) [22], [34], [2].", "It is therefore natural to exploit topological and geometrical aspects of these submanifolds to design data analysis algorithms for data reduction and decomposition, function learning, and other important problems.", "Here a major challenge stems from the fact that the geometry of the attractive manifolds is nonlinear, meaning that linear variance-optimizing algorithms such as principal components analysis (PCA) [3], [32] or singular spectrum analysis (SSA) [11], [54], [24] are likely to perform suboptimally.", "Indeed, it has been documented in the literature (e.g., [4], [18]) that the dynamically significant modes in nonlinear systems are not necessarily those carrying high variance.", "Moreover, in practical applications involving finite sample counts and short observation intervals, the data manifolds are inherently discrete, and represent only coarse-grained aspects of the underlying geometry at the infinite-sample limit.", "Kernel methods have been extensively used as alternatives to classical linear algorithms to take advantage of nonlinear geometric structures of data.", "Prominent applications include dimension reduction and feature extraction [48], [5], [14], [15], [6], [33], learning and regularization of scalar or vector-valued functions [7], [42], [13], and out-of-sample extension of these functions [16].", "Here, a common theme is that suitably constructed kernels, i.e., functions measuring a notion of pairwise similarity between data points, lead naturally to diffusion operators which are closely related to the manifold structure of the data.", "More specifically, because every elliptical diffusion operator induces a Riemannian metric tensor on the data manifold [46], [23], using that operator (or, as is frequently the case, its eigenfunctions) for tasks such as data representation and function learning is tantamount to performing these tasks in a manner compatible with the induced Riemannian geometry.", "Applied to datasets with nonlinear manifold structure, such as those generated by complex dynamical systems, this approach has been found to yield more efficient algorithms and meaningful results than linear PCA-type algorithms (e.g., [37] and references therein).", "A major advantage of constructing diffusion operators through kernels is that kernels are defined in the ambient data space, and thus can be used to control the induced Riemannian metric [9] exploiting features of data space which are available for the problem at hand.", "In the context of dynamical system data, an important feature which is not present in general point clouds is that the samples occur with a time ordering which is the outcome of dynamical flow in phase space.", "Equivalently, we think of these datasets as being associated with a nowhere-vanishing vector field $ v $ on the attractor, which is intrinsic to the system under study in the sense that it does not depend on how the data manifold is embedded in ambient space.", "These observations provide a motivation to seek aspects of $ v $ which are empirically accessible in ambient data space, and incorporate them in dynamics-adapted kernels.", "Efforts in that direction have been made in a series of papers on so-called nonlinear Laplacian spectral analysis (NLSA) algorithms for decomposition of spatiotemporal data [25], [28], [29], and in independent work by Berry and collaborators [10], [9].", "In both of these works, dynamical flow is incorporated in kernels by first embedding the data in a higher-dimensional space (hereafter called embedding space) through Takens' method of delays [44], [51], [11], [47].", "Because each point in embedding space corresponds to a segment of dynamical evolution observed over a temporal window, distances in that space depend on the dynamical flow generating the data.", "This means in turn that the induced Riemannian metric associated with a kernel formulated in embedding space will depend (albeit indirectly) on the dynamical system generating the data.", "In NLSA algorithms, the embedding-space distances are scaled by a factor which is proportional to the distance between temporal nearest neighbors.", "As will be made precise below, the displacement vector between temporal nearest neighbors is a representation of the dynamical vector field, pushed forward to embedding space.", "Therefore, in addition to time-lagged embedding, the NLSA kernel also depends directly on the dynamical vector field through its norm.", "Qualitatively, the result of this dynamics-dependent scaling factor is to assign higher weights to transitory states with large phase space velocity.", "This feature was found to be particularly useful in extracting dynamically significant modes in systems with metastability [28].", "Moreover, because of the “non-dimensionalization” produced by the scaling factor, the NLSA kernel can be naturally extended to process multivariate datasets consisting of components with different physical units [12].", "Yet, in spite of these attractive features, theoretical understanding of the role of the scaling factor in the induced Riemannian metric has so far been lacking.", "Building on the existing work in [25], [28], [29], in this paper we introduce a one-parameter family of kernels which, besides the norm of the dynamical vector field also depend on the angle between that vector field and the displacement vector between the points at which the kernel is evaluated.", "In particular, the new kernels assign higher weight to pairs of points whose relative displacement lies within a cone with axis parallel to the dynamical vector field.", "For this reason we refer to the new kernels as cone kernels.", "The angular influence is controlled by a parameter $ \\zeta \\in [ 0, 1 ) $ , such that the existing NLSA kernel with no angular dependence occurs as the special case $ \\zeta = 0 $ .", "Here, however, our main focus is the limiting behavior $ \\zeta \\rightarrow 1 $ , where the angular influence is maximal.", "In that limit, the induced Riemannian metric becomes degenerate, and diffusion takes place along the integral curves of the dynamical vector field $ v $ (in the sense of [23]).", "Because the dynamical integral curves are intrinsic to the system (i.e., do not depend on the ambient-space induced metric), the along-$ v $ property of the $ \\zeta \\rightarrow 1 $ diffusion operator is invariant under arbitrary diffeomorphisms of the data manifold.", "Moreover, the associated Dirichlet form depends on the directional derivative of functions along $ v $ , which is also metric independent.", "In fact, the only dependence of the limit operator on the ambient-space metric is through a ratio involving its volume form and the norm of $ v $ .", "By virtue of this property, as $ \\zeta \\rightarrow 1 $ , the diffusion operators constructed from cone kernels become invariant under a weakly restrictive class of transformations of the data, which includes conformal transformations as a special case.", "In addition, due to the structure of the Dirichlet form, the associated diffusion eigenfunctions are expected to vary predominantly along directions transverse to $ v $ .", "These eigenfunctions can therefore be employed to carry out tasks such as dimension reduction and feature extraction in a dynamics-adapted manner.", "We demonstrate the utility of cone kernels in two applications involving analytically solvable nonlinear flows on the 2-torus and dynamical evolution of sea surface temperature (SST) in the North Pacific ocean in a comprehensive climate model.", "In the torus application, we explicitly demonstrate the adaptation of the eigenfunctions of the associated diffusion operator to the dynamical flow, and invariance of these eigenfunctions under a non-conformal deformation of the torus.", "The climate model results suggest that the cone-kernel eigenfunctions produce more efficient (i.e., using fewer basis functions) representation of the temporal variability compared to the existing NLSA kernel, while achieving better timescale separation.", "The plan of this paper is as follows.", "In section , we lay out the notation, and develop the cone kernel formulation.", "In section , we study the behavior of the induced metric and the associated diffusion operator.", "We present the applications to torus flows and climate model data in sections and , respectively, and conclude in section .", "Technical results and proofs are included in an appendix.", "A movie illustrating the time evolution of SST filtered by diffusion eigenfunctions is provided as supplemental material.", "A Matlab code used to generate the numerical results in sections and is available upon request from the author." ], [ "Formulation of cone kernels", "We consider a scenario where the dynamics is described by a deterministic flow $ \\Phi _t : \\mathcal { F } \\mapsto \\mathcal { F } $ operating in a phase space $ \\mathcal { F } $ , and evolving on a smooth (of class $ C^1 $ ), compact $ m $ -dimensional attractor $ \\mathcal { M } \\subseteq \\mathcal { F } $ without boundary.", "Moreover, observations are taken uniformly in time with a timestep $ \\delta t > 0 $ on the attractor via a $ C^2 $ vector-valued function $ F : \\mathcal { F } \\mapsto \\mathbb { R }^n $ , leading to a dataset $X = \\lbrace X_1, \\ldots , X_s \\rbrace , \\quad \\text{with} \\quad X_i = F( a_i ), \\quad a_i = \\Phi _{t_i} a_0, \\quad t_i = i \\, \\delta t, \\quad a_0 \\in \\mathcal { M }.$ See Figure REF for an illustration.", "We refer to $ \\mathbb { R }^n $ interchangeably as data space or ambient space, and consider that it is equipped with an inner product $ ( \\cdot , \\cdot ) $ .", "Figure: Illustration of the m m -dimensional data manifold ℳ \\mathcal { M } embedded in ℝ n \\mathbb { R }^n through the map F F .", "ℬ δt \\mathcal { B }_{\\delta t } is the open ball used in the asymptotic analysis in Appendix .Throughout, we assume that $ \\Phi _t $ is sufficiently smooth so that the dynamical vector field $ v $ induced on $ \\mathcal { M } $ , defined through $v( f ) = \\lim _{t\\rightarrow 0} ( f( \\Phi _t a ) - f( a ) ) / t, \\quad \\text{with} \\quad \\text{$ a \\in \\mathcal { M } $, $ f \\in C^1 \\mathcal { M } $},$ is at least $ C^1$ .", "Moreover, without loss of generality, we assume that $ F $ is one-to-one, and has full rank on the tangent spaces of $ \\mathcal { M } $ ; i.e., $ F $ is an embedding of $ \\mathcal { M } $ into $ \\mathbb { R }^n $ (if $ F $ is not an embedding, it is generically possible to construct an embedding through delay-coordinate maps [51], [47]).", "Under these conditions, the observation map generates a vector field $ V $ on $ F( \\mathcal { M } ) $ through its derivative $ D F $ , viz.", "$ V = DF \\, v $ .", "We denote by $ g $ the Riemannian metric induced on $ \\mathcal { M } $ by pulling back the ambient-space inner product, giving $g( u, u^{\\prime } ) = ( DF \\, u, DF \\, u^{\\prime } )$ for any two tangent vectors $ u, u^{\\prime } \\in T_a \\mathcal { M } $ .", "Remark 2.1 Even though we do not attempt to extend the analysis presented here to non-smooth manifolds, we note work of Sauer et al.", "[47], who prove embedding theorems for dynamical systems with fractal attractors.", "Moreover, in section REF we pass to an intrinsically discrete formulation, where the existence of an underlying smooth continuous theory is not required.", "Next, consider how linear combinations of $ X_i $ can be used to construct finite-difference (FD) approximations of $ V $ .", "In particular, let $ \\delta _p $ be a $ p $ -th order FD operator for the first derivative of a function, $\\frac{ df }{ d t } = \\frac{ \\delta _p f }{ \\delta t } + O( \\delta t^p ).$ We then have the following Lemma.", "The vector in data space $ \\xi _i = \\delta _p X_i $ corresponds to an $ O( \\delta t^p ) $ -accurate FD approximation of the pushforward of the dynamical vector field $ v $ evaluated at $ a_i \\in \\mathcal { M } $ , in the sense that $DF \\, v \\vert _{a_i} = \\xi _i / \\delta t + O( \\delta t^p ).$ Moreover, the data space norm $ \\Vert \\xi _i \\Vert = \\sqrt{ ( \\xi _i, \\xi _i ) } $ provides an $ O( \\delta t^p ) $ approximation to the norm of $ v \\vert _{a_i} $ with respect to the induced metric, namely $\\Vert v \\Vert _{g,a_i} := \\sqrt{ g( v, v ) } \\vert _{a_i} = \\Vert \\xi _i \\Vert / \\delta t + O( \\delta t^p ).$ A proof of this Lemma is included in Appendix REF .", "Hereafter, we nominally work with a central FD scheme, $\\delta _p X_i = \\sum _{j=-p}^p w_j X_{i+j},$ where $ w_{-p}, \\ldots , w_p $ are standard FD weights for central schemes (e.g., [38]).", "However, the asymptotic analysis in section only depends on the FD accuracy, and also applies, e.g., for backward and forward schemes.", "Recall now that a kernel is a symmetric function $ K : \\mathcal { M } \\times \\mathcal { M } \\mapsto \\mathbb { R }_+ $ which maps pairs of states in $ \\mathcal { M } $ to a positive number, and, in practical applications, depends only on quantities observed in data space through $ F $ .", "A standard choice in this context is the isotropic Gaussian kernel [5], [14], [15], [6], $\\bar{K}_\\epsilon ( a_i, a_j ) = \\exp ( - \\Vert \\omega _{ij} \\Vert ^2 / \\epsilon ), \\quad \\text{with} \\quad \\omega _{ij} = X_j - X_i = F( a_j ) - F( a_i ).$ Here, $ \\epsilon $ and $ \\omega _{ij} $ are a positive parameter and the displacement vector between data samples $ X_i $ and $ X_j $ , respectively.", "Having established the association between $ \\xi _i $ and $ v_i $ in Lemma , we seek to modify $ \\bar{K} $ to incorporate information about the dynamical vector field through (i) its norm $ \\Vert v \\Vert _{g,a_i} $ with respect to the induced metric estimated via (REF ); (ii) its angle relative to $ \\omega _{ij} $ , estimated via $\\cos \\theta _i = \\frac{ ( \\xi _i, \\omega _{ij } ) }{ \\Vert \\xi _i \\Vert \\Vert \\omega _{ij} \\Vert }.$ Specifically, introducing a parameter $ \\zeta \\in [ 0, 1 ) $ , we define $K_{\\delta t, \\zeta }( a_i, a_j ) = \\exp \\left( - \\frac{ \\Vert \\omega _{ij} \\Vert ^2 }{ \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert } [ ( 1 - \\zeta \\cos ^2 \\theta _ i )( 1 - \\zeta \\cos ^2 \\theta _j ) ]^{1/2} \\right).$ For $ \\zeta = 0 $ , the kernel in (REF ) reduces to the locally scaled kernel employed in NLSA algorithms [28], [29], which features no dependence on the angle between $ \\omega _{ij} $ and the phase space velocity vectors, $ \\xi _i $ and $ \\xi _j $ .", "On the other hand, as $ \\zeta $ approaches 1, $ K_{\\delta t, \\zeta } $ assigns higher affinity to data samples whose relative displacement vector is aligned with either $ \\xi _i $ and/or $ \\xi _j $ .", "For this reason, we term this two-parameter family of kernels cone kernels.", "For the remainder of this section we discuss certain properties of cone kernels which will be useful for the asymptotic analysis in section ." ], [ "Qualitative features of cone kernels", "As is evident from the structure of (REF ), the scaling of the pairwise distances $ \\Vert \\omega _{ij} \\Vert $ by $ \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert $ results in greater similarity being ascribed to transitory states characterized by large norm of the dynamical vector field.", "In [28], this feature was found to be crucial for successful dimensional reduction of a dynamical system with chaotic metastability.", "In section , we give a geometrical interpretation of the scaling factor showing that the metric tensor induced on $ \\mathcal { M } $ through $ K_{\\delta t, \\zeta } $ is invariant under conformal transformations of the data as a result of this scaling.", "In addition, cone kernels with $ \\zeta \\approx 1 $ provide superior discrimination by assigning greater similarity to those sample pairs whose relative displacement vector is aligned with the dynamical flow.", "More specifically, given two distinct data samples $ a_i $ and $ a_j $ with $ \\omega _{ij} \\ne 0 $ , the ratio $\\frac{ K_{\\delta t, 1}( a_i, a_j ) }{ K_{\\delta t, 0 }( a_i, a_j ) } = \\exp \\left( \\frac{ \\Vert \\omega _{ij} \\Vert ^2 (1 - C_{ij}) }{ \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert } \\right), \\quad C_{ij} = [ ( 1 - \\cos ^2 \\theta _ i )( 1 - \\cos ^2 \\theta _j ) ]^{1/2} \\le 1$ grows exponentially as $ \\delta t \\rightarrow 0 $ whenever $ \\cos \\theta _i $ and $ \\cos \\theta _j $ are both not equal to unity.", "An outcome of this property is that the corresponding diffusion operator $ _\\zeta $ generates diffusions along the integral curves of the dynamical vector field $ v $ [23] in the sense that, asymptotically as $ \\zeta \\rightarrow 1 $ , $ _\\zeta f $ vanishes whenever the gradient of $ f $ is parallel to $ v $ .", "Because these curves are intrinsic to the dynamical system generating the data, this property does not depend on the observation function $ F $ and the associated induced metric in (REF ), so long as $ F $ meets the conditions of a manifold embedding [47].", "We also mention the utility of cone kernels in situations where $ F $ is a composite map, $ F : \\mathcal { M } \\mapsto \\mathbb { R }^{n_1} \\oplus \\mathbb { R }^{n_2} $ , such that $ F( a ) = ( F_1( a ), F_2( a ) ) $ where both $ F_1 $ and $ F_2 $ are embeddings.", "This scenario arises in practice when one has access to multivariate observations with distinct physical units, but there is no natural way of choosing a norm for the product space $ \\mathbb { R }^{n_1} \\oplus \\mathbb { R }^{n_2} $ .", "Because the ratio $ \\Vert \\omega _{ij} \\Vert ^2 / \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert $ is invariant under scaling of the data by a constant (including change of units), cone kernels computed individually for $ F_1 $ and $ F_2 $ can be combined into a single product kernel without having to introduce additional scaling parameters.", "A climate science application of this technique can be found in [12]." ], [ "Local behavior at the basepoint", "The metric tensor induced at a reference point $ a $ on the data manifold by an exponentially decaying kernel depends strongly on the local rate of decay of the kernel at $ a $ .", "As established by Berry [9], in many cases of interest, including the isotropic Gaussian kernel in (REF ) and the cone kernel in (REF ), that rate is controlled to leading order by the Hessian (second derivative) matrix of the kernel with respect to a coordinate patch covering the reference point, as we now discuss.", "Fixing a basepoint $ a \\in \\mathcal { M } $ , consider $ K_{\\delta t, \\zeta }( a, a^{\\prime } ) $ for $ a^{\\prime } $ lying in an exponential neighborhood of $ a $ : $a^{\\prime } = \\exp _a u, \\quad u \\in T_a \\mathcal { M }, \\quad u = \\sum _{\\mu =1}^m u^\\mu U_\\mu .$ Here, $ U_1, \\ldots U_m $ is a basis of $ T_a \\mathcal { M } $ so that the components $ u^\\mu $ are exponential coordinates centered at $ a $ .", "Taking partial derivatives with respect to $ u^\\mu $ , it is possible to derive the expressions $\\left.", "\\frac{ \\partial K_{\\delta t, \\zeta } }{ \\partial u^\\mu } \\right\|_{u=0} = 0 \\quad \\text{and} \\quad \\left.", "\\frac{ \\partial ^2 K_{\\delta t, \\zeta } }{ \\partial u^\\mu \\, \\partial u^\\nu } \\right\|_{u=0} = -\\frac{ 2 }{ \\Vert \\xi \\Vert ^2 } \\left( g_{\\mu \\nu } - \\zeta \\frac{ \\xi ^_\\mu \\xi ^_\\nu }{ \\Vert \\xi \\Vert ^2 } \\right).$ In (REF ), $ g_{\\mu \\nu } $ are the components of the ambient-space induced metric $ g $ in (REF ) evaluated in the dual basis $ U^{1}, \\ldots , U^{m} $ with $ U^{\\mu }( U_\\nu ) = {\\delta ^\\mu }_\\nu $ [see also (REF )].", "Moreover, $ \\xi ^_\\mu $ are the components of the dual vectors $ \\xi ^* \\in T^_a \\mathcal { M } $ given by pulling back the dual vectors to $ \\xi $ with respect to the canonical inner product $ ( \\cdot , \\cdot ) $ of $ \\mathbb { R }^n $ .", "That is, $\\xi ^ := DF^* \\, \\Xi = \\sum _{\\mu =1}^m \\xi ^_\\mu U^{\\mu },$ where $ DF^* : T^_{F(a)} \\mapsto T^_a \\mathcal { M } $ is the pullback map for dual vectors, and $ \\Xi = ( \\xi , \\cdot ) $ .", "Details of this calculation are provided in Appendix REF .", "Equation (REF ) in conjunction with Lemma leads to an asymptotic expression in the sampling interval $ \\delta t $ connecting the Hessian of cone kernels to the dual of the dynamical vector field $ v^* = g( v, \\cdot ) = \\sum _{\\mu =1}^m v^_\\mu U^{\\mu } $ : $ \\left.", "\\frac{ \\partial ^2 K_{\\delta t, \\zeta } }{ \\partial u^\\mu \\, \\partial u^\\nu } \\right\|_{u=0} = -\\frac{ 2 }{ \\Vert v \\Vert ^2_g \\, \\delta t^2} \\left( g_{\\mu \\nu } - \\zeta \\frac{ v^_\\mu v^_\\nu }{ \\Vert v \\Vert ^2_g } \\right) + O( \\delta t^{p-2} ) \\quad \\text{with} \\quad v^_\\mu = \\sum _{\\nu =1}^m g_{\\mu \\nu } v^\\nu .$ The metric tensor induced on the data manifold by the cone kernel is in fact proportional to the Hessian [9]; in this case $ h_{\\mu \\nu } $ is given by the negative of the $ O( \\delta t^{-2} ) $ coefficient in (REF ).", "Below, we study the geometry induced on the data by $ K_{\\delta t , \\zeta }$ through the associated diffusion operator.", "Remark 2.2 Equations (REF ) and (REF ) are unaltered if one replaces the geometric means involving $ \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert $ and $ \\cos ^2\\theta _i $ and $ \\cos ^2 \\theta _j $ in (REF ) with the corresponding arithmetic and harmonic means, i.e., $K_{\\delta t, \\zeta }( a_i, a_j ) = \\exp \\left[ - \\frac{ \\Vert \\omega _{ij} \\Vert ^2 }{ 4 } \\left( \\frac{ 1 }{ \\Vert \\xi _i \\Vert ^2 } + \\frac{ 1 }{ \\Vert \\xi _j \\Vert ^2 } \\right) ( 2 - \\zeta \\cos ^2 \\theta _ i - \\zeta \\cos ^2 \\theta _j ) \\right].$ For finite $ \\delta t $ , the behavior of the two kernels will generally differ.", "In particular, (REF ) is large if either $ \\Vert \\xi _i \\Vert $ or $ \\Vert \\xi _j \\Vert $ are small, whereas (REF ) is large if both $ \\Vert \\xi _i \\Vert $ and $ \\Vert \\xi _j \\Vert $ are small (similarly for the angular terms).", "Thus, the kernel in (REF ) may have higher discriminating power than (REF ), but at the same time may require a larger number of samples for stable behavior.", "Remark 2.3 In practical applications, it may be desirable to introduce an additional scaling parameter analogous to $ \\epsilon $ in the isotropic Gaussian kernel in (REF ), i.e., $K_{\\delta t, \\epsilon , \\zeta }( a_i, a_j ) = \\exp \\left( - \\frac{ \\Vert \\omega _{ij} \\Vert ^2 }{ \\epsilon \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert } [ ( 1 - \\zeta \\cos ^2 \\theta _ i )( 1 - \\zeta \\cos ^2 \\theta _j ) ]^{1/2} \\right).$ Apart from an unimportant scaling factor in the Hessian, the presence of $ \\epsilon $ in (REF ) does not influence the $ \\delta t \\rightarrow 0 $ asymptotics, but provides additional freedom to tune the kernel in situations where one does not have control of the sampling interval." ], [ "The associated diffusion operator and induced metric tensor", "The classical procedure to construct a diffusion operator for geometric analysis of data from a kernel (e.g., [5], [15], [6], [9]) begins with the introduction of an integral operator acting on scalar functions on the data manifold, which we denote here by $ H_{\\delta t, \\zeta } $ to make explicit the two parameters appearing in cone kernels.", "Specifically, $H_{\\delta t, \\zeta } f( a ) = \\frac{ 1 }{ \\delta t^m } \\int _\\mathcal { M } K_{\\delta t, \\zeta }( a, \\cdot ) f \\mu ,$ where $ f $ is a sufficiently smooth scalar function on $ \\mathcal { M } $ , and $ \\mu $ the volume form of the induced metric $ g $ in (REF ).", "Scaling $ H_{\\delta t, \\zeta } f $ by the normalization factor $\\rho _{\\delta t, \\zeta }( a ) = H_{\\delta t, \\zeta } \\mathbb {1}( a ) = \\frac{ 1 }{ \\delta t^m } \\int _\\mathcal { M } K_{\\delta t, \\zeta }( a, \\cdot ) \\mu ,$ we obtain the integral operator $ \\mathcal { P }_{\\delta t, \\zeta }( a ) = H_{\\delta t, \\zeta } f( a ) / \\rho _{\\delta t, \\zeta }( a ) $ .", "This operator preserves constant functions, i.e., it is an averaging operator.", "Next, let $\\mathcal { L }_{\\delta t, \\zeta } = ( \\operatorname{Id}- \\mathcal { P }_{\\delta t, \\zeta } ) / \\delta t^2.$ This positive-semidefinite operator can be thought of the generator of $ P_{\\delta t, \\zeta } $ .", "An important property of $ \\mathcal { L }_{\\delta t, \\zeta } $ is that it annihilates constant functions, i.e., $ \\mathcal { L }_{\\delta t, \\zeta } \\mathbb { 1} = 0 $ .", "Following [23], we refer to such operators as diffusion operators.", "For suitably-defined kernels, $ \\mathcal { L }_{\\delta t, \\zeta } $ converges to a second-order self-adjoint operator $ _\\zeta = \\lim _{\\delta t \\rightarrow 0} \\mathcal { L }_{\\delta t, \\zeta } $ .", "This operator induces a geometry on the dataset in the sense that it corresponds to a unique codifferential operator $ \\delta _\\zeta $ and Riemannian metric $ h $ such that $\\Delta _{\\zeta } = \\delta _\\zeta d, \\quad \\text{where} \\quad ( w, df )_h = ( \\delta _\\zeta w, f )_h$ for any smooth 1-form field $ w $ and scalar function $ f $ .", "Here, $ ( \\cdot , \\cdot )_h $ are the canonical (Hodge) inner products for $ p $ -form fields associated with $ h $ ; i.e., $( w, df )_h := \\int _{\\mathcal { M }} h^{-1}( w, df ) \\nu , \\quad ( \\delta _\\zeta w, f ) := \\int _{\\mathcal {M} } \\delta _\\zeta ( w ) f \\nu ,$ where $ \\nu $ and $ h^{-1} $ are the volume form and “inverse metric” associated with $ h $ , respectively.", "In particular, we have the following Lemma: The induced metric tensor $ h $ at $ a \\in \\mathcal { M } $ associated with the cone kernels (REF ) with FD accuracy from Lemma $ p \\ge 4 $ is given by $h = \\frac{ 1 }{ \\Vert v \\Vert _g^2 } \\left( g - \\zeta \\frac{ v^ \\otimes v^* }{ \\Vert v \\Vert _g^2 } \\right),$ where $ g $ is the ambient-space induced metric in (REF ), and $ v^* = g( v, \\cdot ) $ the dual dynamical vector field with respect to $ g $ .", "Moreover, the volume forms $ \\nu $ and $ \\mu $ of $ h $ and $ g $ , respectively, are related through the expression $\\nu = ( 1 - \\zeta )^{1/2} \\bar{\\nu }, \\quad \\text{with} \\quad \\bar{\\nu }= \\mu / \\Vert v \\Vert _g^m.$ A proof of this Lemma can be found in Appendix REF .", "A corollary of Lemma is that cone kernels assign a unique metric to equivalence classes of datasets related by conformal transformations.", "In particular, we say that the datasets associated with the embeddings $ F : \\mathcal { M } \\mapsto \\mathbb { R }^n $ and $ \\tilde{F} : \\mathcal { M } \\mapsto \\mathbb { R }^{\\tilde{n} } $ are conformally equivalent if there exists a positive function $ r : \\mathcal { M } \\mapsto \\mathbb { R }_+ $ such that the induced metric $ \\tilde{g} $ associated with $ \\tilde{F} $ is given by $ \\tilde{g} \\vert _a = r( a ) g \\vert _a $ .", "Because conformally equivalent datasets have the properties $ \\Vert v \\Vert ^2_{\\tilde{g}} = r \\Vert v \\Vert _g^2 $ and $ \\tilde{v}^* := \\tilde{g} ( v, \\cdot ) = r v^* $ , it follows from (REF ) that both $ g $ and $ \\tilde{g} $ lead to the same $ h $ metric; an assertion made in section REF .", "Remark 3.1 The volume form $ \\nu $ of $ h $ is invariant under all transformations that preserve the ratio in the right-hand side of (REF ).", "This set of transformations includes conformal transformations as a special case, but also admits more general transformations where changes in Riemannian volume are appropriately compensated by changes in the norm of $ v $ .", "We will return to this point in section REF .", "A further consequence of (REF ) is that $ h $ leads to a contraction of distances in neighborhoods of the data manifold where $ \\Vert v \\Vert _g $ is uncharacteristically small.", "Such regions correspond to metastable dynamical regimes separated by rapid transitions with large $ \\Vert v \\Vert _g $ .", "As remarked in section REF , the ability of locally-scaled kernels (i.e., cone kernels with $ \\zeta = 0 $ ) to discriminate between regimes of this type has been found to be highly beneficial in Galerkin reduced dynamical models with chaotic metastability [28].", "For our purposes, however, of particular interest is the behavior of $ h $ and the associated codifferential and diffusion operators in (REF ) in the limit $ \\zeta \\rightarrow 1 $ , where the directional influence of the dynamical vector field is maximal.", "In that limit $ h( u, v ) $ vanishes for all tangent vectors $ u \\in T_a \\mathcal { M } $ , i.e., the induced metric becomes degenerate.", "Equivalently, the length of the integral curves of $ v $ measured with respect to $ h $ becomes arbitrarily small.", "In consequence, the diffusion generated by $ _1 $ takes place along the integral curves of the dynamical vector field, as we now discuss." ], [ "Behavior in the $ \\zeta \\rightarrow 1 $ limit", "Recall that a diffusion operator $ \\Delta = \\delta d $ acting on $ C^2 $ scalar-valued functions on an $ m $ -dimensional manifold $ \\mathcal { M } $ is said to be along a vector field $ v \\in T \\mathcal { M } $ if the codifferential $ \\delta w $ vanishes for all $ C^1 $ 1-form fields $ w $ lying in the $ ( m - 1 ) $ -dimensional subspace $ S_v \\subset C^1 T^\\mathcal { M } $ with $ w( v ) = 0 $ [23].", "Intuitively, one thinks of the diffusion process generated by $ \\Delta $ to take place along the integral curves of $ v $ .", "A key property of the diffusion operator $ \\Delta _\\zeta $ in (REF ) associated with cone kernels is that it is along the dynamical vector field $ v $ asymptotically as $ \\zeta \\rightarrow 1 $ , in the sense of the following Lemma.", "The codifferential operator $ \\delta _\\zeta $ associated with cone kernels admits the asymptotic expansion $\\delta _\\zeta w = \\frac{ 1 }{ 1 - \\zeta } \\bar{\\delta }w + O((1-\\zeta )^0), \\quad \\bar{\\delta }w = - \\operatorname{div}_{\\bar{\\nu }} [ w( v ) v ],$ where $ \\operatorname{div}_{\\bar{\\nu }} $ is the divergence operator associated with the volume form $ \\bar{\\nu }$ in (REF ).", "This Lemma is proved in Appendix REF .", "An asymptotic expansion for the corresponding diffusion operator in (REF ) follows by setting $ w = df $ in (REF ) for some $ C^2 $ scalar function $ f $ , i.e., $_\\zeta f = \\frac{ 1 }{ 1 - \\zeta } \\bar{}f + O(( 1- \\zeta )^0), \\quad \\bar{}f = - \\operatorname{div}_{\\bar{\\nu }}[ v( f ) v ].$ A consequence of Lemma REF is that $ \\delta _\\zeta w = O( (1-\\zeta )^0 ) $ if $ w \\in S_v $ , but $ \\delta _\\zeta w^{\\prime } = O( ( 1- \\zeta )^{-1} ) $ if $ w^{\\prime } $ lies outside of $ S_v $ .", "As a result, the norm ratio $\\frac{ \\Vert \\delta _\\zeta w \\Vert _h }{ \\Vert \\delta _\\zeta w^{\\prime } \\Vert _h } = \\frac{ ( \\delta _\\zeta w, \\delta _\\zeta w )^{1/2} }{ ( \\delta _\\zeta w^{\\prime }, \\delta _\\zeta w^{\\prime } )^{1/2} } = O(( 1- \\zeta ))$ tends to zero as $ \\zeta \\rightarrow 1 $ for all nonzero square-integrable 1-form fields $ w^{\\prime } \\notin S_v $ for which $ \\bar{\\delta }w^{\\prime } $ is nonzero.", "We interpret (REF ) as an asymptotic along-$ v $ property of $ _\\zeta $ .", "Remark 3.2 In general, $ _\\zeta f = \\delta _\\zeta d f = - \\operatorname{div}_\\nu _h f $ depends on the metric through the volume form $ \\nu $ , as well as explicitly through the gradient $ _h f = h^{-1}( df, \\cdot ) $ .", "In the $ \\zeta \\rightarrow 1 $ limit, the latter is replaced by the directional derivative $ v( f ) $ in (REF ), which is metric-independent.", "Thus, the only metric dependence of the limit operator $ \\bar{}$ is through the volume form $ \\bar{\\nu }$ .", "According to Remark REF , the latter is invariant under a set of transformations of the data which includes conformal transformations as a subset." ], [ "Discrete formulation", "Discrete analogs of the diffusion operator (REF ) arise naturally in the framework of discrete exterior calculus (DEC; e.g., [19], [31], [55], [30]).", "In this setting, the spaces of scalar functions and 1-form fields appearing in (REF ) are replaced by functions $ f( a_i ) = f_i $ and $ w( [a_i a_j ] ) = w_{ij} $ defined on the vertices $ a_i $ and edges $ [ a_i a_j ] $ , respectively, of a graph formed by the $ s $ sampled states in (REF ).", "These function spaces are equipped with weighted inner products, $( f, f^{\\prime } )_P = \\sum _{i=1}^s \\pi _i f_i f^{\\prime }_i, \\quad ( w, w^{\\prime } )_P = \\sum _{i,j=1}^s \\pi _i P_{ij} w_{ij} w^{\\prime }_{ij} / 2,$ which are the discrete counterparts of (REF ).", "Also, a difference operator $ \\hat{d} $ is is introduced mapping vertex to edge functions via $ \\hat{d} f([a_i a_j]) = f_j-f_i$ .", "The associated discrete codifferential $ \\hat{\\delta }_{\\delta t, \\zeta } $ and diffusion operator $ L_{\\delta t, \\zeta } $ (which in the context of cone kernels depend on both $ \\delta t $ and $ \\zeta $ ) are then defined in direct analogy with (REF ): $L_{\\delta t, \\zeta } = \\hat{\\delta }_{\\delta t, \\zeta } \\hat{d}, \\quad ( w, \\hat{d}f )_P = ( \\hat{\\delta }_{\\delta t, \\zeta } w, f )_P.$ Even though the inner products and associated diffusion operator in (REF ) exist independently of a continuous theory, here we seek to construct $ L_{\\delta t, \\zeta } $ such that, asymptotically as $ s \\rightarrow \\infty $ , it inherits the conformal invariance and along-$ v $ properties established in Lemmas and REF .", "To that end, we employ the diffusion map (DM) algorithm of Coifman and Lafon [15].", "In DM, the inner product weights $ P_{ij} $ in (REF ) are the elements of a Markov matrix whose state space is the discrete dataset in (REF ), constructed via the sequence of operations $K_{ij} = K_{\\delta t, \\zeta }( a_i, a_j ), \\quad \\tilde{K}_{ij} = \\frac{ K_{ij} }{ \\left( \\sum _{k=1}^s K_{ik} \\right)^\\alpha \\left( \\sum _{k=1}^s K_{jk} \\right)^\\alpha }, \\quad P_{ij} = \\frac{ \\tilde{K}_{ij} }{ D_{ii} D_{jj} }.$ Here, $ \\alpha $ is a real parameter, and $ D $ a diagonal degree matrix with $ D_{ii} = \\sum _{k=1}^s \\tilde{K}_{ik} $ .", "The inner-product weights $ \\pi _i $ are given by the invariant distribution of that Markov matrix, i.e., $ \\sum _{i=1}^s \\pi _i P_{ij} = \\pi _j $ .", "With these definitions, it follows that $ \\hat{\\delta }_{\\delta t, \\zeta } w( a_i ) = \\sum _{j=1}^s P_{ij} (w([a_j a_i]) - w([a_i a_j ] ) / 2, \\quad L_{\\delta t, \\zeta } = I - P.$ The normalization step to obtain $ \\tilde{K}_{ij} $ in (REF ), which is not present in standard graph Laplacian algorithms (e.g., [6]), controls the influence of the sampling density with respect to the volume form $ \\nu $ in approximations of the manifold integral in (REF ) by discrete sums of the form $ \\sum _{j=1}^s \\tilde{K}_{ij} f( a_j ) $ .", "In particular, under relatively weak assumptions on $ \\mathcal { M } $ and the embedding map $ F $ , it can be shown [15] that for $ \\alpha = 1 $ , $ \\delta t \\rightarrow 0 $ , and $ s^{1/2} \\delta t^{2+m/2} \\rightarrow \\infty $ [49], the discrete diffusion operator converges pointwise to $ _\\zeta $ , in the sense that $ \\vert _\\zeta f( a ) - L_{\\delta t, \\zeta } f( a ) / \\delta t^2 \\vert \\rightarrow 0 $ for all $ a \\in \\mathcal { M } $ and sufficiently smooth $ f $ .", "This feature is particularly desirable for our purposes, for the sampling density is dictated by the dynamical flow $ \\Phi _t $ and observation function $ F $ , and we are interested in targeting $ \\mathcal { L }_{\\delta t, \\zeta } $ without using a priori information about $ \\Phi _t $ and $ F $ .", "We therefore adopt $ \\alpha = 1 $ DM in all of the experiments of sections and ahead." ], [ "Diffusion eigenfunctions", "As mentioned in the introduction, diffusion operators such as $ L_{\\delta t, \\zeta } $ in (REF ) are useful for a wide range of data analysis tasks, including dimension reduction, feature extraction, and regularization [48], [5], [14], [15], [6], [33], [7], [42], [13], [16].", "Below, we focus on a particular aspect of $ L_{\\delta t, \\zeta } $ , namely its eigenfunctions $ \\phi _i $ , defined through $L_{\\delta t, \\zeta } \\phi _i = \\lambda _i \\phi _i, \\quad \\phi _i = ( \\phi _{1i}, \\ldots , \\phi _{si} )^T, \\quad 0 = \\lambda _0 < \\lambda _1 \\le \\lambda _2 \\le \\cdots .$ Diffusion eigenfunctions are traditionally used to create low-dimensional parameterizations of data of the form $ a \\in \\mathcal { M } \\mapsto \\phi ( a ) = ( \\phi _{i_1}( a ), \\ldots , \\phi _{i_l}( a ) ) \\in \\mathbb { R }^l $ , with rigorous embedding results established in the continuous limit [8], [33], [45].", "A somewhat different perspective, adopted in NLSA algorithms [25], [28], [29], is to associate low-dimensional subspaces spanned by the leading $ \\phi _i $ with spaces of temporal patterns through the time series $ \\tilde{\\phi }_i( t_j ) = \\phi _i( a_j ) = \\phi _{ji},$ and (in the spirit of [3]) extract spatiotemporal modes of variability through singular value decomposition of the data projected onto those eigenfunctions.", "Clearly, in both approaches, the relationship of the basis to the underlying dynamical flow plays a major role on algorithm performance.", "To gain insight on the influence of the along-$ v $ property of diffusion operators associated with cone kernels on their eigenfunctions, it is useful to consider the Dirichlet form associated with $ _\\zeta $ , $\\mathcal { E }_\\zeta ( f ) = ( f, _\\zeta f )_h.$ In particular, it follows from the asymptotic expansion in (REF ), in conjunction with the divergence theorem and the assumption in section that $ \\mathcal { M } $ has no boundary, that $\\mathcal { E }_\\zeta ( f ) = ( 1 - \\zeta )^{-1/2 } \\bar{\\mathcal { E }}( f ) + O( ( 1 - \\zeta )^{3/2} ), \\quad \\text{with} \\quad \\bar{\\mathcal {E}}( f ) = \\int _\\mathcal { M } [ v( f ) ]^2 \\bar{\\nu }.$ Therefore, as $ \\zeta \\rightarrow 1 $ , $ \\mathcal { E }_\\zeta $ assigns low energy to functions which (on average) have large directional derivative $ v( f ) $ along the dynamical flow.", "Because $ \\mathcal { E }_\\zeta ( \\phi _i ) $ is equal to the corresponding eigenvalue $ \\lambda _i $ for normalized eigenfunctions with $ \\Vert \\phi _i \\Vert _h = 1 $ , we expect the leading (small-$ \\lambda _i $ ) diffusion eigenfunctions to vary predominantly in directions transverse to the integral curves of $ v $ .", "This property implies strong adaptivity of $ \\phi _i $ to the dynamical flow, and is also independent of the embedding map $ F $ and the associated induced metric in (REF ).", "Below, we demonstrate these properties in numerical experiments." ], [ "Dynamical systems on the 2-torus", "To explicitly illustrate the key features of cone kernels, we begin with a low-dimensional application where the phase space manifold $ \\mathcal { M } $ is the 2-torus.", "Denoting by $ ( \\theta ^1, \\theta ^2 ) $ the azimuthal and polar angles on the 2-torus, respectively, we consider a two-parameter family of dynamical vector fields $v = \\sum _{\\mu =1}^2 v^\\mu \\frac{ \\partial }{ \\partial \\theta ^\\mu }, \\quad \\text{with} \\quad v^1 = 1 + ( 1 - \\beta )^{1/2} \\cos \\theta ^1, \\quad v^2 = \\Omega ( 1 - ( 1 - \\beta )^{1/2} \\sin \\theta ^2 ).$ Here, $ \\Omega $ is a positive frequency parameter which is set to an irrational number to produce a dense cover of the torus.", "Moreover, the parameter $ \\beta \\in ( 0, 1 ] $ controls the nonlinearity of the flow.", "Specifically, $ \\beta = 1 $ corresponds to a linear flow, but when $ \\beta < 1 $ the flow “slows down” at $ (\\theta ^1, \\theta ^2 ) \\sim ( \\pi , \\pi /2 ) $ and “speeds up” at $ ( \\theta ^1, \\theta ^2 ) \\sim ( 0, -\\pi /2 ) $ .", "The orbit $ ( \\dot{\\theta }^1, \\dot{\\theta }^2 ) = ( v^1, v^2 ) $ passing through $ ( \\theta ^1, \\theta ^2 ) = ( 0, 0 ) $ at time $ t = 0 $ is given by $\\tan ( \\theta ^1 / 2 ) = [ 1 + ( 1 - \\beta )^{1/2} ] \\beta ^{-1/2} \\tan ( \\beta t / 2 ), \\quad \\cot ( \\theta ^2 / 2 ) = ( 1 - \\beta )^{1/2} + \\beta ^{1/2} \\cot ( \\beta ^{1/2} t / 2 ).$ Setting $ \\beta = 0.5 $ throughout, we consider the cases $ \\Omega = 30^{1/2} $ and $ 30^{-1/2} $ , referred to here as Models I and II, respectively.", "The resulting trajectories in data space corresponding to the standard embedding of the 2-torus, $F: \\mathcal { M } \\mapsto \\mathbb { R }^3, \\quad F( a ) = ( x^1, x^2, x^3 ), \\\\x^1 = ( 1 + R \\cos \\theta ^2( a ) ) \\cos \\theta ^1( a ), \\quad x^2 = ( 1 + R \\cos \\theta ^2( a ) ) \\sin \\theta ^1( a ), \\quad x^3 = \\sin \\theta ^2( a ),$ with $ R = 1 /2 $ are illustrated in Figures REF (a) and REF (b), respectively.", "In addition to the standard embedding, we also study a non-conformally deformed embedding $ F_\\gamma ( a ) = ( y^1, y^2, y^3 ) $ of the 2-torus into $ \\mathbb { R }^3 $ , where the $ y^3 $ coordinate is a stretched version of $ x^3 $ with a non-uniform scaling factor, i.e., $y^1 = x^1, \\quad \\tilde{y}^2 = x^2, \\quad y^3 = x^3 e^{\\gamma z}, \\quad z = ( 1 + R - x^1 )( 1 + x^3 ).$ The deformed system with $ \\Omega = 30^{1/2} $ and $ \\gamma = 0.3 $ , which we refer to as Model I$^\\prime $ , is shown in Figure REF (c).", "In all cases, we equip the ambient data space with the canonical Euclidean inner product.", "Figure: Sample trajectories for the dynamical systems () on 2-tori.", "(a) Model I: Ω=30 1/2 \\Omega = 30^{1/2} , no deformation.", "(b) ModelII:Ω=30 -1/2 Model~II: \\Omega = 30^{-1/2} , no deformation.", "(c) Model I ' ^{\\prime }: Ω=30 1/2 \\Omega = 30^{1/2} , non-conformal deformation with γ=0.3 \\gamma = 0.3 .For each of the models in Figure REF , we generated datasets with $ s = \\text{64,000} $ samples taken at a sampling interval $ \\delta t = 2 \\pi / ( S \\min \\lbrace 1, \\Omega \\rbrace ) $ where $ S = 500 $ is an integer parameter controlling the number of samples in each quasi-period.", "We computed the FD approximations of $ v $ using the central scheme in (REF ) with fourth-order accuracy as required by Lemma [i.e., the FD weights are $(w_{-2},\\ldots ,w_2) = ( 1/12, -2/3, 0, 2/3, -1/12)$ ].", "With this FD scheme, the phase space velocity norm ratio $ \\max \\lbrace \\Vert \\xi _i \\Vert \\rbrace / \\min \\lbrace \\Vert \\xi _i \\Vert \\rbrace $ for Models I, II, and I$^\\prime $ was of order 20, 35, and 40, respectively.", "Thus, the influence of the $ \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert $ scaling factors in the cone kernels (REF ) is expected to be significant.", "We evaluated the discrete diffusion operator $ L_{\\delta t, \\zeta } $ from cone kernels with $ \\zeta = 0 $ and 0.995 (corresponding to no influence and strong influence of the directionality of $ \\xi $ , respectively) using the DM procedure in (REF ) with $ \\alpha = 1 $ .", "For comparison, we also computed the diffusion operator associated with the isotropic Gaussian kernel in (REF ) with $ \\epsilon = 0.1 $ .", "To limit memory usage, in all cases we truncated the pairwise kernel evaluations to $ b = 2000 $ nearest neighbors [in the sense of $ K_{ij} $ in (REF )] per sample.", "The kernel values at truncation were no greater than $ O( 10^{-7} ) $ , indicating that truncation has negligible impact on the numerical results.", "Representative eigenfunctions for Models I, II, and I' are shown in Figures REF –REF .", "To test for convergence of our results to the continuous limit, we performed a series of long runs, where $ s $ was increased eightfold, $ \\delta t$ decreased twofold, and $ b $ set to 10,000 (in the isotropic Gaussian kernel case $ \\epsilon $ was decreased fourfold).", "The diffusion eigenfunctions from the long runs (not shown here) were in good agreement with those displayed in Figures REF –REF .", "We also performed eigenfunction calculations using a first-order backward FD scheme so that $ \\xi _i = X_i - X_{i-1} $ , and found again only minor changes relative to the results with fourth-order FD accuracy.", "It therefore appears that in this setting with dense sampling the accuracy required by Lemma is not crucial to recover the salient features of the eigenfunctions.", "Figure: Scatterplots of diffusion eigenfunctions for the dynamical system on the 2-torus with Ω=30 1/2 \\Omega = 30^{1/2} obtained using DM with α=1 \\alpha = 1 in conjunction with the isotropic Gaussian kernel (), and the cone kernels () with ζ=0 \\zeta = 0 and 0.995.", "A portion of the dynamical system trajectory is plotted in a black line for reference.Figure: Same as Figure but for the dynamical system with Ω=30 -1/2 \\Omega = 30^{-1/2} .Figure: Same as Figure but for the deformed torus with deformation parameter γ=0.3 \\gamma = 0.3 [see Figure (c)]." ], [ "Properties of the diffusion eigenfunctions", "First, consider the eigenfunctions for Models I and II.", "According to [15], the results of DM with the isotropic Gaussian kernel should only depend on the embedding $ F $ , and should be independent of the sampling density on $ \\mathcal { M } $ induced by the dynamical flow.", "Indeed, as expected from theory, the isotropic-kernel eigenfunctions in Figures REF and REF are essentially equivalent, despite the fact that the underlying dynamical system trajectories are qualitatively different.", "On the other hand, because cone kernels depend explicitly on the dynamical flow through $ \\xi $ [see (REF )], the eigenfunctions for Models I and II differ.", "In the $ \\zeta = 0 $ results for Model I shown in Figure REF , the eigenfunction wavecrests are compressed in the portion of the torus where the flow evolves slowly, and rarefied in the transitory region characterized by large $ \\Vert v \\Vert _g $ (compare, e.g., the $ \\phi _3 $ and $ \\phi _5 $ eigenfunctions obtained via the isotropic kernel and the $ \\zeta = 0 $ cone kernel).", "As a result, these eigenfunctions have higher discriminating power in the regions of $ \\mathcal { M } $ where the system evolves more slowly.", "The Model II eigenfunctions exhibit a qualitatively similar behavior.", "The along-$ v $ property of the diffusion operators with $ \\zeta \\approx 1 $ , and the resulting adaptation of the eigenfunctions to the dynamical flow expected on the basis of (REF ), is manifested in the right-hand columns of Figures REF and REF .", "There, the eigenfunctions vary predominantly in directions transverse to the dynamical flow, resulting in characteristic swirl patterns for Model I and azimuthal streak-like patterns for Model II.", "Because the integral curves of $ v $ are metric independent, this property is robust against changes in the embedding function $ F $ .", "Indeed, in the Model I' results in Figure REF , it is only the $ \\zeta = 0.995 $ eigenfunctions which remain qualitatively similar to those in Figure REF .", "There, the dataset deformation has left a clear imprint on the eigenfunctions.", "On the other hand, the $ \\zeta = 0.995 $ eigenfunctions retain the dynamics-adaptation featuring the characteristic swirl patterns following the dynamical flow.", "The structure of the eigenfunctions transversely to the flow does exhibit some changes relative to Figure REF , but these are significantly weaker compared to the isotropic Gaussian kernel and $ \\zeta = 0 $ cone kernel examples." ], [ "North Pacific SST data from a comprehensive climate model", "In this experiment, we apply cone kernels to SST data in the North Pacific sector of the Community Climate System Model version 3 [17].", "The dataset, which was studied in [26] via NLSA, consists of monthly-averaged samples of SST in the rectangular domain 120$^\\circ $ E–110$^\\circ $ W and 20$^\\circ $ N–65$^\\circ $ N spanning a 900-year interval.The dataset is available at the Earth System Grid repository, http://www.earthsystemgrid.org, where it is designated CCSM3 integration b30.004.", "We work throughout with the model's nonuniform native ocean grid of $ 1^\\circ $ nominal horizontal resolution.", "The number of gridpoints in the analysis domain is $ d = 6671 $ .", "A major component of North Pacific SST variability is due to the annually varying solar forcing (the seasonal cycle).", "The latter is superposed to low-frequency (interannual to decadal) variability patterns, the most prominent of which are the Pacific Decadal Oscillation (PDO) [41] and the North Pacific Gyre Oscillation (NPGO) [21].", "A signature of ENSO (which is most prominent in the tropical Pacific [53]) is also present in this domain.", "We refer the reader to [26] for further details on these modes of variability extracted via NLSA.", "Here, our objective is to compare the diffusion eigenfunctions of cone kernels with $ \\zeta \\approx 1 $ to those with $ \\zeta = 0 $ , which are equivalent to the earlier NLSA kernels (aside from the fourth-order accurate FD scheme used here in place of the first-order backward scheme in [26]).", "Following [26], [29], to “Markovianize” the time-dependent solar forcing in the data, and induce timescale separation in the diffusion eigenfunctions, we first embed the data to a higher-dimensional space via Takens' method of delays [44], [51], [11], [47].", "That is, we map each spatial snapshot $ x_i \\in \\mathbb { R }^d $ to a spatiotemporal sequence $ X_i = ( x_i, x_{i-1}, \\ldots , x_{i-(q-1)} ) \\in \\mathbb { R }^n $ , where $ q $ is an integer parameter measuring the embedding window length (in months).", "Here, we work with a two-year embedding window, $ q = 24 $ , which corresponds to a data space dimension $ n = q d = \\text{160,104} $ .", "Qualitatively similar results can be obtained with embedding windows in the interval 1–5 years.", "Similarly to the torus examples of section , we evaluate $ \\xi $ using a central fourth-order FD scheme.", "The number of samples $ s $ available for analysis after embedding and removal of two samples in the beginning and end of the simulation interval (to compute $ \\xi _1 $ and $ \\xi _s$ ) is 10,773.", "Unlike the torus experiments, $ s $ is small-enough in this case so as not to require nearest-neighbor truncation.", "We employ an area-weighted data space inner product given by $ ( X_i, X_j ) = \\sum _{\\nu =1}^n A_{\\nu \\bmod d } X_i^\\nu X_j^\\nu $ , where $ A_1, \\ldots , A_d $ are the grid cell areas in the analysis domain.", "Prior to analysis we center the data by subtracting the temporal mean $ \\bar{x} = \\sum _i x_i $ from each snapshot $ x_i $ to produce temperature anomaly snapshots.", "Note that centering is done mainly for visualization purposes, and does not influence the kernel values in (REF ) and (REF ).", "Representative diffusion eigenvalues $ \\lambda _i $ and eigenfunctions $ \\phi _i $ obtained via cone kernels with $ \\zeta = 0 $ and $ 0.995 $ are displayed in Figures REF –REF , where the $ \\phi _i $ are represented by the corresponding $ \\tilde{\\phi }_i $ time series from (REF ).", "To visualize subspaces spanned by selected groups of eigenfunctions $ \\phi = ( \\phi _{i_1}, \\ldots , \\phi _{i_l} ) $ as spatiotemporal patterns, we filter the data using the $ s \\times l $ matrix $ \\phi $ as a convolution filter; i.e., $ X \\mapsto X \\pi \\phi \\phi ^ T$ , where $ \\pi = ( \\pi _1, \\ldots , \\pi _s ) $ is the diagonal matrix formed by the inner product weights in (REF ).", "Spatiotemporal patterns of this type are shown in Figure REF and Movie 1, which is much more revealing.", "Note that the eigenfunctions computed using a first-order backward FD scheme (not shown here) exhibit minor changes compared to those in Figures REF and REF .", "Figure: Eigenvalues λ i \\lambda _i for CCSM North Pacific SST data obtained via cone kernels with ζ=0 \\zeta = 0 and 0.995 0.995 , normalized so that 1-λ 1 =1 1 - \\lambda _1 = 1 .", "To highlight the differences between the two spectra, 1-λ i 1 - \\lambda _i values are plotted in a logarithmic scale.", "Periodic, low-frequency, and intermittent eigenfunctions are indicated using ◯\\bigcirc , △ \\bigtriangleup , and □ \\Box markers, respectively.", "Solid markers correspond to eigenfunctions whose temporal character does not belong in these families.Figure: Representative eigenfunctions for CCSM North Pacific SST data obtained via the cone kernel with ζ=0 \\zeta = 0 .", "The left-hand panels show 20-year portions of the time series φ ˜ i (t)=φ i (a t ) \\tilde{\\phi }_i( t ) = \\phi _i( a_t ) .", "The right-hand panels are frequency spectra computed via the discrete Fourier transform of φ ˜ i \\tilde{\\phi }_i .", "The eigenfunctions in this Figure have been chosen so as to qualitatively match as close as possible those of Figure .", "Note the timescale mixing in {φ 21 ,φ 22 } \\lbrace \\phi _{21}, \\phi _{22} \\rbrace compared to the corresponding ζ=0.995 \\zeta = 0.995 eigenfunctions, {φ 13 ,φ 14 } \\lbrace \\phi _{13}, \\phi _{14} \\rbrace , in Figure .Figure: Same as Figure but for the cone kernel with ζ=0.995 \\zeta = 0.995 .", "The displayed eigenfunctions are (a) annual periodic; (b) PDO; (c,d) annual and semiannual intermittent patterns associated with the PDO; (e) ENSO; (f) NPGO; (g) annual intermittent patterns associated with ENSO; (h) annual intermittent patterns associated with the NPGO.Figure: Snapshots of spatiotemporal patterns of SST anomalies (in K) for September of simulation year 207 obtained from the eigenfunctions in Figures and .", "The displayed patterns are the (a) annual periodic; (b) PDO; (c,d) annual and semiannual intermittent patterns associated with the PDO; (e) ENSO; (f) NPGO; (g) annual intermittent patterns associated with ENSO; (h) annual intermittent patterns associated with the NPGO.", "Note the higher amplitude in panels (f–h) for the ζ=0.995 \\zeta = 0.995 cone kernel compared to the ζ=0 \\zeta =0 cases.", "See Movie 1 for the dynamic evolution of these patterns and the raw data.Remark 5.1 In this application, the variations in the phase space velocity velocity norm $ \\Vert \\xi _i \\Vert $ are of order 10% (cf.", "section ).", "As a result, the influence of the local scaling by $ \\Vert \\xi _i \\Vert \\Vert \\xi _j \\Vert $ in this case is rather weak, and correspondingly the eigenfunctions obtained via the $ \\zeta = 0 $ cone kernel and the isotropic Gaussian kernel (not shown here) do not differ significantly.", "Here, the main difference between the isotropic and cone kernels is due to the directionality of the dynamical flow, which is influenced strongly by the annual cycle." ], [ "Spatiotemporal patterns extracted by diffusion eigenfunctions", "In general, the eigenfunctions can be grouped into three families according to the timescales present in the $ \\tilde{\\phi }_i $ time series, which we refer to as periodic, low-frequency, and intermittent [26].", "Figures REF and REF display examples from each family.", "The periodic eigenfunctions come in doubly-degenerate pairs, and are dominated by a single frequency component which is an integer multiple of the annual cycle.", "Due to the presence of strong seasonality, one would expect the data manifold to have the structure of a circle along one of its dimensions—the presence of the periodic eigenfunctions is consistent with this picture.", "The low-frequency family is characterized by red-noise type frequency spectra, featuring significant power at interannual to decadal timescales.", "These basis functions describe familiar low-frequency patterns of North Pacific SST variability, including the PDO, ENSO, and NPGO [Panels (b), (e), and (f) in Figure REF and Movie 1], which are also accessible through SSA algorithms or PCA of seasonally-detrended data.", "The intermittent eigenfunctions have the structure of amplitude-modulated wavetrains consisting of a periodic carrier signal with integer frequency multiples of the seasonal cycle, which is modulated by a low-frequency envelope.", "In certain cases (though not always) the low-frequency envelope is correlated strongly with a low-frequency eigenfunction.", "In Figure REF , for instance, the low-frequency envelopes of the intermittent eigenfunctions $ \\lbrace \\phi _8, \\phi _{10} \\rbrace $ , $ \\phi _{14} $ , and $ \\phi _{16} $ correlate strongly with the low-frequency eigenfunctions $ \\phi _7 $ , $ \\phi _{12} $ , and $ \\phi _{13} $ , respectively.", "Similarly to the periodic eigenfunctions, the intermittent eigenfunctions arise in near-degenerate pairs (see Figure REF ).", "The corresponding spatiotemporal patterns [Panels (c), (d), (g), and (h) in Figure REF and Movie 1] feature propagating structures in regions of the North Pacific with high variability, including the Kuroshio current, the Bering Sea, and the west coast of North America.", "These patterns carry approximately two orders of magnitude less variance of the raw data than the prominent low-frequency modes.", "As a result, they are not accessible to variance-greedy PCA-type algorithms.", "However, these eigenfunctions are crucial in explaining the lagged temporal correlation structure of the data in so-called anomaly reemergence mechanisms [1], [12], and have also been found to have high skill as external factors in regression models for SST variability [27]." ], [ "Influence of the angular term", "The periodic, low-frequency, and intermittent families arise in both of the $ \\zeta = 0 $ and 0.995 kernels.", "As is evident in Figures REF and REF , there exists a reasonably clear qualitative correspondence between the leading eigenfunctions in each case.", "Nevertheless, the two spectra differ in two important aspects.", "First, the number of eigenfunctions required to capture the prominent features described in section REF is significantly smaller in the $ \\zeta = 0.995 $ case than the $ \\zeta = 0 $ example.", "In particular, it takes 17 $ \\zeta = 0.995 $ eigenfunctions to describe the annual and semiannual cycles, the PDO, ENSO, and NPGO low-frequency modes, and the associated intermittent modes, versus 26 $ \\zeta = 0 $ eigenfunctions.", "The extra eigenfunctions contained in the $ \\zeta = 0 $ spectrum are generally associated with higher harmonics of the seasonal cycle, which appear to be of limited physical significance.", "Examples of these harmonics can be seen in the eigenvalue spectrum of Figure REF , where the annual and semiannual periodic eigenfunctions ($ \\phi _1 $ –$ \\phi _4 $ ) are followed by higher harmonics all the way to the Nyquist limit 5 y$^{-1} $ associated with the one-month sampling interval.", "Higher-frequency harmonics are also present in the intermittent eigenfunctions.", "The suppression of the high-frequency harmonics from the $ \\zeta = 0.995 $ spectrum is consistent with the observation made in section REF that large gradients of the eigenfunctions along the direction of the dynamical vector field $ v $ lead to large Dirichlet form in (REF ), and correspondingly large eigenvalues.", "Because a major component of $ v $ is due to the annual cycle, the higher-harmonics incur a large $ v( \\phi _i ) $ penalty, and thus are removed from the leading part of the spectrum.", "We expect this to be a generic feature in datasets with prominent quasi-periodic behavior.", "A further key difference between the $ \\zeta = 0 $ and $ \\zeta = 0.995 $ eigenfunctions pertains to timescale separation.", "In particular, as can be seen in Figure REF , the $ \\phi _{22} $ eigenfunction for $ \\zeta = 0 $ representing the NPGO exhibits a mixture of interannual and annual timescales.", "Likewise, $ \\phi _{21} $ has a mixed intermittent–low-frequency character.", "On the other hand, the corresponding $ \\zeta = 0.995 $ eigenfunctions in Figure REF ($ \\phi _{13} $ and $ \\phi _{14} $ , respectively) have a well-separated temporal character, with $ \\phi _{13} $ acting as a low-frequency envelope of $ \\phi _{16} $ .", "In consequence, the corresponding spatiotemporal patterns have larger amplitude and feature more prominent intermittent coherent structures; see, e.g., the portion of the dynamic evolution around simulation year 207 in Figure REF and Movie 1.", "Again, we attribute this behavior to the superior adaptation of the $ \\zeta = 0.995 $ eigenfunctions to the dynamical flow." ], [ "Conclusions", "In this work, we have developed a family of kernels for data analysis in dynamical systems, which incorporate empirical information about the dynamical flow in phase space.", "Compared to canonical isotropic Gaussian kernels, the kernels presented here assign higher affinity to pairs of data samples whose relative displacement vector lies on a small-angle cone with axis parallel to the dynamical vector field $ v $ on the data manifold.", "The latter is estimated through finite differences of time-ordered samples, i.e., without requiring prior knowledge of the equations of motion.", "Due to the presence of the angular term, we refer to the new kernels as cone kernels.", "Cone kernels also feature a scaling factor introduced heuristically in earlier work on so-called NLSA algorithms [28], [29], whose role is to locally decrease (increase) the rate of decay of the kernel in regions of phase space where the dynamical flow is fast (slow) in the sense of the norm $ \\Vert v \\Vert _g $ of the dynamical vector field in data space.", "Moreover, the strength of the angular dependence is controlled by a parameter $ \\zeta \\in [ 0, 1 ) $ , such that $ \\zeta = 0 $ and $ \\zeta \\rightarrow 1 $ correspond to zero or maximal influence of the angular term.", "Thus, cone kernels include the earlier NLSA kernels as the special case $ \\zeta = 0 $ .", "We evaluated the metric tensor $ h $ induced on the data manifold by cone kernels (Lemma ) using the asymptotic analysis framework of Berry [9], and also studied the associated diffusion operator $ _\\zeta $ .", "By virtue of the $ \\Vert v \\Vert _g $ -dependent scaling, the induced metric is invariant for all $ \\zeta $ under conformal transformations of the original ambient space metric $ g $ .", "Moreover, for $ \\zeta > 0 $ , $ h $ contracts local distances between points on the data manifold whose relative displacement is parallel to $ v $ , becoming degenerate as $ \\zeta $ approaches 1.", "In that regime, $ _\\zeta $ becomes along $ v $ [23], in the sense that the associated codifferential operator asymptotically annihilates all differential 1-forms $ w $ with the property $ w( v ) = 0 $ (Lemma REF ).", "Intuitively, one thinks of $ _\\zeta $ in the $ \\zeta \\rightarrow 1 $ limit as generating diffusions along the integral curves of the dynamical vector field.", "Because $ v $ and its integral curves do not depend on the ambient space metric, this feature is intrinsic to the dynamical system under study.", "More generally, as $ \\zeta \\rightarrow 1 $ , the action of $ _\\zeta $ on functions depends on the ambient space metric only through the ratio $ \\mu / \\Vert v \\Vert _g^m $ for an $ m $ -dimensional manifold.", "The latter is invariant under conformal transformations of $ g $ , as well as more general transformations.", "A further important property arising in the $ \\zeta \\rightarrow 1 $ limit is that the Dirichlet form associated with $ _\\zeta $ depends on the directional derivative of functions along $ v $ , as opposed to the canonical dependence on the gradient operator [see (REF )].", "This property has significant bearing on the structure of the corresponding diffusion eigenfunctions, which are useful in a wide range of dimension reduction, signal processing, and learning problems.", "In particular, the leading eigenfunctions are expected to be adapted to the dynamical system generating the data, in the sense of varying weakly along the integral curves of $ v $ .", "We discussed the utility of cone kernels in a suite of numerical experiments involving nonlinear flows on the 2-torus and North Pacific sea surface temperature (SST) data from a comprehensive climate model.", "In the torus experiments, we explicitly demonstrated the adaptivity of the diffusion eigenfunctions associated with $ \\zeta \\approx 1 $ cone kernels to the dynamics (Figures REF and REF ), as well as the robustness of those eigenfunctions to non-conformal deformations of the data (Figure REF ).", "In the North Pacific SST experiments, cone kernels were found to have superior feature extraction capabilities, in the sense of requiring fewer basis functions than their $ \\zeta = 0 $ counterparts to describe the salient coherent structures of North Pacific SST variability, while also providing better separation between the timescales associated with the annual solar forcing and low-frequency (interannual) variability of the ocean.", "We attribute this improvement of skill to the ability of $ \\zeta \\approx 1 $ cone kernels to take into account changes in the direction of the dynamical flow due to the annual cycle.", "This feature should be generic in datasets with prominent quasiperiodic behavior.", "There are several open questions generated by this work which lie outside the scope of the present paper.", "On the theory side, a more detailed understanding of the diffusion eigenfunctions obtained from cone kernels would be desirable; e.g., their properties as embedding coordinates [8], [33], [45].", "Furthermore, it would be useful to explore generalizations of the deterministic framework adopted here to stochastic dynamical systems.", "Such approaches might involve replacing the ambient-space distances and inner products with suitable statistical metrics [52], retaining the explicit dependence on aspects related to the drift of the system in phase space.", "Even in the deterministic dynamical system context, potential shortcomings of cone kernels may arise due to sensitivity to observational noise and/or poor performance of the directional term in high intrinsic dimensions.", "Such scenarios would warrant modification of the cone kernel formulation put forward here, but we expect the general approach of incorporating empirically accessible information about the dynamics in kernel design to remain fruitful." ], [ "Technical results", "This Appendix contains the proof of Lemmas , , and REF , as well as derivations of a number of results used in the main text.", "Whenever convenient, we use the shorthand notation $ \\partial _\\mu $ to represent partial differentiation $ \\partial / \\partial u^\\mu $ with respect to a manifold coordinate $ u^\\mu $ ." ], [ "Proof of Lemma ", "Let $ ( u^1, \\ldots , u^m ) $ be a coordinate system defined in a neighborhood of $ a_i \\in \\mathcal { M } $ , and $ U_1, \\ldots , U_m $ be the corresponding coordinate basis vectors of $ T_{a_i} \\mathcal { M } $ .", "Denote the coordinates of the curve $ \\Phi _t a_0 $ by $ u^\\mu ( t ) $ , so that $v \\vert _{a_i} = \\sum _{\\mu =1}^m v^\\mu U_\\mu , \\quad \\text{with} \\quad v^\\mu = \\left.", "\\frac{ d u^\\mu }{ d t } \\right\|_{t_i}.$ Moreover, fix a basis $ e_1, \\ldots , e_n $ of $ \\mathbb { R }^n $ so that $ X_i = F( a_i ) = \\sum _{\\nu =1}^n X^\\nu e_\\nu $ .", "The $ \\lbrace e_\\nu \\rbrace $ basis can be identified with a basis of $ T_{X_i} \\mathbb { R }^n $ via the canonical isomorphism $ \\mathbb { R }^n \\simeq T_{X_i} \\mathbb { R }^n $ .", "With this choice of bases, the components of the derivative map $ D F : T_{a_i} \\mathcal { M } \\mapsto T_{X_i} \\mathbb { R }^n $ become $ {DF^\\nu }_\\mu = \\partial X^\\nu / \\partial u^\\mu \\vert _{a_i}.", "$ We then compute $\\xi _i & = \\delta _p X_i = \\left.", "\\frac{ d X }{ d t } \\right\|_{t_i } \\, \\delta t + O( \\delta t^{p+1} ) = \\left.", "\\left( \\sum _{\\mu =1}^m \\frac{ \\partial X }{ \\partial u^\\mu } \\frac{ d u^\\mu }{ dt } \\right) \\right\|_{t_i} \\, \\delta t + O( \\delta t^{p+1} ) \\\\&= \\sum _{\\mu =1}^m \\sum _{\\nu =1}^n \\left.", "\\frac{ \\partial X^\\nu }{ \\partial u^\\mu } \\right\|_{a_i} v^\\mu e_\\nu \\, \\delta t + O( \\delta t^{p+1} ) = DF \\, v \\vert _{a_i} \\delta t+ O( \\delta t^{p+1} ),$ which gives (REF ).", "Equation (REF ) follows immediately: $\\Vert v \\Vert _{g,a_i}^2 = ( D F \\, v, D F \\, v ) \\vert _{a_i} = ( \\xi _i + O( \\delta t^{p+1} ), \\xi _i + O( \\delta t^{p+1} ) ) / \\delta t^2 \\\\\\Rightarrow \\Vert v_i \\Vert _g = \\Vert \\xi _i \\Vert / \\delta t+ O( \\delta t^{p} ).", "\\qquad $" ], [ "Derivatives of cone kernels", "Let $ \\lbrace e_1, \\ldots , e_n \\rbrace $ be a basis of $ \\mathbb { R }^n $ and $ e^{1}, \\ldots , e^{n} $ its dual basis with $ e^{\\mu }( e_\\nu ) = {\\delta ^\\mu }_\\nu $ .", "Let also $ c_{\\mu \\nu } = ( e_\\mu , e_\\nu ) $ be the corresponding matrix elements of the data space inner product, giving the components of $ \\Xi $ in (REF ) via $\\Xi = \\sum _{\\nu =1}^n \\Xi _\\nu e^{\\nu }, \\quad \\Xi _\\nu = \\sum _{\\mu =1}^n c_{\\mu \\nu } \\xi ^\\mu , \\quad \\text{with} \\quad \\xi = \\sum _{\\mu =1}^n \\xi ^\\mu e_\\mu .$ To evaluate the derivatives of cone kernels in (REF ), it is convenient to write down $ K_{\\delta t, \\zeta }( a, \\exp _a u ) = \\exp ( - A( u ) B( u ) ) $ with $A( u ) = \\frac{ 1 }{ \\Vert \\xi \\Vert ^2 } \\left( \\Vert \\omega \\Vert ^2 - \\zeta \\frac{ ( \\xi , \\omega )^2 }{ \\Vert \\xi \\Vert ^2 } \\right), \\quad B( u ) = \\frac{ \\Vert \\xi \\Vert }{ \\Vert \\xi ^{\\prime } \\Vert } \\sqrt{ C( u ) }, \\\\C( u ) = \\frac{ \\Vert \\xi ^{\\prime } \\Vert ^2 \\Vert \\omega \\Vert ^2 - \\zeta ( \\xi ^{\\prime }, \\omega )^2 }{ \\Vert \\xi \\Vert ^2 \\Vert \\omega \\Vert ^2 - \\zeta ( \\xi , \\omega )^2 }, \\quad \\omega = F( a^{\\prime } ) - F( a ), \\quad \\xi ^{\\prime } = \\delta _p F( a^{\\prime } ).$ The function $ A( u ) $ vanishes at $ u = 0 $ , and its leading two derivatives with respect to $ u $ can be computed straightforwardly: $\\frac{ \\partial A }{ \\partial u^\\mu } &= \\frac{ 2 }{ \\Vert \\xi \\Vert ^2 } \\sum _{\\rho ,\\sigma =1}^n \\left( c_{\\rho \\sigma } \\omega ^\\rho \\frac{ \\partial \\omega ^\\sigma }{ \\partial u^\\mu } - \\zeta ( \\xi , \\omega ) c_{\\rho \\sigma } \\xi ^\\rho \\frac{ \\partial \\omega ^\\sigma }{ \\partial u^\\mu } \\right) \\\\\\frac{ \\partial ^2 A }{ \\partial u^\\mu \\, \\partial u^\\nu } &= \\frac{ 2 }{ \\Vert \\xi \\Vert ^2 } \\sum _{\\rho ,\\sigma =1}^n \\left( c_{\\rho \\sigma } \\frac{ \\partial \\omega ^\\rho }{ \\partial u^\\nu } \\frac{ \\partial \\omega ^\\sigma }{ \\partial \\omega ^\\mu } + c_{\\rho \\sigma } \\omega ^\\rho \\frac{ \\partial ^2 \\omega ^ \\sigma }{ \\partial u^\\mu \\, \\partial u^\\nu } \\right) \\\\& \\quad - \\frac{ \\zeta }{ \\Vert \\xi \\Vert ^2 } \\sum _{\\alpha ,\\beta ,\\rho ,\\sigma =1}^n \\left( c_{\\alpha \\beta } c_{\\rho \\sigma } \\xi ^\\alpha \\xi ^\\rho \\frac{ \\partial \\omega ^\\beta }{ \\partial u^\\nu } \\frac{ \\partial \\omega ^\\sigma }{ \\partial u^\\nu } - ( \\xi , \\omega ) c_{\\rho \\sigma } \\xi ^\\rho \\frac{ \\partial ^2\\omega ^\\sigma }{ \\partial u^\\mu \\, \\partial u^\\nu } \\right).$ Noting that $ \\partial \\omega ^\\nu / \\partial u^\\mu \\vert _{u=0} = {DF^\\nu }_\\mu $ , these expressions lead to $\\left.", "\\frac{ \\partial A }{ \\partial u^\\mu } \\right\|_{u=0} = 0, \\quad \\left.", "\\frac{ \\partial ^2 A }{ \\partial u^\\mu \\partial u^\\nu } \\right\|_{u=0} = \\frac{ 2 }{ \\Vert \\xi \\Vert ^2 } \\left( g_{\\mu \\nu } - \\zeta \\frac{ \\xi ^_\\mu \\xi ^_\\nu }{ \\Vert \\xi \\Vert ^2 } \\right),$ where $g_{\\mu \\nu } = \\sum _{\\rho ,\\sigma =1}^n c_{\\rho \\sigma } \\, {DF^\\rho }_\\mu \\, {DF^\\sigma }_\\nu $ are the components of the ambient spaced induced metric $ g $ in (REF ), and $ \\xi ^_\\mu $ are defined in (REF ).", "Therefore, assuming that $ B $ and its first two derivatives are all bounded at $ u = 0 $ , we have $\\begin{aligned}\\left.", "\\frac{ \\partial K_{\\delta t, \\zeta } }{ \\partial u^\\mu } \\right\|_{u=0} &= \\left.", "- K_{\\delta t, \\zeta }( a, a ) \\left( \\frac{ \\partial A }{ \\partial u^\\mu } B + A \\frac{ \\partial B }{ \\partial u^\\mu } \\right) \\right\|_{u=0} = 0, \\\\\\left.", "\\frac{ \\partial ^2 K_{\\delta t, \\zeta } }{ \\partial u^\\mu u^\\nu } \\right\|_{u=0} &= - \\left.", "\\left[ \\frac{ \\partial K_{\\delta t, \\zeta } }{ \\partial u^\\mu } \\left( \\frac{ \\partial A }{ \\partial u^\\mu } B + A \\frac{ \\partial B }{ \\partial u^\\mu } \\right) \\right] \\right\|_{u=0} \\\\& \\quad - \\left.", "K_{\\delta t, \\zeta }( a, a ) \\left( \\frac{ \\partial ^2 A }{ \\partial u^\\mu \\partial u^\\nu } B + \\frac{ \\partial A }{ \\partial u^\\mu } \\frac{ \\partial B }{ \\partial u^\\nu } + \\frac{ \\partial A }{ \\partial u^\\nu } \\frac{ \\partial B }{ \\partial u^\\mu } + A \\frac{ \\partial ^2 B }{ \\partial u^\\mu \\partial u^\\nu } \\right) \\right\|_{u=0} \\\\& \\quad = - \\left.", "\\frac{ \\partial ^2 A }{ \\partial u^\\mu \\partial u^\\nu } B \\right\|_{u=0}.\\end{aligned}$ Thus, in order to obtain (REF ), it suffices to determine the value $ B( 0 ) $ and check that the first two derivatives of $ B $ are bounded at $ u = 0 $ .", "Now, because $ \\xi $ is a smooth function of $ u $ and $ B( 0 ) = \\sqrt{ C( 0 ) } $ , any pathological behavior of $ B $ and its derivatives at the origin would be caused by $ C $ .", "To confirm that this is not the case, consider the non-degenerate type $ ( 0, 2 ) $ tensor $ Q( u ) $ with components $Q( u )_{\\mu \\nu } = ( \\Vert \\xi \\Vert ^2 \\delta _{\\mu \\nu } - \\zeta \\xi ^_\\mu \\xi ^_\\nu ) \\vert _u,$ where $ \\xi ^_\\mu $ is given by (REF ).", "Using $ Q( u ) $ , we expand $ C( u ) $ as follows: $C( u ) = \\frac{ \\sum _{\\mu ,\\nu =1}^m Q( u )_{\\mu \\nu } u^\\mu u^\\nu }{ \\sum _{\\rho ,\\sigma =1}^m Q( 0 )_{\\rho \\sigma } u^\\rho u^\\sigma } = 1 + \\sum _{\\alpha =1}^m u^\\alpha R^{(1)}_\\alpha + \\frac{ 1 }{ 2 } \\sum _{\\alpha ,\\beta =1}^m u^\\alpha u^\\beta R^{(2)}_{\\alpha \\beta } + O( \\Vert u \\Vert ^3 ), \\\\R^{(1)}_\\alpha = \\frac{ \\sum _{\\mu ,\\nu =1}^m u^\\mu u^\\nu \\partial _\\alpha Q_{\\mu \\nu }( 0 ) }{ \\sum _{\\rho ,\\sigma =1}^m u^\\rho u^\\sigma Q_{\\rho \\sigma }(0)}, \\quad R^{(2)}_{\\alpha \\beta } = \\frac{ \\sum _{\\mu ,\\nu =1}^m u^\\mu u^\\nu \\partial _\\alpha \\partial _\\beta Q(0)_{\\mu \\nu } }{ \\sum _{\\rho ,\\sigma =1}^m u^\\rho u^\\sigma Q(0)_{\\rho \\sigma }}.$ The tensor components $ R_\\alpha ^{(1)} $ and $ R^{(2)}_{\\alpha \\beta } $ are bounded in absolute value through the inequalities $\\vert R_\\alpha ^{(1)} \\vert \\le \\vert \\Lambda _\\text{max}^{(1)} / \\Lambda _\\text{min}^{(1)} \\vert , \\quad \\vert R_{\\alpha \\beta }^{(2)} \\vert \\le \\vert \\Lambda _\\text{max}^{(2)} / \\Lambda _\\text{min}^{(2)} \\vert ,$ where $ \\Lambda _\\text{max}^{(i)} $ ($ \\Lambda _\\text{min}^{(i)} $ ) are the largest (smallest) generalized eigenvalues $\\partial _\\alpha Q(0)_{\\mu \\nu } y = \\Lambda ^{(1)} Q(0)_{\\mu \\nu } y, \\quad \\partial _\\alpha \\partial _\\beta Q(0)_{\\mu \\nu } y = \\Lambda ^{(2)} Q(0)_{\\mu \\nu } y$ in absolute value.", "Thus, $ C( 0 ) = B( 0 ) = 1 $ , and the leading two derivatives of $ C $ and $ B $ are all bounded, leading through (REF ) to (REF )." ], [ "Proof of Lemma ", "We prove this Lemma following the analysis of Berry [9] connecting the induced metric to the Hessian of the kernel.", "First, fix a basis $ U_1, \\ldots , U_m $ of $ T_a \\mathcal { M } $ , and consider the corresponding exponential coordinates $ u^1, \\ldots , u^m $ with respect to the ambient-space induced metric $ g $ in (REF ).", "For small-enough sampling interval $ \\delta t $ , the $ m $ -dimensional Euclidean ball $ B_{\\delta t } $ of radius $ \\Vert v \\Vert _{g,a} \\, \\delta t $ centered at the origin maps diffeomorphically to a neighborhood $ \\mathcal { B }_{\\delta t} = \\exp _a B_{\\delta t } \\subset \\mathcal { M } $ of $ a $ .", "In these coordinates, the volume element $ d \\mu $ associated with $ g $ becomes $ \\mu \\vert _{ \\exp _a a } = \\det D \\exp _a \\vert _u \\, dU^{1} \\wedge \\cdots \\wedge dU^{m} $ , where $ dU^{\\mu } $ are the dual basis vectors to $ U_\\mu $ , and $ D \\exp _a \\vert _u : T_u T_a \\mathcal { M } \\mapsto T_a \\mathcal { M } $ is the derivative of the exponential map, $D \\exp _a \\vert _u( u^{\\prime } ) = \\lim _{\\tau \\rightarrow 0} ( \\exp _a( \\tau u^{\\prime } + u ) - \\exp _a( u ) ) / \\tau .$ Moreover, because the cone kernel $ K_{\\delta t, \\zeta }( a, \\exp _a u ) $ in (REF ) decays exponentially away from $ u = 0 $ for $ \\zeta \\in [ 0, 1 ) $ , it is possible to restrict the integral over $ \\mathcal { M } $ in (REF ) to an integral over $ B_{\\delta t } $ , incurring an exponentially small error as $ \\delta t \\rightarrow 0 $ (e.g., Lemma 1 in [6]).", "For our purposes, it suffices to consider terms up to order $ \\delta t^{3/2} $ in the asymptotics [15], [9], i.e., $H_{\\delta t, \\zeta }f( a ) = \\frac{ 1 }{ \\delta t^m } \\int _{B_{\\delta t}} K_{\\delta t, \\zeta } ( a, \\exp _a u ) \\tilde{f}( u ) \\det D \\exp _a \\vert _u \\, du^1 \\cdots du^m + O( \\delta t^{3/2} ),$ where $ \\tilde{f}( u ) = f( \\exp _a u ) $ .", "Taylor-expanding this expression about $ u = 0 $ using (REF ) and (REF ) with $ p \\ge 4 $ , we obtain $H_{\\delta t, \\zeta }f( a ) = \\frac{ 1 }{ \\delta t^m } \\int _{B_{\\delta t}} \\left( 1 - \\frac{ h( u, u ) }{ \\delta t^2 } \\right) \\det D \\exp _a \\vert _u \\, du^1 \\cdots du^m + O( \\delta t^{3/2} ),$ where $ h $ is the Riemannian metric specified in (REF ).", "Note that $ h = \\sum _{\\mu ,\\nu =1}^m h_{\\mu \\nu } U^{\\mu } U^{\\nu } $ , where $h_{\\mu \\nu } = \\frac{ 1 }{ \\Vert v \\Vert _g^2 } \\left( g_{\\mu \\nu } - \\zeta \\frac{ v^_\\mu v^_\\nu }{ \\Vert v \\Vert _g^2 } \\right), \\quad v^_\\mu = \\sum _{\\nu =1}^m g_{\\mu \\nu } v^\\mu ,$ and $ g_{\\mu \\nu } $ are the components of $ g $ in (REF ).", "Next, let $ \\widehat{\\exp }_a $ and $ \\hat{u}^\\mu $ be the exponential map and the corresponding normal coordinates associated with $ h $ , respectively.", "For small-enough $ \\delta t $ , there exists $ \\hat{B}_{\\delta t} \\subset \\mathbb { R }^m $ such that $ \\widehat{ \\exp }_a \\hat{B}_{\\delta t} = \\mathcal { B }_{\\delta t} $ and $du^1 \\cdots du^m = \\det D \\exp _a^{-1} \\widehat{\\exp }_a \\vert _{\\hat{u}} \\, d\\hat{u}^1 \\cdots d\\hat{u}^m.$ Equation (REF ) can then be expressed as $H_{\\delta t, \\zeta } f( a ) = \\frac{ 1 }{ \\delta t^m} \\int _{\\hat{B}_{\\delta t}} \\left( 1 - \\frac{ \\Vert \\hat{u} \\Vert ^2 }{ \\delta t^2 } \\right) \\, \\hat{f}( \\hat{u} ) \\det D \\widehat{\\exp }_a \\vert _{\\hat{u}} \\, d\\hat{u}^1 \\cdots d\\hat{u}^m + O( \\delta t^{3/2}),$ with $ \\Vert \\hat{u} \\Vert ^2 = \\sum _{\\mu =1}^m (\\hat{u}^\\mu )^2 $ and $ \\hat{f}( \\hat{u} ) = f( \\widehat{\\exp }_a \\hat{u} ) $ .", "Following similar arguments as those used to derive (REF ), it is possible to convert (REF ) to an isotropic Gaussian integral over $ \\hat{B}_{\\delta t } $ , i.e., $H_{\\delta t, \\zeta } f( a ) = \\frac{ 1 }{ \\delta t^m } \\int _{\\hat{B}_{\\delta t} } e^{-\\Vert \\hat{u} \\Vert ^2 / \\delta t^2 } \\hat{f}( \\hat{u} ) \\det D \\widehat{\\exp }_a \\vert _{\\hat{u} } \\, d\\hat{u}^1 \\cdots d\\hat{u}^m + O( \\delta t^3 ).$ It then follows from Lemma 8 of [15] (or Lemma 9 of [6]) that $\\nonumber H_{\\delta t, \\zeta } f( a ) &= C_0 f( a ) + \\hat{C}_2 \\, _{\\mathbb { R }^m} \\left( \\hat{f}( 0 ) \\det D \\widehat{\\exp }_a \\vert _{\\hat{u} = 0} \\right) \\, \\delta t^2 + O( \\delta t^3 ) \\\\ &= C_0 ( f( a ) + ( C_2 \\kappa ( a ) f( a ) + _{\\zeta } f( a ) ) \\, \\delta t^2 + O( \\delta t^3 ).$ Here, $ C_0 $ , $ C_2 $ , and $ \\hat{C}_2 $ are constants, $ _{\\mathbb { R}^m} $ the canonical Laplacian in $ \\mathbb { R }^m $ with $ _{\\mathbb { R }^m} \\hat{f} = \\sum _{\\mu =1}^m \\partial ^2 \\hat{f}/ \\partial {u^\\mu }^2 $ , and $ \\kappa ( a ) $ a function proportional to the scalar curvature of the $ h $ metric at $ a $ .", "Inserting (REF ) into (REF ), we then have that asymptotically as $ \\delta t \\rightarrow 0 $ $ \\mathcal { L }_{\\delta t, \\zeta } f( a )= C _{\\zeta } f( a ) + O( \\delta t ) $ for some constant $ C $ (Theorem 2.4.1 in [9]), which implies the result in (REF ).", "To verify (REF ), it suffices to compute the determinants of the matrices formed by the $ g_{\\mu \\nu } $ and $ h_{\\mu \\nu } $ components and the vector field norm $ \\Vert v \\Vert _g $ in any convenient coordinate system.", "In particular, (REF ) follows immediately by evaluating the expressions in (REF ) and (REF ) for $ \\Vert v \\Vert _g $ and $ \\det h $ in terms of exponential normal coordinates for $ g $ at $ u = 0 $ .", "$ \\qquad $" ], [ "Proof of Lemma ", "We work in Riemannian normal coordinates $ u^1, \\ldots , u^m $ associated with an orthonormal basis $ U_1, \\ldots , U_m $ of $ T_a \\mathcal { M } $ such that $ U_1 $ is aligned with the dynamical vector field $ v $ at $ a \\in \\mathcal { M } $ .", "Thus, in this coordinate system we have $\\begin{aligned}v^1 &= v^1( 0 ) + \\sum _{\\mu =1}^m \\partial _\\mu v^1( 0 ) u^\\mu + O( \\vert u \\Vert ^2 ), \\\\v^\\nu &= \\sum _{\\mu =1}^m \\partial _\\mu v^\\nu ( 0 ) u^\\mu + O( \\Vert u \\Vert ^2 ), \\quad \\nu > 1, \\\\g_{\\mu \\nu } &= \\delta _{\\mu \\nu } + O( \\Vert u \\Vert ^2 ).\\end{aligned}$ Let $ w = \\sum _{\\mu =1}^m w_\\mu U^{\\mu } $ be a $ C^1 $ 1-form field with components $ w_\\mu $ in the dual basis $ U^{\\mu } $ to $ U_\\mu $ .", "Our objective is to evaluate $ \\delta _\\zeta w $ from (REF ) at $ a $ using the standard expression for the codifferential in a local coordinate system (e.g., [46]), $\\nonumber \\delta _\\zeta w &= - \\left.", "\\frac{ 1 }{ \\sqrt{\\det h }} \\sum _{\\mu ,\\nu =1}^m \\partial _\\mu ( h^{-1,\\mu \\nu } \\sqrt{ \\det h } w_\\nu ) \\right\|_{u=0} \\\\&= - \\sum _{\\mu ,\\nu =1}^m \\left.", "\\left( h^{-1,\\mu \\nu } \\partial _\\mu w_\\nu + w_\\nu \\partial _\\mu h^{-1,\\mu \\nu } + \\frac{ h^{-1,\\mu \\nu } w_\\nu }{ 2 \\det h } \\partial _\\mu \\det h \\right) \\right\|_{u=0},$ and then collect the dominant terms in $ ( 1 - \\zeta ) $ .", "First, consider the dual vector field $ v^ = g( v, \\cdot ) = \\sum _{\\mu =1}^m v^_\\mu U^{\\mu } $ defined in (REF ) and its norm $ \\Vert v \\Vert _g $ with respect to the data space metric $ g $ .", "It follows from (REF ) that $\\nonumber v^_1 &= v^1( 0 ) + \\sum _{\\mu =1}^m \\partial _\\mu v^1( 0 ) u^\\mu + O( \\Vert u \\Vert ^2 ), \\\\\\nonumber v^_\\nu &= \\sum _{\\mu =1}^m \\partial _\\mu v^\\nu ( 0 ) u^\\mu + O( \\Vert u \\Vert ^2 ), \\quad \\nu > 1 \\\\ \\Vert v \\Vert _g &= v^1( 0 ) + \\sum _{\\mu =1}^m \\partial _\\mu v^1( 0 ) u^\\mu + O( \\Vert u \\Vert ^2 ).$ Moreover, $(v^_1)^2/\\Vert v \\Vert _g^2 &= 1 + O( \\Vert u \\Vert ^2 ), \\\\v^_1 v^_\\mu / \\Vert v \\Vert _g^2 &= \\frac{ 1 }{ v^1( 0 ) } \\sum _{\\rho =1}^m \\partial _\\rho v^\\mu ( 0 ) u^\\nu + O( \\Vert u \\Vert ^2 ), \\quad \\mu > 1, \\\\v^_\\mu v^*_\\nu / \\Vert v \\Vert _g^2 &= O( \\Vert u \\Vert ^2 ), \\quad \\mu ,\\nu >1.$ With these results, the metric components $ h_{\\mu \\nu } $ in (REF ) can be expressed as $h_{\\mu \\nu } = \\bar{h}_{\\mu \\nu } + h^{\\prime }_{\\mu \\nu } + O( \\Vert u \\Vert ^2 ),$ with $\\bar{h}_{11} = \\frac{ 1 - \\zeta }{ ( v^1(0) )^2 }, \\quad h^{\\prime }_{11} = -\\frac{ 2( 1 - \\zeta ) }{ ( v^1(0 )^3 ) } \\sum _{\\rho =1}^\\nu \\partial _\\rho v^1( 0 ) u^\\rho , \\\\\\bar{h}_{1\\nu } = 0, \\quad h^{\\prime }_{1\\nu } = - \\frac{ \\zeta }{ (v^1(0))^3 } \\sum _{\\rho =1}^m \\partial _\\rho v^\\nu ( 0 ) u^\\rho , \\quad \\nu > 1, \\\\\\bar{h}_{\\mu \\nu } = \\frac{ \\delta _{\\mu \\nu } }{ ( v^1( 0 ) )^2 }, \\quad h^{\\prime }_{\\mu \\nu } = - \\frac{ 2 \\delta _{\\mu \\nu } }{ ( v^1( 0 ) )^3 } \\sum _{\\rho =1}^m \\partial _\\rho v^1( 0 ) u^\\rho , \\quad \\mu ,\\nu >1.$ Next, consider the inverse metric $ h^{-1,\\mu \\nu } $ .", "Asymptotically, we have $h^{-1,\\mu \\nu } = \\bar{h}^{-1, \\mu \\nu } - h^{\\prime \\mu \\nu } + O( \\Vert u \\Vert ^2 ), \\quad \\text{with} \\quad h^{\\prime \\mu \\nu } = \\sum _{\\rho ,\\sigma =1}^m \\bar{h}^{-1,\\mu \\rho } h^{\\prime }_{\\rho \\sigma } \\bar{h}^{-1,\\sigma \\nu }.$ The tensor components appearing in (REF ) are $\\bar{h}^{-1,11} = \\frac{ (v^1( 0 ))^2 }{ 1 - \\zeta }, \\quad h^{\\prime 1 1} = - \\frac{ 2 v^1( 0 ) }{ 1 - \\zeta } \\sum _{\\rho =1}^m \\partial _\\rho v^1( 0 ) u^\\rho , \\\\\\bar{h}^{-1,1\\nu } = 0, \\quad h^{\\prime , 1\\nu } = - \\frac{ \\zeta v^1( 0 ) }{ 1 - \\zeta } \\sum _{\\rho =1}^m \\partial _\\rho v^\\nu ( 0 ) u^ \\rho , \\quad \\nu > 1, \\\\\\bar{h}^{-1,\\mu \\nu } = ( v^1( 0 ) )^2 \\delta _{\\mu \\nu }, \\quad h^{\\prime \\mu \\nu } = - 2 v^1( 0 ) \\sum _{\\rho =1}^m \\partial _\\rho v^1( 0 ) u^\\rho , \\quad \\mu ,\\nu > 1.$ We therefore obtain $- \\left.", "\\sum _{\\mu ,\\nu =1}^m h^{-1,\\mu \\nu } \\partial _\\mu w_\\nu \\right\|_{u=0} = - \\sum _{\\mu ,\\nu =1}^m \\bar{h}^{-1,\\mu \\nu } \\partial _\\mu w_\\nu ( 0 ) = - \\frac{ ( v^1( 0 ) )^2 }{ 1 - \\zeta } \\partial _1 w_1( 0 ) - ( v^1( 0 ) )^2 \\sum _{\\nu =1}^m \\partial _1 w_\\nu ( 0 )$ and $\\nonumber - \\left.", "\\sum _{\\mu ,\\nu =1}^m \\partial _\\mu h^{-1,\\mu \\nu } w_\\nu \\right\|_{u=0} &= \\sum _{\\mu ,\\nu =1}^m \\partial _\\mu h^{\\prime \\mu \\nu }( 0 ) w_\\nu ( 0 ) \\\\\\nonumber &= - \\frac{ 2 v^1( 0 ) }{ 1 - \\zeta } \\partial _1 v^1( 0 ) w_1( 0 ) - \\frac{ \\zeta v^1( 0 ) }{ 1 - \\zeta } \\sum _{\\nu =2}^m \\left( \\partial _1 v^\\nu ( 0 ) w_\\nu ( 0 ) + \\partial _\\nu v^\\nu ( 0 ) w_1( 0 ) \\right) \\\\ & \\quad - 2 v^1( 0 ) \\sum _{\\nu =2}^m \\partial _\\nu v^1( 0 ) w_\\nu ( 0 ).$ What remains is to compute the contribution to the codifferential from the derivatives of $ \\det h $ .", "To carry this calculation, we begin from the standard formula for the determinant, $\\det h = \\sum _{\\alpha _1, \\alpha _2, \\ldots , \\alpha _m=1}^m \\epsilon _{\\alpha _1 \\alpha _2 \\cdots \\alpha _m} h_{1\\alpha _1} h_{2\\alpha _2}\\cdots h_{m\\alpha _m},$ where $ \\epsilon _{\\alpha _1 \\alpha _2 \\cdots \\alpha _m} $ is equal to 1 $(-1) $ if $ ( \\alpha _1, \\ldots , \\alpha _m ) $ is an even (odd) permutation of $ ( 1, \\ldots , m ) $ , and vanishes if the $ \\alpha _i $ are not distinct.", "Because the $ O( \\Vert u \\Vert ) $ off-diagonal components $ h_{\\mu \\nu } $ occur for $ \\mu = 1 $ or $ \\nu = 1 $ only, the $ h_{i \\alpha _i} $ factors in (REF ) are $ O( \\Vert u \\Vert ) $ only for $ \\alpha _i \\in \\lbrace 1, \\alpha _i \\rbrace $ .", "Therefore, $\\det h &= \\sum _{\\alpha _1=1}^m \\sum _{\\alpha _2 \\in \\lbrace 1, 2 \\rbrace } \\sum _{\\alpha _3 \\in \\lbrace 1, 3 \\rbrace } \\cdots \\sum _{\\alpha _m \\in \\lbrace 1, m \\rbrace } \\epsilon _{\\alpha _1 \\alpha _2 \\cdots \\alpha _m } h_{1 \\alpha _1} h_{2\\alpha _2} \\cdots h_{m\\alpha _m} + O( \\Vert u \\Vert ^2 ) \\\\& = \\sum _{\\alpha _1=1}^m \\sum _{\\alpha _3 \\in \\lbrace 1, 3 \\rbrace } \\cdots \\sum _{\\alpha _m \\in \\lbrace 1, m \\rbrace }( \\epsilon _{\\alpha _1 1 \\alpha _3 \\cdots \\alpha _m} h_{1 \\alpha _1 } h_{ 21 } h_{3 \\alpha _3} \\cdots h_{m \\alpha _m} \\\\& \\qquad + \\epsilon _{\\alpha _1 2 \\alpha _3 \\cdots \\alpha _m } h_{1\\alpha _1} h_{22} h_{3 \\alpha _3 } \\cdots h_{m \\alpha _m} ) + O( \\Vert u \\Vert ^2 ) \\\\& = \\sum _{\\alpha _1=1}^m \\epsilon _{\\alpha _1 2 3 \\cdots m} h_{1\\alpha _1} h_{21} h_{33} \\cdots h_{mm} \\\\& \\qquad + \\sum _{\\alpha _1=1}^m \\sum _{\\alpha _3 \\in \\lbrace 1, 3 \\rbrace } \\cdots \\sum _{\\alpha _m \\in \\lbrace 1, m \\rbrace } \\epsilon _{\\alpha _1 2 \\alpha _3 \\cdots \\alpha _m} h_{1\\alpha _1} h_{22} h_{3\\alpha _3} \\cdots h_{m\\alpha m} + O( \\Vert u \\Vert ^2 )\\\\ & = - h_{12} h_{21} h_{33} \\cdots h_{mm} + \\sum _{\\alpha _1=1}^m \\sum _{\\alpha _3 \\in \\lbrace 1, 3 \\rbrace } \\cdots \\sum _{\\alpha _m \\in \\lbrace 1, m \\rbrace } \\epsilon _{\\alpha _1 2 \\alpha _3 \\cdots \\alpha _m} h_{1\\alpha _1} h_{22} h_{3\\alpha _3} \\cdots h_{m\\alpha _m} \\\\& \\qquad + O( \\Vert u \\Vert ^2 ).$ Repeating this decomposition $ m -2 $ times, we obtain $\\det h = h_{11} h_{22} \\cdots h_{mm} + ( - h_{12} h_{21} h_{33} \\cdots h_{mm} + h_{13} h_{22} h_{31} h_{44} \\cdots h_{mm} + \\cdots ) + O( \\Vert u \\Vert ^2 ).$ By virtue of (REF ), the first term in the right-hand side of (REF ) is $ O( \\Vert u \\Vert ) $ , whereas each of the terms in the parentheses is $ O( \\Vert u \\Vert ^2 ) $ .", "Specifically, $\\det h = \\frac{ 1 - \\zeta }{ ( v^1( 0 ) )^{2m} } \\left( 1 - \\frac{ 2 }{ v^1( 0 ) }\\sum _{\\rho =1}^m \\partial _\\rho v^1( 0 ) u^\\rho \\right)^m + O( \\Vert u \\Vert ^2 ),$ which implies that $\\left.", "\\frac{ \\partial _\\mu \\det h }{ \\det h } \\right\|_{u=0} = - \\frac{ 2 m }{ v^1( 0 ) } \\partial _\\mu v^1( 0 )$ and $- \\left.", "\\frac{ h^{-1,\\mu \\nu } w_\\nu }{ 2 \\det h } \\partial _\\mu \\det h \\right\|_{u=0} = \\frac{ m v^1( 0 ) }{ 1 - \\zeta } \\partial _1 v^1( 0 ) w_1( 0 ) + m v^1( 0 ) \\sum _{\\nu =2}^m \\partial _\\nu v^1( 0 ) w_\\nu ( 0 ).$ Using (REF ), (REF ), and (REF ), it is possible to write down an expression for the codifferential $ \\delta _\\zeta w $ in the $ u^\\mu $ coordinate basis.", "For the purpose of proving Lemma REF , it suffices to consider the dominant, $ O( ( 1 - \\zeta )^{-1} ) $ , terms, namely $\\nonumber \\delta _\\zeta ( w ) &= - \\frac{ ( v^1( 0 ) )^2 }{ 1 - \\zeta } \\partial _1 w_1( 0 ) - \\frac{ v^1( 0 ) }{ 1 - \\zeta } \\sum _{\\nu =1}^m w_\\nu ( 0 ) \\partial _1 v^\\nu ( 0 ) - \\frac{ v^1( 0 ) }{ 1 - \\zeta } \\sum _{\\nu =1}^m \\partial _\\nu v^\\nu ( 0 ) w_1( 0 ) \\\\ &+ \\frac{ m v^1( 0 ) }{ 1 - \\zeta } \\partial _1 v^1( 0 ) w_1( 0 ) + O( ( 1 - \\zeta )^0 ).$ Noting the relations $w( v ) \\vert _{u=0} = \\sum _{\\mu =1}^m ( v^\\mu w_\\mu ) \\vert _{u=0} = w_1( 0 ) v^1( 0 ),\\\\\\begin{aligned}v( w( v ) ) \\vert _{u=0} &= \\sum _{\\mu ,\\nu =1}^m v^\\mu \\partial _\\mu ( w_\\nu v^\\nu ) \\vert _{u=0} = \\sum _{\\mu ,\\nu =1}^m ( v^\\mu v^\\nu \\partial _\\mu w_\\nu + v^\\mu w_\\nu \\partial _\\mu v^\\nu ) \\vert _{u=0} \\\\&= ( v^1( 0 ) )^2 \\partial _1 w_1( 0 ) + v^1( 0 ) \\sum _{\\nu =1}^m v^1( 0 ) w_\\nu ( 0 ) \\partial _1 v^\\nu ( 0 ),\\end{aligned}\\\\\\operatorname{div}_\\nu v \\vert _{u=0} = \\left.", "\\frac{ 1 }{ \\sqrt{ \\det h } } \\sum _{\\nu =1}^m \\partial _\\nu ( \\sqrt{ \\det h } v^\\mu ) \\right\|_{u=0} = \\sum _{\\nu =1}^m \\partial _\\nu v^\\nu ( 0 ) - m \\partial _1 v^1( 0 ), \\\\\\operatorname{div}_\\nu [ w( v ) v ] = w( v ) \\operatorname{div}_\\nu v + v( w( v ) ),$ the asymptotic expansion (REF ) can be put in the covariant (basis-independent) form (REF ), thus proving the Lemma.", "$ \\qquad $" ], [ "Acknowledgments", "The author would like to thank T. Berry, M. Bushuk, J. Harlim, and A. Majda for stimulating discussions.", "This work was supported by ONR DRI grant N00014-14-1-0150 and ONR MURI grant 25-74200-F7112." ] ]	1403.0361
[ [ "Sleep Analytics and Online Selective Anomaly Detection" ], [ "Abstract We introduce a new problem, the Online Selective Anomaly Detection (OSAD), to model a specific scenario emerging from research in sleep science.", "Scientists have segmented sleep into several stages and stage two is characterized by two patterns (or anomalies) in the EEG time series recorded on sleep subjects.", "These two patterns are sleep spindle (SS) and K-complex.", "The OSAD problem was introduced to design a residual system, where all anomalies (known and unknown) are detected but the system only triggers an alarm when non-SS anomalies appear.", "The solution of the OSAD problem required us to combine techniques from both machine learning and control theory.", "Experiments on data from real subjects attest to the effectiveness of our approach." ], [ "Introduction", "Research in human sleep condition has emerged as a rapidly growing area within medicine, biology and physics.", "A defining aspect of sleep research is the large amount of data that is generated in a typical sleep experiment.", "A sleep experiment consists of a human subject, in a state of sleep, whose neural activity is being recorded with Electroencephalography (EEG) [19], [4].", "A typical full night EEG time-series, recorded between 4-64 locations on the scalp, at 200 Hz, for eight hours, will generate approximately 300MB of data.", "A typical clinical study will have between ten and fifty subjects.", "Surprisingly vast majority of sleep clinics still use a manual process to analyze the recorded EEG time-series.", "Hence there is considerable interest in automating the analysis of EEG generated from sleep experiments.", "Figure: Sleep spindles (SS) along with K-Complexes (KC) are defining characteristics of stage 2 sleep.", "Both SS and KC will show up as residuals inan LDS system.", "The OSAD problem will lead to a new residual time-serieswhere SS will be automatically supressed but KC will remain unaffected.Due to relatively high frequency of SS, there are certain situationswhere sleep scientists only want to be alerted when a non-SS anomaly occursScientists have segmented sleep into several stages based on the responsiveness of the subject and other physiological features.", "Of particular important is what is termed as stage 2 (moderately deep sleep).", "This stage is characterized by two phenomenon that occur in the EEG time series.", "These are sleep spindles, which are transient bursts of neural activity with a characteristic frequency of 12–14 Hz, and K-Complexes, which are short, large-amplitude voltage spikes.", "Both phenomena are implicated in memory consolidation and learning, but the physiology and mechanisms by which they occur are not yet fully understood, see [4], [9], [19], [8].", "In order to study these phenomena, they anomalies must be first located and identified in the EEG data.", "This can be challenging because they occur for an extremely short duration and irregularly.", "For example, sleep spindles and K-Complexes typically last less than 1 s, and there are only on the order of 100 of these events over the course of an entire night.", "Identification of these events is further complicated by the presence of artifacts in the data, often caused by movement of the subject, but which can also occur due to electrical noise or loose electrodes connections.", "These artifacts must be ignored when attempting to identify sleep spindles and K-Complexes.", "Because the electric fields produced by the brain are quite weak (the induced electrical potential is on the order of 50 $\\mu $ V), the signals also contain a significant noise component.", "In this paper we introduce the Online Selective Anomaly Detection (OSAD) problem which captures a particular scenario in sleep research.", "As noted above, around 100 sleep spindles will occur during the course of a night.", "The number of K-Complexes is much fewer.", "For some experiments scientists are interested in identifying both sleep spindles and K-Complexes but only want to be notified with an alert when a non-spindle anomaly occurs (for example K-Complexes).", "The solution of the OSAD problem combines techniques form both data mining and control theory.", "Data Mining is used to model and infer the normal EEG pattern per subject.", "Experiments have shown that model parameters do not transfer accurately across to other subjects.", "In our case we will use a Linear Dynamical System (LDS) to model the EEG time series.", "Then based on frequency analysis, we infer the sleep spindle (SS) pattern and integrate the pattern as a disturbance into the LDS.", "The control theory part is used to design a new residual which supresses SS signals but faithfully represents other errors generated by the LDS model.", "Thus by selectively supressing SS pattern, the objectives of the OSAD problem are achieved.", "For example, consider Figure REF .", "The top frame shows a typical EEG time series with both the SS and KC highlighted.", "The middle frame shows a typical residual time series based on an LDS model.", "The bottom frame shows a new residual designed to solve the OSAD problem.", "Notice that the error due to the presence of SS is suppressed but the residual due to the appearance of KC remains unaffected.", "The main contributions of the paper are: We introduce the Online Selective Anomaly Detection(OSAD) to address the requirement of selectively reporting sleep anomalies based on specifications by domain experts.", "In order to solve OSAD, we combine techniques from data mining and control theory.", "In particular we will use a Linear Dynamic System (LDS) to model the underlying data generating process and use control theory techniques to design an appropriate residual system.", "The rest of the paper is as follows.", "In Section 2, we rigorously define the OSAD problem.", "In Section 3 we present our methodology to infer the parameters of the LDS and use control theory to design a new residual system.", "In Section 4, we apply our approach to real sleep data and evaluate our results.", "We overiew related work in Section 5 and conclude in Section 6 with a summary and potential ideas for future research." ], [ "Problem Definition", "In this section we present our problem statement for selective anomaly detection.", "The starting point is an observed time series of $N$ points $ y = \\lbrace y_{i}\\rbrace _{i=1}^{N}$ where each $y_{i} \\in \\mathbb {R}^m $ .", "Furthemore, we assume that the $ y$ measures the output of a system which is generated from a latent variable $x \\in \\mathbb {R}^{n}$ .", "The relationship between $x$ and $ y$ is governed by a standard Linear Dynamic system (LDS) model [25] which is specified as $\\begin{array}{rl}x(t+1)= &{\\bf A}x(t)\\\\y(t) = & {\\bf C}x(t)\\end{array}$ Here ${\\bf A}$ is an $n \\times n$ state matrix which governs the dynamics of the LDS while ${\\bf C}$ is an $m \\times n$ observation matrix.", "The modern convention is to represent the LDS as graphical model as shown in Figure REF .", "The state of the system, $x$ , evolves according to LDS beginning at time $t = 0$ , with value $x_0$ .", "The standard learning problem is as follows.", "Figure: A linear dynamic system is a model which defines a linear relationshipbetween the latent (or hidden) state of the model and observed outputs.", "The LDS parameters 𝐀{\\bf A} and 𝐂{\\bf C} need to be estimated from data.", "The LDS canalso be used to model the relationship between the latent and the observedresiduals (right figure).Problem 1 (Learning Problem) Given an observable time series $\\lbrace y_{i}\\rbrace _{i=1}^{N}$ and assuming that the observed $ y$ and the latent $x$ are governed by an LDS, infer ${\\bf A}$ and ${\\bf C}$ .", "The standard LDS inference problem has been extensively studied in both the machine learning and control theory literature.", "Several algorithms have been proposed including those based on gradient descent, Expectation Maximization, subspace identification and spectral approaches [26], [28], [16], [3].", "Several extensions of LDS to include non-linear relationships as well as to include stochastic disturbances have been proposed.", "However, for sleep analysis, the above LDS will suffice.", "For the sake of completeness, in the Appendix we will describe a simple but effective approach for inferring ${\\bf A}$ and ${\\bf C}$ based on a spectral method [3].", "The standard approach to detect outliers using an LDS is to use the inferred ${\\bf A}$ and ${\\bf C}$ matrices to compute the latent and observed error variables as: $\\begin{array}{rl}\\varepsilon (t) :=& x(t)-\\hat{x}(t) \\\\e(t):=& y(t)-\\hat{y}(t)\\end{array}$ where $\\hat{x}$ and $\\hat{y}$ are estimated using LDS.", "Then given a threshold parameter $\\delta $ , an anomaly is reported whenever, $ e(t) > \\delta $ .", "However, our objective is not to report all anomalies but suppress some known user-defined patterns or even known anomalous pattern.", "We now formalize the notion of pattern.", "Definition 1 A pattern ${\\bf P}$ is a user-defined matrix which operates in the latent space.", "In our context, we will design a specific matrix ${\\bf P}$ for a sleep spindle.", "The matrix ${\\bf P}$ is integrated into the LDS as $\\begin{array}{rl}x(t+1)= &{\\bf A}x(t)+ {\\bf P}\\zeta (t)\\\\y(t) = & {\\bf C}x(t)\\end{array}$ We are now ready to define the design part of the OSAD problem.", "Problem 2 (Design Problem) Given an LDS, a pattern ${\\bf P}$ in the latent space, design a residual $r(t)$ such that $r(t)=\\left\\lbrace \\begin{array}{ll}0 & \\mbox{ if }\\epsilon (t) = {\\bf P}\\zeta (t) \\\\{\\bf S}e(t) & \\mbox{ otherwise }\\end{array}\\right.$ Here ${\\bf S}$ is suitably defined linear transformation on $e(t)$ .", "Notice that the residual $r(t)$ depends both on the latent error $\\epsilon (t)$ and the observed error $e(t)$ .", "In practice, $r(t)$ will never be exactly zero when the pattern ${\\bf P}$ is active but will have small absolute values." ], [ "The OSAD Method", "In this section we propose a method based on statistical inference and control theory to provide a solution of the OSAD problem.", "Using the LDS, we first develop a Dynamic Residue Model (DRM).", "Then we will show how to adjust the DRM parameters in order to design a residual $r(t)$ which will satisfy the constraints of the problem, i.e.", "the selected anomalous pattern will be canceled (or projected out) in the generated residual space." ], [ "DRM Formulation", "Assume data is generated by an LDS.", "Any deviation of the state from its expected value can be captured by a structured error model.", "Intuitively, the discrepancy between the observed error $e(t)$ and latent error $\\varepsilon (t)$ is modeled by the same LDS (because of linearity): $\\begin{array}{rl}\\varepsilon (t+1)=& {\\bf A}\\varepsilon (t)+ {\\bf P} \\xi (t)\\\\e(t)=& {\\bf C}\\varepsilon (t)\\end{array}$ The above error model can be used to detect changes occurring in the latent space.", "We design a feedback loop (as shown in Figure REF ) to effect the output of the error model.", "In particular a function of the residual will be used to manipulate the changes in the error.", "The design objective will be to map the anomalies generated by the $\\bf {P}$ pattern into the null space of the new residual.", "The DRM based on this feedback design is developed as follows: Figure: Using parameter F a virtual input u(t)u(t) is generated to feed the error back to the latent space.", "The error e(t)e(t) is is then calibrated by W to generate a new residual space r(t)r(t).To design the feedback we define two transformation matrices ${\\bf W}$ and ${\\bf F}$ for error values to be weighted as: $\\begin{array}{rl}r(t):=& {\\bf W}e(t) \\\\u(t):=&{\\bf F}e(t)\\end{array}$ ${\\bf F}$ will be used as the feedback gain matrix and maps the error to the feedback vector $u(t)$ , and ${\\bf W}$ is the residual weighting matrix that generates the new residual $r(t)$ .", "Now feeding back $u(t)$ into the LDS (as shown in Figure REF ), with $u(t),$ the residual dynamic model will be: $\\begin{array}{rl}\\hat{x}(t + 1) =& {\\bf A}\\hat{x}(t) + u(t) \\\\=& {\\bf A}\\hat{x}(t) +{\\bf F}e(t)\\\\=& {\\bf A}\\hat{x}(t) +{\\bf F}(y(t) - \\hat{y}(t))\\\\=& {\\bf A}\\hat{x}(t) +{\\bf F}({\\bf C}x(t) - {\\bf C}\\hat{x}(t))\\\\=& {\\bf A}\\hat{x}(t) +{\\bf F}{\\bf C}x(t) - {\\bf F}{\\bf C}\\hat{x}(t)\\\\=& ({\\bf A}-{\\bf F}{\\bf C})\\hat{x}(t) + {\\bf F}{\\bf C}x(t)\\\\=& ({\\bf A}-{\\bf F}{\\bf C})\\hat{x}(t) + {\\bf F} y(t)\\\\\\end{array}$ Notice that since the residual is a linear transformation of the error, its rank (suppose $r(t) \\in \\mathbb {R}^p$ ) can not be larger than the observation dimension, i.e., $p\\le m$ .", "We are now able to define the dynamic of the latent error as: $\\begin{array}{rl}\\varepsilon (t+1) =&x(t+1)-\\hat{x}(t+1) \\\\=&{\\bf A}x(t)-({\\bf A}-{\\bf F}{\\bf C})\\hat{x}(t)- {\\bf F} y(t) \\\\=&{\\bf A}x(t)-{\\bf A}\\hat{x}(t)-{\\bf F}{\\bf C})\\hat{x}(t)+ {\\bf F}{\\bf C}x(t) \\\\=&({\\bf A}-{\\bf F}{\\bf C})(x(t)-\\hat{x}(t)) \\\\=&({\\bf A}-{\\bf F}{\\bf C})\\varepsilon (t)\\\\\\end{array}$ and the residue $r(t)$ is obtained as: $\\begin{array}{rl}r(t)=& {\\bf W}(y(t) - \\hat{y}(t))\\\\=& {\\bf W}({\\bf C}x(t) - {\\bf C}\\hat{x}(t))\\\\=& {\\bf W}{\\bf C}(x(t) - \\hat{x}(t))\\\\=& {\\bf WC} \\varepsilon (t)\\end{array}$ We therefore have the following dynamic model for the latent error: $\\begin{array}{rl}\\varepsilon (t+1) =&({\\bf A}-{\\bf F}{\\bf C})\\varepsilon (t)\\\\r(t) =& {\\bf WC} \\varepsilon (t)\\end{array}$ $Notice that the observed residue $ (t)$ is governed by state error $ (t)$ through matrix $WC$ while it evolves in time through $A-FC$.$ To simplify the notation, denote ${\\bf C}_f={\\bf WC}$ and ${\\bf A}_f={\\bf A}-{\\bf FC}$ .", "The DRM is then defined as: $\\begin{array}{rl}\\varepsilon (t+1) =&{\\bf A}_f \\varepsilon (t)\\\\r(t) =& {\\bf C}_f \\varepsilon (t)\\end{array}$ $The graphical diagram for this error model is shown in Figure~\\ref {fig:Feedback}.$" ], [ "OSAD Parameter Design", "In this section we address the problem of designing the ${\\bf F}$ and ${\\bf W}$ matrix with objective of making the DRM insensitive to anomalies generated by ${\\bf P}$ .", "The overarching design is shown in Figure REF and is related to the use of control theory for fault diagnosis [22], [23], [5].", "A typical LDS model will output the observed error $e(t)$ .", "However, the OSAD model has a feedback loop which takes ${\\bf W}$ and ${\\bf F}$ matrices as input and return a variable $u(t)$ which is fed back into the model.", "The observed error is also transformed by a ${\\bf W}$ matrix.", "The ${\\bf F}$ and the ${\\bf W}$ matrices satisfy the constraints which involve the ${\\bf A}$ , ${\\bf C}$ and the ${\\bf P}$ matrices.", "Since the model is time-dependent, we follow a standard approach and map the model into the frequency domain using a $\\mathcal {Z}$ -transform to design the ${\\bf W}$ and ${\\bf F}$ matrices.", "In the frequency domain, it will be easier to design matrices ${\\bf W}$ and ${\\bf F}$ such that ${\\bf WC}({\\bf A-FC}) = 0$ and ${\\bf WCP} =0$ .", "Definition 2 The Z-transform of a discrete-time sequence $x(k)$ is the series $X(z)$ defined as $X(z)= \\mathcal {Z}\\lbrace x(k)\\rbrace =\\sum _{0}^{\\infty }x(k)z^{-k}.$ Observation 1 Two important (and well known) properties of the Z-transform are linearity and time shifting: $ax(k)+by(t) \\overset{Z}{\\longleftrightarrow } aX(z)+ b Y(Z)$ $x(k+b) \\overset{Z}{\\longleftrightarrow } z^{b}X(z)$ Applying Z-transform $\\mathcal {Z}()$ to the DRM yields: $\\begin{array}{rl}z\\mathcal {E}(z) =&{\\bf A}_f\\mathcal {E}(z)+ {\\bf P} \\xi (z) \\\\\\mathcal {E}(z)=&(z{\\bf I}-{\\bf A}_f)^{-1}{\\bf P}\\xi (z) \\\\\\end{array}$ and: $\\begin{array}{rl}R(z) =& {\\bf C}_f \\mathcal {E}(z)\\\\=&[ {\\bf C}_f(z{\\bf I}-{\\bf A}_f)^{-1}{\\bf P}] \\xi (z)\\end{array}$ in which $\\xi (z)=\\mathcal {Z}(\\xi (t))$ , $\\vartheta =\\mathcal {Z}(\\vartheta (t))$ , $R(z)=\\mathcal {Z}(r(t))$ .", "The transfer gain between $\\xi $ and $R$ : $\\begin{array}{rl}{\\bf G}_{\\xi }(z) := & {\\bf C}_f(z{\\bf I}-{\\bf A}_f)^{-1}{\\bf P}\\end{array}$ Thus if ${\\bf G}_{\\xi }$ would be zero, the residual $R(z)$ is independent of the $\\xi (z)$ .", "In the other word, to make $R(z)$ independent of $\\xi (z)$ , one must null the space of $ {\\bf G}_{\\xi }(z) $ .", "Then whenever $\\mathcal {P}$ occurs it is transferred by a zero gain to the residual space.", "To find the null space ${\\bf G}_{\\xi }(z)=0$ , we expand it as: $\\begin{array}{rl}{\\bf G}_{\\xi }(z) = & z^{-1}{\\bf C}_f({\\bf I+A}_fz^{-1} + {\\bf A}_f^2z^{-2}+...){\\bf P}\\\\= & 0\\end{array}$ The sufficient conditions for ${\\bf G}_{\\xi }(z)$ to be nulled are ${\\bf C}_f{\\bf P} = 0$ and either ${\\bf C}_f{\\bf A}_f = 0$ or ${\\bf A}_f {\\bf P} = 0$ .", "Thus we have the following result.", "Theorem 1 For a DRM, a sufficient condition for ${\\bf G}_{\\xi }(z) = 0$ is ${\\bf C}_f{\\bf P} = 0 \\mbox{ and }\\lbrace {\\bf C}_f{\\bf A}_f = 0 \\mbox{ or } {\\bf A}_f {\\bf P} = 0\\rbrace $ Now as ${\\bf C}_f= {\\bf WC}$ , for ${\\bf C}_f{\\bf P} = 0$ it is sufficient that ${\\bf WC}$ be orthogonal to ${\\bf P}$ .", "Furthermore for ${\\bf C}_f{\\bf A}_f = 0$ , it is sufficient to design a matrix ${\\bf A}_f$ such that its left eigevectors corresponding to the zero eigenvalue are orthogonal to ${\\bf P}$ .", "Similarly, for ${\\bf A}_f {\\bf P} = 0$ , it is sufficient to design a matrix ${\\bf A}_f$ , such that the right eigenvectors corresponding to the zero eigenvalues are orthogonal to ${\\bf P}$ .", "See Appendix .", "Now, it design a system which operates in an online fashion we proceed as follows.", "From the definition of residue: $\\begin{array}{rl}r(t) =& {\\bf W}[y(t)-\\hat{y}(t)]\\end{array}$ Using the Z-transform, the computational form of the residual will be: $\\begin{array}{rl}R(z) =&[{\\bf W}-{\\bf C}_f(z{\\bf I-A}_f)^{-1}{\\bf F}]Y(z)\\end{array}$ $Since $ CfAf =0$:$$\\begin{array}{rl}{\\bf C}_f(z{\\bf I-A}_f)^{-1}{\\bf F}= &z^{-1}{\\bf C}_f\\end{array}$$Replacing this result to the above $ R(z)$ equation:$$\\begin{array}{rl}R(z) =&({\\bf W}-z^{-1}{\\bf C}_f{\\bf F})Y(z)\\end{array}$$Applying the inverse Z-transform, the equation will be:$$\\begin{array}{rl}r(t) =&\\begin{bmatrix} {\\bf W} \\quad -{\\bf C}_f{\\bf F} \\end{bmatrix} \\begin{bmatrix} y(t)\\\\ y(t-1) \\end{bmatrix}\\end{array}$ $This clearly says that the residual can be represented directly in terms of theobservations.", "This property is crucial to make the anomaly detection system operate in near real-time.$ Figure: The complete diagram of OSAD.", "Using parameters W and F the residue space r(t)r(t) is calibrated to cancel the impact of 𝐏ξ(t){\\bf P}\\xi (t)." ], [ "Eigenpair Assignment and the ${\\bf F}$ Matrix", "In this section we explain the eigenpair assignment problem and its solution which is used for designing the matrix ${\\bf F}$ .", "Recall from Theorem 1, that we require either ${\\bf C}_{f}{\\bf A}_{f} =0$ or ${\\bf A}_{f}{\\bf P} = 0$ .", "Problem 3 Given a set of scalars $\\lbrace \\lambda _i\\rbrace $ and a set of n-vectors $\\lbrace v_i\\rbrace $ (for $ i=1,2,...,n$ ), find a real matrix ${\\bf A}_o$ ($m \\times n$ ) such that the eigenvalues of ${\\bf A}_o$ are precisely those of the set of scalars $\\lbrace \\lambda _i\\rbrace $ with corresponding eigenvectors the set $\\lbrace v_i\\rbrace $ .", "Given the residue model transition matrix ${\\bf A}_f={\\bf A}-{\\bf FC}$ , the problem is to find a matrix ${\\bf F}$ such that this matrix has the eigenvalues $\\lbrace \\lambda _i\\rbrace $ corresponding to eigenvectors $\\lbrace v_i\\rbrace $ ,i.e.,: $( {\\bf A-FC} ) v_i = \\lambda _i v_i$ or: $\\begin{bmatrix}{\\bf A}-\\lambda _i{\\bf I}& &{\\bf C^{\\prime }} \\end{bmatrix} \\begin{bmatrix} v_i \\\\ -{\\bf F} v_i \\end{bmatrix} = 0$ Define $q_i:=-{\\bf F} v_i$ , then: $\\begin{bmatrix} {\\bf A}-\\lambda _i{\\bf I} & &{\\bf C^{\\prime }}\\end{bmatrix} \\begin{bmatrix} v_i \\\\ q_i \\end{bmatrix} = 0$ The implication of the above statement is of great importance: The vectors $\\begin{bmatrix} v_i & q_i \\end{bmatrix}^{\\prime }$ must be in the kernel space of $\\begin{bmatrix} {\\bf A}-\\lambda _i{\\bf I} & &{\\bf C^{\\prime }}\\end{bmatrix}$ , meaning, for $i=1,2,...,n$ : $\\begin{bmatrix} q_1 & q_2& ...& q_n \\end{bmatrix} = \\begin{bmatrix} -{\\bf F}v_1& -{\\bf F}v_2&...& -{\\bf F}v_n \\end{bmatrix}$ The matrix ${\\bf F}$ now can be obtained as: ${\\bf F}=-\\begin{bmatrix} q_1 & q_2& ...& q_n \\end{bmatrix} \\begin{bmatrix} v_1& v_2&...& v_n \\end{bmatrix}^{+}$ where '+' stands for pseudoinverse.", "The whole procedure is summarized in Algorithm REF .", "[!t] Find F such that the set $\\lbrace \\lambda _i, v_i\\rbrace $ be the eigenpairs of ${\\bf A}-{\\bf FC}$ [1] Input A,C, $\\lambda _{i} = 0$ $\\forall i$ and $v_{i} = P(:,i)$ .", "Output ${\\bf F}$ such that ${\\bf (A - FC)P = 0}$ .", "$i=1:n$ $\\phi _i = \\textit {null} \\begin{bmatrix} {\\bf A}-\\lambda _i{\\bf I} & {\\bf C^{\\prime }}\\end{bmatrix}$ Find an element $[v_{i} \\mbox{ }q_{i}]^{\\prime } \\in \\phi _i$ ${\\bf F}=-\\begin{pmatrix} q_1 & q_2& ...& q_n \\end{pmatrix} \\begin{pmatrix} v_1& v_2&...& v_n \\end{pmatrix}^{+}$" ], [ "Degrees of Freedom of ${\\bf P}$", "There is an an important constraint that the matrix ${\\bf P}$ must satisfy for the DRM approach to be valid solution of the OSAD problem.", "As the ${\\bf WCP} = 0$ , a necessary condition is that $\\mbox{rank}({\\bf P})\\le \\mbox{rank}({\\bf C})$ In the other word, the effective number of independent perturbations generated by the matrix ${\\bf P}$ is bounded by the effective number of independent measurements governed by the observation matrix ${\\bf C}$ , see [23].", "For example, if ${\\bf C}$ is the independent matrix on an LDS where the state vector has dimensionality $n$ , then the rank of the P matrix must be less than $(n-1)$ ." ], [ "Inferring the Matrix ${\\bf P}$", "The OSAD model is predicated on the existence of a ${\\bf P}$ matrix.", "This matrix can be provided by a domain expert or can sometimes be inferred from data.", "For example, in the case of sleep spindle, frequency analysis shows that sleep spindles occur in the interval twelve to fourteen Hz.", "The exact frequency can change from one subject to another.", "The signature for K-Complexes is more a function of the amplitude of the signal rather than the frequency.", "We now show how to construct a ${\\bf P}$ matrix from data.", "For example, suppose there exists a frequency/peridicity $\\mathcal {T}=f^{-1}$ in the EEG time series or: $\\begin{array}{rl}x(t+\\mathcal {T})&=x(t)\\end{array}$ Replace this in linear dynamics: $\\begin{array}{rl}x(t+1)&={\\bf A}x(t)\\\\&={\\bf A}x(t+\\mathcal {T})\\end{array}$ Applying z-transform: $\\begin{array}{rl}zX(z)&={\\bf A}z^\\mathcal {T}X(z)\\end{array}$ Using Tailor expansion we expand $z^\\mathcal {T}$ around $z=1$ : $z^\\mathcal {T}\\approx 1+ \\alpha + \\beta z + \\gamma z^2$ where $\\alpha = 0.5 \\mathcal {T}(\\mathcal {T}-3)$ , $\\beta = 0.5\\mathcal {T}(\\mathcal {T}-1) $ and $\\gamma = - \\mathcal {T}(\\mathcal {T}-2)$ .", "An approximation by this expansion will be: $\\begin{array}{rl}zX(z)&\\approx {\\bf A}X(z) + \\alpha {\\bf A}X(z)+ \\beta z {\\bf A}X(z) + \\gamma z^2 X(z)\\end{array}$ Returning to the time-domain, we obtain ${}{!", "}{\\begin{array}{rl}x(t+1) &\\approx {\\bf A}x(t) + \\alpha {\\bf A} x(t) + \\beta {\\bf A} x(t+1) + \\gamma {\\bf A} x(t+2)\\\\&\\approx {\\bf A}x(t) + [ \\alpha {\\bf A} \\quad \\beta {\\bf A} \\quad \\gamma {\\bf A}] [x(t)\\quad x(t+1)\\quad x(t+2)]^{\\prime }\\end{array}}$" ], [ "Summary Example", "To summarize, the solution of the OSAD problem requires the availability of the following matrices: Table: Parameters for learning and designWe will now give a concrete example.", "Assume we have an LDS system given as $\\begin{array}{rl}\\varepsilon (t+1)=& {\\bf A}\\varepsilon (t)+ {\\bf P} \\xi (t)\\\\e(t)=& {\\bf C}\\varepsilon (t)\\end{array}$ Assume have identified the ${\\bf A}$ and ${\\bf C}$ matrices as ${\\bf A}=\\begin{pmatrix}0.5 & 0.3 \\\\0.3 & 0.2\\end{pmatrix}\\mbox{ and }{\\bf C}=\\begin{pmatrix}1 & 0 \\\\0 & 1\\end{pmatrix}\\mbox{ and }{\\bf P}=\\begin{pmatrix}1 & 1 \\\\2 & 2\\end{pmatrix}$ Now, to form the OSAD model, we have to identify ${\\bf W}$ and ${\\bf F}$ such that: ${\\bf W}$ is in the null space of ${\\bf CP}$ and ${\\bf A - FC}$ has its left eigenvectors (corresponding to the ${\\bf 0}$ eigenvalue ), the rows of ${\\bf WC}$ .", "Since ${\\bf C}$ is the identity matrix, an example of ${\\bf W}$ is ${\\bf W} =\\begin{pmatrix}2 & - 1 \\\\2 & -1\\end{pmatrix}$ Similarly, an example of ${\\bf F}$ matrix is ${\\bf F} =\\begin{pmatrix}0.0 & 0.2 \\\\-0.7 & 0\\end{pmatrix}$ As mentioned, the residual matrix is given by $r(t) =\\begin{pmatrix}1.3 & -1.4 \\\\1.3 & -1.4\\end{pmatrix}\\begin{pmatrix}y(t) \\\\y(t-1)\\end{pmatrix}$" ], [ "Experimental Result", "We now report on the experiments that have been carried out to test the effective of the proposed OSAD solution on sleep data.", "Our particular focus will be determining if OSAD can recognize sleep spindle and K-Complex anomalies and selectively raise an alert for non-Spindle anomalies." ], [ "Sleep Data Set", "Our data set consists of EEG time series from four health controls (age 25-36) as described in [10].", "Recordings were made with an Alice-4 system (Respironics, Murraysville PA, USA) at the Woolcock Institute of Medical Research, at Sydney University, using 6 EEG channels with a sampling rate of 200 Hz, and electrodes positioned according to the International 10-20 system [19], [4], see Figure REF .", "In this study we only examine the Cz electrode.", "A notch filter at 50 Hz (as provided by the Alice-4 system) was used to remove mains voltage interference.", "No other hardware filters were used.", "Spindles and K-Complexes were labeled using another automation program and then manually evaluated.", "As previously noted, while data from only four subjects were used, a typical EEG session generates a large amount of personal data.", "Figure: The position of scalp electrodes for EEG experiment follows the International 10-20 system , ." ], [ "Inference of ${\\bf A}$ and {{formula:848c7a58-4e12-4f31-94d9-08beff806764}} Matrices", "Our first task is to learn the ${\\bf A}$ and ${\\bf C}$ matrices from the LDS for each subject.", "Others have reported, and our experiments confirm, that EEG of each subject tends to different and separate models need to learnt per subject.", "For each subject we took a sample of size 2000 (10 seconds) of EEG time series which did not contain either sleep spindle or K-Complex.", "We then formed a $2000 \\times 6$ data matrix, ${\\bf O}$ .", "The columns of the ${\\bf O}$ matrix are time series associated with the six channels of EEG.", "We used both subspace and spectral methods to infer the matrices ${\\bf A}$ and ${\\bf C}$ .", "Both these methods are based on SVD decomposition of the ${\\bf O}$ matrix and require as input the rank required of the inferred matrices.", "We evaluated the inferred matrices using RMSE and the results are shown in Figure REF and Figure REF .", "Both the subspace and spectral methods have similar performance and RMSE goes up significantly when the rank falls below five.", "We selected a rank six matrix (maximum possible rank) for both ${\\bf A}$ and ${\\bf C}$ .", "In terms of running time, the two methods are comparable as we have to carry out an SVD of a relatively small $6 \\times 6$ matrix.", "Figure: The RMSE error obtained from both methods are comparable.", "Notice the RMSE increases as the rank of LDS is reduced." ], [ "Detection of SS and K-Complex", "For each of the four subjects, statistics of the labeled sleep spindles and K-Complexes and those detected by the LDS are shown in Table REF .", "For LDS detection, we used a threshold derived from CUSUM which automatically adjusts for mean and standard deviation of the observed residual time series $e(t)$ .", "To specify a CUSUM threshold we applied the alpha and beta approach in [17] and we set the probabilities of a false positive and a false negative to $10^{-4}$ and the change detection parameter to 1 sigma, in all subjects.", "In all four subjects, the LDS residual slightly under predicts the number of spindles and K-Complexes.", "Since each labeled and predicted SS and K-Complex spans a time-interval, we have modified the definitions of precision and recall to account for the intervals.", "For a given subject, let $\\lbrace [a_{i},b_{i}]\\rbrace _{i=1}^{n}$ be the intervals of the labeled anomalies (spindles or K-Complex).", "Let $\\lbrace [a^{^{\\prime }}_{j},b^{^{\\prime }}_{j}]\\rbrace _{j=1}^{m}$ be the predicted spindles.", "Then ${\\tt precision} = \\frac{\\sum _{i=1}^{n}\\sum _{j=1}^{m}\|[a_{i},b_{i}] \\cap [a^{^{\\prime }}_{j},b^{^{\\prime }}_{j}]\|}{\\sum _{j=1}^{m}\|[a^{^{\\prime }}_{j},b^{^{\\prime }}_{j}]\|}$ and ${\\tt recall} = \\frac{\\sum _{i=1}^{n}\\sum _{j=1}^{m}\|[a_{i},b_{i}] \\cap [a^{^{\\prime }}_{j},b^{^{\\prime }}_{j}]\|}{\\sum _{j=1}^{m}\|[a_{j},b_{j}]\|}$ Here, $\|[a_{i},b_{i}]\|$ , is the number of points in the time interval $[a_{i},b_{i}]$ .", "With these definitions in place, Table REF and Table REF show the precision and recall SS and K-Complex across alls the subjects.", "In general both precision and recall are high across subjects, but precision is significantly more higher than recall.", "For SS, the recall varies more than precision ranging for 71.24% to 97.18%.", "Also notice that the length of detection of both SS and K-Complex is higher compared to their labeled lengths.", "Table: Summary statistics of results.", "LDS is quite accurate but tends to over-predict the numberof anomalies.Table: Summary statistics for spindles.", "LDS has higher precision than recall and total lengthof predicted interval is higher than the length of labeled intervals.Table: Summary statistics for K-Complex.", "Both precision and recall are high.", "Total lengthof predicted interval is higher than labeled intervals." ], [ "Evaluation across Subjects", "We now investigate the transfer properties of the inferred LDS across subjects.", "That is, we learn the ${\\bf A}$ an ${\\bf C}$ matrices on one subject and evaluate it against an another.", "We just focus on the anomaly.", "The recall and precision results are shown in Table REF and Table REF respectively.", "The diagonal of the table corresponds to the results in Table REF and Table REF .", "It is clear that there is a substantial reduction in accuracy and that indeed the EEG of subjects varies substantially.", "We have also computed the \"average\" ${\\bf A}$ and ${\\bf C}$ matrix and evaluated against all the four subjects.", "The results are shown in Table REF .", "While there is an improvement compared to results in Table REF and Table REF , the absolute performance is still quite low compared to the situation where the learning was customized per individual subject.", "Table: Recall across subjects.", "A substantial reduction in accuracy when model of onesubject is evaluated against the EEG of another.Table: Precision across the subjects.", "Again, a substantial reduction in accuracy whenmodel of one subjected is evaluated against another.Table: Recall and Precision on each subject evaluated against an averaged model.", "Again,a substantial reduction in accuracy compared to individual models." ], [ "Performance of Designed Residual", "In this section we evaluate whether the new residual $r(t)$ satisfies the design criterion.", "Recall, $r(t)$ was designed to suppress the signal whenever a sleep spindle (SS) appears and behave like the observed error ${\\bf e(t)}$ in otherwise.", "Figure REF shows the distribution for $\|r(t) - e(t)\|_{2}$ for values of $t$ when $t$ is in (and not in) the predicted SS interval $[a^{^{\\prime }}_{j},b^{^{\\prime }}_{j}]$ for some $j$ .", "It is clear that the distribution when $t$ is in a predicted SS interval is towards the right compared to when it is not in the interval.", "This is because in an SS interval, $r(t)$ will have a small absolute value (by design).", "In a non-SS interval, $r(t)$ will be a linear function of $e(t)$ , as $r(t) = {\\bf W}e(t)$ .", "This behavior is observed across subjects suggesting that in all cases that $r(t)$ is behaving as designed.", "Furthermore in Figure REF , we plot the $\|r(t)\|$ against $\|e(t)\|$ when $t$ is not in a spindle interval.", "Again we observe a straight line behavior, providing further confirmation that $r(t)$ is behaving according to specifications.", "Figure: Comparison of the distribution of the norm of r(t)-e(t)r(t)-e(t) for SS andnon-SS intervals.", "In all four subjects the designed residual suppresses spindles as designed as thenorm is higher for SS intervals.Figure: the \|r(t)\|\|r(t)\| against \|e(t)\|\|e(t)\| when tt is not in a spindle interval as r(t)=𝐖e(t)r(t) = {\\bf W}e(t)" ], [ "Delay in Detection of Anomalies", "OSAD detects anomalies in near real time.", "We now discuss the lag between the appearance of a SS and before it is reported by the LDS.", "Figure REF presents the delay distributions for subject 1 and subject 4 who experienced 164 and 132 labeled sleep spindles, respectively.", "In general, the predicted SS interval are longer and contain the actual intervals.", "This is confirmed in Figure REF which shows one specific example of the location of the labeled sleep spindle and the predicted interval.", "In this case (which is typical), the prediction of SS begins before and ends later than the labeled spindle.", "Table REF shows the results of the mean delay between matched intervals.", "Thus a mean of $(a_{i},a^{\\prime }_{i})$ equal to -0.0678 implies that on average, there was a delay of 1/200 second before LDS reported an anomaly.", "On the other hand for subject 2 there the SS was, on average, reported before it showed up in the labeled sequence.", "As noted in [10], this is consistent with the observation (and confirmed by double-blind scoring) that the labeling of SS is more conservative i.e., SS are labeled for a shorter duration than what they should be.", "Figure: OSAD provides near real time detection.", "The delay between the actual appearanceof a spindle and the predicted appearance is a fraction of a second.", "Similarly the lag betweenwhen the actual spindle disappears and it is reported to disappear is very small too.", "The x-axisis in seconds.Table: Delay statistics.", "The lag between appearance and prediction of SS is, on average, a fraction of a second.Figure: Top: Cz data and a typical sleep spindle labeled.", "Bottom: Residual and detected sleep spindle.In generalthe predicted spindle interval is longer than the labeled interval.", "The predicted interval tends to includethe labeled interval, i.e., it begins earlier and finishes later.", "The EEG shows that the labeledintervals are actually quite conservative." ], [ "Related work", "Automatic detection of sleep spindles is now an important topic in biomedical research.", "Different techniques including FFTs, wavelet analysis and autoregressive time series modeling have been applied for sleep spindle detection [24], [11], [13].", "Attempts to integrate SVM to detect sleep spindles have also been explored [1].", "There seems to be a large variability between sleep EEG across subjects.", "In our experiments we have also observed this phenomenon.", "This combined with the large amount of EEG noise has resulted in low level of agreement on the exact profile of sleep spindle [20].", "The use of Linear Dyamical Systems (LDS) to model time series is ubiquitous both in computer science [25] and control theory [28], [16], [18], [7], [15].", "Expressing LDS in the language of graphical models and connections with HMM have been extensively examined in machine learning.", "The use of LDS for anomaly detection has also been investigated in network anomaly detection, among other areas [27].", "The use of subspace identification methods for inferring the parameters of LDS have been discussed by Overschee [28].", "Subspace methods estimate LDS parameters through a spectral decomposition of a matrix of observations to yield an estimate of the underlying state space.", "Subspace methods have low computational cost, are robust to perturbations and are relatively easy to implement.", "The recently introduced spectral learning methods are variations of the subspace method [3], [12] The use of eigenstructure assignment to alter the residual of an LDS has been investigated in the control theory literature especially in the context of fault diagnosis [2].Our approach closely follows the work Patton et.", "al.", "[21] who have used eigenstructure assignment for altering the LDS model using feedback.", "Other variations of LDS and fault diagnosis are discussed in [6], [22], [5], [23]." ], [ "Conclusion", "In this paper we have introduced a new problem, the Online Selective Anomaly Detection (OSAD) to capture a specific scenario in sleep research.", "Scientists working on sleep EEG data required an alert system, which trigger alerts on selected anomalies.", "For example, sleep stage two is characterized by two known anomalies: sleep spindle and K-complex.", "The requirement was to design a system which detected both anomalies but only generated an alert when a non sleep spindle anomaly appeared.", "We combined methods from data mining, machine learning and control theory to design such a system.", "Experiments on real data set demonstrate that our approach is accurate and produces the required results and is potentially applicable to many other situations.", "We also note that data from sleep EEG provides a fertile ground to apply existing data mining methodologies and potentially design new computational problems and algorithms." ], [ "Acknowledgments", "This work is partially supported by NICTAhttp://nicta.com.au/.", "NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.", "Proof of Theorem 1 Theorem 1.", "For a DRM, a sufficient condition for ${\\bf G}_{\\xi }(z)$ to be zero, is ${\\bf C}_f{\\bf P} = 0 \\mbox{ and }\\lbrace {\\bf C}_f{\\bf A}_f = 0 \\mbox{ or } {\\bf A}_f {\\bf P} = 0\\rbrace $ Proof: Let the set $\\lbrace \\lambda _i=0,v_i\\rbrace $ , for $i=1:n$ , be the left eigenvectors and corresponding eigenvalues of ${\\bf A}_f$ , i.e.", "$\\begin{array}{rl}v_i {\\bf A}_f &= \\lambda _i v_i \\\\&= 0\\\\\\end{array}$ If one chooses $v_1$ as the rows of matrix $[{\\bf WC}]$ , then: $\\begin{array}{c}\\begin{bmatrix} v_1 &...&v_n \\end{bmatrix}^{\\prime } {\\bf A}_f = 0 \\quad \\Rightarrow \\quad {\\bf WC}{\\bf A}_f = 0\\\\\\end{array}$ The matrix ${\\bf A}_f={\\bf A}-{\\bf F}{\\bf C}$ , so it is sufficient to chose ${\\bf F}$ so that the set $\\lbrace \\lambda _i=0, v_i=[{\\bf CW}]^{\\prime }\\rbrace $ to be assigned as left eigenpairs of $({\\bf A}-{\\bf F}{\\bf C})$ .", "In the other side, suppose If the columns of ${\\bf P}$ are the right eigenvectors of ${\\bf A}_f$ corresponding to zero-values eigenvectors, then $\\begin{array}{c}{\\bf A}_f v_i=0 \\quad \\Rightarrow \\quad {\\bf A}_f {\\bf P} = 0\\\\\\end{array}$ So it is sufficient to chose ${\\bf F}$ so that the set $\\lbrace \\lambda _i=0, v_i={\\bf P}\\rbrace $ to be assigned as right eigenpairs of $({\\bf A}-{\\bf F}{\\bf C})$ ." ], [ "Proof of Theorem 1", "Theorem 1.", "For a DRM, a sufficient condition for ${\\bf G}_{\\xi }(z)$ to be zero, is ${\\bf C}_f{\\bf P} = 0 \\mbox{ and }\\lbrace {\\bf C}_f{\\bf A}_f = 0 \\mbox{ or } {\\bf A}_f {\\bf P} = 0\\rbrace $ Proof: Let the set $\\lbrace \\lambda _i=0,v_i\\rbrace $ , for $i=1:n$ , be the left eigenvectors and corresponding eigenvalues of ${\\bf A}_f$ , i.e.", "$\\begin{array}{rl}v_i {\\bf A}_f &= \\lambda _i v_i \\\\&= 0\\\\\\end{array}$ If one chooses $v_1$ as the rows of matrix $[{\\bf WC}]$ , then: $\\begin{array}{c}\\begin{bmatrix} v_1 &...&v_n \\end{bmatrix}^{\\prime } {\\bf A}_f = 0 \\quad \\Rightarrow \\quad {\\bf WC}{\\bf A}_f = 0\\\\\\end{array}$ The matrix ${\\bf A}_f={\\bf A}-{\\bf F}{\\bf C}$ , so it is sufficient to chose ${\\bf F}$ so that the set $\\lbrace \\lambda _i=0, v_i=[{\\bf CW}]^{\\prime }\\rbrace $ to be assigned as left eigenpairs of $({\\bf A}-{\\bf F}{\\bf C})$ .", "In the other side, suppose If the columns of ${\\bf P}$ are the right eigenvectors of ${\\bf A}_f$ corresponding to zero-values eigenvectors, then $\\begin{array}{c}{\\bf A}_f v_i=0 \\quad \\Rightarrow \\quad {\\bf A}_f {\\bf P} = 0\\\\\\end{array}$ So it is sufficient to chose ${\\bf F}$ so that the set $\\lbrace \\lambda _i=0, v_i={\\bf P}\\rbrace $ to be assigned as right eigenpairs of $({\\bf A}-{\\bf F}{\\bf C})$ ." ] ]	1403.0156
[ [ "An improved elliptic guide concept for a homogeneous neutron beam\n without direct line of sight" ], [ "Abstract Ballistic neutron guides are efficient for neutron transport over long distances, and in particular elliptically shaped guides have received much attention lately.", "However, elliptic neutron guides generally deliver an inhomogeneous divergence distribution when used with a small source, and do not allow kinks or curvature to avoid a direct view from source to sample.", "In this article, a kinked double-elliptic solution is found for neutron transport to a small sample from a small (virtual) source, as given e.g.", "for instruments using a pinhole beam extraction with a focusing feeder.", "A guide consisting of two elliptical parts connected by a linear kinked section is shown by VITESS simulations to deliver a high brilliance transfer as well as a homogeneous divergence distribution while avoiding direct line of sight to the source.", "It performs better than a recently proposed ellipse-parabola hybrid when used in a ballistic context with a kinked or curved central part.", "Another recently proposed solution, an analytically determined non-linear focusing guide shape, is applied here for the first time in a kinked and curved ballistic context.", "The latter is shown to yield comparable results for long wavelength neutrons as the guide design found here, with a larger inhomogeneity in the divergence but higher transmission of thermal neutrons.", "It needs however a larger (virtual) source and might be more difficult to build in a real instrument." ], [ "Introduction", "Neutron guides are important tools used to deliver ample flux to samples at large distances from the source.", "Long neutron beamlines lead to low background, and are of particular interest to the planned European Spallation Source (ESS) [1] due to its long pulse and required time of flight resolution.", "Ballistic guides in which an expanding guide section reduces the beam divergence before the neutrons are transported by a straight guide and then focused by a guide section of decreasing spatial extension have been shown to perform better than conventional straight or curved neutron guides [2], and elliptic or parabolic guide shapes can improve the transmission even further [3].", "Hence a currently widely studied guide shape is the elliptic guide profile, which in principle allows neutrons from one focal point to be transmitted to the second focal point with just one reflection.", "This idealized behavior was recently shown to be true only for a negligibly small fraction of neutrons under realistic conditions [4].", "Even though the beam homogeneity after elliptically focusing is in general superior to the one after linear or parabolically focusing guides [5], inhomogeneous divergence distributions are often seen in Monte Carlo simulation studies of elliptical guides [4], [6], [7].", "A further drawback is the difficulty to avoid direct line of sight, which can be solved in certain cases by gravitational bending of the elliptical guide [8], but this solution is limited to long wavelengths in a small waveband around the wavelength for which it was optimized, and cannot be expected to reduce any inhomogeneities of the divergence spectrum.", "An alternative central beamstop to block the direct line of sight will create a hole in the transmitted phase space.", "A double ellipse somewhat improves the divergence profile with only a small loss in transmission [4] if the two ellipses have the same characteristic angle $\\psi =\\arctan {\\left(b/a\\right)}$ , with $a$ and $b$ the long and short half axis length, and share a common central focal point.", "Furthermore, a double elliptic guide design provides a natural narrow point, which constitutes a second eye of the needle for a possible chopper placement [9].", "This principle is used in the Selene [10] and POWTEX [11] guide concepts, achieving a homogeneous divergence in the respective simulations.", "While the latter does not avoid direct line of sight, the former does so by using only quarter ellipses acting as elliptical mirrors rather than guides, which are inclined and combined with slits and shielding equivalent to an effective central beamstop.", "This approach sacrifices intensity on the sample because the design is optimized for low background, using only the small fraction of ideal neutron trajectories from an approximate point source.", "Its use of elliptical guide parts as focusing devices further constrains it to shorter instrument in which gravity effects are small.", "This article describes a kinked double-elliptic guide concept for a 150 m long instrument looking at a small source, which focuses the neutron beam onto a small sample without introduction of beam inhomogeneities.", "The source can either be a real moderator, or a virtual source created by a preceding slit or focusing device as often considered for beamlines at the ESS, where a small beam spot close to the source is advantageous for the use of a pulse shaping chopper as well as for the placement of shielding.", "Guide systems intended for direct transportation from large sources will be discussed elsewhere.", "Based on the ideas of L. D. Cussen [12], [13], [14], the motivation for a kinked ballistic double-ellipse, consisting of two elliptically shaped guide parts in the beginning and end connected by a linear guide section, is recalled in section .", "After a description of boundary conditions and simulation details in section , the principles of the argumentation are verified by simulation and used to design a modified kinked ballistic double-ellipse in section .", "This new guide design is then compared in section to two recently proposed alternative approaches that have been shown to give a homogeneous divergence distribution in a focused neutron beam under different conditions or in theory: First, for the special case of a large source and an ellipse with a large opening, a hybrid guide consisting of an elliptically diverging and a parabolically converging part with roughly equal lengths and a total length of about 50 m has been shown to yield an improved divergence profile compared to a full ellipse [7].", "Second, an analytical calculation using phase space considerations resulted in a non-linear shape for a focusing guide that retains a rectangular phase space [15].", "These approaches are investigated here under the condition of a small (virtual) source, characterizing their performance in a ballistic guide design including a kinked or curved section to avoid direct line of sight." ], [ "Theoretical considerations", "This section derives a double-ellipse with a linear kinked connecting section and central narrow point from ellipse properties in combination with non-ideal behavior of neutrons not coming from a focal point, summarizing considerations made in [12], [13], [14] in context of a guide design for an ESS extreme environment (ESSEX) instrument." ], [ "General guide shape", "The second ellipse in a double-elliptic guide as drawn in figure REF principally reverses neutron trajectories and thereby partly reverses unwanted aberration effects of the first ellipse only under idealized conditions of neutrons emerging from a point source.", "Under realistic conditions, i.e.", "with an extended source (millimeters and larger), gravity altering neutron trajectories (and thereby reflection angles) and a possible non-perfect elliptic guide shape constructed out of straight pieces, most neutrons will undergo multiple reflections already in the first ellipse and the neutron beam cannot be expected to be perfectly focused at the central focal point.", "The transmission compared to a single ellipse will decrease even more than expected from the ideally only doubled number of reflections.", "Hence a straight forward modification to the double elliptic guide design is to replace the converging part of the first ellipse as well as the diverging part of the second ellipse by a straight guide, to obtain the ballistic double-ellipse in figure REF .", "Note that differently to most ballistic guides with elliptically focusing ends, the elliptic guide parts still share a common focal point in the center of the straight guide.", "Figure: Derivation of a kinked ballistic double-ellipse with narrow point, schematic drawing.", "(a) double-ellipse (b) ballistic double-ellipse (c) simple kinked ballistic double-ellipse (d) kinked ballistic double-ellipse with narrow point (e) extended kinked ballistic double-ellipse (f) extended kinked ballistic double-ellipse B" ], [ "Avoiding direct line of sight", "The constant connection between elliptic guide parts in the ballistic double-ellipse allows to introduce two kinks at the connection points.", "Most neutrons enter the guide with a divergence large enough to undergo at least one reflection in the first half of the ellipse, leading to a smaller divergence in the central part of the guide and making this an advantageous position to place a mirror to reflect the neutron beam out of direct line of sight.", "The kink angles are demanded in [12], [13], [14] to be such that the common central focal point lies on the reflecting guide wall, as illustrated in figure REF , and a central narrow point as shown in figure REF is introduced to provide a suitable position for frame overlap choppers and to be further out of line of sight.", "The study in [12], [13], [14] further states that extending the used part of the ellipses while narrowing the maximal guide width improves the divergence profile further.", "An example of such a guide is schematically shown in figure REF and will be referred to as extended kinked ballistic double-ellipse A.", "This design is improved here to an extended kinked ballistic double-ellipse B schematically drawn in REF : The condition of the kinked guide wall going through the central focal point causes kink angles larger than necessary to avoid direct line of sight, therefore this condition is dropped.", "The guide width at the central narrow point, which is fixed in [12], [13], [14] to 4 cm, will here be chosen such that the guide walls opposite to the mirror wall are parallel to the guide axes.", "This is illustrated by the light blue lines in figure REF .", "This way the central guide width as well as the kink angle are determined by the transition point between ellipse and linear guide.", "The optimal transition point is found by simulation in section ." ], [ "Boundary conditions and simulation details", "The distance from the source to the 1$\\times $ 1 cm$^\\mathrm {2}$ sample is fixed to 150 m. A dedicated study of pinhole beam extraction [16] revealed that efficient neutron transport as well as a satisfying beam homogeneity with an ellipse in a 150 m long instrument can be achieved using a 3$\\times $ 3 cm$^\\mathrm {2}$ pinhole for a desired divergence of $\\pm $ 1$^\\circ $ , and a smaller (larger) pinhole size for smaller (larger) divergence.", "In order to cover a somewhat larger divergence here, the source size is initially chosen to be 4$\\times $ 4 cm$^\\mathrm {2}$ .", "The effect of a smaller source is shown in section .", "For general results independent of a certain source spectrum shape, the simulated source emits a normalized rectangular wavelength spectrum of 1 cm$^{\\mathrm {-2}}$ s$^{\\mathrm {-1}}$ str$^{\\mathrm {-1}}$$\\mathring{\\mathrm {A}}$$^{\\mathrm {-1}}$ .", "Guide kinks or curvature are introduced only in the horizontal dimension so their effects can be studied independently of gravity.", "The guides have rectangular cross-sections, and the entry and exit width $w_{in,out}$ is 2 cm and hence sufficiently small compared to the source size, which has been shown to be a precondition for avoiding irregularities in the divergence caused by the elliptical guide shape itself [4] or gravity [17].", "The distance from source to guide entry position $d_s$ is the same as the distance from guide exit to sample position.", "The central narrow point is initially chosen to be of the same size as the (virtual) source.", "The Monte Carlo ray tracing simulation package VITESS [18] version 3.0 was used to simulate the data.", "Elliptic guide profiles are modeled as perfect ellipses without taking guide segmentation into account because segmentation or misalignment effects are not the topic of this work.", "To minimize losses in the thermal neutron range and otherwise keep the study general, all simulations use $m=$ 6 mirror coatings, where the $m$ number refers to the supermirror coating with a critical angle for total reflection of $m$ times the critical angle of nickel.", "In a practical guide design, the coating has to be optimized for the desired wavelength and divergence on the sample, and a much lower coating can be expected to be sufficient in most regions of the guide [11]." ], [ "Simulation results", "In this section, the considerations made in section are first verified by simulation, before an improved guide called the extended kinked ballistic double-ellipse B is designed.", "Figure REF shows the gradual construction of this guide starting from a double-ellipse.", "Its parameters are given in the second row of table REF (ext.", "knkd.", "dbl-ell (B)).", "The same ellipse parameters have been used in all the simulations of the double-elliptic guides shown in figure REF , other parameters are chosen as described in section .", "A single ellipse with two times the major and minor semi-axes has been simulated for comparison.", "Figure: Schematic drawing (a) and values (b) of guide parameters in horizontal plane: maximal guide width w max w_{max}, width at transition point w TP w_{TP} and narrow point w NP w_{NP}, entry and exit width w in,out w_{in,out}, distance from guide to source and sample d s d_s and to focal points d f d_f, length of one elliptic part L ell L_{ell} and linear or constant length between ellipse and central focal point L lin L_{lin}, total kink angle α\\alpha and total horizontal shift Δy\\Delta y.", "In the curved hybrid and non-linear designs shown for comparison, the curvature radius RR and the effective guide width w eff w_{eff} are given instead of the kink angle and guide width at narrow point, L c L_c is the length of the curved central section and L b L_b the length of one ballistic guide section (elliptic, parabolic or analytic non-linear shape).The divergence profile is smoothened when moving from a single elliptic guide to a double ellipse (fig.", "REF ) and further to the ballistic double-ellipse design (fig.", "REF ).", "However, a substantial loss of intensity on the sample (with the settings used here, 35%-50% loss depending on the neutron wavelength) is seen with the double-elliptic design compared to a single ellipse which is only partly re-gained in the ballistic guide, in which especially short wavelengths benefit from the removal of the narrow central part.", "Kinking the ballistic double-ellipse as in figure REF causes deep breaches in the divergence profile, which are shown as solid lines in figure REF for the four different wavelengths 1 $\\mathring{\\mathrm {A}}$ , 3 $\\mathring{\\mathrm {A}}$ , 6 $\\mathring{\\mathrm {A}}$ and 9 $\\mathring{\\mathrm {A}}$ in bands of $\\pm $ 0.05 $\\mathring{\\mathrm {A}}$ .", "Only neutrons on the 1$\\times $ 1 cm$^\\mathrm {2}$ sample are shown.", "The performance of the un-kinked version of this guide (as in figure REF ) is added as dotted lines for comparison, showing hardly any structure in the divergence distributions.", "The brilliance transfer (BT) is shown in figure REF , where brilliance transfer is defined as the ratio of neutron flux per solid angle and wavelength on the sample compared to the source position.", "Four different divergence ranges are treated, where neutrons outside the given divergence limits are ignored in the calculation of BT, i.e.", "suppression of unwanted neutrons with too large divergences is not included here.", "With this definition, a BT of 80 % is reached for neutrons with divergence $<$ 0.5$^\\circ $ (1$^\\circ $ , 1.5$^\\circ $ , 2$^\\circ $ ) and $\\lambda >$ 1.6 $\\mathring{\\mathrm {A}}$ (2.7 $\\mathring{\\mathrm {A}}$ , 3.9 $\\mathring{\\mathrm {A}}$ , 5.9 $\\mathring{\\mathrm {A}}$ ) in the un-kinked version of the ballistic guide (dotted lines).", "Comparing solid and dotted lines in figure REF illustrates the transmission loss caused by the kink.", "Figure: Horizontal divergence in bins of 0.06 ∘ ^\\circ for different wavelengths (a) and brilliance transfer for different divergence regions (b), obtained with the (simple kinked) ballistic double-ellipse in dotted (solid) lines.", "Error bars are drawn but mostly too small to see, and are calculated from the generated statistics of neutron trajectories.In the shown examples, transmission of thermal neutrons has been enhanced by moving the outer focal points slightly beyond the source and sample position, such that for a fair comparison, the ellipse parameters are identical to the ones in the extended kinked ballistic double-ellipse B solution found later.", "The kink angle obtained when placing one guide wall in the common central focal point is not large enough to loose direct line of sight to the source without the central narrow point (fig.", "REF ), therefore a slightly larger angle has to be used.", "The introduction of a narrow point as in figure REF reduces the needed kink angle and allows to place one guide wall on the central focal point.", "However, this smoothens the divergence profile only marginally, and leads to large losses in the short wavelength regime.", "By extending the elliptical shape beyond the position of maximal guide width as in figure REF , the divergence distribution can be further smoothened but still shows some structure.", "Only if the maximal guide width is limited to 10 cm, a greatly improved divergence profile is achieved, which is shown by the dotted lines in figure REF .", "As in [12], [13], [14], the ellipse has been extended up to 80 % here, corresponding to a guide width of 8 cm at the transition point from elliptic to linear shape.", "There is, however, still some structure visible, and the reduction of maximal guide width has to be paid by a decreased transmission.", "Figure: Horizontal divergence in bins of 0.06 ∘ ^\\circ for different wavelengths (a) and brilliance transfer for different divergence regions (b), obtained with the extended kinked ballistic double-ellipse A (dotted) and B (solid).", "Error bars are drawn but mostly too small to see, and are calculated from the generated statistics of neutron trajectories.If the condition of one guide wall passing through the central narrow point is dropped, the kink angle can be reduced to a value needed to prevent direct line of sight.", "As a consequence, the divergence profile becomes smoother.", "The reduction of the maximal guide width can be undone without re-introducing inhomogeneities, thus enhancing transmission.", "The optimal transition point between elliptic and linear guide is found by a simulation scan to be still at 8 cm guide width, which corresponds to building 92 % of the ellipse.", "The guide is furthermore wider at the narrow point due to the construction outlined in section .", "This modified design will be called extended kinked ballistic double-ellipse B.", "Its performance is shown by the solid lines in figures REF and REF : both transmission and divergence profile are improved compared to the former guide design A.", "For 1 $\\mathring{\\mathrm {A}}$ neutrons or those with very small divergence, the result is about the same.", "Only for short wavelength neutrons with small divergence, design A gives a slightly higher transmission, which is however caused by the smaller guide width and not by the design principle.", "With design B, a BT of 80 % is reached for neutrons with divergence $<$ 0.5$^\\circ $ (1$^\\circ $ , 1.5$^\\circ $ , 2$^\\circ $ ) and $\\lambda >$ 2 $\\mathring{\\mathrm {A}}$ (3.5 $\\mathring{\\mathrm {A}}$ , 5.5 $\\mathring{\\mathrm {A}}$ , 9.5 $\\mathring{\\mathrm {A}}$ ).", "The parameters of both extended kinked ballistic double-ellipse versions A and B are given in the first two rows of table REF .", "In the vertical plane, the elliptical guide parts are the same as the horizontal design but instead of the linear kinked guides in between, a constant connection with height $h_{vert}=w_{TP}$ is used." ], [ "Comparison with hybrid and analytic non-linear approach", "Two other solutions to the problem of an inhomogeneous divergence profile of a focused neutron beam have recently been proposed.", "These are adapted to the boundary conditions of a 4$\\times $ 4 cm$^\\mathrm {2}$ source, a 150 m long instrument and the avoidance of direct line of sight in this section.", "Their performance in a ballistic guide design with parameters similar to the ones used in the extended kinked ballistic double-ellipse B solution described above is then compared.", "Both a kinked and a curved central section are studied." ], [ "Ellipse-parabola hybrid", "An elliptically diverging guide followed by a parabolically focusing guide was shown in [7] to yield a smoother divergence distribution than a comparable full ellipse.", "The cited study uses an 18.6 cm wide source, a 4$\\times $ 4 cm$^\\mathrm {2}$ sample and a 50 m long guide.", "In order to match the requirements used here, a guide similar to the kinked ballistic double-ellipse with narrow point (cf.", "figure REF ) is designed which in principal is identical to the solution found above but with a transition point at the maximum of the ellipse.", "The focusing ellipse is replaced by a parabola with the same exit window of 2$\\times $ 2 cm$^\\mathrm {2}$ .", "This places the focal point of the parabola about 50 cm behind the sample.", "The source size is again 4$\\times $ 4 cm$^\\mathrm {2}$ .", "Both kinked and curved version of the ballistic hybrid guide show a structure in the divergence profile in figure REF , with local minima at different divergence values and an overall irregular divergence distribution for short wavelengths.", "In terms of beam homogeneity, the hybrid type solutions cannot compete with the extended kinked ballistic double-ellipse B.", "The transmission is however slightly higher for large divergence ($> \\pm $ 1.5$^\\circ $ ) neutrons and $\\lambda >$ 1 $\\mathring{\\mathrm {A}}$ , as can be seen in figure REF showing the ratio of brilliance on the sample obtained with the kinked hybrid compared to the extended kinked ballistic double-ellipse B solution found above.", "For small divergence up to 1$^\\circ $ , the transmission is slightly lower.", "Apart from very short wavelengths $<$ 1 $\\mathring{\\mathrm {A}}$ , the maximal difference in brilliance transfer is of the order of 10 %, so the overall transmission of the hybrid solutions is comparable to the one of the extended kinked ballistic double-ellipse B, while the beam homogeneity is lower.", "Figure: Horizontal divergence in bins of 0.06 ∘ ^\\circ obtained with curved (solid) and kinked (dotted) hybrid guide design (a) and brilliance for different divergence regions divided by the one obtained with the extended kinked ballistic double-ellipse B solution (b)." ], [ "Analytically calculated non-linear focusing", "An analytical calculation [15] yielded a solution for a focusing guide that retains a rectangular phase space and hence a homogeneous divergence distribution.", "The proposed focusing guide consists of a non-linear section followed by a linear section with wall inclinations identical to the angle at the exit of the non-linear part.", "The shape of the non-linear guide walls depend on the entry width, the desired ratio of guide width change to entry width (compression), the target wavelength as well as the input divergence, the latter being equivalent to the m-number of a preceding straight guide.", "Keeping a desired exit width of 2 cm, the maximal guide width is set to 10 cm since irregularities occur for larger compression factors.", "The target wavelength is set to 0.9 $\\mathring{\\mathrm {A}}$ in the calculation to ensure a good transmission of 1 $\\mathring{\\mathrm {A}}$ neutrons, the coating in the straight section is assumed to be $m=$ 2.", "These settings lead to a necessary coating of $m=$ 5.2 in the non-linear guide section.", "These coating conditions are just used to calculate the shape of the guide; for the sake of comparability with the previous guide simulations, $m=$ 6 is used for the whole guide in the simulation.", "For a ballistic design with $w_{max}=$ 10 cm and a compression of 0.80, the formulae in [15] result in a non-linear guide part of $L_{NL}=$ 17.1 m length followed by a $L_{lin}=$ 2.1 m long linear guide.", "The guide width at the cross-over point is 4.67 cm.", "Such a guide section is used as both the diverging as well as the converging part of the ballistic guide.", "Note that the non-linear focusing guide has only been calculated to retain a rectangular phase space without introducing inhomogeneities; its performance after a kinked or curved guide section, where the phase space cannot be expected to be rectangular, has not been studied before.", "The kinked version is designed using the same principles as in the extended kinked ballistic double-ellipse B solution - with a central narrow point calculated from the kink angle chosen such that direct line of sight is avoided - which leads to a width of 4.1 cm at the center of the guide.", "Simulation shows that the curved option yields a smoother divergence and more intensity on the sample than the kinked version, see figure REF .", "The horizontal divergence of the curved option is comparable to the one obtained with the extended kinked ballistic double-ellipse B solution for 6 $\\mathring{\\mathrm {A}}$ and 9 $\\mathring{\\mathrm {A}}$ , while shorter wavelengths show a slight asymmetry and there is more structure in the 1 $\\mathring{\\mathrm {A}}$ divergence spectrum.", "The brilliance on the sample is up to 70 % higher for very short wavelengths $\\lambda \\sim $ 0.8 $\\mathring{\\mathrm {A}}$ and divergences of 1$^\\circ $ or larger, see figure REF .", "However, the difference in transmission decreases quickly with wavelength to less than 20 % for $\\lambda >$ 1.5 $\\mathring{\\mathrm {A}}$ , and for longer wavelengths for which the beam homogeneity is comparable to the one obtained with the solution found here, the transmission is also not significantly different.", "Figure: Performance of the non-linear guide design: horizontal divergence in bins of 0.06 ∘ ^\\circ of curved (solid) and kinked (dotted) design (a), and brilliance ratio for different divergence regions compared to the extended kinked ballistic double-ellipse B (b).Figure: Two-dimensional divergence (upper left) and position (lower right) as well as horizontal (upper right) and vertical (lower left) phase space of the whole wavelength spectrum at the sample position, obtained with the extended kinked ballistic double-ellipse B (a) or the curved non-linear guide (b).For a better comparison of these two solutions, the two-dimensional phase space diagrams are shown in figure REF for the extended kinked ballistic double-ellipse B and in figure REF for the analytical non-linear solution.", "The spatial focusing is very similar, while in the three other diagrams, the analytical solutions shows a slightly more rectangular distribution especially in the two-dimensional divergence.", "In summary, for long wavelength neutrons the performance of the curved non-linear guide is similar to the one of the extended kinked ballistic double-ellipse B, while short wavelength neutrons can be delivered in higher abundance but with a more inhomogeneous divergence.", "For a (virtual) source not too small to illuminate the whole guide entry as used so far, the preferred guide shape thus depends on the relative importance of intensity versus beam homogeneity (the performance with a smaller source sizes is described in the next section).", "For both the hybrid as well as the non-linear alternative, it should be noted that only straight-forward guide designs, as similar to the solutions found here as possible, have been simulated.", "A numerical optimization might give parameters for slightly better results for all guide concepts studied here, but the overall tendency cannot be expected to change, and accounting for a homogeneous phase space in an optimization is challenging and will be treated elsewhere." ], [ "Source size", "The phase space produced with an elliptically focusing guide at the sample position was shown to become more homogeneous when the guide entrance is decreased [5], and one recently found requirement for a smooth divergence profile with elliptic neutron guides is indeed a large source compared to the guide entry [4].", "A detailed study of gravity effects in elliptic neutron guides found a necessary ratio of source to guide entry width of about 1, with the exact value depending on the neutron wavelength [17].", "In the extended kinked ballistic double-ellipse B, the same effect is observed: the solid lines in figure REF show some minor structure in the divergence for a 2$\\times $ 2 cm$^\\mathrm {2}$ source the same size as the guide entry, while a pronounced structure is seen in the dotted lines which correspond to a 1.5$\\times $ 1.5 cm$^\\mathrm {2}$ source.", "The curved non-linear guide is more susceptible to smaller source sizes: figure REF shows a significant structure already for a 2.0 cm wide source, and with a 1.5$\\times $ 1.5 cm$^\\mathrm {2}$ source an inhomogeneity much larger than with the extended kinked ballistic double-ellipse B, accompanied by a larger transmission loss for long wavelength neutrons.", "Therefore this solution is only compatible for medium sized sources; decreasing the guide entry for smaller sources will either lead to a larger compression factor and hence more structure in the divergence, or in combination with a decreased maximum width to a smaller transmission.", "Figure: Horizontal divergence distribution in bins of 0.06 ∘ ^\\circ with smaller source size for the extended kinked ballistic double-ellipse B (a) and the curved non-linear (b) solutions." ], [ "Discussion", "A 150 m long guide was developed out of a double-elliptic design by replacing the central part with a linear segment while keeping a central narrow point, first motivated by the possibility of a chopper placement but later seen to improve the divergence profile if used in the right way to reduce the kink angle necessary to avoid a direct line of sight from source to sample.", "With a maximal guide width of 15 cm, simulation revealed the ideal transition point between elliptical and linear guide shape to be at a guide width of 8 cm beyond the maximum of the ellipse, corresponding to 92 % of the guide length being elliptic.", "This extended kinked ballistic double-ellipse B shows very good performance in terms of both brilliance transfer and beam homogeneity.", "While the proposed guide is rather simple using standard components, it was shown that recently proposed more complex guide designs don't give a significantly better performance: A ballistic hybrid guide yields lower beam homogeneity and slightly lower transmission of neutrons with a divergence smaller than $\\pm $ 1$^\\circ $ .", "An analytical non-linear focusing guide put in a ballistic context with curved central section shows a similar performance for long wavelength neutrons and an increased transmission accompanied by decreased beam homogeneity for shorter wavelength.", "If the source size is the same value as the guide entry width or smaller, the divergence profile obtained with the curved analytical non-linear guide shows much larger inhomogeneity as the extended kinked ballistic double-ellipse proposed here, and also the transmission decreases more with decreasing source size.", "Therefore for medium sized sources larger than the guide entry this option is compatible and even gives a larger intensity of very short wavelength neutrons, while it is less useful for smaller (virtual) sources.", "A large distance between guide exit and a small sample should however not be desired, since the analytical focusing guide does not have a focal point that can be placed in or behind the sample position, but yields a compressed beam directly at the guide exit.", "In the example used here, the wavelength integrated intensity on a 1$\\times $ 1 cm$^\\mathrm {2}$ sample at the sample position 19 cm behind the guide is still 90% of the intensity at the guide exit.", "In a distance of 50 cm from the guide exit, however, the intensity has dropped to 30%." ], [ "Conclusions", "The 150 m long extended kinked ballistic double-ellipse developed here gives a high brilliance transfer and a homogeneous divergence profile on the sample while avoiding direct line of sight between source and sample.", "If the (virtual) source size is not too small and the sample can be placed close to the guide exit, a curved ballistic guide with analytical non-linear focusing sections is an alternative to consider, in particular if high intensity of short wavelength neutrons is more important than a perfectly homogeneous beam divergence in that part of the wavelength spectrum." ], [ "Acknowledgments", "We thank L. D. Cussen for sharing unpublished ideas and involving in extensive discussions.", "This work was funded by the German BMBF under “Mitwirkung der Zentren der Helmholtz Gemeinschaft und der Technischen Universität München an der Design-Update Phase der ESS, Förderkennzeichen 05E10CB1”." ], [ "On the calculation of line of sight avoidance", "Whenever a kink angle or curvature radius necessary to avoid direct line of sight to the source is calculated, the guide width is increased by 2$t_g$ in the calculation to account for glass substrates on both sides of the guide of $t_g$ thickness each, which are known to be transparent for high energy particles.", "The glass substrate is assumed to be $t_g=$ 1 cm in all calculations." ], [ "Kinked ballistic guides", "The kink angle necessary to avoid direct line of sight is determined by the three points marked red in figure REF : The possible neutron trajectory shown by the red line has an incoming divergence of $\\alpha _{in} = \\arctan {\\left(\\frac{w_{in,out}/2+w_{TP}/2}{L_{ell}+L_{lin}}\\right)}$ (note that guide walls drawn in light blue are parallel to the guide axes), where $w_{TP}$ is the guide width at the transition point between elliptic and linear guide section.", "Hence its horizontal offset at the guide exit is $L_{tot} \\tan {\\left(\\alpha _{in}\\right)}$ with the total guide length $L_{tot}=2(L_{ell}+L_{lin})$ .", "If a transparent glass substrate of thickness $t_g$ is assumed on each side of the guide, the angle changes to $\\alpha _{in} = \\arctan {\\left(\\frac{w_{in,out}/2+w_{TP}/2+2t_g}{L_{ell}+L_{lin}}\\right)} \\mbox{ .", "}$ The offset of the guide $(L_{ell}+L_{lin}) \\tan {\\left(\\alpha \\right)}$ has to exceed that, so with $\\tan {\\left(\\alpha \\right)}\\simeq \\alpha $ , the kink angle $\\alpha $ follows as $\\alpha /2 > \\left( \\frac{w_{in,out}+w_{TP}+4t_g}{L_{tot}} \\right) \\mbox{ .", "}$ Figure: Schematic drawing of kinked and curved guide designs and neutron trajectory (red line) relevant for direct line of sight." ], [ "Curved ballistic guides", "Similar to the kinked guide, the neutron trajectory through the points $p1$ and $p2$ in figure REF determines the curvature needed to reach point $p3$ .", "Due to the symmetry of this design, these points are the same as for a guide with constant cross-section of effective width $w_{eff}=w_{in,out}/2+w_{max}/2+2t_g$ , where the last term again accounts for the glass substrate.", "The horizontal offset at the end of the curved section with respect to p2 is $\\frac{1}{2}R\\theta ^2 =: \\Delta y_c$ , the additional offset by the following elliptic part is $L_b \\tan {\\left(\\theta \\right)} =L_b \\frac{\\Delta y_c}{L_c/4}$ .", "Hence the horizontal offset between points 2 and 3 is $ \\frac{R}{2}\\theta ^2 \\left(1 + \\frac{L_b}{L_c/4} \\right) = w_{eff} \\mbox{ ,}$ and with $\\theta =\\frac{L_c/2}{R}$ follows $ R=\\frac{L_c^2}{8w_{eff}} \\left( 1 + \\frac{4L_b}{L_c} \\right) \\mbox{ .", "}$" ] ]	1403.0373
[ [ "Baryonic matter perturbations in decaying vacuum cosmology" ], [ "Abstract We consider the perturbation dynamics for the cosmic baryon fluid and determine the corresponding power spectrum for a $\\Lambda(t)$CDM model in which a cosmological term decays into dark matter linearly with the Hubble rate.", "The model is tested by a joint analysis of data from supernovae of type Ia (SNIa) (Constitution and Union 2.1), baryonic acoustic oscillation (BAO), the position of the first peak of the anisotropy spectrum of the cosmic microwave background (CMB) and large-scale-structure (LSS) data (SDSS DR7).", "While the homogeneous and isotropic background dynamics is only marginally influenced by the baryons, there are modifications on the perturbative level if a separately conserved baryon fluid is included.", "Considering the present baryon fraction as a free parameter, we reproduce the observed abundance of the order of $5\\%$ independently of the dark-matter abundance which is of the order of $32\\%$ for this model.", "Generally, the concordance between background and perturbation dynamics is improved if baryons are explicitly taken into account." ], [ "Introduction", "Explaining structure formation in the expanding Universe is one of the major topics in cosmology and astrophysics.", "According to the current main-stream understanding, dark matter (DM) and dark energy (DE) are the dynamically dominating components of the Universe [1], [2], [3].", "Baryons contribute only a small fraction of less than 5% to the cosmic energy budget.", "The standard $\\Lambda $ CDM model does well in fitting most observational data but there is an ongoing interest in alternative models within and beyond General Relativity.", "A class of alternative models within General Relativity \"dynamizes\" the cosmological constant, resulting in so-called $\\Lambda (t)$ CDM models.", "Taking the cosmological principle for granted, cosmic structures represent inhomogeneities in the matter distribution on an otherwise spatially homogeneous and isotropic background.", "Dynamical DE models, $\\Lambda (t)$ CDM models are a subclass of them, have to deal with inhomogeneities of the DE component in addition to the matter inhomogeneities to which they are coupled.", "This makes these models technically more complex than the standard model.", "Ignoring perturbations of the DE component altogether may lead to inconsistencies and unreliable conclusions concerning the interpretation of observational data [4].", "Whether or not DE perturbations are relevant has to be decided on a case-by-case basis.", "The directly observed inhomogeneities are of baryonic nature.", "From the outset it is not clear that the inhomogeneities in the baryonic matter coincide with the inhomogeneities of the DM distribution.", "In particular, if DM interacts nongravitationally with DE, which happens in $\\Lambda (t)$ CDM models, while baryonic matter is in geodesic motion, this issue has to be clarified.", "A reliable description of the observed matter distribution has to consider the perturbation dynamics of the baryon fraction even though the latter only marginally influences the homogeneous and isotropic cosmic background dynamics.", "Then, in models with dynamical DE, the perturbations of baryonic matter will necessarily be coupled to the inhomogeneities of both DM and DE.", "In a general context, the importance of including the physics of the baryon component in the cosmic dynamics has been emphasized recently [5].", "In this paper we extend a previously established decaying vacuum model [6], [7], [8], [9], [10] by including a separately conserved baryon fluid with a four-velocity that differs from the four-velocity of the DM component.", "The basic ingredient of this model is a DE component with an energy density proportional to the Hubble rate.", "Moreover, it is characterized by an equation-of-state (EoS) parameter $-1$ for vacuum.", "Equivalently, the resulting dynamics can be understood as a scenario of DM particle production at a constant rate [9] or as the dynamics of a non-adiabatic Chaplygin gas [10].", "DE perturbations for this model are explicitly related to DM perturbations and their first derivative with respect to the scale factor in a scale-dependent way.", "It has been shown that on scales that are relevant for structure formation, DE fluctuations are smaller than the DM fluctuations by several orders of magnitude [8].", "Our analysis will be performed within a gauge-invariant formalism in terms of variables adapted to comoving observers [11].", "We shall derive a set of two second-order equations that couple the total fractional energy-density perturbations of the cosmic medium to the difference between these total perturbations and the fractional baryonic perturbations.", "The perturbations of the baryon fluid are then found as a suitable linear combination.", "As far as the background dynamics is concerned, our updated tests against observations from SNIa, BAO and the position of the first acoustic peak of the CMB spectrum confirm previous results [12].", "Including the LSS data improves the concordance of the model compared with the case without a separately conserved baryon component.", "The joint analysis allows us to predict the baryon abundance of the Universe independently of the DM abundance.", "The corresponding probability density function (PDF) exhibits a pronounced peak at about 5% for this abundance.", "This is a new feature which entirely relies on a separate consideration of the baryon fluid.", "The paper is organized as follows.", "In Sec.", "we establish the basic relations of our three-component model of DE, DM and baryons.", "In Sec.", "we recall the homogeneous and isotropic background dynamics of this model.", "Sec.", "is devoted to a gauge-invariant perturbation analysis which results in an explicit expression for the energy-density perturbations of the baryon fluid.", "In Sec.", "we test the model against observations using both background and LSS data.", "Our results are summarized in Sec.", "." ], [ "The model", "We describe the cosmic medium as a perfect fluid with a conserved energy momentum tensor $T_{ik} = \\rho u_{i}u_{k} + p h_{ik}\\ , \\qquad T_{\\ ;k}^{ik} = 0\\,,$ where $u^{i}$ is the cosmic four-velocity, $h _{ik}=g_{ik} + u_{i}u_{k}$ and $g_{ik}u^{i}u^{k} = -1$ .", "Here, $\\rho $ is the energy density for a comoving (with $u^{i}$ ) observer and $p$ is the fluid pressure.", "Latin indices run from 0 to 3.", "Let us consider a three–component system by assuming a split of the total energy-momentum tensor in (REF ) into a DM component (subindex M), a DE component (subindex X) and a baryonic component (subindex B), $T^{ik} = T_{M}^{ik} + T_{X}^{ik} + T_{B}^{ik}\\,.$ Each of the components is also modeled as a perfect fluid with ($A= M, X$ , B) $T_{A}^{ik} = \\rho _{A} u_A^{i} u^{k}_{A} + p_{A} h_{A}^{ik} \\ ,\\qquad \\ h_{A}^{ik} = g^{ik} + u_A^{i} u^{k}_{A} \\,.$ DM and baryonic matter are assumed to be pressureless.", "In general, each component has its own four-velocity with $g_{ik}u_{A}^{i}u_{A}^{k} = -1$ .", "According to the model to be studied here we include a (so far unspecified) interaction between the dark components: $T_{M\\ ;k}^{ik} = Q^{i}\\qquad T_{X\\ ;k}^{ik} = - Q^{i}\\,.$ Then, the energy-balance equations of the dark components are $-u_{Mi}T^{ik}_{M\\ ;k} = \\rho _{M,a}u_{M}^{a} + \\Theta _{M}\\rho _{M} = -u_{Ma}Q^{a}\\ $ and $-u_{Xi}T^{ik}_{X\\ ;k} = \\rho _{X,a}u_{X}^{a} + \\Theta _{X} \\left(\\rho _{X} + p_{X}\\right) = u_{Xa}Q^{a}\\,.$ The baryonic component is separately conserved, $-u_{Bi}T^{ik}_{B\\ ;k} = \\rho _{B,a}u_{B}^{a} + \\Theta _{B}\\rho _{B} = 0\\,.$ The quantities $\\Theta _{A}$ are defined as $\\Theta _{A} = u^{a}_{A;a}$ .", "For the homogeneous and isotropic background we assume $u_{M}^{a} = u_{X}^{a} = u_{B}^{a} = u^{a}$ .", "Likewise, we have the momentum balances $h_{Mi}^{a}T^{ik}_{M\\ ;k} = \\rho _{M}\\dot{u}_{M}^{a} = h_{M i}^{a} Q^{i}\\,,$ $h_{Xi}^{a}T^{ik}_{X\\ ;k} = \\left(\\rho _{X} + p_{X}\\right)\\dot{u}_{X}^{a} + p_{X,i}h_{X}^{ai} = - h_{X i}^{a} Q^{i}\\,,$ and $h_{Bi}^{a}T^{ik}_{B\\ ;k} = \\rho _{B}\\dot{u}_{B}^{a} = 0\\,,$ where $\\dot{u}_{A}^{a} \\equiv u_{A ;b}^{a}u_{A}^{b}$ .", "The source term $Q^{i}$ is split into parts proportional and perpendicular to the total four-velocity according to $Q^{i} = u^{i}Q + \\bar{Q}^{i}\\,,$ where $Q = - u_{i}Q^{i}$ and $\\bar{Q}^{i} = h^{i}_{a}Q^{a}$ with $u_{i}\\bar{Q}^{i} = 0$ .", "The contribution $T_{X}^{ik}$ is supposed to describe some form of DE.", "In the simple case of an EoS $p_{X} = - \\rho _{X}$ , where $\\rho _{X}$ is not necessarily constant, we have $T_{X}^{ik} = - \\rho _{X}g^{ik}\\,.$ Dynamically, an energy-momentum tensor like this corresponds to a time-dependent cosmological term.", "Various approaches to such type of $\\Lambda (t)$ cosmology term can be found in the literature [13].", "Since the only time scale in a homogeneous and isotropic universe is the Hubble time $H^{-1}$ , the simplest phenomenological guess here is $\\rho _{X} \\propto H$ .", "Interestingly, this guess has some support from particle physics.", "The QCD vacuum condensate associated to the chiral phase transition leads to a vacuum density proportional to $H$ [14].", "It is a dynamics along this line which we intend to study here, albeit in an entirely phenomenological context.", "An obvious covariant generalization of a cosmological term that, in the homogeneous and isotropic background, decays linearly with the Hubble rate $H$ , i.e., $\\rho _{X} \\propto H$ , is $\\rho _{X} = \\frac{\\sigma }{3}\\Theta \\,,\\qquad p_{X} = - \\frac{\\sigma }{3}\\Theta \\,,$ where $\\Theta \\equiv u^{a}_{;a}$ is the expansion scalar and $\\sigma $ is a constant.", "In the homogeneous and isotropic background one has $\\Theta = 3H$ and recovers $\\rho _{X} \\propto H$ ." ], [ "Background dynamics", "The homogeneous and isotropic background dynamics is governed by Friedmann's equation $3 H^{2} = 8\\pi G \\rho = 8\\pi G \\left(\\rho _{M} + \\rho _{X} + \\rho _{B}\\right)= 8\\pi G \\left(\\rho _{M} + \\rho _{B} + \\sigma H\\right)\\ $ and $\\dot{H} = - 4\\pi G \\left(\\rho + p\\right) = - 4\\pi G \\left(\\rho _{M} + \\rho _{B}\\right) \\,.$ Combining Eqs.", "(REF ) and (REF ) we obtain $\\dot{H} = - \\frac{3}{2}H^{2} + 4\\pi G \\sigma H\\,.$ Changing to the scale factor $a$ as independent variable, the solution of Eq.", "(REF ) is $H = \\frac{8\\pi G }{3}\\sigma + \\left(H_{0} - \\frac{8\\pi G }{3}\\sigma \\right) a^{-3/2} \\,,$ where a subindex 0 indicates the present value of the corresponding quantity and where we put $a_{0} = 1$ .", "With $3H_{0}^{2} = 8\\pi G \\rho _{0}\\,,\\quad \\Omega _{M0}\\equiv \\frac{\\rho _{M0}}{\\rho _{0}}\\,,\\quad \\Omega _{B0}\\equiv \\frac{\\rho _{B0}}{\\rho _{0}}\\,,\\quad \\sigma = \\frac{\\rho _{0}}{H_{0}}\\left(1 - \\Omega _{M0} - \\Omega _{B0}\\right)\\,,$ the Hubble rate (REF ) may be written as $H = H_{0}\\left[1 - \\Omega _{M0} - \\Omega _{B0} + \\left(\\Omega _{M0} + \\Omega _{B0}\\right)a^{-3/2}\\right]\\,.$ The existence of the last relation in (REF ) implies that $\\sigma $ is not an additional parameter.", "The limit of a vanishing $\\sigma $ is the Einstein-de Sitter universe, not the $\\Lambda $ CDM model.", "There is no $\\Lambda $ CDM limit of the dynamics described by the Hubble rate (REF ).", "The background source terms are $u_{a}Q^{a} = - Q = -\\dot{p}_{X} = \\sigma \\dot{H}\\quad \\mathrm {and} \\quad \\bar{Q}^{a} = 0\\ $ and the energy densities $\\rho _{M}$ and $\\rho _{X}$ are given by $\\frac{\\rho _{M}}{\\rho _{0}} = \\left(\\Omega _{M0} + \\Omega _{B0}\\right)a^{-3/2}\\left[1 - \\Omega _{M0} - \\Omega _{B0} + \\left(\\Omega _{M0} + \\Omega _{B0} - \\frac{\\Omega _{B0}}{\\Omega _{M0} + \\Omega _{B0}} \\right)a^{-3/2}\\right]\\ $ and $\\frac{\\rho _{X}}{\\rho _{0}} = \\left(1 - \\Omega _{M0} - \\Omega _{B0}\\right)\\left[1 - \\Omega _{M0} - \\Omega _{B0} + \\left(\\Omega _{M0} + \\Omega _{B0}\\right)a^{-3/2}\\right]\\,,$ respectively.", "The baryon energy density is $\\frac{\\rho _{B}}{\\rho _{0}} = \\Omega _{B0}a^{-3} \\,.$ With (REF ) - (REF ) the background dynamics for the three-component system is exactly solved.", "An additional radiation component (subscript R) can be included approximately [15]: $H = H_{0}\\left[\\left[1 - \\Omega _{M0} - \\Omega _{B0} + \\left(\\Omega _{M0} + \\Omega _{B0}\\right)a^{-3/2}\\right]^{2} + \\Omega _{R0}a^{-4}\\right]^{1/2}\\,.$ (Notice that this is an exact solution of the dynamics only for $\\Omega _{R0} = 0$ .)", "It can be shown that for the standard-model values of $\\Omega _{M0}$ , $\\Omega _{B0}$ and $\\Omega _{R0}$ the deviation of (REF ) from the exact numerical solution for the Hubble rate is only of the order of 0.6%." ], [ "Balance and conservation equations", "First-order perturbations will be denoted by a hat symbol.", "While for the background $u_{M}^{a} = u_{B}^{a} = u_{X}^{a} = u^{a}$ is assumed to be valid, the first-order perturbations of these quantities are different, in general.", "The perturbed time components of the four-velocities, however, still coincide: $\\hat{u}_{0} = \\hat{u}^{0} = \\hat{u}_{M}^{0} = \\hat{u}_{B}^{0} = \\hat{u}_{X}^{0} = \\frac{1}{2}\\hat{g}_{00}\\,.$ According to the perfect-fluid structure of both the total energy-momentum tensor (REF ) and the energy-momentum tensors of the components in (REF ), and with $u_{M}^{a} = u_{B}^{a} = u_{X}^{a} = u^{a}$ in the background, we have first-order energy-density perturbations $\\hat{\\rho } = \\hat{\\rho }_{M} + \\hat{\\rho }_{B} + \\hat{\\rho }_{X}$ , pressure perturbations $\\hat{p} = \\hat{p}_{M} + \\hat{p}_{B} + \\hat{p}_{X} = \\hat{p}_{X}$ and $\\hat{T}^{0}_{\\alpha } = \\hat{T}^{0}_{M\\alpha } + \\hat{T}^{0}_{B\\alpha } +\\hat{T}^{0}_{X\\alpha }\\quad \\Rightarrow \\quad \\left(\\rho + p\\right)\\hat{u}_{\\alpha } = \\rho _{M}\\hat{u}_{M\\alpha } + \\rho _{B}\\hat{u}_{B\\alpha } + \\left(\\rho _{X} + p_{X}\\right)\\hat{u}_{X\\alpha }\\,.$ For $p_{X} = - \\rho _{X}$ it follows $p_{X} = - \\rho _{X} \\ \\Rightarrow \\ \\rho + p = \\rho _{M} + \\rho _{B} \\ \\Rightarrow \\ \\hat{u}_{\\alpha } = \\frac{\\rho _{M}}{\\rho _{M} + \\rho _{B}}\\hat{u}_{M\\alpha } + \\frac{\\rho _{B}}{\\rho _{M} + \\rho _{B}}\\hat{u}_{B\\alpha }\\,.$ The perturbations of the time derivatives of the spatial components of the four-velocities differ from the time derivatives of the perturbations by the spatial gradient of $g_{00}$ : $\\hat{\\dot{u}}_{\\alpha } = \\dot{\\hat{u}}_{\\alpha } - \\frac{1}{2}g_{00,\\alpha }\\,, \\qquad \\hat{\\dot{u}}_{M\\alpha } = \\dot{\\hat{u}}_{M\\alpha } - \\frac{1}{2}g_{00,\\alpha }\\,, \\qquad \\hat{\\dot{u}}_{B\\alpha } = \\dot{\\hat{u}}_{B\\alpha } - \\frac{1}{2}g_{00,\\alpha }\\,.$ The total first-order energy conservation reads $\\dot{\\hat{\\rho }}+ \\dot{\\hat{\\rho }}\\hat{u}^{0} + \\hat{\\Theta }\\left(\\rho _{M} + \\rho _{B}\\right)+ \\Theta \\left(\\hat{\\rho } + \\hat{p}\\right) = 0\\,,$ while the separate balances are $\\dot{\\hat{\\rho }}_{M} + \\dot{\\rho }_{M}\\hat{u}^{0} + \\hat{\\Theta }_{M}\\rho _{M}+ \\Theta \\hat{\\rho }_{M} = Q = - \\left(u_{Ma}Q^{a}\\right)^{\\hat{}}\\,,$ $\\dot{\\hat{\\rho }}_{X} + \\dot{\\rho }_{X}\\hat{u}^{0}+ \\Theta \\left(\\hat{\\rho }_{X} + \\hat{p}_{X}\\right) = \\left(u_{Xa}Q^{a}\\right)^{\\hat{}}\\ $ and $\\dot{\\hat{\\rho }}_{B} + \\dot{\\rho }_{B}\\hat{u}^{0} + \\hat{\\Theta }_{B}\\rho _{B}+ \\Theta \\hat{\\rho }_{B} = 0\\,.$ Comparing the total first-order energy conservation (REF ) with the sum of the separate balances (REF ), (REF ) and (REF ) results in $\\hat{\\Theta }\\left(\\rho _{M} + \\rho _{B}\\right) = \\hat{\\Theta }_{M}\\rho _{M} + \\hat{\\Theta }_{B}\\rho _{B}+ \\left(u_{Ma}Q^{a}\\right)^{\\hat{}} - \\left(u_{Xa}Q^{a}\\right)^{\\hat{}}\\,.$ To be consistent with the last equation in (REF ), the last two terms on the right-hand side of (REF ) have to cancel each other.", "This establishes a relation between the perturbations of the projected interaction terms.", "We shall restrict ourselves to scalar perturbations which are described by the line element $\\mbox{d}s^{2} = - \\left(1 + 2 \\phi \\right)\\mbox{d}t^2 + 2 a^2F_{,\\alpha }\\mbox{d}t\\mbox{d}x^{\\alpha } +a^2\\left[\\left(1-2\\psi \\right)\\delta _{\\alpha \\beta } + 2E_{,\\alpha \\beta } \\right] \\mbox{d}x^\\alpha \\mbox{d}x^\\beta \\,.$ We also define the three-scalar quantities $v$ , $v_{M}$ and $v_{B}$ by $a^2\\hat{u}^\\mu + a^2F_{,\\mu } = \\hat{u}_\\mu \\equiv v_{,\\mu }\\,,\\quad a^2\\hat{u}_{M}^\\mu + a^2F_{,\\mu } = \\hat{u}_{M\\mu } \\equiv v_{M,\\mu }\\,,\\quad a^2\\hat{u}_{B}^\\mu + a^2F_{,\\mu } = \\hat{u}_{B\\mu } \\equiv v_{B,\\mu }\\,.$ With the abbreviation $\\chi \\equiv a^2\\left(\\dot{E} -F\\right) \\,,$ the perturbed scalars $\\Theta _{M}$ , $\\Theta _{B}$ and $\\Theta $ are $\\hat{\\Theta }_{M} = \\frac{1}{a^2}\\left(\\Delta v_{M} +\\Delta \\chi \\right) -3\\dot{\\psi } - 3 H\\phi \\ , \\quad \\hat{\\Theta }_{B} = \\frac{1}{a^2}\\left(\\Delta v_{B} +\\Delta \\chi \\right) -3\\dot{\\psi } - 3 H\\phi \\ $ and $\\hat{\\Theta } = \\frac{1}{a^2}\\left(\\Delta v +\\Delta \\chi \\right) -3\\dot{\\psi } - 3 H\\phi \\,,$ respectively, where $\\Delta $ denotes the three-dimensional Laplacian.", "The last relation of (REF ) then implies $\\left(\\rho _{M} + \\rho _{B}\\right)v = \\rho _{M}v_{M} + \\rho _{B}v_{B}\\,.$ Moreover, as already mentioned, consistency with (REF ) requires $\\left(u_{Ma}Q^{a}\\right)^{\\hat{}} = \\left(u_{Xa}Q^{a}\\right)^{\\hat{}}\\,.$ In terms of the fractional quantities $\\delta = \\frac{\\hat{\\rho }}{\\rho }\\,,\\quad \\delta _{M} = \\frac{\\hat{\\rho }_{M}}{\\rho _{M}}\\,,\\quad \\delta _{X} = \\frac{\\hat{\\rho }_{X}}{\\rho _{X}}\\,,\\quad \\delta _{B} = \\frac{\\hat{\\rho }_{B}}{\\rho _{B}}\\,,$ the energy balances (REF ), (REF ), (REF ) and (REF ) transform into $\\dot{\\delta } + \\frac{\\dot{\\rho }}{\\rho }\\hat{u}^{0} + \\hat{\\Theta }\\frac{\\rho _{M} + \\rho _{B}}{\\rho }+ \\Theta \\frac{p}{\\rho }\\left(\\frac{\\hat{p}}{p} - \\delta \\right) = 0\\,,$ $\\dot{\\delta }_{M} + \\frac{\\dot{\\rho }_{M}}{\\rho _{M}}\\hat{u}^{0} + \\hat{\\Theta }_{M}= \\frac{\\hat{Q}}{\\rho _{M}} - \\frac{Q}{\\rho _{M}}\\delta _{M}\\,,$ $\\dot{\\delta }_{X} + \\Theta \\left(\\frac{\\hat{p}_{X}}{\\rho _{X}} + \\delta _{X}\\right)= \\frac{1}{\\rho _{X}}\\left(u_{Xa}Q^{a}\\right)^{\\hat{}} + \\frac{Q}{\\rho _{X}}\\left(\\delta _{X}+ \\hat{u}^{0}\\right)\\ $ and $\\dot{\\delta }_{B} + \\frac{\\dot{\\rho }_{B}}{\\rho _{B}}\\hat{u}^{0} + \\hat{\\Theta }_{B}= 0\\,,$ respectively.", "The total momentum conservation reads (recall that $p_{X} = - \\rho _{X}$ ) $\\left(\\rho _{M} + \\rho _{B}\\right)\\dot{u}^{a} + p_{X,i}h^{ai} = 0\\,.$ The DM and DE momentum balances are given by (REF ) and (REF ), respectively, with $p_{X} = - \\rho _{X}$ .", "The baryon-fluid motion is geodesic according to (REF ).", "Our aim is to calculate the energy-density perturbations of the baryon component.", "In the following subsection we establish, in a first step, an equation for the perturbations of the total energy density.", "Subsequently, we shall derive an equation for the difference between total and baryonic density perturbations.", "From the solutions of this system of coupled second-order equations we then obtain the desired perturbations of the baryon fluid." ], [ "Perturbations of the total energy density", "To obtain an equation for the total energy-density perturbations it is convenient to introduce gauge-invariant quantities, adapted to an observer that is comoving with the total fluid four-velocity, $\\delta ^{c} \\equiv \\delta + \\frac{\\dot{\\rho }}{\\rho } v\\,, \\quad \\hat{\\Theta }^{c} \\equiv \\hat{\\Theta } + \\dot{\\Theta } v\\,, \\quad \\hat{p}^{c} \\equiv \\hat{p} + \\dot{p}v\\,.$ Then, the total energy and momentum conservations (REF ) and (REF ), respectively, can be combined into $\\dot{\\delta }^{c} - \\Theta \\frac{p}{\\rho }\\delta ^{c} + \\hat{\\Theta }^{c}\\left(1+\\frac{p}{\\rho }\\right) = 0\\,.$ The perturbation $\\hat{\\Theta }$ has to be determined from the Raychaudhuri equation $\\dot{\\Theta } + \\frac{1}{3}\\Theta ^{2} - \\dot{u}^{a}_{;a} + 4\\pi G \\left(\\rho + 3p\\right) = 0\\,,$ where we have neglected shear and vorticity.", "At first order we have $\\dot{\\hat{\\Theta }}^{c} + \\frac{2}{3}\\Theta \\hat{\\Theta }^{c} + 4\\pi G\\rho \\delta ^{c}+ \\frac{1}{a^2}\\frac{\\Delta \\hat{p}^{c}}{\\rho + p} = 0\\,.$ Combining Eqs.", "(REF ) and (REF ) and changing to $a$ as independent variable ($\\delta ^{c\\prime } \\equiv \\frac{d \\delta ^{c}}{d a}$ ), we obtain $\\delta ^{c\\prime \\prime } + \\left[\\frac{3}{2}-\\frac{15}{2}\\frac{p}{\\rho }+ 3\\frac{\\dot{p}}{\\dot{\\rho }}\\right]\\frac{\\delta ^{c\\prime }}{a}- \\left[\\frac{3}{2} + 12\\frac{p}{\\rho } - \\frac{9}{2}\\frac{p^{2}}{\\rho ^{2}} - 9\\frac{\\dot{p}}{\\dot{\\rho }}\\right]\\frac{\\delta ^{c}}{a^{2}}+ \\frac{k^{2}}{a^{2}H^{2}}\\frac{\\hat{p}^{c}}{\\rho a^{2}}= 0\\,,$ where $k$ is the comoving wavenumber.", "According to (REF ), for the present model $\\hat{p}^{c} = - \\frac{\\sigma }{3}\\hat{\\Theta }^{c}$ is valid.", "With the help of (REF ) we find that the pressure perturbation is not just proportional to the energy-density perturbation but to the derivative of $\\delta ^{c}$ as well: $\\hat{p}^{c} = - \\frac{1}{3}\\frac{p}{1 + \\frac{p}{\\rho }}\\left[a\\delta ^{c\\prime } - 3\\frac{p}{\\rho }\\delta ^{c}\\right]\\,.$ For the later important gauge-invariant combination $\\hat{p}^{c} - \\frac{\\dot{p}}{\\dot{\\rho }}\\rho \\delta ^{c}$ we have $\\hat{p}_{nad} \\equiv \\hat{p} - \\frac{\\dot{p}}{\\dot{\\rho }}\\rho \\delta = \\hat{p}^{c} - \\frac{\\dot{p}}{\\dot{\\rho }}\\rho \\delta ^{c} =- \\frac{1}{3}\\frac{p}{1 + w}\\left[a\\delta ^{c\\prime } + \\frac{3}{2}\\left(1 - w\\right)\\delta ^{c}\\right]\\,.$ This quantity describes the non-adiabatic pressure perturbations.", "With the expression (REF ) for the pressure perturbations, Eq.", "(REF ) takes the final form $\\delta ^{c\\prime \\prime } + \\left[\\frac{3}{2}- 6w - \\frac{1}{3}\\frac{w}{1+ w}\\frac{k^{2}}{a^{2}H^{2}}\\right]\\frac{\\delta ^{c\\prime }}{a}- \\left[\\frac{3}{2} + \\frac{15}{2}w - \\frac{9}{2}w^{2}- \\frac{w^{2}}{1+ w}\\frac{k^{2}}{a^{2}H^{2}}\\right]\\frac{\\delta ^{c}}{a^{2}}= 0\\,.$ Here, the total EoS parameter $w = \\frac{p}{\\rho }$ is explicitly given by $w = \\frac{p}{\\rho } = - \\frac{\\sigma H}{\\rho } = - \\frac{1}{1 + r a^{-3/2}}\\,,$ where $r \\equiv \\frac{\\Omega _{M0} + \\Omega _{B0}}{1 - \\Omega _{M0} - \\Omega _{B0}}\\ $ is the present-time ratio of total matter (DM and baryonic matter) to DE.", "It is remarkable that there appears a scale-dependence in the $\\delta ^{c\\prime }$ term in Eq.", "(REF ).", "A similar feature holds in bulk-viscous models which are characterized by a non-adiabatic dynamics as well [11].", "At high redshifts with $a\\ll 1$ the EoS parameter $w$ tends to zero and (REF ) approaches $\\delta ^{c\\prime \\prime } + \\frac{3}{2}\\frac{\\delta ^{c\\prime }}{a}- \\frac{3}{2}\\frac{\\delta ^{c}}{a^{2}}= 0 \\qquad (a \\ll 1)\\,,$ i.e., we recover the equation for density perturbations in an Einstein-de Sitter universe." ], [ "Relative energy-density perturbations", "As already mentioned, we shall calculate the baryonic matter perturbations via the total energy-density perturbations, governed by Eq.", "(REF ), and the relative energy perturbations $\\frac{\\hat{\\rho }}{\\rho + p} - \\frac{\\hat{\\rho }_{B}}{\\rho _{B}}$ .", "It is the dynamics of this difference which we shall consider in the present subsection.", "Let us consider to this purpose equations (REF ) and (REF ).", "In (REF ) we introduce $D\\equiv \\frac{\\hat{\\rho }}{\\rho + p}\\quad \\Rightarrow \\quad \\delta = D\\left(1+\\frac{p}{\\rho }\\right)\\,,$ in terms of which Eq.", "(REF ) reads $\\dot{D} + \\Theta \\left(\\frac{\\hat{p}}{\\rho + p} - \\frac{\\dot{p}}{\\dot{\\rho }}D\\right)+ \\hat{\\Theta } - \\Theta \\hat{u}^{0} = 0\\,.$ Combining the conservation equation (REF ) for the total energy with the energy conservation (REF ) of the baryons and defining $S_{B} \\equiv D - \\delta _{B}$ , we obtain $\\dot{S}_{B} + \\left(\\hat{\\Theta } - \\hat{\\Theta }_{B}\\right)+ \\Theta \\left(\\frac{\\hat{p}}{\\rho + p} - \\frac{\\dot{p}}{\\dot{\\rho }}D\\right)= 0\\,.$ In the following we shall derive an equation for $S_{B}$ in which this quantity is coupled to the total energy-density perturbations $\\delta ^{c}$ .", "While the physical meaning of $\\delta ^{c}$ is obvious, the situation seems less clear for $S_{B}$ .", "Simply from the definition one has $S_{B}= \\frac{\\rho _{X}}{\\rho _{M} + \\rho _{B}}\\delta _{X} + \\frac{\\rho _{M}}{\\rho _{M} + \\rho _{B}}\\left(\\delta _{M} - \\delta _{B}\\right)\\,.$ If the DE perturbations can be neglected, which is the case in many situations (cf.", "[8]), one has $S_{B} \\propto \\delta _{M} - \\delta _{B}$ .", "Thus it represents a measure for the difference in the fractional perturbations of DM and baryonic matter.", "It is useful as an auxiliary quantity since both the total energy-momentum and the baryon energy-momentum are conserved.", "According to the expressions (REF ) and (REF ) the difference between the quantities $\\hat{\\Theta }$ and $\\hat{\\Theta }_{B}$ is $\\hat{\\Theta } - \\hat{\\Theta }_{B} = \\frac{1}{a^{2}}\\Delta \\left(v - v_{B}\\right)\\,.$ Differentiating equation (REF ) and using the definition of $\\hat{p}_{nad}$ in (REF ) results in $\\ddot{S}_{B} + \\left(\\hat{\\Theta } - \\hat{\\Theta }_{B}\\right)^{\\displaystyle \\cdot }+ \\left[\\Theta \\frac{\\hat{p}_{nad}}{\\rho + p}\\right]^{\\displaystyle \\cdot } = 0\\,.$ To deal with the time-derivative of expression (REF ) we consider the momentum conservations (REF ) and (REF ) which, at first order, can be written as $\\dot{v} + \\phi = - \\frac{\\hat{p}^{c}}{\\rho + p}\\ \\quad \\mathrm {and } \\quad \\dot{v}_{B} + \\phi = 0\\,,$ respectively.", "It follows that $\\left(v - v_{B}\\right)^{\\displaystyle \\cdot } = - \\frac{\\hat{p}^{c}}{\\rho + p}\\,.$ With (REF ) and (REF ) the resulting $k$ -space equation for $S_{B}$ is $\\ddot{S}_{B}+ 2 H \\dot{S}_{B} +\\frac{k^{2}}{a^{2}}\\frac{\\hat{p}^{c}}{\\rho + p}+ \\left[3H \\frac{\\hat{p}^{c}_{nad}}{\\rho + p}\\right]^{\\displaystyle \\cdot }+ 6H^{2}\\frac{\\hat{p}^{c}_{nad}}{\\rho + p} = 0\\,.$ Introducing the explicit expressions (REF ) and (REF ), use of (REF ) to eliminate the second derivative of $\\delta ^{c}$ provides us with $S_{B}^{\\prime \\prime } + \\frac{3}{2}\\left(1-w\\right)\\frac{S_{B}^{\\prime }}{a}&=& \\frac{w}{\\left(1 + w\\right)^{2}} \\left[\\left(3 + \\frac{3}{2}w + \\frac{1}{3}\\frac{1 + 2w}{1+w}\\frac{k^{2}}{a^{2}H^{2}}\\right)\\frac{\\delta ^{c\\prime }}{a}\\qquad \\qquad \\qquad \\right.", "\\nonumber \\\\&&\\left.", "\\qquad \\qquad + \\left(\\frac{9}{2} - \\frac{9}{4}w - \\frac{9}{4}w^{2} - w\\frac{1 + 2w}{1+w}\\frac{k^{2}}{a^{2}H^{2}}\\right)\\frac{\\delta ^{c}}{a^{2}}\\right]\\,.$ The total density perturbation $\\delta ^{c}$ and its first derivative appear as inhomogeneities in the equation for $S_{B}$ .", "Eqs.", "(REF ) and (REF ) are the key equations of this paper.", "In the next section we demonstrate how a solution of the coupled system (REF ) and (REF ) will allow us to obtain the perturbations of the baryon fluid.", "It is expedient to notice that for $a \\ll 1$ one has $w\\approx 0$ and the total cosmic medium behaves as dust.", "Under this condition the right-hand side of Eq.", "(REF ) vanishes and we can use $S_{B} = $ const $\\approx 0$ as initial condition for the numerical analysis." ], [ "Baryonic energy-density perturbations", "By definition, the fractional baryonic energy-density perturbations $\\delta _{B} = \\frac{\\hat{\\rho }_{B}}{\\rho _{B}}$ are determined by $D$ and $S_{B}$ , $\\delta _{B} = D - S_{B}\\,.$ Since $S_{B}$ is gauge-invariant by itself, we may write $S_{B} = D - \\frac{\\hat{\\rho }_{B}}{\\rho _{B}} = \\frac{\\delta ^{c}}{1+w} - \\delta _{B}^{c}\\,,$ where $\\delta _{B}^{c} = \\delta _{B} + \\frac{\\dot{\\rho }_{B}}{\\rho _{B}}v = \\delta _{B} - \\Theta v\\,.$ Consequently, the comoving (with $v$ ) baryon energy-density perturbations are given by the combination $\\delta _{B}^{c} = \\frac{\\delta ^{c}}{1+w} - S_{B}\\,.$ Figure: (a) Constitution data set with MLCS17 fitter combined with BAO and the position of the first acoustic peak.", "(b) The same as in (a) with LSS data added.", "The dashed and continuous contour lines refer to the 1σ1\\sigma and 2σ2\\sigma confidence regions, respectively.", "The blue regions indicate the results of the joint tests at the 2σ2\\sigma level.It seems more convenient, however, to consider the perturbations of the baryon fluid with respect to the velocity potential $v_{B}$ of the baryon component itself.", "These perturbations are obtained via $\\delta _{B}^{c_{B}} \\equiv \\delta _{B} - \\Theta v_{B} = \\delta _{B}^{c} + \\Theta \\left(v - v_{B}\\right)\\,.$ Use of (REF ) with (REF ) and (REF ) leads to $\\frac{k^{2}}{a^{2}}\\left(v - v_{B}\\right) = \\dot{S}_{B} + 3H \\frac{\\hat{p}_{nad}}{\\rho + p}\\,.$ For $\\delta _{B}^{c_{B}}$ we obtain $\\delta _{B}^{c_{B}} = \\delta _{B}^{c} + 3 \\frac{a^{2}H^{2}}{k^{2}}\\left[aS_{B}^{\\prime }+ \\frac{\\hat{p}_{nad}}{\\rho + p}\\right]\\,.$ Equation (REF ) establishes a relation between perturbations measured by an observer, comoving with the baryon fluid and perturbations measured by an observer, comoving with the total velocity of the cosmic substratum.", "Obviously, the difference between both quantities depends on the perturbation scale.", "On small scales $\\frac{a^{2}H^{2}}{k^{2}} \\ll 1$ one has $\\delta _{B}^{c_{B}} \\approx \\delta _{B}^{c}$ , i.e., the difference is negligible.", "Explicitly, $\\delta _{B}^{c_{B}}$ is given in terms of $\\delta ^{c}$ and $S_{B}$ and their first derivatives by $\\delta _{B}^{c_{B}} = \\frac{\\delta ^{c}}{1+w} - S_{B}+ 3 \\frac{a^{2}H^{2}}{k^{2}}\\left[aS_{B}^{\\prime }- \\frac{w}{3}\\frac{1}{\\left(1+w\\right)^{2}}\\left(a\\delta ^{c\\prime } + \\frac{3}{2}\\left(1-w\\right)\\delta ^{c}\\right)\\right]\\,.$ One has to solve now Eq.", "(REF ) for $\\delta ^{c}$ and afterwards equation (REF ) for $S_{B}$ , in which $\\delta ^{c}$ and its first derivative appear as inhomogeneities.", "The coefficients are given by (REF ) and (REF ).", "The initial conditions at high redshift are determined by the Einstein - de Sitter type behavior (REF ) with $S_{B} \\approx 0$ , equivalent to an almost adiabatic behavior.", "The perturbations of the baryonic component then are found by the combinations (REF ) or (REF ).", "As already mentioned, because of the factor $\\frac{a^{2}H^{2}}{k^{2}}$ in front of the last term on the right-hand side of (REF ) one expects negligible differences between $\\delta _{B}^{c_{B}}$ and $\\delta _{B}^{c}$ on sub-horizon scales $k^{2} \\ll a^{2}H^{2}$ .", "Figure: Data as in Fig.", "1, here with SALT II fitter." ], [ "Observational analysis", "As far as the background dynamics is concerned, the explicit inclusion of a baryon component does not significantly change the Hubble rate (REF ).", "It is only the combination $\\Omega _{M0} + \\Omega _{B0}$ which matters.", "For our background tests, which in part are updates of previous studies, we have considered data from SNIa (Constitution [16] and Union 2.1 [17]), BAO [18], [19], [20] and the position of the first acoustic peak of the CMB spectrum [21], [22].", "For a more complete analysis of the SNIa samples and to test the robustness of the results, we use both the fitters Multicolor Light Curve Shapes (MLCS) [23] and Spectral Adaptive Lightcurve Template (SALT II) [24], [25].", "As is well known, SNIa tests are using the luminosity distance modulus $\\mu =5\\log d_L(z)+\\mu _0 \\ $ with $\\mu _0=42.384-5\\log h $ , where $d_{L}=\\left(z+1\\right)H_0\\int _{0}^{z}\\frac{dz^{\\prime }}{H\\left(z^{\\prime }\\right)}\\,$ and $h$ is given by $H_0 = 100 h \\mathrm {km s^{-1} Mpc^{-1}}$ .", "Tests against BAO data are based on the geometric quantity [18], [19], [20] $D_{v}\\left(z\\right)=\\left[\\left(1+z\\right)^{2}d_{A}^{2}\\frac{z}{H\\left(z\\right)}\\right]^{1/3} z\\,,$ where $d_{A}$ is the angular-diameter distance.", "Concerning the position of the first acoustic peak of the CMB anisotropy spectrum, we rely on the distance scale [26], [27], Figure: (a) Union 2.1 data set with SALT II fitter combined with BAO and the position of the first acoustic peak.", "(b) The same as in (a) with LSS data added.", "The dashed and continuous contour lines refer to the 1σ1\\sigma and 2σ2\\sigma confidence regions, respectively.", "The blue regions indicate the results of the joint tests at the 2σ2\\sigma level.$l_{1}=l_{A}\\left(1-\\delta _{1}\\right)\\,.$ Here, $l_{A}$ is the acoustic scale ($c_{s}$ is the sound speed) $l_{A}=\\pi \\frac{\\int \\frac{dz}{H(z)}}{\\int c_{s}^{2}\\frac{dz}{H(z)}}, \\qquad c_{s}^{2}=\\sqrt{3+\\frac{9\\Omega _{B0}}{4\\Omega _{R0}}z^{-1}}\\,$ and $\\delta _{1}\\approx 0.267\\left(\\frac{10\\Omega _{R_{0}}}{3\\Omega ^{2}_{m_{0}}}\\right)^{1/10}$ is a correction term, adapted to the decaying vacuum model [12].", "At the perturbative level we consider the LSS data of Ref.", "[28] and calculate the baryonic power spectrum $P_k\\propto \|\\delta _B\|^2$ .", "Figure: Left panel: baryonic matter-power spectrum with different values of Ω M0 \\Omega _{M0}.Values between 0.280.28 and 0.360.36 are in reasonable agreement with the LSS data (SDSS DR7).Notice that these values are considerably lower than those found in , , (∼0.37-0.43\\sim 0.37 -0.43)without a separate baryon component.Right panel: best-fit power spectra for the Λ(t)\\Lambda (t)CDM and Λ\\Lambda CDM models.For our tests we perform a $\\chi ^2$ analysis, using $\\chi ^{2}(\\theta )=\\sum \\limits _{i=1}^{N}\\frac{\\left[y_{i}-y\\left(x_{i}\\vert \\theta \\right)\\right]^{2}}{\\sigma _{i}^{2}}\\,.$ Here, the $y_{i}$ are the observational data (SNIa, CMB, BAO, LSS) which are compared with the theoretical predictions $y(x_{i}\\vert \\theta )$ , where $\\theta $ represents a set of model parameters and $\\sigma _i$ denotes the error bars.", "Out of $\\chi ^{2}$ in (REF ) one defines the probability distribution function (PDF) $\\mathcal {P}\\propto \\exp \\left(-\\frac{\\chi ^{2}(\\theta )}{2}\\right)$ .", "For the present model the set of parameters is $\\theta =(h, \\Omega _{B0}, \\Omega _{M0})$ .", "In a first step, however, we fix the baryon abundance in agreement with primordial nucleosynthesis.", "Under this condition the free parameters are the same as in the $\\Lambda $ CDM model, namely $h$ and the DM abundance $\\Omega _{M0}$ .", "Our results are presented in figures 1 - 3.", "The dashed and continuous contour lines in all these figures refer to the $1\\sigma $ and $2\\sigma $ confidence levels (CL), respectively.", "Fig.", "1(a) shows the $h$ - $\\Omega _{M_{0}}$ plane based on the Constitution data with MLCS17 fitter combined with data from BAO and the position of the first acoustic peak of the CMB.", "In Fig.", "1(b) we have added LSS data to the background tests of Fig. 1(a).", "In both cases blue regions mark the results of the joint tests at the $2\\sigma $ CL.", "Figures 2(a) and 2(b) visualize the $h$ - $\\Omega _{M_{0}}$ plane for the same data as in Figs.", "1(a) and 1(b), but with SALT II fitter.", "In Figs.", "3(a) and 3(b) the corresponding curves for the Union 2.1 sample are presented.", "Again, in both cases blue regions indicate the results of the joint tests at 2$\\sigma $ CL.", "Our background tests largely reproduce previous results [12].", "Only that our value for the position of the first acoustic peak differs slightly from the result of [12].", "In our case the baryon abundance is fixed both in the Hubble rate and in the expression for the sound speed, in [12] it is fixed only for calculating the sound speed.", "The best-fit values for the background tests alone are summarized in Table I where we compare our model with the $\\Lambda $ CDM model via their $\\chi _{\\nu }^{2}$ values (reduced $\\chi ^{2}$ values).", "For the joint background and LSS tests we find the best-fit values in Table II.", "Table: Best fit values at the 2σ\\sigma CL using background tests (SNIa, BAO, CMB).Figure: PDFs for the baryon fraction Ω B0 \\Omega _{B0} (left panel) and the DM fraction Ω M0 \\Omega _{M0} (central panel) based on the LSS data.The right panel shows the Ω B0 \\Omega _{B0}-Ω M0 \\Omega _{M0} plane with the 1σ1\\sigma , 2σ2\\sigma and 3σ3\\sigma contour lines.The dot indicates the best-fit values Ω B0 =0.05±0.02\\Omega _{B0}= 0.05\\pm 0.02 and Ω M0 =0.35±0.03\\Omega _{M0}=0.35\\pm 0.03 at the2σ2\\sigma CL.Our analysis confirms that the decaying $\\Lambda $ model predicts a higher value of the current DM abundance than the $\\Lambda $ CDM model.", "Interestingly, from the LSS data alone we find (at the $2\\sigma $ CL) $\\Omega _{M0}=0.32\\pm 0.04$ , a lower value than for the model without a separate baryon component [12], [8], [10], although still higher than in the $\\Lambda $ CDM model.", "The $\\chi ^2_{\\nu }$ values in Table I reveal that, as far as the background dynamics is concerned, our $\\Lambda (t)$ CDM model is competitive with the $\\Lambda $ CDM model.", "On the other hand, comparing the results for the baryon power spectrum, the situation changes.", "While for the data from the 2dFGRS project [29] we find $\\chi _{\\nu }^{2} \\approx 0.91$ for the $\\Lambda (t)$ CDM model and $\\chi _{\\nu }^{2} \\approx 0.96$ for $\\Lambda $ CDM, the SDSS DR7 data with their much smaller error bars clearly favor the $\\Lambda $ CDM model with $\\chi _{\\nu }^{2} \\approx 0.93$ compared with $\\chi _{\\nu }^{2} = 3.63$ of the decaying $\\Lambda $ model.", "The left panel of Fig.", "4 visualizes the baryonic power spectrum confronted with the SDSS DR7 data for different values of $\\Omega _{M0}$ .", "The best-fit power spectra for both models are shown in Fig.", "4.", "One should keep in mind here that in obtaining the spectrum the BBKS transfer function [30] was used which naturally favors the $\\Lambda $ CDM model.", "Figure: Two-dimensional contour plots for the abundances of baryons and DM.", "(a) Joint analysis with data from LSS, CMB, BAO and Constitution SNIa data with SALT II fitter.", "(b) Same data as in (a) with MLSCk2 fitter.", "(c) Joint analysis with data from LSS, CMB, BAO and Union 2.1 SNIa data.", "The results for the baryon abundance(see Table III) are in agreement with primordial nucleosynthesis.In the tests so far the baryon fraction $\\Omega _{B0}$ was assumed to be given.", "Now we relax this assumption and consider $\\Omega _{B0}$ and $\\Omega _{M0}$ to be free parameters.", "Performing a statistical analysis of the LSS data with $h=0.7$ as a prior (in concordance with our result for the Union2.1 based background test in Tab.", "I), we obtain the the two-dimensional curves in the right panel of Fig.", "5 with the best-fit values $\\Omega _{B0}= 0.05\\pm 0.02$ ($2\\sigma $ CL) and $\\Omega _{M0}= 0.35\\pm 0.03$ ($2\\sigma $ CL).", "The one-dimensional PDF for $\\Omega _{B0}$ (left panel of Fig.", "5) is then found by fixing $\\Omega _{M0}= 0.35$ , the corresponding plot for $\\Omega _{M0}$ (central panel) by fixing $\\Omega _{B0}= 0.05$ .", "The same PDFs follow for a prior $h=0.65$ , indicating that these results do not depend strongly on the specific choice of the prior.", "Remarkably, the best-fit value $\\Omega _{B0}= 0.05\\pm 0.02$ ($2\\sigma $ CL) is found to be in agreement with the result from nucleosynthesis and, at the same time, also demonstrates the consistency of our approach.", "In a next step we performed an enlarged analysis using the entire set of data (SNIa, CMB, BAO and LSS).", "This enlarged analysis (see Fig.", "6) confirms the LSS-based results of Fig. 5.", "The left panel of Fig.", "6 shows the two-dimensional contour plots resulting from a joint test with LSS, CMB, BAO and the Constitution data with SALT II fitter.", "Figure 6(b) was obtained with the same data but now with MLSC17 fitter.", "On the basis of the Union 2.1 data we found the results in Fig. 6(c).", "The best-fit values for the baryon and DM abundances are summarized in Table III.", "We conclude that our results for the baryon abundance are in agreement with the results from nucleosynthesis at the 2$\\sigma $ CL.", "The consistent reproduction of the cosmic baryon abundance on the basis of data from LSS and background tests is a main achievement of this paper.", "Table: Best fit values at the 2σ\\sigma CL using joint tests (SNIa, BAO, CMB, LSS).Table: Best-fit values for the Λ\\Lambda (t)CDM model at the 2σ\\sigma CL using data from SNIa, CMB, BAO and LSS, considering DM and baryon abundances as free parameters." ], [ "Conclusions", "The components of the cosmological dark sector, DM and DE, are dominating the overall dynamics of the Universe.", "The small baryonic fraction of presently less than 5% of the energy budget does only marginally influence the homogeneous and isotropic expansion history.", "With the help of data from SNIa, BAO and the position of the first peak of the CMB anisotropy spectrum we updated and confirmed previous results for the background.", "But as far as structure formation is concerned, the situation is different.", "The directly observed inhomogeneous matter distribution in the Universe is the distribution of visible, i.e., baryonic matter.", "While the standard scenario according to which the baryons after radiation decoupling are falling into the potential wells created by the DM inhomogeneities may suggest a similar distribution of DM and baryonic matter, the situation less clear if DM is in (non-gravitational) interaction with DE, while the (directly) observed baryon component is separately conserved.", "We have carried out a detailed gauge-invariant perturbation analysis for the baryon fluid in a $\\Lambda (t)$ CDM cosmology in which a cosmological term is decaying into DM linearly with the Hubble rate.", "Our key result is an expression for the fractional baryon energy-density perturbation for an observer comoving with the baryon fluid.", "Using the LSS data of the SDSS DR7 project we obtained the PDF for the baryon abundance of the Universe independently of the DM abundance.", "The best-fit value of this abundance is $\\Omega _{B0} = 0.05 \\pm 0.02$ (2$\\sigma $ CL) in remarkable agreement with the result from primordial nucleosnthesis.", "A combined analysis, including also data from SNIa, BAO and CMB confirms this result.", "For the best-fit value of the DM abundance we found $\\Omega _{M0}=0.32\\pm 0.02$ ($2\\sigma $ CL) from the combined analysis (LSS+BAO+SNIa(Union2.1)+CMB) and $\\Omega _{M0}= 0.35\\pm 0.03$ ($2\\sigma $ CL) from the LSS data alone.", "These values are higher than those for the standard model but smaller than the corresponding value for a $\\Lambda (t)$ CDM model without a separately conserved baryon component.", "Generally, the explicit inclusion of the baryon fluid improves the concordance between background and perturbation dynamics.", "Our results indicate that the investigated $\\Lambda (t)$ CDM cosmology, which does not have a $\\Lambda $ CDM limit, has a competitive background dynamics but as far as the baryon matter power spectrum is concerned, the $\\Lambda $ CDM model is clearly favored.", "We thank Saulo Carneiro and Júlio Fabris for helpful discussions.", "Financial support by CAPES, FAPES and CNPq is gratefully acknowledged.", "WSHR is thankful to FAPES for the grant (BPC No 476/2013), under which this work was carried out." ] ]	1403.0427
[ [ "Measurement of event-plane correlations in sqrt(s_NN)=2.76 TeV lead-lead\n collisions with the ATLAS detector" ], [ "Abstract A measurement of event-plane correlations involving two or three event planes of different order is presented as a function of centrality for 7 ub-1 Pb+Pb collision data at sqrt(s_NN)=2.76 TeV, recorded by the ATLAS experiment at the LHC.", "Fourteen correlators are measured using a standard event-plane method and a scalar-product method, and the latter method is found to give a systematically larger correlation signal.", "Several different trends in the centrality dependence of these correlators are observed.", "These trends are not reproduced by predictions based on the Glauber model, which includes only the correlations from the collision geometry in the initial state.", "Calculations that include the final-state collective dynamics are able to describe qualitatively, and in some cases also quantitatively, the centrality dependence of the measured correlators.", "These observations suggest that both the fluctuations in the initial geometry and non-linear mixing between different harmonics in the final state are important for creating these correlations in momentum space." ], [ "Introduction", "Heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) create hot and dense matter that is thought to be composed of strongly interacting quarks and gluons.", "One striking observation that supports this picture is the large momentum anisotropy of particle emission in the transverse plane.", "This anisotropy is believed to be the result of anisotropic expansion of the created matter driven by the pressure gradients, with more particles emitted in the direction of the largest gradients [1].", "The collective expansion of the matter can be modeled by relativistic viscous hydrodynamic theory [2].", "The magnitude of the azimuthal anisotropy is sensitive to transport properties of the matter, such as the ratio of the shear viscosity to the entropy density and the equation of state [3].", "The anisotropy of the particle distribution ($\\frac{dN}{d\\phi }$ ) in azimuthal angle $\\phi $ is customarily characterized by a Fourier series: $\\frac{dN}{d\\phi }\\propto 1+2\\sum _{n=1}^{\\infty }v_{n}\\cos n(\\phi -\\Phi _{n})\\;,$ where $v_n$ and $\\Phi _n$ represent the magnitude and phase (referred to as the event plane) of the $n^{\\mathrm {th}}$ -order azimuthal anisotropy (or flow) at the corresponding angular scale.", "These quantities can also be conveniently represented in a two-dimensional vector format or in the standard complex form [4], [5]: ${v}_n=(v_n\\cos n\\Phi _{n},v_n\\sin n\\Phi _{n})\\;\\; {\\mathrm {or}}\\;\\;v_n e^{i n\\Phi _n}\\;.$ In non-central collisions, the overlap region of the initial geometry has an almost elliptic shape.", "The anisotropy is therefore dominated by the second harmonic term, $v_2$ .", "However, first-order ($n=1$ ) and higher-order ($n>2$ ) $v_n$ coefficients have also been observed [6], [7], [8].", "These coefficients have been related to additional shape components arising from the fluctuations of the positions of nucleons in the overlap region.", "The amplitude and the directions of these shape components can be estimated via a simple Glauber model [9] from the transverse positions $(r,\\phi )$ of the participating nucleons relative to their center of mass [10]: $\\epsilon _n &=& \\frac{\\sqrt{\\langle r^n\\cos n\\phi \\rangle ^2+\\langle r^n\\sin n\\phi \\rangle ^2}}{\\langle r^n\\rangle },\\\\n\\Phi _n^&=&\\arctan \\left(\\frac{\\langle r^n\\sin n\\phi \\rangle }{\\langle r^n\\cos n\\phi \\rangle }\\right) +\\pi \\;,$ where $\\epsilon _n$ is the eccentricity and the angle $\\Phi ^_n$ is commonly referred to as the participant-plane (PP) angle.", "These shape components are transferred via hydrodynamic evolution into higher-order azimuthal anisotropy in momentum space.", "For small $\\epsilon _n$ values, one expects $v_n\\propto \\epsilon _n$ , and the $\\Phi _n$ to be correlated with the minor-axis direction given by $\\Phi _n^$ .", "However, model calculations show that the values of $\\epsilon _n$ are large, and the alignment between $\\Phi _n$ and $\\Phi _n^$ is strongly violated for $n>3$ due to non-linear effects in the hydrodynamic evolution [11].", "Detailed measurements of $v_n$ have been performed at RHIC and the LHC, and non-zero $v_n$ values are observed for $n\\le 6$ [12], [13], [14], [6], [8], [15], [7], [16], consistent with the existence of sizable fluctuations in the initial state.", "Further information on these fluctuations can be obtained by studying the correlations between $\\Phi _{n}$ of different order.", "If the fluctuations in $\\epsilon _n$ are small and totally random, the orientations of $\\Phi _{n}$ of different order are expected to be uncorrelated.", "Calculations based on the Glauber model reveal strong correlations between some PP angles such as $\\Phi _2^$ and $\\Phi _4^$ or $\\Phi _2^$ and $\\Phi _6^$ [10], and weak correlations between others such as $\\Phi _2^$ and $\\Phi _3^$ or $\\Phi _2^$ and $\\Phi _5^$ [17].", "Previous measurements at RHIC and the LHC support a weak correlation between $\\Phi _2$ and $\\Phi _3$ [19], [20] and a strong correlation between $\\Phi _2$ and $\\Phi _4$ [18].", "The former is consistent with no strong correlation between $\\Phi _2^$ and $\\Phi _3^$ and the dominance of linear response for elliptic flow and triangular flow, i.e.", "$\\Phi _2\\approx \\Phi _2^$ and $\\Phi _3\\approx \\Phi _3^$ .", "The latter is consistent with a significant non-linear hydrodynamic response for quadrangular flow, which couples $v_4$ to $v_2^2$ .", "The correlations among three event planes of different order have also been investigated in a model framework and several significant correlators have been identified [10], [21], [22], [23].", "However, no published experimental measurements on three-plane correlations exist to date.", "A measurement of the correlations between two and three event planes can shed light on the patterns of the fluctuations of the initial-state geometry and non-linear effects in the final state." ], [ "ATLAS Detector and trigger", "The ATLAS detector [24] provides nearly full solid angle coverage of the collision point with tracking detectors, calorimeters and muon chambers, which are well suited for measurements of azimuthal anisotropies over a large pseudorapidity range.", "ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the $z$ -axis along the beam pipe.", "The $x$ -axis points from the IP to the center of the LHC ring, and the $y$ -axis points upward.", "Cylindrical coordinates $(r,\\phi )$ are used in the transverse plane, $\\phi $ being the azimuthal angle around the beam pipe.", "The pseudorapidity is defined in terms of the polar angle $\\theta $ as $\\eta =-\\ln \\tan (\\theta /2)$ .", "This analysis primarily uses three subsystems to measure the event plane: the inner detector (ID), the barrel and endcap electromagnetic calorimeters (ECal) and the forward calorimeter (FCal).", "The ID is contained within the 2 T field of a superconducting solenoid magnet, and measures the trajectories of charged particles in the pseudorapidity range $\|\\eta \|<2.5$ and over the full azimuth.", "A charged particle passing through the ID typically traverses three modules of the silicon pixel detector (Pixel), four double-sided silicon strip modules of the semiconductor tracker (SCT), and a transition radiation tracker for $\|\\eta \|<2$ .", "The electromagnetic energy measurement in the ECal is based on a liquid-argon sampling technology.", "The FCal uses tungsten and copper absorbers with liquid argon as the active medium, and has a total thickness of about ten interaction lengths.", "The ECal covers the pseudorapidity range $\|\\eta \|<3.2$ , and the FCal extends the calorimeter coverage to $\|\\eta \|< 4.9$ .", "The energies in the ECal and FCal are reconstructed and grouped into towers with segmentation in pseudorapidity and azimuthal angle of $\\Delta \\eta \\times \\Delta \\phi =0.1\\times 0.1$ to $0.2\\times 0.2$ , which are then used to calculate the event plane.", "The procedure for obtaining the event-plane correlations is found to be insensitive to the segmentation and energy calibration of the calorimeters.", "The minimum-bias Level-1 trigger [25] used for this analysis requires a signal in each of two zero-degree calorimeters (ZDC) or a signal in either one of the two minimum-bias trigger scintillator (MBTS) counters.", "The ZDC is positioned at 140 m from the collision point, detecting neutrons and photons with $\|\\eta \|>8.3$ , and the MBTS covers $2.1<\|\\eta \|<3.9$ .", "The ZDC Level-1 trigger thresholds on each side are set below the peak corresponding to a single neutron.", "A Level-2 timing requirement based on signals from each side of the MBTS is imposed to suppress beam backgrounds [25]." ], [ "Event and track selections", "This paper is based on Pb+Pb collision data collected in 2010 at the LHC with a nucleon–nucleon center-of-mass energy $\\sqrt{s_{_{\\mathrm {NN}}}}=2.76$ TeV.", "The data correspond to an integrated luminosity of approximately 7 $\\mu \\mathrm {b}^{-1}$ .", "In order to suppress non-collision backgrounds, an offline event selection requires a reconstructed primary vertex with at least three associated charged tracks reconstructed in the ID and a time difference $\|\\Delta t\| < 3$ ns between the MBTS trigger counters on either side of the interaction point.", "A coincidence between the two ZDCs at forward and backward pseudorapidity is required to reject a variety of background processes, while maintaining high efficiency for non-Coulomb processes.", "Events satisfying these conditions are required to have a reconstructed primary vertex with $z_{\\mathrm {vtx}}$ within 150 mm of the nominal center of the ATLAS detector.", "The pile-up probability is estimated to be at the $10^{-4}$ level and is therefore negligible.", "About 48 million events pass the requirements for the analysis.", "The Pb+Pb event centrality is characterized using the total transverse energy ($\\Sigma $ ) deposited in the FCal over the pseudorapidity range $3.2 < \|\\eta \| < 4.9$ and measured at the electromagnetic energy scale [26].", "A larger $\\Sigma $ value corresponds to a more central collision.", "From an analysis of the $\\Sigma $ distribution after applying all trigger and event selection criteria, the sampled fraction of the total inelastic cross section has been estimated to be (98$\\pm $ 2)% in a previous analysis [27].", "The uncertainty in this estimate is evaluated by varying the trigger criteria, event selection and background rejection requirements on the FCal $\\Sigma $ distribution [27].", "The FCal $\\Sigma $ distribution is divided into a set of 5%-wide percentile bins, together with five 1%-wide bins for the most central 5% of the events.", "A centrality interval refers to a percentile range, starting at 0% for the most central collisions.", "Thus the 0%–1% centrality interval corresponds to the most central 1% of the events.", "A standard Glauber model Monte Carlo analysis [9] is used to estimate the average number of participating nucleons, $\\langle N_{\\mathrm {part}}\\rangle $ , and its associated systematic uncertainties for each centrality interval [27].", "These numbers are summarized in Table REF .", "Table: The list of centrality intervals and associated 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle values used in this paper.", "The systematic uncertainties are taken from Ref.", ".The event plane is also measured by the ID, using reconstructed tracks with $>0.5$ GeV and $\|\\eta \|<2.5$ [8].", "To improve the robustness of track reconstruction in the high-multiplicity environment of heavy-ion collisions, more stringent requirements on track quality, compared to those defined for proton–proton collisions [28], are used.", "At least nine hits in the silicon detectors are required for each track, with no missing Pixel hits and not more than one missing SCT hit, excluding the known non-operational modules.", "In addition, at its point of closest approach the track is required to be within 1 mm of the primary vertex in both the transverse and longitudinal directions [29].", "The track reconstruction performance is studied by comparing data to Monte Carlo calculations based on the HIJING event generator [30] and a full GEANT4 simulation of the detector [31], [32].", "The track reconstruction efficiency ranges from 72% at $\\eta = 0$ to 51% for $\|\\eta \| > 2$ in peripheral collisions, while it ranges from 72% at $\\eta = 0$ to about 42% for $\|\\eta \| > 2$ in central collisions [33].", "However, the event-plane correlation results are found to be insensitive to the reconstruction efficiency (see Sec.", "REF )." ], [ "Experimental observables", "The $n^{\\mathrm {th}}$ -order harmonic has a $n$ -fold symmetry in azimuth, and is thus invariant under the transformation $\\Phi _n\\rightarrow \\Phi _n+2\\pi /n$ .", "Therefore, a general definition of the relative angle between two event planes, $a_n\\Phi _n+a_m\\Phi _m$ , has to be invariant under a phase shift $\\Phi _l\\rightarrow \\Phi _l+2\\pi /l$ .", "It should also be invariant under a global rotation by any angle.", "The first condition requires $a_n$ ($a_m$ ) to be multiple of $n$ ($m$ ), while the second condition requires the sum of the coefficients to vanish: $a_n+a_m=0$ .", "The relative angle $\\Phi _{n,m}=k(\\Phi _n-\\Phi _m)$ , with $k$ being the least common multiple (LCM) of $n$ and $m$ , satisfies these constraints, as does any integer multiple of $\\Phi _{n,m}$ .", "The correlation between $\\Phi _n$ and $\\Phi _m$ is completely described by the differential distribution of the event yield $dN_{\\mathrm {evts}}/(d\\left(k(\\Phi _n-\\Phi _m)\\right)$ .", "This distribution must be an even function due to the symmetry of the underlying physics, and hence can be expanded into the following Fourier series: $\\frac{dN_{\\mathrm {evts}}}{d\\left(k(\\Phi _n-\\Phi _m)\\right)}&\\propto & 1+2\\sum _{j=1}^{\\infty } V_{n,m}^j \\cos jk(\\Phi _n-\\Phi _m)\\;,\\\\V_{n,m}^j &=& \\langle \\cos jk(\\Phi _n-\\Phi _m)\\rangle \\;.$ The measurement of the two-plane correlation is thus equivalent to measuring a set of cosine functions $\\langle \\cos jk(\\Phi _n-\\Phi _m)\\rangle $ averaged over many events [22].", "This discussion can be generalized for correlations involving three or more event planes.", "The multi-plane correlators can be written as $\\langle \\cos (c_1\\Phi _{1}+2c_2\\Phi _{2}...+lc_l\\Phi _{l})\\rangle $ with the constraint [21], [23]: $c_1+2c_2...+lc_l=0\\;,$ where the coefficients $c_n$ are integers.", "The two-plane correlators defined in Eq.", "() satisfy this constraint.", "For convenience, correlation involving two event planes $\\Phi _n$ and $\\Phi _m$ is referred to as “$n$ -$m$ ” correlation, and one involving three event planes $\\Phi _n$ , $\\Phi _m$ and $\\Phi _h$ as “$n$ -$m$ -$h$ ” correlation.", "The multi-plane correlators can always be decomposed into a linear combination of several two-plane relative angles and they carry additional information not accessible through two-plane correlators [22].", "Experimentally the $\\Phi _n$ angles are estimated from the observed event-plane angles, $\\Psi _{n}$ , defined as the directions of the “flow vectors” ${q}_n$ , which in turn are calculated from the azimuthal distribution of particles in the calorimeter or the ID: $\\nonumber {q}_n =(q_{x,n},q_{y,n}) &=& \\frac{1}{\\Sigma u_i}\\left(\\textstyle \\Sigma [u_i\\cos n\\phi _i]-\\langle \\Sigma [u_i\\cos n\\phi _i]\\rangle _{\\mathrm {evts}}, \\Sigma [u_i\\sin n\\phi _i]-\\langle \\Sigma [u_i\\sin n\\phi _i]\\rangle _{\\mathrm {evts}}\\right)\\;,\\\\ \\tan n\\Psi _n &=& \\frac{q_{y,n}}{q_{x,n}}\\;.$ Here the weight $u_i$ is either the $$ of the $i^{\\mathrm {th}}$ tower in the ECal and the FCal or the $$ of the $i^{\\mathrm {th}}$ reconstructed track in the ID.", "Subtraction of the event-averaged centroid, $(\\langle \\Sigma [u_i\\cos n\\phi _i]\\rangle _{\\mathrm {evts}},\\langle \\Sigma [u_i\\sin n\\phi _i]\\rangle _{\\mathrm {evts}})$ , in Eq.", "(REF ) removes biases due to detector effects [34].", "For example, a localized inefficiency over a $\\phi $ region in the detector would lead to a non-zero average $\\protect {q}_n$ .", "The subtraction corrects this bias.", "A standard flattening technique [35] is then used to remove the small residual non-uniformities in the distribution of $\\Psi _{n}$ .", "The ${q}_n$ defined this way, when averaged over events with the same $\\Phi _n$ , is insensitive to the energy scale in the calorimeter or the momentum scale in the ID and to any random smearing effect.", "In the limit of infinite multiplicity it approaches the single-particle flow weighted by $u$ : ${q}_n\\rightarrow \\left({v}_n\\right)_u= \\Sigma u_i({v}_n)_i/\\Sigma u_i$ .", "The correlators in terms of $\\Phi _n$ can be obtained from the correlations between the measured angles $\\Psi _{n}$ divided by a resolution term [22]: $\\nonumber \\langle \\cos (c_1\\Phi _{1}+2c_2\\Phi _{2}+...+lc_l\\Phi _{l})\\rangle &=& \\frac{\\langle \\cos (c_1\\Psi _{1}+2c_2\\Psi _{2}+...+lc_l\\Psi _{l})\\rangle }{\\mathrm {Res}\\lbrace c_1\\Psi _1\\rbrace \\mathrm {Res}\\lbrace c_22\\Psi _2\\rbrace ...\\mathrm {Res}\\lbrace c_ll\\Psi _l\\rbrace }\\\\\\mathrm {Res}\\lbrace c_nn\\Psi _n\\rbrace &=& \\sqrt{\\left\\langle \\left(\\cos c_nn(\\Psi _n-\\Phi _n)\\right)^2\\right\\rangle }\\;.$ The resolution factors $\\mathrm {Res}\\lbrace c_nn\\Psi _n\\rbrace $ can be determined using the standard two-subevent or three-subevent methods [4] as discussed in Sec.", "REF .", "To avoid auto-correlations, each $\\Psi _n$ needs to be measured using subevents covering different $\\eta $ ranges, preferably with a gap in between.", "Here a subevent refers to a collection of particles over a certain $\\eta $ range in the event.", "This method of obtaining the correlator is referred to as the event-plane or EP method.", "In Eq.", "(REF ), all events are given equal weights in both the numerator (raw correlator) and the denominator (resolution).", "It was recently proposed [36], [37] that the potential bias in the EP method arising from the effects of event-by-event fluctuations of the flow and multiplicity can be removed by applying additional weight factors: $\\nonumber \\langle \\cos (c_1\\Phi _{1}+2c_2\\Phi _{2}+...+lc_l\\Phi _{l})\\rangle _{\\mathrm {w}} &=& \\frac{\\langle \\cos (c_1\\Psi _{1}+2c_2\\Psi _{2}+...+lc_l\\Psi _{l})\\rangle _{\\mathrm {w}}}{\\mathrm {Res}\\lbrace c_1\\Psi _1\\rbrace _{\\mathrm {w}}\\mathrm {Res}\\lbrace c_22\\Psi _2\\rbrace _{\\mathrm {w}}...\\mathrm {Res}\\lbrace c_ll\\Psi _l\\rbrace _{\\mathrm {w}}}\\\\\\nonumber \\langle \\cos (c_1\\Psi _{1}+2c_2\\Psi _{2}+...+lc_l\\Psi _{l})\\rangle _{\\mathrm {w}} &=& \\langle q_1^{c_1} q_2^{c_2}... q_l^{c_l}\\cos (c_1\\Psi _{1}+2c_2\\Psi _{2}+...+lc_l\\Psi _{l})\\rangle \\\\\\mathrm {Res}\\lbrace c_nn\\Psi _n\\rbrace _{\\mathrm {w}} &=& \\sqrt{\\left\\langle \\left({q_n^{c_n}\\cos c_nn(\\Psi _n-\\Phi _n)}\\right)^2\\right\\rangle }\\;,$ where the $q_n=\|{q}_n\|$ represents the magnitude of the flow vector of the subevent used to calculate the $\\Psi _n$ (Eq.", "(REF )), and the subscript “$\\rm w$ ” is used to indicate the $q_n$ -weighting.", "This weighting method is often referred to as the “scalar-product” or SP method [38].", "Correspondingly, the weighted version of the PP correlators can be obtained by using the eccentricity $\\epsilon _n$ defined in Eq.", "(REF ) as the weight [21]: $\\langle \\cos (c_1\\Phi ^_{1}+2c_2\\Phi ^_{2}+...+lc_l\\Phi ^_{l})\\rangle _{\\mathrm {w}}= \\frac{\\langle \\epsilon _1^{c_1} \\epsilon _2^{c_2}... \\epsilon _l^{c_l}\\cos (c_1\\Phi ^_{1}+2c_2\\Phi ^_{2}+...+lc_l\\Phi ^_{l})\\rangle }{\\sqrt{\\langle \\epsilon _1^{2c_1}\\rangle \\langle \\epsilon _2^{2c_2}\\rangle ...\\langle \\epsilon _l^{2c_l}\\rangle }}\\;.$ In Eq.", "(REF ), events with larger flow have bigger weights in the calculation of the raw correlation and the resolution factors.", "Other than the weighting, the procedure for obtaining the raw signal and resolution factors is identical in the EP and SP methods.", "Hence the discussion in the remainder of the paper should be regarded as applicable to both methods and the subscript “$\\rm w$ ” is dropped in all formulae, unless required for clarity.", "It is worth emphasizing that the expression for the correlators in Eq.", "REF is constructed to be insensitive to the details of the detector performance, such as the $\\eta $ -coverage, segmentation, energy calibration or the efficiency [4], [34].", "This is because the angle $\\Phi _n$ is a global property of the event that can be estimated from the $\\Psi _n$ from independent detectors, and the procedure for obtaining the correlators is “self-correcting”.", "A poor segmentation or energy calibration of the calorimeter, for example, increases the smearing of $\\Psi _n$ about $\\Phi _n$ , and hence reduces the raw correlation (numerator of Eq.", "REF ).", "This reduction in the raw correlation, however, is expected to be mostly compensated by smaller resolution terms $\\mathrm {Res}\\lbrace c_nn\\Psi _n\\rbrace $ in the denominators.", "A very large number of correlators could be studied.", "However, the measurability of these correlators is dictated by the values of ${\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace }$ ($c_n$ replaced by $j$ for simplicity).", "A detailed study in this analysis shows that the values of ${\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace }$ decrease very quickly for increasing $n$ , but they decrease more slowly with $j$ for fixed $n$ [23].", "The resolution factors are sufficiently good for ${\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace }$ for $n=2$ to 6 and $j$ values up to $j=6$ for $n=2$ .", "This defines the two- and three-plane correlators that can be measured.", "Table REF gives a summary of the set of two-plane correlators and resolution terms that need to be measured in this analysis for each centrality interval.", "The corresponding information for the three-plane correlators is shown in Table REF .", "The first three correlators in Table REF correspond to the first three Fourier coefficients ($j=$ 1,2,3) in Eq.", "(), and are derived from the observed distribution $dN_{\\mathrm {evts}}/d\\left(4(\\Psi _2-\\Psi _4)\\right)$ .", "All the other correlators in Tables REF and REF only correspond to the first Fourier coefficient of the observed distribution.", "The two-plane and three-plane correlators are listed separately because different subdetectors are used (see Sec.", "REF ), and this requires separate evaluation of the resolution corrections.", "Table: The list of two-plane correlators and associated event-plane resolution factors that need to be measured.Table: The list of three-plane correlators and associated event-plane resolution factors that need to be measured." ], [ "Analysis method", "For two-plane correlation (2PC) measurements, the event is divided into two subevents symmetric around $\\eta =0$ with a gap in between, so they nominally have the same resolution.", "Each subevent provides its own estimate of the event plane via Eq.", "(REF ): $\\Psi _n^{\\mathrm {P}}$ and $\\Psi _m^{\\mathrm {P}}$ for positive $\\eta $ and $\\Psi _n^{\\mathrm {N}}$ and $\\Psi _m^{\\mathrm {N}}$ for negative $\\eta $ .", "This leads to two statistically independent estimates of the correlator, which are averaged to obtain the final signal.", "Because of the symmetry of the subevents, the product of resolution factors in the denominator is identical for each measurement, and the event-averaged correlator can be written as: $\\langle \\cos k(\\Phi _n-\\Phi _m)\\rangle = \\frac{\\left\\langle \\cos k(\\Psi _n^{\\mathrm {P}}-\\Psi _m^{\\mathrm {N}})\\right\\rangle +\\left\\langle \\cos k(\\Psi _n^{\\mathrm {N}}-\\Psi _m^{\\mathrm {P}})\\right\\rangle }{\\mathrm {Res}\\lbrace k\\Psi _n^{\\mathrm {P}}\\rbrace \\mathrm {Res}\\lbrace k\\Psi _m^{\\mathrm {N}}\\rbrace +\\mathrm {Res}\\lbrace k\\Psi _n^{\\mathrm {N}}\\rbrace \\mathrm {Res}\\lbrace k\\Psi _m^{\\mathrm {P}}\\rbrace }\\;.$ To measure a three-plane correlation (3PC), three non-overlapping subevents, labeled as A, B and C, are chosen to have approximately the same $\\eta $ coverage.", "In this analysis, subevents A and C are chosen to be symmetric about $\\eta =0$ , and hence have identical resolution, while the resolution of subevent B in general is different.", "There are $3!=6$ independent ways of obtaining the same three-plane correlator.", "But the symmetry between A and C reduces this to three pairs of measurements, which are labeled as Type1, Type2 and Type3.", "For example, the Type1 measurement of the correlation $2\\Phi _2+3\\Phi _3-5\\Phi _5$ is obtained from $2\\Psi _2^{\\mathrm {B}}+3\\Psi _3^{\\mathrm {A}}-5\\Psi _5^{\\mathrm {C}}$ and $2\\Psi _2^{\\mathrm {B}}+3\\Psi _3^{\\mathrm {C}}-5\\Psi _5^{\\mathrm {A}}$ , i.e.", "by requiring the $\\Psi _2$ angle to be given by subevent B: $\\left\\langle \\cos (2\\Phi _2+3\\Phi _3-5\\Phi _5)\\right\\rangle _{\\mathrm {Type1}}=\\frac{\\left\\langle \\cos (2\\Psi _2^{\\mathrm {B}}+3\\Psi _3^{\\mathrm {A}}-5\\Psi _5^{\\mathrm {C}})\\right\\rangle +\\left\\langle \\cos (2\\Psi _2^{\\mathrm {B}}+3\\Psi _3^{\\mathrm {C}}-5\\Psi _5^{\\mathrm {A}})\\right\\rangle }{\\mathrm {Res}\\lbrace 2\\Psi _2^{B}\\rbrace \\mathrm {Res}\\lbrace 3\\Psi _3^{A}\\rbrace \\mathrm {Res}\\lbrace 5\\Psi _5^{C}\\rbrace +\\mathrm {Res}\\lbrace 2\\Psi _2^{B}\\rbrace \\mathrm {Res}\\lbrace 3\\Psi _3^{C}\\rbrace \\mathrm {Res}\\lbrace 5\\Psi _5^{A}\\rbrace }\\;.$ Similarly, the Type2 (Type3) measurement is obtained by requiring the $\\Psi _3$ ($\\Psi _5$ ) to be measured by subevent B.", "Since the three angles in each detector, e.g.", "$\\Psi ^{\\mathrm {A}}_2$ , $\\Psi ^{\\mathrm {A}}_3$ and $\\Psi ^{\\mathrm {A}}_5$ , are obtained from orthogonal Fourier modes, the different types of estimates for a given correlator are expected to be statistically independent.", "The resolution factors $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace $ are obtained from a two-subevent method (2SE) and a three-subevent method (3SE) [4].", "The 2SE method follows almost identically the 2PC procedure described above: two subevents symmetric about $\\eta =0$ are chosen and used to make two measurements of the event plane at the same order $n$ : $\\Psi _n^{\\mathrm {P}}$ and $\\Psi _n^{\\mathrm {N}}$ .", "The correlator $\\langle \\cos jn(\\Psi _n^{\\mathrm {P}}-\\Psi _n^{\\mathrm {N}})\\rangle $ is then calculated, and the square-root yields the desired resolution [8]: $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace = \\sqrt{\\langle \\cos jn(\\Psi _n^{\\mathrm {P}}-\\Psi _n^{\\mathrm {N}})\\rangle }\\equiv \\mathrm {Res}\\lbrace jn\\Psi _n^{\\mathrm {P}}\\rbrace \\equiv \\mathrm {Res}\\lbrace jn\\Psi _n^{\\mathrm {N}}\\rbrace \\;.$ In the 3SE method, the value of $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace $ for a given subevent A is determined from angle correlations with two subevents B and C covering different regions in $\\eta $ : $\\nonumber {\\mathrm {Res}}\\lbrace jn\\Psi ^{\\mathrm {A}}_{n}\\rbrace =\\sqrt{\\frac{\\left\\langle {\\cos jn \\left(\\Psi _n^{\\mathrm {A}} - \\Psi _n^{\\mathrm {B}}\\right)} \\right\\rangle \\left\\langle {\\cos jn \\left(\\Psi _n^{\\mathrm {A}} - \\Psi _n^{\\mathrm {C}}\\right)} \\right\\rangle }{\\left\\langle {\\cos jn \\left(\\Psi _n^{\\mathrm {B}} - \\Psi _n^{\\mathrm {C}}\\right)} \\right\\rangle }}.\\\\$ The 3SE method does not rely on equal resolutions for the subevents, and hence there are many ways of choosing subevents B and C. In the case of the weighted correlators given by the SP method, the resolution terms defined by Eqs.", "REF and REF are instead calculated as [36]: $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace _{\\mathrm {w}}=\\sqrt{\\langle (q_n^{\\mathrm {P}}q_n^{\\mathrm {N}})^j\\cos jn(\\Psi _n^{\\mathrm {P}}-\\Psi _n^{\\mathrm {N}})\\rangle }\\;,$ and ${\\mathrm {Res}}\\lbrace jn\\Psi ^{\\mathrm {A}}_{n}\\rbrace _{\\mathrm {w}}= \\sqrt{\\frac{\\left\\langle { (q_n^{\\mathrm {A}}q_n^{\\mathrm {B}})^j \\cos jn \\left(\\Psi _n^{\\mathrm {A}} - \\Psi _n^{\\mathrm {B}}\\right)} \\right\\rangle \\left\\langle { (q_n^{\\mathrm {A}}q_n^{\\mathrm {C}})^j\\cos jn \\left(\\Psi _n^{\\mathrm {A}} - \\Psi _n^{\\mathrm {C}}\\right)} \\right\\rangle }{\\left\\langle { (q_n^{\\mathrm {B}}q_n^{\\mathrm {C}})^j\\cos jn \\left(\\Psi _n^{\\mathrm {B}} - \\Psi _n^{\\mathrm {C}}\\right)} \\right\\rangle }}\\;.$" ], [ "Analysis procedure", "The large $\\eta $ coverage of the ID, ECal and FCal, with their fine segmentation, allows many choices of subevents for estimating the event planes and studying their correlations over about ten units in $\\eta $ .", "The edge towers of the FCal (approximately $4.8<\|\\eta \|<4.9$ ) are excluded to minimize the non-uniformity of $$ in azimuth, as in a previous analysis [8].", "These detectors are divided into a set of small segments in $\\eta $ , and the subevents are constructed by combining these segments.", "A large number of subevents can be used for measuring both the raw correlation signal and the resolution corrections.", "A detailed set of cross-checks and estimations of systematic uncertainties can therefore be performed.", "The guiding principle for choosing the subevents is that they should have large $\\eta $ acceptance, but still have a sufficiently large $\\eta $ gap from each other.", "For two-plane correlations, the default subevents are ECal+FCal at negative ($-4.8<\\eta <-0.5$ ) and positive ($0.5<\\eta <4.8$ ) $\\eta $ , with a gap of one unit in between.", "For three-plane correlations, the default subevents are ECal$_{\\mathrm {P}}$ ($0.5<\\eta <2.7$ ), FCal ($3.3<\|\\eta \|<4.8$ ), and ECal$_{\\mathrm {N}}$ ($-2.7<\\eta <-0.5$ ).", "As an important consistency cross-check for the 2PC and 3PC analyses, subevents are also chosen only from the ID.", "These combinations are listed in Table REF .", "The resolution for each of these subevents is determined via the 2SE method and the 3SE method, and the latter typically involves measuring correlations with many other subevents not listed in Table REF , for example using smaller sections of the ECal or ID.", "Table: Combinations of subevents used in two-plane and three-plane correlation analysis.", "The calorimter-based analysis is the default, while the ID-based result provides an important cross-check.Figure REF shows the two-plane relative angle distributions for the 20%–30% centrality interval.", "The signal or “foreground” distributions are calculated by combining event-plane angles from the same event.", "The “background” distributions are calculated from mixed events by combining the event-plane angles obtained from different events with similar centrality (matched within 5%) and $z_{\\mathrm {vtx}}$ (matched within 3 cm).", "Ten mixed events are constructed for each foreground event.", "Both distributions are normalized so that the average of the entries is one.", "The background distributions provide an estimate of detector effects, while the foreground distributions contain both the detector effects and physics.", "The background distributions are almost flat, but do indicate some small variations at a level of about $10^{-3}$ .", "To cancel these non-physical structures, the correlation functions are obtained by dividing the foreground (S) by the background distributions (B): $C(k(\\Psi _{n}-\\Psi _{m})) &=& \\frac{S(k(\\Psi _{n}-\\Psi _{m}))}{B(k(\\Psi _{n}-\\Psi _{m}))}\\;.$ The correlation functions show significant positive signals for $4(\\Psi _2-\\Psi _4)$ , $6(\\Psi _2-\\Psi _3)$ , $6(\\Psi _2-\\Psi _6)$ and $6(\\Psi _3-\\Psi _6)$ .", "The observed correlation signals (not corrected by resolution) in terms of the cosine average are calculated directly from these correlation functions.", "Figure: (Color online) Relative angle distributions between two raw event planes from ECalFCal N _{\\mathrm {N}} and ECalFCal P _{\\mathrm {P}} defined in Table for the 20%–30% centrality interval for the foreground (open circles), background (open squares) and correlation function (filled circles) based on the EP method.", "The correlation functions give (via Eq.", "()) the two-plane correlators defined in Table .", "The yy-axis scales are not the same for all panels.Figure REF shows the centrality dependence of the observed correlation signals for various two-plane correlators.", "The systematic uncertainty, shown as shaded bands, is estimated as the values of the sine terms $\\left\\langle \\sin jk\\left(\\Psi _n-\\Psi _m\\right)\\right\\rangle $ .", "Non-zero sine terms may arise from detector effects which lead to non-physical correlations between the two subevents.", "This uncertainty is calculated by averaging sine terms across the measured centrality range, giving uncertainties of (0.2–1.5)$\\times 10^{-3}$ depending on the type of the correlator.", "This uncertainty is correlated with centrality and is significant only when the $\\left\\langle \\cos jk\\left(\\Psi _n-\\Psi _m\\right)\\right\\rangle $ term is itself small, as in the rightmost four panels of Fig.", "REF .", "This uncertainty is included in the final results (see Sec.", "REF ).", "Figure: (Color online) Observed correlation signals based on the EP method, 〈cosjk(Ψ n -Ψ m )〉\\langle \\cos jk(\\Psi _n-\\Psi _m)\\rangle , calculated from the correlation functions such as those in Fig.", "as a function of 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle .", "The middle two panels in the top row have j=2j=2 and j=3j=3, while all other panels have j=1j=1.", "The error bars and shaded bands indicate the statistical and systematic uncertainties, respectively.A large number of resolution factors $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace $ needs to be determined using the 2SE and the 3SE methods, separately for each subevent listed in Table REF .", "For example, the resolution of ECalFCal$_{\\mathrm {N}}$ can be obtained from its correlation with ECalFCal$_{\\mathrm {P}}$ via Eq.", "(REF ) (2SE method), or from its correlation with any two non-overlapping reference subevents at $\\eta >0$ via Eq.", "(REF ) (3SE method) such as $0.5<\\eta <1.5$ and $3.3<\\eta <4.8$ .", "Therefore, for a particular subevent in Table REF , there are usually several determinations of $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace $ , one from the 2SE method and several from the 3SE method.", "The default value used is obtained from the 2SE method where available, or from the 3SE combination with the smallest uncertainty.", "The spread of these values is included in the systematic uncertainty, separately for each centrality interval.", "The relative differences between most of these estimates are found to be independent of the event centrality, except for the 50%–75% centrality range, where weak centrality dependences are observed in some cases.", "All the cosine terms in the 2SE and 3SE formulae are calculated from the distributions similar to those in Eq.", "(REF ), but at the same order $n$ : $C(n(\\Psi _{n}^{\\mathrm {A}}-\\Psi _{n}^{\\mathrm {B}}))=\\frac{S(n(\\Psi _{n}^{\\mathrm {A}}-\\Psi _{n}^{\\mathrm {B}}))}{B(n(\\Psi _{n}^{\\mathrm {A}}-\\Psi _{n}^{\\mathrm {B}}))},$ where the background distribution is obtained by combining the $\\Psi _{n}$ of subevent A in one event with $\\Psi _{n}$ of subevent B from a different event with similar centrality and $z_{\\mathrm {vtx}}$ .", "Furthermore, the non-zero sine values $\\left\\langle \\sin jn\\left(\\Psi _n^{\\mathrm {A}}-\\Psi _n^{\\mathrm {B}}\\right)\\right\\rangle $ arising from the 2SE and 3SE analyses are also included in the uncertainty in the resolution factor.", "Once the individual resolution factors are determined for each subevent, the combined resolution factors are then calculated by multiplying the relevant individual $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace $ terms.", "They are shown in Fig.", "REF as a function of centrality for the eight two-plane correlators listed in Table REF .", "The systematic uncertainty is calculated via a simple error propagation from the individual resolution terms, and is nearly independent of the event centrality.", "This uncertainty is included in the final results (see Sec.", "REF ).", "Figure: (Color online) The combined resolution factors based on the EP method for two-plane correlators, Res{jk(Ψ n -Ψ m )}≡ Res {jkΨ n } Res {jkΨ m }\\lbrace jk(\\Psi _n-\\Psi _{m})\\rbrace \\equiv \\mathrm {Res}\\lbrace jk\\Psi _n\\rbrace \\mathrm {Res}\\lbrace jk\\Psi _m\\rbrace , as a function of 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle .", "The middle two panels in the top row have j=2j=2 and j=3j=3, while all other panels have j=1j=1.", "The error bars and shaded bands indicate the statistical and systematic uncertainties, respectively.The analysis procedure and the systematic uncertainties discussed above are also valid for the three-plane correlation analysis.", "However, the 3PC is slightly more complicated because it has three independent measurements for each correlator, which also need to be combined.", "Figure REF shows the relative angle distributions for various three-plane correlators from the Type1 measurement in the 20%–30% centrality interval.", "The observed correlation signals are calculated as cosine averages of the correlation functions in an obvious generalization of Eq.", "(REF ): $C(c_nn\\Psi _{n}+c_mm\\Psi _{m}+c_hh\\Psi _{h}) &=& \\frac{S(c_nn\\Psi _{n}+c_mm\\Psi _{m}+c_hh\\Psi _{h})}{B(c_nn\\Psi _{n}+c_mm\\Psi _{m}+c_hh\\Psi _{h})}\\;,$ where the background distribution is constructed by requiring that all three angles $\\Psi _{n}$ , $\\Psi _{m}$ , and $\\Psi _{h}$ are from different events.", "The correlation functions show significant positive signals for $2\\Psi _{2}+3\\Psi _{3}-5\\Psi _5$ , $2\\Psi _{2}+4\\Psi _{4}-6\\Psi _6$ , and $-10\\Psi _{2}+4\\Psi _{4}+6\\Psi _6$ , while the signal for $2\\Psi _{2}-6\\Psi _{3}+4\\Psi _4$ is negative, and the signals for the remaining correlators are consistent with zero.", "Figure: (Color online) Relative angle distributions between three event planes from ECal N _{\\mathrm {N}}, FCal and ECal P _{\\mathrm {P}} defined in Table for Type1 correlation in the 20%–30% centrality interval for foreground (open circles), background (open squares) and correlation function (filled circles) based on the EP method.", "The correlation functions give (via equations similar to Eq.", "()) the three-plane correlators defined in Table .", "The yy-axis scales are not the same for all panels.Figure REF shows the centrality dependence of the observed correlation signals (left panel), combined resolutions (middle panel) and corrected signals (right panel) for Type1, Type2 and Type3 combinations of $\\left\\langle \\cos (2\\Psi _{2}+3\\Psi _{3}-5\\Psi _5)\\right\\rangle $ .", "The systematic uncertainty in the observed correlation signals is estimated from the values of $\\left\\langle \\sin \\left(c_nn\\Psi _n+c_mm\\Psi _m+c_hh\\Psi _h\\right)\\right\\rangle $ , and is calculated by averaging these sine terms over the measured centrality range.", "This uncertainty is (0.2–1.5)$\\times 10^{-3}$ in absolute variation, depending on the type of three-plane correlator.", "The uncertainty in the combined resolution is obtained by propagation from those for the individual resolution factors.", "Both sources of uncertainties are strongly correlated with centrality, and they are included in the final results (see Sec.", "REF ).", "Figure REF shows that all three types of measurements (Type1, Type2 and Type3) have similar values for the observed signal and the combined resolution.", "This behavior is expected since the three subevents cover similar rapidity ranges.", "The three corrected results are statistically combined, and the spreads between them are included in the total systematic uncertainty.", "Figure: (Color online) The 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle dependence of the observed correlation signals (left panel), combined resolutions (middle panel) and corrected signals (right panel) based on the EP method for the three types of event plane combinations for 2Φ 2 +3Φ 3 -5Φ 5 2\\Phi _2+3\\Phi _{3}-5\\Phi _5 using the ATLAS calorimeters.", "The error bars and shaded bands indicate the statistical and systematic uncertainties, respectively.The same analysis procedure is repeated for event-plane correlations obtained via the SP method.", "The performance of the SP method is found to be very similar to that of the EP method.", "The magnitudes of the sine terms relative to the cosine terms for both the signal distributions in $k(\\Psi _{n}-\\Psi _{m})$ and $c_nn\\Psi _{n}+c_mm\\Psi _{m}+c_hh\\Psi _{h}$ , as well as the distributions in $n(\\Psi _{n}^{\\mathrm {A}}-\\Psi _{n}^{\\mathrm {B}})$ for calculating the resolution factors are found to be nearly the same as those for the EP method.", "This behavior is quite natural as the effects of detector acceptance are expected to be independent of the strength of the flow signal.", "The resolution factors and their associated systematic uncertainties are calculated with the same detector combinations as those used for the EP method.", "The spreads of the results between various detector combinations are included in the systematic uncertainty for the resolution factors.", "These uncertainties are also found to be strongly correlated between the two methods.", "The uncorrelated systematic uncertainties between the two methods are evaluated by calculating a double ratio for each detector “X” listed in Table REF : $R_{\\mathrm {X}}= \\frac{\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace _{\\mathrm {w}}(\\mathrm {X,other})/\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace _{\\mathrm {w}}(\\mathrm {X,ref})}{\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace (\\mathrm {X,other})/\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace (\\mathrm {X,ref})}\\;,$ where the “ref” refers to the default detector combination used to calculate the resolution of “X”, and “other” refers to other detector combinations used to evaluate the systematic uncertainties in the resolution of “X” via the 2SE and 3SE methods as discussed above (see the paragraph before Eq.", "(REF )).", "The spread of the $R_{\\mathrm {X}}$ values provides an estimate of the uncorrelated uncertainty between the two methods for resolution factor $\\mathrm {Res}\\lbrace jn\\Psi _{n}\\rbrace $ .", "This uncorrelated uncertainty is typically much smaller than the total systematic uncertainty in the resolution factor in either method." ], [ "Systematic uncertainties", "The main systematic uncertainties in the result are introduced and discussed in Sec.", "REF at various key steps of the analysis.", "This section gives a summary of these uncertainties, and then discusses any additional systematic uncertainties and cross-checks.", "The systematic uncertainties associated with the analysis procedure are dominated by contributions from residual detector acceptance effects and uncertainties in the resolution factors.", "Most detector acceptance effects are expected to cancel in the raw correlation function by dividing the foreground and background distributions (Eqs.", "REF –REF ).", "The residual acceptance effects, estimated by the sine terms of the distributions, are found to be (0.2–1.5)$\\times 10^{-3}$ of the average amplitude of the correlation functions, and are found to be independent of the event centrality.", "The uncertainties in the resolution factors are calculated from the differences between the 2SE estimate and various 3SE estimates, which are then propagated to give the total uncertainties for the combined resolution factor.", "These uncertainties are found to be quite similar in the EP and SP methods since they both rely on the same subevent correlations; the larger of the two is quoted as the total systematic uncertainty.", "The uncorrelated uncertainties are evaluated separately via Eq.", "(REF ), and are used for comparison between the two methods.", "The uncertainties in the resolution factors are found to depend only weakly on event centrality.", "Additional systematic uncertainties include those associated with the trigger and event selections, as well as variations of resolution-corrected signals between different running periods.", "The former is evaluated by varying the full centrality range by $\\pm $ 2% according to the estimated efficiency of (98$\\pm 2$ )% for selecting minimum-bias Pb+Pb events.", "The latter is evaluated by comparing the results obtained independently from three running periods each with 1/3 of the total event statistics.", "All these uncertainties are generally small, and are quite similar between the EP method and the SP method.", "Both types of uncertainties are found to be independent of the event centrality.", "Tables REF and REF summarize the sources of systematic uncertainties for two-plane and three-plane correlations.", "The total systematic uncertainties are the quadrature sum of the three sources listed in these tables and the uncertainties associated with residual detector effects.", "The total uncertainties are found to be nearly independent of the event centrality over the 0%–50% centrality range, although a small increase is observed for some of the correlators in the 50%–75% centrality range.", "In most cases, the total systematic uncertainties are dominated by uncertainties associated with the resolution factors.", "The uncertainties in the resolution correction can become quite sizable when the angles $\\Psi _5$ and $\\Psi _6$ are involved.", "This is expected since the higher-order flow signals $v_5$ and $v_6$ are weak, leading to small values of $\\mathrm {Res}\\lbrace 6\\Psi _{6}\\rbrace $ , $\\mathrm {Res}\\lbrace 5\\Psi _{5}\\rbrace $ and $\\mathrm {Res}\\lbrace 10\\Psi _{5}\\rbrace $ with large uncertainties.", "One important issue in this analysis is the extent to which the measured correlations are biased by short-range correlations such as jet fragmentation, resonance decays and Bose–Einstein correlations.", "These short-range correlations may contribute to the observed correlation signals and the resolution factors and hence affect the measured correlations.", "The potential influence of these short-range correlations is studied for the eight two-plane correlators with the EP method.", "The $\\eta $ gap between the two symmetric subevents from ECalFCal, $\\eta _{\\mathrm {min}}$ , is varied in the range of 0 to 8.", "Seventeen symmetric pairs of subevents are chosen, each corresponding to a different $\\eta $ separation.", "For each case, the observed correlation signals and the resolution factors are obtained using the correlations between these two subevents.", "Both the observed correlation signals and combined resolutions decrease significantly (by up to a factor of four) as $\\eta _{\\mathrm {min}}$ is increased.", "However, the final corrected correlation signals are relatively stable.", "For example, a gradual change of a few percent is observed for $\\eta _{\\mathrm {min}}<4$ where the statistical and systematic uncertainties are not very large.", "This observation strongly suggests that the measurement indeed reflects long-range correlations between the event planes.", "In most cases, the raw correlation signals decrease smoothly with $\\eta _{\\mathrm {min}}$ .", "In contrast, the estimated resolution factors have a sharp increase towards small $\\eta _{\\mathrm {min}}$ in many cases, leading to a suppression of the corrected correlation signals at small $\\eta _{\\mathrm {min}}$ .", "This behavior suggests that short-range correlations can influence individual harmonics, and hence the resolution factors, but their influences are weak for correlations between EP angles of different order.", "In all cases, the influences of these short-range correlations are negligible for $\\eta _{\\mathrm {min}}>0.4$ .", "The choices of the subevents in Table REF have a minimum $\\eta $ gap of 0.6, and hence are sufficient to suppress these short-range correlations.", "The event-plane correlators measured by the calorimeters are also compared with those obtained independently from the ID for both the EP method and the SP method (see Table REF for the definition of the subevents).", "Despite the larger fluctuations due to the limited $\\eta $ range of its subevents, the results from the ID are consistent with those from the calorimeters (see Appendix).", "Since the ID is an entirely different type of detector and measures only charged particles, this consistency gives confidence that the measured results are robust.", "It is argued in Ref.", "[37] that the SP method as defined in Eq.", "(REF ) is insensitive to various smearing effects on the weighting factors, such as energy or momentum resolution or multiplicity fluctuations, and as long as these smearings are random and isotropic, they should cancel after averaging over events in the numerator and denominator of Eq.", "(REF ).", "This behavior was checked explicitly in the ID by calculating ${q}_n$ given by Eq.", "(REF ) in several different ways: (1) instead of $u=$ as in the default calculation, the charged particles are set to have equal weight $u=1$ , (2) the weight $u$ is randomly set to be zero for half of the charged particles, or (3) the ${q}_n$ is redefined as ${q}_n\\Sigma u_i$ to include explicitly the event-by-event multiplicity fluctuations.", "The results of all of these cross-checks are consistent with the results of the default calculation.", "Table: Three sources of uncertainties for the two-plane correlators, cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle , where ΣΦ=jk(Φ n -Φ m )\\Sigma \\Phi =jk(\\Phi _n-\\Phi _{m}).", "They are given as percentage uncertainties.Table: Three sources of uncertainties for the three-plane correlators, cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle , where ΣΦ=c n nΦ n +c m mΦ m +c h hΦ h \\Sigma \\Phi =c_nn\\Phi _{n}+c_mm\\Phi _{m}+c_hh\\Phi _{h}.", "They are given as percentage uncertainties." ], [ "Results and discussions", "Figures REF and REF show the centrality dependence of the two-plane and three-plane correlators, respectively.", "The results from both the EP method and SP method are shown with their respective systematic uncertainties.", "These systematic uncertainties are similar in the two methods and are strongly correlated across the centrality range.", "Strong positive values are observed in most cases and their magnitudes usually decrease with increasing $\\langle N_{\\mathrm {part}}\\rangle $ , such as $\\left\\langle \\cos 4(\\Phi _2-\\Phi _4)\\right\\rangle $ , $\\left\\langle \\cos 8(\\Phi _2-\\Phi _4)\\right\\rangle $ , $\\left\\langle \\cos 12(\\Phi _2-\\Phi _4)\\right\\rangle $ , $\\left\\langle \\cos 6(\\Phi _2-\\Phi _6)\\right\\rangle $ , $\\left\\langle \\cos (2\\Phi _2+3\\Phi _3-5\\Phi _5)\\right\\rangle $ , $\\left\\langle \\cos (2\\Phi _2+4\\Phi _4-6\\Phi _6)\\right\\rangle $ and $\\left\\langle \\cos (-10\\Phi _2+4\\Phi _4+6\\Phi _6)\\right\\rangle $ .", "The value of $\\left\\langle \\cos 6(\\Phi _2-\\Phi _3)\\right\\rangle $ is small ($<0.02$ ), yet exhibits a similar dependence on $\\langle N_{\\mathrm {part}}\\rangle $ .", "A small $\\left\\langle \\cos 6(\\Phi _2-\\Phi _{3}) \\right\\rangle $ value in this analysis is a consequence of dividing a small $\\left\\langle \\cos 6(\\Psi _2-\\Psi _{3}) \\right\\rangle $ signal (Figure REF ) by a relatively large combined resolution factor (Fig.", "REF ).", "Two other correlators show very different trends: the value of $\\left\\langle \\cos 6(\\Phi _3-\\Phi _6)\\right\\rangle $ increases with $\\langle N_{\\mathrm {part}}\\rangle $ , and the value of $\\left\\langle \\cos (2\\Phi _2-6\\Phi _3+4\\Phi _4)\\right\\rangle $ is negative and its magnitude decreases with $\\langle N_{\\mathrm {part}}\\rangle $ .", "The values of the remaining correlators are consistent with zero.", "Figure: (Color online) The centrality dependence of eight two-plane correlators, cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle with ΣΦ=jk(Φ n -Φ m )\\Sigma \\Phi =jk(\\Phi _n-\\Phi _{m}) obtained via the SP method (solid symbols) and the EP method (open symbols).", "The middle two panels in the top row have j=2j=2 and j=3j=3, respectively, while all other panels have j=1j=1.", "The error bars and shaded bands indicate the statistical uncertainties and total systematic uncertainties, respectively.", "The expected correlations among participant-plane angles Φ n * \\Phi ^{}_n from a Glauber model are indicated by the solid curves for weighted case (Eq.", "()) and dashed lines for the unweighted case.Figure: (Color online) The centrality dependence of six three-plane correlators, cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle with ΣΦ=c n nΦ n +c m mΦ m +c h hΦ h \\Sigma \\Phi =c_nn\\Phi _{n}+c_mm\\Phi _{m}+c_hh\\Phi _{h} obtained via the SP method (solid symbols) and the EP method (open symbols).", "The error bars and shaded bands indicate the statistical uncertainty and total systematic uncertainty, respectively.", "The expected correlations among participant-plane angles Φ n \\Phi ^{}_n from a Glauber model are indicated by the solid curves for weighted case (Eq.", "()) and dashed lines for the unweighted case.Figures REF and REF also suggest that the magnitude of the correlations from the SP method is always larger than that from the EP method.", "To better quantify their differences, Figures REF and REF show the ratio (SP/EP) for some selected two-plane and three-plane correlators, respectively.", "As discussed in Sec.", "REF , the nature of the systematic uncertainties is very similar in the EP and SP methods, and hence these uncertainties mostly cancel in the ratio.", "The results from the SP method are larger than those from the EP method, and their ratios reach a maximum at around $100<\\langle N_{\\mathrm {part}}\\rangle <300$ range or 10%–40% centrality range.", "The maximum difference is about 10–15% for most two-plane correlators, but reaches 20–30% in mid-central collisions for $\\left\\langle \\cos 8(\\Phi _2-\\Phi _{4}) \\right\\rangle $ and $\\left\\langle \\cos 6(\\Phi _2-\\Phi _{6}) \\right\\rangle $ .", "The differences are smaller for the three-plane correlators, except for $\\left\\langle \\cos (-10\\Phi _2+4\\Phi _4+6\\Phi _6)\\right\\rangle $ .", "Figure: (Color online) The ratios of the SP-method correlators to the EP-method correlators, cos(ΣΦ) w /cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle _{\\mathrm {w}}/\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle for several two-plane correlators i.e with ΣΦ=jk(Φ n -Φ m )\\Sigma \\Phi =jk(\\Phi _n-\\Phi _{m}).", "The error bars and shaded bands indicate the statistical uncertainties and total systematic uncertainties, respectively.Figure: (Color online) The ratios of the SP-method correlators to the EP-method correlators, cos(ΣΦ) w /cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle _{\\mathrm {w}}/\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle for several three-plane correlators i.e with ΣΦ=c n nΦ n +c m mΦ m +c h hΦ h \\Sigma \\Phi =c_nn\\Phi _{n}+c_mm\\Phi _{m}+c_hh\\Phi _{h}.", "The error bars and shaded bands indicate the statistical uncertainties and total systematic uncertainties, respectively.Figures REF and REF also compare the data with the correlators calculated using the participant-plane angles defined in Eq.", "() from the Glauber model [9].", "Thirty million events were generated and grouped into centrality intervals according to the impact parameter.", "If each flow harmonic is driven solely by the corresponding geometric component and the $\\Phi _n$ aligns with the $\\Phi _n^$ , then the event-plane correlation and participant-plane correlation are expected to have the same sign and show similar centrality dependence.", "The results in Figs.", "REF and REF show that for several correlators the centrality dependence of the Glauber model predictions show trends similar to the data, although in some cases the sign is opposite, e.g.", "$\\left\\langle \\cos 4(\\Phi _2-\\Phi _4)\\right\\rangle $ and $\\left\\langle \\cos (2\\Phi _2+3\\Phi _3-5\\Phi _5)\\right\\rangle $ .", "In some cases, even the magnitudes of the correlators show opposite centrality dependence between the Glauber model and the data in addition to the sign-flip, such as $\\left\\langle \\cos 6(\\Phi _3-\\Phi _6)\\right\\rangle $ .", "These discrepancies suggest that in general $\\Phi _n$ may not align with $\\Phi _n^$ .", "Indeed, large misalignments between $\\Phi _n$ and $\\Phi _n^$ have been observed in event-by-event hydrodynamic model calculations for flow harmonics with $n>3$ , and these have been ascribed to the non-linear response of the medium to the fluctuations in the initial geometry [11], [39].", "The non-linear effects are found to be small for lower-order harmonics [11], [40], such that $\\Phi _n\\approx \\Phi _n^$ and $v_n\\propto \\epsilon _n$ for $n=2$ and 3 or equivalently in the form introduced in Eq.", "(REF ): $v_2e^{i2\\Phi _2} \\propto \\epsilon _2e^{i2\\Phi ^_2},\\;\\; v_3e^{i3\\Phi _3} \\propto \\epsilon _3e^{i3\\Phi ^_3}\\;.$ Recently, motivated by the preliminary version [41] of the results presented in this paper, several theory groups calculated the centrality dependence of EP correlators based on hydrodynamic models [5], [42], [43], [44], [45].", "The results of these calculations are in qualitative agreement with the experimental data.", "The dynamical origin of these correlators has been explained using the so-called single-shot hydrodynamics [44], [42], [45], where small fluctuations are imposed on a smooth average geometry profile, and the hydrodynamic response to these small fluctuations is then derived analytically using a cumulant expansion method.", "In this analytical approach, the $v_4$ signal comprises a term proportional to the $\\epsilon _4$ (linear response term) and a leading non-linear term that is proportional to $\\epsilon _2^2$ [5], [44]: $\\nonumber v_4e^{i4\\Phi _4} &=& \\alpha _{_4}\\; \\epsilon _4e^{i4\\Phi ^_4} + \\alpha _{_{2,4}}\\; \\left(\\epsilon _2e^{i2\\Phi ^_2}\\right)^2 +...\\\\&=& \\alpha _{_4}\\; \\epsilon _4e^{i4\\Phi ^_4} + \\beta _{_{2,4}}\\; v_2^2e^{i4\\Phi _2} +...\\;,$ where the second line of the equation is derived from Eq.", "(REF ), and the coefficients $\\alpha _4$ , $\\alpha _{2,4}$ and $\\beta _{2,4}$ are all weak functions of centrality.", "Since $v_2$ increases rapidly for smaller $\\langle N_{\\mathrm {part}}\\rangle $ [8], the angle $\\Phi _4$ becomes more closely aligned with $\\Phi _2$ .", "Hence the centrality dependence of $\\left\\langle \\cos j4(\\Phi _2-\\Phi _{4}) \\right\\rangle $ reflects mainly the increase of the $v_2$ as $\\langle N_{\\mathrm {part}}\\rangle $ decreases.", "Similarly, the correlations between $\\Phi _2$ and $\\Phi _6$ or between $\\Phi _3$ and $\\Phi _6$ have been explained by the following decomposition of the $v_6$ signal [5], [44]: $\\nonumber v_6e^{i6\\Phi _6} &=& \\alpha _6 \\epsilon _6e^{i6\\Phi ^_6} + \\alpha _{_{2,6}} \\left(\\epsilon _2e^{i2\\Phi ^_2}\\right)^3 + \\alpha _{_{3,6}} \\left(\\epsilon _3e^{i3\\Phi ^_3}\\right)^2+....\\\\&=& \\alpha _6 \\epsilon _6e^{i6\\Phi ^_6} + \\beta _{_{2,6}} v_2^3e^{i6\\Phi _2} + \\beta _{_{3,6}} v_3^2e^{i6\\Phi _3}+....\\;.$ Due to the non-linear contributions, $\\Phi _6$ becomes correlated with $\\Phi _2$ and $\\Phi _3$ , even though $\\Phi _2$ and $\\Phi _3$ are only very weakly correlated.", "The centrality dependences of $\\left\\langle \\cos 6(\\Phi _2-\\Phi _{6}) \\right\\rangle $ and $\\left\\langle \\cos 6(\\Phi _3-\\Phi _{6}) \\right\\rangle $ are strongly influenced by the centrality dependence of $v_2$ and $v_3$ : since $v_2$ increases for smaller $\\langle N_{\\mathrm {part}}\\rangle $ and $v_3$ is relatively independent of $\\langle N_{\\mathrm {part}}\\rangle $ [8], the relative contribution of the second term increases and that of the third term decreases for smaller $\\langle N_{\\mathrm {part}}\\rangle $ , i.e.", "the collisions become more peripheral.", "This behavior explains the opposite centrality dependence of $\\left\\langle \\cos 6(\\Phi _2-\\Phi _{6}) \\right\\rangle $ and $\\left\\langle \\cos 6(\\Phi _3-\\Phi _{6}) \\right\\rangle $ .", "In the same manner, the correlation between $\\Phi _2, \\Phi _3$ and $\\Phi _5$ has been explained by the following decomposition of the $v_5$ signal [5], [44]: $\\nonumber v_5e^{i5\\Phi _5} &=& \\alpha _5 \\epsilon _5e^{i5\\Phi ^_5} + \\alpha _{_{2,3,5}} \\epsilon _2e^{i2\\Phi ^_2}\\epsilon _3e^{i3\\Phi ^_3}+....\\\\&=& \\alpha _5 \\epsilon _5e^{i5\\Phi ^_5} + \\beta _{_{2,3,5}} v_2v_3e^{i(2\\Phi _2+3\\Phi _3)}+...\\;.$ The coupling between $v_5$ and $v_2v_3$ explains qualitatively the centrality dependence of the correlator $\\left\\langle \\cos (2\\Phi _2+3\\Phi _3-5\\Phi _5)\\right\\rangle $ .", "A multi-phase transport (AMPT) model [46] is frequently used to study the harmonic flow coefficients $v_n$ and to study the relation of $v_n$ to the initial geometry.", "The AMPT model combines the initial-state geometry fluctuations of the Glauber model and final-state interactions through a parton and hadron transport model.", "The AMPT model generates collective flow by elastic scatterings in the partonic and hadronic phase and was shown to reproduce the $v_n$ values [47] and the particle multiplicity [48] reasonably well.", "As a full event generator, the AMPT model allows the generated events to be analyzed with the same procedures as in the data.", "Figures REF and REF compare some selected correlators (six two-plane correlators and four three-plane correlators) with a prediction [37] from the AMPT model.", "Good agreement is observed between the data and the calculation, and in particular the model predicts correctly the stronger signal observed with the SP method.", "Figure: (Color online) Comparison of six two-plane correlators, cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle with ΣΦ=jk(Φ n -Φ m )\\Sigma \\Phi =jk(\\Phi _n-\\Phi _{m}), with results from the AMPT model calculated via the SP method (solid lines) and the EP method (dashed lines) from Ref. .", "The error bars on the lines represent the statistical uncertainties in the calculation.Figure: (Color online) Comparison of four three-plane correlators, cos(ΣΦ)\\left\\langle {\\cos (\\Sigma \\Phi ) }\\right\\rangle with ΣΦ=c n nΦ n +c m mΦ m +c h hΦ h \\Sigma \\Phi =c_nn\\Phi _{n}+c_mm\\Phi _{m}+c_hh\\Phi _{h}, with results from the AMPT model calculated via the SP method (solid lines) and the EP method (dashed lines) from Ref. .", "The error bars on the curves represent the statistical uncertainties in the calculation." ], [ "Conclusions", "Measurements of fourteen correlators between two and three event planes, $\\left\\langle {\\cos jk(\\Phi _n-\\Phi _{m}) } \\right\\rangle $ and $\\left\\langle \\cos \\left(c_nn\\Psi _n+c_mm\\Psi _m+c_hh\\Psi _h\\right)\\right\\rangle $ , respectively, are presented using 7 $\\mu $ b$^{-1}$ of Pb+Pb collision data at $\\sqrt{s_{_{\\mathrm {NN}}}}=2.76$ TeV collected by the ATLAS experiment at the LHC.", "These correlations are estimated from correlations of observed event-plane angles measured in the calorimeters over a large pseudorapidity range $\|\\eta \|<4.8$ using both a standard event-plane method and a scalar-product method.", "Significant positive correlation signals are observed for $4(\\Phi _2-\\Phi _4)$ , $6(\\Phi _2-\\Phi _6)$ , $6(\\Phi _3-\\Phi _6)$ , $2\\Phi _2+3\\Phi _3-5\\Phi _5$ , $2\\Phi _2+4\\Phi _4-6\\Phi _6$ and $-10\\Phi _2+4\\Phi _4+6\\Phi _6$ .", "The correlation signals are negative for $2\\Phi _2-6\\Phi _3+4\\Phi _4$ .", "The magnitudes of the correlations from the scalar-product method are observed to be systematically larger than those obtained from the event-plane method.", "The centrality dependence of most correlators is found to be very different from that predicted by a Glauber model.", "However, calculations based on the same Glauber model, but including the final-state collective dynamics, are able to describe qualitatively, and in many cases also quantitatively, the centrality dependence of the measured correlators.", "These observations suggest that both the fluctuations in the initial geometry and non-linear mixing between different harmonics in the final state are important for creating these correlations in momentum space.", "A detailed theoretical description of these correlations can improve our present understanding of the space-time evolution of the hot and dense matter created in heavy-ion collisions." ], [ "Acknowledgments", "We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.", "We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWF and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET, ERC and NSRF, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT and NSRF, Greece; ISF, MINERVA, GIF, I-CORE and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; BRF and RCN, Norway; MNiSW and NCN, Poland; GRICES and FCT, Portugal; MNE/IFA, Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America.", "The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.", "Figures REF –REF compare results between the calorimeter and the ID for the two-plane and three-plane correlations.", "As discussed at the end of Sec.", "REF , the results are consistent between the calorimeter and the ID within their respective systematic uncertainties.", "Figure: (Color online) The comparison of the eight two-plane correlators between the calorimeters (default) and ID (cross-check) as a function of 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle , both obtained from the EP method.", "The error bars and the shaded bands indicate the statistical and total systematic uncertainties, respectively.Figure: The comparison of the eight two-plane correlators between the calorimeters (default) and ID (cross-check) as a function of 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle , both obtained from the SP method.", "The error bars and the shaded bands indicate the statistical and total systematic uncertainties, respectively.Figure: (Color online) The comparison of the six three-plane correlators between the calorimeters (default) and ID (cross-check) as a function of 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle , both obtained from the EP method.", "The error bars and the shaded bands indicate the statistical and total systematic uncertainty, respectively.Figure: (Color online) The comparison of the six three-plane correlators between the calorimeters (default) and ID (cross-check) as a function of 〈N part 〉\\langle N_{\\mathrm {part}}\\rangle , both obtained from the SP method.", "The error bars and the shaded bands indicate the statistical and total systematic uncertainties, respectively.The ATLAS Collaboration G. Aad$^{\\rm 84}$ , B. Abbott$^{\\rm 112}$ , J. Abdallah$^{\\rm 152}$ , S. Abdel Khalek$^{\\rm 116}$ , O. Abdinov$^{\\rm 11}$ , R. Aben$^{\\rm 106}$ , B. Abi$^{\\rm 113}$ , M. Abolins$^{\\rm 89}$ , O.S.", "AbouZeid$^{\\rm 159}$ , H. Abramowicz$^{\\rm 154}$ , H. Abreu$^{\\rm 137}$ , R. Abreu$^{\\rm 30}$ , Y. Abulaiti$^{\\rm 147a,147b}$ , B.S.", "Acharya$^{\\rm 165a,165b}$$^{,a}$ , L. Adamczyk$^{\\rm 38a}$ , D.L.", "Adams$^{\\rm 25}$ , J. Adelman$^{\\rm 177}$ , S. Adomeit$^{\\rm 99}$ , T. Adye$^{\\rm 130}$ , T. Agatonovic-Jovin$^{\\rm 13b}$ , J.A.", "Aguilar-Saavedra$^{\\rm 125f,125a}$ , M. Agustoni$^{\\rm 17}$ , S.P.", "Ahlen$^{\\rm 22}$ , A. Ahmad$^{\\rm 149}$ , F. Ahmadov$^{\\rm 64}$$^{,b}$ , G. Aielli$^{\\rm 134a,134b}$ , T.P.A.", "Åkesson$^{\\rm 80}$ , G. Akimoto$^{\\rm 156}$ , A.V.", "Akimov$^{\\rm 95}$ , J. Albert$^{\\rm 170}$ , S. Albrand$^{\\rm 55}$ , M.J. Alconada Verzini$^{\\rm 70}$ , M. Aleksa$^{\\rm 30}$ , I.N.", "Aleksandrov$^{\\rm 64}$ , C. Alexa$^{\\rm 26a}$ , G. Alexander$^{\\rm 154}$ , G. Alexandre$^{\\rm 49}$ , T. Alexopoulos$^{\\rm 10}$ , M. Alhroob$^{\\rm 165a,165c}$ , G. Alimonti$^{\\rm 90a}$ , L. Alio$^{\\rm 84}$ , J. Alison$^{\\rm 31}$ , B.M.M.", "Allbrooke$^{\\rm 18}$ , L.J.", "Allison$^{\\rm 71}$ , P.P.", "Allport$^{\\rm 73}$ , S.E.", "Allwood-Spiers$^{\\rm 53}$ , J. Almond$^{\\rm 83}$ , A. Aloisio$^{\\rm 103a,103b}$ , A. Alonso$^{\\rm 36}$ , F. Alonso$^{\\rm 70}$ , C. Alpigiani$^{\\rm 75}$ , A. Altheimer$^{\\rm 35}$ , B. Alvarez Gonzalez$^{\\rm 89}$ , M.G.", "Alviggi$^{\\rm 103a,103b}$ , K. Amako$^{\\rm 65}$ , Y. Amaral Coutinho$^{\\rm 24a}$ , C. Amelung$^{\\rm 23}$ , D. Amidei$^{\\rm 88}$ , S.P.", "Amor Dos Santos$^{\\rm 125a,125c}$ , A. Amorim$^{\\rm 125a,125b}$ , S. Amoroso$^{\\rm 48}$ , N. Amram$^{\\rm 154}$ , G. Amundsen$^{\\rm 23}$ , C. Anastopoulos$^{\\rm 140}$ , L.S.", "Ancu$^{\\rm 49}$ , N. Andari$^{\\rm 30}$ , T. Andeen$^{\\rm 35}$ , C.F.", "Anders$^{\\rm 58b}$ , G. Anders$^{\\rm 30}$ , K.J.", "Anderson$^{\\rm 31}$ , A. Andreazza$^{\\rm 90a,90b}$ , V. Andrei$^{\\rm 58a}$ , X.S.", "Anduaga$^{\\rm 70}$ , S. Angelidakis$^{\\rm 9}$ , I. Angelozzi$^{\\rm 106}$ , P. Anger$^{\\rm 44}$ , A. Angerami$^{\\rm 35}$ , F. Anghinolfi$^{\\rm 30}$ , A.V.", "Anisenkov$^{\\rm 108}$ , N. Anjos$^{\\rm 125a}$ , A. Annovi$^{\\rm 47}$ , A. Antonaki$^{\\rm 9}$ , M. Antonelli$^{\\rm 47}$ , A. Antonov$^{\\rm 97}$ , J. Antos$^{\\rm 145b}$ , F. Anulli$^{\\rm 133a}$ , M. Aoki$^{\\rm 65}$ , L. Aperio Bella$^{\\rm 18}$ , R. Apolle$^{\\rm 119}$$^{,c}$ , G. Arabidze$^{\\rm 89}$ , I. Aracena$^{\\rm 144}$ , Y. Arai$^{\\rm 65}$ , J.P. Araque$^{\\rm 125a}$ , A.T.H.", "Arce$^{\\rm 45}$ , J-F. Arguin$^{\\rm 94}$ , S. Argyropoulos$^{\\rm 42}$ , M. Arik$^{\\rm 19a}$ , A.J.", "Armbruster$^{\\rm 30}$ , O. Arnaez$^{\\rm 82}$ , V. Arnal$^{\\rm 81}$ , H. Arnold$^{\\rm 48}$ , O. Arslan$^{\\rm 21}$ , A. Artamonov$^{\\rm 96}$ , G. Artoni$^{\\rm 23}$ , S. Asai$^{\\rm 156}$ , N. Asbah$^{\\rm 94}$ , A. Ashkenazi$^{\\rm 154}$ , S. Ask$^{\\rm 28}$ , B. Åsman$^{\\rm 147a,147b}$ , L. Asquith$^{\\rm 6}$ , K. Assamagan$^{\\rm 25}$ , R. Astalos$^{\\rm 145a}$ , M. Atkinson$^{\\rm 166}$ , N.B.", "Atlay$^{\\rm 142}$ , B. Auerbach$^{\\rm 6}$ , K. Augsten$^{\\rm 127}$ , M. Aurousseau$^{\\rm 146b}$ , G. Avolio$^{\\rm 30}$ , G. Azuelos$^{\\rm 94}$$^{,d}$ , Y. Azuma$^{\\rm 156}$ , M.A.", "Baak$^{\\rm 30}$ , C. Bacci$^{\\rm 135a,135b}$ , H. Bachacou$^{\\rm 137}$ , K. Bachas$^{\\rm 155}$ , M. Backes$^{\\rm 30}$ , M. Backhaus$^{\\rm 30}$ , J. Backus Mayes$^{\\rm 144}$ , E. Badescu$^{\\rm 26a}$ , P. Bagiacchi$^{\\rm 133a,133b}$ , P. Bagnaia$^{\\rm 133a,133b}$ , Y. Bai$^{\\rm 33a}$ , T. Bain$^{\\rm 35}$ , J.T.", "Baines$^{\\rm 130}$ , O.K.", "Baker$^{\\rm 177}$ , S. Baker$^{\\rm 77}$ , P. Balek$^{\\rm 128}$ , F. Balli$^{\\rm 137}$ , E. Banas$^{\\rm 39}$ , Sw. Banerjee$^{\\rm 174}$ , D. Banfi$^{\\rm 30}$ , A. Bangert$^{\\rm 151}$ , A.A.E.", "Bannoura$^{\\rm 176}$ , V. Bansal$^{\\rm 170}$ , H.S.", "Bansil$^{\\rm 18}$ , L. Barak$^{\\rm 173}$ , S.P.", "Baranov$^{\\rm 95}$ , E.L. Barberio$^{\\rm 87}$ , D. Barberis$^{\\rm 50a,50b}$ , M. Barbero$^{\\rm 84}$ , T. Barillari$^{\\rm 100}$ , M. Barisonzi$^{\\rm 176}$ , T. Barklow$^{\\rm 144}$ , N. Barlow$^{\\rm 28}$ , B.M.", "Barnett$^{\\rm 130}$ , R.M.", "Barnett$^{\\rm 15}$ , Z. Barnovska$^{\\rm 5}$ , A. Baroncelli$^{\\rm 135a}$ , G. Barone$^{\\rm 49}$ , A.J.", "Barr$^{\\rm 119}$ , F. Barreiro$^{\\rm 81}$ , J. Barreiro Guimarães da Costa$^{\\rm 57}$ , R. Bartoldus$^{\\rm 144}$ , A.E.", "Barton$^{\\rm 71}$ , P. Bartos$^{\\rm 145a}$ , V. Bartsch$^{\\rm 150}$ , A. Bassalat$^{\\rm 116}$ , A. Basye$^{\\rm 166}$ , R.L.", "Bates$^{\\rm 53}$ , L. Batkova$^{\\rm 145a}$ , J.R. Batley$^{\\rm 28}$ , M. Battistin$^{\\rm 30}$ , F. Bauer$^{\\rm 137}$ , H.S.", "Bawa$^{\\rm 144}$$^{,e}$ , T. Beau$^{\\rm 79}$ , P.H.", "Beauchemin$^{\\rm 162}$ , R. Beccherle$^{\\rm 123a,123b}$ , P. Bechtle$^{\\rm 21}$ , H.P.", "Beck$^{\\rm 17}$ , K. Becker$^{\\rm 176}$ , S. Becker$^{\\rm 99}$ , M. Beckingham$^{\\rm 139}$ , C. Becot$^{\\rm 116}$ , A.J.", "Beddall$^{\\rm 19c}$ , A. Beddall$^{\\rm 19c}$ , S. Bedikian$^{\\rm 177}$ , V.A.", "Bednyakov$^{\\rm 64}$ , C.P.", "Bee$^{\\rm 149}$ , L.J.", "Beemster$^{\\rm 106}$ , T.A.", "Beermann$^{\\rm 176}$ , M. Begel$^{\\rm 25}$ , K. Behr$^{\\rm 119}$ , C. Belanger-Champagne$^{\\rm 86}$ , P.J.", "Bell$^{\\rm 49}$ , W.H.", "Bell$^{\\rm 49}$ , G. Bella$^{\\rm 154}$ , L. Bellagamba$^{\\rm 20a}$ , A. Bellerive$^{\\rm 29}$ , M. Bellomo$^{\\rm 85}$ , A. Belloni$^{\\rm 57}$ , O.L.", "Beloborodova$^{\\rm 108}$$^{,f}$ , K. Belotskiy$^{\\rm 97}$ , O. Beltramello$^{\\rm 30}$ , O. Benary$^{\\rm 154}$ , D. Benchekroun$^{\\rm 136a}$ , K. Bendtz$^{\\rm 147a,147b}$ , N. Benekos$^{\\rm 166}$ , Y. Benhammou$^{\\rm 154}$ , E. Benhar Noccioli$^{\\rm 49}$ , J.A.", "Benitez Garcia$^{\\rm 160b}$ , D.P.", "Benjamin$^{\\rm 45}$ , J.R. Bensinger$^{\\rm 23}$ , K. Benslama$^{\\rm 131}$ , S. Bentvelsen$^{\\rm 106}$ , D. Berge$^{\\rm 106}$ , E. Bergeaas Kuutmann$^{\\rm 16}$ , N. Berger$^{\\rm 5}$ , F. Berghaus$^{\\rm 170}$ , E. Berglund$^{\\rm 106}$ , J. Beringer$^{\\rm 15}$ , C. Bernard$^{\\rm 22}$ , P. Bernat$^{\\rm 77}$ , C. Bernius$^{\\rm 78}$ , F.U.", "Bernlochner$^{\\rm 170}$ , T. Berry$^{\\rm 76}$ , P. Berta$^{\\rm 128}$ , C. Bertella$^{\\rm 84}$ , F. Bertolucci$^{\\rm 123a,123b}$ , M.I.", "Besana$^{\\rm 90a}$ , G.J.", "Besjes$^{\\rm 105}$ , O. Bessidskaia$^{\\rm 147a,147b}$ , N. Besson$^{\\rm 137}$ , C. Betancourt$^{\\rm 48}$ , S. Bethke$^{\\rm 100}$ , W. Bhimji$^{\\rm 46}$ , R.M.", "Bianchi$^{\\rm 124}$ , L. Bianchini$^{\\rm 23}$ , M. Bianco$^{\\rm 30}$ , O. Biebel$^{\\rm 99}$ , S.P.", "Bieniek$^{\\rm 77}$ , K. Bierwagen$^{\\rm 54}$ , J. Biesiada$^{\\rm 15}$ , M. Biglietti$^{\\rm 135a}$ , J. Bilbao De Mendizabal$^{\\rm 49}$ , H. Bilokon$^{\\rm 47}$ , M. Bindi$^{\\rm 54}$ , S. Binet$^{\\rm 116}$ , A. Bingul$^{\\rm 19c}$ , C. Bini$^{\\rm 133a,133b}$ , C.W.", "Black$^{\\rm 151}$ , J.E.", "Black$^{\\rm 144}$ , K.M.", "Black$^{\\rm 22}$ , D. Blackburn$^{\\rm 139}$ , R.E.", "Blair$^{\\rm 6}$ , J.-B.", "Blanchard$^{\\rm 137}$ , T. Blazek$^{\\rm 145a}$ , I. Bloch$^{\\rm 42}$ , C. Blocker$^{\\rm 23}$ , W. Blum$^{\\rm 82}$$^{,}$ , U. Blumenschein$^{\\rm 54}$ , G.J.", "Bobbink$^{\\rm 106}$ , V.S.", "Bobrovnikov$^{\\rm 108}$ , S.S. Bocchetta$^{\\rm 80}$ , A. Bocci$^{\\rm 45}$ , C.R.", "Boddy$^{\\rm 119}$ , M. Boehler$^{\\rm 48}$ , J. Boek$^{\\rm 176}$ , T.T.", "Boek$^{\\rm 176}$ , J.A.", "Bogaerts$^{\\rm 30}$ , A.G. Bogdanchikov$^{\\rm 108}$ , A. Bogouch$^{\\rm 91}$$^{,}$ , C. Bohm$^{\\rm 147a}$ , J. Bohm$^{\\rm 126}$ , V. Boisvert$^{\\rm 76}$ , T. Bold$^{\\rm 38a}$ , V. Boldea$^{\\rm 26a}$ , A.S. Boldyrev$^{\\rm 98}$ , M. Bomben$^{\\rm 79}$ , M. Bona$^{\\rm 75}$ , M. Boonekamp$^{\\rm 137}$ , A. Borisov$^{\\rm 129}$ , G. Borissov$^{\\rm 71}$ , M. Borri$^{\\rm 83}$ , S. Borroni$^{\\rm 42}$ , J. 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Canepa$^{\\rm 160a}$ , J. Cantero$^{\\rm 81}$ , R. Cantrill$^{\\rm 76}$ , T. Cao$^{\\rm 40}$ , M.D.M.", "Capeans Garrido$^{\\rm 30}$ , I. Caprini$^{\\rm 26a}$ , M. Caprini$^{\\rm 26a}$ , M. Capua$^{\\rm 37a,37b}$ , R. Caputo$^{\\rm 82}$ , R. Cardarelli$^{\\rm 134a}$ , T. Carli$^{\\rm 30}$ , G. Carlino$^{\\rm 103a}$ , L. Carminati$^{\\rm 90a,90b}$ , S. Caron$^{\\rm 105}$ , E. Carquin$^{\\rm 32a}$ , G.D. Carrillo-Montoya$^{\\rm 146c}$ , A.A. Carter$^{\\rm 75}$ , J.R. Carter$^{\\rm 28}$ , J. Carvalho$^{\\rm 125a,125c}$ , D. Casadei$^{\\rm 77}$ , M.P.", "Casado$^{\\rm 12}$ , E. Castaneda-Miranda$^{\\rm 146b}$ , A. Castelli$^{\\rm 106}$ , V. Castillo Gimenez$^{\\rm 168}$ , N.F.", "Castro$^{\\rm 125a}$ , P. Catastini$^{\\rm 57}$ , A. Catinaccio$^{\\rm 30}$ , J.R. Catmore$^{\\rm 118}$ , A. Cattai$^{\\rm 30}$ , G. Cattani$^{\\rm 134a,134b}$ , S. Caughron$^{\\rm 89}$ , V. Cavaliere$^{\\rm 166}$ , D. Cavalli$^{\\rm 90a}$ , M. Cavalli-Sforza$^{\\rm 12}$ , V. Cavasinni$^{\\rm 123a,123b}$ , F. 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Clemens$^{\\rm 84}$ , C. Clement$^{\\rm 147a,147b}$ , Y. Coadou$^{\\rm 84}$ , M. Cobal$^{\\rm 165a,165c}$ , A. Coccaro$^{\\rm 139}$ , J. Cochran$^{\\rm 63}$ , L. Coffey$^{\\rm 23}$ , J.G.", "Cogan$^{\\rm 144}$ , J. Coggeshall$^{\\rm 166}$ , B. Cole$^{\\rm 35}$ , S. Cole$^{\\rm 107}$ , A.P.", "Colijn$^{\\rm 106}$ , C. Collins-Tooth$^{\\rm 53}$ , J. Collot$^{\\rm 55}$ , T. Colombo$^{\\rm 58c}$ , G. Colon$^{\\rm 85}$ , G. Compostella$^{\\rm 100}$ , P. Conde Muiño$^{\\rm 125a,125b}$ , E. Coniavitis$^{\\rm 167}$ , M.C.", "Conidi$^{\\rm 12}$ , S.H.", "Connell$^{\\rm 146b}$ , I.A.", "Connelly$^{\\rm 76}$ , S.M.", "Consonni$^{\\rm 90a,90b}$ , V. Consorti$^{\\rm 48}$ , S. Constantinescu$^{\\rm 26a}$ , C. Conta$^{\\rm 120a,120b}$ , G. Conti$^{\\rm 57}$ , F. Conventi$^{\\rm 103a}$$^{,h}$ , M. Cooke$^{\\rm 15}$ , B.D.", "Cooper$^{\\rm 77}$ , A.M. Cooper-Sarkar$^{\\rm 119}$ , N.J. Cooper-Smith$^{\\rm 76}$ , K. Copic$^{\\rm 15}$ , T. Cornelissen$^{\\rm 176}$ , M. Corradi$^{\\rm 20a}$ , F. Corriveau$^{\\rm 86}$$^{,i}$ , A. Corso-Radu$^{\\rm 164}$ , A. Cortes-Gonzalez$^{\\rm 12}$ , G. Cortiana$^{\\rm 100}$ , G. Costa$^{\\rm 90a}$ , M.J. Costa$^{\\rm 168}$ , D. Costanzo$^{\\rm 140}$ , D. Côté$^{\\rm 8}$ , G. Cottin$^{\\rm 28}$ , G. Cowan$^{\\rm 76}$ , B.E.", "Cox$^{\\rm 83}$ , K. Cranmer$^{\\rm 109}$ , G. Cree$^{\\rm 29}$ , S. Crépé-Renaudin$^{\\rm 55}$ , F. Crescioli$^{\\rm 79}$ , M. Crispin Ortuzar$^{\\rm 119}$ , M. Cristinziani$^{\\rm 21}$ , V. Croft$^{\\rm 105}$ , G. Crosetti$^{\\rm 37a,37b}$ , C.-M. Cuciuc$^{\\rm 26a}$ , C. Cuenca Almenar$^{\\rm 177}$ , T. Cuhadar Donszelmann$^{\\rm 140}$ , J. Cummings$^{\\rm 177}$ , M. Curatolo$^{\\rm 47}$ , C. Cuthbert$^{\\rm 151}$ , H. Czirr$^{\\rm 142}$ , P. Czodrowski$^{\\rm 3}$ , Z. Czyczula$^{\\rm 177}$ , S. D'Auria$^{\\rm 53}$ , M. D'Onofrio$^{\\rm 73}$ , M.J. Da Cunha Sargedas De Sousa$^{\\rm 125a,125b}$ , C. Da Via$^{\\rm 83}$ , W. Dabrowski$^{\\rm 38a}$ , A. Dafinca$^{\\rm 119}$ , T. Dai$^{\\rm 88}$ , O. Dale$^{\\rm 14}$ , F. Dallaire$^{\\rm 94}$ , C. Dallapiccola$^{\\rm 85}$ , M. Dam$^{\\rm 36}$ , A.C. Daniells$^{\\rm 18}$ , M. Dano Hoffmann$^{\\rm 137}$ , V. Dao$^{\\rm 105}$ , G. Darbo$^{\\rm 50a}$ , G.L.", "Darlea$^{\\rm 26c}$ , S. Darmora$^{\\rm 8}$ , J.A.", "Dassoulas$^{\\rm 42}$ , A. Dattagupta$^{\\rm 60}$ , W. Davey$^{\\rm 21}$ , C. David$^{\\rm 170}$ , T. Davidek$^{\\rm 128}$ , E. Davies$^{\\rm 119}$$^{,c}$ , M. Davies$^{\\rm 154}$ , O. Davignon$^{\\rm 79}$ , A.R.", "Davison$^{\\rm 77}$ , P. Davison$^{\\rm 77}$ , Y. Davygora$^{\\rm 58a}$ , E. Dawe$^{\\rm 143}$ , I. Dawson$^{\\rm 140}$ , R.K. Daya-Ishmukhametova$^{\\rm 23}$ , K. De$^{\\rm 8}$ , R. de Asmundis$^{\\rm 103a}$ , S. De Castro$^{\\rm 20a,20b}$ , S. De Cecco$^{\\rm 79}$ , J. de Graat$^{\\rm 99}$ , N. De Groot$^{\\rm 105}$ , P. de Jong$^{\\rm 106}$ , H. De la Torre$^{\\rm 81}$ , F. De Lorenzi$^{\\rm 63}$ , L. De Nooij$^{\\rm 106}$ , D. De Pedis$^{\\rm 133a}$ , A.", "De Salvo$^{\\rm 133a}$ , U.", "De Sanctis$^{\\rm 165a,165b}$ , A.", "De Santo$^{\\rm 150}$ , J.B. De Vivie De Regie$^{\\rm 116}$ , G. De Zorzi$^{\\rm 133a,133b}$ , W.J.", "Dearnaley$^{\\rm 71}$ , R. Debbe$^{\\rm 25}$ , C. Debenedetti$^{\\rm 46}$ , B. Dechenaux$^{\\rm 55}$ , D.V.", "Dedovich$^{\\rm 64}$ , J. Degenhardt$^{\\rm 121}$ , I. Deigaard$^{\\rm 106}$ , J. Del Peso$^{\\rm 81}$ , T. Del Prete$^{\\rm 123a,123b}$ , F. Deliot$^{\\rm 137}$ , C.M.", "Delitzsch$^{\\rm 49}$ , M. Deliyergiyev$^{\\rm 74}$ , A. Dell'Acqua$^{\\rm 30}$ , L. Dell'Asta$^{\\rm 22}$ , M. Dell'Orso$^{\\rm 123a,123b}$ , M. Della Pietra$^{\\rm 103a}$$^{,h}$ , D. della Volpe$^{\\rm 49}$ , M. Delmastro$^{\\rm 5}$ , P.A.", "Delsart$^{\\rm 55}$ , C. Deluca$^{\\rm 106}$ , S. Demers$^{\\rm 177}$ , M. Demichev$^{\\rm 64}$ , A. Demilly$^{\\rm 79}$ , S.P.", "Denisov$^{\\rm 129}$ , D. Derendarz$^{\\rm 39}$ , J.E.", "Derkaoui$^{\\rm 136d}$ , F. Derue$^{\\rm 79}$ , P. Dervan$^{\\rm 73}$ , K. Desch$^{\\rm 21}$ , C. Deterre$^{\\rm 42}$ , P.O.", "Deviveiros$^{\\rm 106}$ , A. Dewhurst$^{\\rm 130}$ , S. Dhaliwal$^{\\rm 106}$ , A.", "Di Ciaccio$^{\\rm 134a,134b}$ , L. Di Ciaccio$^{\\rm 5}$ , A.", "Di Domenico$^{\\rm 133a,133b}$ , C. Di Donato$^{\\rm 103a,103b}$ , A.", "Di Girolamo$^{\\rm 30}$ , B.", "Di Girolamo$^{\\rm 30}$ , A.", "Di Mattia$^{\\rm 153}$ , B.", "Di Micco$^{\\rm 135a,135b}$ , R. Di Nardo$^{\\rm 47}$ , A.", "Di Simone$^{\\rm 48}$ , R. Di Sipio$^{\\rm 20a,20b}$ , D. Di Valentino$^{\\rm 29}$ , M.A.", "Diaz$^{\\rm 32a}$ , E.B.", "Diehl$^{\\rm 88}$ , J. Dietrich$^{\\rm 42}$ , T.A.", "Dietzsch$^{\\rm 58a}$ , S. Diglio$^{\\rm 84}$ , A. Dimitrievska$^{\\rm 13a}$ , J. Dingfelder$^{\\rm 21}$ , C. Dionisi$^{\\rm 133a,133b}$ , P. Dita$^{\\rm 26a}$ , S. Dita$^{\\rm 26a}$ , F. Dittus$^{\\rm 30}$ , F. Djama$^{\\rm 84}$ , T. Djobava$^{\\rm 51b}$ , M.A.B.", "do Vale$^{\\rm 24c}$ , A.", "Do Valle Wemans$^{\\rm 125a,125g}$ , T.K.O.", "Doan$^{\\rm 5}$ , D. Dobos$^{\\rm 30}$ , E. Dobson$^{\\rm 77}$ , C. Doglioni$^{\\rm 49}$ , T. Doherty$^{\\rm 53}$ , T. Dohmae$^{\\rm 156}$ , J. Dolejsi$^{\\rm 128}$ , Z. Dolezal$^{\\rm 128}$ , B.A.", "Dolgoshein$^{\\rm 97}$$^{,}$ , M. Donadelli$^{\\rm 24d}$ , S. Donati$^{\\rm 123a,123b}$ , P. Dondero$^{\\rm 120a,120b}$ , J. Donini$^{\\rm 34}$ , J. Dopke$^{\\rm 30}$ , A. Doria$^{\\rm 103a}$ , A. Dos Anjos$^{\\rm 174}$ , M.T.", "Dova$^{\\rm 70}$ , A.T. Doyle$^{\\rm 53}$ , M. Dris$^{\\rm 10}$ , J. Dubbert$^{\\rm 88}$ , S. Dube$^{\\rm 15}$ , E. Dubreuil$^{\\rm 34}$ , E. Duchovni$^{\\rm 173}$ , G. Duckeck$^{\\rm 99}$ , O.A.", "Ducu$^{\\rm 26a}$ , D. Duda$^{\\rm 176}$ , A. Dudarev$^{\\rm 30}$ , F. Dudziak$^{\\rm 63}$ , L. Duflot$^{\\rm 116}$ , L. Duguid$^{\\rm 76}$ , M. Dührssen$^{\\rm 30}$ , M. Dunford$^{\\rm 58a}$ , H. Duran Yildiz$^{\\rm 4a}$ , M. Düren$^{\\rm 52}$ , A. Durglishvili$^{\\rm 51b}$ , M. Dwuznik$^{\\rm 38a}$ , M. Dyndal$^{\\rm 38a}$ , J. Ebke$^{\\rm 99}$ , W. Edson$^{\\rm 2}$ , N.C. Edwards$^{\\rm 46}$ , W. Ehrenfeld$^{\\rm 21}$ , T. Eifert$^{\\rm 144}$ , G. Eigen$^{\\rm 14}$ , K. Einsweiler$^{\\rm 15}$ , T. Ekelof$^{\\rm 167}$ , M. El Kacimi$^{\\rm 136c}$ , M. Ellert$^{\\rm 167}$ , S. Elles$^{\\rm 5}$ , F. Ellinghaus$^{\\rm 82}$ , N. Ellis$^{\\rm 30}$ , J. Elmsheuser$^{\\rm 99}$ , M. Elsing$^{\\rm 30}$ , D. Emeliyanov$^{\\rm 130}$ , Y. Enari$^{\\rm 156}$ , O.C.", "Endner$^{\\rm 82}$ , M. Endo$^{\\rm 117}$ , R. Engelmann$^{\\rm 149}$ , J. Erdmann$^{\\rm 177}$ , A. Ereditato$^{\\rm 17}$ , D. Eriksson$^{\\rm 147a}$ , G. Ernis$^{\\rm 176}$ , J. Ernst$^{\\rm 2}$ , M. Ernst$^{\\rm 25}$ , J. Ernwein$^{\\rm 137}$ , D. Errede$^{\\rm 166}$ , S. Errede$^{\\rm 166}$ , E. Ertel$^{\\rm 82}$ , M. Escalier$^{\\rm 116}$ , H. Esch$^{\\rm 43}$ , C. Escobar$^{\\rm 124}$ , B. Esposito$^{\\rm 47}$ , A.I.", "Etienvre$^{\\rm 137}$ , E. Etzion$^{\\rm 154}$ , H. Evans$^{\\rm 60}$ , L. Fabbri$^{\\rm 20a,20b}$ , G. Facini$^{\\rm 30}$ , R.M.", "Fakhrutdinov$^{\\rm 129}$ , S. Falciano$^{\\rm 133a}$ , Y. Fang$^{\\rm 33a}$ , M. Fanti$^{\\rm 90a,90b}$ , A. Farbin$^{\\rm 8}$ , A. Farilla$^{\\rm 135a}$ , T. Farooque$^{\\rm 12}$ , S. Farrell$^{\\rm 164}$ , S.M.", "Farrington$^{\\rm 171}$ , P. Farthouat$^{\\rm 30}$ , F. Fassi$^{\\rm 168}$ , P. Fassnacht$^{\\rm 30}$ , D. Fassouliotis$^{\\rm 9}$ , A. Favareto$^{\\rm 50a,50b}$ , L. Fayard$^{\\rm 116}$ , P. Federic$^{\\rm 145a}$ , O.L.", "Fedin$^{\\rm 122}$ , W. Fedorko$^{\\rm 169}$ , M. Fehling-Kaschek$^{\\rm 48}$ , S. Feigl$^{\\rm 30}$ , L. Feligioni$^{\\rm 84}$ , C. Feng$^{\\rm 33d}$ , E.J.", "Feng$^{\\rm 6}$ , H. Feng$^{\\rm 88}$ , A.B.", "Fenyuk$^{\\rm 129}$ , S. Fernandez Perez$^{\\rm 30}$ , S. Ferrag$^{\\rm 53}$ , J. Ferrando$^{\\rm 53}$ , A. Ferrari$^{\\rm 167}$ , P. Ferrari$^{\\rm 106}$ , R. Ferrari$^{\\rm 120a}$ , D.E.", "Ferreira de Lima$^{\\rm 53}$ , A. Ferrer$^{\\rm 168}$ , D. Ferrere$^{\\rm 49}$ , C. Ferretti$^{\\rm 88}$ , A. Ferretto Parodi$^{\\rm 50a,50b}$ , M. Fiascaris$^{\\rm 31}$ , F. Fiedler$^{\\rm 82}$ , A. Filipčič$^{\\rm 74}$ , M. Filipuzzi$^{\\rm 42}$ , F. Filthaut$^{\\rm 105}$ , M. Fincke-Keeler$^{\\rm 170}$ , K.D.", "Finelli$^{\\rm 151}$ , M.C.N.", "Fiolhais$^{\\rm 125a,125c}$ , L. Fiorini$^{\\rm 168}$ , A. Firan$^{\\rm 40}$ , J. Fischer$^{\\rm 176}$ , W.C. Fisher$^{\\rm 89}$ , E.A.", "Fitzgerald$^{\\rm 23}$ , M. Flechl$^{\\rm 48}$ , I. Fleck$^{\\rm 142}$ , P. Fleischmann$^{\\rm 175}$ , S. Fleischmann$^{\\rm 176}$ , G.T.", "Fletcher$^{\\rm 140}$ , G. Fletcher$^{\\rm 75}$ , T. Flick$^{\\rm 176}$ , A. Floderus$^{\\rm 80}$ , L.R.", "Flores Castillo$^{\\rm 174}$ , A.C. Florez Bustos$^{\\rm 160b}$ , M.J. Flowerdew$^{\\rm 100}$ , A. Formica$^{\\rm 137}$ , A. Forti$^{\\rm 83}$ , D. Fortin$^{\\rm 160a}$ , D. Fournier$^{\\rm 116}$ , H. Fox$^{\\rm 71}$ , S. Fracchia$^{\\rm 12}$ , P. Francavilla$^{\\rm 79}$ , M. Franchini$^{\\rm 20a,20b}$ , S. Franchino$^{\\rm 30}$ , D. Francis$^{\\rm 30}$ , M. Franklin$^{\\rm 57}$ , S. Franz$^{\\rm 61}$ , M. Fraternali$^{\\rm 120a,120b}$ , S.T.", "French$^{\\rm 28}$ , C. Friedrich$^{\\rm 42}$ , F. Friedrich$^{\\rm 44}$ , D. Froidevaux$^{\\rm 30}$ , J.A.", "Frost$^{\\rm 28}$ , C. Fukunaga$^{\\rm 157}$ , E. Fullana Torregrosa$^{\\rm 82}$ , B.G.", "Fulsom$^{\\rm 144}$ , J. Fuster$^{\\rm 168}$ , C. Gabaldon$^{\\rm 55}$ , O. Gabizon$^{\\rm 173}$ , A. Gabrielli$^{\\rm 20a,20b}$ , A. Gabrielli$^{\\rm 133a,133b}$ , S. Gadatsch$^{\\rm 106}$ , S. Gadomski$^{\\rm 49}$ , G. Gagliardi$^{\\rm 50a,50b}$ , P. Gagnon$^{\\rm 60}$ , C. Galea$^{\\rm 105}$ , B. Galhardo$^{\\rm 125a,125c}$ , E.J.", "Gallas$^{\\rm 119}$ , V. Gallo$^{\\rm 17}$ , B.J.", "Gallop$^{\\rm 130}$ , P. Gallus$^{\\rm 127}$ , G. Galster$^{\\rm 36}$ , K.K.", "Gan$^{\\rm 110}$ , R.P.", "Gandrajula$^{\\rm 62}$ , J. Gao$^{\\rm 33b}$$^{,g}$ , Y.S.", "Gao$^{\\rm 144}$$^{,e}$ , F.M.", "Garay Walls$^{\\rm 46}$ , F. Garberson$^{\\rm 177}$ , C. García$^{\\rm 168}$ , J.E.", "García Navarro$^{\\rm 168}$ , M. Garcia-Sciveres$^{\\rm 15}$ , R.W.", "Gardner$^{\\rm 31}$ , N. Garelli$^{\\rm 144}$ , V. Garonne$^{\\rm 30}$ , C. Gatti$^{\\rm 47}$ , G. Gaudio$^{\\rm 120a}$ , B. Gaur$^{\\rm 142}$ , L. Gauthier$^{\\rm 94}$ , P. Gauzzi$^{\\rm 133a,133b}$ , I.L.", "Gavrilenko$^{\\rm 95}$ , C. Gay$^{\\rm 169}$ , G. Gaycken$^{\\rm 21}$ , E.N.", "Gazis$^{\\rm 10}$ , P. Ge$^{\\rm 33d}$ , Z. Gecse$^{\\rm 169}$ , C.N.P.", "Gee$^{\\rm 130}$ , D.A.A.", "Geerts$^{\\rm 106}$ , Ch.", "Geich-Gimbel$^{\\rm 21}$ , K. Gellerstedt$^{\\rm 147a,147b}$ , C. Gemme$^{\\rm 50a}$ , A. Gemmell$^{\\rm 53}$ , M.H.", "Genest$^{\\rm 55}$ , S. Gentile$^{\\rm 133a,133b}$ , M. George$^{\\rm 54}$ , S. George$^{\\rm 76}$ , D. Gerbaudo$^{\\rm 164}$ , A. Gershon$^{\\rm 154}$ , H. Ghazlane$^{\\rm 136b}$ , N. Ghodbane$^{\\rm 34}$ , B. Giacobbe$^{\\rm 20a}$ , S. Giagu$^{\\rm 133a,133b}$ , V. Giangiobbe$^{\\rm 12}$ , P. Giannetti$^{\\rm 123a,123b}$ , F. Gianotti$^{\\rm 30}$ , B. Gibbard$^{\\rm 25}$ , S.M.", "Gibson$^{\\rm 76}$ , M. 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Harper$^{\\rm 88}$ , R.D.", "Harrington$^{\\rm 46}$ , O.M.", "Harris$^{\\rm 139}$ , P.F.", "Harrison$^{\\rm 171}$ , F. Hartjes$^{\\rm 106}$ , S. Hasegawa$^{\\rm 102}$ , Y. Hasegawa$^{\\rm 141}$ , A. Hasib$^{\\rm 112}$ , S. Hassani$^{\\rm 137}$ , S. Haug$^{\\rm 17}$ , M. Hauschild$^{\\rm 30}$ , R. Hauser$^{\\rm 89}$ , M. Havranek$^{\\rm 126}$ , C.M.", "Hawkes$^{\\rm 18}$ , R.J. Hawkings$^{\\rm 30}$ , A.D. Hawkins$^{\\rm 80}$ , T. Hayashi$^{\\rm 161}$ , D. Hayden$^{\\rm 89}$ , C.P.", "Hays$^{\\rm 119}$ , H.S.", "Hayward$^{\\rm 73}$ , S.J.", "Haywood$^{\\rm 130}$ , S.J.", "Head$^{\\rm 18}$ , T. Heck$^{\\rm 82}$ , V. Hedberg$^{\\rm 80}$ , L. Heelan$^{\\rm 8}$ , S. Heim$^{\\rm 121}$ , T. Heim$^{\\rm 176}$ , B. Heinemann$^{\\rm 15}$ , L. Heinrich$^{\\rm 109}$ , S. Heisterkamp$^{\\rm 36}$ , J. Hejbal$^{\\rm 126}$ , L. Helary$^{\\rm 22}$ , C. Heller$^{\\rm 99}$ , M. Heller$^{\\rm 30}$ , S. Hellman$^{\\rm 147a,147b}$ , D. Hellmich$^{\\rm 21}$ , C. Helsens$^{\\rm 30}$ , J. 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Hohlfeld$^{\\rm 82}$ , T.R.", "Holmes$^{\\rm 15}$ , T.M.", "Hong$^{\\rm 121}$ , L. Hooft van Huysduynen$^{\\rm 109}$ , J-Y.", "Hostachy$^{\\rm 55}$ , S. Hou$^{\\rm 152}$ , A. Hoummada$^{\\rm 136a}$ , J. Howard$^{\\rm 119}$ , J. Howarth$^{\\rm 42}$ , M. Hrabovsky$^{\\rm 114}$ , I. Hristova$^{\\rm 16}$ , J. Hrivnac$^{\\rm 116}$ , T. Hryn'ova$^{\\rm 5}$ , P.J.", "Hsu$^{\\rm 82}$ , S.-C. Hsu$^{\\rm 139}$ , D. Hu$^{\\rm 35}$ , X. Hu$^{\\rm 25}$ , Y. Huang$^{\\rm 42}$ , Z. Hubacek$^{\\rm 30}$ , F. Hubaut$^{\\rm 84}$ , F. Huegging$^{\\rm 21}$ , T.B.", "Huffman$^{\\rm 119}$ , E.W.", "Hughes$^{\\rm 35}$ , G. Hughes$^{\\rm 71}$ , M. Huhtinen$^{\\rm 30}$ , T.A.", "Hülsing$^{\\rm 82}$ , M. Hurwitz$^{\\rm 15}$ , N. Huseynov$^{\\rm 64}$$^{,b}$ , J. Huston$^{\\rm 89}$ , J. Huth$^{\\rm 57}$ , G. Iacobucci$^{\\rm 49}$ , G. Iakovidis$^{\\rm 10}$ , I. Ibragimov$^{\\rm 142}$ , L. Iconomidou-Fayard$^{\\rm 116}$ , J. Idarraga$^{\\rm 116}$ , E. Ideal$^{\\rm 177}$ , P. Iengo$^{\\rm 103a}$ , O. Igonkina$^{\\rm 106}$ , T. Iizawa$^{\\rm 172}$ , Y. Ikegami$^{\\rm 65}$ , K. Ikematsu$^{\\rm 142}$ , M. Ikeno$^{\\rm 65}$ , D. Iliadis$^{\\rm 155}$ , N. Ilic$^{\\rm 159}$ , Y. Inamaru$^{\\rm 66}$ , T. Ince$^{\\rm 100}$ , P. Ioannou$^{\\rm 9}$ , M. Iodice$^{\\rm 135a}$ , K. Iordanidou$^{\\rm 9}$ , V. Ippolito$^{\\rm 57}$ , A. Irles Quiles$^{\\rm 168}$ , C. Isaksson$^{\\rm 167}$ , M. Ishino$^{\\rm 67}$ , M. Ishitsuka$^{\\rm 158}$ , R. Ishmukhametov$^{\\rm 110}$ , C. Issever$^{\\rm 119}$ , S. Istin$^{\\rm 19a}$ , J.M.", "Iturbe Ponce$^{\\rm 83}$ , J. Ivarsson$^{\\rm 80}$ , A.V.", "Ivashin$^{\\rm 129}$ , W. Iwanski$^{\\rm 39}$ , H. Iwasaki$^{\\rm 65}$ , J.M.", "Izen$^{\\rm 41}$ , V. Izzo$^{\\rm 103a}$ , B. Jackson$^{\\rm 121}$ , J.N.", "Jackson$^{\\rm 73}$ , M. Jackson$^{\\rm 73}$ , P. Jackson$^{\\rm 1}$ , M.R.", "Jaekel$^{\\rm 30}$ , V. Jain$^{\\rm 2}$ , K. Jakobs$^{\\rm 48}$ , S. Jakobsen$^{\\rm 30}$ , T. Jakoubek$^{\\rm 126}$ , J. 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Ju$^{\\rm 174}$ , C.A.", "Jung$^{\\rm 43}$ , R.M.", "Jungst$^{\\rm 30}$ , P. Jussel$^{\\rm 61}$ , A. Juste Rozas$^{\\rm 12}$$^{,l}$ , M. Kaci$^{\\rm 168}$ , A. Kaczmarska$^{\\rm 39}$ , M. Kado$^{\\rm 116}$ , H. Kagan$^{\\rm 110}$ , M. Kagan$^{\\rm 144}$ , E. Kajomovitz$^{\\rm 45}$ , S. Kama$^{\\rm 40}$ , N. Kanaya$^{\\rm 156}$ , M. Kaneda$^{\\rm 30}$ , S. Kaneti$^{\\rm 28}$ , T. Kanno$^{\\rm 158}$ , V.A.", "Kantserov$^{\\rm 97}$ , J. Kanzaki$^{\\rm 65}$ , B. Kaplan$^{\\rm 109}$ , A. Kapliy$^{\\rm 31}$ , D. Kar$^{\\rm 53}$ , K. Karakostas$^{\\rm 10}$ , N. Karastathis$^{\\rm 10}$ , M. Karnevskiy$^{\\rm 82}$ , S.N.", "Karpov$^{\\rm 64}$ , K. Karthik$^{\\rm 109}$ , V. Kartvelishvili$^{\\rm 71}$ , A.N.", "Karyukhin$^{\\rm 129}$ , L. Kashif$^{\\rm 174}$ , G. Kasieczka$^{\\rm 58b}$ , R.D.", "Kass$^{\\rm 110}$ , A. Kastanas$^{\\rm 14}$ , Y. Kataoka$^{\\rm 156}$ , A. Katre$^{\\rm 49}$ , J. Katzy$^{\\rm 42}$ , V. Kaushik$^{\\rm 7}$ , K. Kawagoe$^{\\rm 69}$ , T. Kawamoto$^{\\rm 156}$ , G. 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Liu$^{\\rm 33b}$$^{,t}$ , L. Liu$^{\\rm 88}$ , M. Liu$^{\\rm 45}$ , M. Liu$^{\\rm 33b}$ , Y. Liu$^{\\rm 33b}$ , M. Livan$^{\\rm 120a,120b}$ , S.S.A.", "Livermore$^{\\rm 119}$ , A. Lleres$^{\\rm 55}$ , J. Llorente Merino$^{\\rm 81}$ , S.L.", "Lloyd$^{\\rm 75}$ , F. Lo Sterzo$^{\\rm 152}$ , E. Lobodzinska$^{\\rm 42}$ , P. Loch$^{\\rm 7}$ , W.S.", "Lockman$^{\\rm 138}$ , T. Loddenkoetter$^{\\rm 21}$ , F.K.", "Loebinger$^{\\rm 83}$ , A.E.", "Loevschall-Jensen$^{\\rm 36}$ , A. Loginov$^{\\rm 177}$ , C.W.", "Loh$^{\\rm 169}$ , T. Lohse$^{\\rm 16}$ , K. Lohwasser$^{\\rm 48}$ , M. Lokajicek$^{\\rm 126}$ , V.P.", "Lombardo$^{\\rm 5}$ , B.A.", "Long$^{\\rm 22}$ , J.D.", "Long$^{\\rm 88}$ , R.E.", "Long$^{\\rm 71}$ , L. Lopes$^{\\rm 125a}$ , D. Lopez Mateos$^{\\rm 57}$ , B. Lopez Paredes$^{\\rm 140}$ , J. Lorenz$^{\\rm 99}$ , N. Lorenzo Martinez$^{\\rm 60}$ , M. Losada$^{\\rm 163}$ , P. Loscutoff$^{\\rm 15}$ , X. Lou$^{\\rm 41}$ , A. Lounis$^{\\rm 116}$ , J. Love$^{\\rm 6}$ , P.A.", "Love$^{\\rm 71}$ , A.J.", "Lowe$^{\\rm 144}$$^{,e}$ , F. Lu$^{\\rm 33a}$ , H.J.", "Lubatti$^{\\rm 139}$ , C. Luci$^{\\rm 133a,133b}$ , A. Lucotte$^{\\rm 55}$ , F. Luehring$^{\\rm 60}$ , W. Lukas$^{\\rm 61}$ , L. Luminari$^{\\rm 133a}$ , O. Lundberg$^{\\rm 147a,147b}$ , B. Lund-Jensen$^{\\rm 148}$ , M. Lungwitz$^{\\rm 82}$ , D. Lynn$^{\\rm 25}$ , R. Lysak$^{\\rm 126}$ , E. Lytken$^{\\rm 80}$ , H. Ma$^{\\rm 25}$ , L.L.", "Ma$^{\\rm 33d}$ , G. Maccarrone$^{\\rm 47}$ , A. Macchiolo$^{\\rm 100}$ , J. Machado Miguens$^{\\rm 125a,125b}$ , D. Macina$^{\\rm 30}$ , D. Madaffari$^{\\rm 84}$ , R. Madar$^{\\rm 48}$ , H.J.", "Maddocks$^{\\rm 71}$ , W.F.", "Mader$^{\\rm 44}$ , A. Madsen$^{\\rm 167}$ , M. Maeno$^{\\rm 8}$ , T. Maeno$^{\\rm 25}$ , E. Magradze$^{\\rm 54}$ , K. Mahboubi$^{\\rm 48}$ , J. Mahlstedt$^{\\rm 106}$ , S. Mahmoud$^{\\rm 73}$ , C. Maiani$^{\\rm 137}$ , C. Maidantchik$^{\\rm 24a}$ , A. Maio$^{\\rm 125a,125b,125d}$ , S. Majewski$^{\\rm 115}$ , Y. Makida$^{\\rm 65}$ , N. Makovec$^{\\rm 116}$ , P. Mal$^{\\rm 137}$$^{,u}$ , B. Malaescu$^{\\rm 79}$ , Pa. Malecki$^{\\rm 39}$ , V.P.", "Maleev$^{\\rm 122}$ , F. Malek$^{\\rm 55}$ , U. Mallik$^{\\rm 62}$ , D. Malon$^{\\rm 6}$ , C. Malone$^{\\rm 144}$ , S. Maltezos$^{\\rm 10}$ , V.M.", "Malyshev$^{\\rm 108}$ , S. Malyukov$^{\\rm 30}$ , J. Mamuzic$^{\\rm 13b}$ , B. Mandelli$^{\\rm 30}$ , L. Mandelli$^{\\rm 90a}$ , I. Mandić$^{\\rm 74}$ , R. Mandrysch$^{\\rm 62}$ , J. Maneira$^{\\rm 125a,125b}$ , A. Manfredini$^{\\rm 100}$ , L. Manhaes de Andrade Filho$^{\\rm 24b}$ , J.A.", "Manjarres Ramos$^{\\rm 160b}$ , A. Mann$^{\\rm 99}$ , P.M. Manning$^{\\rm 138}$ , A. Manousakis-Katsikakis$^{\\rm 9}$ , B. Mansoulie$^{\\rm 137}$ , R. Mantifel$^{\\rm 86}$ , L. Mapelli$^{\\rm 30}$ , L. March$^{\\rm 168}$ , J.F.", "Marchand$^{\\rm 29}$ , G. Marchiori$^{\\rm 79}$ , M. Marcisovsky$^{\\rm 126}$ , C.P.", "Marino$^{\\rm 170}$ , C.N.", "Marques$^{\\rm 125a}$ , F. Marroquim$^{\\rm 24a}$ , S.P.", "Marsden$^{\\rm 83}$ , Z. Marshall$^{\\rm 15}$ , L.F. Marti$^{\\rm 17}$ , S. Marti-Garcia$^{\\rm 168}$ , B. Martin$^{\\rm 30}$ , B. Martin$^{\\rm 89}$ , J.P. Martin$^{\\rm 94}$ , T.A.", "Martin$^{\\rm 171}$ , V.J.", "Martin$^{\\rm 46}$ , B. Martin dit Latour$^{\\rm 14}$ , H. Martinez$^{\\rm 137}$ , M. Martinez$^{\\rm 12}$$^{,l}$ , S. Martin-Haugh$^{\\rm 130}$ , A.C. Martyniuk$^{\\rm 77}$ , M. Marx$^{\\rm 139}$ , F. Marzano$^{\\rm 133a}$ , A. Marzin$^{\\rm 30}$ , L. Masetti$^{\\rm 82}$ , T. Mashimo$^{\\rm 156}$ , R. Mashinistov$^{\\rm 95}$ , J. Masik$^{\\rm 83}$ , A.L.", "Maslennikov$^{\\rm 108}$ , I. Massa$^{\\rm 20a,20b}$ , N. Massol$^{\\rm 5}$ , P. Mastrandrea$^{\\rm 149}$ , A. Mastroberardino$^{\\rm 37a,37b}$ , T. Masubuchi$^{\\rm 156}$ , P. Matricon$^{\\rm 116}$ , H. Matsunaga$^{\\rm 156}$ , T. Matsushita$^{\\rm 66}$ , P. Mättig$^{\\rm 176}$ , S. Mättig$^{\\rm 42}$ , J. Mattmann$^{\\rm 82}$ , J. Maurer$^{\\rm 26a}$ , S.J.", "Maxfield$^{\\rm 73}$ , D.A.", "Maximov$^{\\rm 108}$$^{,f}$ , R. Mazini$^{\\rm 152}$ , L. Mazzaferro$^{\\rm 134a,134b}$ , G. Mc Goldrick$^{\\rm 159}$ , S.P.", "Mc Kee$^{\\rm 88}$ , A. McCarn$^{\\rm 88}$ , R.L.", "McCarthy$^{\\rm 149}$ , T.G.", "McCarthy$^{\\rm 29}$ , N.A.", "McCubbin$^{\\rm 130}$ , K.W.", "McFarlane$^{\\rm 56}$$^{,}$ , J.A.", "Mcfayden$^{\\rm 77}$ , G. Mchedlidze$^{\\rm 54}$ , T. Mclaughlan$^{\\rm 18}$ , S.J.", "McMahon$^{\\rm 130}$ , R.A. McPherson$^{\\rm 170}$$^{,i}$ , A. Meade$^{\\rm 85}$ , J. Mechnich$^{\\rm 106}$ , M. Medinnis$^{\\rm 42}$ , S. Meehan$^{\\rm 31}$ , S. Mehlhase$^{\\rm 36}$ , A. Mehta$^{\\rm 73}$ , K. Meier$^{\\rm 58a}$ , C. Meineck$^{\\rm 99}$ , B. Meirose$^{\\rm 80}$ , C. Melachrinos$^{\\rm 31}$ , B.R.", "Mellado Garcia$^{\\rm 146c}$ , F. Meloni$^{\\rm 90a,90b}$ , A. Mengarelli$^{\\rm 20a,20b}$ , S. Menke$^{\\rm 100}$ , E. Meoni$^{\\rm 162}$ , K.M.", "Mercurio$^{\\rm 57}$ , S. Mergelmeyer$^{\\rm 21}$ , N. Meric$^{\\rm 137}$ , P. Mermod$^{\\rm 49}$ , L. Merola$^{\\rm 103a,103b}$ , C. Meroni$^{\\rm 90a}$ , F.S.", "Merritt$^{\\rm 31}$ , H. Merritt$^{\\rm 110}$ , A. Messina$^{\\rm 30}$$^{,v}$ , J. Metcalfe$^{\\rm 25}$ , A.S. Mete$^{\\rm 164}$ , C. Meyer$^{\\rm 82}$ , C. Meyer$^{\\rm 31}$ , J-P. Meyer$^{\\rm 137}$ , J. Meyer$^{\\rm 30}$ , R.P.", "Middleton$^{\\rm 130}$ , S. Migas$^{\\rm 73}$ , L. Mijović$^{\\rm 137}$ , G. Mikenberg$^{\\rm 173}$ , M. Mikestikova$^{\\rm 126}$ , M. Mikuž$^{\\rm 74}$ , D.W. Miller$^{\\rm 31}$ , C. Mills$^{\\rm 46}$ , A. Milov$^{\\rm 173}$ , D.A.", "Milstead$^{\\rm 147a,147b}$ , D. Milstein$^{\\rm 173}$ , A.A. Minaenko$^{\\rm 129}$ , M. Miñano Moya$^{\\rm 168}$ , I.A.", "Minashvili$^{\\rm 64}$ , A.I.", "Mincer$^{\\rm 109}$ , B. Mindur$^{\\rm 38a}$ , M. Mineev$^{\\rm 64}$ , Y. Ming$^{\\rm 174}$ , L.M.", "Mir$^{\\rm 12}$ , G. Mirabelli$^{\\rm 133a}$ , T. Mitani$^{\\rm 172}$ , J. Mitrevski$^{\\rm 99}$ , V.A.", "Mitsou$^{\\rm 168}$ , S. Mitsui$^{\\rm 65}$ , A. Miucci$^{\\rm 49}$ , P.S.", "Miyagawa$^{\\rm 140}$ , J.U.", "Mjörnmark$^{\\rm 80}$ , T. Moa$^{\\rm 147a,147b}$ , K. Mochizuki$^{\\rm 84}$ , V. Moeller$^{\\rm 28}$ , S. Mohapatra$^{\\rm 35}$ , W. Mohr$^{\\rm 48}$ , S. Molander$^{\\rm 147a,147b}$ , R. Moles-Valls$^{\\rm 168}$ , K. Mönig$^{\\rm 42}$ , C. Monini$^{\\rm 55}$ , J. Monk$^{\\rm 36}$ , E. Monnier$^{\\rm 84}$ , J. Montejo Berlingen$^{\\rm 12}$ , F. Monticelli$^{\\rm 70}$ , S. Monzani$^{\\rm 133a,133b}$ , R.W.", "Moore$^{\\rm 3}$ , A. Moraes$^{\\rm 53}$ , N. Morange$^{\\rm 62}$ , J. Morel$^{\\rm 54}$ , D. Moreno$^{\\rm 82}$ , M. Moreno Llácer$^{\\rm 54}$ , P. Morettini$^{\\rm 50a}$ , M. Morgenstern$^{\\rm 44}$ , M. Morii$^{\\rm 57}$ , S. Moritz$^{\\rm 82}$ , A.K.", "Morley$^{\\rm 148}$ , G. Mornacchi$^{\\rm 30}$ , J.D.", "Morris$^{\\rm 75}$ , L. Morvaj$^{\\rm 102}$ , H.G.", "Moser$^{\\rm 100}$ , M. Mosidze$^{\\rm 51b}$ , J. Moss$^{\\rm 110}$ , R. Mount$^{\\rm 144}$ , E. Mountricha$^{\\rm 25}$ , S.V.", "Mouraviev$^{\\rm 95}$$^{,}$ , E.J.W.", "Moyse$^{\\rm 85}$ , S.G. Muanza$^{\\rm 84}$ , R.D.", "Mudd$^{\\rm 18}$ , F. Mueller$^{\\rm 58a}$ , J. Mueller$^{\\rm 124}$ , K. Mueller$^{\\rm 21}$ , T. Mueller$^{\\rm 28}$ , T. Mueller$^{\\rm 82}$ , D. Muenstermann$^{\\rm 49}$ , Y. Munwes$^{\\rm 154}$ , J.A.", "Murillo Quijada$^{\\rm 18}$ , W.J.", "Murray$^{\\rm 171,130}$ , H. Musheghyan$^{\\rm 54}$ , E. Musto$^{\\rm 153}$ , A.G. Myagkov$^{\\rm 129}$$^{,w}$ , M. Myska$^{\\rm 127}$ , O. Nackenhorst$^{\\rm 54}$ , J. Nadal$^{\\rm 54}$ , K. Nagai$^{\\rm 61}$ , R. Nagai$^{\\rm 158}$ , Y. Nagai$^{\\rm 84}$ , K. Nagano$^{\\rm 65}$ , A. Nagarkar$^{\\rm 110}$ , Y. Nagasaka$^{\\rm 59}$ , M. Nagel$^{\\rm 100}$ , A.M. Nairz$^{\\rm 30}$ , Y. Nakahama$^{\\rm 30}$ , K. Nakamura$^{\\rm 65}$ , T. Nakamura$^{\\rm 156}$ , I. Nakano$^{\\rm 111}$ , H. Namasivayam$^{\\rm 41}$ , G. Nanava$^{\\rm 21}$ , R. Narayan$^{\\rm 58b}$ , T. Nattermann$^{\\rm 21}$ , T. Naumann$^{\\rm 42}$ , G. Navarro$^{\\rm 163}$ , R. Nayyar$^{\\rm 7}$ , H.A.", "Neal$^{\\rm 88}$ , P.Yu.", "Nechaeva$^{\\rm 95}$ , T.J. Neep$^{\\rm 83}$ , A. Negri$^{\\rm 120a,120b}$ , G. Negri$^{\\rm 30}$ , M. Negrini$^{\\rm 20a}$ , S. Nektarijevic$^{\\rm 49}$ , A. Nelson$^{\\rm 164}$ , T.K.", "Nelson$^{\\rm 144}$ , S. Nemecek$^{\\rm 126}$ , P. Nemethy$^{\\rm 109}$ , A.A. Nepomuceno$^{\\rm 24a}$ , M. Nessi$^{\\rm 30}$$^{,x}$ , M.S.", "Neubauer$^{\\rm 166}$ , M. Neumann$^{\\rm 176}$ , R.M.", "Neves$^{\\rm 109}$ , P. Nevski$^{\\rm 25}$ , F.M.", "Newcomer$^{\\rm 121}$ , P.R.", "Newman$^{\\rm 18}$ , D.H. Nguyen$^{\\rm 6}$ , R.B.", "Nickerson$^{\\rm 119}$ , R. Nicolaidou$^{\\rm 137}$ , B. Nicquevert$^{\\rm 30}$ , J. Nielsen$^{\\rm 138}$ , N. Nikiforou$^{\\rm 35}$ , A. Nikiforov$^{\\rm 16}$ , V. Nikolaenko$^{\\rm 129}$$^{,w}$ , I. Nikolic-Audit$^{\\rm 79}$ , K. Nikolics$^{\\rm 49}$ , K. Nikolopoulos$^{\\rm 18}$ , P. Nilsson$^{\\rm 8}$ , Y. Ninomiya$^{\\rm 156}$ , A. Nisati$^{\\rm 133a}$ , R. Nisius$^{\\rm 100}$ , T. Nobe$^{\\rm 158}$ , L. Nodulman$^{\\rm 6}$ , M. Nomachi$^{\\rm 117}$ , I. Nomidis$^{\\rm 155}$ , S. Norberg$^{\\rm 112}$ , M. Nordberg$^{\\rm 30}$ , J. Novakova$^{\\rm 128}$ , S. Nowak$^{\\rm 100}$ , M. Nozaki$^{\\rm 65}$ , L. Nozka$^{\\rm 114}$ , K. Ntekas$^{\\rm 10}$ , G. Nunes Hanninger$^{\\rm 87}$ , T. Nunnemann$^{\\rm 99}$ , E. Nurse$^{\\rm 77}$ , F. Nuti$^{\\rm 87}$ , B.J.", "O'Brien$^{\\rm 46}$ , F. O'grady$^{\\rm 7}$ , D.C. O'Neil$^{\\rm 143}$ , V. O'Shea$^{\\rm 53}$ , F.G. Oakham$^{\\rm 29}$$^{,d}$ , H. Oberlack$^{\\rm 100}$ , T. Obermann$^{\\rm 21}$ , J. Ocariz$^{\\rm 79}$ , A. Ochi$^{\\rm 66}$ , M.I.", "Ochoa$^{\\rm 77}$ , S. Oda$^{\\rm 69}$ , S. Odaka$^{\\rm 65}$ , H. Ogren$^{\\rm 60}$ , A. Oh$^{\\rm 83}$ , S.H.", "Oh$^{\\rm 45}$ , C.C.", "Ohm$^{\\rm 30}$ , H. Ohman$^{\\rm 167}$ , T. Ohshima$^{\\rm 102}$ , W. Okamura$^{\\rm 117}$ , H. Okawa$^{\\rm 25}$ , Y. Okumura$^{\\rm 31}$ , T. Okuyama$^{\\rm 156}$ , A. Olariu$^{\\rm 26a}$ , A.G. Olchevski$^{\\rm 64}$ , S.A. Olivares Pino$^{\\rm 46}$ , D. Oliveira Damazio$^{\\rm 25}$ , E. Oliver Garcia$^{\\rm 168}$ , A. Olszewski$^{\\rm 39}$ , J. Olszowska$^{\\rm 39}$ , A. Onofre$^{\\rm 125a,125e}$ , P.U.E.", "Onyisi$^{\\rm 31}$$^{,y}$ , C.J.", "Oram$^{\\rm 160a}$ , M.J. Oreglia$^{\\rm 31}$ , Y. Oren$^{\\rm 154}$ , D. Orestano$^{\\rm 135a,135b}$ , N. Orlando$^{\\rm 72a,72b}$ , C. Oropeza Barrera$^{\\rm 53}$ , R.S.", "Orr$^{\\rm 159}$ , B. Osculati$^{\\rm 50a,50b}$ , R. Ospanov$^{\\rm 121}$ , G. Otero y Garzon$^{\\rm 27}$ , H. Otono$^{\\rm 69}$ , M. Ouchrif$^{\\rm 136d}$ , E.A.", "Ouellette$^{\\rm 170}$ , F. Ould-Saada$^{\\rm 118}$ , A. Ouraou$^{\\rm 137}$ , K.P.", "Oussoren$^{\\rm 106}$ , Q. Ouyang$^{\\rm 33a}$ , A. Ovcharova$^{\\rm 15}$ , M. Owen$^{\\rm 83}$ , V.E.", "Ozcan$^{\\rm 19a}$ , N. Ozturk$^{\\rm 8}$ , K. Pachal$^{\\rm 119}$ , A. Pacheco Pages$^{\\rm 12}$ , C. Padilla Aranda$^{\\rm 12}$ , M. Pagáčová$^{\\rm 48}$ , S. Pagan Griso$^{\\rm 15}$ , E. Paganis$^{\\rm 140}$ , C. Pahl$^{\\rm 100}$ , F. Paige$^{\\rm 25}$ , P. Pais$^{\\rm 85}$ , K. Pajchel$^{\\rm 118}$ , G. Palacino$^{\\rm 160b}$ , S. Palestini$^{\\rm 30}$ , D. Pallin$^{\\rm 34}$ , A. Palma$^{\\rm 125a,125b}$ , J.D.", "Palmer$^{\\rm 18}$ , Y.B.", "Pan$^{\\rm 174}$ , E. Panagiotopoulou$^{\\rm 10}$ , J.G.", "Panduro Vazquez$^{\\rm 76}$ , P. Pani$^{\\rm 106}$ , N. Panikashvili$^{\\rm 88}$ , S. Panitkin$^{\\rm 25}$ , D. Pantea$^{\\rm 26a}$ , L. Paolozzi$^{\\rm 134a,134b}$ , Th.D.", "Papadopoulou$^{\\rm 10}$ , K. Papageorgiou$^{\\rm 155}$$^{,j}$ , A. Paramonov$^{\\rm 6}$ , D. Paredes Hernandez$^{\\rm 34}$ , M.A.", "Parker$^{\\rm 28}$ , F. Parodi$^{\\rm 50a,50b}$ , J.A.", "Parsons$^{\\rm 35}$ , U. Parzefall$^{\\rm 48}$ , E. Pasqualucci$^{\\rm 133a}$ , S. Passaggio$^{\\rm 50a}$ , A. Passeri$^{\\rm 135a}$ , F. Pastore$^{\\rm 135a,135b}$$^{,}$ , Fr.", "Pastore$^{\\rm 76}$ , G. Pásztor$^{\\rm 49}$$^{,z}$ , S. Pataraia$^{\\rm 176}$ , N.D. Patel$^{\\rm 151}$ , J.R. Pater$^{\\rm 83}$ , S. Patricelli$^{\\rm 103a,103b}$ , T. Pauly$^{\\rm 30}$ , J. Pearce$^{\\rm 170}$ , M. Pedersen$^{\\rm 118}$ , S. Pedraza Lopez$^{\\rm 168}$ , R. Pedro$^{\\rm 125a,125b}$ , S.V.", "Peleganchuk$^{\\rm 108}$ , D. Pelikan$^{\\rm 167}$ , H. Peng$^{\\rm 33b}$ , B. Penning$^{\\rm 31}$ , J. Penwell$^{\\rm 60}$ , D.V.", "Perepelitsa$^{\\rm 25}$ , E. Perez Codina$^{\\rm 160a}$ , M.T.", "Pérez García-Estañ$^{\\rm 168}$ , V. Perez Reale$^{\\rm 35}$ , L. Perini$^{\\rm 90a,90b}$ , H. Pernegger$^{\\rm 30}$ , R. Perrino$^{\\rm 72a}$ , R. Peschke$^{\\rm 42}$ , V.D.", "Peshekhonov$^{\\rm 64}$ , K. Peters$^{\\rm 30}$ , R.F.Y.", "Peters$^{\\rm 83}$ , B.A.", "Petersen$^{\\rm 87}$ , J. Petersen$^{\\rm 30}$ , T.C.", "Petersen$^{\\rm 36}$ , E. Petit$^{\\rm 42}$ , A. Petridis$^{\\rm 147a,147b}$ , C. Petridou$^{\\rm 155}$ , E. Petrolo$^{\\rm 133a}$ , F. Petrucci$^{\\rm 135a,135b}$ , M. Petteni$^{\\rm 143}$ , N.E.", "Pettersson$^{\\rm 158}$ , R. Pezoa$^{\\rm 32b}$ , P.W.", "Phillips$^{\\rm 130}$ , G. Piacquadio$^{\\rm 144}$ , E. Pianori$^{\\rm 171}$ , A. Picazio$^{\\rm 49}$ , E. Piccaro$^{\\rm 75}$ , M. Piccinini$^{\\rm 20a,20b}$ , R. Piegaia$^{\\rm 27}$ , D.T.", "Pignotti$^{\\rm 110}$ , J.E.", "Pilcher$^{\\rm 31}$ , A.D. Pilkington$^{\\rm 77}$ , J. Pina$^{\\rm 125a,125b,125d}$ , M. Pinamonti$^{\\rm 165a,165c}$$^{,aa}$ , A. Pinder$^{\\rm 119}$ , J.L.", "Pinfold$^{\\rm 3}$ , A. Pingel$^{\\rm 36}$ , B. Pinto$^{\\rm 125a}$ , S. Pires$^{\\rm 79}$ , M. Pitt$^{\\rm 173}$ , C. Pizio$^{\\rm 90a,90b}$ , M.-A.", "Pleier$^{\\rm 25}$ , V. Pleskot$^{\\rm 128}$ , E. Plotnikova$^{\\rm 64}$ , P. Plucinski$^{\\rm 147a,147b}$ , S. Poddar$^{\\rm 58a}$ , F. Podlyski$^{\\rm 34}$ , R. Poettgen$^{\\rm 82}$ , L. Poggioli$^{\\rm 116}$ , D. Pohl$^{\\rm 21}$ , M. Pohl$^{\\rm 49}$ , G. Polesello$^{\\rm 120a}$ , A. Policicchio$^{\\rm 37a,37b}$ , R. Polifka$^{\\rm 159}$ , A. Polini$^{\\rm 20a}$ , C.S.", "Pollard$^{\\rm 45}$ , V. Polychronakos$^{\\rm 25}$ , K. Pommès$^{\\rm 30}$ , L. Pontecorvo$^{\\rm 133a}$ , B.G.", "Pope$^{\\rm 89}$ , G.A.", "Popeneciu$^{\\rm 26b}$ , D.S.", "Popovic$^{\\rm 13a}$ , A. Poppleton$^{\\rm 30}$ , X. Portell Bueso$^{\\rm 12}$ , G.E.", "Pospelov$^{\\rm 100}$ , S. Pospisil$^{\\rm 127}$ , K. Potamianos$^{\\rm 15}$ , I.N.", "Potrap$^{\\rm 64}$ , C.J.", "Potter$^{\\rm 150}$ , C.T.", "Potter$^{\\rm 115}$ , G. Poulard$^{\\rm 30}$ , J. Poveda$^{\\rm 60}$ , V. Pozdnyakov$^{\\rm 64}$ , P. Pralavorio$^{\\rm 84}$ , A. Pranko$^{\\rm 15}$ , S. Prasad$^{\\rm 30}$ , R. Pravahan$^{\\rm 8}$ , S. Prell$^{\\rm 63}$ , D. Price$^{\\rm 83}$ , J. Price$^{\\rm 73}$ , L.E.", "Price$^{\\rm 6}$ , D. Prieur$^{\\rm 124}$ , M. Primavera$^{\\rm 72a}$ , M. Proissl$^{\\rm 46}$ , K. Prokofiev$^{\\rm 109}$ , F. Prokoshin$^{\\rm 32b}$ , E. Protopapadaki$^{\\rm 137}$ , S. Protopopescu$^{\\rm 25}$ , J. Proudfoot$^{\\rm 6}$ , M. Przybycien$^{\\rm 38a}$ , H. Przysiezniak$^{\\rm 5}$ , E. Ptacek$^{\\rm 115}$ , E. Pueschel$^{\\rm 85}$ , D. Puldon$^{\\rm 149}$ , M. Purohit$^{\\rm 25}$$^{,ab}$ , P. Puzo$^{\\rm 116}$ , J. Qian$^{\\rm 88}$ , G. Qin$^{\\rm 53}$ , Y. Qin$^{\\rm 83}$ , A. Quadt$^{\\rm 54}$ , D.R.", "Quarrie$^{\\rm 15}$ , W.B.", "Quayle$^{\\rm 165a,165b}$ , D. Quilty$^{\\rm 53}$ , A. Qureshi$^{\\rm 160b}$ , V. Radeka$^{\\rm 25}$ , V. Radescu$^{\\rm 42}$ , S.K.", "Radhakrishnan$^{\\rm 149}$ , P. Radloff$^{\\rm 115}$ , P. Rados$^{\\rm 87}$ , F. Ragusa$^{\\rm 90a,90b}$ , G. Rahal$^{\\rm 179}$ , S. Rajagopalan$^{\\rm 25}$ , M. Rammensee$^{\\rm 30}$ , A.S. Randle-Conde$^{\\rm 40}$ , C. Rangel-Smith$^{\\rm 167}$ , K. Rao$^{\\rm 164}$ , F. Rauscher$^{\\rm 99}$ , T.C.", "Rave$^{\\rm 48}$ , T. Ravenscroft$^{\\rm 53}$ , M. Raymond$^{\\rm 30}$ , A.L.", "Read$^{\\rm 118}$ , D.M.", "Rebuzzi$^{\\rm 120a,120b}$ , A. Redelbach$^{\\rm 175}$ , G. Redlinger$^{\\rm 25}$ , R. Reece$^{\\rm 138}$ , K. Reeves$^{\\rm 41}$ , L. Rehnisch$^{\\rm 16}$ , A. Reinsch$^{\\rm 115}$ , H. Reisin$^{\\rm 27}$ , M. Relich$^{\\rm 164}$ , C. Rembser$^{\\rm 30}$ , Z.L.", "Ren$^{\\rm 152}$ , A. Renaud$^{\\rm 116}$ , M. Rescigno$^{\\rm 133a}$ , S. Resconi$^{\\rm 90a}$ , B. Resende$^{\\rm 137}$ , P. Reznicek$^{\\rm 128}$ , R. Rezvani$^{\\rm 94}$ , R. Richter$^{\\rm 100}$ , M. Ridel$^{\\rm 79}$ , P. Rieck$^{\\rm 16}$ , M. Rijssenbeek$^{\\rm 149}$ , A. Rimoldi$^{\\rm 120a,120b}$ , L. Rinaldi$^{\\rm 20a}$ , E. Ritsch$^{\\rm 61}$ , I. Riu$^{\\rm 12}$ , F. Rizatdinova$^{\\rm 113}$ , E. Rizvi$^{\\rm 75}$ , S.H.", "Robertson$^{\\rm 86}$$^{,i}$ , A. Robichaud-Veronneau$^{\\rm 119}$ , D. Robinson$^{\\rm 28}$ , J.E.M.", "Robinson$^{\\rm 83}$ , A. Robson$^{\\rm 53}$ , C. Roda$^{\\rm 123a,123b}$ , L. Rodrigues$^{\\rm 30}$ , S. Roe$^{\\rm 30}$ , O. Røhne$^{\\rm 118}$ , S. Rolli$^{\\rm 162}$ , A. Romaniouk$^{\\rm 97}$ , M. Romano$^{\\rm 20a,20b}$ , G. Romeo$^{\\rm 27}$ , E. Romero Adam$^{\\rm 168}$ , N. Rompotis$^{\\rm 139}$ , L. Roos$^{\\rm 79}$ , E. Ros$^{\\rm 168}$ , S. Rosati$^{\\rm 133a}$ , K. Rosbach$^{\\rm 49}$ , M. Rose$^{\\rm 76}$ , P.L.", "Rosendahl$^{\\rm 14}$ , O. Rosenthal$^{\\rm 142}$ , V. Rossetti$^{\\rm 147a,147b}$ , E. Rossi$^{\\rm 103a,103b}$ , L.P. Rossi$^{\\rm 50a}$ , R. Rosten$^{\\rm 139}$ , M. Rotaru$^{\\rm 26a}$ , I. Roth$^{\\rm 173}$ , J. Rothberg$^{\\rm 139}$ , D. Rousseau$^{\\rm 116}$ , C.R.", "Royon$^{\\rm 137}$ , A. Rozanov$^{\\rm 84}$ , Y. Rozen$^{\\rm 153}$ , X. Ruan$^{\\rm 146c}$ , F. Rubbo$^{\\rm 12}$ , I. Rubinskiy$^{\\rm 42}$ , V.I.", "Rud$^{\\rm 98}$ , C. Rudolph$^{\\rm 44}$ , M.S.", "Rudolph$^{\\rm 159}$ , F. Rühr$^{\\rm 48}$ , A. Ruiz-Martinez$^{\\rm 30}$ , Z. Rurikova$^{\\rm 48}$ , N.A.", "Rusakovich$^{\\rm 64}$ , A. Ruschke$^{\\rm 99}$ , J.P. Rutherfoord$^{\\rm 7}$ , N. Ruthmann$^{\\rm 48}$ , Y.F.", "Ryabov$^{\\rm 122}$ , M. Rybar$^{\\rm 128}$ , G. Rybkin$^{\\rm 116}$ , N.C. Ryder$^{\\rm 119}$ , A.F.", "Saavedra$^{\\rm 151}$ , S. Sacerdoti$^{\\rm 27}$ , A. Saddique$^{\\rm 3}$ , I. Sadeh$^{\\rm 154}$ , H.F-W. Sadrozinski$^{\\rm 138}$ , R. Sadykov$^{\\rm 64}$ , F. Safai Tehrani$^{\\rm 133a}$ , H. Sakamoto$^{\\rm 156}$ , Y. Sakurai$^{\\rm 172}$ , G. Salamanna$^{\\rm 75}$ , A. Salamon$^{\\rm 134a}$ , M. Saleem$^{\\rm 112}$ , D. Salek$^{\\rm 106}$ , P.H.", "Sales De Bruin$^{\\rm 139}$ , D. Salihagic$^{\\rm 100}$ , A. Salnikov$^{\\rm 144}$ , J. Salt$^{\\rm 168}$ , B.M.", "Salvachua Ferrando$^{\\rm 6}$ , D. Salvatore$^{\\rm 37a,37b}$ , F. Salvatore$^{\\rm 150}$ , A. Salvucci$^{\\rm 105}$ , A. Salzburger$^{\\rm 30}$ , D. Sampsonidis$^{\\rm 155}$ , A. Sanchez$^{\\rm 103a,103b}$ , J. Sánchez$^{\\rm 168}$ , V. Sanchez Martinez$^{\\rm 168}$ , H. Sandaker$^{\\rm 14}$ , R.L.", "Sandbach$^{\\rm 75}$ , H.G.", "Sander$^{\\rm 82}$ , M.P.", "Sanders$^{\\rm 99}$ , M. Sandhoff$^{\\rm 176}$ , T. Sandoval$^{\\rm 28}$ , C. Sandoval$^{\\rm 163}$ , R. Sandstroem$^{\\rm 100}$ , D.P.C.", "Sankey$^{\\rm 130}$ , A. Sansoni$^{\\rm 47}$ , C. Santoni$^{\\rm 34}$ , R. Santonico$^{\\rm 134a,134b}$ , H. Santos$^{\\rm 125a}$ , I. Santoyo Castillo$^{\\rm 150}$ , K. Sapp$^{\\rm 124}$ , A. Sapronov$^{\\rm 64}$ , J.G.", "Saraiva$^{\\rm 125a,125d}$ , B. Sarrazin$^{\\rm 21}$ , G. Sartisohn$^{\\rm 176}$ , O. Sasaki$^{\\rm 65}$ , Y. Sasaki$^{\\rm 156}$ , I. Satsounkevitch$^{\\rm 91}$ , G. Sauvage$^{\\rm 5}$$^{,}$ , E. Sauvan$^{\\rm 5}$ , P. Savard$^{\\rm 159}$$^{,d}$ , D.O.", "Savu$^{\\rm 30}$ , C. Sawyer$^{\\rm 119}$ , L. Sawyer$^{\\rm 78}$$^{,k}$ , D.H. Saxon$^{\\rm 53}$ , J. Saxon$^{\\rm 121}$ , C. Sbarra$^{\\rm 20a}$ , A. Sbrizzi$^{\\rm 3}$ , T. Scanlon$^{\\rm 30}$ , D.A.", "Scannicchio$^{\\rm 164}$ , M. Scarcella$^{\\rm 151}$ , J. Schaarschmidt$^{\\rm 173}$ , P. Schacht$^{\\rm 100}$ , D. Schaefer$^{\\rm 121}$ , R. Schaefer$^{\\rm 42}$ , S. Schaepe$^{\\rm 21}$ , S. Schaetzel$^{\\rm 58b}$ , U. Schäfer$^{\\rm 82}$ , A.C. Schaffer$^{\\rm 116}$ , D. Schaile$^{\\rm 99}$ , R.D.", "Schamberger$^{\\rm 149}$ , V. Scharf$^{\\rm 58a}$ , V.A.", "Schegelsky$^{\\rm 122}$ , D. Scheirich$^{\\rm 128}$ , M. Schernau$^{\\rm 164}$ , M.I.", "Scherzer$^{\\rm 35}$ , C. Schiavi$^{\\rm 50a,50b}$ , J. Schieck$^{\\rm 99}$ , C. Schillo$^{\\rm 48}$ , M. Schioppa$^{\\rm 37a,37b}$ , S. Schlenker$^{\\rm 30}$ , E. Schmidt$^{\\rm 48}$ , K. Schmieden$^{\\rm 30}$ , C. Schmitt$^{\\rm 82}$ , C. Schmitt$^{\\rm 99}$ , S. Schmitt$^{\\rm 58b}$ , B. Schneider$^{\\rm 17}$ , Y.J.", "Schnellbach$^{\\rm 73}$ , U. Schnoor$^{\\rm 44}$ , L. Schoeffel$^{\\rm 137}$ , A. Schoening$^{\\rm 58b}$ , B.D.", "Schoenrock$^{\\rm 89}$ , A.L.S.", "Schorlemmer$^{\\rm 54}$ , M. Schott$^{\\rm 82}$ , D. Schouten$^{\\rm 160a}$ , J. Schovancova$^{\\rm 25}$ , M. Schram$^{\\rm 86}$ , S. Schramm$^{\\rm 159}$ , M. Schreyer$^{\\rm 175}$ , C. Schroeder$^{\\rm 82}$ , N. Schuh$^{\\rm 82}$ , M.J. Schultens$^{\\rm 21}$ , H.-C. Schultz-Coulon$^{\\rm 58a}$ , H. Schulz$^{\\rm 16}$ , M. Schumacher$^{\\rm 48}$ , B.A.", "Schumm$^{\\rm 138}$ , Ph.", "Schune$^{\\rm 137}$ , A. Schwartzman$^{\\rm 144}$ , Ph.", "Schwegler$^{\\rm 100}$ , Ph.", "Schwemling$^{\\rm 137}$ , R. Schwienhorst$^{\\rm 89}$ , J. Schwindling$^{\\rm 137}$ , T. Schwindt$^{\\rm 21}$ , M. Schwoerer$^{\\rm 5}$ , F.G. Sciacca$^{\\rm 17}$ , E. Scifo$^{\\rm 116}$ , G. Sciolla$^{\\rm 23}$ , W.G.", "Scott$^{\\rm 130}$ , F. Scuri$^{\\rm 123a,123b}$ , F. Scutti$^{\\rm 21}$ , J. Searcy$^{\\rm 88}$ , G. Sedov$^{\\rm 42}$ , E. Sedykh$^{\\rm 122}$ , S.C. Seidel$^{\\rm 104}$ , A. Seiden$^{\\rm 138}$ , F. Seifert$^{\\rm 127}$ , J.M.", "Seixas$^{\\rm 24a}$ , G. Sekhniaidze$^{\\rm 103a}$ , S.J.", "Sekula$^{\\rm 40}$ , K.E.", "Selbach$^{\\rm 46}$ , D.M.", "Seliverstov$^{\\rm 122}$$^{,}$ , G. Sellers$^{\\rm 73}$ , N. Semprini-Cesari$^{\\rm 20a,20b}$ , C. Serfon$^{\\rm 30}$ , L. Serin$^{\\rm 116}$ , L. Serkin$^{\\rm 54}$ , T. Serre$^{\\rm 84}$ , R. Seuster$^{\\rm 160a}$ , H. Severini$^{\\rm 112}$ , F. Sforza$^{\\rm 100}$ , A. Sfyrla$^{\\rm 30}$ , E. Shabalina$^{\\rm 54}$ , M. Shamim$^{\\rm 115}$ , L.Y.", "Shan$^{\\rm 33a}$ , J.T.", "Shank$^{\\rm 22}$ , Q.T.", "Shao$^{\\rm 87}$ , M. Shapiro$^{\\rm 15}$ , P.B.", "Shatalov$^{\\rm 96}$ , K. Shaw$^{\\rm 165a,165b}$ , P. Sherwood$^{\\rm 77}$ , S. Shimizu$^{\\rm 66}$ , C.O.", "Shimmin$^{\\rm 164}$ , M. Shimojima$^{\\rm 101}$ , T. Shin$^{\\rm 56}$ , M. Shiyakova$^{\\rm 64}$ , A. Shmeleva$^{\\rm 95}$ , M.J. Shochet$^{\\rm 31}$ , D. Short$^{\\rm 119}$ , S. Shrestha$^{\\rm 63}$ , E. Shulga$^{\\rm 97}$ , M.A.", "Shupe$^{\\rm 7}$ , S. Shushkevich$^{\\rm 42}$ , P. Sicho$^{\\rm 126}$ , D. Sidorov$^{\\rm 113}$ , A. Sidoti$^{\\rm 133a}$ , F. Siegert$^{\\rm 44}$ , Dj.", "Sijacki$^{\\rm 13a}$ , O. Silbert$^{\\rm 173}$ , J. Silva$^{\\rm 125a,125d}$ , Y. Silver$^{\\rm 154}$ , D. Silverstein$^{\\rm 144}$ , S.B.", "Silverstein$^{\\rm 147a}$ , V. Simak$^{\\rm 127}$ , O. Simard$^{\\rm 5}$ , Lj.", "Simic$^{\\rm 13a}$ , S. Simion$^{\\rm 116}$ , E. Simioni$^{\\rm 82}$ , B. Simmons$^{\\rm 77}$ , R. Simoniello$^{\\rm 90a,90b}$ , M. Simonyan$^{\\rm 36}$ , P. Sinervo$^{\\rm 159}$ , N.B.", "Sinev$^{\\rm 115}$ , V. Sipica$^{\\rm 142}$ , G. Siragusa$^{\\rm 175}$ , A. Sircar$^{\\rm 78}$ , A.N.", "Sisakyan$^{\\rm 64}$$^{,}$ , S.Yu.", "Sivoklokov$^{\\rm 98}$ , J. Sjölin$^{\\rm 147a,147b}$ , T.B.", "Sjursen$^{\\rm 14}$ , H.P.", "Skottowe$^{\\rm 57}$ , K.Yu.", "Skovpen$^{\\rm 108}$ , P. Skubic$^{\\rm 112}$ , M. Slater$^{\\rm 18}$ , T. Slavicek$^{\\rm 127}$ , K. Sliwa$^{\\rm 162}$ , V. Smakhtin$^{\\rm 173}$ , B.H.", "Smart$^{\\rm 46}$ , L. Smestad$^{\\rm 14}$ , S.Yu.", "Smirnov$^{\\rm 97}$ , Y. Smirnov$^{\\rm 97}$ , L.N.", "Smirnova$^{\\rm 98}$$^{,ac}$ , O. Smirnova$^{\\rm 80}$ , K.M.", "Smith$^{\\rm 53}$ , M. Smizanska$^{\\rm 71}$ , K. Smolek$^{\\rm 127}$ , A.A. Snesarev$^{\\rm 95}$ , G. Snidero$^{\\rm 75}$ , J. Snow$^{\\rm 112}$ , S. Snyder$^{\\rm 25}$ , R. Sobie$^{\\rm 170}$$^{,i}$ , F. Socher$^{\\rm 44}$ , J. Sodomka$^{\\rm 127}$ , A. Soffer$^{\\rm 154}$ , D.A.", "Soh$^{\\rm 152}$$^{,r}$ , C.A.", "Solans$^{\\rm 30}$ , M. Solar$^{\\rm 127}$ , J. Solc$^{\\rm 127}$ , E.Yu.", "Soldatov$^{\\rm 97}$ , U. Soldevila$^{\\rm 168}$ , E. Solfaroli Camillocci$^{\\rm 133a,133b}$ , A.A. Solodkov$^{\\rm 129}$ , O.V.", "Solovyanov$^{\\rm 129}$ , V. Solovyev$^{\\rm 122}$ , P. Sommer$^{\\rm 48}$ , H.Y.", "Song$^{\\rm 33b}$ , N. Soni$^{\\rm 1}$ , A. Sood$^{\\rm 15}$ , A. Sopczak$^{\\rm 127}$ , V. Sopko$^{\\rm 127}$ , B. Sopko$^{\\rm 127}$ , V. Sorin$^{\\rm 12}$ , M. Sosebee$^{\\rm 8}$ , R. Soualah$^{\\rm 165a,165c}$ , P. Soueid$^{\\rm 94}$ , A.M. Soukharev$^{\\rm 108}$ , D. South$^{\\rm 42}$ , S. Spagnolo$^{\\rm 72a,72b}$ , F. Spanò$^{\\rm 76}$ , W.R. Spearman$^{\\rm 57}$ , R. Spighi$^{\\rm 20a}$ , G. Spigo$^{\\rm 30}$ , M. Spousta$^{\\rm 128}$ , T. Spreitzer$^{\\rm 159}$ , B. Spurlock$^{\\rm 8}$ , R.D. St.", "Denis$^{\\rm 53}$ , S. Staerz$^{\\rm 44}$ , J. Stahlman$^{\\rm 121}$ , R. Stamen$^{\\rm 58a}$ , E. Stanecka$^{\\rm 39}$ , R.W.", "Stanek$^{\\rm 6}$ , C. Stanescu$^{\\rm 135a}$ , M. Stanescu-Bellu$^{\\rm 42}$ , M.M.", "Stanitzki$^{\\rm 42}$ , S. Stapnes$^{\\rm 118}$ , E.A.", "Starchenko$^{\\rm 129}$ , J. Stark$^{\\rm 55}$ , P. Staroba$^{\\rm 126}$ , P. Starovoitov$^{\\rm 42}$ , R. Staszewski$^{\\rm 39}$ , P. Stavina$^{\\rm 145a}$$^{,}$ , G. Steele$^{\\rm 53}$ , P. Steinberg$^{\\rm 25}$ , I. Stekl$^{\\rm 127}$ , B. Stelzer$^{\\rm 143}$ , H.J.", "Stelzer$^{\\rm 30}$ , O. Stelzer-Chilton$^{\\rm 160a}$ , H. Stenzel$^{\\rm 52}$ , S. Stern$^{\\rm 100}$ , G.A.", "Stewart$^{\\rm 53}$ , J.A.", "Stillings$^{\\rm 21}$ , M.C.", "Stockton$^{\\rm 86}$ , M. Stoebe$^{\\rm 86}$ , G. Stoicea$^{\\rm 26a}$ , P. Stolte$^{\\rm 54}$ , S. Stonjek$^{\\rm 100}$ , A.R.", "Stradling$^{\\rm 8}$ , A. Straessner$^{\\rm 44}$ , J. Strandberg$^{\\rm 148}$ , S. Strandberg$^{\\rm 147a,147b}$ , A. Strandlie$^{\\rm 118}$ , E. Strauss$^{\\rm 144}$ , M. Strauss$^{\\rm 112}$ , P. Strizenec$^{\\rm 145b}$ , R. Ströhmer$^{\\rm 175}$ , D.M.", "Strom$^{\\rm 115}$ , R. Stroynowski$^{\\rm 40}$ , S.A. Stucci$^{\\rm 17}$ , B. Stugu$^{\\rm 14}$ , N.A.", "Styles$^{\\rm 42}$ , D. Su$^{\\rm 144}$ , J. Su$^{\\rm 124}$ , HS.", "Subramania$^{\\rm 3}$ , R. Subramaniam$^{\\rm 78}$ , A. Succurro$^{\\rm 12}$ , Y. Sugaya$^{\\rm 117}$ , C. Suhr$^{\\rm 107}$ , M. Suk$^{\\rm 127}$ , V.V.", "Sulin$^{\\rm 95}$ , S. Sultansoy$^{\\rm 4c}$ , T. Sumida$^{\\rm 67}$ , X. Sun$^{\\rm 33a}$ , J.E.", "Sundermann$^{\\rm 48}$ , K. Suruliz$^{\\rm 140}$ , G. Susinno$^{\\rm 37a,37b}$ , M.R.", "Sutton$^{\\rm 150}$ , Y. Suzuki$^{\\rm 65}$ , M. Svatos$^{\\rm 126}$ , S. Swedish$^{\\rm 169}$ , M. Swiatlowski$^{\\rm 144}$ , I. Sykora$^{\\rm 145a}$ , T. Sykora$^{\\rm 128}$ , D. Ta$^{\\rm 89}$ , K. Tackmann$^{\\rm 42}$ , J. Taenzer$^{\\rm 159}$ , A. Taffard$^{\\rm 164}$ , R. Tafirout$^{\\rm 160a}$ , N. Taiblum$^{\\rm 154}$ , Y. Takahashi$^{\\rm 102}$ , H. Takai$^{\\rm 25}$ , R. Takashima$^{\\rm 68}$ , H. Takeda$^{\\rm 66}$ , T. Takeshita$^{\\rm 141}$ , Y. Takubo$^{\\rm 65}$ , M. Talby$^{\\rm 84}$ , A.A. Talyshev$^{\\rm 108}$$^{,f}$ , J.Y.C.", "Tam$^{\\rm 175}$ , M.C.", "Tamsett$^{\\rm 78}$$^{,ad}$ , K.G.", "Tan$^{\\rm 87}$ , J. Tanaka$^{\\rm 156}$ , R. Tanaka$^{\\rm 116}$ , S. Tanaka$^{\\rm 132}$ , S. Tanaka$^{\\rm 65}$ , A.J.", "Tanasijczuk$^{\\rm 143}$ , K. Tani$^{\\rm 66}$ , N. Tannoury$^{\\rm 84}$ , S. Tapprogge$^{\\rm 82}$ , S. Tarem$^{\\rm 153}$ , F. Tarrade$^{\\rm 29}$ , G.F. Tartarelli$^{\\rm 90a}$ , P. Tas$^{\\rm 128}$ , M. Tasevsky$^{\\rm 126}$ , T. Tashiro$^{\\rm 67}$ , E. Tassi$^{\\rm 37a,37b}$ , A. Tavares Delgado$^{\\rm 125a,125b}$ , Y. Tayalati$^{\\rm 136d}$ , F.E.", "Taylor$^{\\rm 93}$ , G.N.", "Taylor$^{\\rm 87}$ , W. Taylor$^{\\rm 160b}$ , F.A.", "Teischinger$^{\\rm 30}$ , M. Teixeira Dias Castanheira$^{\\rm 75}$ , P. Teixeira-Dias$^{\\rm 76}$ , K.K.", "Temming$^{\\rm 48}$ , H. Ten Kate$^{\\rm 30}$ , P.K.", "Teng$^{\\rm 152}$ , S. Terada$^{\\rm 65}$ , K. Terashi$^{\\rm 156}$ , J. Terron$^{\\rm 81}$ , S. Terzo$^{\\rm 100}$ , M. Testa$^{\\rm 47}$ , R.J. Teuscher$^{\\rm 159}$$^{,i}$ , J. Therhaag$^{\\rm 21}$ , T. Theveneaux-Pelzer$^{\\rm 34}$ , S. Thoma$^{\\rm 48}$ , J.P. Thomas$^{\\rm 18}$ , J. Thomas-Wilsker$^{\\rm 76}$ , E.N.", "Thompson$^{\\rm 35}$ , P.D.", "Thompson$^{\\rm 18}$ , P.D.", "Thompson$^{\\rm 159}$ , A.S. Thompson$^{\\rm 53}$ , L.A. Thomsen$^{\\rm 36}$ , E. Thomson$^{\\rm 121}$ , M. Thomson$^{\\rm 28}$ , W.M.", "Thong$^{\\rm 87}$ , R.P.", "Thun$^{\\rm 88}$$^{,}$ , F. Tian$^{\\rm 35}$ , M.J. Tibbetts$^{\\rm 15}$ , V.O.", "Tikhomirov$^{\\rm 95}$$^{,ae}$ , Yu.A.", "Tikhonov$^{\\rm 108}$$^{,f}$ , S. Timoshenko$^{\\rm 97}$ , E. Tiouchichine$^{\\rm 84}$ , P. Tipton$^{\\rm 177}$ , S. Tisserant$^{\\rm 84}$ , T. Todorov$^{\\rm 5}$ , S. Todorova-Nova$^{\\rm 128}$ , B. Toggerson$^{\\rm 164}$ , J. Tojo$^{\\rm 69}$ , S. Tokár$^{\\rm 145a}$ , K. Tokushuku$^{\\rm 65}$ , K. Tollefson$^{\\rm 89}$ , L. Tomlinson$^{\\rm 83}$ , M. Tomoto$^{\\rm 102}$ , L. Tompkins$^{\\rm 31}$ , K. Toms$^{\\rm 104}$ , N.D. Topilin$^{\\rm 64}$ , E. Torrence$^{\\rm 115}$ , H. Torres$^{\\rm 143}$ , E. Torró Pastor$^{\\rm 168}$ , J. Toth$^{\\rm 84}$$^{,z}$ , F. Touchard$^{\\rm 84}$ , D.R.", "Tovey$^{\\rm 140}$ , H.L.", "Tran$^{\\rm 116}$ , T. Trefzger$^{\\rm 175}$ , L. Tremblet$^{\\rm 30}$ , A. Tricoli$^{\\rm 30}$ , I.M.", "Trigger$^{\\rm 160a}$ , S. Trincaz-Duvoid$^{\\rm 79}$ , M.F.", "Tripiana$^{\\rm 70}$ , N. Triplett$^{\\rm 25}$ , W. Trischuk$^{\\rm 159}$ , B. Trocmé$^{\\rm 55}$ , C. Troncon$^{\\rm 90a}$ , M. Trottier-McDonald$^{\\rm 143}$ , M. Trovatelli$^{\\rm 135a,135b}$ , P. True$^{\\rm 89}$ , M. Trzebinski$^{\\rm 39}$ , A. Trzupek$^{\\rm 39}$ , C. Tsarouchas$^{\\rm 30}$ , J.C-L. Tseng$^{\\rm 119}$ , P.V.", "Tsiareshka$^{\\rm 91}$ , D. Tsionou$^{\\rm 137}$ , G. Tsipolitis$^{\\rm 10}$ , N. Tsirintanis$^{\\rm 9}$ , S. Tsiskaridze$^{\\rm 12}$ , V. Tsiskaridze$^{\\rm 48}$ , E.G.", "Tskhadadze$^{\\rm 51a}$ , I.I.", "Tsukerman$^{\\rm 96}$ , V. Tsulaia$^{\\rm 15}$ , S. Tsuno$^{\\rm 65}$ , D. Tsybychev$^{\\rm 149}$ , A. Tudorache$^{\\rm 26a}$ , V. Tudorache$^{\\rm 26a}$ , A.N.", "Tuna$^{\\rm 121}$ , S.A. Tupputi$^{\\rm 20a,20b}$ , S. Turchikhin$^{\\rm 98}$$^{,ac}$ , D. Turecek$^{\\rm 127}$ , I. Turk Cakir$^{\\rm 4d}$ , R. Turra$^{\\rm 90a,90b}$ , P.M. Tuts$^{\\rm 35}$ , A. Tykhonov$^{\\rm 74}$ , M. Tylmad$^{\\rm 147a,147b}$ , M. Tyndel$^{\\rm 130}$ , K. Uchida$^{\\rm 21}$ , I. Ueda$^{\\rm 156}$ , R. Ueno$^{\\rm 29}$ , M. Ughetto$^{\\rm 84}$ , M. Ugland$^{\\rm 14}$ , M. Uhlenbrock$^{\\rm 21}$ , F. Ukegawa$^{\\rm 161}$ , G. Unal$^{\\rm 30}$ , A. Undrus$^{\\rm 25}$ , G. Unel$^{\\rm 164}$ , F.C.", "Ungaro$^{\\rm 48}$ , Y. Unno$^{\\rm 65}$ , D. Urbaniec$^{\\rm 35}$ , P. Urquijo$^{\\rm 21}$ , G. Usai$^{\\rm 8}$ , A. Usanova$^{\\rm 61}$ , L. Vacavant$^{\\rm 84}$ , V. Vacek$^{\\rm 127}$ , B. Vachon$^{\\rm 86}$ , N. Valencic$^{\\rm 106}$ , S. Valentinetti$^{\\rm 20a,20b}$ , A. Valero$^{\\rm 168}$ , L. Valery$^{\\rm 34}$ , S. Valkar$^{\\rm 128}$ , E. Valladolid Gallego$^{\\rm 168}$ , S. Vallecorsa$^{\\rm 49}$ , J.A.", "Valls Ferrer$^{\\rm 168}$ , R. Van Berg$^{\\rm 121}$ , P.C.", "Van Der Deijl$^{\\rm 106}$ , R. van der Geer$^{\\rm 106}$ , H. van der Graaf$^{\\rm 106}$ , R. Van Der Leeuw$^{\\rm 106}$ , D. van der Ster$^{\\rm 30}$ , N. van Eldik$^{\\rm 30}$ , P. van Gemmeren$^{\\rm 6}$ , J.", "Van Nieuwkoop$^{\\rm 143}$ , I. van Vulpen$^{\\rm 106}$ , M.C.", "van Woerden$^{\\rm 30}$ , M. Vanadia$^{\\rm 133a,133b}$ , W. Vandelli$^{\\rm 30}$ , R. Vanguri$^{\\rm 121}$ , A. Vaniachine$^{\\rm 6}$ , P. Vankov$^{\\rm 42}$ , F. Vannucci$^{\\rm 79}$ , G. Vardanyan$^{\\rm 178}$ , R. Vari$^{\\rm 133a}$ , E.W.", "Varnes$^{\\rm 7}$ , T. Varol$^{\\rm 85}$ , D. Varouchas$^{\\rm 79}$ , A. Vartapetian$^{\\rm 8}$ , K.E.", "Varvell$^{\\rm 151}$ , V.I.", "Vassilakopoulos$^{\\rm 56}$ , F. Vazeille$^{\\rm 34}$ , T. Vazquez Schroeder$^{\\rm 54}$ , J. Veatch$^{\\rm 7}$ , F. Veloso$^{\\rm 125a,125c}$ , S. Veneziano$^{\\rm 133a}$ , A. Ventura$^{\\rm 72a,72b}$ , D. Ventura$^{\\rm 85}$ , M. Venturi$^{\\rm 48}$ , N. Venturi$^{\\rm 159}$ , A. Venturini$^{\\rm 23}$ , V. Vercesi$^{\\rm 120a}$ , M. Verducci$^{\\rm 139}$ , W. Verkerke$^{\\rm 106}$ , J.C. Vermeulen$^{\\rm 106}$ , A. Vest$^{\\rm 44}$ , M.C.", "Vetterli$^{\\rm 143}$$^{,d}$ , O. Viazlo$^{\\rm 80}$ , I. Vichou$^{\\rm 166}$ , T. Vickey$^{\\rm 146c}$$^{,af}$ , O.E.", "Vickey Boeriu$^{\\rm 146c}$ , G.H.A.", "Viehhauser$^{\\rm 119}$ , S. Viel$^{\\rm 169}$ , R. Vigne$^{\\rm 30}$ , M. Villa$^{\\rm 20a,20b}$ , M. Villaplana Perez$^{\\rm 168}$ , E. Vilucchi$^{\\rm 47}$ , M.G.", "Vincter$^{\\rm 29}$ , V.B.", "Vinogradov$^{\\rm 64}$ , J. Virzi$^{\\rm 15}$ , I. Vivarelli$^{\\rm 150}$ , F. Vives Vaque$^{\\rm 3}$ , S. Vlachos$^{\\rm 10}$ , D. Vladoiu$^{\\rm 99}$ , M. Vlasak$^{\\rm 127}$ , A. Vogel$^{\\rm 21}$ , P. Vokac$^{\\rm 127}$ , G. Volpi$^{\\rm 47}$ , M. Volpi$^{\\rm 87}$ , H. von der Schmitt$^{\\rm 100}$ , H. von Radziewski$^{\\rm 48}$ , E. von Toerne$^{\\rm 21}$ , V. Vorobel$^{\\rm 128}$ , K. Vorobev$^{\\rm 97}$ , M. Vos$^{\\rm 168}$ , R. Voss$^{\\rm 30}$ , J.H.", "Vossebeld$^{\\rm 73}$ , N. Vranjes$^{\\rm 137}$ , M. Vranjes Milosavljevic$^{\\rm 106}$ , V. Vrba$^{\\rm 126}$ , M. Vreeswijk$^{\\rm 106}$ , T. Vu Anh$^{\\rm 48}$ , R. Vuillermet$^{\\rm 30}$ , I. Vukotic$^{\\rm 31}$ , Z. Vykydal$^{\\rm 127}$ , W. Wagner$^{\\rm 176}$ , P. Wagner$^{\\rm 21}$ , S. Wahrmund$^{\\rm 44}$ , J. Wakabayashi$^{\\rm 102}$ , J. Walder$^{\\rm 71}$ , R. Walker$^{\\rm 99}$ , W. Walkowiak$^{\\rm 142}$ , R. Wall$^{\\rm 177}$ , P. Waller$^{\\rm 73}$ , B. Walsh$^{\\rm 177}$ , C. Wang$^{\\rm 152}$ , C. Wang$^{\\rm 45}$ , F. Wang$^{\\rm 174}$ , H. Wang$^{\\rm 15}$ , H. Wang$^{\\rm 40}$ , J. Wang$^{\\rm 42}$ , J. Wang$^{\\rm 33a}$ , K. Wang$^{\\rm 86}$ , R. Wang$^{\\rm 104}$ , S.M.", "Wang$^{\\rm 152}$ , T. Wang$^{\\rm 21}$ , X. Wang$^{\\rm 177}$ , C. Wanotayaroj$^{\\rm 115}$ , A. Warburton$^{\\rm 86}$ , C.P.", "Ward$^{\\rm 28}$ , D.R.", "Wardrope$^{\\rm 77}$ , M. Warsinsky$^{\\rm 48}$ , A. Washbrook$^{\\rm 46}$ , C. Wasicki$^{\\rm 42}$ , I. Watanabe$^{\\rm 66}$ , P.M. Watkins$^{\\rm 18}$ , A.T. Watson$^{\\rm 18}$ , I.J.", "Watson$^{\\rm 151}$ , M.F.", "Watson$^{\\rm 18}$ , G. Watts$^{\\rm 139}$ , S. Watts$^{\\rm 83}$ , B.M.", "Waugh$^{\\rm 77}$ , S. Webb$^{\\rm 83}$ , M.S.", "Weber$^{\\rm 17}$ , S.W.", "Weber$^{\\rm 175}$ , J.S.", "Webster$^{\\rm 31}$ , A.R.", "Weidberg$^{\\rm 119}$ , P. Weigell$^{\\rm 100}$ , B. Weinert$^{\\rm 60}$ , J. Weingarten$^{\\rm 54}$ , C. Weiser$^{\\rm 48}$ , H. Weits$^{\\rm 106}$ , P.S.", "Wells$^{\\rm 30}$ , T. Wenaus$^{\\rm 25}$ , D. Wendland$^{\\rm 16}$ , Z. Weng$^{\\rm 152}$$^{,r}$ , T. Wengler$^{\\rm 30}$ , S. Wenig$^{\\rm 30}$ , N. Wermes$^{\\rm 21}$ , M. Werner$^{\\rm 48}$ , P. Werner$^{\\rm 30}$ , M. Wessels$^{\\rm 58a}$ , J. Wetter$^{\\rm 162}$ , K. Whalen$^{\\rm 29}$ , A. White$^{\\rm 8}$ , M.J. White$^{\\rm 1}$ , R. White$^{\\rm 32b}$ , S. White$^{\\rm 123a,123b}$ , D. Whiteson$^{\\rm 164}$ , D. Wicke$^{\\rm 176}$ , F.J. Wickens$^{\\rm 130}$ , W. Wiedenmann$^{\\rm 174}$ , M. Wielers$^{\\rm 130}$ , P. Wienemann$^{\\rm 21}$ , C. Wiglesworth$^{\\rm 36}$ , L.A.M.", "Wiik-Fuchs$^{\\rm 21}$ , P.A.", "Wijeratne$^{\\rm 77}$ , A. Wildauer$^{\\rm 100}$ , M.A.", "Wildt$^{\\rm 42}$$^{,ag}$ , H.G.", "Wilkens$^{\\rm 30}$ , J.Z.", "Will$^{\\rm 99}$ , H.H.", "Williams$^{\\rm 121}$ , S. Williams$^{\\rm 28}$ , C. Willis$^{\\rm 89}$ , S. Willocq$^{\\rm 85}$ , J.A.", "Wilson$^{\\rm 18}$ , A. Wilson$^{\\rm 88}$ , I. Wingerter-Seez$^{\\rm 5}$ , F. Winklmeier$^{\\rm 115}$ , M. Wittgen$^{\\rm 144}$ , T. Wittig$^{\\rm 43}$ , J. Wittkowski$^{\\rm 99}$ , S.J.", "Wollstadt$^{\\rm 82}$ , M.W.", "Wolter$^{\\rm 39}$ , H. Wolters$^{\\rm 125a,125c}$ , B.K.", "Wosiek$^{\\rm 39}$ , J. Wotschack$^{\\rm 30}$ , M.J. Woudstra$^{\\rm 83}$ , K.W.", "Wozniak$^{\\rm 39}$ , M. Wright$^{\\rm 53}$ , M. Wu$^{\\rm 55}$ , S.L.", "Wu$^{\\rm 174}$ , X. Wu$^{\\rm 49}$ , Y. Wu$^{\\rm 88}$ , E. Wulf$^{\\rm 35}$ , T.R.", "Wyatt$^{\\rm 83}$ , B.M.", "Wynne$^{\\rm 46}$ , S. Xella$^{\\rm 36}$ , M. Xiao$^{\\rm 137}$ , D. Xu$^{\\rm 33a}$ , L. Xu$^{\\rm 33b}$$^{,ah}$ , B. Yabsley$^{\\rm 151}$ , S. Yacoob$^{\\rm 146b}$$^{,ai}$ , M. Yamada$^{\\rm 65}$ , H. Yamaguchi$^{\\rm 156}$ , Y. Yamaguchi$^{\\rm 156}$ , A. Yamamoto$^{\\rm 65}$ , K. Yamamoto$^{\\rm 63}$ , S. Yamamoto$^{\\rm 156}$ , T. Yamamura$^{\\rm 156}$ , T. Yamanaka$^{\\rm 156}$ , K. Yamauchi$^{\\rm 102}$ , Y. Yamazaki$^{\\rm 66}$ , Z. Yan$^{\\rm 22}$ , H. Yang$^{\\rm 33e}$ , H. Yang$^{\\rm 174}$ , U.K. Yang$^{\\rm 83}$ , Y. Yang$^{\\rm 110}$ , S. Yanush$^{\\rm 92}$ , L. Yao$^{\\rm 33a}$ , W-M. Yao$^{\\rm 15}$ , Y. Yasu$^{\\rm 65}$ , E. Yatsenko$^{\\rm 42}$ , K.H.", "Yau Wong$^{\\rm 21}$ , J. Ye$^{\\rm 40}$ , S. Ye$^{\\rm 25}$ , A.L.", "Yen$^{\\rm 57}$ , E. Yildirim$^{\\rm 42}$ , M. Yilmaz$^{\\rm 4b}$ , R. Yoosoofmiya$^{\\rm 124}$ , K. Yorita$^{\\rm 172}$ , R. Yoshida$^{\\rm 6}$ , K. Yoshihara$^{\\rm 156}$ , C. Young$^{\\rm 144}$ , C.J.S.", "Young$^{\\rm 30}$ , S. Youssef$^{\\rm 22}$ , D.R.", "Yu$^{\\rm 15}$ , J. Yu$^{\\rm 8}$ , J.M.", "Yu$^{\\rm 88}$ , J. Yu$^{\\rm 113}$ , L. Yuan$^{\\rm 66}$ , A. Yurkewicz$^{\\rm 107}$ , B. Zabinski$^{\\rm 39}$ , R. Zaidan$^{\\rm 62}$ , A.M. Zaitsev$^{\\rm 129}$$^{,w}$ , A. Zaman$^{\\rm 149}$ , S. Zambito$^{\\rm 23}$ , L. Zanello$^{\\rm 133a,133b}$ , D. Zanzi$^{\\rm 100}$ , A. Zaytsev$^{\\rm 25}$ , C. Zeitnitz$^{\\rm 176}$ , M. Zeman$^{\\rm 127}$ , A. Zemla$^{\\rm 38a}$ , K. Zengel$^{\\rm 23}$ , O. Zenin$^{\\rm 129}$ , T. Ženiš$^{\\rm 145a}$ , D. Zerwas$^{\\rm 116}$ , G. Zevi della Porta$^{\\rm 57}$ , D. Zhang$^{\\rm 88}$ , F. Zhang$^{\\rm 174}$ , H. Zhang$^{\\rm 89}$ , J. Zhang$^{\\rm 6}$ , L. Zhang$^{\\rm 152}$ , X. Zhang$^{\\rm 33d}$ , Z. Zhang$^{\\rm 116}$ , Z. Zhao$^{\\rm 33b}$ , A. Zhemchugov$^{\\rm 64}$ , J. Zhong$^{\\rm 119}$ , B. Zhou$^{\\rm 88}$ , L. Zhou$^{\\rm 35}$ , N. Zhou$^{\\rm 164}$ , C.G.", "Zhu$^{\\rm 33d}$ , H. Zhu$^{\\rm 33a}$ , J. Zhu$^{\\rm 88}$ , Y. Zhu$^{\\rm 33b}$ , X. Zhuang$^{\\rm 33a}$ , A. Zibell$^{\\rm 175}$ , D. Zieminska$^{\\rm 60}$ , N.I.", "Zimine$^{\\rm 64}$ , C. Zimmermann$^{\\rm 82}$ , R. Zimmermann$^{\\rm 21}$ , S. Zimmermann$^{\\rm 21}$ , S. Zimmermann$^{\\rm 48}$ , Z. Zinonos$^{\\rm 54}$ , M. Ziolkowski$^{\\rm 142}$ , G. Zobernig$^{\\rm 174}$ , A. Zoccoli$^{\\rm 20a,20b}$ , M. zur Nedden$^{\\rm 16}$ , G. Zurzolo$^{\\rm 103a,103b}$ , V. Zutshi$^{\\rm 107}$ , L. Zwalinski$^{\\rm 30}$ .", "$^{1}$ Department of Physics, University of Adelaide, Adelaide, Australia $^{2}$ Physics Department, SUNY Albany, Albany NY, United States of America $^{3}$ Department of Physics, University of Alberta, Edmonton AB, Canada $^{4}$ $^{(a)}$ Department of Physics, Ankara University, Ankara; $^{(b)}$ Department of Physics, Gazi University, Ankara; $^{(c)}$ Division of Physics, TOBB University of Economics and Technology, Ankara; $^{(d)}$ Turkish Atomic Energy Authority, Ankara, Turkey $^{5}$ LAPP, CNRS/IN2P3 and Université de Savoie, Annecy-le-Vieux, France $^{6}$ High Energy Physics Division, Argonne National Laboratory, Argonne IL, United States of America $^{7}$ Department of Physics, University of Arizona, Tucson AZ, United States of America $^{8}$ Department of Physics, The University of Texas at Arlington, Arlington TX, United States of America $^{9}$ Physics Department, University of Athens, Athens, Greece $^{10}$ Physics Department, National Technical University of Athens, Zografou, Greece $^{11}$ Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{12}$ Institut de Física d'Altes Energies and Departament de Física de la Universitat Autònoma de Barcelona, Barcelona, Spain $^{13}$ $^{(a)}$ Institute of Physics, University of Belgrade, Belgrade; $^{(b)}$ Vinca Institute of Nuclear Sciences, University of Belgrade, Belgrade, Serbia $^{14}$ Department for Physics and Technology, University of Bergen, Bergen, Norway $^{15}$ Physics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley CA, United States of America $^{16}$ Department of Physics, Humboldt University, Berlin, Germany $^{17}$ Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Bern, Switzerland $^{18}$ School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom $^{19}$ $^{(a)}$ Department of Physics, Bogazici University, Istanbul; $^{(b)}$ Department of Physics, Dogus University, Istanbul; $^{(c)}$ Department of Physics Engineering, Gaziantep University, Gaziantep, Turkey $^{20}$ $^{(a)}$ INFN Sezione di Bologna; $^{(b)}$ Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna, Italy $^{21}$ Physikalisches Institut, University of Bonn, Bonn, Germany $^{22}$ Department of Physics, Boston University, Boston MA, United States of America $^{23}$ Department of Physics, Brandeis University, Waltham MA, United States of America $^{24}$ $^{(a)}$ Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; $^{(b)}$ Federal University of Juiz de Fora (UFJF), Juiz de Fora; $^{(c)}$ Federal University of Sao Joao del Rei (UFSJ), Sao Joao del Rei; $^{(d)}$ Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil $^{25}$ Physics Department, Brookhaven National Laboratory, Upton NY, United States of America $^{26}$ $^{(a)}$ National Institute of Physics and Nuclear Engineering, Bucharest; $^{(b)}$ National Institute for Research and Development of Isotopic and Molecular Technologies, Physics Department, Cluj Napoca; $^{(c)}$ University Politehnica Bucharest, Bucharest; $^{(d)}$ West University in Timisoara, Timisoara, Romania $^{27}$ Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina $^{28}$ Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom $^{29}$ Department of Physics, Carleton University, Ottawa ON, Canada $^{30}$ CERN, Geneva, Switzerland $^{31}$ Enrico Fermi Institute, University of Chicago, Chicago IL, United States of America $^{32}$ $^{(a)}$ Departamento de Física, Pontificia Universidad Católica de Chile, Santiago; $^{(b)}$ Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile $^{33}$ $^{(a)}$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; $^{(b)}$ Department of Modern Physics, University of Science and Technology of China, Anhui; $^{(c)}$ Department of Physics, Nanjing University, Jiangsu; $^{(d)}$ School of Physics, Shandong University, Shandong; $^{(e)}$ Physics Department, Shanghai Jiao Tong University, Shanghai, China $^{34}$ Laboratoire de Physique Corpusculaire, Clermont Université and Université Blaise Pascal and CNRS/IN2P3, Clermont-Ferrand, France $^{35}$ Nevis Laboratory, Columbia University, Irvington NY, United States of America $^{36}$ Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark $^{37}$ $^{(a)}$ INFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati; $^{(b)}$ Dipartimento di Fisica, Università della Calabria, Rende, Italy $^{38}$ $^{(a)}$ AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow; $^{(b)}$ Marian Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland $^{39}$ The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland $^{40}$ Physics Department, Southern Methodist University, Dallas TX, United States of America $^{41}$ Physics Department, University of Texas at Dallas, Richardson TX, United States of America $^{42}$ DESY, Hamburg and Zeuthen, Germany $^{43}$ Institut für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany $^{44}$ Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany $^{45}$ Department of Physics, Duke University, Durham NC, United States of America $^{46}$ SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom $^{47}$ INFN Laboratori Nazionali di Frascati, Frascati, Italy $^{48}$ Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg, Germany $^{49}$ Section de Physique, Université de Genève, Geneva, Switzerland $^{50}$ $^{(a)}$ INFN Sezione di Genova; $^{(b)}$ Dipartimento di Fisica, Università di Genova, Genova, Italy $^{51}$ $^{(a)}$ E. Andronikashvili Institute of Physics, Iv.", "Javakhishvili Tbilisi State University, Tbilisi; $^{(b)}$ High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia $^{52}$ II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany $^{53}$ SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom $^{54}$ II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany $^{55}$ Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier and CNRS/IN2P3 and Institut National Polytechnique de Grenoble, Grenoble, France $^{56}$ Department of Physics, Hampton University, Hampton VA, United States of America $^{57}$ Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA, United States of America $^{58}$ $^{(a)}$ Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(b)}$ Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(c)}$ ZITI Institut für technische Informatik, Ruprecht-Karls-Universität Heidelberg, Mannheim, Germany $^{59}$ Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima, Japan $^{60}$ Department of Physics, Indiana University, Bloomington IN, United States of America $^{61}$ Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität, Innsbruck, Austria $^{62}$ University of Iowa, Iowa City IA, United States of America $^{63}$ Department of Physics and Astronomy, Iowa State University, Ames IA, United States of America $^{64}$ Joint Institute for Nuclear Research, JINR Dubna, Dubna, Russia $^{65}$ KEK, High Energy Accelerator Research Organization, Tsukuba, Japan $^{66}$ Graduate School of Science, Kobe University, Kobe, Japan $^{67}$ Faculty of Science, Kyoto University, Kyoto, Japan $^{68}$ Kyoto University of Education, Kyoto, Japan $^{69}$ Department of Physics, Kyushu University, Fukuoka, Japan $^{70}$ Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET, La Plata, Argentina $^{71}$ Physics Department, Lancaster University, Lancaster, United Kingdom $^{72}$ $^{(a)}$ INFN Sezione di Lecce; $^{(b)}$ Dipartimento di Matematica e Fisica, Università del Salento, Lecce, Italy $^{73}$ Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom $^{74}$ Department of Physics, Jožef Stefan Institute and University of Ljubljana, Ljubljana, Slovenia $^{75}$ School of Physics and Astronomy, Queen Mary University of London, London, United Kingdom $^{76}$ Department of Physics, Royal Holloway University of London, Surrey, United Kingdom $^{77}$ Department of Physics and Astronomy, University College London, London, United Kingdom $^{78}$ Louisiana Tech University, Ruston LA, United States of America $^{79}$ Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and Université Paris-Diderot and CNRS/IN2P3, Paris, France $^{80}$ Fysiska institutionen, Lunds universitet, Lund, Sweden $^{81}$ Departamento de Fisica Teorica C-15, Universidad Autonoma de Madrid, Madrid, Spain $^{82}$ Institut für Physik, Universität Mainz, Mainz, Germany $^{83}$ School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom $^{84}$ CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{85}$ Department of Physics, University of Massachusetts, Amherst MA, United States of America $^{86}$ Department of Physics, McGill University, Montreal QC, Canada $^{87}$ School of Physics, University of Melbourne, Victoria, Australia $^{88}$ Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{89}$ Department of Physics and Astronomy, Michigan State University, East Lansing MI, United States of America $^{90}$ $^{(a)}$ INFN Sezione di Milano; $^{(b)}$ Dipartimento di Fisica, Università di Milano, Milano, Italy $^{91}$ B.I.", "Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Republic of Belarus $^{92}$ National Scientific and Educational Centre for Particle and High Energy Physics, Minsk, Republic of Belarus $^{93}$ Department of Physics, Massachusetts Institute of Technology, Cambridge MA, United States of America $^{94}$ Group of Particle Physics, University of Montreal, Montreal QC, Canada $^{95}$ P.N.", "Lebedev Institute of Physics, Academy of Sciences, Moscow, Russia $^{96}$ Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia $^{97}$ Moscow Engineering and Physics Institute (MEPhI), Moscow, Russia $^{98}$ D.V.Skobeltsyn Institute of Nuclear Physics, M.V.Lomonosov Moscow State University, Moscow, Russia $^{99}$ Fakultät für Physik, Ludwig-Maximilians-Universität München, München, Germany $^{100}$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München, Germany $^{101}$ Nagasaki Institute of Applied Science, Nagasaki, Japan $^{102}$ Graduate School of Science and Kobayashi-Maskawa Institute, Nagoya University, Nagoya, Japan $^{103}$ $^{(a)}$ INFN Sezione di Napoli; $^{(b)}$ Dipartimento di Fisica, Università di Napoli, Napoli, Italy $^{104}$ Department of Physics and Astronomy, University of New Mexico, Albuquerque NM, United States of America $^{105}$ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen, Netherlands $^{106}$ Nikhef National Institute for Subatomic Physics and University of Amsterdam, Amsterdam, Netherlands $^{107}$ Department of Physics, Northern Illinois University, DeKalb IL, United States of America $^{108}$ Budker Institute of Nuclear Physics, SB RAS, Novosibirsk, Russia $^{109}$ Department of Physics, New York University, New York NY, United States of America $^{110}$ Ohio State University, Columbus OH, United States of America $^{111}$ Faculty of Science, Okayama University, Okayama, Japan $^{112}$ Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK, United States of America $^{113}$ Department of Physics, Oklahoma State University, Stillwater OK, United States of America $^{114}$ Palacký University, RCPTM, Olomouc, Czech Republic $^{115}$ Center for High Energy Physics, University of Oregon, Eugene OR, United States of America $^{116}$ LAL, Université Paris-Sud and CNRS/IN2P3, Orsay, France $^{117}$ Graduate School of Science, Osaka University, Osaka, Japan $^{118}$ Department of Physics, University of Oslo, Oslo, Norway $^{119}$ Department of Physics, Oxford University, Oxford, United Kingdom $^{120}$ $^{(a)}$ INFN Sezione di Pavia; $^{(b)}$ Dipartimento di Fisica, Università di Pavia, Pavia, Italy $^{121}$ Department of Physics, University of Pennsylvania, Philadelphia PA, United States of America $^{122}$ Petersburg Nuclear Physics Institute, Gatchina, Russia $^{123}$ $^{(a)}$ INFN Sezione di Pisa; $^{(b)}$ Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa, Italy $^{124}$ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA, United States of America $^{125}$ $^{(a)}$ Laboratorio de Instrumentacao e Fisica Experimental de Particulas - LIP, Lisboa; $^{(b)}$ Faculdade de Ciências, Universidade de Lisboa, Lisboa; $^{(c)}$ Department of Physics, University of Coimbra, Coimbra; $^{(d)}$ Centro de Física Nuclear da Universidade de Lisboa, Lisboa; $^{(e)}$ Departamento de Fisica, Universidade do Minho, Braga; $^{(f)}$ Departamento de Fisica Teorica y del Cosmos and CAFPE, Universidad de Granada, Granada (Spain); $^{(g)}$ Dep Fisica and CEFITEC of Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal $^{126}$ Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic $^{127}$ Czech Technical University in Prague, Praha, Czech Republic $^{128}$ Faculty of Mathematics and Physics, Charles University in Prague, Praha, Czech Republic $^{129}$ State Research Center Institute for High Energy Physics, Protvino, Russia $^{130}$ Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{131}$ Physics Department, University of Regina, Regina SK, Canada $^{132}$ Ritsumeikan University, Kusatsu, Shiga, Japan $^{133}$ $^{(a)}$ INFN Sezione di Roma; $^{(b)}$ Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italy $^{134}$ $^{(a)}$ INFN Sezione di Roma Tor Vergata; $^{(b)}$ Dipartimento di Fisica, Università di Roma Tor Vergata, Roma, Italy $^{135}$ $^{(a)}$ INFN Sezione di Roma Tre; $^{(b)}$ Dipartimento di Matematica e Fisica, Università Roma Tre, Roma, Italy $^{136}$ $^{(a)}$ Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies - Université Hassan II, Casablanca; $^{(b)}$ Centre National de l'Energie des Sciences Techniques Nucleaires, Rabat; $^{(c)}$ Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; $^{(d)}$ Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; $^{(e)}$ Faculté des sciences, Université Mohammed V-Agdal, Rabat, Morocco $^{137}$ DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat à l'Energie Atomique et aux Energies Alternatives), Gif-sur-Yvette, France $^{138}$ Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA, United States of America $^{139}$ Department of Physics, University of Washington, Seattle WA, United States of America $^{140}$ Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom $^{141}$ Department of Physics, Shinshu University, Nagano, Japan $^{142}$ Fachbereich Physik, Universität Siegen, Siegen, Germany $^{143}$ Department of Physics, Simon Fraser University, Burnaby BC, Canada $^{144}$ SLAC National Accelerator Laboratory, Stanford CA, United States of America $^{145}$ $^{(a)}$ Faculty of Mathematics, Physics & Informatics, Comenius University, Bratislava; $^{(b)}$ Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic $^{146}$ $^{(a)}$ Department of Physics, University of Cape Town, Cape Town; $^{(b)}$ Department of Physics, University of Johannesburg, Johannesburg; $^{(c)}$ School of Physics, University of the Witwatersrand, Johannesburg, South Africa $^{147}$ $^{(a)}$ Department of Physics, Stockholm University; $^{(b)}$ The Oskar Klein Centre, Stockholm, Sweden $^{148}$ Physics Department, Royal Institute of Technology, Stockholm, Sweden $^{149}$ Departments of Physics & Astronomy and Chemistry, Stony Brook University, Stony Brook NY, United States of America $^{150}$ Department of Physics and Astronomy, University of Sussex, Brighton, United Kingdom $^{151}$ School of Physics, University of Sydney, Sydney, Australia $^{152}$ Institute of Physics, Academia Sinica, Taipei, Taiwan $^{153}$ Department of Physics, Technion: Israel Institute of Technology, Haifa, Israel $^{154}$ Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel $^{155}$ Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece $^{156}$ International Center for Elementary Particle Physics and Department of Physics, The University of Tokyo, Tokyo, Japan $^{157}$ Graduate School of Science and Technology, Tokyo Metropolitan University, Tokyo, Japan $^{158}$ Department of Physics, Tokyo Institute of Technology, Tokyo, Japan $^{159}$ Department of Physics, University of Toronto, Toronto ON, Canada $^{160}$ $^{(a)}$ TRIUMF, Vancouver BC; $^{(b)}$ Department of Physics and Astronomy, York University, Toronto ON, Canada $^{161}$ Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan $^{162}$ Department of Physics and Astronomy, Tufts University, Medford MA, United States of America $^{163}$ Centro de Investigaciones, Universidad Antonio Narino, Bogota, Colombia $^{164}$ Department of Physics and Astronomy, University of California Irvine, Irvine CA, United States of America $^{165}$ $^{(a)}$ INFN Gruppo Collegato di Udine, Sezione di Trieste, Udine; $^{(b)}$ ICTP, Trieste; $^{(c)}$ Dipartimento di Chimica, Fisica e Ambiente, Università di Udine, Udine, Italy $^{166}$ Department of Physics, University of Illinois, Urbana IL, United States of America $^{167}$ Department of Physics and Astronomy, University of Uppsala, Uppsala, Sweden $^{168}$ Instituto de Física Corpuscular (IFIC) and Departamento de Física Atómica, Molecular y Nuclear and Departamento de Ingeniería Electrónica and Instituto de Microelectrónica de Barcelona (IMB-CNM), University of Valencia and CSIC, Valencia, Spain $^{169}$ Department of Physics, University of British Columbia, Vancouver BC, Canada $^{170}$ Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada $^{171}$ Department of Physics, University of Warwick, Coventry, United Kingdom $^{172}$ Waseda University, Tokyo, Japan $^{173}$ Department of Particle Physics, The Weizmann Institute of Science, Rehovot, Israel $^{174}$ Department of Physics, University of Wisconsin, Madison WI, United States of America $^{175}$ Fakultät für Physik und Astronomie, Julius-Maximilians-Universität, Würzburg, Germany $^{176}$ Fachbereich C Physik, Bergische Universität Wuppertal, Wuppertal, Germany $^{177}$ Department of Physics, Yale University, New Haven CT, United States of America $^{178}$ Yerevan Physics Institute, Yerevan, Armenia $^{179}$ Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), Villeurbanne, France $^{a}$ Also at Department of Physics, King's College London, London, United Kingdom $^{b}$ Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{c}$ Also at Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{d}$ Also at TRIUMF, Vancouver BC, Canada e Also at Department of Physics, California State University, Fresno CA, United States of America $^{f}$ Also at Novosibirsk State University, Novosibirsk, Russia $^{g}$ Also at CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{h}$ Also at Università di Napoli Parthenope, Napoli, Italy $^{i}$ Also at Institute of Particle Physics (IPP), Canada $^{j}$ Also at Department of Financial and Management Engineering, University of the Aegean, Chios, Greece $^{k}$ Also at Louisiana Tech University, Ruston LA, United States of America $^{l}$ Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona, Spain $^{m}$ Also at CERN, Geneva, Switzerland $^{n}$ Also at Ochadai Academic Production, Ochanomizu University, Tokyo, Japan $^{o}$ Also at Manhattan College, New York NY, United States of America $^{p}$ Also at Institute of Physics, Academia Sinica, Taipei, Taiwan $^{q}$ Also at Department of Physics, Nanjing University, Jiangsu, China $^{r}$ Also at School of Physics and Engineering, Sun Yat-sen University, Guangzhou, China $^{s}$ Also at Academia Sinica Grid Computing, Institute of Physics, Academia Sinica, Taipei, Taiwan $^{t}$ Also at Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and Université Paris-Diderot and CNRS/IN2P3, Paris, France $^{u}$ Also at School of Physical Sciences, National Institute of Science Education and Research, Bhubaneswar, India $^{v}$ Also at Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italy $^{w}$ Also at Moscow Institute of Physics and Technology State University, Dolgoprudny, Russia $^{x}$ Also at Section de Physique, Université de Genève, Geneva, Switzerland $^{y}$ Also at Department of Physics, The University of Texas at Austin, Austin TX, United States of America $^{z}$ Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary $^{aa}$ Also at International School for Advanced Studies (SISSA), Trieste, Italy $^{ab}$ Also at Department of Physics and Astronomy, University of South Carolina, Columbia SC, United States of America $^{ac}$ Also at Faculty of Physics, M.V.Lomonosov Moscow State University, Moscow, Russia $^{ad}$ Also at Physics Department, Brookhaven National Laboratory, Upton NY, United States of America $^{ae}$ Also at Moscow Engineering and Physics Institute (MEPhI), Moscow, Russia $^{af}$ Also at Department of Physics, Oxford University, Oxford, United Kingdom $^{ag}$ Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany $^{ah}$ Also at Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{ai}$ Also at Discipline of Physics, University of KwaZulu-Natal, Durban, South Africa $^{*}$ Deceased" ] ]	1403.0489
[ [ "Deep $\\Pi^0_1$ Classes" ], [ "Abstract A set of infinite binary sequences $\\mathcal{C}\\subseteq2^\\omega$ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability.", "The study of negligibility is of particular interest in the context of $\\Pi^0_1$ classes.", "In this paper, we introduce the notion of depth for $\\Pi^0_1$ classes, which is a stronger form of negligibility.", "Whereas a negligible $\\Pi^0_1$ class $\\mathcal{C}$ has the property that one cannot probabilistically compute a member of $\\mathcal{C}$ with positive probability, a deep $\\Pi^0_1$ class $\\mathcal{C}$ has the property that one cannot probabilistically compute an initial segment of a member of $\\mathcal{C}$ with high probability.", "That is, the probability of computing a length $n$ initial segment of a deep $\\Pi^0_1$ class converges to 0 effectively in $n$.", "We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call $\\mathit{tt}$-negligibility.", "We also provide a number of examples of deep $\\Pi^0_1$ classes that occur naturally in computability theory and algorithmic randomness.", "We also study deep classes in the context of mass problems, we examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin's $\\Omega$." ], [ "Introduction", "A vast majority of the work carried out in computability theory since its inception has been concerned with deterministic computation, that is, computational procedures with the property that after any finite number of steps have been carried out, there is one unique step that follows from them.", "However, by generalizing the Turing model of computation, one can also study various formalizations of non-deterministic computation.", "In particular, there are a number of models of probabilistic computation that have been well studied in the context of computational complexity.", "By contrast, no systematic study of such models of probabilistic computation in the study of structures such as the degrees of unsolvability has been carried out.", "Here we undertake in such a study.", "The basic objects of our investigation are $\\Pi ^0_1$ classes, that is, effectively closed subclasses of $2^\\mathbb {N}$ .", "Our goal is to study the properties of those $\\Pi ^0_1$ classes whose members cannot be probabilistically computed with positive probability.", "For our purposes, the model of probabilistic computation that is most convenient is given by oracle Turing machines equipped with algorithmically random oracles.", "Thus, for a given $\\Pi ^0_1$ class $2^\\mathbb {N}$ , the question Can one probabilistically compute some member of $ with positive probability?$ amounts to asking Does the class of random oracles that compute some member of $ have positive Lebesgue measure?$ We will discuss two specific types of $\\Pi ^0_1$ classes, negligible classes and deep classes, focussing in particular on the latter type, which is a special case of the former.", "Roughly, a $\\Pi ^0_1$ class $ is \\emph {negligible} if the probability of producing a member of~$ via any probabilistic algorithm is 0.", "Rephrased in terms of the model of probabilistic computation we use here, $ is negligible if the collection$ eNe-1($ has Lebesgue measure~0, where $ (e)eN$ is an effective enumeration of all Turing functionals.", "The first example of a negligible class appeared in \\cite {JockuschS1972}, in which it was proved that the $ 01$ class of consistent completions of $ PA$ is negligible.", "Negligibility was further studied in \\cite {LevinV1977} and \\cite {Vyugin1982}.$ Depth can be viewed as a strong form of negligibility.", "A deep $\\Pi ^0_1$ class $ has the property that producing an initial segment of some member of $ is maximally difficult: the probability of obtaining such an initial segment not only converges to 0 as we consider longer initial segments of members of $, but this convergence occurs quickly, as we can effectively bound the rate of convergence.", "That initial segments of members of deep classes are so difficult to produce via probabilistic computation reflects the fact that members of deep classes are highly structured: Initial segments of these sequences cannot be successfully produced by any combination of Turing machine and random oracle.", "As we will see, one consequence of this latter fact is that no $ 01$ class containing a computable member or an algorithmically random member can be deep.$ Although the notion of depth is isolated for the first time in the present study, it is implicitly used in the work of both Levin and Stephan .", "In fact, in each of these papers one can extract a proof that the consistent, completions of Peano arithmetic form a deep class, a result that we reprove in $§$.", "The primary goals of this paper are (1) to prove a number of basic results about deep $\\Pi ^0_1$ classes, (2) to determine the exact relationship between negligibility and depth (and a related notion we call $\\mathit {tt}$ -negligibility), and (3) to provide a number of examples of deep $\\Pi ^0_1$ classes that occur naturally in computability theory and algorithmic randomness.", "Carrying out these tasks is just a first step towards establishing depth as a notion that merits further study in computability theory and algorithmic randomness.", "The outline of this paper is as follows: In $§$, we provide the technical background for the remainder of the study.", "The notions of negligibility of $\\mathit {tt}$ -negligibility are introduced and compared in $§$ while the notions of depth and $\\mathit {tt}$ -depth are introduced in $§$.", "In $§$, we also separate depth from negligibility, show the equivalence between $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility, and prove several basic facts about these various kinds of classes.", "$§$ contains a brief discussion of the levels of randomness that guarantee that a random sequence cannot compute any member of any $\\mathit {tt}$ -negligible, negligible, and deep class.", "We consider the notions of depth and negligibility in the context of Medvedev and Muchnik reducibility in $§$.", "In $§$ we present six examples of families of deep classes, each of which is defined in terms of some well-studied notion from computability theory and algorithmic randomness: completions of arithmetic, shift-complex sequences, diagonally non-computable functions, compression functions, finite sets of maximally incompressible strings, and dominating martingales related to the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ .", "In $§$, we establish connections between depth, $\\mathit {tt}$ -depth and several lowness notions.", "Lastly, in $§$, we apply the notion of mutual information to deep classes, generalizing a result of Levin's from .", "Background Notation Let us fix some notation and terminology.", "We denote by $2^\\mathbb {N}$ the set of infinite binary sequences (which we often refer to as “sequences\"), also known as Cantor space.", "We denote the set of finite strings by $2^{<\\mathbb {N}}$ and the empty string by $\\Lambda $ .", "$\\mathbb {Q}_2$ is the set of dyadic rationals, i.e., multiples of a negative power of 2.", "Given $X\\in 2^\\mathbb {N}$ and an integer $n$ , $X {\\upharpoonright }n$ is the string that consists of the first $n$ bits of $X$ , and $X(n)$ is the $(n+1)$ st bit of $X$ (so that $X(0)$ is the first bit of $X$ ).", "For integers $n$ and $k$ , $X {\\upharpoonright }[n,n+k]$ denotes the subword $\\sigma $ of $X$ of length $k+1$ such that for $i \\le k$ , $\\sigma (i)=X(i+n)$ .", "If $\\sigma $ is a string and $x\\in 2^{<\\mathbb {N}}\\cup 2^\\mathbb {N}$ , then $\\sigma \\preceq x$ means that $\\sigma $ is a prefix of $x$ .", "A prefix-free set of strings is a set of strings such that none of its elements is a strict prefix of another one.", "For strings $\\sigma ,\\tau \\in 2^{<\\mathbb {N}}$ , $\\sigma ^\\frown \\tau $ denotes string obtained by concatenating $\\sigma $ and $\\tau $ .", "Similarly, for $\\sigma \\in 2^{<\\mathbb {N}}$ and $X\\in 2^\\mathbb {N}$ , $\\sigma ^\\frown X$ is the sequence obtained by concatenating $\\sigma $ and $X$ .", "For $X,Y\\in 2^\\mathbb {N}$ , $X\\oplus Y\\in 2^\\mathbb {N}$ satisfies $X\\oplus Y(2n)=X(n)$ and $X\\oplus Y(2n+1)=Y(n)$ .", "Given a string $\\sigma $ , the cylinder $\\llbracket \\sigma \\rrbracket $ is the set of elements of $2^\\mathbb {N}$ having $\\sigma $ as a prefix.", "Moreover, given $S\\subseteq 2^{<\\mathbb {N}}$ , $\\llbracket S\\rrbracket $ is defined to be the set $\\bigcup _{\\sigma \\in S}\\llbracket \\sigma \\rrbracket $ .", "When we refer to the topology of the Cantor space, we implicitly mean the product topology, i.e., the topology whose open sets are exactly those of type $\\llbracket S\\rrbracket $ for some $S$ .", "For this topology, some sets are both open and closed (clopen): these are the sets of type $\\llbracket S\\rrbracket $ when $S$ is finite.", "A tree is a set of strings that is closed downwards under the prefix relation.", "A path through a tree $T$ is a member of $2^\\mathbb {N}$ all of whose prefixes are in $T$ .", "The set of paths of a tree $T$ is denoted by $[T]$ .", "The $n^\\mathrm {th}$ level of a tree $T$ , denoted $T_n$ , is the set of members of $T$ of length $n$ .", "An effectively open set, also called $\\Sigma ^0_1$ , is a set of type $\\llbracket S \\rrbracket $ for some c.e.", "set of strings $S$ .", "An effectively closed set, or $\\Pi ^0_1$ class, is the complement of some effectively open set.", "It is well-known that a class is $\\Pi ^0_1$ if and only if it is of the form $[T]$ for some co-c.e.", "(or computable) tree $T$ .", "Given a $\\Pi ^0_1$ class $, its \\emph {canonical co-c.e.\\ tree} is the tree $ T={[] = }$.", "The arithmetic hierarchy is defined inductively: a set is $ 0n+1$ if it is a uniform union of $ 0n$ sets, and an set is $ 0n+1$ if it is a uniform intersection of~$ 0n$ sets.$ A order function is a function $h: \\mathbb {N}\\rightarrow \\mathbb {N}$ that is non-decreasing and unbounded.", "Given an order $h$ , the inverse of $h$, denoted $h^{-1}$ is the order function defined as follows: For all $k$ , $h^{-1}(k)$ is the smallest $n$ such that $h(n) \\ge k$ .", "Note that $h^{-1}$ is computable if $h$ is.", "We adopt standard computability notation: $\\le _T$ denotes Turing reducibility, $\\le _{tt}$ denotes $\\mathit {tt}$ -reducibility, $A^{\\prime }$ is the Turing jump of $A$ (and $\\emptyset ^{\\prime }$ denotes the jump of the zero sequence).", "Throughout the paper, when an object $A$ has a computable enumeration or limit approximation, we use the notation $A[s]$ to denote the approximation of the object at stage $s$ .", "Moreover, if $A$ contains several expressions that have a computable approximation, the notation $A[s]$ means that all of these expressions are approximated up to stage $s$ .", "For example, if $T$ is a tree and $f$ is a function, both of which have computable approximation, $T_{f(n)}[s]$ is equal to $(T[s])_{f(n)[s]}$ .", "Finally, we adopt the following asymptotic notation.", "For two functions $f,g: \\mathbb {N}\\rightarrow \\mathbb {N}$ , we sometimes write $f \\le ^+ g$ to abbreviate $f \\le g + O(1)$ and $f \\le ^* g$ to abbreviate $f = O(g)$ .", "Measures on $2^\\mathbb {N}$ Recall that by Caratheodory's Theorem, a measure $\\mu $ on $2^\\mathbb {N}$ is uniquely determined by specifying the values of $\\mu $ on the basic open sets of $2^\\mathbb {N}$ , where $\\mu (\\llbracket \\sigma \\rrbracket )=\\mu (\\llbracket \\sigma 0\\rrbracket )+\\mu (\\llbracket \\sigma 1\\rrbracket )$ for every $\\sigma \\in 2^{<\\mathbb {N}}$ .", "If we further require that $\\mu (2^\\mathbb {N})=1$ , then $\\mu $ is a probability measure.", "Hereafter, we will write $\\mu (\\llbracket \\sigma \\rrbracket )$ as $\\mu (\\sigma )$ .", "In addition, for a set $S\\subseteq 2^{<\\mathbb {N}}$ , we will write $\\mu (S)$ as shorthand for $\\mu (\\llbracket S\\rrbracket )$ .", "In the case that $S$ is prefix-free, we will have $\\mu (S)=\\sum _{\\sigma \\in 2^{<\\mathbb {N}}} \\mu (\\sigma )$ .", "The uniform (or Lebesgue) measure $\\lambda $ is the unique Borel measure such that $\\lambda (\\sigma )=2^{-\|\\sigma \|}$ for all strings $\\sigma $ .", "A measure $\\mu $ on $2^\\mathbb {N}$ is computable if $\\sigma \\mapsto \\mu (\\sigma )$ is computable as a real-valued function, i.e., if there is a computable function $\\tilde{\\mu }:2^{<\\mathbb {N}}\\times \\mathbb {N}\\rightarrow \\mathbb {Q}_2$ such that $\|\\mu (\\sigma )-\\tilde{\\mu }(\\sigma ,i)\|\\le 2^{-i}$ for every $\\sigma \\in 2^{<\\mathbb {N}}$ and $i\\in \\mathbb {N}$ .", "One family of examples of computable measures is given by the collection of Dirac measures concentrated on some computable point.", "That is, if $X\\in 2^\\mathbb {N}$ is a computable sequence, then the Dirac measure concentrated on $X$, denoted $\\delta _X$ , is defined as follows: $\\delta _X(\\sigma )=\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if } \\sigma \\prec X\\\\0 & \\mbox{if } \\sigma \\lnot \\prec X\\end{array}.\\right.$ More generally, for a measure $\\mu $ , we say that $X\\in 2^\\mathbb {N}$ is an atom of $\\mu $ or a $\\mu $ -atom, denoted $X\\in \\mathsf {Atom}_\\mu $ , if $\\mu (\\lbrace X\\rbrace )>0$ .", "Kautz proved the following: Lemma 2.1 (Kautz ) $X\\in 2^\\mathbb {N}$ is computable if and only if $X$ is an atom of some computable measure.", "There is a close connection between computable measures and a certain class of Turing functionals.", "Recall that a Turing functional $\\Phi :\\subseteq 2^\\mathbb {N}\\rightarrow 2^\\mathbb {N}$ may be defined as a c.e.", "set $\\Phi $ of pairs of strings $(\\sigma ,\\tau )$ such that if $(\\sigma ,\\tau ),(\\sigma ^{\\prime },\\tau ^{\\prime })\\in \\Phi $ and $\\sigma \\preceq \\sigma ^{\\prime }$ , then $\\tau \\preceq \\tau ^{\\prime }$ or $\\tau ^{\\prime }\\preceq \\tau $ .", "For each $\\sigma \\in 2^{<\\mathbb {N}}$ , we define $\\Phi ^\\sigma $ to be the maximal string in $\\lbrace \\tau : (\\exists \\sigma ^{\\prime }\\preceq \\sigma )((\\sigma ^{\\prime },\\tau )\\in \\Phi )\\rbrace $ in the order given by $\\preceq $ .", "To obtain a map defined on $2^\\mathbb {N}$ from the c.e.", "set of pairs $\\Phi $ , for each $X\\in 2^\\mathbb {N}$ , we let $\\Phi ^X$ be the maximal $y\\in 2^{<\\mathbb {N}}\\cup 2^\\mathbb {N}$ in the order given by $\\preceq $ such that $\\Phi ^{X {\\upharpoonright }n}$ is a prefix of $y$ for all $n$ .", "We will thus set $\\mathsf {dom}(\\Phi )=\\lbrace X\\in 2^\\mathbb {N}:\\Phi ^X\\in 2^\\mathbb {N}\\rbrace $ .", "When $\\Phi ^X\\in 2^\\mathbb {N}$ , we will often write $\\Phi ^X$ as $\\Phi (X)$ to emphasize the functional $\\Phi $ as a map from $2^\\mathbb {N}$ to $2^\\mathbb {N}$ .", "We also use the notation $\\Phi ^X {\\upharpoonright }n \\downarrow $ to emphasize that $\\Phi ^X$ has length at least $n$ .", "For $\\tau \\in 2^{<\\mathbb {N}}$ let $\\Phi ^{-1}(\\tau )$ be the set $\\lbrace \\sigma \\in 2^{<\\mathbb {N}}: \\exists \\tau ^{\\prime } \\succeq \\tau \\colon (\\sigma ,\\tau ^{\\prime })\\in \\Phi \\rbrace $ .", "Similarly, for $S \\subseteq 2^{<\\mathbb {N}}$ we define $\\Phi ^{-1}(S) = \\bigcup _{\\tau \\in S} \\Phi ^{-1}(\\tau )$ .", "When $\\mathcal {A}$ is a subset of $2^\\mathbb {N}$ , we denote by $\\Phi ^{-1}(\\mathcal {A})$ the set $\\lbrace X\\in \\mathsf {dom}(\\Phi ):\\Phi (X) \\in \\mathcal {A}\\rbrace $ .", "Note in particular that $\\Phi ^{-1}(\\llbracket \\tau \\rrbracket )= \\llbracket \\Phi ^{-1}(\\tau )\\rrbracket \\cap \\mathsf {dom}(\\Phi )$ .", "Remark 2.2 The Turing functionals that induce computable measures are precisely the almost total Turing functionals, where a Turing functional $\\Phi $ is almost total if $\\lambda (\\mathsf {dom}(\\Phi ))=1.$ Given an almost total Turing functional $\\Phi $ , the measure induced by $\\Phi $ , denoted $\\lambda _\\Phi $ , is defined by $\\lambda _\\Phi (\\sigma )=\\lambda (\\llbracket \\Phi ^{-1}(\\sigma )\\rrbracket )=\\lambda (\\lbrace X:\\Phi ^X\\succeq \\sigma \\rbrace ).$ It is not difficult to verify that $\\lambda _\\Phi $ is a computable measure.", "Moreover, one can easily show that given a computable measure $\\mu $ , there is some almost total functional $\\Phi $ such that $\\mu =\\lambda _\\Phi $ .", "Lower semi-computable semi-measures A discrete semi-measure is a map $m:2^{<\\mathbb {N}}\\rightarrow [0,1]$ such that $\\sum _{\\sigma \\in 2^{<\\mathbb {N}}}m(\\sigma )\\le 1$ .", "Moreover, if $S$ is a set of strings, then $m(S)$ is defined to be $\\sum _{\\sigma \\in S} m(\\sigma )$ .", "Henceforth, we will restrict our attention to the class of lower semi-computable discrete semi-measures, where a discrete semi-measure $m$ is lower semi-computable if there is a computable function $\\widetilde{m} : 2^{<\\mathbb {N}}\\times \\mathbb {N}\\rightarrow \\mathbb {Q}_2$ , non-decreasing in its second argument such that for all $\\sigma $ : $\\lim _{i \\rightarrow +\\infty } \\widetilde{m}(\\sigma ,i) = m(\\sigma ).$ Levin showed that there is a universal lower semi-computable discrete semi-measure $\\mathbf {m}$ ; that is, for every lower semi-computable discrete semi-measure $m$ , there is some constant $c$ such that $m\\le c\\cdot \\mathbf {m}$ .", "This universal discrete semi-measure $\\mathbf {m}$ is closely related to the notion of prefix-free Kolmogorov complexity.", "Recall that $\\mathrm {K}(\\sigma )$ denotes the prefix-free Kolmogorov complexity of $\\sigma $ , i.e.", "$\\mathrm {K}(\\sigma )=\\min \\lbrace \|\\tau \|:U(\\tau )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free Turing machine.", "Then by the coding theorem (see ), we have $\\mathrm {K}(\\sigma )=-\\log \\mathbf {m}(\\sigma )+O(1)$ .", "In particular, since $\\mathrm {K}(n)\\le ^+ 2\\log (n)$ , it follows that $\\mathbf {m}(n)\\ge ^* n^{-2}$ , a fact we will make use of below.", "A continuous semi-measure $\\rho :2^{<\\mathbb {N}}\\rightarrow [0,1]$ satisfies (i) $\\rho (\\epsilon ) = 1$ and (ii) $\\rho (\\sigma )\\ge \\rho (\\sigma 0)+\\rho (\\sigma 1)$ .", "If $S$ is a set of strings, then $\\rho (S)$ denotes the sum $\\sum _{\\sigma \\in S} \\rho (\\sigma )$ .", "As in the case of discrete semi-measures, we will restrict our attention to the class of lower semi-computable continuous semi-measures (the values of which are effectively approximable from below as defined above in the case of discrete lower semi-computable semi-measures).", "One particularly important property of lower semi-computable continuous semi-measures is their connection to Turing functionals.", "Just as computable measures are precisely the measures that are induced by almost total Turing functionals (as discussed at the end of the previous section), lower semi-computable continuous semi-measures are precisely the semi-measures that are induced by Turing functionals: Theorem 2.1 (Levin, Zvonkin ) (i) For every Turing functional $\\Phi $ , the function $\\lambda _\\Phi (\\sigma )=\\lambda (\\llbracket \\Phi ^{-1}(\\sigma )\\rrbracket )=\\lambda (\\lbrace X:\\Phi ^X\\succeq \\sigma \\rbrace )$ is a left-c.e.", "semi-measure.", "(ii) For every left-c.e.", "semi-measure $\\rho $ , there is a Turing functional $\\Phi $ such that $\\rho =\\lambda _\\Phi $ .", "As there is a universal lower semi-computable discrete semi-measure, so too is there a universal lower semi-computable continuous semi-measure.", "That is, there exists a lower semi-computable continuous semi-measure $\\mathbf {M}$ such that, for every lower semi-computable continuous semi-measure $\\rho $ , there exists a $c\\in \\mathbb {N}$ such that $\\rho \\le c\\cdot \\mathbf {M}$ .", "One way to obtain a universal lower semi-computable continuous semi-measure is to effectively list all lower semi-computable continuous semi-measures $(\\rho _e)_{e \\in \\mathbb {N}}$ (which can be obtained from an effective list of all Turing functionals by appealing to Theorem REF ) and set $\\mathbf {M}= \\sum _{e \\in \\mathbb {N}} 2^{-e-1} \\rho _e$ .", "Alternatively, one can induce it by means of a universal Turing functional: Let $(\\Phi _i)_{i\\in \\mathbb {N}}$ be an effective enumeration of all Turing functionals.", "Then the functional $\\widehat{\\Phi }$ such that $\\widehat{\\Phi }(1^e0X)=\\Phi _e(X)$ for every $e\\in \\mathbb {N}$ and $X\\in 2^\\mathbb {N}$ is a universal Turing functional and we can set $\\mathbf {M}=\\lambda _{\\widehat{\\Phi }}$ .", "One can readily verify that $\\mathbf {M}$ is an universal lower semi-computable continuous semi-measure, which is sometimes called a priori probability (see for example Gács for the basic properties of $\\mathbf {M}$ ).", "Another feature of continuous semi-measures that we will make use of throughout this study is that there is a canonical measure on $2^\\mathbb {N}$ that can be obtained from a continuous semi-measure.", "To motivate the definition of this measure, it is helpful to think of a semi-measure as a network flow through the full binary tree $2^{<\\mathbb {N}}$ seen as a directed graph (see, for instance, or ).", "First we give the node at the root of the tree flow equal to 1 (corresponding to the condition $\\rho (\\epsilon )=1$ ).", "Some amount of this flow at each node $\\sigma $ is passed along to the node corresponding to $\\sigma 0$ , some is passed along to the node corresponding to $\\sigma 1$ , and, potentially, some of the flow is lost (corresponding to the condition that $\\rho (\\sigma )\\ge \\rho (\\sigma 0)+\\rho (\\sigma 1)$ ).", "We obtain a measure $\\overline{\\rho }$ from $\\rho $ if we ignore all of the flow that is lost below a given node and just consider the behavior of the flow that never leaves the network below this node.", "We will refer to $\\overline{\\rho }$ as the measure derived from $\\rho $ .", "This can be formalized as follows.", "Definition 2.3 Let $\\rho $ be a semi-measure.", "The canonical measure obtained from $\\rho $ is defined to be $\\overline{\\rho }(\\sigma ):=\\inf _{n\\ge \|\\sigma \|}\\sum _{\\tau \\succeq \\sigma \\;\\&\\;\|\\tau \|=n}\\rho (\\tau ) = \\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\succeq \\sigma \\;\\&\\;\|\\tau \|=n}\\rho (\\tau ).$ Several important facts about these canonical measures are the following: Proposition 2.4 Let $\\rho $ be a semi-measure and let $\\overline{\\rho }$ be the canonical measure obtained from $\\rho $ .", "(i) $\\overline{\\rho }$ is the largest measure $\\mu $ such that $\\mu \\le \\rho $ .", "(ii) If $\\rho (\\sigma )=\\lambda (\\lbrace X:\\Phi ^X\\succeq \\sigma \\rbrace )$ , then $\\overline{\\rho }(\\sigma )=\\lambda (\\lbrace X \\in \\mathsf {dom}(\\Phi ): \\Phi ^X\\succeq \\sigma \\rbrace )$ .", "Thus, in replacing $\\rho $ with $\\overline{\\rho }$ , this amounts to restricting the Turing functional $\\Phi $ that induces $\\rho $ to those inputs on which $\\Phi $ is total.", "The proof of part $(i)$ is straightforward; for a proof of $(\\mathit {ii})$ , see .", "Remark 2.5 Using Proposition REF ($\\mathit {ii}$ ) and the universality of $\\mathbf {M}$ , one can readily verify that for every lower-semi-computable continuous semi-measure $\\rho $ , there is some $c\\in \\mathbb {N}$ such that $\\overline{\\rho }\\le c\\cdot \\overline{\\mathbf {M}}$ .", "Thus $\\overline{\\mathbf {M}}$ can be seen as a measure that is universal for the class of canonical measures obtained from some lower-semi-computable continuous semi-measure (a class that contains all computable measures).", "Notions of algorithmic randomness The primary notion of algorithmic randomness that we will consider here is Martin-Löf randomness.", "Although the standard definition is given in terms of the Lebesgue measure, we will consider Martin-Löf randomness with respect to any computable measure.", "Definition 2.6 Let $\\mu $ be a computable measure on $2^\\mathbb {N}$ .", "(i) A $\\mu $ -Martin-Löf test is a sequence $(\\mathcal {U}_i)_{i\\in \\mathbb {N}}$ of uniformly effectively open subsets of $2^\\mathbb {N}$ such that for each $i$ , $\\mu (\\mathcal {U}_i)\\le 2^{-i}.$ (ii) $X\\in 2^\\mathbb {N}$ passes the $\\mu $ -Martin-Löf test $(\\mathcal {U}_i)_{i\\in \\mathbb {N}}$ if $X\\notin \\bigcap _{i \\in \\mathbb {N}}\\mathcal {U}_i$ .", "(iii) $X\\in 2^\\mathbb {N}$ is $\\mu $ -Martin-Löf random, denoted $X\\in \\mathsf {MLR}_\\mu $ , if $X$ passes every $\\mu $ -Martin-Löf test.", "When $\\mu $ is the uniform (or Lebesgue) measure $\\lambda $ , we often abbreviate $\\mathsf {MLR}_\\mu $ by $\\mathsf {MLR}$ .", "An important feature of Martin-Löf randomness is the existence of a universal test: For every computable measure $\\mu $ , there is a single $\\mu $ -Martin-Löf test $(\\hat{\\mathcal {U}}_i)_{i\\in \\mathbb {N}}$ , having the property that $X\\in \\mathsf {MLR}_\\mu $ if and only if $X\\notin \\bigcap _{i\\in \\mathbb {N}}\\hat{\\mathcal {U}}_i$ .", "Remark 2.7 As we saw earlier, some computable measures have atoms, such as the Dirac measure $\\delta _X$ concentrated on some computable sequence $X$ .", "Moreover, given a computable measure $\\mu $ , if $X$ is a $\\mu $ -atom, it immediately follows that $X\\in \\mathsf {MLR}_\\mu $ .", "Four additional notions of algorithmic randomness that will be considered in this study are difference randomness, Kurtz randomness, weak 2-randomness, and computable randomness.", "Definition 2.8 (i) A difference test is a computable sequence $((\\mathcal {U}_i,\\mathcal {V}_i))_{i\\in \\mathbb {N}}$ of pairs of $\\Sigma ^0_1$ classes such that for each $i$ , $\\lambda (\\mathcal {U}_i\\setminus \\mathcal {V}_i)\\le 2^{-i}.$ (ii) A sequence $X\\in 2^\\mathbb {N}$ passes a difference test $((\\mathcal {U}_i,\\mathcal {V}_i))_{i\\in \\mathbb {N}}$ if $X\\notin \\bigcap _i(\\mathcal {U}_i\\setminus \\mathcal {V}_i)$ .", "(iii) $X\\in 2^\\mathbb {N}$ is difference random if $X$ passes every difference test.", "We denote by $\\mathsf {DiffR}$ the class of difference random reals.", "Franklin and Ng proved the following remarkable theorem about difference randomness: Theorem 2.2 (Franklin and Ng ) A sequence $X$ is difference random if and only if $X$ is Martin-Löf random and $X \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Recall that a sequence $X$ has $\\mathsf {PA}$ degree if $X$ can compute a consistent completion of Peano arithmetic (hereafter, $\\mathsf {PA}$ ).", "This is equivalent to requiring that $X$ compute a total function extending a universal partial computable $\\lbrace 0,1\\rbrace $ -valued function, a fact that will be useful in $§$.", "A related result is the following: Theorem 2.3 (Stephan ) A Martin-Löf random sequence $X$ has $\\mathsf {PA}$ degree if and only if $X\\ge _T\\emptyset ^{\\prime }$ .", "It follows from the previous two results that a Martin-Löf random sequence is difference random if and only if it does not have $\\mathsf {PA}$ degree.", "Another approach to defining randomness is to take the collection of all null sets definable at the same level of complexity and require that a random sequence avoid all such null sets.", "For instance, if we take all $\\Pi ^0_1$ definable null sets or all $\\Pi ^0_2$ definable null sets, we have the following two notions of randomness, first introduced by Kurtz in .", "Definition 2.9 Let $X\\in 2^\\mathbb {N}$ .", "(i) $X$ is Kurtz random (or weakly 1-random) if and only if $X$ is not contained in any $\\Pi ^0_1$ class of Lebesgue measure 0 (equivalently, if and only if it is not contained in any $\\Sigma ^0_2$ class of measure 0).", "(ii) $X$ is weakly 2-random if and only if $X$ is not contained in any $\\Pi ^0_2$ class of Lebesgue measure 0.", "Let $\\mathsf {KR}$ denoted the collection of Kurtz random sequences and $\\mathsf {W2R}$ denote the collection of weakly 2-random sequences.", "The last definition of randomness we will consider in the study is defined in terms of certain effective betting strategies called martingales.", "Definition 2.10 (i) A martingale is a function $d:2^{<\\mathbb {N}}\\rightarrow \\mathbb {R}^{\\ge 0} \\cup \\lbrace +\\infty \\rbrace $ such that for every $\\sigma \\in 2^{<\\mathbb {N}}$ , $2d(\\sigma )=d(\\sigma 0)+d(\\sigma 1).$ (ii) A martingale $d$ succeeds on $X\\in 2^\\mathbb {N}$ if $\\limsup _{n\\rightarrow \\infty }\\, d(X{\\upharpoonright }n)=+\\infty .$ (iii) A sequence $X\\in 2^\\mathbb {N}$ is computably random if there is no computable martingale $d$ that succeeds on $X$ .", "The collection of computably random sequences will be written as $\\mathsf {CR}$ .", "The different notions of randomness discussed in this section form a strict hierarchy.", "Namely, the following relations hold: $\\mathsf {W2R}\\subsetneq \\mathsf {DiffR}\\subsetneq \\mathsf {MLR}\\subsetneq \\mathsf {CR}\\subsetneq \\mathsf {KR}$ Lastly, we note that each of these notions of randomness can be relativized to an oracle $A\\in 2^\\mathbb {N}$ .", "In the sequel, we will consider relative versions of each of the above-listed notions of randomness except for weak 2-randomness.", "Below we list the relevant modifications for each notion of relative randomness.", "(i) $A$ -Martin-Löf randomness ($\\mathsf {MLR}^A$ ) is defined in terms of an $A$ -computable sequence of $\\Sigma ^{0,A}_1$ classes $(\\mathcal {U}_i^A)_{i\\in \\mathbb {N}}$ such that $\\lambda (\\mathcal {U}_i^A)\\le 2^{-i}$ for every $i$ ; (ii) $A$ -difference randomness ($\\mathsf {DiffR}^A$ ) is defined in terms of an $A$ -computable sequence of pairs $\\Sigma ^{0,A}_1$ classes $((\\mathcal {U}_i^A,i^A))_{i\\in \\mathbb {N}}$ such that $\\lambda (\\mathcal {U}_i^A\\setminus i^A)\\le 2^{-i}$ for every $i$ ; (iii) $A$ -Kurtz randomness ($\\mathsf {KR}^A$ ) is defined in terms of $\\Pi ^{0,A}_1$ classes of Lebesgue measure 0; (iv) $A$ -computable randomness ($\\mathsf {CR}^A$ ) is defined in terms of $A$ -computable martingales.", "Negligibility and $\\mathit {tt}$ -Negligibility We are now in a position to define negligibility and $\\mathit {tt}$ -negligibility, two notions that are central to this study.", "As discussed in the introduction, the intuitive idea behind negligibility is that a set $2^\\mathbb {N}$ is negligible if no member of $ can be produced with positive probability by means of any Turing functional with a random oracle.", "Similarly, a set $ 2N$ is $ tt$-negligible if no member of $ can be produced with positive probability by means of any total Turing functional with a random oracle.", "However, we will primarily work with the following measure-theoretic definition of these two notions.", "Definition 3.1 Let $2^\\mathbb {N}$ .", "(i) $ is \\emph {negligible} if $M(=0$.\\item [(ii)] $ is tt-negligible if $\\mu (=0$ for every computable measure $\\mu $ .", "The intuitive description of negligibility and $\\mathit {tt}$ -negligibility given above is justified by the following proposition.", "For $2^\\mathbb {N}$ , we define $(^{\\le _T}$ to be the set $\\lbrace Y \\in 2^\\mathbb {N}: (\\exists X\\in [X\\le _T Y]\\rbrace $ .", "Furthermore, $(^{\\le _{tt}}$ denotes the set $\\lbrace Y \\in 2^\\mathbb {N}: (\\exists X\\in [X\\le _{tt} Y]\\rbrace .$ Proposition 3.2 Let $2^\\mathbb {N}$ .", "(i) $ is negligible if and only if $ ((T)=0$.\\item [(ii)] $ is $\\mathit {tt}$ -negligible if and only if $\\lambda ((^{\\le _{tt}})=0$ .", "($\\mathit {i}$ ) ($\\Rightarrow $ ) If $ is negligible, then $(=0$ for every lower semi-computable continuous semi-measure $$ by Remark \\ref {rmk-Mbar}.", "In particular, by Theorem \\ref {thm-MachinesInduceSemiMeasures} ($ ii$) and Proposition \\ref {prop-rhobar-properties} ($ ii$), $(=0$ for every Turing functional $$ and hence$$\\lambda ((^{\\le _T})=\\sum _{i\\in \\mathbb {N}}\\lambda (\\lbrace Y \\in 2^\\mathbb {N}: \\Phi _i(Y)\\in )=\\sum _{i\\in \\mathbb {N}}\\overline{\\lambda }_{\\Phi _i}(=0.$$($$) Since $ ((T)=0$, it follows that $(=0$, where $$ is a universal Turing functional.", "Thus $M(=(=0$.$ ($\\mathit {ii}$ ) ($\\Rightarrow $ ) If $ is $ tt$-negligible, then $ (=0$ for every $ tt$-functional $$.", "The result clearly follows.\\\\($$) Now suppose that $ (>0$ for some computable measure $$.", "Then by Remark \\ref {rmk-functionals-measures}, there is some almost total Turing functional $$ such that $ =$.", "Since the domain of a Turing functional is $ 02$, if $ Xdom()$, it follows that $ XMLR$ (in fact, $ X$ is not even Kurtz random).", "Indeed, the complement of $ dom()$ is a $ 02$ class of measure~$ 0$, so $ Xdom()$ implies that $ X$ is contained in a $ 01$ class of measure~$ 0$.$ Let $i$ be the least such that $\\mu (>2^{-i}$ .", "Then $\\lambda (\\Phi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i)>0$ , where $\\hat{\\mathcal {U}}_i$ is the $i^{\\mathrm {th}}$ level of the universal Martin-Löf test (so that $\\lambda (\\hat{\\mathcal {U}}_i^c)>1-2^{-i}$ ).", "Now since $\\Phi $ is total on $\\hat{\\mathcal {U}}_i^c$ , which is a $\\Pi ^0_1$ class, we can define a $\\mathit {tt}$ -functional $\\Psi $ as follows: $\\Psi (X)=\\left\\lbrace \\begin{array}{ll}\\Phi (X) & \\mbox{if } X\\in \\hat{\\mathcal {U}}^c_i\\\\0^\\infty & \\mbox{if } X\\in \\hat{\\mathcal {U}}_i\\end{array}.\\right.$ Since $\\Phi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i=\\Psi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i$ , it follows that $\\lambda (\\Psi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i)>0$ and hence $\\lambda ((^{\\le _{tt}})>0$ .", "By the following proposition, there is a simple characterization of negligible and $\\mathit {tt}$ -negligible singletons.", "Proposition 3.3 For $X\\in 2^\\mathbb {N}$ , the following are equivalent: (i) $\\lbrace X\\rbrace $ is negligible.", "(ii) $\\lbrace X\\rbrace $ is $\\mathit {tt}$ -negligible.", "(iii) $X$ is non-computable.", "$(\\mathit {i})\\Rightarrow (\\mathit {ii})$ is immediate.", "$(\\mathit {ii})\\Rightarrow (\\mathit {iii})$ follows from Lemma REF , which tells us that $X$ is computable if $\\mu (\\lbrace X\\rbrace )>0$ for some computable measure $\\mu $ .", "Lastly, $(\\mathit {iii})\\Rightarrow (\\mathit {i})$ is a theorem due to Sacks (see, for instance ).", "Further, it is clear that for a $\\Sigma ^0_1$ class $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , $\\mathcal {S}$ is negligible if and only if $\\mathcal {S}$ is $\\mathit {tt}$ -negligible if and only if $\\mathcal {S}$ is empty.", "However, the notions of negligibility and $\\mathit {tt}$ -negligibility are non-trivial for $\\Pi ^0_1$ classes.", "For instance, if we let $\\mathcal {PA}$ be the $\\Pi ^0_1$ class of consistent completions of $\\mathsf {PA}$ , then as shown by Jockusch and Soare in , $\\mathcal {PA}$ is negligible (and thus $tt$ -negligible).", "Although every negligible $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible, the converse does not hold.", "Theorem 3.1 There exists a tt-negligible $\\Pi ^0_1$ class that is not negligible.", "The proof of Theorem REF draws upon a theorem of Downey, Greenberg and Miller : there exists a non-negligible perfect thin $\\Pi ^0_1$ class, where a $\\Pi ^0_1$ class $ is \\emph {thin} if for every $ 01$ subclass $$ of $ , there exists a clopen set $D$ such that $ = D$ .", "We argue that any perfect thin class must be $\\mathit {tt}$ -negligible, which extends a result of Simpson who proved that every thin class must have Lebesgue measure 0.", "However, our proof strategy is very different from Simpson's.", "We first need the following lemma, which is folklore.", "Lemma 3.4 Let $ be a $ 01$ class.", "If for some computable probability measure~$$ the value of $ ($ is a positive computable real number, then $ contains a computable member.", "Consider the measure $\\nu : 2^{<\\mathbb {N}}\\rightarrow [0,1]$ defined by $\\nu (\\sigma )=\\mu (\\llbracket \\sigma \\rrbracket )$ .", "Since $ is $ 01$, the measure~$$ is upper semi-computable.", "However, for every $$, we have$$\\nu (\\sigma )=\\nu (\\Lambda )-\\sum _{{\|\\tau \|=\|\\sigma \|\\\\\\tau \\ne \\sigma }} \\nu (\\tau )$$and since $ ()$ is computable (it is equal to $ ($), this shows that $ ()$ is also lower semi-computable uniformly in~$$, therefore $$ is computable.", "It is then easy to computably build by induction a sequence of strings $ 0 1 ...$ with $ \|i\|=i$ and such that $ (i) () 4-i >0$.", "In particular $ i= $.", "Therefore, the sequence $ X$ extending all~$ i$ is computable and must be an element of $ .", "Lemma 3.5 If $ is a perfect thin $ 01$ class, then it is $ tt$-negligible.$ First, observe that if a thin class $\\mathcal {P}$ contains a computable member $X$ , then since $\\lbrace X\\rbrace $ is a $\\Pi ^0_1$ subclass of $\\mathcal {P}$ , there is some $\\sigma \\in 2^{<\\mathbb {N}}$ such that $\\lbrace X\\rbrace =\\mathcal {P}\\cap \\llbracket \\sigma \\rrbracket $ .", "Thus, $X$ is isolated in $\\mathcal {P}$ .", "It thus follows that a perfect thin $\\Pi ^0_1$ class contains no computable members.", "Now, for the sake of contradiction, suppose that there exists a computable measure $\\mu $ such that $\\mu (>q$ for some positive rational $q$ .", "Identifying $2^\\mathbb {N}$ with the unit interval $[0,1]$ in the usual way, let $\\alpha = \\sup \\big \\lbrace r \\in [0,1] \\cap \\mathbb {Q}\\, : \\, \\mu ([0,r]) < q \\big \\rbrace .$ Since $ \\mu ([0,r]) < q$ is a $\\Sigma ^0_1$ predicate in $r$ , $\\alpha $ is a lower semi-computable real.", "Moreover, since $ has no computable member, it contains no atom of $$, and thus the function $ x ([0,x])$ is continuous.", "Therefore, by definition of $$, $ ([0,))=([0,])=q$.", "We now consider two cases.", "\\\\$ Case 1: The real $\\alpha $ is dyadic.", "In this case, $[0,\\alpha ]$ is a $\\Pi ^0_1$ class of measure $q$ , which by Lemma REF implies that $[0,\\alpha ]$ has a computable member, which contradicts our above observation.", "Case 2: The real $\\alpha $ is not dyadic.", "Consider the class $[\\alpha ,1]$ , which is a $\\Pi ^0_1$ subclass of $ since $$ is lower-semicomputable.", "By thinness of $ , there exists a dyadic number $\\beta \\ge \\alpha $ (thus $\\beta > \\alpha $ since $\\alpha $ is not dyadic) such that $[\\alpha ,1]=[\\beta ,1]$ .", "This means that $(\\alpha ,\\beta ) = \\emptyset $ , and thus, $\\mu ([0,\\beta ])=\\mu ([0,\\alpha ))=q$ .", "Thus we are back in Case 1 with $\\beta $ in place of $\\alpha $ , and we get a contradiction in the same way.", "Depth and $\\mathit {tt}$ -depth Members of negligible and $\\mathit {tt}$ -negligible classes are difficult to produce, in the sense that their members cannot be computed from random oracles with positive probability.", "Given a $\\Pi ^0_1$ class $\\mathcal {P}$ , we can instead consider the probability of producing an initial segment of some member of $\\mathcal {P}$ .", "Thus, by looking at local versions of negligibility and $\\mathit {tt}$ -negligibility, i.e., given in terms of initial segments of members of the classes in question, we obtain the notions of depth and $\\mathit {tt}$ -depth, respectively.", "To compute an initial segment of a member of a $\\Pi ^0_1$ class $, we must represent~$ as the collection of paths through a tree $T\\subseteq 2^{<\\mathbb {N}}$ .", "Recall from $§$ that there are two primary ways to do so.", "First, for any $\\Pi ^0_1$ class $, there is a computable tree~$ T$ such that $ [T]=, i.e., $ consists of precisely the infinite paths through~$ T$.", "Second, given a $ 01$ class~$ 2N$, the \\emph {canonical co-c.e.\\ tree associated to $ } is the tree consisting of the strings $\\sigma $ that are prefixes of some path of $ (by the standard compactness argument, this is a co-c.e.\\ set of strings).", "It is this latter formulation that we use in the definition of depth (a choice that is justified by Proposition \\ref {prop:no-deep-comp-tree} below).$ Recall our convention from $§$REF : if $S$ is a set of strings, then $\\mathbf {M}(S)$ and $\\mathbf {m}(S)$ are defined to be $\\sum _{\\sigma \\in S} \\mathbf {M}(\\sigma )$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma )$ , respectively.", "Similarly, if $\\mu $ is a measure, $\\mu (S)$ denotes $\\sum _{\\sigma \\in S} \\mu (\\sigma )$ .", "Furthermore, recall from $§$REF that given a tree $T\\subseteq 2^{<\\mathbb {N}}$ , $T_n$ denotes the set of all members of $T$ of length $n$ .", "We now define the central notion of this paper, namely the notion of deep $\\Pi ^0_1$ class.", "Depth strengthens the notion of negligibility.", "It is easy to see that a class $2^\\mathbb {N}$ of canonical co-c.e.", "tree $T$ is negligible if and only if $\\mathbf {M}(T_n)$ converges to 0 as $n$ grows without bound.", "When this convergence to 0 is effective, $ is said to be deep.$ Definition 4.1 Let $2^\\mathbb {N}$ be a $\\Pi ^0_1$ class and $T$ its associated co-c.e.", "tree.", "(i) $ is \\emph {deep} if there is some computable order~$ h$ such that $ M(Tn) < 2-h(n)$ for all~$ n$.\\item [(ii)] $ is $\\mathit {tt}$ -deep if for every computable measure $\\mu $ there exists a computable order $h$ such that $\\mu (T_n) < 2^{-h(n)}$ .", "Our choice of the term “deep\" is due to the similarity between the notions of a deep class and that of a logically deep sequence, introduced by Bennett in .", "Logically deep sequences are highly structured and thus it is difficult to produce initial segments of a deep sequence via probabilistic computation; deep classes can thus be seen as an analogue of logically deep sequences for $\\Pi ^0_1$ classes.", "Remark 4.2 Note that depth and can be equivalently defined as follows: $ is deep if for some computable function~$ f$ one has $ M(Tf(k)) < 2-k$ for all~$ k$.", "Indeed, if $ M(Tn)<2-h(n)$, then, setting $ f=h-1$, we have $ M(Tf(k)) < 2-n$, and conversely, if $ M(Tf(n)) < 2-n$, we can assume that~$ f$ is increasing and then taking $ h=f-1$, we have $ M(Tn)<2-h(n)$ (here we use the fact that $ M(Tn)$ is non-increasing in $ n$).$ The same argument shows that $ is $ tt$-deep if and only if for every computable measure~$$, there exists a computable function~$ f$ such that for all~$ n$, $ (Tf(n)) < 2-n$.", "\\\\$ Remark 4.3 Another alternative way to define depth is to use $\\mathbf {m}$ instead of $\\mathbf {M}$ : a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree~$ T$ is deep if and only if there is a computable order~$ h$ such that $ m(Tn) < 2-h(n)$ for all~$ n$, if and only if there is a computable function~$ f$ such that $ m(Tf(n))< 2-n$ for all~$ n$.", "Indeed, the following inequality holds for all strings~$$ (see for example \\cite {Gacs-notes}):$$\\mathbf {m}(\\sigma ) \\le ^* \\mathbf {M}(\\sigma ) \\le ^* \\mathbf {m}(\\sigma )/ \\mathbf {m}(\|\\sigma \|)$$Thus, for~$ h$ a computable order, if $ M(Tn) < 2-h(n)$ for all~$ n$, we also have $ m(Tn) < 2-h(n)$ for all~$ n$.", "Conversely, if $ m(Tn) < 2-h(n)$, then $ m(Tf(n)) < 2-n$ for $ f=h-1$, and by the above inequality,$$\\mathbf {M}(T_{f(n)}) \\le ^* 2^{-n}/ \\mathbf {m}(f(n)) \\le ^* 2^{-n} \\cdot n^2.$$Thus, taking $ g(n)=f(2n+c)$ for some large enough constant~$ c$, we get $ M(Tg(n)) < 2-n$, and thus $ is deep.", "Since every $\\Pi ^0_1$ class $ is the set of paths through a computable tree, why can^{\\prime }t we simply define depth and $ tt$-depth in terms of this tree and not the canonical co-c.e.\\ tree associated to $ ?", "In the case of depth, there are two reasons to restrict to the canonical co-c.e.", "trees associated to $\\Pi ^0_1$ classes.", "First, the idea behind a deep class is that it is difficult to produce initial segments of some member of the class.", "In general, for a $\\Pi ^0_1$ class $, any computable tree $ T$ contains non-extendible nodes, and so if we have a procedure that can compute these non-extendible nodes of $ T$ with high probability, this tells us nothing about the difficulty of computing the extendible nodes of~$ T$.$ Second, if we were to use any tree $T$ representing a $\\Pi ^0_1$ class $ in the definition of depth, then depth would become a void notion, by the following proposition.$ Proposition 4.4 If $ is a non-empty $ 01$ class and~$ T$ is a computable tree such that $ [T]$, then there is no computable order~$ f$ such that $ m(Tf(n)) < 2-n$.$ If $T$ is a computable infinite tree and $f$ a computable order, there is a computable sequence $(\\sigma _n)_{n \\in \\mathbb {N}}$ of strings such that $\\sigma _n \\in T_{f(n)}$ for every $n$ .", "Thus $\\mathbf {m}(T_{f(n)}) \\ge \\mathbf {m}(\\sigma _n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2.$ By contrast with the notion of depth, for the definition of $\\mathit {tt}$ -depth, it does not matter whether we work with the canonical co-c.e.", "tree associated to $ or some computable tree $ T$ such that $ [T]$ (a direct consequence of Theorem~\\ref {prop:carac-tt-negl}).$ An important question concerns the relationship between depth and negligibility (and $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility).", "Clearly every deep ($\\mathit {tt}$ -deep) $\\Pi ^0_1$ class is negligible ($\\mathit {tt}$ -negligible), but does the converse hold?", "In the case of $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility, the answer is positive.", "We also identify two other equivalent formulations of $\\mathit {tt}$ -depth.", "Theorem 4.1 Let $ be a $ 01$ class.", "The following are equivalent:\\\\$ (i)$ $ is $\\mathit {tt}$ -deep.", "$(ii)$ $ is $ tt$-negligible.", "\\\\$ (iii)$ For every computable measure~$$, $ contains no $\\mu $ -Kurtz random element.", "$(iv)$ For every computable measure $\\mu $ , $ contains no $$-Martin-Löf random element.$ $(i) \\Rightarrow (ii)$ .", "Let $T$ be the canonical co-c.e.", "tree associated to $.", "If $ (Tn) < 2-h(n)$ for some computable order $ h$, then $ (=(n Tn) = 0$.", "\\\\$ (ii) (iii)$.", "This follows directly from the definition of Kurtz randomness.\\\\$ (iii) (iv)$.", "This follows from the fact that Martin-Löf randomness implies Kurtz randomness.\\\\$ (iv) (ii)$.", "If $ (>0$ for some computable measure $$, then $ must contain some $\\mu $ -Martin-Löf random element since the set of $\\mu $ -Martin-Löf random sequences has $\\mu $ -measure 1.", "$(ii) \\Rightarrow (i)$ .", "Suppose that $ is $ tt$-negligible and let $$ be a computable measure.", "Let~$ T$ be the co-c.e.\\ tree associated to~$ .", "By $\\mathit {tt}$ -negligibility, $\\mu (=0$ , or equivalently, $\\mu (T_n)$ tends to 0.", "Since the $T_n$ are co-c.e.", "sets of strings, $\\mu (T_n)$ is upper semi-computable uniformly in $n$ .", "Thus, given $k$ , it is possible to effectively find an $n$ such that $\\mu (T_n) < 2^{-k}$ .", "Setting $f(k)=n$ , we have a computable function $f$ such that $\\mu (T_{f(k)})<2^{-k}$ for all $k$ , therefore $ is $ tt$-deep by Remark~\\ref {rem:deep-order}.$ Significantly, negligibility and depth do not coincide when $\\mathit {tt}$ -reducibility is replaced by Turing reducibility.", "This is due to a fundamental aspect of depth, namely that it is not — unlike negligibility — invariant under Turing equivalence.", "Suppose that $ and $ are two classes such that for every $X \\in there exists a $ Y such that $X \\equiv _T Y$ , and vice-versa.", "Then by Proposition REF , $ is negligible if and only if~$ is negligible.", "This invariance does not hold in general for deep classes, as the next proposition shows.", "Theorem 4.2 For any $\\Pi ^0_1$ class $, there is a $ 01$ class~$ such that: the elements of $ are the same as the elements of~$ , modulo deletion of a finite prefix (which in particular guarantees that the elements of $ and $ have the same Turing degrees); and $ is not deep.$ Let $ be a $ 01$ class.", "Let $ T$ be a computable tree such that $ [T]=, and let $S$ be the canonical co-c.e.", "tree associated to $.", "Consider the tree~$ U$ obtained by appending a copy of~$ S$ to each terminal node of~$ T$.", "Formally,$$U = T \\cup \\lbrace \\sigma ^ \\frown \\tau \\mid \\sigma \\in T_{term} ~\\text{and}~ \\tau \\in S\\rbrace ,$$where~$ Tterm$ is the set of terminal nodes of~$ T$, which is a computable set.", "One can readily verify that $ U$ is the canonical co-c.e.\\ tree associated to some $ 01$ class.", "Indeed, suppose $$ is a node of~$ U$.", "Then,\\begin{itemize}\\item either \\rho \\in T, in which case \\rho is either a prefix of some~X\\in [T] (and thus~X\\in [U]), or \\rho is a prefix of a terminal node \\rho ^{\\prime }, which can then be extended to some~(\\rho ^{\\prime })^\\frown X\\in [U], where X\\in [S]; or\\item \\rho is of the form \\sigma ^\\frown \\tau , with \\sigma \\in T_{term} and \\tau \\in S. Since S is a canonical co-c.e.\\ tree, all its nodes extend to an infinite path, and thus \\sigma ^\\frown \\tau has an extension \\sigma ^\\frown X\\in [U] for some path~X\\in [S].\\end{itemize}$ The fact that $U$ is co-c.e.", "follows directly from the fact that $T_{term}$ is computable and $S$ is co-c.e.", "Let $ be the $ 01$ class whose canonical co-c.e.\\ tree is~$ U$.", "Then the elements of~$ are either elements of $[T]=, or are of the form $ X$ for some finite string~$$ and $ X , which gives us the first part of the conclusion.", "Finally, the canonical co-c.e.", "tree of $ contains an infinite computable tree, namely, the tree~$ T$, and therefore, by the argument as in the proof of Proposition~\\ref {prop:no-deep-comp-tree}, $ is not deep.", "It is now straightforward to get a $\\Pi ^0_1$ class that is negligible but not deep: it suffices to take a negligible class $ and apply the above theorem.", "The resulting class~$ is also negligible as its elements have the same Turing degrees as the elements of $, but it is not deep.", "\\\\$ The following summarizes the implications between the different concepts introduced above for $\\Pi ^0_1$ classes.", "depth $\\Rightarrow $ negligibility $\\Rightarrow $ $\\mathit {tt}$ -negligibility $\\Leftrightarrow $ $\\mathit {tt}$ -depth $\\Rightarrow $ having no computable member Computational limits of randomness As we have seen, negligible $\\Pi ^0_1$ classes (and thus deep classes) have the property that one cannot compute a member of them with positive probability.", "Although some random sequences can compute a member in any negligible $\\Pi ^0_1$ class (namely the Martin-Löf random sequences of $\\mathsf {PA}$ degree, as sequences of $\\mathsf {PA}$ degree compute a member of every $\\Pi ^0_1$ class), by the definition of negligibility, almost every random sequence fails to compute a member of a negligible class.", "Similarly, one cannot $\\mathit {tt}$ -compute a member of a $\\mathit {tt}$ -negligible $\\Pi ^0_1$ class with positive probability.", "The definition of $\\mathit {tt}$ -negligibility implies that the random sequences that can compute a member of a $\\mathit {tt}$ -negligible class form a set of Lebesgue measure zero.", "In this section, we specify a precise level of randomness at which computing a member of a $\\mathit {tt}$ -negligible, negligible, or deep class fails.", "First, we consider the case for $\\mathit {tt}$ -negligible classes.", "Theorem 5.1 If $X\\in 2^\\mathbb {N}$ is Kurtz random, it cannot $\\mathit {tt}$ -compute any member of a $\\mathit {tt}$ -negligible class.", "Let $ be $ tt$-negligible $ 01$ class and $$ be a $ tt$-functional.", "The set $ -1($ is a $ 01$ class that, by $ tt$-negligibility, has Lebesgue measure~$ 0$.", "Thus, it contains no Kurtz random.$ A similar proof can be used to prove the following: Theorem 5.2 If $X\\in 2^\\mathbb {N}$ is weakly 2-random, it cannot compute any member of a negligible class.", "Let $ be negligible $ 01$ class, $$ be a Turing functional, and $ T$ a computable tree such that $ [T]$.", "The set $ -1($ is a $ 02$ class, since$$X\\in \\Phi ^{-1}(\\;\\text{if and only if}\\;(\\forall k)(\\exists \\sigma )(\\exists n)[\\sigma \\in T\\;\\&\\;\|\\sigma \|=k\\;\\&\\; \\sigma \\preceq \\Phi ^{X {\\upharpoonright }n}]$$By negligibility, $ -1($ has Lebesgue measure~$ 0$ and thus contains no weakly 2-random sequence.$ Our next result, despite its simplicity, is probably the most interesting of this section.", "It will help us unify a number of theorems that have appeared in the literature.", "These are theorems of form ($$ ) If $X$ is difference random, then it cannot compute an element of $,$ where $ is a given $ 01$ class.", "Theorem~\\ref {thm:diff-pa} is an example of such a theorem, with $ the class of consistent completions of $\\mathsf {PA}$ .", "The same result has been obtained with $ the class of shift-complex sequences (Khan \\cite {Khan2013}), the set of compression functions (Greenberg, Miller, Nies \\cite {GreenbergMN-ip}), and the set $ DNCq$ functions for some orders~$ q$ (Miller, unpublished).", "We will give the precise definition of these classes in $ §$\\ref {sec:examples} but the important fact is that all of these classes are deep, and indeed, showing the depth of a $ 01$ class is sufficient to obtain a theorem of the form ($ $).", "$ Theorem 5.3 If a sequence $X$ is difference random, it cannot compute any member of a deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with associated co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ M(Tf(n)) < 2-n$.", "Let~$ X$ be a sequence that computes a member of $ via a Turing functional $\\Phi $ .", "Let $\\mathcal {Z}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace $ The set $\\mathcal {Z}_n$ can be written as the difference $\\mathcal {U}_n \\setminus n$ of two effectively open sets (uniformly in $n$ ) with $\\mathcal {U}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\rbrace $ and $n= \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\notin T_{f(n)}\\rbrace $ Moreover, by definition of the semi-measure induced by $\\Phi $ , $\\lambda (\\mathcal {Z}_n) \\le \\lambda _\\Phi (T_{f(n)}) \\le ^* \\mathbf {M}(T_{f(n)}) < 2^{-n}.$ The sequence $(\\mathcal {Z}_n)_{n\\in \\mathbb {N}}$ thus yields a difference test.", "Therefore, the sequence $X$ , which by assumption belongs to all $\\mathcal {Z}_n$ , is not difference random.", "We remark that the converse does not hold: i.e., there is a class $ such that (i) $ is not deep but (ii) no difference random real can compute an element of $.", "Indeed, take a deep class~$ and apply Theorem REF to get a class $ that is not deep but whose members have the same Turing degrees as the elements of~$ .", "Thus, no difference random real can compute an element of $.$ Depth, negligibility, and mass problems In this section we discuss depth and negligibility in the context of mass problems, i.e., in the context of the Muchnik and Medvedev reducibility.", "Both Muchnik and Medvedev reducibility are generalizations of Turing reducibility.", "Whereas Turing reducibility is defined in terms of a pair of sequences, both Muchnik and Medevedev reducibility are defined in term of a pair of collections of sequences.", "In what follows, we will consider these two reducibilities when restricted to $\\Pi ^0_1$ subclasses of $2^\\mathbb {N}$ .", "We will follow the notation of the survey , to which we refer the reader for a thorough exposition of mass problems in the context of $\\Pi ^0_1$ classes.", "Let $2^\\mathbb {N}$ be $\\Pi ^0_1$ classes.", "We say that $ is \\emph {Muchnik reducible} to $ , denoted $_w, if for every $ X there exists a Turing functional $\\Phi $ such that (i) $X\\in \\mathsf {dom}(\\Phi )$ and $\\Phi (X)\\in .", "Moreover, $ is Medvedev reducible to $, denoted $ s, if $ is Muchnik reducible to $ via a single Turing functional, i.e., there exists a Turing functional $\\Phi $ such that (i) $\\mathsf {dom}(\\Phi )$ and $\\Phi (\\subseteq .$ Just as Turing reducibility gives rise to a degree structure, we can define degree structures from $\\le _w$ and $\\le _s$ .", "We say that $ and $ are Muchnik equivalent (resp.", "Medvedev equivalent), denoted $_w (resp.\\ $ s) if and only if $_w and $ w (resp.", "$_s and $ s).", "The collections of Muchnik and Medvedev degrees given by the equivalence classes under $\\equiv _w$ and $\\equiv _s$ are denoted $\\mathcal {E}_w$ and $\\mathcal {E}_s$ , respectively.", "Both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are lattices, unlike the Turing degrees, which only form an upper semi-lattice.", "We define the meet and join operations as follows.", "Given $\\Pi ^0_1$ classes $2^\\mathbb {N}$ , $\\sup ($ is the $\\Pi ^0_1$ class $\\lbrace X\\oplus Y:X\\in \\&\\;Y\\in $ .", "Furthermore, we define $\\inf ($ to be the $\\Pi ^0_1$ class $\\lbrace 0^\\frown X: X\\in \\cup \\lbrace 1^\\frown Y:Y\\in $ .", "One can readily check that the least upper bound of $ and $ in $\\mathcal {E}_w$ is the Muchnik degree of $\\sup ($ while their greatest lower bound is the Muchnik degree $\\inf ($ , and similarly for $\\mathcal {E}_s$ .", "Recall that a filter $\\mathcal {F}$ in a lattice $(\\mathcal {L},\\le ,\\inf ,\\sup )$ is a subset that satisfies the following two conditions: (i) for all $x,y\\in \\mathcal {L}$ , if $x\\in \\mathcal {F}$ and $x\\le y$ , then $y\\in \\mathcal {F}$ , and (ii) for all $x,y\\in \\mathcal {F}$ , $\\inf (x,y)\\in \\mathcal {F}$ .", "The goal of this section is to study the role of depth and negligibility in the structures $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let us start with an easy result.", "Theorem 6.1 The collection of negligible $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let $ and $ be negligible $\\Pi ^0_1$ classes.", "By Proposition REF , we have that $\\lambda ((^{\\le _T})=0$ and $\\lambda ((^{\\le _T})=0$ .", "But since $\\inf (^{\\le _T}=(^{\\le _T}\\cup (^{\\le _T},$ it follows that $\\lambda (\\inf (^{\\le _T})=0$ , which shows that $\\inf ($ is negligible.", "Thus, the degrees of negligible classes in both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are closed under $\\inf $ .", "Let $\\mathcal {D}$ be non-negligible $\\Pi ^0_1$ and $\\mathcal {C}\\le _w\\mathcal {D}$ .", "For each $i$ , we define $i:=\\lbrace X\\in X\\in \\mathsf {dom}(\\Phi _i)\\;\\&\\;\\Phi _i(X)\\in .$ Since $_w, it follows that $ ii$.", "Furthermore, we have $ (T=(ii)T=i(i)T$.", "Since $ is non-negligible, we have $0<\\lambda ((^{\\le _T})\\le \\sum _i\\lambda \\big (({i})^{\\le _T}\\big ),$ and thus $\\lambda (({k})^{\\le _T})>0$ for some $k$ .", "But since $\\Phi _{k}({k})\\subseteq , it follows that $ ((T)>0$, thus $ is non-negligible.", "Thus, negligibility is closed upwards under $\\le _w$ (and a fortiori, under $\\le _s$ as well).", "Remark 6.1 Simpson proved in that in $\\mathcal {E}_s$ , the complement of the filter of negligible classes is in fact a principal ideal, namely the ideal generated by the class $\\inf (\\mathcal {PA},2\\mathcal {RAN})$ , where $2\\mathcal {RAN}$ is the class of 2-random sequences, i.e.", "the sequences that are Martin-Löf random relative to $\\emptyset ^{\\prime }$ .", "For the next two theorems, we need the following fact.", "Fact 6.2 Let $ and $ be $\\Pi ^0_1$ classes such that $_s.", "Then there is a total Turing functional $$ such that $ (.", "This holds because for every $\\Pi ^0_1$ class $ and every Turing functional~$$ that is total on $ , there is a functional $\\Psi $ that is total on $2^\\mathbb {N}$ and coincides with $\\Phi $ on $.$ Theorem 6.2 The collection of deep $\\Pi ^0_1$ classes forms a filter in $\\mathcal {E}_s$ .", "Let $ and $ be deep $\\Pi ^0_1$ classes with associated co-c.e.", "trees $S$ and $T$ , respectively.", "Moreover, let $g$ and $h$ be computable orders such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ and $\\mathbf {M}(T_n) < 2^{-h(n)}$ .", "We define $f(n)=\\min \\lbrace g(n),h(n)\\rbrace $ , which is clearly a computable order.", "It follows immediately that $\\mathbf {M}(S_n)\\le 2^{-f(n)}$ and $\\mathbf {M}(T_n) < 2^{-f(n)}$ .", "Now the co-c.e.", "tree associated with $\\inf ($ is $R=\\lbrace 0^\\frown \\sigma :\\sigma \\in S\\rbrace \\cup \\lbrace 1^\\frown \\tau :\\tau \\in T\\rbrace .$ Consider the class $\\inf ($ .", "Setting $0^\\frown S_n=\\lbrace 0^\\frown \\sigma :\\sigma \\in S_n\\rbrace $ and $1^\\frown T_n=\\lbrace 1^\\frown \\sigma :\\sigma \\in T_n\\rbrace $ for each $n$ , the co-c.e.", "tree of $\\inf ($ is $0^\\frown S_n \\cup 1^\\frown T_n$ .", "Moreover, $\\mathbf {M}(0^\\frown S_n \\cup 1^\\frown T_n) = \\mathbf {M}(0^\\frown S_n) + \\mathbf {M}(1^\\frown T_n) \\le ^* 2^{-f(n)} + 2^{-f(n)}$ which shows that $\\inf ($ is deep.", "Next, suppose that $\\mathcal {C}$ is a deep $\\Pi ^0_1$ class and $\\mathcal {D}$ is a $\\Pi ^0_1$ class satisfying $\\mathcal {C} \\le _s \\mathcal {D}$ via the Turing functional $\\Psi $ , which we can assume to be total by Fact REF .", "Let $S$ and $T$ be the co-c.e.", "trees associated to $ and $ , respectively, and let $g$ be a computable order such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ .", "Since $\\Psi $ is total, its use is bounded by some computable function $f$ .", "It follows that for every $\\sigma \\in T_{f(n)}$ , there is some $\\tau \\in S_n$ such that $(\\sigma ,\\tau )\\in \\Psi $ .", "Thus $\\llbracket T_{f(n)}\\rrbracket \\subseteq \\Psi ^{-1}(\\llbracket S_n\\rrbracket )$ .", "Now let $\\Phi $ be a universal Turing functional such that $\\mathbf {M}=\\lambda _\\Phi $ .", "Then: $\\mathbf {M}(T_{f(n)})=\\lambda (\\Phi ^{-1}( T_{f(n)}))\\le \\lambda (\\Phi ^{-1}(\\llbracket T_{f(n)}\\rrbracket )\\le \\lambda (\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket )))\\le \\lambda _{\\Psi \\circ \\Phi }(S_n),$ where the last inequality holds because $\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket ))=(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq \\mathsf {dom}(\\Psi \\circ \\Phi )$ and hence $(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq (\\Psi \\circ \\Phi )^{-1}(S_n)$ , $\\lambda _{\\Psi \\circ \\Phi }\\le ^* \\mathbf {M}$ .", "Thus we have $\\mathbf {M}(T_{f(n)})\\le \\lambda _{\\Psi \\circ \\Phi }(S_n)\\le ^* \\mathbf {M}(S_n)\\le 2^{-g(n)}.$ Thus, $\\mathbf {M}(T_{f \\circ g^{-1}(n)}) \\le 2^{-n}$ , which shows that $ is deep.$ The invariance of the notion of depth under Medvedev-equivalence is of importance for the next section.", "There, we prove that certain classes of objects, which are not necessarily infinite binary sequences, are deep.", "To do so, we fix a certain encoding of these objects by infinite binary sequences and prove the depth of the corresponding encoded class.", "By the above theorem, the particular choice of encoding is irrelevant: if the class is deep for an encoding, it will be deep for another encoding, as long as switching from the first encoding to the second one can be done computably and uniformly.", "One could ask whether we have a similar result in the lattice $\\mathcal {E}_w$ .", "However, Theorem REF shows that depth is not invariant under Muchnik equivalence: if we apply this theorem to a deep class $, we get a $ 01$ class~$ which is clearly Muchnik equivalent to $ but is not deep itself.", "Thus depth does not marry well with Muchnik reducibility and is only a `Medvedev notion^{\\prime }.", "\\\\$ More surprisingly, $\\mathit {tt}$ -depth is a `Muchnik notion' as well as a `Medvedev notion' and the $\\mathit {tt}$ -deep classes form a filter in both lattices.", "Theorem 6.3 The collection of $\\mathit {tt}$ -deep $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Suppose that $_w and $ is not $\\mathit {tt}$ -deep.", "By definition, this means that $ has positive $$-measure for some computable probability measure~$$.", "Let $ {Uk}k N$ be the universal $$-Martin-Löf test and define $ Rk$ to be the complement of $ Uk$ (which is a $ 01$ class).", "Since $ (Rk) > 1- 2-k$ for every $ k$, there must be a~$ j$ such that $ (Rj) > 0$ and in particular $ Rj = $.", "By the hyperimmune-free basis theorem, there is some $ XRj$ of hyperimmune-free Turing degree.", "Since $ w , $X$ must compute some element $Y$ of $.", "But since~$ X$ is of hyperimmune-free degree, $ X$ in fact $ tt$-computes $ Y$, i.e., $ (X)=Y$ for some total functional~$$.$ Since $X$ is $\\mu $ -Martin-Löf random, by the preservation of randomness theorem (see, for instance, Theorem 3.2 in ), $Y$ is Martin-Löf random with respect to the computable measure $\\mu _\\Psi $ defined by $\\mu _\\Psi (\\sigma ):=\\mu (\\Psi ^{-1}(\\sigma ))$ for every $\\sigma $ .", "Thus, by Proposition REF , $ is not $ tt$-deep.", "This shows, a fortiori, that if $ s and $ is not $ tt$-deep, then $ is not $\\mathit {tt}$ -deep.", "Thus $\\mathit {tt}$ -depth is closed upwards in $\\mathcal {E}_w$ and $\\mathcal {E}_s$ and in particular is compatible with the equivalence relations $\\equiv _s$ and $\\equiv _w$ .", "Next, suppose that $ and $ are $\\mathit {tt}$ -deep classes but that $\\inf ($ is not $\\mathit {tt}$ -deep (recall that the $\\inf $ operator is the same for both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ ).", "Then by Theorem REF , $\\inf ($ contains a sequence $X$ that is $\\mu $ -Martin-Löf random for some computable measure $\\mu $ .", "Then for $i=0,1$ we define computable measures $\\mu _i$ such that $\\mu _i(\\sigma )=\\mu (i^\\frown \\sigma )$ for every $\\sigma $ .", "It is routine to check that $Y$ is $\\mu _i$ -Martin-Löf random if and only if $i^\\frown Y$ is $\\mu $ -Martin-Löf random for $i=0,1$ .", "Since $X=i^\\frown Z$ for some $i=0,1$ and $Z\\in 2^\\mathbb {N}$ , it follows that $Z$ is $\\mu _i$ -Martin-Löf random.", "But then $Z$ is contained in either $ or $ , which contradicts our hypothesis that $ and $ are both $\\mathit {tt}$ -deep.", "Examples of deep $\\Pi ^0_1$ classes In this section, we provide a number of examples of deep $\\Pi ^0_1$ classes that naturally occur in computability theory and algorithmic randomness.", "We give a uniform treatment of all these classes, i.e., we give a generic method to prove the depth of $\\Pi ^0_1$ classes.", "Consistent completions of Peano Arithmetic As mentioned in $§$, Jockusch and Soare proved in that the $\\Pi ^0_1$ class $\\mathcal {PA}$ of consistent completions of $\\mathsf {PA}$ is negligible.", "However, as shown by implicitly by Levin in and Stephan in , $\\mathcal {PA}$ is also deep.", "We will reproduce this result here.", "Following both Levin and Stephan, we will use fact that the class of consistent completions of $\\mathsf {PA}$ is Medvedev equivalent to the class of total extensions of a universal, partial-computable $\\lbrace 0,1\\rbrace $ -valued function.", "Thus, by showing the latter class is deep, we thereby establish that the former class is deep (via Theorem REF ).", "Theorem 7.1 (Levin , Stephan ) Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of all $\\lbrace 0,1\\rbrace $ -valued partial computable functions.", "Let $u$ be a function that is universal for this collection, e.g., defined by $u(\\langle e,x\\rangle ) = \\phi _e(x)$ .", "Then the class $ of total extensions of~$ u$ is a deep $ 01$ class.$ We build a partial computable function $\\phi _e$ , whose index we know in advance by the recursion theorem.", "This means that we control the value of $u(\\langle e,x\\rangle )$ for all $x$ .", "First, we partition $\\mathbb {N}$ into consecutive intervals $I_1, I_2, ...$ such that we control $2^{k+1}$ values of $u$ inside $I_k$ .", "For each $k$ in parallel, we define $\\phi _e$ on $I_k$ as follows.", "Step 1: Wait for a stage $s$ such that the set $E_k[s]=\\lbrace \\sigma \\mid \\sigma {\\upharpoonright }I_k ~ \\text{extends}~ u[s] {\\upharpoonright }I_k \\rbrace $ is such that $\\mathbf {M}(E_k)[s] \\ge 2^{-k}$ .", "Step 2: Find a $y \\in I_k$ that we control and on which $u_s$ is not defined.", "Consider the two “halves\" $E^0_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=0\\rbrace $ and $E^1_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=1\\rbrace $ of $E_k[s]$ .", "Note that either $\\mathbf {M}(E^0_k[s])\\ge 2^{-k-1}$ or $\\mathbf {M}(E^1_k)[s] \\ge 2^{-k-1}$ .", "If the first holds, set $u(y)[s+1]=1$ , otherwise set $u(y)[s+1]=0$ .", "Go back to Step 1.", "The co-c.e.", "tree $T$ associated to the class $ is the set of strings~$$ such that~$$ is an extension of $ u \|\|$.", "The construction works because every time we pass by Step 2, we remove from~$ T[s]$ a set $ Eik[s]$ (for some $ i{0,1}$ and $ k,sN$) such that $ M(Eik)[s]>2-k-1$.", "Therefore, Step 2 can be executed at most $ 2k+1$ times, and by the definition of $ Ik$ we do not run out of values~$ y Ik$ on which we control~$ u$.", "Therefore, the algorithm eventually reaches Step 1 and waits there forever.", "Setting $ f(k)=(Ik)$, this implies that the $ M$-weight of the set $ {f(k) extends u}$ is bounded by $ 2-k$, or equivalently,$$\\mathbf {M}(T_{f(k)}) < 2^{-k},$$which proves that~$ is deep.", "The proof provided here gives us a general template to prove the depth of a $\\Pi ^0_1$ class.", "First of all, the definition of the $\\Pi ^0_1$ class should allow us to control parts of it in some way, either because we are defining the class ourselves or because, as in the above proof, the definition of the class involves some universal object which we can assume to partially control due to the recursion theorem.", "All the other examples of deep $\\Pi ^0_1$ class we will see below belong to this second category.", "Let us take a step back and analyze more closely the structure of the proof of Theorem REF .", "Given a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree $ T$, the proof consists of the following steps.\\\\$ (1) For a given $k$ , we identify a level $N=f(k)$ at which we wish to ensure $\\mathbf {M}(T_N) < 2^{-k}$ .", "The choice of $N$ will depend on the particular class $.", "\\\\$ (2) Next we implement a two-step strategy to ensure that $\\mathbf {M}(T_N)<2^{-k}$ .", "Such a strategy will be called a $k$ -strategy.", "(2.1) First, we wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] > 2^{-k}$ .", "(2.2) If at some stage $s$ this occurs, we remove one or several of the nodes of $T[s]$ at level $N$ such that the total $\\mathbf {M}[s]$ -weight of these nodes is at least $\\delta (k)$ for a certain function $\\delta $ , and then go back to Step 2.", "(3) Each execution of Step 2 (removing nodes from $T[s]$ for some $s$ ) comes at a cost $\\gamma (k)$ for some $k$ , and we need to make sure that we do not go over some maximal total cost $\\Gamma (k)$ throughout the execution of the $k$ -strategy.", "In the above example, $\\delta (k)=2^{-k-1}$ and the only cost for us is to define one value of $u(y)$ out of the $2^{k+1}$ we control, so we can, for example, set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ .", "By definition of $\\mathbf {M}$ , the $k$ -strategy can only go through Step 2 at most $1/\\delta (k)$ times, and thus the total cost of the $k$ -strategy will be at most $\\gamma (k)/\\delta (k)$ .", "All we have to do is to make sure that $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ (which is the case in the above example).", "Again, we have some flexibility on the choice of $N=f(k)$ , so it will suffice to choose an appropriate $N$ to ensure $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ .", "In some cases, there will not be a predefined maximum for each $k$ , but rather a global maximum $\\Gamma $ that the sum of the the costs of the $k$ -strategies should not exceed, i.e., we will want to have $\\sum _k \\gamma (k)/\\delta (k) \\le \\Gamma $ .", "Let us now proceed to more examples of $\\Pi ^0_1$ classes.", "Shift-complex sequences Definition 7.1 Let $\\alpha \\in (0,1)$ and $c>0$ .", "(i) $\\sigma \\in 2^{<\\mathbb {N}}$ (resp.", "$X\\in 2^\\mathbb {N}$ ) is said to be $(\\alpha ,c)$ -shift complex if $\\mathrm {K}(\\tau ) \\ge \\alpha \|\\tau \| - c$ for every substring $\\tau $ of $\\sigma $ (resp.", "of $X$ ).", "(ii) $X\\in 2^\\mathbb {N}$ is said to be $\\alpha $ -shift complex if it is $(\\alpha ,c)$ -shift complex for some $c$ .", "The very existence of $\\alpha $ -shift complex sequences is by no means obvious.", "Such sequences were first constructed by Durand et al.", "who showed that there exist $\\alpha $ -shift complex sequences for all $\\alpha \\in (0,1)$ .", "It is easy to see that for every computable pair $(\\alpha ,c)$ , the class of $(\\alpha ,c)$ -shift complex sequences is a $\\Pi ^0_1$ class.", "Rumyantsev proved that the class of $(\\alpha ,c)$ -complex sequences is always negligible, but in fact, his proof essentially shows that it is even deep.", "The cornerstone of Rumyantsev's theorem is the following lemma.", "It relies on an ingenious combinatorial argument that we do not reproduce here.", "We refer the reader to for the full proof.", "Lemma 7.2 (Rumyantsev ) Let $\\beta \\in (0,1)$ .", "For every rational $\\eta \\in (0,1)$ and integer $d$ , there exist two integers $n$ and $N$ , with $n<N$ , such that the following holds.", "For every probability distribution $P$ on $\\lbrace 0,1\\rbrace ^N$ , there exist finite sets of strings $A_n, A_{n+1}, ..., A_N$ such that for all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements; and the $P$ -probability that a sequence $X$ has some substring in $\\cup _{i=n}^N A_i$ is at least $1-\\eta $ .", "Moreover, $n$ and $N$ can be effectively computed from $\\eta $ and $d$ and can be chosen to be arbitrarily large.", "Once $n$ and $N$ are fixed, the sets $A_i$ can be computed uniformly in $P$ .", "This is Lemma 6 of (Rumyantsev does not explicitly state that the conclusion holds for all computable probability measures, but nothing in his proof makes use of a particular measure).", "Theorem 7.2 For any computable $\\alpha \\in (0,1)$ and integer $c>0$ , the $\\Pi ^0_1$ class of $(\\alpha ,c)$ -shift complex sequences is deep.", "Let $ be the $ 01$ class of $ (,c)$-shift complex sequences and let~$ T$ be its canonical co-c.e.\\ tree.", "We shall build a discrete semi-measure~$ m$ whose coding constant~$ e$ we know in advance.", "This means that whenever we will set a string~$$ to be such that $ m() > 2-\|\| + c + e +1 $, then automatically we will have $ m() > 2-\|\| + c +1$, and thus $ K() < \|\| - c$.", "This will \\textit {de facto} remove $$ from~$ T$.$ Let us now turn to the construction.", "First, we pick some $\\beta $ such that $0 < \\beta < \\alpha $ .", "For each $k$ , we apply the above Lemma REF to $\\beta $ and $\\eta = 1/2$ to obtain a pair $(n,N)$ with the above properties as described in the statement of the lemma (we will also make use of the fact that $n$ can be chosen arbitrarily large, see below).", "Then the two step strategy is the following: Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Let $P$ be the (computable) probability distribution on $\\lbrace 0,1\\rbrace ^N$ whose support is contained in $T_N[s]$ and such that for all $\\sigma \\in T_N[s]$ , $P(\\sigma )=2^k \\cdot \\mathbf {M}(\\sigma )[s]$ .", "By Lemma REF , we can compute a collection of finite sets $A_i$ such that: (i) For all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements.", "(ii) The $P$ -probability that a sequence $X$ has some substring in $F = \\cup _{i=n}^N A_i$ is at least $1-\\eta =1/2$ .", "Thus, $\\mathbf {M}(T_N \\cap F)[s] > 2^{-k-1}$ .", "Then, for each $i$ and each $\\sigma \\in A_i$ , we ensure, increasing $m$ if necessary, that $m(\\sigma ) > 2^{-\\alpha \|\\sigma \| + c + e +1 }$ .", "As explained above, this ensures that all strings in $F$ are removed from $T_N$ at that stage, and therefore we have removed a $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ .", "Then we go back to Step 1.", "To finish the proof, we need to make sure that this algorithm does not cost us too much.", "The constraint here is that we need to make sure that $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau ) \\le 1$ , so we are trying to stay under a global cost of $\\Gamma =1$ .", "For a given $k$ , our cost at each execution of Step 2 is the increase of $m$ on the strings in $F$ .", "The total $m$ -weight we add at during one such execution of Step 2 is at most $\\gamma (k) = \\sum _{i=n}^N \|A_i\| \\cdot 2^{-\\alpha i + c + e +1 } \\le \\sum _{i=n}^N 2^{(\\beta -\\alpha ) i + c+ e + 1} \\le \\frac{2^{(\\beta -\\alpha ) n + c+ e + 1}}{1-2^{\\beta -\\alpha }}$ But since $n=n(k)$ can be chosen arbitrarily large, therefore, with an appropriate choice of $n$ , we can make $\\gamma (k) \\le 2^{-2k-2}$ .", "Therefore, since the $k$ -strategy executes Step 2 at most $1/\\delta (k)$ times, we have $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau )\\le \\sum _{k\\in \\mathbb {N}} \\gamma (k)/\\delta (k) \\le \\sum _{k\\in \\mathbb {N}} 2^{-2k-2}/2^{-k-1} \\le 1$ and therefore $m$ is indeed a discrete semi-measure.", "DNC$_q$ functions Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of partial computable functions from $\\mathbb {N}$ to $\\mathbb {N}$ .", "Let $J$ be the universal partial computable function defined as follows.", "For all $(e,x) \\in \\mathbb {N}^2$ : $J(2^e(2x+1))=\\phi _e(x)$ The following notion was studied in .", "Definition 7.3 Let $q: \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable order.", "The set $\\mathcal {DNC}_q$ is the set of total functions $f: \\mathbb {N}\\rightarrow \\mathbb {N}$ such that for all $n$ : (i) $f(n) \\ne J(n)$ (where this condition is trivially satisfied if $J(n)$ is undefined), and (ii) $f(n) < q(n)$ .", "We should note that this is a slight variation of the standard definition of diagonal non-computability, which is formulated in terms of the condition (i$^{\\prime }$ ) $f(n)\\ne \\phi _n(n)$ instead of the condition (i) given above.", "However, this condition would make the class $\\mathcal {DNC}_q$ and the results below dependent on the particular choice of enumeration of the partial computable functions, while with condition $(i)$ , the results of the section are independent of the chosen enumeration.", "Note that for every fixed computable order $q$ , the class $\\mathcal {DNC}_q$ is a $\\Pi ^0_1$ subset of $\\mathbb {N}^\\mathbb {N}$ , but due to the bound $q$ , $\\mathcal {DNC}_q$ can be viewed as a $\\Pi ^0_1$ subset of $2^\\mathbb {N}$ .", "Whether the class $\\mathcal {DNC}_q$ is deep turns out to depend on $q$ .", "Indeed, we have the following interesting dichotomy theorem.", "Theorem 7.3 Let $q$ be a computable order.", "(i) If $\\sum _n 1/q(n) < \\infty $ , then $\\mathcal {DNC}_q$ is not $\\mathit {tt}$ -negligible.", "(ii) If $\\sum _n 1/q(n) = \\infty $ , then $\\mathcal {DNC}_q$ is deep.", "Part $(i)$ is a straightforward adaptation of Kučera's proof that every Martin-Löf random element computes a DNC function.", "Pick an $X \\in 2^\\mathbb {N}$ at random and use it as a random source to randomly pick the value of $f(i)$ uniformly among $\\lbrace 0,...,q(i)-1\\rbrace $ , independently of the other values of $f$ .", "The details are as follows.", "First, we can assume that $q(n)$ is a power of 2 for all $n$ .", "Indeed, since for $q^{\\prime } \\le q$ the class $\\mathcal {DNC}_{q^{\\prime }}$ is contained in $\\mathcal {DNC}_{q}$ , so if we take $q^{\\prime }(n)$ to be the largest power of 2, we have that $q^{\\prime }$ is computable, $q^{\\prime } \\le q$ , and $\\sum _n 1/q(n) < \\infty $ because $q^{\\prime }$ is equal to $q$ up to factor 2.", "Now, set $q(n)=2^{r(n)}$ .", "Split $\\mathbb {N}$ into intervals where interval $I_n$ has length $r(n)$ .", "One can now interpret any infinite binary sequence $X$ as a funtion $f_X: \\mathbb {N}\\rightarrow \\mathbb {N}$ , where $X(n)$ is the index of $X {\\upharpoonright }I_n$ in the lexicographic ordering of strings of length $r(n)$ .", "For $X$ taken at random with respect to the uniform measure, the event $f_X(n) = \\phi _n(n)$ has probability at most $1/q(n)$ and is independent of all such events for $n^{\\prime } \\ne n$ .", "Thus the total probability over $X$ such that $X(n)\\ne \\phi _n(n)$ for all $n$ is at least $\\prod _n (1-1/q(n))$ an expression that is positive if and only if $\\sum 1/q(n)< \\infty $ , which is satisfied by hypothesis.", "Thus, the class $\\mathcal {DNC}_q$ , encoded as above, has positive uniform measure, and thus is not $\\mathit {tt}$ -negligible.", "For $(ii)$ , let $q$ be a computable order such that $\\sum _n 1/q(n) = \\infty $ .", "Let $T$ be the canonical co-c.e.", "binary tree in which the elements of $\\mathcal {DNC}_q$ are encoded.", "By the recursion theorem, we will build a partial recursive function $\\phi _e$ whose index $e$ we know in advance and therefore will be able to define $J$ on the set of values $D = \\lbrace 2^e (2x+1) : x \\in \\mathbb {N}\\rbrace $ .", "Note that the set $D$ has positive (lower) density in $\\mathbb {N}$ : for every interval $I$ of length at least $2^{e+1}$ , $\|D \\cap I\| \\ge 2^{-e-2} \|I\|$ .", "This in particular implies that $\\sum _{n \\in D} 1/q(n) = \\infty .$ To show this, we appeal to Cauchy's condensation test, according to which for any positive non-increasing sequence $(a_n)$ , $\\sum _n a_n < \\infty $ if and only if $\\sum _k 2^k a_{2^k} < \\infty $ .", "Since the sum $\\sum _n1/q(n)$ diverges, so does $\\sum _k 2^k/q(2^k)$ and thus $\\sum _{n \\in D} \\frac{1}{q(n)} \\ge \\sum _{k>e} \\frac{\|D \\cap \\lbrace 2^{k-1},...,2^k-1\\rbrace \|}{q(2^k)} \\ge \\sum _{k>e} \\frac{2^{-e-2} \\cdot 2^{k-1}}{q(2^k)} = \\infty .$ Now that we have established that $\\sum _{n \\in D} 1/q(n) = \\infty $ , we remark that this is equivalent to having $\\prod _{n \\in D} (1-1/q(n)) = 0$ .", "Thus, we can effectively partition $D$ into countably many finite sets $D_j$ such that $\\prod _{n \\in D_j} (1-1/q(n)) < 1/2$ .", "We are ready to describe the construction.", "For each $k$ , reserve some finite collection $D_{j_1}, D_{j_2}, ..., D_{j_{k+1}}$ of sets $D_j$ .", "Let $N$ be a level of the binary tree $T$ sufficiently large such that the encoding of each path $f$ up to length $N$ is enough to recover the values of $f$ on $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , which can be found effectively in $k$ .", "The $k$ -strategy then works as follows.", "Initialisation.", "Set $i=1$ .", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Since each $\\sigma \\in T_N[s]$ can be viewed as a function from some initial segment of $\\mathbb {N}$ that contains $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , take the first value $x \\in D_{j_i}$ .", "For all $\\sigma \\in T_N$ , $\\sigma (x) < q(x)$ , thus by the pigeonhole principle there must be at least one value $v < q(x)$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x)=v\\rbrace )[s] \\ge 2^{-k}/q(x).$ Set $J(x)$ to be the least such value $v$ .", "This thus gives $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v\\rbrace )[s] < 2^{-k}(1-1/q(x)).$ Now take another $x^{\\prime }$ in $D_{j_i}$ on which $J$ has not been defined yet.", "By the same reasoning, there must be a value $v^{\\prime } < q(x^{\\prime })$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v \\wedge \\sigma (x^{\\prime }) \\ne v^{\\prime } \\rbrace ) < 2^{-k}(1-1/q(x))(1-1/q(x^{\\prime }))[s].$ We then set $J(x^{\\prime })$ to be equal to $v^{\\prime }$ .", "Continuing in this fashion, we can assign all the values of $J$ on $D_{j_i}$ in such a way that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\forall x \\in D_{j_i}, \\ \\sigma (x) \\ne J(x) \\rbrace \\big )[s] < 2^{-k} \\prod _{n \\in D_j} (1-1/q(n)) < 2^{-k}/2.$ Then we increment $i$ by 1 and go back to Step 1.", "Once again, at each execution of Step 2 we remove an $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ , since ensuring $\\sigma (x)=J(x)$ for some $x$ immediately ensures that no extension of $\\sigma $ is in $\\mathcal {DNC}_q$ .", "Moreover, each execution of Step 2 requires us to define $J$ on all values in some $D_{j_i}$ for $1\\le j\\le k+1$ .", "That is, each such execution costs us one $D_{j_i}$ , so as in the case of consistent completions of $\\mathsf {PA}$ , we can set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ , which ensures that $\\gamma (k)/\\delta (k)\\le \\Gamma (k)$ .", "Therefore, $\\mathcal {DNC}_q$ is a deep $\\Pi ^0_1$ class.", "Finite sets of maximally complex strings For a constant $c>0$ , generating a long string $\\sigma $ of high Kolmogorov complexity, for example $\\mathrm {K}(\\sigma ) > \|\\sigma \|-c$ , can be easily achieved with high probability if one has access to a random source (just repeatedly flip a fair coin and output the raw result).", "One can even use this technique to generate a sequence of strings $\\sigma _1, \\sigma _2, \\sigma _3, \\ldots , $ such that $\|\\sigma _n\| = n$ and $\\mathrm {K}(\\sigma _n) \\ge n-c$ .", "Indeed, the measure of the sequences $X$ such that $\\mathrm {K}(X {\\upharpoonright }n) \\ge n -d$ for all $n$ is at least $2^{-d}$ , and thus the $\\Pi ^0_1$ class of such sequences of strings $(\\sigma _1,\\sigma _2,...,)$ of high complexity (encoded as elements of $2^\\mathbb {N}$ ) is not even $\\mathit {tt}$ -negligible.", "The situation changes dramatically if one wishes to obtain many distinct complex strings at a given length.", "Let $\\ell : \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable increasing function, $f,d:\\mathbb {N}\\rightarrow \\mathbb {N}$ be any computable functions, and $c>0$ .", "Consider the $\\Pi ^0_1$ class $\\mathcal {K}_{f,\\ell ,d}$ whose members are sequences $\\vec{F}=(F_1, F_2, F_3, ...)$ where for all $i$ , $F_i$ is a finite set of $f(i)$ strings $\\sigma $ of length $\\ell (i)$ such that $\\mathrm {K}(\\sigma ) \\ge \\ell (i)-d(i)$ .", "Note once again that $\\mathcal {K}_{f,\\ell ,d}$ can be viewed, modulo encoding, as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ .", "Theorem 7.4 For all computable functions $f,\\ell ,d$ such that $f(i)/2^{d(i)}$ takes arbitrarily large values and $\\ell $ is increasing, the class $\\mathcal {K}_{f,\\ell ,d}$ is deep.", "Let $T$ be the canonical co-c.e.", "tree associated to the class $\\mathcal {K}_{f,\\ell ,d}$ .", "Just like in the proof of Theorem REF , we will build a discrete semi-measure $m$ whose coding constant $e$ we know in advance and thus, by setting $m(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) + e +1 }$ , we will force $\\mathbf {m}(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) +1}$ , and thus $\\mathrm {K}(\\sigma ) < \|\\sigma \| - d(\|\\sigma \|)$ , which will consequently remove from $T$ every sequence of sets $(F_i)_{i\\in \\mathbb {N}}$ such that $\\sigma $ belongs to some $F_i$ .", "Fix a $k$ , and let us pick some well-chosen $i=i(k)$ , to be determined later.", "For readability, let $f=f(i(k))$ , $\\ell =\\ell (i(k))$ and $d=d(i(k))$ .", "Let $N$ be the level of $T$ at which the first $i$ sets $F_1\\cdots F_i$ of the sequence $\\vec{F}$ of $\\mathcal {K}_{f,\\ell ,d}$ are encoded.", "The $k$ -strategy does the following.", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: The members of $T_N[s]$ consist of finite sequences $F_1, ..., F_i$ of sets of strings where $F_i$ contains $f$ distinct strings of length $\\ell $ .", "By the pigeonhole principle, there must be a string $\\sigma $ of length $\\ell $ such that $\\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N : \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ Indeed, there is a total $\\mathbf {M}$ -weight $2^{-k}$ of possible sets $F_i$ , each of which contains $f$ strings of length $\\ell $ , thus $\\sum _{\|\\sigma \|=\\ell } \\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N \\mid \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f$ , and there are $2^\\ell $ strings of length $\\ell $ in total.", "Effectively find such a string $\\sigma $ , and set $m(\\sigma ) > 2^{-\\ell + d + e +1 }$ .", "By the choice of $\\sigma $ , this causes an $\\mathbf {M}$ -weight of at least $2^{-k} \\cdot f \\cdot 2^{-\\ell }$ of nodes of $T_N[s]$ to leave the tree.", "Then go back to Step 1.", "As in the previous examples, it remains to conduct the “cost analysis\".", "The resource here again is the weight we are allowed to assign to $m$ , which has to be bounded by $\\Gamma =1$ .", "At each execution of Step 2 of the $k$ -strategy, our cost is the increase of $m$ , which is at most of $\\gamma (k)=2^{-\\ell + d + e +1 }$ , while an $\\mathbf {M}$ -weight of at least $\\delta (k) = 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ leaves the tree $T_N$ .", "This gives a ratio $\\gamma (k)/\\delta (k) = 2^{d+e+k+1}/f$ , which bounds the total cost of the $k$ -strategy, so we want $\\sum _k \\gamma (k)/\\delta (k) \\le 1$ , i.e., we want: $\\sum _k 2^{d(i(k)) + e +k+1}/f(i(k))\\le 1$ By assumption, $f(i)/2^{d(i)}$ takes arbitrarily large values, it thus suffices to choose $i(k)$ such that $2^{d(i(k)) + e+1}/f(i(k)) \\le 2^{-2k-1}$ .", "Compression functions Our next example of a family of deep classes is given in terms of compression functions, which were introduced by Nies, Stephan, and Terwijn in to provide a characterization of 2-randomness (that is, $\\emptyset ^{\\prime }$ -Martin-Löf randomness) in terms of incompressibility.", "Definition 7.4 A $\\mathrm {K}$ -compression function with constant $c>0$ is a function $g: 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ such that $g \\le \\mathrm {K}+c$ and $\\sum _\\sigma 2^{-g(\\sigma )} \\le 1$ .", "We denote by $\\mathcal {CF}_c$ be the class of compression functions with constant $c$ .", "The condition $g(\\sigma ) \\le \\mathrm {K}(\\sigma )+c$ implies that $g(\\sigma ) \\le 2\|\\sigma \| + c +c^{\\prime } $ for some fixed constant $c^{\\prime }$ , and therefore $\\mathcal {CF}_c$ can be seen as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ , modulo encoding the functions as binary sequences.", "Of course $\\mathcal {CF}_c$ contains the function $\\mathrm {K}$ itself so it is a non-empty class.", "We now show the following.", "Theorem 7.5 For all $c\\ge 0$ , the class $\\mathcal {CF}_c$ is deep.", "Although we could give a direct proof following the same template as the previous examples, we will instead show that for all $c$ , we have $\\mathcal {K}_{f,\\ell ,d} \\le _s \\mathcal {CF}_c$ for some computable $f, \\ell , d$ (where $\\mathcal {K}_{f,\\ell ,d}$ is the class we defined in the previous section) such that $\\ell $ is increasing and $f/2^d$ takes arbitrarily large values, which by Theorem REF implies the depth of $\\mathcal {CF}_c$ .", "Let $g \\in \\mathcal {CF}_c$ .", "For all $n$ , since $\\sum _{\|\\sigma \|=n} 2^{-g(\\sigma )} \\le 1$ , there are at most $2^{n-1}$ strings of length $n$ such that $g(\\sigma ) < n - 1$ , and thus at least $2^{n-1}$ strings $\\sigma $ of length $n$ such that $g(\\sigma ) \\ge n - 1$ .", "Since $g \\in \\mathcal {CF}_c$ , for each $\\sigma $ such that $g(\\sigma ) \\ge \|\\sigma \| - 1$ , we also have $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Thus, given $g \\in \\mathcal {CF}_c$ as an oracle we can find, for each $n \\ge 3$ , $2^{n-1}$ strings $\\sigma $ of length $n$ such that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Setting $f(i)=2^{i+2}$ , $\\ell (i)=i+3$ and $d(i)=c+1$ , we have uniformly reduced $\\mathcal {K}_{f,\\ell ,d}$ to $\\mathcal {CF}_{c}$ .", "Since $d$ is a constant function it is obvious that $f(i)/2^d$ is unbounded, thus $\\mathcal {K}_{f,\\ell ,d}$ is deep (by Theorem REF ) and by Theorem REF so is $\\mathcal {CF}_c$ .", "The above result is not tight: the proof in fact shows that if $d$ is not a constant function but is such that $2^n/d(n)$ takes arbitrarily large values, then the class of functions $g$ such that $g(\\sigma ) \\le \\mathrm {K}(\\sigma ) + d(\|\\sigma \|)$ for all $\\sigma $ is a deep class.", "A notion related to $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ In $§$, we discuss lowness for randomness notions.", "Here we look at a dual notion, highness for randomness, specifically the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , whose precise characterization is an outstanding open question in algorithmic randomness.", "We shall see that this class is tightly connected to the notion of depth.", "Definition 7.5 A sequence $A\\in 2^\\mathbb {N}$ is in the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ if $\\mathsf {MLR}\\subseteq \\mathsf {CR}^A$ .", "The class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ itself not $\\Pi ^0_1$ (as it is closed under finite change of prefixes), but all its members have a “deep property\".", "Bienvenu and Miller proved that when $A$ is in $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , then for every c.e.", "set of strings $S$ such that $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ , $A$ computes a martingale $d$ such that $d(\\Lambda )=1$ and for some fixed rational $r>0$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ .", "It is straightforward to show that real-valued martingales can be approximated by dyadic-valued martingales with arbitrary precision; in particular one can assume that $d(\\sigma )$ is dyadic for all $\\sigma $ (and thus can be coded using $f(\|\\sigma \|)$ bits for some computable function $f$ ), still keeping the property $\\sigma \\in S \\Rightarrow d(\\sigma )>1+r$ .", "For a well-chosen $S$ , such martingales form a deep class.", "Theorem 7.6 Let $S$ be the set of strings $\\sigma $ such that $\\mathrm {K}(\\sigma ) < \|\\sigma \|$ (so that we have $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ ).", "Let $g$ be a computable function and $r>0$ be a rational number.", "Define the class $\\mathcal {W}_{g,r}$ to be the set of dyadic-valued martingales $d$ such that $d(\\Lambda )=1$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ and such that for all $\\sigma $ , $d(\\sigma )$ can be coded using $g(\|\\sigma \|)$ bits.", "Then $\\mathcal {W}_{g,r}$ can be viewed as a $\\Pi ^0_1$ class of $2^\\mathbb {N}$ , and this class is deep.", "Again, we are going to show the depth of $\\mathcal {W}_{g,r}$ by a Medvedev reduction from $\\mathcal {K}_{f,\\ell ,h}$ for some increasing computable function $\\ell $ and computable functions $f$ and $h$ such that $f/2^h$ is unbounded.", "More precisely, we will take $f(n)=n$ , $\\ell (n)=n + c$ (for a well-chosen $c$ given below) and $h(n)=0$ .", "Now suppose we are given a martingale $d \\in \\mathcal {W}_{f,r}$ .", "For any given length $n$ , we have, by the martingale fairness condition, $\\sum _{\\sigma : \|\\sigma \|=n} d(\\sigma ) = 2^n.$ It follows that there are at most $2^n/(1+r)$ strings $\\sigma $ of length $n$ such that $d(\\sigma )>1+r$ , and therefore at least $2^n-2^n/(1+r)$ strings such that $d(\\sigma ) \\le 1+r$ .", "Having oracle access to $d$ , such strings can be effectively found and listed.", "Since $2^n-2^n/(1+r) > n -c$ for all $n$ and some fixed constant $c$ , for each $n$ we can use $d$ to list $n$ strings $\\sigma _1,\\cdots ,\\sigma _n$ of length $n+c$ such that $d(\\sigma _i) \\le 1+r$ for $1\\le i\\le n$ .", "But by definition of $d$ , $d(\\sigma )\\le 1+r$ implies that $\\sigma \\notin S$ , which further implies that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \|$ .", "This shows that $\\mathcal {W}_{g,r}$ is above $\\mathcal {K}_{f,\\ell ,d}$ in the Medvedev degrees.", "By Theorem REF , $\\mathcal {K}_{f,\\ell ,h}$ is deep, and hence by Theorem REF , $\\mathcal {W}_{g,r}$ is deep as well.", "The examples of deep classes provided in this section, combined with Theorem REF , give us the results mentioned page : If $X$ is a difference random sequence, it does not compute any shift-complex sequence (Khan) it does not compute any $\\mathcal {DNC}_q$ function when $q$ is a computable order such that $\\sum _n 1/q(n) = \\infty $ (Miller) it does not compute any compression function (Greenberg, Miller, Nies) it is not $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ (Greenberg, Miller, Nies) The fact that a difference random cannot compute large sets of complex strings (in the sense of Theorem REF ) appears to be new.", "Lowness and depth Various lowness notions have been well-studied in algorithmic randomness, where a lowness notion is given by a collection of sequences that are in some sense computationally weak.", "Many lowness notions take the following form: For a relativizable collection $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , we say that $A$ is low for $\\mathcal {S}$ if $\\mathcal {S}\\subseteq \\mathcal {S}^A$ .", "For instance, if we let $\\mathcal {S}=\\mathsf {MLR}$ , then the resulting lowness notion consists of the sequences that are low for Martin-Löf random, a collection we write as $\\mathrm {Low}(\\mathsf {MLR})$ .", "Lowness notions need not be given in terms of relativizable classes.", "For instance, if we let $\\mathbf {m}^A$ be a universal $A$ -lower semi-computable discrete semi-measure, we can defined $A$ to be low for $\\mathbf {m}$ if $\\mathbf {m}^A(\\sigma )\\le ^\\mathbf {m}(\\sigma )$ for every $\\sigma $ .", "In addition, some lowness notions are not given in terms of relativization, such as the notion of $\\mathrm {K}$ -triviality, where $A\\in 2^\\mathbb {N}$ is $\\mathrm {K}$ -trivial if and only if $\\mathrm {K}(A{\\upharpoonright }n)\\le \\mathrm {K}(n)+O(1)$ (where we take $n$ to be $1^n$ ).", "Surprisingly, we have the following result (see for a detailed survey of results in this direction).", "Theorem 8.1 Let $A\\in 2^\\mathbb {N}$ .", "The following are equivalent: (i) $A\\in \\mathrm {Low}(\\mathsf {MLR})$ ; (ii) $A$ is low for $\\mathbf {m}$ ; (iii) $A$ is $\\mathrm {K}$ -trivial.", "The cupping problem, a longstanding open problem in algorithmic randomness involving $\\mathrm {K}$ -triviality, is to determine whether there exists a $\\mathrm {K}$ -trivial sequence $A$ and some Martin-Löf random sequence $X\\lnot \\ge _T\\emptyset ^{\\prime }$ such that $X\\oplus A\\ge _T\\emptyset ^{\\prime }$ .", "A negative answer was recently provided by Day and Miller : Theorem 8.2 (Day-Miller ) A sequence $A$ is K-trivial if and only if for every difference random sequence $X$ , $X \\oplus A \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Using the notion of depth, we can strengthen the Day-Miller result.", "First, we need to partially relativize the notion of depth: for $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $2^\\mathbb {N}$ is deep relative to $A$ if there is some computable order $f$ such that $\\mathbf {m}^A(T_{f(n)})\\le 2^{-n}$ if and only if there is some $A$ -computable order $g$ such that $\\mathbf {M}^A(T_{g(n)})\\le 2^{-n}$ (where $\\mathbf {M}^A$ is a universal $A$ -lower semi-computable continuous semi-measure).", "Theorem 8.3 Let $X$ be an incomplete Martin-Löf random sequence and $A$ be $K$ -trivial.", "Then $X \\oplus A$ does not compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with canonical co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ m(Tf(n)) 2-n$.", "Since~$ A$ is low for~$ m$, we have also have $ mA(Tf(n)) 2-n$, and thus $ is deep relative to $A$ .", "Let $X$ be a sequence such that $X \\oplus A$ computes a member of $ via a Turing functional $$ and suppose, for the sake of contradiction, that $ X$ is difference random.", "Let$$n = \\lbrace Z \\, : \\, \\Phi ^{Z \\oplus A} {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace .$$The set $ n$ can be written as the difference $ Un n$ of two $ A$-effectively open sets (uniformly in~$ n$) with $ Un = {Z : Z A f(n) }$ and $ n= {Z : Z A f(n) Tf(n)}$$ We can see the functional $Z \\mapsto \\Phi ^{Z \\oplus A}$ as an $A$ -Turing functional $\\Psi $ , and thus by the univerality of $\\mathbf {M}^A$ for the class of $A$ -lower semi-computable continuous semi-measures, we have $\\mathbf {M}^A \\ge ^* \\lambda _\\Psi $ .", "By definition of $n$ , we therefore obtain: $\\lambda (n) \\le \\lambda _\\Psi (T_{f(n)}) \\le ^* \\mathbf {M}^A(T_{f(n)}) < 2^{-n}.$ This shows that the sequence $X$ , which by assumption belongs to all $n$ , is not $A$ -difference random.", "It is, however, $A$ -Martin-Löf random as $A$ is low for Martin-Löf randomness.", "Relativizing Theorem REF to $A$ , this shows that $X \\oplus A \\ge _T A^{\\prime }$ .", "But this contradicts the Day-Miller theorem (Theorem REF ).", "As we cannot compute any members of a deep class by joining a Martin-Löf random sequence with a low for Martin-Löf random sequence, it is not unreasonable to ask if there is a notion of randomness $\\mathcal {R}$ such that we cannot $\\mathit {tt}$ -compute any members of a $\\mathit {tt}$ -deep class by joining an $\\mathcal {R}$ -random sequence with a low for $\\mathcal {R}$ sequence.", "We obtain a partial answer to this question using Kurtz randomness.", "From the discussion of lowness at the beginning of this section, we have $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $\\mathsf {KR}\\subseteq \\mathsf {KR}^A$ .", "Moreover, we define the class $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ to be the collection of sequences $A$ such that $\\mathsf {MLR}\\subseteq \\mathsf {KR}^A$ .", "Since $\\mathsf {MLR}\\subseteq \\mathsf {KR}$ , it follows that $\\mathrm {Low}(\\mathsf {KR})\\subseteq \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Miller and Greenberg obtained the following characterization of $\\mathrm {Low}(\\mathsf {KR})$ and $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Recall that $A\\in 2^\\mathbb {N}$ is computably dominated if every $f\\le _T A$ is dominated by some computable function.", "Theorem 8.4 (Greenberg-Miller ) Let $A\\in 2^\\mathbb {N}$ .", "(i) $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ if and only if $A$ is of non-DNC degree.", "(ii) $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ and computably dominated.", "For $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $ is \\emph {$ tt$-negligible relative to $ A$} if $ A(=0$ for every $ A$-computable measure $$.", "We first prove the following.$ Proposition 8.1 If $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ , then every deep $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible relative to $A$ .", "Let $ be a deep $ 01$ class and $ T$ its associated canonical co-c.e.\\ tree.", "Let $ ALow(MLR,KR)$, which by Theorem \\ref {thm:greenberg-miller-kr} (i) is equivalent to being of non-DNC degree.", "We appeal to a useful characterization of non-DNC degrees due to Hölzl and Merkle~\\cite {HolzlM2010}: $ A$ is of non-DNC degree if and only if it is \\emph {infinitely often c.e.\\ traceable} (hereafter, i.o.\\ c.e.\\ traceable).", "This means that there exists a computable order~$ h$, such that the following holds: for every total $ A$-computable function~$ s: NN$, there exists a family $ (Sn)nN$ of uniformly c.e.\\ finite sets such that $ \|Sn\|<h(n)$ for all~$ n$ and $ s(n) Sn$ for infinitely many~$ n$.$ Let $h$ be an order witnessing the i.o.", "c.e.", "traceability of $A$ .", "Since $ is deep, let~$ f$ be a computable function such that $ M(Tf(n)) < 2-2n/h(n)$.Suppose for the sake of contradiction that $ is not $\\mathit {tt}$ -negligible relative to $A$ , which means that there exists an $A$ -computable measure $\\mu ^A$ such that $\\mu ^A(>r$ for some rational $r>0$ .", "Let $s\\le _TA$ be the function that on input $n$ gives a rational lower-approximation, with precision $1/2$ , of the values of $\\mu ^A$ on all strings of length $f(n)$ (encoded as an integer).", "By this we mean that $s(n)$ gives us for all strings $\\sigma $ of length $f(n)$ a rational value $s(n,\\sigma )$ such that $\\mu ^A(\\sigma )/2 \\le s(n,\\sigma ) \\le \\mu ^A(\\sigma )$ .", "Let $(S_n)_{n\\in \\mathbb {N}}$ witness the traceability of $s$ , i.e., the $S_n$ 's are uniformly c.e., $\|S_n\|<h(n)$ for every $n$ , and $s(n) \\in S_n$ for infinitely many $n$ .", "We now build a lower semi-computable continuous semi-measure $\\rho $ as follows.", "For all $n$ , enumerate $S_n$ .", "For each member $z \\in S_n$ , interpret $z$ as a mass distribution $\\nu $ on the collection of strings of length $f(n)$ .", "Then, for each string $\\sigma $ of length $f(n)$ , increase $\\rho (\\sigma )$ (as well as strings comparable with $\\sigma $ in any way that ensures that $\\rho $ remains a semi-measure) by $2^{-n-1}\\nu (\\sigma )/h(n)$ .", "This has a total cost of $2^{-n-1}/h(n)$ , and since there are at most $h(n)$ elements in $S_n$ , the total cost at level $f(n)$ is at most $2^{-n-1}$ .", "Therefore the total cost of the construction of $\\rho $ is bounded by 1 and thus $\\rho $ is indeed a lower semi-computable continuous semi-measure Now, for any $n$ such that $s(n) \\in S_n$ (as there are infinitely many such $n$ ), $\\rho $ distributes an amount of at least $(\\mu ^A(T_{f(n)})/2) \\cdot (2^{-n-1}/h(n))$ on $T_{f(n)}$ , and since $\\mu ^A(T_n)>r$ , this gives $\\rho (T_{f(n)}) \\ge 2^{-n-O(1)}/h(n).$ However, we had assumed that $\\mathbf {M}(T_{f(n)}) \\le 2^{-2n}/h(n),$ for all $n$ .", "But since $\\rho \\le ^* \\mathbf {M}$ , we get a contradiction.", "The following result involves the notion of relative $\\mathit {tt}$ -reducibility.", "For a fixed $A\\in 2^\\mathbb {N}$ , a $\\mathit {tt}(A)$ -functional is a total $A$ -computable Turing functional.", "Equivalently, we can define a $\\mathit {tt}(A)$ -functional $\\Psi ^A$ in terms of a Turing functional $\\Phi $ as follows: Let $\\Phi $ be defined on all inputs of the form $X\\oplus A$ .", "Then we set $\\Psi ^A(X)=\\Phi (X\\oplus A)$ .", "Furthermore, one can show that there is an $A$ -computable bound on the use of $X$ in the computation (just as there is a computable bound in the use function for unrelativized $\\mathit {tt}$ -computations).", "Theorem 8.5 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class and $ ALow(KR)$.", "By Proposition \\ref {prop:low-for-tt-depth}, $ is also $\\mathit {tt}$ -deep relative to $A$ .", "Let $\\Phi ^A$ be a $\\mathit {tt}(A)$ -functional.", "The pre-image $ of $ under $\\Phi ^A$ is a $\\Pi ^0_1(A)$ class, which must be $\\mathit {tt}$ -deep relative to $A$ as well, by Theorem REF relativized to $A$ .", "Now, applying Proposition REF relativized to $A$ , $ contains no $ A$-Kurtz random sequence.", "But since $ A$ is low for Kurtz randomness, $ contains no Kurtz-random sequence as well.", "We now obtain a partial analogue of Theorem REF .", "Corollary 8.2 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $\\Phi $ be a $\\mathit {tt}$ -functional.", "Since $\\Phi $ is total, it is certainly total on all sequences of the form $X\\oplus A$ for $X\\in 2^\\mathbb {N}$ .", "Thus $\\Psi ^A(X)=\\Phi (X\\oplus A)$ is a $\\mathit {tt}(A)$ -functional.", "By Theorem REF , it follows that $\\Phi (X\\oplus A)$ cannot be contained in any deep class.", "Question 1 Does Corollary REF still hold if we replace “deep\" with “$\\mathit {tt}$ -deep\"?", "We can extend Theorem REF to the following result, which proceeds by almost the same proof, the details of which are left to the reader.", "Theorem 8.6 Let $X$ be Martin-Löf random and $A$ be $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "(in particular, $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class).", "Depth, mutual information, and the Independence Postulate In this final section, we introduce the notion of mutual information and apply it to the notion of depth.", "Roughly, what we prove is that every member of every deep class has infinite mutual information with Chaitin's $\\Omega $ , a Martin-Löf random sequence that encodes the halting problem.", "This generalizes a result of Levin's, that every consistent completion of $\\mathsf {PA}$ has infinite mutual information with $\\Omega $ .", "We conclude with a discussion of the Independence Postulate, a principle introduced by Levin to derive the statement that that no consistent completion of arithmetic is physically obtainable.", "The definition of mutual information First we review the definitions of Kolmogorov complexity of a pair and the universal conditional discrete semi-measure $\\mathbf {m}(\\cdot \\mid \\cdot )$ .", "Let $\\langle \\cdot ,\\cdot \\rangle :2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ be a computable bijection.", "Then we define $\\mathrm {K}(\\sigma ,\\tau ):=\\mathrm {K}(\\langle \\sigma ,\\tau \\rangle )$ .", "Similarly, we set $\\mathbf {m}(\\sigma ,\\tau ):=\\mathbf {m}(\\langle \\sigma ,\\tau \\rangle )$ .", "A conditional lower semi-computable discrete semi-measure $m(\\cdot \\mid \\cdot ):2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow [0,1]$ is a function satisfying $\\sum _\\sigma m(\\sigma \\mid \\tau )\\le 1$ for every $\\tau $ .", "Then $\\mathbf {m}(\\cdot \\mid \\cdot )$ is defined to be a universal conditional lower semi-computable discrete semi-measure, so that for every conditional lower semi-computable discrete semi-measure, there is some $c$ such that $m(\\sigma \\mid \\tau )\\le c\\cdot \\mathbf {m}(\\sigma \\mid \\tau )$ for every $\\sigma $ and $\\tau $ .", "Lastly, we define the conditional prefix-free Kolmogorov complexity $\\mathrm {K}(\\sigma \\mid \\tau )$ to be $\\mathrm {K}(\\sigma \\mid \\tau )=\\min \\lbrace \|\\xi \|:U(\\langle \\xi ,\\tau \\rangle )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free machine.", "The mutual information of two strings $\\sigma $ and $\\tau $ , denoted by $I(\\sigma :\\tau )$ , is defined by $I(\\sigma :\\tau ) = \\mathrm {K}(\\sigma ) + \\mathrm {K}(\\tau ) - \\mathrm {K}(\\sigma ,\\tau )$ or equivalently by $2^{I(\\sigma :\\tau )} = \\frac{\\mathbf {m}(\\sigma ,\\tau )}{\\mathbf {m}(\\sigma ) \\cdot \\mathbf {m}(\\tau )}.$ By the symmetry of information (see Gács ), we also have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} =^* \\frac{\\mathbf {m}(\\tau \\mid \\sigma ,\\mathrm {K}(\\sigma ))}{\\mathbf {m}(\\tau )}.$ Levin extends mutual information to infinite sequences by setting $2^{I(X:Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^X(\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}.$ Recall that Chaitin's $\\Omega $ can be obtained as the probability that a universal prefix-free machine will halt on a given input, that is, $\\Omega =\\sum _{U(\\sigma ){\\downarrow }}2^{-\|\\sigma \|}$ , where $U$ is a fixed universal prefix-free machine.", "Generalizing a result of Levin's from , we have: Theorem 9.1 Let $ be a $ 01$ class and $ T$ its associated co-c.e.\\ tree.", "Suppose~$ is deep, witnessed by a computable order $f$ such that $\\mathbf {m}(T_{f(n)}) < 2^{-n}$ .", "Then for every $Y \\in and all~$ n$,$$I\\Big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\Big ) \\ge n-O(\\log n).$$In particular,$$I(\\Omega :Y)=\\infty .$$$ Our proof follows the same idea Levin uses for consistent completions of $\\mathsf {PA}$ (see ), although some extra care is needed for arbitrary deep classes.", "Suppose for a given $n$ we have an exact description $\\tau $ of $T_{f(n)}$ ; that is, on input $\\tau $ , the universal machine outputs a code for the finite set $T_{f(n)}$ .", "By the definition of $f$ , $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\le 2^{-n}$ or equivalently $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\cdot 2^{n} \\le 1$ Therefore, the quantity $\\mathbf {m}(\\sigma ) \\cdot 2^{n} \\cdot \\mathbf {1}_{\\sigma \\in T_{f(n)}}$ is a discrete semi-measure, but it is not lower semi-computable since $T_{f(n)}$ is merely co-c.e.", "(and, in general, not c.e.", "by Proposition REF ).", "However, it is a lower semi-computable semi-measure relative to the exact description $\\tau $ of $T_{f(n)}$ .", "Thus, for every $\\sigma \\in T_{f(n)}$ , by the universality of $\\mathbf {m}(\\cdot \\mid \\tau )$ , $\\mathbf {m}(\\sigma \\mid \\tau ) \\ge ^* \\mathbf {m}(\\sigma ) \\cdot 2^n.$ By the symmetry of information, we have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} \\ge ^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau )}{\\mathbf {m}(\\sigma )} \\ge ^2^n,$ and hence $I(\\tau :\\sigma ) \\ge ^+ n$ .", "We would like to apply this fact to the case where $\\sigma = Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ .", "But this is not technically sufficient, as $\\Omega {\\upharpoonright }n$ does not necessarily contain enough information to exactly describe $T_{f(n)}$ .", "This is not an obstacle in Levin's argument for completions of $\\mathsf {PA}$ , but it is for arbitrary deep classes.", "However, $\\Omega {\\upharpoonright }n$ contains enough information to get a “good enough\" approximation of $T_{f(n)}$ .", "Let us refine the idea above: suppose now that $\\tau $ is no longer an exact description of $T_{f(n)}$ , but is a description of a set of strings $S$ of length $f(n)$ such that $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^ 2^{-n} n^2.$ Then, by following the same reasoning as above, we would have $I(\\sigma : \\tau ) \\ge ^+ n-2\\log n$ for all $\\sigma \\in S$ (and thus all $\\sigma \\in T_{f(n)}$ ).", "We shall prove that $\\Omega {\\upharpoonright }n$ contains enough information to recover such a set $S$ , thus proving the theorem.", "The real number $\\Omega $ is lower semi-computable and Solovay complete (see ).", "As a consequence, for every other lower semi-computable real $\\alpha $ , knowing the first $k$ bits of $\\Omega $ allows us to compute the first $k-O(1)$ bits of $\\alpha $ .", "For all $n$ , define: $a_n = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}}} \\mathbf {m}(\\sigma )$ and observe that $a_n$ is lower semi-computable uniformly in $n$ (because $T_{f(n)}$ is co-c.e.", "uniformly in $n$ ), and belongs to $[0,1]$ .", "Define now $\\alpha = \\sum _n \\frac{a_n}{n^2},$ which is a lower semi-computable real.", "Thus, knowing the first $n$ bits of $\\Omega $ gives us the first $n-O(1)$ bits of $\\alpha $ , i.e., an approximation of $\\alpha $ with precision $2^{-n}$ .", "In particular this gives us an approximation of $a_n$ with precision $2^{-n} \\cdot n^2 \\cdot O(1)$ , which we can assume to be a lower approximation, which we will write as $a^{\\prime }_n$ .", "Now, using $a^{\\prime }_n$ , one can enumerate $a_n$ until we find a stage $s_n$ such that $a_n[s_n] = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}[s_n]}} \\mathbf {m}(\\sigma )[s_n] \\ge a^{\\prime }_n.$ Since $\|a_n - a^{\\prime }_n\| \\le 2^{-n} \\cdot n^2 \\cdot O(1)$ , this implies $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}[s_n]} \\setminus T_{f(n)}}} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n^2.$ But recall from above that $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}}}} \\mathbf {m}(\\sigma ) \\le 2^{-n}.$ Combining these two facts, and taking $S$ to be the set $T_{f(n)}[s_n]$ , we have $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n ^2$ which establishes the first part of the theorem.", "To see that the second part of the statement follows from the first, take $\\sigma =Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ and observe that $\\mathbf {m}^Y(\\sigma ) =^* \\mathbf {m}^Y(n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ $\\mathbf {m}^\\Omega (\\tau ) =^* \\mathbf {m}^\\Omega (n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ and $2^{I(\\sigma :\\tau )}=2^n/n^{O(1)}$ (since $I(\\sigma :\\tau )\\ge ^+n-2\\log n$ as established above).", "Then we have $\\begin{split}2^{I(\\Omega :Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^\\Omega (\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}&\\ge \\sum _n \\mathbf {m}^\\Omega (Y{\\upharpoonright }f(n)) \\cdot \\mathbf {m}^Y(\\Omega {\\upharpoonright }n) \\cdot 2^{I(Y{\\upharpoonright }f(n),\\Omega {\\upharpoonright }n)}\\\\&\\ge \\sum _n 2^n/n^{O(1)}=\\infty .\\end{split}$ Remark 9.1 The converse of this theorem does not hold, i.e., there is a $\\Pi ^0_1$ class that is not deep but all of whose elements have infinite mutual information with $\\Omega $ .", "This follows from Theorem REF and the fact that having infinite mutual information with $\\Omega $ is a property that is invariant under addition or deletion of a finite prefix.", "Levin's proof of Theorem REF restricted to the particular case of completions of $\\mathsf {PA}$ is the mathematical part of a more general discussion, the other part of which is philosophical in nature.", "While Gödel's theorem asserts that no completion of $\\mathsf {PA}$ can be computably obtained, Levin's goal is to show that no completion of $\\mathsf {PA}$ can be obtained by any physical means whatsoever (computationally or otherwise), thus generalizing Gödel's theorem.", "Levin does not fully specify what he means by physically obtainable (the exact term he uses is “located in the physical world\"), but nonetheless he makes the following postulate, which he dubs the “Independence Postulate\": if $\\sigma $ is a mathematically definable string that has an $n$ bit description and $\\tau $ can be located in the physical world with a $k$ bit description, then for a fixed small constant $c$ , one has $I(\\sigma : \\tau ) < n+k+c$ .", "In particular, if one admits that some infinite sequences can be physically obtained, the Independence Postulate for infinite sequences says that if $X$ and $Y$ are two infinite sequences with $X$ mathematically definable and $Y$ physically obtainable, then $I(X:Y) < \\infty $ .", "Being $\\Delta ^0_2$ , $\\Omega $ is mathematically definable, and, as Levin shows, $I(\\Omega :Y)=\\infty $ for any completion $Y$ of $\\mathsf {PA}$ .", "Thus, assuming the Independence Postulate, no completion of $\\mathsf {PA}$ is physically obtainable.", "Our Theorem REF extends Levin's theorem and, assuming the Independence Postulate, shows that no member of a deep class (shift-complex sequences, compression functions, etc.)", "is physically obtainable.", "Of course, evaluating the validity of the Independence Postulate would require an extended philosophical discussion that would take us well beyond the scope of this paper.", "In any case, whether or not the reader accepts the Independence Postulate, Theorem REF is interesting in its own right.", "In fact, it is quite surprising because it seems to contradict the “basis for randomness theorem\" (see ), which states that if $X$ is a Martin-Löf random sequence and $\\mathcal {C}$ is a $\\Pi ^0_1$ class, then there exists a member $Y$ of $\\mathcal {C}$ such that $X$ is random relative to $Y$ .", "If a sequence $X$ is random relative to another sequence $Y$ , the intuition is that $Y$ “knows nothing about $X$ \", and thus one could conjecture that $I(X : Y) < \\infty $ .", "However, this cannot always be the case, since by Theorem REF , $I(\\Omega :Y) = \\infty $ for all members $Y$ of a deep $\\Pi ^0_1$ class $, even though $$ is random relative to some $ Y.", "This apparent paradox can be explained by taking a closer look at the definition of mutual information.", "Let $ be a deep $ 01$ class, whose canonical co-c.e.\\ tree~$ T$ satisfies $ m(Tf(n)) < 2-n$ for some computable function~$ f$.", "By Theorem~\\ref {thm:mutual-info} and the symmetry of information, for every $ Y we have $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big ) = I\\big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\big ) \\ge n-O(\\log n)$ where $k_n$ stands for $\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "Take a $Y \\in such that $$ is random relative to~$ Y$.", "It is well-known that a sequence~$ Z$ is random if and only if $ K(Z n n) n-O(1)$ (see for example Gács~\\cite {Gacs1980}).", "Applying this fact (relativized to~$ Y$) to $$, we have$$\\mathrm {K}^Y(\\Omega {\\upharpoonright }n \\mid n) \\ge n-O(1)$$and thus in particular that$$\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\ge n-O(1).$$Since $ K(n) n+O(n)$, it follows that$ $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\le O(\\log n)$ The only difference between (REF ) and (REF ) is the term $k_n=\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "But it makes a big difference, as one can verify that $\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n))-\\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big )\\ge n-O(\\log n).$ Informally, while $Y$ “knows nothing\" about $\\Omega $ , the complexity of its initial segments, seen as a function, does.", "In particular, the change of complexity caused by $k_n$ implies that $\\mathrm {K}(\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge n - O(\\log n)$ and thus $\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge 2^n/n^{O(1)}$ .", "Acknowledgements We would like to thank Noam Greenberg, Rupert Hölzl, Mushfeq Khan, Leonid Levin, Joseph Miller, André Nies, Paul Shafer, and Antoine Taveneaux for many fruitful discussions on the subject.", "Particular thanks go to Steve Simpson who provided very detailed feedback on the first arXiv version of this paper.", "Let us fix some notation and terminology.", "We denote by $2^\\mathbb {N}$ the set of infinite binary sequences (which we often refer to as “sequences\"), also known as Cantor space.", "We denote the set of finite strings by $2^{<\\mathbb {N}}$ and the empty string by $\\Lambda $ .", "$\\mathbb {Q}_2$ is the set of dyadic rationals, i.e., multiples of a negative power of 2.", "Given $X\\in 2^\\mathbb {N}$ and an integer $n$ , $X {\\upharpoonright }n$ is the string that consists of the first $n$ bits of $X$ , and $X(n)$ is the $(n+1)$ st bit of $X$ (so that $X(0)$ is the first bit of $X$ ).", "For integers $n$ and $k$ , $X {\\upharpoonright }[n,n+k]$ denotes the subword $\\sigma $ of $X$ of length $k+1$ such that for $i \\le k$ , $\\sigma (i)=X(i+n)$ .", "If $\\sigma $ is a string and $x\\in 2^{<\\mathbb {N}}\\cup 2^\\mathbb {N}$ , then $\\sigma \\preceq x$ means that $\\sigma $ is a prefix of $x$ .", "A prefix-free set of strings is a set of strings such that none of its elements is a strict prefix of another one.", "For strings $\\sigma ,\\tau \\in 2^{<\\mathbb {N}}$ , $\\sigma ^\\frown \\tau $ denotes string obtained by concatenating $\\sigma $ and $\\tau $ .", "Similarly, for $\\sigma \\in 2^{<\\mathbb {N}}$ and $X\\in 2^\\mathbb {N}$ , $\\sigma ^\\frown X$ is the sequence obtained by concatenating $\\sigma $ and $X$ .", "For $X,Y\\in 2^\\mathbb {N}$ , $X\\oplus Y\\in 2^\\mathbb {N}$ satisfies $X\\oplus Y(2n)=X(n)$ and $X\\oplus Y(2n+1)=Y(n)$ .", "Given a string $\\sigma $ , the cylinder $\\llbracket \\sigma \\rrbracket $ is the set of elements of $2^\\mathbb {N}$ having $\\sigma $ as a prefix.", "Moreover, given $S\\subseteq 2^{<\\mathbb {N}}$ , $\\llbracket S\\rrbracket $ is defined to be the set $\\bigcup _{\\sigma \\in S}\\llbracket \\sigma \\rrbracket $ .", "When we refer to the topology of the Cantor space, we implicitly mean the product topology, i.e., the topology whose open sets are exactly those of type $\\llbracket S\\rrbracket $ for some $S$ .", "For this topology, some sets are both open and closed (clopen): these are the sets of type $\\llbracket S\\rrbracket $ when $S$ is finite.", "A tree is a set of strings that is closed downwards under the prefix relation.", "A path through a tree $T$ is a member of $2^\\mathbb {N}$ all of whose prefixes are in $T$ .", "The set of paths of a tree $T$ is denoted by $[T]$ .", "The $n^\\mathrm {th}$ level of a tree $T$ , denoted $T_n$ , is the set of members of $T$ of length $n$ .", "An effectively open set, also called $\\Sigma ^0_1$ , is a set of type $\\llbracket S \\rrbracket $ for some c.e.", "set of strings $S$ .", "An effectively closed set, or $\\Pi ^0_1$ class, is the complement of some effectively open set.", "It is well-known that a class is $\\Pi ^0_1$ if and only if it is of the form $[T]$ for some co-c.e.", "(or computable) tree $T$ .", "Given a $\\Pi ^0_1$ class $, its \\emph {canonical co-c.e.\\ tree} is the tree $ T={[] = }$.", "The arithmetic hierarchy is defined inductively: a set is $ 0n+1$ if it is a uniform union of $ 0n$ sets, and an set is $ 0n+1$ if it is a uniform intersection of~$ 0n$ sets.$ A order function is a function $h: \\mathbb {N}\\rightarrow \\mathbb {N}$ that is non-decreasing and unbounded.", "Given an order $h$ , the inverse of $h$, denoted $h^{-1}$ is the order function defined as follows: For all $k$ , $h^{-1}(k)$ is the smallest $n$ such that $h(n) \\ge k$ .", "Note that $h^{-1}$ is computable if $h$ is.", "We adopt standard computability notation: $\\le _T$ denotes Turing reducibility, $\\le _{tt}$ denotes $\\mathit {tt}$ -reducibility, $A^{\\prime }$ is the Turing jump of $A$ (and $\\emptyset ^{\\prime }$ denotes the jump of the zero sequence).", "Throughout the paper, when an object $A$ has a computable enumeration or limit approximation, we use the notation $A[s]$ to denote the approximation of the object at stage $s$ .", "Moreover, if $A$ contains several expressions that have a computable approximation, the notation $A[s]$ means that all of these expressions are approximated up to stage $s$ .", "For example, if $T$ is a tree and $f$ is a function, both of which have computable approximation, $T_{f(n)}[s]$ is equal to $(T[s])_{f(n)[s]}$ .", "Finally, we adopt the following asymptotic notation.", "For two functions $f,g: \\mathbb {N}\\rightarrow \\mathbb {N}$ , we sometimes write $f \\le ^+ g$ to abbreviate $f \\le g + O(1)$ and $f \\le ^* g$ to abbreviate $f = O(g)$ ." ], [ "Measures on $2^\\mathbb {N}$", "Recall that by Caratheodory's Theorem, a measure $\\mu $ on $2^\\mathbb {N}$ is uniquely determined by specifying the values of $\\mu $ on the basic open sets of $2^\\mathbb {N}$ , where $\\mu (\\llbracket \\sigma \\rrbracket )=\\mu (\\llbracket \\sigma 0\\rrbracket )+\\mu (\\llbracket \\sigma 1\\rrbracket )$ for every $\\sigma \\in 2^{<\\mathbb {N}}$ .", "If we further require that $\\mu (2^\\mathbb {N})=1$ , then $\\mu $ is a probability measure.", "Hereafter, we will write $\\mu (\\llbracket \\sigma \\rrbracket )$ as $\\mu (\\sigma )$ .", "In addition, for a set $S\\subseteq 2^{<\\mathbb {N}}$ , we will write $\\mu (S)$ as shorthand for $\\mu (\\llbracket S\\rrbracket )$ .", "In the case that $S$ is prefix-free, we will have $\\mu (S)=\\sum _{\\sigma \\in 2^{<\\mathbb {N}}} \\mu (\\sigma )$ .", "The uniform (or Lebesgue) measure $\\lambda $ is the unique Borel measure such that $\\lambda (\\sigma )=2^{-\|\\sigma \|}$ for all strings $\\sigma $ .", "A measure $\\mu $ on $2^\\mathbb {N}$ is computable if $\\sigma \\mapsto \\mu (\\sigma )$ is computable as a real-valued function, i.e., if there is a computable function $\\tilde{\\mu }:2^{<\\mathbb {N}}\\times \\mathbb {N}\\rightarrow \\mathbb {Q}_2$ such that $\|\\mu (\\sigma )-\\tilde{\\mu }(\\sigma ,i)\|\\le 2^{-i}$ for every $\\sigma \\in 2^{<\\mathbb {N}}$ and $i\\in \\mathbb {N}$ .", "One family of examples of computable measures is given by the collection of Dirac measures concentrated on some computable point.", "That is, if $X\\in 2^\\mathbb {N}$ is a computable sequence, then the Dirac measure concentrated on $X$, denoted $\\delta _X$ , is defined as follows: $\\delta _X(\\sigma )=\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if } \\sigma \\prec X\\\\0 & \\mbox{if } \\sigma \\lnot \\prec X\\end{array}.\\right.$ More generally, for a measure $\\mu $ , we say that $X\\in 2^\\mathbb {N}$ is an atom of $\\mu $ or a $\\mu $ -atom, denoted $X\\in \\mathsf {Atom}_\\mu $ , if $\\mu (\\lbrace X\\rbrace )>0$ .", "Kautz proved the following: Lemma 2.1 (Kautz ) $X\\in 2^\\mathbb {N}$ is computable if and only if $X$ is an atom of some computable measure.", "There is a close connection between computable measures and a certain class of Turing functionals.", "Recall that a Turing functional $\\Phi :\\subseteq 2^\\mathbb {N}\\rightarrow 2^\\mathbb {N}$ may be defined as a c.e.", "set $\\Phi $ of pairs of strings $(\\sigma ,\\tau )$ such that if $(\\sigma ,\\tau ),(\\sigma ^{\\prime },\\tau ^{\\prime })\\in \\Phi $ and $\\sigma \\preceq \\sigma ^{\\prime }$ , then $\\tau \\preceq \\tau ^{\\prime }$ or $\\tau ^{\\prime }\\preceq \\tau $ .", "For each $\\sigma \\in 2^{<\\mathbb {N}}$ , we define $\\Phi ^\\sigma $ to be the maximal string in $\\lbrace \\tau : (\\exists \\sigma ^{\\prime }\\preceq \\sigma )((\\sigma ^{\\prime },\\tau )\\in \\Phi )\\rbrace $ in the order given by $\\preceq $ .", "To obtain a map defined on $2^\\mathbb {N}$ from the c.e.", "set of pairs $\\Phi $ , for each $X\\in 2^\\mathbb {N}$ , we let $\\Phi ^X$ be the maximal $y\\in 2^{<\\mathbb {N}}\\cup 2^\\mathbb {N}$ in the order given by $\\preceq $ such that $\\Phi ^{X {\\upharpoonright }n}$ is a prefix of $y$ for all $n$ .", "We will thus set $\\mathsf {dom}(\\Phi )=\\lbrace X\\in 2^\\mathbb {N}:\\Phi ^X\\in 2^\\mathbb {N}\\rbrace $ .", "When $\\Phi ^X\\in 2^\\mathbb {N}$ , we will often write $\\Phi ^X$ as $\\Phi (X)$ to emphasize the functional $\\Phi $ as a map from $2^\\mathbb {N}$ to $2^\\mathbb {N}$ .", "We also use the notation $\\Phi ^X {\\upharpoonright }n \\downarrow $ to emphasize that $\\Phi ^X$ has length at least $n$ .", "For $\\tau \\in 2^{<\\mathbb {N}}$ let $\\Phi ^{-1}(\\tau )$ be the set $\\lbrace \\sigma \\in 2^{<\\mathbb {N}}: \\exists \\tau ^{\\prime } \\succeq \\tau \\colon (\\sigma ,\\tau ^{\\prime })\\in \\Phi \\rbrace $ .", "Similarly, for $S \\subseteq 2^{<\\mathbb {N}}$ we define $\\Phi ^{-1}(S) = \\bigcup _{\\tau \\in S} \\Phi ^{-1}(\\tau )$ .", "When $\\mathcal {A}$ is a subset of $2^\\mathbb {N}$ , we denote by $\\Phi ^{-1}(\\mathcal {A})$ the set $\\lbrace X\\in \\mathsf {dom}(\\Phi ):\\Phi (X) \\in \\mathcal {A}\\rbrace $ .", "Note in particular that $\\Phi ^{-1}(\\llbracket \\tau \\rrbracket )= \\llbracket \\Phi ^{-1}(\\tau )\\rrbracket \\cap \\mathsf {dom}(\\Phi )$ .", "Remark 2.2 The Turing functionals that induce computable measures are precisely the almost total Turing functionals, where a Turing functional $\\Phi $ is almost total if $\\lambda (\\mathsf {dom}(\\Phi ))=1.$ Given an almost total Turing functional $\\Phi $ , the measure induced by $\\Phi $ , denoted $\\lambda _\\Phi $ , is defined by $\\lambda _\\Phi (\\sigma )=\\lambda (\\llbracket \\Phi ^{-1}(\\sigma )\\rrbracket )=\\lambda (\\lbrace X:\\Phi ^X\\succeq \\sigma \\rbrace ).$ It is not difficult to verify that $\\lambda _\\Phi $ is a computable measure.", "Moreover, one can easily show that given a computable measure $\\mu $ , there is some almost total functional $\\Phi $ such that $\\mu =\\lambda _\\Phi $ ." ], [ "Lower semi-computable semi-measures", "A discrete semi-measure is a map $m:2^{<\\mathbb {N}}\\rightarrow [0,1]$ such that $\\sum _{\\sigma \\in 2^{<\\mathbb {N}}}m(\\sigma )\\le 1$ .", "Moreover, if $S$ is a set of strings, then $m(S)$ is defined to be $\\sum _{\\sigma \\in S} m(\\sigma )$ .", "Henceforth, we will restrict our attention to the class of lower semi-computable discrete semi-measures, where a discrete semi-measure $m$ is lower semi-computable if there is a computable function $\\widetilde{m} : 2^{<\\mathbb {N}}\\times \\mathbb {N}\\rightarrow \\mathbb {Q}_2$ , non-decreasing in its second argument such that for all $\\sigma $ : $\\lim _{i \\rightarrow +\\infty } \\widetilde{m}(\\sigma ,i) = m(\\sigma ).$ Levin showed that there is a universal lower semi-computable discrete semi-measure $\\mathbf {m}$ ; that is, for every lower semi-computable discrete semi-measure $m$ , there is some constant $c$ such that $m\\le c\\cdot \\mathbf {m}$ .", "This universal discrete semi-measure $\\mathbf {m}$ is closely related to the notion of prefix-free Kolmogorov complexity.", "Recall that $\\mathrm {K}(\\sigma )$ denotes the prefix-free Kolmogorov complexity of $\\sigma $ , i.e.", "$\\mathrm {K}(\\sigma )=\\min \\lbrace \|\\tau \|:U(\\tau )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free Turing machine.", "Then by the coding theorem (see ), we have $\\mathrm {K}(\\sigma )=-\\log \\mathbf {m}(\\sigma )+O(1)$ .", "In particular, since $\\mathrm {K}(n)\\le ^+ 2\\log (n)$ , it follows that $\\mathbf {m}(n)\\ge ^* n^{-2}$ , a fact we will make use of below.", "A continuous semi-measure $\\rho :2^{<\\mathbb {N}}\\rightarrow [0,1]$ satisfies (i) $\\rho (\\epsilon ) = 1$ and (ii) $\\rho (\\sigma )\\ge \\rho (\\sigma 0)+\\rho (\\sigma 1)$ .", "If $S$ is a set of strings, then $\\rho (S)$ denotes the sum $\\sum _{\\sigma \\in S} \\rho (\\sigma )$ .", "As in the case of discrete semi-measures, we will restrict our attention to the class of lower semi-computable continuous semi-measures (the values of which are effectively approximable from below as defined above in the case of discrete lower semi-computable semi-measures).", "One particularly important property of lower semi-computable continuous semi-measures is their connection to Turing functionals.", "Just as computable measures are precisely the measures that are induced by almost total Turing functionals (as discussed at the end of the previous section), lower semi-computable continuous semi-measures are precisely the semi-measures that are induced by Turing functionals: Theorem 2.1 (Levin, Zvonkin ) (i) For every Turing functional $\\Phi $ , the function $\\lambda _\\Phi (\\sigma )=\\lambda (\\llbracket \\Phi ^{-1}(\\sigma )\\rrbracket )=\\lambda (\\lbrace X:\\Phi ^X\\succeq \\sigma \\rbrace )$ is a left-c.e.", "semi-measure.", "(ii) For every left-c.e.", "semi-measure $\\rho $ , there is a Turing functional $\\Phi $ such that $\\rho =\\lambda _\\Phi $ .", "As there is a universal lower semi-computable discrete semi-measure, so too is there a universal lower semi-computable continuous semi-measure.", "That is, there exists a lower semi-computable continuous semi-measure $\\mathbf {M}$ such that, for every lower semi-computable continuous semi-measure $\\rho $ , there exists a $c\\in \\mathbb {N}$ such that $\\rho \\le c\\cdot \\mathbf {M}$ .", "One way to obtain a universal lower semi-computable continuous semi-measure is to effectively list all lower semi-computable continuous semi-measures $(\\rho _e)_{e \\in \\mathbb {N}}$ (which can be obtained from an effective list of all Turing functionals by appealing to Theorem REF ) and set $\\mathbf {M}= \\sum _{e \\in \\mathbb {N}} 2^{-e-1} \\rho _e$ .", "Alternatively, one can induce it by means of a universal Turing functional: Let $(\\Phi _i)_{i\\in \\mathbb {N}}$ be an effective enumeration of all Turing functionals.", "Then the functional $\\widehat{\\Phi }$ such that $\\widehat{\\Phi }(1^e0X)=\\Phi _e(X)$ for every $e\\in \\mathbb {N}$ and $X\\in 2^\\mathbb {N}$ is a universal Turing functional and we can set $\\mathbf {M}=\\lambda _{\\widehat{\\Phi }}$ .", "One can readily verify that $\\mathbf {M}$ is an universal lower semi-computable continuous semi-measure, which is sometimes called a priori probability (see for example Gács for the basic properties of $\\mathbf {M}$ ).", "Another feature of continuous semi-measures that we will make use of throughout this study is that there is a canonical measure on $2^\\mathbb {N}$ that can be obtained from a continuous semi-measure.", "To motivate the definition of this measure, it is helpful to think of a semi-measure as a network flow through the full binary tree $2^{<\\mathbb {N}}$ seen as a directed graph (see, for instance, or ).", "First we give the node at the root of the tree flow equal to 1 (corresponding to the condition $\\rho (\\epsilon )=1$ ).", "Some amount of this flow at each node $\\sigma $ is passed along to the node corresponding to $\\sigma 0$ , some is passed along to the node corresponding to $\\sigma 1$ , and, potentially, some of the flow is lost (corresponding to the condition that $\\rho (\\sigma )\\ge \\rho (\\sigma 0)+\\rho (\\sigma 1)$ ).", "We obtain a measure $\\overline{\\rho }$ from $\\rho $ if we ignore all of the flow that is lost below a given node and just consider the behavior of the flow that never leaves the network below this node.", "We will refer to $\\overline{\\rho }$ as the measure derived from $\\rho $ .", "This can be formalized as follows.", "Definition 2.3 Let $\\rho $ be a semi-measure.", "The canonical measure obtained from $\\rho $ is defined to be $\\overline{\\rho }(\\sigma ):=\\inf _{n\\ge \|\\sigma \|}\\sum _{\\tau \\succeq \\sigma \\;\\&\\;\|\\tau \|=n}\\rho (\\tau ) = \\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\succeq \\sigma \\;\\&\\;\|\\tau \|=n}\\rho (\\tau ).$ Several important facts about these canonical measures are the following: Proposition 2.4 Let $\\rho $ be a semi-measure and let $\\overline{\\rho }$ be the canonical measure obtained from $\\rho $ .", "(i) $\\overline{\\rho }$ is the largest measure $\\mu $ such that $\\mu \\le \\rho $ .", "(ii) If $\\rho (\\sigma )=\\lambda (\\lbrace X:\\Phi ^X\\succeq \\sigma \\rbrace )$ , then $\\overline{\\rho }(\\sigma )=\\lambda (\\lbrace X \\in \\mathsf {dom}(\\Phi ): \\Phi ^X\\succeq \\sigma \\rbrace )$ .", "Thus, in replacing $\\rho $ with $\\overline{\\rho }$ , this amounts to restricting the Turing functional $\\Phi $ that induces $\\rho $ to those inputs on which $\\Phi $ is total.", "The proof of part $(i)$ is straightforward; for a proof of $(\\mathit {ii})$ , see .", "Remark 2.5 Using Proposition REF ($\\mathit {ii}$ ) and the universality of $\\mathbf {M}$ , one can readily verify that for every lower-semi-computable continuous semi-measure $\\rho $ , there is some $c\\in \\mathbb {N}$ such that $\\overline{\\rho }\\le c\\cdot \\overline{\\mathbf {M}}$ .", "Thus $\\overline{\\mathbf {M}}$ can be seen as a measure that is universal for the class of canonical measures obtained from some lower-semi-computable continuous semi-measure (a class that contains all computable measures)." ], [ "Notions of algorithmic randomness", "The primary notion of algorithmic randomness that we will consider here is Martin-Löf randomness.", "Although the standard definition is given in terms of the Lebesgue measure, we will consider Martin-Löf randomness with respect to any computable measure.", "Definition 2.6 Let $\\mu $ be a computable measure on $2^\\mathbb {N}$ .", "(i) A $\\mu $ -Martin-Löf test is a sequence $(\\mathcal {U}_i)_{i\\in \\mathbb {N}}$ of uniformly effectively open subsets of $2^\\mathbb {N}$ such that for each $i$ , $\\mu (\\mathcal {U}_i)\\le 2^{-i}.$ (ii) $X\\in 2^\\mathbb {N}$ passes the $\\mu $ -Martin-Löf test $(\\mathcal {U}_i)_{i\\in \\mathbb {N}}$ if $X\\notin \\bigcap _{i \\in \\mathbb {N}}\\mathcal {U}_i$ .", "(iii) $X\\in 2^\\mathbb {N}$ is $\\mu $ -Martin-Löf random, denoted $X\\in \\mathsf {MLR}_\\mu $ , if $X$ passes every $\\mu $ -Martin-Löf test.", "When $\\mu $ is the uniform (or Lebesgue) measure $\\lambda $ , we often abbreviate $\\mathsf {MLR}_\\mu $ by $\\mathsf {MLR}$ .", "An important feature of Martin-Löf randomness is the existence of a universal test: For every computable measure $\\mu $ , there is a single $\\mu $ -Martin-Löf test $(\\hat{\\mathcal {U}}_i)_{i\\in \\mathbb {N}}$ , having the property that $X\\in \\mathsf {MLR}_\\mu $ if and only if $X\\notin \\bigcap _{i\\in \\mathbb {N}}\\hat{\\mathcal {U}}_i$ .", "Remark 2.7 As we saw earlier, some computable measures have atoms, such as the Dirac measure $\\delta _X$ concentrated on some computable sequence $X$ .", "Moreover, given a computable measure $\\mu $ , if $X$ is a $\\mu $ -atom, it immediately follows that $X\\in \\mathsf {MLR}_\\mu $ .", "Four additional notions of algorithmic randomness that will be considered in this study are difference randomness, Kurtz randomness, weak 2-randomness, and computable randomness.", "Definition 2.8 (i) A difference test is a computable sequence $((\\mathcal {U}_i,\\mathcal {V}_i))_{i\\in \\mathbb {N}}$ of pairs of $\\Sigma ^0_1$ classes such that for each $i$ , $\\lambda (\\mathcal {U}_i\\setminus \\mathcal {V}_i)\\le 2^{-i}.$ (ii) A sequence $X\\in 2^\\mathbb {N}$ passes a difference test $((\\mathcal {U}_i,\\mathcal {V}_i))_{i\\in \\mathbb {N}}$ if $X\\notin \\bigcap _i(\\mathcal {U}_i\\setminus \\mathcal {V}_i)$ .", "(iii) $X\\in 2^\\mathbb {N}$ is difference random if $X$ passes every difference test.", "We denote by $\\mathsf {DiffR}$ the class of difference random reals.", "Franklin and Ng proved the following remarkable theorem about difference randomness: Theorem 2.2 (Franklin and Ng ) A sequence $X$ is difference random if and only if $X$ is Martin-Löf random and $X \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Recall that a sequence $X$ has $\\mathsf {PA}$ degree if $X$ can compute a consistent completion of Peano arithmetic (hereafter, $\\mathsf {PA}$ ).", "This is equivalent to requiring that $X$ compute a total function extending a universal partial computable $\\lbrace 0,1\\rbrace $ -valued function, a fact that will be useful in $§$.", "A related result is the following: Theorem 2.3 (Stephan ) A Martin-Löf random sequence $X$ has $\\mathsf {PA}$ degree if and only if $X\\ge _T\\emptyset ^{\\prime }$ .", "It follows from the previous two results that a Martin-Löf random sequence is difference random if and only if it does not have $\\mathsf {PA}$ degree.", "Another approach to defining randomness is to take the collection of all null sets definable at the same level of complexity and require that a random sequence avoid all such null sets.", "For instance, if we take all $\\Pi ^0_1$ definable null sets or all $\\Pi ^0_2$ definable null sets, we have the following two notions of randomness, first introduced by Kurtz in .", "Definition 2.9 Let $X\\in 2^\\mathbb {N}$ .", "(i) $X$ is Kurtz random (or weakly 1-random) if and only if $X$ is not contained in any $\\Pi ^0_1$ class of Lebesgue measure 0 (equivalently, if and only if it is not contained in any $\\Sigma ^0_2$ class of measure 0).", "(ii) $X$ is weakly 2-random if and only if $X$ is not contained in any $\\Pi ^0_2$ class of Lebesgue measure 0.", "Let $\\mathsf {KR}$ denoted the collection of Kurtz random sequences and $\\mathsf {W2R}$ denote the collection of weakly 2-random sequences.", "The last definition of randomness we will consider in the study is defined in terms of certain effective betting strategies called martingales.", "Definition 2.10 (i) A martingale is a function $d:2^{<\\mathbb {N}}\\rightarrow \\mathbb {R}^{\\ge 0} \\cup \\lbrace +\\infty \\rbrace $ such that for every $\\sigma \\in 2^{<\\mathbb {N}}$ , $2d(\\sigma )=d(\\sigma 0)+d(\\sigma 1).$ (ii) A martingale $d$ succeeds on $X\\in 2^\\mathbb {N}$ if $\\limsup _{n\\rightarrow \\infty }\\, d(X{\\upharpoonright }n)=+\\infty .$ (iii) A sequence $X\\in 2^\\mathbb {N}$ is computably random if there is no computable martingale $d$ that succeeds on $X$ .", "The collection of computably random sequences will be written as $\\mathsf {CR}$ .", "The different notions of randomness discussed in this section form a strict hierarchy.", "Namely, the following relations hold: $\\mathsf {W2R}\\subsetneq \\mathsf {DiffR}\\subsetneq \\mathsf {MLR}\\subsetneq \\mathsf {CR}\\subsetneq \\mathsf {KR}$ Lastly, we note that each of these notions of randomness can be relativized to an oracle $A\\in 2^\\mathbb {N}$ .", "In the sequel, we will consider relative versions of each of the above-listed notions of randomness except for weak 2-randomness.", "Below we list the relevant modifications for each notion of relative randomness.", "(i) $A$ -Martin-Löf randomness ($\\mathsf {MLR}^A$ ) is defined in terms of an $A$ -computable sequence of $\\Sigma ^{0,A}_1$ classes $(\\mathcal {U}_i^A)_{i\\in \\mathbb {N}}$ such that $\\lambda (\\mathcal {U}_i^A)\\le 2^{-i}$ for every $i$ ; (ii) $A$ -difference randomness ($\\mathsf {DiffR}^A$ ) is defined in terms of an $A$ -computable sequence of pairs $\\Sigma ^{0,A}_1$ classes $((\\mathcal {U}_i^A,i^A))_{i\\in \\mathbb {N}}$ such that $\\lambda (\\mathcal {U}_i^A\\setminus i^A)\\le 2^{-i}$ for every $i$ ; (iii) $A$ -Kurtz randomness ($\\mathsf {KR}^A$ ) is defined in terms of $\\Pi ^{0,A}_1$ classes of Lebesgue measure 0; (iv) $A$ -computable randomness ($\\mathsf {CR}^A$ ) is defined in terms of $A$ -computable martingales." ], [ "Negligibility and $\\mathit {tt}$ -Negligibility", "We are now in a position to define negligibility and $\\mathit {tt}$ -negligibility, two notions that are central to this study.", "As discussed in the introduction, the intuitive idea behind negligibility is that a set $2^\\mathbb {N}$ is negligible if no member of $ can be produced with positive probability by means of any Turing functional with a random oracle.", "Similarly, a set $ 2N$ is $ tt$-negligible if no member of $ can be produced with positive probability by means of any total Turing functional with a random oracle.", "However, we will primarily work with the following measure-theoretic definition of these two notions.", "Definition 3.1 Let $2^\\mathbb {N}$ .", "(i) $ is \\emph {negligible} if $M(=0$.\\item [(ii)] $ is tt-negligible if $\\mu (=0$ for every computable measure $\\mu $ .", "The intuitive description of negligibility and $\\mathit {tt}$ -negligibility given above is justified by the following proposition.", "For $2^\\mathbb {N}$ , we define $(^{\\le _T}$ to be the set $\\lbrace Y \\in 2^\\mathbb {N}: (\\exists X\\in [X\\le _T Y]\\rbrace $ .", "Furthermore, $(^{\\le _{tt}}$ denotes the set $\\lbrace Y \\in 2^\\mathbb {N}: (\\exists X\\in [X\\le _{tt} Y]\\rbrace .$ Proposition 3.2 Let $2^\\mathbb {N}$ .", "(i) $ is negligible if and only if $ ((T)=0$.\\item [(ii)] $ is $\\mathit {tt}$ -negligible if and only if $\\lambda ((^{\\le _{tt}})=0$ .", "($\\mathit {i}$ ) ($\\Rightarrow $ ) If $ is negligible, then $(=0$ for every lower semi-computable continuous semi-measure $$ by Remark \\ref {rmk-Mbar}.", "In particular, by Theorem \\ref {thm-MachinesInduceSemiMeasures} ($ ii$) and Proposition \\ref {prop-rhobar-properties} ($ ii$), $(=0$ for every Turing functional $$ and hence$$\\lambda ((^{\\le _T})=\\sum _{i\\in \\mathbb {N}}\\lambda (\\lbrace Y \\in 2^\\mathbb {N}: \\Phi _i(Y)\\in )=\\sum _{i\\in \\mathbb {N}}\\overline{\\lambda }_{\\Phi _i}(=0.$$($$) Since $ ((T)=0$, it follows that $(=0$, where $$ is a universal Turing functional.", "Thus $M(=(=0$.$ ($\\mathit {ii}$ ) ($\\Rightarrow $ ) If $ is $ tt$-negligible, then $ (=0$ for every $ tt$-functional $$.", "The result clearly follows.\\\\($$) Now suppose that $ (>0$ for some computable measure $$.", "Then by Remark \\ref {rmk-functionals-measures}, there is some almost total Turing functional $$ such that $ =$.", "Since the domain of a Turing functional is $ 02$, if $ Xdom()$, it follows that $ XMLR$ (in fact, $ X$ is not even Kurtz random).", "Indeed, the complement of $ dom()$ is a $ 02$ class of measure~$ 0$, so $ Xdom()$ implies that $ X$ is contained in a $ 01$ class of measure~$ 0$.$ Let $i$ be the least such that $\\mu (>2^{-i}$ .", "Then $\\lambda (\\Phi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i)>0$ , where $\\hat{\\mathcal {U}}_i$ is the $i^{\\mathrm {th}}$ level of the universal Martin-Löf test (so that $\\lambda (\\hat{\\mathcal {U}}_i^c)>1-2^{-i}$ ).", "Now since $\\Phi $ is total on $\\hat{\\mathcal {U}}_i^c$ , which is a $\\Pi ^0_1$ class, we can define a $\\mathit {tt}$ -functional $\\Psi $ as follows: $\\Psi (X)=\\left\\lbrace \\begin{array}{ll}\\Phi (X) & \\mbox{if } X\\in \\hat{\\mathcal {U}}^c_i\\\\0^\\infty & \\mbox{if } X\\in \\hat{\\mathcal {U}}_i\\end{array}.\\right.$ Since $\\Phi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i=\\Psi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i$ , it follows that $\\lambda (\\Psi ^{-1}(\\cap \\hat{\\mathcal {U}}^c_i)>0$ and hence $\\lambda ((^{\\le _{tt}})>0$ .", "By the following proposition, there is a simple characterization of negligible and $\\mathit {tt}$ -negligible singletons.", "Proposition 3.3 For $X\\in 2^\\mathbb {N}$ , the following are equivalent: (i) $\\lbrace X\\rbrace $ is negligible.", "(ii) $\\lbrace X\\rbrace $ is $\\mathit {tt}$ -negligible.", "(iii) $X$ is non-computable.", "$(\\mathit {i})\\Rightarrow (\\mathit {ii})$ is immediate.", "$(\\mathit {ii})\\Rightarrow (\\mathit {iii})$ follows from Lemma REF , which tells us that $X$ is computable if $\\mu (\\lbrace X\\rbrace )>0$ for some computable measure $\\mu $ .", "Lastly, $(\\mathit {iii})\\Rightarrow (\\mathit {i})$ is a theorem due to Sacks (see, for instance ).", "Further, it is clear that for a $\\Sigma ^0_1$ class $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , $\\mathcal {S}$ is negligible if and only if $\\mathcal {S}$ is $\\mathit {tt}$ -negligible if and only if $\\mathcal {S}$ is empty.", "However, the notions of negligibility and $\\mathit {tt}$ -negligibility are non-trivial for $\\Pi ^0_1$ classes.", "For instance, if we let $\\mathcal {PA}$ be the $\\Pi ^0_1$ class of consistent completions of $\\mathsf {PA}$ , then as shown by Jockusch and Soare in , $\\mathcal {PA}$ is negligible (and thus $tt$ -negligible).", "Although every negligible $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible, the converse does not hold.", "Theorem 3.1 There exists a tt-negligible $\\Pi ^0_1$ class that is not negligible.", "The proof of Theorem REF draws upon a theorem of Downey, Greenberg and Miller : there exists a non-negligible perfect thin $\\Pi ^0_1$ class, where a $\\Pi ^0_1$ class $ is \\emph {thin} if for every $ 01$ subclass $$ of $ , there exists a clopen set $D$ such that $ = D$ .", "We argue that any perfect thin class must be $\\mathit {tt}$ -negligible, which extends a result of Simpson who proved that every thin class must have Lebesgue measure 0.", "However, our proof strategy is very different from Simpson's.", "We first need the following lemma, which is folklore.", "Lemma 3.4 Let $ be a $ 01$ class.", "If for some computable probability measure~$$ the value of $ ($ is a positive computable real number, then $ contains a computable member.", "Consider the measure $\\nu : 2^{<\\mathbb {N}}\\rightarrow [0,1]$ defined by $\\nu (\\sigma )=\\mu (\\llbracket \\sigma \\rrbracket )$ .", "Since $ is $ 01$, the measure~$$ is upper semi-computable.", "However, for every $$, we have$$\\nu (\\sigma )=\\nu (\\Lambda )-\\sum _{{\|\\tau \|=\|\\sigma \|\\\\\\tau \\ne \\sigma }} \\nu (\\tau )$$and since $ ()$ is computable (it is equal to $ ($), this shows that $ ()$ is also lower semi-computable uniformly in~$$, therefore $$ is computable.", "It is then easy to computably build by induction a sequence of strings $ 0 1 ...$ with $ \|i\|=i$ and such that $ (i) () 4-i >0$.", "In particular $ i= $.", "Therefore, the sequence $ X$ extending all~$ i$ is computable and must be an element of $ .", "Lemma 3.5 If $ is a perfect thin $ 01$ class, then it is $ tt$-negligible.$ First, observe that if a thin class $\\mathcal {P}$ contains a computable member $X$ , then since $\\lbrace X\\rbrace $ is a $\\Pi ^0_1$ subclass of $\\mathcal {P}$ , there is some $\\sigma \\in 2^{<\\mathbb {N}}$ such that $\\lbrace X\\rbrace =\\mathcal {P}\\cap \\llbracket \\sigma \\rrbracket $ .", "Thus, $X$ is isolated in $\\mathcal {P}$ .", "It thus follows that a perfect thin $\\Pi ^0_1$ class contains no computable members.", "Now, for the sake of contradiction, suppose that there exists a computable measure $\\mu $ such that $\\mu (>q$ for some positive rational $q$ .", "Identifying $2^\\mathbb {N}$ with the unit interval $[0,1]$ in the usual way, let $\\alpha = \\sup \\big \\lbrace r \\in [0,1] \\cap \\mathbb {Q}\\, : \\, \\mu ([0,r]) < q \\big \\rbrace .$ Since $ \\mu ([0,r]) < q$ is a $\\Sigma ^0_1$ predicate in $r$ , $\\alpha $ is a lower semi-computable real.", "Moreover, since $ has no computable member, it contains no atom of $$, and thus the function $ x ([0,x])$ is continuous.", "Therefore, by definition of $$, $ ([0,))=([0,])=q$.", "We now consider two cases.", "\\\\$ Case 1: The real $\\alpha $ is dyadic.", "In this case, $[0,\\alpha ]$ is a $\\Pi ^0_1$ class of measure $q$ , which by Lemma REF implies that $[0,\\alpha ]$ has a computable member, which contradicts our above observation.", "Case 2: The real $\\alpha $ is not dyadic.", "Consider the class $[\\alpha ,1]$ , which is a $\\Pi ^0_1$ subclass of $ since $$ is lower-semicomputable.", "By thinness of $ , there exists a dyadic number $\\beta \\ge \\alpha $ (thus $\\beta > \\alpha $ since $\\alpha $ is not dyadic) such that $[\\alpha ,1]=[\\beta ,1]$ .", "This means that $(\\alpha ,\\beta ) = \\emptyset $ , and thus, $\\mu ([0,\\beta ])=\\mu ([0,\\alpha ))=q$ .", "Thus we are back in Case 1 with $\\beta $ in place of $\\alpha $ , and we get a contradiction in the same way." ], [ "Depth and $\\mathit {tt}$ -depth", "Members of negligible and $\\mathit {tt}$ -negligible classes are difficult to produce, in the sense that their members cannot be computed from random oracles with positive probability.", "Given a $\\Pi ^0_1$ class $\\mathcal {P}$ , we can instead consider the probability of producing an initial segment of some member of $\\mathcal {P}$ .", "Thus, by looking at local versions of negligibility and $\\mathit {tt}$ -negligibility, i.e., given in terms of initial segments of members of the classes in question, we obtain the notions of depth and $\\mathit {tt}$ -depth, respectively.", "To compute an initial segment of a member of a $\\Pi ^0_1$ class $, we must represent~$ as the collection of paths through a tree $T\\subseteq 2^{<\\mathbb {N}}$ .", "Recall from $§$ that there are two primary ways to do so.", "First, for any $\\Pi ^0_1$ class $, there is a computable tree~$ T$ such that $ [T]=, i.e., $ consists of precisely the infinite paths through~$ T$.", "Second, given a $ 01$ class~$ 2N$, the \\emph {canonical co-c.e.\\ tree associated to $ } is the tree consisting of the strings $\\sigma $ that are prefixes of some path of $ (by the standard compactness argument, this is a co-c.e.\\ set of strings).", "It is this latter formulation that we use in the definition of depth (a choice that is justified by Proposition \\ref {prop:no-deep-comp-tree} below).$ Recall our convention from $§$REF : if $S$ is a set of strings, then $\\mathbf {M}(S)$ and $\\mathbf {m}(S)$ are defined to be $\\sum _{\\sigma \\in S} \\mathbf {M}(\\sigma )$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma )$ , respectively.", "Similarly, if $\\mu $ is a measure, $\\mu (S)$ denotes $\\sum _{\\sigma \\in S} \\mu (\\sigma )$ .", "Furthermore, recall from $§$REF that given a tree $T\\subseteq 2^{<\\mathbb {N}}$ , $T_n$ denotes the set of all members of $T$ of length $n$ .", "We now define the central notion of this paper, namely the notion of deep $\\Pi ^0_1$ class.", "Depth strengthens the notion of negligibility.", "It is easy to see that a class $2^\\mathbb {N}$ of canonical co-c.e.", "tree $T$ is negligible if and only if $\\mathbf {M}(T_n)$ converges to 0 as $n$ grows without bound.", "When this convergence to 0 is effective, $ is said to be deep.$ Definition 4.1 Let $2^\\mathbb {N}$ be a $\\Pi ^0_1$ class and $T$ its associated co-c.e.", "tree.", "(i) $ is \\emph {deep} if there is some computable order~$ h$ such that $ M(Tn) < 2-h(n)$ for all~$ n$.\\item [(ii)] $ is $\\mathit {tt}$ -deep if for every computable measure $\\mu $ there exists a computable order $h$ such that $\\mu (T_n) < 2^{-h(n)}$ .", "Our choice of the term “deep\" is due to the similarity between the notions of a deep class and that of a logically deep sequence, introduced by Bennett in .", "Logically deep sequences are highly structured and thus it is difficult to produce initial segments of a deep sequence via probabilistic computation; deep classes can thus be seen as an analogue of logically deep sequences for $\\Pi ^0_1$ classes.", "Remark 4.2 Note that depth and can be equivalently defined as follows: $ is deep if for some computable function~$ f$ one has $ M(Tf(k)) < 2-k$ for all~$ k$.", "Indeed, if $ M(Tn)<2-h(n)$, then, setting $ f=h-1$, we have $ M(Tf(k)) < 2-n$, and conversely, if $ M(Tf(n)) < 2-n$, we can assume that~$ f$ is increasing and then taking $ h=f-1$, we have $ M(Tn)<2-h(n)$ (here we use the fact that $ M(Tn)$ is non-increasing in $ n$).$ The same argument shows that $ is $ tt$-deep if and only if for every computable measure~$$, there exists a computable function~$ f$ such that for all~$ n$, $ (Tf(n)) < 2-n$.", "\\\\$ Remark 4.3 Another alternative way to define depth is to use $\\mathbf {m}$ instead of $\\mathbf {M}$ : a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree~$ T$ is deep if and only if there is a computable order~$ h$ such that $ m(Tn) < 2-h(n)$ for all~$ n$, if and only if there is a computable function~$ f$ such that $ m(Tf(n))< 2-n$ for all~$ n$.", "Indeed, the following inequality holds for all strings~$$ (see for example \\cite {Gacs-notes}):$$\\mathbf {m}(\\sigma ) \\le ^* \\mathbf {M}(\\sigma ) \\le ^* \\mathbf {m}(\\sigma )/ \\mathbf {m}(\|\\sigma \|)$$Thus, for~$ h$ a computable order, if $ M(Tn) < 2-h(n)$ for all~$ n$, we also have $ m(Tn) < 2-h(n)$ for all~$ n$.", "Conversely, if $ m(Tn) < 2-h(n)$, then $ m(Tf(n)) < 2-n$ for $ f=h-1$, and by the above inequality,$$\\mathbf {M}(T_{f(n)}) \\le ^* 2^{-n}/ \\mathbf {m}(f(n)) \\le ^* 2^{-n} \\cdot n^2.$$Thus, taking $ g(n)=f(2n+c)$ for some large enough constant~$ c$, we get $ M(Tg(n)) < 2-n$, and thus $ is deep.", "Since every $\\Pi ^0_1$ class $ is the set of paths through a computable tree, why can^{\\prime }t we simply define depth and $ tt$-depth in terms of this tree and not the canonical co-c.e.\\ tree associated to $ ?", "In the case of depth, there are two reasons to restrict to the canonical co-c.e.", "trees associated to $\\Pi ^0_1$ classes.", "First, the idea behind a deep class is that it is difficult to produce initial segments of some member of the class.", "In general, for a $\\Pi ^0_1$ class $, any computable tree $ T$ contains non-extendible nodes, and so if we have a procedure that can compute these non-extendible nodes of $ T$ with high probability, this tells us nothing about the difficulty of computing the extendible nodes of~$ T$.$ Second, if we were to use any tree $T$ representing a $\\Pi ^0_1$ class $ in the definition of depth, then depth would become a void notion, by the following proposition.$ Proposition 4.4 If $ is a non-empty $ 01$ class and~$ T$ is a computable tree such that $ [T]$, then there is no computable order~$ f$ such that $ m(Tf(n)) < 2-n$.$ If $T$ is a computable infinite tree and $f$ a computable order, there is a computable sequence $(\\sigma _n)_{n \\in \\mathbb {N}}$ of strings such that $\\sigma _n \\in T_{f(n)}$ for every $n$ .", "Thus $\\mathbf {m}(T_{f(n)}) \\ge \\mathbf {m}(\\sigma _n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2.$ By contrast with the notion of depth, for the definition of $\\mathit {tt}$ -depth, it does not matter whether we work with the canonical co-c.e.", "tree associated to $ or some computable tree $ T$ such that $ [T]$ (a direct consequence of Theorem~\\ref {prop:carac-tt-negl}).$ An important question concerns the relationship between depth and negligibility (and $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility).", "Clearly every deep ($\\mathit {tt}$ -deep) $\\Pi ^0_1$ class is negligible ($\\mathit {tt}$ -negligible), but does the converse hold?", "In the case of $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility, the answer is positive.", "We also identify two other equivalent formulations of $\\mathit {tt}$ -depth.", "Theorem 4.1 Let $ be a $ 01$ class.", "The following are equivalent:\\\\$ (i)$ $ is $\\mathit {tt}$ -deep.", "$(ii)$ $ is $ tt$-negligible.", "\\\\$ (iii)$ For every computable measure~$$, $ contains no $\\mu $ -Kurtz random element.", "$(iv)$ For every computable measure $\\mu $ , $ contains no $$-Martin-Löf random element.$ $(i) \\Rightarrow (ii)$ .", "Let $T$ be the canonical co-c.e.", "tree associated to $.", "If $ (Tn) < 2-h(n)$ for some computable order $ h$, then $ (=(n Tn) = 0$.", "\\\\$ (ii) (iii)$.", "This follows directly from the definition of Kurtz randomness.\\\\$ (iii) (iv)$.", "This follows from the fact that Martin-Löf randomness implies Kurtz randomness.\\\\$ (iv) (ii)$.", "If $ (>0$ for some computable measure $$, then $ must contain some $\\mu $ -Martin-Löf random element since the set of $\\mu $ -Martin-Löf random sequences has $\\mu $ -measure 1.", "$(ii) \\Rightarrow (i)$ .", "Suppose that $ is $ tt$-negligible and let $$ be a computable measure.", "Let~$ T$ be the co-c.e.\\ tree associated to~$ .", "By $\\mathit {tt}$ -negligibility, $\\mu (=0$ , or equivalently, $\\mu (T_n)$ tends to 0.", "Since the $T_n$ are co-c.e.", "sets of strings, $\\mu (T_n)$ is upper semi-computable uniformly in $n$ .", "Thus, given $k$ , it is possible to effectively find an $n$ such that $\\mu (T_n) < 2^{-k}$ .", "Setting $f(k)=n$ , we have a computable function $f$ such that $\\mu (T_{f(k)})<2^{-k}$ for all $k$ , therefore $ is $ tt$-deep by Remark~\\ref {rem:deep-order}.$ Significantly, negligibility and depth do not coincide when $\\mathit {tt}$ -reducibility is replaced by Turing reducibility.", "This is due to a fundamental aspect of depth, namely that it is not — unlike negligibility — invariant under Turing equivalence.", "Suppose that $ and $ are two classes such that for every $X \\in there exists a $ Y such that $X \\equiv _T Y$ , and vice-versa.", "Then by Proposition REF , $ is negligible if and only if~$ is negligible.", "This invariance does not hold in general for deep classes, as the next proposition shows.", "Theorem 4.2 For any $\\Pi ^0_1$ class $, there is a $ 01$ class~$ such that: the elements of $ are the same as the elements of~$ , modulo deletion of a finite prefix (which in particular guarantees that the elements of $ and $ have the same Turing degrees); and $ is not deep.$ Let $ be a $ 01$ class.", "Let $ T$ be a computable tree such that $ [T]=, and let $S$ be the canonical co-c.e.", "tree associated to $.", "Consider the tree~$ U$ obtained by appending a copy of~$ S$ to each terminal node of~$ T$.", "Formally,$$U = T \\cup \\lbrace \\sigma ^ \\frown \\tau \\mid \\sigma \\in T_{term} ~\\text{and}~ \\tau \\in S\\rbrace ,$$where~$ Tterm$ is the set of terminal nodes of~$ T$, which is a computable set.", "One can readily verify that $ U$ is the canonical co-c.e.\\ tree associated to some $ 01$ class.", "Indeed, suppose $$ is a node of~$ U$.", "Then,\\begin{itemize}\\item either \\rho \\in T, in which case \\rho is either a prefix of some~X\\in [T] (and thus~X\\in [U]), or \\rho is a prefix of a terminal node \\rho ^{\\prime }, which can then be extended to some~(\\rho ^{\\prime })^\\frown X\\in [U], where X\\in [S]; or\\item \\rho is of the form \\sigma ^\\frown \\tau , with \\sigma \\in T_{term} and \\tau \\in S. Since S is a canonical co-c.e.\\ tree, all its nodes extend to an infinite path, and thus \\sigma ^\\frown \\tau has an extension \\sigma ^\\frown X\\in [U] for some path~X\\in [S].\\end{itemize}$ The fact that $U$ is co-c.e.", "follows directly from the fact that $T_{term}$ is computable and $S$ is co-c.e.", "Let $ be the $ 01$ class whose canonical co-c.e.\\ tree is~$ U$.", "Then the elements of~$ are either elements of $[T]=, or are of the form $ X$ for some finite string~$$ and $ X , which gives us the first part of the conclusion.", "Finally, the canonical co-c.e.", "tree of $ contains an infinite computable tree, namely, the tree~$ T$, and therefore, by the argument as in the proof of Proposition~\\ref {prop:no-deep-comp-tree}, $ is not deep.", "It is now straightforward to get a $\\Pi ^0_1$ class that is negligible but not deep: it suffices to take a negligible class $ and apply the above theorem.", "The resulting class~$ is also negligible as its elements have the same Turing degrees as the elements of $, but it is not deep.", "\\\\$ The following summarizes the implications between the different concepts introduced above for $\\Pi ^0_1$ classes.", "depth $\\Rightarrow $ negligibility $\\Rightarrow $ $\\mathit {tt}$ -negligibility $\\Leftrightarrow $ $\\mathit {tt}$ -depth $\\Rightarrow $ having no computable member Computational limits of randomness As we have seen, negligible $\\Pi ^0_1$ classes (and thus deep classes) have the property that one cannot compute a member of them with positive probability.", "Although some random sequences can compute a member in any negligible $\\Pi ^0_1$ class (namely the Martin-Löf random sequences of $\\mathsf {PA}$ degree, as sequences of $\\mathsf {PA}$ degree compute a member of every $\\Pi ^0_1$ class), by the definition of negligibility, almost every random sequence fails to compute a member of a negligible class.", "Similarly, one cannot $\\mathit {tt}$ -compute a member of a $\\mathit {tt}$ -negligible $\\Pi ^0_1$ class with positive probability.", "The definition of $\\mathit {tt}$ -negligibility implies that the random sequences that can compute a member of a $\\mathit {tt}$ -negligible class form a set of Lebesgue measure zero.", "In this section, we specify a precise level of randomness at which computing a member of a $\\mathit {tt}$ -negligible, negligible, or deep class fails.", "First, we consider the case for $\\mathit {tt}$ -negligible classes.", "Theorem 5.1 If $X\\in 2^\\mathbb {N}$ is Kurtz random, it cannot $\\mathit {tt}$ -compute any member of a $\\mathit {tt}$ -negligible class.", "Let $ be $ tt$-negligible $ 01$ class and $$ be a $ tt$-functional.", "The set $ -1($ is a $ 01$ class that, by $ tt$-negligibility, has Lebesgue measure~$ 0$.", "Thus, it contains no Kurtz random.$ A similar proof can be used to prove the following: Theorem 5.2 If $X\\in 2^\\mathbb {N}$ is weakly 2-random, it cannot compute any member of a negligible class.", "Let $ be negligible $ 01$ class, $$ be a Turing functional, and $ T$ a computable tree such that $ [T]$.", "The set $ -1($ is a $ 02$ class, since$$X\\in \\Phi ^{-1}(\\;\\text{if and only if}\\;(\\forall k)(\\exists \\sigma )(\\exists n)[\\sigma \\in T\\;\\&\\;\|\\sigma \|=k\\;\\&\\; \\sigma \\preceq \\Phi ^{X {\\upharpoonright }n}]$$By negligibility, $ -1($ has Lebesgue measure~$ 0$ and thus contains no weakly 2-random sequence.$ Our next result, despite its simplicity, is probably the most interesting of this section.", "It will help us unify a number of theorems that have appeared in the literature.", "These are theorems of form ($$ ) If $X$ is difference random, then it cannot compute an element of $,$ where $ is a given $ 01$ class.", "Theorem~\\ref {thm:diff-pa} is an example of such a theorem, with $ the class of consistent completions of $\\mathsf {PA}$ .", "The same result has been obtained with $ the class of shift-complex sequences (Khan \\cite {Khan2013}), the set of compression functions (Greenberg, Miller, Nies \\cite {GreenbergMN-ip}), and the set $ DNCq$ functions for some orders~$ q$ (Miller, unpublished).", "We will give the precise definition of these classes in $ §$\\ref {sec:examples} but the important fact is that all of these classes are deep, and indeed, showing the depth of a $ 01$ class is sufficient to obtain a theorem of the form ($ $).", "$ Theorem 5.3 If a sequence $X$ is difference random, it cannot compute any member of a deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with associated co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ M(Tf(n)) < 2-n$.", "Let~$ X$ be a sequence that computes a member of $ via a Turing functional $\\Phi $ .", "Let $\\mathcal {Z}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace $ The set $\\mathcal {Z}_n$ can be written as the difference $\\mathcal {U}_n \\setminus n$ of two effectively open sets (uniformly in $n$ ) with $\\mathcal {U}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\rbrace $ and $n= \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\notin T_{f(n)}\\rbrace $ Moreover, by definition of the semi-measure induced by $\\Phi $ , $\\lambda (\\mathcal {Z}_n) \\le \\lambda _\\Phi (T_{f(n)}) \\le ^* \\mathbf {M}(T_{f(n)}) < 2^{-n}.$ The sequence $(\\mathcal {Z}_n)_{n\\in \\mathbb {N}}$ thus yields a difference test.", "Therefore, the sequence $X$ , which by assumption belongs to all $\\mathcal {Z}_n$ , is not difference random.", "We remark that the converse does not hold: i.e., there is a class $ such that (i) $ is not deep but (ii) no difference random real can compute an element of $.", "Indeed, take a deep class~$ and apply Theorem REF to get a class $ that is not deep but whose members have the same Turing degrees as the elements of~$ .", "Thus, no difference random real can compute an element of $.$ Depth, negligibility, and mass problems In this section we discuss depth and negligibility in the context of mass problems, i.e., in the context of the Muchnik and Medvedev reducibility.", "Both Muchnik and Medvedev reducibility are generalizations of Turing reducibility.", "Whereas Turing reducibility is defined in terms of a pair of sequences, both Muchnik and Medevedev reducibility are defined in term of a pair of collections of sequences.", "In what follows, we will consider these two reducibilities when restricted to $\\Pi ^0_1$ subclasses of $2^\\mathbb {N}$ .", "We will follow the notation of the survey , to which we refer the reader for a thorough exposition of mass problems in the context of $\\Pi ^0_1$ classes.", "Let $2^\\mathbb {N}$ be $\\Pi ^0_1$ classes.", "We say that $ is \\emph {Muchnik reducible} to $ , denoted $_w, if for every $ X there exists a Turing functional $\\Phi $ such that (i) $X\\in \\mathsf {dom}(\\Phi )$ and $\\Phi (X)\\in .", "Moreover, $ is Medvedev reducible to $, denoted $ s, if $ is Muchnik reducible to $ via a single Turing functional, i.e., there exists a Turing functional $\\Phi $ such that (i) $\\mathsf {dom}(\\Phi )$ and $\\Phi (\\subseteq .$ Just as Turing reducibility gives rise to a degree structure, we can define degree structures from $\\le _w$ and $\\le _s$ .", "We say that $ and $ are Muchnik equivalent (resp.", "Medvedev equivalent), denoted $_w (resp.\\ $ s) if and only if $_w and $ w (resp.", "$_s and $ s).", "The collections of Muchnik and Medvedev degrees given by the equivalence classes under $\\equiv _w$ and $\\equiv _s$ are denoted $\\mathcal {E}_w$ and $\\mathcal {E}_s$ , respectively.", "Both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are lattices, unlike the Turing degrees, which only form an upper semi-lattice.", "We define the meet and join operations as follows.", "Given $\\Pi ^0_1$ classes $2^\\mathbb {N}$ , $\\sup ($ is the $\\Pi ^0_1$ class $\\lbrace X\\oplus Y:X\\in \\&\\;Y\\in $ .", "Furthermore, we define $\\inf ($ to be the $\\Pi ^0_1$ class $\\lbrace 0^\\frown X: X\\in \\cup \\lbrace 1^\\frown Y:Y\\in $ .", "One can readily check that the least upper bound of $ and $ in $\\mathcal {E}_w$ is the Muchnik degree of $\\sup ($ while their greatest lower bound is the Muchnik degree $\\inf ($ , and similarly for $\\mathcal {E}_s$ .", "Recall that a filter $\\mathcal {F}$ in a lattice $(\\mathcal {L},\\le ,\\inf ,\\sup )$ is a subset that satisfies the following two conditions: (i) for all $x,y\\in \\mathcal {L}$ , if $x\\in \\mathcal {F}$ and $x\\le y$ , then $y\\in \\mathcal {F}$ , and (ii) for all $x,y\\in \\mathcal {F}$ , $\\inf (x,y)\\in \\mathcal {F}$ .", "The goal of this section is to study the role of depth and negligibility in the structures $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let us start with an easy result.", "Theorem 6.1 The collection of negligible $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let $ and $ be negligible $\\Pi ^0_1$ classes.", "By Proposition REF , we have that $\\lambda ((^{\\le _T})=0$ and $\\lambda ((^{\\le _T})=0$ .", "But since $\\inf (^{\\le _T}=(^{\\le _T}\\cup (^{\\le _T},$ it follows that $\\lambda (\\inf (^{\\le _T})=0$ , which shows that $\\inf ($ is negligible.", "Thus, the degrees of negligible classes in both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are closed under $\\inf $ .", "Let $\\mathcal {D}$ be non-negligible $\\Pi ^0_1$ and $\\mathcal {C}\\le _w\\mathcal {D}$ .", "For each $i$ , we define $i:=\\lbrace X\\in X\\in \\mathsf {dom}(\\Phi _i)\\;\\&\\;\\Phi _i(X)\\in .$ Since $_w, it follows that $ ii$.", "Furthermore, we have $ (T=(ii)T=i(i)T$.", "Since $ is non-negligible, we have $0<\\lambda ((^{\\le _T})\\le \\sum _i\\lambda \\big (({i})^{\\le _T}\\big ),$ and thus $\\lambda (({k})^{\\le _T})>0$ for some $k$ .", "But since $\\Phi _{k}({k})\\subseteq , it follows that $ ((T)>0$, thus $ is non-negligible.", "Thus, negligibility is closed upwards under $\\le _w$ (and a fortiori, under $\\le _s$ as well).", "Remark 6.1 Simpson proved in that in $\\mathcal {E}_s$ , the complement of the filter of negligible classes is in fact a principal ideal, namely the ideal generated by the class $\\inf (\\mathcal {PA},2\\mathcal {RAN})$ , where $2\\mathcal {RAN}$ is the class of 2-random sequences, i.e.", "the sequences that are Martin-Löf random relative to $\\emptyset ^{\\prime }$ .", "For the next two theorems, we need the following fact.", "Fact 6.2 Let $ and $ be $\\Pi ^0_1$ classes such that $_s.", "Then there is a total Turing functional $$ such that $ (.", "This holds because for every $\\Pi ^0_1$ class $ and every Turing functional~$$ that is total on $ , there is a functional $\\Psi $ that is total on $2^\\mathbb {N}$ and coincides with $\\Phi $ on $.$ Theorem 6.2 The collection of deep $\\Pi ^0_1$ classes forms a filter in $\\mathcal {E}_s$ .", "Let $ and $ be deep $\\Pi ^0_1$ classes with associated co-c.e.", "trees $S$ and $T$ , respectively.", "Moreover, let $g$ and $h$ be computable orders such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ and $\\mathbf {M}(T_n) < 2^{-h(n)}$ .", "We define $f(n)=\\min \\lbrace g(n),h(n)\\rbrace $ , which is clearly a computable order.", "It follows immediately that $\\mathbf {M}(S_n)\\le 2^{-f(n)}$ and $\\mathbf {M}(T_n) < 2^{-f(n)}$ .", "Now the co-c.e.", "tree associated with $\\inf ($ is $R=\\lbrace 0^\\frown \\sigma :\\sigma \\in S\\rbrace \\cup \\lbrace 1^\\frown \\tau :\\tau \\in T\\rbrace .$ Consider the class $\\inf ($ .", "Setting $0^\\frown S_n=\\lbrace 0^\\frown \\sigma :\\sigma \\in S_n\\rbrace $ and $1^\\frown T_n=\\lbrace 1^\\frown \\sigma :\\sigma \\in T_n\\rbrace $ for each $n$ , the co-c.e.", "tree of $\\inf ($ is $0^\\frown S_n \\cup 1^\\frown T_n$ .", "Moreover, $\\mathbf {M}(0^\\frown S_n \\cup 1^\\frown T_n) = \\mathbf {M}(0^\\frown S_n) + \\mathbf {M}(1^\\frown T_n) \\le ^* 2^{-f(n)} + 2^{-f(n)}$ which shows that $\\inf ($ is deep.", "Next, suppose that $\\mathcal {C}$ is a deep $\\Pi ^0_1$ class and $\\mathcal {D}$ is a $\\Pi ^0_1$ class satisfying $\\mathcal {C} \\le _s \\mathcal {D}$ via the Turing functional $\\Psi $ , which we can assume to be total by Fact REF .", "Let $S$ and $T$ be the co-c.e.", "trees associated to $ and $ , respectively, and let $g$ be a computable order such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ .", "Since $\\Psi $ is total, its use is bounded by some computable function $f$ .", "It follows that for every $\\sigma \\in T_{f(n)}$ , there is some $\\tau \\in S_n$ such that $(\\sigma ,\\tau )\\in \\Psi $ .", "Thus $\\llbracket T_{f(n)}\\rrbracket \\subseteq \\Psi ^{-1}(\\llbracket S_n\\rrbracket )$ .", "Now let $\\Phi $ be a universal Turing functional such that $\\mathbf {M}=\\lambda _\\Phi $ .", "Then: $\\mathbf {M}(T_{f(n)})=\\lambda (\\Phi ^{-1}( T_{f(n)}))\\le \\lambda (\\Phi ^{-1}(\\llbracket T_{f(n)}\\rrbracket )\\le \\lambda (\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket )))\\le \\lambda _{\\Psi \\circ \\Phi }(S_n),$ where the last inequality holds because $\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket ))=(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq \\mathsf {dom}(\\Psi \\circ \\Phi )$ and hence $(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq (\\Psi \\circ \\Phi )^{-1}(S_n)$ , $\\lambda _{\\Psi \\circ \\Phi }\\le ^* \\mathbf {M}$ .", "Thus we have $\\mathbf {M}(T_{f(n)})\\le \\lambda _{\\Psi \\circ \\Phi }(S_n)\\le ^* \\mathbf {M}(S_n)\\le 2^{-g(n)}.$ Thus, $\\mathbf {M}(T_{f \\circ g^{-1}(n)}) \\le 2^{-n}$ , which shows that $ is deep.$ The invariance of the notion of depth under Medvedev-equivalence is of importance for the next section.", "There, we prove that certain classes of objects, which are not necessarily infinite binary sequences, are deep.", "To do so, we fix a certain encoding of these objects by infinite binary sequences and prove the depth of the corresponding encoded class.", "By the above theorem, the particular choice of encoding is irrelevant: if the class is deep for an encoding, it will be deep for another encoding, as long as switching from the first encoding to the second one can be done computably and uniformly.", "One could ask whether we have a similar result in the lattice $\\mathcal {E}_w$ .", "However, Theorem REF shows that depth is not invariant under Muchnik equivalence: if we apply this theorem to a deep class $, we get a $ 01$ class~$ which is clearly Muchnik equivalent to $ but is not deep itself.", "Thus depth does not marry well with Muchnik reducibility and is only a `Medvedev notion^{\\prime }.", "\\\\$ More surprisingly, $\\mathit {tt}$ -depth is a `Muchnik notion' as well as a `Medvedev notion' and the $\\mathit {tt}$ -deep classes form a filter in both lattices.", "Theorem 6.3 The collection of $\\mathit {tt}$ -deep $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Suppose that $_w and $ is not $\\mathit {tt}$ -deep.", "By definition, this means that $ has positive $$-measure for some computable probability measure~$$.", "Let $ {Uk}k N$ be the universal $$-Martin-Löf test and define $ Rk$ to be the complement of $ Uk$ (which is a $ 01$ class).", "Since $ (Rk) > 1- 2-k$ for every $ k$, there must be a~$ j$ such that $ (Rj) > 0$ and in particular $ Rj = $.", "By the hyperimmune-free basis theorem, there is some $ XRj$ of hyperimmune-free Turing degree.", "Since $ w , $X$ must compute some element $Y$ of $.", "But since~$ X$ is of hyperimmune-free degree, $ X$ in fact $ tt$-computes $ Y$, i.e., $ (X)=Y$ for some total functional~$$.$ Since $X$ is $\\mu $ -Martin-Löf random, by the preservation of randomness theorem (see, for instance, Theorem 3.2 in ), $Y$ is Martin-Löf random with respect to the computable measure $\\mu _\\Psi $ defined by $\\mu _\\Psi (\\sigma ):=\\mu (\\Psi ^{-1}(\\sigma ))$ for every $\\sigma $ .", "Thus, by Proposition REF , $ is not $ tt$-deep.", "This shows, a fortiori, that if $ s and $ is not $ tt$-deep, then $ is not $\\mathit {tt}$ -deep.", "Thus $\\mathit {tt}$ -depth is closed upwards in $\\mathcal {E}_w$ and $\\mathcal {E}_s$ and in particular is compatible with the equivalence relations $\\equiv _s$ and $\\equiv _w$ .", "Next, suppose that $ and $ are $\\mathit {tt}$ -deep classes but that $\\inf ($ is not $\\mathit {tt}$ -deep (recall that the $\\inf $ operator is the same for both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ ).", "Then by Theorem REF , $\\inf ($ contains a sequence $X$ that is $\\mu $ -Martin-Löf random for some computable measure $\\mu $ .", "Then for $i=0,1$ we define computable measures $\\mu _i$ such that $\\mu _i(\\sigma )=\\mu (i^\\frown \\sigma )$ for every $\\sigma $ .", "It is routine to check that $Y$ is $\\mu _i$ -Martin-Löf random if and only if $i^\\frown Y$ is $\\mu $ -Martin-Löf random for $i=0,1$ .", "Since $X=i^\\frown Z$ for some $i=0,1$ and $Z\\in 2^\\mathbb {N}$ , it follows that $Z$ is $\\mu _i$ -Martin-Löf random.", "But then $Z$ is contained in either $ or $ , which contradicts our hypothesis that $ and $ are both $\\mathit {tt}$ -deep.", "Examples of deep $\\Pi ^0_1$ classes In this section, we provide a number of examples of deep $\\Pi ^0_1$ classes that naturally occur in computability theory and algorithmic randomness.", "We give a uniform treatment of all these classes, i.e., we give a generic method to prove the depth of $\\Pi ^0_1$ classes.", "Consistent completions of Peano Arithmetic As mentioned in $§$, Jockusch and Soare proved in that the $\\Pi ^0_1$ class $\\mathcal {PA}$ of consistent completions of $\\mathsf {PA}$ is negligible.", "However, as shown by implicitly by Levin in and Stephan in , $\\mathcal {PA}$ is also deep.", "We will reproduce this result here.", "Following both Levin and Stephan, we will use fact that the class of consistent completions of $\\mathsf {PA}$ is Medvedev equivalent to the class of total extensions of a universal, partial-computable $\\lbrace 0,1\\rbrace $ -valued function.", "Thus, by showing the latter class is deep, we thereby establish that the former class is deep (via Theorem REF ).", "Theorem 7.1 (Levin , Stephan ) Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of all $\\lbrace 0,1\\rbrace $ -valued partial computable functions.", "Let $u$ be a function that is universal for this collection, e.g., defined by $u(\\langle e,x\\rangle ) = \\phi _e(x)$ .", "Then the class $ of total extensions of~$ u$ is a deep $ 01$ class.$ We build a partial computable function $\\phi _e$ , whose index we know in advance by the recursion theorem.", "This means that we control the value of $u(\\langle e,x\\rangle )$ for all $x$ .", "First, we partition $\\mathbb {N}$ into consecutive intervals $I_1, I_2, ...$ such that we control $2^{k+1}$ values of $u$ inside $I_k$ .", "For each $k$ in parallel, we define $\\phi _e$ on $I_k$ as follows.", "Step 1: Wait for a stage $s$ such that the set $E_k[s]=\\lbrace \\sigma \\mid \\sigma {\\upharpoonright }I_k ~ \\text{extends}~ u[s] {\\upharpoonright }I_k \\rbrace $ is such that $\\mathbf {M}(E_k)[s] \\ge 2^{-k}$ .", "Step 2: Find a $y \\in I_k$ that we control and on which $u_s$ is not defined.", "Consider the two “halves\" $E^0_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=0\\rbrace $ and $E^1_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=1\\rbrace $ of $E_k[s]$ .", "Note that either $\\mathbf {M}(E^0_k[s])\\ge 2^{-k-1}$ or $\\mathbf {M}(E^1_k)[s] \\ge 2^{-k-1}$ .", "If the first holds, set $u(y)[s+1]=1$ , otherwise set $u(y)[s+1]=0$ .", "Go back to Step 1.", "The co-c.e.", "tree $T$ associated to the class $ is the set of strings~$$ such that~$$ is an extension of $ u \|\|$.", "The construction works because every time we pass by Step 2, we remove from~$ T[s]$ a set $ Eik[s]$ (for some $ i{0,1}$ and $ k,sN$) such that $ M(Eik)[s]>2-k-1$.", "Therefore, Step 2 can be executed at most $ 2k+1$ times, and by the definition of $ Ik$ we do not run out of values~$ y Ik$ on which we control~$ u$.", "Therefore, the algorithm eventually reaches Step 1 and waits there forever.", "Setting $ f(k)=(Ik)$, this implies that the $ M$-weight of the set $ {f(k) extends u}$ is bounded by $ 2-k$, or equivalently,$$\\mathbf {M}(T_{f(k)}) < 2^{-k},$$which proves that~$ is deep.", "The proof provided here gives us a general template to prove the depth of a $\\Pi ^0_1$ class.", "First of all, the definition of the $\\Pi ^0_1$ class should allow us to control parts of it in some way, either because we are defining the class ourselves or because, as in the above proof, the definition of the class involves some universal object which we can assume to partially control due to the recursion theorem.", "All the other examples of deep $\\Pi ^0_1$ class we will see below belong to this second category.", "Let us take a step back and analyze more closely the structure of the proof of Theorem REF .", "Given a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree $ T$, the proof consists of the following steps.\\\\$ (1) For a given $k$ , we identify a level $N=f(k)$ at which we wish to ensure $\\mathbf {M}(T_N) < 2^{-k}$ .", "The choice of $N$ will depend on the particular class $.", "\\\\$ (2) Next we implement a two-step strategy to ensure that $\\mathbf {M}(T_N)<2^{-k}$ .", "Such a strategy will be called a $k$ -strategy.", "(2.1) First, we wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] > 2^{-k}$ .", "(2.2) If at some stage $s$ this occurs, we remove one or several of the nodes of $T[s]$ at level $N$ such that the total $\\mathbf {M}[s]$ -weight of these nodes is at least $\\delta (k)$ for a certain function $\\delta $ , and then go back to Step 2.", "(3) Each execution of Step 2 (removing nodes from $T[s]$ for some $s$ ) comes at a cost $\\gamma (k)$ for some $k$ , and we need to make sure that we do not go over some maximal total cost $\\Gamma (k)$ throughout the execution of the $k$ -strategy.", "In the above example, $\\delta (k)=2^{-k-1}$ and the only cost for us is to define one value of $u(y)$ out of the $2^{k+1}$ we control, so we can, for example, set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ .", "By definition of $\\mathbf {M}$ , the $k$ -strategy can only go through Step 2 at most $1/\\delta (k)$ times, and thus the total cost of the $k$ -strategy will be at most $\\gamma (k)/\\delta (k)$ .", "All we have to do is to make sure that $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ (which is the case in the above example).", "Again, we have some flexibility on the choice of $N=f(k)$ , so it will suffice to choose an appropriate $N$ to ensure $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ .", "In some cases, there will not be a predefined maximum for each $k$ , but rather a global maximum $\\Gamma $ that the sum of the the costs of the $k$ -strategies should not exceed, i.e., we will want to have $\\sum _k \\gamma (k)/\\delta (k) \\le \\Gamma $ .", "Let us now proceed to more examples of $\\Pi ^0_1$ classes.", "Shift-complex sequences Definition 7.1 Let $\\alpha \\in (0,1)$ and $c>0$ .", "(i) $\\sigma \\in 2^{<\\mathbb {N}}$ (resp.", "$X\\in 2^\\mathbb {N}$ ) is said to be $(\\alpha ,c)$ -shift complex if $\\mathrm {K}(\\tau ) \\ge \\alpha \|\\tau \| - c$ for every substring $\\tau $ of $\\sigma $ (resp.", "of $X$ ).", "(ii) $X\\in 2^\\mathbb {N}$ is said to be $\\alpha $ -shift complex if it is $(\\alpha ,c)$ -shift complex for some $c$ .", "The very existence of $\\alpha $ -shift complex sequences is by no means obvious.", "Such sequences were first constructed by Durand et al.", "who showed that there exist $\\alpha $ -shift complex sequences for all $\\alpha \\in (0,1)$ .", "It is easy to see that for every computable pair $(\\alpha ,c)$ , the class of $(\\alpha ,c)$ -shift complex sequences is a $\\Pi ^0_1$ class.", "Rumyantsev proved that the class of $(\\alpha ,c)$ -complex sequences is always negligible, but in fact, his proof essentially shows that it is even deep.", "The cornerstone of Rumyantsev's theorem is the following lemma.", "It relies on an ingenious combinatorial argument that we do not reproduce here.", "We refer the reader to for the full proof.", "Lemma 7.2 (Rumyantsev ) Let $\\beta \\in (0,1)$ .", "For every rational $\\eta \\in (0,1)$ and integer $d$ , there exist two integers $n$ and $N$ , with $n<N$ , such that the following holds.", "For every probability distribution $P$ on $\\lbrace 0,1\\rbrace ^N$ , there exist finite sets of strings $A_n, A_{n+1}, ..., A_N$ such that for all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements; and the $P$ -probability that a sequence $X$ has some substring in $\\cup _{i=n}^N A_i$ is at least $1-\\eta $ .", "Moreover, $n$ and $N$ can be effectively computed from $\\eta $ and $d$ and can be chosen to be arbitrarily large.", "Once $n$ and $N$ are fixed, the sets $A_i$ can be computed uniformly in $P$ .", "This is Lemma 6 of (Rumyantsev does not explicitly state that the conclusion holds for all computable probability measures, but nothing in his proof makes use of a particular measure).", "Theorem 7.2 For any computable $\\alpha \\in (0,1)$ and integer $c>0$ , the $\\Pi ^0_1$ class of $(\\alpha ,c)$ -shift complex sequences is deep.", "Let $ be the $ 01$ class of $ (,c)$-shift complex sequences and let~$ T$ be its canonical co-c.e.\\ tree.", "We shall build a discrete semi-measure~$ m$ whose coding constant~$ e$ we know in advance.", "This means that whenever we will set a string~$$ to be such that $ m() > 2-\|\| + c + e +1 $, then automatically we will have $ m() > 2-\|\| + c +1$, and thus $ K() < \|\| - c$.", "This will \\textit {de facto} remove $$ from~$ T$.$ Let us now turn to the construction.", "First, we pick some $\\beta $ such that $0 < \\beta < \\alpha $ .", "For each $k$ , we apply the above Lemma REF to $\\beta $ and $\\eta = 1/2$ to obtain a pair $(n,N)$ with the above properties as described in the statement of the lemma (we will also make use of the fact that $n$ can be chosen arbitrarily large, see below).", "Then the two step strategy is the following: Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Let $P$ be the (computable) probability distribution on $\\lbrace 0,1\\rbrace ^N$ whose support is contained in $T_N[s]$ and such that for all $\\sigma \\in T_N[s]$ , $P(\\sigma )=2^k \\cdot \\mathbf {M}(\\sigma )[s]$ .", "By Lemma REF , we can compute a collection of finite sets $A_i$ such that: (i) For all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements.", "(ii) The $P$ -probability that a sequence $X$ has some substring in $F = \\cup _{i=n}^N A_i$ is at least $1-\\eta =1/2$ .", "Thus, $\\mathbf {M}(T_N \\cap F)[s] > 2^{-k-1}$ .", "Then, for each $i$ and each $\\sigma \\in A_i$ , we ensure, increasing $m$ if necessary, that $m(\\sigma ) > 2^{-\\alpha \|\\sigma \| + c + e +1 }$ .", "As explained above, this ensures that all strings in $F$ are removed from $T_N$ at that stage, and therefore we have removed a $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ .", "Then we go back to Step 1.", "To finish the proof, we need to make sure that this algorithm does not cost us too much.", "The constraint here is that we need to make sure that $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau ) \\le 1$ , so we are trying to stay under a global cost of $\\Gamma =1$ .", "For a given $k$ , our cost at each execution of Step 2 is the increase of $m$ on the strings in $F$ .", "The total $m$ -weight we add at during one such execution of Step 2 is at most $\\gamma (k) = \\sum _{i=n}^N \|A_i\| \\cdot 2^{-\\alpha i + c + e +1 } \\le \\sum _{i=n}^N 2^{(\\beta -\\alpha ) i + c+ e + 1} \\le \\frac{2^{(\\beta -\\alpha ) n + c+ e + 1}}{1-2^{\\beta -\\alpha }}$ But since $n=n(k)$ can be chosen arbitrarily large, therefore, with an appropriate choice of $n$ , we can make $\\gamma (k) \\le 2^{-2k-2}$ .", "Therefore, since the $k$ -strategy executes Step 2 at most $1/\\delta (k)$ times, we have $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau )\\le \\sum _{k\\in \\mathbb {N}} \\gamma (k)/\\delta (k) \\le \\sum _{k\\in \\mathbb {N}} 2^{-2k-2}/2^{-k-1} \\le 1$ and therefore $m$ is indeed a discrete semi-measure.", "DNC$_q$ functions Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of partial computable functions from $\\mathbb {N}$ to $\\mathbb {N}$ .", "Let $J$ be the universal partial computable function defined as follows.", "For all $(e,x) \\in \\mathbb {N}^2$ : $J(2^e(2x+1))=\\phi _e(x)$ The following notion was studied in .", "Definition 7.3 Let $q: \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable order.", "The set $\\mathcal {DNC}_q$ is the set of total functions $f: \\mathbb {N}\\rightarrow \\mathbb {N}$ such that for all $n$ : (i) $f(n) \\ne J(n)$ (where this condition is trivially satisfied if $J(n)$ is undefined), and (ii) $f(n) < q(n)$ .", "We should note that this is a slight variation of the standard definition of diagonal non-computability, which is formulated in terms of the condition (i$^{\\prime }$ ) $f(n)\\ne \\phi _n(n)$ instead of the condition (i) given above.", "However, this condition would make the class $\\mathcal {DNC}_q$ and the results below dependent on the particular choice of enumeration of the partial computable functions, while with condition $(i)$ , the results of the section are independent of the chosen enumeration.", "Note that for every fixed computable order $q$ , the class $\\mathcal {DNC}_q$ is a $\\Pi ^0_1$ subset of $\\mathbb {N}^\\mathbb {N}$ , but due to the bound $q$ , $\\mathcal {DNC}_q$ can be viewed as a $\\Pi ^0_1$ subset of $2^\\mathbb {N}$ .", "Whether the class $\\mathcal {DNC}_q$ is deep turns out to depend on $q$ .", "Indeed, we have the following interesting dichotomy theorem.", "Theorem 7.3 Let $q$ be a computable order.", "(i) If $\\sum _n 1/q(n) < \\infty $ , then $\\mathcal {DNC}_q$ is not $\\mathit {tt}$ -negligible.", "(ii) If $\\sum _n 1/q(n) = \\infty $ , then $\\mathcal {DNC}_q$ is deep.", "Part $(i)$ is a straightforward adaptation of Kučera's proof that every Martin-Löf random element computes a DNC function.", "Pick an $X \\in 2^\\mathbb {N}$ at random and use it as a random source to randomly pick the value of $f(i)$ uniformly among $\\lbrace 0,...,q(i)-1\\rbrace $ , independently of the other values of $f$ .", "The details are as follows.", "First, we can assume that $q(n)$ is a power of 2 for all $n$ .", "Indeed, since for $q^{\\prime } \\le q$ the class $\\mathcal {DNC}_{q^{\\prime }}$ is contained in $\\mathcal {DNC}_{q}$ , so if we take $q^{\\prime }(n)$ to be the largest power of 2, we have that $q^{\\prime }$ is computable, $q^{\\prime } \\le q$ , and $\\sum _n 1/q(n) < \\infty $ because $q^{\\prime }$ is equal to $q$ up to factor 2.", "Now, set $q(n)=2^{r(n)}$ .", "Split $\\mathbb {N}$ into intervals where interval $I_n$ has length $r(n)$ .", "One can now interpret any infinite binary sequence $X$ as a funtion $f_X: \\mathbb {N}\\rightarrow \\mathbb {N}$ , where $X(n)$ is the index of $X {\\upharpoonright }I_n$ in the lexicographic ordering of strings of length $r(n)$ .", "For $X$ taken at random with respect to the uniform measure, the event $f_X(n) = \\phi _n(n)$ has probability at most $1/q(n)$ and is independent of all such events for $n^{\\prime } \\ne n$ .", "Thus the total probability over $X$ such that $X(n)\\ne \\phi _n(n)$ for all $n$ is at least $\\prod _n (1-1/q(n))$ an expression that is positive if and only if $\\sum 1/q(n)< \\infty $ , which is satisfied by hypothesis.", "Thus, the class $\\mathcal {DNC}_q$ , encoded as above, has positive uniform measure, and thus is not $\\mathit {tt}$ -negligible.", "For $(ii)$ , let $q$ be a computable order such that $\\sum _n 1/q(n) = \\infty $ .", "Let $T$ be the canonical co-c.e.", "binary tree in which the elements of $\\mathcal {DNC}_q$ are encoded.", "By the recursion theorem, we will build a partial recursive function $\\phi _e$ whose index $e$ we know in advance and therefore will be able to define $J$ on the set of values $D = \\lbrace 2^e (2x+1) : x \\in \\mathbb {N}\\rbrace $ .", "Note that the set $D$ has positive (lower) density in $\\mathbb {N}$ : for every interval $I$ of length at least $2^{e+1}$ , $\|D \\cap I\| \\ge 2^{-e-2} \|I\|$ .", "This in particular implies that $\\sum _{n \\in D} 1/q(n) = \\infty .$ To show this, we appeal to Cauchy's condensation test, according to which for any positive non-increasing sequence $(a_n)$ , $\\sum _n a_n < \\infty $ if and only if $\\sum _k 2^k a_{2^k} < \\infty $ .", "Since the sum $\\sum _n1/q(n)$ diverges, so does $\\sum _k 2^k/q(2^k)$ and thus $\\sum _{n \\in D} \\frac{1}{q(n)} \\ge \\sum _{k>e} \\frac{\|D \\cap \\lbrace 2^{k-1},...,2^k-1\\rbrace \|}{q(2^k)} \\ge \\sum _{k>e} \\frac{2^{-e-2} \\cdot 2^{k-1}}{q(2^k)} = \\infty .$ Now that we have established that $\\sum _{n \\in D} 1/q(n) = \\infty $ , we remark that this is equivalent to having $\\prod _{n \\in D} (1-1/q(n)) = 0$ .", "Thus, we can effectively partition $D$ into countably many finite sets $D_j$ such that $\\prod _{n \\in D_j} (1-1/q(n)) < 1/2$ .", "We are ready to describe the construction.", "For each $k$ , reserve some finite collection $D_{j_1}, D_{j_2}, ..., D_{j_{k+1}}$ of sets $D_j$ .", "Let $N$ be a level of the binary tree $T$ sufficiently large such that the encoding of each path $f$ up to length $N$ is enough to recover the values of $f$ on $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , which can be found effectively in $k$ .", "The $k$ -strategy then works as follows.", "Initialisation.", "Set $i=1$ .", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Since each $\\sigma \\in T_N[s]$ can be viewed as a function from some initial segment of $\\mathbb {N}$ that contains $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , take the first value $x \\in D_{j_i}$ .", "For all $\\sigma \\in T_N$ , $\\sigma (x) < q(x)$ , thus by the pigeonhole principle there must be at least one value $v < q(x)$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x)=v\\rbrace )[s] \\ge 2^{-k}/q(x).$ Set $J(x)$ to be the least such value $v$ .", "This thus gives $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v\\rbrace )[s] < 2^{-k}(1-1/q(x)).$ Now take another $x^{\\prime }$ in $D_{j_i}$ on which $J$ has not been defined yet.", "By the same reasoning, there must be a value $v^{\\prime } < q(x^{\\prime })$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v \\wedge \\sigma (x^{\\prime }) \\ne v^{\\prime } \\rbrace ) < 2^{-k}(1-1/q(x))(1-1/q(x^{\\prime }))[s].$ We then set $J(x^{\\prime })$ to be equal to $v^{\\prime }$ .", "Continuing in this fashion, we can assign all the values of $J$ on $D_{j_i}$ in such a way that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\forall x \\in D_{j_i}, \\ \\sigma (x) \\ne J(x) \\rbrace \\big )[s] < 2^{-k} \\prod _{n \\in D_j} (1-1/q(n)) < 2^{-k}/2.$ Then we increment $i$ by 1 and go back to Step 1.", "Once again, at each execution of Step 2 we remove an $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ , since ensuring $\\sigma (x)=J(x)$ for some $x$ immediately ensures that no extension of $\\sigma $ is in $\\mathcal {DNC}_q$ .", "Moreover, each execution of Step 2 requires us to define $J$ on all values in some $D_{j_i}$ for $1\\le j\\le k+1$ .", "That is, each such execution costs us one $D_{j_i}$ , so as in the case of consistent completions of $\\mathsf {PA}$ , we can set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ , which ensures that $\\gamma (k)/\\delta (k)\\le \\Gamma (k)$ .", "Therefore, $\\mathcal {DNC}_q$ is a deep $\\Pi ^0_1$ class.", "Finite sets of maximally complex strings For a constant $c>0$ , generating a long string $\\sigma $ of high Kolmogorov complexity, for example $\\mathrm {K}(\\sigma ) > \|\\sigma \|-c$ , can be easily achieved with high probability if one has access to a random source (just repeatedly flip a fair coin and output the raw result).", "One can even use this technique to generate a sequence of strings $\\sigma _1, \\sigma _2, \\sigma _3, \\ldots , $ such that $\|\\sigma _n\| = n$ and $\\mathrm {K}(\\sigma _n) \\ge n-c$ .", "Indeed, the measure of the sequences $X$ such that $\\mathrm {K}(X {\\upharpoonright }n) \\ge n -d$ for all $n$ is at least $2^{-d}$ , and thus the $\\Pi ^0_1$ class of such sequences of strings $(\\sigma _1,\\sigma _2,...,)$ of high complexity (encoded as elements of $2^\\mathbb {N}$ ) is not even $\\mathit {tt}$ -negligible.", "The situation changes dramatically if one wishes to obtain many distinct complex strings at a given length.", "Let $\\ell : \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable increasing function, $f,d:\\mathbb {N}\\rightarrow \\mathbb {N}$ be any computable functions, and $c>0$ .", "Consider the $\\Pi ^0_1$ class $\\mathcal {K}_{f,\\ell ,d}$ whose members are sequences $\\vec{F}=(F_1, F_2, F_3, ...)$ where for all $i$ , $F_i$ is a finite set of $f(i)$ strings $\\sigma $ of length $\\ell (i)$ such that $\\mathrm {K}(\\sigma ) \\ge \\ell (i)-d(i)$ .", "Note once again that $\\mathcal {K}_{f,\\ell ,d}$ can be viewed, modulo encoding, as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ .", "Theorem 7.4 For all computable functions $f,\\ell ,d$ such that $f(i)/2^{d(i)}$ takes arbitrarily large values and $\\ell $ is increasing, the class $\\mathcal {K}_{f,\\ell ,d}$ is deep.", "Let $T$ be the canonical co-c.e.", "tree associated to the class $\\mathcal {K}_{f,\\ell ,d}$ .", "Just like in the proof of Theorem REF , we will build a discrete semi-measure $m$ whose coding constant $e$ we know in advance and thus, by setting $m(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) + e +1 }$ , we will force $\\mathbf {m}(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) +1}$ , and thus $\\mathrm {K}(\\sigma ) < \|\\sigma \| - d(\|\\sigma \|)$ , which will consequently remove from $T$ every sequence of sets $(F_i)_{i\\in \\mathbb {N}}$ such that $\\sigma $ belongs to some $F_i$ .", "Fix a $k$ , and let us pick some well-chosen $i=i(k)$ , to be determined later.", "For readability, let $f=f(i(k))$ , $\\ell =\\ell (i(k))$ and $d=d(i(k))$ .", "Let $N$ be the level of $T$ at which the first $i$ sets $F_1\\cdots F_i$ of the sequence $\\vec{F}$ of $\\mathcal {K}_{f,\\ell ,d}$ are encoded.", "The $k$ -strategy does the following.", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: The members of $T_N[s]$ consist of finite sequences $F_1, ..., F_i$ of sets of strings where $F_i$ contains $f$ distinct strings of length $\\ell $ .", "By the pigeonhole principle, there must be a string $\\sigma $ of length $\\ell $ such that $\\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N : \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ Indeed, there is a total $\\mathbf {M}$ -weight $2^{-k}$ of possible sets $F_i$ , each of which contains $f$ strings of length $\\ell $ , thus $\\sum _{\|\\sigma \|=\\ell } \\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N \\mid \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f$ , and there are $2^\\ell $ strings of length $\\ell $ in total.", "Effectively find such a string $\\sigma $ , and set $m(\\sigma ) > 2^{-\\ell + d + e +1 }$ .", "By the choice of $\\sigma $ , this causes an $\\mathbf {M}$ -weight of at least $2^{-k} \\cdot f \\cdot 2^{-\\ell }$ of nodes of $T_N[s]$ to leave the tree.", "Then go back to Step 1.", "As in the previous examples, it remains to conduct the “cost analysis\".", "The resource here again is the weight we are allowed to assign to $m$ , which has to be bounded by $\\Gamma =1$ .", "At each execution of Step 2 of the $k$ -strategy, our cost is the increase of $m$ , which is at most of $\\gamma (k)=2^{-\\ell + d + e +1 }$ , while an $\\mathbf {M}$ -weight of at least $\\delta (k) = 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ leaves the tree $T_N$ .", "This gives a ratio $\\gamma (k)/\\delta (k) = 2^{d+e+k+1}/f$ , which bounds the total cost of the $k$ -strategy, so we want $\\sum _k \\gamma (k)/\\delta (k) \\le 1$ , i.e., we want: $\\sum _k 2^{d(i(k)) + e +k+1}/f(i(k))\\le 1$ By assumption, $f(i)/2^{d(i)}$ takes arbitrarily large values, it thus suffices to choose $i(k)$ such that $2^{d(i(k)) + e+1}/f(i(k)) \\le 2^{-2k-1}$ .", "Compression functions Our next example of a family of deep classes is given in terms of compression functions, which were introduced by Nies, Stephan, and Terwijn in to provide a characterization of 2-randomness (that is, $\\emptyset ^{\\prime }$ -Martin-Löf randomness) in terms of incompressibility.", "Definition 7.4 A $\\mathrm {K}$ -compression function with constant $c>0$ is a function $g: 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ such that $g \\le \\mathrm {K}+c$ and $\\sum _\\sigma 2^{-g(\\sigma )} \\le 1$ .", "We denote by $\\mathcal {CF}_c$ be the class of compression functions with constant $c$ .", "The condition $g(\\sigma ) \\le \\mathrm {K}(\\sigma )+c$ implies that $g(\\sigma ) \\le 2\|\\sigma \| + c +c^{\\prime } $ for some fixed constant $c^{\\prime }$ , and therefore $\\mathcal {CF}_c$ can be seen as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ , modulo encoding the functions as binary sequences.", "Of course $\\mathcal {CF}_c$ contains the function $\\mathrm {K}$ itself so it is a non-empty class.", "We now show the following.", "Theorem 7.5 For all $c\\ge 0$ , the class $\\mathcal {CF}_c$ is deep.", "Although we could give a direct proof following the same template as the previous examples, we will instead show that for all $c$ , we have $\\mathcal {K}_{f,\\ell ,d} \\le _s \\mathcal {CF}_c$ for some computable $f, \\ell , d$ (where $\\mathcal {K}_{f,\\ell ,d}$ is the class we defined in the previous section) such that $\\ell $ is increasing and $f/2^d$ takes arbitrarily large values, which by Theorem REF implies the depth of $\\mathcal {CF}_c$ .", "Let $g \\in \\mathcal {CF}_c$ .", "For all $n$ , since $\\sum _{\|\\sigma \|=n} 2^{-g(\\sigma )} \\le 1$ , there are at most $2^{n-1}$ strings of length $n$ such that $g(\\sigma ) < n - 1$ , and thus at least $2^{n-1}$ strings $\\sigma $ of length $n$ such that $g(\\sigma ) \\ge n - 1$ .", "Since $g \\in \\mathcal {CF}_c$ , for each $\\sigma $ such that $g(\\sigma ) \\ge \|\\sigma \| - 1$ , we also have $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Thus, given $g \\in \\mathcal {CF}_c$ as an oracle we can find, for each $n \\ge 3$ , $2^{n-1}$ strings $\\sigma $ of length $n$ such that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Setting $f(i)=2^{i+2}$ , $\\ell (i)=i+3$ and $d(i)=c+1$ , we have uniformly reduced $\\mathcal {K}_{f,\\ell ,d}$ to $\\mathcal {CF}_{c}$ .", "Since $d$ is a constant function it is obvious that $f(i)/2^d$ is unbounded, thus $\\mathcal {K}_{f,\\ell ,d}$ is deep (by Theorem REF ) and by Theorem REF so is $\\mathcal {CF}_c$ .", "The above result is not tight: the proof in fact shows that if $d$ is not a constant function but is such that $2^n/d(n)$ takes arbitrarily large values, then the class of functions $g$ such that $g(\\sigma ) \\le \\mathrm {K}(\\sigma ) + d(\|\\sigma \|)$ for all $\\sigma $ is a deep class.", "A notion related to $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ In $§$, we discuss lowness for randomness notions.", "Here we look at a dual notion, highness for randomness, specifically the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , whose precise characterization is an outstanding open question in algorithmic randomness.", "We shall see that this class is tightly connected to the notion of depth.", "Definition 7.5 A sequence $A\\in 2^\\mathbb {N}$ is in the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ if $\\mathsf {MLR}\\subseteq \\mathsf {CR}^A$ .", "The class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ itself not $\\Pi ^0_1$ (as it is closed under finite change of prefixes), but all its members have a “deep property\".", "Bienvenu and Miller proved that when $A$ is in $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , then for every c.e.", "set of strings $S$ such that $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ , $A$ computes a martingale $d$ such that $d(\\Lambda )=1$ and for some fixed rational $r>0$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ .", "It is straightforward to show that real-valued martingales can be approximated by dyadic-valued martingales with arbitrary precision; in particular one can assume that $d(\\sigma )$ is dyadic for all $\\sigma $ (and thus can be coded using $f(\|\\sigma \|)$ bits for some computable function $f$ ), still keeping the property $\\sigma \\in S \\Rightarrow d(\\sigma )>1+r$ .", "For a well-chosen $S$ , such martingales form a deep class.", "Theorem 7.6 Let $S$ be the set of strings $\\sigma $ such that $\\mathrm {K}(\\sigma ) < \|\\sigma \|$ (so that we have $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ ).", "Let $g$ be a computable function and $r>0$ be a rational number.", "Define the class $\\mathcal {W}_{g,r}$ to be the set of dyadic-valued martingales $d$ such that $d(\\Lambda )=1$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ and such that for all $\\sigma $ , $d(\\sigma )$ can be coded using $g(\|\\sigma \|)$ bits.", "Then $\\mathcal {W}_{g,r}$ can be viewed as a $\\Pi ^0_1$ class of $2^\\mathbb {N}$ , and this class is deep.", "Again, we are going to show the depth of $\\mathcal {W}_{g,r}$ by a Medvedev reduction from $\\mathcal {K}_{f,\\ell ,h}$ for some increasing computable function $\\ell $ and computable functions $f$ and $h$ such that $f/2^h$ is unbounded.", "More precisely, we will take $f(n)=n$ , $\\ell (n)=n + c$ (for a well-chosen $c$ given below) and $h(n)=0$ .", "Now suppose we are given a martingale $d \\in \\mathcal {W}_{f,r}$ .", "For any given length $n$ , we have, by the martingale fairness condition, $\\sum _{\\sigma : \|\\sigma \|=n} d(\\sigma ) = 2^n.$ It follows that there are at most $2^n/(1+r)$ strings $\\sigma $ of length $n$ such that $d(\\sigma )>1+r$ , and therefore at least $2^n-2^n/(1+r)$ strings such that $d(\\sigma ) \\le 1+r$ .", "Having oracle access to $d$ , such strings can be effectively found and listed.", "Since $2^n-2^n/(1+r) > n -c$ for all $n$ and some fixed constant $c$ , for each $n$ we can use $d$ to list $n$ strings $\\sigma _1,\\cdots ,\\sigma _n$ of length $n+c$ such that $d(\\sigma _i) \\le 1+r$ for $1\\le i\\le n$ .", "But by definition of $d$ , $d(\\sigma )\\le 1+r$ implies that $\\sigma \\notin S$ , which further implies that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \|$ .", "This shows that $\\mathcal {W}_{g,r}$ is above $\\mathcal {K}_{f,\\ell ,d}$ in the Medvedev degrees.", "By Theorem REF , $\\mathcal {K}_{f,\\ell ,h}$ is deep, and hence by Theorem REF , $\\mathcal {W}_{g,r}$ is deep as well.", "The examples of deep classes provided in this section, combined with Theorem REF , give us the results mentioned page : If $X$ is a difference random sequence, it does not compute any shift-complex sequence (Khan) it does not compute any $\\mathcal {DNC}_q$ function when $q$ is a computable order such that $\\sum _n 1/q(n) = \\infty $ (Miller) it does not compute any compression function (Greenberg, Miller, Nies) it is not $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ (Greenberg, Miller, Nies) The fact that a difference random cannot compute large sets of complex strings (in the sense of Theorem REF ) appears to be new.", "Lowness and depth Various lowness notions have been well-studied in algorithmic randomness, where a lowness notion is given by a collection of sequences that are in some sense computationally weak.", "Many lowness notions take the following form: For a relativizable collection $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , we say that $A$ is low for $\\mathcal {S}$ if $\\mathcal {S}\\subseteq \\mathcal {S}^A$ .", "For instance, if we let $\\mathcal {S}=\\mathsf {MLR}$ , then the resulting lowness notion consists of the sequences that are low for Martin-Löf random, a collection we write as $\\mathrm {Low}(\\mathsf {MLR})$ .", "Lowness notions need not be given in terms of relativizable classes.", "For instance, if we let $\\mathbf {m}^A$ be a universal $A$ -lower semi-computable discrete semi-measure, we can defined $A$ to be low for $\\mathbf {m}$ if $\\mathbf {m}^A(\\sigma )\\le ^\\mathbf {m}(\\sigma )$ for every $\\sigma $ .", "In addition, some lowness notions are not given in terms of relativization, such as the notion of $\\mathrm {K}$ -triviality, where $A\\in 2^\\mathbb {N}$ is $\\mathrm {K}$ -trivial if and only if $\\mathrm {K}(A{\\upharpoonright }n)\\le \\mathrm {K}(n)+O(1)$ (where we take $n$ to be $1^n$ ).", "Surprisingly, we have the following result (see for a detailed survey of results in this direction).", "Theorem 8.1 Let $A\\in 2^\\mathbb {N}$ .", "The following are equivalent: (i) $A\\in \\mathrm {Low}(\\mathsf {MLR})$ ; (ii) $A$ is low for $\\mathbf {m}$ ; (iii) $A$ is $\\mathrm {K}$ -trivial.", "The cupping problem, a longstanding open problem in algorithmic randomness involving $\\mathrm {K}$ -triviality, is to determine whether there exists a $\\mathrm {K}$ -trivial sequence $A$ and some Martin-Löf random sequence $X\\lnot \\ge _T\\emptyset ^{\\prime }$ such that $X\\oplus A\\ge _T\\emptyset ^{\\prime }$ .", "A negative answer was recently provided by Day and Miller : Theorem 8.2 (Day-Miller ) A sequence $A$ is K-trivial if and only if for every difference random sequence $X$ , $X \\oplus A \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Using the notion of depth, we can strengthen the Day-Miller result.", "First, we need to partially relativize the notion of depth: for $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $2^\\mathbb {N}$ is deep relative to $A$ if there is some computable order $f$ such that $\\mathbf {m}^A(T_{f(n)})\\le 2^{-n}$ if and only if there is some $A$ -computable order $g$ such that $\\mathbf {M}^A(T_{g(n)})\\le 2^{-n}$ (where $\\mathbf {M}^A$ is a universal $A$ -lower semi-computable continuous semi-measure).", "Theorem 8.3 Let $X$ be an incomplete Martin-Löf random sequence and $A$ be $K$ -trivial.", "Then $X \\oplus A$ does not compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with canonical co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ m(Tf(n)) 2-n$.", "Since~$ A$ is low for~$ m$, we have also have $ mA(Tf(n)) 2-n$, and thus $ is deep relative to $A$ .", "Let $X$ be a sequence such that $X \\oplus A$ computes a member of $ via a Turing functional $$ and suppose, for the sake of contradiction, that $ X$ is difference random.", "Let$$n = \\lbrace Z \\, : \\, \\Phi ^{Z \\oplus A} {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace .$$The set $ n$ can be written as the difference $ Un n$ of two $ A$-effectively open sets (uniformly in~$ n$) with $ Un = {Z : Z A f(n) }$ and $ n= {Z : Z A f(n) Tf(n)}$$ We can see the functional $Z \\mapsto \\Phi ^{Z \\oplus A}$ as an $A$ -Turing functional $\\Psi $ , and thus by the univerality of $\\mathbf {M}^A$ for the class of $A$ -lower semi-computable continuous semi-measures, we have $\\mathbf {M}^A \\ge ^* \\lambda _\\Psi $ .", "By definition of $n$ , we therefore obtain: $\\lambda (n) \\le \\lambda _\\Psi (T_{f(n)}) \\le ^* \\mathbf {M}^A(T_{f(n)}) < 2^{-n}.$ This shows that the sequence $X$ , which by assumption belongs to all $n$ , is not $A$ -difference random.", "It is, however, $A$ -Martin-Löf random as $A$ is low for Martin-Löf randomness.", "Relativizing Theorem REF to $A$ , this shows that $X \\oplus A \\ge _T A^{\\prime }$ .", "But this contradicts the Day-Miller theorem (Theorem REF ).", "As we cannot compute any members of a deep class by joining a Martin-Löf random sequence with a low for Martin-Löf random sequence, it is not unreasonable to ask if there is a notion of randomness $\\mathcal {R}$ such that we cannot $\\mathit {tt}$ -compute any members of a $\\mathit {tt}$ -deep class by joining an $\\mathcal {R}$ -random sequence with a low for $\\mathcal {R}$ sequence.", "We obtain a partial answer to this question using Kurtz randomness.", "From the discussion of lowness at the beginning of this section, we have $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $\\mathsf {KR}\\subseteq \\mathsf {KR}^A$ .", "Moreover, we define the class $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ to be the collection of sequences $A$ such that $\\mathsf {MLR}\\subseteq \\mathsf {KR}^A$ .", "Since $\\mathsf {MLR}\\subseteq \\mathsf {KR}$ , it follows that $\\mathrm {Low}(\\mathsf {KR})\\subseteq \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Miller and Greenberg obtained the following characterization of $\\mathrm {Low}(\\mathsf {KR})$ and $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Recall that $A\\in 2^\\mathbb {N}$ is computably dominated if every $f\\le _T A$ is dominated by some computable function.", "Theorem 8.4 (Greenberg-Miller ) Let $A\\in 2^\\mathbb {N}$ .", "(i) $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ if and only if $A$ is of non-DNC degree.", "(ii) $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ and computably dominated.", "For $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $ is \\emph {$ tt$-negligible relative to $ A$} if $ A(=0$ for every $ A$-computable measure $$.", "We first prove the following.$ Proposition 8.1 If $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ , then every deep $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible relative to $A$ .", "Let $ be a deep $ 01$ class and $ T$ its associated canonical co-c.e.\\ tree.", "Let $ ALow(MLR,KR)$, which by Theorem \\ref {thm:greenberg-miller-kr} (i) is equivalent to being of non-DNC degree.", "We appeal to a useful characterization of non-DNC degrees due to Hölzl and Merkle~\\cite {HolzlM2010}: $ A$ is of non-DNC degree if and only if it is \\emph {infinitely often c.e.\\ traceable} (hereafter, i.o.\\ c.e.\\ traceable).", "This means that there exists a computable order~$ h$, such that the following holds: for every total $ A$-computable function~$ s: NN$, there exists a family $ (Sn)nN$ of uniformly c.e.\\ finite sets such that $ \|Sn\|<h(n)$ for all~$ n$ and $ s(n) Sn$ for infinitely many~$ n$.$ Let $h$ be an order witnessing the i.o.", "c.e.", "traceability of $A$ .", "Since $ is deep, let~$ f$ be a computable function such that $ M(Tf(n)) < 2-2n/h(n)$.Suppose for the sake of contradiction that $ is not $\\mathit {tt}$ -negligible relative to $A$ , which means that there exists an $A$ -computable measure $\\mu ^A$ such that $\\mu ^A(>r$ for some rational $r>0$ .", "Let $s\\le _TA$ be the function that on input $n$ gives a rational lower-approximation, with precision $1/2$ , of the values of $\\mu ^A$ on all strings of length $f(n)$ (encoded as an integer).", "By this we mean that $s(n)$ gives us for all strings $\\sigma $ of length $f(n)$ a rational value $s(n,\\sigma )$ such that $\\mu ^A(\\sigma )/2 \\le s(n,\\sigma ) \\le \\mu ^A(\\sigma )$ .", "Let $(S_n)_{n\\in \\mathbb {N}}$ witness the traceability of $s$ , i.e., the $S_n$ 's are uniformly c.e., $\|S_n\|<h(n)$ for every $n$ , and $s(n) \\in S_n$ for infinitely many $n$ .", "We now build a lower semi-computable continuous semi-measure $\\rho $ as follows.", "For all $n$ , enumerate $S_n$ .", "For each member $z \\in S_n$ , interpret $z$ as a mass distribution $\\nu $ on the collection of strings of length $f(n)$ .", "Then, for each string $\\sigma $ of length $f(n)$ , increase $\\rho (\\sigma )$ (as well as strings comparable with $\\sigma $ in any way that ensures that $\\rho $ remains a semi-measure) by $2^{-n-1}\\nu (\\sigma )/h(n)$ .", "This has a total cost of $2^{-n-1}/h(n)$ , and since there are at most $h(n)$ elements in $S_n$ , the total cost at level $f(n)$ is at most $2^{-n-1}$ .", "Therefore the total cost of the construction of $\\rho $ is bounded by 1 and thus $\\rho $ is indeed a lower semi-computable continuous semi-measure Now, for any $n$ such that $s(n) \\in S_n$ (as there are infinitely many such $n$ ), $\\rho $ distributes an amount of at least $(\\mu ^A(T_{f(n)})/2) \\cdot (2^{-n-1}/h(n))$ on $T_{f(n)}$ , and since $\\mu ^A(T_n)>r$ , this gives $\\rho (T_{f(n)}) \\ge 2^{-n-O(1)}/h(n).$ However, we had assumed that $\\mathbf {M}(T_{f(n)}) \\le 2^{-2n}/h(n),$ for all $n$ .", "But since $\\rho \\le ^* \\mathbf {M}$ , we get a contradiction.", "The following result involves the notion of relative $\\mathit {tt}$ -reducibility.", "For a fixed $A\\in 2^\\mathbb {N}$ , a $\\mathit {tt}(A)$ -functional is a total $A$ -computable Turing functional.", "Equivalently, we can define a $\\mathit {tt}(A)$ -functional $\\Psi ^A$ in terms of a Turing functional $\\Phi $ as follows: Let $\\Phi $ be defined on all inputs of the form $X\\oplus A$ .", "Then we set $\\Psi ^A(X)=\\Phi (X\\oplus A)$ .", "Furthermore, one can show that there is an $A$ -computable bound on the use of $X$ in the computation (just as there is a computable bound in the use function for unrelativized $\\mathit {tt}$ -computations).", "Theorem 8.5 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class and $ ALow(KR)$.", "By Proposition \\ref {prop:low-for-tt-depth}, $ is also $\\mathit {tt}$ -deep relative to $A$ .", "Let $\\Phi ^A$ be a $\\mathit {tt}(A)$ -functional.", "The pre-image $ of $ under $\\Phi ^A$ is a $\\Pi ^0_1(A)$ class, which must be $\\mathit {tt}$ -deep relative to $A$ as well, by Theorem REF relativized to $A$ .", "Now, applying Proposition REF relativized to $A$ , $ contains no $ A$-Kurtz random sequence.", "But since $ A$ is low for Kurtz randomness, $ contains no Kurtz-random sequence as well.", "We now obtain a partial analogue of Theorem REF .", "Corollary 8.2 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $\\Phi $ be a $\\mathit {tt}$ -functional.", "Since $\\Phi $ is total, it is certainly total on all sequences of the form $X\\oplus A$ for $X\\in 2^\\mathbb {N}$ .", "Thus $\\Psi ^A(X)=\\Phi (X\\oplus A)$ is a $\\mathit {tt}(A)$ -functional.", "By Theorem REF , it follows that $\\Phi (X\\oplus A)$ cannot be contained in any deep class.", "Question 1 Does Corollary REF still hold if we replace “deep\" with “$\\mathit {tt}$ -deep\"?", "We can extend Theorem REF to the following result, which proceeds by almost the same proof, the details of which are left to the reader.", "Theorem 8.6 Let $X$ be Martin-Löf random and $A$ be $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "(in particular, $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class).", "Depth, mutual information, and the Independence Postulate In this final section, we introduce the notion of mutual information and apply it to the notion of depth.", "Roughly, what we prove is that every member of every deep class has infinite mutual information with Chaitin's $\\Omega $ , a Martin-Löf random sequence that encodes the halting problem.", "This generalizes a result of Levin's, that every consistent completion of $\\mathsf {PA}$ has infinite mutual information with $\\Omega $ .", "We conclude with a discussion of the Independence Postulate, a principle introduced by Levin to derive the statement that that no consistent completion of arithmetic is physically obtainable.", "The definition of mutual information First we review the definitions of Kolmogorov complexity of a pair and the universal conditional discrete semi-measure $\\mathbf {m}(\\cdot \\mid \\cdot )$ .", "Let $\\langle \\cdot ,\\cdot \\rangle :2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ be a computable bijection.", "Then we define $\\mathrm {K}(\\sigma ,\\tau ):=\\mathrm {K}(\\langle \\sigma ,\\tau \\rangle )$ .", "Similarly, we set $\\mathbf {m}(\\sigma ,\\tau ):=\\mathbf {m}(\\langle \\sigma ,\\tau \\rangle )$ .", "A conditional lower semi-computable discrete semi-measure $m(\\cdot \\mid \\cdot ):2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow [0,1]$ is a function satisfying $\\sum _\\sigma m(\\sigma \\mid \\tau )\\le 1$ for every $\\tau $ .", "Then $\\mathbf {m}(\\cdot \\mid \\cdot )$ is defined to be a universal conditional lower semi-computable discrete semi-measure, so that for every conditional lower semi-computable discrete semi-measure, there is some $c$ such that $m(\\sigma \\mid \\tau )\\le c\\cdot \\mathbf {m}(\\sigma \\mid \\tau )$ for every $\\sigma $ and $\\tau $ .", "Lastly, we define the conditional prefix-free Kolmogorov complexity $\\mathrm {K}(\\sigma \\mid \\tau )$ to be $\\mathrm {K}(\\sigma \\mid \\tau )=\\min \\lbrace \|\\xi \|:U(\\langle \\xi ,\\tau \\rangle )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free machine.", "The mutual information of two strings $\\sigma $ and $\\tau $ , denoted by $I(\\sigma :\\tau )$ , is defined by $I(\\sigma :\\tau ) = \\mathrm {K}(\\sigma ) + \\mathrm {K}(\\tau ) - \\mathrm {K}(\\sigma ,\\tau )$ or equivalently by $2^{I(\\sigma :\\tau )} = \\frac{\\mathbf {m}(\\sigma ,\\tau )}{\\mathbf {m}(\\sigma ) \\cdot \\mathbf {m}(\\tau )}.$ By the symmetry of information (see Gács ), we also have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} =^* \\frac{\\mathbf {m}(\\tau \\mid \\sigma ,\\mathrm {K}(\\sigma ))}{\\mathbf {m}(\\tau )}.$ Levin extends mutual information to infinite sequences by setting $2^{I(X:Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^X(\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}.$ Recall that Chaitin's $\\Omega $ can be obtained as the probability that a universal prefix-free machine will halt on a given input, that is, $\\Omega =\\sum _{U(\\sigma ){\\downarrow }}2^{-\|\\sigma \|}$ , where $U$ is a fixed universal prefix-free machine.", "Generalizing a result of Levin's from , we have: Theorem 9.1 Let $ be a $ 01$ class and $ T$ its associated co-c.e.\\ tree.", "Suppose~$ is deep, witnessed by a computable order $f$ such that $\\mathbf {m}(T_{f(n)}) < 2^{-n}$ .", "Then for every $Y \\in and all~$ n$,$$I\\Big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\Big ) \\ge n-O(\\log n).$$In particular,$$I(\\Omega :Y)=\\infty .$$$ Our proof follows the same idea Levin uses for consistent completions of $\\mathsf {PA}$ (see ), although some extra care is needed for arbitrary deep classes.", "Suppose for a given $n$ we have an exact description $\\tau $ of $T_{f(n)}$ ; that is, on input $\\tau $ , the universal machine outputs a code for the finite set $T_{f(n)}$ .", "By the definition of $f$ , $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\le 2^{-n}$ or equivalently $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\cdot 2^{n} \\le 1$ Therefore, the quantity $\\mathbf {m}(\\sigma ) \\cdot 2^{n} \\cdot \\mathbf {1}_{\\sigma \\in T_{f(n)}}$ is a discrete semi-measure, but it is not lower semi-computable since $T_{f(n)}$ is merely co-c.e.", "(and, in general, not c.e.", "by Proposition REF ).", "However, it is a lower semi-computable semi-measure relative to the exact description $\\tau $ of $T_{f(n)}$ .", "Thus, for every $\\sigma \\in T_{f(n)}$ , by the universality of $\\mathbf {m}(\\cdot \\mid \\tau )$ , $\\mathbf {m}(\\sigma \\mid \\tau ) \\ge ^* \\mathbf {m}(\\sigma ) \\cdot 2^n.$ By the symmetry of information, we have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} \\ge ^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau )}{\\mathbf {m}(\\sigma )} \\ge ^2^n,$ and hence $I(\\tau :\\sigma ) \\ge ^+ n$ .", "We would like to apply this fact to the case where $\\sigma = Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ .", "But this is not technically sufficient, as $\\Omega {\\upharpoonright }n$ does not necessarily contain enough information to exactly describe $T_{f(n)}$ .", "This is not an obstacle in Levin's argument for completions of $\\mathsf {PA}$ , but it is for arbitrary deep classes.", "However, $\\Omega {\\upharpoonright }n$ contains enough information to get a “good enough\" approximation of $T_{f(n)}$ .", "Let us refine the idea above: suppose now that $\\tau $ is no longer an exact description of $T_{f(n)}$ , but is a description of a set of strings $S$ of length $f(n)$ such that $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^ 2^{-n} n^2.$ Then, by following the same reasoning as above, we would have $I(\\sigma : \\tau ) \\ge ^+ n-2\\log n$ for all $\\sigma \\in S$ (and thus all $\\sigma \\in T_{f(n)}$ ).", "We shall prove that $\\Omega {\\upharpoonright }n$ contains enough information to recover such a set $S$ , thus proving the theorem.", "The real number $\\Omega $ is lower semi-computable and Solovay complete (see ).", "As a consequence, for every other lower semi-computable real $\\alpha $ , knowing the first $k$ bits of $\\Omega $ allows us to compute the first $k-O(1)$ bits of $\\alpha $ .", "For all $n$ , define: $a_n = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}}} \\mathbf {m}(\\sigma )$ and observe that $a_n$ is lower semi-computable uniformly in $n$ (because $T_{f(n)}$ is co-c.e.", "uniformly in $n$ ), and belongs to $[0,1]$ .", "Define now $\\alpha = \\sum _n \\frac{a_n}{n^2},$ which is a lower semi-computable real.", "Thus, knowing the first $n$ bits of $\\Omega $ gives us the first $n-O(1)$ bits of $\\alpha $ , i.e., an approximation of $\\alpha $ with precision $2^{-n}$ .", "In particular this gives us an approximation of $a_n$ with precision $2^{-n} \\cdot n^2 \\cdot O(1)$ , which we can assume to be a lower approximation, which we will write as $a^{\\prime }_n$ .", "Now, using $a^{\\prime }_n$ , one can enumerate $a_n$ until we find a stage $s_n$ such that $a_n[s_n] = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}[s_n]}} \\mathbf {m}(\\sigma )[s_n] \\ge a^{\\prime }_n.$ Since $\|a_n - a^{\\prime }_n\| \\le 2^{-n} \\cdot n^2 \\cdot O(1)$ , this implies $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}[s_n]} \\setminus T_{f(n)}}} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n^2.$ But recall from above that $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}}}} \\mathbf {m}(\\sigma ) \\le 2^{-n}.$ Combining these two facts, and taking $S$ to be the set $T_{f(n)}[s_n]$ , we have $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n ^2$ which establishes the first part of the theorem.", "To see that the second part of the statement follows from the first, take $\\sigma =Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ and observe that $\\mathbf {m}^Y(\\sigma ) =^* \\mathbf {m}^Y(n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ $\\mathbf {m}^\\Omega (\\tau ) =^* \\mathbf {m}^\\Omega (n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ and $2^{I(\\sigma :\\tau )}=2^n/n^{O(1)}$ (since $I(\\sigma :\\tau )\\ge ^+n-2\\log n$ as established above).", "Then we have $\\begin{split}2^{I(\\Omega :Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^\\Omega (\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}&\\ge \\sum _n \\mathbf {m}^\\Omega (Y{\\upharpoonright }f(n)) \\cdot \\mathbf {m}^Y(\\Omega {\\upharpoonright }n) \\cdot 2^{I(Y{\\upharpoonright }f(n),\\Omega {\\upharpoonright }n)}\\\\&\\ge \\sum _n 2^n/n^{O(1)}=\\infty .\\end{split}$ Remark 9.1 The converse of this theorem does not hold, i.e., there is a $\\Pi ^0_1$ class that is not deep but all of whose elements have infinite mutual information with $\\Omega $ .", "This follows from Theorem REF and the fact that having infinite mutual information with $\\Omega $ is a property that is invariant under addition or deletion of a finite prefix.", "Levin's proof of Theorem REF restricted to the particular case of completions of $\\mathsf {PA}$ is the mathematical part of a more general discussion, the other part of which is philosophical in nature.", "While Gödel's theorem asserts that no completion of $\\mathsf {PA}$ can be computably obtained, Levin's goal is to show that no completion of $\\mathsf {PA}$ can be obtained by any physical means whatsoever (computationally or otherwise), thus generalizing Gödel's theorem.", "Levin does not fully specify what he means by physically obtainable (the exact term he uses is “located in the physical world\"), but nonetheless he makes the following postulate, which he dubs the “Independence Postulate\": if $\\sigma $ is a mathematically definable string that has an $n$ bit description and $\\tau $ can be located in the physical world with a $k$ bit description, then for a fixed small constant $c$ , one has $I(\\sigma : \\tau ) < n+k+c$ .", "In particular, if one admits that some infinite sequences can be physically obtained, the Independence Postulate for infinite sequences says that if $X$ and $Y$ are two infinite sequences with $X$ mathematically definable and $Y$ physically obtainable, then $I(X:Y) < \\infty $ .", "Being $\\Delta ^0_2$ , $\\Omega $ is mathematically definable, and, as Levin shows, $I(\\Omega :Y)=\\infty $ for any completion $Y$ of $\\mathsf {PA}$ .", "Thus, assuming the Independence Postulate, no completion of $\\mathsf {PA}$ is physically obtainable.", "Our Theorem REF extends Levin's theorem and, assuming the Independence Postulate, shows that no member of a deep class (shift-complex sequences, compression functions, etc.)", "is physically obtainable.", "Of course, evaluating the validity of the Independence Postulate would require an extended philosophical discussion that would take us well beyond the scope of this paper.", "In any case, whether or not the reader accepts the Independence Postulate, Theorem REF is interesting in its own right.", "In fact, it is quite surprising because it seems to contradict the “basis for randomness theorem\" (see ), which states that if $X$ is a Martin-Löf random sequence and $\\mathcal {C}$ is a $\\Pi ^0_1$ class, then there exists a member $Y$ of $\\mathcal {C}$ such that $X$ is random relative to $Y$ .", "If a sequence $X$ is random relative to another sequence $Y$ , the intuition is that $Y$ “knows nothing about $X$ \", and thus one could conjecture that $I(X : Y) < \\infty $ .", "However, this cannot always be the case, since by Theorem REF , $I(\\Omega :Y) = \\infty $ for all members $Y$ of a deep $\\Pi ^0_1$ class $, even though $$ is random relative to some $ Y.", "This apparent paradox can be explained by taking a closer look at the definition of mutual information.", "Let $ be a deep $ 01$ class, whose canonical co-c.e.\\ tree~$ T$ satisfies $ m(Tf(n)) < 2-n$ for some computable function~$ f$.", "By Theorem~\\ref {thm:mutual-info} and the symmetry of information, for every $ Y we have $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big ) = I\\big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\big ) \\ge n-O(\\log n)$ where $k_n$ stands for $\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "Take a $Y \\in such that $$ is random relative to~$ Y$.", "It is well-known that a sequence~$ Z$ is random if and only if $ K(Z n n) n-O(1)$ (see for example Gács~\\cite {Gacs1980}).", "Applying this fact (relativized to~$ Y$) to $$, we have$$\\mathrm {K}^Y(\\Omega {\\upharpoonright }n \\mid n) \\ge n-O(1)$$and thus in particular that$$\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\ge n-O(1).$$Since $ K(n) n+O(n)$, it follows that$ $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\le O(\\log n)$ The only difference between (REF ) and (REF ) is the term $k_n=\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "But it makes a big difference, as one can verify that $\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n))-\\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big )\\ge n-O(\\log n).$ Informally, while $Y$ “knows nothing\" about $\\Omega $ , the complexity of its initial segments, seen as a function, does.", "In particular, the change of complexity caused by $k_n$ implies that $\\mathrm {K}(\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge n - O(\\log n)$ and thus $\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge 2^n/n^{O(1)}$ .", "Acknowledgements We would like to thank Noam Greenberg, Rupert Hölzl, Mushfeq Khan, Leonid Levin, Joseph Miller, André Nies, Paul Shafer, and Antoine Taveneaux for many fruitful discussions on the subject.", "Particular thanks go to Steve Simpson who provided very detailed feedback on the first arXiv version of this paper.", "Members of negligible and $\\mathit {tt}$ -negligible classes are difficult to produce, in the sense that their members cannot be computed from random oracles with positive probability.", "Given a $\\Pi ^0_1$ class $\\mathcal {P}$ , we can instead consider the probability of producing an initial segment of some member of $\\mathcal {P}$ .", "Thus, by looking at local versions of negligibility and $\\mathit {tt}$ -negligibility, i.e., given in terms of initial segments of members of the classes in question, we obtain the notions of depth and $\\mathit {tt}$ -depth, respectively.", "To compute an initial segment of a member of a $\\Pi ^0_1$ class $, we must represent~$ as the collection of paths through a tree $T\\subseteq 2^{<\\mathbb {N}}$ .", "Recall from $§$ that there are two primary ways to do so.", "First, for any $\\Pi ^0_1$ class $, there is a computable tree~$ T$ such that $ [T]=, i.e., $ consists of precisely the infinite paths through~$ T$.", "Second, given a $ 01$ class~$ 2N$, the \\emph {canonical co-c.e.\\ tree associated to $ } is the tree consisting of the strings $\\sigma $ that are prefixes of some path of $ (by the standard compactness argument, this is a co-c.e.\\ set of strings).", "It is this latter formulation that we use in the definition of depth (a choice that is justified by Proposition \\ref {prop:no-deep-comp-tree} below).$ Recall our convention from $§$REF : if $S$ is a set of strings, then $\\mathbf {M}(S)$ and $\\mathbf {m}(S)$ are defined to be $\\sum _{\\sigma \\in S} \\mathbf {M}(\\sigma )$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma )$ , respectively.", "Similarly, if $\\mu $ is a measure, $\\mu (S)$ denotes $\\sum _{\\sigma \\in S} \\mu (\\sigma )$ .", "Furthermore, recall from $§$REF that given a tree $T\\subseteq 2^{<\\mathbb {N}}$ , $T_n$ denotes the set of all members of $T$ of length $n$ .", "We now define the central notion of this paper, namely the notion of deep $\\Pi ^0_1$ class.", "Depth strengthens the notion of negligibility.", "It is easy to see that a class $2^\\mathbb {N}$ of canonical co-c.e.", "tree $T$ is negligible if and only if $\\mathbf {M}(T_n)$ converges to 0 as $n$ grows without bound.", "When this convergence to 0 is effective, $ is said to be deep.$ Definition 4.1 Let $2^\\mathbb {N}$ be a $\\Pi ^0_1$ class and $T$ its associated co-c.e.", "tree.", "(i) $ is \\emph {deep} if there is some computable order~$ h$ such that $ M(Tn) < 2-h(n)$ for all~$ n$.\\item [(ii)] $ is $\\mathit {tt}$ -deep if for every computable measure $\\mu $ there exists a computable order $h$ such that $\\mu (T_n) < 2^{-h(n)}$ .", "Our choice of the term “deep\" is due to the similarity between the notions of a deep class and that of a logically deep sequence, introduced by Bennett in .", "Logically deep sequences are highly structured and thus it is difficult to produce initial segments of a deep sequence via probabilistic computation; deep classes can thus be seen as an analogue of logically deep sequences for $\\Pi ^0_1$ classes.", "Remark 4.2 Note that depth and can be equivalently defined as follows: $ is deep if for some computable function~$ f$ one has $ M(Tf(k)) < 2-k$ for all~$ k$.", "Indeed, if $ M(Tn)<2-h(n)$, then, setting $ f=h-1$, we have $ M(Tf(k)) < 2-n$, and conversely, if $ M(Tf(n)) < 2-n$, we can assume that~$ f$ is increasing and then taking $ h=f-1$, we have $ M(Tn)<2-h(n)$ (here we use the fact that $ M(Tn)$ is non-increasing in $ n$).$ The same argument shows that $ is $ tt$-deep if and only if for every computable measure~$$, there exists a computable function~$ f$ such that for all~$ n$, $ (Tf(n)) < 2-n$.", "\\\\$ Remark 4.3 Another alternative way to define depth is to use $\\mathbf {m}$ instead of $\\mathbf {M}$ : a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree~$ T$ is deep if and only if there is a computable order~$ h$ such that $ m(Tn) < 2-h(n)$ for all~$ n$, if and only if there is a computable function~$ f$ such that $ m(Tf(n))< 2-n$ for all~$ n$.", "Indeed, the following inequality holds for all strings~$$ (see for example \\cite {Gacs-notes}):$$\\mathbf {m}(\\sigma ) \\le ^* \\mathbf {M}(\\sigma ) \\le ^* \\mathbf {m}(\\sigma )/ \\mathbf {m}(\|\\sigma \|)$$Thus, for~$ h$ a computable order, if $ M(Tn) < 2-h(n)$ for all~$ n$, we also have $ m(Tn) < 2-h(n)$ for all~$ n$.", "Conversely, if $ m(Tn) < 2-h(n)$, then $ m(Tf(n)) < 2-n$ for $ f=h-1$, and by the above inequality,$$\\mathbf {M}(T_{f(n)}) \\le ^* 2^{-n}/ \\mathbf {m}(f(n)) \\le ^* 2^{-n} \\cdot n^2.$$Thus, taking $ g(n)=f(2n+c)$ for some large enough constant~$ c$, we get $ M(Tg(n)) < 2-n$, and thus $ is deep.", "Since every $\\Pi ^0_1$ class $ is the set of paths through a computable tree, why can^{\\prime }t we simply define depth and $ tt$-depth in terms of this tree and not the canonical co-c.e.\\ tree associated to $ ?", "In the case of depth, there are two reasons to restrict to the canonical co-c.e.", "trees associated to $\\Pi ^0_1$ classes.", "First, the idea behind a deep class is that it is difficult to produce initial segments of some member of the class.", "In general, for a $\\Pi ^0_1$ class $, any computable tree $ T$ contains non-extendible nodes, and so if we have a procedure that can compute these non-extendible nodes of $ T$ with high probability, this tells us nothing about the difficulty of computing the extendible nodes of~$ T$.$ Second, if we were to use any tree $T$ representing a $\\Pi ^0_1$ class $ in the definition of depth, then depth would become a void notion, by the following proposition.$ Proposition 4.4 If $ is a non-empty $ 01$ class and~$ T$ is a computable tree such that $ [T]$, then there is no computable order~$ f$ such that $ m(Tf(n)) < 2-n$.$ If $T$ is a computable infinite tree and $f$ a computable order, there is a computable sequence $(\\sigma _n)_{n \\in \\mathbb {N}}$ of strings such that $\\sigma _n \\in T_{f(n)}$ for every $n$ .", "Thus $\\mathbf {m}(T_{f(n)}) \\ge \\mathbf {m}(\\sigma _n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2.$ By contrast with the notion of depth, for the definition of $\\mathit {tt}$ -depth, it does not matter whether we work with the canonical co-c.e.", "tree associated to $ or some computable tree $ T$ such that $ [T]$ (a direct consequence of Theorem~\\ref {prop:carac-tt-negl}).$ An important question concerns the relationship between depth and negligibility (and $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility).", "Clearly every deep ($\\mathit {tt}$ -deep) $\\Pi ^0_1$ class is negligible ($\\mathit {tt}$ -negligible), but does the converse hold?", "In the case of $\\mathit {tt}$ -depth and $\\mathit {tt}$ -negligibility, the answer is positive.", "We also identify two other equivalent formulations of $\\mathit {tt}$ -depth.", "Theorem 4.1 Let $ be a $ 01$ class.", "The following are equivalent:\\\\$ (i)$ $ is $\\mathit {tt}$ -deep.", "$(ii)$ $ is $ tt$-negligible.", "\\\\$ (iii)$ For every computable measure~$$, $ contains no $\\mu $ -Kurtz random element.", "$(iv)$ For every computable measure $\\mu $ , $ contains no $$-Martin-Löf random element.$ $(i) \\Rightarrow (ii)$ .", "Let $T$ be the canonical co-c.e.", "tree associated to $.", "If $ (Tn) < 2-h(n)$ for some computable order $ h$, then $ (=(n Tn) = 0$.", "\\\\$ (ii) (iii)$.", "This follows directly from the definition of Kurtz randomness.\\\\$ (iii) (iv)$.", "This follows from the fact that Martin-Löf randomness implies Kurtz randomness.\\\\$ (iv) (ii)$.", "If $ (>0$ for some computable measure $$, then $ must contain some $\\mu $ -Martin-Löf random element since the set of $\\mu $ -Martin-Löf random sequences has $\\mu $ -measure 1.", "$(ii) \\Rightarrow (i)$ .", "Suppose that $ is $ tt$-negligible and let $$ be a computable measure.", "Let~$ T$ be the co-c.e.\\ tree associated to~$ .", "By $\\mathit {tt}$ -negligibility, $\\mu (=0$ , or equivalently, $\\mu (T_n)$ tends to 0.", "Since the $T_n$ are co-c.e.", "sets of strings, $\\mu (T_n)$ is upper semi-computable uniformly in $n$ .", "Thus, given $k$ , it is possible to effectively find an $n$ such that $\\mu (T_n) < 2^{-k}$ .", "Setting $f(k)=n$ , we have a computable function $f$ such that $\\mu (T_{f(k)})<2^{-k}$ for all $k$ , therefore $ is $ tt$-deep by Remark~\\ref {rem:deep-order}.$ Significantly, negligibility and depth do not coincide when $\\mathit {tt}$ -reducibility is replaced by Turing reducibility.", "This is due to a fundamental aspect of depth, namely that it is not — unlike negligibility — invariant under Turing equivalence.", "Suppose that $ and $ are two classes such that for every $X \\in there exists a $ Y such that $X \\equiv _T Y$ , and vice-versa.", "Then by Proposition REF , $ is negligible if and only if~$ is negligible.", "This invariance does not hold in general for deep classes, as the next proposition shows.", "Theorem 4.2 For any $\\Pi ^0_1$ class $, there is a $ 01$ class~$ such that: the elements of $ are the same as the elements of~$ , modulo deletion of a finite prefix (which in particular guarantees that the elements of $ and $ have the same Turing degrees); and $ is not deep.$ Let $ be a $ 01$ class.", "Let $ T$ be a computable tree such that $ [T]=, and let $S$ be the canonical co-c.e.", "tree associated to $.", "Consider the tree~$ U$ obtained by appending a copy of~$ S$ to each terminal node of~$ T$.", "Formally,$$U = T \\cup \\lbrace \\sigma ^ \\frown \\tau \\mid \\sigma \\in T_{term} ~\\text{and}~ \\tau \\in S\\rbrace ,$$where~$ Tterm$ is the set of terminal nodes of~$ T$, which is a computable set.", "One can readily verify that $ U$ is the canonical co-c.e.\\ tree associated to some $ 01$ class.", "Indeed, suppose $$ is a node of~$ U$.", "Then,\\begin{itemize}\\item either \\rho \\in T, in which case \\rho is either a prefix of some~X\\in [T] (and thus~X\\in [U]), or \\rho is a prefix of a terminal node \\rho ^{\\prime }, which can then be extended to some~(\\rho ^{\\prime })^\\frown X\\in [U], where X\\in [S]; or\\item \\rho is of the form \\sigma ^\\frown \\tau , with \\sigma \\in T_{term} and \\tau \\in S. Since S is a canonical co-c.e.\\ tree, all its nodes extend to an infinite path, and thus \\sigma ^\\frown \\tau has an extension \\sigma ^\\frown X\\in [U] for some path~X\\in [S].\\end{itemize}$ The fact that $U$ is co-c.e.", "follows directly from the fact that $T_{term}$ is computable and $S$ is co-c.e.", "Let $ be the $ 01$ class whose canonical co-c.e.\\ tree is~$ U$.", "Then the elements of~$ are either elements of $[T]=, or are of the form $ X$ for some finite string~$$ and $ X , which gives us the first part of the conclusion.", "Finally, the canonical co-c.e.", "tree of $ contains an infinite computable tree, namely, the tree~$ T$, and therefore, by the argument as in the proof of Proposition~\\ref {prop:no-deep-comp-tree}, $ is not deep.", "It is now straightforward to get a $\\Pi ^0_1$ class that is negligible but not deep: it suffices to take a negligible class $ and apply the above theorem.", "The resulting class~$ is also negligible as its elements have the same Turing degrees as the elements of $, but it is not deep.", "\\\\$ The following summarizes the implications between the different concepts introduced above for $\\Pi ^0_1$ classes.", "depth $\\Rightarrow $ negligibility $\\Rightarrow $ $\\mathit {tt}$ -negligibility $\\Leftrightarrow $ $\\mathit {tt}$ -depth $\\Rightarrow $ having no computable member Computational limits of randomness As we have seen, negligible $\\Pi ^0_1$ classes (and thus deep classes) have the property that one cannot compute a member of them with positive probability.", "Although some random sequences can compute a member in any negligible $\\Pi ^0_1$ class (namely the Martin-Löf random sequences of $\\mathsf {PA}$ degree, as sequences of $\\mathsf {PA}$ degree compute a member of every $\\Pi ^0_1$ class), by the definition of negligibility, almost every random sequence fails to compute a member of a negligible class.", "Similarly, one cannot $\\mathit {tt}$ -compute a member of a $\\mathit {tt}$ -negligible $\\Pi ^0_1$ class with positive probability.", "The definition of $\\mathit {tt}$ -negligibility implies that the random sequences that can compute a member of a $\\mathit {tt}$ -negligible class form a set of Lebesgue measure zero.", "In this section, we specify a precise level of randomness at which computing a member of a $\\mathit {tt}$ -negligible, negligible, or deep class fails.", "First, we consider the case for $\\mathit {tt}$ -negligible classes.", "Theorem 5.1 If $X\\in 2^\\mathbb {N}$ is Kurtz random, it cannot $\\mathit {tt}$ -compute any member of a $\\mathit {tt}$ -negligible class.", "Let $ be $ tt$-negligible $ 01$ class and $$ be a $ tt$-functional.", "The set $ -1($ is a $ 01$ class that, by $ tt$-negligibility, has Lebesgue measure~$ 0$.", "Thus, it contains no Kurtz random.$ A similar proof can be used to prove the following: Theorem 5.2 If $X\\in 2^\\mathbb {N}$ is weakly 2-random, it cannot compute any member of a negligible class.", "Let $ be negligible $ 01$ class, $$ be a Turing functional, and $ T$ a computable tree such that $ [T]$.", "The set $ -1($ is a $ 02$ class, since$$X\\in \\Phi ^{-1}(\\;\\text{if and only if}\\;(\\forall k)(\\exists \\sigma )(\\exists n)[\\sigma \\in T\\;\\&\\;\|\\sigma \|=k\\;\\&\\; \\sigma \\preceq \\Phi ^{X {\\upharpoonright }n}]$$By negligibility, $ -1($ has Lebesgue measure~$ 0$ and thus contains no weakly 2-random sequence.$ Our next result, despite its simplicity, is probably the most interesting of this section.", "It will help us unify a number of theorems that have appeared in the literature.", "These are theorems of form ($$ ) If $X$ is difference random, then it cannot compute an element of $,$ where $ is a given $ 01$ class.", "Theorem~\\ref {thm:diff-pa} is an example of such a theorem, with $ the class of consistent completions of $\\mathsf {PA}$ .", "The same result has been obtained with $ the class of shift-complex sequences (Khan \\cite {Khan2013}), the set of compression functions (Greenberg, Miller, Nies \\cite {GreenbergMN-ip}), and the set $ DNCq$ functions for some orders~$ q$ (Miller, unpublished).", "We will give the precise definition of these classes in $ §$\\ref {sec:examples} but the important fact is that all of these classes are deep, and indeed, showing the depth of a $ 01$ class is sufficient to obtain a theorem of the form ($ $).", "$ Theorem 5.3 If a sequence $X$ is difference random, it cannot compute any member of a deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with associated co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ M(Tf(n)) < 2-n$.", "Let~$ X$ be a sequence that computes a member of $ via a Turing functional $\\Phi $ .", "Let $\\mathcal {Z}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace $ The set $\\mathcal {Z}_n$ can be written as the difference $\\mathcal {U}_n \\setminus n$ of two effectively open sets (uniformly in $n$ ) with $\\mathcal {U}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\rbrace $ and $n= \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\notin T_{f(n)}\\rbrace $ Moreover, by definition of the semi-measure induced by $\\Phi $ , $\\lambda (\\mathcal {Z}_n) \\le \\lambda _\\Phi (T_{f(n)}) \\le ^* \\mathbf {M}(T_{f(n)}) < 2^{-n}.$ The sequence $(\\mathcal {Z}_n)_{n\\in \\mathbb {N}}$ thus yields a difference test.", "Therefore, the sequence $X$ , which by assumption belongs to all $\\mathcal {Z}_n$ , is not difference random.", "We remark that the converse does not hold: i.e., there is a class $ such that (i) $ is not deep but (ii) no difference random real can compute an element of $.", "Indeed, take a deep class~$ and apply Theorem REF to get a class $ that is not deep but whose members have the same Turing degrees as the elements of~$ .", "Thus, no difference random real can compute an element of $.$ Depth, negligibility, and mass problems In this section we discuss depth and negligibility in the context of mass problems, i.e., in the context of the Muchnik and Medvedev reducibility.", "Both Muchnik and Medvedev reducibility are generalizations of Turing reducibility.", "Whereas Turing reducibility is defined in terms of a pair of sequences, both Muchnik and Medevedev reducibility are defined in term of a pair of collections of sequences.", "In what follows, we will consider these two reducibilities when restricted to $\\Pi ^0_1$ subclasses of $2^\\mathbb {N}$ .", "We will follow the notation of the survey , to which we refer the reader for a thorough exposition of mass problems in the context of $\\Pi ^0_1$ classes.", "Let $2^\\mathbb {N}$ be $\\Pi ^0_1$ classes.", "We say that $ is \\emph {Muchnik reducible} to $ , denoted $_w, if for every $ X there exists a Turing functional $\\Phi $ such that (i) $X\\in \\mathsf {dom}(\\Phi )$ and $\\Phi (X)\\in .", "Moreover, $ is Medvedev reducible to $, denoted $ s, if $ is Muchnik reducible to $ via a single Turing functional, i.e., there exists a Turing functional $\\Phi $ such that (i) $\\mathsf {dom}(\\Phi )$ and $\\Phi (\\subseteq .$ Just as Turing reducibility gives rise to a degree structure, we can define degree structures from $\\le _w$ and $\\le _s$ .", "We say that $ and $ are Muchnik equivalent (resp.", "Medvedev equivalent), denoted $_w (resp.\\ $ s) if and only if $_w and $ w (resp.", "$_s and $ s).", "The collections of Muchnik and Medvedev degrees given by the equivalence classes under $\\equiv _w$ and $\\equiv _s$ are denoted $\\mathcal {E}_w$ and $\\mathcal {E}_s$ , respectively.", "Both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are lattices, unlike the Turing degrees, which only form an upper semi-lattice.", "We define the meet and join operations as follows.", "Given $\\Pi ^0_1$ classes $2^\\mathbb {N}$ , $\\sup ($ is the $\\Pi ^0_1$ class $\\lbrace X\\oplus Y:X\\in \\&\\;Y\\in $ .", "Furthermore, we define $\\inf ($ to be the $\\Pi ^0_1$ class $\\lbrace 0^\\frown X: X\\in \\cup \\lbrace 1^\\frown Y:Y\\in $ .", "One can readily check that the least upper bound of $ and $ in $\\mathcal {E}_w$ is the Muchnik degree of $\\sup ($ while their greatest lower bound is the Muchnik degree $\\inf ($ , and similarly for $\\mathcal {E}_s$ .", "Recall that a filter $\\mathcal {F}$ in a lattice $(\\mathcal {L},\\le ,\\inf ,\\sup )$ is a subset that satisfies the following two conditions: (i) for all $x,y\\in \\mathcal {L}$ , if $x\\in \\mathcal {F}$ and $x\\le y$ , then $y\\in \\mathcal {F}$ , and (ii) for all $x,y\\in \\mathcal {F}$ , $\\inf (x,y)\\in \\mathcal {F}$ .", "The goal of this section is to study the role of depth and negligibility in the structures $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let us start with an easy result.", "Theorem 6.1 The collection of negligible $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let $ and $ be negligible $\\Pi ^0_1$ classes.", "By Proposition REF , we have that $\\lambda ((^{\\le _T})=0$ and $\\lambda ((^{\\le _T})=0$ .", "But since $\\inf (^{\\le _T}=(^{\\le _T}\\cup (^{\\le _T},$ it follows that $\\lambda (\\inf (^{\\le _T})=0$ , which shows that $\\inf ($ is negligible.", "Thus, the degrees of negligible classes in both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are closed under $\\inf $ .", "Let $\\mathcal {D}$ be non-negligible $\\Pi ^0_1$ and $\\mathcal {C}\\le _w\\mathcal {D}$ .", "For each $i$ , we define $i:=\\lbrace X\\in X\\in \\mathsf {dom}(\\Phi _i)\\;\\&\\;\\Phi _i(X)\\in .$ Since $_w, it follows that $ ii$.", "Furthermore, we have $ (T=(ii)T=i(i)T$.", "Since $ is non-negligible, we have $0<\\lambda ((^{\\le _T})\\le \\sum _i\\lambda \\big (({i})^{\\le _T}\\big ),$ and thus $\\lambda (({k})^{\\le _T})>0$ for some $k$ .", "But since $\\Phi _{k}({k})\\subseteq , it follows that $ ((T)>0$, thus $ is non-negligible.", "Thus, negligibility is closed upwards under $\\le _w$ (and a fortiori, under $\\le _s$ as well).", "Remark 6.1 Simpson proved in that in $\\mathcal {E}_s$ , the complement of the filter of negligible classes is in fact a principal ideal, namely the ideal generated by the class $\\inf (\\mathcal {PA},2\\mathcal {RAN})$ , where $2\\mathcal {RAN}$ is the class of 2-random sequences, i.e.", "the sequences that are Martin-Löf random relative to $\\emptyset ^{\\prime }$ .", "For the next two theorems, we need the following fact.", "Fact 6.2 Let $ and $ be $\\Pi ^0_1$ classes such that $_s.", "Then there is a total Turing functional $$ such that $ (.", "This holds because for every $\\Pi ^0_1$ class $ and every Turing functional~$$ that is total on $ , there is a functional $\\Psi $ that is total on $2^\\mathbb {N}$ and coincides with $\\Phi $ on $.$ Theorem 6.2 The collection of deep $\\Pi ^0_1$ classes forms a filter in $\\mathcal {E}_s$ .", "Let $ and $ be deep $\\Pi ^0_1$ classes with associated co-c.e.", "trees $S$ and $T$ , respectively.", "Moreover, let $g$ and $h$ be computable orders such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ and $\\mathbf {M}(T_n) < 2^{-h(n)}$ .", "We define $f(n)=\\min \\lbrace g(n),h(n)\\rbrace $ , which is clearly a computable order.", "It follows immediately that $\\mathbf {M}(S_n)\\le 2^{-f(n)}$ and $\\mathbf {M}(T_n) < 2^{-f(n)}$ .", "Now the co-c.e.", "tree associated with $\\inf ($ is $R=\\lbrace 0^\\frown \\sigma :\\sigma \\in S\\rbrace \\cup \\lbrace 1^\\frown \\tau :\\tau \\in T\\rbrace .$ Consider the class $\\inf ($ .", "Setting $0^\\frown S_n=\\lbrace 0^\\frown \\sigma :\\sigma \\in S_n\\rbrace $ and $1^\\frown T_n=\\lbrace 1^\\frown \\sigma :\\sigma \\in T_n\\rbrace $ for each $n$ , the co-c.e.", "tree of $\\inf ($ is $0^\\frown S_n \\cup 1^\\frown T_n$ .", "Moreover, $\\mathbf {M}(0^\\frown S_n \\cup 1^\\frown T_n) = \\mathbf {M}(0^\\frown S_n) + \\mathbf {M}(1^\\frown T_n) \\le ^* 2^{-f(n)} + 2^{-f(n)}$ which shows that $\\inf ($ is deep.", "Next, suppose that $\\mathcal {C}$ is a deep $\\Pi ^0_1$ class and $\\mathcal {D}$ is a $\\Pi ^0_1$ class satisfying $\\mathcal {C} \\le _s \\mathcal {D}$ via the Turing functional $\\Psi $ , which we can assume to be total by Fact REF .", "Let $S$ and $T$ be the co-c.e.", "trees associated to $ and $ , respectively, and let $g$ be a computable order such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ .", "Since $\\Psi $ is total, its use is bounded by some computable function $f$ .", "It follows that for every $\\sigma \\in T_{f(n)}$ , there is some $\\tau \\in S_n$ such that $(\\sigma ,\\tau )\\in \\Psi $ .", "Thus $\\llbracket T_{f(n)}\\rrbracket \\subseteq \\Psi ^{-1}(\\llbracket S_n\\rrbracket )$ .", "Now let $\\Phi $ be a universal Turing functional such that $\\mathbf {M}=\\lambda _\\Phi $ .", "Then: $\\mathbf {M}(T_{f(n)})=\\lambda (\\Phi ^{-1}( T_{f(n)}))\\le \\lambda (\\Phi ^{-1}(\\llbracket T_{f(n)}\\rrbracket )\\le \\lambda (\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket )))\\le \\lambda _{\\Psi \\circ \\Phi }(S_n),$ where the last inequality holds because $\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket ))=(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq \\mathsf {dom}(\\Psi \\circ \\Phi )$ and hence $(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq (\\Psi \\circ \\Phi )^{-1}(S_n)$ , $\\lambda _{\\Psi \\circ \\Phi }\\le ^* \\mathbf {M}$ .", "Thus we have $\\mathbf {M}(T_{f(n)})\\le \\lambda _{\\Psi \\circ \\Phi }(S_n)\\le ^* \\mathbf {M}(S_n)\\le 2^{-g(n)}.$ Thus, $\\mathbf {M}(T_{f \\circ g^{-1}(n)}) \\le 2^{-n}$ , which shows that $ is deep.$ The invariance of the notion of depth under Medvedev-equivalence is of importance for the next section.", "There, we prove that certain classes of objects, which are not necessarily infinite binary sequences, are deep.", "To do so, we fix a certain encoding of these objects by infinite binary sequences and prove the depth of the corresponding encoded class.", "By the above theorem, the particular choice of encoding is irrelevant: if the class is deep for an encoding, it will be deep for another encoding, as long as switching from the first encoding to the second one can be done computably and uniformly.", "One could ask whether we have a similar result in the lattice $\\mathcal {E}_w$ .", "However, Theorem REF shows that depth is not invariant under Muchnik equivalence: if we apply this theorem to a deep class $, we get a $ 01$ class~$ which is clearly Muchnik equivalent to $ but is not deep itself.", "Thus depth does not marry well with Muchnik reducibility and is only a `Medvedev notion^{\\prime }.", "\\\\$ More surprisingly, $\\mathit {tt}$ -depth is a `Muchnik notion' as well as a `Medvedev notion' and the $\\mathit {tt}$ -deep classes form a filter in both lattices.", "Theorem 6.3 The collection of $\\mathit {tt}$ -deep $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Suppose that $_w and $ is not $\\mathit {tt}$ -deep.", "By definition, this means that $ has positive $$-measure for some computable probability measure~$$.", "Let $ {Uk}k N$ be the universal $$-Martin-Löf test and define $ Rk$ to be the complement of $ Uk$ (which is a $ 01$ class).", "Since $ (Rk) > 1- 2-k$ for every $ k$, there must be a~$ j$ such that $ (Rj) > 0$ and in particular $ Rj = $.", "By the hyperimmune-free basis theorem, there is some $ XRj$ of hyperimmune-free Turing degree.", "Since $ w , $X$ must compute some element $Y$ of $.", "But since~$ X$ is of hyperimmune-free degree, $ X$ in fact $ tt$-computes $ Y$, i.e., $ (X)=Y$ for some total functional~$$.$ Since $X$ is $\\mu $ -Martin-Löf random, by the preservation of randomness theorem (see, for instance, Theorem 3.2 in ), $Y$ is Martin-Löf random with respect to the computable measure $\\mu _\\Psi $ defined by $\\mu _\\Psi (\\sigma ):=\\mu (\\Psi ^{-1}(\\sigma ))$ for every $\\sigma $ .", "Thus, by Proposition REF , $ is not $ tt$-deep.", "This shows, a fortiori, that if $ s and $ is not $ tt$-deep, then $ is not $\\mathit {tt}$ -deep.", "Thus $\\mathit {tt}$ -depth is closed upwards in $\\mathcal {E}_w$ and $\\mathcal {E}_s$ and in particular is compatible with the equivalence relations $\\equiv _s$ and $\\equiv _w$ .", "Next, suppose that $ and $ are $\\mathit {tt}$ -deep classes but that $\\inf ($ is not $\\mathit {tt}$ -deep (recall that the $\\inf $ operator is the same for both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ ).", "Then by Theorem REF , $\\inf ($ contains a sequence $X$ that is $\\mu $ -Martin-Löf random for some computable measure $\\mu $ .", "Then for $i=0,1$ we define computable measures $\\mu _i$ such that $\\mu _i(\\sigma )=\\mu (i^\\frown \\sigma )$ for every $\\sigma $ .", "It is routine to check that $Y$ is $\\mu _i$ -Martin-Löf random if and only if $i^\\frown Y$ is $\\mu $ -Martin-Löf random for $i=0,1$ .", "Since $X=i^\\frown Z$ for some $i=0,1$ and $Z\\in 2^\\mathbb {N}$ , it follows that $Z$ is $\\mu _i$ -Martin-Löf random.", "But then $Z$ is contained in either $ or $ , which contradicts our hypothesis that $ and $ are both $\\mathit {tt}$ -deep.", "Examples of deep $\\Pi ^0_1$ classes In this section, we provide a number of examples of deep $\\Pi ^0_1$ classes that naturally occur in computability theory and algorithmic randomness.", "We give a uniform treatment of all these classes, i.e., we give a generic method to prove the depth of $\\Pi ^0_1$ classes.", "Consistent completions of Peano Arithmetic As mentioned in $§$, Jockusch and Soare proved in that the $\\Pi ^0_1$ class $\\mathcal {PA}$ of consistent completions of $\\mathsf {PA}$ is negligible.", "However, as shown by implicitly by Levin in and Stephan in , $\\mathcal {PA}$ is also deep.", "We will reproduce this result here.", "Following both Levin and Stephan, we will use fact that the class of consistent completions of $\\mathsf {PA}$ is Medvedev equivalent to the class of total extensions of a universal, partial-computable $\\lbrace 0,1\\rbrace $ -valued function.", "Thus, by showing the latter class is deep, we thereby establish that the former class is deep (via Theorem REF ).", "Theorem 7.1 (Levin , Stephan ) Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of all $\\lbrace 0,1\\rbrace $ -valued partial computable functions.", "Let $u$ be a function that is universal for this collection, e.g., defined by $u(\\langle e,x\\rangle ) = \\phi _e(x)$ .", "Then the class $ of total extensions of~$ u$ is a deep $ 01$ class.$ We build a partial computable function $\\phi _e$ , whose index we know in advance by the recursion theorem.", "This means that we control the value of $u(\\langle e,x\\rangle )$ for all $x$ .", "First, we partition $\\mathbb {N}$ into consecutive intervals $I_1, I_2, ...$ such that we control $2^{k+1}$ values of $u$ inside $I_k$ .", "For each $k$ in parallel, we define $\\phi _e$ on $I_k$ as follows.", "Step 1: Wait for a stage $s$ such that the set $E_k[s]=\\lbrace \\sigma \\mid \\sigma {\\upharpoonright }I_k ~ \\text{extends}~ u[s] {\\upharpoonright }I_k \\rbrace $ is such that $\\mathbf {M}(E_k)[s] \\ge 2^{-k}$ .", "Step 2: Find a $y \\in I_k$ that we control and on which $u_s$ is not defined.", "Consider the two “halves\" $E^0_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=0\\rbrace $ and $E^1_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=1\\rbrace $ of $E_k[s]$ .", "Note that either $\\mathbf {M}(E^0_k[s])\\ge 2^{-k-1}$ or $\\mathbf {M}(E^1_k)[s] \\ge 2^{-k-1}$ .", "If the first holds, set $u(y)[s+1]=1$ , otherwise set $u(y)[s+1]=0$ .", "Go back to Step 1.", "The co-c.e.", "tree $T$ associated to the class $ is the set of strings~$$ such that~$$ is an extension of $ u \|\|$.", "The construction works because every time we pass by Step 2, we remove from~$ T[s]$ a set $ Eik[s]$ (for some $ i{0,1}$ and $ k,sN$) such that $ M(Eik)[s]>2-k-1$.", "Therefore, Step 2 can be executed at most $ 2k+1$ times, and by the definition of $ Ik$ we do not run out of values~$ y Ik$ on which we control~$ u$.", "Therefore, the algorithm eventually reaches Step 1 and waits there forever.", "Setting $ f(k)=(Ik)$, this implies that the $ M$-weight of the set $ {f(k) extends u}$ is bounded by $ 2-k$, or equivalently,$$\\mathbf {M}(T_{f(k)}) < 2^{-k},$$which proves that~$ is deep.", "The proof provided here gives us a general template to prove the depth of a $\\Pi ^0_1$ class.", "First of all, the definition of the $\\Pi ^0_1$ class should allow us to control parts of it in some way, either because we are defining the class ourselves or because, as in the above proof, the definition of the class involves some universal object which we can assume to partially control due to the recursion theorem.", "All the other examples of deep $\\Pi ^0_1$ class we will see below belong to this second category.", "Let us take a step back and analyze more closely the structure of the proof of Theorem REF .", "Given a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree $ T$, the proof consists of the following steps.\\\\$ (1) For a given $k$ , we identify a level $N=f(k)$ at which we wish to ensure $\\mathbf {M}(T_N) < 2^{-k}$ .", "The choice of $N$ will depend on the particular class $.", "\\\\$ (2) Next we implement a two-step strategy to ensure that $\\mathbf {M}(T_N)<2^{-k}$ .", "Such a strategy will be called a $k$ -strategy.", "(2.1) First, we wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] > 2^{-k}$ .", "(2.2) If at some stage $s$ this occurs, we remove one or several of the nodes of $T[s]$ at level $N$ such that the total $\\mathbf {M}[s]$ -weight of these nodes is at least $\\delta (k)$ for a certain function $\\delta $ , and then go back to Step 2.", "(3) Each execution of Step 2 (removing nodes from $T[s]$ for some $s$ ) comes at a cost $\\gamma (k)$ for some $k$ , and we need to make sure that we do not go over some maximal total cost $\\Gamma (k)$ throughout the execution of the $k$ -strategy.", "In the above example, $\\delta (k)=2^{-k-1}$ and the only cost for us is to define one value of $u(y)$ out of the $2^{k+1}$ we control, so we can, for example, set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ .", "By definition of $\\mathbf {M}$ , the $k$ -strategy can only go through Step 2 at most $1/\\delta (k)$ times, and thus the total cost of the $k$ -strategy will be at most $\\gamma (k)/\\delta (k)$ .", "All we have to do is to make sure that $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ (which is the case in the above example).", "Again, we have some flexibility on the choice of $N=f(k)$ , so it will suffice to choose an appropriate $N$ to ensure $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ .", "In some cases, there will not be a predefined maximum for each $k$ , but rather a global maximum $\\Gamma $ that the sum of the the costs of the $k$ -strategies should not exceed, i.e., we will want to have $\\sum _k \\gamma (k)/\\delta (k) \\le \\Gamma $ .", "Let us now proceed to more examples of $\\Pi ^0_1$ classes.", "Shift-complex sequences Definition 7.1 Let $\\alpha \\in (0,1)$ and $c>0$ .", "(i) $\\sigma \\in 2^{<\\mathbb {N}}$ (resp.", "$X\\in 2^\\mathbb {N}$ ) is said to be $(\\alpha ,c)$ -shift complex if $\\mathrm {K}(\\tau ) \\ge \\alpha \|\\tau \| - c$ for every substring $\\tau $ of $\\sigma $ (resp.", "of $X$ ).", "(ii) $X\\in 2^\\mathbb {N}$ is said to be $\\alpha $ -shift complex if it is $(\\alpha ,c)$ -shift complex for some $c$ .", "The very existence of $\\alpha $ -shift complex sequences is by no means obvious.", "Such sequences were first constructed by Durand et al.", "who showed that there exist $\\alpha $ -shift complex sequences for all $\\alpha \\in (0,1)$ .", "It is easy to see that for every computable pair $(\\alpha ,c)$ , the class of $(\\alpha ,c)$ -shift complex sequences is a $\\Pi ^0_1$ class.", "Rumyantsev proved that the class of $(\\alpha ,c)$ -complex sequences is always negligible, but in fact, his proof essentially shows that it is even deep.", "The cornerstone of Rumyantsev's theorem is the following lemma.", "It relies on an ingenious combinatorial argument that we do not reproduce here.", "We refer the reader to for the full proof.", "Lemma 7.2 (Rumyantsev ) Let $\\beta \\in (0,1)$ .", "For every rational $\\eta \\in (0,1)$ and integer $d$ , there exist two integers $n$ and $N$ , with $n<N$ , such that the following holds.", "For every probability distribution $P$ on $\\lbrace 0,1\\rbrace ^N$ , there exist finite sets of strings $A_n, A_{n+1}, ..., A_N$ such that for all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements; and the $P$ -probability that a sequence $X$ has some substring in $\\cup _{i=n}^N A_i$ is at least $1-\\eta $ .", "Moreover, $n$ and $N$ can be effectively computed from $\\eta $ and $d$ and can be chosen to be arbitrarily large.", "Once $n$ and $N$ are fixed, the sets $A_i$ can be computed uniformly in $P$ .", "This is Lemma 6 of (Rumyantsev does not explicitly state that the conclusion holds for all computable probability measures, but nothing in his proof makes use of a particular measure).", "Theorem 7.2 For any computable $\\alpha \\in (0,1)$ and integer $c>0$ , the $\\Pi ^0_1$ class of $(\\alpha ,c)$ -shift complex sequences is deep.", "Let $ be the $ 01$ class of $ (,c)$-shift complex sequences and let~$ T$ be its canonical co-c.e.\\ tree.", "We shall build a discrete semi-measure~$ m$ whose coding constant~$ e$ we know in advance.", "This means that whenever we will set a string~$$ to be such that $ m() > 2-\|\| + c + e +1 $, then automatically we will have $ m() > 2-\|\| + c +1$, and thus $ K() < \|\| - c$.", "This will \\textit {de facto} remove $$ from~$ T$.$ Let us now turn to the construction.", "First, we pick some $\\beta $ such that $0 < \\beta < \\alpha $ .", "For each $k$ , we apply the above Lemma REF to $\\beta $ and $\\eta = 1/2$ to obtain a pair $(n,N)$ with the above properties as described in the statement of the lemma (we will also make use of the fact that $n$ can be chosen arbitrarily large, see below).", "Then the two step strategy is the following: Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Let $P$ be the (computable) probability distribution on $\\lbrace 0,1\\rbrace ^N$ whose support is contained in $T_N[s]$ and such that for all $\\sigma \\in T_N[s]$ , $P(\\sigma )=2^k \\cdot \\mathbf {M}(\\sigma )[s]$ .", "By Lemma REF , we can compute a collection of finite sets $A_i$ such that: (i) For all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements.", "(ii) The $P$ -probability that a sequence $X$ has some substring in $F = \\cup _{i=n}^N A_i$ is at least $1-\\eta =1/2$ .", "Thus, $\\mathbf {M}(T_N \\cap F)[s] > 2^{-k-1}$ .", "Then, for each $i$ and each $\\sigma \\in A_i$ , we ensure, increasing $m$ if necessary, that $m(\\sigma ) > 2^{-\\alpha \|\\sigma \| + c + e +1 }$ .", "As explained above, this ensures that all strings in $F$ are removed from $T_N$ at that stage, and therefore we have removed a $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ .", "Then we go back to Step 1.", "To finish the proof, we need to make sure that this algorithm does not cost us too much.", "The constraint here is that we need to make sure that $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau ) \\le 1$ , so we are trying to stay under a global cost of $\\Gamma =1$ .", "For a given $k$ , our cost at each execution of Step 2 is the increase of $m$ on the strings in $F$ .", "The total $m$ -weight we add at during one such execution of Step 2 is at most $\\gamma (k) = \\sum _{i=n}^N \|A_i\| \\cdot 2^{-\\alpha i + c + e +1 } \\le \\sum _{i=n}^N 2^{(\\beta -\\alpha ) i + c+ e + 1} \\le \\frac{2^{(\\beta -\\alpha ) n + c+ e + 1}}{1-2^{\\beta -\\alpha }}$ But since $n=n(k)$ can be chosen arbitrarily large, therefore, with an appropriate choice of $n$ , we can make $\\gamma (k) \\le 2^{-2k-2}$ .", "Therefore, since the $k$ -strategy executes Step 2 at most $1/\\delta (k)$ times, we have $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau )\\le \\sum _{k\\in \\mathbb {N}} \\gamma (k)/\\delta (k) \\le \\sum _{k\\in \\mathbb {N}} 2^{-2k-2}/2^{-k-1} \\le 1$ and therefore $m$ is indeed a discrete semi-measure.", "DNC$_q$ functions Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of partial computable functions from $\\mathbb {N}$ to $\\mathbb {N}$ .", "Let $J$ be the universal partial computable function defined as follows.", "For all $(e,x) \\in \\mathbb {N}^2$ : $J(2^e(2x+1))=\\phi _e(x)$ The following notion was studied in .", "Definition 7.3 Let $q: \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable order.", "The set $\\mathcal {DNC}_q$ is the set of total functions $f: \\mathbb {N}\\rightarrow \\mathbb {N}$ such that for all $n$ : (i) $f(n) \\ne J(n)$ (where this condition is trivially satisfied if $J(n)$ is undefined), and (ii) $f(n) < q(n)$ .", "We should note that this is a slight variation of the standard definition of diagonal non-computability, which is formulated in terms of the condition (i$^{\\prime }$ ) $f(n)\\ne \\phi _n(n)$ instead of the condition (i) given above.", "However, this condition would make the class $\\mathcal {DNC}_q$ and the results below dependent on the particular choice of enumeration of the partial computable functions, while with condition $(i)$ , the results of the section are independent of the chosen enumeration.", "Note that for every fixed computable order $q$ , the class $\\mathcal {DNC}_q$ is a $\\Pi ^0_1$ subset of $\\mathbb {N}^\\mathbb {N}$ , but due to the bound $q$ , $\\mathcal {DNC}_q$ can be viewed as a $\\Pi ^0_1$ subset of $2^\\mathbb {N}$ .", "Whether the class $\\mathcal {DNC}_q$ is deep turns out to depend on $q$ .", "Indeed, we have the following interesting dichotomy theorem.", "Theorem 7.3 Let $q$ be a computable order.", "(i) If $\\sum _n 1/q(n) < \\infty $ , then $\\mathcal {DNC}_q$ is not $\\mathit {tt}$ -negligible.", "(ii) If $\\sum _n 1/q(n) = \\infty $ , then $\\mathcal {DNC}_q$ is deep.", "Part $(i)$ is a straightforward adaptation of Kučera's proof that every Martin-Löf random element computes a DNC function.", "Pick an $X \\in 2^\\mathbb {N}$ at random and use it as a random source to randomly pick the value of $f(i)$ uniformly among $\\lbrace 0,...,q(i)-1\\rbrace $ , independently of the other values of $f$ .", "The details are as follows.", "First, we can assume that $q(n)$ is a power of 2 for all $n$ .", "Indeed, since for $q^{\\prime } \\le q$ the class $\\mathcal {DNC}_{q^{\\prime }}$ is contained in $\\mathcal {DNC}_{q}$ , so if we take $q^{\\prime }(n)$ to be the largest power of 2, we have that $q^{\\prime }$ is computable, $q^{\\prime } \\le q$ , and $\\sum _n 1/q(n) < \\infty $ because $q^{\\prime }$ is equal to $q$ up to factor 2.", "Now, set $q(n)=2^{r(n)}$ .", "Split $\\mathbb {N}$ into intervals where interval $I_n$ has length $r(n)$ .", "One can now interpret any infinite binary sequence $X$ as a funtion $f_X: \\mathbb {N}\\rightarrow \\mathbb {N}$ , where $X(n)$ is the index of $X {\\upharpoonright }I_n$ in the lexicographic ordering of strings of length $r(n)$ .", "For $X$ taken at random with respect to the uniform measure, the event $f_X(n) = \\phi _n(n)$ has probability at most $1/q(n)$ and is independent of all such events for $n^{\\prime } \\ne n$ .", "Thus the total probability over $X$ such that $X(n)\\ne \\phi _n(n)$ for all $n$ is at least $\\prod _n (1-1/q(n))$ an expression that is positive if and only if $\\sum 1/q(n)< \\infty $ , which is satisfied by hypothesis.", "Thus, the class $\\mathcal {DNC}_q$ , encoded as above, has positive uniform measure, and thus is not $\\mathit {tt}$ -negligible.", "For $(ii)$ , let $q$ be a computable order such that $\\sum _n 1/q(n) = \\infty $ .", "Let $T$ be the canonical co-c.e.", "binary tree in which the elements of $\\mathcal {DNC}_q$ are encoded.", "By the recursion theorem, we will build a partial recursive function $\\phi _e$ whose index $e$ we know in advance and therefore will be able to define $J$ on the set of values $D = \\lbrace 2^e (2x+1) : x \\in \\mathbb {N}\\rbrace $ .", "Note that the set $D$ has positive (lower) density in $\\mathbb {N}$ : for every interval $I$ of length at least $2^{e+1}$ , $\|D \\cap I\| \\ge 2^{-e-2} \|I\|$ .", "This in particular implies that $\\sum _{n \\in D} 1/q(n) = \\infty .$ To show this, we appeal to Cauchy's condensation test, according to which for any positive non-increasing sequence $(a_n)$ , $\\sum _n a_n < \\infty $ if and only if $\\sum _k 2^k a_{2^k} < \\infty $ .", "Since the sum $\\sum _n1/q(n)$ diverges, so does $\\sum _k 2^k/q(2^k)$ and thus $\\sum _{n \\in D} \\frac{1}{q(n)} \\ge \\sum _{k>e} \\frac{\|D \\cap \\lbrace 2^{k-1},...,2^k-1\\rbrace \|}{q(2^k)} \\ge \\sum _{k>e} \\frac{2^{-e-2} \\cdot 2^{k-1}}{q(2^k)} = \\infty .$ Now that we have established that $\\sum _{n \\in D} 1/q(n) = \\infty $ , we remark that this is equivalent to having $\\prod _{n \\in D} (1-1/q(n)) = 0$ .", "Thus, we can effectively partition $D$ into countably many finite sets $D_j$ such that $\\prod _{n \\in D_j} (1-1/q(n)) < 1/2$ .", "We are ready to describe the construction.", "For each $k$ , reserve some finite collection $D_{j_1}, D_{j_2}, ..., D_{j_{k+1}}$ of sets $D_j$ .", "Let $N$ be a level of the binary tree $T$ sufficiently large such that the encoding of each path $f$ up to length $N$ is enough to recover the values of $f$ on $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , which can be found effectively in $k$ .", "The $k$ -strategy then works as follows.", "Initialisation.", "Set $i=1$ .", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Since each $\\sigma \\in T_N[s]$ can be viewed as a function from some initial segment of $\\mathbb {N}$ that contains $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , take the first value $x \\in D_{j_i}$ .", "For all $\\sigma \\in T_N$ , $\\sigma (x) < q(x)$ , thus by the pigeonhole principle there must be at least one value $v < q(x)$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x)=v\\rbrace )[s] \\ge 2^{-k}/q(x).$ Set $J(x)$ to be the least such value $v$ .", "This thus gives $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v\\rbrace )[s] < 2^{-k}(1-1/q(x)).$ Now take another $x^{\\prime }$ in $D_{j_i}$ on which $J$ has not been defined yet.", "By the same reasoning, there must be a value $v^{\\prime } < q(x^{\\prime })$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v \\wedge \\sigma (x^{\\prime }) \\ne v^{\\prime } \\rbrace ) < 2^{-k}(1-1/q(x))(1-1/q(x^{\\prime }))[s].$ We then set $J(x^{\\prime })$ to be equal to $v^{\\prime }$ .", "Continuing in this fashion, we can assign all the values of $J$ on $D_{j_i}$ in such a way that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\forall x \\in D_{j_i}, \\ \\sigma (x) \\ne J(x) \\rbrace \\big )[s] < 2^{-k} \\prod _{n \\in D_j} (1-1/q(n)) < 2^{-k}/2.$ Then we increment $i$ by 1 and go back to Step 1.", "Once again, at each execution of Step 2 we remove an $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ , since ensuring $\\sigma (x)=J(x)$ for some $x$ immediately ensures that no extension of $\\sigma $ is in $\\mathcal {DNC}_q$ .", "Moreover, each execution of Step 2 requires us to define $J$ on all values in some $D_{j_i}$ for $1\\le j\\le k+1$ .", "That is, each such execution costs us one $D_{j_i}$ , so as in the case of consistent completions of $\\mathsf {PA}$ , we can set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ , which ensures that $\\gamma (k)/\\delta (k)\\le \\Gamma (k)$ .", "Therefore, $\\mathcal {DNC}_q$ is a deep $\\Pi ^0_1$ class.", "Finite sets of maximally complex strings For a constant $c>0$ , generating a long string $\\sigma $ of high Kolmogorov complexity, for example $\\mathrm {K}(\\sigma ) > \|\\sigma \|-c$ , can be easily achieved with high probability if one has access to a random source (just repeatedly flip a fair coin and output the raw result).", "One can even use this technique to generate a sequence of strings $\\sigma _1, \\sigma _2, \\sigma _3, \\ldots , $ such that $\|\\sigma _n\| = n$ and $\\mathrm {K}(\\sigma _n) \\ge n-c$ .", "Indeed, the measure of the sequences $X$ such that $\\mathrm {K}(X {\\upharpoonright }n) \\ge n -d$ for all $n$ is at least $2^{-d}$ , and thus the $\\Pi ^0_1$ class of such sequences of strings $(\\sigma _1,\\sigma _2,...,)$ of high complexity (encoded as elements of $2^\\mathbb {N}$ ) is not even $\\mathit {tt}$ -negligible.", "The situation changes dramatically if one wishes to obtain many distinct complex strings at a given length.", "Let $\\ell : \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable increasing function, $f,d:\\mathbb {N}\\rightarrow \\mathbb {N}$ be any computable functions, and $c>0$ .", "Consider the $\\Pi ^0_1$ class $\\mathcal {K}_{f,\\ell ,d}$ whose members are sequences $\\vec{F}=(F_1, F_2, F_3, ...)$ where for all $i$ , $F_i$ is a finite set of $f(i)$ strings $\\sigma $ of length $\\ell (i)$ such that $\\mathrm {K}(\\sigma ) \\ge \\ell (i)-d(i)$ .", "Note once again that $\\mathcal {K}_{f,\\ell ,d}$ can be viewed, modulo encoding, as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ .", "Theorem 7.4 For all computable functions $f,\\ell ,d$ such that $f(i)/2^{d(i)}$ takes arbitrarily large values and $\\ell $ is increasing, the class $\\mathcal {K}_{f,\\ell ,d}$ is deep.", "Let $T$ be the canonical co-c.e.", "tree associated to the class $\\mathcal {K}_{f,\\ell ,d}$ .", "Just like in the proof of Theorem REF , we will build a discrete semi-measure $m$ whose coding constant $e$ we know in advance and thus, by setting $m(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) + e +1 }$ , we will force $\\mathbf {m}(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) +1}$ , and thus $\\mathrm {K}(\\sigma ) < \|\\sigma \| - d(\|\\sigma \|)$ , which will consequently remove from $T$ every sequence of sets $(F_i)_{i\\in \\mathbb {N}}$ such that $\\sigma $ belongs to some $F_i$ .", "Fix a $k$ , and let us pick some well-chosen $i=i(k)$ , to be determined later.", "For readability, let $f=f(i(k))$ , $\\ell =\\ell (i(k))$ and $d=d(i(k))$ .", "Let $N$ be the level of $T$ at which the first $i$ sets $F_1\\cdots F_i$ of the sequence $\\vec{F}$ of $\\mathcal {K}_{f,\\ell ,d}$ are encoded.", "The $k$ -strategy does the following.", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: The members of $T_N[s]$ consist of finite sequences $F_1, ..., F_i$ of sets of strings where $F_i$ contains $f$ distinct strings of length $\\ell $ .", "By the pigeonhole principle, there must be a string $\\sigma $ of length $\\ell $ such that $\\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N : \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ Indeed, there is a total $\\mathbf {M}$ -weight $2^{-k}$ of possible sets $F_i$ , each of which contains $f$ strings of length $\\ell $ , thus $\\sum _{\|\\sigma \|=\\ell } \\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N \\mid \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f$ , and there are $2^\\ell $ strings of length $\\ell $ in total.", "Effectively find such a string $\\sigma $ , and set $m(\\sigma ) > 2^{-\\ell + d + e +1 }$ .", "By the choice of $\\sigma $ , this causes an $\\mathbf {M}$ -weight of at least $2^{-k} \\cdot f \\cdot 2^{-\\ell }$ of nodes of $T_N[s]$ to leave the tree.", "Then go back to Step 1.", "As in the previous examples, it remains to conduct the “cost analysis\".", "The resource here again is the weight we are allowed to assign to $m$ , which has to be bounded by $\\Gamma =1$ .", "At each execution of Step 2 of the $k$ -strategy, our cost is the increase of $m$ , which is at most of $\\gamma (k)=2^{-\\ell + d + e +1 }$ , while an $\\mathbf {M}$ -weight of at least $\\delta (k) = 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ leaves the tree $T_N$ .", "This gives a ratio $\\gamma (k)/\\delta (k) = 2^{d+e+k+1}/f$ , which bounds the total cost of the $k$ -strategy, so we want $\\sum _k \\gamma (k)/\\delta (k) \\le 1$ , i.e., we want: $\\sum _k 2^{d(i(k)) + e +k+1}/f(i(k))\\le 1$ By assumption, $f(i)/2^{d(i)}$ takes arbitrarily large values, it thus suffices to choose $i(k)$ such that $2^{d(i(k)) + e+1}/f(i(k)) \\le 2^{-2k-1}$ .", "Compression functions Our next example of a family of deep classes is given in terms of compression functions, which were introduced by Nies, Stephan, and Terwijn in to provide a characterization of 2-randomness (that is, $\\emptyset ^{\\prime }$ -Martin-Löf randomness) in terms of incompressibility.", "Definition 7.4 A $\\mathrm {K}$ -compression function with constant $c>0$ is a function $g: 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ such that $g \\le \\mathrm {K}+c$ and $\\sum _\\sigma 2^{-g(\\sigma )} \\le 1$ .", "We denote by $\\mathcal {CF}_c$ be the class of compression functions with constant $c$ .", "The condition $g(\\sigma ) \\le \\mathrm {K}(\\sigma )+c$ implies that $g(\\sigma ) \\le 2\|\\sigma \| + c +c^{\\prime } $ for some fixed constant $c^{\\prime }$ , and therefore $\\mathcal {CF}_c$ can be seen as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ , modulo encoding the functions as binary sequences.", "Of course $\\mathcal {CF}_c$ contains the function $\\mathrm {K}$ itself so it is a non-empty class.", "We now show the following.", "Theorem 7.5 For all $c\\ge 0$ , the class $\\mathcal {CF}_c$ is deep.", "Although we could give a direct proof following the same template as the previous examples, we will instead show that for all $c$ , we have $\\mathcal {K}_{f,\\ell ,d} \\le _s \\mathcal {CF}_c$ for some computable $f, \\ell , d$ (where $\\mathcal {K}_{f,\\ell ,d}$ is the class we defined in the previous section) such that $\\ell $ is increasing and $f/2^d$ takes arbitrarily large values, which by Theorem REF implies the depth of $\\mathcal {CF}_c$ .", "Let $g \\in \\mathcal {CF}_c$ .", "For all $n$ , since $\\sum _{\|\\sigma \|=n} 2^{-g(\\sigma )} \\le 1$ , there are at most $2^{n-1}$ strings of length $n$ such that $g(\\sigma ) < n - 1$ , and thus at least $2^{n-1}$ strings $\\sigma $ of length $n$ such that $g(\\sigma ) \\ge n - 1$ .", "Since $g \\in \\mathcal {CF}_c$ , for each $\\sigma $ such that $g(\\sigma ) \\ge \|\\sigma \| - 1$ , we also have $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Thus, given $g \\in \\mathcal {CF}_c$ as an oracle we can find, for each $n \\ge 3$ , $2^{n-1}$ strings $\\sigma $ of length $n$ such that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Setting $f(i)=2^{i+2}$ , $\\ell (i)=i+3$ and $d(i)=c+1$ , we have uniformly reduced $\\mathcal {K}_{f,\\ell ,d}$ to $\\mathcal {CF}_{c}$ .", "Since $d$ is a constant function it is obvious that $f(i)/2^d$ is unbounded, thus $\\mathcal {K}_{f,\\ell ,d}$ is deep (by Theorem REF ) and by Theorem REF so is $\\mathcal {CF}_c$ .", "The above result is not tight: the proof in fact shows that if $d$ is not a constant function but is such that $2^n/d(n)$ takes arbitrarily large values, then the class of functions $g$ such that $g(\\sigma ) \\le \\mathrm {K}(\\sigma ) + d(\|\\sigma \|)$ for all $\\sigma $ is a deep class.", "A notion related to $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ In $§$, we discuss lowness for randomness notions.", "Here we look at a dual notion, highness for randomness, specifically the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , whose precise characterization is an outstanding open question in algorithmic randomness.", "We shall see that this class is tightly connected to the notion of depth.", "Definition 7.5 A sequence $A\\in 2^\\mathbb {N}$ is in the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ if $\\mathsf {MLR}\\subseteq \\mathsf {CR}^A$ .", "The class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ itself not $\\Pi ^0_1$ (as it is closed under finite change of prefixes), but all its members have a “deep property\".", "Bienvenu and Miller proved that when $A$ is in $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , then for every c.e.", "set of strings $S$ such that $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ , $A$ computes a martingale $d$ such that $d(\\Lambda )=1$ and for some fixed rational $r>0$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ .", "It is straightforward to show that real-valued martingales can be approximated by dyadic-valued martingales with arbitrary precision; in particular one can assume that $d(\\sigma )$ is dyadic for all $\\sigma $ (and thus can be coded using $f(\|\\sigma \|)$ bits for some computable function $f$ ), still keeping the property $\\sigma \\in S \\Rightarrow d(\\sigma )>1+r$ .", "For a well-chosen $S$ , such martingales form a deep class.", "Theorem 7.6 Let $S$ be the set of strings $\\sigma $ such that $\\mathrm {K}(\\sigma ) < \|\\sigma \|$ (so that we have $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ ).", "Let $g$ be a computable function and $r>0$ be a rational number.", "Define the class $\\mathcal {W}_{g,r}$ to be the set of dyadic-valued martingales $d$ such that $d(\\Lambda )=1$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ and such that for all $\\sigma $ , $d(\\sigma )$ can be coded using $g(\|\\sigma \|)$ bits.", "Then $\\mathcal {W}_{g,r}$ can be viewed as a $\\Pi ^0_1$ class of $2^\\mathbb {N}$ , and this class is deep.", "Again, we are going to show the depth of $\\mathcal {W}_{g,r}$ by a Medvedev reduction from $\\mathcal {K}_{f,\\ell ,h}$ for some increasing computable function $\\ell $ and computable functions $f$ and $h$ such that $f/2^h$ is unbounded.", "More precisely, we will take $f(n)=n$ , $\\ell (n)=n + c$ (for a well-chosen $c$ given below) and $h(n)=0$ .", "Now suppose we are given a martingale $d \\in \\mathcal {W}_{f,r}$ .", "For any given length $n$ , we have, by the martingale fairness condition, $\\sum _{\\sigma : \|\\sigma \|=n} d(\\sigma ) = 2^n.$ It follows that there are at most $2^n/(1+r)$ strings $\\sigma $ of length $n$ such that $d(\\sigma )>1+r$ , and therefore at least $2^n-2^n/(1+r)$ strings such that $d(\\sigma ) \\le 1+r$ .", "Having oracle access to $d$ , such strings can be effectively found and listed.", "Since $2^n-2^n/(1+r) > n -c$ for all $n$ and some fixed constant $c$ , for each $n$ we can use $d$ to list $n$ strings $\\sigma _1,\\cdots ,\\sigma _n$ of length $n+c$ such that $d(\\sigma _i) \\le 1+r$ for $1\\le i\\le n$ .", "But by definition of $d$ , $d(\\sigma )\\le 1+r$ implies that $\\sigma \\notin S$ , which further implies that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \|$ .", "This shows that $\\mathcal {W}_{g,r}$ is above $\\mathcal {K}_{f,\\ell ,d}$ in the Medvedev degrees.", "By Theorem REF , $\\mathcal {K}_{f,\\ell ,h}$ is deep, and hence by Theorem REF , $\\mathcal {W}_{g,r}$ is deep as well.", "The examples of deep classes provided in this section, combined with Theorem REF , give us the results mentioned page : If $X$ is a difference random sequence, it does not compute any shift-complex sequence (Khan) it does not compute any $\\mathcal {DNC}_q$ function when $q$ is a computable order such that $\\sum _n 1/q(n) = \\infty $ (Miller) it does not compute any compression function (Greenberg, Miller, Nies) it is not $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ (Greenberg, Miller, Nies) The fact that a difference random cannot compute large sets of complex strings (in the sense of Theorem REF ) appears to be new.", "Lowness and depth Various lowness notions have been well-studied in algorithmic randomness, where a lowness notion is given by a collection of sequences that are in some sense computationally weak.", "Many lowness notions take the following form: For a relativizable collection $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , we say that $A$ is low for $\\mathcal {S}$ if $\\mathcal {S}\\subseteq \\mathcal {S}^A$ .", "For instance, if we let $\\mathcal {S}=\\mathsf {MLR}$ , then the resulting lowness notion consists of the sequences that are low for Martin-Löf random, a collection we write as $\\mathrm {Low}(\\mathsf {MLR})$ .", "Lowness notions need not be given in terms of relativizable classes.", "For instance, if we let $\\mathbf {m}^A$ be a universal $A$ -lower semi-computable discrete semi-measure, we can defined $A$ to be low for $\\mathbf {m}$ if $\\mathbf {m}^A(\\sigma )\\le ^\\mathbf {m}(\\sigma )$ for every $\\sigma $ .", "In addition, some lowness notions are not given in terms of relativization, such as the notion of $\\mathrm {K}$ -triviality, where $A\\in 2^\\mathbb {N}$ is $\\mathrm {K}$ -trivial if and only if $\\mathrm {K}(A{\\upharpoonright }n)\\le \\mathrm {K}(n)+O(1)$ (where we take $n$ to be $1^n$ ).", "Surprisingly, we have the following result (see for a detailed survey of results in this direction).", "Theorem 8.1 Let $A\\in 2^\\mathbb {N}$ .", "The following are equivalent: (i) $A\\in \\mathrm {Low}(\\mathsf {MLR})$ ; (ii) $A$ is low for $\\mathbf {m}$ ; (iii) $A$ is $\\mathrm {K}$ -trivial.", "The cupping problem, a longstanding open problem in algorithmic randomness involving $\\mathrm {K}$ -triviality, is to determine whether there exists a $\\mathrm {K}$ -trivial sequence $A$ and some Martin-Löf random sequence $X\\lnot \\ge _T\\emptyset ^{\\prime }$ such that $X\\oplus A\\ge _T\\emptyset ^{\\prime }$ .", "A negative answer was recently provided by Day and Miller : Theorem 8.2 (Day-Miller ) A sequence $A$ is K-trivial if and only if for every difference random sequence $X$ , $X \\oplus A \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Using the notion of depth, we can strengthen the Day-Miller result.", "First, we need to partially relativize the notion of depth: for $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $2^\\mathbb {N}$ is deep relative to $A$ if there is some computable order $f$ such that $\\mathbf {m}^A(T_{f(n)})\\le 2^{-n}$ if and only if there is some $A$ -computable order $g$ such that $\\mathbf {M}^A(T_{g(n)})\\le 2^{-n}$ (where $\\mathbf {M}^A$ is a universal $A$ -lower semi-computable continuous semi-measure).", "Theorem 8.3 Let $X$ be an incomplete Martin-Löf random sequence and $A$ be $K$ -trivial.", "Then $X \\oplus A$ does not compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with canonical co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ m(Tf(n)) 2-n$.", "Since~$ A$ is low for~$ m$, we have also have $ mA(Tf(n)) 2-n$, and thus $ is deep relative to $A$ .", "Let $X$ be a sequence such that $X \\oplus A$ computes a member of $ via a Turing functional $$ and suppose, for the sake of contradiction, that $ X$ is difference random.", "Let$$n = \\lbrace Z \\, : \\, \\Phi ^{Z \\oplus A} {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace .$$The set $ n$ can be written as the difference $ Un n$ of two $ A$-effectively open sets (uniformly in~$ n$) with $ Un = {Z : Z A f(n) }$ and $ n= {Z : Z A f(n) Tf(n)}$$ We can see the functional $Z \\mapsto \\Phi ^{Z \\oplus A}$ as an $A$ -Turing functional $\\Psi $ , and thus by the univerality of $\\mathbf {M}^A$ for the class of $A$ -lower semi-computable continuous semi-measures, we have $\\mathbf {M}^A \\ge ^* \\lambda _\\Psi $ .", "By definition of $n$ , we therefore obtain: $\\lambda (n) \\le \\lambda _\\Psi (T_{f(n)}) \\le ^* \\mathbf {M}^A(T_{f(n)}) < 2^{-n}.$ This shows that the sequence $X$ , which by assumption belongs to all $n$ , is not $A$ -difference random.", "It is, however, $A$ -Martin-Löf random as $A$ is low for Martin-Löf randomness.", "Relativizing Theorem REF to $A$ , this shows that $X \\oplus A \\ge _T A^{\\prime }$ .", "But this contradicts the Day-Miller theorem (Theorem REF ).", "As we cannot compute any members of a deep class by joining a Martin-Löf random sequence with a low for Martin-Löf random sequence, it is not unreasonable to ask if there is a notion of randomness $\\mathcal {R}$ such that we cannot $\\mathit {tt}$ -compute any members of a $\\mathit {tt}$ -deep class by joining an $\\mathcal {R}$ -random sequence with a low for $\\mathcal {R}$ sequence.", "We obtain a partial answer to this question using Kurtz randomness.", "From the discussion of lowness at the beginning of this section, we have $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $\\mathsf {KR}\\subseteq \\mathsf {KR}^A$ .", "Moreover, we define the class $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ to be the collection of sequences $A$ such that $\\mathsf {MLR}\\subseteq \\mathsf {KR}^A$ .", "Since $\\mathsf {MLR}\\subseteq \\mathsf {KR}$ , it follows that $\\mathrm {Low}(\\mathsf {KR})\\subseteq \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Miller and Greenberg obtained the following characterization of $\\mathrm {Low}(\\mathsf {KR})$ and $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Recall that $A\\in 2^\\mathbb {N}$ is computably dominated if every $f\\le _T A$ is dominated by some computable function.", "Theorem 8.4 (Greenberg-Miller ) Let $A\\in 2^\\mathbb {N}$ .", "(i) $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ if and only if $A$ is of non-DNC degree.", "(ii) $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ and computably dominated.", "For $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $ is \\emph {$ tt$-negligible relative to $ A$} if $ A(=0$ for every $ A$-computable measure $$.", "We first prove the following.$ Proposition 8.1 If $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ , then every deep $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible relative to $A$ .", "Let $ be a deep $ 01$ class and $ T$ its associated canonical co-c.e.\\ tree.", "Let $ ALow(MLR,KR)$, which by Theorem \\ref {thm:greenberg-miller-kr} (i) is equivalent to being of non-DNC degree.", "We appeal to a useful characterization of non-DNC degrees due to Hölzl and Merkle~\\cite {HolzlM2010}: $ A$ is of non-DNC degree if and only if it is \\emph {infinitely often c.e.\\ traceable} (hereafter, i.o.\\ c.e.\\ traceable).", "This means that there exists a computable order~$ h$, such that the following holds: for every total $ A$-computable function~$ s: NN$, there exists a family $ (Sn)nN$ of uniformly c.e.\\ finite sets such that $ \|Sn\|<h(n)$ for all~$ n$ and $ s(n) Sn$ for infinitely many~$ n$.$ Let $h$ be an order witnessing the i.o.", "c.e.", "traceability of $A$ .", "Since $ is deep, let~$ f$ be a computable function such that $ M(Tf(n)) < 2-2n/h(n)$.Suppose for the sake of contradiction that $ is not $\\mathit {tt}$ -negligible relative to $A$ , which means that there exists an $A$ -computable measure $\\mu ^A$ such that $\\mu ^A(>r$ for some rational $r>0$ .", "Let $s\\le _TA$ be the function that on input $n$ gives a rational lower-approximation, with precision $1/2$ , of the values of $\\mu ^A$ on all strings of length $f(n)$ (encoded as an integer).", "By this we mean that $s(n)$ gives us for all strings $\\sigma $ of length $f(n)$ a rational value $s(n,\\sigma )$ such that $\\mu ^A(\\sigma )/2 \\le s(n,\\sigma ) \\le \\mu ^A(\\sigma )$ .", "Let $(S_n)_{n\\in \\mathbb {N}}$ witness the traceability of $s$ , i.e., the $S_n$ 's are uniformly c.e., $\|S_n\|<h(n)$ for every $n$ , and $s(n) \\in S_n$ for infinitely many $n$ .", "We now build a lower semi-computable continuous semi-measure $\\rho $ as follows.", "For all $n$ , enumerate $S_n$ .", "For each member $z \\in S_n$ , interpret $z$ as a mass distribution $\\nu $ on the collection of strings of length $f(n)$ .", "Then, for each string $\\sigma $ of length $f(n)$ , increase $\\rho (\\sigma )$ (as well as strings comparable with $\\sigma $ in any way that ensures that $\\rho $ remains a semi-measure) by $2^{-n-1}\\nu (\\sigma )/h(n)$ .", "This has a total cost of $2^{-n-1}/h(n)$ , and since there are at most $h(n)$ elements in $S_n$ , the total cost at level $f(n)$ is at most $2^{-n-1}$ .", "Therefore the total cost of the construction of $\\rho $ is bounded by 1 and thus $\\rho $ is indeed a lower semi-computable continuous semi-measure Now, for any $n$ such that $s(n) \\in S_n$ (as there are infinitely many such $n$ ), $\\rho $ distributes an amount of at least $(\\mu ^A(T_{f(n)})/2) \\cdot (2^{-n-1}/h(n))$ on $T_{f(n)}$ , and since $\\mu ^A(T_n)>r$ , this gives $\\rho (T_{f(n)}) \\ge 2^{-n-O(1)}/h(n).$ However, we had assumed that $\\mathbf {M}(T_{f(n)}) \\le 2^{-2n}/h(n),$ for all $n$ .", "But since $\\rho \\le ^* \\mathbf {M}$ , we get a contradiction.", "The following result involves the notion of relative $\\mathit {tt}$ -reducibility.", "For a fixed $A\\in 2^\\mathbb {N}$ , a $\\mathit {tt}(A)$ -functional is a total $A$ -computable Turing functional.", "Equivalently, we can define a $\\mathit {tt}(A)$ -functional $\\Psi ^A$ in terms of a Turing functional $\\Phi $ as follows: Let $\\Phi $ be defined on all inputs of the form $X\\oplus A$ .", "Then we set $\\Psi ^A(X)=\\Phi (X\\oplus A)$ .", "Furthermore, one can show that there is an $A$ -computable bound on the use of $X$ in the computation (just as there is a computable bound in the use function for unrelativized $\\mathit {tt}$ -computations).", "Theorem 8.5 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class and $ ALow(KR)$.", "By Proposition \\ref {prop:low-for-tt-depth}, $ is also $\\mathit {tt}$ -deep relative to $A$ .", "Let $\\Phi ^A$ be a $\\mathit {tt}(A)$ -functional.", "The pre-image $ of $ under $\\Phi ^A$ is a $\\Pi ^0_1(A)$ class, which must be $\\mathit {tt}$ -deep relative to $A$ as well, by Theorem REF relativized to $A$ .", "Now, applying Proposition REF relativized to $A$ , $ contains no $ A$-Kurtz random sequence.", "But since $ A$ is low for Kurtz randomness, $ contains no Kurtz-random sequence as well.", "We now obtain a partial analogue of Theorem REF .", "Corollary 8.2 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $\\Phi $ be a $\\mathit {tt}$ -functional.", "Since $\\Phi $ is total, it is certainly total on all sequences of the form $X\\oplus A$ for $X\\in 2^\\mathbb {N}$ .", "Thus $\\Psi ^A(X)=\\Phi (X\\oplus A)$ is a $\\mathit {tt}(A)$ -functional.", "By Theorem REF , it follows that $\\Phi (X\\oplus A)$ cannot be contained in any deep class.", "Question 1 Does Corollary REF still hold if we replace “deep\" with “$\\mathit {tt}$ -deep\"?", "We can extend Theorem REF to the following result, which proceeds by almost the same proof, the details of which are left to the reader.", "Theorem 8.6 Let $X$ be Martin-Löf random and $A$ be $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "(in particular, $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class).", "Depth, mutual information, and the Independence Postulate In this final section, we introduce the notion of mutual information and apply it to the notion of depth.", "Roughly, what we prove is that every member of every deep class has infinite mutual information with Chaitin's $\\Omega $ , a Martin-Löf random sequence that encodes the halting problem.", "This generalizes a result of Levin's, that every consistent completion of $\\mathsf {PA}$ has infinite mutual information with $\\Omega $ .", "We conclude with a discussion of the Independence Postulate, a principle introduced by Levin to derive the statement that that no consistent completion of arithmetic is physically obtainable.", "The definition of mutual information First we review the definitions of Kolmogorov complexity of a pair and the universal conditional discrete semi-measure $\\mathbf {m}(\\cdot \\mid \\cdot )$ .", "Let $\\langle \\cdot ,\\cdot \\rangle :2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ be a computable bijection.", "Then we define $\\mathrm {K}(\\sigma ,\\tau ):=\\mathrm {K}(\\langle \\sigma ,\\tau \\rangle )$ .", "Similarly, we set $\\mathbf {m}(\\sigma ,\\tau ):=\\mathbf {m}(\\langle \\sigma ,\\tau \\rangle )$ .", "A conditional lower semi-computable discrete semi-measure $m(\\cdot \\mid \\cdot ):2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow [0,1]$ is a function satisfying $\\sum _\\sigma m(\\sigma \\mid \\tau )\\le 1$ for every $\\tau $ .", "Then $\\mathbf {m}(\\cdot \\mid \\cdot )$ is defined to be a universal conditional lower semi-computable discrete semi-measure, so that for every conditional lower semi-computable discrete semi-measure, there is some $c$ such that $m(\\sigma \\mid \\tau )\\le c\\cdot \\mathbf {m}(\\sigma \\mid \\tau )$ for every $\\sigma $ and $\\tau $ .", "Lastly, we define the conditional prefix-free Kolmogorov complexity $\\mathrm {K}(\\sigma \\mid \\tau )$ to be $\\mathrm {K}(\\sigma \\mid \\tau )=\\min \\lbrace \|\\xi \|:U(\\langle \\xi ,\\tau \\rangle )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free machine.", "The mutual information of two strings $\\sigma $ and $\\tau $ , denoted by $I(\\sigma :\\tau )$ , is defined by $I(\\sigma :\\tau ) = \\mathrm {K}(\\sigma ) + \\mathrm {K}(\\tau ) - \\mathrm {K}(\\sigma ,\\tau )$ or equivalently by $2^{I(\\sigma :\\tau )} = \\frac{\\mathbf {m}(\\sigma ,\\tau )}{\\mathbf {m}(\\sigma ) \\cdot \\mathbf {m}(\\tau )}.$ By the symmetry of information (see Gács ), we also have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} =^* \\frac{\\mathbf {m}(\\tau \\mid \\sigma ,\\mathrm {K}(\\sigma ))}{\\mathbf {m}(\\tau )}.$ Levin extends mutual information to infinite sequences by setting $2^{I(X:Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^X(\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}.$ Recall that Chaitin's $\\Omega $ can be obtained as the probability that a universal prefix-free machine will halt on a given input, that is, $\\Omega =\\sum _{U(\\sigma ){\\downarrow }}2^{-\|\\sigma \|}$ , where $U$ is a fixed universal prefix-free machine.", "Generalizing a result of Levin's from , we have: Theorem 9.1 Let $ be a $ 01$ class and $ T$ its associated co-c.e.\\ tree.", "Suppose~$ is deep, witnessed by a computable order $f$ such that $\\mathbf {m}(T_{f(n)}) < 2^{-n}$ .", "Then for every $Y \\in and all~$ n$,$$I\\Big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\Big ) \\ge n-O(\\log n).$$In particular,$$I(\\Omega :Y)=\\infty .$$$ Our proof follows the same idea Levin uses for consistent completions of $\\mathsf {PA}$ (see ), although some extra care is needed for arbitrary deep classes.", "Suppose for a given $n$ we have an exact description $\\tau $ of $T_{f(n)}$ ; that is, on input $\\tau $ , the universal machine outputs a code for the finite set $T_{f(n)}$ .", "By the definition of $f$ , $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\le 2^{-n}$ or equivalently $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\cdot 2^{n} \\le 1$ Therefore, the quantity $\\mathbf {m}(\\sigma ) \\cdot 2^{n} \\cdot \\mathbf {1}_{\\sigma \\in T_{f(n)}}$ is a discrete semi-measure, but it is not lower semi-computable since $T_{f(n)}$ is merely co-c.e.", "(and, in general, not c.e.", "by Proposition REF ).", "However, it is a lower semi-computable semi-measure relative to the exact description $\\tau $ of $T_{f(n)}$ .", "Thus, for every $\\sigma \\in T_{f(n)}$ , by the universality of $\\mathbf {m}(\\cdot \\mid \\tau )$ , $\\mathbf {m}(\\sigma \\mid \\tau ) \\ge ^* \\mathbf {m}(\\sigma ) \\cdot 2^n.$ By the symmetry of information, we have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} \\ge ^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau )}{\\mathbf {m}(\\sigma )} \\ge ^2^n,$ and hence $I(\\tau :\\sigma ) \\ge ^+ n$ .", "We would like to apply this fact to the case where $\\sigma = Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ .", "But this is not technically sufficient, as $\\Omega {\\upharpoonright }n$ does not necessarily contain enough information to exactly describe $T_{f(n)}$ .", "This is not an obstacle in Levin's argument for completions of $\\mathsf {PA}$ , but it is for arbitrary deep classes.", "However, $\\Omega {\\upharpoonright }n$ contains enough information to get a “good enough\" approximation of $T_{f(n)}$ .", "Let us refine the idea above: suppose now that $\\tau $ is no longer an exact description of $T_{f(n)}$ , but is a description of a set of strings $S$ of length $f(n)$ such that $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^ 2^{-n} n^2.$ Then, by following the same reasoning as above, we would have $I(\\sigma : \\tau ) \\ge ^+ n-2\\log n$ for all $\\sigma \\in S$ (and thus all $\\sigma \\in T_{f(n)}$ ).", "We shall prove that $\\Omega {\\upharpoonright }n$ contains enough information to recover such a set $S$ , thus proving the theorem.", "The real number $\\Omega $ is lower semi-computable and Solovay complete (see ).", "As a consequence, for every other lower semi-computable real $\\alpha $ , knowing the first $k$ bits of $\\Omega $ allows us to compute the first $k-O(1)$ bits of $\\alpha $ .", "For all $n$ , define: $a_n = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}}} \\mathbf {m}(\\sigma )$ and observe that $a_n$ is lower semi-computable uniformly in $n$ (because $T_{f(n)}$ is co-c.e.", "uniformly in $n$ ), and belongs to $[0,1]$ .", "Define now $\\alpha = \\sum _n \\frac{a_n}{n^2},$ which is a lower semi-computable real.", "Thus, knowing the first $n$ bits of $\\Omega $ gives us the first $n-O(1)$ bits of $\\alpha $ , i.e., an approximation of $\\alpha $ with precision $2^{-n}$ .", "In particular this gives us an approximation of $a_n$ with precision $2^{-n} \\cdot n^2 \\cdot O(1)$ , which we can assume to be a lower approximation, which we will write as $a^{\\prime }_n$ .", "Now, using $a^{\\prime }_n$ , one can enumerate $a_n$ until we find a stage $s_n$ such that $a_n[s_n] = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}[s_n]}} \\mathbf {m}(\\sigma )[s_n] \\ge a^{\\prime }_n.$ Since $\|a_n - a^{\\prime }_n\| \\le 2^{-n} \\cdot n^2 \\cdot O(1)$ , this implies $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}[s_n]} \\setminus T_{f(n)}}} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n^2.$ But recall from above that $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}}}} \\mathbf {m}(\\sigma ) \\le 2^{-n}.$ Combining these two facts, and taking $S$ to be the set $T_{f(n)}[s_n]$ , we have $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n ^2$ which establishes the first part of the theorem.", "To see that the second part of the statement follows from the first, take $\\sigma =Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ and observe that $\\mathbf {m}^Y(\\sigma ) =^* \\mathbf {m}^Y(n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ $\\mathbf {m}^\\Omega (\\tau ) =^* \\mathbf {m}^\\Omega (n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ and $2^{I(\\sigma :\\tau )}=2^n/n^{O(1)}$ (since $I(\\sigma :\\tau )\\ge ^+n-2\\log n$ as established above).", "Then we have $\\begin{split}2^{I(\\Omega :Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^\\Omega (\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}&\\ge \\sum _n \\mathbf {m}^\\Omega (Y{\\upharpoonright }f(n)) \\cdot \\mathbf {m}^Y(\\Omega {\\upharpoonright }n) \\cdot 2^{I(Y{\\upharpoonright }f(n),\\Omega {\\upharpoonright }n)}\\\\&\\ge \\sum _n 2^n/n^{O(1)}=\\infty .\\end{split}$ Remark 9.1 The converse of this theorem does not hold, i.e., there is a $\\Pi ^0_1$ class that is not deep but all of whose elements have infinite mutual information with $\\Omega $ .", "This follows from Theorem REF and the fact that having infinite mutual information with $\\Omega $ is a property that is invariant under addition or deletion of a finite prefix.", "Levin's proof of Theorem REF restricted to the particular case of completions of $\\mathsf {PA}$ is the mathematical part of a more general discussion, the other part of which is philosophical in nature.", "While Gödel's theorem asserts that no completion of $\\mathsf {PA}$ can be computably obtained, Levin's goal is to show that no completion of $\\mathsf {PA}$ can be obtained by any physical means whatsoever (computationally or otherwise), thus generalizing Gödel's theorem.", "Levin does not fully specify what he means by physically obtainable (the exact term he uses is “located in the physical world\"), but nonetheless he makes the following postulate, which he dubs the “Independence Postulate\": if $\\sigma $ is a mathematically definable string that has an $n$ bit description and $\\tau $ can be located in the physical world with a $k$ bit description, then for a fixed small constant $c$ , one has $I(\\sigma : \\tau ) < n+k+c$ .", "In particular, if one admits that some infinite sequences can be physically obtained, the Independence Postulate for infinite sequences says that if $X$ and $Y$ are two infinite sequences with $X$ mathematically definable and $Y$ physically obtainable, then $I(X:Y) < \\infty $ .", "Being $\\Delta ^0_2$ , $\\Omega $ is mathematically definable, and, as Levin shows, $I(\\Omega :Y)=\\infty $ for any completion $Y$ of $\\mathsf {PA}$ .", "Thus, assuming the Independence Postulate, no completion of $\\mathsf {PA}$ is physically obtainable.", "Our Theorem REF extends Levin's theorem and, assuming the Independence Postulate, shows that no member of a deep class (shift-complex sequences, compression functions, etc.)", "is physically obtainable.", "Of course, evaluating the validity of the Independence Postulate would require an extended philosophical discussion that would take us well beyond the scope of this paper.", "In any case, whether or not the reader accepts the Independence Postulate, Theorem REF is interesting in its own right.", "In fact, it is quite surprising because it seems to contradict the “basis for randomness theorem\" (see ), which states that if $X$ is a Martin-Löf random sequence and $\\mathcal {C}$ is a $\\Pi ^0_1$ class, then there exists a member $Y$ of $\\mathcal {C}$ such that $X$ is random relative to $Y$ .", "If a sequence $X$ is random relative to another sequence $Y$ , the intuition is that $Y$ “knows nothing about $X$ \", and thus one could conjecture that $I(X : Y) < \\infty $ .", "However, this cannot always be the case, since by Theorem REF , $I(\\Omega :Y) = \\infty $ for all members $Y$ of a deep $\\Pi ^0_1$ class $, even though $$ is random relative to some $ Y.", "This apparent paradox can be explained by taking a closer look at the definition of mutual information.", "Let $ be a deep $ 01$ class, whose canonical co-c.e.\\ tree~$ T$ satisfies $ m(Tf(n)) < 2-n$ for some computable function~$ f$.", "By Theorem~\\ref {thm:mutual-info} and the symmetry of information, for every $ Y we have $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big ) = I\\big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\big ) \\ge n-O(\\log n)$ where $k_n$ stands for $\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "Take a $Y \\in such that $$ is random relative to~$ Y$.", "It is well-known that a sequence~$ Z$ is random if and only if $ K(Z n n) n-O(1)$ (see for example Gács~\\cite {Gacs1980}).", "Applying this fact (relativized to~$ Y$) to $$, we have$$\\mathrm {K}^Y(\\Omega {\\upharpoonright }n \\mid n) \\ge n-O(1)$$and thus in particular that$$\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\ge n-O(1).$$Since $ K(n) n+O(n)$, it follows that$ $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\le O(\\log n)$ The only difference between (REF ) and (REF ) is the term $k_n=\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "But it makes a big difference, as one can verify that $\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n))-\\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big )\\ge n-O(\\log n).$ Informally, while $Y$ “knows nothing\" about $\\Omega $ , the complexity of its initial segments, seen as a function, does.", "In particular, the change of complexity caused by $k_n$ implies that $\\mathrm {K}(\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge n - O(\\log n)$ and thus $\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge 2^n/n^{O(1)}$ .", "Acknowledgements We would like to thank Noam Greenberg, Rupert Hölzl, Mushfeq Khan, Leonid Levin, Joseph Miller, André Nies, Paul Shafer, and Antoine Taveneaux for many fruitful discussions on the subject.", "Particular thanks go to Steve Simpson who provided very detailed feedback on the first arXiv version of this paper." ], [ "Computational limits of randomness", "As we have seen, negligible $\\Pi ^0_1$ classes (and thus deep classes) have the property that one cannot compute a member of them with positive probability.", "Although some random sequences can compute a member in any negligible $\\Pi ^0_1$ class (namely the Martin-Löf random sequences of $\\mathsf {PA}$ degree, as sequences of $\\mathsf {PA}$ degree compute a member of every $\\Pi ^0_1$ class), by the definition of negligibility, almost every random sequence fails to compute a member of a negligible class.", "Similarly, one cannot $\\mathit {tt}$ -compute a member of a $\\mathit {tt}$ -negligible $\\Pi ^0_1$ class with positive probability.", "The definition of $\\mathit {tt}$ -negligibility implies that the random sequences that can compute a member of a $\\mathit {tt}$ -negligible class form a set of Lebesgue measure zero.", "In this section, we specify a precise level of randomness at which computing a member of a $\\mathit {tt}$ -negligible, negligible, or deep class fails.", "First, we consider the case for $\\mathit {tt}$ -negligible classes.", "Theorem 5.1 If $X\\in 2^\\mathbb {N}$ is Kurtz random, it cannot $\\mathit {tt}$ -compute any member of a $\\mathit {tt}$ -negligible class.", "Let $ be $ tt$-negligible $ 01$ class and $$ be a $ tt$-functional.", "The set $ -1($ is a $ 01$ class that, by $ tt$-negligibility, has Lebesgue measure~$ 0$.", "Thus, it contains no Kurtz random.$ A similar proof can be used to prove the following: Theorem 5.2 If $X\\in 2^\\mathbb {N}$ is weakly 2-random, it cannot compute any member of a negligible class.", "Let $ be negligible $ 01$ class, $$ be a Turing functional, and $ T$ a computable tree such that $ [T]$.", "The set $ -1($ is a $ 02$ class, since$$X\\in \\Phi ^{-1}(\\;\\text{if and only if}\\;(\\forall k)(\\exists \\sigma )(\\exists n)[\\sigma \\in T\\;\\&\\;\|\\sigma \|=k\\;\\&\\; \\sigma \\preceq \\Phi ^{X {\\upharpoonright }n}]$$By negligibility, $ -1($ has Lebesgue measure~$ 0$ and thus contains no weakly 2-random sequence.$ Our next result, despite its simplicity, is probably the most interesting of this section.", "It will help us unify a number of theorems that have appeared in the literature.", "These are theorems of form ($$ ) If $X$ is difference random, then it cannot compute an element of $,$ where $ is a given $ 01$ class.", "Theorem~\\ref {thm:diff-pa} is an example of such a theorem, with $ the class of consistent completions of $\\mathsf {PA}$ .", "The same result has been obtained with $ the class of shift-complex sequences (Khan \\cite {Khan2013}), the set of compression functions (Greenberg, Miller, Nies \\cite {GreenbergMN-ip}), and the set $ DNCq$ functions for some orders~$ q$ (Miller, unpublished).", "We will give the precise definition of these classes in $ §$\\ref {sec:examples} but the important fact is that all of these classes are deep, and indeed, showing the depth of a $ 01$ class is sufficient to obtain a theorem of the form ($ $).", "$ Theorem 5.3 If a sequence $X$ is difference random, it cannot compute any member of a deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with associated co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ M(Tf(n)) < 2-n$.", "Let~$ X$ be a sequence that computes a member of $ via a Turing functional $\\Phi $ .", "Let $\\mathcal {Z}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace $ The set $\\mathcal {Z}_n$ can be written as the difference $\\mathcal {U}_n \\setminus n$ of two effectively open sets (uniformly in $n$ ) with $\\mathcal {U}_n = \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\rbrace $ and $n= \\lbrace Z \\, : \\, \\Phi ^Z {\\upharpoonright }f(n) \\downarrow \\notin T_{f(n)}\\rbrace $ Moreover, by definition of the semi-measure induced by $\\Phi $ , $\\lambda (\\mathcal {Z}_n) \\le \\lambda _\\Phi (T_{f(n)}) \\le ^* \\mathbf {M}(T_{f(n)}) < 2^{-n}.$ The sequence $(\\mathcal {Z}_n)_{n\\in \\mathbb {N}}$ thus yields a difference test.", "Therefore, the sequence $X$ , which by assumption belongs to all $\\mathcal {Z}_n$ , is not difference random.", "We remark that the converse does not hold: i.e., there is a class $ such that (i) $ is not deep but (ii) no difference random real can compute an element of $.", "Indeed, take a deep class~$ and apply Theorem REF to get a class $ that is not deep but whose members have the same Turing degrees as the elements of~$ .", "Thus, no difference random real can compute an element of $.$" ], [ "Depth, negligibility, and mass problems", "In this section we discuss depth and negligibility in the context of mass problems, i.e., in the context of the Muchnik and Medvedev reducibility.", "Both Muchnik and Medvedev reducibility are generalizations of Turing reducibility.", "Whereas Turing reducibility is defined in terms of a pair of sequences, both Muchnik and Medevedev reducibility are defined in term of a pair of collections of sequences.", "In what follows, we will consider these two reducibilities when restricted to $\\Pi ^0_1$ subclasses of $2^\\mathbb {N}$ .", "We will follow the notation of the survey , to which we refer the reader for a thorough exposition of mass problems in the context of $\\Pi ^0_1$ classes.", "Let $2^\\mathbb {N}$ be $\\Pi ^0_1$ classes.", "We say that $ is \\emph {Muchnik reducible} to $ , denoted $_w, if for every $ X there exists a Turing functional $\\Phi $ such that (i) $X\\in \\mathsf {dom}(\\Phi )$ and $\\Phi (X)\\in .", "Moreover, $ is Medvedev reducible to $, denoted $ s, if $ is Muchnik reducible to $ via a single Turing functional, i.e., there exists a Turing functional $\\Phi $ such that (i) $\\mathsf {dom}(\\Phi )$ and $\\Phi (\\subseteq .$ Just as Turing reducibility gives rise to a degree structure, we can define degree structures from $\\le _w$ and $\\le _s$ .", "We say that $ and $ are Muchnik equivalent (resp.", "Medvedev equivalent), denoted $_w (resp.\\ $ s) if and only if $_w and $ w (resp.", "$_s and $ s).", "The collections of Muchnik and Medvedev degrees given by the equivalence classes under $\\equiv _w$ and $\\equiv _s$ are denoted $\\mathcal {E}_w$ and $\\mathcal {E}_s$ , respectively.", "Both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are lattices, unlike the Turing degrees, which only form an upper semi-lattice.", "We define the meet and join operations as follows.", "Given $\\Pi ^0_1$ classes $2^\\mathbb {N}$ , $\\sup ($ is the $\\Pi ^0_1$ class $\\lbrace X\\oplus Y:X\\in \\&\\;Y\\in $ .", "Furthermore, we define $\\inf ($ to be the $\\Pi ^0_1$ class $\\lbrace 0^\\frown X: X\\in \\cup \\lbrace 1^\\frown Y:Y\\in $ .", "One can readily check that the least upper bound of $ and $ in $\\mathcal {E}_w$ is the Muchnik degree of $\\sup ($ while their greatest lower bound is the Muchnik degree $\\inf ($ , and similarly for $\\mathcal {E}_s$ .", "Recall that a filter $\\mathcal {F}$ in a lattice $(\\mathcal {L},\\le ,\\inf ,\\sup )$ is a subset that satisfies the following two conditions: (i) for all $x,y\\in \\mathcal {L}$ , if $x\\in \\mathcal {F}$ and $x\\le y$ , then $y\\in \\mathcal {F}$ , and (ii) for all $x,y\\in \\mathcal {F}$ , $\\inf (x,y)\\in \\mathcal {F}$ .", "The goal of this section is to study the role of depth and negligibility in the structures $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let us start with an easy result.", "Theorem 6.1 The collection of negligible $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Let $ and $ be negligible $\\Pi ^0_1$ classes.", "By Proposition REF , we have that $\\lambda ((^{\\le _T})=0$ and $\\lambda ((^{\\le _T})=0$ .", "But since $\\inf (^{\\le _T}=(^{\\le _T}\\cup (^{\\le _T},$ it follows that $\\lambda (\\inf (^{\\le _T})=0$ , which shows that $\\inf ($ is negligible.", "Thus, the degrees of negligible classes in both $\\mathcal {E}_w$ and $\\mathcal {E}_s$ are closed under $\\inf $ .", "Let $\\mathcal {D}$ be non-negligible $\\Pi ^0_1$ and $\\mathcal {C}\\le _w\\mathcal {D}$ .", "For each $i$ , we define $i:=\\lbrace X\\in X\\in \\mathsf {dom}(\\Phi _i)\\;\\&\\;\\Phi _i(X)\\in .$ Since $_w, it follows that $ ii$.", "Furthermore, we have $ (T=(ii)T=i(i)T$.", "Since $ is non-negligible, we have $0<\\lambda ((^{\\le _T})\\le \\sum _i\\lambda \\big (({i})^{\\le _T}\\big ),$ and thus $\\lambda (({k})^{\\le _T})>0$ for some $k$ .", "But since $\\Phi _{k}({k})\\subseteq , it follows that $ ((T)>0$, thus $ is non-negligible.", "Thus, negligibility is closed upwards under $\\le _w$ (and a fortiori, under $\\le _s$ as well).", "Remark 6.1 Simpson proved in that in $\\mathcal {E}_s$ , the complement of the filter of negligible classes is in fact a principal ideal, namely the ideal generated by the class $\\inf (\\mathcal {PA},2\\mathcal {RAN})$ , where $2\\mathcal {RAN}$ is the class of 2-random sequences, i.e.", "the sequences that are Martin-Löf random relative to $\\emptyset ^{\\prime }$ .", "For the next two theorems, we need the following fact.", "Fact 6.2 Let $ and $ be $\\Pi ^0_1$ classes such that $_s.", "Then there is a total Turing functional $$ such that $ (.", "This holds because for every $\\Pi ^0_1$ class $ and every Turing functional~$$ that is total on $ , there is a functional $\\Psi $ that is total on $2^\\mathbb {N}$ and coincides with $\\Phi $ on $.$ Theorem 6.2 The collection of deep $\\Pi ^0_1$ classes forms a filter in $\\mathcal {E}_s$ .", "Let $ and $ be deep $\\Pi ^0_1$ classes with associated co-c.e.", "trees $S$ and $T$ , respectively.", "Moreover, let $g$ and $h$ be computable orders such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ and $\\mathbf {M}(T_n) < 2^{-h(n)}$ .", "We define $f(n)=\\min \\lbrace g(n),h(n)\\rbrace $ , which is clearly a computable order.", "It follows immediately that $\\mathbf {M}(S_n)\\le 2^{-f(n)}$ and $\\mathbf {M}(T_n) < 2^{-f(n)}$ .", "Now the co-c.e.", "tree associated with $\\inf ($ is $R=\\lbrace 0^\\frown \\sigma :\\sigma \\in S\\rbrace \\cup \\lbrace 1^\\frown \\tau :\\tau \\in T\\rbrace .$ Consider the class $\\inf ($ .", "Setting $0^\\frown S_n=\\lbrace 0^\\frown \\sigma :\\sigma \\in S_n\\rbrace $ and $1^\\frown T_n=\\lbrace 1^\\frown \\sigma :\\sigma \\in T_n\\rbrace $ for each $n$ , the co-c.e.", "tree of $\\inf ($ is $0^\\frown S_n \\cup 1^\\frown T_n$ .", "Moreover, $\\mathbf {M}(0^\\frown S_n \\cup 1^\\frown T_n) = \\mathbf {M}(0^\\frown S_n) + \\mathbf {M}(1^\\frown T_n) \\le ^* 2^{-f(n)} + 2^{-f(n)}$ which shows that $\\inf ($ is deep.", "Next, suppose that $\\mathcal {C}$ is a deep $\\Pi ^0_1$ class and $\\mathcal {D}$ is a $\\Pi ^0_1$ class satisfying $\\mathcal {C} \\le _s \\mathcal {D}$ via the Turing functional $\\Psi $ , which we can assume to be total by Fact REF .", "Let $S$ and $T$ be the co-c.e.", "trees associated to $ and $ , respectively, and let $g$ be a computable order such that $\\mathbf {M}(S_n)\\le 2^{-g(n)}$ .", "Since $\\Psi $ is total, its use is bounded by some computable function $f$ .", "It follows that for every $\\sigma \\in T_{f(n)}$ , there is some $\\tau \\in S_n$ such that $(\\sigma ,\\tau )\\in \\Psi $ .", "Thus $\\llbracket T_{f(n)}\\rrbracket \\subseteq \\Psi ^{-1}(\\llbracket S_n\\rrbracket )$ .", "Now let $\\Phi $ be a universal Turing functional such that $\\mathbf {M}=\\lambda _\\Phi $ .", "Then: $\\mathbf {M}(T_{f(n)})=\\lambda (\\Phi ^{-1}( T_{f(n)}))\\le \\lambda (\\Phi ^{-1}(\\llbracket T_{f(n)}\\rrbracket )\\le \\lambda (\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket )))\\le \\lambda _{\\Psi \\circ \\Phi }(S_n),$ where the last inequality holds because $\\Phi ^{-1}(\\Psi ^{-1}(\\llbracket S_n\\rrbracket ))=(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq \\mathsf {dom}(\\Psi \\circ \\Phi )$ and hence $(\\Psi \\circ \\Phi )^{-1}(\\llbracket S_n\\rrbracket )\\subseteq (\\Psi \\circ \\Phi )^{-1}(S_n)$ , $\\lambda _{\\Psi \\circ \\Phi }\\le ^* \\mathbf {M}$ .", "Thus we have $\\mathbf {M}(T_{f(n)})\\le \\lambda _{\\Psi \\circ \\Phi }(S_n)\\le ^* \\mathbf {M}(S_n)\\le 2^{-g(n)}.$ Thus, $\\mathbf {M}(T_{f \\circ g^{-1}(n)}) \\le 2^{-n}$ , which shows that $ is deep.$ The invariance of the notion of depth under Medvedev-equivalence is of importance for the next section.", "There, we prove that certain classes of objects, which are not necessarily infinite binary sequences, are deep.", "To do so, we fix a certain encoding of these objects by infinite binary sequences and prove the depth of the corresponding encoded class.", "By the above theorem, the particular choice of encoding is irrelevant: if the class is deep for an encoding, it will be deep for another encoding, as long as switching from the first encoding to the second one can be done computably and uniformly.", "One could ask whether we have a similar result in the lattice $\\mathcal {E}_w$ .", "However, Theorem REF shows that depth is not invariant under Muchnik equivalence: if we apply this theorem to a deep class $, we get a $ 01$ class~$ which is clearly Muchnik equivalent to $ but is not deep itself.", "Thus depth does not marry well with Muchnik reducibility and is only a `Medvedev notion^{\\prime }.", "\\\\$ More surprisingly, $\\mathit {tt}$ -depth is a `Muchnik notion' as well as a `Medvedev notion' and the $\\mathit {tt}$ -deep classes form a filter in both lattices.", "Theorem 6.3 The collection of $\\mathit {tt}$ -deep $\\Pi ^0_1$ classes forms a filter in both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ .", "Suppose that $_w and $ is not $\\mathit {tt}$ -deep.", "By definition, this means that $ has positive $$-measure for some computable probability measure~$$.", "Let $ {Uk}k N$ be the universal $$-Martin-Löf test and define $ Rk$ to be the complement of $ Uk$ (which is a $ 01$ class).", "Since $ (Rk) > 1- 2-k$ for every $ k$, there must be a~$ j$ such that $ (Rj) > 0$ and in particular $ Rj = $.", "By the hyperimmune-free basis theorem, there is some $ XRj$ of hyperimmune-free Turing degree.", "Since $ w , $X$ must compute some element $Y$ of $.", "But since~$ X$ is of hyperimmune-free degree, $ X$ in fact $ tt$-computes $ Y$, i.e., $ (X)=Y$ for some total functional~$$.$ Since $X$ is $\\mu $ -Martin-Löf random, by the preservation of randomness theorem (see, for instance, Theorem 3.2 in ), $Y$ is Martin-Löf random with respect to the computable measure $\\mu _\\Psi $ defined by $\\mu _\\Psi (\\sigma ):=\\mu (\\Psi ^{-1}(\\sigma ))$ for every $\\sigma $ .", "Thus, by Proposition REF , $ is not $ tt$-deep.", "This shows, a fortiori, that if $ s and $ is not $ tt$-deep, then $ is not $\\mathit {tt}$ -deep.", "Thus $\\mathit {tt}$ -depth is closed upwards in $\\mathcal {E}_w$ and $\\mathcal {E}_s$ and in particular is compatible with the equivalence relations $\\equiv _s$ and $\\equiv _w$ .", "Next, suppose that $ and $ are $\\mathit {tt}$ -deep classes but that $\\inf ($ is not $\\mathit {tt}$ -deep (recall that the $\\inf $ operator is the same for both $\\mathcal {E}_s$ and $\\mathcal {E}_w$ ).", "Then by Theorem REF , $\\inf ($ contains a sequence $X$ that is $\\mu $ -Martin-Löf random for some computable measure $\\mu $ .", "Then for $i=0,1$ we define computable measures $\\mu _i$ such that $\\mu _i(\\sigma )=\\mu (i^\\frown \\sigma )$ for every $\\sigma $ .", "It is routine to check that $Y$ is $\\mu _i$ -Martin-Löf random if and only if $i^\\frown Y$ is $\\mu $ -Martin-Löf random for $i=0,1$ .", "Since $X=i^\\frown Z$ for some $i=0,1$ and $Z\\in 2^\\mathbb {N}$ , it follows that $Z$ is $\\mu _i$ -Martin-Löf random.", "But then $Z$ is contained in either $ or $ , which contradicts our hypothesis that $ and $ are both $\\mathit {tt}$ -deep." ], [ "Examples of deep $\\Pi ^0_1$ classes", "In this section, we provide a number of examples of deep $\\Pi ^0_1$ classes that naturally occur in computability theory and algorithmic randomness.", "We give a uniform treatment of all these classes, i.e., we give a generic method to prove the depth of $\\Pi ^0_1$ classes." ], [ "Consistent completions of Peano Arithmetic", "As mentioned in $§$, Jockusch and Soare proved in that the $\\Pi ^0_1$ class $\\mathcal {PA}$ of consistent completions of $\\mathsf {PA}$ is negligible.", "However, as shown by implicitly by Levin in and Stephan in , $\\mathcal {PA}$ is also deep.", "We will reproduce this result here.", "Following both Levin and Stephan, we will use fact that the class of consistent completions of $\\mathsf {PA}$ is Medvedev equivalent to the class of total extensions of a universal, partial-computable $\\lbrace 0,1\\rbrace $ -valued function.", "Thus, by showing the latter class is deep, we thereby establish that the former class is deep (via Theorem REF ).", "Theorem 7.1 (Levin , Stephan ) Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of all $\\lbrace 0,1\\rbrace $ -valued partial computable functions.", "Let $u$ be a function that is universal for this collection, e.g., defined by $u(\\langle e,x\\rangle ) = \\phi _e(x)$ .", "Then the class $ of total extensions of~$ u$ is a deep $ 01$ class.$ We build a partial computable function $\\phi _e$ , whose index we know in advance by the recursion theorem.", "This means that we control the value of $u(\\langle e,x\\rangle )$ for all $x$ .", "First, we partition $\\mathbb {N}$ into consecutive intervals $I_1, I_2, ...$ such that we control $2^{k+1}$ values of $u$ inside $I_k$ .", "For each $k$ in parallel, we define $\\phi _e$ on $I_k$ as follows.", "Step 1: Wait for a stage $s$ such that the set $E_k[s]=\\lbrace \\sigma \\mid \\sigma {\\upharpoonright }I_k ~ \\text{extends}~ u[s] {\\upharpoonright }I_k \\rbrace $ is such that $\\mathbf {M}(E_k)[s] \\ge 2^{-k}$ .", "Step 2: Find a $y \\in I_k$ that we control and on which $u_s$ is not defined.", "Consider the two “halves\" $E^0_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=0\\rbrace $ and $E^1_k[s]=\\lbrace \\sigma \\in E_k[s] \\mid \\sigma (y)=1\\rbrace $ of $E_k[s]$ .", "Note that either $\\mathbf {M}(E^0_k[s])\\ge 2^{-k-1}$ or $\\mathbf {M}(E^1_k)[s] \\ge 2^{-k-1}$ .", "If the first holds, set $u(y)[s+1]=1$ , otherwise set $u(y)[s+1]=0$ .", "Go back to Step 1.", "The co-c.e.", "tree $T$ associated to the class $ is the set of strings~$$ such that~$$ is an extension of $ u \|\|$.", "The construction works because every time we pass by Step 2, we remove from~$ T[s]$ a set $ Eik[s]$ (for some $ i{0,1}$ and $ k,sN$) such that $ M(Eik)[s]>2-k-1$.", "Therefore, Step 2 can be executed at most $ 2k+1$ times, and by the definition of $ Ik$ we do not run out of values~$ y Ik$ on which we control~$ u$.", "Therefore, the algorithm eventually reaches Step 1 and waits there forever.", "Setting $ f(k)=(Ik)$, this implies that the $ M$-weight of the set $ {f(k) extends u}$ is bounded by $ 2-k$, or equivalently,$$\\mathbf {M}(T_{f(k)}) < 2^{-k},$$which proves that~$ is deep.", "The proof provided here gives us a general template to prove the depth of a $\\Pi ^0_1$ class.", "First of all, the definition of the $\\Pi ^0_1$ class should allow us to control parts of it in some way, either because we are defining the class ourselves or because, as in the above proof, the definition of the class involves some universal object which we can assume to partially control due to the recursion theorem.", "All the other examples of deep $\\Pi ^0_1$ class we will see below belong to this second category.", "Let us take a step back and analyze more closely the structure of the proof of Theorem REF .", "Given a $\\Pi ^0_1$ class $ with canonical co-c.e.\\ tree $ T$, the proof consists of the following steps.\\\\$ (1) For a given $k$ , we identify a level $N=f(k)$ at which we wish to ensure $\\mathbf {M}(T_N) < 2^{-k}$ .", "The choice of $N$ will depend on the particular class $.", "\\\\$ (2) Next we implement a two-step strategy to ensure that $\\mathbf {M}(T_N)<2^{-k}$ .", "Such a strategy will be called a $k$ -strategy.", "(2.1) First, we wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] > 2^{-k}$ .", "(2.2) If at some stage $s$ this occurs, we remove one or several of the nodes of $T[s]$ at level $N$ such that the total $\\mathbf {M}[s]$ -weight of these nodes is at least $\\delta (k)$ for a certain function $\\delta $ , and then go back to Step 2.", "(3) Each execution of Step 2 (removing nodes from $T[s]$ for some $s$ ) comes at a cost $\\gamma (k)$ for some $k$ , and we need to make sure that we do not go over some maximal total cost $\\Gamma (k)$ throughout the execution of the $k$ -strategy.", "In the above example, $\\delta (k)=2^{-k-1}$ and the only cost for us is to define one value of $u(y)$ out of the $2^{k+1}$ we control, so we can, for example, set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ .", "By definition of $\\mathbf {M}$ , the $k$ -strategy can only go through Step 2 at most $1/\\delta (k)$ times, and thus the total cost of the $k$ -strategy will be at most $\\gamma (k)/\\delta (k)$ .", "All we have to do is to make sure that $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ (which is the case in the above example).", "Again, we have some flexibility on the choice of $N=f(k)$ , so it will suffice to choose an appropriate $N$ to ensure $\\gamma (k)/\\delta (k) \\le \\Gamma (k)$ .", "In some cases, there will not be a predefined maximum for each $k$ , but rather a global maximum $\\Gamma $ that the sum of the the costs of the $k$ -strategies should not exceed, i.e., we will want to have $\\sum _k \\gamma (k)/\\delta (k) \\le \\Gamma $ .", "Let us now proceed to more examples of $\\Pi ^0_1$ classes." ], [ "Shift-complex sequences", "Definition 7.1 Let $\\alpha \\in (0,1)$ and $c>0$ .", "(i) $\\sigma \\in 2^{<\\mathbb {N}}$ (resp.", "$X\\in 2^\\mathbb {N}$ ) is said to be $(\\alpha ,c)$ -shift complex if $\\mathrm {K}(\\tau ) \\ge \\alpha \|\\tau \| - c$ for every substring $\\tau $ of $\\sigma $ (resp.", "of $X$ ).", "(ii) $X\\in 2^\\mathbb {N}$ is said to be $\\alpha $ -shift complex if it is $(\\alpha ,c)$ -shift complex for some $c$ .", "The very existence of $\\alpha $ -shift complex sequences is by no means obvious.", "Such sequences were first constructed by Durand et al.", "who showed that there exist $\\alpha $ -shift complex sequences for all $\\alpha \\in (0,1)$ .", "It is easy to see that for every computable pair $(\\alpha ,c)$ , the class of $(\\alpha ,c)$ -shift complex sequences is a $\\Pi ^0_1$ class.", "Rumyantsev proved that the class of $(\\alpha ,c)$ -complex sequences is always negligible, but in fact, his proof essentially shows that it is even deep.", "The cornerstone of Rumyantsev's theorem is the following lemma.", "It relies on an ingenious combinatorial argument that we do not reproduce here.", "We refer the reader to for the full proof.", "Lemma 7.2 (Rumyantsev ) Let $\\beta \\in (0,1)$ .", "For every rational $\\eta \\in (0,1)$ and integer $d$ , there exist two integers $n$ and $N$ , with $n<N$ , such that the following holds.", "For every probability distribution $P$ on $\\lbrace 0,1\\rbrace ^N$ , there exist finite sets of strings $A_n, A_{n+1}, ..., A_N$ such that for all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements; and the $P$ -probability that a sequence $X$ has some substring in $\\cup _{i=n}^N A_i$ is at least $1-\\eta $ .", "Moreover, $n$ and $N$ can be effectively computed from $\\eta $ and $d$ and can be chosen to be arbitrarily large.", "Once $n$ and $N$ are fixed, the sets $A_i$ can be computed uniformly in $P$ .", "This is Lemma 6 of (Rumyantsev does not explicitly state that the conclusion holds for all computable probability measures, but nothing in his proof makes use of a particular measure).", "Theorem 7.2 For any computable $\\alpha \\in (0,1)$ and integer $c>0$ , the $\\Pi ^0_1$ class of $(\\alpha ,c)$ -shift complex sequences is deep.", "Let $ be the $ 01$ class of $ (,c)$-shift complex sequences and let~$ T$ be its canonical co-c.e.\\ tree.", "We shall build a discrete semi-measure~$ m$ whose coding constant~$ e$ we know in advance.", "This means that whenever we will set a string~$$ to be such that $ m() > 2-\|\| + c + e +1 $, then automatically we will have $ m() > 2-\|\| + c +1$, and thus $ K() < \|\| - c$.", "This will \\textit {de facto} remove $$ from~$ T$.$ Let us now turn to the construction.", "First, we pick some $\\beta $ such that $0 < \\beta < \\alpha $ .", "For each $k$ , we apply the above Lemma REF to $\\beta $ and $\\eta = 1/2$ to obtain a pair $(n,N)$ with the above properties as described in the statement of the lemma (we will also make use of the fact that $n$ can be chosen arbitrarily large, see below).", "Then the two step strategy is the following: Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Let $P$ be the (computable) probability distribution on $\\lbrace 0,1\\rbrace ^N$ whose support is contained in $T_N[s]$ and such that for all $\\sigma \\in T_N[s]$ , $P(\\sigma )=2^k \\cdot \\mathbf {M}(\\sigma )[s]$ .", "By Lemma REF , we can compute a collection of finite sets $A_i$ such that: (i) For all $i \\in [n,N]$ , $A_i$ contains only strings of length $i$ and has at most $2^{\\beta i}$ elements.", "(ii) The $P$ -probability that a sequence $X$ has some substring in $F = \\cup _{i=n}^N A_i$ is at least $1-\\eta =1/2$ .", "Thus, $\\mathbf {M}(T_N \\cap F)[s] > 2^{-k-1}$ .", "Then, for each $i$ and each $\\sigma \\in A_i$ , we ensure, increasing $m$ if necessary, that $m(\\sigma ) > 2^{-\\alpha \|\\sigma \| + c + e +1 }$ .", "As explained above, this ensures that all strings in $F$ are removed from $T_N$ at that stage, and therefore we have removed a $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ .", "Then we go back to Step 1.", "To finish the proof, we need to make sure that this algorithm does not cost us too much.", "The constraint here is that we need to make sure that $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau ) \\le 1$ , so we are trying to stay under a global cost of $\\Gamma =1$ .", "For a given $k$ , our cost at each execution of Step 2 is the increase of $m$ on the strings in $F$ .", "The total $m$ -weight we add at during one such execution of Step 2 is at most $\\gamma (k) = \\sum _{i=n}^N \|A_i\| \\cdot 2^{-\\alpha i + c + e +1 } \\le \\sum _{i=n}^N 2^{(\\beta -\\alpha ) i + c+ e + 1} \\le \\frac{2^{(\\beta -\\alpha ) n + c+ e + 1}}{1-2^{\\beta -\\alpha }}$ But since $n=n(k)$ can be chosen arbitrarily large, therefore, with an appropriate choice of $n$ , we can make $\\gamma (k) \\le 2^{-2k-2}$ .", "Therefore, since the $k$ -strategy executes Step 2 at most $1/\\delta (k)$ times, we have $\\sum _{\\tau \\in 2^{<\\mathbb {N}}} m(\\tau )\\le \\sum _{k\\in \\mathbb {N}} \\gamma (k)/\\delta (k) \\le \\sum _{k\\in \\mathbb {N}} 2^{-2k-2}/2^{-k-1} \\le 1$ and therefore $m$ is indeed a discrete semi-measure." ], [ "DNC$_q$ functions", "Let $(\\phi _e)_{e\\in \\mathbb {N}}$ be a standard enumeration of partial computable functions from $\\mathbb {N}$ to $\\mathbb {N}$ .", "Let $J$ be the universal partial computable function defined as follows.", "For all $(e,x) \\in \\mathbb {N}^2$ : $J(2^e(2x+1))=\\phi _e(x)$ The following notion was studied in .", "Definition 7.3 Let $q: \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable order.", "The set $\\mathcal {DNC}_q$ is the set of total functions $f: \\mathbb {N}\\rightarrow \\mathbb {N}$ such that for all $n$ : (i) $f(n) \\ne J(n)$ (where this condition is trivially satisfied if $J(n)$ is undefined), and (ii) $f(n) < q(n)$ .", "We should note that this is a slight variation of the standard definition of diagonal non-computability, which is formulated in terms of the condition (i$^{\\prime }$ ) $f(n)\\ne \\phi _n(n)$ instead of the condition (i) given above.", "However, this condition would make the class $\\mathcal {DNC}_q$ and the results below dependent on the particular choice of enumeration of the partial computable functions, while with condition $(i)$ , the results of the section are independent of the chosen enumeration.", "Note that for every fixed computable order $q$ , the class $\\mathcal {DNC}_q$ is a $\\Pi ^0_1$ subset of $\\mathbb {N}^\\mathbb {N}$ , but due to the bound $q$ , $\\mathcal {DNC}_q$ can be viewed as a $\\Pi ^0_1$ subset of $2^\\mathbb {N}$ .", "Whether the class $\\mathcal {DNC}_q$ is deep turns out to depend on $q$ .", "Indeed, we have the following interesting dichotomy theorem.", "Theorem 7.3 Let $q$ be a computable order.", "(i) If $\\sum _n 1/q(n) < \\infty $ , then $\\mathcal {DNC}_q$ is not $\\mathit {tt}$ -negligible.", "(ii) If $\\sum _n 1/q(n) = \\infty $ , then $\\mathcal {DNC}_q$ is deep.", "Part $(i)$ is a straightforward adaptation of Kučera's proof that every Martin-Löf random element computes a DNC function.", "Pick an $X \\in 2^\\mathbb {N}$ at random and use it as a random source to randomly pick the value of $f(i)$ uniformly among $\\lbrace 0,...,q(i)-1\\rbrace $ , independently of the other values of $f$ .", "The details are as follows.", "First, we can assume that $q(n)$ is a power of 2 for all $n$ .", "Indeed, since for $q^{\\prime } \\le q$ the class $\\mathcal {DNC}_{q^{\\prime }}$ is contained in $\\mathcal {DNC}_{q}$ , so if we take $q^{\\prime }(n)$ to be the largest power of 2, we have that $q^{\\prime }$ is computable, $q^{\\prime } \\le q$ , and $\\sum _n 1/q(n) < \\infty $ because $q^{\\prime }$ is equal to $q$ up to factor 2.", "Now, set $q(n)=2^{r(n)}$ .", "Split $\\mathbb {N}$ into intervals where interval $I_n$ has length $r(n)$ .", "One can now interpret any infinite binary sequence $X$ as a funtion $f_X: \\mathbb {N}\\rightarrow \\mathbb {N}$ , where $X(n)$ is the index of $X {\\upharpoonright }I_n$ in the lexicographic ordering of strings of length $r(n)$ .", "For $X$ taken at random with respect to the uniform measure, the event $f_X(n) = \\phi _n(n)$ has probability at most $1/q(n)$ and is independent of all such events for $n^{\\prime } \\ne n$ .", "Thus the total probability over $X$ such that $X(n)\\ne \\phi _n(n)$ for all $n$ is at least $\\prod _n (1-1/q(n))$ an expression that is positive if and only if $\\sum 1/q(n)< \\infty $ , which is satisfied by hypothesis.", "Thus, the class $\\mathcal {DNC}_q$ , encoded as above, has positive uniform measure, and thus is not $\\mathit {tt}$ -negligible.", "For $(ii)$ , let $q$ be a computable order such that $\\sum _n 1/q(n) = \\infty $ .", "Let $T$ be the canonical co-c.e.", "binary tree in which the elements of $\\mathcal {DNC}_q$ are encoded.", "By the recursion theorem, we will build a partial recursive function $\\phi _e$ whose index $e$ we know in advance and therefore will be able to define $J$ on the set of values $D = \\lbrace 2^e (2x+1) : x \\in \\mathbb {N}\\rbrace $ .", "Note that the set $D$ has positive (lower) density in $\\mathbb {N}$ : for every interval $I$ of length at least $2^{e+1}$ , $\|D \\cap I\| \\ge 2^{-e-2} \|I\|$ .", "This in particular implies that $\\sum _{n \\in D} 1/q(n) = \\infty .$ To show this, we appeal to Cauchy's condensation test, according to which for any positive non-increasing sequence $(a_n)$ , $\\sum _n a_n < \\infty $ if and only if $\\sum _k 2^k a_{2^k} < \\infty $ .", "Since the sum $\\sum _n1/q(n)$ diverges, so does $\\sum _k 2^k/q(2^k)$ and thus $\\sum _{n \\in D} \\frac{1}{q(n)} \\ge \\sum _{k>e} \\frac{\|D \\cap \\lbrace 2^{k-1},...,2^k-1\\rbrace \|}{q(2^k)} \\ge \\sum _{k>e} \\frac{2^{-e-2} \\cdot 2^{k-1}}{q(2^k)} = \\infty .$ Now that we have established that $\\sum _{n \\in D} 1/q(n) = \\infty $ , we remark that this is equivalent to having $\\prod _{n \\in D} (1-1/q(n)) = 0$ .", "Thus, we can effectively partition $D$ into countably many finite sets $D_j$ such that $\\prod _{n \\in D_j} (1-1/q(n)) < 1/2$ .", "We are ready to describe the construction.", "For each $k$ , reserve some finite collection $D_{j_1}, D_{j_2}, ..., D_{j_{k+1}}$ of sets $D_j$ .", "Let $N$ be a level of the binary tree $T$ sufficiently large such that the encoding of each path $f$ up to length $N$ is enough to recover the values of $f$ on $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , which can be found effectively in $k$ .", "The $k$ -strategy then works as follows.", "Initialisation.", "Set $i=1$ .", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: Since each $\\sigma \\in T_N[s]$ can be viewed as a function from some initial segment of $\\mathbb {N}$ that contains $D_{j_1} \\cup D_{j_2} \\cup ...\\cup D_{j_{k+1}}$ , take the first value $x \\in D_{j_i}$ .", "For all $\\sigma \\in T_N$ , $\\sigma (x) < q(x)$ , thus by the pigeonhole principle there must be at least one value $v < q(x)$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x)=v\\rbrace )[s] \\ge 2^{-k}/q(x).$ Set $J(x)$ to be the least such value $v$ .", "This thus gives $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v\\rbrace )[s] < 2^{-k}(1-1/q(x)).$ Now take another $x^{\\prime }$ in $D_{j_i}$ on which $J$ has not been defined yet.", "By the same reasoning, there must be a value $v^{\\prime } < q(x^{\\prime })$ such that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\sigma (x) \\ne v \\wedge \\sigma (x^{\\prime }) \\ne v^{\\prime } \\rbrace ) < 2^{-k}(1-1/q(x))(1-1/q(x^{\\prime }))[s].$ We then set $J(x^{\\prime })$ to be equal to $v^{\\prime }$ .", "Continuing in this fashion, we can assign all the values of $J$ on $D_{j_i}$ in such a way that $\\mathbf {M}\\big ( \\lbrace \\sigma \\in T_N : \\forall x \\in D_{j_i}, \\ \\sigma (x) \\ne J(x) \\rbrace \\big )[s] < 2^{-k} \\prod _{n \\in D_j} (1-1/q(n)) < 2^{-k}/2.$ Then we increment $i$ by 1 and go back to Step 1.", "Once again, at each execution of Step 2 we remove an $\\mathbf {M}$ -weight of at least $\\delta (k)=2^{-k-1}$ from $T_N$ , since ensuring $\\sigma (x)=J(x)$ for some $x$ immediately ensures that no extension of $\\sigma $ is in $\\mathcal {DNC}_q$ .", "Moreover, each execution of Step 2 requires us to define $J$ on all values in some $D_{j_i}$ for $1\\le j\\le k+1$ .", "That is, each such execution costs us one $D_{j_i}$ , so as in the case of consistent completions of $\\mathsf {PA}$ , we can set $\\gamma (k)=1$ and $\\Gamma (k)=2^{k+1}$ , which ensures that $\\gamma (k)/\\delta (k)\\le \\Gamma (k)$ .", "Therefore, $\\mathcal {DNC}_q$ is a deep $\\Pi ^0_1$ class." ], [ "Finite sets of maximally complex strings", "For a constant $c>0$ , generating a long string $\\sigma $ of high Kolmogorov complexity, for example $\\mathrm {K}(\\sigma ) > \|\\sigma \|-c$ , can be easily achieved with high probability if one has access to a random source (just repeatedly flip a fair coin and output the raw result).", "One can even use this technique to generate a sequence of strings $\\sigma _1, \\sigma _2, \\sigma _3, \\ldots , $ such that $\|\\sigma _n\| = n$ and $\\mathrm {K}(\\sigma _n) \\ge n-c$ .", "Indeed, the measure of the sequences $X$ such that $\\mathrm {K}(X {\\upharpoonright }n) \\ge n -d$ for all $n$ is at least $2^{-d}$ , and thus the $\\Pi ^0_1$ class of such sequences of strings $(\\sigma _1,\\sigma _2,...,)$ of high complexity (encoded as elements of $2^\\mathbb {N}$ ) is not even $\\mathit {tt}$ -negligible.", "The situation changes dramatically if one wishes to obtain many distinct complex strings at a given length.", "Let $\\ell : \\mathbb {N}\\rightarrow \\mathbb {N}$ be a computable increasing function, $f,d:\\mathbb {N}\\rightarrow \\mathbb {N}$ be any computable functions, and $c>0$ .", "Consider the $\\Pi ^0_1$ class $\\mathcal {K}_{f,\\ell ,d}$ whose members are sequences $\\vec{F}=(F_1, F_2, F_3, ...)$ where for all $i$ , $F_i$ is a finite set of $f(i)$ strings $\\sigma $ of length $\\ell (i)$ such that $\\mathrm {K}(\\sigma ) \\ge \\ell (i)-d(i)$ .", "Note once again that $\\mathcal {K}_{f,\\ell ,d}$ can be viewed, modulo encoding, as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ .", "Theorem 7.4 For all computable functions $f,\\ell ,d$ such that $f(i)/2^{d(i)}$ takes arbitrarily large values and $\\ell $ is increasing, the class $\\mathcal {K}_{f,\\ell ,d}$ is deep.", "Let $T$ be the canonical co-c.e.", "tree associated to the class $\\mathcal {K}_{f,\\ell ,d}$ .", "Just like in the proof of Theorem REF , we will build a discrete semi-measure $m$ whose coding constant $e$ we know in advance and thus, by setting $m(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) + e +1 }$ , we will force $\\mathbf {m}(\\sigma ) > 2^{-\|\\sigma \| + d(\|\\sigma \|) +1}$ , and thus $\\mathrm {K}(\\sigma ) < \|\\sigma \| - d(\|\\sigma \|)$ , which will consequently remove from $T$ every sequence of sets $(F_i)_{i\\in \\mathbb {N}}$ such that $\\sigma $ belongs to some $F_i$ .", "Fix a $k$ , and let us pick some well-chosen $i=i(k)$ , to be determined later.", "For readability, let $f=f(i(k))$ , $\\ell =\\ell (i(k))$ and $d=d(i(k))$ .", "Let $N$ be the level of $T$ at which the first $i$ sets $F_1\\cdots F_i$ of the sequence $\\vec{F}$ of $\\mathcal {K}_{f,\\ell ,d}$ are encoded.", "The $k$ -strategy does the following.", "Step 1: Wait for a stage $s$ at which $\\mathbf {M}(T_N)[s] \\ge 2^{-k}$ .", "Up to delaying the increase of $\\mathbf {M}$ , we can assume that when such a stage occurs, one in fact has $\\mathbf {M}(T_N)[s] = 2^{-k}$ .", "Step 2: The members of $T_N[s]$ consist of finite sequences $F_1, ..., F_i$ of sets of strings where $F_i$ contains $f$ distinct strings of length $\\ell $ .", "By the pigeonhole principle, there must be a string $\\sigma $ of length $\\ell $ such that $\\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N : \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ Indeed, there is a total $\\mathbf {M}$ -weight $2^{-k}$ of possible sets $F_i$ , each of which contains $f$ strings of length $\\ell $ , thus $\\sum _{\|\\sigma \|=\\ell } \\mathbf {M}\\bigl ( \\lbrace (F_1,\\cdots ,F_i) \\in T_N \\mid \\sigma \\in F_i \\rbrace \\bigr )[s] \\ge 2^{-k} \\cdot f$ , and there are $2^\\ell $ strings of length $\\ell $ in total.", "Effectively find such a string $\\sigma $ , and set $m(\\sigma ) > 2^{-\\ell + d + e +1 }$ .", "By the choice of $\\sigma $ , this causes an $\\mathbf {M}$ -weight of at least $2^{-k} \\cdot f \\cdot 2^{-\\ell }$ of nodes of $T_N[s]$ to leave the tree.", "Then go back to Step 1.", "As in the previous examples, it remains to conduct the “cost analysis\".", "The resource here again is the weight we are allowed to assign to $m$ , which has to be bounded by $\\Gamma =1$ .", "At each execution of Step 2 of the $k$ -strategy, our cost is the increase of $m$ , which is at most of $\\gamma (k)=2^{-\\ell + d + e +1 }$ , while an $\\mathbf {M}$ -weight of at least $\\delta (k) = 2^{-k} \\cdot f \\cdot 2^{-\\ell }$ leaves the tree $T_N$ .", "This gives a ratio $\\gamma (k)/\\delta (k) = 2^{d+e+k+1}/f$ , which bounds the total cost of the $k$ -strategy, so we want $\\sum _k \\gamma (k)/\\delta (k) \\le 1$ , i.e., we want: $\\sum _k 2^{d(i(k)) + e +k+1}/f(i(k))\\le 1$ By assumption, $f(i)/2^{d(i)}$ takes arbitrarily large values, it thus suffices to choose $i(k)$ such that $2^{d(i(k)) + e+1}/f(i(k)) \\le 2^{-2k-1}$ ." ], [ "Compression functions", "Our next example of a family of deep classes is given in terms of compression functions, which were introduced by Nies, Stephan, and Terwijn in to provide a characterization of 2-randomness (that is, $\\emptyset ^{\\prime }$ -Martin-Löf randomness) in terms of incompressibility.", "Definition 7.4 A $\\mathrm {K}$ -compression function with constant $c>0$ is a function $g: 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ such that $g \\le \\mathrm {K}+c$ and $\\sum _\\sigma 2^{-g(\\sigma )} \\le 1$ .", "We denote by $\\mathcal {CF}_c$ be the class of compression functions with constant $c$ .", "The condition $g(\\sigma ) \\le \\mathrm {K}(\\sigma )+c$ implies that $g(\\sigma ) \\le 2\|\\sigma \| + c +c^{\\prime } $ for some fixed constant $c^{\\prime }$ , and therefore $\\mathcal {CF}_c$ can be seen as a $\\Pi ^0_1$ subclass of $2^\\mathbb {N}$ , modulo encoding the functions as binary sequences.", "Of course $\\mathcal {CF}_c$ contains the function $\\mathrm {K}$ itself so it is a non-empty class.", "We now show the following.", "Theorem 7.5 For all $c\\ge 0$ , the class $\\mathcal {CF}_c$ is deep.", "Although we could give a direct proof following the same template as the previous examples, we will instead show that for all $c$ , we have $\\mathcal {K}_{f,\\ell ,d} \\le _s \\mathcal {CF}_c$ for some computable $f, \\ell , d$ (where $\\mathcal {K}_{f,\\ell ,d}$ is the class we defined in the previous section) such that $\\ell $ is increasing and $f/2^d$ takes arbitrarily large values, which by Theorem REF implies the depth of $\\mathcal {CF}_c$ .", "Let $g \\in \\mathcal {CF}_c$ .", "For all $n$ , since $\\sum _{\|\\sigma \|=n} 2^{-g(\\sigma )} \\le 1$ , there are at most $2^{n-1}$ strings of length $n$ such that $g(\\sigma ) < n - 1$ , and thus at least $2^{n-1}$ strings $\\sigma $ of length $n$ such that $g(\\sigma ) \\ge n - 1$ .", "Since $g \\in \\mathcal {CF}_c$ , for each $\\sigma $ such that $g(\\sigma ) \\ge \|\\sigma \| - 1$ , we also have $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Thus, given $g \\in \\mathcal {CF}_c$ as an oracle we can find, for each $n \\ge 3$ , $2^{n-1}$ strings $\\sigma $ of length $n$ such that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \| - c - 1$ .", "Setting $f(i)=2^{i+2}$ , $\\ell (i)=i+3$ and $d(i)=c+1$ , we have uniformly reduced $\\mathcal {K}_{f,\\ell ,d}$ to $\\mathcal {CF}_{c}$ .", "Since $d$ is a constant function it is obvious that $f(i)/2^d$ is unbounded, thus $\\mathcal {K}_{f,\\ell ,d}$ is deep (by Theorem REF ) and by Theorem REF so is $\\mathcal {CF}_c$ .", "The above result is not tight: the proof in fact shows that if $d$ is not a constant function but is such that $2^n/d(n)$ takes arbitrarily large values, then the class of functions $g$ such that $g(\\sigma ) \\le \\mathrm {K}(\\sigma ) + d(\|\\sigma \|)$ for all $\\sigma $ is a deep class." ], [ "A notion related to $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$", "In $§$, we discuss lowness for randomness notions.", "Here we look at a dual notion, highness for randomness, specifically the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , whose precise characterization is an outstanding open question in algorithmic randomness.", "We shall see that this class is tightly connected to the notion of depth.", "Definition 7.5 A sequence $A\\in 2^\\mathbb {N}$ is in the class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ if $\\mathsf {MLR}\\subseteq \\mathsf {CR}^A$ .", "The class $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ itself not $\\Pi ^0_1$ (as it is closed under finite change of prefixes), but all its members have a “deep property\".", "Bienvenu and Miller proved that when $A$ is in $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ , then for every c.e.", "set of strings $S$ such that $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ , $A$ computes a martingale $d$ such that $d(\\Lambda )=1$ and for some fixed rational $r>0$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ .", "It is straightforward to show that real-valued martingales can be approximated by dyadic-valued martingales with arbitrary precision; in particular one can assume that $d(\\sigma )$ is dyadic for all $\\sigma $ (and thus can be coded using $f(\|\\sigma \|)$ bits for some computable function $f$ ), still keeping the property $\\sigma \\in S \\Rightarrow d(\\sigma )>1+r$ .", "For a well-chosen $S$ , such martingales form a deep class.", "Theorem 7.6 Let $S$ be the set of strings $\\sigma $ such that $\\mathrm {K}(\\sigma ) < \|\\sigma \|$ (so that we have $\\sum _{\\sigma \\in S} 2^{-\|\\sigma \|} < 1$ ).", "Let $g$ be a computable function and $r>0$ be a rational number.", "Define the class $\\mathcal {W}_{g,r}$ to be the set of dyadic-valued martingales $d$ such that $d(\\Lambda )=1$ , $d(\\sigma )>1+r$ for all $\\sigma \\in S$ and such that for all $\\sigma $ , $d(\\sigma )$ can be coded using $g(\|\\sigma \|)$ bits.", "Then $\\mathcal {W}_{g,r}$ can be viewed as a $\\Pi ^0_1$ class of $2^\\mathbb {N}$ , and this class is deep.", "Again, we are going to show the depth of $\\mathcal {W}_{g,r}$ by a Medvedev reduction from $\\mathcal {K}_{f,\\ell ,h}$ for some increasing computable function $\\ell $ and computable functions $f$ and $h$ such that $f/2^h$ is unbounded.", "More precisely, we will take $f(n)=n$ , $\\ell (n)=n + c$ (for a well-chosen $c$ given below) and $h(n)=0$ .", "Now suppose we are given a martingale $d \\in \\mathcal {W}_{f,r}$ .", "For any given length $n$ , we have, by the martingale fairness condition, $\\sum _{\\sigma : \|\\sigma \|=n} d(\\sigma ) = 2^n.$ It follows that there are at most $2^n/(1+r)$ strings $\\sigma $ of length $n$ such that $d(\\sigma )>1+r$ , and therefore at least $2^n-2^n/(1+r)$ strings such that $d(\\sigma ) \\le 1+r$ .", "Having oracle access to $d$ , such strings can be effectively found and listed.", "Since $2^n-2^n/(1+r) > n -c$ for all $n$ and some fixed constant $c$ , for each $n$ we can use $d$ to list $n$ strings $\\sigma _1,\\cdots ,\\sigma _n$ of length $n+c$ such that $d(\\sigma _i) \\le 1+r$ for $1\\le i\\le n$ .", "But by definition of $d$ , $d(\\sigma )\\le 1+r$ implies that $\\sigma \\notin S$ , which further implies that $\\mathrm {K}(\\sigma ) \\ge \|\\sigma \|$ .", "This shows that $\\mathcal {W}_{g,r}$ is above $\\mathcal {K}_{f,\\ell ,d}$ in the Medvedev degrees.", "By Theorem REF , $\\mathcal {K}_{f,\\ell ,h}$ is deep, and hence by Theorem REF , $\\mathcal {W}_{g,r}$ is deep as well.", "The examples of deep classes provided in this section, combined with Theorem REF , give us the results mentioned page : If $X$ is a difference random sequence, it does not compute any shift-complex sequence (Khan) it does not compute any $\\mathcal {DNC}_q$ function when $q$ is a computable order such that $\\sum _n 1/q(n) = \\infty $ (Miller) it does not compute any compression function (Greenberg, Miller, Nies) it is not $\\mathrm {High}(\\mathsf {CR},\\mathsf {MLR})$ (Greenberg, Miller, Nies) The fact that a difference random cannot compute large sets of complex strings (in the sense of Theorem REF ) appears to be new." ], [ "Lowness and depth", "Various lowness notions have been well-studied in algorithmic randomness, where a lowness notion is given by a collection of sequences that are in some sense computationally weak.", "Many lowness notions take the following form: For a relativizable collection $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , we say that $A$ is low for $\\mathcal {S}$ if $\\mathcal {S}\\subseteq \\mathcal {S}^A$ .", "For instance, if we let $\\mathcal {S}=\\mathsf {MLR}$ , then the resulting lowness notion consists of the sequences that are low for Martin-Löf random, a collection we write as $\\mathrm {Low}(\\mathsf {MLR})$ .", "Lowness notions need not be given in terms of relativizable classes.", "For instance, if we let $\\mathbf {m}^A$ be a universal $A$ -lower semi-computable discrete semi-measure, we can defined $A$ to be low for $\\mathbf {m}$ if $\\mathbf {m}^A(\\sigma )\\le ^\\mathbf {m}(\\sigma )$ for every $\\sigma $ .", "In addition, some lowness notions are not given in terms of relativization, such as the notion of $\\mathrm {K}$ -triviality, where $A\\in 2^\\mathbb {N}$ is $\\mathrm {K}$ -trivial if and only if $\\mathrm {K}(A{\\upharpoonright }n)\\le \\mathrm {K}(n)+O(1)$ (where we take $n$ to be $1^n$ ).", "Surprisingly, we have the following result (see for a detailed survey of results in this direction).", "Theorem 8.1 Let $A\\in 2^\\mathbb {N}$ .", "The following are equivalent: (i) $A\\in \\mathrm {Low}(\\mathsf {MLR})$ ; (ii) $A$ is low for $\\mathbf {m}$ ; (iii) $A$ is $\\mathrm {K}$ -trivial.", "The cupping problem, a longstanding open problem in algorithmic randomness involving $\\mathrm {K}$ -triviality, is to determine whether there exists a $\\mathrm {K}$ -trivial sequence $A$ and some Martin-Löf random sequence $X\\lnot \\ge _T\\emptyset ^{\\prime }$ such that $X\\oplus A\\ge _T\\emptyset ^{\\prime }$ .", "A negative answer was recently provided by Day and Miller : Theorem 8.2 (Day-Miller ) A sequence $A$ is K-trivial if and only if for every difference random sequence $X$ , $X \\oplus A \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Using the notion of depth, we can strengthen the Day-Miller result.", "First, we need to partially relativize the notion of depth: for $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $2^\\mathbb {N}$ is deep relative to $A$ if there is some computable order $f$ such that $\\mathbf {m}^A(T_{f(n)})\\le 2^{-n}$ if and only if there is some $A$ -computable order $g$ such that $\\mathbf {M}^A(T_{g(n)})\\le 2^{-n}$ (where $\\mathbf {M}^A$ is a universal $A$ -lower semi-computable continuous semi-measure).", "Theorem 8.3 Let $X$ be an incomplete Martin-Löf random sequence and $A$ be $K$ -trivial.", "Then $X \\oplus A$ does not compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with canonical co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ m(Tf(n)) 2-n$.", "Since~$ A$ is low for~$ m$, we have also have $ mA(Tf(n)) 2-n$, and thus $ is deep relative to $A$ .", "Let $X$ be a sequence such that $X \\oplus A$ computes a member of $ via a Turing functional $$ and suppose, for the sake of contradiction, that $ X$ is difference random.", "Let$$n = \\lbrace Z \\, : \\, \\Phi ^{Z \\oplus A} {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace .$$The set $ n$ can be written as the difference $ Un n$ of two $ A$-effectively open sets (uniformly in~$ n$) with $ Un = {Z : Z A f(n) }$ and $ n= {Z : Z A f(n) Tf(n)}$$ We can see the functional $Z \\mapsto \\Phi ^{Z \\oplus A}$ as an $A$ -Turing functional $\\Psi $ , and thus by the univerality of $\\mathbf {M}^A$ for the class of $A$ -lower semi-computable continuous semi-measures, we have $\\mathbf {M}^A \\ge ^* \\lambda _\\Psi $ .", "By definition of $n$ , we therefore obtain: $\\lambda (n) \\le \\lambda _\\Psi (T_{f(n)}) \\le ^* \\mathbf {M}^A(T_{f(n)}) < 2^{-n}.$ This shows that the sequence $X$ , which by assumption belongs to all $n$ , is not $A$ -difference random.", "It is, however, $A$ -Martin-Löf random as $A$ is low for Martin-Löf randomness.", "Relativizing Theorem REF to $A$ , this shows that $X \\oplus A \\ge _T A^{\\prime }$ .", "But this contradicts the Day-Miller theorem (Theorem REF ).", "As we cannot compute any members of a deep class by joining a Martin-Löf random sequence with a low for Martin-Löf random sequence, it is not unreasonable to ask if there is a notion of randomness $\\mathcal {R}$ such that we cannot $\\mathit {tt}$ -compute any members of a $\\mathit {tt}$ -deep class by joining an $\\mathcal {R}$ -random sequence with a low for $\\mathcal {R}$ sequence.", "We obtain a partial answer to this question using Kurtz randomness.", "From the discussion of lowness at the beginning of this section, we have $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $\\mathsf {KR}\\subseteq \\mathsf {KR}^A$ .", "Moreover, we define the class $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ to be the collection of sequences $A$ such that $\\mathsf {MLR}\\subseteq \\mathsf {KR}^A$ .", "Since $\\mathsf {MLR}\\subseteq \\mathsf {KR}$ , it follows that $\\mathrm {Low}(\\mathsf {KR})\\subseteq \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Miller and Greenberg obtained the following characterization of $\\mathrm {Low}(\\mathsf {KR})$ and $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Recall that $A\\in 2^\\mathbb {N}$ is computably dominated if every $f\\le _T A$ is dominated by some computable function.", "Theorem 8.4 (Greenberg-Miller ) Let $A\\in 2^\\mathbb {N}$ .", "(i) $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ if and only if $A$ is of non-DNC degree.", "(ii) $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ and computably dominated.", "For $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $ is \\emph {$ tt$-negligible relative to $ A$} if $ A(=0$ for every $ A$-computable measure $$.", "We first prove the following.$ Proposition 8.1 If $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ , then every deep $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible relative to $A$ .", "Let $ be a deep $ 01$ class and $ T$ its associated canonical co-c.e.\\ tree.", "Let $ ALow(MLR,KR)$, which by Theorem \\ref {thm:greenberg-miller-kr} (i) is equivalent to being of non-DNC degree.", "We appeal to a useful characterization of non-DNC degrees due to Hölzl and Merkle~\\cite {HolzlM2010}: $ A$ is of non-DNC degree if and only if it is \\emph {infinitely often c.e.\\ traceable} (hereafter, i.o.\\ c.e.\\ traceable).", "This means that there exists a computable order~$ h$, such that the following holds: for every total $ A$-computable function~$ s: NN$, there exists a family $ (Sn)nN$ of uniformly c.e.\\ finite sets such that $ \|Sn\|<h(n)$ for all~$ n$ and $ s(n) Sn$ for infinitely many~$ n$.$ Let $h$ be an order witnessing the i.o.", "c.e.", "traceability of $A$ .", "Since $ is deep, let~$ f$ be a computable function such that $ M(Tf(n)) < 2-2n/h(n)$.Suppose for the sake of contradiction that $ is not $\\mathit {tt}$ -negligible relative to $A$ , which means that there exists an $A$ -computable measure $\\mu ^A$ such that $\\mu ^A(>r$ for some rational $r>0$ .", "Let $s\\le _TA$ be the function that on input $n$ gives a rational lower-approximation, with precision $1/2$ , of the values of $\\mu ^A$ on all strings of length $f(n)$ (encoded as an integer).", "By this we mean that $s(n)$ gives us for all strings $\\sigma $ of length $f(n)$ a rational value $s(n,\\sigma )$ such that $\\mu ^A(\\sigma )/2 \\le s(n,\\sigma ) \\le \\mu ^A(\\sigma )$ .", "Let $(S_n)_{n\\in \\mathbb {N}}$ witness the traceability of $s$ , i.e., the $S_n$ 's are uniformly c.e., $\|S_n\|<h(n)$ for every $n$ , and $s(n) \\in S_n$ for infinitely many $n$ .", "We now build a lower semi-computable continuous semi-measure $\\rho $ as follows.", "For all $n$ , enumerate $S_n$ .", "For each member $z \\in S_n$ , interpret $z$ as a mass distribution $\\nu $ on the collection of strings of length $f(n)$ .", "Then, for each string $\\sigma $ of length $f(n)$ , increase $\\rho (\\sigma )$ (as well as strings comparable with $\\sigma $ in any way that ensures that $\\rho $ remains a semi-measure) by $2^{-n-1}\\nu (\\sigma )/h(n)$ .", "This has a total cost of $2^{-n-1}/h(n)$ , and since there are at most $h(n)$ elements in $S_n$ , the total cost at level $f(n)$ is at most $2^{-n-1}$ .", "Therefore the total cost of the construction of $\\rho $ is bounded by 1 and thus $\\rho $ is indeed a lower semi-computable continuous semi-measure Now, for any $n$ such that $s(n) \\in S_n$ (as there are infinitely many such $n$ ), $\\rho $ distributes an amount of at least $(\\mu ^A(T_{f(n)})/2) \\cdot (2^{-n-1}/h(n))$ on $T_{f(n)}$ , and since $\\mu ^A(T_n)>r$ , this gives $\\rho (T_{f(n)}) \\ge 2^{-n-O(1)}/h(n).$ However, we had assumed that $\\mathbf {M}(T_{f(n)}) \\le 2^{-2n}/h(n),$ for all $n$ .", "But since $\\rho \\le ^* \\mathbf {M}$ , we get a contradiction.", "The following result involves the notion of relative $\\mathit {tt}$ -reducibility.", "For a fixed $A\\in 2^\\mathbb {N}$ , a $\\mathit {tt}(A)$ -functional is a total $A$ -computable Turing functional.", "Equivalently, we can define a $\\mathit {tt}(A)$ -functional $\\Psi ^A$ in terms of a Turing functional $\\Phi $ as follows: Let $\\Phi $ be defined on all inputs of the form $X\\oplus A$ .", "Then we set $\\Psi ^A(X)=\\Phi (X\\oplus A)$ .", "Furthermore, one can show that there is an $A$ -computable bound on the use of $X$ in the computation (just as there is a computable bound in the use function for unrelativized $\\mathit {tt}$ -computations).", "Theorem 8.5 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class and $ ALow(KR)$.", "By Proposition \\ref {prop:low-for-tt-depth}, $ is also $\\mathit {tt}$ -deep relative to $A$ .", "Let $\\Phi ^A$ be a $\\mathit {tt}(A)$ -functional.", "The pre-image $ of $ under $\\Phi ^A$ is a $\\Pi ^0_1(A)$ class, which must be $\\mathit {tt}$ -deep relative to $A$ as well, by Theorem REF relativized to $A$ .", "Now, applying Proposition REF relativized to $A$ , $ contains no $ A$-Kurtz random sequence.", "But since $ A$ is low for Kurtz randomness, $ contains no Kurtz-random sequence as well.", "We now obtain a partial analogue of Theorem REF .", "Corollary 8.2 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $\\Phi $ be a $\\mathit {tt}$ -functional.", "Since $\\Phi $ is total, it is certainly total on all sequences of the form $X\\oplus A$ for $X\\in 2^\\mathbb {N}$ .", "Thus $\\Psi ^A(X)=\\Phi (X\\oplus A)$ is a $\\mathit {tt}(A)$ -functional.", "By Theorem REF , it follows that $\\Phi (X\\oplus A)$ cannot be contained in any deep class.", "Question 1 Does Corollary REF still hold if we replace “deep\" with “$\\mathit {tt}$ -deep\"?", "We can extend Theorem REF to the following result, which proceeds by almost the same proof, the details of which are left to the reader.", "Theorem 8.6 Let $X$ be Martin-Löf random and $A$ be $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "(in particular, $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class)." ], [ "Depth, mutual information, and the Independence Postulate", "In this final section, we introduce the notion of mutual information and apply it to the notion of depth.", "Roughly, what we prove is that every member of every deep class has infinite mutual information with Chaitin's $\\Omega $ , a Martin-Löf random sequence that encodes the halting problem.", "This generalizes a result of Levin's, that every consistent completion of $\\mathsf {PA}$ has infinite mutual information with $\\Omega $ .", "We conclude with a discussion of the Independence Postulate, a principle introduced by Levin to derive the statement that that no consistent completion of arithmetic is physically obtainable." ], [ "The definition of mutual information", "First we review the definitions of Kolmogorov complexity of a pair and the universal conditional discrete semi-measure $\\mathbf {m}(\\cdot \\mid \\cdot )$ .", "Let $\\langle \\cdot ,\\cdot \\rangle :2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ be a computable bijection.", "Then we define $\\mathrm {K}(\\sigma ,\\tau ):=\\mathrm {K}(\\langle \\sigma ,\\tau \\rangle )$ .", "Similarly, we set $\\mathbf {m}(\\sigma ,\\tau ):=\\mathbf {m}(\\langle \\sigma ,\\tau \\rangle )$ .", "A conditional lower semi-computable discrete semi-measure $m(\\cdot \\mid \\cdot ):2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow [0,1]$ is a function satisfying $\\sum _\\sigma m(\\sigma \\mid \\tau )\\le 1$ for every $\\tau $ .", "Then $\\mathbf {m}(\\cdot \\mid \\cdot )$ is defined to be a universal conditional lower semi-computable discrete semi-measure, so that for every conditional lower semi-computable discrete semi-measure, there is some $c$ such that $m(\\sigma \\mid \\tau )\\le c\\cdot \\mathbf {m}(\\sigma \\mid \\tau )$ for every $\\sigma $ and $\\tau $ .", "Lastly, we define the conditional prefix-free Kolmogorov complexity $\\mathrm {K}(\\sigma \\mid \\tau )$ to be $\\mathrm {K}(\\sigma \\mid \\tau )=\\min \\lbrace \|\\xi \|:U(\\langle \\xi ,\\tau \\rangle )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free machine.", "The mutual information of two strings $\\sigma $ and $\\tau $ , denoted by $I(\\sigma :\\tau )$ , is defined by $I(\\sigma :\\tau ) = \\mathrm {K}(\\sigma ) + \\mathrm {K}(\\tau ) - \\mathrm {K}(\\sigma ,\\tau )$ or equivalently by $2^{I(\\sigma :\\tau )} = \\frac{\\mathbf {m}(\\sigma ,\\tau )}{\\mathbf {m}(\\sigma ) \\cdot \\mathbf {m}(\\tau )}.$ By the symmetry of information (see Gács ), we also have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} =^* \\frac{\\mathbf {m}(\\tau \\mid \\sigma ,\\mathrm {K}(\\sigma ))}{\\mathbf {m}(\\tau )}.$ Levin extends mutual information to infinite sequences by setting $2^{I(X:Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^X(\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}.$ Recall that Chaitin's $\\Omega $ can be obtained as the probability that a universal prefix-free machine will halt on a given input, that is, $\\Omega =\\sum _{U(\\sigma ){\\downarrow }}2^{-\|\\sigma \|}$ , where $U$ is a fixed universal prefix-free machine.", "Generalizing a result of Levin's from , we have: Theorem 9.1 Let $ be a $ 01$ class and $ T$ its associated co-c.e.\\ tree.", "Suppose~$ is deep, witnessed by a computable order $f$ such that $\\mathbf {m}(T_{f(n)}) < 2^{-n}$ .", "Then for every $Y \\in and all~$ n$,$$I\\Big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\Big ) \\ge n-O(\\log n).$$In particular,$$I(\\Omega :Y)=\\infty .$$$ Our proof follows the same idea Levin uses for consistent completions of $\\mathsf {PA}$ (see ), although some extra care is needed for arbitrary deep classes.", "Suppose for a given $n$ we have an exact description $\\tau $ of $T_{f(n)}$ ; that is, on input $\\tau $ , the universal machine outputs a code for the finite set $T_{f(n)}$ .", "By the definition of $f$ , $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\le 2^{-n}$ or equivalently $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\cdot 2^{n} \\le 1$ Therefore, the quantity $\\mathbf {m}(\\sigma ) \\cdot 2^{n} \\cdot \\mathbf {1}_{\\sigma \\in T_{f(n)}}$ is a discrete semi-measure, but it is not lower semi-computable since $T_{f(n)}$ is merely co-c.e.", "(and, in general, not c.e.", "by Proposition REF ).", "However, it is a lower semi-computable semi-measure relative to the exact description $\\tau $ of $T_{f(n)}$ .", "Thus, for every $\\sigma \\in T_{f(n)}$ , by the universality of $\\mathbf {m}(\\cdot \\mid \\tau )$ , $\\mathbf {m}(\\sigma \\mid \\tau ) \\ge ^* \\mathbf {m}(\\sigma ) \\cdot 2^n.$ By the symmetry of information, we have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} \\ge ^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau )}{\\mathbf {m}(\\sigma )} \\ge ^2^n,$ and hence $I(\\tau :\\sigma ) \\ge ^+ n$ .", "We would like to apply this fact to the case where $\\sigma = Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ .", "But this is not technically sufficient, as $\\Omega {\\upharpoonright }n$ does not necessarily contain enough information to exactly describe $T_{f(n)}$ .", "This is not an obstacle in Levin's argument for completions of $\\mathsf {PA}$ , but it is for arbitrary deep classes.", "However, $\\Omega {\\upharpoonright }n$ contains enough information to get a “good enough\" approximation of $T_{f(n)}$ .", "Let us refine the idea above: suppose now that $\\tau $ is no longer an exact description of $T_{f(n)}$ , but is a description of a set of strings $S$ of length $f(n)$ such that $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^ 2^{-n} n^2.$ Then, by following the same reasoning as above, we would have $I(\\sigma : \\tau ) \\ge ^+ n-2\\log n$ for all $\\sigma \\in S$ (and thus all $\\sigma \\in T_{f(n)}$ ).", "We shall prove that $\\Omega {\\upharpoonright }n$ contains enough information to recover such a set $S$ , thus proving the theorem.", "The real number $\\Omega $ is lower semi-computable and Solovay complete (see ).", "As a consequence, for every other lower semi-computable real $\\alpha $ , knowing the first $k$ bits of $\\Omega $ allows us to compute the first $k-O(1)$ bits of $\\alpha $ .", "For all $n$ , define: $a_n = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}}} \\mathbf {m}(\\sigma )$ and observe that $a_n$ is lower semi-computable uniformly in $n$ (because $T_{f(n)}$ is co-c.e.", "uniformly in $n$ ), and belongs to $[0,1]$ .", "Define now $\\alpha = \\sum _n \\frac{a_n}{n^2},$ which is a lower semi-computable real.", "Thus, knowing the first $n$ bits of $\\Omega $ gives us the first $n-O(1)$ bits of $\\alpha $ , i.e., an approximation of $\\alpha $ with precision $2^{-n}$ .", "In particular this gives us an approximation of $a_n$ with precision $2^{-n} \\cdot n^2 \\cdot O(1)$ , which we can assume to be a lower approximation, which we will write as $a^{\\prime }_n$ .", "Now, using $a^{\\prime }_n$ , one can enumerate $a_n$ until we find a stage $s_n$ such that $a_n[s_n] = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}[s_n]}} \\mathbf {m}(\\sigma )[s_n] \\ge a^{\\prime }_n.$ Since $\|a_n - a^{\\prime }_n\| \\le 2^{-n} \\cdot n^2 \\cdot O(1)$ , this implies $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}[s_n]} \\setminus T_{f(n)}}} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n^2.$ But recall from above that $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}}}} \\mathbf {m}(\\sigma ) \\le 2^{-n}.$ Combining these two facts, and taking $S$ to be the set $T_{f(n)}[s_n]$ , we have $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n ^2$ which establishes the first part of the theorem.", "To see that the second part of the statement follows from the first, take $\\sigma =Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ and observe that $\\mathbf {m}^Y(\\sigma ) =^* \\mathbf {m}^Y(n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ $\\mathbf {m}^\\Omega (\\tau ) =^* \\mathbf {m}^\\Omega (n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ and $2^{I(\\sigma :\\tau )}=2^n/n^{O(1)}$ (since $I(\\sigma :\\tau )\\ge ^+n-2\\log n$ as established above).", "Then we have $\\begin{split}2^{I(\\Omega :Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^\\Omega (\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}&\\ge \\sum _n \\mathbf {m}^\\Omega (Y{\\upharpoonright }f(n)) \\cdot \\mathbf {m}^Y(\\Omega {\\upharpoonright }n) \\cdot 2^{I(Y{\\upharpoonright }f(n),\\Omega {\\upharpoonright }n)}\\\\&\\ge \\sum _n 2^n/n^{O(1)}=\\infty .\\end{split}$ Remark 9.1 The converse of this theorem does not hold, i.e., there is a $\\Pi ^0_1$ class that is not deep but all of whose elements have infinite mutual information with $\\Omega $ .", "This follows from Theorem REF and the fact that having infinite mutual information with $\\Omega $ is a property that is invariant under addition or deletion of a finite prefix.", "Levin's proof of Theorem REF restricted to the particular case of completions of $\\mathsf {PA}$ is the mathematical part of a more general discussion, the other part of which is philosophical in nature.", "While Gödel's theorem asserts that no completion of $\\mathsf {PA}$ can be computably obtained, Levin's goal is to show that no completion of $\\mathsf {PA}$ can be obtained by any physical means whatsoever (computationally or otherwise), thus generalizing Gödel's theorem.", "Levin does not fully specify what he means by physically obtainable (the exact term he uses is “located in the physical world\"), but nonetheless he makes the following postulate, which he dubs the “Independence Postulate\": if $\\sigma $ is a mathematically definable string that has an $n$ bit description and $\\tau $ can be located in the physical world with a $k$ bit description, then for a fixed small constant $c$ , one has $I(\\sigma : \\tau ) < n+k+c$ .", "In particular, if one admits that some infinite sequences can be physically obtained, the Independence Postulate for infinite sequences says that if $X$ and $Y$ are two infinite sequences with $X$ mathematically definable and $Y$ physically obtainable, then $I(X:Y) < \\infty $ .", "Being $\\Delta ^0_2$ , $\\Omega $ is mathematically definable, and, as Levin shows, $I(\\Omega :Y)=\\infty $ for any completion $Y$ of $\\mathsf {PA}$ .", "Thus, assuming the Independence Postulate, no completion of $\\mathsf {PA}$ is physically obtainable.", "Our Theorem REF extends Levin's theorem and, assuming the Independence Postulate, shows that no member of a deep class (shift-complex sequences, compression functions, etc.)", "is physically obtainable.", "Of course, evaluating the validity of the Independence Postulate would require an extended philosophical discussion that would take us well beyond the scope of this paper.", "In any case, whether or not the reader accepts the Independence Postulate, Theorem REF is interesting in its own right.", "In fact, it is quite surprising because it seems to contradict the “basis for randomness theorem\" (see ), which states that if $X$ is a Martin-Löf random sequence and $\\mathcal {C}$ is a $\\Pi ^0_1$ class, then there exists a member $Y$ of $\\mathcal {C}$ such that $X$ is random relative to $Y$ .", "If a sequence $X$ is random relative to another sequence $Y$ , the intuition is that $Y$ “knows nothing about $X$ \", and thus one could conjecture that $I(X : Y) < \\infty $ .", "However, this cannot always be the case, since by Theorem REF , $I(\\Omega :Y) = \\infty $ for all members $Y$ of a deep $\\Pi ^0_1$ class $, even though $$ is random relative to some $ Y.", "This apparent paradox can be explained by taking a closer look at the definition of mutual information.", "Let $ be a deep $ 01$ class, whose canonical co-c.e.\\ tree~$ T$ satisfies $ m(Tf(n)) < 2-n$ for some computable function~$ f$.", "By Theorem~\\ref {thm:mutual-info} and the symmetry of information, for every $ Y we have $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big ) = I\\big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\big ) \\ge n-O(\\log n)$ where $k_n$ stands for $\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "Take a $Y \\in such that $$ is random relative to~$ Y$.", "It is well-known that a sequence~$ Z$ is random if and only if $ K(Z n n) n-O(1)$ (see for example Gács~\\cite {Gacs1980}).", "Applying this fact (relativized to~$ Y$) to $$, we have$$\\mathrm {K}^Y(\\Omega {\\upharpoonright }n \\mid n) \\ge n-O(1)$$and thus in particular that$$\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\ge n-O(1).$$Since $ K(n) n+O(n)$, it follows that$ $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\le O(\\log n)$ The only difference between (REF ) and (REF ) is the term $k_n=\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "But it makes a big difference, as one can verify that $\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n))-\\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big )\\ge n-O(\\log n).$ Informally, while $Y$ “knows nothing\" about $\\Omega $ , the complexity of its initial segments, seen as a function, does.", "In particular, the change of complexity caused by $k_n$ implies that $\\mathrm {K}(\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge n - O(\\log n)$ and thus $\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge 2^n/n^{O(1)}$ .", "Acknowledgements We would like to thank Noam Greenberg, Rupert Hölzl, Mushfeq Khan, Leonid Levin, Joseph Miller, André Nies, Paul Shafer, and Antoine Taveneaux for many fruitful discussions on the subject.", "Particular thanks go to Steve Simpson who provided very detailed feedback on the first arXiv version of this paper." ], [ "Lowness and depth", "Various lowness notions have been well-studied in algorithmic randomness, where a lowness notion is given by a collection of sequences that are in some sense computationally weak.", "Many lowness notions take the following form: For a relativizable collection $\\mathcal {S}\\subseteq 2^\\mathbb {N}$ , we say that $A$ is low for $\\mathcal {S}$ if $\\mathcal {S}\\subseteq \\mathcal {S}^A$ .", "For instance, if we let $\\mathcal {S}=\\mathsf {MLR}$ , then the resulting lowness notion consists of the sequences that are low for Martin-Löf random, a collection we write as $\\mathrm {Low}(\\mathsf {MLR})$ .", "Lowness notions need not be given in terms of relativizable classes.", "For instance, if we let $\\mathbf {m}^A$ be a universal $A$ -lower semi-computable discrete semi-measure, we can defined $A$ to be low for $\\mathbf {m}$ if $\\mathbf {m}^A(\\sigma )\\le ^\\mathbf {m}(\\sigma )$ for every $\\sigma $ .", "In addition, some lowness notions are not given in terms of relativization, such as the notion of $\\mathrm {K}$ -triviality, where $A\\in 2^\\mathbb {N}$ is $\\mathrm {K}$ -trivial if and only if $\\mathrm {K}(A{\\upharpoonright }n)\\le \\mathrm {K}(n)+O(1)$ (where we take $n$ to be $1^n$ ).", "Surprisingly, we have the following result (see for a detailed survey of results in this direction).", "Theorem 8.1 Let $A\\in 2^\\mathbb {N}$ .", "The following are equivalent: (i) $A\\in \\mathrm {Low}(\\mathsf {MLR})$ ; (ii) $A$ is low for $\\mathbf {m}$ ; (iii) $A$ is $\\mathrm {K}$ -trivial.", "The cupping problem, a longstanding open problem in algorithmic randomness involving $\\mathrm {K}$ -triviality, is to determine whether there exists a $\\mathrm {K}$ -trivial sequence $A$ and some Martin-Löf random sequence $X\\lnot \\ge _T\\emptyset ^{\\prime }$ such that $X\\oplus A\\ge _T\\emptyset ^{\\prime }$ .", "A negative answer was recently provided by Day and Miller : Theorem 8.2 (Day-Miller ) A sequence $A$ is K-trivial if and only if for every difference random sequence $X$ , $X \\oplus A \\lnot \\ge _T \\emptyset ^{\\prime }$ .", "Using the notion of depth, we can strengthen the Day-Miller result.", "First, we need to partially relativize the notion of depth: for $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $2^\\mathbb {N}$ is deep relative to $A$ if there is some computable order $f$ such that $\\mathbf {m}^A(T_{f(n)})\\le 2^{-n}$ if and only if there is some $A$ -computable order $g$ such that $\\mathbf {M}^A(T_{g(n)})\\le 2^{-n}$ (where $\\mathbf {M}^A$ is a universal $A$ -lower semi-computable continuous semi-measure).", "Theorem 8.3 Let $X$ be an incomplete Martin-Löf random sequence and $A$ be $K$ -trivial.", "Then $X \\oplus A$ does not compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class with canonical co-c.e.\\ tree~$ T$ and let~$ f$ be a computable function such that $ m(Tf(n)) 2-n$.", "Since~$ A$ is low for~$ m$, we have also have $ mA(Tf(n)) 2-n$, and thus $ is deep relative to $A$ .", "Let $X$ be a sequence such that $X \\oplus A$ computes a member of $ via a Turing functional $$ and suppose, for the sake of contradiction, that $ X$ is difference random.", "Let$$n = \\lbrace Z \\, : \\, \\Phi ^{Z \\oplus A} {\\upharpoonright }f(n) \\downarrow \\, \\in T_{f(n)}\\rbrace .$$The set $ n$ can be written as the difference $ Un n$ of two $ A$-effectively open sets (uniformly in~$ n$) with $ Un = {Z : Z A f(n) }$ and $ n= {Z : Z A f(n) Tf(n)}$$ We can see the functional $Z \\mapsto \\Phi ^{Z \\oplus A}$ as an $A$ -Turing functional $\\Psi $ , and thus by the univerality of $\\mathbf {M}^A$ for the class of $A$ -lower semi-computable continuous semi-measures, we have $\\mathbf {M}^A \\ge ^* \\lambda _\\Psi $ .", "By definition of $n$ , we therefore obtain: $\\lambda (n) \\le \\lambda _\\Psi (T_{f(n)}) \\le ^* \\mathbf {M}^A(T_{f(n)}) < 2^{-n}.$ This shows that the sequence $X$ , which by assumption belongs to all $n$ , is not $A$ -difference random.", "It is, however, $A$ -Martin-Löf random as $A$ is low for Martin-Löf randomness.", "Relativizing Theorem REF to $A$ , this shows that $X \\oplus A \\ge _T A^{\\prime }$ .", "But this contradicts the Day-Miller theorem (Theorem REF ).", "As we cannot compute any members of a deep class by joining a Martin-Löf random sequence with a low for Martin-Löf random sequence, it is not unreasonable to ask if there is a notion of randomness $\\mathcal {R}$ such that we cannot $\\mathit {tt}$ -compute any members of a $\\mathit {tt}$ -deep class by joining an $\\mathcal {R}$ -random sequence with a low for $\\mathcal {R}$ sequence.", "We obtain a partial answer to this question using Kurtz randomness.", "From the discussion of lowness at the beginning of this section, we have $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $\\mathsf {KR}\\subseteq \\mathsf {KR}^A$ .", "Moreover, we define the class $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ to be the collection of sequences $A$ such that $\\mathsf {MLR}\\subseteq \\mathsf {KR}^A$ .", "Since $\\mathsf {MLR}\\subseteq \\mathsf {KR}$ , it follows that $\\mathrm {Low}(\\mathsf {KR})\\subseteq \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Miller and Greenberg obtained the following characterization of $\\mathrm {Low}(\\mathsf {KR})$ and $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Recall that $A\\in 2^\\mathbb {N}$ is computably dominated if every $f\\le _T A$ is dominated by some computable function.", "Theorem 8.4 (Greenberg-Miller ) Let $A\\in 2^\\mathbb {N}$ .", "(i) $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ if and only if $A$ is of non-DNC degree.", "(ii) $A\\in \\mathrm {Low}(\\mathsf {KR})$ if and only if $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ and computably dominated.", "For $A\\in 2^\\mathbb {N}$ , a $\\Pi ^0_1$ class $ is \\emph {$ tt$-negligible relative to $ A$} if $ A(=0$ for every $ A$-computable measure $$.", "We first prove the following.$ Proposition 8.1 If $A\\in \\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ , then every deep $\\Pi ^0_1$ class is $\\mathit {tt}$ -negligible relative to $A$ .", "Let $ be a deep $ 01$ class and $ T$ its associated canonical co-c.e.\\ tree.", "Let $ ALow(MLR,KR)$, which by Theorem \\ref {thm:greenberg-miller-kr} (i) is equivalent to being of non-DNC degree.", "We appeal to a useful characterization of non-DNC degrees due to Hölzl and Merkle~\\cite {HolzlM2010}: $ A$ is of non-DNC degree if and only if it is \\emph {infinitely often c.e.\\ traceable} (hereafter, i.o.\\ c.e.\\ traceable).", "This means that there exists a computable order~$ h$, such that the following holds: for every total $ A$-computable function~$ s: NN$, there exists a family $ (Sn)nN$ of uniformly c.e.\\ finite sets such that $ \|Sn\|<h(n)$ for all~$ n$ and $ s(n) Sn$ for infinitely many~$ n$.$ Let $h$ be an order witnessing the i.o.", "c.e.", "traceability of $A$ .", "Since $ is deep, let~$ f$ be a computable function such that $ M(Tf(n)) < 2-2n/h(n)$.Suppose for the sake of contradiction that $ is not $\\mathit {tt}$ -negligible relative to $A$ , which means that there exists an $A$ -computable measure $\\mu ^A$ such that $\\mu ^A(>r$ for some rational $r>0$ .", "Let $s\\le _TA$ be the function that on input $n$ gives a rational lower-approximation, with precision $1/2$ , of the values of $\\mu ^A$ on all strings of length $f(n)$ (encoded as an integer).", "By this we mean that $s(n)$ gives us for all strings $\\sigma $ of length $f(n)$ a rational value $s(n,\\sigma )$ such that $\\mu ^A(\\sigma )/2 \\le s(n,\\sigma ) \\le \\mu ^A(\\sigma )$ .", "Let $(S_n)_{n\\in \\mathbb {N}}$ witness the traceability of $s$ , i.e., the $S_n$ 's are uniformly c.e., $\|S_n\|<h(n)$ for every $n$ , and $s(n) \\in S_n$ for infinitely many $n$ .", "We now build a lower semi-computable continuous semi-measure $\\rho $ as follows.", "For all $n$ , enumerate $S_n$ .", "For each member $z \\in S_n$ , interpret $z$ as a mass distribution $\\nu $ on the collection of strings of length $f(n)$ .", "Then, for each string $\\sigma $ of length $f(n)$ , increase $\\rho (\\sigma )$ (as well as strings comparable with $\\sigma $ in any way that ensures that $\\rho $ remains a semi-measure) by $2^{-n-1}\\nu (\\sigma )/h(n)$ .", "This has a total cost of $2^{-n-1}/h(n)$ , and since there are at most $h(n)$ elements in $S_n$ , the total cost at level $f(n)$ is at most $2^{-n-1}$ .", "Therefore the total cost of the construction of $\\rho $ is bounded by 1 and thus $\\rho $ is indeed a lower semi-computable continuous semi-measure Now, for any $n$ such that $s(n) \\in S_n$ (as there are infinitely many such $n$ ), $\\rho $ distributes an amount of at least $(\\mu ^A(T_{f(n)})/2) \\cdot (2^{-n-1}/h(n))$ on $T_{f(n)}$ , and since $\\mu ^A(T_n)>r$ , this gives $\\rho (T_{f(n)}) \\ge 2^{-n-O(1)}/h(n).$ However, we had assumed that $\\mathbf {M}(T_{f(n)}) \\le 2^{-2n}/h(n),$ for all $n$ .", "But since $\\rho \\le ^* \\mathbf {M}$ , we get a contradiction.", "The following result involves the notion of relative $\\mathit {tt}$ -reducibility.", "For a fixed $A\\in 2^\\mathbb {N}$ , a $\\mathit {tt}(A)$ -functional is a total $A$ -computable Turing functional.", "Equivalently, we can define a $\\mathit {tt}(A)$ -functional $\\Psi ^A$ in terms of a Turing functional $\\Phi $ as follows: Let $\\Phi $ be defined on all inputs of the form $X\\oplus A$ .", "Then we set $\\Psi ^A(X)=\\Phi (X\\oplus A)$ .", "Furthermore, one can show that there is an $A$ -computable bound on the use of $X$ in the computation (just as there is a computable bound in the use function for unrelativized $\\mathit {tt}$ -computations).", "Theorem 8.5 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $ be a deep $ 01$ class and $ ALow(KR)$.", "By Proposition \\ref {prop:low-for-tt-depth}, $ is also $\\mathit {tt}$ -deep relative to $A$ .", "Let $\\Phi ^A$ be a $\\mathit {tt}(A)$ -functional.", "The pre-image $ of $ under $\\Phi ^A$ is a $\\Pi ^0_1(A)$ class, which must be $\\mathit {tt}$ -deep relative to $A$ as well, by Theorem REF relativized to $A$ .", "Now, applying Proposition REF relativized to $A$ , $ contains no $ A$-Kurtz random sequence.", "But since $ A$ is low for Kurtz randomness, $ contains no Kurtz-random sequence as well.", "We now obtain a partial analogue of Theorem REF .", "Corollary 8.2 Let $X$ be Kurtz random and $A\\in \\mathrm {Low}(\\mathsf {KR})$ .", "Then $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class.", "Let $\\Phi $ be a $\\mathit {tt}$ -functional.", "Since $\\Phi $ is total, it is certainly total on all sequences of the form $X\\oplus A$ for $X\\in 2^\\mathbb {N}$ .", "Thus $\\Psi ^A(X)=\\Phi (X\\oplus A)$ is a $\\mathit {tt}(A)$ -functional.", "By Theorem REF , it follows that $\\Phi (X\\oplus A)$ cannot be contained in any deep class.", "Question 1 Does Corollary REF still hold if we replace “deep\" with “$\\mathit {tt}$ -deep\"?", "We can extend Theorem REF to the following result, which proceeds by almost the same proof, the details of which are left to the reader.", "Theorem 8.6 Let $X$ be Martin-Löf random and $A$ be $\\mathrm {Low}(\\mathsf {MLR},\\mathsf {KR})$ .", "Then $X$ does not $\\mathit {tt}(A)$ -compute any member of any deep $\\Pi ^0_1$ class.", "(in particular, $X \\oplus A$ does not $\\mathit {tt}$ -compute any member of any deep $\\Pi ^0_1$ class)." ], [ "Depth, mutual information, and the Independence Postulate", "In this final section, we introduce the notion of mutual information and apply it to the notion of depth.", "Roughly, what we prove is that every member of every deep class has infinite mutual information with Chaitin's $\\Omega $ , a Martin-Löf random sequence that encodes the halting problem.", "This generalizes a result of Levin's, that every consistent completion of $\\mathsf {PA}$ has infinite mutual information with $\\Omega $ .", "We conclude with a discussion of the Independence Postulate, a principle introduced by Levin to derive the statement that that no consistent completion of arithmetic is physically obtainable." ], [ "The definition of mutual information", "First we review the definitions of Kolmogorov complexity of a pair and the universal conditional discrete semi-measure $\\mathbf {m}(\\cdot \\mid \\cdot )$ .", "Let $\\langle \\cdot ,\\cdot \\rangle :2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow 2^{<\\mathbb {N}}$ be a computable bijection.", "Then we define $\\mathrm {K}(\\sigma ,\\tau ):=\\mathrm {K}(\\langle \\sigma ,\\tau \\rangle )$ .", "Similarly, we set $\\mathbf {m}(\\sigma ,\\tau ):=\\mathbf {m}(\\langle \\sigma ,\\tau \\rangle )$ .", "A conditional lower semi-computable discrete semi-measure $m(\\cdot \\mid \\cdot ):2^{<\\mathbb {N}}\\times 2^{<\\mathbb {N}}\\rightarrow [0,1]$ is a function satisfying $\\sum _\\sigma m(\\sigma \\mid \\tau )\\le 1$ for every $\\tau $ .", "Then $\\mathbf {m}(\\cdot \\mid \\cdot )$ is defined to be a universal conditional lower semi-computable discrete semi-measure, so that for every conditional lower semi-computable discrete semi-measure, there is some $c$ such that $m(\\sigma \\mid \\tau )\\le c\\cdot \\mathbf {m}(\\sigma \\mid \\tau )$ for every $\\sigma $ and $\\tau $ .", "Lastly, we define the conditional prefix-free Kolmogorov complexity $\\mathrm {K}(\\sigma \\mid \\tau )$ to be $\\mathrm {K}(\\sigma \\mid \\tau )=\\min \\lbrace \|\\xi \|:U(\\langle \\xi ,\\tau \\rangle )=\\sigma \\rbrace ,$ where $U$ is a universal prefix-free machine.", "The mutual information of two strings $\\sigma $ and $\\tau $ , denoted by $I(\\sigma :\\tau )$ , is defined by $I(\\sigma :\\tau ) = \\mathrm {K}(\\sigma ) + \\mathrm {K}(\\tau ) - \\mathrm {K}(\\sigma ,\\tau )$ or equivalently by $2^{I(\\sigma :\\tau )} = \\frac{\\mathbf {m}(\\sigma ,\\tau )}{\\mathbf {m}(\\sigma ) \\cdot \\mathbf {m}(\\tau )}.$ By the symmetry of information (see Gács ), we also have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} =^* \\frac{\\mathbf {m}(\\tau \\mid \\sigma ,\\mathrm {K}(\\sigma ))}{\\mathbf {m}(\\tau )}.$ Levin extends mutual information to infinite sequences by setting $2^{I(X:Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^X(\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}.$ Recall that Chaitin's $\\Omega $ can be obtained as the probability that a universal prefix-free machine will halt on a given input, that is, $\\Omega =\\sum _{U(\\sigma ){\\downarrow }}2^{-\|\\sigma \|}$ , where $U$ is a fixed universal prefix-free machine.", "Generalizing a result of Levin's from , we have: Theorem 9.1 Let $ be a $ 01$ class and $ T$ its associated co-c.e.\\ tree.", "Suppose~$ is deep, witnessed by a computable order $f$ such that $\\mathbf {m}(T_{f(n)}) < 2^{-n}$ .", "Then for every $Y \\in and all~$ n$,$$I\\Big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\Big ) \\ge n-O(\\log n).$$In particular,$$I(\\Omega :Y)=\\infty .$$$ Our proof follows the same idea Levin uses for consistent completions of $\\mathsf {PA}$ (see ), although some extra care is needed for arbitrary deep classes.", "Suppose for a given $n$ we have an exact description $\\tau $ of $T_{f(n)}$ ; that is, on input $\\tau $ , the universal machine outputs a code for the finite set $T_{f(n)}$ .", "By the definition of $f$ , $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\le 2^{-n}$ or equivalently $\\sum _{\\sigma \\in T_{f(n)}} \\mathbf {m}(\\sigma ) \\cdot 2^{n} \\le 1$ Therefore, the quantity $\\mathbf {m}(\\sigma ) \\cdot 2^{n} \\cdot \\mathbf {1}_{\\sigma \\in T_{f(n)}}$ is a discrete semi-measure, but it is not lower semi-computable since $T_{f(n)}$ is merely co-c.e.", "(and, in general, not c.e.", "by Proposition REF ).", "However, it is a lower semi-computable semi-measure relative to the exact description $\\tau $ of $T_{f(n)}$ .", "Thus, for every $\\sigma \\in T_{f(n)}$ , by the universality of $\\mathbf {m}(\\cdot \\mid \\tau )$ , $\\mathbf {m}(\\sigma \\mid \\tau ) \\ge ^* \\mathbf {m}(\\sigma ) \\cdot 2^n.$ By the symmetry of information, we have $2^{I(\\sigma :\\tau )} =^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau ,\\mathrm {K}(\\tau ))}{\\mathbf {m}(\\sigma )} \\ge ^* \\frac{\\mathbf {m}(\\sigma \\mid \\tau )}{\\mathbf {m}(\\sigma )} \\ge ^2^n,$ and hence $I(\\tau :\\sigma ) \\ge ^+ n$ .", "We would like to apply this fact to the case where $\\sigma = Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ .", "But this is not technically sufficient, as $\\Omega {\\upharpoonright }n$ does not necessarily contain enough information to exactly describe $T_{f(n)}$ .", "This is not an obstacle in Levin's argument for completions of $\\mathsf {PA}$ , but it is for arbitrary deep classes.", "However, $\\Omega {\\upharpoonright }n$ contains enough information to get a “good enough\" approximation of $T_{f(n)}$ .", "Let us refine the idea above: suppose now that $\\tau $ is no longer an exact description of $T_{f(n)}$ , but is a description of a set of strings $S$ of length $f(n)$ such that $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^ 2^{-n} n^2.$ Then, by following the same reasoning as above, we would have $I(\\sigma : \\tau ) \\ge ^+ n-2\\log n$ for all $\\sigma \\in S$ (and thus all $\\sigma \\in T_{f(n)}$ ).", "We shall prove that $\\Omega {\\upharpoonright }n$ contains enough information to recover such a set $S$ , thus proving the theorem.", "The real number $\\Omega $ is lower semi-computable and Solovay complete (see ).", "As a consequence, for every other lower semi-computable real $\\alpha $ , knowing the first $k$ bits of $\\Omega $ allows us to compute the first $k-O(1)$ bits of $\\alpha $ .", "For all $n$ , define: $a_n = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}}} \\mathbf {m}(\\sigma )$ and observe that $a_n$ is lower semi-computable uniformly in $n$ (because $T_{f(n)}$ is co-c.e.", "uniformly in $n$ ), and belongs to $[0,1]$ .", "Define now $\\alpha = \\sum _n \\frac{a_n}{n^2},$ which is a lower semi-computable real.", "Thus, knowing the first $n$ bits of $\\Omega $ gives us the first $n-O(1)$ bits of $\\alpha $ , i.e., an approximation of $\\alpha $ with precision $2^{-n}$ .", "In particular this gives us an approximation of $a_n$ with precision $2^{-n} \\cdot n^2 \\cdot O(1)$ , which we can assume to be a lower approximation, which we will write as $a^{\\prime }_n$ .", "Now, using $a^{\\prime }_n$ , one can enumerate $a_n$ until we find a stage $s_n$ such that $a_n[s_n] = \\sum _{{\|\\sigma \|=f(n) \\\\ \\sigma \\notin T_{f(n)}[s_n]}} \\mathbf {m}(\\sigma )[s_n] \\ge a^{\\prime }_n.$ Since $\|a_n - a^{\\prime }_n\| \\le 2^{-n} \\cdot n^2 \\cdot O(1)$ , this implies $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}[s_n]} \\setminus T_{f(n)}}} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n^2.$ But recall from above that $\\sum _{{\|\\sigma \|=f(n) \\\\ {\\sigma \\in T_{f(n)}}}} \\mathbf {m}(\\sigma ) \\le 2^{-n}.$ Combining these two facts, and taking $S$ to be the set $T_{f(n)}[s_n]$ , we have $T_{f(n)} \\subseteq S$ and $\\sum _{\\sigma \\in S} \\mathbf {m}(\\sigma ) \\le ^* 2^{-n} \\cdot n ^2$ which establishes the first part of the theorem.", "To see that the second part of the statement follows from the first, take $\\sigma =Y {\\upharpoonright }f(n)$ and $\\tau = \\Omega {\\upharpoonright }n$ and observe that $\\mathbf {m}^Y(\\sigma ) =^* \\mathbf {m}^Y(n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ $\\mathbf {m}^\\Omega (\\tau ) =^* \\mathbf {m}^\\Omega (n) \\ge ^* \\mathbf {m}(n) \\ge ^* 1/n^2,$ and $2^{I(\\sigma :\\tau )}=2^n/n^{O(1)}$ (since $I(\\sigma :\\tau )\\ge ^+n-2\\log n$ as established above).", "Then we have $\\begin{split}2^{I(\\Omega :Y)} = \\sum _{\\sigma , \\tau \\in 2^{<\\mathbb {N}}} \\mathbf {m}^\\Omega (\\sigma ) \\cdot \\mathbf {m}^Y(\\tau ) \\cdot 2^{I(\\sigma ,\\tau )}&\\ge \\sum _n \\mathbf {m}^\\Omega (Y{\\upharpoonright }f(n)) \\cdot \\mathbf {m}^Y(\\Omega {\\upharpoonright }n) \\cdot 2^{I(Y{\\upharpoonright }f(n),\\Omega {\\upharpoonright }n)}\\\\&\\ge \\sum _n 2^n/n^{O(1)}=\\infty .\\end{split}$ Remark 9.1 The converse of this theorem does not hold, i.e., there is a $\\Pi ^0_1$ class that is not deep but all of whose elements have infinite mutual information with $\\Omega $ .", "This follows from Theorem REF and the fact that having infinite mutual information with $\\Omega $ is a property that is invariant under addition or deletion of a finite prefix.", "Levin's proof of Theorem REF restricted to the particular case of completions of $\\mathsf {PA}$ is the mathematical part of a more general discussion, the other part of which is philosophical in nature.", "While Gödel's theorem asserts that no completion of $\\mathsf {PA}$ can be computably obtained, Levin's goal is to show that no completion of $\\mathsf {PA}$ can be obtained by any physical means whatsoever (computationally or otherwise), thus generalizing Gödel's theorem.", "Levin does not fully specify what he means by physically obtainable (the exact term he uses is “located in the physical world\"), but nonetheless he makes the following postulate, which he dubs the “Independence Postulate\": if $\\sigma $ is a mathematically definable string that has an $n$ bit description and $\\tau $ can be located in the physical world with a $k$ bit description, then for a fixed small constant $c$ , one has $I(\\sigma : \\tau ) < n+k+c$ .", "In particular, if one admits that some infinite sequences can be physically obtained, the Independence Postulate for infinite sequences says that if $X$ and $Y$ are two infinite sequences with $X$ mathematically definable and $Y$ physically obtainable, then $I(X:Y) < \\infty $ .", "Being $\\Delta ^0_2$ , $\\Omega $ is mathematically definable, and, as Levin shows, $I(\\Omega :Y)=\\infty $ for any completion $Y$ of $\\mathsf {PA}$ .", "Thus, assuming the Independence Postulate, no completion of $\\mathsf {PA}$ is physically obtainable.", "Our Theorem REF extends Levin's theorem and, assuming the Independence Postulate, shows that no member of a deep class (shift-complex sequences, compression functions, etc.)", "is physically obtainable.", "Of course, evaluating the validity of the Independence Postulate would require an extended philosophical discussion that would take us well beyond the scope of this paper.", "In any case, whether or not the reader accepts the Independence Postulate, Theorem REF is interesting in its own right.", "In fact, it is quite surprising because it seems to contradict the “basis for randomness theorem\" (see ), which states that if $X$ is a Martin-Löf random sequence and $\\mathcal {C}$ is a $\\Pi ^0_1$ class, then there exists a member $Y$ of $\\mathcal {C}$ such that $X$ is random relative to $Y$ .", "If a sequence $X$ is random relative to another sequence $Y$ , the intuition is that $Y$ “knows nothing about $X$ \", and thus one could conjecture that $I(X : Y) < \\infty $ .", "However, this cannot always be the case, since by Theorem REF , $I(\\Omega :Y) = \\infty $ for all members $Y$ of a deep $\\Pi ^0_1$ class $, even though $$ is random relative to some $ Y.", "This apparent paradox can be explained by taking a closer look at the definition of mutual information.", "Let $ be a deep $ 01$ class, whose canonical co-c.e.\\ tree~$ T$ satisfies $ m(Tf(n)) < 2-n$ for some computable function~$ f$.", "By Theorem~\\ref {thm:mutual-info} and the symmetry of information, for every $ Y we have $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big ) = I\\big (\\Omega {\\upharpoonright }n : Y {\\upharpoonright }f(n)\\big ) \\ge n-O(\\log n)$ where $k_n$ stands for $\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "Take a $Y \\in such that $$ is random relative to~$ Y$.", "It is well-known that a sequence~$ Z$ is random if and only if $ K(Z n n) n-O(1)$ (see for example Gács~\\cite {Gacs1980}).", "Applying this fact (relativized to~$ Y$) to $$, we have$$\\mathrm {K}^Y(\\Omega {\\upharpoonright }n \\mid n) \\ge n-O(1)$$and thus in particular that$$\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\ge n-O(1).$$Since $ K(n) n+O(n)$, it follows that$ $\\mathrm {K}(\\Omega {\\upharpoonright }n) - \\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n)) \\le O(\\log n)$ The only difference between (REF ) and (REF ) is the term $k_n=\\mathrm {K}(Y {\\upharpoonright }f(n))$ .", "But it makes a big difference, as one can verify that $\\mathrm {K}(\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n))-\\mathrm {K}\\big (\\Omega {\\upharpoonright }n \\mid Y {\\upharpoonright }f(n), k_n\\big )\\ge n-O(\\log n).$ Informally, while $Y$ “knows nothing\" about $\\Omega $ , the complexity of its initial segments, seen as a function, does.", "In particular, the change of complexity caused by $k_n$ implies that $\\mathrm {K}(\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge n - O(\\log n)$ and thus $\\mathrm {K}(Y {\\upharpoonright }f(n)) \\ge 2^n/n^{O(1)}$ .", "Acknowledgements We would like to thank Noam Greenberg, Rupert Hölzl, Mushfeq Khan, Leonid Levin, Joseph Miller, André Nies, Paul Shafer, and Antoine Taveneaux for many fruitful discussions on the subject.", "Particular thanks go to Steve Simpson who provided very detailed feedback on the first arXiv version of this paper." ] ]	1403.0450
[ [ "Phase Field Crystal Modeling as a Unified Atomistic Approach to Defect\n Dynamics" ], [ "Abstract Material properties controlled by evolving defect structures, such as mechanical response, often involve processes spanning many length and time scales which cannot be modeled using a single approach.", "We present a variety of new results that demonstrate the ability of phase field crystal (PFC) models to describe complex defect evolution phenomena on atomistic length scales and over long, diffusive time scales.", "Primary emphasis is given to the unification of conservative and non- conservative dislocation creation mechanisms in three-dimensional FCC and BCC materials.", "These include Frank-Read-type glide mechanisms involving closed dislocation loops or grain boundaries as well as Bardeen-Herring-type climb mechanisms involving precipitates, inclusions, and/or voids.", "Both source classes are naturally and simultaneously captured at the atomistic level by PFC de- scriptions, with arbitrarily complex defect configurations, types, and environments.", "An unexpected dipole-to-quadrupole source transformation is identified, as well as various new and complex geomet- rical features of loop nucleation via climb from spherical particles.", "Results for the strain required to nucleate a dislocation loop from such a particle are in agreement with analytic continuum theories.", "Other basic features of FCC and BCC dislocation structure and dynamics are also outlined, and initial results for dislocation-stacking fault tetrahedron interactions are presented.", "These findings together highlight various capabilities of the PFC approach as a coarse-grained atomistic tool for the study of three-dimensional crystal plasticity." ], [ "Introduction", "The macroscopic mechanical properties of crystals and polycrystals are primarily consequences of complex collective interactions between atomic level defects.", "The characteristic scales of these interaction processes can span many orders of magnitude in length and time, presenting major fundamental challenges to the development of a unified modeling approach.", "In particular, line defects or dislocations, which are the central mediators of plasticity in many systems, can evolve rapidly via conservative mechanisms (e.g., glide and cross-slip) or relatively slowly via nonconservative mechanisms mediated by interactions with point defects (e.g., climb).", "Characteristic length scales of dislocation structures range from atomic dimensions to micron-level and up for collective, organized arrays.", "A complete model of dislocation dynamics would therefore ideally include the fundamental physics of dislocation creation, interaction, annihilation/absorption, etc, via both conservative and nonconservative mechanisms, accessible across all relevant length and time scales.", "This is not feasible with any presently available model.", "Other classes of defects (point, planar, and bulk) should also be considered in a more general model of crystalline materials subjected to driving forces.", "Elements of conservative dislocation processes are often quite readily modeled at the atomic level using approaches such as molecular dynamics (MD) [1], [2].", "These conservative mechanisms have also been built into larger length scale mesoscopic modeling approaches such as discrete dislocation dynamics (DDD) [3], [4], [1] and continuum phase field (PF) dislocation models [5], [6], [1].", "Basic conservative processes are generally implemented in DDD with detailed rule-based formulations that consider some number of the innumerable possible defect interaction scenarios.", "PF models avoid this complexity by treating select dislocation lines as interfaces in a continuum field description.", "These interfaces interact and evolve automatically in response to local driving forces.", "The cost is greater computational demand, since the PF equations must be solved throughout the entire system, not only at localized dislocation positions.", "Nonconservative dislocation processes present another set of challenges to plasticity models, as these motions are mediated by vacancy diffusion and are therefore inherently difficult to access on conventional atomistic simulation time scales.", "Climb becomes relevant and often dominant at high temperatures or large vacancy concentrations, and is fundamental to such phenomena as creep, annealing, recrystallization, and irradiation damage.", "Elements of climb have been built into DDD [7], [8], [9], [10] and PF [11], [12] models, though this issue is still in many ways under development.", "Meaningful coarse-grained input parameters and their values, for example, which would ideally be extracted from microscopic simulations, are lacking.", "No atomistic modeling approach has yet proven capable of consistently describing both conservative and nonconservative dislocation processes over both nano and mesoscales.", "In this work, we present results for a method that unifies both types of dislocation motion on atomistic length scales.", "Phase field crystal (PFC) models [13], [14], [15] describe diffusive dynamics in condensed matter systems with atomistic resolution, and are therefore potentially capable of bridging the gap between fast glide plasticity and slow climb plasticity at the nanoscale.", "Defect superstructures of arbitrary complexity can be studied, including polycrystals with nearly any variation/combination of dislocation, grain boundary, precipitate, and stacking fault configurations, for example.", "The ability to naturally describe arbitrary defect structures is a feature of atomistic approaches that is inherently absent from mesoscale approaches.", "The larger length scales described by DDD and PF models cannot currently be reached by PFC, though one may imagine using PFC simulations to generate input parameters for such models or numerically coupling PFC with DDD or PF.", "Coarse-grained complex amplitude PFC models also provide an interesting means of self-consistently reaching larger length scales [16], [17], [18], [19], [20], [21].", "The goals of the present work are to demonstrate that simple PFC models naturally capture well-established conservative dislocation creation mechanisms as well as new elements of the relatively poorly understood nonconservative dislocation creation mechanisms, which cannot be easily studied with other methods.", "These goals are part of a larger effort to exploit the novel features of the PFC approach within traditional areas of materials science, including crystal plasticity, structural phase transformations, and microstructure evolution [22], [23], [24], [25], [26], [21].", "Some initial groundwork covering fundamental dislocation properties in FCC materials was reported by the present authors in Ref.", "[25].", "Perfect dislocations and simple grain boundaries in 2D triangular and 3D BCC crystals have also been examined in various contexts [14], [27], [28], [29], [30], [31], [15], [32], [33], [34], [35], [36], [37], [38].", "The rest of this paper is organized as follows.", "In Section , the basic model equations, numerical solution methods, and strain application procedures are outlined.", "In Section , some qualitative and quantitative features of the specific dislocations relevant to FCC and BCC plasticity are surveyed in the PFC framework.", "In Section , conservative Frank-Read-type dislocation sources in FCC materials are studied in two contexts.", "The first considers controlled nucleation of dislocation loop dipoles, and a new mechanism whereby a dipole source transforms into a quadrupole source is reported.", "The second context considers uncontrolled nucleation of partial and perfect dislocations from grain boundaries in nanopolycrystalline samples.", "In Section , nonconservative Bardeen-Herring-type dislocation sources in BCC materials are studied.", "The case of uniform loop nucleation from spherical inclusions or precipitates is considered.", "A range of complex nucleation behaviors caused by non-trivial interactions between interface structure, strain orientation, and dislocation energetics are examined.", "Selected results are compared with earlier analytic predictions and shown to agree well when the analytic theories are adapted to the scenario considered in our simulations.", "Finally, a brief presentation of stacking fault tetrahedron (SFT) formation and SFT-dislocation interactions is provided in Section ." ], [ "Model and Methods", "The standard PFC free energy functional, modified to stabilize FCC stacking faults as described in Ref.", "[25], is used for all FCC systems studied in the following sections.", "It is written $\\tilde{F}&=&\\int d\\vec{r}~\\left[\\frac{1}{2}n^2(\\vec{r})-\\frac{w}{6}n^3(\\vec{r}) +\\frac{u}{12}n^4(\\vec{r}) \\right]-\\nonumber \\\\& &\\frac{1}{2} \\int \\int d\\vec{r}~d\\vec{r}_2~n(\\vec{r})C_2(\|\\vec{r}-\\vec{r}_2\|) n(\\vec{r}_2).$ where $\\tilde{F}=F/(k_B T \\rho _{\\ell })$ , $\\rho _{\\ell }$ is a constant reference density, $n(\\vec{r})=\\rho (\\vec{r})/\\rho _{\\ell }-1$ is the rescaled atomic density field, $\\rho (\\vec{r})$ is the unscaled atomic number density field, $w$ and $u$ are coefficients treated as free parameters to provide additional model flexibility, and $C_2(\|\\vec{r}-\\vec{r}_2\|)$ is the two-point direct correlation function of the fluid, assumed isotropic.", "The modified standard PFC kernel (which approximates the full $C_2(\|\\vec{r}-\\vec{r}_2\|)$ ) after Fourier transformation reads $\\hat{C}_2(k) = -r + 1 - B^x (1-\\tilde{k}^2)^2 -H_0 e^{-(k-k_0)^2/(2\\alpha _0^2)},$ where $r$ is a constant proportional to temperature, $B^x$ is a constant proportional to the solid-phase elastic moduli, $\\tilde{k}=k/(2\\pi )$ is the normalized wavenumber, $H_0$ is a constant related to stacking fault energy $\\gamma _{\\rm ISF}$ , $k_0=2\\pi \\sqrt{41/12}/a$ , $a$ is the equilibrium lattice constant, and $\\alpha _0$ is an additional constant related to $\\gamma _{\\rm ISF}$ .", "$n(\\vec{r})$ will be allowed to assume nonzero average values $n_0$ , and it will be implied that $w=0$ and $u=3$ for all FCC simulations.", "The structural or XPFC free energy functional class [22], [23], [24] is used for all BCC systems studied in the following sections.", "It is written as Eq.", "(REF ) with the Fourier transformed kernel $\\hat{C}_2(k)_i =e^{-(k-k_i)^2/(2\\alpha _i^2)}e^{-\\sigma ^2 k_i^2/(2\\rho _i \\beta _i)}$ where $i$ denotes a family of lattice planes at wavenumber $k_i$ , and $\\sigma $ is a temperature parameter.", "The constants $\\alpha _i$ , $\\rho _i$ , and $\\beta _i$ are the Gaussian width (which sets the elastic constants), planar atomic density, and number of planes, respectively, associated with the $i$ th family of lattice planes.", "The envelope of all selected Gaussians $i$ composes the final $\\hat{C}_2(k)$ .", "Only one reflection at $k_1=2\\sqrt{3}\\pi $ will be used here, as this is all that is necessary to produce equilibrium BCC structures with $a \\simeq \\sqrt{2/3}$ .", "Two dynamic equations for $n(\\vec{r})$ will be considered.", "The first is a purely diffusive Model B form, $\\frac{\\partial n(\\vec{r})}{\\partial t} = \\nabla ^2\\frac{\\delta \\tilde{F}}{\\delta n(\\vec{r})}$ where $t$ is dimensionless time.", "The second equation of motion introduces a faster inertial, quasi-phonon dynamic component in addition to diffusive dynamics [28], $\\frac{\\partial ^2 n(\\vec{r})}{\\partial t^2}+\\beta \\frac{\\partial n(\\vec{r})}{\\partial t}= \\alpha ^2 \\nabla ^2 \\frac{\\delta \\tilde{F}}{\\delta n(\\vec{r})}$ where $\\alpha $ and $\\beta $ are constants related to sound speed and damping rate, respectively.", "All simulations were performed in 3D using a pseudo-spectral algorithm with semi-implicit time stepping and periodic boundary conditions.", "Deformation was applied via strain-controlled methods for both simple shear and uniaxial tension or compression simulations.", "Constant strain rates were employed for both deformation types." ], [ "Some Fundamental Elements of FCC & BCC Crystal Plasticity", "Many aspects of the distinct plastic response of FCC and BCC crystals can be understood in terms of structural and dynamic differences between the dominant carriers of plasticity in either lattice.", "The primary dislocation type in FCC materials is that with total Burgers vector $a/2\\langle 110\\rangle $ , dissociated into two $a/6\\langle 112\\rangle $ Shockley partials connected by a stacking fault.", "Such dissociated dislocations provide FCC crystals with 12 active primary slip systems, all of type $\\lbrace 111\\rbrace \\langle 110\\rangle $ .", "The very small Peierls stress of the Shockley partials can explain the low yield stress of FCC materials, and the subsequent formation of large numbers of dislocation junctions and stacking faults can explain their excellent work-hardening properties, ductility, and the commonly observed formation of mesoscale dislocation patterns [39].", "The primary dislocation type in BCC materials is that with Burgers vector $a/2\\langle 111\\rangle $ .", "This dislocation is glissile within 48 potential primary slip systems of type $\\lbrace 110\\rbrace \\langle 111\\rangle $ , $\\lbrace 112\\rbrace \\langle 111\\rangle $ , or $\\lbrace 123\\rbrace \\langle 111\\rangle $ .", "The non-planar core structure of the $a/2\\langle 111\\rangle $ screw dislocation in particular at low temperatures leads to a much smaller glide mobility than that in edge orientation.", "The $a/2\\langle 111\\rangle $ screw dislocation therefore controls plastic flow in BCC crystals at $T \\lesssim 0.15T_{melt}$ .", "Its high Peierls stress can explain the large yield and flow stresses of BCC crystals, as well as the absence of mesoscale dislocation patterning at low to moderate stresses [39].", "It can therefore be argued that these dislocation types must be stabilized and their basic core features reproduced if one wishes to perform atomistic PFC simulations of FCC and BCC plasticity.", "We have found that it is indeed possible to stabilize both of these dislocation types within a given PFC model.", "Dissociated $a/2\\langle 110\\rangle $ FCC dislocations were previously studied by the current authors [25].", "With proper selection of the model parameters, static properties (dissociation width, Peierls strains, etc) and glide dynamics of these dislocations were found to be in good agreement with other atomistic calculations and continuum elastic theories.", "For the primary BCC $a/2\\langle 111\\rangle $ screw dislocation, we have found that the PFC model used in this study reproduces the same nondegenerate nonpolarized core configuration obtained from density functional theory calculations and MD simulations employing various empirical potentials [40] (see Fig.", "REF ).", "The central core features of the primary dislocation types in both FCC and BCC crystals can therefore be well-captured by PFC models.", "This level of accuracy in terms of core structure should not always be expected, especially for more complex, directionally bonded materials such as diamond cubic Si or Ge crystals.", "Nonetheless, when sufficient accuracy is achieved, we find as a general consequence that the correct slip systems naturally emerge during simulations of plastic flow, and that atomistically detailed plasticity mechanisms also often follow.", "Figure: (Color online)The BCC a/2〈111〉a/2\\langle 111\\rangle screw dislocationfrom a single-peaked XPFC model.A cross-section of n(r →)n(\\vec{r}) is shown on the left, andthe differential displacement maparound the core is shown on the right.Results were generated usingEqs.", "(), (), and ()with model parametersw=1.4w=1.4, u=1u=1, n 0 =0.05n_0=0.05, α 1 =0.25\\alpha _1=0.25, σ=0.12\\sigma =0.12,ρ 1 =1\\rho _1=1, and β 1 =8\\beta _1=8.Nearly identical results are obtained at α 1 =1\\alpha _1=1 and α 1 =2\\alpha _1=2.For example, we have confirmed that the glide mobility of the BCC $a/2\\langle 111\\rangle $ dislocation in screw orientation is significantly lower than that in edge orientation, as expected based on the non-planar screw core structure.", "This mobility difference was inferred from zero-strain simulations of glide-mediated dipole annihilation, in which it was observed that screw dislocation dipoles take roughly one to two orders of magnitude longer to annihilate than edge dipoles at equal initial separations.", "These same basic structural and energetic features also influence climb processes.", "For example, compressive and tensile strains along the axis of the dislocation Burgers vector induce the correct positive and negative climb directions in PFC simulations.", "Jogged dislocations subjected to the same type of strain climb in the expected fashion in which jogs diffusively translate along the line direction.", "The result is a net motion in the climb direction perpendicular to the glide plane (see Supplemental Material [42] for animations).", "If no jogs are present, climb proceeds by a more uniform simultaneous translation of larger line segments or of the entire line, though the energy barrier for this type of climb is larger than that of diffusive jog translation.", "As shown previously for 2D triangular crystals in PFC [27], [28], we find in 3D that uniform climb velocities of undissociated dislocations in the limit of low dislocation density ($\\rho _d \\lesssim 10^{13}m^{-2}$ ) follow a power law as a function of applied stress, $v \\sim \\sigma ^m$ , with $m \\sim 1$ for both dynamic equations, Eqs.", "(REF ) and (REF ).", "Apparent exponents $m$ as large as 4 can appear at higher dislocation densities.", "Qualitatively similar behaviors as a function of $\\rho _d$ have been observed in kinetic Monte Carlo simulations [43].", "We also note that screw dislocation cross-slip readily occurs in 3D PFC simulations, and we find that the cross-slip barrier for dissociated FCC screw dislocations increases strongly with dissociation width and therefore with inverse stacking fault energy (see Supplemental Material [42] for animations).", "Such results, though far from a complete survey, demonstrate that relatively simple PFC models are capable of capturing many of the central atomistic features of plasticity in FCC and BCC materials over diffusive time scales." ], [ "Frank-Read-Type Glide Sources", "The Frank-Read dislocation multiplication mechanism is an important element of crystal plasticity that has been widely observed and studied both experimentally and via computer simulations [45], [46], [47], [44].", "In the most commonly discussed scenario, a dislocation line pinned at two points within its glide plane bows out under stress until it meets itself on the opposite side of the pinning points.", "The contacting segments annihilate, resulting in a single loop and the original pinned segment, which then repeats the process if sufficient strain energy is still available.", "The many possible manifestations of this basic mechanism, including pinning of existing dislocation lines and emission of new line segments from grain boundaries (which act as pinning centers), contribute centrally to the large increases in dislocation density that occur during plastic deformation.", "Though Frank-Read mechanisms occur via conservative dislocation motion and are therefore readily studied with conventional MD [44] and/or mesoscale continuum methods [46], [47], we consider them here because of their general importance in plastic deformation processes, to demonstrate that the basic physics of such sources is well-captured by PFC models.", "Non-conservative dislocation creation methods are separately considered in the following section.", "Unless noted otherwise, all simulations presented in this section describe FCC materials and employ Eqs.", "(REF ), (REF ), and (REF )." ], [ "Prismatic sources", "We first examine a Frank-Read dipole source consisting of a single rectangular, prismatic edge dislocation loop in which the $a/2\\langle 110\\rangle $ lines along the $[1\\bar{1}1]$ direction are relatively immobile (sessile), while the dissociated lines along the $[\\bar{1}12]$ direction are mobile (glissile) (see Figs.", "REF and REF a).", "The $[1\\bar{1}1]$ lines act as pinning points, such that under applied shear strain $\\epsilon _{\\rm zy}$ the glissile lines bow out via glide in opposite directions.", "This setup was used in the MD simulations of Ref.", "[44].", "Such a loop could in principle be formed, for example, by vacancy agglomeration following plastic deformation or irradiation, but it more generally provides a convenient source configuration that is entirely analogous to that of a longer jogged or locally pinned line segment.", "Figure: (Color online)Schematic of the simulation setup used for Frank-Read source operation.Figure: (Color online)Operationof a Frank-Read dipole source under shear strain ϵ zy \\epsilon _{\\rm zy}.", "(a) A dislocation loop composed of four a/2〈110〉a/2\\langle 110\\rangle edge dislocations.The two mobile horizontal segments are dissociated in the (11 ¯1)(1\\bar{1}1) plane,the two vertical segments are relatively immobile.", "(b) 3D view of the source near pinch-off at 560t560t.", "(c), (d), (e), and (f) show xyxy-plane views att=375t=375, 500, 595, and 625, respectively.For analysis and visualization purposes,local peaks in n(r →)n(\\vec{r}),which represent the most probable atomic positions,are taken to correspond to atomic sites.Density peaks with HCP coordination (stacking faults)are shown in orange (light) or red (dark) dependingon position along the out of plane zz axis.Those with irregularcoordination (dislocation core) are shown in gray.See Supplemental Material for the associated animation.To allow application of simple shear strain $\\epsilon _{\\rm zy}$ , the standard penalty function approach was used [28].", "The penalty function is written as an additional free energy term of the form $M(\\vec{r})(n(\\vec{r})-n_p(\\vec{r}))^2$ , where $M(\\vec{r})$ controls the strength of the penalty field and $n_p(\\vec{r})$ is the configuration of the penalty field.", "We specify $M(\\vec{r}), n_p(\\vec{r}) ={\\left\\lbrace \\begin{array}{ll}M_{\\ell }, n_0 & \\text{if } z \\in L_{\\ell }\\\\M_P, n^{\\rm EQ}_{\\rm FCC}(\\vec{r}) & \\text{if } z \\in L_P\\\\0, 0 & \\text{otherwise}\\end{array}\\right.", "}$ where $M_{\\ell }$ , $M_P$ , $L_{\\ell }$ , and $L_P$ are constants (see Fig.", "REF ) and $n^{\\rm EQ}_{\\rm FCC}(\\vec{r})$ is the commensurate equilibrium FCC $n(\\vec{r})$ .", "The resulting system is a thin infinite slab of FCC bounded in the $z$ -direction by a homogeneous quasi-liquid phase of width $2L_{\\ell }$ .", "The quasi-liquid layer simply circumvents any unphysical strains that would otherwise be caused by the large shear disregistry at the periodic $z$ boundary.", "Simple shear strain $\\epsilon _{\\rm zy}$ can then be applied to the crystalline slab by translating the upper pinned region $L_p$ along $+\\vec{y}$ and the lower pinned region $L_p$ along $-\\vec{y}$ at some constant velocity.", "When Eq.", "(REF ) is employed to allow rapid elastic relaxations, a nearly uniform shear profile is produced across the sample.", "Various initial loop sizes and stacking fault energies were considered, with two representative results shown in Figs.", "REF and REF (see Supplemental Material [42] for the associated animations).", "In both cases, model parameters $n_0=-0.48$ , $r=-0.63$ , $B^x=1$ , $\\alpha _0=1/2$ , $k_0=6.2653$ , $H_0=0.025$ , $\\beta =0.01$ , and $\\alpha =1$ were used.", "Additional simulation details are given here [48].", "To estimate simulation time scales, we match the numerically measured FCC vacancy diffusion constant $D_v \\simeq 1.0 a^2/t$ to that of Cu at $1063^{\\circ }$ C ($D_v \\simeq 10^{-13}m^2/s, a \\simeq 0.36nm$ ) [27].", "The time unit $t$ is then found to correspond to $\\sim 1.3\\mu s$ , and the shear rate $0.000235/t$ converts to $\\sim 180/s$ , which is roughly five orders of magnitude lower than that of a typical and comparable MD simulation (applying $1.8\\%$ strain in $1ns$ produces a rate of $1.8\\times 10^7/s$ or see, e.g.", "Ref.", "[49]).", "Other FCC shear rates used in this study range from $\\sim 10/s - 800/s$ .", "Figure: (Color online)Operationof a Frank-Read dipole-to-quadrupole source.The initial loop is shorter in the xx-direction and longer in thezz-direction than that of Fig.", ".", "(a), (b), (c), (d), (e), (f), and (g) show xyxy-plane views att=450t=450, 520, 580, 680, 750, 840 and 1030, respectively.A 3D view at t=750t=750 is also shown in (e).The source emits two initial loops near 520t520t, then reorientsand converts into a quadrupole source before being restored to aclosed loop form near 1030t1030t.Density peaks with HCP coordination (stacking faults)are shown in white, yellow (light gray), orange (gray), or red-orange(dark gray), dependingon position along the out of plane zz axis.Those with irregularcoordination (dislocation cores) are shown in gray.", "(h) An unzipped stacking fault tetrahedron acting as a Frank-Read-typesource.Density peaks with HCP coordination (stacking faults)are shown in red (dark gray), those with irregularcoordination (dislocation cores) are shown in gray.See Supplemental Material for the associated animations.The operation of the dipole source shown in Fig.", "REF closely follows the basic Frank-Read mechanism, with slight asymmetries in loop shape caused by image stresses in the $z$ -direction and interactions between opposing loops.", "Also, the $[1\\bar{1}1]$ line segments, though sessile, do not respond purely rigidly to local stresses, they are not perfectly pinned to their initial lattice locations.", "As the shear strain $\\epsilon _{zy}$ increases and the Frank-Read loops begin to bow out, the growing forces exerted on the $[1\\bar{1}1]$ segments stretch these lines such their angle from vertical $\\theta $ becomes larger than that of the applied shear $\\theta _A=\\tan {\\epsilon _{zy}}$ .", "These small strain relief mechanisms also contribute to the asymmetric shape of the growing loops, and can lead to more complex effects when the prismatic loop dimensions and strain rate are varied.", "One such effect is shown in Fig.", "REF .", "Here the initial $[\\bar{1}12]$ line segments are shorter and the initial $[1\\bar{1}1]$ segments are longer.", "When the $[1\\bar{1}1]$ segments stretch under the influence of the loop bow-out stresses, they begin to approach alignment with the nearest of the four $\\lbrace 111\\rbrace $ planes.", "They are then able to lower their energy by taking a periodically jogged configuration with dissociated glissile segments in adjacent $\\lbrace 111\\rbrace $ planes, each connected by a single jog where the lines are locally constricted.", "After the maximum bow-out stress has been overcome and the growing loops become nearly circular (Fig.", "REF a), the now jogged pinning lines begin to hinge back toward vertical.", "Rather than returning to their original undissociated, unjogged configuration, the most highly strained segments near the loop ends cross-slip at their constricted jog sites onto the $(1\\bar{1}1)$ glide plane normal to the $z$ -direction.", "These segments cannot easily cross-slip back to the original configuration, they instead begin to bow out on the $(1\\bar{1}1)$ plane with the same jog sites now serving as pinning points for the new sources.", "Depending on $L_z$ and $\\dot{\\epsilon }_{zy}$ , the number of new intermediate loops nucleated can vary.", "In the case of Fig.", "REF , two additional loops are formed and the dipole is converted into a quadrupole.", "The stresses exerted by the new loops cause the $[1\\bar{1}1]$ lines to reorient such that all segment pairs align with the shear direction and the bow-out direction becomes perpendicular to the shear direction (Fig.", "REF d,e).", "Once all four loops are pinched-off, the initial loop configuration is restored (Fig.", "REF g).", "If the maximum or activation stress of the initial source is low, then the $[1\\bar{1}1]$ segments may never reach the angle required to attain a stepped configuration or may attain a stepped configuration with only very short glissile segments.", "In such a case, no additional sources are formed.", "As the maximum or activation stress of the initial source increases, the length of the resulting glissile segments in the stepped configuration also increases.", "If the length of any of these segments becomes large enough to activate its operation as a new source for a given stress, then it will bow out and begin forming a new loop.", "This type of transformation is therefore most likely to occur when such sources have a large activation stress, which for Frank-Read [45] sources $\\sim 1/L_x$ , corresponding to small $L_x$ .", "Greater potential segmentation lengths (large $L_z$ ) also favor this behavior.", "These expectations are in agreement with our results.", "Such a transformation from dipole to multi-dipole also requires that the pinning points have some small, non-zero mobility, i.e., the pinning is not absolute.", "This condition is perhaps more applicable to soft crystalline materials such as colloidal crystals than to metals, for example, but similar imperfect pinning behaviors have been reported in MD and atomistic quasicontinuum studies of metallic crystals [50], [51], [52].", "Such transformations may therefore be observable in MD simulations.", "Analogous situations should also become more probable, for example, following a rapid quench or at high temperatures where vacancy concentrations are large and climb is active.", "The general behaviors discussed in these two examples exhibit some dependence on stacking fault energy $\\gamma _{\\rm ISF}$ .", "When $\\gamma _{\\rm ISF}$ is small, the strain required to operate the source tends to be lowest, and the pinning segments remain relatively immobile throughout operation.", "When $\\gamma _{\\rm ISF}$ is large, the strain required to operate the source increases, and the forces exerted on the pinning segments by the bowing loops increase such that the pinning segments may be dragged through the crystal, effectively destroying the source.", "It was also observed that stacking fault tetrahedra, created under low $\\gamma _{\\rm ISF}$ conditions, can act somewhat similarly as multi-polar Frank-Read-type sources at high strains [42].", "The $a/6\\langle 011\\rangle $ Lomer-Cottrell (LC) stair-rod junctions that make up the SFT edges are sessile, but under sufficient stress they may unzip and emit $a/6\\langle 112\\rangle $ Shockley partials that are pinned to neighboring vertices of the tetrahedron.", "The emitted segments bow out and can eventually form new, freely growing loops (see Fig.", "REF h).", "The observed critical strain for unzipping and activation of the leading Shockley partial bow-out ($\\sim \\mu /25$ ) was found to decrease as either $\\gamma _{\\rm ISF}$ or the SFT size is increased." ], [ "Grain boundary sources", "Dislocations can also be emitted from or absorbed into grain boundaries during plastic deformation.", "The initiation of this type of nucleation process is more structurally complex than those already discussed due to the disordered nature of high angle grain boundaries, but the basic elements of the Frank-Read mechanism are still present.", "If stress builds up near or within a grain boundary, it may be relieved by grain boundary sliding or migration, for example, but in many cases strain is most readily relieved by spontaneously nucleating new dislocation lines which are then translated into the grain interior.", "These lines may begin as small half-loops pinned to the boundary at either end, which then bow out much like a Frank-Read source.", "Rather than sweeping around the pinning points and pinching-off a complete loop (the grain boundary generally prevents this) the pinning points are more likely to migrate along the grain boundary as the half-loop grows.", "Examples of such grain boundary nucleation processes in FCC nanopolycrystals are shown in Fig.", "REF a-d (see Supplemental Material [42] for the associated animation).", "This system has the same model parameters as those of Figs.", "REF and REF , and the grain structures were formed using Voronoi tessellation.", "Tensile strain $\\epsilon _{\\rm zz}$ was then applied at a constant rate under constant volume conditions, such that $\\epsilon _{\\rm xx}=\\epsilon _{\\rm yy}=(\\epsilon _{\\rm zz}+1)^{-1/2}-1$ .", "Shear strain $\\epsilon _{\\rm zy}$ was found to produce similar results.", "As a point of reference, the simulation shown, which contains $\\sim 1.15 \\times 10^6$ atoms or density peaks, required 57 hours of wall-clock time to execute $2.4 \\times 10^4$ time steps (or $\\sim 3 ms$ estimated duration for Cu at $1063^{\\circ }$ C) using 48 CPU cores.", "A direct comparison with LAMMPS benchmark data for Cu with an EAM potential [53] indicates that the same computation of $\\sim 1.15 \\times 10^6$ atoms for $3 ms$ on 48 CPU cores would require $\\sim 1.25\\times 10^8$ hours of wall-clock time, more than six orders of magnitude greater than the PFC time.", "Figure: (Color online)Emissionof dissociated a/2〈110〉a/2\\langle 110\\rangle dislocationsand a/6〈121〉a/6\\langle 121\\rangle partials from grain boundaries.", "(a)-(d) Nanopolycrystalline FCC sample with average grain sized ¯≃46a\\bar{d}\\simeq 46a under tensile strainϵ zz =0.082\\epsilon _{\\rm zz}=0.082, 0.0860.086, 0.0920.092, and 0.10.1, respectively.Both full dislocations (leading and trailing partials with stacking faults,two of which are tracked by green (light gray) arrows) and leading partialswith stacking faults are emitted from / absorbed into grain boundaries,beginning near the yield point.Density peaks with HCP (stacking faults), irregular (dislocation cores),and FCC coordinationare shown in red (dark gray), gray, and green (light gray), respectively.", "(e) Stress-strain curves for FCC polycrystals withvarious average grain sizes d ¯\\bar{d} and applied strain rates(constant strain rate tensile deformation).Model parameters are the same as those of Figs.", "and.", "(f)As (e), but for BCC polycrystalswith model parameters identicalto those reported in Section , exceptα 1 =2\\alpha _1=2, and Eq.", "was employed with β=0.01\\beta =0.01,α=1\\alpha =1.See Supplemental Material for the associated animations.Examples of the spontaneous nucleation of complete dissociated $a/2\\langle 110\\rangle $ dislocations are highlighted with green arrows in Fig.", "REF .", "These half-loops traverse the grain and are eventually absorbed into the opposite grain boundary as no fixed obstacles are present.", "Numerous examples of leading partial nucleation and heavy faulting are also apparent.", "Each grain has four available $\\lbrace 111\\rbrace $ planes in which the partials may glide, resulting in faulting in some or all of these planes within a given grain.", "The complex interactions of the various partials and stacking faults lead to varying intra-grain textures and structures with results very similar to those produced by MD simulations and consistent with experimental observations [54], [55].", "The stress-strain curves shown in Fig.", "REF e also confirm this qualitative agreement.", "Stresses $\\sigma _{\\rm ij}$ were periodically quantified by measuring the rate of change in average free energy of an instantaneous $n(\\vec{r})$ configuration as the appropriate deformation is statically applied.", "For example, $\\sigma _{\\rm zz}$ was measured by varying the grid spacing in the $z$ -direction and quantifying $\\sigma _{\\rm zz} = \\delta \\bar{F}/\\delta \\epsilon _{\\rm zz}$ in the small strain limit every ten time steps.", "Upper and lower yield points are observed for FCC systems with both yield stresses decreasing with decreasing rate, as observed in MD studies [54].", "The upper yield point is associated with initiation of plastic flow / dislocation nucleation within the initial clean grains, while the lower yield point is associated with steady-state dislocation and grain boundary driven plasticity within the resulting dislocated systems.", "Larger grain sizes produce higher yield points, indicating that these systems are in the reverse Hall-Petch regime [56].", "This dominance of dislocation nucleation, glide, and annihilation, and a lack of visible pile-ups suggests that this reverse Hall-Petch behavior is associated with an absence of pile-up-induced hardening at these small grain sizes.", "Other mechanisms such as grain boundary sliding and dislocation source starvation may contribute to this behavior as well, but we have not quantified the contributions of such effects.", "We only note that they are visually less evident than the described dislocation activity.", "Analogous simulations of BCC polycrystals produced the results shown in Fig.", "REF f (see Supplemental Material [42] for the associated animation).", "A simpler yielding behavior is observed in this case, as the BCC nanopolycrystals within this parameter range deform plastically via grain boundary migration.", "Very little dislocation nucleation occurs, leaving essentially a network of sliding and creeping grain boundaries.", "Grain boundary mechanisms therefore appear to be entirely responsible for the inverse Hall-Petch behavior in these systems.", "Such behavior is consistent with intermediate/high temperature experiments on nanopolycrystalline BCC metals in which grain boundary mechanisms are found to dominate plasticity [55].", "The greater dependence of BCC stress-strain behavior on strain rate also indicates a larger diffusive creep component than in FCC.", "This is in agreement with general observations of higher creep rates in BCC materials, an effect ascribed to the higher self diffusivity of non-close-packed structures [57].", "These simulations of FCC and BCC polycrystals demonstrate that the qualitative features of conservative dislocation emission from grain boundaries and of overall stress-strain response in elastic and plastic regimes can be well-captured by PFC models." ], [ "Bardeen-Herring-Type Climb Sources", "Climb-mediated or Bardeen-Herring-type sources can become active at high temperatures or following rapid quenches when the excess vacancy concentration is large.", "The basic principles are in many ways analogous to those of Frank-Read sources, except that the dislocation motion is mediated by vacancy diffusion rather than by glide.", "Nonetheless such sources are not as well understood as Frank-Read sources, at least partly because they are not easily modeled at the atomistic level using conventional approaches, and because mesoscale models must account for the often complex nature of vacancy diffusion around dislocation cores and among other heterogeneous strain fields.", "We show here that PFC simulations permit the study of such sources with atomistic resolution, and reveal a range of complex nucleation behaviors caused by non-trivial interactions between interface structure, strain orientation, and dislocation energetics." ], [ "Critical strain for loop nucleation", "The specific phenomenon considered in this section is nucleation of dislocation loops from spherical objects such as precipitates, inclusions, or voids in BCC crystals under uniaxial tension or compression $\\epsilon _{ii}$ .", "The case of loop nucleation and coherency loss at precipitates has been studied in early theoretical and experimental work [58], [59], [60], [61], as precipitate coherency can have a significant impact on the mechanical properties of metal alloys.", "Concentric dislocation loops centered on precipitates or impurities have also been observed in various metals [62], [63], [64], [65], [66].", "These may be formed by Bardeen-Herring-type mechanisms similar to those described here.", "To our knowledge this problem has not been examined via numerical simulations nor at the atomistic level due to the long, diffusive time scales involved.", "As strain is applied to a system containing a spherical precipitate, the free energy eventually becomes higher than that of the same system with a dislocation loop that is able to grow and relieve strain energy.", "An energy barrier for the nucleation of such a loop will generally exist such that its appearance in a dynamic simulation may be delayed to higher strains.", "Nonetheless, a loop eventually appears at the sphere-matrix interface, and when the applied strain is purely uniaxial, the nucleation and growth of the loop is largely mediated by climb.", "Other strain types can lead to different, relatively well-characterized conservative loop formation processes such as prismatic punching [67], [68], [69].", "In a linear elastic isotropic continuum, the critical strain $\\epsilon ^_{ii}$ at which loop nucleation becomes favorable is approximately $\\epsilon ^_{ii} =\\frac{b}{8\\pi (1-\\nu )(\\frac{B_S}{B_M}-1)R_0}\\left[ \\ln {\\frac{8R_0}{b}}+\\frac{2\\nu -1}{4(1-\\nu )} \\right]$ where $b$ is the magnitude of the dislocation Burgers vector, $\\nu $ is the Poisson's ratio of the matrix, $B_S$ and $B_M$ are the bulk moduli of the sphere and matrix phases, respectively, and $R_0$ is the sphere radius [58], [61].", "We note that anisotropy may play a role in the BCC system studied here.", "Homogeneous spherical inclusions were introduced into the present simulations by adding a uniform penalty function over some predefined spherical volume in the center of a simulation cell with initially perfect BCC crystal structure (see Fig.", "REF ).", "The resulting spherical body has a larger elastic modulus than that of the bulk crystal, but since $n(\\vec{r})$ within the sphere is uniform rather than periodic, the issue of coherent vs. incoherent interface structure is not relevant.", "After the system was equilibrated, uniaxial strain $\\epsilon _{ii}$ was applied at a constant rate by uniformly increasing or decreasing the numerical grid spacing along one axis of the periodic simulation cell by a small amount at every time step.", "Equations (REF ), (REF ), and (REF ), were employed in all simulations discussed in this section.", "Parameter values used were $w=1.4$ , $u=1$ , $n_0=0$ , $\\alpha _1=1$ , $\\sigma =0.1$ , $\\rho _1=1$ , and $\\beta _1=8$ .", "Other simulation details are given here [70].", "Following the procedure used for FCC simulations, if we match the numerically measured BCC vacancy diffusion constant $D_v \\simeq 1.5 a^2/t$ to that of vanadium at $1842^{\\circ }$ C ($D_v \\simeq 1.36\\times 10^{-13}m^2/s, a \\simeq 0.302nm$ ), then the time unit $t$ is found to correspond to $\\sim 1\\mu s$ .", "The range of shear rates used in BCC simulations then converts to $\\sim 20/s-2500/s$ , values again roughly four to six orders of magnitude lower than those of typical and comparable MD simulations.", "Compiled results for the critical nucleation strain $\\epsilon ^_{ii}$ at all sphere sizes and strain orientations are shown in Fig.", "REF .", "The agreement between the low strain rate results and the static energy minimization results indicates that any rate dependence is minimal at the slower rate considered.", "The general trend is a decrease in $\\epsilon ^_{ii}$ as $R_0$ increases.", "The form of the decrease is well described by Eq.", "(REF ) after an additional constant strain $\\epsilon ^{\\rm min}_{ii}$ is added to the right hand side.", "This constant is discussed further in the following paragraph.", "The adjustable parameters in the fits shown are therefore $\\epsilon ^{\\rm min}_{ii}$ and $B_S$ , as there are some ambiguities in the effective value of the bulk modulus of the sphere as modeled.", "Nonetheless, the fits are quite good for $B_S/B_M \\simeq 4$ , which seems to be a reasonable estimate of the ratio produced by our simulations.", "Figure: (Color online)Critical strain ϵ ii * \\epsilon ^_{ii} for Bardeen-Herringclimb source activation vs sphere radius R 0 R_0.The points represent simulation data at different effective strain rates,and the lines are predictions of the theory of Brown et al.", ", without and with a finite minimum critical strain.The fits employ fixed parametersb=1/2b = 1/\\sqrt{2} and ν=1/3\\nu =1/3,and adjustable parametersϵ ii min \\epsilon ^{\\rm min}_{ii} and B S /B M B_S/B_M,which are in all cases close to 0.0160.016 and 4, respectively.It is not conclusive whether $\\epsilon ^_{ii}$ will continue to slowly decrease as $R_0$ becomes very large or whether it levels off to some minimum value.", "Our expectation is that $\\epsilon ^_{ii}$ will plateau in the PFC simulations due to finite size effects as well as the existence of a threshold Eckhaus-like strain for activation of the wavelength selection or climb `instability' [71].", "Both of these effects are driven by lattice periodicity.", "An integer number of unit cells must fit into the simulation box, such that the energy of a perfect crystal will not be reduced by the addition or removal of a plane of atoms or density peaks until $\\epsilon _{ii} > 1/(2N_i)$ , where $N_i$ is the number of unit cells in the $i$ direction.", "Furthermore, there will be an energy barrier for this removal process associated with an Eckhaus-like instability strain, which quantifies the strain at which this barrier goes to zero [71].", "Thus, without thermal fluctuations we expect to observe $\\epsilon ^_{ii} > 1/(2N_i) \\simeq 0.009$ for the system size used in this study.", "As the elastic moduli become large, $\\epsilon ^*_{ii} \\rightarrow 1/(2N_i)$ since the perfect crystal Eckhaus strain is roughly proportional to $\\alpha _i$ .", "The single data point at $\\alpha _1=1/5$ , $R_0=15.8b$ is consistent with this expectation." ], [ "Loop geometry and evolution", "The critical strain for nucleation is therefore in general agreement with continuum elastic predictions, but it is worthwhile to examine the nucleation and growth process in greater detail.", "A typical result from the dynamic simulations is shown in Fig.", "REF (see Supplemental Material [42] for the associated animation).", "The sphere radius in this case is $R_0=11.3a$ , and the strain is compressive along the $x$ axis, $\\epsilon _{\\rm xx}$ .", "A loop first begins to form at the sphere-matrix interface with a slightly serpentine shape due to the variations in local line energy around the surface of the sphere.", "Essentially, the nature in which the spherical surface intersects the various crystallographic planes of the matrix creates a quasi-2D energy landscape on the spherical surface which the dislocation loop must navigate to minimize its total energy with the constraint of fixed total Burgers vector.", "Certain planes and line directions will be preferred over others.", "This effect involves not only crystallographically-dependent dislocation energies, but also atomic-level core structure effects as well as elastic anisotropy, which together are beyond the scope of continuum elastic theories.", "The impact on initial loop shape tends to be small for small $R_0$ but increases considerably for larger $R_0$ values, as will be shown.", "Figure: (Color online)Operationof a Bardeen-Herring-type spherical climb source with R 0 =11.3aR_0=11.3ain a BCC crystal under uniaxial compression ϵ xx \\epsilon _{\\rm xx}.Time evolution shown at (a) t=110t=110, (b) t=126t=126, (c) t=135t=135,(d) t=145t=145, (e) t=160t=160, and (f) t=180t=180.Only density peaks with irregular coordination (interface and dislocation coresites)are displayed.", "Those at the sphere-matrix interface are shown in gray (dark),those inside the dislocation core are shown in blue-green (light) depending onposition along the yy-axis.See Supplemental Material for the associated animation.The initial loop nucleation and its subsequent growth both require vacancy diffusion to or away from the surface of the sphere.", "In the PFC approach, this process is mediated by local diffusive modes of the density wave amplitudes.", "After the loop detaches from the sphere, its shape continues to evolve as the effective local energy landscape changes with loop radius.", "Non-planar, non-circular shapes are common as the competition between minimum static energy and lowest energy pathway to continued growth can be delicate and non-trivial.", "This is somewhat analogous to the cross-slip of dissociated screw dislocations, during which local segments must constrict at large energy cost to permit cross-slip into the next local energy minimum.", "Any instantaneous configuration may not be the lowest energy static configuration for the given loop radius, but it should facilitate evolution toward an even lower energy state with larger radius.", "A more complex loop nucleation process is shown in Fig.", "REF (see Supplemental Material [42] for the associated animation).", "The only difference from Fig.", "REF is the increased sphere size of $R_0=28.3a$ .", "The result is a much more pronounced serpentine shape due to the larger areas on the surface of the sphere that nearly coincide with low energy lattice planes of the matrix.", "Within the $(110)$ plane for example, the preferred line direction for the nucleated $a/2[1\\bar{1}1]$ dislocation is along the nearest $\\langle 001\\rangle $ vector rather than the $[\\bar{1}12]$ vector normal to the strain axis.", "The dislocation therefore forms with mixed edge-screw character along the $[001]$ direction within the uppermost $(110)$ plane for example, taking the loop locally out of alignment with the strain-normal $(1\\bar{1}1)$ plane.", "The line must then wind back toward the strain-normal plane in the areas where it curves out of the top and bottom $(110)$ planes.", "Figure: (Color online)Operationof a Bardeen-Herring-type spherical climb source with R 0 =28.3aR_0=28.3ain a BCC crystal under uniaxial compression ϵ xx \\epsilon _{\\rm xx}.Time evolution shown at (a) t=110t=110, (b) t=138t=138, (c) t=150t=150, and (d) t=162t=162.Only density peaks with irregular coordination (interface and dislocation coresites)are displayed.", "Those at the sphere-matrix interface are shown in gray (dark),those inside the dislocation core are shown in blue-green (light) depending onposition along the yy-axis.See Supplemental Material for the associated animation.The same arguments hold as the line crosses through the other intersecting $\\lbrace 110\\rbrace $ planes.", "The $a/2[1\\bar{1}1](011)$ segments, for example, prefer alignment with the $[\\bar{1}00]$ direction.", "The $a/2[1\\bar{1}1](\\bar{1}12)$ segments on the other hand prefer alignment with the $[110]$ direction (pure edge character).", "Thus a serpentine winding pattern is produced as a result of the dynamic competition between loop energy minimization, which tends to promote a certain degree of winding, and strain relief maximization, which tends to suppress winding in favor of maximizing the outward growth (climb) velocity.", "Detachment and growth away from the sphere occurs first in regions that do not align with any low energy dislocation slip planes.", "Segments in low energy $\\lbrace 110\\rbrace $ planes are observed to detach last, as these have the lowest local energy and the highest barrier for out of plane motion.", "Terrace sites at the edges of faceted low energy planes in particular appear to have the maximum detachment barrier.", "The extra half loops protruding from the sphere in Fig.", "REF are a dynamic effect that disappears at low applied strain rates.", "Other variations of climb-mediated loop nucleation processes are shown in Fig.", "REF (see Supplemental Material [42] for the associated animations).", "Uniaxial strain along a $\\langle 110\\rangle $ direction produces two disjointed $a/2\\langle 111\\rangle $ half-loops that grow symmetrically at a $12.5^{\\circ }$ angle to the strain-normal $(110)$ plane (Fig.", "REF a).", "An $a\\langle 100\\rangle $ edge line segment appears at the intersection of these half-loops, and can link to a second, concentric inner pair of half loops that allows complete detachment from the sphere.", "Uniaxial strain along a $\\langle 112\\rangle $ direction produces two separate $a/2\\langle 111\\rangle $ edge dislocation half-loops, as displayed in Fig.", "REF b.", "These half-loops climb until their terminal ends meet and merge into a single, nearly circular loop.", "Dual loop nucleation as shown in Fig.", "REF c is also possible for certain $R_0$ , elastic moduli, and strain rates.", "Finally, arrays of spheres simultaneously nucleate complex networks of dislocation lines via mixed climb-glide processes.", "Effects from dislocation-dislocation, dislocation-grain boundary, and dislocation-inclusion interactions, for example, are naturally incorporated into the evolution of such networks in PFC simulations.", "Figure: (Color online)Examples of various other Bardeen-Herring-type climb sourcesin BCC crystals.", "(a) Linked, concentric, biplanar loop pairs under ϵ zz \\epsilon _{\\rm zz}with R 0 =11.3aR_0=11.3a.", "(b) Offset nonconcentric loop pairs under ϵ yy \\epsilon _{\\rm yy}with R 0 =28.3aR_0=28.3a.", "(c) Offset concentric loop pairs under ϵ xx \\epsilon _{\\rm xx}with R 0 =9aR_0=9a and α=2\\alpha =2.", "(d) Dislocation network / tangle formation from an array of 10 sourceswith R 0 =9aR_0=9a in a bicrystal under constant volume tension ϵ zz \\epsilon _{\\rm zz}.Only density peaks with irregular coordination (interface and dislocation coresites)are shown.", "Those at the sphere-matrix interface are shown in gray (dark),those inside the dislocation core are shown in blue-green (light) depending onposition along the out of plane axis.See Supplemental Material for the associated animations." ], [ "Dislocation-SFT Interactions", "Impediments to dislocation motion, including other dislocations, planar faults, and 3D obstacles, play a central role in the mechanical response and work-hardening properties of metals.", "The dominant irradiation-induced defect in FCC materials is the SFT, and dislocation-SFT interactions are therefore believed to largely control the mechanical response of FCC materials in nuclear applications [72].", "Such interactions have been widely studied in MD (and DDD) simulations [49], [72], [73] providing for our purposes a potentially useful body of benchmark results with which PFC simulations can be compared.", "A few selected results are presented in this section.", "SFTs were formed in the present simulations via the Silcox-Hirsch mechanism [74].", "A triangular Frank loop is initiated on a $\\lbrace 111\\rbrace $ plane, after which it spontaneously relaxes into the local energy minimum corresponding to a perfect SFT with base prescribed by the initial Frank loop (see Supplemental Material [42] for the associated animation).", "Dissociated $a/2\\langle 110\\rangle $ edge or screw dislocations were equilibrated some lateral distance away from the SFT and some vertical distance relative to the SFT base.", "Shear strain $\\epsilon _{zx}$ was then applied at a constant rate to cause the dislocation to glide toward and through the SFT (see Fig.", "REF ).", "Only two cases will be reported here, given in the notation of Ref.", "[49] as ($ED/Down, 4/13, 0.0001/t$ ) and ($SD/Edge, 4/13, 0.0001/t$ ).", "In the first case, this notation indicates that an edge dislocation ($ED$ ) intersects a SFT with apex oriented in the $[\\bar{1}1\\bar{1}]$ direction ($Down$ ), at the fourth $\\lbrace 111\\rbrace $ plane from the base of the SFT which is 13 $\\lbrace 111\\rbrace $ planes tall ($4/13$ ), and with shear rate $0.0001/t$ .", "The second case is the same except that it considers a screw dislocation ($SD$ ) intersecting a SFT with one edge oriented along the $SD$ line direction ($Edge$ ).", "Additional simulations details are given here [75].", "Results from the ($ED/Down, 4/13, 0.0001/t$ ) simulation are shown in Fig.", "REF .", "The accompanying animation, along with that of the SFT-screw dislocation interaction, can be found in the Supplemental Material [42].", "Figure: (Color online)Interaction between a SFT and a gliding dissociated edge dislocation in aFCC crystal,(ED/Down,4/13,0.0001/tED/Down, 4/13, 0.0001/t).Perspective views of the SFT and dislocation are shown in (a).The upper image shows both at t=10100t=10100,while the lower left and right imagesshow the SFT at t=15800t=15800 and t=20100t=20100, respectively.Only the leading partial has passed the SFT at t=15800t=15800, both partials havepassed by t=20100t=20100.xyxy-plane views are shown in (b), (c), (d), (e), and (f) att=12100t=12100, 15100, 16600, 18600 and 19600, respectively.", "(g) The damaged SFT following the (SD/Edge,4/13,0.0001/tSD/Edge, 4/13, 0.0001/t) interaction.Density peaks with HCP coordination (stacking faults)are shown in blue (dark gray), those with irregularcoordination (dislocation cores) are shown in gray (light gray).See Supplemental Material for the associated animations.The general sequence of events includes dislocation pinning at SFT Lomer-Cottrell stair rod junctions, bow out of the gliding dislocation line between the image SFTs, Orowan looping of the SFT by the leading partial, and damage of the SFT after the trailing partial has passed through.", "The pinning and bow-out effects are of course expected.", "The Orowan loop created by the leading partial is also consistent with MD simulations of individual Shockley partial-SFT interactions [73].", "In the present simulations, the trailing partial eventually shears the SFT as well, clearing the Orowan loop and leaving either one or two ledges on the SFT faces.", "The ledge structures appear to be consistent with those observed in MD [49].", "The height of the SFT apex above is also reduced by one $\\lbrace 111\\rbrace $ interplanar distance in the $ED$ case.", "We have not yet observed other possible outcomes reported in MD simulations, such as partial SFT absorption and jog formation, but we have considered only a very small subset of the conditions examined via MD.", "We therefore argue that these results provide partial but strong qualitative evidence that PFC simulations can correctly reproduce complex defect phenomena of this type." ], [ "Conclusions", "Basic dislocation properties in FCC and BCC crystals have been examined in the context of phase field crystal models, and extended into simulations of conservative and nonconservative dislocation creation mechanisms and obstacle flow processes.", "Core structures of dissociated $a/2\\langle 110\\rangle $ FCC dislocations and $a/2\\langle 111\\rangle $ BCC dislocations have now been reproduced in PFC with sufficient accuracy to capture many aspects of plastic flow that derive from such structures.", "These include the known anisotropy in BCC screw-edge glide mobility as well as the effect of FCC dissociation width on cross-slip and climb barriers.", "Classical Frank-Read-type sources have been simulated for the first time with such models, and a new mechanism by which dislocation lines or superjogs under strain can segment onto multiple glide planes, converting local monopole or dipole sources into multipole sources, has been identified.", "Stacking fault tetrahedra under high strain have also been shown to reconstruct and emit dislocations via a Frank-Read-type mechanism.", "Basic features of 3D polycrystal plasticity and dislocation emission from grain boundaries have also been examined and shown to be consistent with MD simulation results.", "Nonconservative dislocation creation mechanisms associated with spherical precipitates, inclusions, or voids have been studied for the first time using atomistic simulations.", "Results for the critical strain to nucleate a loop from a spherical body are in agreement with predictions of continuum elastic theory after accounting for finite size effects and moduli-dependent climb barriers present in our simulations.", "A range of complex nucleation behaviors caused by non-trivial interactions between interface structure, strain orientation, and dislocation energetics have been revealed.", "Observed loop geometries have been rationalized for a few select cases, but the results in general highlight the sometimes unexpected complexity that can emerge when atomistic effects associated with crystal structure, dislocation cores, and climb dynamics are simultaneously considered.", "The Silcox-Hirsch SFT formation mechanism has also been reproduced, as well as qualitative features of SFT-dislocation interactions observed in MD and DDD simulations.", "Such processes will require further study to gain a fuller understanding of the similarities and differences between PFC and other atomistic simulation methods.", "But there appears to be promise in the possibility of simulating features of obstacle flow involving, for example, climb bypass mechanisms that cannot be accessed with other conventional methods.", "In a wider sense, it is hoped that these results convey the potential of the PFC approach as applied to solid-state materials phenomena in three dimensions.", "This type of description unifies conservative and nonconservative plastic flow mechanisms with atomistic resolution, enabling the study of complex high temperature diffusive evolution processes in the nanoscale size regime.", "Many such processes are inaccessible to conventional atomistic approaches.", "Applications to pure or multicomponent systems and phenomena such as creep, recovery, recrystallization, grain growth, structural phase transformations, and strain-hardening have already been reported or are currently underway.", "The additional, coupled effect of solute diffusion in alloy materials is naturally incorporated into PFC-type descriptions.", "Issues that we believe require further development or should be kept in mind include choice of ensemble, control of stress-strain-volume relations, quantification of vacancy concentration, and its connection to climb rates.", "This work has been supported by the Natural Science and Engineering Research Council of Canada (NSERC), and supercomputing resources have been provided by CLUMEQ/Compute Canada.", "The atomic visualization and analysis packages Ovito [76] and the Dislocation Extraction Algorithm (DXA) [77] were used in this work." ] ]	1403.0601
[ [ "Moment Determinacy of Powers and Products of Nonnegative Random\n Variables" ], [ "Abstract We find conditions which guarantee moment (in)determinacy of powers and products of nonnegative random variables.", "We establish new and general results which are based either on the rate of growth of the moments of a random variable or on conditions about the distribution itself.", "For the class of generalized gamma random variables we show that the power and the product of such variables share the same moment determinacy property.", "A similar statement holds for half-logistic random variables.", "Besides answering new questions in this area, we either extend some previously known results or provide new and transparent proofs of existing results." ], [ "J Theor Probab (2015) 28:1337-1353 Moment Determinacy of Powers and Products of Nonnegative Random Variables Gwo Dong Lin$^{1}$ $\\bullet $ Jordan Stoyanov$^{2}$ $^{1}$ Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan (ROC) e-mail: [email protected] $^{2}$ School of Mathematics $\\&$ Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK e-mail: [email protected] Abstract: We find conditions which guarantee moment (in)determinacy of powers and products of nonnegative random variables.", "We establish new and general results which are based either on the rate of growth of the moments of a random variable or on conditions about the distribution itself.", "For the class of generalized gamma random variables we show that the power and the product of such variables share the same moment determinacy property.", "A similar statement holds for half-logistic random variables.", "Besides answering new questions in this area, we either extend some previously known results or provide new and transparent proofs of existing results.", "Mathematics Subject Classification (2010) 60E05, 44A60 Keywords Stieltjes moment problem, powers, products, Carleman's condition, Cramér's condition, Hardy's condition, Krein's condition, generalized gamma distribution, half-logistic distribution 1.", "Introduction Throughout the paper we assume that $\\xi $ is a nonnegative random variable defined on a given probability space $(\\Omega , {\\cal F}, {\\bf P})$ with finite moments ${\\bf E}[\\xi ^k], \\ k=1,2,\\ldots .$ Let further $\\xi _1,\\xi _2, \\ldots ,\\xi _n$ be independent copies of $\\xi .$ Here $n\\ge 1$ is a fixed integer number.", "Of interest to us are the following two random variables, the power and the product: $X_n=\\xi ^n \\ \\mbox{ and } \\ Y_n=\\xi _1\\xi _2 \\cdots \\xi _n.$ Each of the variables $X_n$ and $Y_n$ also has all moments finite.", "Thus the natural problem arising here is to study, characterize and compare the moment (in)determinacy of these two variables.", "Since $X_n$ and $Y_n$ take values in ${\\mathbb {R}}^+=[0,\\infty )$ , this means that we deal with the Stieltjes moment problem.", "We find conditions on $\\xi $ and $n$ guaranteeing that $X_n$ and $Y_n$ are M-determinate (uniquely determined by the moments), and other conditions when they are M-indeterminate (nonunique in terms of the moments).", "In these two cases we use the abbreviations M-det and M-indet for both random variables and their distributions.", "In our reasonings we use classical or new conditions such as Cramér's condition, Carleman's condition, Hardy's condition and Krein's condition.", "The reader may find it useful to consult available sources, among them are [10], [12], [17], [19], [22], [23] and [24].", "For reader's convenience, we have included these conditions when formulating our results.", "To study powers, products, etc., or other nonlinear transformations of random data (called sometimes Box-Cox transformations), is a challenging probabilistic problem which is of independent interest.", "Note however that products and powers of random variables considered in this paper and the results established are definitely related to contemporary stochastic models of real and complex phenomena; see, e.g., [3], [5], [7] and [20].", "In this paper we deal with new problems and present new results with their proofs.", "We establish new and general criteria which are then applied to describe the moment (in)determinacy of the above transformations.", "We also provide new proofs of some known results with reference to the original papers.", "Our results complement previous studies or represent different aspects of existing studies on this topic; see, e.g., [2], [4], [11], [16], [20] and [22].", "The approach and the results in this paper can be further extended to distributions on the whole real line (Hamburger moment problem, see [25]).", "Also, they can be used to characterize the moment determinacy properties of nonlinear transformations of some important sub-classes of distributions such as, e.g., the subexponential distributions; see [6].", "The material is divided into relatively short sections each dealing with a specific question related to a general or specific distribution.", "General results are included in Sections 2, 4, 6, 7 and 9.", "Sections 3, 5, 8 and 10 deal with powers and products based on the generalized gamma distribution while Section 11 is based on half-logistic distribution.", "All statements are followed by detailed proofs.", "2.", "Comparing the moment determinacy of powers and products The power $X_n=\\xi ^n$ and the product $Y_n=\\xi _1\\xi _2 \\cdots \\xi _n$ have some `similarity'.", "They both are defined in terms of $n$ and $\\xi $ or of $n$ independent copies of $\\xi $ , and both have all moments finite.", "Thus we arrive naturally to the question: Is it true that the random variables $X_n$ and $Y_n$ share the same moment determinacy property?", "If the generic random variable $\\xi $ has a bounded support, then so does each of $X_n$ and $Y_n$ , and hence both $X_n$ and $Y_n$ have all moments finite and both are M-det.", "This simple observation shows that interesting is to study powers and products based on a random variable $\\xi $ with unbounded support contained in ${\\mathbb {R}}^+$ and such that $\\xi $ has all moments finite.", "Let us mention first a special case.", "Suppose $\\xi \\sim Exp(1),$ the standard exponential distribution.", "Then the power $X_n=\\xi ^n$ is M-det iff the product $Y_n=\\xi _1\\xi _2\\cdots \\xi _n$ is M-det and this is true iff $n\\le 2$ (see, e.g., [2] and [16]).", "This means that for any $n=1,2,\\ldots ,$ the power $X_n$ and the product $Y_n$ share the same moment determinacy property.", "Since Weibull random variable is just a power of the exponential one, it follows that if $\\xi $ obeys a Weibull distribution, then for any $n=1,2,\\ldots ,$ the power $X_n$ and the product $Y_n$ also have the same moment determinacy property.", "Therefore, the answer to the above question is positive for at least some special distributions including Weibull distributions.", "In this paper we will explore more distributions (see Theorem 6 and Section 11 below).", "Note that in general, we have, by Lyapunov's inequality, ${\\bf E}[X_n^s]={\\bf E}[\\xi ^{ns}] \\ge ({\\bf E}[\\xi ^s])^n={\\bf E}[Y_n^s]~~\\hbox{for all real}~ s>0.$ We use this moment inequality to establish a result which involves three of the most famous conditions for moment determinacy (Carleman's, Cramér's and Hardy's).", "For more details about Hardy's condition, see [24].", "Proposition 1 (i) If $X_n$ satisfies Carleman's condition (and hence is M-det), i.e., $\\sum _{k=1}^{\\infty }({\\bf E}[X_n^k])^{-1/(2k)}=\\infty ,$ then so does $Y_n$ .", "(ii) If $X_n$ satisfies Cramér's condition (and hence is M-det), i.e., ${\\bf E}[\\exp ({cX_n})]<\\infty $ for some constant $c>0,$ then so does $Y_n$ .", "(iii) If $X_n$ satisfies Hardy's condition (and hence is M-det), i.e., ${\\bf E}[\\exp ({c\\sqrt{X_n}})]<\\infty $ for some constant $c>0,$ then so does $Y_n$ .", "Proof Part (i) follows immediately from (1).", "Parts (ii) and (iii) follow from the fact that for each real $s>0$ , ${\\bf E}[\\exp (cX_n^s)]=\\sum _{k=0}^{\\infty }\\frac{c^k}{k!", "}{\\bf E}[(X_n^s)^{k}]\\ge \\sum _{k=0}^{\\infty } \\frac{c^k}{k!", "}{\\bf E}[(Y_n^s)^{k}]={\\bf E}[\\exp (cY_n^s)].$ $\\Box $ Corollary 1 If $\\xi $ satisfies Cramér's condition and if $n=2$ , then both $X_2=\\xi ^2$ and $Y_2=\\xi _1\\xi _2$ are M-det, and hence $X_2$ and $Y_2$ share the same moment determinacy property.", "Proof Note that $\\xi $ satisfies Cramér's condition iff $X_2$ satisfies Hardy's condition.", "Then by Proposition 1(iii), both $X_2$ and $Y_2$ are M-det as claimed above.", "$\\Box $ 3.", "Generalized gamma distributions.", "Part (a) Some of our results can be well illustrated if we assume that the generic random variable $\\xi $ has a generalized gamma distribution.", "We write $\\xi \\sim GG(\\alpha , \\beta , \\gamma )$ if $\\xi $ has the following density function $f$ : $f(x)=cx^{\\gamma -1}e^{-\\alpha x^{\\beta }},~x\\ge 0,$ where $\\alpha , \\beta , \\gamma >0$ , $f(0)=0$ if $\\gamma \\ne 1$ , and $c=\\beta \\alpha ^{\\gamma /\\beta }/\\Gamma (\\gamma /\\beta )$ is the norming constant.", "Note that $GG(\\alpha , \\beta , \\gamma )$ is a rich class containing several commonly used distributions such as exponential, Weibull, half-normal and chi-square.", "It is known that the power $X_n=\\xi ^n$ is M-det iff $n\\le 2\\beta $ (see, e.g., [18] and [23]).", "We claim now that for $n\\le 2\\beta $ , the product $Y_n=\\xi _1\\xi _2\\cdots \\xi _n$ is also M-det.", "To see this, we note first that the density function $h_n$ of the random variable $\\sqrt{X_n}$ is $h_n(z)=\\frac{2c}{n}z^{2\\gamma /n-1}e^{-\\alpha z^{2\\beta /n}},~z\\ge 0.$ This in turn implies that $X_n$ satisfies Hardy's condition if $2\\beta /n\\ge 1$ , hence so does $Y_n$ for $n\\le 2\\beta $ by Proposition 1(iii).", "To obtain further results, it is quite useful to write the explicit form of the density of the product $Y_2=\\xi _1\\xi _2$ when $\\xi $ has the generalized gamma distribution.", "This involves the function $K_0(x), \\ x >0$ , the modified Bessel function of the second kind.", "Its definition and approximation are given as follows: $K_0(x)&=& \\frac{1}{2}\\int _0^{\\infty }t^{-1}e^{-t-x^2/(4t)}dt,~~x>0, \\\\&=&\\left(\\frac{\\pi }{2x}\\right)^{1/2}e^{-x}\\left[1-\\frac{1}{8x}\\left(1-\\frac{9}{16x}\\left(1-\\frac{25}{24x}\\right)\\right)+o(x^{-3})\\right]~~\\hbox{as}~x\\rightarrow \\infty $ (see, e.g., [8] and [13], pp.", "37–38).", "Lemma 1 (See also [14]) Let $Y_2=\\xi _1\\xi _2$ , where $\\xi _1$ and $\\xi _2$ are independent random variables having the same distribution $GG(\\alpha , \\beta , \\gamma ).$ Then the density function $g_2$ of $Y_2$ is $g_2(x)&=&\\frac{2c^2}{\\beta }x^{\\gamma -1}K_0\\left(2\\alpha x^{\\beta /2}\\right),~~x>0,\\\\&\\approx & Cx^{\\gamma -\\beta /4-1}e^{-2\\alpha x^{\\beta /2}},~~\\hbox{as}~x\\rightarrow \\infty .$ Proof (Method I) Let $G_2$ be the distribution function of $Y_2$ .", "Then $\\overline{G}_2(x):=1-G_2(x)={\\bf P}[Y_2>x]=\\int _0^{\\infty }{\\bf P}[\\xi _1>x/y]cy^{\\gamma -1}e^{-\\alpha y^{\\beta }}dy,~~x>0,$ and hence the density of $Y_2$ is $g_2(x)&=&c^2x^{\\gamma -1}\\int _0^{\\infty }y^{-1}e^{-\\alpha x^{\\beta }/y^{\\beta } -\\alpha y^{\\beta }}dy=\\frac{c^2}{\\beta }x^{\\gamma -1}\\int _0^{\\infty }t^{-1}e^{-t-(\\alpha ^2x^{\\beta })/t}dt\\\\&=&\\frac{2c^2}{\\beta }x^{\\gamma -1}K_0\\left(2\\alpha x^{\\beta /2}\\right),~~x>0.$ (Method II) We can use the moment function (or Mellin transform) usually denoted by $\\cal M$ , because it uniquely determines the corresponding distribution.", "To do this, we note that ${\\cal M}(s) =: {\\bf E}[Y_2^s]=({\\bf E}[\\xi _1^s])^2, ~~{\\bf E}[\\xi _1^s]=c\\Gamma ((\\gamma +s)/\\beta )\\left(\\beta \\alpha ^{(\\gamma +s)/\\beta }\\right)^{-1},~~\\hbox{and}$ $\\int _0^{\\infty }x^sK_0(x)dx=2^{s-1}(\\Gamma ((s+1)/2))^2~~\\hbox{for all}~~s>0$ (see, e.g., [9], p. 676, Formula 6.561(16)).", "We omit the detailed calculation.", "$\\Box $ It may look surprising, but it is well-known, that several commonly used distributions are related to the Bessel function in such a natural way as in Lemma 1.", "For example, if $\\xi $ is a half-normal random variable, i.e., $\\xi \\sim GG(\\frac{1}{2}, 2, 1)$ with the density $f(x)=\\sqrt{2/\\pi }e^{-x^2/2},~x\\ge 0$ , then $Y_2$ has the density function $g_2(x)=(2/{\\pi })K_0(x)\\approx C_2x^{-1/2}e^{-x}~~\\hbox{as}~x\\rightarrow \\infty $ , with the moment function ${\\cal M}(s)={\\bf E}[Y_2^s]=(2^s/\\pi )\\Gamma ^2((s+1)/2),~s>-1$ .", "The distribution of $Y_2=\\xi _1\\xi _2$ may be called the half-Bessel distribution and its symmetric counterpart with density $h_2(x)=(1/{\\pi })K_0(x),~x\\in {\\mathbb {R}}= (-\\infty ,\\infty )$ , is called the standard Bessel distribution.", "Note that $K_0$ is an even function and $h_2$ happens to be the density of the product of two independent standard normal random variables; see also [4].", "It can be checked that for real $s>0$ we have $({\\bf E}[(Y_2^{s})^n])^{-1/(2n)}\\approx C_sn^{-s/2}$ as $n\\rightarrow \\infty $ , and hence $Y_2^s$ satisfies Carleman's condition iff $s\\le 2$ .", "Actually, it follows from the density $g_2$ and its asymptotic behavior that $Y_2$ satisfies Cramér's condition.", "Therefore, by Hardy's criterion, the square of $Y_2$ , i.e., $Y_2^2=\\xi _1^2\\xi _2^2$ , is M-det.", "Let us express the latter by words: The square of the product of two independent half-normal random variables is M-det.", "Since $\\xi ^2 = \\chi ^2_1$ , we conclude also that the product of two independent $\\chi ^2$ -distributed random variables is M-det.", "In addition, these properties can be compared with the known fact that the power 4 of a normal random variable is M-det (see, e.g, [1] or [22]).", "4.", "Slow growth rate of the moments implies moment determinacy It is known and well understood that the moment determinacy of a distribution depends on the rate of growth of the moments.", "Let us establish first results which are of a general and independent interest.", "Later we will apply them and make conclusions about powers and products of random variables.", "Suppose $X$ is a nonnegative random variable with finite moments $m_k={\\bf E}[X^k]$ , $k=1,2,\\ldots .$ To avoid trivial cases, we assume that $m_1>0$ , meaning that $X$ is not a degenerate random variable at 0.", "Lemma 2 For each $k\\ge 1$ , we have the following properties: (i) $m_k\\le m_{k+1}$ if $m_1\\ge 1$ , and (ii) $m_1m_k\\le m_{k+1}.$ Proof By Lyapunov's inequality, we have $(m_k)^{1/k}\\le (m_{k+1})^{1/(k+1)}.$ Therefore, $\\frac{1}{k}\\log m_k\\le \\frac{1}{k+1}\\log m_{k+1}\\le \\frac{1}{k}\\log m_{k+1}, ~\\hbox{if} ~m_1\\ge 1,$ and hence $m_k\\le m_{k+1}$ if $m_1\\ge 1$ .", "This proves claim (i).", "To prove claim (ii), we use the relations $m_1\\le (m_k)^{1/k}\\le (m_{k+1})^{1/(k+1)},$ implying that $m_1m_k\\le (m_k)^{1/k}m_k=m_k^{(k+1)/k}\\le m_{k+1}.$ $\\Box $ In Lemma 2, claim (i) tells us that the moment sequence $\\lbrace m_k, k=1,2,\\ldots \\rbrace $ is nondecreasing if $m_1\\ge 1$ , while claim (ii) shows that the ratio $m_{k+1}/m_k$ has a lower bound $m_1$ whatever the nonnegative random variable $X$ is.", "The next theorem provides the upper bound of the ratio $m_{k+1}/m_k$ , or, equivalently, of the growth rate of the moments $m_k$ for which $X$ is M-det.", "Theorem 1 Let $m_{k+1}/m_k={\\cal O}((k+1)^2)$ as $k\\rightarrow \\infty $ .", "Then $X$ satisfies Carleman's condition and is M-det.", "(We refer to the constant 2, the exponent in the term ${\\cal O}((k+1)^2)$ , as the rate of growth of the moments of $X$ .)", "Proof By the assumption, there exists a constant $C>0$ such that $m_k^{(k+1)/k}\\le m_{k+1}\\le C(k+1)^2m_k~~\\hbox{for alllarge}~k,$ which implies $m_k^{1/k}\\le C(k+1)^2~~\\hbox{for alllarge}~k,$ and hence $m_k^{-1/(2k)}\\ge C^{-1/2}(k+1)^{-1}~~\\hbox{forlarge}~k.$ Therefore, $X$ satisfies Carleman's condition $\\sum _{k=1}^{\\infty }m_k^{-1/(2k)}=\\infty $ , and is M-det.", "$\\Box $ We can slightly extend Theorem 1 as follows.", "For a real number $a$ we denote by $\\lfloor a \\rfloor $ the largest integer which is less than or equal to $a$ .", "Theorem 1$^{\\prime }$ Suppose there is a real number $a \\ge 1$ such that the moments of the random variable $X$ satisfy the condition $m_{k+1}/m_k={\\cal O}((k+1)^{2/a})$ as $k\\rightarrow \\infty $ .", "Then the power $X^{\\lfloor a \\rfloor }$ satisfies Carleman's condition and is M-det.", "Proof Note that $\\frac{{\\bf E}[(X^{\\lfloor a\\rfloor })^{k+1}]}{{\\bf E}[(X^{\\lfloor a \\rfloor })^k]}&=&\\frac{{\\bf E}[X^{{\\lfloor a \\rfloor }k+{\\lfloor a \\rfloor }}]}{{\\bf E}[X^{{\\lfloor a \\rfloor }k+{\\lfloor a \\rfloor }-1}]}\\frac{{\\bf E}[X^{{\\lfloor a \\rfloor }k+{\\lfloor a \\rfloor }-1}]}{{\\bf E}[X^{{\\lfloor a \\rfloor }k+{\\lfloor a\\rfloor }-2}]}\\cdots \\frac{{\\bf E}[X^{{\\lfloor a \\rfloor }k+1}]}{{\\bf E}[X^{{\\lfloor a \\rfloor }k}]}\\\\&=&{\\cal O}((k+1)^{(2/a)\\lfloor a\\rfloor })={\\cal O}((k+1)^2)~~\\hbox{as}~~k\\rightarrow \\infty .$ Hence, by Theorem 1, $X^{\\lfloor a \\rfloor }$ satisfies Carleman's condition and is M-det.", "$\\Box $ Theorem 2 Let $\\xi ,~\\xi _i,~i=1,2,\\ldots ,n$ , be defined as before and $Y_n=\\xi _1\\cdots \\xi _n.$ If $\\xi $ and the index $n$ are such that ${\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^k]={\\cal O}((k+1)^{2/n}) \\ \\mbox{ as } k\\rightarrow \\infty ,$ then $Y_n$ satisfies Carleman's condition and is M-det.", "Proof By the assumption, we have ${\\bf E}[Y_n^{k+1}]/{\\bf E}[Y_n^k]=({\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^k])^n={\\cal O}((k+1)^2)\\ \\mbox{ as }~ k\\rightarrow \\infty .$ This, according to Theorem 1, implies the validity of Carleman's condition for $Y_n,$ hence $Y_n$ is M-det as stated above.", "$\\Box $ Theorem 2$^{\\prime }$ Let $a\\ge 1$ .", "If ${\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^k]={\\cal O}((k+1)^{2/a}) \\ \\mbox{ as }k\\rightarrow \\infty ,$ then $Y_{\\lfloor a \\rfloor }$ satisfies Carleman's condition and is M-det.", "Proof Note that $&~&{\\bf E}[Y_{\\lfloor a \\rfloor }^{k+1}]/{\\bf E}[Y_{\\lfloor a\\rfloor }^k]=({\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^k])^{\\lfloor a\\rfloor }\\\\ &=&{\\cal O}((k+1)^{(2/a){\\lfloor a \\rfloor }})={\\cal O}((k+1)^2)\\ \\mbox{ as }k\\rightarrow \\infty .", "$ The conclusions follow from Theorem 1.", "$\\Box $ 5.", "Generalized gamma distributions.", "Part (b) We now apply the general results, Theorems 1 and 2 in Section 4, to give an alternative proof of the moment determinacy established in Section 3.", "Let, as before, $\\xi \\sim GG(\\alpha , \\beta , \\gamma ).$ We claim that for $n\\le 2\\beta $ , both $X_n=\\xi ^n$ and $Y_n=\\xi _1\\xi _2\\cdots \\xi _n$ are M-det.", "To see this, we first calculate that $\\frac{{\\bf E}[X_n^{k+1}]}{{\\bf E}[X_n^{k}]}=\\frac{{\\bf E}[\\xi ^{n(k+1)}]}{{\\bf E}[\\xi ^{nk}]}=\\frac{\\Gamma ((\\gamma +n(k+1))/\\beta )}{\\alpha ^{n/\\beta }\\Gamma ((\\gamma +nk)/\\beta )}\\approx (n/\\alpha \\beta )^{n/\\beta } (k+1)^{n/\\beta }~~ \\hbox{as}~k\\rightarrow \\infty .$ For this relation we have used the approximation of the gamma function: $\\Gamma (x)\\approx \\sqrt{2\\pi }x^{x-1/2}e^{-x}~~\\hbox{as}~x\\rightarrow \\infty $ (see, e.g., [26], p. 253).", "Then by Theorem 1, $X_n$ is M-det if $n\\le 2\\beta $ , and by Theorem 2, $Y_n$ is M-det if $1/\\beta \\le 2/n$ , i.e., if $n\\le 2\\beta $ , because ${\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^k]={\\cal O}((k+1)^{1/\\beta })$ as $k\\rightarrow \\infty $ .", "For example, if $\\xi \\sim Exp(1) = GG(1, 1, 1)$ , then the product $Y_2=\\xi _1\\xi _2$ is M-det.", "In fact, by Lemma 1, the density $g_2$ of $Y_2$ is $g_2(x)=2K_0(2\\sqrt{x})\\approx Cx^{-1/4}e^{-2\\sqrt{x}}$ as $x\\rightarrow \\infty $ , where $K_0$ is the modified Bessel function of the second kind (see also [15], p. 417, and [9], p. 917, Formula 8.432(8)).", "If $\\xi \\sim GG(1/2, 2, 1)$ , the half-normal distribution, then $Y_n=\\xi _1\\xi _2\\cdots \\xi _n$ is M-det for $n\\le 4$ .", "As mentioned before, the density function of the product of two half-normals is $g_2(x)=(2/{\\pi })K_0(x)\\approx C_2x^{-1/2}e^{-x}\\hbox{ as }~x\\rightarrow \\infty $ .", "6.", "More results related to Theorems 1 and 2 Under the same assumption as that in Theorem 1, we even have a stronger statement; see Theorem 3 below.", "Note that its proof does not use Lyapunov's inequality, and that Hardy's condition implies Carleman's condition.", "For convenience, we recall in the next lemma a characterization of Hardy's condition in terms of the moments (see [24], Theorem 3).", "Lemma 3 Let $a\\in (0,1]$ and let $X$ be a nonnegative random variable.", "Then ${\\bf E}[\\exp ({c{X}^a})]<\\infty $ for some constant $c>0$ iff ${\\bf E}[X^k]\\le c_0^k\\,\\Gamma (k/a+1), ~k=1,2,\\ldots ,$ for some constant $c_0>0$ (independent of $k$ ).", "In particular, $X$ satisfies Hardy's condition, i.e., ${\\bf E}[\\exp ({c\\sqrt{X}})]<\\infty $ for some constant $c>0$ , iff ${\\bf E}[X^k]\\le c_0^k\\,(2k)!, ~k=1,2,\\ldots ,$ for some constant $c_0>0$ (independent of $k$ ).", "Theorem 3 Suppose $X$ is a nonnegative random variable with finite moments $m_k={\\bf E}[X^k],\\ k=1,2,\\ldots $ , such that the condition in Theorem 1 holds: $m_{k+1}/m_k = {\\cal O}((k+1)^2)$ as $k \\rightarrow \\infty .$ Then $X$ satisfies Hardy's condition, and is M-det.", "Proof By the assumption, there exists a constant $c_\\ge m_1>0$ such that $m_{k+1}\\le c_(k+1)^2m_k~~\\hbox{for}~k=0,1,2,\\ldots ,$ where $m_0\\equiv 1$ .", "This implies that $m_{k+1}\\le (c_/2)(2k+2)(2k+1)m_k~~\\hbox{for}~k=0,1,2,\\ldots ,$ and hence $m_{k+1}\\le (c_/2)^{k+1}\\Gamma (2k+3)m_0~~\\hbox{for}~k=0,1,2,\\ldots .$ Taking $c_0=c_/2$ , $m_{k+1}\\le c_0^{k+1}\\Gamma (2k+3)~~\\hbox{for}~k=0,1,2,\\ldots ,$ or, equivalently, $m_{k}\\le c_0^{k}\\Gamma (2k+1)~~\\hbox{for}~k=1,2,\\ldots .$ Hence $X$ satisfies Hardy's condition by Lemma 3.", "$\\Box $ Remark 1 The constant 2 (the growth rate of the moments) in the condition of Theorem 1 is the best possible in the following sense.", "For each $\\varepsilon >0$ , there exists a random variable $X$ such that $m_{k+1}/m_k={\\cal O}( (k+1)^{2+\\varepsilon })$ as $k\\rightarrow \\infty $ , and $X$ is M-indet.", "To see this, let us consider $X=\\xi \\sim GG(1, \\beta , 1),$ which has density $f(x)=c\\exp (-x^{\\beta }), ~x>0.$ We have $\\frac{{\\bf E}[\\xi ^{k+1}]}{{\\bf E}[\\xi ^{k}]}=\\frac{\\Gamma ((k+2)/\\beta )}{{\\Gamma ((k+1)/\\beta )}}\\approx \\beta ^{-1/\\beta }(k+1)^{1/\\beta } ~~ \\hbox{as}~ k\\rightarrow \\infty .$ If for $\\varepsilon >0$ we take $\\beta =\\frac{1}{2+\\varepsilon }<\\frac{1}{2}$ , then ${{\\bf E}[\\xi ^{k+1}]}/{{\\bf E}[\\xi ^{k}]}=$ ${\\cal O}( (k+1)^{2+\\varepsilon }) ~~ \\hbox{as}~k\\rightarrow \\infty .$ However, as mentioned before, $X$ is M-indet.", "Remark 2 The constant $2/n$ in the condition of Theorem 2 is the best possible.", "Indeed, we can show that for each $\\varepsilon >0$ , there exists a random variable $\\xi $ such that ${\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^k]={\\cal O}((k+1)^{2/n+\\varepsilon })$ as $k\\rightarrow \\infty $ , but $Y_n$ is M-indet.", "To see this, let us consider $X=\\xi \\sim GG(1, \\beta ,1)$ .", "For each $\\varepsilon >0$ , take $\\beta =1/(2/n+\\varepsilon )$ , then $\\frac{{\\bf E}[\\xi ^{k+1}]}{{\\bf E}[\\xi ^{k}]}=\\frac{\\Gamma ((k+2)/\\beta )}{{\\Gamma ((k+1)/\\beta )}}={\\cal O}\\left((k+1)^{2/n+\\varepsilon }\\right)~~ \\hbox{as}~ k\\rightarrow \\infty .$ However, since $n>2\\beta $ , $Y_n$ is M-indet (compare this with the statement in Section 10).", "7.", "Faster growth rate of the moments implies moment indeterminacy We now establish a result which is converse to Theorem 1.", "Theorem 4 Suppose $X$ is a nonnegative random variables whose moments $m_k, \\ k=1,2,\\ldots $ , are such that $m_{k+1}/m_k\\ge C(k+1)^{2+\\varepsilon }$ for all large $k$ , where $C$ and $\\varepsilon $ are positive constants.", "Assume further that $X$ has a density $f$ satisfying the condition: for some $x_0>0$ , $f$ is positive and differentiable on $[x_0,\\infty )$ and $L_{f}(x):=-\\frac{xf^{\\prime }(x)}{f(x)}\\nearrow \\infty ~~\\hbox{as}~~x_0<x\\rightarrow \\infty .$ Then $X$ is M-indet.", "Proof Without loss of generality we can assume that $m_{k+1}/m_k\\ge C(k+1)^{2+\\varepsilon }$ for each $k\\ge 1$ .", "Therefore, $m_{k+1}\\ge C^k ((k+1)!", ")^{2+\\varepsilon }m_1~~\\hbox{for}~k=1,2,\\ldots .$ Taking $C_0=\\min \\lbrace C,m_1\\rbrace $ , we have $m_{k+1}\\ge C_0^{k+1} ((k+1)!", ")^{2+\\varepsilon }~~\\hbox{for}~k=1,2,\\ldots ,$ or, equivalently, $m_{k}\\ge C_0^{k} (k!", ")^{2+\\varepsilon }=C_0^{k}(\\Gamma (k+1))^{2+\\varepsilon }~~\\hbox{for}~k=2,3,\\ldots .$ Since $\\Gamma (x+1)=x\\Gamma (x)\\approx \\sqrt{2\\pi }\\,x^{x+1/2}\\,e^{-x}$ as $x\\rightarrow \\infty $ , we have that for some constant $c>0$ , $m_{k}^{-1/(2k)}\\le C_0^{-1/2}(\\Gamma (k+1))^{-(2+\\varepsilon )/(2k)}\\approx ck^{-1-\\varepsilon /2}~~\\hbox{for all large}~k.$ This implies that the Carleman quantity for the moments of $f$ is finite: ${\\bf C}[f]:=\\sum _{k=1}^{\\infty }m_k^{-1/(2k)}<\\infty .$ We sketch the rest of the proof.", "Following the proof of Theorem 3 in [10], we first construct a symmetric distribution $G$ on ${\\mathbb {R}}$ , obeyed by a random variable $Y$ , such that ${\\bf E}[Y^{2k}]={\\bf E}[X^k],$ $~{\\bf E}[Y^{2k-1}]$ $=0$ for $k=1,2,\\ldots $ .", "Let $g$ be the density of $G$ .", "Then for the Carleman quantity of the moments of $g$ we have: ${\\bf C}[g]:=\\sum _{k=1}^{\\infty }\\left({\\bf E}[Y^{2k}]\\right)^{-1/(2k)}=\\sum _{k=1}^{\\infty }\\left({\\bf E}[X^{k}]\\right)^{-1/(2k)}={\\bf C}[f]<\\infty .$ This implies that for some $x_0^>x_0$ , the logarithmic normalized integral (called also Krein quantity of $g$ ) over the domain $\\lbrace x:\|x\|\\ge x_0^\\rbrace $ is finite: ${\\bf K}[g]:=\\int _{\|x\|\\ge x_0^}\\frac{-\\log g(x)}{1+x^2}dx<\\infty ,$ as shown in the proof of Theorem 2 in [10].", "Finally, according to Theorem 2.2 in [19], this is a sufficient condition for $Y$ to be M-indet on ${\\mathbb {R}}$ and we conclude that $X$ is M-indet on ${\\mathbb {R}}^+$ by mimicking the proof of Corollary 1 in [21] (see also [17], Proposition 1 and Theorem 3).", "$\\Box $ 8.", "Generalized gamma distributions.", "Part (c) Let us see how Theorem 4 in Section 7 works for a random variable $\\xi \\sim GG(\\alpha , \\beta , \\gamma ).$ We claim that for $n>2\\beta $ , the power $X_n=\\xi ^n$ is M-indet.", "To see this, recall that $\\frac{{\\bf E}[X_n^{k+1}]}{{\\bf E}[X_n^{k}]}\\approx (n/\\alpha \\beta )^{n/\\beta } (k+1)^{n/\\beta }~~ \\hbox{as}~k\\rightarrow \\infty ,$ where $n/\\beta >2$ .", "Thus the moments of $X_n$ grow at a rate more than 2.", "Let us check that the density $h$ of $X_n$ satisfies the condition (2).", "Indeed, we have $L_{h}(x):=-\\frac{xh^{\\prime }(x)}{h(x)}=1-\\frac{\\gamma }{n}+\\frac{\\alpha \\beta }{n}x^{\\beta /n} \\nearrow \\infty ~~~\\hbox{ultimately as}~ x\\rightarrow \\infty .$ Therefore, for $n>2\\beta $ , $X_n$ is M-indet by Theorem 4.", "Remark 3 To use Theorem 4 is another way to prove some known facts, for example, that the log-normal distribution and the cube of the exponential distribution are M-indet.", "Indeed, for $X\\sim LogN(0,1),$ we have the moment recurrence $m_{k+1}=e^{k+1/2}m_k,~k=1,2,\\ldots ,$ and for $X=\\xi ^3$ , where $\\xi \\sim Exp (1)$ , we have $m_{k+1}=(3k+1)(3k+2)(3k+3)m_k, ~k=1,2,\\ldots .$ It is easily seen that in both cases the growth rates of the moments are quite fast.", "For the cube of $Exp(1)$ we have $m_{k+1}/m_k\\ge C(k+1)^3,~k=1,2,\\ldots $ , for some constant $C>0$ , so the rate is more than 2.", "For $LogN$ the rate is exponential, hence much larger than 2.", "It remains to check that condition (2) is satisfied for the density of $\\xi ^3$ and the density of $LogN.$ Details are omitted.", "We can make one step more by considering the logarithmic skew-normal distributions with density $f_{\\lambda }(x)=(2/x)\\varphi (\\ln x)\\Phi (\\lambda \\ln x),$ $~x>0$ , where $\\lambda $ is a real number.", "(When $\\lambda =0$ , $f_{\\lambda }$ reduces to the standard log-normal density.)", "Then we have the moment relationship $m_{k+1}\\approx e^{(k+1/2)\\rho }m_k,~~\\hbox{as}~~k\\rightarrow \\infty ,$ where $\\rho \\in (0,1]$ is a constant (see, e.g., [12], Proposition 3).", "Thus the moments grow very fast, exponentially, and it remains to check that the density function $f_{\\lambda }$ satisfies the condition (2): $L_{f_{\\lambda }}(x):=-\\frac{xf_{\\lambda }^{\\prime }(x)}{f_{\\lambda }(x)}\\nearrow \\infty ~~\\hbox{ultimately as}~~x\\rightarrow \\infty .$ Therefore, by the above Theorem 4, we conclude that all logarithmic skew-normal distributions are M-indet.", "This is one of the results in [12] where a different proof is given.", "9.", "General result on the M-indet property of the product $Y_n=\\xi _1\\xi _2 \\cdots \\xi _n$ In the next theorem we describe conditions on the distribution of $\\xi $ under which the product $Y_n=\\xi _1\\xi _2\\cdots \\xi _n$ is M-indet.", "Theorem 5 Let $\\xi \\sim F,$ where $F$ is absolutely continuous with density $f>0$ on ${\\mathbb {R}}^+$ .", "Assume further that: (i) $f(x)$ is decreasing in $x\\ge 0,$ and (ii) there exist two constants $x_0\\ge 1$ and $A>0$ such that $f(x)/\\overline{F}(x)\\ge A/x~~\\hbox{for}~~x\\ge x_0,$ and some constants $B>0,~ \\alpha >0,$ $\\beta >0$ and a real $\\gamma $ such that $\\overline{F}(x)\\ge Bx^{\\gamma }e^{-\\alpha x^{\\beta }}~~ \\hbox{for}~~ x\\ge x_0.", "$ Then, for $n>2\\beta $ , the product $Y_n$ has a finite Krein quantity and is M-indet.", "Corollary 2 Let $\\xi \\sim F$ satisfy the conditions in Theorem 5 with $\\beta <\\frac{1}{2}$ .", "Then $F$ itself is M-indet.", "Lemma 4 Under the condition (3), we have $\\int _x^{\\infty }\\frac{f(u)}{u}du\\ge \\frac{A}{1+A}\\frac{\\overline{F}(x)}{x} \\ \\mbox{ and } \\ \\overline{F}(x)\\le \\frac{C}{x^A}, ~ x>x_0,~\\hbox{for someconstant}~ C>0.$ Proof Note that for $x>x_0$ , $\\int _x^{\\infty }\\frac{f(u)}{u}du=-\\int _x^{\\infty }\\frac{1}{u}d\\overline{F}(u)=\\frac{\\overline{F}(x)}{x}-\\int _x^{\\infty }\\frac{\\overline{F}(u)}{u^2}du\\ge \\frac{\\overline{F}(x)}{x}-\\frac{1}{A}\\int _x^{\\infty }\\frac{f(u)}{u}du.$ The last inequality is due to (3).", "Hence $\\left(1+\\frac{1}{A}\\right)\\int _x^{\\infty }\\frac{f(u)}{u}du\\ge \\frac{\\overline{F}(x)}{x}.$ On the other hand, for $x>x_0$ , $\\log \\overline{F}(x)&=&{-\\int _0^xf(t)/\\overline{F}(t)dt}={-\\int _0^{x_0}f(t)/\\overline{F}(t)dt-\\int _{x_0}^xf(t)/\\overline{F}(t)dt}\\\\&\\equiv &C_0-\\int _{x_0}^xf(t)/\\overline{F}(t)dt\\le C_0-\\int _{x_0}^xA/tdt= C_0+A\\log x_0 -A\\log x.$ Therefore, $\\overline{F}(x)\\le C/x^A,~x>x_0,$ where $C=x_0^Ae^{C_0}.$ $\\Box $ Remark 4 After deriving in Lemma 4 a lower bound for $\\int _x^{\\infty }(f(u)/u)du$ we have the following upper bound for arbitrary density $f$ on ${\\mathbb {R}}^+$ : $\\int _x^{\\infty }\\frac{f(u)}{u}du\\le \\frac{1}{x}\\int _x^{\\infty }{f(u)}du= \\frac{\\overline{F}(x)}{x},~x>0.$ Proof of Theorem 5 The density $g_n$ of $Y_n$ is expressed as follows: $g_n(x)=\\int _0^{\\infty }\\!\\!\\int _0^{\\infty }\\!\\!\\cdots \\!\\!\\int _0^{\\infty }\\frac{f(u_1)}{u_1}\\frac{f(u_2)}{u_2}\\cdots \\frac{f(u_{n-1})}{u_{n-1}}f\\left(\\frac{x}{u_1u_2\\cdots u_{n-1}}\\right)du_1du_2\\cdots du_{n-1}$ for $x>0$ .", "Hence $g_n(x)>0$ and decreases in $x \\in (0,\\infty )$ .", "For any $a>0$ , we have $g_n(x)\\!\\!&\\ge &\\!\\!\\int _a^{\\infty }\\!\\!\\int _a^{\\infty }\\!\\!\\cdots \\!\\!\\int _a^{\\infty }\\frac{f(u_1)}{u_1}\\frac{f(u_2)}{u_2}\\cdots \\frac{f(u_{n-1})}{u_{n-1}}f\\left(\\frac{x}{u_1u_2\\cdots u_{n-1}}\\right)du_1du_2\\cdots du_{n-1}\\nonumber \\\\&\\ge &\\int _a^{\\infty }\\!\\!\\int _a^{\\infty }\\!\\!\\cdots \\!\\!\\int _a^{\\infty }\\frac{f(u_1)}{u_1}\\frac{f(u_2)}{u_2}\\cdots \\frac{f(u_{n-1})}{u_{n-1}}f\\left(\\frac{x}{a^{n-1}}\\right)du_1du_2\\cdots du_{n-1}\\nonumber \\\\&=&f\\left(\\frac{x}{a^{n-1}}\\right)\\left(\\int _a^{\\infty }\\frac{f(u)}{u}du\\right)^{n-1},~~x>0.$ The above second inequality follows from the monotone property of $f$ .", "Taking $a=x^{1/n}>x_0$ , we have, by (3)–(5) and Lemma 4, that $g_n(x)&\\ge &f\\left(x^{1/n}\\right)\\left(\\int _{x^{1/n}}^{\\infty }\\frac{f(u)}{u}du\\right)^{n-1}\\ge f\\left(x^{1/n}\\right)\\left(\\frac{A}{1+A}\\frac{\\overline{F}(x^{1/n})}{x^{1/n}}\\right)^{n-1}\\\\&\\ge &\\left(\\frac{A}{1+A}\\right)^{n-1}x^{-(1-1/n)}\\frac{f\\left(x^{1/n}\\right)}{\\overline{F}(x^{1/n})}\\left(\\overline{F}(x^{1/n})\\right)^n\\\\&\\ge &C_nx^{\\gamma -1}e^{-n\\alpha x^{\\beta /n}},$ where $C_n=\\left(\\frac{A}{1+A}\\right)^{n-1}AB^n$ .", "Therefore, the Krein quantity for $g_n$ is as follows: ${\\bf K}[g_n]&=&\\int _0^{\\infty }\\frac{-\\log g_n(x^2)}{1+x^2}dx=\\int _0^{x_0^n}\\frac{-\\log g_n(x^2)}{1+x^2}dx+\\int _{x_0^n}^{\\infty }\\frac{-\\log g_n(x^2)}{1+x^2}dx\\\\&\\le &\\left(-\\log g_n(x_0^{2n})\\right)\\int _0^{x_0^n}\\frac{1}{1+x^2}dx+\\int _{x_0^n}^{\\infty }\\frac{-\\log g_n(x^2)}{1+x^2}dx<\\infty ~~\\hbox{if}~ n>2\\beta .$ This in turn implies that $Y_n$ is M-indet for $n>2\\beta $ (see, e.g., [10], Theorem 3).", "$\\Box $ 10.", "Generalized gamma distributions.", "Part (d) Let us see how the general result from Section 9 can be used to establish the moment indeterminacy of products of independent copies of a random variable $\\xi \\sim GG(\\alpha , \\beta ,1).$ Here $\\gamma =1$ and the density is $f(x)=ce^{-\\alpha x^{\\beta }}, \\ x \\ge 0.$ We claim that for $n>2\\beta $ , the product $Y_n=\\xi _1\\xi _2\\cdots \\xi _n$ is M-indet.", "To see this, note that $f(x)/\\overline{F}(x)\\approx \\alpha \\beta x^{\\beta -1}$ and $\\overline{F}(x)\\approx [c/(\\alpha \\beta )]x^{1-\\beta }e^{-\\alpha x^{\\beta }}$ as $x\\rightarrow \\infty $ .", "Then the density $f$ satisfies the conditions (i) and (ii) in Theorem 5 and hence $Y_n$ is M-indet if $n>2\\beta $ .", "For example, if $\\xi \\sim Exp(1),$ then, as mentioned before, the product $Y_n=\\xi _1\\xi _2 \\cdots \\xi _n$ is M-indet for $n\\ge 3.$ If $\\xi $ has the half-normal distribution, its density is $f(x)=\\sqrt{{2}/{\\pi }}e^{-x^2/2},$ $~x\\ge 0$ , then $Y_n=\\xi _1\\xi _2 \\cdots \\xi _n$ is M-indet for $n\\ge 5$ (recall from Section 5 that $Y_n$ is M-det for $n\\le 4$ ).", "By words: The product of two, three or four half-normal random variables is M-det, while the product of five or more such variables is M-indet.", "In summary, we have the following result about $GG(\\alpha , \\beta , \\gamma )$ with $\\gamma =1$ .", "Lemma 5 Let $n\\ge 2,~X_n=\\xi ^n$ and $Y_n=\\xi _1 \\cdots \\xi _n$ , where $\\xi _1,\\ldots ,\\xi _n$ are independent copies of $\\xi \\sim GG(\\alpha , \\beta ,1)$ .", "Then the power $X_n$ is M-det iff the product $Y_n$ is M-det and this is true iff $n\\le 2\\beta $ .", "We now consider the general case $\\gamma >0$ .", "Theorem 6 Let $n\\ge 2,~X_n=\\xi ^n$ and $Y_n=\\xi _1\\cdots \\xi _n$ , where $\\xi _1,\\ldots ,\\xi _n$ are independent copies of $\\xi \\sim GG(\\alpha ,\\beta , \\gamma )$ .", "Then $X_n$ is M-det iff $Y_n$ is M-det and this is true iff $n\\le 2\\beta $ .", "In other words, both $X_n$ and $Y_n$ have the same moment determinacy property.", "Proof Define $\\eta =\\xi ^{\\gamma }$ , $\\eta _i=\\xi _i^{\\gamma }$ , $i=1,2,\\ldots ,n$ , $X_n^=\\eta ^n=(\\xi ^n)^{\\gamma }=X_n^{\\gamma }$ and $Y_n^=\\eta _1\\eta _2\\cdots \\eta _n=(\\xi _1\\xi _2\\cdots \\xi _n)^{\\gamma }=Y_n^{\\gamma }$ .", "Since $\\eta \\sim GG(\\alpha , \\beta /\\gamma , 1)$ , we have, by Lemma 5, $X_n^$ is M-det iff $Y_n^$ is M-det iff $n\\le 2\\beta /\\gamma $ .", "Next, note that for each $x>0$ , we have ${\\bf P}[X_n^>x]={\\bf P}[X_n>x^{1/\\gamma }]$ and ${\\bf P}[Y_n^>x]={\\bf P}[Y_n>x^{1/\\gamma }]$ .", "This implies that any distributional property shared by $X_n^$ and $Y_n^$ can be transferred to a similar property shared by $X_n$ and $Y_n$ , and vice versa.", "Therefore, $X_n$ is M-det iff $Y_n$ is M-det iff $n\\le 2\\beta $ , because $X_n$ is M-det iff $n\\le 2\\beta $ (see, e.g., [18]).", "$\\Box $ 11.", "Half-logistic distribution Some of the above results and illustrations involve the generalized gamma distribution $GG(\\alpha , \\beta ,\\gamma )$ .", "It is useful to have a moment determinacy characterization for powers and products based on non-GG distributions.", "Here is an example based on the half-logistic distribution, which clearly is not in the class $GG$ .", "Statement We say that the random variable $\\xi $ has the half-logistic distribution if its density is $f(x)=\\frac{2e^{-x}}{(1+e^{-x})^2},~~x\\ge 0.$ The power $X_n=\\xi ^n$ and the product $Y_n=\\xi _1\\xi _2 \\cdots \\xi _n$ are defined as above.", "Then $X_n$ is M-det iff $Y_n$ is M-det and this is true iff $n\\le 2$ .", "This means that for each $n$ , the two random variables $X_n$ and $Y_n$ share the same moment determinacy property.", "Proof (i) The claim that $X_n$ is M-det iff $n\\le 2$ follows from results in [11].", "Actually, in [11] it is proved that for any real $s>0$ , the power $\\xi ^s$ is M-det iff $s\\le 2$ .", "Let us give here an alternative proof.", "The density $h_s$ of $\\xi ^s$ is $h_s(z)=\\frac{2}{s}z^{1/s-1}\\frac{e^{-z^{1/s}}}{(1+e^{-z^{1/s}})^{2}},~z\\ge 0.$ Using the inequality: $\\frac{1}{4}\\le (1+e^{-x})^{-2}\\le 1$ for $x\\ge 0$ , we find two-sided bounds for the moments of $\\xi ^s$ : $\\frac{1}{2}\\Gamma (ks+1)\\le {\\bf E}[(\\xi ^s)^k]\\le \\int _0^{\\infty }\\frac{2}{s}z^{k+1/s-1}{e^{-z^{1/s}}}dz=2\\Gamma (ks+1).$ Therefore the growth rate of the moments of $\\xi ^s$ is $\\frac{{\\bf E}[(\\xi ^s)^{k+1}]}{{\\bf E}[(\\xi ^s)^k]}\\le 4\\cdot \\frac{\\Gamma ((k+1)s+1)}{\\Gamma (ks+1)}\\approx 4s^s(k+1)^s~~\\hbox{as}~k\\rightarrow \\infty .$ By Theorem 1, this implies that $\\xi ^s$ is M-det for $s\\le 2$ .", "On the other hand, we have $\\frac{{\\bf E}[(\\xi ^s)^{k+1}]}{{\\bf E}[(\\xi ^s)^k]}\\ge \\frac{1}{4}\\cdot \\frac{\\Gamma ((k+1)s+1)}{\\Gamma (ks+1)}\\approx \\frac{1}{4}s^s (k+1)^s~~\\hbox{as}~k\\rightarrow \\infty .$ The moment condition in Theorem 4 is satisfied if $s>2$ .", "It remains now to check the validity of condition (2) for the density $h_s$ .", "We have $&&L_{h_s}(z):=-\\frac{zh_s^{\\prime }(z)}{h_s(z)}\\\\&=&1-\\frac{1}{s}+\\frac{1}{s}z^{1/s}-\\frac{2}{s}\\,z^{1/s}\\frac{e^{-z^{1/s}}}{1+e^{-z^{1/s}}}~~\\nearrow \\infty ~\\hbox{ultimately~~as}~~z\\rightarrow \\infty .$ Hence, if $s>2$ , $\\xi ^s$ is M-indet.", "(ii) It remains to prove that $Y_n$ is M-det iff $n\\le 2$ .", "(Sufficiency) As in part (i), we have $\\frac{1}{2}\\Gamma (k+1)\\le {\\bf E}[\\xi ^k]= 2\\Gamma (k+1).$ Therefore, ${\\bf E}[\\xi ^{k+1}]/{\\bf E}[\\xi ^{k}]={\\cal O}(k+1)$ as $k\\rightarrow \\infty $ .", "By Theorem 2, we conclude that $Y_n$ is M-det if $n\\le 2$ .", "(Necessity) Note that $\\overline{F}(x)={\\bf P}[\\xi >x]=2e^{-x}/(1+e^{-x})\\ge e^{-x},~~x\\ge 0,$ and $f(x)/\\overline{F}(x)=1/(1+e^{-x})\\ge 1/2,~x\\ge 0$ .", "Therefore, taking $\\beta =1$ in Theorem 5, we conclude that $Y_n$ is M-indet if $n> 2$ .", "Let us express this conclusion by words: The product of three or more half-logistic random variables is M-indet.", "$\\Box $ Acknowledgments We would like to thank the Editors and the Referees for their valuable suggestions which helped us shorten the original manuscript and improve the presentation.", "We thank also Prof. M.C.", "Jones (Open University, UK) for showing interest in our work and making useful comments.", "This paper was basically completed during the visit of JS to Academia Sinica, Taipei, December 2012 – January 2013.", "The support and the hospitality provided by the Institute of Statistical Science are greatly appreciated.", "The work of GDL was partly supported by the National Science Council of ROC (Taiwan) under Grant NSC 102-2118-M-001-008-MY2, and that of JS by the Emeritus Fellowship provided by the Leverhulme Trust (UK)." ] ]	1403.0301
[ [ "Analysis of high-order dispersion on ultrabroadband microresonator-based\n frequency combs" ], [ "Abstract We numerically investigate the influence of high-order dispersion on both temporal and spectral characterizations of microresonator-based optical frequency combs.", "The moment method is utilized to study the temporal dynamics of intracavity solitons.", "The theoretical and numerical results indicate the temporal shifts are induced by high-odd-order dispersion rather than high-even-order dispersion.", "The role of high-order dispersion on the frequency comb envelopes is carefully elucidated through analyzing the intracavity Cherenkov radiations.", "We further demonstrate that the spectra envelope of an ultrabroadband optical frequency comb can be engineered by using dispersion profiles with multiple zero dispersion wavelengths." ], [ "Analysis of high-order dispersion on ultrabroadband microresonator-based frequency combs Shaofei Wang The Key Lab of Specialty Fiber Optics and Optical Access Network, Shanghai University, 200072 Shanghai, China Hairun Guo Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kgs.", "Lyngby, Denmark Xuekun Bai The Key Lab of Specialty Fiber Optics and Optical Access Network, Shanghai University, 200072 Shanghai, China Xianglong ZengCorresponding author: [email protected] The Key Lab of Specialty Fiber Optics and Optical Access Network, Shanghai University, 200072 Shanghai, China We numerically investigate the influence of high-order dispersion on both temporal and spectral characterizations of microresonator-based optical frequency combs.", "The moment method is utilized to study the temporal dynamics of intracavity solitons.", "The theoretical and numerical results indicate the temporal shifts are induced by high-odd-order dispersion rather than high-even-order dispersion.", "The role of high-order dispersion on the frequency comb envelopes is carefully elucidated through analyzing the intracavity Cherenkov radiations.", "We further demonstrate that the spectra envelope of an ultrabroadband optical frequency comb can be engineered by using dispersion profiles with multiple zero dispersion wavelengths.", "(230.5750) Resonators; (190.4410) Nonlinear optics, parametric processes; (190.5530) Pulse propagation and temporal solitons.", "Frequency combs generated in a monolithic micro-ring resonator have attracted lots of interests soon after its proposal [1], [2].", "With short cavity lengths, high fineness and robust platforms, micro-resonators based on Kerr materials, such as $\\rm {CaF_2}$ or $\\rm {MgF_2}$ crystallines, high-index silica microspheres and silicon nitride micro-rings, can support a frequency comb with high repetition rate, high signal-to-noise ratio and high stabilization.", "The cavity is pumped with a monochromatic CW laser.", "Applications include precise frequency metrology, spectroscopy, telecommunications and optical clocks.", "One promising regime to generate such a frequency comb relies on the four-wave mixing (FWM) parametric process induced by Kerr nonlinearities, also called Kerr frequency comb generation [3].", "The steady state of a frequency comb exhibits mode-locking and the intracavity pulse formation [4], which promises the comb to be highly coherent.", "Dynamics of the intracavity pulse formation [5], [6] and transient regime of the Kerr frequency comb [7] have been illustrated, which in fact are analogous to pulse propagation dynamics in a Kerr material [8].", "The intracavity pulse can form a soliton pattern if the net cavity dispersion is anomalous, or the pulse turns to wave breaking under normal dispersion [9].", "Significantly, octave-spanning Kerr frequency combs were demonstrated as a result of the ultra-short soliton formation inside the cavity where group velocity dispersion (GVD) is quite weak over a wide wavelength span [10], [11].", "The dispersive wave could also be generated accompanied with the intracavity soliton in the opposite GVD regions, which helps to extend the comb bandwidth just as it does in the well-known supercontinuum generation [12].", "On the other hand, effects such as pulse self-steepening, material Raman scattering and high-order dispersion (HOD) perturbations may influence the intracavity pulse formation as well as the frequency comb generation, which unfortunately were rarely reported so far.", "In this letter we numerically investigate the HOD influence on the intracavity pulse dynamics.", "Effects of odd-order dispersion (third-order, fifth-order and so on) are revealed to cause temporal delays of the intracavity solitons.", "Since the HOD effects also work to tailor the cavity dispersion profile with determining one or multiple zero dispersion wavelengths (ZDWs), we present that group-velocity-matched dispersion wave generations [13] are evoked during the intracavity soliton formation, and Kerr frequency combs are tuned to have different spectral envelopes.", "Figure: Temporal evolution dynamics of intracavity solitons.", "(a) only β 2 \\beta _2 (-2×10 -2 ps 2 /m-2\\times {10^{-2}}{\\rm {~p}}{{\\rm {s}}^2{/\\rm m}}).", "(b) β 2 \\beta _2 and β 3 \\beta _3 (-4×10 -4 ps 3 /m-4\\times {10^{-4}}{\\rm {~p}}{{\\rm {s}}^3/{\\rm m}}).", "(c) β 2 \\beta _2 and β 4 \\beta _4 (-16×10 -6 ps 4 /m-16\\times {10^{-6}}{\\rm {~p}}{{\\rm {s}}^4/{\\rm m}}).", "(d) β 2 \\beta _2 and β 5 \\beta _5 (8×10 -8 ps 5 /m8\\times {10^{-8}}{\\rm {~p}}{{\\rm {s}}^5/{\\rm m}}).The numerical model is the Lugiato-Lefever (LL) equation [14], [15] which is actually a periodically damped nonlinear Schr$\\ddot{\\rm o}$dinger equation driven by an external pump source, describing the intracavity pulse dynamics ${t_R}\\frac{{\\partial E\\left( {t,\\tau } \\right)}}{{\\partial t}} = \\left[ { - \\alpha - i{\\delta _0} + iL\\sum \\limits _{k \\ge 2} {\\frac{{{\\beta _k}}}{{k!", "}}{{\\left( {i\\frac{\\partial }{{\\partial \\tau }}} \\right)}^k}} } \\right.", "\\\\\\left.", "{ + i\\gamma L{{\\left\| E \\right\|}^2}} \\right]E\\left( {t,\\tau } \\right) + \\sqrt{\\theta }{E_{in}},$ where $E\\left( {t,\\tau } \\right)$ is the intracavity electric field.", "$t$ denotes the slow time with respect to consecutive roundtrips.", "$\\tau $ is a fast time defined in a reference frame traveling at the group velocity of $E$ .", "$t_R$ is the round-trip time relate to the cavity length $L$ .", "$\\alpha $ represents the linear cavity loss, $\\theta $ is the transmission coefficient, and the critical coupling state occurs when $\\alpha = \\theta $ .", "The phase detuning parameter ${\\delta _0}$ is chosen by scanning the CW laser wavelength.", "${\\beta _k}$ accounts for the HOD coefficients at the central frequency of the driving field.", "$\\gamma $ is the nonlinear coefficient.", "The moment method is an extensively used approach to describe pulse dynamics in optical fibers governed by the nonlinear Schr$\\ddot{\\rm o}$dinger equation [16], [17].", "Since Eq.", "(REF ) is formally similar to the Schr$\\ddot{\\rm o}$dinger equation, we herein also use this method to analyze the comb dynamics.", "For an optical resonator, the dynamics of final steady-state pulses are mainly focused because the initial evolution stage is fairly chaotic [18].", "In the moment method,pulse energy ($E_p$ ), temporal position ($T_p$ ) and frequency shift ($\\Omega _p$ ) are calculated with the following set of moments $&{E_p} = \\int _{ - \\infty }^{ + \\infty } {{{\\left\| E \\right\|}^2}d\\tau }, {T_p} = \\frac{1}{{{E_p}}}\\int _{ - \\infty }^{ + \\infty } {\\tau {{\\left\| E \\right\|}^2}d\\tau }, \\\\&{\\Omega _p} = \\frac{i}{{2{E_p}}}\\int _{ - \\infty }^{ + \\infty } {\\left( {{E^}\\frac{{\\partial E}}{{\\partial \\tau }} - E\\frac{{\\partial {E^}}}{{\\partial \\tau }}} \\right)} d\\tau .", "$ Substituting Eqs.", "(REF ) and () into Eq.", "(REF ) and after some algebra, we get $&\\frac{{d{E_p}}}{{dt}} = 0, \\frac{{d{\\Omega _p}}}{{dt}} = 0, \\\\&\\frac{{d{T_p}}}{{dt}} = \\sum \\limits _{k \\ge 1} {\\left( {\\int _{ - \\infty }^{ + \\infty } {\\frac{{L{\\beta _{2k + 1}}}}{{\\left( {2k + 1} \\right)!", "}}{{\\left\| {\\frac{{{\\partial ^k}E}}{{\\partial \\tau }}} \\right\|}^{2k}}d\\tau } } \\right)} + \\\\&{\\rm {~~~~~~~~~~}}\\sum \\limits _{k \\ge 1} {\\left( {\\int _{ - \\infty }^{ + \\infty } {\\frac{{iL{\\beta _{2k}}}}{{\\left( {2k} \\right)!", "}}\\left( {{\\Theta _k} \\left( {t,\\tau } \\right)} \\right)d\\tau } } \\right)}, $ where ${\\Theta _k} \\left( {t,\\tau } \\right) = \\frac{{{\\partial ^{k - 1}}{E^}}}{{\\partial \\tau }}\\frac{{{\\partial ^k}E}}{{\\partial \\tau }} - \\frac{{{\\partial ^{k - 1}}E}}{{\\partial \\tau }}\\frac{{{\\partial ^k}{E^}}}{{\\partial \\tau }}.$ Evidently, the frequency shift with respect to the slow time $t$ should be a constant from Eq.", "(REF ).", "Hence, $\\Omega _p = 0$ must be fulfilled for a stable frequency comb generation with fixed frequencies.", "Furthermore, using the following relations $&{\\mathcal {F}}\\left[ {\\frac{{{\\partial ^k}E\\left( {t,\\tau } \\right)}}{{\\partial {\\tau ^k}}}} \\right] = {\\left( { - i\\omega } \\right)^k}\\tilde{E}\\left( {t,\\omega } \\right),k = 1,2,3, \\ldots .\\, \\\\&\\frac{{\\partial \\left[ {{{\\left( { - i\\omega } \\right)}^k}\\tilde{E}\\left( {t,\\omega } \\right)} \\right]}}{{\\partial t}} = 0, $ the frequency shift of the k-th order derivation of $E\\left( {t,\\tau } \\right)$ should be zero as well, therefore yields ${\\Theta _k}\\left( {t,\\tau } \\right) = 0$ .", "Eventually, Eq.", "() can be reduced to $\\frac{{d{T_p}}}{{dt}} = \\sum \\limits _{k \\ge 1} {\\left( {\\int _{ - \\infty }^{ + \\infty } {\\frac{{L{\\beta _{2k + 1}}}}{{\\left( {2k + 1} \\right)!", "}}{{\\left\| {\\frac{{{\\partial ^k}E}}{{\\partial \\tau }}} \\right\|}^{2k}}d\\tau } } \\right)}.$ Equation (REF ) clearly indicates that the temporal shifts of intracavity solitons are exclusively determined by high-odd-order dispersion (HOOD) rather than high-even-order dispersion (HEOD).", "In order to validate the above theoretical predictions, we numerically solve Eq.", "(REF ) by exploiting the split-step Fourier method.", "The resonator parameters used here are same with that of a recently common used silicon nitride microresonator [10], [15].", "Typical simulation results are shown in Fig.", "REF , in which the resonators are all operated in the anomalous GVD regime, and the HOD terms are expanded up to five order.", "Due to the intracavity initial modulation instability, the incident field firstly undergoes a chaotic period, then periodic pulse series are generated, and a Kerr frequency comb with arbitrary phase relation is formed in frequency domain [19].", "If we suitably tune the parameter $\\delta _0$ , the overall field will evolve towards the final stage consisting of a single soliton superimposed upon a weak CW background.", "As a consequence, the steady-state frequency comb is formed with definite phase relation, however, from the perspective of temporal domain, the intracavity solitons continuously drift along the fast time axis $\\tau $ due to the HOD influence.", "Specifically, if we consider only GVD term in Fig.", "REF (a) or GVD together with $\\beta _4$ in Fig.", "REF (c), the solitons evolve without any temporal drift.", "Whereas the temporal shifts can be clearly observed when the HOODs are taken into account, as shown in Fig.", "REF (b) ($\\beta _2$ and $\\beta _3$ ) and Fig.", "REF (d) ($\\beta _2$ and $\\beta _5$ ).", "These features are well understood from the theoretical predictions.", "Figure: Schematic PM relations in different regions in (β 3 \\beta _3, β 4 \\beta _4) space.", "The gray line in the inset denotes the boundary of different PM features.", "β 2 \\beta _2 is -2×10 -2 ps 2 /m-2\\times {10^{-2}}{\\rm {~p}}{{\\rm {s}}^2/{\\rm m}}; \|β 3 \|\|\\beta _3\|, \|β 4 \|\|\\beta _4\| are 4×10 -4 ps 3 /m4\\times {10^{-4}}{\\rm {~p}}{{\\rm {s}}^3/{\\rm m}} and 8×10 -6 ps 4 /m8\\times {10^{-6}}{\\rm {~p}}{{\\rm {s}}^4/{\\rm m}}, respectively.", "A, B, D and E locate in (β 3 \\beta _3, β 4 \\beta _4) with different signs and C is (β 3 \\beta _3, 0).", "All PM lines in different colors are corresponding with the colors of A, B, C, D, E in the inset.Next we focus the HOD influence on the envelopes of Kerr frequency combs.", "The ubiquitous phenomenon behind the HOD effect is dispersive wave generation, also called optical Cherenkov radiation (OCR), resulting in frequency-shifting spectra generated from solitons [20], [21], [22], [23].", "As reported in [15], OCR can also be achieved in microresonators when the phase matching (PM) is satisfied between the intracavity soliton and the linear radiation.", "The PM condition becomes ${\\rm PM}\\left( \\Omega \\right) = \\sum \\limits _{k = 2}^4 {\\frac{{{\\beta _k}\\left( {{\\omega _0}} \\right)}}{{k!", "}}{{\\left( {\\omega - {\\omega _0}} \\right)}^k}}$ with a negligible soliton wave number, i.e., only considering $\\beta _2$ , $\\beta _3$ and $\\beta _4$ and removing the nonlinear term.", "Here $\\Omega $ is the frequency difference between the pump frequency $\\omega $ and the frequency of dispersive wave $\\omega _0$ .", "Actually, the possible wavelengths of the OCRs are the real roots satisfying ${\\rm PM} = 0$ .", "The generated PM topology shown in the inset of Fig.", "REF indicating that two different regions are divided by the gray line in ($\\beta _3$ , $\\beta _4$ ) plane.", "Specifically, on the right side of the gray line, single or double real roots exist, such as A, C and D, while no zero PM wavelength exists on the left side, such as points of B and E. Their corresponding colored PM lines are depicted in Fig.", "REF .", "By solving Eq.", "REF employing the dispersion parameters in Fig.", "REF , the stable frequency combs and intracavity solitons are obtained, as shown in Fig.", "REF .", "Obviously, the smooth temporal and spectral envelopes can be seen in the cases of B and E in Fig.", "REF due to no existence of OCRs.", "Instead, For the dispersion curves of A, C and D in Fig.", "REF , both the complex spectra and temporal sidebands arise.", "Either advance or delay of a sideband in the temporal domain is governed by the perturbations of the generated OCRs to the intracavity soliton.", "Double sideband oscillations appear for B and D and single oscillation can be found in C due to the PM conditions.", "Figure: Final steady-state (a) temporal and (b) spectral envelopes calculated based on same-color dispersion curves shown in Fig.", ".On the other hand, the opposite sign of $\\beta _3$ (e.g., from A (B) to D (E) in Fig.", "REF ) leads to the mirror-symmetry envelopes in both temporal and spectral domains.", "Nevertheless, comparing A (D) with B (E), we find that the sign of $\\beta _4$ dominates the switching of the OCRs.", "Further expanding the dispersion to higher orders are introduced in [22], they found that the HEODs cause either a pair of conjugate OCRs or no OCR at all, whilst the HOODs merely lead to a single OCR.", "As a result, the HEODs unlikely induce any asymmetry to the temporal profile of a soliton, but the HOODs are possible.", "The asymmetry of temporal pulse shape leads to a temporal shift of the intracavity solitons, which is same behaviors with an asymmetric Airy-shaped pulse accelerating during propagation [24].", "It should be noted here that all of the final intracavity solitons shift somehow along $\\tau $ axis, but all of them are shifted to the peak positions for a well comparison of the envelopes in Fig.", "REF (a).", "Figure: Engineering the spectra of ultrabroadband frequency combs by tuning the GVD profiles with (a) three and (b) four ZDWs.", "The corresponding group velocity profiles are (c) and (d) as well as the PM conditions in (e) and (f), respectively.", "(g) and (h) are the resultant comb spectra.Furthermore, the spectrum of an ultrabroadband frequency comb can be engineered by suitably tailoring the dispersion profiles if including more HOD terms.", "Two typical GVD profiles with multiple ZDWs are presented here due to their rich PM relations, as shown in Figs.", "REF (a) and REF (b).", "Note here the HOD terms are expanded up to 10-th order.", "Three and four ZDWs can be found due to the strong waveguide dispersion counterbalancing the material dispersion, as seen in Ref.", "[11], [25].", "The group-velocity (GV) matching between the soliton and the radiation waves are given in Figs.", "REF (c) and REF (d) and their corresponding PM curves in Figs.", "REF (e) and REF (f), respectively.", "For a GVD profile with three ZDWs, the GV of the intracavity soliton (i.e., pink point) is equal to that of the OCR at longer wavelength (i.e., blue point), while it is faster than another OCR at shorter wavelength (i.e, black point).", "The GV matching between the soliton and the OCR allows sufficient interaction between the soliton and the OCR, leading to a class of broadband, highly efficient OCR, as shown in Fig.", "REF (g).", "The OCR at longer wavelength is broader, more efficient than that at shorter wavelength.", "Similar features are found in case of a GVD profile exhibiting four ZDWs, in which a GV-matching OCR promotes a wide range of spectra between the pump wavelength and narrow band OCR beyond $2.2~\\mu m$ .", "In both cases, considerable energy from the first anomalous GVD range is transferred into the second anomalous GVD region at longer wavelength, so that the overall comb envelopes expand to even an octave spectral range with high flatness.", "In summary, the HOD influence on both temporal intracavity solitons and spectral characterizations of the ultrabroadband microresonator-based optical frequency combs have been studied.", "We find that the temporal shifts of the intracavity solitons are merely determined by the HOOD terms, which are well understood by exploiting the moment method.", "Roles of the HOD terms in affecting the envelopes of the Kerr frequency combs are also analyzed by explaining the intracavity OCRs.", "Finally, two typical GVD profiles with multiple ZDWs by tuning the HODs are given to engineer the overall frequency envelopes of Kerr combs.", "Our analysis will give insight for future exploring spectral properties of ultrabroadband optical frequency combs.", "This work was supported in part by National Natural Science Foundation of China (11274224), Shanghai Shuguang Program (10SG38) and the open program (2013GZKF031307) from State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University.", "Xianglong Zeng acknowledges the Research support by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning." ] ]	1403.0183
[ [ "Why the Quantitative Condition Fails to Reveal Quantum Adiabaticity" ], [ "Abstract The quantitative adiabatic condition (QAC), or quantitative condition, is a convenient (a priori) tool for estimating the adiabaticity of quantum evolutions.", "However, the range of the applicability of QAC is not well understood.", "It has been shown that QAC can become insufficient for guaranteeing the validity of the adiabatic approximation, but under what conditions the QAC would become necessary has become controversial.", "Furthermore, it is believed that the inability for the QAC to reveal quantum adiabaticity is due to induced resonant transitions.", "However, it is not clear how to quantify these transitions in general.", "Here we present a progress to this problem by finding an exact relation that can reveal how transition amplitudes are related to QAC directly.", "As a posteriori condition for quantum adiabaticity, our result is universally applicable to any (nondegenerate) quantum system and gives a clear picture on how QAC could become insufficient or unnecessary for the adiabatic approximation, which is a problem that has gained considerable interest in the literature in recent years." ], [ "Introduction", "The quantum adiabatic theorem (QAT) , suggests that a physical system initialized in an eigenstate $\\left\| {{E_n}( t =0 )} \\right\\rangle $ (commonly the ground state) of a certain gapped time-dependent Hamiltonian $H(t)$ , with an eigenvalue $E_n$ , at time $t$ remains in the same instantaneous eigenstate (up to a multiplicative phase factor), provided that the Hamiltonian $H(t)$ varies in a continuous and sufficiently slow way.", "The adiabatic theorem was first proposed by Born and Fock at the dawn of quantum mechanics , who were motivated by the idea of adiabatic invariants of Ehrenfest .", "Born and Fock's result is restricted to bounded Hamiltonians with discrete energy levels, e.g.", "1D harmonic oscillators; their result is not applicable to systems with a continuous spectrum e.g.", "Hydrogen atom.", "This restriction was relaxed by Kato in 1950 , who found that in the adiabatic limit, the time evolution of a time-dependent Hamiltonian is equivalent to a geometric evolution.", "Kato's result is applicable to systems including Hydrogen atom, where the ground state is unique and has a gap from the excited states that can have degeneracy.", "Later, the requirement of the existence of a gap for proving the adiabatic theorem was found to be unnecessary .", "This intriguing physical property of quantum adiabaticity finds many interesting applications, including but not limited to quantum field theory , geometric phase , stimulated Raman adiabatic passage (STIRAP) , energy level crossings in molecules , , adiabatic quantum computation , , , , , , , quantum simulation (see e.g.", "the review ), and other applications .", "Despite its long history, the study of the QAT is still a very active field of research.", "Many works have been performed aiming to achieve a better understanding of the adiabatic theorem.", "In particular, the problem of quantifying the slowness of adiabatic evolution has not been completely solved.", "Traditionally , , , the so-called (e.g.", "see Ref. )", "quantitative adiabatic condition (QAC) or simply quantitative condition (for all $m\\ne n$ ): $\\left\|\\frac{\\langle E_{m}(t)\|\\dot{E}_{n}(t)\\rangle }{E_{m}(t)-E_{n}(t)}\\right\|\\ll 1, $ was meant to quantify the slowness of $H(t)$ (see for details on the definitions of the Hamiltonian and eigenvectors).", "However, QAC was numerically shown to be not a good indicator for revealing the fidelity of the final state .", "Furthermore, it has been shown that QAC is inconsistent with the QAT and insufficient for maintaining the validity of the adiabatic approximation , except for some special cases .", "The arguments for showing the inconsistency and insufficiency of QAC were constructed from a comparison between two systems, A and B, where A was evolved under a Hamiltonian $H_a(t)$ .", "The Hamiltonian ${H_b}\\left( t \\right) = - {U_a}{\\left( t \\right)^ \\dagger }{H_a}\\left( t \\right)U\\left( t \\right)$ of system B is related to that of system A through a unitary transformation ${U_a}\\left( t \\right) = T\\exp ( { - i\\int _0^t {{H_a}\\left( {t^{\\prime }} \\right)dt^{\\prime }} } )$ that corresponds to the exact propagator of $H_a(t)$ .", "It was shown that both systems A and B satisfy the QAC, but only one of them can fulfill the adiabatic approximation.", "This conclusion is consistent with the results performed in an NMR experiment ." ], [ "Related studies in the literature", "Many studies (e.g.", ", , , , , , ) have been made trying to understand the inconsistency raised by Marzlin and Sanders .", "It was argued , , , , that resonant transitions between energy levels are responsible for the violations of the adiabatic theorem.", "A refined adiabatic condition has been found , which takes into account the effects of resonant energy-level transitions.", "On the other hand, the validity of the adiabatic theorem was analyzed from a perturbative-expansion approach , , which provides a diagrammatic representation for adiabatic dynamics and yields the quantitative condition (in Eq.", "(REF )) as the first-order approximation.", "It was argued that the quantitative condition is insufficient for the adiabatic approximation when the Hamiltonian varies rapidly but with a small amplitude.", "Furthermore, generalizing QAC for open quantum systems , and many-body systems have been achieved.", "Efforts for finding conditions that can replace QAC were made , , .", "Another line of research related to the adiabatic theorem is to estimate or bound the scaling of the final-state fidelity.", "Under some general conditions for a gapped Hamiltonian, it was found that the transition probability scales as $O(1/T^2)$ for a total evolution time $T$ .", "When the total time is fixed, it was shown , that both the minimum eigenvalue gap $\\Delta $ and the length of the traversed path $L \\equiv \\int _0^1 {\\left\\Vert {\\left\| {{\\partial _r}\\psi \\left( r \\right)} \\right\\rangle } \\right\\Vert } dr$ , where $r(t)$ is a time-varying parameter in the Hamiltonian, are important.", "Instead of questioning the validity of the quantitative condition as an indicator for quantum adiabaticity, we are interested in the question “Under what additional conditions would QAC become necessary?\".", "The answer to this question has not been clear , , .", "Our work is motivated by a recent development achieved in Ref.", ", where QAC is argued to be necessary under certain additional assumptions related to the adiabatic state $\|\\psi _{n}^{adi}(t)\\rangle $ , , which is defined by attaching a time-dependent phase factor (essentially the Berry phase ) $e^{i\\beta _{n}(t)}$ to the energy eigenstate $\\left\|{{E_{n}}\\left( t\\right) }\\right\\rangle $ , i.e., $\|\\psi _{n}^{adi}(t)\\rangle \\equiv e^{i\\beta _{n}(t)}\|E_{n}(t)\\rangle ,$ where $\\beta _{n}(t)\\equiv -\\int _{0}^{t}E_{n}(x)dx+i\\int _{0}^{t}\\langle E_{n}(x)\| \\dot{E}_{n}(x)\\rangle dx.$ The key result obtained in Ref.", "is that (in our notations) the probability amplitude ${c_{m}}\\left( t\\right) =\\left\\langle {{E_{m}}\\left( t\\right) }\\right\|\\left.", "{\\psi \\left( t\\right) }\\right\\rangle $ for the eigenstate $\|{{E_{m}}\\left( t\\right) }\\rangle $ at time $t$ is given by the following expression: ${c_{m}}\\left( t\\right) \\approx i{e^{i\\beta _{n}\\left( t\\right) }}\\frac{{\\langle {E_{m}(t)}\|{{{\\dot{E}}_{n}(t)}}\\rangle }}{{{E_{m}(t)}-{E_{n}(t)}}},$ which leads to the conclusion that if the adiabatic approximation is valid, i.e., the probability amplitude $c_{m}$ for all eigenstates $m\\ne n$ are small, $\\left\|{{c_{m}}\\left(t\\right) }\\right\|\\ll 1$ , then the quantitative adiabatic condition (cf Eq.", "(REF )) necessarily holds.", "For comparison, a similar expression (in our notation) was given by Schiff as ${c_m}\\left( t \\right) & \\approx & \\frac{{\\langle {E_m}(t)\|\\dot{H}\|{E_n}(t)\\rangle }}{{i{{\\left( {{E_m}(t) - {E_n}(t)} \\right)}^2}}}\\left( {{e^{i\\left( {{E_m} - {E_n}} \\right)t}} - 1} \\right),\\\\& = & i\\frac{{\\left\\langle {{E_m}\\left( t \\right)} \\right\| {{{\\dot{E}}_n}\\left( t \\right)} \\rangle }}{{{E_m}(t) - {E_n}(t)}}\\left( {{e^{i\\left( {{E_m} - {E_n}} \\right)t}} - 1} \\right).$ The derivation from the first line to the second line is provided in .", "These two expressions (in Eq.", "(REF ) and Eq.", "REF ) predict the validity of the adiabatic approximation when the quantitative condition (cf Eq.", "(REF )) is satisfied.", "However, the result in Ref.", "was not uncontroversial , .", "Zhao and Wu argued that the contribution of the missing term in the result in Ref.", "is underestimated.", "Comparat pointed out that the non-rigorous use of the approximation sign `$\\approx $ ' in Ref.", "leads to an obscure meaning for quantum adiabaticity.", "This problem is avoided in our derivation.", "Tong's reply emphasized the connection with the adiabatic state in his result, but did not resolve the oppositions completely.", "Table: References" ] ]	1403.0069
[ [ "A New Model for Solving Narrow Escape Problem in Domain with Long Neck" ], [ "Abstract The narrow escape problem arises in deriving the asymptotic expansion of the solution of an inhomogeneous mixed Dirichlet-Neumann boundary value problem.", "In this paper, we mainly deal with narrow escape problem in a smooth domain connected to a long neck-Dendritic spine shape domain, which has a certain significance in biology.", "Since the special geometry of dendritic spine, we develop a new model for solving this narrow escape problem which is Neumann-Robin Boundary Model.", "This model transform spine singular domain to smooth spine head domain by inserting Robin boundary condition to the connection part between spine head and neck.", "We rigorously find the high-order asymptotic expansion of Neumann-Robin Boundary Model and apply it to the solution of narrow escape problem in a dendritic spine shape domain.", "Our results show that the asymptotic expansion of the Neumann-Robin Boundary Model can be easily applied to the narrow escape problem for any smooth spine head domain with straight spine neck.", "By numerical simulations, we show that there is great agreement between the results of our Neumann-Robin Boundary Model and the original escape problem.", "In this paper, we also get some results for non-straight long spine neck case by considering curvature of spine neck." ], [ "Introduction", "When a Brownian particle is confined in a bounded domain with a small absorbing windows on an otherwise reflecting boundary, it attempts to escape from this domain through this small absorbing windows.", "Narrow escape problem is to calculate the mean first passage time Brownian particle takes to get to the absorbing window.", "From the biological point of view, the Brownian particles could be diffusing ions, globular proteins or cell-surface receptors.", "It is then of interest to determine, for example, the mean time that an ion requires to find an open ion channel located in the cell membrane or the mean time of a receptor to hit a certain target binding site.", "In two dimension, the results of narrow escape problem in smooth bounded domain with one absorbing window was relatively complete , , , , , , , .", "When there are several absorbing windows on the boundary, interaction of multiple absorbing windows are discussed in , .", "In paper , several kinds of singular domains have been discussed.", "In three dimension, the case that bounded domain is a ball with spherical boundary has been discussed in , .", "While, our interest is different from these talked above, but another example of singular domain that a smooth domain connected by a long neck, such as dendritic spine(Fig.REF ).", "Dendritic spines serve as a storage site for synaptic strength and help transmit electrical signals to the neuron's cell body.", "As the important site of excitatory synaptic interaction, dendritic spines play an important role in neural plasticity, and their ability to regulate calcium attracts interests of many mathematicians and biologists , , , , .", "Each spine has a bulbous head, and a thin neck that connects the head of the spine to the shaft of the dendrite.", "We consider simplified model of calcium diffusion in dendritic spines, which is discussed in .", "That is, first we consider the calcium irons to be point charges, furthermore, we assume the motion of irons is free Brownian motion; second, the interaction between two electrostatic ions is neglected; third, we shall simply ignore impenetrable obstacles to the ionic motion posed by the presence of proteins.", "Thus, the iron motion inside the dendritic spine is geometrically unrestricted.", "In this paper, we regard the iron as calcium molecule.", "The calcium diffusion problem is narrow escape problem that is approximated by free Brownian motion in a domain which consists of a spherical head whose length is $L$ and a long cylindrical neck whose radius is $a$ , where the radius of the neck is sufficiently small relative to that of the spine head(Fig.REF ).", "In this paper, we only talk about two dimensional case, where spine head is a bounded domain with smooth boundary and spine neck is rectangle, while, Neumann-Robin Boundary Model(REF ) can be easily applied to three dimensions.", "The narrow escape problem can be explained explicitly in the following way.", "Let $\\Omega $ be a bounded simply connected domain in $\\mathbb {R}^{2}$ .", "Suppose that $\\partial \\Omega $ is decomposed into the reflecting part $\\partial \\Omega _{r}$ and the absorbing part $\\partial \\Omega _{a}$ .", "We assume that $\\varepsilon =\|\\partial \\Omega _{a}\|/2$ is much smaller than the whole boundary(Fig.REF ).", "The narrow escape problem is to calculate the mean first passage time which is the solution $u_{\\varepsilon }$ to (REF ), Figure: Abstract graph of dendritic spineFigure: The modeling shape of dendritic spine with disk spine head and long spine neck, where Ω\\Omega is the domain with long neck, ∂Ω r \\partial \\Omega _{r} is the reflection part, ∂Ω a \\partial \\Omega _{a} is the absorbing part.${\\left\\lbrace \\begin{array}{ll}\\triangle u_{\\varepsilon }=-1,\\quad &\\mbox{in}~\\Omega ,\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }=0,&\\mbox{on}~\\partial \\Omega _{r},\\\\u_{\\varepsilon }=0,&\\mbox{on}~\\partial \\Omega _{a}.\\end{array}\\right.", "}$ The asymptotic analysis for narrow escape problem arises in deriving the asymptotic expansion of $u_{\\varepsilon }$ as $\\varepsilon \\rightarrow 0$ , from which one can estimate the escape time of the Brownian particle.", "In this work, instead of (REF ), we develop another proper model to solve narrow escape problem in dendrite spine shape domain which we call it Neumann-Robin Boundary Model.", "The model is described by the following equations in domain $\\Omega _h$ (Fig.REF ), ${\\left\\lbrace \\begin{array}{ll}\\triangle u_{\\varepsilon }=-1,\\quad &\\mbox{in}~\\Omega _h,\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }=0,&\\mbox{on}~\\partial \\Omega _{r},\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }+\\alpha u_{\\varepsilon }=\\beta ,&\\mbox{on}~\\Gamma _\\varepsilon :=\\partial \\Omega _h \\setminus \\overline{\\partial \\Omega _{r}}.\\end{array}\\right.", "}$ where $\\Omega _h$ is the spine head of $\\Omega $ mentioned in Fig.REF , the size of $\\Gamma _\\varepsilon $ is still small with $\|\\Gamma _\\varepsilon \|=2\\varepsilon $ , but it is not just an absorbing boundary any more.", "Note that, the domain $\\Omega $ we considered here is only the head domain, without the neck.", "Figure: The domain considered in Neumann-Robin model, which omit the long neck by adding Robin boundary condition to small arc Γ ε \\Gamma _\\varepsilon , ∂Ω r \\partial \\Omega _{r} still represents reflecting boundary.In this paper, we analyze the asymptotic behavior of the solution of Neumann-Robin Boundary Model in the domain $\\Omega _h$ (see Fig.REF ) in two dimensions.", "Actually, the asymptotic behavior can be applied to any smooth bounded domain in two dimensions, not only disk case.", "We rigorously derived the expansion formula for (1.2) up to order $O(\\varepsilon )$ $u_{\\varepsilon }(x)=\\frac{\|\\Omega \|}{2\\alpha \\varepsilon }+\\frac{\|\\Omega \|}{\\pi }\\left(\\frac{3}{2}+\\ln \\frac{1}{2\\varepsilon }\\right)+ \\frac{\\beta }{\\alpha }+\\Phi _{\\Omega }(x,x^)+O(\\varepsilon ).$ where $\\Phi _{\\Omega }(x,x^)$ can be referred to (REF ).", "By assigning specific $\\alpha =1/L$ , $\\beta =L/2$ to our Neumann-Robin Boundary Model(see the reason for choice of $\\alpha ,\\beta $ in section 4), the solution formula (1.3) can approximate the mean first passing time of narrow escape problem (1.1) in spine domain (Fig.REF ) up to order $O(\\varepsilon )$ .", "The numerical results show a great agreement between them.", "This paper is organized as follows.", "In Section 2, we review the Neumann function for Laplacian and introduce an integral operator for further calculations.", "In section 3 the asymptotic formula for the solution to Neumann-Robin Boundary Model has been rigorously derived by using layer potential techniques.", "In section 4 and 5, we discuss how the Neumann-Robin Boundary Model corresponds to the original escape problem theoretically and numerically.", "The paper ends with a short conclusion." ], [ "Preliminaries", "Let $N(x,z)$ be the Neumann function for $-\\triangle $ in $\\Omega $ corresponding to a Dirac mass at $z\\in \\Omega $ .", "We assume $\\partial \\Omega $ is $C^2$ smooth.", "$N(x,z)$ is the solution to ${\\left\\lbrace \\begin{array}{ll}\\triangle _{x} N(x,z)=-\\delta _{z}, &x\\in \\Omega ,\\\\\\displaystyle \\frac{\\partial N}{\\partial \\nu _{x}}=-\\frac{1}{\|\\partial \\Omega \|}, & x\\in \\partial \\Omega ,\\\\\\end{array}\\right.", "}$ For uniqueness, we assume $\\int _{\\partial \\Omega }N(x,z)d\\sigma (x)=0$ .", "If $z\\in \\Omega $ , $N(x,z)$ can be written in the form $N(x,z)=-\\frac{1}{2\\pi }\\ln \|x-z\|+R_{\\Omega }(x,z),\\quad x\\in \\Omega ,$ where $R_{\\Omega }(x,z)$ is the regular part which belongs to $H^{3/2}(\\Omega )$ , and solves ${\\left\\lbrace \\begin{array}{ll}-\\triangle _{x} R_{\\Omega }(x,z)=0,&x\\in \\Omega ,\\\\\\displaystyle \\frac{\\partial R_{\\Omega }}{\\partial \\nu _{x}}\\Big \|_{x\\in \\partial \\Omega }=-\\frac{1}{\|\\partial \\Omega \|}+\\frac{1}{2\\pi }\\frac{\\langle x-z,\\nu _{x}\\rangle }{\|x-z\|^{2}},&x\\in \\partial \\Omega .\\end{array}\\right.", "}$ If $z\\in \\partial \\Omega $ , Neumann function on the boundary $N_{\\partial \\Omega }$ , can be written as $ N_{\\partial \\Omega }(x,z)=-\\frac{1}{\\pi }\\ln \|x-z\|+R_{\\partial \\Omega }(x,z),\\quad x \\in \\Omega ,z \\in \\partial \\Omega , $ where the singularity of $N_\\partial \\Omega (x,z)$ is $-\\frac{1}{\\pi }\\ln \|x-z\|$ (See ), $R_{\\partial \\Omega }(x,z)$ solves the problem ${\\left\\lbrace \\begin{array}{ll}\\triangle _{x}R_{\\partial \\Omega }(x,z)=0,& x\\in \\Omega ,\\\\\\displaystyle \\frac{\\partial R_{\\Omega }}{\\partial \\nu _{x}}\\Big \|_{x\\in \\partial \\Omega }=-\\frac{1}{\|\\partial \\Omega \|}+\\frac{1}{\\pi }\\frac{\\langle x-z,\\nu _{x}\\rangle }{\|x-z\|^{2}},&x\\in \\partial \\Omega , ~z\\in \\partial \\Omega .\\\\\\end{array}\\right.", "}$ Note that the Neumann data above is bounded on $\\partial \\Omega $ uniformly in $z\\in \\partial \\Omega $ since $\\partial \\Omega $ is $C^{2}$ -smooth, and hence $R_{\\partial \\Omega }(x,z)$ belongs to $H^{3/2}(\\Omega )$ uniformly in $z\\in \\partial \\Omega $ .", "(See ) For later use, we introduce the integral operator $L:L^2[-1,1]\\rightarrow L^2[-1,1]$ , defined by $L[\\phi ](x)=\\int \\limits _{-1}^{1}\\ln \|x-y\|\\phi (y)dy.$ We can see operator $L$ is bounded (see Lemma 2.1 in )." ], [ "Neumann-Robin boundary value problem", "In this section we rigorously give the asymptotic analysis of our Neumann-Robin Boundary Model in a more general spine head domain $\\Omega _h$ (Fig.", "REF ), where $\\Omega _h\\in C^2(R^2)$ , and derive full expansion solution formula for (REF ) in domain $\\Omega _h$ up to order $O(\\varepsilon )$ .", "Figure: Any smooth domain with small opening Γ ε \\Gamma _\\varepsilon .", "This domain is where Neumann-Robin Boundary Model considered.We consider the Laplace equation in $\\Omega _h$ with the mixed Neumann-Robin boundary condition.", "The Robin boundary condition is imposed on $\\Gamma _{\\varepsilon }$ ($\\Gamma _{\\varepsilon }$ is a very small part) and the Neumann boundary condition on the part $\\partial \\Omega _{r}:= \\partial \\Omega _h \\setminus \\overline{\\Gamma _\\varepsilon }$ : ${\\left\\lbrace \\begin{array}{ll}\\triangle u_{\\varepsilon }=-1, &\\mbox{in}~\\Omega _h,\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }=0,&\\mbox{on}~\\partial \\Omega _{r},\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }+\\alpha u_{\\varepsilon }=\\beta ,&\\mbox{on}~\\Gamma _{\\varepsilon }.\\end{array}\\right.", "}$ Here, $\\alpha >0$ and $\\beta $ are given constants.", "We assume that $\\varepsilon $ is sufficiently small so that $\\alpha \\varepsilon \\ll 1$ and $\\alpha < \\alpha _0$ for some constant $\\alpha _0>0$ .", "The goal in this section is to derive the asymptotic expansion of $u_{\\varepsilon }$ as $\\varepsilon \\rightarrow 0$ , from which one can estimate the exit time of the calcium iron in the spine head.", "By integrating the first equation in (3.1) over $\\Omega _h$ using the divergence theorem we get $\\int _{\\Gamma _{\\varepsilon }}\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }d\\sigma =-\|\\Omega _h\|.$ Let us define $g(x)$ by $g(x)=\\int _{\\Omega _h}N(x,z)dz,\\quad x\\in \\Omega _h,$ which satisfies ${\\left\\lbrace \\begin{array}{ll}\\triangle g=-1,& \\mbox{in}~\\Omega _h,\\\\\\displaystyle \\frac{\\partial g}{\\partial \\nu }=-\\frac{\|\\Omega _h\|}{\|\\partial \\Omega _h\|}, & \\mbox{on}~ \\partial \\Omega _h,\\\\\\displaystyle \\int _{\\partial \\Omega _h} g d\\sigma =0.&\\end{array}\\right.", "}$ Therefore, applying the Green's formula to $u_\\varepsilon $ and the Neumann function $N$ and using (REF ) and (REF ), we get $u_{\\varepsilon }(x)=g(x) + \\int _{\\Gamma _{\\varepsilon }}N_{\\partial \\Omega _{h}}(x,z)\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu _{z}}d\\sigma (z)+C_{\\varepsilon },$ where $C_{\\varepsilon }=\\frac{1}{\|\\partial \\Omega _h\|}\\int _{\\partial \\Omega _h}u_{\\varepsilon }(z)d\\sigma (z).$ By (REF ), the equation (REF ) becomes $u_{\\varepsilon }(x)=g(x)-\\frac{1}{\\pi }\\int _{\\Gamma _{\\varepsilon }}\\ln \|x-z\|\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu }d\\sigma (z)+ \\int _{\\Gamma _{\\varepsilon }}R_{\\partial \\Omega _h}(x,z)\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu }d\\sigma (z)+C_{\\varepsilon }.\\\\$ On $\\Gamma _\\varepsilon $ , by Robin boundary condition, (3.5) can be written as $\\frac{1}{\\pi }\\int _{\\Gamma _{\\varepsilon }}\\ln \|x-z\|\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu _{z}}d\\sigma (z)- \\int _{\\Gamma _{\\varepsilon }}R_{\\partial \\Omega _h}(x,z)\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu _{z}}d\\sigma (z)=\\frac{1}{\\alpha }\\frac{\\partial u_{\\varepsilon }(x)}{\\partial \\nu _{x}}-\\frac{\\beta }{\\alpha }+g(x)+C_{\\varepsilon }.$ Let $x(t):[-\\varepsilon ,\\varepsilon ]\\rightarrow \\mathbb {R}^{2}$ be the arc-length parametrization of $\\Gamma _{\\varepsilon }$ , i.e., $\|x^{\\prime }(t)\|=1$ for all $t \\in [-\\varepsilon ,\\varepsilon ]$ and $\\Gamma _{\\varepsilon }=\\lbrace \\,x(t)~\|~t\\in [-\\varepsilon ,\\varepsilon ]\\,\\rbrace .$ For simplicity, we let $f(t)=g(x(t)), \\quad \\phi _{\\varepsilon }(t)=\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }(x(t)), \\quad r(t,s)=R_{\\partial \\Omega _h}(x(t),x(s)).$ Then it follows from (REF ) that $\\frac{1}{\\pi }\\int _{-\\varepsilon }^{\\varepsilon }\\ln \|\\,x(t)-x(s)\|\\phi _{\\varepsilon }(s)ds- \\int _{-\\varepsilon }^{\\varepsilon }r(t,s)\\phi _{\\varepsilon }(s)ds=\\frac{1}{\\alpha }\\phi _{\\varepsilon }(t)+f(t)+C_{\\varepsilon }-\\frac{\\beta }{\\alpha }.$ By the change of variable, we obtain $\\frac{1}{\\alpha \\varepsilon }\\widetilde{\\phi }_{\\varepsilon }(t)-\\frac{1}{\\pi }\\int _{-1}^{1}\\ln \|x(\\varepsilon t)-x(\\varepsilon s)\|\\widetilde{\\phi }_{\\varepsilon }(s)ds+ \\int _{-1}^{1}r(\\varepsilon t,\\varepsilon s)\\widetilde{\\phi }_{\\varepsilon }(s)ds=-f(\\varepsilon t)-C_{\\varepsilon }+\\frac{\\beta }{\\alpha },$ where $\\widetilde{\\phi }_{\\varepsilon }(t)=\\varepsilon \\phi _{\\varepsilon }(\\varepsilon t)$ .", "We define two bounded integral operators $L,~ L_1: L^2[-1,1]\\rightarrow L^2[-1,1]$ by $ L[\\phi ]&=\\int _{-1}^1 \\ln \|t-s\|\\phi (s)ds,\\\\L_{1}[\\phi ]&=\\frac{1}{\\varepsilon }\\int _{-1}^{1}\\left(\\ln \\frac{\|x(\\varepsilon t)-x(\\varepsilon s)\|}{\\varepsilon \|t-s\|}+\\pi r(0,0)-\\pi r(\\varepsilon t,\\varepsilon s)\\right)\\phi (s)ds.$ Since $\|x(\\varepsilon t)-x(\\varepsilon s)\|=\\varepsilon \|t-s\|(1+O(\\varepsilon )),$ one can see that $L_{1}$ is bounded independently of $\\varepsilon $ .", "Using the compatibility condition $ \\int _{-1}^{1}\\widetilde{\\phi }_{\\varepsilon }(t)dt=-\|\\Omega _h\|,$ we may write (REF ) as $\\frac{1}{\\alpha \\varepsilon }\\widetilde{\\phi }_{\\varepsilon }(t)-\\frac{1}{\\pi }(L+\\varepsilon L_{1})[\\widetilde{\\phi }_{\\varepsilon }](t)=-\\frac{\|\\Omega _h\|\\ln \\varepsilon }{\\pi }+r(0,0)\|\\Omega _h\|-f(\\varepsilon t)-C_{\\varepsilon }+\\frac{\\beta }{\\alpha }.$ Assume $\\alpha < \\alpha _0$ and $\\alpha \\varepsilon \\ll 1$ .", "Then we have $(I-\\frac{\\alpha \\varepsilon }{\\pi }(L+\\varepsilon L_{1}))^{-1}=I+\\frac{\\alpha \\varepsilon }{\\pi }(L+\\varepsilon L_{1})+O(\\alpha ^2\\varepsilon ^{2}).$ Therefore we have $\\widetilde{\\phi }_{\\varepsilon }(t)&={-\\alpha \\varepsilon }\\left[I+\\frac{\\alpha \\varepsilon }{\\pi }(L+\\varepsilon L_{1})+O(\\alpha ^2\\varepsilon ^{2})\\right]\\left(C_{\\varepsilon }+\\frac{\|\\Omega _h\|\\ln \\varepsilon }{\\pi }- r(0,0)\|\\Omega _h\|+ f(\\varepsilon t)-\\frac{\\beta }{\\alpha }\\right)\\\\&={-\\alpha \\varepsilon }\\left[I+\\frac{\\alpha \\varepsilon }{\\pi }(L+\\varepsilon L_{1})+O(\\alpha ^2\\varepsilon ^{2})\\right]\\left(\\tilde{C}_{\\varepsilon }+O(\\varepsilon )\\right),$ where $\\tilde{C}_\\varepsilon =C_{\\varepsilon }+\\frac{\|\\Omega _h\|\\ln \\varepsilon }{\\pi }- r(0,0)\|\\Omega _h\|+ f(0)-\\frac{\\beta }{\\alpha }$ .", "By (REF ), we see $ \\tilde{C}_\\varepsilon = O((\\alpha \\varepsilon )^{-1})$ .", "Then collecting terms we have $\\widetilde{\\phi }_{\\varepsilon }(t)=-{\\alpha \\varepsilon }\\tilde{C}_\\varepsilon -\\frac{(\\alpha \\varepsilon )^2}{\\pi }\\tilde{C}_\\varepsilon L[1](t)+O(\\alpha \\varepsilon ^{2}).$ Plugging it into (REF ) we obtain $2{\\alpha \\varepsilon }\\tilde{C}_\\varepsilon +\\frac{(\\alpha \\varepsilon )^2}{\\pi }\\tilde{C}_\\varepsilon \\int _{-1}^1 L[1](t)dt = {\|\\Omega _h\|}+O(\\alpha \\varepsilon ^{2}).$ Then we get $\\tilde{C}_\\varepsilon &= \\left(1 +\\frac{\\alpha \\varepsilon }{2\\pi }\\int _{-1}^1 L[1](t)dt \\right)^{-1}\\left(\\frac{\|\\Omega _h\|}{2\\alpha \\varepsilon }+O(\\varepsilon )\\right)\\\\&=\\frac{\|\\Omega _h\|}{2\\alpha \\varepsilon }-\\frac{\|\\Omega _h\|}{4\\pi }\\int _{-1}^1 L[1](t)dt+O(\\varepsilon ).$ The direct calculation shows us $\\int _{-1}^{1}L[1](t)dt=\\int _{-1}^{1}\\int _{-1}^{1}\\ln \|t-y\|dtdy=4\\ln 2-6.$ Therefore we arrive at $\\tilde{C}_\\varepsilon =\\frac{\|\\Omega _h\|}{2\\alpha \\varepsilon }+\\frac{\|\\Omega _h\|}{\\pi }(\\frac{3}{2}-\\ln 2)+O(\\varepsilon ),$ and hence $C_{\\varepsilon }=\\frac{\|\\Omega _h\|}{2\\alpha \\varepsilon }+\\frac{\|\\Omega _h\|}{\\pi }(\\frac{3}{2}+\\ln \\frac{1}{2\\varepsilon })+ \\frac{\\beta }{\\alpha }+r(0,0)\|\\Omega _h\|-f(0)+O(\\varepsilon ).$ Substituting (REF ) into (REF ), we have $\\widetilde{\\phi }_{\\varepsilon }(t)=-\\frac{\|\\Omega _h\|}{2}-\\frac{\|\\Omega _h\|}{\\pi }\\alpha \\varepsilon \\left(\\frac{3}{2}-\\ln 2+\\frac{1}{2}L[1](t)\\right)+O(\\alpha \\varepsilon ^{2}),$ and $\\phi _{\\varepsilon }=\\frac{1}{\\varepsilon }\\widetilde{\\phi }_{\\varepsilon }(\\frac{t}{\\varepsilon })=-\\frac{\|\\Omega \|}{2\\varepsilon }-\\frac{\|\\Omega \|}{\\pi }\\alpha \\left(\\frac{3}{2}-\\ln 2+\\frac{1}{2}L[1]\\left(\\frac{t}{\\varepsilon }\\right)\\right)+O(\\alpha \\varepsilon ^{3/2}),$ where $O(\\alpha \\varepsilon ^{2})$ and $O(\\alpha \\varepsilon ^{3/2})$ are measured in $\\parallel \\cdot \\parallel _{L^2[-1,1]}$ and $\\parallel \\cdot \\parallel _{L^2[-\\varepsilon ,\\varepsilon ]}$ , respectively.", "If $x$ is away from $\\Gamma _{\\varepsilon }$ , i.e., $\\mbox{dist}(x,\\Gamma _{\\varepsilon })\\ge c$ for some constant $c>0$ , then $&\\int _{\\Gamma _{\\varepsilon }}N_{\\partial \\Omega _h}(x,z)\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu _{z}}d\\sigma (z)\\nonumber \\\\&=\\int _{-\\varepsilon }^\\varepsilon N_{\\partial \\Omega _h}(x,z(t))\\left(-\\frac{\|\\Omega _h\|}{2\\varepsilon }-\\frac{\|\\Omega _h\|}{\\pi }\\alpha \\left(\\frac{3}{2}-\\ln 2+\\frac{1}{2}L[1]\\left(\\frac{t}{\\varepsilon }\\right)\\right)+O(\\alpha \\varepsilon ^{3/2})\\right)dt\\nonumber \\\\&=-\|\\Omega _h\| N_{\\partial \\Omega _h}(x,x^)+O(\\epsilon ).$ Finally, combining (REF ), (REF ) and (REF ) yields $\\begin{split}u_{\\varepsilon }(x) &=g(x)+\\int _{\\Gamma _{\\varepsilon }}N_{\\partial \\Omega _h}(x,z)\\frac{\\partial u_{\\varepsilon }(z)}{\\partial \\nu _{z}}d\\sigma (z)+C_{\\varepsilon }\\\\&=g(x)-\|\\Omega \|N_{\\partial \\Omega _h}(x,x^)+\\frac{\|\\Omega _h\|}{2\\alpha \\varepsilon }+\\frac{\|\\Omega _h\|}{\\pi }(\\frac{3}{2}+\\ln \\frac{1}{2\\varepsilon })+ \\frac{\\beta }{\\alpha }+r(0,0)\|\\Omega _h\|-f(0)+O(\\varepsilon )\\end{split}$ for $x\\in \\Omega _h$ provided that dist(x,$\\Gamma _{\\varepsilon }$ )$\\ge c$ for some constant $c>0$ .", "Thus we have the following theorem.", "Theorem 3.1 Suppose that $\\Gamma _{\\varepsilon }$ is an arc of center $x^{}$ and length $2\\epsilon $ .", "Then the following asymptotic expansion of $u_{\\epsilon }$ for (3.1) holds $u_{\\varepsilon }(x)=\\frac{\|\\Omega _h\|}{2\\alpha \\varepsilon }+\\frac{\|\\Omega _h\|}{\\pi }\\left(\\frac{3}{2}+\\ln \\frac{1}{2\\varepsilon }\\right)+ \\frac{\\beta }{\\alpha }+\\Phi _{\\Omega _h}(x,x^)+O(\\varepsilon ),$ where $\\Phi _{\\Omega _h}(x,x^)=\\int _{\\Omega _h}N(x,z)dz-\|\\Omega _h\|N_{\\partial \\Omega _h}(x,x^)-\\int _{\\Omega _h}N(x^,z)dz+\|\\Omega _h\|R_{\\partial \\Omega _h}(x^,x^).$ The remainder $O(\\varepsilon )$ is uniform in $x\\in \\Omega _h$ satisfying dist$(x,\\Gamma _{\\varepsilon })\\ge c$ for some constant $c>0$ .", "Moreover, if $x(t),-\\epsilon <t<\\epsilon $ , is the arclength parameterization of $\\Gamma _{\\varepsilon }$ , then, $\\frac{\\partial u_{\\epsilon }}{\\partial \\nu }(x(t))=-\\frac{\|\\Omega _h\|}{2\\varepsilon }-\\frac{\|\\Omega _h\|}{\\pi }\\alpha \\left(\\frac{3}{2}-\\ln 2+\\frac{1}{2}L[1](\\frac{t}{\\varepsilon })\\right)+O(\\alpha \\varepsilon ^{3/2}),\\\\$ where $O(\\alpha \\varepsilon ^{3/2})$ is with respect to $\\parallel $ $\\parallel _{L^2[-\\varepsilon ,\\varepsilon ]}$ .", "We note that the function $\\Phi _{\\Omega _h}(x,x^)$ solves the following problem ${\\left\\lbrace \\begin{array}{ll}\\triangle _{x}\\Phi _{\\Omega _h}(x,x^) =0, &x\\in \\Omega _h,\\\\\\frac{\\partial \\Phi _{\\Omega _h}(x,x^)}{\\partial \\nu _{x}}=-\|\\Omega _h\|\\delta _{x^{}}, &x\\in \\partial \\Omega _h.\\\\\\end{array}\\right.", "}$ If $\\Omega _h$ is a unit disk centered at 0, one can easily see from (2.6) and (2.7) that $\\Phi _{\\Omega _h}(x,x^{})=\\ln \|x-x^{}\|+\\frac{1}{4}(1-\|x\|^{2}).$" ], [ "Calcium diffusion in dendritic spines", "In this section we use our new Neumann-Robin Boundary Model to solve the narrow escape problem in dendritic spine shape domain(Fig.", "REF ) which is calcium diffusion problem.", "That is, we calculate how long a single calcium molecule stays in the spine before it escapes from it.", "The usual way to calculate the solution in smooth domain requires boundary layer expansions for small window size and the asymptotic of the Neumann function that worked for nonsingular problems failed for calcium diffusion model since there are two singular points on the conneting part of spine head and spine neck.", "A quite different approach to the asymptotic problem is required which are much different from those reported in the cited reviews.", "In order to approximate the escape time for a particle in spine head, we approach a new method, different from those dealt with in , , that we use the Neumann-Robin Boundary Model in the spine head domain(Fig.REF ), but with the specific $\\alpha =\\frac{1}{L},\\beta =\\frac{L}{2},$ on the boundary $\\Gamma _{\\epsilon }$ which is the opening part of the big head.", "Here $L$ is the length of the neck.", "Note that we have changed the domain by dropping the long neck and assigning Robin boundary condition to the connecting arc $\\Gamma _{\\varepsilon }$ between the spine head and the long neck.", "Instead of dealing with the singular part on spine domain, we put a proper Robin boundary to the connecting part between spine head and neck.", "Figure: Decompose spine domain into two parts Ω=Ω h +Ω n \\Omega =\\Omega _{h}+\\Omega _{n}.", "One is spine head domain Ω h \\Omega _{h}, with reflecting boundary ∂Ω r \\partial \\Omega _r and Robin boundary Γ ε \\Gamma _\\varepsilon (where the Neumann-Robin Model can be solved using layer potential techniques in this smooth domain), the other part is long neck Ω n \\Omega _{n}.The heuristic reason for this specific choice of $\\alpha $ and $\\beta $ comes from the following: In the spine domain, we decompose the domain into two parts, one is spine head which has smooth boundary, the other is the long neck domain(Fig.REF ).", "Since the spine neck radius is small enough, we assume the escape time on the small part $\\Gamma _{\\varepsilon }$ (which connects head and long neck) is constant.", "Thus, in the spine neck domain $\\Omega _n$ (Fig.REF ), escape time $u_\\varepsilon $ satisfies the following equation ${\\left\\lbrace \\begin{array}{ll}\\triangle u_\\varepsilon =-1, &\\mbox{in}~\\Omega _{n},\\\\\\frac{\\partial u_\\varepsilon }{\\partial \\nu } = 0, &\\mbox{on}~\\partial \\Omega _{r},\\\\u_\\varepsilon =0, &\\mbox{on}~\\partial \\Omega _{a},\\\\u_\\varepsilon =C, &\\mbox{on}~\\Gamma _{\\varepsilon },\\end{array}\\right.", "}$ where $C$ is constant whose value means the escape time for the point initiated on $\\Gamma _{\\varepsilon }$ .", "Taking the center point of $\\Gamma _{\\varepsilon }$ to be original point $(0,0)$ , by separation of variables, we can solve this partial differential equation in $\\Omega _{n}$ .", "The solution of (4.1) is $u_\\varepsilon (x,y)=-\\frac{1}{2}(L-x)^2+(\\frac{C}{L}+\\frac{L}{2})(L-x),$ where $x\\in [0,L]$ , $y\\in (-\\varepsilon ,\\varepsilon )$ .", "Since $\\Omega _h$ and $\\Omega _n$ are connected by $\\Gamma _\\varepsilon $ , they share the same boundary value.", "The above solution $u_\\varepsilon (x,y)$ satisfies the Robin boundary condition $\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }+\\alpha u_{\\varepsilon }=\\beta ,$ with $\\alpha =\\frac{1}{L},\\beta =\\frac{L}{2}.$ Then the approximated Neumann-Robin Boundary Model for narrow escape problem is: ${\\left\\lbrace \\begin{array}{ll}\\triangle u_{\\varepsilon }=-1, &\\mbox{in}~\\Omega _{h},\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }=0, &\\mbox{on}~\\partial \\Omega _{r},\\\\\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }+\\frac{u_{\\varepsilon }}{L}=\\frac{L}{2}, &\\mbox{on}~\\Gamma _{\\varepsilon },\\end{array}\\right.", "}$ where $u_{\\varepsilon }(x)$ is the escape time of the calcium iron which initiated at $x$ position in the spine head.", "$\\Omega _{h}$ is the domain of the dendritic spine head and $\\partial \\Omega _{r}$ means the boundary where calcium molecule is reflected.", "According to Theorem 3.1 in the last section, the solution to (4.2) is: $u_{\\epsilon }(x)=\\frac{\|\\Omega _{h}\|L}{2\\varepsilon }+\\frac{\|\\Omega _{h}\|}{\\pi }(\\frac{3}{2}+\\ln \\frac{1}{2\\varepsilon })+\\frac{L^2}{2}+\\Phi _{\\Omega _{h}}(x,x^)+O(\\varepsilon ),$ where $\\Phi _{\\Omega _{h}}(x,x^)$ is the same as (3.24) for the domain $\\Omega _h$ .", "Suppose that $\\Omega $ is the spine with straight spine neck domain.", "The length of the neck is $L$ , and $\\partial \\Omega _a$ is the exiting arc(See Fig.REF ).", "$\\Omega _{h}$ is the spine head, and $\\Gamma _{\\varepsilon }$ is the arc of center $x^$ which connects spine head and spine neck.", "The first mean passage time $u_\\varepsilon (x)$ of a Brownian particle confined in $\\Omega _h$ exiting through $\\partial \\Omega _a$ can be approximated by the following formula $u_\\varepsilon (x)\\approx \\frac{\|\\Omega _{h}\|L}{2\\varepsilon }+\\frac{\|\\Omega _{h}\|}{\\pi }(\\frac{3}{2}+\\ln \\frac{1}{2\\varepsilon })+ \\frac{L^{2}}{2}+\\Phi _{\\Omega _{h}}(x,x^),$ where $\\Phi _{\\Omega _{h}}(x,x^)$ is the same as (3.24) for the domain $\\Omega _h$ .", "The error between formula (4.6) and the exact solution to (1.1) in spine domain is of order $O(\\varepsilon )$ , which can be seen by the numerical experiment data in the next section.", "The method using Neumann-Robin Boundary Model to solve narrow escape problem in domain with long neck is quite different from what has been discussed in .", "Their idea is to calculate the exit time by separating the exiting process of the particle into two processes.", "One is the time from the head to the interface $\\Gamma _{\\varepsilon }$ between head and neck, the other is the time from the interface to the absorbing arc.", "The mean first passage time can be obtained by adding the time of these two processes together.", "Their approximated formulation for planar spine connected to the neck at a right angle is $u_{\\varepsilon }(x)=\\frac{\|\\Omega _h\|}{\\pi }\\ln \\frac{\|\\partial \\Omega _h\|}{2\\varepsilon }+O(1)+\\frac{L^2}{2}+\\frac{\|\\Omega _h\|L}{2\\varepsilon },$ where $O(1)$ is the error term.", "From (4.6) and (4.7), we can see that the results from these two methods have the same first leading order term $\\frac{L^2}{2}+\\frac{\|\\Omega _1\|L}{2\\varepsilon }+\\frac{\|\\Omega _h\|}{\\pi }\\ln \\frac{1}{2\\varepsilon }$ .", "Thus, using our method we can obtain the exact formula for $O(1)$ in (4.7)." ], [ "Numerical experiment", "In order to check whether the asymptotic formula (4.6) can solve the narrow escape problem, we compare the numerical results of (4.6) denoted by $u_\\varepsilon $ in spine head domain, with the numerical solutions obtained by solving the two dimensional narrow escape problem (1.1) by using Matlab, which is denoted in this section by $u$ .", "Without loss of generality, we use the approximated spine geometry with unit disk spine head, and a rectangle neck in two dimension.", "Figure: The spine geometry is approximated by a unit disk head and a rectangle neck with width 2ε=0.22\\varepsilon =0.2 and length L=1L=1.", "Coordinates represent the position of the iron where it is initiated.", "The color represents the exit time of particle initiated at this point.", "Left figure: the numerical result uu of (1.1).", "Right figure: exit time u ε u_\\varepsilon computed by asymptotic formula (4.6).The first goal in this section is to compare the expansion formula (4.6) and the numerical solution to (1.1).", "The domain of the escape problem (1.1) is given in Fig.REF .", "As one example, we choose the spine head to be unit disk, with the neck length $L=1$ , and with the exit arc length $\|\\partial \\Omega _a\|=2\\varepsilon $ , $\\varepsilon =0.1$ .", "Then, the Neumann-Robin model solve the narrow escape problem in the spine head domain(Fig.REF ).", "Instead of considering the neck, we put the Robin boundary condition $\\frac{\\partial u_{\\varepsilon }}{\\partial \\nu }+\\frac{u_{\\varepsilon }}{L}=\\frac{L}{2}$ on $\\Gamma _\\varepsilon $ , where the arc length of $\\Gamma _\\varepsilon $ is $2\\varepsilon $ , $\\varepsilon =0.1$ .", "Note that these two problems have the same spine head domain.", "The numerical results are given in Fig.", "REF .", "The figure on the left side gives the numerical solution to (1.1) in the spine head domain.", "The value at each point shows the exit time of the particle initiated at this point.", "We can easily see that if the particle is initiated near the small arc, then it takes less time to escape.", "On the other hand, the figure on the right side is the numerical result of the expansion formula (4.6).", "The value at every point represents the exit time of the particle initiated at this point.", "From these two figures, we can easily see the results in both situations coincide perfectly.", "The numerical data show that the error between these two situations are of order $O(\\varepsilon )$ .", "This will be seen in Table.REF .", "Figure: Error between uu and u ε u_\\varepsilon in the above case, where L=1L=1, ε=0.1\\varepsilon =0.1.Meanwhile, Fig.", "REF shows the difference between $u$ and $u_\\varepsilon $ .", "The graph shows that away from the exit arc of small distance, $u_\\varepsilon $ can approximate $u$ with small error of order $O(\\varepsilon )$ .", "The second goal of the experiment is to see whether the Neumann-Robin model can be perfectly applied to solve the escape problem with different radius and different neck length.", "Figure: Left figure: Comparison between numerical results of original narrow escape problem (1.1) and expansion formula (4.6) derived from Neumann-Robin model, with different neck radius.", "We choose head to be unit disk, the neck length to be L=2L=2, vary the neck radius ε\\varepsilon from 0.01 to 0.1.", "Right figure: Comparison between numerical results of original narrow escape problem (1.1) and asymptotic formula (4.6) derived from Neumann-Robin model, with different neck length.", "Fix head to be unit disk, radius of neck to be ε=0.1\\varepsilon =0.1, vary neck length LL from 1 to 4.From Fig.", "REF , we can see the Neumann-Robin model perfectly solves the escape problem.", "The numerical solution of these two problems match with each other within error of order $O(\\varepsilon )$ .", "The figure on the left side is the case with different radius, 'o' represents the numerical solution of the original narrow escape problem, '+' represents the asymptotic formula (4.6) for Neumann-Robin model.", "Easy to see they are coincide.", "The figure on the right side is the case with different neck length.", "Similarly, the results coincide with each other.", "Table: Acknowledgement" ] ]	1403.0179
[ [ "Frobenius manifolds and Frobenius algebra-valued integrable systems" ], [ "Abstract The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields.", "In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved.", "In this paper a new theory of Frobenius-algebra valued integrable systems is developed.", "This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann, Kontsevich and Manin \\cite{Kaufmann,KMK}.", "By specializing this construction, using a fixed Frobenius algebra $\\mathcal{A},$ one can arrive at such a theory.", "More generally one can apply the same idea to construct an $\\mathcal{A}$-valued Topological Quantum Field Theory.", "The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations.", "Application of these ideas are discussed and, as an example, an $\\mathcal{A}$-valued modified Camassa-Holm equation is constructed." ], [ "Introduction", "Of the many ways to generalize the Korteweg-de Vries equation $u_t=u_{xxx}+6 u u_x\\,,$ the one that will be of most relevance to this paper is the matrix generalization (see, for example, [3], [4]) $\\mathcal {U}_t = \\mathcal {U}_{xxx} + 3 \\mathcal {U} \\mathcal {U}_x+ 3 \\mathcal {U}_x \\mathcal {U}\\,,$ where the two first derivative terms are required due to the non-commutativity of matrix multiplication.", "If one restricts such an equation to the space of commuting matrices one arrives at the equation $\\mathcal {U}_t =\\mathcal {U}_{xxx} + 6 \\mathcal {U} \\mathcal {U}_x$ which is identical in form to the original KdV equation but with a matrix-valued, as opposed to a scalar-valued, field (see, for example, [15], [26], [27]).", "The purpose of this paper is to construct $\\mathcal {A}$ -valued, where $\\mathcal {A}$ is a Frobenius algebra, generalizations of integrable systems, starting with those associated to an underlying Frobenius manifold and related dispersionless hierarchies, and extending the ideas to topological quantum field theories and dispersive hierarchies.", "The structure of this paper may be summarized in the following diagram: $\\begin{array}{ccc}\\left\\lbrace \\begin{array}{c}\\mathcal {A}-{\\rm valued}\\\\{\\rm Frobenius~manifold~(§2)}\\end{array}\\right\\rbrace & \\longrightarrow &\\left\\lbrace \\begin{array}{c}\\mathcal {A}-{\\rm valued}\\\\{\\rm TQFT~(§3)}\\end{array}\\right\\rbrace \\\\&&\\\\\\Big \\downarrow && \\\\&&\\\\\\left\\lbrace \\begin{array}{c}\\mathcal {A}-{\\rm valued~bi-Hamiltonian}\\\\{\\rm dispersionless~systems~(§4)}\\end{array}\\right\\rbrace & \\longrightarrow &\\left\\lbrace \\begin{array}{c}\\mathcal {A}-{\\rm valued~bi-Hamiltonian}\\\\{\\rm dispersive~systems~(§5)}\\end{array}\\right\\rbrace \\\\\\end{array}$ The full reconstruction of a dispersive hierarchy (the missing vertical arrow in the above diagram) remains an open problem, even before one considers $\\mathcal {A}$ -valued systems.", "The starting point (Section 2) for the study of such $\\mathcal {A}$ -valued hierarchies is the classical construction of Dubrovin [5] which associates to a Frobenius manifold a bi-Hamiltonian hierarchy of hydrodynamic type.", "By constructing the tensor product [13], [14] of such a manifold with a trivial Frobenius manifold (i.e.", "a fixed algebra) one automatically obtains a new Frobenius manifold and hence a bi-Hamiltonian hierarchy.", "The component fields of this new hierarchy can then be reassembled to form an $\\mathcal {A}$ -valued hierarchy.", "The important feature of this construction is a simple, explicit, form of the new prepotential that defines the $\\mathcal {A}$ -valued hierarchies.", "More explicitly, given a Frobenius algebra $\\mathcal {A}$ with basis $e_i\\,,i=1\\,,\\ldots \\,,n\\,,$ one can replace the flat coordinates of a Frobenius manifold $\\mathcal {M}$ with $\\mathcal {A}$ -valued fields via the map $\\hat{}\\,:\\,t^\\alpha \\mapsto \\widehat{t^\\alpha } = t^{(\\alpha i)} e_i,\\quad \\alpha =1\\,,\\ldots \\,, m,\\quad i=1\\,,\\ldots \\,,n\\nonumber $ and this action can be extended to functions, at least in the case of analytic Frobenius manifolds (and to wider classes of functions - see the Appendix).", "Conversely, an $\\mathcal {A}$ -valued field can be reduced to a scalar field via the Frobenius form (or trace form) $\\omega \\,.$ This construction is described in Section .", "The main result is the following: Main Theorem 1 (Theorem REF ) Let $F$ be the prepotential of a Frobenius manifold $\\mathcal {M}$ and let $\\mathcal {A}$ be a trivial Frobenius algebra with 1-form $\\omega \\,.$ The function ${F}^\\mathcal {A} = \\omega \\left(\\widehat{F}\\right)$ defines a Frobenius manifold, namely the manifold $\\mathcal {M}\\otimes \\mathcal {A}\\,.$ Normally the prepotential of a tensor product of Frobenius manifolds bears little resemblance to the underlying prepotentials, and in any case is only defined implicitly from the original prepotentials.", "However when one of the manifolds is trivial, the above closed form of the new prepotential exists and this enables the resulting hierarchies to be constructed explicitly.", "In Section we extend these ideas to a full Topological Quantum Field Theory on the big phase space $\\mathcal {M}^\\infty $ , i.e.", "with gravitational descendent fields.", "Main Theorem 2 (Theorem REF ) Let $\\mathcal {F}_{g\\ge 0}$ be the prepotentials defining a TQFT, $\\mathcal {S}$ and $\\mathcal {D}$ the corresponding String and Dilaton vector fields and $\\mathcal {A}$ be a trivial Frobenius algebra.", "Let $f$ be an analytic function on $\\mathcal {M}^\\infty $ and define the $\\mathcal {A}$ -valued function $\\hat{f}$ to be: ${\\hat{f}} = \\left.", "f\\right\|_{t^\\alpha _N \\mapsto t^{(\\alpha i)}_N e_i}\\,,\\qquad N\\in \\mathbb {Z}_{\\ge 0}\\,,\\quad \\alpha =1\\,,\\ldots \\,, m,\\quad i=1\\,,\\ldots \\,,n\\,.\\nonumber $ Then the functions $\\mathcal {F}^{\\mathcal {A}}_{g\\ge 0} = \\omega \\left( {\\widehat{ \\mathcal {F}}_{g\\ge 0} }\\right)$ and vector fields $\\mathcal {S}^\\mathcal {A} = -\\sum _{N,(\\alpha i)} \\tilde{t}^{(\\alpha i)}_N \\tau _{N-1,(\\alpha i)},\\quad \\mathcal {D}^\\mathcal {A} = -\\sum _{N,(\\alpha i)} \\tilde{t}^{(\\alpha i)}_N \\tau _{N,(\\alpha i)}$ satisfy the axioms of a Topological Quantum Field Theory.", "In the remaining sections a theory of $\\mathcal {A}$ -valued integrable systems is developed, first for dispersionless systems and then for certain dispersive systems.", "More specifically, in section the construction of the $\\mathcal {A}$ -valued dispersionless (or hydrodynamic) hierarchies is given.", "The deformed flat coordinates can be described very simply, and these form the Hamiltonian densities for the new evolution equations.", "By reassembling the fields these equations can be written as $\\mathcal {A}$ -valued evolution equations.", "To write these in Hamiltonian form requires the definition of a functional derivative with respect to an $\\mathcal {A}$ -valued field, and such a derivative was defined in [19] and with this one can write the flow equations as $\\mathcal {A}$ -valued bi-Hamiltonian evolution equations.", "These ideas are then extended to the dispersive case in section ." ], [ "Frobenius algebras and manifolds", "We begin with the definition of a Frobenius algebra [5].", "Definition 2.1 A Frobenius algebra $\\lbrace \\mathcal {A},\\circ ,e,\\omega \\rbrace $ over $\\mathbb {R}$ satisfies the following conditions: (i) $\\circ \\,: \\mathcal {A} \\times \\mathcal {A} \\rightarrow \\mathcal {A}$ is a commutative, associative algebra with unity $e$ ; (ii) $\\omega \\in \\mathcal {A}^\\star $ defines a non-degenerate inner product $\\langle a,b \\rangle = \\omega (a \\circ b)\\,.$ Since $\\omega (a) = \\langle e,a \\rangle $ the inner product determines the form $\\omega $ and visa-versa.", "This linear form $\\omega $ is often called a trace form (or Frobenius form).", "One dimensional Frobenius algebras are trivial: the requirement of an identity and the non-degeneracy of the inner product determines the algebra uniquely and the inner product up to a non-zero constant.", "Two dimensional algebra are easily classified.", "Example 2.2 Let $\\mathcal {A}$ be a 2-dimensional commutative and associative algebra with a basis $e=e_1, e_2$ satisfying $e_1\\circ e_1=e_1,\\quad e_1\\circ e_2=e_2, \\quad e_2\\circ e_2=\\varepsilon e_1+\\mu e_2,\\quad \\varepsilon ,\\mu \\in \\mathbb {R}.$ Obviously, the algebra $\\mathcal {A}$ has a matrix representation as follows $e_1 \\mapsto \\mathrm {I}_2=\\left(\\begin{array}{cc}1&0\\\\0&1\\end{array}\\right),\\quad e_2 \\mapsto \\left(\\begin{array}{cc}0& {\\varepsilon }\\\\1&\\mu \\end{array}\\right).\\nonumber $ It is easy to show that: (1) if $\\mu ^2=-4\\varepsilon $ , $\\mathcal {A}$ is nonsemisimple, i.e., $\\exists \\, \\widetilde{e}=\\mu e_1-2 e_2$ such that $\\widetilde{e}\\circ \\widetilde{e}=0$ ; (2) if $\\mu ^2\\ne -4\\varepsilon $ , then $\\mathcal {A}$ is semisimple, i.e., for any nonzero element $\\widetilde{e}=x e_1+y e_2$ , $\\widetilde{e}\\circ \\widetilde{e}\\ne 0$ .", "Furthermore, we introduce two “basic\" trace-type forms for $a=a_1e_1+a_2e_2\\in \\mathcal {A}$ as follows $\\omega _k(a)=a_k+a_2(1-\\delta _{k,2})\\delta _{\\varepsilon ,0}, \\quad k=1,\\, 2,$ which induce two nondegenerate inner products on $\\mathcal {A}$ given by $\\langle a, b\\rangle _k:=\\omega _k(a\\circ b),\\quad a,\\,b\\in \\mathcal {A},\\quad k=1,2.$ The two Frobenius algebras $\\left\\lbrace \\mathcal {A},\\circ , e, \\omega _k\\right\\rbrace $ will be denoted by $\\mathcal {Z}_{2,k}^{\\varepsilon ,\\mu }$ for $k=1,2$ .", "Example 2.3 Let $\\mathcal {A}$ be an $n$ -dimensional nonsemisimple commutative associative algebra $\\mathcal {Z}_n$ over $\\mathbb {R}$ with a unity $e$ and a basis $e_1=e, \\cdots ,e_n$ satisfying $e_i \\circ e_j=\\left\\lbrace \\begin{array}{ll}e_{i+j-1}, & i+j\\le n+1,\\\\0,& i+j=n+2.", "\\end{array}\\right.$ Taking $\\Lambda =(\\delta _{i,j+1})\\in gl(m,\\mathbb {R})$ , one obtains a matrix representation of $\\mathcal {A}$ as $e_j \\mapsto \\Lambda ^{j-1}, \\quad j=1,\\cdots , n.\\nonumber $ Similarly, for any $a=\\displaystyle \\sum _{k=1}^{n} a_k e_k\\in \\mathcal {A}$ , we introduce $n$ trace-type forms, called “basic\" trace-type forms, as follows $\\omega _{k-1} (a)=a_k+a_{n}(1-\\delta _{k,n}),\\quad k=1,\\cdots ,n. $ Every trace map $\\omega _k$ induces a nondegenerate symmetric bilinear form on $\\mathcal {A}$ given by $\\langle a, b\\rangle _k:=\\omega _k (a\\circ b),\\quad a,\\,b\\in \\mathcal {A},\\quad k=0,\\cdots ,n-1.$ Thus all of $\\left\\lbrace \\mathcal {A}, \\circ , e, \\omega _{k-1}\\right\\rbrace $ are nonsemisimple Frobenius algebras, denoted by $\\mathcal {Z}_{n,k-1}$ for $k=1,\\cdots ,n$ .", "We remark that if we consider a linear combination of $n$ “basic\" trace-type forms as $\\mathrm {tr}_n:=\\displaystyle \\sum _{s=0}^{n-1}\\omega _{s}-(n-1)\\,\\omega _{n-1},\\nonumber $ then $\\left\\lbrace \\mathcal {A}, \\circ , e, \\mathrm {tr}_n\\right\\rbrace $ is also a Frobenius algebra which is exactly the algebra $\\left\\lbrace \\mathcal {Z}_n,\\mathrm {tr}_n\\right\\rbrace $ used in [26]It was the realization that the matrix algebra used in this paper was a specific example of a Frobenius algebra that led to the development of the current paper.. A Frobenius manifold has such a structure on each tangent space.", "Definition 2.4 [5] The set $\\lbrace \\mathcal {M},\\circ ,e,\\langle ~,~\\rangle ,E \\rbrace $ is a Frobenius manifold if each tangent space $T_t\\mathcal {M}$ carries a smoothly varying Frobenius algebra with the properties: (i) $\\langle ~,~\\rangle $ is a flat metric on $\\mathcal {M}$ ; (ii) $\\nabla e=0$ , where $\\nabla $ is the Levi-Civita connection of $\\langle \\,,\\rangle $ ; (iii) the tensors $c(u,v,w):=\\langle u\\circ v,w\\rangle $ and $\\nabla _zc(u,v,w)$ are totally symmetric; (iv) A vector field $E$ exists, linear in the flat-variables, such that the corresponding group of diffeomorphisms acts by conformal transformation on the metric and by rescalings on the algebra on $T_t\\mathcal {M}\\,.$ These axioms imply the existence of the prepotential $F$ which satifies the WDVV-equations of associativity in the flat-coordinates of the metric (strictly speaking only a complex, non-degenerate bilinear form) on $\\mathcal {M}\\,.$ The multiplication is then defined by the third derivatives of the prepotential: $\\frac{\\partial ~}{\\partial t^\\alpha } \\circ \\frac{\\partial ~}{\\partial t^\\beta }= c_{\\alpha \\beta }^{\\phantom{\\alpha \\beta }\\gamma }({\\bf t}) \\frac{\\partial ~}{\\partial t^\\gamma }$ where $c_{\\alpha \\beta \\gamma } = \\frac{ \\partial ^3 F}{\\partial t^\\alpha \\partial t^\\beta \\partial t^\\gamma }$ and indices are raised and lowered using the metric $\\eta _{\\alpha \\beta }= \\langle \\dfrac{\\partial ~}{\\partial t^\\alpha },\\dfrac{\\partial ~}{\\partial t^\\beta } \\rangle $ .", "Example 2.5 Suppose $c_{ij}^{~~k}$ are the structure constants for the Frobenius algebra $\\mathcal {A}$ , so $e_i \\circ e_j = c_{ij}^{~~k} e_k$ and $\\eta _{ij} = \\langle e_i,e_j \\rangle \\,.$ For such an algebra one obtains a cubic prepotential $F & = & \\frac{1}{6} c_{ijk} t^i t^j t^k\\,,\\\\& = & \\frac{1}{6} \\omega ( {\\bf t}\\circ {\\bf t}\\circ {\\bf t})\\,,\\qquad {\\bf t} = t^i e_i\\,.$ The Euler vector field takes the form $E=\\displaystyle \\sum _{i} t^i\\frac{\\partial }{\\partial t^i}$ and $E(F)=3F\\,.$ The notation $\\mathcal {A}$ will be used for both the algebra and the corresponding manifold.", "Motivated by the classical Künneth formula in cohomology, Kaufmann, Kontsevich and Manin [13], [14] constructed the tensor product of two Frobenius manifolds $\\mathcal {M}^\\prime $ and $\\mathcal {M}^{\\prime \\prime }$ , denoted $\\mathcal {M}^\\prime \\otimes \\mathcal {M}^{\\prime \\prime }\\,.$ The following formulation of this construction is taken from [6].", "This formulation also gives criteria to check if a particular manifold is the tensor product of two more basic manifolds.", "For simplicity we use the notation $\\partial _\\alpha =\\dfrac{\\partial }{\\partial t^\\alpha }$ and $\\partial _{\\alpha \\beta }=\\dfrac{\\partial ~~}{\\partial t^{(\\alpha \\beta )}}$ .", "Proposition 2.6 Let $\\mathcal {M}^\\prime $ and $\\mathcal {M}^{\\prime \\prime }$ be two Frobenius manifolds of dimension $n^\\prime $ and $n^{\\prime \\prime }\\,.$ A Frobenius manifold $\\mathcal {M}$ of dimension $n^\\prime n^{\\prime \\prime }$ is the tensor product $\\mathcal {M} = \\mathcal {M}^\\prime \\otimes \\mathcal {M}^{\\prime \\prime }$ if the following conditions hold: (i) $\\lbrace T\\mathcal {M}, \\langle \\,,\\rangle ,e\\rbrace = \\lbrace T\\mathcal {M}^\\prime \\otimes T\\mathcal {M}^{\\prime \\prime }, \\langle \\,,\\rangle ^\\prime \\otimes \\langle \\,,\\rangle ^{\\prime \\prime },e^\\prime \\otimes e^{\\prime \\prime } \\rbrace \\,.$ Flat coordinates are labeled by pairs $t^{(\\alpha ^\\prime \\alpha ^{\\prime \\prime })}\\,,\\alpha ^\\prime =1\\,,\\ldots \\,,n^\\prime \\,,\\alpha ^{\\prime \\prime } =1\\,,\\ldots \\,,n^{\\prime \\prime }\\,,$ and the unity vector field is $e=\\dfrac{\\partial ~}{\\partial t^{(1 1)}}$ and the metric $\\langle \\,,\\rangle $ has the form $\\eta _{(\\alpha ^\\prime \\alpha ^{\\prime \\prime })(\\beta ^\\prime \\beta ^{\\prime \\prime })}= \\eta _{\\alpha ^\\prime \\beta ^\\prime } \\, \\eta _{\\alpha ^{\\prime \\prime } \\beta ^{\\prime \\prime }}\\,.$ (ii) At a point $t^{(\\alpha ^\\prime \\alpha ^{\\prime \\prime })}=0\\,,\\alpha ^\\prime >1\\,,\\alpha ^{\\prime \\prime }>1$ the algebra $T_t\\mathcal {M}$ is a tensor product $T_t\\mathcal {M} = T_{t^\\prime } \\mathcal {M}^\\prime \\otimes T_{t^{\\prime \\prime }} \\mathcal {M}^{\\prime \\prime }\\,,$ that is: $c_{(\\alpha ^\\prime \\alpha ^{\\prime \\prime })(\\beta ^\\prime \\beta ^{\\prime \\prime })}^{\\phantom{(\\alpha ^\\prime \\alpha ^{\\prime \\prime })(\\beta ^\\prime \\beta ^{\\prime \\prime })}(\\gamma ^\\prime \\gamma ^{\\prime \\prime })}(t)=c_{\\alpha ^\\prime \\beta ^\\prime }^{\\phantom{{\\alpha ^\\prime \\beta ^\\prime }}\\gamma ^\\prime }(t^\\prime ) \\,c_{\\alpha ^{\\prime \\prime }\\beta ^{\\prime \\prime }}^{\\phantom{{\\alpha ^{\\prime \\prime }\\beta ^{\\prime \\prime }}}\\gamma ^{\\prime \\prime }}(t^{\\prime \\prime })\\,.$ (iii) If the Euler vector fields of the two manifolds $\\mathcal {M}$ and $\\mathcal {M}^{\\prime \\prime }$ take the form $E^\\prime & = & \\sum _{\\alpha ^\\prime } \\left[ (1-q_{\\alpha ^\\prime }) t^{\\alpha ^\\prime }+ r_{\\alpha ^\\prime }\\right] \\partial _{\\alpha ^\\prime }\\,,\\\\E^{\\prime \\prime } & = & \\sum _{\\alpha ^{\\prime \\prime }} \\left[ (1-q_{\\alpha ^{\\prime \\prime }})t^{\\alpha ^{\\prime \\prime }} + r_{\\alpha ^{\\prime \\prime }}\\right] \\partial _{\\alpha ^{\\prime \\prime }}\\,,\\\\$ with scaling dimensions $d^\\prime $ and $d^{\\prime \\prime }$ respectively, then the Euler vector field on $\\mathcal {M}$ takes the form $E=\\sum _{\\alpha ^\\prime ,\\alpha ^{\\prime \\prime }} (1-q_{\\alpha ^\\prime }-q_{\\alpha ^{\\prime \\prime }})\\partial _{(\\alpha ^\\prime \\alpha ^{\\prime \\prime })} +\\sum _{\\alpha ^\\prime } r_{\\alpha ^\\prime }\\partial _{\\alpha ^\\prime 1^{\\prime \\prime }} +\\sum _{\\alpha ^{\\prime \\prime }} r_{\\alpha ^{\\prime \\prime }} \\partial _{{1^\\prime \\alpha ^{\\prime \\prime }}}$ and $d=d^\\prime +d^{\\prime \\prime }\\,.$ Such products describe the quantum cohomology of a product of varieties, and within singularity theory it appears when one takes the direct sum of singularities." ], [ "Tensor products with trivial algebras", "We now take the tensor product of a Frobenius manifold $\\mathcal {M}$ with a trivial manifold $\\mathcal {A}$ defined by a Frobenius algebra (Example REF ).", "To emphasize the different roles played by $\\mathcal {M}$ and $\\mathcal {A}$ we alter the general notation for tensor products as described above.", "The tensor product will be written as $\\mathcal {M}_{\\mathcal {A}}\\,,$ (so $\\mathcal {M}_{\\mathcal {A}}= \\mathcal {M}\\otimes {\\mathcal {A}}$ ).", "The basis $e_i$ for $\\mathcal {A}$ will be retained and the unity element denoted by $e_1\\,.$ Thus notation such as $e=\\partial _{1}$ will not be used.", "Latin indices will be reserved for $\\mathcal {A}$ -related objects, and Greek indices will be reserved for $\\mathcal {M}$ -related objects.", "Thus $c_{\\alpha \\beta }^{~~\\gamma }$ will denote the structure functions for the multiplication on $\\mathcal {M}$ and $c_{ij}^{~~k}$ will denote the structure constants for the multiplication on $\\mathcal {A}\\,.$ Coordinates on $\\mathcal {M}_{\\mathcal {A}}$ are denoted $\\lbrace t^{(\\alpha i)}\\,, \\alpha = 1 \\,,\\ldots \\,, m=dim \\mathcal {M}\\,,\\quad i=1\\,,\\ldots \\,, n=dim \\mathcal {A} \\rbrace \\,.$ No confusion should arise with this notation.", "We begin by constructing a lift of a scalar valued function to an $\\mathcal {A}$ -valued function and visa-versa.", "Definition 2.7 Let $f$ be an analytic function on $\\mathcal {M}$ (that is, analytic in the flat coordinates for $\\mathcal {M}$ ).", "The $\\mathcal {A}$ -valued function $\\hat{f}$ is defined to be: ${\\hat{f}} = \\left.", "f\\right\|_{t^\\alpha \\mapsto t^{(\\alpha i)} e_i}$ with $\\widehat{fg}=\\hat{f}\\circ \\hat{g}$ and $\\hat{1}=e_1\\,.$ The evaluation $f^\\mathcal {A}$ of $\\hat{f}$ is defined by $f^\\mathcal {A}=\\omega \\left(\\hat{f}\\right)\\,,$ where $\\omega \\in \\mathcal {A}^\\star \\,.$ Since the function is analytic and the algebra $\\mathcal {A}$ is commutative and associative the above construction is well-defined.", "Remark 2.8 This definition requires the existence of a distinguished coordinate system on $\\mathcal {M}$ in which the function $f$ is analytic.", "In the case of analytic Frobenius manifolds one automatically has such a distinguished system of coordinates, namely the flat coordinates of the metric.", "With these definitions one may construct a new prepotential from the original one.", "Theorem 2.9 Let $F$ be the prepotential of a Frobenius manifold $\\mathcal {M}$ and let $\\mathcal {A}$ be a Frobenius algebra with 1-form $\\omega \\,.$ The function $F^\\mathcal {A} = \\omega \\left(\\widehat{F}\\right)$ defines a Frobenius manifold, namely the manifold $\\mathcal {M}_\\mathcal {A}\\,.$ Note, one could `straighten out' the coordinates $t^{(\\alpha i)}$ via the map $ v^{i+(\\alpha -1) n}=t^{(\\alpha i)}\\,, \\quad 1\\le i \\le n,\\quad 1\\le \\alpha \\le m,$ and hence $F^\\mathcal {A}=F^\\mathcal {A}(v^1,\\cdots ,v^{mn})$ .", "However such a map is not unique and the tensor structure is lost.", "The proof is in two parts: we first show that the prepotential $F^\\mathcal {A}$ defines a Frobenius manifold, and then identify this with the tensor product $\\mathcal {M} \\otimes \\mathcal {A}\\,.$ By construction we have an $nm$ -dimensional manifold with coordinates $t^{(\\alpha i)}\\,,\\alpha =1\\,,\\ldots \\,,m=dim\\mathcal {M}\\,,i=1\\,,\\ldots \\,,n=dim\\mathcal {A}\\,.$ We begin with two simple results: $\\bullet $ Because $\\eta _{ij} = \\omega (e_i \\circ e_j)$ it follows, since by definition, $(\\eta ^{ij})=(\\eta _{ij})^{-1}\\,,$ that $\\omega (e_i \\circ e_r) \\eta ^{rs} \\omega (e_s\\circ e_j) = \\omega (e_i\\circ e_j)\\,.$ More generally, using the properties of the multiplication on $\\mathcal {A}\\,,$ $\\omega (\\ldots \\circ e_i \\circ e_r) \\eta ^{rs} \\omega (e_s\\circ e_j\\circ \\ldots )= \\omega (\\ldots \\circ e_i\\circ e_j \\circ \\ldots )\\,.$ $\\bullet $ The fundamental result that will be used extensively in the rest of the paper is the following: $\\frac{\\partial \\hat{f}}{\\partial t^{(\\alpha i)}}= \\widehat{\\frac{\\partial f}{\\partial t^\\alpha }} \\circ e_i\\,.$ We introduce the notation $\\hat{f} = [\\hat{f}]^p e_p\\,,$ so $\\frac{\\partial \\hat{f}}{\\partial t^{(\\alpha i)}}= \\left[\\widehat{\\frac{\\partial f}{\\partial t^\\alpha }}\\right]^p e_p \\circ e_i\\,.$ This will be used to separate out the $\\mathcal {A}$ -valued part of various expressions.", "With these, $\\frac{\\partial ^3 \\hat{F}}{\\partial t^{(\\alpha i)} \\partial t^{(\\beta j)} \\partial t^{(\\gamma k)}}= \\widehat{\\left( \\frac{\\partial ^3 F}{\\partial t^\\alpha \\partial t^\\beta \\partial t^\\gamma }\\right)}\\circ e_i \\circ e_j \\circ e_k\\,,$ so $\\frac{\\partial ^3 F^\\mathcal {A}}{\\partial t^{(\\alpha i)} \\partial t^{(\\beta j)} \\partial t^{(\\gamma k)}}& = & \\omega \\left( \\widehat{c_{\\alpha \\beta \\gamma }}\\circ e_i \\circ e_j \\circ e_k\\right)\\,,\\\\& = & \\left[ \\widehat{c_{\\alpha \\beta \\gamma }} \\right]^p \\omega (e_p\\circ e_i \\circ e_j \\circ e_k)\\,,\\\\& = & c_{(\\alpha i)(\\beta j)(\\gamma k)}\\,.$ Normalization We define $\\eta _{(\\alpha i)(\\beta j)}$ by $\\eta _{(\\alpha i)(\\beta j)} & = & c_{(11)(\\alpha i)(\\beta j)}\\,,\\\\& = & \\omega \\left( \\widehat{c_{1\\alpha \\beta }}\\circ e_1 \\circ e_i \\circ e_j\\right)\\,,\\\\& = & \\eta _{\\alpha \\beta }\\, \\eta _{ij}$ since $\\widehat{c_{1\\alpha \\beta }} = \\widehat{\\eta _{\\alpha \\beta }}= \\eta _{\\alpha \\beta } e_1\\,,$ and $e_1$ is the unity for the multiplication on $\\mathcal {A}\\,.$ This is non-degenerate (since by assumption $\\eta _{\\alpha \\beta }$ and $\\eta _{ij}$ are non-degenerate) and this will be taken to be the metric and used to raise and lower indices.", "In particular, $\\eta ^{(\\alpha i)(\\beta j)}=\\eta ^{\\alpha \\beta }\\, \\eta ^{ij}\\,.$ Associativity Using the metric to raise an index one obtains $c_{(\\alpha i)(\\beta j)}^{\\phantom{{(\\alpha i)(\\beta j)}}(\\gamma k)}= \\left[ \\widehat{c_{\\alpha \\beta }^{~~\\gamma }} \\right]^p \\, c_{ij}^{~~q} c_{pq}^{~~k}$ and this defines a multiplication on $\\mathcal {M}_{\\mathcal {A}}\\,.$ The structure of this multiplication may be made more transparent if one writes the basis for $T\\mathcal {M}_{\\mathcal {A}}$ as a tensor product: $\\frac{\\partial ~}{\\partial t^{(\\alpha i)}} = \\partial _\\alpha \\otimes e_i\\,.$ With this, the multiplication may be written as: $\\left( \\partial _\\alpha \\otimes e_i\\right) \\circ \\left( \\partial _\\beta \\otimes e_j\\right)= \\left[ \\widehat{ \\partial _\\alpha \\circ \\partial _\\beta }\\right]^p \\otimes e_p \\circ e_i\\circ e_j\\,,$ where $ \\widehat{f\\partial _\\alpha }=[\\hat{f}]^p \\partial _\\alpha \\otimes e_p\\,,$ and hence $\\left[\\widehat{f\\partial _\\alpha }\\right]^p=[\\hat{f}]^p \\partial _\\alpha \\,.$ By construction this multiplication defines a commutative multiplication with unity $e=\\dfrac{\\partial ~}{\\partial t^{(1 1)}}=\\partial _1\\otimes e_1\\,.$ To prove associativity we first rewrite the equation that has to be satisfied by $F^\\mathcal {A}$ , namely the WDVV equation: $\\dfrac{\\partial ^3F^\\mathcal {A}}{\\partial t^{(\\gamma k)}\\,\\partial t^{(\\sigma s)}\\, \\partial t^{(\\alpha i)}} {\\eta }^{(\\alpha i)(\\beta j)}\\dfrac{\\partial ^3F^\\mathcal {A}}{\\partial t^{(\\beta j)}\\,\\partial t^{(\\delta p)}\\, \\partial t^{(\\mu q)}}=\\dfrac{\\partial ^3F^\\mathcal {A}}{\\partial t^{(\\mu q)}\\,\\partial t^{(\\sigma s)}\\, \\partial t^{(\\alpha i)}} {\\eta }^{(\\alpha i)(\\beta j)}\\dfrac{\\partial ^3F^\\mathcal {A}}{\\partial t^{(\\beta j)}\\,\\partial t^{(\\delta p)}\\, \\partial t^{(\\gamma k)}}\\,.$ This is equivalent to $&&\\left[\\widehat{c_{\\gamma \\sigma \\alpha }}\\right]^a \\omega (e_a\\circ e_k\\circ e_s\\circ e_i)\\eta ^{\\alpha \\beta } \\eta ^{ij} \\omega (e_j\\circ e_p\\circ e_q\\circ e_b)\\left[\\widehat{c_{\\beta \\delta \\mu }}\\right]^b\\nonumber \\\\&=&\\left[\\widehat{c_{ \\mu \\sigma \\alpha }}\\right]^a \\omega (e_a\\circ e_q\\circ e_s\\circ e_i) \\eta ^{\\alpha \\beta } \\eta ^{ij}\\omega (e_j\\circ e_p\\circ e_k\\circ e_b)\\left[\\widehat{c_{\\beta \\delta \\gamma }}\\right]^b\\,,\\nonumber $ which becomes, on using equation (REF ), $&&\\left[\\widehat{c_{\\gamma \\sigma \\alpha }}\\right]^a \\eta ^{\\alpha \\beta }\\omega (e_a\\circ e_k\\circ e_s\\circ e_p\\circ e_q\\circ e_b)\\left[\\widehat{c_{\\beta \\delta \\mu }}\\right]^b\\nonumber \\\\&=&\\left[\\widehat{c_{ \\mu \\sigma \\alpha }}\\right]^a \\eta ^{\\alpha \\beta } \\omega (e_a\\circ e_q\\circ e_s\\circ e_p\\circ e_k\\circ e_b)\\left[\\widehat{c_{\\beta \\delta \\gamma }}\\right]^b\\,.", "$ Since the prepotential $F$ for the Frobenius manifold $\\mathcal {M}$ satisfies the WDVV equation $ \\dfrac{\\partial ^3 F}{\\partial t^\\gamma \\,\\partial t^\\sigma \\, \\partial t^\\alpha } {\\eta }^{\\alpha \\beta }\\dfrac{\\partial ^3 {F}}{\\partial t^\\beta \\,\\partial t^\\delta \\, \\partial t^\\mu }=\\dfrac{\\partial ^3 {F}}{\\partial t^\\mu \\,\\partial t^\\sigma \\, \\partial t^\\alpha } {\\eta }^{\\alpha \\beta }\\dfrac{\\partial ^3 {F}}{\\partial t^\\beta \\,\\partial t^\\delta \\, \\partial t^\\gamma }\\,,$ it follows that $ \\widehat{\\dfrac{\\partial ^3 F}{\\partial t^\\gamma \\,\\partial t^\\sigma \\, \\partial t^\\alpha }} \\circ \\widehat{ {\\eta }^{\\alpha \\beta }} \\circ \\widehat{\\dfrac{\\partial ^3 {F}}{\\partial t^\\beta \\,\\partial t^\\delta \\, \\partial t^\\mu }}=\\widehat{\\dfrac{\\partial ^3 {F}}{\\partial t^\\mu \\,\\partial t^\\sigma \\, \\partial t^\\alpha }} \\circ \\widehat{ {\\eta }^{\\alpha \\beta }} \\circ \\widehat{\\dfrac{\\partial ^3 {F}}{\\partial t^\\beta \\,\\partial t^\\delta \\, \\partial t^\\gamma }},$ where $\\widehat{ {\\eta }^{\\alpha \\beta }}={\\eta }^{\\alpha \\beta }\\, e_1$ .", "This reduces to $\\left[\\widehat{c_{\\gamma \\sigma \\alpha }}\\right]^a \\eta ^{\\alpha \\beta }e_a\\circ e_b\\left[\\widehat{c_{\\beta \\delta \\mu }}\\right]^b=\\left[\\widehat{c_{ \\mu \\sigma \\alpha }}\\right] \\eta ^{\\alpha \\beta } e_a \\circ e_b\\left[\\widehat{c_{\\beta \\delta \\gamma }}\\right]^b\\,.$ Thus we have, by multiplying by $e_q\\circ e_s\\circ e_p\\circ e_k$ , $\\left[\\widehat{c_{\\gamma \\sigma \\alpha }}\\right]^a \\eta ^{\\alpha \\beta }e_a\\circ e_k\\circ e_s\\circ e_p\\circ e_q\\circ e_b\\left[\\widehat{c_{\\beta \\delta \\mu }}\\right]^b =\\left[\\widehat{c_{ \\mu \\sigma \\alpha }}\\right]^a \\eta ^{\\alpha \\beta }e_a\\circ e_q\\circ e_s\\circ e_p\\circ e_k\\circ e_b\\left[\\widehat{c_{\\beta \\delta \\gamma }}\\right]^b,\\nonumber \\nonumber $ and evaluating the function with $\\omega \\,,$ gives the identity (REF ).", "Hence $F^\\mathcal {A}$ satisfies the WDVV equation in the flat coordinates of the metric $\\eta _{(\\alpha i)(\\beta j)}\\,.$ Quasi-homogeneity This follows immediately from the definition of $F^\\mathcal {A}$ , but one can also derive the result by direct computation.", "The quasi-homogeneity of $F$ is expressed by the equation $\\sum _\\alpha \\left[(1-q_\\alpha ) t^\\alpha + r_\\alpha \\right] \\frac{\\partial F}{\\partial t^\\alpha } = (3-d) F$ where quadratic terms will be ignored.", "On lifting this and using the evaluation map defined by $\\omega $ one obtains $\\sum _{(\\alpha i)} (1-q_\\alpha ) t^{(\\alpha i)} \\omega \\left(\\widehat{\\left(\\frac{\\partial F}{\\partial t^\\alpha }\\right)} \\circ e_i\\right) + \\sum _\\alpha r_\\alpha \\omega \\left( \\widehat{\\frac{\\partial F}{\\partial t^\\alpha }} \\right) = (3-d) F^\\mathcal {A}\\,.$ Using (REF ) yields the result ${E}^\\mathcal {A}\\left(F^\\mathcal {A}\\right) = (3-d) F^\\mathcal {A}$ (again, up to quadratic terms) where $E^\\mathcal {A} = \\sum _{(\\alpha i)} (1-q_\\alpha ) t^{(\\alpha i)} \\frac{\\partial ~}{\\partial t^{(\\alpha i)}}+ \\sum _\\alpha r_\\alpha \\frac{\\partial ~}{\\partial t^{(\\alpha 1)}}\\,.$ These show that $F^\\mathcal {A}$ defines a Frobenius manifold.", "It remains to show that this is the tensor product $\\mathcal {M} \\otimes \\mathcal {A}\\,.$ In fact this is straightforward.", "Parts (i) and (iii) of Proposition REF are immediate from above (since for the trivial Frobenius manifold $\\mathcal {A}$ , $q_i=r_i=d=0$ ), so it just remains to verify condition (ii).", "Since $c_{\\alpha \\beta }^{\\phantom{\\alpha \\beta }\\gamma }$ is independent of $t^1$ it follows that at points $t^{(\\alpha i)}=0\\,,\\alpha >1\\,,i>1$ that $\\widehat{c_{\\alpha \\beta }^{\\phantom{\\alpha \\beta }\\gamma }}=c_{\\alpha \\beta }^{\\phantom{\\alpha \\beta }\\gamma }\\left(t^{(\\sigma 1)}\\right) e_1$ and the result follows from equation (REF ).", "Hence the prepotential $F^\\mathcal {A}=\\omega (\\widehat{F})$ defines the Frobenius manifold structure on the tensor product $\\mathcal {M}_\\mathcal {A} = \\mathcal {M}\\otimes \\mathcal {A}\\,.$ If the multiplications on $\\mathcal {M}$ and $\\mathcal {A}$ are semisimple then the multiplication on $\\mathcal {M}_\\mathcal {A}$ is also semisimple [13], [14].", "Remark 2.10 Note the existence of such a prepotential $F^\\mathcal {A}$ for such a tensor product follows from the original work of Kaufmann, Kontsevich and Manin.", "However the explicit form for such an $F^\\mathcal {A}$ is not immediate from their construction.", "The above result gives an explicit and easily computable prepotential in the case when one of the manifolds is trivial.", "Example 2.11 Let $\\mathcal {M}$ be a one-dimensional Frobenius manifold $F(t^1)=\\dfrac{1}{6}(t^1)^3,\\, e=\\partial _1, \\, E=t^1\\partial _1,$ so $\\mathcal {M}_{\\mathcal {A}}=\\mathcal {A}$ given in Example REF .", "Example 2.12 Suppose $\\mathcal {A}$ is a Frobenius algebra $\\mathcal {Z}_{2,2}^{\\varepsilon , 0}$ defined in Example REF .", "When $\\varepsilon \\ne 0$ , $\\mathcal {A}$ is semisimple.", "When $\\varepsilon =0$ , $\\mathcal {A}$ is nonsemisimple and exactly the algebra $\\mathcal {Z}_{2,2}$ given in Example REF .", "Let $\\mathcal {M}$ be a 2-dimensional Frobenius manifold with the flat coordinate $(t^1,t^2)$ .", "We denote $\\widehat{t^1}=v^1e_1+v^2 e_2,\\quad \\widehat{t^2}= v^3e_1+v^4 e_2.$ Case 1.", "$\\mathcal {M}=\\mathbb {C}^2/W(A_2)$ , i.e., $F(t)=\\frac{1}{2}(t^1)^2t^2-\\frac{1}{72}(t^2)^4, \\quad e=\\frac{\\partial }{\\partial t^1},\\quad E=t^1 \\frac{\\partial }{\\partial t^1}+ \\frac{2}{3}t^2 \\frac{\\partial }{\\partial t^2}.\\nonumber $ The unity vector field and the Euler vector field of $\\mathcal {M}_\\mathcal {A}$ are given by, respectively, $e=\\frac{\\partial }{\\partial v^1}, \\quad E^\\mathcal {A}=v^1\\frac{\\partial }{\\partial v^1}+v^2\\frac{\\partial }{\\partial v^2}+\\frac{2}{3}v^3\\frac{\\partial }{\\partial v^3}+\\frac{2}{3}v^4\\frac{\\partial }{\\partial v^4}$ and the potential of $\\mathcal {M}_\\mathcal {A}$ is given by $F^\\mathcal {A}(v)=\\frac{1}{2}(v^1)^2v^4+v^1v^2v^3-\\frac{1}{18}(v^3)^3v^4+\\varepsilon \\left(\\frac{1}{2}(v^2)^2v^4-\\frac{1}{18}v^3(v^4)^3\\right).\\nonumber $ We remark that when $\\varepsilon \\ne 0$ , $\\mathcal {M}_\\mathcal {A}$ is a polynomial semisimple Frobenius manifold.", "By a result of Hertling [11], the manifold $\\mathcal {M}_\\mathcal {A}$ decomposes into a product of $A_2$ -Frobenius manifolds.", "The algebra $\\mathcal {A}$ can be seen as controlling this decomposition.", "Case 2.", "$\\mathcal {M}=\\mathrm {QH^(\\mathrm {CP}^1)}$ , i.e., $F(t)=\\frac{1}{2}(t^1)^2t^2+e^{t^2}, \\quad e=\\dfrac{\\partial }{\\partial t^1},\\quad E=t^1\\dfrac{\\partial }{\\partial t^1}+2\\dfrac{\\partial }{\\partial t^2}.$ The unity vector field and the Euler vector field of $\\mathcal {M}_\\mathcal {A}$ are given by, respectively, $e=\\dfrac{\\partial }{\\partial v^1},\\quad E^\\mathcal {A}=v^1\\dfrac{\\partial }{\\partial v^1}+v^2\\dfrac{\\partial }{\\partial v^2}+2\\dfrac{\\partial }{\\partial v^3}+2\\dfrac{\\partial }{\\partial v^4}$ and the potential of $\\mathcal {M}_\\mathcal {A}$ is given by $F^\\mathcal {A}(v)=\\left\\lbrace \\begin{array}{ll}\\dfrac{1}{2}(v^1)^2v^4+v^1v^2v^3+\\varepsilon (v^2)^2v^4+\\dfrac{\\sinh (\\sqrt{\\varepsilon } v^4)}{\\sqrt{\\varepsilon }}\\,e^{v^3},&\\varepsilon \\ne 0,\\\\\\dfrac{1}{2}(v^1)^2v^4+v^1v^2v^3+v^4\\,e^{v^3}, &\\varepsilon =0.\\end{array}\\right.\\nonumber $" ], [ "$\\mathcal {A}$ -valued Topological Quantum Field Theories", "The ideas developed in the last section may be applied to the construction of $\\mathcal {A}$ -valued Topological Quantum Fields Theories on a suitably defined big-phase space (i.e.", "with gravitational descendent fields).", "In fact one could have started with this larger construction and obtained the results of the last section by restriction to the small-phase space.", "Conversely, the reconstruction theorems which give big-phase space structures from Frobenius manifold structures could be used to construct these $\\mathcal {A}$ -valued TQFTs from the Frobenius manifold $\\mathcal {M}_\\mathcal {A}\\,.$" ], [ "Background", "A topological quantum field theory (or TQFT) is defined in terms of properties of certain correlators which are themselves defined in terms of prepotential $\\mathcal {F}_{g\\ge 0}$ .", "For example, consider a smooth projective variety $V$ with $H^{\\rm odd}(V;\\mathbb {C})=0$ , $\\lbrace \\gamma _1\\,,\\ldots \\,,\\gamma _N\\rbrace $ a basis for the cohomology ring $M:=H^{}(V;\\mathbb {C})$ and let $\\eta _{\\alpha \\beta } = \\eta (\\gamma _\\alpha ,\\gamma _\\beta ) = \\int _V\\gamma _\\alpha \\cup \\gamma _\\beta $ be the Poincaré pairing which defines a non-degenerate metric which may be used to raise and lower indices.", "Following the conventions of Liu and Tian [16], [17], a flat coordinate system $\\lbrace t^\\alpha _0\\,, \\alpha =1\\,,\\ldots \\,,N\\rbrace $ may be found on $M$ so $\\gamma _\\alpha =\\frac{\\partial ~}{\\partial t^\\alpha _0}$ , and in which the components of $\\eta $ are constant.", "The big phase space consists of an infinite number of copies of the $M\\,,$ the small phase space, so $M^\\infty = \\prod _{n\\ge 0} H^{*}(V;\\mathbb {C})\\,.$ The coordinate system $\\lbrace t_{0}^{\\alpha }\\rbrace $ induces, in a canonical way, a coordinate system $\\lbrace t^\\alpha _n\\,,n \\in \\mathbb {Z}_{\\ge 0}\\,,\\alpha =1\\,,\\ldots \\,,N\\rbrace $ on $M^{\\infty }.$ We denote by $\\tau _n(\\gamma _\\alpha ) = \\frac{\\partial ~}{\\partial t^\\alpha _n}$ (also abbreviated to $\\tau _{n,\\alpha }\\,)$ the associated fundamental vector fields, which represent various tautological line bundles over the moduli space of curves.", "The descendant Gromov-Witten invariants $\\langle \\tau _{n_1}(\\gamma _{a_1}) \\ldots \\tau _{n_k}(\\gamma _{a_k})\\rangle _g$ may be combined into generating functions, called prepotentials, labeled by the genus $g\\,,$ ${\\mathcal {F}}_g=\\sum _{k\\ge 0} \\frac{1}{k!}", "\\sum _{n_1 ,\\alpha _1\\ldots n_k,\\alpha _k} t^{\\alpha _1}_{n_1} \\ldots t^{\\alpha _k}_{n_k} \\langle \\tau _{n_1}(\\gamma _{\\alpha _1}) \\ldots \\tau _{n_k}(\\gamma _{\\alpha _k})\\rangle _g\\,,$ and these in turn may be used to define $k$ -tensor fields on the big phase space, via the formula $\\langle \\langle {\\mathcal {W}}_{1}\\cdots {\\mathcal {W}}_{k}\\rangle \\rangle _{g} = \\sum _{m_{1},\\alpha _{1},\\cdots , m_{k},\\alpha _{k}} f^{1}_{m_{1},\\alpha _{1}}\\cdots f^{k}_{m_{k},\\alpha _{k}} \\frac{\\partial ^{k} {\\mathcal {F}_{g}}}{\\partial t^{\\alpha _{1}}_{m_{1}}\\cdots \\partial t^{\\alpha _{k}}_{m_{k}} },$ for any vector fields ${\\mathcal {W}}_{i} = \\sum _{m,\\alpha } f^{i}_{m,\\alpha } \\frac{\\partial }{\\partial t^{\\alpha }_{m}}$ .", "The tensor field (REF ) has a physical interpretation as the $k$ -point correlation function of the TQFT.", "The basic relationships between these correlators may then be encapsulated in the following: Definition 3.1 Let $\\tilde{t}^\\alpha _n=t^\\alpha _n - \\delta _{n,1} \\delta _{\\alpha ,1}$ and let $\\mathcal {S} & = & -\\sum _{n,\\alpha } \\tilde{t}^\\alpha _n \\tau _{n-1}(\\gamma _{\\alpha } )\\,,\\\\\\mathcal {D} & = & -\\sum _{n,\\alpha } \\tilde{t}^\\alpha _n\\tau _{n}(\\gamma _{\\alpha })$ be the string and dilaton vector fields respectively.", "Then the prepotentials ${\\mathcal {F}}_g$ satisfy the following relations: String Equation: $\\langle \\langle \\mathcal {S} \\rangle \\rangle _g = \\frac{1}{2} \\delta _{g,0} \\sum _{\\alpha ,\\beta } \\eta _{\\alpha \\beta } t^\\alpha _0 t^\\beta _0\\,;$ Dilaton Equation: $\\langle \\langle \\mathcal {D} \\rangle \\rangle _g = (2g-2) {\\mathcal {F}}_g - \\frac{1}{24} \\chi (V) \\delta _{g,1}\\,;$ Genus-zero Topological Recursion Relation: $\\langle \\langle \\tau _{m+1}(\\gamma _\\alpha ) \\tau _n(\\gamma _\\beta ) \\tau _k(\\gamma _\\sigma ) \\rangle \\rangle _{{}_0} = \\langle \\langle \\tau _{m}(\\gamma _\\alpha )\\gamma _\\mu \\rangle \\rangle _{{}_0} \\langle \\langle \\gamma ^\\mu \\tau _n(\\gamma _\\beta ) \\tau _k(\\gamma _\\sigma ) \\rangle \\rangle _{{}_0}\\,.$ By restricting such theories to primary vector fields with coefficients in the small phase space one recovers a Frobenius manifold structure [5], [6] on the small phase space, with $F_0(t_0^1\\,,\\ldots \\,,t_0^N) = \\left.\\mathcal {F}_0( {\\bf t})\\right\|_{t^\\alpha _n=0\\,,\\,n>0}$ becoming the prepotential for the Frobenius manifold and multiplication given by $\\tau _{0,\\alpha }\\circ \\tau _{0,\\beta } = \\langle \\langle \\tau _{0,\\alpha }\\tau _{0,\\beta }\\gamma ^{\\sigma }\\rangle \\rangle _{{}_0}\\vert _{M} \\gamma _{\\sigma }.$" ], [ "$\\mathcal {A}$ -TQFT", "Given such a theory one may extend the previous construction to obtain a new TQFT.", "Again, the existence of such a result follows from various reconstruction theorems, but explicit formulae may be obtained when one tensors by a constant Frobenius algebra.", "Theorem 3.2 Let $\\mathcal {F}_{g\\ge 0}$ be the prepotentials defining a TQFT, $\\mathcal {S}$ and $\\mathcal {D}$ the corresponding String and Dilaton vector fields and $\\mathcal {A}$ be a trivial Frobenius algebra.", "Let $f$ be an analytic function on $\\mathcal {M}^\\infty $ (that is, analytic in the flat coordinates $t_N^\\alpha $ for $\\mathcal {M}^\\infty $ ) and define the $\\mathcal {A}$ -valued function $\\hat{f}$ to be: ${\\hat{f}} = \\left.", "f\\right\|_{t^\\alpha _N \\mapsto t^{(\\alpha i)}_N e_i}\\,,\\qquad N\\in \\mathbb {Z}_{\\ge 0}\\,,\\quad \\alpha =1\\,,\\ldots \\,, m,\\quad i=1\\,,\\ldots \\,,n\\,.$ Then the functions $\\mathcal {F}^{\\mathcal {A}}_{g\\ge 0} = \\omega \\left( {\\widehat{ \\mathcal {F}}_{g\\ge 0} }\\right)$ and vector fields $\\mathcal {S}^\\mathcal {A} & = & -\\sum _{N,(\\alpha i)} \\tilde{t}^{(\\alpha ,i)}_N \\tau _{N-1,(\\alpha i)}\\,,\\\\\\mathcal {D}^\\mathcal {A} & = & -\\sum _{N,(\\alpha i)} \\tilde{t}^{(\\alpha ,i)}_N \\tau _{N,(\\alpha i)}$ satisfy the axioms of a Topological Quantum Field Theory.", "Genus-zero Topological Recursion Relation By repeating the construction in Theorem REF (essentially using (REF )) one easily obtains the equation $\\langle \\langle \\tau _{M+1,(\\alpha i)} \\tau _{N,(\\beta j)} \\tau _{K,(\\sigma k)}\\rangle \\rangle _{{}_0} = \\omega \\left(\\langle \\langle \\tau _{M+1,\\alpha } \\tau _{N,\\beta } \\tau _{K,\\sigma }\\rangle \\rangle _{{}_0}^{\\hat{}} \\circ e_i \\circ e_j \\circ e_k \\right)$ (where we displace the $\\hat{}$ symbol for notational convenience, so $f\\,{}^{\\hat{}}=\\hat{f}$ ).", "On using the topological recursion relation this decomposes as $\\langle \\langle \\tau _{M+1,(\\alpha i)} \\tau _{N,(\\beta j)} \\tau _{K,(\\sigma k)}\\rangle \\rangle _{{}_0} =\\eta ^{\\mu \\nu } \\omega \\left(\\langle \\langle \\tau _{M,\\alpha } \\gamma _\\mu \\rangle \\rangle _{{}_0}^{\\hat{}} \\circ e_i \\circ e_j \\circ \\langle \\langle \\gamma _\\mu \\tau _{N,\\beta } \\tau _{K,\\sigma } \\rangle \\rangle _{{}_0}^{\\hat{}}\\circ e_k\\right)\\,$ $=\\eta ^{\\mu \\nu } \\omega \\left(\\langle \\langle \\tau _{M,\\alpha } \\gamma _\\mu \\rangle \\rangle _{{}_0}^{\\hat{}} \\circ e_i \\circ e_r\\right) \\eta ^{rs}\\omega \\left(e_s \\circ e_j \\circ \\langle \\langle \\gamma _\\mu \\tau _{N,\\beta } \\tau _{K,\\sigma } \\rangle \\rangle _{{}_0}^{\\hat{}}\\circ e_k \\right)$ on using (REF ).", "Since $\\langle \\langle \\tau _{M,(\\alpha i)} \\gamma _{(\\mu r)} \\rangle \\rangle _0 & = & \\omega \\left(\\langle \\langle \\tau _{M,\\alpha } \\gamma _\\mu \\rangle \\rangle _{{}_0}^{\\hat{}} \\circ e_i \\circ e_r\\right)\\,,\\\\\\langle \\langle \\gamma _{(\\mu s)} \\tau _{N,(\\beta j)} \\tau _{K,(\\sigma k)} \\rangle \\rangle _{{}_0} & = &\\omega \\left(e_s \\circ \\langle \\langle \\gamma _\\mu \\tau _{N,\\beta } \\tau _{K,\\sigma } \\rangle \\rangle _{{}_0}^{\\hat{}}\\circ e_s \\circ e_j \\circ e_k \\right),$ the result follows.", "String Equation Again, on using (REF ) it follows that $\\langle \\langle \\mathcal {S}^\\mathcal {A} \\rangle \\rangle _g & = & - \\sum _{M,(\\alpha i)} {\\tilde{t}}^{(\\alpha ,i)}_M\\omega \\left[\\widehat{\\frac{\\partial \\mathcal {F}_g}{\\partial t^\\alpha _{M-1}}} \\circ e_i\\right]\\,,\\\\& = & \\omega \\left( \\langle \\langle \\mathcal {S} \\rangle \\rangle ^{\\hat{}}_g \\right)\\,.$ Since $\\mathcal {S}$ satisfies the string equation, $\\langle \\langle \\mathcal {S}^\\mathcal {A} \\rangle \\rangle _g & = & \\frac{1}{2} \\delta _{g,0} \\omega \\left[ \\sum _{\\alpha ,\\beta } \\hat{t^\\alpha _0} \\circ \\hat{t^\\beta _0} \\right]\\,,\\\\&=&\\frac{1}{2} \\delta _{g,0} \\sum _{(\\alpha ,i),(\\beta ,j)} \\eta _{(\\alpha i)(\\beta j)} t^{(\\alpha i)}_0 t^{(\\beta j)}_0\\,,$ using the definition of the lifting map and the fundamental property $\\omega (e_i \\circ e_j)=\\eta _{ij}\\,.$ Dilaton Equation Similarly, since $\\mathcal {D}$ satisfies the Dilaton equation, $\\langle \\langle \\mathcal {D}^\\mathcal {A} \\rangle \\rangle _g & = & \\omega \\left( \\langle \\langle \\mathcal {D} \\rangle \\rangle _g^{\\hat{}} \\right)\\,,\\\\& = & (2g-2) \\omega (\\hat{\\mathcal {F}}_g) - \\frac{1}{24} \\delta _{g,1} \\chi (V) \\omega (e_1)\\,,\\\\& = & (2g-2) \\mathcal {F}^\\mathcal {A}_g - \\frac{1}{24} \\delta _{g,1} \\chi ^\\mathcal {A}(V),$ where $\\chi ^\\mathcal {A}(V)=\\chi (V) \\omega (e_1)\\,.$ Remark 3.3 The above axioms do not include the big-phase space counterpart to the Euler vector field, but the same ideas may be applied if such a field exists on the original TQFT.", "The individual prepotentials may be combined into a single $\\tau $ -function $\\tau (t^\\alpha _N) = e^{\\sum \\hbar ^{g-1} \\mathcal {F}_g}.$ In the simplest case, when $dim \\mathcal {M}=1$ this defines a specific solution of the KdV hierarchy.", "The full connection between such $\\tau $ -functions and corresponding integrable hierarchies remains an important open problem.", "Since each prepotential $\\mathcal {F}_g$ lifts to prepotentials $\\mathcal {F}^\\mathcal {A}_g$ one may define a corresponding $\\tau $ -function $\\tau ^\\mathcal {A}(t^\\alpha _N) = e^{\\sum \\hbar ^{g-1} \\mathcal {F}^\\mathcal {A}_g}$ and it is clear that $\\tau ^\\mathcal {A} = \\omega \\left[\\hat{\\tau }\\right]\\,.$ It seems natural to conjecture that such a function should define a solution to a corresponding $\\mathcal {A}$ -valued dispersive integrable hierarchy.", "However, this first requires the development of a theory of such $\\mathcal {A}$ -valued hierarchies." ], [ "The role of the Frobenius form $\\omega $", "The Frobenius form $\\omega $ plays a vital role in the above constructions; without it one only has $\\mathcal {A}$ -valued objects.", "However, one can dispense with it and deal directly with such $\\mathcal {A}$ -valued objects and derive relations satisfied by them.", "For example, using the lifting map (REF ), one can define $\\mathcal {A}$ -valued `correlators' : $\\langle \\langle \\tau _{N,(\\alpha i)} \\ldots \\tau _{M,(\\beta j)} \\rangle \\rangle _g^{\\mathcal {A}} & = & \\left[\\frac{\\partial ~}{\\partial t^\\alpha _N} \\ldots \\frac{\\partial ~}{\\partial t^\\beta _M} \\mathcal {F}_g\\right]^{\\hat{}} \\circ e_i \\circ \\ldots \\circ e_j\\,,\\\\&=& \\langle \\langle \\tau _{N,\\alpha } \\ldots \\tau _{M,\\beta } \\rangle \\rangle _g^{\\hat{}}\\circ e_i \\circ \\ldots \\circ e_j\\,.$ It is straightforward to derive the following recursion relation: $\\Omega \\circ \\langle \\langle \\tau _{M+1, (\\alpha i)} \\tau _{N, (\\beta j)} \\tau _{K, (\\sigma k)} \\rangle \\rangle ^{\\mathcal {A}}_{{}_0} \\\\= \\eta ^{(\\mu r)(\\gamma s)}\\langle \\langle \\tau _{M,(\\alpha i)} \\tau _{0,(\\mu r)} \\rangle \\rangle ^\\mathcal {A}_{{}_0} \\circ \\langle \\langle \\tau _{0, (\\gamma s)} \\tau _{N, (\\beta j)} \\tau _{K, (\\sigma k)} \\rangle \\rangle ^{\\mathcal {A}}_{{}_0}\\,,$ where $\\Omega = \\eta ^{rs} e_r \\circ e_s\\,.$ If this element is invertible, then one can obtain a bona fide $\\mathcal {A}$ -valued recursion relation.", "We will not further develop such a theory here." ], [ "$\\mathcal {A}$ -valued dispersionless integrable systems", "It was shown by Dubrovin that, given a Frobenius manifold $\\mathcal {M}$ , one can construct an associated bi-Hamiltonian hierarchy of hydrodynamic type, known as the principal hierarchy, with the geometry of the manifold encoding the various components required in its construction.", "This hierarchy may be written as $\\begin{array}{lcr}\\displaystyle {\\frac{\\partial t^\\alpha }{\\partial T^{(N,\\sigma )}}} &= & \\displaystyle {\\mathcal {P}_1^{\\alpha \\beta } \\, \\frac{\\partial h_{(N,\\sigma )}}{\\partial t^\\beta }}\\,,\\\\&&\\\\& = &\\displaystyle {\\mathcal {P}_2^{\\alpha \\beta } \\, \\frac{\\partial h_{(N-1,\\sigma )}}{\\partial t^\\beta }}\\end{array}$ with (compatible) Hamiltonian operators $\\mathcal {P}^{\\alpha \\beta }_1=\\eta ^{\\alpha \\beta } \\frac{d~}{dX}\\,, \\qquad \\mathcal {P}^{\\alpha \\beta }_2= \\, g^{\\alpha \\beta } \\frac{d~}{dX} + \\Gamma ^{\\alpha \\beta }_\\gamma t^\\gamma _X\\,,$ where $g^{\\alpha \\beta }=c^{\\alpha \\beta }_{\\phantom{\\alpha \\beta }\\gamma } E^\\gamma $ is the intersection form on $\\mathcal {M}$ (and $\\Gamma ^{\\alpha \\beta }_\\gamma =-g^{\\alpha \\mu } \\Gamma ^\\beta _{\\mu \\gamma }$ ).", "The Hamiltonian densities $h_{(N,\\sigma )}$ come from the coefficients in the expansion of the deformed flat connection for the Dubrovin connection, $t_\\alpha (\\lambda ) = \\sum _{N=0}^\\infty h_{(N,\\alpha )} \\lambda ^N\\,,\\qquad h_{(0,\\alpha )}= \\eta _{\\alpha \\beta } t^\\beta \\,,$ and these satisfy the recursion relation $\\frac{\\partial ^2 h_{(N,\\sigma )}}{\\partial t^\\alpha \\partial t^\\beta }= c_{\\alpha \\beta }^{\\phantom{\\alpha \\beta }\\mu }({\\bf t}) \\frac{\\partial h_{(N-1,\\sigma )}}{\\partial t^\\mu }$ (together with certain normalization conditions).", "The Frobenius manifold $\\mathcal {M}_\\mathcal {A}$ will automatically inherit such a hierarchy by the very nature of it being a Frobenius manifold.", "However such a hierarchy is best written as an $\\mathcal {A}$ -valued system, with $m$ -$\\mathcal {A}$ -valued dependent fields rather than $mn$ -scalar-valued dependent fields.", "We begin by showing how the deformed flat variables on $\\mathcal {M}_\\mathcal {A}$ may be constructed from those on $\\mathcal {M}\\,.$ This is achieved by lifting and evaluation the Hamiltonian densities for $\\mathcal {M}\\,.$ Lemma 4.1 Let $h_{N,\\sigma }$ be the coefficients in the deformed flat connection on $\\mathcal {M}\\,.$ Then the functions $\\mathfrak {h}_{(N,\\sigma ,r)} = \\omega \\left( \\widehat{h_{(N,\\sigma )}} \\circ e_r \\right)$ satisfy the recursion relation $\\frac{\\partial ^2 \\mathfrak {h}_{(N,\\sigma ,r)}}{\\partial t^{(\\alpha i)} \\partial t^{(\\beta j)}} =c_{(\\alpha i)(\\beta j)}^{\\phantom{{(\\alpha i)(\\beta j)}}(\\gamma k)}\\frac{\\partial \\mathfrak {h}_{(N-1,\\sigma ,r)}}{\\partial t^{(\\gamma k)}}\\,.$ and the initial conditions $\\mathfrak {h}_{(0,\\sigma ,r)} = \\eta _{(\\sigma r)(\\mu s)} t^{(\\mu s)}$ and hence define the deformed flat coordinates on $\\mathcal {M}_\\mathcal {A}\\,.$ This is a straightforward calculation (we drop the $\\sigma $ -label on the various $h$ 's for clarity): We have $\\frac{\\partial \\widehat{h_N}}{\\partial t^{(\\alpha i)}}= \\widehat{\\left(\\frac{\\partial h_N}{\\partial t^\\alpha }\\right)} \\circ e_i$ and hence $\\frac{\\partial ^2 \\widehat{h_N}}{\\partial t^{(\\alpha i)} \\partial t^{(\\beta j)}}& = & \\widehat{\\left(\\frac{\\partial ^2 h_N}{\\partial t^\\alpha \\partial t^\\beta }\\right)} \\circ e_i \\circ e_j\\,,\\\\& = &\\widehat{\\left( c_{\\alpha \\beta }^{~~\\gamma }\\right)}\\circ \\widehat{ \\frac{\\partial h_{N-1}}{\\partial t^\\gamma } } \\circ e_i \\circ e_j\\,.$ Thus using $\\omega $ to evaluate this $\\mathcal {A}$ -valued expression gives $\\frac{\\partial ^2 \\mathfrak {h}_{(N,r)}}{\\partial t^{(\\alpha i)} \\partial t^{(\\beta j)}} & = &\\omega \\left(\\widehat{\\left(\\frac{\\partial ^2 h_N}{\\partial t^\\alpha \\partial t^\\beta }\\right)} \\circ e_i\\circ e_j \\circ e_r\\right)\\,,\\\\& = & \\left[ \\widehat{ c_{\\alpha \\beta }^{~~\\gamma } }\\right]^p c_{ij}^{~~q}\\omega \\left(\\widehat{ \\left(\\frac{\\partial h_{N-1}}{\\partial t^\\gamma }\\right) } \\circ e_p \\circ e_q \\circ e_r\\right)\\,,\\\\& = & \\underbrace{\\left[ \\widehat{ c_{\\alpha \\beta }^{~~\\gamma } }\\right]^pc_{ij}^{~~q} c_{pq}^{~~k} }_{c_{(\\alpha i)(\\beta j)}^{\\phantom{(\\alpha i)(\\beta j)} (\\gamma k)}}\\omega \\left(\\frac{\\partial \\widehat{h_{N-1}}}{\\partial t^{(\\gamma k)}} \\circ e_r\\right)\\,,\\\\& = & c_{(\\alpha i)(\\beta j)}^{\\phantom{(\\alpha i)(\\beta j)} (\\gamma k)}\\frac{\\partial \\mathfrak {h}_{(N-1,r)}}{\\partial t^{(\\gamma k)}}\\,.$ If $N=0$ , then, since $\\widehat{t^\\mu }=t^{(\\mu s)} e_s\\,,$ $\\mathfrak {h}_{(0,\\sigma ,r)} & = & \\omega \\left( \\widehat{ h_{(0,\\sigma )}} \\circ e_r\\right)\\,,\\\\& = & \\eta _{\\sigma \\mu } \\eta _{rs} t^{(\\mu s)} \\omega \\left(e_s \\circ e_r\\right)\\,,\\\\& = & \\eta _{(\\sigma r)(\\mu s)} t^{(\\mu s)}\\,,$ which is, as required, a Casimir function on $\\mathcal {M}_{\\mathcal {A}}\\,.$ In the obvious way, one can lift the operators $\\mathcal {P}_1\\,,\\mathcal {P}_2$ to $\\mathcal {A}$ -valued operators and obtain the following theorem: Theorem 4.2 The principal hierarchy on $\\mathcal {M}_\\mathcal {A}$ may be written in terms of $\\mathcal {A}$ -valued fields, densities and operators, as $\\begin{array}{lcr}\\displaystyle {\\frac{\\partial {\\widehat{t}^\\alpha }}{\\partial T^{(N,\\sigma , r)}}} & = &\\displaystyle {\\widehat{\\mathcal {P}_1^{\\alpha \\beta }} \\circ \\,\\frac{\\partial \\widehat{h_{(N,\\sigma )}}}{\\partial t^{(\\beta r)}}}\\,,\\\\&&\\\\& = & \\displaystyle {\\widehat{\\mathcal {P}_2^{\\alpha \\beta }} \\circ \\,\\frac{\\partial \\widehat{h_{(N-1,\\sigma )}}}{\\partial t^{(\\beta r)}}}\\,.\\end{array}$ First Hamiltonian Structure By definition, and on using previous results, $\\frac{\\partial t^{(\\alpha i)}}{\\partial T^{(N,\\sigma ,r)}} & = &\\eta ^{(\\alpha i)(\\beta j)} \\frac{d~}{dX} \\frac{\\partial \\mathfrak {h}_{(N,\\sigma ,r)}}{\\partial t^{(\\beta j)}}\\,,\\\\& = & \\eta ^{\\alpha \\beta } \\eta ^{ij} \\frac{d~}{dX}\\left[ \\widehat{ \\frac{\\partial h_{(N,\\sigma )}}{\\partial t^\\beta }}\\right]^k \\omega (e_k\\circ e_j\\circ e_r)\\,.$ Since $\\widehat{t^\\alpha } = t^{(\\alpha i)} e_i$ by definition, one obtains $\\frac{\\partial \\widehat{t^\\alpha } }{\\partial T^{(N,\\sigma ,r)}} & = & \\eta ^{\\alpha \\beta }\\frac{d~}{dX} \\left[ \\widehat{ \\frac{\\partial h_{(N,\\sigma )}}{\\partial t^\\beta }}\\right]^k \\,\\eta ^{ij} \\omega (e_k\\circ e_j\\circ e_r) e_i\\,,\\\\& = & \\eta ^{\\alpha \\beta } \\frac{d~}{dX} \\left\\lbrace \\widehat{ \\frac{\\partial h_{(N,\\sigma )}}{\\partial t^\\beta }}\\circ e_r \\right\\rbrace \\,,\\\\& = & \\widehat{\\eta ^{\\alpha \\beta }}\\circ \\frac{d~}{dX}\\widehat{ \\frac{\\partial h_{(N,\\sigma )}}{\\partial t^{(\\beta r)}}}\\,,$ since as the components of $\\eta $ are constants, $\\widehat{\\eta ^{\\alpha \\beta }} = \\eta ^{\\alpha \\beta } e_1\\,.$ Second Hamiltonian Structure The second Hamiltonian operator $\\mathcal {P}^{\\alpha \\beta }_2$ on $\\mathcal {M}$ takes the formWe ignore the precise normalization of the second Hamiltonian structure.", "We also assume here that the manifold $\\mathcal {M}$ is non-resonant.", "It is easy to show that if $\\mathcal {M}$ is non-resonant, then so is $\\mathcal {M}_\\mathcal {A}\\,.$ $\\mathcal {P}^{\\alpha \\beta }_2 = g^{\\alpha \\beta } \\frac{d~}{dX} + \\left(\\frac{d+1}{2} - q_\\beta \\right)c^{\\alpha \\beta }_{\\phantom{\\alpha \\beta }\\gamma } t^\\gamma _X$ and hence on $\\mathcal {M}_\\mathcal {A}\\,,$ $\\frac{\\partial t^{(\\alpha i)}}{\\partial T^{(N,\\sigma ,r)}} = \\left[g^{(\\alpha i)(\\beta j)} \\frac{d~}{dX} + \\left(\\frac{d+1}{2} - q_\\beta \\right)c^{(\\alpha i)(\\beta j)}_{\\phantom{(\\alpha i)(\\beta j)}(\\gamma k)} t^{(\\gamma k)}_X\\right]\\frac{ \\partial \\mathfrak {h}_{(N-1,\\sigma , r)}}{\\partial t^{(\\beta j)}}\\,.$ Note, since the Euler vector field on $\\mathcal {A}$ is trivial ($q_i=r_i=d_\\mathcal {A}=0$ ) it follows that $q_{(\\beta j)} = q_\\beta $ and $d$ is the same on both $\\mathcal {M}$ and $\\mathcal {M}_\\mathcal {A}\\,.$ Also, by definition, $g^{(\\alpha i)(\\beta j)} & = & c^{(\\alpha i)(\\beta j)}_{\\phantom{(\\alpha i)(\\beta j)}(\\gamma k)} E^{(\\gamma k)} \\,,\\\\& = & \\eta ^{\\beta \\mu } \\eta ^{js} \\left[ \\widehat{ c_{\\mu \\gamma }^{\\phantom{\\mu \\gamma }\\alpha } } \\right]^pc_{sk}^{\\phantom{sk}q} c_{pq}^{\\phantom{pq}i} (1-q_\\gamma ) t^{(\\gamma k)}\\,.$ For simplicity we will consider the first term in (REF ) only, the corresponding proof of the second term follows practically verbatim the proof of the first.", "Thus $g^{(\\alpha i)(\\beta j)} \\frac{d~}{dX} \\frac{ \\partial \\mathfrak {h}_{(N-1,\\sigma , r)}}{\\partial t^{(\\beta j)}} & = &\\left[ c^{\\alpha \\beta }_{\\phantom{\\alpha \\beta }\\gamma }\\right]^p c_{pk}^{\\phantom{pk}q} (1-q_\\gamma ) t^{(\\gamma k)}\\frac{d~}{dX}\\left[ \\widehat{ \\frac{\\partial h_{(N-1,\\sigma )}}{\\partial t^\\beta }} \\right]^d \\omega (e_d \\circ e_j \\circ e_r)\\,,\\\\& = & \\left[ \\widehat{g^{\\alpha \\beta }} \\right]^q c_q^{\\phantom{q}ij} \\frac{d~}{dX}\\left[ \\widehat{ \\frac{\\partial h_{(N-1,\\sigma )}}{\\partial t^\\beta }} \\right]^d c_{dr}^{\\phantom{dr}s} \\eta _{sj}\\,,$ since $\\widehat{g^{\\alpha \\beta }} = \\widehat{c^{\\alpha \\beta }_{\\phantom{\\alpha \\beta }\\gamma }} \\circ (1-q_\\gamma )t^{(\\gamma q)} e_q\\,.$ On using the associative and commutative properties of the multiplication, and on contracting with $e_i$ one obtains $g^{(\\alpha i)(\\beta j)} \\frac{d~}{dX} \\frac{ \\partial \\mathfrak {h}_{(N-1,\\sigma , r)}}{\\partial t^{(\\beta j)}} e_i& = &\\left[ \\widehat{g^{\\alpha \\beta }} \\right]^qc_{qs}^{\\phantom{qs}i} \\frac{d~}{dX}\\left[ \\widehat{ \\frac{\\partial h_{(N-1,\\sigma )}}{\\partial t^\\beta }} \\circ e_r \\right]^s e_i\\,,\\\\& = & \\widehat{g^{\\alpha \\beta }} \\circ \\frac{d~}{dX}\\left[ \\widehat{ \\frac{ \\partial h_{(N-1,\\sigma )}}{\\partial t^\\beta }} \\circ e_r \\right]\\,,\\\\& = & \\widehat{g^{\\alpha \\beta }} \\circ \\frac{d~}{dX}\\widehat{ \\frac{\\partial h_{(N-1,\\sigma )}}{\\partial t^{(\\beta r)}} }\\,.$ Note that these flows on $\\mathcal {M}_\\mathcal {A}$ simplify if $r=1\\,.$ Example 4.3 If $\\dim \\mathcal {M}=1$ and $r=1$ one obtains the bi-Hamiltonian structures from the $\\mathcal {A}$ -valued Mongé equation $\\mathcal {U}_T= \\mathcal {U}\\circ \\mathcal {U}_X$ with conserved densities $\\mathfrak {h}_N = \\frac{1}{(N+1)!}", "\\omega ( \\underbrace{\\mathcal {U} \\circ \\cdots \\circ \\mathcal {U}}_{N+1\\rm {~terms}} )\\,.$ The form of the flows in Theorem REF is somewhat hybrid in nature and to rewrite them as a genuine $\\mathcal {A}$ -valued bi-Hamiltonian system one must introduce the variational derivative with respect to an $\\mathcal {A}$ -valued field.", "Such a derivative was introduced in [19] and is defined by the equation $\\langle \\delta \\mathcal {H}; v \\rangle = \\left.\\frac{d~}{d\\epsilon } \\mathcal {H}\\left[ \\widehat{u^\\alpha } + \\epsilon \\widehat{v^\\alpha } \\right]\\right\|_{\\epsilon =0},$ where $\\mathcal {H}= \\int \\omega (\\widehat{h}) \\,dX\\,.$ With this the flows may be written as an $\\mathcal {A}$ -valued bi-Hamiltonian system.", "Corollary 4.4 The flows given in Theorem REF may be written as $\\begin{array}{lcr}\\displaystyle {\\frac{\\partial {\\widehat{t}^\\alpha }}{\\partial T^{(N,\\sigma , r)}}} & = &\\displaystyle {\\widehat{\\mathcal {P}_1^{\\alpha \\beta }} \\circ \\,\\frac{\\delta \\mathcal {H}_{(N,\\sigma ,r)}}{\\delta \\widehat{t^\\beta }}}\\,,\\\\&&\\\\& = & \\displaystyle {\\widehat{\\mathcal {P}_2^{\\alpha \\beta }} \\circ \\,\\frac{\\delta \\mathcal {H}_{(N-1,\\sigma ,r)}}{\\delta \\widehat{t^\\beta }}}\\,,\\end{array}$ where $\\mathcal {H}_{(N,\\sigma ,r)} = \\int \\omega \\left(\\widehat{h_{(N,\\sigma ,r)}}\\right) \\, dX\\,.$ From (REF ) , $\\langle \\delta \\mathcal {H}_{(N,\\sigma ,r)} ; \\widehat{v^\\beta } \\rangle & = & \\int \\omega \\left( \\frac{ \\partial \\widehat{h_{(N,\\sigma ,r)}}}{\\partial t^{(\\beta j)}} v^{(\\beta j)} \\circ e_r \\right) \\, dX\\,,\\\\& = &\\int \\omega \\left(\\widehat{ \\frac{\\partial h_{(N,\\sigma ,r)}}{\\partial t^\\beta } }\\circ e_r \\circ \\underbrace{v^{(\\beta j)} e_j}_{\\widehat{v^\\beta }} \\right) \\, dX\\,,$ and hence $\\frac{\\delta \\mathcal {H}_{(N,\\sigma ,r)}}{\\delta \\widehat{t^\\beta }} =\\frac{ \\partial \\widehat{h_{(N,\\sigma )}}}{\\partial t^{\\beta }} \\circ e_r \\,.$ With this, the result follows immediately." ], [ "Polynomial (inverse)-metrics and bi-Hamiltonian structures", "Since all 1-dimensional metrics are flat, it follows immediately from the Dubrovin-Novikov [7] Theorem that the operator ${\\mathcal {P}} = f(u) \\frac{d~}{dX} + \\frac{1}{2} f^\\prime (u)$ is Hamiltonian.", "In this section we study the case where $f$ is a polynomial.", "Example 4.5 Applying the lifting procedures to the operator $\\mathcal {P}$ defined by the linear function $f(u)=u+\\lambda $ results in the linear operator ${\\mathcal {P}^{ij}} = \\left\\lbrace c^{ij}_k u^k_X \\frac{d~}{dX} + \\frac{1}{2} c^{ij}_k u^k_X \\right\\rbrace + \\lambda \\frac{d~}{dX}$ defined on the Frobenius algebra $\\mathcal {A}\\,.$ This is the Hamiltonian operator first constructed by Balinski and Novikov [2].", "Similarly, more complicated examples by be obtains by starting with more general polynomials and applying the same procedure.", "These more general examples appear to be in contradiction to an alternative method of constructing Hamiltonian operators via bi-Hamiltonian recursion.", "The recursion operator constructed from the bi-Hamiltonian pencil (REF ) takes the form ${\\mathcal {R}}^i_j = c^i_{jk} u^k + \\frac{1}{2} c^i_{jk} u^k_X \\left( \\frac{d~}{dX} \\right)^{-1}\\,.$ Suppose one has a (local) Hamiltonian operator $\\mathcal {P}_n = g^{ij}_{(n)}(u) \\frac{d~}{dX} + \\Gamma ^{ij}_{(n)k}(u) u^k_X$ with $g^{ij}_{(0)} = \\eta ^{ij}\\,, \\Gamma ^{ij}_{(0)k}=0\\,.$ Applying the operator $\\mathcal {R}$ gives $\\left(\\mathcal {R}\\mathcal {P}_{(n)}\\right)^{ij} = \\left\\lbrace g^{ij}_{(n+1)}(u) \\frac{d~}{dX} + \\Gamma ^{ij}_{(n+1)k}(u) u^k_X\\right\\rbrace + {\\rm {non-local~terms}}$ and we now define $\\mathcal {P}_{(n+1)}$ to be the local-term in the above expression.", "This gives the recursion scheme: $g^{ij}_{(n+1)} & = & 2 c^{ip}_r u^r \\eta _{pq} g^{qj}_{(n)}\\,,\\\\\\Gamma ^{ij}_{(n+1)k} & = & 2 c^{ip}_r u^r \\eta _{pq} \\Gamma ^{qj}_{(n)k} + c^{ip}_k \\eta _{pq} g^{qj}_{(n)}\\,.$ It is a tedious, through straightforward exercise to show that, if the pair $\\lbrace g_{(n)},\\Gamma _{(n)}\\rbrace $ defines a flat metric, then so does $\\lbrace g_{(n+1)},\\Gamma _{(n+1)}\\rbrace $ , and hence $\\mathcal {P}_{(n)}$ is a local Hamiltonian operator for all $n\\,.$ The above lifting procedure circumvents such a direct computational approach.", "The fact that the local and non-local parts of the Hamiltonian operator define separate, compatible, Hamiltonian operator is of course, well known (see, for example, [10])." ], [ "$\\mathcal {A}$ -valued dispersive integrable systems", "In this section the above ideas are extended to include dispersive, higher-order, dispersive systems." ], [ "$\\mathcal {A}$ -valued dispersive integrable systems", "The main result of this section is the following theorem: Theorem 5.1 Let $u=\\lbrace u^\\alpha (x,t)\|\\alpha =1,\\cdots ,n \\rbrace $ .", "Let $u^\\alpha _t=K^\\alpha (u,u_x,\\cdots ) $ be a Hamiltonian system with the Hamiltonian $H[u]$ , then the corresponding $\\mathcal {A}$ -valued system $\\widehat{u^\\alpha _t}=\\widehat{K^\\alpha (u,u_x,\\cdots )} $ is also Hamiltonian with the Hamiltonian $\\mathcal {H}[\\widehat{u}]=\\omega \\,\\left(\\widehat{H[u]}\\right)$ .", "The proof is very similar to those done in section .", "Without loss of generality, we assume that the system (REF ) can be written as $u^\\alpha _t=\\lbrace u^\\alpha , H[u]\\rbrace =\\mathcal {P}^{\\alpha \\beta }\\dfrac{\\delta h}{\\delta u^\\beta }, \\quad H[u]=\\int h(u)dx,$ where $\\mathcal {P}^{\\alpha \\beta }$ is a Hamiltonian operator.", "So the system (REF ) reads $\\widehat{u^\\alpha _t}=\\widehat{\\mathcal {P}^{\\alpha \\beta }}\\circ \\widehat{\\dfrac{\\delta h}{\\delta u^\\beta }}.$ Let $\\mathcal {H}[\\widehat{u}]=\\int \\mathfrak {h}(\\widehat{u})dx,\\quad \\mathfrak {h}(\\widehat{u})=\\omega \\,\\left(\\widehat{h(u)}\\right).$ With respect to an $\\mathcal {A}$ -valued field, the variational derivative $\\dfrac{\\delta \\mathfrak {h}}{\\delta \\widehat{u^\\beta }}$ is defined by the formula, essentially due to [19], $\\omega \\,\\int \\left(\\dfrac{\\delta \\mathfrak {h}}{\\delta \\widehat{u^\\beta }}\\circ \\widehat{\\delta u^\\beta } \\right)dx= \\left.\\frac{d}{d\\epsilon }\\right\|_{\\epsilon =0}\\mathcal {H}\\left[ \\widehat{u^\\beta } + \\epsilon \\widehat{\\delta u^\\beta } \\right].$ Observe that $&& \\left.\\frac{d}{d\\epsilon }\\mathcal {H}\\left[ \\widehat{u^\\beta }+ \\epsilon \\widehat{\\delta u^\\beta } \\right]\\right\|_{\\epsilon =0}=\\left.\\frac{d}{d\\epsilon }\\right\|_{\\epsilon =0}\\omega \\,\\left(\\int h(\\widehat{u^\\beta }+ \\epsilon \\widehat{\\delta u^\\beta }) dx\\right)\\,,\\\\&=&\\omega \\,\\left(\\widehat{\\left.\\frac{d}{d\\epsilon }\\right\|_{\\epsilon =0} H[{u^\\beta }+ \\epsilon {\\delta u^\\beta }]}\\right)=\\omega \\,\\left(\\int \\left(\\widehat{\\dfrac{\\delta h}{\\delta u^\\beta }}\\circ \\widehat{\\delta u^\\beta }\\right)dx \\right) \\nonumber $ from which follows $\\dfrac{\\delta \\mathfrak {h}}{\\delta \\widehat{u^\\beta }}=\\widehat{\\dfrac{\\delta h}{\\delta u^\\beta }}.$ For two functionals $\\mathcal {F}[\\widehat{u}]=\\int \\mathfrak {f}(\\widehat{u})dx,\\quad \\mathcal {G}[\\widehat{u}]=\\int \\mathfrak {g}(\\widehat{u})dx,$ with $\\mathfrak {f}(\\widehat{u})=\\omega \\,\\left(\\widehat{f(u)}\\right)$ and $\\mathfrak {g}(\\widehat{u})=\\omega \\,\\left(\\widehat{g(u)}\\right)$ , we define a bilinear bracket as $\\left\\lbrace \\mathcal {F}[\\widehat{u}], \\mathcal {G}[\\widehat{u}]\\right\\rbrace _{\\mathcal {A}}=\\omega \\,\\left(\\int \\dfrac{\\delta \\mathfrak {g}}{\\delta \\widehat{u^\\alpha }}\\circ \\widehat{\\mathcal {P}^{\\alpha \\beta }} \\circ \\dfrac{\\delta \\mathfrak {g}}{\\delta \\widehat{u^\\beta }} dx\\right).$ By using the definition of the hat map and (REF ), we rewrite the bracket (REF ) as $\\left\\lbrace \\mathcal {F}[\\widehat{u}], \\mathcal {G}[\\widehat{u}]\\right\\rbrace _{\\mathcal {A}}=\\omega \\,\\widehat{\\lbrace F[u], G[u]\\rbrace },$ where $F[u]=\\int f(u)dx$ and $G[u]=\\int g(u)dx$ .", "Consequently, we conclude that the bracket $\\lbrace ~,~\\rbrace _{\\mathcal {A}}$ is also a Poisson bracket.", "Furthermore using (REF ), the system (REF ) could be written as $u^{(\\alpha ,i)}_t=\\lbrace u^{(\\alpha ,i)}, \\mathcal {H}[\\widehat{u}]\\rbrace _{\\mathcal {A}},\\quad \\mathcal {H}[\\widehat{u}]=\\int \\omega \\,\\left(\\widehat{h(u)}\\right) dx.\\nonumber $ We thus complete the proof of the theorem.", "Corollary 5.2 The $\\mathcal {A}$ -valued version of the Hamiltonian system $u^\\alpha _t=\\lbrace u^\\alpha , H[u]\\rbrace $ is also Hamiltonian and given by $u^{(\\alpha ,i)}_t=\\lbrace u^{(\\alpha ,i)}, \\mathcal {H}[\\widehat{u}]\\rbrace _{\\mathcal {A}},\\quad \\mathcal {H}[\\widehat{u}]=\\omega \\,\\left(\\widehat{H[u]}\\right).\\nonumber $ These results extend naturally to the lifts of bi-Hamiltonian structures, yielding $\\mathcal {A}$ -valued bi-Hamiltonian operators." ], [ "mKdV and (modified)-Camassa-Holm bi-Hamiltonian structures", "The celebrated Muira transformation maps the second Hamiltonian operator of the KdV hierarchy to constant form.", "Explicitly, if $\\mathcal {H}^{KdV}_1 = -D\\,, \\qquad \\mathcal {H}^{KdV}_2 = -D^3 + 2 u D + u_X$ (in this section we write $D$ in place of $\\frac{d~}{dX}$ ).", "Then applying the Miura map $u=-v_X+\\frac{1}{2} v^2$ gives $\\mathcal {H}^{KdV}_2 = \\mathcal {H}^{mKdV}_1=D\\,,$ and the second mKdV structure is then obtained by applying the same map to the third KdV Hamiltonian structure defined by bi-Hamiltonian recursion ($\\mathcal {H}_3=\\mathcal {H}_2 \\mathcal {H}^{-1} \\mathcal {H}_2$ ), yielding the non-local operator $\\mathcal {H}_2^{mKdV} = D^3 -D v D^{-1} v D\\,.$ Just as the Balinski-Novikov structures on the Frobenius algebra $\\mathcal {A}$ may be obtained by lifting, so $\\mathcal {A}$ -valued non-local operators may be found by using the above results.", "Proposition 5.3 The $\\mathcal {A}$ -valued operators, defined by lifting $\\mathcal {H}^{mKdV}_1$ and $\\mathcal {H}^{mKdV}_2$ to the Frobenius algebra $\\mathcal {A}$ are: $\\left(\\mathcal {H}^{mKdV}_1\\right)^{ij} & = & \\eta ^{ij} D\\,,\\\\\\left(\\mathcal {H}^{mKdV}_2\\right)^{ij} & = & \\eta ^{ij} D^3 - c^{ij}_p c^p_{mn} D v^m D^{-1} v^n D\\,.$ These may also be obtained using the $\\mathcal {A}$ -valued Miura map $u = -v_x + \\frac{1}{2} v \\circ v\\,.$ These results follow directly by applying the results in section REF .", "They may also be obtained by direct (but tedious) calculation.", "The form of the $\\mathcal {A}$ -valued Miura map is obvious, and again can be verified by direct calculations.", "While not developed here, one should be able to applying lifting results directly to scalar-Miura maps, with all the actions commuting.", "$\\mathcal {A}$ -valued KdV and mKdV equations can now easily be constructed, the KdV examples coinciding with the examples constructed in [21].", "Here we construct $\\mathcal {A}$ -valued modified Camassa-Holm equations.", "Example 5.4 One may apply the standard tri-Hamiltonian `tricks' [9] to obtain the $\\mathcal {A}$ -valued bi-Hamiltonian pair: $\\mathcal {C}^{ij}_1 & = & \\eta ^{ij} (D^3+D)\\,,\\\\\\mathcal {C}^{ij}_2 & = & c^{ij}_p c^p_{mn} D v^m D^{-1} v^n D\\,.$ Starting with the lifted Casimir of the scalar operator $\\mathcal {C}_1$ one obtains the multi-component modified Camassa-Holm equation $v_T + v_{XXT} & = &\\phantom{+} \\frac{1}{2} v_{XXX} \\circ v_X \\circ v_X + v_{XX} \\circ v_{XX} \\circ v_x\\\\&& + \\frac{1}{2} v_{XXX} \\circ v \\circ v + 2 v_{XX} \\circ v_X \\circ v + \\frac{1}{2} v_X \\circ v_X \\circ v_X\\\\&& + \\frac{3}{2} v_X \\circ v \\circ v\\,.$ Note we use the adjective `modified' in the original, strict, sense of equations obtained from an original, unmodified, equation via the action of a Miura map, rather than in the looser sense of just modifying `by-hand' the terms that appear in the equation.", "Two-component examples may easily be found using one of the algebras constructed in example REF ." ], [ "Conclusions", "Central to the results of this paper is the use of a distinguished coordinate system, namely the flat coordinates of the Frobenius manifold $\\mathcal {M}\\,.$ But the lifting procedure may be applied to any geometric structure which is analytic in some fixed coordinate system.", "However, such results loose some of their coordinate free character: one is using a specific coordinate system to define new objects then relying in their tensorial properties to define then properly in an arbitrary system of coordinates.", "As an example of this, one can apply the idea to $F$ -manifolds defined by Hertling and Manin [12].", "Proposition 6.1 Consider an $F$ -manifold with structure functions $c_{\\alpha \\beta }^{\\phantom{\\alpha \\beta }\\gamma }(t)$ analytic in the coordinates $\\lbrace t^\\alpha \\rbrace \\,.$ Let $\\mathcal {A}$ be an arbitrary Frobenius algebra.", "Then the structure functions defined by the lifted multiplication (REF ) $c_{(\\alpha i)(\\beta j)}^{\\phantom{{(\\alpha i)(\\beta j)}}(\\gamma k)}= \\left[ \\widehat{c_{\\alpha \\beta }^{~~\\gamma }} \\right]^p \\, c_{ij}^{~~q} c_{pq}^{~~k}$ define an $F$ -manifold.", "The proof is straightforward and will be omitted.", "The link between $F$ -manifolds and equations of hydrodynamic type has been explored by a number of authors [22], [18] so one should be able to apply the idea of this paper to construct their $\\mathcal {A}$ -valued counterparts.", "In quantum cohomology, the tensor product of Frobenius manifolds generalizes the classical Künneth product formula.", "In singularity theory it corresponds to the direct sum of singularities.", "If one of the manifolds is trivial then this descriptions degenerates - there is no parameter space of versal deformations.", "However, one could try to construct an $\\mathcal {A}$ -valued singularity theory.", "This is purely speculative, but Arnold has constructed a theory of versal deformations of matrices [1] but it remains to see if this is what would be required.", "As remarked earlier, since $\\mathcal {M}_\\mathcal {A}$ is a Frobenius manifold in its own right, one can apply the deformation theory developed by Dubrovin and Zhang [8] directly to the hydrodynamic flows given in Theorem REF .", "But central to this approach is the existence of a single $\\tau $ -function.", "However the deformations/dispersive systems constructed in sections and have $\\mathcal {A}$ -valued $\\tau $ -functions.", "Thus we have two distinct deformation procedures, unless they are connected by some set of transformations.", "It may be possible to construct a deformation theory along the lines of [8] but with an $\\mathcal {A}$ -valued $\\tau $ -function.", "This paper has concentrated on Frobenius algebra-valued integrable systems, via their Hamiltonian structure.", "Other approaches to integrability - the structure of $\\mathcal {A}$ -valued Lax equations, for example, have not been considered here.", "Part of such a theory have been constructed by the authors in [23] where an $\\mathcal {A}$ -valued KP hierarchy is constructed via such $\\mathcal {A}$ -valued Lax equations and operators.", "In a different direction there are many other algebra valued generalizations of KdV equation, from Jordan algebra to Novikov algebra-valued systems [20], [21], [24], [25].", "Whether such algebra-valued systems can be combined with the theory of Frobenius manifolds remains an open question.", "Developing a theory which encompasses the non-commutative/non-local hierarchies, such as the original matrix KdV equation (REF ) would be of considerable interest and would encompass the theory developed in this paper.", "Acknowledgments.", "D. Zuo is grateful to Professors Qing Chen, Yi Cheng and Youjin Zhang for constant supports and also thanks the University of Glasgow for the hospitality.", "The research of D. Zuo is partially supported by NSFC(11271345, 11371338), NCET-13-0550 and the Fundamental Research Funds for the Central Universities." ], [ "Appendix", "The lifting operation (Definition REF ) was defined only for analytic functions.", "However this may be extended to a wider class of functions, in particular rational functions.", "This observation is based on the following: Lemma 6.2 A generic element $\\kappa \\in \\mathcal {A}$ is invertible.", "By a similar argument laid out in [6], the Frobenius algebra $\\mathcal {A}$ is isomorphic to orthogonal direct sum of a semi-simple and a nilpotent algebra, $\\mathcal {A}=\\mathcal {A}_s \\oplus \\mathcal {A}_n$ with $\\mathcal {A}_s$ having a basis $\\pi _1,\\cdots ,\\pi _s\\,,$ with $\\pi _j\\circ \\pi _l=\\delta _{jl}\\pi _j\\,.$ Suppose the unity element of the algebra takes the form $e=\\sum _{i=1}^s a_i \\pi _i + n$ where $n\\in \\mathcal {A}_n$ and so $n^N=0\\,.$ Then $n \\circ \\pi _i = (1-a_i) \\pi _i$ and hence $(1-a_i)^N \\pi _i = 0\\,.$ Thus $a_i=1$ and $n \\circ \\pi _i=0$ .", "Since $e=e^N$ it follows that the unity element takes the form $e=\\sum _{i=1}^s \\pi _i\\,.$ Writing a generic element $\\kappa \\in \\mathcal {A}$ as $\\kappa = \\pi + \\mu $ (with $\\pi \\in \\mathcal {A}_s\\,,\\mu \\in \\mathcal {A}_n$ ) then $(\\kappa -\\pi )^N=0$ for some $N\\,.$ Expanding this yields $\\pi ^N = \\kappa \\circ \\Xi (\\kappa ,\\pi )$ for some funtion $\\Xi \\in \\mathcal {A}\\,.$ Since $\\pi ^N\\in \\mathcal {A}_s$ is invertible (generically) it follow that $\\kappa \\circ \\left\\lbrace \\Xi (\\kappa ,\\pi ) \\circ \\pi ^{-N} \\right\\rbrace = e\\,.$ Hence the result." ] ]	1403.0021
[ [ "On group gradings on PI-algebras" ], [ "Abstract We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \\leq exp(W)^K$.", "A G-grading $W = \\bigoplus_{g \\in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \\neq 0$ for any $r \\geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$." ], [ "Introduction", "In the last two decades there were significant efforts to extend important results in the theory of polynomial identities for (ordinary) associative algebras to $G$ -graded algebras, where $G$ is a finite group, and more generally to $H$ -comodule algebras where $H$ is a finite dimensional Hopf algebra.", "For instance Kemer's representability theorem and the solution of the Specht problem were established for $G$ -graded associative algebras over a field of characteristic zero (see [5], [24]).", "Recall that Kemer's representability theorem says that any associative PI-algebra over a field $F$ of characteristic zero is PI-equivalent to the Grassmann envelope of a finite dimensional $\\mathbb {Z}_{2}$ -graded algebra $A$ over some field extension $L$ of $F$ (see below the precise statement and Proposition ).", "Another instance of these efforts is the proof of Amitsur's conjecture which was originally proved for ungraded associative algebras over $F$ by Giambruno and Zaicev [14], and was extended to the context of $G$ -graded algebras by Giambruno, La Mattina and the first named author of this article (see [4], [12]) and considerable more generally for $H$ -comodule algebras by Gordienko [17].", "Amitsur's conjecture states that the sequence $c_{n}^{1/n}$ , where $c_{n}=c_{n}(W)$ is the $n$ th term of the codimension sequence of $W$ , has an integer limit (denoted by $\\exp (W)$ ).", "In [2] a different point of view was considered (in combining PI-theory and $G$ -gradings, still under the condition that $G$ is finite), namely asymptotic PI-theory was applied in order to prove invariance of the order of the grading group on an associative algebra whenever the grading is minimal regular (as conjectured by Bahturin and Regev [6]).", "In fact, it is shown there that the order of the grading group coincides with $\\exp (W)$ .", "Suppose now $G$ is arbitrary (i.e.", "not necessarily finite).", "Our goal in this paper, roughly speaking, is to exploit the invariant $\\exp (W)$ of the algebra $W$ , in order to put an effective bound on the minimal index of an abelian subgroup of $G$ whenever the algebra $W$ admits a $G$ -gradings satisfying a natural condition which we call “nondegenerate” (see Definition ).", "Our results extend considerable known results for PI group algebras (which are obviously nondegenerately $G$ -graded).", "Let us remark here that a big part of our analysis is devoted to the case where the group $G$ is finite (a case where Kemer and asymptotic PI theory can be applied) and then we pass to infinite groups.", "In this paper we only consider fields of characteristic zero.", "Let $W$ be an associative PI-algebra over a field $F$ .", "Suppose $W\\cong \\bigoplus _{g \\in G}W_{g}$ is $G$ -graded where $G$ is arbitrary.", "Definition 1.1 We say the $G$ -grading on $W$ is nondegenerate if for any positive integer $r$ and any tuple $(g_{1},\\ldots , g_{r}) \\in G^{(r)}$ , we have $W_{g_{1}}W_{g_{2}}\\cdots W_{g_{r}}\\ne 0$ .", "Theorem 1.2 $(Main$ $theorem)$ There exists an integer $K$ such that for any PI-algebra $W$ and for any \"nondegenerate” $G$ -grading on $W$ by any group $G$ , there exists an abelian subgroup $U$ of $G$ with $[G:U] \\le \\exp (W)^{K}$ .", "It is known (and not difficult to prove; see Lemma REF ) that if a group $G$ has an abelian subgroup of index $n$ , then it contains a characteristic abelian subgroup whose index is bounded by a function of $n$ .", "We therefore have the following corollary.", "Corollary 1.3 There exists a function $f:\\mathbb {N}\\rightarrow \\mathbb {N}$ such that for any PI-algebra $W$ and for any \"nondegenerate” $G$ -grading on $W$ by any group $G$ , there exists a characteristic abelian subgroup $U$ of $G$ with $[G:U] \\le f(\\exp (W))$ .", "In order to put our main result in an “appropriate” context, we recall $(i)$ different type of $G$ -gradings on associative algebras $(ii)$ three conditions on groups which are closely related to the content of the main theorem, namely $n$ -permutability, $n$ -rewritability and $PI_{n}$ (the group algebra $FG$ satisfies a polynomial identity of degree $n$ , $char(F)=0$ ).", "A $G$ -grading on $W\\ne 0$ is called strong if $W_{g}W_{h}=W_{gh}$ for every $g,h \\in G$ .", "Note that this condition is considerable stronger than a nondegenerate grading.", "For instance, the well known $\\mathbb {Z}_{2}$ -grading on the infinite dimensional Grassmann algebra is nondegenerate but not strong.", "The fact that the $\\mathbb {Z}_{2}$ -grading on the Grassmann algebra is nondegenerate will play an important role in the proof of the main theorem.", "Strong grading is considerably weaker than crossed product grading which requires that every homogeneous component has an invertible element (e.g.", "group algebras).", "In the other direction we may consider conditions on $G$ -gradings which are weaker than nondegenerate $G$ -gradings as $G$ -gradings where $W_{g}\\ne 0$ for every $g \\in G$ (call it connected grading).", "A somewhat stronger condition to the latter but yet weaker than nondegenerate grading is a condition which we call bounded nondegenerate: by definition a $G$ -grading on an algebra $W$ is bounded nondegenerate if any product of homogeneous components $W_{g_{1}}W_{g_{2}}\\cdots W_{g_{r}}$ does not vanish unless $r > r_{0}$ for some (large) fixed integer $r_{0}$ .", "We thus have crossed product grading $\\Rightarrow $ strong grading $\\Rightarrow $ nondegenerate grading $\\Rightarrow $ bounded nondegenerate grading $\\Rightarrow $ connected grading.", "In section we show that if a PI algebra $W$ is “bounded nondegeneratly” $G$ -graded than the main theorem is false in general.", "Definition 1.4 (see [10], [11], [21]) Let $n>1$ be an integer.", "We say that a group $G$ is $n$ -permutable (resp.", "$n$ -rewritable), denoted by $P_{n}$ (resp.", "$Q_{n})$ , if for any $n$ -tuple $(g_{1},\\ldots , g_{n}) \\in G^{(n)}$ there exists a nontrivial permutation $\\sigma \\in Sym(n)$ (resp.", "distinct permutations $\\sigma , \\tau \\in Sym(n)$ ) such that $g_{1}g_{2}\\cdots g_{n}= g_{\\sigma (1)}g_{\\sigma (2)}\\cdots g_{\\sigma (n)} \\in G$ (resp.", "$g_{\\sigma (1)}g_{\\sigma (2)}\\cdots g_{\\sigma (n)}= g_{\\tau (1)}g_{\\tau (2)}\\cdots g_{\\tau (n)} \\in G).$ We say that a group $G$ satisfies $PI_{n}$ if the group algebra $FG$ satisfies a (multilinear) identity of degree $n$ (it is well known that since $F$ is a field of characteristic zero, the $T$ -ideal of identities is generated by multilinear polynomials).", "Clearly, $P_{n} \\Rightarrow P_{n+1}$ , $Q_{n} \\Rightarrow Q_{n+1}$ and $P_{n} \\Rightarrow Q_{n}$ .", "We say that group is permutable (resp.", "rewritable, PI), if it is $n$ -permutable (resp.", "$n$ -rewritable, $PI_{n}$ ) for some $n$ .", "We denote (with a slight abuse of notation) by $P$ , $Q$ , $PI$ the families of all permutable, rewritable or PI groups.", "It was proved in [11] that if a group is $n$ -rewritable then it is $m$ -permutable where $m$ is bounded by a function of $n$ .", "As for the condition $PI_{n}$ , it is easy to show that if $FG$ satisfies a (multilinear) polynomial identity of degree $n$ then the group $G$ is $n$ -permutable and in particular $n$ -rewritable (indeed, if $f(x_1,\\ldots ,x_n)=x_1\\cdots x_n + \\sum _{e\\ne \\sigma \\in Sym(n)}\\alpha _{\\sigma }x_\\sigma (1)\\cdots x_\\sigma (n)$ , $\\alpha _{\\sigma }\\in F$ , is a multilinear identity of $FG$ and $(g_1,\\ldots ,g_{n}) \\in G^{(n)}$ is any $n$ th tuple, the evaluation $x_{i}=g_{i}$ , $i=1,\\ldots ,n$ , yields $g_1\\cdots g_n =g_{\\sigma (1)}\\cdots g_{\\sigma (n)}$ for some $e \\ne \\sigma \\in Sym(n)$ ).", "Thus we have that $PI_{n} \\Rightarrow P_{n} \\Rightarrow Q_{n}$ .", "As for the reverse direction of arrows the following is known (see [11]).", "$Q_{n}$ is strictly weaker than $P_{n}$ (although, as mentioned above, there exists a function $f$ such that $Q_{n}\\Rightarrow P_{f(n)}$ ).", "$P_{n} \\nRightarrow PI_{m}$ for any $n$ and $m$ .", "In particular it is known that if $G$ satisfies $PI_{m}$ , then $G$ has a finite index abelian subgroup whose index is bounded by a function of $m$ whereas for any $n > 2$ there exists an infinite family of finite groups $\\lbrace G_{i}\\rbrace _i$ which satisfy $P_{n}$ , whose PI degree is $d_{i}$ and $\\lim d_{i}=\\infty $ (the PI degree of $G$ is the minimal degree of a nontrivial polynomial identity of $FG$ ).", "Remark 1.5 As for the existence of a finite index abelian subgroup in $G$ and the permutability or rewritability conditions there is an interesting distinction between finitely/nonfinitely generated groups.", "If $G$ is finitely generated and satisfies $P_{n}$ (or $Q_{n}$ ) then it has an abelian subgroup of finite index (note however, as mentioned above, the index is not bounded by a function of $n$ ; see example in section ).", "If $G$ is not finitely generated, it may not have a finite index abelian subgroup.", "However it does have a characteristic subgroup $H$ whose index $[G:H]$ is bounded by a function of $n$ and whose commutator subgroup $H^{\\prime }$ is finite, and its order is bounded by a function of $n$ .", "In view of the above considerations it is natural to introduce the following condition on a group $G$ .", "Definition 1.6 Let $G$ be any group.", "We say that $G$ satisfies $T_{n}$ if there exists a PI algebra $W$ of PI degree $n$ which admits a nondegenerate $G$ -grading.", "We say that $G$ has $T$ if it has $T_{n}$ for some $n$ .", "It is easy to see that the argument which shows $PI_{n}\\Rightarrow P_{n}$ shows also that $T_{n} \\Rightarrow P_{n}$ .", "This simple fact will play an important role while extending the proof of the main theorem from finitely generated residually finite groups to arbitrary finitely generated groups.", "Note that since the group algebra $FG$ is nondegenerately $G$ -graded we have $PI_{n} \\Rightarrow T_{n}$ .", "In the other direction it follows from our main theorem that if $G$ satisfies $T_{n}$ , then $G$ has $PI_{m}$ for some $m$ (indeed, $G$ is abelian by finite and hence, by [20], the group algebra $FG$ is PI).", "As for the relation between $m$ and $n$ we have the following result.", "Theorem 1.7 Let $G$ be any group and suppose it grades nondegenerately a PI algebra $W$ of PI degree $n$ .", "Then the group algebra $FG$ is PI and its PI degree is bounded by $n^{2}$ .", "Similarly, $\\exp (FG) \\le \\exp (W)^{2}$ .", "It is somewhat surprising that $T_{n} \\nRightarrow PI_{n}$ (intuitively, the group algebra $FG$ seems to be the “smallest or simplest” $G$ -graded algebra whose grading is nondegenerate).", "The following example shows that twisted group algebras may have lower PI degree.", "Example 1.8 Let $A_{4}$ be the alternating group of order 12.", "It is known that the largest irreducible complex representation is of degree 3 and hence by Amitsur-Levitzky theorem the PI degree is 6.", "On the other hand, the group $A_{4}$ admits a nontrivial cohomology class $\\alpha \\in H^{2}(A_{4}, \\mathbb {C}^{})$ (corresponding to the binary tetrahedral group of order 24).", "Since twisted group algebras with nontrivial cohomology class cannot admit the trivial representation, we have $\\mathbb {C}^{\\alpha }A_{4}\\cong M_{2}(\\mathbb {C}) \\oplus M_{2}(\\mathbb {C}) \\oplus M_{2}(\\mathbb {C})$ and hence the PI degree is 4.", "Conjecture 1.9 Let $W$ be a an algebra over an algebraically closed field $F$ of characteristic zero satisfying a PI of degree $n$ .", "Suppose $W$ is nondegenerately graded by a group $G$ .", "Then there exists a class $\\alpha \\in H^{2}(G,F^{})$ such that the twisted group algebra $F^{\\alpha }G$ has PI degree bounded by the same integer $n$ .", "Theorem 1.10 Notation as above.", "The conjecture holds whenever the group $G$ is finite.", "The main tools used in the proof of the main theorem are the representability theorem for $G$ -graded algebras where $G$ is a finite group [5] and Giambruno and Zaicev's result on the exponent of $W$ [14].", "The representability theorem allows us to replace the $G$ -graded algebra $W$ by a finite dimensional $G$ -graded algebra $A$ (or the Grassmann envelope of a finite dimensional $\\mathbb {Z}_{2}\\times G$ -graded algebra $A$ ) whereas Giambruno and Zaicev's result provides an interpretation of $\\exp (W)$ in terms of the dimension of a certain subalgebra of $A$ .", "The proof of Theorem REF in case the group $G$ is finite is presented in section 3.", "In section 4 we show how to pass from finite groups to arbitrary groups and by this we complete the proof of Theorem REF .", "In section 2 we recall some background on group gradings and PI theory needed for the proofs of the main results of the paper.", "In the last section of the paper, section $\\ref {sec:examples}$ , we present (1) a family of $n$ -permutable with no uniform bound on the index of abelian subgroups and (2) an example which shows that we cannot replace in the main theorem nondegenerate $G$ -gradings with bounded nondegenerate $G$ -grading.", "We close the introduction by explaining why one would prefer bounding the index of an abelian subgroup by a function of the $\\exp (W)$ (as in the main theorem) rather than by the PI degree of $W$ .", "It is known that $\\exp (W)$ is bounded by a function of the PI degree (e.g.", "$\\exp (W) \\le (d(W)-1)^{2}$ , see Theorem 4.2.4 [15]) but such function does not exist in the reverse direction.", "Indeed, since $\\exp (W)$ is an asymptotic invariant it remains invariant if we consider the $G$ -graded $T$ -ideal generated by all polynomials in $\\operatorname{Id}_{G}(W)$ of degree at least $m$ (any $m$ ) whereas the $G$ -graded PI degree and hence the ordinary PI degree is at least $m$ ." ], [ "Background and some preliminary reductions", "We start by recalling some facts on $G$ -graded algebras $W$ over a field $F$ of characteristic zero and their corresponding $G$ -graded identities.", "We refer the reader to [5] for a detailed account on this topic.", "Remark 2.1 In this section we consider only finite groups.", "Although some of the basic results in $G$ -graded PI theory hold for arbitrary groups, one of our main tools, namely the “representability theorem” for $G$ -graded PI algebras, is false for infinite groups." ], [ "$G$ -graded identities", "Let $W$ be a PI-algebra over $F$ .", "Suppose $W$ is $G$ -graded where $G$ is a finite group.", "Denote by $I=\\operatorname{Id}_{G}(W)$ the ideal of $G$ -graded polynomial identities of $W$ .", "It consists of all elements in the free $G$ -graded algebra $F\\langle X_{G} \\rangle $ over $F$ , that vanish upon any admissible evaluation on $W$ .", "Here, $X_{G}=\\bigcup _{g \\in G} X_{g}$ and $X_{g}$ is a set of countably many variables of degree $g$ .", "An evaluation on $W$ is admissible if the variables from $X_{g}$ are replaced only by elements of $W_{g}$ .", "The ideal $I$ is a $G$ -graded $T$ -ideal, i.e.", "it is invariant under all $G$ -graded endomorphisms of $F \\langle X_{G} \\rangle $ .", "We recall from [5] that the $G$ -graded $T$ -ideal $I$ is generated by multilinear polynomials.", "Consequently, it remains invariant when passing to any field extension $L$ of $F$ , that is $\\operatorname{Id}_{G}(W\\otimes _{F}L)=\\operatorname{Id}_{G}(W)\\otimes _{F}L$ .", "The following observations play an important role in the proofs.", "Observation 2.2 The condition nondegenerate $G$ -grading on $W$ can be easily translated into the language of $G$ -graded polynomial identities.", "Indeed a $G$ -grading on $W$ is nondegenerate if and only if for any integer $r$ and any tuple $(g_{1},\\ldots , g_{r}) \\in G^{(r)}$ , the $G$ -graded multilinear monomial $x_{g_{1},1}\\cdots x_{g_{r},r}$ is a $G$ -graded nonidentity of $W$ (in short we say that $\\operatorname{Id}_{G}(W)$ contains no multilinear $G$ -graded monomials).", "Consequently, if $G$ -graded algebras $W_{1}$ and $W_{2}$ are $G$ -graded PI-equivalent (i.e.", "have the same $T$ -ideal of $G$ -graded identities), then the grading on $W_{1}$ is nondegenerate if and only if the grading on $W_{2}$ is nondegenerate.", "Observation 2.3 If $W_1, W_2$ are two $G$ -graded algebras with $\\operatorname{Id}_G(W_1)=\\operatorname{Id}_G(W_2)$ , then $\\operatorname{Id}(W_1)=\\operatorname{Id}(W_2)$ (the ungraded identities).", "In particular we have $\\exp (W_1)=\\exp (W_2)$ .", "Indeed, this follows easily from the fact that a polynomial $p(x_{1},\\ldots ,x_{n})$ is an ungraded identity of an algebra $W$ with a $G$ -grading if and only if the polynomial $p(\\sum _{g\\in G}x_{g,1}\\ldots ,\\sum _{g\\in G}x_{g,n})$ is a graded identity of $W$ as a $G$ -graded algebra.", "As noted above, the nondegeneracy condition satisfied by a $G$ -grading on $W$ depends only on the $T$ -ideal of $G$ -graded identities, hence if the grading on a $G$ -graded algebra $W$ over a field $F$ is nondegenerate, the same holds for the $G$ -graded algebra $W_{L}=W \\otimes _{F}L$ .", "Similarly, the numerical invariant $\\exp (W)$ of the algebra $W$ remains unchanged if we extend scalars.", "Remark 2.4 In the main steps of the proof (in case $G$ is a finite group), roughly speaking, we ”pass” to simpler algebras without increasing too much the exponent the PI degree.", "More precisely, given an arbitrary $G$ -grading on a PI algebra $W$ we first pass to a finite dimensional $G$ -graded algebra $A$ , then to a $G$ -simple algebra and finally to a group algebra, a case which was solved by Gluck using the classification of finite simple groups (see [16]).", "Let us recall some terminology and some facts from Kemer's theory extended to the context of $G$ -graded algebras as they appear in [5].", "Let $W$ be a $G$ -graded algebra over $F$ .", "Suppose that $W$ is PI (as an ungraded algebra).", "Kemer's representability theorem for $G$ -graded algebras assures that there exists a field extension $L/F$ and a finite dimensional $\\mathbb {Z}_{2} \\times G$ -graded algebra $A$ over $L$ such that the Grassmann envelope $E(A)$ (with respect to the $\\mathbb {Z}_{2}$ -grading) yields a $G$ -graded algebra which is $G$ -graded PI-equivalent to $W_L$ (see Proposition REF ).", "In case the algebra $W$ is affine, or more generally in case it satisfies a Capelli identity (it is known that any affine PI algebra satisfies a Capelli identity), there exists a field extension $L/F$ such that the algebra $W_L$ is $G$ -graded PI-equivalent to a finite dimensional $G$ -graded algebra $A$ over $L$ .", "This result will be used to reduce our discussion from infinite dimensional algebras to finite dimensional ones in case the group $G$ is finite.", "As extensions of scalars do not change the exponent (nor the PI-degree) we assume that the field $L$ is algebraically closed." ], [ "$G$ -simple algebras", "The next ingredient we need is a result of Bahturin, Sehgal and Zaicev, which determines the $G$ -graded structure of finite dimensional $G$ -simple algebra over an algebraically closed field of characteristic zero.", "Let $A$ be the algebra of $r \\times r$ -matrices over $F$ and let $G$ be any group (here, $G$ may be infinite).", "Fix an $r$ -tuple $\\alpha =(g_{1},\\ldots ,g_{r}) \\in G^{(r)}$ .", "Consider the $G$ -grading on $A$ given by $A_{g}=\\operatorname{span}_{F}\\lbrace e_{i,j}: g=g_{i}^{-1}g_{j}\\rbrace .$ One checks easily that this indeed determines a $G$ -grading on $A$ .", "Clearly, since the algebra $A$ is simple, it is $G$ -simple as a $G$ -graded algebra.", "Next we present a different type of $G$ -gradings on semisimple algebras which turn out to be $G$ -simple.", "Let $H$ be any finite subgroup of $G$ and consider the group algebra $FH$ .", "By Maschke's theorem $FH$ is semisimple and of course $H$ -simple (any nonzero homogeneous element is invertible).", "More generally we consider twisted group algebras $F^{\\alpha }H$ as $H$ -graded algebras, where $\\alpha $ is a 2-cocycle in $Z^{2}(H, F^{})$ ($H$ acts trivially on $F$ ).", "Recall that $F^{\\alpha }H = \\operatorname{span}_{F} \\lbrace U_{h}: h \\in H\\rbrace $ , $U_{h_{1}}U_{h_{2}}=\\alpha (h_{1},h_{2})U_{h_{1}h_{2}}$ , for all $h_{1},h_{2} \\in H$ .", "We say that the basis $\\lbrace U_{h}:h \\in H\\rbrace $ corresponds to the 2-cocycle $\\alpha $ .", "Finally, we may view the twisted group algebra $F^{\\alpha }H$ as a $G$ -graded algebra by setting $A_{g} = 0$ for $g\\in G \\setminus H$ and as such it is $G$ -simple.", "We refer to the $G$ -grading on $F^{\\alpha }H$ as a fine grading (i.e.", "every homogeneous component is of dimension $\\le 1$ ).", "Remark 2.5 In the sequel, whenever we say that $\\lbrace U_{h}: h\\in H\\rbrace $ is a basis of $F^{\\alpha }H$ , we mean that the basis corresponds to the cocycle $\\alpha $ .", "One knows that in general an homogeneous basis of that kind corresponds to a cocycle $\\alpha {^{\\prime }}$ cohomologous to $\\alpha $ .", "In case the field $F$ is algebraically closed of characteristic zero, we have that these two gradings (elementary and fine) are the building blocks of any $G$ -grading on a finite dimensional algebra so that it is $G$ -simple.", "This is a theorem of Bahturin, Sehgal and Zaicev.", "Theorem 2.6 [7] Let $A$ be a finite dimensional $G$ -graded simple algebra.", "Then there exists a finite subgroup $H$ of $G$ , a 2-cocycle $\\alpha :H\\times H\\rightarrow F^{}$ where the action of $H$ on $F$ is trivial, an integer $r$ and an $r$ -tuple $(g_{1},g_{2},\\ldots ,g_{r})\\in G^{(r)}$ such that $A$ is $G$ -graded isomorphic to $\\Lambda =F^{\\alpha }H\\otimes M_{r}(F)$ where $\\Lambda _{g}=span_{F}\\lbrace U_{h}\\otimes e_{i,j} \\mid g=g_{i}^{-1}hg_{j}\\rbrace $ .", "Here $U_{h}\\in F^{\\alpha }H$ is a representative of $h\\in H$ and $e_{i,j}\\in M_{r}(F)$ is the $(i,j)$ elementary matrix.", "In particular the idempotents $1\\otimes e_{i,i}$ as well as the identity element of $A$ are homogeneous of degree $e\\in G$ ." ], [ "Asymptotic PI-theory", "The last ingredient we need is Regev, Giambruno and Zaicev's PI-asymptotic theory.", "Let $W$ be an ordinary PI-algebra over an algebraically closed field $F$ of characteristic zero and let $\\operatorname{Id}(W)$ be its $T$ -ideal of identities.", "Consider the $n!$ -dimensional vector space $P_{n}= span_{F}\\lbrace x_{\\sigma (1)}\\cdots x_{\\sigma (n)}: \\sigma \\in Sym(n) \\rbrace $ and let $c_{n}(W) = \\dim _{F}(P_{n}/P_{n} \\cap \\operatorname{Id}(W))$ be the $n$ -th term of the codimension sequence of the algebra $W$ .", "It was proved by Regev in 72 [22] that the sequence $\\lbrace c_{n}(W)\\rbrace $ is exponentially bounded and conjectured by Amitsur that the limit $lim_{n\\rightarrow \\infty } c_{n}^{1/n}$ exists (the exponent of $W$ ) and is a nonnegative integer.", "The conjecture was established by Giambruno and Zaicev in the late 90's by showing that the limit coincides, roughly speaking, with the dimension of a certain subspace “attached” to $W$ .", "In particular, for a matrix algebra $M_d(F)$ we have $exp(M_d(F))=d^2$ and by the Amitsur Levitzky theorem it has PI-degree $2d$ .", "Any finite dimensional $G$ -simple algebra is a direct product of matrix algebra (as an ungraded algebra), hence its $T$ -ideal of identities coincides with the ideal of identities (and therefore the exponent and PI-degree) of the largest matrix algebra appearing in its decomposition.", "Remark 2.7 It follows from the Amitsur-Levitzki theorem that if $A$ is a finite dimensional $G$ -simple algebra $A$ we have $\\exp (A)=\\frac{1}{4}(PIdeg(A))^2$ .", "For an arbitrary PI-algebra we only have the bound $\\exp (A)\\le (PIdeg(A)-1)^2$ .", "Recall (from the last paragraph of the introduction) that the PI-degree cannot be bounded from above by any function of $\\exp (A)$ ." ], [ "Proof of main theorem-Finite groups", "All groups considered in this section are finite.", "For a PI-algebra $W$ over a field $F$ of characteristic zero we denote by $\\exp (W)$ its exponent.", "Proposition 3.1 Let $W$ be a PI-algebra over a field $F$ .", "Suppose $W$ is graded nondegenerately by a group $G$ .", "Then there exists a field extension $L$ of $F$ and a finite dimensional $L$ -algebra $W_0$ which is nondegenerately $G$ -graded, such that $exp(W_0)\\le exp(W)$ .", "Let us consider first the case where $W$ is affine.", "Applying [5] there exists a finite dimensional $G$ -graded algebra $B$ over a field extension $L$ of $F$ such that $\\operatorname{Id}_{G}(W\\otimes _{F} L)=\\operatorname{Id}_{G}(B)$ .", "Clearly, we may assume that $L$ is algebraically closed by further extending the scalars if needed.", "Next, by Observations REF and REF we know that the $G$ -grading on $B$ is nondegenerate and $\\exp (W_{L})=\\exp (B)$ , thus proving the proposition for this case.", "Suppose now that $W$ is arbitrary (i.e., not necessarily affine).", "By [5] there exists a finite dimensional $\\mathbb {Z}_{2} \\times G$ -graded algebra $C\\cong \\bigoplus _{(\\epsilon , g)\\in \\mathbb {Z}_{2} \\times G}C_{(\\epsilon , g)}$ over an (algebraically closed) field extension $L$ of $F$ such that $W_L$ is $G$ -PI-equivalent to $E(C)=(E_{0} \\otimes C_{0}) \\oplus (E_{1} \\otimes C_{1})$ (the Grassmann envelope of $C$ ) where $C_0=\\bigoplus _{g \\in G}C_{(0,g)}$ and $C_1=\\bigoplus _{g \\in G}C_{(1,g)}$ .", "The $G$ -grading on $E(C)$ is given by $E(C)_{g}=(E_{0}\\otimes C_{(0,g)})\\oplus (E_{1}\\otimes C_{(1,g)}).$ We claim that the $G$ -grading on $C$ is nondegenerate (where $C_g=C_{(0,g)}\\oplus C_{(1,g)}$ ).", "To this end fix an $n$ -th tuple $(g_{1}, \\ldots , g_{n}) \\in G^{(n)}$ .", "By linearity we need to show that at least one of the $2^{n}$ monomials of the form $x_{(\\epsilon _{1}, g_{1}),1}x_{(\\epsilon _{2}, g_{2}),2}\\cdots x_{(\\epsilon _{n}, g_{n}),n}$ is not in $\\operatorname{Id}_{\\mathbb {Z}_{2} \\times G}(C)$ .", "Let us show that if this is not the case, then the monomial $x_{g_{1},1}\\cdots x_{g_{n},n}$ is a $G$ -graded identity of $E(C)$ , contradicting the fact that the $G$ -grading on $E(C)$ and hence on $W$ is nondegenerate.", "To see this consider the evaluation $x_{g_i,i}=z_{0,i}\\otimes a_{0,i} + z_{ 1,i}\\otimes a_{1,i}$ for $i=1,\\ldots ,n$ where $z_{\\epsilon ,i}\\in E_{\\epsilon }$ and $a_{\\epsilon ,i} \\in C_{(\\epsilon ,g_i)}$ .", "This evaluation yields an expression with $2^{n}$ summands of the form $z_{\\epsilon _1,i_1}z_{\\epsilon _2,i_2}\\cdots z_{\\epsilon _n,i_n} \\otimes a_{(\\epsilon _1,i_1)}a_{(\\epsilon _2,i_2)}\\cdots a_{(\\epsilon _n,i_n)}$ which are all zero and the claim follows.", "Finally, by a theorem of Giambruno and Zaicev (see [14], proof of main theorem or [1], Theorem 2.3) we have that $\\exp (C)\\le \\exp _{\\mathbb {Z}_2}(C)=\\exp (E(C))=\\exp (W)$ , which is precisely what we need.", "Our next step is to reduce the main theorem from finite dimensional algebras to $G$ -simple algebras.", "Proposition 3.2 Let $W$ be a finite dimensional PI $F$ -algebra graded nondegenerately by a group $G$ .", "Then there exists a $G$ -simple algebra $W_0$ such that $Id_G(W)\\subseteq Id_G(W_0)$ and the grading on $W_0$ is nondegenerate $($ in fact $W_{0}$ is an homomorphic image of $W$$)$ .", "In particular $\\exp (W_0)\\le \\exp (W)$ .", "Denote by $J=J(W)$ the Jacobson radical of $W$ .", "Since the characteristic of the field is zero, it is known that $J$ is $G$ -graded and so $W/J$ is a semi-simple $G$ -graded algebra (see [8]).", "We claim that the $G$ -grading on $W/J$ is still nondegenerate.", "Indeed, if $W/J$ satisfies a monomial identity $f$ , then any evaluation of this monomial on $W$ yields an element in $J$ .", "Since $J$ is nilpotent (say of nilpotency degree is $k$ ) we have that the product of $k$ copies of $f$ (with distinct variables) is a monomial identity of $W$ .", "This contradicts the assumption the $G$ -grading on $W$ is nondegenerate and the claim is proved.", "The algebra $W/J$ is $G$ -semisimple and therefore a direct product of $G$ -simple algebra $\\prod _1^n A_i$ .", "If for each $i$ there is a multilinear monomial identity $f_i$ of $A_i$ , then the product $\\prod f_i$ is a multilinear monomial identity of $W/J$ , contradicting our assumption on the $G$ -grading on $W/J$ .", "Consequently, there is an $i$ such that $A_{i}$ is nondegenerately $G$ -graded.", "Letting $W_{0}=A_{i}$ we have $Id_G(W)\\subseteq Id_G(W/J)\\subseteq Id_G(W_0)$ as desired.", "In the next lemma we characterize (in terms of Bahturin, Sehgal and Zaicev's theorem) when the grading on a $G$ -simple algebra is nondegenerate.", "Recall that a $G$ -grading on $A$ is strong if for any $g, h \\in G$ we have $A_{g}A_{h}=A_{gh}$ .", "Lemma 3.3 Let $A\\ne 0$ be a finite dimensional $G$ -simple algebra.", "Then the following conditions are equivalent.", "The $G$ -grading on $A$ is nondegenerate.", "The $G$ -grading on $A$ is strong.", "In particular $A_{g} \\ne 0$ , for every $g\\in G$ .", "Let $F^{\\alpha }H \\otimes M_{r}(F)$ be a presentation of the $G$ -grading on $A$ $($ as given by Theorem REF $)$ where $H$ is a finite subgroup of $G$ and $(g_1,\\ldots ,g_{r})\\in G^{(r)}$ is the $r$ -tuple which determines the elementary grading on $M_{r}(F)$ .", "Then every right coset of $H$ in $G$ is represented in the $r$ -tuple.", "Remark 3.4 Note that in general (i.e.", "in case the algebra $A$ is not necessarily $G$ -simple) the first two conditions are not equivalent.", "For instance (as mentioned in the introduction), the $\\mathbb {Z}_{2}$ -grading on the infinite dimensional Grassmann algebra is nondegenerate but not strong.", "Indeed, $E_{1}E_{1}\\subsetneq E_{0}$ (or $E_{0}E_{0}\\subsetneq E_{0}$ in case the algebra $E$ is assumed to have no identity element).", "Note that since $A$ is assumed to be finite dimensional $G$ -simple, each one of the conditions (1)-(3) implies that $G$ is finite.", "As for the 3rd condition of the lemma we replace (as we may by [3], Lemma 1.3) the given presentation with another so that the $r$ -tuple has the following form $(g_{(1,1)},\\dots ,g_{(1,d_1)},g_{(2,1)},\\dots ,g_{(2,d_2)},\\ldots ,\\ldots g_{(s,1)},\\dots ,g_{(s,d_s)})$ where $r=d_{1}+\\cdots + d_{s}$ .", "$g_{i,1}=g_{i,2}=\\ldots =g_{i,d_{i}}$ (denoted by $z_{i}$ ), and for $i \\ne k$ the elements $g_{i,j}, g_{k,l}$ represent different right $H$ -cosets in $G$ .", "$g_{1,j}=e$ for $j=1,\\ldots ,d_1$ .", "$(2)\\rightarrow (1):$ This is clear.", "$(1)\\rightarrow (3):$ Suppose $(3)$ does not hold.", "We claim there exists a multilinear monomial of degree at most $r$ which is a $G$ -graded identity of $F^{\\alpha }H\\otimes M_{r}(F)=\\operatorname{span}_{F}\\lbrace U_{h}\\otimes e_{i,j}: h \\in H, 1\\le i,j \\le r\\rbrace $ .", "It is convenient to view the matrices in $M_{r}(F)$ as $s\\times s$ block matrices corresponding to the decomposition $d_1+\\cdots +d_s = r$ .", "More precisely, let $D_k = d_1 + \\cdots +d_k$ and decompose $M_r(F)=\\bigoplus _{i,j=1}^s M_{[i,j]}$ into the direct sum of vector spaces $M_{[i,j]}=span \\lbrace e_{k,l} \\mid D_{i-1}<k\\le D_i,\\; D_{j-1}<l\\le D_j \\rbrace $ .", "Note that $M_{[i,j]}$ are submatrices supported on a single block of size $d_i \\times d_j$ .", "This decomposition is natural in the sense that $(F^{\\alpha }H\\otimes M_{r}(F))_g$ is the direct sum of the vector spaces $U_h\\otimes M_{[i,j]}$ such that $z_i^{-1}h z_j = g$ .", "For a fixed index $i \\in \\lbrace 1,\\ldots ,s\\rbrace $ and an element $g \\in G$ , consider the equation $h z_j = z_i g$ .", "It has a solution if and only if $H z_i g$ has a representative in $(z_1 ,\\ldots , z_s)$ .", "It follows that if $U_h \\otimes B$ is homogeneous of degree $g$ and $H z_i g$ has no representative in $(z_1 ,\\ldots , z_s)$ , then the $i$ -th row of blocks in $B$ must be zero.", "Consider the multilinear monomial $x_{w_{1},1}x_{w_{2},2}\\cdots x_{w_{n},n}$ where $x_{w_{i},i}$ is homogeneous of degree $w_{i} \\in G$ .", "We will show there exist $w_{i} \\in G, i=1,\\ldots ,n$ , so that the monomial above is a $G$ -graded identity.", "To this end, note that such a monomial (being multilinear) is a $G$ -graded identity if and only if it is zero on graded assignments of the form $x_{w_{i},i}=U_{h_{i}}\\otimes A_{i}$ which span the algebra.", "In particular the value of $x_{w_{1},1}x_{w_{2},2}\\cdots x_{w_{n},n}$ under this assignment is $U_{h_{1}}\\cdots U_{h_{n}}\\otimes A_{1}\\cdots A_{n}$ which is zero if and only if $A=A_{1}\\cdots A_{n}=0$ .", "It follows that if for any such homogeneous assignment, the $i$ -th row of blocks in the matrix $B_i = A_1 A_2 \\cdots A_i$ is zero, then $A$ must be zero (since each of its blocks rows is zero).", "Following the argument above we choose $w_i \\in G$ such that for each $i$ the right coset $H z_i w_1 \\cdots w_i $ (i.e.", "the right coset of $H$ represented by $z_i$ times the homogeneous degree of $U_{h_1}\\cdots U_{h_i}\\otimes B_i$ ) has no representative in $(z_1,\\ldots , z_s)$ .", "Now, by assumption, there is some $z\\in G$ such that $Hz$ has no representative in $(z_1 ,\\ldots , z_s)$ .", "Thus, choosing $w_i= (z_i w_1 \\cdots w_{i-1})^{-1} z $ we obtain the required result.", "$(3)\\rightarrow (2):$ Suppose that all right $H$ -cosets are represented in the tuple $(g_{1}, \\ldots , g_{r})$ .", "To show that the grading is strong, it is enough to show that any basis element $U_h\\otimes e_{i,j}$ can be written as a product in $A_{w_1}A_{w_2}$ where $w_1\\cdot w_2=g_i ^{-1} h g_j$ .", "Indeed, since each right coset has a representative in the tuple $(g_1,\\ldots ,g_{r})\\in G^{(r)}$ , we can find $k$ such that $g_{k} \\in H g_i w_1=Hg_j w_2^{-1}$ .", "Letting $h_1=g_{i} w_1g_k^{-1}$ and $h_2 = g_k w_2 g_j ^{-1}$ , we get that $a =U_{h_1}\\otimes e_{i,k},\\; b = U_{h_2} \\otimes e_{k,j}$ are in $A_{w_1},A_{w_2}$ respectively and $a\\cdot b= \\alpha (h_1,h_2) U_h\\otimes e_{i,j}$ .", "The lemma is now proved.", "Our next step is to pass from $G$ -simple algebras to the group algebra $FG$ .", "Let $V=\\bigoplus V_g$ be a $G$ -graded $F$ -vector space.", "Then the algebra of endomorphisms $End_{F}(V)$ has a natural $G$ -grading where an endomorphism $\\psi \\in End(V)$ has homogeneous degree $g$ if $\\psi (V_h)\\subseteq V_{gh}$ for every $h\\in G$ .", "In particular, this grading on $End(FG)$ is isomorphic to the elementary grading by a tuple $(g_1,\\ldots ,g_n)$ where each element of $G$ appears exactly once.", "It is clear that the left regular action of $G$ on $FG$ induces a natural $G$ -graded embedding of $FG$ in $End(FG)\\cong M_{\|G\|}(F)$ .", "This statement can be generalized as follows.", "Lemma 3.5 Let $G$ be a finite group, $H$ a subgroup and $\\lbrace w_1,\\ldots ,w_k\\rbrace $ a complete set of representatives for the right cosets of $H$ in $G$ .", "Then the group algebra $FG$ can be embedded in $FH\\otimes M_k(F)$ where the tuple of the elementary grading is $(w_1,\\ldots ,w_k)$ .", "For any $g\\in G$ and any $H$ -right coset representative $w\\in \\lbrace w_1,\\ldots ,w_k\\rbrace $ , there are $h\\in H$ and $w^{\\prime }\\in \\lbrace w_1,\\ldots ,w_k\\rbrace $ such that $wg=hw^{\\prime }$ .", "We denote these elements by $h:=h_{w,g}$ and $w^{\\prime }:=w^g$ .", "From associativity of $G$ , we get that $h_{w,g_1 g_2} = h_{w,g_1} h_{w^{g_1},g_2},\\; \\; w^{g_1 g_2}=(w^{g_1})^{g_2}.$ Define a map $\\psi :FG\\rightarrow FH\\otimes M_k(F)$ by $\\psi (U_g) = \\sum _{i=1}^{k} V_{h_{w_i,g}}\\otimes E_{i,j(i)}$ where $\\lbrace U_g\\rbrace _g$ and $\\lbrace V_h\\rbrace _h$ are the corresponding bases of $FG$ and $FH$ , $E_{i,j(i)}$ is the $(i,j)$ elementary matrix and $j(i)$ is determined by the equation $w_{j(i)}= w_i^g$ .", "It is easy to show (left to the reader) that $\\psi $ is a homomorphism.", "Furthermore, by definition of the $G$ -grading on $FH\\otimes M_k(F)$ (see Theorem REF ), we have that the homogeneous degree of $V_{h_{w_i,g}}\\otimes E_{i,j(i)}$ is $w_i^{-1}h_{w_i,g}w_i^g=w_i^{-1}w_ig=g$ , and hence $\\psi $ is a $G$ -graded map.", "Finally, since $FG$ is $G$ -simple and $\\psi \\ne 0$ , it follows that $\\psi $ is an embedding.", "Returning to our proof, we have a $G$ -simple algebra $F^\\alpha H\\otimes M_k(F)$ , nondegenerately $G$ -graded.", "Recall that this means that any right coset of $H$ in $G$ appears at least once in the tuple corresponding to the elementary grading $\\lbrace w_1,\\ldots ,w_k\\rbrace $ .", "If $\\alpha =1$ , then by the previous lemma the group algebra $FG$ embeds in $FH\\otimes M_k(F)$ and hence $\\operatorname{Id}(FG)\\supseteq \\operatorname{Id}(FH\\otimes M_k(F))$ .", "This proves the reduction to $FG$ in that case.", "In general (i.e.", "$\\alpha $ not necessarily trivial), $FG$ may not be $G$ -graded embedded in $F^\\alpha H\\otimes M_k(F)$ .", "We might hope however, that even if such an embedding is not possible, still $\\exp (FG)\\le \\exp (F^\\alpha H\\otimes M_k(F))$ .", "It turns out that this is also false as Example REF shows.", "The next lemma shows how to get rid of the 2-cocycle $\\alpha $ .", "Lemma 3.6 Let $A=F^\\alpha H \\otimes M_k(F)$ be a nondegenerate $G$ -simple graded algebra.", "Let $\\rho :F^\\alpha H\\rightarrow M_d(F)$ be a nonzero $($ ungraded$)$ representation and denote by $B=M_d(F)$ the trivially $G$ -graded algebra $($ and therefore trivially $H$ -graded$)$ .", "Then $FH$ can be embedded in $F^\\alpha H\\otimes B$ and $FG$ can be embedded in $A\\otimes B$ as $H$ and $G$ -graded algebras respectively.", "Define the map $\\psi :FH\\rightarrow F^\\alpha H\\otimes M_d(F)$ by $\\psi (U_h)=V_h\\otimes \\rho (V_h^{-1})^t$ , where $\\lbrace U_h\\rbrace _h$ and $\\lbrace V_h\\rbrace _h$ are the corresponding bases of $FH$ and $F^\\alpha H$ .", "This is easily checked to be an $H$ -graded homomorphism, and it is an embedding since $FH$ is $H$ -simple.", "This proves the first claim of the lemma.", "The second claim follows from the last lemma using the graded embeddings $FG\\hookrightarrow FH\\otimes M_k(F)\\hookrightarrow F^\\alpha H\\otimes M_d(F) \\otimes M_k(F)\\cong A\\otimes B.$ Corollary 3.7 Let $A=F^\\alpha H \\otimes M_k(F)$ be a nondegenerate $G$ -simple graded algebra.", "Then $\\exp (FG)\\le \\exp (A)^2$ .", "Recall that the exponent of $F^\\alpha H$ is $d^{2}$ , where $d$ is the dimension of the its largest irreducible representation.", "It follows that $FG$ can be embedded in $A\\otimes M_d(F)$ where $d^2=\\exp (F^\\alpha H)\\le \\exp (A)$ and therefore $\\exp (FG)\\le \\exp (A\\otimes M_d(F))= \\exp (A) \\exp (M_d(F))\\le \\exp (A)^2.$ Corollary 3.8 Let $W$ be an associative PI $F$ -algebra nondegenerately $G$ -graded.", "Then the following hold.", "$\\exp (FG) \\le \\exp (W)^{2}$ and $d(FG)\\le 2(d(W)-1)^{2}.$ $($$d(W)$ denotes the PI degree$)$ The first inequality follows from Propositions REF , REF and Corollary REF .", "For the proof of the second inequality recall that in general $\\exp (W)\\le (d(W)-1)^2$ (see Theorem 4.2.4 in [15]).", "Now, since $FG$ is semisimple, it follows by Amitsur-Levitzky theorem that $\\frac{1}{2}d(FG)=\\sqrt{\\exp (FG)}$ and so, we conclude that $d(FG)=2\\sqrt{\\exp (FG)}\\le 2\\exp (W)\\le 2(d(W)-1)^2.$ The last step in our analysis concerns with group algebras.", "Here we refer to the following result of D. Gluck (see [16]) in which he bounds the minimal index of an abelian subgroup $U$ in $G$ in terms of the maximal character degree of $G$ .", "We emphasize that the proof uses the classification of finite simple groups.", "Theorem 3.9 $($ D. Gluck$)$ There exists a constant $m$ with the following property.", "For any finite group $G$ there exists an abelian subgroup $U$ of $G$ such that $[G:U] \\le b(G)^{m}$ , where $b(G)$ is the largest irreducible character degree of $G$ .", "We can now complete the proof of the main theorem for finite groups.", "We note that by Giambruno and Zaicev's result $\\exp (FG)=b(G)^{2}$ and hence, any finite group has an abelian subgroup $U$ with $[G:U]\\le b(G)^{m} = \\exp (FG)^{m/2}$ .", "Combining with our results above, we see that if a PI-algebra $W$ admits a nondegenerate $G$ -grading where $G$ is a finite group, then $\\exp (FG)\\le \\exp (W)^2$ , hence then there is an abelian subgroup $U$ with $[G:U] \\le \\exp (W)^{m}$ .", "In particular, taking $K=m$ where $m$ is determined by the theorem above, will do." ], [ " Proof of main theorem-Infinite groups", "In this section we prove the main theorem for arbitrary groups.", "Let us sketch briefly the structure of our proof.", "In the preceding section we proved the main theorem for arbitrary finite groups.", "Our first step in this section is to prove the main theorem for groups which are finitely generated and residually finite.", "Next, we pass to finitely generated groups (not necessarily residually finite) by the following argument.", "Any group $G$ which grades nondegenerately a PI algebra is permutable and hence being finitely generated, it is abelian by finite (see [10] or Remark REF ) and hence residually finite.", "Finally we show how to pass from finitely generated groups to arbitrary groups.", "We emphasize that the constant $K$ (which appears in the main theorem) remains unchanged when passing from finite groups to arbitrary group.", "Proposition 4.1 Suppose the main theorem holds for arbitrary finite groups with the constant $K$ , that is, for any finite group $G$ and any PI algebra $W$ which is nondegenerately $G$ -graded, there exists an abelian subgroup $U \\subseteq G$ with $[G:U] \\le \\exp (W)^{K}$ .", "Then the main theorem holds for finitely generated residually finite groups with the same constant $K$ .", "Since $G$ is finitely generated, by Hall's theorem [18] there are finitely many subgroups of index $\\le \\exp (A)^{K}$ .", "Denoting these groups by $U_{1},\\ldots ,U_{n}$ , we wish to show that one of them is abelian.", "Suppose the contrary holds.", "Hence we can find $g_{i},h_{i}\\in U_{i}$ such that $e\\ne \\left[g_{i},h_{i}\\right]$ for any $i=1,\\ldots ,n$ and we let $N$ be a normal subgroup of finite index which doesn't contain any of the $\\left[g_{i},h_{i}\\right]$ .", "Define an induced $G/N$ grading on $A$ by setting $A_{gN}=\\bigoplus _{h\\in N} A_{gh}$ .", "Clearly, the induced ${G}/{N}$ -grading on $A$ is nondegenerate, thus by the main theorem there is some $U\\le G$ (containing $N$ ) such that $\\left[G:U\\right]\\le \\exp (A)^{K}$ and ${U}/{N}$ is abelian.", "By the construction, $U=U_{i}$ for some $i$ , and we get that $\\left[g_{i},h_{i}\\right]\\in N$ - a contradiction.", "The next step is to remove the condition of residually finiteness.", "Proposition 4.2 Suppose the main theorem holds for finitely generated residually finite groups with the constant $K$ .", "Then the main theorem holds for arbitrary finitely generated groups with the same constant $K$ .", "As mentioned above this is obtained using permutability.", "Let $G$ be a finitely generated group and suppose it grades nondegenerately a PI algebra $A$ .", "Let us show that $G$ must be permutable.", "To this end let $f=\\sum c_\\sigma x_{\\sigma (1),1}\\cdots x_{\\sigma (n),n}$ be a nonzero ordinary identity of $A$ and assume that $c_{id}=1$ .", "Fix a tuple $g_1,\\ldots ,g_n\\in G$ and consider the graded identity $\\tilde{f}=f(x_{g_1,1},\\ldots ,x_{g_n,n})=\\sum _{h\\in G} f_h(x_{g_1,1},\\ldots ,x_{g_n,n})$ where ${f}_h$ is the $h$ homogenous part of $\\tilde{f}$ .", "Since $\\tilde{f}$ is a graded identity, its homogenous parts are also graded identities.", "Letting $g=g_1 \\cdots g_n$ , the polynomial ${f}_g$ contains the monomial $x_{g_1,1}\\cdots x_{g_n,n}$ (with coefficient 1).", "Since the grading is nondegenerate, ${f}_g$ is not a monomial and therefore has another monomial with nonzero coefficient corresponding to some permutation $\\sigma \\ne id$ , hence $g_1\\cdots g_n=g_{\\sigma (1)}\\cdots g_{\\sigma (n)}$ .", "This can be done for any tuple of length $n$ , so it follows that $G$ is $n$ -permutable.", "The main theorem now follows from the following proposition.", "Proposition 4.3 Let $G$ be any group and $d$ be a positive integer.", "Suppose that any finitely generated subgroup $H$ of $G$ contains an abelian subgroup $U_{H}$ with $[H:U_{H}] \\le d$ .", "Then there exists an abelian subgroup $U$ of $G$ with $[G:U] \\le d$ .", "Remark 4.4 The proposition above generalizes a statement which appears in [19] but the proof is basically the same (see Lemma 3.5 and the proof of Theorem II).", "We believe the result of the proposition is well known but we were unable to find an appropriate reference in the literature.", "Let $A\\le F\\le G$ .", "We say that $(F,A)$ is a pair if $F$ is f.g., $A$ is abelian and $[F:A]\\le d$ .", "We write $(F,A) \\le (F_1,A_1)$ if $F \\cap A_1 = A$ .", "Note in particular that $ [F:A] \\le [F_1:A_1]$ .", "A pair $(F,A)$ is called good if whenever $F\\le F_1 \\le G$ with $F_1$ finitely generated, there is a pair $(F_1,A_1)$ with $(F,A) \\le (F_1,A_1)$ .", "Note that the assumption of the proposition says that $({e},{e})$ is a good pair.", "We claim that if $(F,A)$ is good pair and $F\\le H \\le G$ with $H$ finitely generated, we can find $B\\le H$ such that $(H,B)$ is a good pair and $(F,A)\\le (H,B)$ .", "Indeed, since $(F,A)$ is a good pair, there are pairs $(H,B_i)$ with $(F,A)\\le (H,B_i)$ , and by Hall's theorem there exist only finitely many such pairs.", "Suppose by negation that none of them are good pairs.", "Thus we can find $H\\le F_i \\le G$ ($F_i$ -f.g.) such that there are no abelian subgroups $A_i$ with $(H,B_i)\\le (F_i,A_i)$ .", "The group $K=\\left\\langle F_1, ..., F_n \\right\\rangle $ is f.g. so there is some abelian subgroup $A_K\\le K$ of index $\\le d$ such that $(F,A)\\le (K,A_K)$ .", "Clearly, there is some $i$ such that $A_K\\cap H=B_i$ , but then $(H,B_i)\\le (F_i,F_i\\cap A_K)$ - contradiction.", "Let $(F,A)$ be a good pair with $s=[F:A]$ maximal.", "Note that if $(F,A)\\le (H,B)$ are good pairs, then we must have $[F:A]=[H:B]$ .", "Claim: for any such $B$ we have $[G:C_G(B)] \\le d$ .", "Let us show that if $g_1,\\ldots ,g_s$ represent the left cosets of $A$ in $F$ then they also represent the left cosets of $C_G(B)$ in $G$ .", "Fix an element $g\\in G$ .", "Then, by (1) above, $\\langle H,g \\rangle $ has an abelian subgroup $C$ such that $(H,B)\\le (\\langle H,g\\rangle ,C)$ are good pairs.", "It follows that $g_1,\\ldots ,g_s$ represent also the left cosets of $C$ in $\\langle H, g \\rangle $ and hence $g \\in g_i C$ for some $i$ .", "Since $B\\le C$ are abelian groups we get that $g_i C \\subseteq g_i C_G(B)$ and the claim follows.", "Assume now that $(F,A)$ is a good pair with $[F:A]=s$ and $[G:C_G(A)]$ maximal.", "Define $J= \\langle B \\mid (H,B)\\ge (F,A) \\mbox{ is a good pair} \\rangle .", "$ We claim that $J$ is abelian and $[G:J]\\le d$ .", "Let $(H_i,B_i)\\ge (F,A)$ , $i=1,2$ , be good pairs, and let $b_i\\in B_i$ .", "Since $A\\le B_1$ , we have that $C_G(B_1)\\le C_G(A)$ , but from the maximality of $[G:C_G(A)]$ , it follows that there is an equality.", "Similarly, we have that $C_G(B_2) = C_G(A)$ and since $B_2$ is abelian we get that $b_2\\in B_2\\subseteq C_G(B_2)=C_G(B_1)$ , so that $b_1,b_2$ commute.", "This proves $J$ is abelian.", "Suppose now that $[G:J]>d$ , and let $g_0,\\ldots ,g_d$ different coset representatives of $J$ in $G$ .", "The group $F_1 = \\langle F,g_1,\\ldots ,g_d \\rangle $ is finitely generated and so we can find $A_1 \\le F_1$ such that $(F_1,A_1)$ is a good pair larger than $(F,A)$ , and in particular $[F_1:A_1]\\le d$ .", "But this means that there are some $0\\le i<j\\le d$ with $g_i ^{-1} g_j \\in A_1\\subseteq J$ which is a contradiction.", "Thus, $[G:J] \\le d$ and we are done.", "As mentioned in the introduction, once a group has an abelian subgroup of finite index (say $d$ ), then it also has a characteristic abelian subgroup of (finite) index bounded by a function of $d$ .", "For completeness of the article we provide a simple proof here (shown to us by Uri Bader).", "Lemma 4.5 There is a function $f:\\mathbb {N}\\rightarrow \\mathbb {N}$ such that if a group $G$ contains an abelian subgroup $A$ of index at most $n$ , then $G$ contains a characteristic abelian subgroup of index $\\le f(n)$ .", "Let $N$ be the characteristic subgroup of $G$ generated by $A$ (the group generated by all images of $A$ under all automorphisms of $G$ ).", "Let $Z=Z(N)$ (the center of $N$ ).", "We claim $[N:Z]$ (and hence $[G:Z]$ ) is bounded by a function of $n$ .", "Indeed, there are $n$ images of $A$ which already generate $N$ and $Z$ contains their intersection.", "This proves the lemma.", "In the preceding section, we proved that for a finite group and a nondegenerate $G$ -graded algebra $A$ , we have $\\exp (FG)\\le \\exp (A)^2$ and $d(FG)\\le 2 (d(A)-1)^2$ .", "The rest of this section is dedicated to generalize these results for infinite groups.", "Lemma 4.6 Let $G$ be a finitely generated group such that $FG$ is PI.", "Then there exists a finite index normal subgroup $N$ of $G$ such that $Id(FG)=Id(FG/N)$ .", "In particular $Id(FG)=Id(M_k(F))$ for some integer $k$ .", "We know that $G$ is abelian by finite.", "Furthermore, since it is finitely generated it is residually finite.", "If $N$ is any finite index normal subgroup of $G$ , then $Id(FG)\\subseteq Id(FG/N)$ and hence $I:=\\bigcap _{[G:N]<\\infty } Id(FG/N)\\supseteq Id(FG)$ .", "On the other hand, the algebra $FG/N$ is semisimple, so that $Id(FG/N)=Id(M_k(F))$ where $k^2=\\exp (FG/N)\\le \\exp (FG)$ and in particular the set $\\lbrace \\exp (FG/N)\\rbrace _{N}$ is bounded.", "It follows that $I=Id(M_k(F))$ for some $k$ and there is some finite index normal subgroup $N$ with $I=Id(FG/N)$ .", "The lemma will follow if we can show that $Id(FG)=I$ .", "Let $f(x_1,\\ldots ,x_m)\\in I$ be any multilinear polynomial.", "If $f$ is not an identity of $FG$ , we can find some $g_1,\\ldots ,g_m\\in G$ such that $f(U_{g_1},\\ldots ,U_{g_m})\\ne 0$ ($U_{g}$ represents $g$ in $FG$ ).", "Let $h_1,\\ldots ,h_k\\in G$ and $a_1,\\ldots ,a_k\\in F^\\times $ such that $f(U_{g_1},\\ldots ,U_{g_m})=\\sum a_i U_{h_i}$ .", "Since $G$ is residually finite, there is some finite index normal subgroup $N$ not containing $h_i^{-1}h_j$ for any $i\\ne j$ .", "Reducing the equation above modulo $N$ , the elements $U_{h_i}$ remain linearly independent by the choice of $N$ , so in particular $f$ is not an identity of $FG/N$ .", "We obtain that $I \\subseteq \\operatorname{Id}(FG)$ and the result follows.", "Lemma 4.7 Let $G$ be any group such that $FG$ is PI.", "Then there exists some finitely generated subgroup $H$ of $G$ such that $Id(FG)=Id(FH)$ .", "Consequently, $Id(FG)=Id(M_k(F))$ for some integer $k$ .", "By the preceding lemma, for each finitely generated subgroup $H$ of $G$ we have $Id(FH)=Id(M_k(F))$ for some integer $k$ which is uniformly bounded over the finitely generated subgroups (by $\\exp (FG)^{1/2})$ .", "Since any multilinear nonidentity of $FG$ is already a nonidentity of $FH$ for some finitely generated subgroup $H$ of $G$ , we have that $Id(FG)=\\bigcap _{H\\le G} Id(FH)$ , $H$ is f.g., and the lemma follows.", "We can now generalize to arbitrary groups the result of the previous section.", "Theorem 4.8 Let $G$ be any group and $A$ be a nondegenerate $G$ -graded algebra.", "Then $\\exp (FG)\\le \\exp (A)^2$ and $d(FG)\\le 2 (d(A)-1)^2$ .", "Since $G$ grades nondegenerately the algebra $A$ , it is abelian by finite and therefore the group algebra $FG$ is PI.", "It follows from the previous two lemmas that there is some finitely generated subgroup $H$ and a finite index normal subgroup $N$ in $H$ such that $Id(FG)=Id(FH/N)$ , hence it is enough to bound the exponent and PI-degree of $FH/N$ .", "If $A_H$ is the subalgebra of $A$ supported on the $H$ homogeneous components of $A$ , we have $\\operatorname{Id}(A_{H}) \\supseteq \\operatorname{Id}(A)$ , $\\exp (A_{H})\\le \\exp (A)$ and $d(A_{H}) \\le d(A)$ .", "Moreover, since $A$ is nondegenerately $G$ -graded, $A_{H}$ is nondegenerately $H$ -graded and hence $A_{H}$ is also $H/N$ -nondegenerately graded where $N$ is any normal subgroup of $H$ (by the induced grading).", "By Corollary REF we have $\\exp (FH/N)\\le \\exp (A_H)^2 \\le \\exp (A)^2$ and $d(FH/N) \\le 2(d(A_{H})-1)^{2} \\le 2(d(A_{H})-1)^{2}$ and the result follows." ], [ " Some examples", "Let $G$ be a finitely generated group and suppose it grades nondegenerately a PI algebra $A$ .", "We know that $G$ is $n$ permutable for some $n \\in \\mathbb {N}$ .", "While $G$ must be abelian by finite, the minimal index of an abelian subgroup is not bounded by a function of the permutablity index.", "Indeed, if there was such a function $f(n)$ , then given an arbitrary $n$ -permutable group $H$ , its finitely generated subgroup would be $n$ -permutable as well.", "By the assumption, each such subgroup has an abelian subgroup of index $\\le f(n)$ , and hence, by Proposition REF , the group $G$ would contain an abelian subgroup of index bounded by $f(n)$ .", "This is known to be false.", "In fact $G$ need not have an abelian subgroup of finite index (see [10]).", "Let us give a concrete example, i.e.", "a family of (finite) $n$ -permutable groups $\\lbrace G_{k}\\rbrace _{k\\in \\mathbb {N}}$ , with $d_{k}=min \\lbrace [G_{k}:U_{k}] \\mid U_{k} $ abelian subgroup$\\rbrace $ and $\\lim d_{k}= \\infty $ .", "Example 5.1 Let $G=C_p^{2n}$ for some $n$ and let $\\alpha \\in Z^2(G,\\mathbb {C}^{})$ be a nontrivial two cocycle.", "It is well known that up to a coboundary $\\alpha $ takes values which are roots of unity, and for $G$ above, the values must be $p$ -roots of unity.", "Thus, we may consider $\\alpha $ as a cocycle in $Z^2(G,C_p)$ which corresponds to a central extension $1\\rightarrow C_{p}\\rightarrow H\\rightarrow G\\rightarrow 1.$ Since the group $G$ is abelian and the cocycle $\\alpha $ is nontrivial we have that $[H,H]=Z(H)\\cong C_{p}$ and hence the group $H$ is $p+1$ permutable (see [10], (3.3)).", "Let $B=\\mathbb {C}^\\alpha G$ be the corresponding twisted group algebra with basis $\\lbrace U_g\\rbrace _{g\\in G}$ .", "If $A\\le H$ is an abelian group of minimal index, we have that $[H,H]\\le A$ , and thus we have $\\tilde{A}=A/[H,H] \\le G$ .", "Clearly, the group $A$ is abelian if and only if $[U_{g_1},U_{g_2}]=1$ (the multiplicative commutator) for any $g_1,g_2 \\in \\tilde{A}$ .", "For $g,h\\in G$ , set $\\mu (g,h)=\\frac{\\alpha (g,h)}{\\alpha (h,g)}$ , namely the scalar satisfying $U_g U_h = \\mu (g,h) U_h U_g$ .", "It is easily seen that $\\mu :G\\times G\\rightarrow \\mathbb {C}^{}$ is a bicharacter, i.e.", "$\\mu (g_1 g_2,h)=\\mu (g_1,h)\\mu (g_2,h)$ and $\\mu (g,h_1 h_2)=\\mu (g,h_1)\\mu (g,h_2)$ .", "With this notation we have that $\\mu (g_1,g_2)=1$ for any $g_1,g_2 \\in \\tilde{A}$ .", "Identifying $C_p$ with the additive group of the field $F_p$ with $p$ elements, we see that $\\mu $ is a bilinear map.", "In particular, if $g \\in G$ , then $dim\\lbrace h \\in G\\mid \\mu (h,g)=1\\rbrace \\ge n-1$ .", "If $dim_{F_p}(\\tilde{A})>\\frac{1}{2} \\dim _{F_p} (G)$ , or equivalently $[G:\\tilde{A}]<p^n$ , then there is some $e\\ne u\\in \\tilde{A}$ such that $\\mu (u,g)=1$ for all $g\\in G$ (by dimension counting).", "Thus, if $\\mu $ is nondegenerate, i.e.", "for any $e\\ne h\\in G$ there is some $g\\in G$ such that $\\mu (h,g)\\ne 1$ , then $[H:A]=[G:\\tilde{A}]\\ge p^n$ .", "Note that to say that $\\mu $ is nondegenerate is equivalent to saying that $U_g$ is in the center of the twisted group algebra if and only if $g=e$ , which in turn is equivalent to the twisted group algebras $\\mathbb {C}^{\\alpha }C_p ^{2n}$ being isomorphic to a matrix algebra $M_{p^{n}}(\\mathbb {C})$ .", "Fix a prime $p$ and let $\\sigma , \\tau $ be generators for $C_p\\times C_p$ .", "Let $B$ be the twisted group algebra $B=\\bigoplus _{0\\le i,j \\le p-1} \\mathbb {C}U_{\\sigma ^i \\tau ^j}$ where the multiplication is defined by $ U_{\\sigma ^i \\tau ^j} = {U_\\sigma }^i{U_\\tau }^j, \\; U_\\sigma U_\\tau = \\zeta U_\\tau U_\\sigma $ and $\\zeta $ is a primitive $p$ -root of unity.", "It is well known that $B\\cong M_p(\\mathbb {C})$ , and hence $\\bigotimes _1^n B$ is on one hand isomorphic to a twisted group algebra with the group $C_p^{2n}$ and on the other hand isomorphic to $M_{p^{n}}(\\mathbb {C})$ .", "This completes the construction of the required family of groups.", "We remark here that the function $\\mu $ defined above plays a central role in the theory of twisted group algebras and their polynomial identities (see [2]).", "Remark 5.2 Let $\\alpha _n \\in Z^2(C_p^{2n},\\left\\langle \\zeta \\right\\rangle )$ be the nondegenerate 2-cocycle as constructed in the previous example, and let $H_n$ be the central extensions defined by such cocycle.", "The last example shows that the group algebra $\\mathbb {C}H_n$ has an irreducible representation of degree $p^n$ .", "On the other hand, Kaplansky's theorem [20] states that if a group has an abelian subgroup of index $m$ , then all of its irreducible representations are finite with degree at most $m$ .", "This provides another proof that the minimal index of an abelian subgroup of $H_n$ tends to infinity.", "Next we provide some examples/counter examples to statements that are related to the main theorem.", "Example 5.3 Let F be an algebraically closed field of characteristic zero.", "For any finite abelian group $G$ , the group algebra $FG$ is isomorphic to a product of $\|G\|$ copies of $F$ .", "In particular, we get that $\\exp (FG)=1$ .", "Hence, we cannot hope to get an inequality of the form $\|G\| \\le \\exp (A)^K $ for any constant $K$ .", "More generally, given an $H$ -graded algebra $A$ with a nondegenerate grading, the algebra $B=FG\\otimes A$ has a natural $G\\times H$ grading which is also nondegenerate.", "In addition we have that $\\exp (B)=\\exp (A)$ .", "While the grading group is of course larger, the index of the largest abelian group remains the same.", "Example 5.4 Suppose we omit the requirement that $\\operatorname{Id}_{G}(A)$ has no $G$ -graded monomials and only assume that $\\operatorname{Id}_{G}(A)$ has $G$ -graded monomials of high degrees (as a function of $\\dim (A)$ or the cardinality of $G$ ).", "In other words we drop the assumption that $A$ is nondegenerately $G$ -graded and we only assume that the $G$ -grading on $A$ is nondegenerately bounded.", "We show that the consequence of the main theorem does not hold in general.", "Consider the algebras $A_{m}$ of upper triangular matrices $m\\times m$ where the diagonal matrices consist only of scalar matrices.", "Note that by Giambruno and Zaicev's theorem (see [13]) we have $\\exp (A)=1$ .", "Let $G$ be a group of order $n$ and assume that $m=n^{2}+1$ .", "Let $s^{\\prime }=(g_{1},\\ldots ,g_{n})\\in G^{n}$ be a tuple such that each element of $G$ appears in $s^{\\prime }$ exactly once and let $s\\in G^{\\left(n^{2}+1\\right)}$ be $n$ copies of $s^{\\prime }$ with additional $g_{1}$ at the end.", "Consider the algebra $A_{m}$ with the elementary grading corresponding to the tuple $s$ .", "We claim that $A_{m}$ has no graded multilinear monomial identities of degree $\\le n$ .", "Fix $1\\le i\\le n^{2}+1-n$ and $h\\in G$ .", "We first note that by the definition of the grading we have that $e_{i,j}$ is homogeneous of degree $s_{i}^{-1}s_{j}$ for each $1\\le i<j\\le n^{2}+1$ .", "By the choice of the tuple $s$ , the elements $\\left\\lbrace s_{i}^{-1}s_{i+1},s_{i}^{-1}s_{i+2},\\ldots ,s_{i}^{-1}s_{i+n}\\right\\rbrace $ are all distinct, and therefore, for any $h \\in G$ and $i\\le n^{2}+1-n$ we can choose $j=j(i,h)$ such that $i<j\\le i+n$ and $e_{i,j}\\in A_{h}$ .", "Let $x_{h_{1},1}\\cdots x_{h_{n},n}$ be any multilinear monomial, $h_{1},\\ldots ,h_{n}\\in G$ .", "Set $i_{1}=1$ .", "Given $i_{k}$ , define $i_{k+1}$ to be $j(i_{k},h_{k})$ so that $e_{i_{k},i_{k+1}}$ is homogeneous of degree $h_{k}$ and $i_{k}<i_{k+1}\\le i_{k}+n$ .", "It is now easy to see by induction that $i_{k}\\le 1+(k-1)n\\le 1+n^{2}$ for all $1\\le k\\le n$ so that $e_{i_{1},i_{2}}\\cdots e_{i_{n},i_{n+1}}$ is well defined as an element of $A$ and it is a nonzero evaluation of $x_{h_{1},1}\\cdots x_{h_{n},n}$ .", "For a finite group $G$ , denote by $\\gamma (G)$ the smallest index of an abelian subgroup in $G$ .", "Let $G_{n}$ be any sequence of groups where $\\gamma (G_{n})$ goes to infinity with $n$ .", "By the above construction, the algebras $B_{n}=A_{\|G_{n}\|^{2}+1}$ have $G_{n}$ gradings such that $\\dim (B_{n})$ and $\\gamma (G_{n})$ tend to infinity with $n$ .", "$B_{n}$ has no multilinear monomial identities of degrees smaller then $\|G_{n}\|$ .", "$\\exp (B_{n})=1$ .", "Example 5.5 Suppose we have a sequence of algebras $A_{n}$ with $d_{n}=\\exp (A_{n})$ monotonically increasing (i.e.", "to infinity).", "Can we necessarily find groups $G_{n}$ and nondegenerate $G_{n}$ -gradings such that the index of any abelian subgroups $U_{n}$ of $G_{n}$ tends to infinity?", "The answer is negative as the algebras of upper triangular matrices show.", "More precisely, let $UT_n(F)$ be the algebra of $n\\times n$ upper triangular matrices, which have exponent $\\exp (UT_n(F))=n$ .", "By a theorem of Valenti and Zaicev [25], every $G$ -grading on $UT_n(F)$ is isomorphic to an elementary grading.", "Unless the grading is trivial, the grading cannot be nondegenerate since $UT_n(F)_g$ contains only upper triangular matrices with zero on the diagonal for every $e\\ne g\\in G$ , so $x_{g,1} \\cdots x_{g,n}$ is an identity.", "We conclude that the only nondegenerate grading is with the trivial group, so in particular there are no nondegenerate grading such that the index of the largest abelian subgroup tends to infinity." ] ]	1403.0200
[ [ "The Scalar Field Effective Action for The Spontaneous Symmetry Breaking\n in Gravity" ], [ "Abstract We calculate the quantum effective action for a scalar field which has been recently used for a specific kind of symmetry breaking in gravity.", "Our study consists of calculating the 1-loop path integral of canonical momentum and determining the renormalization conditions.", "We will also discuss on the new renormalization conditions to redefine the new degrees of freedom corresponding to a massive vector field." ], [ "Introduction", "Studying the creation and formation of the universe is an important area in gravity where the spontaneous symmetry breaking can have useful application in[1].", "Indeed, production of the particles without loss of energy and arising the gauge bosons can be regarded as the results of the symmetry breaking method.", "There is also a sample of symmetry breaking which gives mass to the graviton.", "A massive graviton leads the newtonian gravity force to fall down at large distances.", "In addition, studying of the massive gravity and the mutual effects between the mass and the cosmological constant and also investigating the stability and renormalizability problems are another points of interest in the massive gravity framework.", "A covariant higgs mechanism method in gravity and its unitatrity problem have also been explained by Gerard t'Hooft[2].", "In the t'Hooft formalism, four scalar fields $\\phi ^A(x), A=1,...,4$ , have been used to break the gauge symmetry.", "These fields which have the global symmetry $so(3,1)$ , enter in the action through the usual quadratic form: $\\sqrt{-g}g^{\\mu \\nu }\\partial _{\\mu }\\phi ^A(x)\\partial _{\\nu }\\phi ^A(x)\\nonumber $ In the presence of the cosmological constant, we can choose the solution in the form of $\\phi ^A(x)\\propto m\\delta ^A_{\\mu }x^{\\mu }$ (as a dynamical vacuum) to produce the mass terms for all gravity degrees of freedom.", "In this way the symmetry of diffeomorphism reduces to the global $so(3,1)$ .", "One of the massive degrees of freedom has a kinetic term with different sign in the Ricci scalar.", "Such a difference makes the classical solution unstable and also the unitarity failed.", "There has been made some attempts to study the massive gravity solution which none of them needed to solve the unitatrity problem[3], [4].", "However in the t'Hooft version of massive gravity, the unitatrity problem has to be solved.", "In the t'Hooft method, there are six massive degrees of freedom: Five traceless $\\overline{h}_{ij}(x)$ with definite kinetic terms.", "They play the role of degrees of freedom for a spin 2 particle.", "The remaining degree of freedom is a scalar $u(x)$ corresponding to a scalar particle.", "As have pointed out above, its kinetic term is indefinite and the solution is unstable.", "In order to restore the untarity, $u(x)$ has to be decoupled from the other matter fields.", "Although the mass of $u(x)$ in the Fierz-Pauli approach is infinite[4], it is finite in the t'Hooft mechanism and therefore has to be decoupled through a different way.", "The proposal is to define a coupling between the matter fields and a new metric $g_{\\mu \\nu }^{matter}$ which is constructed from the original metric and the Higgs scalar field.", "Such a coupling is defined so that $u(x)$ doesn't couple to any other matter field.", "By using this proposal, $u(x)$ is finally decoupled from the matter fields only in the linear terms of $g_{\\mu \\nu }^{matter}$ .", "By use of the original metric and scalar fields, Chamseddin and Mukhanov, defined a new metric as the following[5], $H^{AB}(x)=g^{\\mu \\nu }(x)\\partial _{\\mu }\\phi ^A(x)\\partial _{\\nu }\\phi ^B(x)\\nonumber $ They added a new term functional of $H^{AB}$ to the Einstein and Hilbert action such that creates the Fierz and pauli mass terms.", "Their higgs mechanism is clean of unitarity problem and also is linearly ghost free.", "The action for the scalar fields that they considered is: $S(\\phi )\\propto \\int (d^4x\\sqrt{-g})\\,\\,\\,3(\\frac{1}{16}H^2-1)^2-\\overline{H}^A_{B}\\overline{H}^B_{A}\\nonumber $ Such that, $H=H^A_A$ and $\\overline{H}^A_B=H^A_B-\\frac{\\delta ^A_B}{4}H$ .", "In such an action, since there exist objects like $\\dot{\\phi }^n$ , $n>2$ , we require the new canonical process to have hamiltonian and poisson brackets[6] and also to quantize the theory.", "R. Fukuda and E. Kyriakopoulos, derived the effective potential through the path integral with a constraint on the zero mode field[7].", "The effective potential is function of the zero mode field, and the extremum of the effective potential is the classical solution of the action.", "By using their method, the symmetry breaking could be understood more clearly[8], [9].", "In this paper in section 2, we will use Fukuda and Kyriakopoulos method to define the spontaneous symmetry breaking in the t'Hooft mechanism.", "In section 3, the one-loop effective action will be obtained from the actions with different functionality of canonical momentum, like Chamseddine and Mukhanov sample.", "In section 4, firstly we will consider the normal renormalization conditions to practically calculate the effective action.", "Secondly In subsection 4.1, through the new renormalization conditions we will construct the New degrees of freedom.", "Such degrees of freedom which represent a non symmetric vector field in the IR scale." ], [ "Spontaneous symmetry breaking", "In the quantum field theory, the spontaneous symmetry breaking puts the quantum modes on a solution of the classical equation of motion.", "In fact we exchange the quantum vacuum with a macroscopic state in which the classical field could be measured in.", "A quantum vacuum is a linear combination of all eigne states of the field.", "By expelling the classical solution from the linear combination, we can define the effective vacuum.", "By this definition, the symmetry in the effective vacuum state will be missed.", "For example, considering a lagrangian with even function of scalar field, we have: $\\langle \\Omega \\mid \\phi \\mid \\Omega \\rangle \\,\\propto \\int \\mathcal {D}\\phi \\,\\phi \\, e^{iS(\\phi ^2)}=0 $ While by expelling the zero momentum mode from the path integral, the symmetry $\\phi \\rightarrow -\\phi $ will be broken.", "$\\langle \\Omega ^\\prime \\mid \\phi \\mid \\Omega ^\\prime \\rangle \\propto \\int \\mathcal {D}\\phi \\,\\phi \\, e^{iS(\\phi ^2)}\\delta (\\frac{1}{V}\\int dx\\phi (x)-\\phi _{0}) \\ne 0 $ As a result,$\\langle \\Omega ^\\prime \\mid \\phi \\mid \\Omega ^\\prime \\rangle $ is the classical solution of the equation of motion of the scalar field, which could be measured in the macroscopic scale.", "In fact such a macroscopic scale also determines the renormalization conditions.", "These conditions define the observable effective objects from the fundamental unscreened objects.", "Through the averaging of all macroscopic states, the expectation value of the field in the original vacuum state will be obtained: $\\langle \\Omega \\mid \\phi \\mid \\Omega \\rangle \\,\\propto \\int d\\phi _{0}\\,\\langle \\Omega ^\\prime \\mid \\phi \\mid \\Omega ^\\prime \\rangle =0$ For the diffeomorphism symmetry breaking in gravity, the scalar field lagrangian would be defined as a function of $H^{AB}$ : $H^{AB}=g^{\\mu \\nu }\\partial _{\\mu }\\phi ^{A}\\partial _{\\nu }\\phi ^{B}\\nonumber $ By choosing the classical solution $\\phi ^{A}\\propto x^{A}$ as an effective vacuum, the diffeomorphism will be broken and the gravity will be massive.", "The Riemannian space $x^{\\mu }$ becomes to a flat one $x^{A}$ , and the symmetry of diffeomorphism reduces to the lorentz symmetry.", "To consider the separation between the quantum modes and the classical solution, the partition function will be written like this: $\\mathcal {Z}=\\int da^{A}db_{\\mu }^{B}\\int \\mathcal {D}\\phi \\mathcal {D}\\pi \\delta (\\frac{1}{V}\\int dx \\phi ^{A}-a^{A})\\delta (\\frac{1}{V}\\int dx\\partial _{\\mu }\\phi ^{B}-b_{\\mu }^{B})\\nonumber \\\\\\exp ({i\\int dx(\\dot{\\phi }\\pi -\\mathcal {H}(\\phi ,\\pi ))})$ By considering $\\phi ^{A}(x)=a^{A}+b_{\\mu }^{A}x^{\\mu }+\\chi ^{A}(x)$ , we can define $\\chi ^A(x)$ as a fourier expansion.", "The fourier coefficients are analytical functions of momentum like $\\chi ^{A}(p=0)=0$ , $\\chi ^{A}(p\\ne 0)<\\infty $ .", "By defining a Hilbert space constructed of the states that $a$ and $b$ are measured in, a usual quantum theory will be created for the fields $\\chi ^{A}$ .", "Of course, if the lagrangian has only the gravitational probes, the measurable classical values in the macroscopic scale are $\\partial _{\\mu }\\phi ^{A}$ which are coupled to the gravitational metric.", "And in such a situation, $\\phi (x)$ is not measurable.", "But if the zero mode of $\\phi (x)$ to be determined in an experiment, it is a result of a little term (functional of $\\phi (x)$ ) that is attached to the real lagrangian.", "Such a little term, is omitted at the first order of approximation in the macroscopic scale.", "Anyhow in the measured states with values $b_{\\mu }^{A}$ , we can use a generating function for obtaining the quantum quantities: $e^{iF(J)}=\\int \\mathcal {D}\\phi \\mathcal {D}\\pi \\,\\delta (\\frac{1}{V}\\int dx\\partial _{\\mu }\\phi ^{A}-b_{\\mu }^{A})\\exp ({i\\int dx(\\dot{\\phi }\\pi -\\mathcal {H}+iJ\\phi )}) $" ], [ "Effective action", "In all lagrangians that are quadratic functions of time derivative of fields, it can be shown that: $\\int \\mathcal {D}\\pi e^{i\\int dx(\\dot{\\phi }\\pi -\\mathcal {H}(\\phi ,\\pi ))}=e^{iS(\\phi ,\\dot{\\phi })} $ While the lagrangians are used to break the gauge symmetry in the gravity are mainly functions of the higher power of time derivative, therefore equation (3.1) loses its efficiency.", "In such lagrangians to use the path integral method, The hamiltonian and the canonical momentum have to be obtained and the integration over the canonical momentum has to be calculated.", "In the general form, we can calculate the path integral via the perturbation method around the extremum point of the term inside the exponent: $\\dot{\\phi }(\\phi (x),\\pi (x))-\\frac{\\delta }{\\delta \\pi _{A}(x)}\\int dx\\mathcal {H}(\\phi ,\\pi )=0 $ That the extremum point of $\\pi _{A}(x)$ is arrived from the time derivative of the scalar fields.", "Through expansion of $\\pi _{A}(x)$ around the extremum point, we can obtain the generating function at the one-loop order: $e^{iF(J)}= \\int \\mathcal {D}\\phi \\, e^{S(\\phi ,\\dot{\\phi })+i\\int dx J(x)\\phi (x)}\\int \\mathcal {D}\\pi \\nonumber \\\\\\exp {-i\\int dx dy(\\pi _{A}(x)-\\pi _{A}(\\phi ,\\dot{\\phi }))\\frac{\\delta ^2 \\int dz\\mathcal {H}(\\phi ,\\pi )}{\\delta \\pi _{A}(x)\\delta \\pi _{B}(x)}\\mid _{\\pi (\\phi ,\\dot{\\phi })}(\\pi _{B}(x)-\\pi _{B}(\\phi ,\\dot{\\phi }))}$ It has to be attended to the first integral that has been written only in the space of the fourier modes of $\\phi (x)$ .", "In the equation (3.3), the term $Lndet(\\frac{\\delta ^2\\int dz\\mathcal {H}(\\phi (z),\\pi (z))}{\\delta \\pi _{A}(x)\\delta \\pi _{B}(y)})$ will be added to the action.", "We can derive this term, through the action and its functionality of the fields: $\\frac{\\delta ^2}{\\delta \\pi _{A}(x)\\delta \\pi _{B}(y)}\\int dz\\mathcal {H}(\\phi ,\\pi )\\mid _{\\pi (\\phi ,\\dot{\\phi })}=\\frac{\\delta }{\\delta \\pi _{A}(x)}\\dot{\\phi }^B(\\phi (y),\\pi (y))\\mid _{\\pi (\\phi ,\\dot{\\phi })}$ Take attention to $\\dot{\\phi }^B(x)$ in the right hand side that is defined independent of $\\phi ^A(x)$ and is not time derivative.", "And also we have the definition of the independent canonical momentum: $\\pi _{A}(\\phi ,\\dot{\\phi })=\\frac{\\delta S(\\phi ,\\dot{\\phi })}{\\delta \\dot{\\phi }^A} $ As regards, $\\dot{\\phi }^A(x)$ and $\\pi _{A}(x)$ are supposed to be independent of the scalar fields, therefore by use of the chain rule of the functional derivatives we can write: $\\int dz\\frac{\\delta \\dot{\\phi }^B(\\phi (x),\\pi (x))}{\\delta \\pi _{A}(z)}\\mid _{\\pi (\\phi ,\\dot{\\phi })}\\times \\frac{\\delta \\pi _{A}(\\phi (z),\\dot{\\pi }(z))}{\\delta \\dot{\\phi }^C(y)}=\\delta ^B_{C}\\delta (x-y)$ and therefore: $Ln det(\\frac{\\delta ^2\\int dz\\mathcal {H}(\\phi ,\\pi )}{\\delta \\pi _{A}(x)\\delta \\pi _{B}(y)})=-Lndet(\\frac{\\delta ^2S(\\phi ,\\dot{\\phi })}{\\delta \\dot{\\phi }^A(x)\\delta \\dot{\\phi }^B(y)}) $ If in the left hand side of the present equation, $\\pi _{A}(x)$ goes to the extremum point which is defined in the equation (3.2), thus $\\dot{\\phi }_{A}(x)$ in the right hand side presents as the time derivative of the scalar field.", "Then by defining the effective action: $\\Gamma (\\varphi (x))=F(J(x))-\\int dx J(x)\\varphi (x) $ and using the equations (3.7) and (3.3), we will find the effective action at one-loop order: $\\Gamma ^{(1)}(\\varphi )=S(\\varphi )+\\frac{i}{2}trLn(\\frac{\\delta ^2S(\\phi )}{\\delta \\phi ^2})\\mid _{\\varphi }-\\frac{i}{2}trLn(\\frac{\\delta ^2S(\\phi ,\\dot{\\phi })}{\\delta \\dot{\\phi }^2})\\mid _{\\dot{\\varphi }}+\\delta S^{(1)}(\\varphi ) $ The first trace, is defined in the fourier space as already reminded.", "And $\\delta S^{(1)}(\\varphi )$ is the one-loop ordered counterterm added to the renormalized action." ], [ "Renormalization conditions", "Usually, for calculation of the effective action, people have used the assumption that the effective fields are constant.", "And as a result the continuum matrix inside the trace has been diagonalized.", "While in our paper the scalar fields cannot be constant values and the derivation of the effective action seems to be difficult.", "By the way, as an important property, the effective action is a generating function.", "And we can derive the n-point functions through the functional derivatives of the effective action.", "An important problem that we encounter in calculating the effective action and n-point functions is renormalization, particularly in the lagrangians which are functions of $H^{AB}$ and have the nonrenormalizable monomials.", "The first renormalization condition could be assumed is accepting the common equation of motion between the effective action and the renormalized action: $\\frac{\\delta \\Gamma (\\varphi (x))}{\\delta \\varphi (x)}\\mid _{\\partial \\varphi =b}=0$ Inserting (3.9) in (4.1), the present equation will be produced: $\\frac{i}{2}tr[(\\frac{\\delta ^2S}{\\delta \\varphi ^2})^{-1}\\frac{\\delta }{\\delta \\varphi (x)}(\\frac{\\delta ^2S}{\\delta \\varphi ^2})]\\mid _{\\partial \\varphi =b}\\nonumber \\\\-\\frac{i}{2}tr[(\\frac{\\delta ^2S}{\\delta \\dot{\\varphi }^2})^{-1}\\frac{\\delta }{\\delta \\varphi (x)}(\\frac{\\delta ^2S}{\\delta \\dot{\\varphi }^2})]\\mid _{\\partial \\varphi =b}+\\frac{\\delta }{\\delta \\varphi (x)}\\delta S^{(1)}\\mid _{\\partial \\varphi =b}=0 $ Since the renormalized action is a local function, standing continuum matrices like $G^{AB}_{xy}=(\\frac{\\delta ^2S}{\\delta \\varphi ^A(x)\\delta \\varphi ^B(y)})^{-1}$ on the point $\\partial \\varphi =b$ , leads them to be diagonalized in the fourier space.", "for example: $\\alpha (b)p^2G(p)=1 $ And the term $\\frac{\\delta }{\\delta \\varphi }\\delta S^{(1)}\\mid _{\\partial \\varphi =b}$ , simply will be calculated in the fourier space.", "And it seems that for the actions with even function of fields, this term is trivially zero, $\\beta (b)\\delta ^{\\mu }_{A}\\int dp\\,p_{\\mu }=0$ .", "But there are many different monomials that are not considered in the condition (4.1), and to determine them we need more conditions like: $\\int dx_{1}\\cdots dx_{n-1}\\frac{\\delta ^n\\Gamma (\\varphi )}{\\delta \\varphi (x_{n})\\cdots \\delta \\varphi (x_{1})}\\mid _{\\partial \\varphi =b}=\\int dx_{1}\\cdots dx_{n-1}\\frac{\\delta ^nS(\\varphi )}{\\delta \\varphi (x_{n})\\cdots \\delta \\varphi (x_{1})}\\mid _{\\partial \\varphi =b}$ And the written equations are trivial for odd n. These equations can be solved in the fourier space because the continuum matrices on the point $\\partial \\varphi =b$ are diagonal as before.", "And the point $x_n$ has remaind outside integrals because of the translational symmetry of the connected and 1PI n-point functions.", "It means the same as the momentum conservation in the fourier space.", "In fact these conditions are written for the 1PI n-point functions with zero momentum legs in the fourier space.", "Therefore these conditions determine the Feynman vertices between interacting particles which has been probed by an observer in the IR scale.", "If the bare action has the linear symmetry $\\phi ^A(x)\\longrightarrow \\phi ^A(x)+a^A$ , it is trivial that the effective action has the same symmetry too.", "And also we can find the effective action as a functional of $\\partial _{\\mu }\\varphi ^A$ , and the observed object is $\\partial _{\\mu }\\varphi ^A$ which is coupled to the metric." ], [ "Broken gauge symmetry action for the new degrees of freedom", "Assume a bare action that the renormalization conditions cause the counterterm monomials which are constructed by both $\\varphi ^A$ and $\\partial _{\\mu }\\varphi ^A$ .", "In this case, in the points other than the classical solution, the effective action loses the symmetry $\\varphi (x)\\rightarrow \\varphi (x)+a$ .", "And even if we have $\\partial _{\\mu }\\varphi ^A$ as classical measurable values, but the quantum degrees of freedom of the effective action are values of the scalar fields in all points.", "And now if we want to have the quantum degrees of freedom analogous to the classical measurable values in the IR scale, it has been required that the effective action to be functional of only $\\partial _{\\mu }\\varphi $ and also to have the symmetry $\\varphi \\rightarrow \\varphi +a$ .", "For this purpose, the renormalization conditions have to be redefined.", "By adding an integral in the free coordinate $x_{n}$ to the equations (4.4) and choosing a $\\varphi $ other than the classical solution, we will have these conditions: $\\int dx_{1}\\cdots dx_{n}\\frac{\\delta ^n\\Gamma (\\varphi )}{\\delta \\varphi (x_{n})\\cdots \\delta \\varphi (x_{1})}=\\int dx_{1}\\cdots dx_{n}\\frac{\\delta ^nS(\\varphi )}{\\delta \\varphi (x_{n})\\cdots \\delta \\varphi (x_{1})}=0 $ In which are derived from the Taylor series of $\\Gamma (\\varphi +a)$ around $a=0$ .", "Indeed we can use the equation $\\int dx\\,\\frac{\\delta \\Gamma (\\varphi )}{\\delta \\varphi (x)}=0$ which applied for the all possible $\\varphi (x)$ , instead of using the equations(4.5) that are used for a selected $\\varphi (x)$ .", "On this approach, the symmetry $\\varphi \\rightarrow \\varphi +a$ will be appear in the effective action again.", "Now we can write $\\Gamma (\\varphi )$ as a functional of $v^A_{\\mu }(x)=\\partial _{\\mu }\\varphi ^A(x)$ and define $v^A_{\\mu }(x)$ as the effective degrees of freedom.", "By this definition, we can have: $\\frac{\\delta ^n\\Gamma (\\varphi )}{\\delta \\varphi (x_{1})\\cdots \\delta \\varphi (x_{n})}=\\partial _{1\\mu }\\cdots \\partial _{n\\nu }\\frac{\\delta ^n\\Gamma (v)}{\\delta v_{\\mu }(x_{1})\\cdots \\delta v_{\\nu }(x_{n})} $ In which are consistent to the equations (4.5).", "It has to be attended to the renormalization conditions (4.5) that eliminate only the nonsymmetric monomials.", "To renormalize the remain monomials, we need more conditions.", "Assume that an observer probes the effective vector fields $v^A_{\\mu }(x)$ as the experimental objects.", "By considering $v^A_{\\mu }(x)=0$ as the expectation values in the effective vacuum state, we can define the new action for such vector fields.", "At first we define the mass $m$ for such vector fields(without gauge symmetry): $\\int dx e^{i p (x-y)}\\frac{\\delta ^2\\Gamma (v)}{\\delta v_{\\mu }(x)\\delta v_{\\nu }(y)}\\mid _{p^2=m^2}=0$ At second, the coupling constants $g_{n}$ in the new action (interaction amplitudes between recent experimental objects) will be obtained through the following equations: $g_{n}= \\int dx_{1}\\cdots dx_{n-1}e^{ip_{1}x_{1}}\\cdot e^{ip_{n-1}x_{n-1}}e^{ip_{n}x_{n}}\\frac{\\delta ^n\\Gamma (v)}{\\delta v_{\\mu }(x_{1})\\cdots \\delta v_{\\nu }(x_{n})}\\mid _{on-shell}\\,\\,\\,\\,n\\ge 3 $ Therefore we have the scalar fields that behave like the massive vector fields in the IR scale.", "If the degrees of freedom like $g_{\\mu \\nu }:\\eta _{AB}\\partial _{\\mu }\\varphi ^A\\partial _{\\nu }\\varphi ^B$ (one of the metric $g^{matter}_{\\mu \\nu }$ introduced by t Hooft) can be defined in a more advanced way, we can have an arbitrary massive gravity action.", "Through choosing the correct mass terms, we can conserve the unitarity.", "In fact the one particle state of such an effective gravity theory will be occurred in the bounding states of the defined vector field theory." ], [ "Conclusion", "The first purpose in this letter, was studding the quantum theory of the symmetry breaking in the t'Hooft method[2].", "In the path integrals, we separated the usual fourier modes from the classical solution.", "By solving the new form of the path integral of the canonical momentum, we obtained the effective action.", "Then the renormalization conditions were implemented on the effective action.", "The next purpose, was finding the new degrees of freedom corresponded to the coordinate derivative of the fields.", "Such new degrees of freedom define a quantum massive vector field theory in the IR scale.", "Through studding the new renormalization conditions, we discussed on the recent purpose.", "From now on, one can select a real sample in the actions to continue more practically these present computations.", "And also, the effective action of the gravity could be calculated and added to the present effective action of the scalar fields.", "As well, one can obtain the new degrees of freedom from the original scalar fields to offers them as the quantum massive gravity in the IR scale.", "Acknowledgments I would like to thank dear Navid Abbasi for his useful discussions and comments." ] ]	1403.0198
[ [ "Generation of an optimal target list for the Exoplanet Characterisation\n Observatory (EChO)" ], [ "Abstract The Exoplanet Characterisation Observatory (EChO) has been studied as a space mission concept by the European Space Agency in the context of the M3 selection process.", "Through direct measurement of the atmospheric chemical composition of hundreds of exoplanets, EChO would address fundamental questions such as: What are exoplanets made of?", "How do planets form and evolve?", "What is the origin of exoplanet diversity?", "More specifically, EChO is a dedicated survey mission for transit and eclipse spectroscopy capable of observing a large, diverse and well-defined planetary sample within its four to six year mission lifetime.", "In this paper we use the end-to-end instrument simulator EChOSim to model the currently discovered targets, to gauge which targets are observable and assess the EChO performances obtainable for each observing tier and time.", "We show that EChO would be capable of observing over 170 relativity diverse planets if it were launched today, and the wealth of optimal targets for EChO expected to be discovered in the next 10 years by space and ground-based facilities is simply overwhelming.", "In addition, we build on previous molecular detectability studies to show what molecules and abundances will be detectable by EChO for a selection of real targets with various molecular compositions and abundances.", "EChO's unique contribution to exoplanetary science will be in identifying the main constituents of hundreds of exoplanets in various mass/temperature regimes, meaning that we will be looking no longer at individual cases but at populations.", "Such a universal view is critical if we truly want to understand the processes of planet formation and evolution in various environments.", "In this paper we present a selection of key results.", "The full results are available online (http://www.ucl.ac.uk/exoplanets/echotargetlist/)." ], [ "Introduction", "Within the last two decades, the field of exoplanetary science has made breathtaking advances, both in the number of systems known and in the wealth of information.", "Recently we marked the 1000th extrasolar planet discovered, of which over 400 are transiting [56], [52].", "Such numbers are impressive in themselves but dwarfed by the 3000+ transiting exoplanet candidates [66], [25], [5] obtained by the Kepler mission [10], [33], as well as the predicted tens of thousands of planets to be discovered by the GAIA mission [59].", "The large number of detections suggests that planet formation is the norm in our own galaxy [31], [25], [5], [14], [20], [83].", "Through the measurement of the planets' masses and radii we can estimate their bulk properties and get a first insight into their compositions and potential formation histories [77], [1], [27], [13].", "To take this characterisation work to the next level, we must gain an understanding of the planet's chemical composition.", "The best way to probe their chemical composition is through the study of their atmospheres.", "For transiting planets this is feasible when the planet transits its host star in our line of sight.", "This allows some of the stellar light to shine through the terminator region of the planet (transmission spectroscopy).", "Similarly, when the star eclipses the planet (i.e.", "it passes behind its host star in our line of sight) we can measure the flux difference resulting from the planet's dayside emissions (emission spectroscopy).", "In the last decade, a large body of work has accumulated on the atmospheric spectroscopy of transiting extrasolar planets [8], [9], [15], [12], [6], [63], [62], [64], [16], [19], [28], [71], [76], [47], [61], [36], [58], [74], [18], [7], [60] also see [75] for a comprehensive review.", "Given these large and ever increasing numbers of detections, it is important to understand which of these systems lend themselves to be characterised further by the use of transmission and emission spectroscopy.", "In the light of the mission concept The Exoplanet Characterisation Observatory, which has been studied by the European Space Agency as one of the M3 mission candidates, this question becomes critical.", "In this paper, we aim to quantify the number, as well as the time required to characterise spectroscopically the transiting extrasolar planets known to date.", "An overview of our results with examples for specific cases is given here with the full results available online (http://www.ucl.ac.uk/exoplanets/echotargetlist/).", "In the frame of ESA's Cosmic Vision programme, EChO has been studied as a medium-sized M3 mission candidate for launch in the 2022 - 2024 timeframehttp://sci.esa.int/echo/ [73], [72].", "During the `Phase-A study' EChO has been designed as a 1 metre class telescope, passively cooled to $\\sim $ 50 K and orbiting around the second Lagrangian Point (L2).", "The baseline for the payload consists of four integrated spectrographs providing continuous spectral coverage from 0.5 to 11$\\mu $ m (goal 0.4 to 16$\\mu $ m) at a resolving power ranging from R $\\sim $ 300 ($\\lambda < 5\\mu m$ ) to 30 ($\\lambda > 5\\mu m$ ).", "For a detailed description of the telescope and payload design, we refer the reader to the literature [49], [50], [73], [72], [65], [21], [51], [2], [84], [42], [24], [67], [80].", "The EChO science case can be best achieved by splitting the mission lifetime into three surveys, where the instrument capabilties are optimally suited to address different classes of question.", "The studied targets range from super-Earth to gas giant, temperate to very hot and stellar classes M to F. The aims of these three tiers described in the EChO Assessment Study Reporthttp://sci.esa.int/echo/53446-echo-yellow-book/ are as follows; Chemical Census: Statistically complete sample to explore the key atmospheric features: albedo, bulk thermal properties, most abundant atomic and molecular species, clouds.", "Origin: Addresses the question of the origin of exoplanet diversity by enabling the retrieval of the vertical thermal profiles and molecular abundances, including key and trace gases.", "Rosetta Stone: Benchmark cases to get insight into the key classes of planets.", "This tier will provide high signal-to-noise observations yielding very refined molecular abundances, chemical gradients and atmospheric structure.", "Spatial and temporal resolution will enable the study of weather and climate.", "The spectral resolving power (R) and signal-to-noise (SNR) target for each mission is shown in table REF .", "Table: The spectral resolving power (R=λ/ΔλR=\\lambda /\\Delta \\lambda ) and SNR requirements of each survey mode.", "The SNR target is the average in a chosen spectral element (see §).", "Target number of planets refers to the number of planets expected to be observed in each mode by the mission" ], [ "Method", "Given the instrument characteristics and the transiting planets known today, we have developed a series of models and simulators to assess the EChO capabilities and optimise the science return of the mission.", "An overview of this generation process is shown in Fig.", "REF and detailed below.", "Figure: Simplified flow chart of the workings of ETLOS and how it links with the Open Exoplanet Catalogue, OECPy and EChOSim.", "OECPy loads the catalogue values, performs the necessary calculations and assumptions and then creates a planet object containing them.", "ETLOS takes this planet object and calls the EChO end-to-end simulator (EChOSim) that simulates the observation.", "ETLOS then interprets and plots the results" ], [ "Planet Catalogue and OECPy", "We adopted the Open Exoplanet Catalogue (OEC, [52]) for the exoplanet database which generally cites original papers as sources and is kept up-to-date as an open source community project.", "We have further verified most targets using SIMBAD, the 2MASS catalogue and exoplanet.eu [56] where appropriate.", "To facilitate the EChO Target List Simulator (ETLOS) we developed the Open Exoplanet Catalogue Python Interface (OECPy, [78]) to load the exoplanet database into ETLOS.", "OECPy is a general package for exoplanet research which retrieves exoplanet parameters from databases and performs common equations.", "It is the package where most of our assumptions (such as planetary classification) are handled, along with the calculations of orbital parameters and estimations for any missing values.", "OECPy is freely available todayhttps://github.com/ryanvarley/open-exoplanet-catalogue-python." ], [ "EChOSim", "EChOSim [80], [43] is the EChO mission end-to-end simulator.", "EChOSim implements a detailed simulation of the major observational and instrumental effects, and associated systematics.", "It also allows sensitivity studies for the parameters used and thus it represents a key tool in the optimisation of the instrument design.", "Observation and calibration strategies, data reduction pipelines and analysis tools can all be designed effectively using the realistic outputs produced by EChOSim.", "The simulation output closely mimics standard STScihttp://archive.stsci.edu/hst/ FITS files, allowing for a high degree of compatibility with standard astronomical data reduction routines.", "In addition to EChOSim, we provide an observation pipeline performing the `observed' data reduction in an optimal way [79]." ], [ "EChO Target List Simulator (ETLOS)", "ETLOS works by generating the parameter files (containing the planet system information and simulation run conditions) and then simulating the observations at the spectral resolving power for each observing tier using EChOSim (version 3.0) and the associated observation pipeline (which analyses the EChOSim generated data to obtain transmission/emission spectra).", "EChOSim outputs spectra as ascii files which are used to calculate the signal to noise of the exoplanetary observation by ETLOS.", "Along with the full simulations for each tier, a single transit and eclipse for each target was simulated and binned per spectrometer channel giving the total SNR of each channel individually.", "This offers a powerful way to assess the observability of a planet in each instrument module and is useful for studies of the albedo, thermal emission and orbital phase curves.", "ETLOS is designed for the long term needs of the mission.", "New targets can be simulated as they are discovered and parameters can be changed to gauge the impact of instrument changes or optimised observation strategies.", "Changes can be made both globally, to targets meeting certain conditions and on a target by target basis making it a powerful informative tool.", "Version 1.0 of the target list has been generated using ETLOS and The Open Exoplanet CatalogueOEC Version (commit SHA-1) 305b90f (25th February 2014), future versions will be published onlinehttp://www.ucl.ac.uk/exoplanets/echotargetlist/." ], [ "Selection and Run Conditions", "We were able to simulate 404 transiting targets from the catalogue as of 25th February 2014.", "The following conservative assumptions were used in our simulations to account for contingency.", "The Phase A instrument design, see [48] and [22] The contamination due to the zodical light is a strong function of viewing direction.", "The background model (eqn.", "REF ) is evaluated based on the Hubble model out to 2.5 micron, and the DIRBE model at wavelengths beyond [34].", "The value we use is 3 times this expression for all targets.", "This is above the average value of 2.5 (the actual value varies between 0.9 at the ecliptic poles to 8 in a small number of extreme cases).", "$Zodi(\\lambda )= B_\\lambda (5500 K)\\times \\text{3.5E-14} + B_\\lambda (270 K)\\times \\text{3.58E-8} \\times I$ in units of $W/m^2/sr/m$ , where $B_\\lambda (T)$ is Planck's law written in terms of wavelength at a temperature of T $K$ .", "No flux correlation along wavelengths is assumed.", "A theoretical SNR gain of up to $\\sqrt{2}$ is achievable given that lightcurves are fully correlated over wavelength.", "Telescope jitter of 20 mas-rms at 2.8$\\times $ 10$^{-2}$ - 1 Hz and 50 mas-rms at 1 - 300 Hz [80] Circular orbits are assumed for all targets in this initial study.", "Exotic targets with eccentricity $>0.5$ , like HD 80606 b along with targets around binary stars, cannot be automatically simulated in the current iteration and need to be modelled separately.", "Future versions will include these targets automatically through improved calculations." ], [ "Missing information in catalogues", "When measured parameters such as mass and inclination are unknown, we infer them using OECPy.", "Our assumptions for these cases are described below.", "Inclination = $90\\deg $ Planetary effective temperature ($T_{\\text{pl}}$ ) is estimated using the equation: $T_{pl} = T_\\star \\left( \\sqrt{\\frac{(1-A)}{\\epsilon }\\frac{R_\\star }{2a}}\\right)^{1/2}$ where the planetary albedo is given in table REF and a greenhouse effect contribution of $\\epsilon = 0.7$ is assumed [68], [57].", "Planetary mass ($M_\\text{p}$ ) was estimated using the density of planet classes included in Table REF and the measured planetary radius ($R_\\text{p}$ ).", "The distance to the star is estimated first by using an absolute magnitude lookup table based on spectral typehttp://xoomer.virgilio.it/hrtrace/Sk.htm_SK3 from Schmid-Kaler (1982) and then using the distance magnitude relationship (eqn.", "REF ) to calculate the distance.", "We do not correct for absorption as we adopt a conservative estimate.", "$m-M = 5 \\log _{10}{d} -5$ Apparent magnitude (in K band) is calculated by converting the K band magnitude using table A5 of [35].", "This is only used for distance estimation when the V band magnitude is unavailable.", "The mean molecular weight has been estimated according to the planetary classes given in table REF .", "In particular we assumed molecular hydrogen for gaseous Jupiters and Neptunes and water vapour for super-Earths.", "Table: Our assumptions for target values based on type.", "Note that we use mass to classify a planet first, if mass is missing we then use the radius.", "In this case the mass is estimated from the densities from which are given in this table and is used in the calculation of the scale height within EChOSimTable: Planet assumptions based on planetary effective temperature" ], [ "SNR Calculation", "To assess the observability of the atmospheres, we first simulated featureless transmission spectra and black body emission spectra.", "We then assessed the detectability of specific molecular features at different abundances for a selection of planets in transit and eclipse (see §REF ).", "Each of the spectrographs in EChO are simulated (VNIR $0.55$ -$2.5 \\mu m$ ; SWIR $2.5$ -$5.0 \\mu m$ ; MW1IR $5.0$ -$8.5 \\mu m$ ; MW2IR $8.5$ -$11.0 \\mu m$ ; LWIR $11.0$ -$16.0 \\mu m$ ).", "The observability of a target is assessed by taking the mean SNR of the SWIR and two MWIR channels (covering 2.5-11$\\mu m$ ).", "The next section shows EChOSim simulated spectra with varying molecular species simulated using the number of transits determined by the original (featureless) calculations." ], [ "Molecular observability and SNR Validation", "To verify our SNR choices for the different tiers are indeed optimal, we ran three planets (GJ 1214b, GJ 436b, HD 189733b) in transmission for $CO$ , $CO_2$ , $H_2O$ , $CH_4$ at abundances of $10^{-3}$ , $10^{-5}$ , $10^{-7}$ and three planets (55 Cnc e, GJ 436b, HD 189733b) in emission with the same compositions and abundancesOnly $CO_2$ and $H_2O$ were ran for 55 Cnc b as the other cases are unrealistic given its temperature.", "The spectral files were generated using TAU [30] for transmission and as described by [69] for emission.", "The simulations were ran using the number of transits calculated in our original featureless simulations." ], [ "Results", "We find that as of 25th February 2014, 173 planets are observable in the EChO Chemical Census observation tier within the proposed mission lifetime of 4 years in transit or eclipse of which 162 are observable in both.", "In Origin 165 targets are observable in transit or eclipse with 148 in both.", "In Rossetta Stone 132 targets are observable in transit or eclipse with 78 in both.", "We note the recent discovery of over 700 planets [37], [54] are not included in our results and will be added in a future version.", "We generated three types of plot per target to show the signal-to-noise of each target with wavelength: SNR of the cumulative observations of transits required to fulfil the requirements for each observing tier (from table REF ), Fig.", "REF Multiple broadband photometry for a single transit and eclipse, Fig.", "REF For a subset of targets simulations of transit and eclipse spectra indicating the strength of molecular features when the appropriate SNR and R are reached.", "Fig.", "REF In addition to the examples shown here, the plots for all observable targets are available online (http://www.ucl.ac.uk/exoplanets/echotargetlist/).", "Our simulations of molecular features show that even for low abundances (e.g.", "mixing ratios $<10^{-5}$ ) the Chemical Census tier is sufficient to detect most of the trace gases (see also [69] [70]) with Origin and Rossetta Stone tiers being increasingly able to constrain spectral features.", "Figures REF & REF show the parameter space that EChO could explore if launched today.", "Observable systems cover a very broad parameter space in terms of stellar types, eccentricity, temperatures and densities.", "Fig.", "REF and REF show the sample of todays targets that are observable with EChO compared with the expected yield of the TESS mission [53]; by stellar magnitude and planetary radius (Fig.", "REF ) and orbital period and planetary radius (Fig.", "REF ).", "Fig.", "REF shows how the number of observable planets changes with mission lifetime (based on how long is needed for the required number of transit or eclipse events to occur).", "Overheads and scheduling are not considered here, see [40] and [26].", "Note that the targets given here are the current sample which each observation tiers can handpick from based on the scientific benefit and scheduling.", "Figure: Example of the signal-to-noise plots generated by ETLOS (through EChOSim) for Gilese 3470 b in transmission (left) and Gilese 436 b emission (right).", "The upper plots show the SNR per bin at the number of transits required for each EChO tier.", "The lower plots demonstrate the differences of each tier by showing a simulated molecular case for each planet (offset for clarity) at the number of transits per mode given in the upper plots.", "Gliese 3470 b is simulated using a composition of 8E-3 H 2 OH_2O, 4E-3 CH 4 CH_4, 2E-3 COCO, 2E-5 CO 2 CO_2, 1E-4 NH 3 NH_3 generated by TauRex .", "Depending on the radius and temperature ratios of the planet and the star, some spectral bands are more appropriate where others may be less informative.", "By covering a broad wavelength range we can cover a large range of planets from Jupiter to super-Earth, hot to temperate.Figure: Example of the per instrument module signal-to-noise plots generated by ETLOS (through EChOSim) for Gilese 3470 b in transmission (left) and Gilese 436 b emission (right).", "The plots show the SNR of each instrument module as a single bin for a single transit and eclipse.Figure: Examples of the molecular cases simulated.", "The plots show the types of features detectable in each tier with Rosseta Stone being able to constrain features in models with much lower abundances.", "See http://www.ucl.ac.uk/exoplanets/echotargetlist/ for the other simulations.Figure: Planetary parameter space probed today by the Chemical Census tier.Figure: Stellar parameter space probed today by the Chemical Census tier.Figure: Plots showing the stellar magnitude with planetary radii for current observable population for Left: The current observable population with EChO.", "Right: the predicted yield of planets from the Transiting Exoplanet Survey Satellite (TESS, figure published with permission from the TESS Concept Study Report submitted in September 2012).Figure: Top Left: The predicted yield of planets from the Transiting Exoplanet Survey Satellite (TESS, figure published with permission from ).", "The others show the current observable population (in the Chemical Census tier) with EChO with stars brighter than J=10 and J=11.", "Note that differences in the EChO and TESS plots are explained by differing catalogue versions, the TESS plot showing all targets (not just transiting) and our calculation of missing parameters (such as J magnitude) as described in section Figure: Plot showing how the length of the mission affects the number of observable targets.", "The mission length required per target is calculated by taking the number of transits required meet the requirements of each tier times the period.", "Scheduling, overheads and total observation time are not taken into account (see discussion)" ], [ "Discussion", "In this first iteration (V1.0) of the EChO target list we show that a large number of characteristically diverse targets are observable today.", "As mentioned in our run conditions (§REF ), in this version we used a circular orbit assumption (transiting planets found so far normally have an eccentricity of less than 0.1).", "Eccentric targets have more complex orbital parameters (eg the planet temperature, transit duration, flux from the star and transit depth are all affected) which are not modelled automatically in the current versions of our simulators.", "Future software updates will incorporate the tools needed to simulate these planets; as more exoplanets are being discovered it is likely more targets will fall into this category increasing its importance.", "We do not consider scheduling or overheads in this exercise as we only generate the sample from which targets can be selected.", "Efficiency due to overheads is expected to be 80% but scheduling conflicts and required observation time per target will be the major factors determining what can be observed (see [40] and [26]).", "The Rosetta Stone tier in particular requires a high number of transits for many of its targets which can be very demanding on the EChO schedule.", "In binary systems in which the planet orbits just one of the stars we can simulate these as normal, where the planet orbits the binary (ie the Kepler-47b system [41]) additional modelling will be required which will be included in a future version.", "This work shows only the science achievable with today's (catalogue as of 25th February 2014) target sample.", "Survey missions from space like Gaia [45], TESS [53], CHEOPS [11] and K2 [32] and ground based surveys such as NGTS [82], ESPRESSO [44], WASP [46] and HAT [3] are expected to yield thousands of transiting planets in the next five years.", "With many additional targets the choice of ideal candidates will be much greater in the future, adding to the efficiency of the EChO mission.", "TESS in particular (launching 2017) is observing brighter targets than Kepler which are ideal candidates for EChO (Fig.", "REF ).", "See [39] for a detailed summary of each survey with respect to EChO target selection.", "Both JWST (James Webb Space Telescope) and E-ELT (European Extremely Large Telescope) are expected on-line in the next decade and will take high resolution spectra of exoplanets.", "These are highly complementary and mutually beneficial to EChO with JWST observing a few tens of planets at mid to high resolution and E-ELT providing ultra-high resolution spectroscopy of a few tens of planets in narrow bands.", "EChO has been designed to look at the broader picture, surveying hundreds of exoplanets with instantaneous broad wavelength coverage.", "This instantaneous wavelength coverage allows EChO observations to be corrected for stellar activity effects (like star-spots and faculae) which have a strong chromatic dependence [29], [38], [17], [55].", "Additionally some instrument on-board JWST and EChO are designed for different regimes e.g.", "JWST NIRSpec is particularly sensitive allowing high resolution spectroscopy in its spectral coverage but is therefore restricted to fainter targets (see Fig.", "REF ).", "Whilst EChO is optimised for brighter stars around K $<$ 9 [23] JWST NIRSpec is optimised for targets roughly K $>$ 8.5 magnitude.", "Figure: The J-band limiting magnitudes for the different NIRSpec modes as a function of host star temperature.", "The lines represent the magnitude limit at which a source can be observed in the full wavelength range of the given mode.", "The dashed lines are for the high resolution gratings and solid lines for the medium resolution gratings.", "The plots are taken from the ESA JWST page on exoplanet transit spectroscopy with NIRSpec (http://www.cosmos.esa.int/web/jwst/exoplanets, retrieved 10th December 2014)." ], [ "Conclusion", "EChO has been designed as a 1 metre dedicated survey mission for transit and eclipse spectroscopy capable of observing a large, diverse and well-defined planet sample within its four year mission lifetime, our results show that the majority of this diversity can be achieved with today's target sample.", "173 of today's targets can be observed in EChO's broadest survey tier (Chemical Census, R = 50 at $\\lambda < 5 \\mu m$ , SNR = 5) in transit and/or eclipse, 162 are observable in both.", "This sample covers a wide range of planetary and stellar sizes, temperatures, metallicities and semi-major axes.", "This excludes the recent discovery of over 700 planets [37], [54] that will be added in a future version.", "Out of these 173, the majority (165) can be observed in transit or eclipse (148 in both) at the higher spectral resolving power and SNR of the Origin tier (R = 100 at $\\lambda < 5 \\mu m$ , SNR = 10).", "Dedicated studies show that an accurate retrieval can be performed out of origin targets, so that the physical causes of said diversity can be identified ([70], [4]).", "For a subset of these, we can push the spectral resolving power to R = 300 at $\\lambda < 5 \\mu m$ at SNR = 20 so that a very detailed knowledge of the planets can be achieved (Rosetta Stone).", "Said knowledge will include spatial and temporal resolution enabling studies of weather and climate, as well as very refined chemical composition of these atmospheres to penetrate the intricacies of equilibrium and non-equilibrium chemistry and formation.", "While today there are 132 targets capable of being observed in this tier in transit or eclipse (of which 78 can be done in both), as the Rosetta Stone tier is very demanding of the EChO schedule, this is the large sample from which the target 10-20 planets can be chosen from.", "EChO's unique contribution to exoplanetary science is in identifying the main constituents of hundreds of exoplanets in various mass/temperature regimes, meaning that we will be looking no longer at individual cases but at populations of planets.", "Such a universal view is critical if we truly want to understand the processes of planet formation and evolution and how they behave in various environments.", "R. Varley is Funded by a UCL IMPACT Studentship, I. Waldmann is funded by the UK Space Agency and STFC, J. C. Morales is funded by a CNES fellowship.", "G. Tinetti is a Royal Society URF, G. Micela acknowledges support by the ASI/INAF contract I/022/12/0.", "We would like to thank Vincent Coudé du Foresto and Jean-Philippe Beaulieu for their help with this work." ] ]	1403.0357
[ [ "Shaping topological properties of the band structures in a shaken\n optical lattice" ], [ "Abstract To realize band structures with non-trivial topological properties in an optical lattice is an exciting topic in current studies on ultra cold atoms.", "Here we point out that this lofty goal can be achieved by using a simple scheme of shaking an optical lattice, which is directly applicable in current experiments.", "The photon-assistant band hybridization leads to the production of an effective spin-orbit coupling, in which the band index represents the pseudospin.", "When this spin-orbit coupling has finite strengths along multiple directions, non-trivial topological structures emerge in the Brillouin zone, such as topological defects with a winding number 1 or 2 in a shaken square lattice.", "The shaken lattice also allows one to study the transition between two band structures with distinct topological properties." ], [ "=1 Shaping topological properties of the band structures in a shaken optical lattice Shao-Liang Zhang, Qi Zhou Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, HK To realize band structures with non-trivial topological properties in an optical lattice is an exciting topic in current studies on ultra cold atoms.", "Here we point out that this lofty goal can be achieved by using a simple scheme of shaking an optical lattice, which is directly applicable in current experiments.", "The photon-assistant band hybridization leads to the production of an effective spin-orbit coupling, in which the band index represents the pseudospin.", "When this spin-orbit coupling has finite strengths along multiple directions, non-trivial topological structures emerge in the Brillouin zone, such as topological defects with a winding number 1 or 2 in a shaken square lattice.", "The shaken lattice also allows one to study the transition between two band structures with distinct topological properties.", "The study on topological matters is one of the most important themes in condense matter physics in the past few years[1], [2].", "When non-trivial topology exists in the band structures of certain solid materials, a wide range of novel topological matters arise.", "Whereas the effort of searching for such materials in solids has been continuously growing, there have been great interests of realizing topological matters using ultra cold atoms[3], [4], [6], [5], [7].", "In such highly controllable atomic systems, it is easy to manipulate the interaction between atoms and external fields so that topological properties of quantum matters could be engineered using standard experimental techniques.", "It is hoped that ultra cold atoms will not only provide a perfect simulator of electronic systems, but also opportunities to create new types of topological matters with no counterpart in solids.", "As SOC is a key ingredient in many topological matters, the realization of synthetic spin-orbit coupling(SOC) using the Raman scheme[8], [9], [10], [11], [12], [13] opens the door for accessing topological matters in ultra cold atoms.", "However, a shortcoming of the current scheme is that SOC exists along only one spatial direction.", "This has become one of the bottlenecks for an experimental realization of topological matters in ultra cold atoms.", "Both theoretical and experimental interests on shaken optical lattices have been arising recently[14], [15], [16], [17], [18], [19], [20].", "It has been shown that such a scheme allows one to manipulate both the magnitude and the sign of tunneling constants.", "In this Letter, we point out that shaken lattices provide physicists an unprecedented opportunity to explore topological matters.", "We will show that (I) one could use shaken lattices to create a fully controllable “SOC\" with finite strengths along multiple spatial directions, where band indices play the role of the “spin\" degree of freedom; (II) such an effective SOC allows one to create band structures with non-trivial topological properties using currently available experimental techniques; (III) varying these microscopic parameters, including the frequency, amplitude and phase shift of the shaken lattice, physicists could study the evolution between two band structures with different topological properties.", "As an example, we show that a two-dimensional shaken square lattice, where photon-assistant band hybridization creates an effective SOC near the $\\Gamma $ (${\\vec{k}}=(0,0)$ ) and $M$ ($\\vec{k}=(\\pi ,\\pi )$ ) point in the Brillouin zone(BZ), $H=A(k_x^2-k_y^2)\\sigma _z+(B k_xk_y+C)\\sigma _x+D\\sigma _y,$ where $A, B, C, D$ are momentum-independent constants.", "$A\\sim (t_p+ t_s)/4$ is determined by the static lattice, $B$ can be regarded as the strength of a momentum-dependent magnetic field in the transverse direction, $C, D$ correspond to the strengths of a constant magnetic field.", "Depending on the choices of these parameters, which are well tunable in shaken lattices, the Hamiltonian in Eq.", "(REF ) can be classified to two categories.", "Case 1 $B=0$ .", "Eq.", "(REF ) reduces to $H=A(k_x^2-k_y^2)\\sigma _z+C\\sigma _x+D\\sigma _y$ , where spin-momentum locking exists along only one direction, similar to SOC realized by the Raman scheme in continuum[8], [9], [10], [11], [12], [13].", "This type of SOC does not give rise to interesting topological properties of the band structure.", "Case 2 $B\\ne 0$ .", "SOC exists along multiple spatial directions and leads to nontrivial topological properties of band structures.", "A special case is $C=D=0$ , which corresponds to a SOC of $d$ -wave nature, as $H=Ak^2 \\cos (2\\theta _{\\bf k})\\sigma _z+Bk^2\\sin (2\\theta _{\\bf k})\\sigma _x$ , where $\\theta _{\\bf k}=\\arg \\lbrace k_x+ik_y\\rbrace $ .", "It could be used to produce a topological semimetal[21], and is also relevant in the studies of crystalline topological insulators[22].", "The lattice we consider is written as $V({\\bf r}, t)=V\\sum _{i=x,y} \\cos ^2(k_0r_i+f\\cos (\\omega t+\\varphi _i)/{2})+V^{\\prime }({\\bf r})$ where $r_x=x$ , $r_y=y$ , $k_0=\\pi /d$ , $d$ is the lattice spacing.", "$f$ is the shaking amplitude, $\\omega $ is the frequency, and $\\varphi _i$ is the phase of the shaking along the $x,y$ directions, as shown in Fig.(1A).", "An additional lattice $V^{\\prime }({\\bf r})=\\alpha V\\cos (2k_0x)\\cos (2k_0y)$ is introduced to make the external potential to be inseparable(See Supplementary Materials), where $\\alpha $ is a small number.", "We set $\\varphi _x=0$ and $\\varphi _y=\\varphi $ .", "$\\varphi =0$ and $\\varphi =\\pi /2$ correspond to a linear shaking along the diagonal direction and a cyclic mode respectively.", "Eq.", "(REF ) can be regarded as Kramer-Henneberger representation of an irradiated lattice[23], [24].", "Using standard Floquet-Bloch theorem, the solution of the Schrodinger equation could be written as $\\Psi ({\\bf r},t)=e^{-i\\epsilon t}\\Phi _{\\bf k}({\\bf r},t)=e^{-i\\epsilon t}e^{i{\\bf k}\\cdot {\\bf r}}u_{\\bf k}({\\bf r},t)$ , where $\\epsilon $ is the quasienergy, $u_{{\\bf k}}({\\bf r},t)$ satisfies $u_{\\bf k}({\\bf r+R},t+2\\pi /\\omega )=u_{\\bf k}({\\bf r},t)$ , and ${\\bf R}$ is the lattice vector of the static one.", "Eq.", "(REF ) possesses a certain spatio-temporal(dynamical) symmetry, which allows one to conclude that for the circular and linear shaking, Floquet-Bloch bands have a four- and two-fold symmetry respectively (See Supplementary Materials).", "Figure: (A) Photon-assistant band hybridization.", "Solid and dashed curves represent bands in the static lattice and side bands produced by shaking.", "In the left panel, ω\\omega is close to the separation between the ss and pp bands of the static lattice, while the frequency ω ' \\omega ^{\\prime } for the right pannel is half of ω\\omega .", "Inset is a schematic of the shaken lattice.", "The arrow and circle represent the linear and cyclic shaking respectively.", "(B) Due to parity conservation, 1-1- photon resonance couples orbital with different parities at the same lattice site, whereas 2-2- photon resonance couples orbital with different parities at the nearest neighbor sites.", "Red and blue colors represent signs of the wave functions.Applying the identify for Bessel function $\\exp [{x(\\zeta -\\zeta ^{-1})/2}]=\\sum _{n=-\\infty }^{\\infty } J_n(x)\\zeta ^n$ , we obtain $V({\\bf r}, t)=V_0({\\bf r})+V^{\\prime }({\\bf r})+\\sum _{n\\ne 0}V_n({\\bf r})e^{in\\omega t}$ , $V_n({\\bf r})=\\bigg \\lbrace \\begin{array}{cc}\\frac{i^n}{2} VJ_n(f)&\\left( \\cos (2k_0 x)+e^{in\\varphi } \\cos (2k_0y)\\right), \\\\& n\\in even\\\\\\frac{i^{n+1}}{2} VJ_n(f)& \\left(\\sin (2k_0 x)+e^{in\\varphi }\\sin (2k_0y)\\right).", "\\\\& n\\in odd\\end{array}$ where $V_0({\\bf r})+V^{\\prime }({\\bf r})$ is time-independent, and ${V}_n({\\bf r})$ is a dynamically induced lattice potential that excites the system by a multiple-photon energy $n\\hbar \\omega $ , as shown in Fig (1B).", "Eq.", "(REF ) shows that $\\omega $ controls which bands shall be hybridized at resonance, i.e., which ${V}_n({\\bf r})$ is dominant.", "Moreover, it shows that the parity of $V_n({\\bf r})$ is $(-1)^n$ , which gives rise to distinct properties of the band hybridization for even and odd values of $n$ .", "This can be directly seen in the tight-binding picture.", "As shown in Fig (1B), due to the parity conservation, ${V}_n({\\bf r})$ with an even $n$ cannot couple Wannier wave functions with different parities, for instance, $s$ and $p_x$ orbitals, at the same lattice site.", "Its leading contribution is to couple them at nearest neighbor sites.", "Moreover, the coupling between a $s$ orbital at site ${\\bf R}_i $ with the two $p_x$ orbitals at ${\\bf R}_i\\pm d\\hat{x}$ , where $\\hat{x}$ is a unit vector along the x direction, differ by a minus sign.", "This provides a momentum-dependent inter-band coupling for accessing nontrivial topological band structures.", "In contrast, for ${V}_n({\\bf r})$ with an odd $n$ , it couples Wannier wave functions with different parities at the same site.", "Its leading contribution is a momentum-independent inter-band coupling, which could not produce non-trivial topological bands.", "Figure: (A) A typical band structure, where V=16E R V=16E_R, E R =ℏ 2 π 2 /(2md 2 )E_R=\\hbar ^2\\pi ^2/(2md^2), α=0.2\\alpha =0.2, f=0.1df=0.1d, ω=3.7E R \\omega =3.7E_R.", "(B) Schematic of the effective SOC produced by photon-assistent band hybridization.", "The (p x ,0)(p_x, 0), (p y ,0)(p_y, 0) bands (highlighted in the dashed box) form a pseudo–spin-1/2 system.", "Each of the four bands (s,1)(s,1), (s,2)(s,2), (d xy ,-1)(d_{xy},-1) and (d xy ,-2)(d_{xy},-2) serves as an intermediate state as the one in a typical Λ\\Lambda transitions.", "Due to the parity conservation as shown in Fig (1), (s,2)(s,2) and (d xy,-2 )(d_{xy,-2}) band naturally provide a momentum dependent coupling between (p x ,0)(p_x, 0) and (p y ,0)(p_y, 0) while (s,1)(s,1) and (d xy ,-1)(d_{xy},-1) only provide a constant coupling.We expand the Floquet mode as $\\Phi _{\\bf k}({\\bf r},t)=\\sum _{m{\\bf k},n} c_{m {\\bf k}, n} \\phi _{m {\\bf k}}({\\bf r}) e^{in\\omega t}$ , where $c_{m {\\bf k}, n} $ are time-independent constants, and $\\phi _{m {\\bf k}}({\\bf r})$ is the Bloch wave function of the static lattice $V_0({\\bf r})$ with band index $m$ and crystal momentum ${\\bf k}$ .", "The standard Floquet-matrix representation may be expressed as $\\begin{split}\\sum _{m^{\\prime }, n^{\\prime }}(\\mathcal {V}^{m,m^{\\prime }}_{n-n^{\\prime },{\\bf k}}+\\mathcal {V^{\\prime }}^{m,m^{\\prime }}_{{\\bf k}}\\delta _{n,n^{\\prime }} +(\\epsilon _{m^{\\prime }{\\bf k}}^0\\\\+n^{\\prime }\\hbar \\omega )\\delta _{n,n^{\\prime }}\\delta _{m,m^{\\prime }}) c_{m^{\\prime } {\\bf k}, n^{\\prime }} =\\epsilon c_{m {\\bf k}, n},\\end{split}$ where $\\mathcal {V}^{m,m^{\\prime }}_{n-n^{\\prime },{\\bf k}}=\\int d{\\bf r}\\phi _{m{\\bf k}}^({\\bf r})V_{n-n^{\\prime }}({\\bf r})\\phi _{m^{\\prime }{\\bf k}}({\\bf r})$ and $\\mathcal {V^{\\prime }}^{m,m^{\\prime }}_{{\\bf k}}=\\int d{\\bf r}\\phi _{m{\\bf k}}^({\\bf r})V^{\\prime }({\\bf r})\\phi _{m^{\\prime }{\\bf k}}({\\bf r})$ .", "The physical meaning of Eq.", "(REF ) is apparent.", "A band of the static lattice could absorb or emit $n$ photons and form a sequence of side bands.", "This photon-assistant process make a resonance between certain side bands possible.", "For convenience, we use the notation $(m, n)$ to represent the dynamically generated $n$ th side band of the band $m$ of the static lattice.", "The coupling between two side bands through $\\mathcal {V}^{m,m^{\\prime }}_{n,{\\bf k}}$ will be referred as to a $n$ -photon process.", "As discussed before, the matrix elements $\\mathcal {V}^{m,m^{\\prime }}_{n,{\\bf k}}$ is either a constant or ${\\bf k}$ -dependent, depending on $n$ and the parity difference of these two bands .", "For instance, $\\begin{split}&\\mathcal {V}^{s,p_x}_{2l,{\\bf k}}=i^{2l+1}\\Omega _{2l}\\sin {k_x}, \\mathcal {V}^{s,p_y}_{2l,{\\bf k}}=i^{2l+1} e^{2il \\varphi }\\Omega _{2l}\\sin {k_y},\\\\&\\mathcal {V}^{s,p_x}_{2l+1,{\\bf k}}=i^{2l+2}\\Omega _{2l+1}, \\mathcal {V}^{s,p_y}_{2l+1,{\\bf k}}=i^{2l+2}e^{i(2l+1)\\varphi }\\Omega _{2l+1}\\\\\\end{split}$ where $l$ is an integer, and $d$ has been absorbed to $k_{i=x,y}$ , $\\Omega _{2l}=VJ_{2l}(f) \\langle W_{s, {\\bf R}_i} \|\\cos (2k_0x)\|W_{p_x, {\\bf R}_i+d\\hat{x}}\\rangle $ , $\\Omega _{2l+1}= \\frac{V}{2}J_{2l+1}(f)\\langle W_{s, {\\bf R}_i} \|\\sin (2k_0x)\|W_{p_x, {\\bf R}_i}\\rangle $ are constants, $W_{m, {\\bf R}_i}$ is a Wannier function for the band $m$ at the lattice site ${\\bf R}_i$ .", "Other matrix elements are provided in the Supplementary Materials.", "We perform a numerical calculation on the Floquet-matrix by including up to the $g$ bands, each of which contains 9 side bands.", "Tight-binding model for the dispersions in the static lattice has been used, i.e., $\\epsilon ^0_{s, \\bf k}=-t_s(\\cos {k_x}+\\cos {k_y})-\\Delta $ , $\\epsilon ^0_{p_x, \\bf k}=t_p\\cos {k_x}-t_s\\cos {k_y}$ , $\\epsilon ^0_{p_y, \\bf k}=-t_s\\cos {k_x}+t_p\\cos {k_y}$ , $\\epsilon ^0_{d_{xy}, \\bf k}=t_p(\\cos {k_x}+\\cos {k_y})+\\Delta $ where $t_s$ , $t_p$ are the tunneling amplitudes.", "A typical band structure is shown in Fig (2 A).", "At $\\Gamma $ and $M$ point, there are two nearly degenerate bands, the main contributions to which come from $(p_x, 0 )$ and $(p_y, 0)$ .", "The hybridization with the other side bands lifts the degeneracy at $\\Gamma $ and $M$ point present in an ordinary static square lattice.", "From the numerical solutions, we have found that the qualitative physics of the Floquet-matrix is captured by a six-band model.", "As shown in Fig (2 B), each of the four bands $(s, -2)$ , $(s, -1)$ , $(d_{xy},-1)$ and $(d_{xy},-2)$ , couples to both $(p_x,0)$ and $(p_y,0)$ , producing a second order virtual hopping processes between the latter two bands, similar to the standard $\\Lambda $ process in atomic physics.", "Including other bands only leads to quantitative changes of the results.", "If one treats the nearly degenerate $p_y$ and $p_x$ bands as spin-up and spin-down, an effective SOC Hamiltonian can be formulated, $H={\\bf B_k}\\cdot \\vec{\\sigma }$ , where ${\\sigma }_{x,y,z}$ are the Pauli matrices.", "To be explicit, we obtain $\\begin{split}H&=-(t_p+t_s)(\\cos {k_x}-\\cos {k_y})\\sigma _z/2\\\\&+(B_{x,e}\\sin {k_x}\\sin {k_y}+B_{x,o})\\sigma _x\\\\&+(B_{y,e}\\sin {k_x}\\sin {k_y}+B_{y,o})\\sigma _y,\\end{split}$ where the subscript $e$ and $o$ represent the effective magnetic field induced by processes of even and odd number of photons.", "For the $\\sigma _z$ term, the main contribution comes from the energy difference between $(p_x, 0)$ and $(p_y, 0)$ bands, and a small correction $B_z^{\\prime }$ from hybridization with other bands does not affect the results(See Supplementary Materials).", "For the transverse fields, we define $\\bar{B}_{i=e,o}=B_{x,i}-i B_{y,i}$ for convenience, where $\\begin{split}\\bar{B}_{e}=\\Omega _0+\\sum _{n\\in even}\\frac{-e^{-in\\varphi }\\Omega ^2_{n}}{\\epsilon _{s, {\\bf k}}+n\\hbar \\omega -(\\epsilon _{p_x, {\\bf k}}+\\epsilon _{p_y, {\\bf k}})/2},\\\\-\\frac{e^{i n\\varphi }\\Omega ^2_{n}}{\\epsilon _{d_{xy}, {\\bf k}}+n\\hbar \\omega -(\\epsilon _{p_x, {\\bf k}}+\\epsilon _{p_y, {\\bf k}})/2}.\\\\\\end{split}$ where $\\Omega _0=\\langle W_{p_x,{\\bf R}_i+d\\hat{y}}\|V^{\\prime }({\\bf r})\|W_{p_y,{\\bf R}_i+d\\hat{x}}\\rangle $ , $\\epsilon _{m, {\\bf k}}=\\epsilon ^0_{m, {\\bf k}}+\\langle \\phi _{m{\\bf k}}({\\bf r}) \|V^{\\prime }({\\bf r})\|\\phi _{m{\\bf k}}({\\bf r}) \\rangle $ has taken into account the shift of each bands due to $V^{\\prime }({\\bf r})$ .", "As for $B_{x,o}$ and $B_{y,o}$ , the expressions are identical, with the summation over odd integers.", "In the leading order, $\\bar{B}_{e,o}$ are momentum independent constants, as $\\epsilon _{m, {\\bf k}}$ in the denominators of Eq.", "(REF ) may be replaced by their values at the $\\Gamma $ and $M$ point in the numerator.", "To simplify the expressions, we apply a spin rotation about the $z$ axis, $ e^{i\\theta \\sigma _z/2} \\sigma _x e^{-i\\theta \\sigma _z/2}=\\cos \\theta \\sigma _x+\\sin \\theta \\sigma _y $ , $ e^{i\\theta \\sigma _z/2} \\sigma _y e^{-i\\theta \\sigma _z/2}=-\\sin \\theta \\sigma _x+\\cos \\theta \\sigma _y$ , where $\\tan (\\theta )=-B_{y,e}/B_{x,e}$ , so that $B_{x, e}\\sigma _x+B_{y, e}\\sigma _y \\rightarrow B_{e}\\sigma _x$ , where $B_e=\\sqrt{B^{2}_{x,e}+B^2_{y,e}}$ .", "Near the $\\Gamma $ point, $\\cos {k_x}-\\cos {k_y}\\sim (k_x^2-k_y^2)$ , the effective magnetic field ${\\bf B_k}$ in Eq.", "(REF )becomes $\\begin{split}{\\bf B_k}=\\Big (B_{e}k_xk_y+\\tilde{B}_{x,o}, \\tilde{B}_{y,o}, \\frac{t_p+t_s}{4}(k_x^2-k_y^2)\\Big )\\end{split}$ where $\\tilde{B}_{x,o}=B_{x,o}\\cos \\theta -B_{y,o}\\sin \\theta $ and $\\tilde{B}_{y,o}=B_{x,o}\\sin \\theta +B_{y,o}\\cos \\theta $ .", "This leads to the expression for the Hamiltonian in Eq.", "($\\ref {MO}$ ).", "The formalism of the Hamiltonian near the $M$ point is the same, with quantitatively different values of the three components of ${\\bf B_k}$ .", "Depending on the choice of $\\omega $ and $f$ , both Case 1 and Case 2 of Eq.", "(REF ) can be realized.", "1-photon process If one tunes the $(s,1)$ band to be closest to the $(p_x,0), (p_y,0)$ bands, the 1-photon process is dominant, which leads to $\\tilde{B}_{x,o}\\gg B_{e}$ .", "Eq.", "(REF ) becomes $H=\\pm \\frac{t_s+t_p}{4}(k_x^2-k_y^2)\\sigma _z+\\tilde{B}_{x,o}\\sigma _x+\\tilde{B}_{y,o}\\sigma _y,$ where $\\pm $ corresponds to the $\\Gamma $ and $M$ points respectively.", "Case 1 of Eq.", "(REF ) is then achieved.", "2-photon process Choosing a proper frequency so that $(s, 2)$ is the closest one to $(p_x,0)$ and $(p_y,0)$ bands, 2-photon process is dominate.", "It is worth pointing out that $\\Omega _2$ can be further amplified by significantly enlarging the overlap integral for the Wannier wave functions in the nearest neighbor sites in double-well lattices[25].", "Eq.", "(REF ) then becomes $H=\\pm \\frac{t_s+t_p}{4}(k_x^2-k_y^2)\\sigma _z+B_{e}k_xk_y\\sigma _x$ Topological defects then emerge at the $\\Gamma $ and $M$ points where the effective magnetic field ${\\bf B_k}$ vanishes.", "For a closed loop in the momentum space around one of these two points , a winding number of $\\pm 2$ of ${\\bf B_k}$ is evident, as ${\\bf B_k}\\sim (\\sin (2\\theta _{\\bf k}),0, \\cos (2\\theta _{\\bf k}))$ .", "In general, both 1- and 2-photon processes contribute to the effective Hamiltonian.", "Eq.", "(REF ) allows one to investigate how the two band structures with distinct topological properties may evolve from one to the other when $\\omega $ continuously changes.", "From the numerical solution of the Floquet-matrix, we find that if the $(p_x, 0)$ and $(p_y,0)$ bands are not degenerate with other side bands at the $\\Gamma $ and $M$ point, a spin-$1/2$ description is sufficient for describing the eigenstates near these two points, as they are dominated by $(p_x, 0)$ and $(p_y,0)$ bands.", "The spin eigen state is written as $(\\cos (\\alpha _{\\bf k}/2), e^{i\\beta _{\\bf k}}\\sin (\\alpha _{\\bf k}/2))$ , from which an effective magnetic field ${\\bf B_k}$ is constructed, as $\\alpha _{\\bf k}$ and $\\beta _{\\bf k}$ correspond to the direction of the unit vector ${\\bf B_k}/\|{\\bf B_k}\|$ on the Bloch sphere and the energy splitting gives rise to the strength of ${\\bf B_k}$ .", "Figure: Topological defects in the band structure.", "Small arrows represent the strength and direction of the xx and zz components of the 𝐁 𝐤 {\\bf B_k} field in Eq.().", "Such a momentum-dependent magnetic field determines the orientation of the pseudo-spin-1/2 composed by (p x ,0)(p_x,0) and (p y ,0)(p_y, 0).", "Topological defects emerge at places where 𝐁 𝐤 {\\bf B_k} vanishes.", "Small filled and empty dots represent defects with winding number 1 and -1-1 respectively.", "The number and directions of big arrows on closed loops represent the winding number of the 𝐁 𝐤 {\\bf B_k} field along these loops in the Brillouin zone.", "(A-B) For linear shaking with ϕ=0\\varphi =0, a finite constant component B ˜ x,o \\tilde{B}_{x,o} in Eq.", "() leads to the splitting of the defects of winding number 2 to two defects with winding number 1.", "(C) Topological defects with opposite winding numbers have annihilated each other when the shaken frequency excesses a certain critical value.", "(D) For circular shaking with ϕ=π/2\\varphi =\\pi /2, B ˜ x,o =0\\tilde{B}_{x,o}=0, the spin is tilted towards the normal direction of the k x -k y k_x-k_y plane due to a finite B ˜ y,o \\tilde{B}_{y,o} (not shown).", "The projection of the spin on the plane has a winding number of 2.", "With increasing B ˜ y,o \\tilde{B}_{y,o}, all spins eventually become perpendicular to the plane.", "For all figures, V=20E R V=20E_R, α=0.05\\alpha =0.05, f=0.03df=0.03d.", "ω/E R =3.9,4.0,4.2,4.2\\omega /E_R=3.9, 4.0,4.2,4.2 for (A-D).Fig.", "(3) shows a few typical topological structures of ${\\bf B_k}$ .", "When $\\varphi =0$ , ${\\bf B_k}$ has only the $x$ and $z$ components.", "When $(s, 2)$ is the closest side band to $(p_x, 0)$ and $(p_y,0)$ bands, topological defects are present.", "Interestingly, we find that, due to a finite $\\tilde{B}_{x, o}$ induced by the $(s, 1)$ band, the topological defects of winding number 2 at the $\\Gamma $ point splits to two ones with winding number 1 as shown in Fig (3 A).", "The same phenomenon occurs at the $M$ point.", "This can be seen from the fact that ${\\bf B}$ now vanishes at $(\\pm k^, \\mp k^)$ , where $k^=\\sqrt{\\tilde{B}_{x,o}/B_e}$ and $\\tilde{B}_{x,o}/B_e>0$ in our case.", "Near these two points, ${\\bf B_k} \\sim (\\tilde{k}_x+\\tilde{k}_y, 0, \\tilde{k}_x-\\tilde{k}_y)$ that corresponds to a winding number 1, where $\\tilde{k}_{i=x,y}=k_{i}\\pm k^$ .", "We have also verified that if $\\Omega _1=0$ , the splitting is absent and only defects of winding number 2 show up at the $\\Gamma $ and $M$ points.", "In general cases with a finite $\\tilde{B}_{x, o}$ , the winding number of the ${\\bf B}$ field on a closed loop in the BZ depends on how many defects it encloses, as shown in Fig.", "(3 A).", "If one makes $(s, 1)$ to be more close to the $p$ bands with changing $\\omega $ , $\\tilde{B}_{x, o}$ and $k^*$ increases, and the defect with winding number 1 split from the $\\Gamma $ point gradually approaches the defect of winding number $-1$ from the $M$ point, as shown Fig.", "(3 B, C), and the topological structures eventually disappears.", "This establishes the evolution between two band structures with distinct topological properties.", "As the spin corresponds to the band index, the topological structure here and its evolution can be visualized in experiments using a variety of schemes[26], [27], [28].", "It is worth pointing out that $\\tilde{B}_{x,o}$ relies on the phase shift $\\varphi $ .", "For the cyclic shaking, $\\varphi =\\pi /2$ , $\\tilde{B}_{x,o}=0$ .", "Changing the value of $\\omega $ only leads to a tilting of the spin along the $y$ direction, and the winding number of the spin on the $\\sigma _x-\\sigma _z$ plane is not affected, as shown in Fig.", "(3 D).", "This also indicates that for the cyclic shaking, there is a four-fold symmetry in the Brillouin zone.", "We have verified that figure 3 (A-D) are consistent with general understandings of the symmetry of the Floquet-Bloch bands for linear and circularly shaking lattice as discussed before.", "Whereas we have been focusing on two dimensions in this Letter, all discussions can be directly applied to one dimension, where a shaken lattice produces nearly flat bands with a Zak phase $\\pm \\pi $ (See Supplementary Materials).", "The general principle of producing a multi-dimensional SOC using dynamically generated band hybridization could be straightforwardly generalized to other lattice geometries.", "In practice, to minimize heating effects, one should choose a small shaken amplitude, and a shaking frequency that is off-resonance with characteristic energy scales in a single Floquet-Bloch band such as $t_s$ and $t_p$ .", "In C. Chin's recent experiment[16], both these requirements have been fulfilled, and atoms can be prepared in a desired Floquet-Bloch band with long lifetime up to 1s.", "Whereas a one-photon resonance has been used in current experiments, one could straightforwardly generalize such a shaken scheme to two-photon resonance.", "It is very promising that the interplay between tunable lattice geometry and the well controllable shaking scheme will lead to fruitful results on shaping the topology of band structures in optical lattices in the near future.", "Acknowledgement This work is supported by NSFC-RGC( NCUHK453/13).", "Note Near the completion of this manuscript, two preprints (arXiv:1402.3295, arXiv:1402.4034) on topological band structures in shaken optical lattices have just appeared.", "Supplementary Material Spatio-temporal (dynamical) symmetry of shaken lattices Consider a static lattice with a four-fold symmetry, i.e., the one discussed in the main text.", "If one circularly shakes it, i.e., $\\varphi =\\pi /2$ , the Floquet-Bloch Hamiltonian $e^{-i{\\bf k}\\cdot {\\bf r}}(\\hat{P}^2/2m+V({\\bf r}, t))e^{i{\\bf k}\\cdot {\\bf r}}$ remains unchanged under a simultaneous $\\pi /2$ rotation in the real and momentum space and a time-translation, $ x\\rightarrow y, y\\rightarrow -x, k_x\\rightarrow k_y, k_y\\rightarrow -k_x, t\\rightarrow t-\\pi /(2\\omega ).", "$ Floquet-Bloch bands are four-fold symmetry.", "In contrast, for linear shaking with $\\varphi =0$ , the Floquet-Bloch Hamiltonian is no longer invariant under the above transformation.", "Instead, it is unchanged only if ${\\bf r}\\rightarrow -{\\bf r}$ , ${\\bf k}\\rightarrow -{\\bf k}$ and $t\\rightarrow t-\\pi /\\omega $ , which shows that Floquet-Bloch bands have two-fold symmetry under linear shaking.", "Different symmetries of Floquet-Bloch bands indicate different topological structures under circularly or linearly shaking, as shown in Fig.", "(3) of the main text.", "Whereas the above discussions apply to any lattices with a finite value of $\\alpha $ , the case that $\\alpha =0 $ and $V^{\\prime }({\\bf r})=0$ needs a special consideration.", "Under this situation, the lattice potential becomes separable, i.e., $V({\\bf r}, t)=V(x, t)+V(y,t)$ , the Floquet equation $(H(x,y,t)-i\\partial _t)\\Phi _{\\bf k}(x,y,t)=\\epsilon \\Phi _{\\bf k}(x,y,t)$ can be decoupled to two independent equations $\\begin{split}\\left(-\\frac{\\hbar }{2m}\\partial _x^2+V\\cos ^2(k_0x+f\\cos (\\omega t))-i\\partial _t\\right)\\Phi _{k_x}(x,t)\\\\=\\epsilon _{k_x} \\Phi _{k_x}(x,t)\\\\\\left(-\\frac{\\hbar }{2m}\\partial _y^2+V\\cos ^2(k_0x+f\\cos (\\omega t+\\varphi )-i\\partial _t)\\right)\\Phi _{k_y}(y,t)\\\\=\\epsilon _{k_y} \\Phi _{k_y}(y,t),\\end{split}$ where $&\\Phi _{\\bf k}(x,y,t)=\\Phi _{k_x}(x,t)\\Phi _{k_y}(y,t), \\,\\,\\,\\,\\,\\,\\,\\,\\, \\epsilon _{\\bf k}=\\epsilon _{k_x}+\\epsilon _{k_y}.$ Figure: The energy difference δ\\delta between the (p x ,0)(p_x,0) and the (p y ,0)(p_y,0) bands, where V=16E R V=16E_R.", "As a demonstration, δ\\delta is evaluated at (π/2,π/2)(\\pi /2, \\pi /2).", "When α=0\\alpha =0, the lattice potential becomes separable, δ\\delta vanishes at any points along the (π,±π)(\\pi , \\pm \\pi ) directions.", "For any infinitesimal α\\alpha , δ\\delta becomes finite.Figure: Topological band structures as a function of α\\alpha .", "When α=0\\alpha =0(left column), the band structure is independent on ϕ\\varphi .", "The two bands, (p x ,0)(p_x, 0) and (p y ,0)(p_y,0) become degenerate along the (π,±π)(\\pi , \\pm \\pi ) directions, as indicated by the black solid lines.", "For any finite values of α\\alpha (middle and right column), δ\\delta becomes finite away from the location of defects.", "For all figures, V=20E R V=20E_R, f=0.01df=0.01d and ω=8.4E R \\omega =8.4E_R.It is clear that the phase difference $\\varphi $ can be absorbed by applying a time-translation $t\\rightarrow t-\\varphi /\\omega $ to Eq.", "(2) alone.", "The Floquet-Bloch spectrum is therefore independent on $\\varphi $ .", "In particular, due to Eq.", "(4), $p_x$ and $p_y$ bands become degenerate if $k_x=\\pm k_y$ , i.e., along the $(\\pi , \\pi )$ and $(\\pi , -\\pi )$ direction in the BZ, as shown in Fig(1) of this supplementary material.", "For any infinitesimal $\\alpha \\ne 0$ , the Floquet equation is no longer separable, and a finite gap emerges along the $(\\pi , \\pi )$ and $(\\pi , -\\pi )$ directions in the BZ.", "As shown in Fig (2) of this supplementary material, depending on the choice of the phase shift $\\varphi $ , Floquet-Bloch bands exhibit different symmetries and topological structures.", "Such a dynamical symmetry was not studied in shaken lattice before.", "Matrix elements $\\mathcal {V}^{i,j}_{n,{\\bf k}}$ Take $s$ and $p_x$ band as an example, we expand the Bloch wave functions in the basis of Wannier functions $W_m({\\bf r})$ , and rewrite $\\mathcal {V}^{s,p_x}_{n,{\\bf k}}$ as $\\mathcal {V}^{s,p_x}_{n,{\\bf k}}=\\sum _{{\\bf R}_i{\\bf R}_j}\\int d{\\bf r}W_s({\\bf r-R}_i)W_{p_x}({\\bf r-R}_j)V_n({\\bf r})e^{i{\\bf k\\cdot }({\\bf R}_j-{\\bf R}_i)}.$ As the static state $V_0({\\bf r})$ is separable along the $x$ and $y$ directions, the two Wannier wave function can be written as $W_s({\\bf r})=w_0(x)w_0(y)$ and $W_{p_x}({\\bf r})=w_1(x)w_0(y)$ , where $w_0(x)$ and $w_1(x)$ are the lowest two Wannier functions for a one dimensional lattice $V_0(x,0)$ or $V_0(0,x)$ respectively.", "Apparently, $W_s({\\bf r})=W_s(-{\\bf r}) $ and $W_{p_x}(x, y)=-W_{p_x}(-x, y)=W_{p_x}(x, -y)$ .", "If $n$ is odd, one sees that the integral in Eq.", "(REF ) is finite when taking $i=j$ , due to the fact that $V_{2l+1}({\\bf r})=-V_{2l+1}(-{\\bf r})$ .", "This means that $V_{2l+1}({\\bf r})$ is able to couple the Wannier orbital $W_s({\\bf r-R}_i)$ and $W_p({\\bf r-R}_i)$ at the same lattice site ${\\bf R}_i$ .", "Meanwhile, the integral in Eq.", "(REF ) is much smaller if $i\\ne j$ because of the the small overlap of the Wannier wave functions at different lattice site.", "Therefore, $\\mathcal {V}^{s,p_x}_{2l+1,{\\bf k}}$ becomes a constant in the leading order, $\\mathcal {V}^{s,p_x}_{2l+1,{\\bf k}}=i^{2l+2}\\Omega _{2l+1},\\,\\,\\,\\,\\,\\,\\, \\mathcal {V}^{s,p_y}_{2l+1,{\\bf k}}=i^{2l+2}e^{i(2l+1)\\varphi }\\Omega _{2l+1}$ where $\\Omega _{2l+1}= \\frac{V}{2}J_{2l+1}(f)\\langle W_{s, {\\bf R}_i} \|\\sin (2k_0x)\|W_{p_x, {\\bf R}_i}\\rangle $ .", "If $n$ is even, the situation is very different.", "It is clear that $V_{2l}({\\bf r})$ is not able to couple the two Wannier orbital $W_s({\\bf r})$ and $W_p({\\bf r})$ at the same lattice site.", "The leading contribution to $\\mathcal {V}^{s,p_x}_{2l,{\\bf k}}$ therefore must come from the nearest neighbor ones.", "Through a simple calculation, one sees that $\\mathcal {V}^{s,p_x}_{2l,{\\bf k}}=i^{2l+1}\\Omega _{2l}\\sin (k_xd), \\,\\,\\,\\,\\,\\,\\, \\mathcal {V}^{s,p_y}_{2l,{\\bf k}}=i^{2l+1} e^{2il \\varphi }\\Omega _{2l}\\sin (k_yd),$ where $\\Omega _{2l}=VJ_{2l}(f) \\langle W_{s, {\\bf R}_i} \|\\cos (2k_0x)\|W_{p_x, {\\bf R}_i+d\\hat{x}}\\rangle $ , $d$ is the lattice spacing and $\\hat{\\bf x}$ is the unit vector along the $x$ axis.", "Similarly, we have $\\mathcal {V}^{p_x,d_{xy}}_{2l,{\\bf k}}&=&i^{2l+1}e^{2il \\varphi }\\Omega _{2l}\\sin {k_yd}, \\mathcal {V}^{p_y,d_{xy}}_{2l,{\\bf k}}=i^{2l+1} \\Omega _{2l}\\sin {k_xd}\\nonumber \\\\ \\mathcal {V}^{p_x,d_{xy}}_{2l+1,{\\bf k}}&=&i^{2l+2}e^{i(2l+1)\\varphi }\\Omega _{2l+1}, \\mathcal {V}^{p_y,d_{xy}}_{2l+1,{\\bf k}}=i^{2l+2}\\Omega _{2l+1}$ $\\mathcal {V}^{d_x,p_x}_{2l+1,{\\bf k}}&=&i^{2l+2}\\Omega ^{\\prime }_{2l+1},\\,\\,\\,\\,\\,\\,\\, \\mathcal {V}^{d_x,p_y}_{2l+1,{\\bf k}}=0\\nonumber \\\\ \\mathcal {V}^{d_x,p_x}_{2l,{\\bf k}}&=&i^{2l+1}\\Omega ^{\\prime }_{2l}\\sin (k_xd), \\,\\,\\,\\,\\,\\,\\, \\mathcal {V}^{d_x,p_y}_{2l,{\\bf k}}=0\\nonumber \\\\ \\mathcal {V}^{d_y,p_x}_{2l+1,{\\bf k}}&=&0,\\,\\,\\,\\,\\,\\,\\, \\mathcal {V}^{d_y,p_y}_{2l+1,{\\bf k}}=i^{2l+2}e^{i(2l+1)\\varphi } \\Omega ^{\\prime }_{2l+1}\\nonumber \\\\ \\mathcal {V}^{d_y,p_x}_{2l,{\\bf k}}&=&0, \\,\\,\\,\\,\\,\\,\\, \\mathcal {V}^{d_y,p_y}_{2l,{\\bf k}}=i^{2l+1} e^{2il \\varphi }\\Omega ^{\\prime }_{2l}\\sin (k_yd)$ where $\\Omega ^{\\prime }_{2l+1}=\\frac{V}{2}J_{2l+1}(f)\\langle W_{d_x, {\\bf R}_i} \|\\sin (2k_0x)\|W_{p_x, {\\bf R}_i}\\rangle $ and $\\Omega ^{\\prime }_{2l}=VJ_{2l}(f) \\langle W_{d_x, {\\bf R}_i} \|\\cos (2k_0x)\|W_{p_x, {\\bf R}_i+d\\hat{x}}\\rangle $ .", "$&\\mathcal {V}^{s,d_{xy}}_{n,{\\bf k}}=0\\\\ \\nonumber &\\mathcal {V}^{s,d_x}_{2l+1,{\\bf k}}=i^{2l+2}\\Omega _{d,2l+1}\\sin (k_xd),\\\\ \\nonumber &\\mathcal {V}^{s,d_y}_{2l+1,{\\bf k}}=i^{2l+2}e^{i(2l+1)\\varphi }\\Omega _{d,2l+1}\\sin (k_yd)\\nonumber \\\\ \\nonumber &\\mathcal {V}^{s,d_x}_{2l,{\\bf k}}=i^{2l+1}\\Omega _{d,2l}, \\\\ \\nonumber &\\mathcal {V}^{s,d_y}_{2l,{\\bf k}}=i^{2l+1} e^{2il \\varphi }\\Omega _{d,2l}$ where $\\Omega _{d,2l+1}= VJ_{2l+1}(f)\\langle W_{s, {\\bf R}_i} \|\\sin (2k_0x)\|W_{d_x, {\\bf R}_i+d\\hat{x}}\\rangle $ and $\\Omega _{d,2l}=\\frac{V}{2}J_{2l}(f) \\langle W_{s, {\\bf R}_i} \|\\cos (2k_0x)\|W_{d_x, {\\bf R}_i}\\rangle $ Matrix elements $\\mathcal {V^{\\prime }}^{i,j}_{{\\bf k}}$ We can define $&\\Omega _s=\\langle W_{s,{\\bf R}_i}\|\\cos (2k_0x)\|W_{s,{\\bf _R}_i}\\rangle , \\\\ \\nonumber &\\Omega ^{\\prime }_s=\\langle W_{s,{\\bf R}_i}\|\\cos (2k_0x)\|W_{s,{\\bf _R}+d\\hat{x}}\\rangle , \\\\ \\nonumber &\\Omega _p=\\langle W_{p_x,{\\bf R}_i}\|\\cos (2k_0x)\|W_{p_x,{\\bf _R}_i}\\rangle , \\\\ \\nonumber &\\Omega ^{\\prime }_p=\\langle W_{p_x,{\\bf R}_i}\|\\cos (2k_0x)\|W_{p_x,{\\bf _R}+d\\hat{x}}\\rangle , \\\\ \\nonumber &\\Omega _{sp}=\\langle W_{s,{\\bf R}_i}\|\\cos (2k_0x)\|W_{p_x,{\\bf _R}+d\\hat{x}}\\rangle .$ Then $&\\mathcal {V^{\\prime }}^{s,s}_{\\bf k}=\\alpha V \\Omega ^2_s,\\,\\,\\,\\,\\,\\,\\mathcal {V^{\\prime }}^{p_x,p_x}_{\\bf k}=\\alpha V \\Omega _s\\Omega _p, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{p_y,p_y}_{\\bf k}=\\alpha V \\Omega _s\\Omega _p,\\,\\,\\,\\,\\,\\,\\mathcal {V^{\\prime }}^{d_{xy},d_{xy}}_{\\bf k}=\\alpha V \\Omega ^2_p, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{s,p_x}_{\\bf k}=i\\alpha V \\Omega _{sp}\\Omega _s \\sin k_xd, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{s,p_y}_{\\bf k}=i\\alpha V \\Omega _{sp}\\Omega _s \\sin k_yd, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{p_x,d_{xy}}_{\\bf k}=i\\alpha V \\Omega _{sp}\\Omega _p \\sin k_yd, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{p_y,d_{xy}}_{\\bf k}=i\\alpha V \\Omega _{sp}\\Omega _p \\sin k_xd, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{p_x,p_y}_{\\bf k}=\\alpha V \\Omega ^2_{sp}\\sin k_xd \\sin k_yd, \\\\ \\nonumber &\\mathcal {V^{\\prime }}^{s,d_{xy}}_{\\bf k}=-\\alpha V \\Omega ^2_{sp}\\sin k_xd \\sin k_yd.$ Correction to $B_z$ $\\begin{split}\\Delta E^{\\prime }=\\sum _{n\\in even}\\left(\\frac{\\Omega ^2_{n}\\sin ^2(k_y d)}{\\epsilon _{s, {\\bf k},n}-\\epsilon _{p_y, {\\bf k}}}-\\frac{\\Omega ^2_{n}\\sin ^2(k_x d)}{\\epsilon _{s, {\\bf k},n}-\\epsilon _{p_x, {\\bf k}}}\\right)+\\sum _{n\\in odd}\\left(\\frac{\\Omega ^2_{n}}{\\epsilon _{s, {\\bf k},n}-\\epsilon _{p_y, {\\bf k}}}-\\frac{\\Omega ^2_{n}}{\\epsilon _{s, {\\bf k},n}-\\epsilon _{p_x, {\\bf k}}}\\right)\\nonumber \\\\+\\sum _{n\\in even}\\left(\\frac{\\Omega ^2_{n}\\sin ^2(k_x d)}{\\epsilon _{d_{xy}, {\\bf k},n}-\\epsilon _{p_y, {\\bf k}}}-\\frac{\\Omega ^2_{n}\\sin ^2(k_y d)}{\\epsilon _{d_{xy}, {\\bf k},n}-\\epsilon _{p_x, {\\bf k}}}\\right)+\\sum _{n\\in odd}\\left(\\frac{\\Omega ^2_{n}}{\\epsilon _{d_{xy}, {\\bf k},n}-\\epsilon _{p_y, {\\bf k}}}-\\frac{\\Omega ^2_{n}}{\\epsilon _{d_{xy}, {\\bf k},n}-\\epsilon _{p_x, {\\bf k}}}\\right)\\nonumber \\\\+\\left(\\frac{\\Omega ^2_{0s}\\sin ^2(k_y d)}{\\epsilon _{s, {\\bf k}}-\\epsilon _{p_y, {\\bf k}}}-\\frac{\\Omega ^2_{0s}\\sin ^2(k_x d)}{\\epsilon _{s, {\\bf k}}-\\epsilon _{p_x, {\\bf k}}}\\right)+\\left(\\frac{\\Omega ^2_{0p}\\sin ^2(k_x d)}{\\epsilon _{d_{xy}, {\\bf k}}-\\epsilon _{p_y, {\\bf k}}}-\\frac{\\Omega ^2_{0p}\\sin ^2(k_y d)}{\\epsilon _{d_{xy}, {\\bf k}}-\\epsilon _{p_x, {\\bf k}}}\\right)\\end{split}$ where $n$ in the subscript of $\\epsilon _{s, {\\bf k},n}$ and $\\epsilon _{p_{xy}, {\\bf k},n}$ is the side band index, $\\Omega _{0s}=\\alpha V\\Omega _{sp}\\Omega _s$ and $\\Omega _{0p}=\\alpha V\\Omega _{sp}\\Omega _p$ .", "As both these terms $\\sim k_x^2-k_y^2$ near the $\\Gamma $ and $M$ points, they contribute a correction to the expression of $B_z$ .", "Their coefficients in the parentheses are much smaller than $t_p+t_s$ in the small $\\Omega _n$ limit and can be ignored.", "One-dimensional shaken lattices Figure: Contours of pseudo-spin when the momentum k x k_x changes from -π/d-\\pi /d to π/d\\pi /d.", "From left to right, the detuning is 0, =Δ ˜ c =\\tilde{\\Delta }_c and >Δ ˜ c >\\tilde{\\Delta }_c respectively.All discussions in the main text can be directly generalized to one dimension.", "For the 1-photon process that hybridizes $(s,1)$ and $(p, 0)$ , the Hamiltonian can be written as $H=(t_s+t_p)\\cos (2k_x)\\sigma _z/2-\\Omega _1\\sigma _x$ .", "This is what has been realized in C. Chin's experiment[16].", "The Zak phase[29], which characterizes the winding number of the spin when $k_x$ changes from $-\\pi $ to $\\pi $ , is zero.", "In contrast, when the 2-photon process is dominant, the Hamiltonian can then be written as $H=\\left((t_s+t_p)\\cos (2k_x)/2+\\tilde{\\Delta }/2\\right)\\sigma _z-\\Omega _2\\sin (2k_x) \\sigma _y,$ where $\\tilde{\\Delta }=\\Delta -2\\hbar \\omega $ is the detuning.", "Such Hamiltonian is equivalent to that obtained in a tilted double-well lattice[7].", "This Hamiltonian could produce two flat bands in the limit $t_s=t_p=\\Omega $ and $\\tilde{\\Delta }=0$ .", "Moreover, in certain parameter regions, it provides a finite Zak phases $\\pm \\pi $ .", "As shown in Fig (3) of this supplementary material, when $\|\\tilde{\\Delta }\|<\\tilde{\\Delta }_c=t_s+t_p$ , it corresponds to a $2\\pi $ rotation of the spin on the $y-z$ plane when $k_x$ changes from $-\\pi $ to $\\pi $ , which leads to a Zak phase of $\\pm \\pi $ for this pseudo-spin-1/2 system.", "With changing the value of $\\omega $ , when $\\tilde{\\Delta }>\\tilde{\\Delta }_c$ , the contour of spin does not encloses the origin and the Zak phase vanishes." ] ]	1403.0210
[ [ "Commensurations and Metric Properties of Houghton's Groups" ], [ "Abstract We describe the automorphism groups and the abstract commensurators of Houghton's groups.", "Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups.", "As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays." ], [ "Introduction", "The family of Houghton groups $\\mathcal {H}_n$ was introduced by Houghton [7].", "These groups serve as an interesting family of groups, studied by Brown [2], who described their homological finiteness properties, by Röver [10], who showed that these groups are all subgroups of Thompson's group $V$ , and by Lehnert [9] who described the metric for ${\\mathcal {H}}_2$ .", "Lee [8] described isoperimetric bounds, and de Cornulier, Guyot, and Pitsch [4] showed that they are isolated points in the space of groups.", "Here, we classify automorphisms and determine the abstract commensurator of $\\mathcal {H}_n$ .", "We also give sharp estimates for the word metric which are sufficient to show that the map from the abstract commensurator to the group of quasi-isometries of $\\mathcal {H}_n$ is an injection." ], [ "Definitions and background", "Let ${\\mathbb {N}}$ be the set of natural numbers (positive integers) and $n\\ge 1$ be an integer.", "We write ${\\mathbb {Z}}_n$ for the integers modulo $n$ with addition and put $R_n={\\mathbb {Z}}_n\\times {\\mathbb {N}}$ .", "We interpret $R_n$ as the graph of $n$ pairwise disjoint rays; each vertex $(i,k)$ is connected to $(i,k+1)$ .", "We denote by $\\mathsf {Sym}_n$ , $\\mathsf {FSym}_n$ and $\\mathsf {FAlt}_n$ , or simply $\\mathsf {Sym}$ , $\\mathsf {FSym}$ and $\\mathsf {FAlt}$ if $n$ is understood, the full symmetric group, the finitary symmetric group and the finitary alternating group on the set $R_n$ , respectively.", "The Houghton group $\\mathcal {H}_n$ is the subgroup of $\\mathsf {Sym}$ consisting of those permutations that are eventually translations (of each of the rays).", "In other words, the permutation $\\sigma $ of the set $R_n$ is in $\\mathcal {H}_n$ if there exist integers $N\\ge 0$ and $t_i=t_i(\\sigma )$ for $i\\in {\\mathbb {Z}}_n$ such that for all $k \\ge N$ , $(i,k)\\sigma = (i, k+t_i)$ ; throughout we will use right actions.", "Note that necessarily the sum of the translations $t_i$ must be zero because the permutation needs of course to be a bijection.", "This implies that ${\\mathcal {H}}_1\\cong \\mathsf {FSym}$ .", "For $i,j\\in {\\mathbb {Z}}_n$ with $i\\ne j$ let $g_{ij}\\in \\mathcal {H}_n$ be the element which translates the line obtained by joining rays $i$ and $j$ , given by $(i,n)g_{ij} &= (i,n-1) \\text{ if }n > 1,\\\\(i,1)g_{ij} &= (j,1),\\\\(j,n)g_{ij} &= (j,n+1) \\text{ if } n\\ge 1\\text{ and}\\\\(k,n)g_{ij} &= (k,n)\\text{ if } k\\notin \\lbrace i,j\\rbrace .$ We also write $g_i$ instead of $g_{i\\,i+1}$ .", "It is easy to see that $\\lbrace g_i\\mid i\\in {\\mathbb {Z}}_n\\rbrace $ , as well as $\\lbrace g_{ij}\\mid i,j\\in {\\mathbb {Z}}_n,\\,i\\ne j\\rbrace $ , are generating sets for $\\mathcal {H}_n$ if $n\\ge 3$ as we can simply check that the commutator $[g_0,g_1]=g_0^{-1}g_{1}^{-1}g_0g_{1}$ transposes $(1,1)$ and $(2,1)$ .", "In the special case of ${\\mathcal {H}}_2$ , $g_1$ is redundant as $g_1=g_0^{-1}$ .", "Further, an additional generator to $g_0$ is required to generate the group; we choose $\\tau $ which fixes all points except for transposing $(0,1)$ and $(1,1)$ .", "It is now clear that the commutator subgroup of $\\mathcal {H}_n$ is given by $\\mathcal {H}_n^{\\prime }=\\left\\lbrace \\begin{array}{ll}\\mathsf {FAlt},& \\text{if } n\\le 2\\\\\\mathsf {FSym},& \\text{if } n\\ge 3\\end{array}\\right.$ For $n\\ge 3$ , we thus have a short exact sequence $1\\longrightarrow \\mathsf {FSym}\\stackrel{\\hbox{}}{\\longrightarrow }\\mathcal {H}_n\\stackrel{\\pi }{\\longrightarrow }{\\mathbb {Z}}^{n-1}\\longrightarrow 1$ where $\\pi (\\sigma )=(t_0(\\sigma ),\\ldots ,t_{n-2}(\\sigma ))$ is the abelianization homomorphism.", "We note that as the sum of all the eventual translations must be zero, we have the last translation is determined by the preceding ones: $t_{n-1}(\\sigma )=-\\sum _{i=0}^{n-2}t_i(\\sigma ).$ We will use the following facts freely throughout this paper, see Dixon and Mortimer [5] or Scott [11].", "Lemma 1.1 The group $\\mathsf {FAlt}$ is simple and equal to the commutator subgroup of $\\mathsf {FSym}$ , and $\\operatorname{Aut}(\\mathsf {FAlt})=\\operatorname{Aut}(\\mathsf {FSym})=\\mathsf {Sym}$ ." ], [ "Automorphisms of $\\mathcal {H}_n$", "Here we determine the automorphism group of $\\mathcal {H}_n$ .", "First we establish that we have to look no further than $\\mathsf {Sym}$ .", "We let $N_G(H)$ denote the normalizer, in $G$ , of the subgroup $H$ of $G$ .", "Proposition 2.1 Every automorphism of $\\mathcal {H}_n$ , $n\\ge 1$ , is given by conjugation by an element of $\\mathsf {Sym}$ , that is to say $\\operatorname{Aut}(\\mathcal {H}_n)=N_{\\mathsf {Sym}}(\\mathcal {H}_n)$ .", "From the above, the finitary alternating group $\\mathsf {FAlt}$ is the second derived subgroup of $\\mathcal {H}_n$ , and hence characteristic in $\\mathcal {H}_n$ .", "So every automorphism of $\\mathcal {H}_n$ restricts to an automorphism of $\\mathsf {FAlt}$ .", "Since $\\operatorname{Aut}(\\mathsf {FAlt})=\\mathsf {Sym}$ , this restriction yields a homomorphism $\\operatorname{Aut}(\\mathcal {H}_n)\\longrightarrow \\mathsf {Sym}$ and we need to show that it is injective with image equal to $N_{\\mathsf {Sym}}(\\mathcal {H}_n)$ .", "In order to see this let $\\psi \\in \\operatorname{Aut}(\\mathcal {H}_n)$ be an automorphism.", "Compose this with an inner automorphism (of $\\mathsf {Sym}$ ) so that the result is an (injective) homomorphism $\\alpha : \\mathcal {H}_n\\rightarrow \\mathsf {Sym}$ whose restriction to $\\mathsf {FAlt}$ is trivial.", "We let $k\\in {\\mathbb {N}}$ and consider the following six consecutive points $a_\\ell =(i,k+\\ell )$ of $R_n$ for $\\ell \\in \\lbrace 0,1,\\ldots ,5\\rbrace $ .", "We denote by $g_i^\\alpha $ the image of $g_i$ under $\\alpha $ , and by $(x\\,y\\,z)$ the 3-cycle of the points $x$ , $y$ and $z$ .", "Using the identities $g_i^{-1} (a_1\\,a_2\\,a_3)g_i = (a_0\\,a_1\\,a_2) \\text{ and }g_i^{-1} (a_3\\,a_4\\,a_5)g_i = (a_2\\,a_3\\,a_4)$ and applying $\\alpha $ , which is trivial on $\\mathsf {FAlt}$ , we get $(g_i^\\alpha )^{-1} (a_1\\,a_2\\,a_3) g_i^\\alpha = (a_0\\,a_1\\,a_2)\\text{ and }(g_i^\\alpha )^{-1} (a_3\\,a_4\\,a_5) g_i^\\alpha = (a_2\\,a_3\\,a_4).$ Hence, we must have that $g_i^\\alpha $ maps $\\lbrace a_1,a_2,a_3\\rbrace $ to $\\lbrace a_0,a_1,a_2\\rbrace $ , and then also $\\lbrace a_3,a_4,a_5\\rbrace $ to $\\lbrace a_2,a_3,a_4\\rbrace $ .", "The conclusion is that it maps $a_3$ to $a_2$ .", "Applying a similar argument to all points in the branches $i$ and $i+1$ , it follows that $g_i^\\alpha =g_i$ , and since $i$ was arbitrary, this means that $\\alpha $ is the identity map.", "With Lemma REF in mind we now present the complete description of $\\operatorname{Aut}(\\mathcal {H}_n)$ .", "Theorem 2.2 For $n\\ge 2$ , the automorphism group $\\operatorname{Aut}(\\mathcal {H}_n)$ of the Houghton group $\\mathcal {H}_n$ is isomorphic to the semidirect product $\\mathcal {H}_n\\rtimes \\mathcal {S}_n$ , where $\\mathcal {S}_n$ is the symmetric group that permutes the $n$ rays.", "By the proposition, it suffices to prove that every $\\alpha \\in \\mathsf {Sym}$ which normalizes $\\mathcal {H}_n$ must map $(i,k+m)$ to $(j,l+m)$ for some $k,l\\ge 1$ and all $m\\ge 0$ .", "To this end, we pick $\\alpha \\in N_\\mathsf {Sym}(\\mathcal {H}_n)$ and $\\sigma \\in \\mathcal {H}_n$ .", "Since $\\sigma ^\\alpha $ ($=\\alpha ^{-1}\\sigma \\alpha )$ is in $\\mathcal {H}_n$ and maps the point $x\\alpha $ to $(x\\sigma )\\alpha $ , these two points must lie on the same ray for all but finitely many $x\\in R_n$ .", "Similarly, $x$ and $x\\sigma $ lie on the same ray for all but finitely many $x\\in R_n$ , as $\\sigma \\in \\mathcal {H}_n$ .", "In fact, given a ray, we can choose $\\sigma $ so that whenever $x$ lies on that ray, then $x$ and $x \\sigma $ are successors on the same ray.", "Combining these facts, we see that $\\alpha $ maps all but finitely many points of ray $i$ to one and the same ray, say ray $j$ .", "This defines a homomorphism from $\\operatorname{Aut}(\\mathcal {H}_n)$ onto $\\mathcal {S}_n$ , which is obviously split, since given a permutation of the $n$ rays, it clearly defines an automorphism of $\\mathcal {H}_n$ .", "So assume that $\\alpha $ is now in the kernel of that map, so it does not permute the rays, and take $\\sigma $ a $g_{ji}$ generator of $\\mathcal {H}_n$ , i.e., an infinite cycle inside $\\mathsf {Sym}$ .", "Then, since conjugating inside $\\mathsf {Sym}$ preserves cycle types, the element $\\sigma ^\\alpha \\in \\mathcal {H}_n$ is also a single infinite cycle.", "This means that $\\sigma ^\\alpha $ has nonzero translations in only two rays, and these translations are 1 and $-1$ .", "For all points in the support of $\\sigma $ , we have that $\\sigma ^k$ sends this point into the $i$ -th ray for all sufficiently large $k$ .", "Therefore, as $\\alpha $ sends almost all points in the $i$ -th ray into the $i$ -th ray, the same is true for $\\alpha ^\\sigma $ .", "Hence $t_i(\\sigma ^\\alpha )$ is positive, so it must be $t_i(\\sigma ^\\alpha )=1$ .", "It is quite clear now that $\\alpha $ translates by an integer in the ray $i$ , sufficiently far out.", "This finishes the proof since this could be done for any $i$ , and hence $\\alpha \\in \\mathcal {H}_n$ ." ], [ "Commensurations of $\\mathcal {H}_n$", "First, we recall that a commensuration of a group $G$ is an isomorphism $A\\stackrel{\\phi }{\\longrightarrow }B$ , where $A$ and $B$ are subgroups of finite index in $G$ .", "Two commensurations $\\phi $ and $\\psi $ of $G$ are equivalent if there exists a subgroup $A$ of finite index in $G$ , such that the restrictions of $\\phi $ and $\\psi $ to $A$ are equal.", "The set of all commensurations of $G$ modulo this equivalence relation forms a group, known as the (abstract) commensurator of $G$, and is denoted $\\operatorname{Com}(G)$ .", "In this section we will determine the commensurator of $\\mathcal {H}_n$ .", "For a moment, we let $H$ be a subgroup of a group $G$ .", "The relative commensurator of $H$ in $G$ is denoted $\\operatorname{Com}_G(H)$ and consists of those $g\\in G$ such that $H\\cap H^g$ has finite index in both $H$ and $H^g$ .", "Similar to the homomorphism from $N_G(H)$ to $\\operatorname{Aut}(H)$ , there is a homomorphism from $\\operatorname{Com}_G(H)$ to $\\operatorname{Com}(H)$ ; Its kernel consists of those elements of $G$ that centralize a finite index subgroup of $H$ .", "In order to pin down $\\operatorname{Com}(\\mathcal {H}_n)$ , we first establish that every commensuration of $\\mathcal {H}_n$ can be seen as conjugation by an element of $\\mathsf {Sym}$ , and then characterize $\\operatorname{Com}_\\mathsf {Sym}(\\mathcal {H}_n)$ .", "Since a commensuration $\\phi $ and its restriction to a subgroup of finite index in its domain are equivalent, we can restrict our attention to the following family of subgroups of finite index in $\\mathcal {H}_n$ , in order to exhibit $\\operatorname{Com}(\\mathcal {H}_n)$ .", "For every integer $p\\ge 1$ , we define the subgroup $U_p$ of $\\mathcal {H}_n$ by $ U_p=\\langle \\mathsf {FAlt}, g_i^p\\mid i\\in {\\mathbb {Z}}_n\\rangle .$ We collect some useful properties of these subgroup first, where $A\\subset _f B$ means that $A$ is a subgroup of finite index in $B$ .", "Lemma 3.1 Let $n\\ge 3$ .", "For every $p$ , the group $U_p$ coincides with $\\mathcal {H}_n^p$ , the subgroup generated by all $p^\\mathrm {th}$ powers in $\\mathcal {H}_n$ , and hence is characteristic in $\\mathcal {H}_n$ .", "$\\displaystyle U_p^{\\prime }=\\left\\lbrace \\begin{array}{cl}\\mathsf {FAlt}, & p \\text{even}\\\\\\mathsf {FSym}, & p \\text{ odd}\\end{array}\\right.$ and $\\displaystyle \\vert \\mathcal {H}_n\\colon U_p\\vert =\\left\\lbrace \\begin{array}{cl} 2p^{n-1}, & p \\text{even}\\\\ p^{n-1}, & p \\text{ odd}\\end{array}\\right.$ .", "For every finite index subgroup $U$ of $\\mathcal {H}_n$ , there exists a $p\\ge 1$ with $\\mathsf {FAlt}= U_p^{\\prime } \\subset U_p \\subset _f U \\subset _f \\mathcal {H}_n$ .", "The same is essentially true for the case $n=2$ , except that $U_p^{\\prime }$ is always equal to $\\mathsf {FAlt}$ in this case, with the appropriate change in the index.", "First we establish (ii).", "We know that $[g_i,g_j]$ is either trivial, when $j\\notin \\lbrace i-1,i+1\\rbrace $ , or an odd permutation.", "So the commutator identities $[ab,c]=[a,c]^b[b,c]$ and $[a,bc]=[a,c][a,b]^c$ imply the first part, and the second part follows immediately using the short exact sequence (REF ) from Section .", "Part (i) is now an exercise, using that $\\mathsf {FAlt}^p=\\mathsf {FAlt}$ .", "In order to show (iii), let $U$ be a subgroup of finite index in $\\mathcal {H}_n$ .", "The facts that $\\mathsf {FAlt}$ is simple and $U$ contains a normal finite index subgroup of $\\mathcal {H}_n$ , imply that $\\mathsf {FAlt}\\subset U$ .", "Let $p$ be the smallest even integer such that $(p{\\mathbb {Z}})^{n-1}$ is contained in the image of $U$ in the abelianisation of $\\mathcal {H}_n$ .", "It is now clear that $U_p$ is contained in $U$ .", "Noting that $\\operatorname{Com}({\\mathcal {H}}_1) = \\operatorname{Aut}({\\mathcal {H}}_1) = \\mathsf {Sym}$ , we now characterize the commensurators of Houghton's groups.", "Theorem 3.2 Let $n\\ge 2$ .", "Every commensuration of $\\mathcal {H}_n$ normalizes $U_p$ for some even integer $p$ .", "The group $N_p=N_\\mathsf {Sym}(U_p)$ is isomorphic to $\\mathcal {H}_{np}\\rtimes (\\mathcal {S}_p\\wr \\mathcal {S}_n)$ , and $\\operatorname{Com}(\\mathcal {H}_n)$ is the direct limit of $N_p$ with even $p$ under the natural embeddings $N_p\\longrightarrow N_{pq}$ for $q \\in {\\mathbb {N}}$ .", "Let $\\phi \\in \\operatorname{Com}(\\mathcal {H}_n)$ .", "By Lemma REF , we can assume that $U_p$ is contained in the domain of both $\\phi $ and $\\phi ^{-1}$ for some even $p$ .", "Let $V$ be the image of $U_p$ under $\\phi $ .", "Then $V$ has finite index in $\\mathcal {H}_n$ and so contains $\\mathsf {FAlt}$ , by Lemma REF .", "However, the set of elements of finite order in $V$ equals $[V,V]$ , whence $[V,V]=\\mathsf {FAlt}$ , as $\\mathsf {FAlt}$ and $\\mathsf {FSym}$ are not isomorphic.", "This means that the restriction of $\\phi $ to $\\mathsf {FAlt}$ is an automorphism of $\\mathsf {FAlt}$ , and hence yields a homomorphism $\\iota \\colon \\operatorname{Com}(\\mathcal {H}_n) \\longrightarrow \\mathsf {Sym}.$ That $\\iota $ is an injective homomorphism to $\\operatorname{Com}_\\mathsf {Sym}(\\mathcal {H}_n)$ follows from a similar argument to the one in Proposition REF applied to $g_i\\,^p$ and six points of the form $a_\\ell = (i, k+p\\ell )$ with $\\ell \\in \\lbrace 0,1,\\ldots ,5\\rbrace $ .", "Since the centralizer in $\\mathsf {Sym}$ of $\\mathsf {FAlt}$ , and hence of any finite index subgroup of $\\mathcal {H}_n$ , is trivial, the natural homomorphism from $\\operatorname{Com}_\\mathsf {Sym}(\\mathcal {H}_n)$ to $\\operatorname{Com}(\\mathcal {H}_n)$ mentioned above is also injective, and we conclude that $\\operatorname{Com}(\\mathcal {H}_n)$ is isomorphic to $\\operatorname{Com}_\\mathsf {Sym}(\\mathcal {H}_n)$ , and that $\\iota $ is the isomorphism: $\\iota \\colon \\operatorname{Com}(\\mathcal {H}_n) \\longrightarrow \\operatorname{Com}_\\mathsf {Sym}(\\mathcal {H}_n).$ From now on, we assume that $\\phi \\in \\operatorname{Com}_\\mathsf {Sym}(\\mathcal {H}_n)$ .", "In particular, the action of $\\phi $ is given by conjugation, and our hypothesis is that $U_p^\\phi \\subset \\mathcal {H}_n$ .", "Now we can apply the argument from the proof of Theorem REF to $\\sigma \\in U$ and $\\sigma ^\\phi $ (instead of $\\sigma ^\\alpha $ ).", "Namely, consider the $i^{th}$ ray, and choose a $\\sigma =g_{ji}^p \\in U_p$ .", "So we have $t_i(\\sigma )=p$ , $t_j(\\sigma )=-p$ and zero translation elsewhere.", "Now, except for finitely many points, $\\sigma ^\\phi $ preserves the rays and sends $x \\phi $ to $x \\sigma \\phi $ .", "Thus there is an infinite subset of the $i^{th}$ ray which is sent to the same ray by $\\phi $ , say ray $k$ .", "The infinite subset should be thought of as a union of congruence classes modulo $p$ , except for finitely many points.", "We claim that no infinite subset of ray $j$ can be mapped by $\\phi $ to ray $k$ .", "This is because, if it were, the infinite subset would contain a congruence class modulo $p$ (except for finitely many points) from which we would be able to produce two points, $x,y$ , in the support of $\\sigma ^\\phi $ such that all sufficiently large positive powers of $\\sigma ^\\phi $ send $x$ into ray $k$ and all sufficiently large negative powers of $\\sigma ^\\phi $ send $y$ into ray $k$ , and this is not possible for an element of $\\mathcal {H}_n$ .", "This means that $\\phi $ maps an infinite subset of ray $i$ onto almost all of ray $k$ (observe that ray $k$ is almost contained in the support of $\\sigma ^\\phi $ so must be almost contained in the image of the unions of rays $i$ and $j$ ).", "Now applying similar arguments to $\\phi ^{-1}$ we get that $\\phi $ is a bijection between rays $i$ and $k$ except for finitely many points.", "In fact, $\\phi $ must induce bijections between the congruence classes modulo $p$ (except for finitely many points) inside the two ray systems.", "Thus $\\phi $ induces a permutation of the ray system.", "Again, looking at large positive powers we deduce that $t_k(\\sigma ^\\phi ) > 0 $ and since the support of $\\sigma ^\\phi $ is almost equal to the union of two rays, we must have that $t_k(\\sigma ^\\phi )=p$ , as $\\sigma $ and hence $\\sigma ^\\phi $ have exactly $p$ orbits.", "In particular, we may deduce that $\\phi $ normalises $U_p$ .", "So $U_p^\\phi = U_p$ .", "In order to proceed, it will be useful to change the ray system.", "Specifically, each ray can be split into $p$ rays, preserving the order.", "This realises $U_p$ as a (normal) subgroup of $\\mathcal {H}_{np}$ .", "We say two of these new rays are equivalent if they came from the same old ray.", "Thus there are $n$ equivalence classes, each having $p$ elements.", "The group $U_p$ acts on this new ray system as the subgroup of $\\mathcal {H}_{np}$ consisting of all $\\sigma \\in \\mathcal {H}_{np}$ such that $t_i(\\sigma )=t_j(\\sigma )$ whenever the $i^{th}$ and $j^{th}$ rays are equivalent.", "Note that because we have split the rays, these translation amounts can be arbitrary in $U_p$ (as a subgroup of $\\mathcal {H}_{np}$ ) and not just multiples of $p$ .", "In particular, we have that $U_p= U_p^\\phi \\subset \\mathcal {H}_{np}$ and the previous arguments imply that $\\phi $ induces a bijection on the new ray system and sends equivalent rays to equivalent rays (since it is actually permuting the old ray system).", "Since $\\phi $ permutes the rays, but must preserve equivalence classes, we get a homomorphism from $N_\\mathsf {Sym}(U_p)$ to the subgroup of $\\mathcal {S}_{np}$ which preserves the equivalence classes - this is easily seen to be $(\\mathcal {S}_p\\wr \\mathcal {S}_n)$ and the above homomorphism is split.", "As in Theorem REF we now conclude that the kernel of this homomorphism is $\\mathcal {H}_{np}$ and hence we get that $N_p=N_\\mathsf {Sym}(U_p)$ is $\\mathcal {H}_{np}\\rtimes (\\mathcal {S}_p\\wr \\mathcal {S}_n)$ .", "We note that $\\operatorname{Com}(\\mathcal {H}_n)$ is not finitely generated, for if it were, it would lie in some maximal $N_p$ ." ], [ "Metric estimates for $\\mathcal {H}_n$", "In this section we will give sharp estimates for the word length of elements of Houghton's groups.", "This makes no sense for 1 which is not finitely generated.", "As mentioned in the introduction, the metric in 2 was described by Lehnert [9].", "In order to deal with $\\mathcal {H}_n$ for $n\\ge 3$ , we introduce the following measure of complexity of an element.", "Given $\\sigma \\in \\mathcal {H}_n$ , we define $p_i(\\sigma )$ , for $i\\in {\\mathbb {Z}}_n$ , to be the largest integer such that $(i,p_i(\\sigma ))\\sigma \\ne (i,p_i(\\sigma )+t_i(\\sigma ))$ .", "Note that if $t_i(\\sigma ) < 0$ , then necessarily $p_i(\\sigma )\\ge \|t_i(\\sigma )\|$ , as the first element in each ray is numbered 1.", "The complexity of $\\sigma \\in \\mathcal {H}_n$ is the natural number $P(\\sigma )$ , defined by $P(\\sigma ) = \\sum _{i\\in {\\mathbb {Z}}_n} p_i(\\sigma ).$ And the translation amount of $\\sigma $ is $T(\\sigma ) = \\frac{1}{2}\\sum _{i\\in {\\mathbb {Z}}_n} \|t_i(\\sigma )\|.$ The above remark combined with (REF ) immediately implies $P(\\sigma )\\ge T(\\sigma )$ .", "It is easy to see that an element with complexity zero is trivial, and only the generators $g_{ij}$ have complexity one.", "Theorem 4.1 Let $n\\ge 3$ and $\\sigma \\in \\mathcal {H}_n$ , with complexity $P=P(\\sigma )\\ge 2$ .", "Then the word length $\|\\sigma \|$ of $\\sigma $ with respect to any finite generating set satisfies $P/C \\le \|\\sigma \| \\le KP\\log P,$ where the constants $C$ and $K$ only depend on the choice of generating set.", "Since the word length with respect to two different finite generating sets differs only by a multiplicative constant, we can and will choose $\\lbrace \\,g_{ij}\\mid i,j\\in {\\mathbb {Z}}_n,\\,i\\ne j\\rbrace $ as generating set to work with, and show that the statement holds with $C=1$ and $K=7$ .", "The lower bound is established by examining how multiplication by a generator can change the complexity.", "Suppose $\\sigma $ has complexity $P$ and consider $\\sigma g_{ij}$ .", "It is not difficult to see that $p_k(\\sigma g_{ij}) = \\left\\lbrace \\begin{array}{cl}p_k(\\sigma ) + 1, & \\text{ if } k=i \\text{ and } (i,p_i(\\sigma )+1)\\sigma = (i,1)\\\\p_k(\\sigma ) - 1, & \\text{ if } k=j,\\ (j,p_j(\\sigma )+1)\\sigma = (j,1) \\text{ and }\\\\& (j,p_j(\\sigma ))\\sigma = (i,1)\\\\p_k(\\sigma ), & \\text{ otherwise}\\end{array}\\right.$ where the first two cases are mutually exclusive, as $i\\ne j$ .", "Thus $\|P(\\sigma g_{ij})- P(\\sigma )\|\\le 1$ , which establishes the lower bound.", "The upper bound is obtained as follows.", "Suppose $\\sigma \\in \\mathcal {H}_n$ has complexity $P$ .", "First we show by induction on $T=T(\\sigma )$ that there is a word $\\rho $ of length at most $T\\le P$ such that the complexity of $\\sigma \\rho $ is $\\bar{P}$ with $\\bar{P}\\le P$ and $T(\\sigma \\rho ) =0$ .", "The case $T=0$ is trivial.", "If $T > 0$ , then there are $i,j\\in {\\mathbb {Z}}_n$ with $t_i(\\sigma ) > 0$ and $t_j(\\sigma ) < 0$ .", "So $T(\\sigma g_{ij}) =T-1$ .", "Moreover, $P(\\sigma g_{ij})\\le P$ , because the first case of (REF ) is excluded, as it implies that $t_i(\\sigma ) =-p_i(\\sigma )\\le 0$ , contrary to our assumption.", "This completes the induction step.", "We are now in the situation that $\\sigma \\rho \\in \\mathsf {FSym}$ and loosely speaking we proceed as follows.", "We push all irregularities into ray 0, i.e.", "multiply by $\\prod g_{i0}\\,^{p_i(\\sigma \\rho )}$ .", "We push all points back into the ray to which they belong, except for those from ray 0 which we mix into ray 1, say.", "We push out of ray 1 separating the points belonging to rays 0 and 1 into ray 0 and any other ray, say ray 2, respectively.", "We push the points belonging to ray 1 back from ray 2 into it.", "These four steps can be achieved by multiplying by an element $\\mu $ of length at most $4\\bar{P}$ , such that $\\sigma \\rho \\mu $ is an element which, for each $i$ , permutes an initial segment $I_i$ of ray $i$ .", "Notice that $\\sigma \\rho \\mu $ is now an element of $\\mathcal {H}_n$ which maps each ray to itself, and hence $t_i(\\sigma \\rho \\mu )=0$ for all $i$ .", "It is clear that $\\mu $ can be chosen so that the total length of the moved intervals, $\\sum \|I_i\| \\le \\bar{P}$ .", "Finally, we sort each of these intervals using a recursive procedure, modeled on standard merge sort.", "In order to sort the interval $I=I_2$ say, we push each of its points out of ray 2 and into either ray 0 if it belongs to the lower half, or to ray 1 if it belongs to the upper half of $I$ .", "If each of the two halves occurs in the correct order, then we only have to push them back into ray 2 and are done, having used $2\|I\|$ generators.", "If the two halves are not yet sorted, then we use the same “separate the upper and lower halves” approach on each of them recursively in order to sort them.", "In total this takes at most $2\|I\|\\log _2 \|I\|$ steps.", "Altogether we have used at most $P + 4\\bar{P} + 2\\sum _{i\\in {\\mathbb {Z}}_n} \|I_i\|\\log _2 \|I_i\| \\le 7P\\log _2P$ generators to represent the inverse of $\\sigma $ ; we used the hypothesis $P\\ge 2$ in the last inequality.", "We note that because there are many permutations, the fraction of elements which are close to the lower bound goes to zero in much the same way as shown for Thompson's group $V$ by Birget [1] and its generalization $nV$ by Burillo and Cleary [3].", "Lemma 4.2 Let $n\\ge 3$ .", "For $\\mathcal {H}_n$ take the generating set $g_1,\\ldots , g_{n-1}$ with $n-1$ elements.", "Consider the following sets: $B_k$ is the ball of radius $k$ , $C_k$ is the set of elements in $\\mathcal {H}_n$ which have complexity $P=k$ , $D_k\\subset C_k$ is the set of elements of $C_k$ which have word length at most $k\\log _{2n-2}k$ Then, we have that $\\lim _{k\\rightarrow \\infty }\\frac{\|D_k\|}{\|C_k\|}=0$ An element of complexity $P$ , according to the metric estimates proved above, has word length between $P$ and $P\\log P$ .", "What this lemma means is that most elements with complexity $P$ will have word length closer to $P\\log P$ than to $P$ .", "Observe that $\\frac{\|D_k\|}{\|B_{k\\log _{2n-2}k}\|}\\le 1$ because it is a subset.", "Now, introduce the $C_k$ as $\\frac{\|D_k\|}{\|C_k\|}\\frac{\|C_k\|}{\|B_{k\\log _{2n-2}k}\|}$ and the proof will be complete if we show that $\\lim _{k\\rightarrow \\infty }\\frac{\|C_k\|}{\|B_{k\\log _{2n-2}k}\|}=\\infty .$ In $C_k$ there are at least $(nk-2)!$ elements.", "This is because we can take a transposition involving a point at distance $k$ down one of the rays, with another point.", "Since this already ensures $P=k$ , we are free to choose any permutation of the other $nk-2$ points which are at one of the first $k$ positions in each ray.", "And inside $\|B_{k\\log _{2n-2}k}\|$ , counting grossly according to the number of generators, there are at most $(2n-2)^{k\\log _{2n-2}k}=k^k$ elements.", "Now the limit becomes $\\lim _{k\\rightarrow \\infty }\\frac{(nk-2)!", "}{k^k}$ which is easily seen that it approaches infinity using Stirling's formula and the fact that $n\\ge 3$ .", "Consequentially, these estimates give an easy way to see that the group has exponential growth.", "We note that exponential growth also follows easily from the fact that $g_{01}$ and $g_{02}$ generate a free subsemigroup.", "Proposition 4.3 Let $n\\ge 3$ .", "Then $\\mathcal {H}_n$ has exponential growth.", "Consider a finitary permutation of complexity $P$ , and observe that there are at least $P!$ of those.", "By the metric estimate, its word length is at most $KP\\log P$ .", "Using again as in the previous lemma the notation $B_k$ for a ball, we will have that the group has exponential growth if $\\lim _{k\\rightarrow \\infty }\\frac{\\log \|B_k\|}{k}>0.$ In our case, this amounts to $\\lim _{P\\rightarrow \\infty }\\frac{\\log \|B_{KP\\log P}\|}{KP\\log P}\\ge \\lim _{P\\rightarrow \\infty }\\frac{\\log P!", "}{KP\\log P}=\\frac{1}{K}.$" ], [ "Subgroup embeddings", "We note that each $\\mathcal {H}_n$ is a subgroup of ${\\mathcal {H}}_m$ for $n < m$ and that our estimates together with work of Lehnert are enough to give at least quadratic distortion for some of these embeddings.", "Theorem 5.1 The group ${\\mathcal {H}}_2$ is at least quadratically distorted in ${\\mathcal {H}}_m$ for $m\\ge 3$ .", "We consider the element $\\sigma _n$ of 2 which has $T(\\sigma _n)=0$ and transposes $(0,k)$ and $(1,k)$ for all $k \\le n$ .", "Then $\\sigma _n$ corresponds to the word $g_n$ defined in Theorem 8 of [9], where it is shown to have length of the order of $n^2$ with respect to the generators of 2 in Lemma 10 there, which are exactly the generators for 2 given in the introduction.", "One easily checks that $\\sigma _n = g_{02}\\,^n g_{1 2}\\,^n g_{0 2}\\,^{-n} g_{12}\\,^{-n}$ in 3.", "Thus a family of words of quadratically growing length in 2 has linearly growing length in 3, which proves the theorem.", "A natural, but seemingly difficult, question is whether $\\mathcal {H}_n$ is distorted in $m$ for $3\\le n < m$ .", "Another question, which also seems difficult, is to ask whether $\\mathcal {H}_n$ is distorted in Thompson's group $V$ , under the embeddings mentioned in the introduction, [10]." ], [ "Some quasi-isometries of $\\mathcal {H}_n$", "Commensurations give rise to quasi-isometries and are often a rich source of examples of quasi-isometries.", "Here we show that the natural map from the commensurator of $\\mathcal {H}_n$ to the quasi-isometry group of $\\mathcal {H}_n$ , which we denote by $\\operatorname{QI}(\\mathcal {H}_n)$ , is an injection.", "That is, we show that each commensuration is not within a bounded distance of the identity.", "That this is an injection also follows from the more general argument of Whyte which appears as Proposition 7.5 in Farb-Mosher [6].", "Theorem 6.1 The natural homomorphism from $\\operatorname{Com}(\\mathcal {H}_n)$ to $\\operatorname{QI}(\\mathcal {H}_n)$ is an embedding for $n \\ge 2$ .", "We will show that for each non-trivial $\\phi \\in \\operatorname{Com}(\\mathcal {H}_n)$ and every $N\\in {\\mathbb {N}}$ we can find a $\\sigma \\in \\mathcal {H}_n$ such that $d(\\sigma ,\\sigma ^\\phi )\\ge N$ , so none of the non-trivial images are within a bounded distance of the identity.", "By Theorem REF , we can and will view $\\phi $ as a non-trivial element of $N_p\\subset \\mathsf {Sym}$ for some even $p$ .", "If $\\phi $ eventually translates a ray $i$ non-trivially to a possibly different ray $j$ , then we let $\\sigma =((i,N)\\,(i,N+1))$ , a transposition in the translated ray.", "The image of $\\sigma $ under conjugation by $\\phi $ is the transposition $((j,N+t),(j,N+t^{\\prime }+1))$ , and the distance $d(\\sigma ,\\sigma ^\\phi )$ is the length of $\\sigma ^{-1}\\sigma ^\\phi $ , which is at least $N$ since it moves at least one point at distance $N$ down one of the rays.", "If $\\phi $ does not eventually translate a ray but eventually non-trivially permutes ray $i$ with another ray $j$ , then we can show boundedness away from the identity by taking $\\sigma =((j,N)\\,(j,N+1))$ .", "The point $(i,N)$ is fixed by $\\sigma $ but is moved to $(i, N+1)$ under $\\sigma ^\\phi $ ensuring that the length of $\\sigma ^{-1}\\sigma ^\\phi $ is at least $N$ .", "Finally, if $\\phi $ does not have the preceding two properties, then $\\phi $ is a non-trivial finitary permutation.", "Since Houghton's group is $k$ -transitive, for every $k$ , we can find a $\\sigma \\in \\mathcal {H}_n$ such that $\\phi ^\\sigma $ has support disjoint from that of $\\phi $ , and at distance at least $N$ down one of the rays.", "Hence $\\sigma ^{-1} \\phi ^{-1} \\sigma \\phi = \\sigma ^{-1} \\sigma ^\\phi $ has length at least $N$ ." ], [ "Co-Hopficity", "Houghton's groups are long known to be Hopfian although they are not residually finite, see [4].", "In this section we will prove that $\\mathcal {H}_n$ is not co-Hopfian, by exhibiting a map which is injective but not surjective.", "The map is the following: $\\begin{array}cf\\colon \\mathcal {H}_n\\longrightarrow \\mathcal {H}_n\\\\s\\mapsto f(s)\\end{array}$ defined by: if $(i,n)s=(j,m)$ , then: $(i,2n-1)f(s)=(j,2m-1)\\qquad \\text{and}\\qquad (i,2n)f(s)=(j,2m).$ It is straightforward to show that $f$ is a homomorphism.", "It is injective, because if $s$ is not the identity with $(i,n)s\\ne (i,n)$ , then $(i,2n)f(s)\\ne (i,2n)$ .", "And clearly the map is not surjective, because the permutation always sends adjacent points $(i,2n-1)$ , $(i,2n)$ to adjacent points, and a permutation which does not do this cannot be in the image.", "In fact, $\\mathcal {H}_n$ has many proper subgroups isomorphic to the whole group.", "The following argument was pointed out to us by Peter Kropholler.", "One can well-order the ray system by taking a lexicographic order.", "The group $\\mathcal {H}_n$ is then the group of all almost order preserving bijections of the well-ordered ray system.", "It is then clear that the ray system minus a point is order isomorphic to the original ray system, which demonstrates that a point stabiliser is a subgroup isomorphic to $\\mathcal {H}_n$ .", "Theorem 7.1 Houghton's groups $\\mathcal {H}_n$ are not co-Hopfian." ] ]	1403.0026
[ [ "The Lukacs-Olkin-Rubin theorem on symmetric cones without invariance of\n the \"quotient\"" ], [ "Abstract We prove the Lukacs-Olkin-Rubin theorem without invariance of the distribution of the \"quotient\", which was the key assumption in the original proof of [Olkin--Rubin, Ann.", "Math.", "Stat.", "33 (1962), 1272--1280].", "Instead we assume existence of strictly positive continuous densities of respective random variables.", "We consider the (cone variate) \"quotient\" for any division algorithm satisfying some natural conditions.", "For that purpose, the new proof of the Olkin--Baker functional equation on symmetric cones is given." ], [ "Introduction", "The [17] theorem is one of the most celebrated characterizations of probability distributions.", "It states that if $X$ and $Y$ are independent, positive, non-degenerate random variables such that their sum and quotient are also independent then $X$ and $Y$ have gamma distributions with the same scale parameter.", "This theorem has many generalizations.", "The most important in the multivariate setting were given by [21] and [7], where the authors extended characterization to matrix and symmetric cones variate distributions, respectively.", "There is no unique way of defining the quotient of elements of the cone of positive definite symmetric matrices $ \\Omega _+$ and in these papers the authors have considered very general form $U=g(X+Y)\\cdot X\\cdot g^T(X+Y)$ , where $g$ is the so called division algorithm, that is, $g( {\\textbf {a}} )\\cdot {\\textbf {a}} \\cdot g^T( {\\textbf {a}} )=I$ for any $ {\\textbf {a}} \\in \\Omega _+$ , where $I$ is the identity matrix and $g( {\\textbf {a}} )$ is invertible for any $ {\\textbf {a}} \\in \\Omega _+$ (later on, abusing notation we will write $g( {\\textbf {x}} ) {\\textbf {y}} =g( {\\textbf {x}} )\\cdot {\\textbf {y}} \\cdot g^{T}( {\\textbf {x}} )$ , that is, in this case $g( {\\textbf {x}} )$ denotes the linear operator acting on $ \\Omega _+$ ).", "The drawback of their extension was the additional strong assumption of invariance of the distribution of $U$ under a group of automorphisms.", "This result was generalized to homogeneous cones in [6].", "There were successful attempts in replacing the invariance of the “quotient” assumption with the existence of regular densities of random variables $X$ and $Y$ .", "[2] assuming existence of strictly positive, twice differentiable densities proved a characterization of Wishart distribution on the cone $ \\Omega _+$ for division algorithm $g_1( {\\textbf {a}} )= {\\textbf {a}} ^{-1/2}$ , where $ {\\textbf {a}} ^{1/2}$ denotes the unique positive definite symmetric root of $ {\\textbf {a}} \\in \\Omega _+$ .", "This results was generalized to all non-octonion symmetric cones of rank greater than 2 and to the Lorentz cone for strictly positive and continuous densities by [12], [13].", "Exploiting the same approach, with the same technical assumptions on densities as in [2] it was proven by [11] that the independence of $X+Y$ and the quotient defined through the Cholesky decomposition, i.e.", "$g_2( {\\textbf {a}} )=T_ {\\textbf {a}} ^{-1}$ , where $T_ {\\textbf {a}} $ is a lower triangular matrix such that $ {\\textbf {a}} =T_ {\\textbf {a}} \\cdot T_ {\\textbf {a}} ^T\\in \\Omega _+$ , characterizes a wider family of distributions called Riesz (or sometimes called Riesz-Wishart).", "This fact shows that the invariance property assumed in [21] and [7] is not of technical nature only.", "Analogous results for homogeneous cones were obtained by [4], [5].", "In this paper we deal with the density version of Lukacs-Olkin-Rubin theorem on symmetric cones for division algorithm satisfying some natural properties.", "We assume that the densities of $X$ and $Y$ are strictly positive and continuous.", "We consider quotient $U$ for an arbitrary, fixed division algorithm $g$ as in the original paper of [21], additionally satisfying some natural conditions.", "In the known cases ($g=g_1$ and $g=g_2$ ) this improves the results obtained in [2], [11], [13].", "In general case, the densities of $X$ and $Y$ are given in terms of, so called, $w$ -multiplicative Cauchy functions, that is functions satisfying $f( {\\textbf {x}} )f\\left(w(I) {\\textbf {y}} \\right)=f\\left(w( {\\textbf {x}} ) {\\textbf {y}} \\right),\\quad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega _+^2,$ where $w( {\\textbf {x}} ) {\\textbf {y}} =w( {\\textbf {x}} )\\cdot {\\textbf {y}} \\cdot w^T( {\\textbf {x}} )$ (i.e.", "$g( {\\textbf {x}} )=w( {\\textbf {x}} )^{-1}$ is a division algorithm).", "Consistently, we will call $w$ a multiplication algorithm.", "Such functions were recently considered in [14].", "Unfortunately we can't answer the question whether there exists division (or equivalently multiplication) algorithm resulting in characterizing other distribution than Riesz or Wishart.", "Moreover, the simultaneous removal of the assumptions of the invariance of the “quotient” and the existence of densities remains a challenge.", "This paper is organized as follows.", "We start in the next section with basic definitions and theorems regarding analysis on symmetric cones.", "The statement and proof of the main result are given in Section .", "Section is devoted to consideration of $w$ -logarithmic Cauchy functions and the Olkin–Baker functional equation.", "In that section we offer much shorter, simpler and covering more general cones proof of the Olkin–Baker functional equation than given in [2], [11], [13]." ], [ "Preliminaries", "In this section we give a short introduction to the theory of symmetric cones.", "For further details we refer to [8].", "A Euclidean Jordan algebra is a Euclidean space $\\mathbb {E}$ (endowed with scalar product denoted $\\left\\langle {\\textbf {x}} , {\\textbf {y}} \\right\\rangle $ ) equipped with a bilinear mapping (product) $\\mathbb {E}\\times \\mathbb {E}\\ni \\left( {\\textbf {x}} , {\\textbf {y}} \\right)\\mapsto {\\textbf {x}} {\\textbf {y}} \\in \\mathbb {E}$ and a neutral element $ {\\textbf {e}} $ in $\\mathbb {E}$ such that for all $ {\\textbf {x}} $ , $ {\\textbf {y}} $ , $ {\\textbf {z}} $ in $\\mathbb {E}$ : $ {\\textbf {x}} {\\textbf {y}} = {\\textbf {y}} {\\textbf {x}} $ , $ {\\textbf {x}} ( {\\textbf {x}} ^2 {\\textbf {y}} )= {\\textbf {x}} ^2( {\\textbf {x}} {\\textbf {y}} )$ , $ {\\textbf {x}} {\\textbf {e}} = {\\textbf {x}} $ , $\\left\\langle {\\textbf {x}} , {\\textbf {y}} {\\textbf {z}} \\right\\rangle =\\left\\langle {\\textbf {x}} {\\textbf {y}} , {\\textbf {z}} \\right\\rangle $ .", "For $ {\\textbf {x}} \\in \\mathbb {E}$ let $\\mathbb {L}( {\\textbf {x}} )\\colon \\mathbb {E}\\rightarrow \\mathbb {E}$ be linear map defined by $\\mathbb {L}( {\\textbf {x}} ) {\\textbf {y}} = {\\textbf {x}} {\\textbf {y}} ,$ and define $\\mathbb {P}( {\\textbf {x}} )=2\\mathbb {L}^2( {\\textbf {x}} )-\\mathbb {L}\\left( {\\textbf {x}} ^2\\right).$ The map $\\mathbb {P}\\colon \\mathbb {E}\\mapsto End(\\mathbb {E})$ is called the quadratic representation of $\\mathbb {E}$ .", "An element $ {\\textbf {x}} $ is said to be invertible if there exists an element $ {\\textbf {y}} $ in $\\mathbb {E}$ such that $\\mathbb {L}( {\\textbf {x}} ) {\\textbf {y}} = {\\textbf {e}} $ .", "Then $ {\\textbf {y}} $ is called the inverse of $ {\\textbf {x}} $ and is denoted by $ {\\textbf {y}} = {\\textbf {x}} ^{-1}$ .", "Note that the inverse of $ {\\textbf {x}} $ is unique.", "It can be shown that $ {\\textbf {x}} $ is invertible if and only if $\\mathbb {P}( {\\textbf {x}} )$ is invertible and in this case $\\left(\\mathbb {P}( {\\textbf {x}} )\\right)^{-1} =\\mathbb {P}\\left( {\\textbf {x}} ^{-1}\\right)$ .", "Euclidean Jordan algebra $\\mathbb {E}$ is said to be simple if it is not a Cartesian product of two Euclidean Jordan algebras of positive dimensions.", "Up to linear isomorphism there are only five kinds of Euclidean simple Jordan algebras.", "Let $\\mathbb {K}$ denote either the real numbers $\\mathbb {R}$ , the complex ones $\\mathbb {C}$ , quaternions $\\mathbb {H}$ or the octonions $\\mathbb {O}$ , and write $S_r(\\mathbb {K})$ for the space of $r\\times r$ Hermitian matrices with entries valued in $\\mathbb {K}$ , endowed with the Euclidean structure $\\left\\langle {\\textbf {x}} , {\\textbf {y}} \\right\\rangle =\\mathrm {Trace}\\,( {\\textbf {x}} \\cdot \\bar{ {\\textbf {y}} })$ and with the Jordan product $ {\\textbf {x}} {\\textbf {y}} =\\tfrac{1}{2}( {\\textbf {x}} \\cdot {\\textbf {y}} + {\\textbf {y}} \\cdot {\\textbf {x}} ),$ where $ {\\textbf {x}} \\cdot {\\textbf {y}} $ denotes the ordinary product of matrices and $\\bar{ {\\textbf {y}} }$ is the conjugate of $ {\\textbf {y}} $ .", "Then $S_r(\\mathbb {R})$ , $r\\ge 1$ , $S_r(\\mathbb {C})$ , $r\\ge 2$ , $S_r(\\mathbb {H})$ , $r\\ge 2$ , and the exceptional $S_3(\\mathbb {O})$ are the first four kinds of Euclidean simple Jordan algebras.", "Note that in this case $\\mathbb {P}( {\\textbf {y}} ) {\\textbf {x}} = {\\textbf {y}} \\cdot {\\textbf {x}} \\cdot {\\textbf {y}} .$ The fifth kind is the Euclidean space $\\mathbb {R}^{n+1}$ , $n\\ge 2$ , with Jordan product $\\begin{split}\\left(x_0,x_1,\\dots , x_n\\right)\\left(y_0,y_1,\\dots ,y_n\\right) =\\left(\\sum _{i=0}^n x_i y_i,x_0y_1+y_0x_1,\\dots ,x_0y_n+y_0x_n\\right).\\end{split}$ To each Euclidean simple Jordan algebra one can attach the set of Jordan squares $\\bar{ \\Omega }=\\left\\lbrace {\\textbf {x}} ^2\\colon {\\textbf {x}} \\in \\mathbb {E}\\right\\rbrace .$ The interior $ \\Omega $ is a symmetric cone.", "Moreover $ \\Omega $ is irreducible, i.e.", "it is not the Cartesian product of two convex cones.", "One can prove that an open convex cone is symmetric and irreducible if and only if it is the cone $ \\Omega $ of some Euclidean simple Jordan algebra.", "Each simple Jordan algebra corresponds to a symmetric cone, hence there exist up to linear isomorphism also only five kinds of symmetric cones.", "The cone corresponding to the Euclidean Jordan algebra $\\mathbb {R}^{n+1}$ equipped with Jordan product (REF ) is called the Lorentz cone.", "We denote by $G(\\mathbb {E})$ the subgroup of the linear group $GL(\\mathbb {E})$ of linear automorphisms which preserves $ \\Omega $ , and we denote by $G$ the connected component of $G(\\mathbb {E})$ containing the identity.", "Recall that if $\\mathbb {E}=S_r(\\mathbb {R})$ and $GL(r,\\mathbb {R})$ is the group of invertible $r\\times r$ matrices, elements of $G(\\mathbb {E})$ are the maps $g\\colon \\mathbb {E}\\rightarrow \\mathbb {E}$ such that there exists $ {\\textbf {a}} \\in GL(r,\\mathbb {R})$ with $g( {\\textbf {x}} )= {\\textbf {a}} \\cdot {\\textbf {x}} \\cdot {\\textbf {a}} ^T.$ We define $K=G\\cap O(\\mathbb {E})$ , where $O(\\mathbb {E})$ is the orthogonal group of $\\mathbb {E}$ .", "It can be shown that $K=\\lbrace k\\in G\\colon k {\\textbf {e}} = {\\textbf {e}} \\rbrace .$ A multiplication algorithm is a map $ \\Omega \\rightarrow G\\colon {\\textbf {x}} \\mapsto w( {\\textbf {x}} )$ such that $w( {\\textbf {x}} ) {\\textbf {e}} = {\\textbf {x}} $ for all $ {\\textbf {x}} \\in \\Omega $ .", "This concept is consistent with, so called, division algorithm $g$ , which was introduced by [21] and [7], that is a mapping $ \\Omega \\ni {\\textbf {x}} \\mapsto g( {\\textbf {x}} )\\in G$ such that $g( {\\textbf {x}} ) {\\textbf {x}} = {\\textbf {e}} $ for any $ {\\textbf {x}} \\in \\Omega $ .", "If $w$ is a multiplication algorithm then $g=w^{-1}$ (that is, $g( {\\textbf {x}} )w( {\\textbf {x}} )=w( {\\textbf {x}} )g( {\\textbf {x}} )=Id_ \\Omega $ for any $ {\\textbf {x}} \\in \\Omega $ ) is a division algorithm and vice versa, if $g$ is a division algorithm then $w=g^{-1}$ is a multplication algorithm.", "One of two important examples of multiplication algorithms is the map $w_1( {\\textbf {x}} )=\\mathbb {P}\\left( {\\textbf {x}} ^{1/2}\\right)$ .", "We will now introduce a very useful decomposition in $\\mathbb {E}$ , called spectral decomposition.", "An element $ {\\textbf {c}} \\in \\mathbb {E}$ is said to be a idempotent if $ {\\textbf {c}} {\\textbf {c}} = {\\textbf {c}} \\ne 0$ .", "Idempotents $ {\\textbf {a}} $ and $ {\\textbf {b}} $ are orthogonal if $ {\\textbf {a}} {\\textbf {b}} =0$ .", "Idempotent $ {\\textbf {c}} $ is primitive if $ {\\textbf {c}} $ is not a sum of two non-null idempotents.", "A complete system of primitive orthogonal idempotents is a set $\\left( {\\textbf {c}} _1,\\dots , {\\textbf {c}} _r\\right)$ such that $\\sum _{i=1}^r {\\textbf {c}} _i= {\\textbf {e}} \\quad \\mbox{and}\\quad {\\textbf {c}} _i {\\textbf {c}} _j=\\delta _{ij} {\\textbf {c}} _i\\quad \\mbox{for } 1\\le i\\le j\\le r.$ The size $r$ of such system is a constant called the rank of $\\mathbb {E}$ .", "Any element $ {\\textbf {x}} $ of a Euclidean simple Jordan algebra can be written as $ {\\textbf {x}} =\\sum _{i=1}^r\\lambda _i {\\textbf {c}} _i$ for some complete system of primitive orthogonal idempotents $\\left( {\\textbf {c}} _1,\\dots , {\\textbf {c}} _r\\right)$ .", "The real numbers $\\lambda _i$ , $i=1,\\dots ,r$ are the eigenvalues of $ {\\textbf {x}} $ .", "One can then define trace and determinant of $ {\\textbf {x}} $ by, respectively, $\\mathrm {tr}\\, {\\textbf {x}} =\\sum _{i=1}^r\\lambda _i$ and $\\det {\\textbf {x}} =\\prod _{i=1}^r\\lambda _i$ .", "An element $ {\\textbf {x}} \\in \\mathbb {E}$ belongs to $ \\Omega $ if and only if all its eigenvalues are strictly positive.", "The rank $r$ and $\\dim \\Omega $ of irreducible symmetric cone are connected through relation $\\dim \\Omega =r+\\frac{d r(r-1)}{2},$ where $d$ is an integer called the Peirce constant.", "If $ {\\textbf {c}} $ is a primitive idempotent of $\\mathbb {E}$ , the only possible eigenvalues of $\\mathbb {L}( {\\textbf {c}} )$ are 0, $\\tfrac{1}{2}$ and 1.", "We denote by $\\mathbb {E}( {\\textbf {c}} ,0)$ , $\\mathbb {E}( {\\textbf {c}} ,\\tfrac{1}{2})$ and $\\mathbb {E}( {\\textbf {c}} ,1)$ the corresponding eigenspaces.", "The decomposition $\\mathbb {E}=\\mathbb {E}( {\\textbf {c}} ,0)\\oplus \\mathbb {E}( {\\textbf {c}} ,\\tfrac{1}{2})\\oplus \\mathbb {E}( {\\textbf {c}} ,1)$ is called the Peirce decomposition of $\\mathbb {E}$ with respect to $ {\\textbf {c}} $.", "Note that $\\mathbb {P}( {\\textbf {c}} )$ is the orthogonal projection of $\\mathbb {E}$ onto $\\mathbb {E}( {\\textbf {c}} ,1)$ .", "Fix a complete system of orthogonal idempotents $\\left( {\\textbf {c}} _i\\right)_{i=1}^r$ .", "Then for any $i,j\\in \\left\\lbrace 1,2,\\dots ,r\\right\\rbrace $ we write $\\begin{split}\\mathbb {E}_{ii} & =\\mathbb {E}( {\\textbf {c}} _i,1)=\\mathbb {R} {\\textbf {c}} _i, \\\\\\mathbb {E}_{ij} & = \\mathbb {E}\\left( {\\textbf {c}} _i,\\frac{1}{2}\\right) \\cap \\mathbb {E}\\left( {\\textbf {c}} _j,\\frac{1}{2}\\right) \\mbox{ if }i\\ne j.\\end{split}$ It can be proved (see [8]) that $\\mathbb {E}=\\bigoplus _{i\\le j}\\mathbb {E}_{ij}$ and $\\begin{split}\\mathbb {E}_{ij}\\cdot \\mathbb {E}_{ij} & \\subset \\mathbb {E}_{ii}+\\mathbb {E}_{ij}, \\\\\\mathbb {E}_{ij}\\cdot \\mathbb {E}_{jk} & \\subset \\mathbb {E}_{ik},\\mbox{ if }i\\ne k, \\\\\\mathbb {E}_{ij}\\cdot \\mathbb {E}_{kl} & =\\lbrace 0\\rbrace ,\\mbox{ if }\\lbrace i,j\\rbrace \\cap \\lbrace k,l\\rbrace =\\emptyset .\\end{split}$ Moreover ([8]), if $ {\\textbf {x}} \\in \\mathbb {E}_{ij}$ , $ {\\textbf {y}} \\in \\mathbb {E}_{jk}$ , $i\\ne k$ , then $ {\\textbf {x}} ^2 & =\\tfrac{1}{2}\\Vert {\\textbf {x}} \\Vert ^2( {\\textbf {c}} _i+ {\\textbf {c}} _j),\\\\\\Vert {\\textbf {x}} {\\textbf {y}} \\Vert ^2 & =\\tfrac{1}{8}\\Vert {\\textbf {x}} \\Vert ^2\\Vert {\\textbf {y}} \\Vert ^2.$ The dimension of $\\mathbb {E}_{ij}$ is the Peirce constant $d$ for any $i\\ne j$ .", "When $\\mathbb {E}$ is $S_r(\\mathbb {K})$ , if $(e_1,\\dots ,e_r)$ is an orthonormal basis of $\\mathbb {R}^r$ , then $\\mathbb {E}_{ii}=\\mathbb {R}e_i e_i^T$ and $\\mathbb {E}_{ij}=\\mathbb {K}\\left(e_i e_j^T+e_je_i^T\\right)$ for $i<j$ and $d$ is equal to $dim_{\|\\mathbb {R}}\\mathbb {K}$ .", "For $1\\le k \\le r$ let $P_k$ be the orthogonal projection onto $\\mathbb {E}^{(k)}=\\mathbb {E}( {\\textbf {c}} _1+\\ldots + {\\textbf {c}} _k,1)$ , $\\det ^{(k)}$ the determinant in the subalgebra $\\mathbb {E}^{(k)}$ , and, for $ {\\textbf {x}} \\in \\Omega $ , $\\Delta _k( {\\textbf {x}} )=\\det ^{(k)}(P_k( {\\textbf {x}} ))$ .", "Then $\\Delta _k$ is called the principal minor of order $k$ with respect to the Jordan frame $( {\\textbf {c}} _k)_{k=1}^r$ .", "Note that $\\Delta _r( {\\textbf {x}} )=\\det {\\textbf {x}} $ .", "For $s=(s_1,\\ldots ,s_r)\\in \\mathbb {R}^r$ and $ {\\textbf {x}} \\in \\Omega $ , we write $\\Delta _s( {\\textbf {x}} )=\\Delta _1( {\\textbf {x}} )^{s_1-s_2}\\Delta _2( {\\textbf {x}} )^{s_2-s_3}\\ldots \\Delta _r( {\\textbf {x}} )^{s_r}.$ $\\Delta _s$ is called a generalized power function.", "If $ {\\textbf {x}} =\\sum _{i=1}^r\\alpha _i {\\textbf {c}} _i$ , then $\\Delta _s( {\\textbf {x}} )=\\alpha _1^{s_1}\\alpha _2^{s_2}\\ldots \\alpha _r^{s_r}$ .", "We will now introduce some basic facts about triangular group.", "For $ {\\textbf {x}} $ and $ {\\textbf {y}} $ in $ \\Omega $ , let $ {\\textbf {x}} \\Box {\\textbf {y}} $ denote the endomorphism of $\\mathbb {E}$ defined by $ {\\textbf {x}} \\Box {\\textbf {y}} =\\mathbb {L}( {\\textbf {x}} {\\textbf {y}} )+\\mathbb {L}( {\\textbf {x}} )\\mathbb {L}( {\\textbf {y}} )-\\mathbb {L}( {\\textbf {y}} )\\mathbb {L}( {\\textbf {x}} ).$ If $ {\\textbf {c}} $ is an idempotent and $ {\\textbf {z}} \\in \\mathbb {E}( {\\textbf {c}} ,\\frac{1}{2})$ we define the Frobenius transformation $\\tau _ {\\textbf {c}} ( {\\textbf {z}} )$ in $G$ by $\\tau _ {\\textbf {c}} ( {\\textbf {z}} )=\\exp (2 {\\textbf {z}} \\Box {\\textbf {c}} ).$ Since $2 {\\textbf {z}} \\Box {\\textbf {c}} $ is nilpotent of degree 3 (see [8]) we get $\\tau _{ {\\textbf {c}} }( {\\textbf {z}} )=I+(2 {\\textbf {z}} \\Box {\\textbf {c}} )+\\frac{1}{2}(2 {\\textbf {z}} \\Box {\\textbf {c}} )^2.$ Given a Jordan frame $( {\\textbf {c}} _i)_{i=1}^r$ , the subgroup of $G$ , $\\mathcal {T}=\\left\\lbrace \\tau _{ {\\textbf {c}} _1}( {\\textbf {z}} ^{(1)})\\ldots \\tau _{ {\\textbf {c}} _{r-1}}( {\\textbf {z}} ^{(r-1)})\\mathbb {P}\\left(\\sum _{i=1}^r \\alpha _i {\\textbf {c}} _i\\right)\\colon \\alpha _i>0, {\\textbf {z}} ^{(j)}\\in \\bigoplus _{k=j+1}^r\\mathbb {E}_{jk}\\right\\rbrace $ is called the triangular group corresponding to the Jordan frame $( {\\textbf {c}} _i)_{i=1}^r$.", "For any $ {\\textbf {x}} $ in $ \\Omega $ there exists a unique $t_{ {\\textbf {x}} }$ in $\\mathcal {T}$ such that $ {\\textbf {x}} =t_{ {\\textbf {x}} } {\\textbf {e}} $ , that is, there exist (see [8]) elements $ {\\textbf {z}} ^{(j)}\\in \\bigoplus _{k=j+1}^r \\mathbb {E}_{jk}$ , $1\\le j\\le r-1$ and positive numbers $\\alpha _1, \\ldots ,\\alpha _r$ such that $ {\\textbf {x}} =\\tau _{ {\\textbf {c}} _1}( {\\textbf {z}} ^{(1)})\\tau _{ {\\textbf {c}} _2}( {\\textbf {z}} ^{(2)})\\ldots \\tau _{ {\\textbf {c}} _{r-1}}( {\\textbf {z}} ^{(r-1)})\\left(\\sum _{k=1}^r \\alpha _k {\\textbf {c}} _k \\right).$ Mapping $w_2\\colon \\Omega \\rightarrow \\mathcal {T}, {\\textbf {x}} \\mapsto w_2( {\\textbf {x}} )=t_{ {\\textbf {x}} }$ realizes a multiplication algorithm.", "For $\\mathbb {E}=S_r(\\mathbb {R})$ we have $ \\Omega = \\Omega _+$ .", "Let us define for $1\\le i,j\\le r$ matrix $\\mu _{ij}=\\left(\\gamma _{kl}\\right)_{1\\le k,l\\le r}$ such that $\\gamma _{ij}=1$ and all other entries are equal 0.", "Then for Jordan frame $\\left( {\\textbf {c}} _i\\right)_{i=1}^r$ , where $ {\\textbf {c}} _k=\\mu _{kk}$ , $k=1,\\ldots ,r$ , we have $ {\\textbf {z}} _{jk}=(\\mu _{jk}+\\mu _{kj})\\in \\mathbb {E}_{jk}$ oraz $\\Vert {\\textbf {z}} _{jk}\\Vert ^2=2$ , $1\\le j,k\\le r$ , $j\\ne k$ .", "if $ {\\textbf {z}} ^{(i)}\\in \\bigoplus _{j=i+1}^r \\mathbb {E}_{ij}$ , $i=1,\\ldots ,r-1$ , then there exists $\\alpha ^{(i)}=(\\alpha _{i+1},\\ldots ,\\alpha _r)\\in \\mathbb {R}^{r-i}$ such that $ {\\textbf {z}} ^{(i)}=\\sum _{j=i+1}^r \\alpha _j {\\textbf {z}} _{ij}$ .", "Then the Frobenius transformation reads $\\tau _{ {\\textbf {c}} _i}( {\\textbf {z}} ^{(i)}) {\\textbf {x}} =\\mathcal {F}_i(\\alpha ^{(i)})\\cdot {\\textbf {x}} \\cdot \\mathcal {F}_i(\\alpha ^{(i)})^T,$ where $\\mathcal {F}_i(\\alpha ^{(i)})$ is so called Frobenius matrix: $\\mathcal {F}_i(\\alpha ^{(i)})=I+\\sum _{j=i+1}^r \\alpha _j \\mu _{ji},$ i.e.", "bellow $i$ th one of identity matrix there is a vector $\\alpha ^{(i)}$ , particularly $\\mathcal {F}_2(\\alpha ^{(2)})=\\begin{pmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 1 & 0 & \\cdots & 0 \\\\0 & \\alpha _{3} & 1 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & \\alpha _{r} & 0 & \\cdots & 1\\end{pmatrix}.$ It can be shown ([8]) that for each $t\\in \\mathcal {T}$ , $ {\\textbf {x}} \\in \\Omega $ and $s\\in \\mathbb {R}^r$ , $\\Delta _s(t {\\textbf {x}} )=\\Delta _s(t {\\textbf {e}} )\\Delta _s( {\\textbf {x}} )$ and for any $ {\\textbf {z}} \\in \\mathbb {E}( {\\textbf {c}} _i,\\frac{1}{2})$ , $i=1,\\ldots ,r$ , $\\Delta _s(\\tau _{ {\\textbf {c}} _i}( {\\textbf {z}} ) {\\textbf {e}} )=1,$ if only $\\Delta _s$ and $\\mathcal {T}$ are associated with the same Jordan frame $\\left( {\\textbf {c}} _i\\right)_{i=1}^r$ .", "We will now introduce some necessary basics regarding certain probability distribution on symmetric cones.", "Absolutely continuous Riesz distribution $R_{s, {\\textbf {a}} }$ on $ \\Omega $ is defined for any $ {\\textbf {a}} \\in \\Omega $ and $s=(s_1,\\ldots ,s_r)\\in \\mathbb {R}^r$ such that $s_i>(i-1)d/2$ , $i=1,\\ldots ,r$ , though its density $R_{s, {\\textbf {a}} }(d {\\textbf {x}} )=\\frac{\\Delta _s( {\\textbf {a}} )}{\\Gamma _ \\Omega (s)} \\Delta _{s-\\dim \\Omega /r}( {\\textbf {x}} )e^{-\\left\\langle {\\textbf {a}} , {\\textbf {x}} \\right\\rangle }I_ \\Omega ( {\\textbf {x}} )\\,d {\\textbf {x}} ,\\quad {\\textbf {x}} \\in \\Omega ,$ where $\\Delta _s$ is the generalized power function with respect to a Jordan frame $( {\\textbf {c}} _i)_{i=1}^r$ and $\\Gamma _ \\Omega $ is the Gamma function of the symmetric cone $ \\Omega $ .", "It can be shown that $\\Gamma _ \\Omega (s)=(2\\pi )^{(\\dim \\Omega -r)/2}\\prod _{j=1}^r \\Gamma (s_j-(j-1)\\tfrac{d}{2})$ (see [8]).", "Riesz distribution was introduced in [10].", "Absolutely continuous Wishart distribution $\\gamma _{p, {\\textbf {a}} }$ on $ \\Omega $ is a special case of Riesz distribution for $s_1=\\ldots =s_r=p$ .", "If $ {\\textbf {a}} \\in \\Omega $ and $p>\\dim \\Omega /r-1$ it has density $\\gamma _{p, {\\textbf {a}} }(d {\\textbf {x}} )=\\frac{(\\det {\\textbf {a}} )^p}{\\Gamma _ \\Omega (p)} (\\det {\\textbf {x}} )^{p-\\dim \\Omega /r}e^{-\\left\\langle {\\textbf {a}} , {\\textbf {x}} \\right\\rangle }I_ \\Omega ( {\\textbf {x}} )\\,d {\\textbf {x}} ,\\quad {\\textbf {x}} \\in \\Omega ,$ where $\\Gamma _ \\Omega (p):=\\Gamma _ \\Omega (p,\\ldots ,p)$ .", "Wishart distribution is a generalization of gamma distribution (case $r=1$ ).", "In generality, Riesz and Wishart distributions does not always have densities, but due to the assumption of existence of densities in Theorem REF , we are not interested in other cases." ], [ "Logarithmic Cauchy functions", "As will be seen, the densities of respective random variables will be given in terms of $w$ -logarithmic Cauchy functions, ie.", "functions $f\\colon \\Omega \\rightarrow \\mathbb {R}$ that satisfy the following functional equation $f( {\\textbf {x}} )+f(w( {\\textbf {e}} ) {\\textbf {y}} )=f(w( {\\textbf {x}} ) {\\textbf {y}} ),\\quad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2,$ where $w$ is a multiplication algorithm.", "If $f$ is $w$ -logarithmic, then $e^f$ is called $w$ -multiplicative.", "In the following section we will give the form of $w$ -logarithmic Cauchy functions for two basic multiplication algorithms, one connected with the quadratic representation $w_1( {\\textbf {x}} )=\\mathbb {P}( {\\textbf {x}} ^{1/2}),$ and the other related to a triangular group $\\mathcal {T}$ , $w_2( {\\textbf {x}} )=t_ {\\textbf {x}} \\in \\mathcal {T}.$ Such functions were recently considered without any regularity assumptions in [14].", "It should be stressed that there exist infinite number of multiplication algorithms.", "If $ w $ is a multiplication algorithm, then trivial extensions are given by $ w ^{(k)}( {\\textbf {x}} )= w ( {\\textbf {x}} ) k$ , where $k\\in K$ is fixed (Remark REF explains why this extension is trivial when it comes to multiplicative functions).", "One may consider also multiplication algorithms of the form $\\mathbb {P}( {\\textbf {x}} ^\\alpha )t_{ {\\textbf {x}} ^{1-2\\alpha }}$ , which interpolate between the two main examples: $ w _1$ (which is $\\alpha =1/2$ ) and $ w _2$ (which is $\\alpha =0$ ).", "In general any multiplication algorithm may be written in the form $ w (x)=\\mathbb {P}( {\\textbf {x}} ^{1/2})k_x$ , where $ {\\textbf {x}} \\mapsto k_ {\\textbf {x}} \\in K$ .", "Functional equation (REF ) for $w_1$ were already considered by [3] for differentiable functions and by [19] for continuous functions of real or complex Hermitian positive definite matrices of rank greater than 2.", "Without any regularity assumptions it was solved on the Lorentz cone by [23].", "Case of $w_2( {\\textbf {x}} )=t_ {\\textbf {x}} \\in \\mathcal {T}$ for a triangular group $\\mathcal {T}$ , perhaps a bit surprisingly, leads to a different solution.", "It was indirectly solved for differentiable functions by [11].", "By [8], for any $g$ in the group $G$ , $\\det (g {\\textbf {x}} )=(\\mathrm {Det}\\,g)^{r/\\dim \\Omega }\\det {\\textbf {x}} ,$ where $\\mathrm {Det}$ denotes the determinant in the space of endomorphisms on $ \\Omega $ .", "Inserting a multiplication algorithm $g=w( {\\textbf {y}} )$ , $ {\\textbf {y}} \\in \\Omega $ , and $ {\\textbf {x}} = {\\textbf {e}} $ we obtain $\\mathrm {Det}\\left(w( {\\textbf {y}} )\\right) =(\\det {\\textbf {y}} )^{\\dim \\Omega /r}$ and hence $\\det (w( {\\textbf {y}} ) {\\textbf {x}} ) =\\det {\\textbf {y}} \\det {\\textbf {x}} $ for any $ {\\textbf {x}} , {\\textbf {y}} \\in \\Omega $ .", "This means that $f( {\\textbf {x}} )=H(\\det {\\textbf {x}} )$ , where $H$ is generalized logarithmic function, ie.", "$H(ab)=H(a)+H(b)$ for $a,b>0$ , is always a solution to (REF ), regardless of the choice of multiplication algorithm $w$ .", "If a $w$ -logarithmic functions $f$ is additionally $K$ -invariant ($f( {\\textbf {x}} )=f(k {\\textbf {x}} )$ for any $k\\in K$ ), then $H(\\det {\\textbf {x}} )$ is the only possible solution (Theorem REF ).", "In [14] the following theorems have been proved.", "They will be useful in the proof of the main theorems in this paper.", "Theorem 3.1 ($w_1$ -logarithmic Cauchy functional equation) Let $f\\colon \\Omega \\rightarrow \\mathbb {R}$ be a function such that $f( {\\textbf {x}} )+f( {\\textbf {y}} )=f\\left(\\mathbb {P}\\left( {\\textbf {x}} ^{1/2}\\right) {\\textbf {y}} \\right),\\quad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2.$ Then there exists a logarithmic function $H$ such that for any $ {\\textbf {x}} \\in \\Omega $ , $f( {\\textbf {x}} )=H(\\det {\\textbf {x}} ).$ Theorem 3.2 ($w_2$ -logarithmic Cauchy functional equation) Let $f\\colon \\Omega \\rightarrow \\mathbb {R}$ be a function satisfying $f( {\\textbf {x}} )+f( {\\textbf {y}} )=f(t_{ {\\textbf {y}} } {\\textbf {x}} )$ for any $ {\\textbf {x}} $ and $ {\\textbf {y}} $ in the cone $ \\Omega $ of rank $r$ , $t_{ {\\textbf {y}} }\\in \\mathcal {T}$ , where $\\mathcal {T}$ is the triangular group with respect to the Jordan frame $\\left( {\\textbf {c}} _i\\right)_{i=1}^r$ .", "Then there exist generalized logarithmic functions $H_1,\\ldots , H_r$ such that for any $ {\\textbf {x}} \\in \\Omega $ , $f( {\\textbf {x}} )=\\sum _{k=1}^r H_k(\\Delta _k( {\\textbf {x}} )),$ where $\\Delta _k$ is the principal minor of order $k$ with respect to $\\left( {\\textbf {c}} _i\\right)_{i=1}^r$ .", "If we assume in Theorem REF that $f$ is additionally measurable, then functions $H_k$ are measurable.", "This implies that there exists constants $s_k\\in \\mathbb {R}$ such that $H_k(\\alpha )=s_k\\log \\alpha $ and $f( {\\textbf {x}} )=\\sum _{k=1}^r s_k\\log (\\Delta _k( {\\textbf {x}} ))=\\log \\prod _{k=1}^r \\Delta ^{s_k}_k( {\\textbf {x}} ).$ Thus, we obtain the following Remark 3.3 If we impose on $f$ in Theorem REF some mild conditions (eg.", "measurability), then there exists $s\\in \\mathbb {R}^r$ such that for any $ {\\textbf {x}} \\in \\Omega $ , $f( {\\textbf {x}} )=\\log \\Delta _s( {\\textbf {x}} ).$ Theorem 3.4 Let $f\\colon \\Omega \\rightarrow \\mathbb {R}$ be a function satisfying (REF ).", "Assume additionally that $f$ is $K$ -invariant, ie.", "$f(k {\\textbf {x}} )=f( {\\textbf {x}} )$ for any $k\\in K$ and $ {\\textbf {x}} \\in \\Omega $ .", "Then there exists a logarithmic function $H$ such that for any $ {\\textbf {x}} \\in \\Omega $ , $f( {\\textbf {x}} )=H(\\det {\\textbf {x}} ).$ Lemma 3.5 ($w$ -logarithmic Pexider functional equation) Assume that $a$ , $b$ , $c$ defined on the cone $\\Omega $ satisfy following functional equation $a( {\\textbf {x}} )+b( {\\textbf {y}} )=c(w( {\\textbf {x}} ) {\\textbf {y}} ),\\quad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2.$ Then there exist $w$ -logarithmic function $f$ and real constants $a_0, b_0$ such that for any $ {\\textbf {x}} \\in \\Omega $ , $a( {\\textbf {x}} ) & =f( {\\textbf {x}} )+a_0,\\\\b( {\\textbf {x}} ) & =f(w( {\\textbf {e}} ) {\\textbf {x}} )+b_0,\\\\c( {\\textbf {x}} ) & =f( {\\textbf {x}} )+a_0+b_0.$" ], [ "The Olkin–Baker functional equation", "In the following section we deal with the Olkin-Baker functional equation on irreducible symmetric cones, which is related to the Lukacs independence condition (see proof of the Theorem REF ).", "Henceforth we will assume that multiplication algorithm $ w $ additionally is homogeneous of degree 1, that is $ w (s {\\textbf {x}} )=s w ( {\\textbf {x}} )$ for any $s>0$ and $ {\\textbf {x}} \\in \\Omega $ .", "It is easy to create a multiplication algorithm without this property, for example: $ w ( {\\textbf {x}} )={\\left\\lbrace \\begin{array}{ll} w _1( {\\textbf {x}} ), &\\mbox{ if } \\det {\\textbf {x}} >1, \\\\ w _2( {\\textbf {x}} ), &\\mbox{ if } \\det {\\textbf {x}} \\le 1.\\end{array}\\right.", "}$ The problem of solving $f(x)g(y)=p(x+y)q(x/y),\\quad (x,y)\\in (0,\\infty )^2$ for unknown positive functions $f$ , $g$ , $p$ and $q$ was first posed in [20].", "Note that in one dimensional case it does not matter whether one considers $q(x/y)$ or $q(x/(x+y))$ on the right hand side of (REF ).", "Its general solution was given in [1], and later analyzed in [15] using a different approach.", "Recently, in [18] and [16] the equation (REF ) was solved assuming that it is satisfied almost everywhere on $(0,\\infty )^2$ for measurable functions which are non-negative on its domain or positive on some sets of positive Lebesgue measure, respectively.", "Finally, a new derivation of solution to (REF ), when the equation holds almost everywhere on $(0,\\infty )^2$ and no regularity assumptions on unknown positive functions are imposed, was given in [9].", "The following theorem is concerned with an adaptation of (REF ) (after taking logarithm) to the symmetric cone case.", "Theorem 3.6 (Olkin–Baker functional equation on symmetric cones) Let $a$ , $b$ , $c$ and $d$ be real continuous functions on an irreducible symmetric cone $ \\Omega $ of rank $r$ .", "Assume $a( {\\textbf {x}} )+b( {\\textbf {y}} )=c( {\\textbf {x}} + {\\textbf {y}} )+d\\left( g \\left( {\\textbf {x}} + {\\textbf {y}} \\right) {\\textbf {x}} \\right),\\qquad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2,$ where $ g ^{-1}= w $ is a homogeneous of degree 1 multiplication algorithm.", "Then there exist constants $C_i\\in \\mathbb {R}$ , $i=1,\\ldots ,4$ , $\\Lambda \\in \\mathbb {E}$ such that for any $ {\\textbf {x}} \\in \\Omega $ and $ {\\textbf {u}} \\in \\mathcal {D}=\\left\\lbrace {\\textbf {x}} \\in \\Omega \\colon {\\textbf {e}} - {\\textbf {x}} \\in \\Omega \\right\\rbrace $ , $a( {\\textbf {x}} )&=\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle +e( {\\textbf {x}} )+C_1,\\\\b( {\\textbf {x}} )&=\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle +f( {\\textbf {x}} )+C_2,\\\\c( {\\textbf {x}} )&=\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle +e( {\\textbf {x}} )+f( {\\textbf {x}} )+C_3,\\\\d( {\\textbf {u}} )&=e(w( {\\textbf {e}} ) {\\textbf {u}} )+f( {\\textbf {e}} -w( {\\textbf {e}} ) {\\textbf {u}} )+C_4,$ where $e$ and $f$ are continuous $ w $ -logarithmic Cauchy functions and $C_1+C_2=C_3+C_4$ .", "We will need following simple lemma.", "For the elementary proof we refer to [13].", "Lemma 3.7 (Additive Pexider functional equation on symmetric cones) Let $a$ , $b$ and $c$ be measurable functions on a symmetric cone $ \\Omega $ satisfying $a( {\\textbf {x}} )+b( {\\textbf {y}} )=c( {\\textbf {x}} + {\\textbf {y}} ),\\qquad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2.$ Then there exist constants $\\alpha , \\beta \\in \\mathbb {R}$ and $\\lambda \\in \\mathbb {E}$ such that for all $ {\\textbf {x}} \\in \\Omega $ , $\\begin{split}a( {\\textbf {x}} )&=\\left\\langle \\lambda , {\\textbf {x}} \\right\\rangle +\\alpha , \\\\b( {\\textbf {x}} )&=\\left\\langle \\lambda , {\\textbf {x}} \\right\\rangle +\\beta , \\\\c( {\\textbf {x}} )&=\\left\\langle \\lambda , {\\textbf {x}} \\right\\rangle +\\alpha +\\beta .\\end{split}$ Now we can come back and give a new proof the Olkin–Baker functional equation.", "In the first part of the proof we adapt the argument given in [9], where the analogous result on $(0,\\infty )$ was analyzed, to the symmetric cone setting.", "For any $s>0$ and $( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2$ we get $a(s {\\textbf {x}} )+b(s {\\textbf {y}} )=c(s( {\\textbf {x}} + {\\textbf {y}} ))+d\\left(g(s {\\textbf {x}} +s {\\textbf {y}} )s {\\textbf {x}} \\right).$ Since $w$ is homogeneous of degree 1 we have $g(s {\\textbf {x}} )=\\tfrac{1}{s}g( {\\textbf {x}} )$ and so $g(s {\\textbf {x}} +s {\\textbf {y}} )s {\\textbf {x}} =g( {\\textbf {x}} + {\\textbf {y}} ) {\\textbf {x}} $ for any $s>0$ .", "Subtracting now (REF ) from (REF ) for any $s>0$ we arrive at the additive Pexider equation on symmetric cone $ \\Omega $ , $a_s( {\\textbf {x}} )+b_s( {\\textbf {y}} )=c_s( {\\textbf {x}} + {\\textbf {y}} ),\\qquad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2,$ where $a_s$ , $b_s$ and $c_s$ are functions defined by $a_s( {\\textbf {x}} ):=a(s {\\textbf {x}} )-a( {\\textbf {x}} )$ , $b_s( {\\textbf {x}} ):=b(s {\\textbf {x}} )-b( {\\textbf {x}} )$ and $c_s( {\\textbf {x}} ):=c(s {\\textbf {x}} )-c( {\\textbf {x}} )$ .", "Due to continuity of $a$ , $b$ and $c$ and Lemma REF it follows that for any $s>0$ there exist constants $\\lambda (s)\\in \\mathbb {E}$ , $\\alpha (s)\\in \\mathbb {R}$ and $\\beta (s)\\in \\mathbb {R}$ such that for any $ {\\textbf {x}} \\in \\Omega $ , $a_s( {\\textbf {x}} ) &= \\left\\langle \\lambda (s), {\\textbf {x}} \\right\\rangle +\\alpha (s),\\\\b_s( {\\textbf {x}} ) &= \\left\\langle \\lambda (s), {\\textbf {x}} \\right\\rangle +\\beta (s),\\\\c_s( {\\textbf {x}} ) &= \\left\\langle \\lambda (s), {\\textbf {x}} \\right\\rangle +\\alpha (s)+\\beta (s).$ By the definition of $a_s$ and the above observation it follows that for any $(s,t)\\in (0,\\infty )^2$ and $ {\\textbf {z}} \\in \\Omega $ $a_{st}( {\\textbf {z}} )=a_t(s {\\textbf {z}} )+a_s( {\\textbf {z}} ).$ Hence, $\\left\\langle \\lambda (st), {\\textbf {z}} \\right\\rangle +\\alpha (st)=\\left\\langle \\lambda (t),s {\\textbf {z}} \\right\\rangle +\\alpha (t)+\\left\\langle \\lambda (s), {\\textbf {z}} \\right\\rangle +\\alpha (s).$ Since (REF ) holds for any $ {\\textbf {z}} \\in \\Omega $ we see that $\\alpha (st)=\\alpha (s)+\\alpha (t)$ for all $(s,t)\\in (0,\\infty )^2$ .", "That is $\\alpha (s)=k_1\\log \\,s$ for $s\\in (0,\\infty )$ , where $k_1$ is a real constant.", "On the other hand $\\left\\langle \\lambda (st), {\\textbf {z}} \\right\\rangle =\\left\\langle \\lambda (s), {\\textbf {z}} \\right\\rangle +\\left\\langle \\lambda (t),s {\\textbf {z}} \\right\\rangle =\\left\\langle \\lambda (t), {\\textbf {z}} \\right\\rangle +\\left\\langle \\lambda (s),t {\\textbf {z}} \\right\\rangle $ since one can interchange $s$ and $t$ on the left hand side.", "Putting $s=2$ and denoting $\\Lambda =\\lambda (2)$ we obtain $\\left\\langle \\lambda (t), {\\textbf {z}} \\right\\rangle =\\left\\langle \\Lambda , {\\textbf {z}} \\right\\rangle (t-1)$ for $t>0$ and $ {\\textbf {z}} \\in \\Omega $ .", "It then follows that for all $s\\in (0,\\infty )$ and $ {\\textbf {z}} \\in \\Omega $ , $a_s( {\\textbf {z}} )=a(s {\\textbf {z}} )-a( {\\textbf {z}} )=\\left\\langle \\Lambda , {\\textbf {z}} \\right\\rangle (s-1)+k_1\\log \\,s.$ Let us define function $\\bar{a}$ by formula $\\bar{a}( {\\textbf {x}} )=a( {\\textbf {x}} )-\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle .$ From (REF ) we get $\\bar{a}(s {\\textbf {x}} )=\\bar{a}( {\\textbf {x}} )+k_1\\log s$ for $s>0$ and $ {\\textbf {x}} \\in \\Omega $ .", "Analogous considerations for function $b_s$ gives existence of constant $k_2$ such that $\\bar{b}(s {\\textbf {x}} )=\\bar{b}( {\\textbf {x}} )+k_2\\log s$ , where $\\bar{b}( {\\textbf {x}} )=b( {\\textbf {x}} )-\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle ,$ hence $\\bar{c}(s {\\textbf {x}} )=\\bar{c}( {\\textbf {x}} )+(k_1+k_2)\\log s$ and $\\bar{c}( {\\textbf {x}} )=c( {\\textbf {x}} )-\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle $ for any $s>0$ and $ {\\textbf {x}} \\in \\Omega $ .", "Functions $\\bar{a}$ , $\\bar{b}$ , $\\bar{c}$ and $d$ satisfy original Olkin-Baker functional equation: $\\bar{a}( {\\textbf {x}} )+\\bar{b}( {\\textbf {y}} )=\\bar{c}( {\\textbf {x}} + {\\textbf {y}} )+d\\left( g \\left( {\\textbf {x}} + {\\textbf {y}} \\right) {\\textbf {x}} \\right),\\quad ( {\\textbf {x}} , {\\textbf {y}} )\\in \\Omega ^2.$ Taking $ {\\textbf {x}} = {\\textbf {y}} = {\\textbf {v}} \\in \\Omega $ in (REF ), we arrive at $\\bar{a}( {\\textbf {v}} )+\\bar{b}( {\\textbf {v}} )=\\bar{c}(2 {\\textbf {v}} )+d(\\tfrac{1}{2} {\\textbf {e}} )=\\bar{c}( {\\textbf {v}} )+(k_1+k_2)\\log 2+d(\\tfrac{1}{2} {\\textbf {e}} ).$ Insert $ {\\textbf {x}} =\\alpha w ( {\\textbf {v}} ) {\\textbf {u}} $ and $ {\\textbf {y}} = w ( {\\textbf {v}} )( {\\textbf {e}} -\\alpha {\\textbf {u}} )$ into (REF ) for $0<\\alpha <1$ and $( {\\textbf {u}} , {\\textbf {v}} )\\in (\\mathcal {D}, \\Omega )$ .", "Using (REF ) we obtain $\\bar{a}( w ( {\\textbf {v}} ) {\\textbf {u}} )+\\bar{b}( w ( {\\textbf {v}} )( {\\textbf {e}} -\\alpha {\\textbf {u}} ))=\\bar{c}( {\\textbf {v}} )+d\\left(\\alpha {\\textbf {u}} \\right)-k_1\\log \\alpha ,\\quad ( {\\textbf {u}} , {\\textbf {v}} )\\in (\\mathcal {D}, \\Omega ).$ Let us observe, that due to continuity of $\\bar{b}$ on $ \\Omega $ and $\\lim _{\\alpha \\rightarrow 0}\\left\\lbrace w ( {\\textbf {v}} )( {\\textbf {e}} -\\alpha {\\textbf {u}} )\\right\\rbrace = w ( {\\textbf {v}} ) {\\textbf {e}} = {\\textbf {v}} \\in \\Omega $ (convergence in the norm generated by scalar product $\\left\\langle \\cdot ,\\cdot \\right\\rangle $ ), limit as $\\alpha \\rightarrow 0$ of the left hand side of the above equality exists.", "Hence, the limit of the right hand side also exists and $\\bar{a}( w ( {\\textbf {v}} ) {\\textbf {u}} )+\\bar{b}( {\\textbf {v}} )=\\bar{c}( {\\textbf {v}} )+\\lim _{\\alpha \\rightarrow 0}\\left\\lbrace d(\\alpha {\\textbf {u}} ) -k_1\\log \\alpha \\right\\rbrace ,\\quad ( {\\textbf {u}} , {\\textbf {v}} )\\in (\\mathcal {D}, \\Omega ).$ Subtracting (REF ) from (REF ) we have $\\bar{a}( w ( {\\textbf {v}} ) {\\textbf {u}} )=\\bar{a}( {\\textbf {v}} )+g( {\\textbf {u}} )$ for $ {\\textbf {u}} \\in \\mathcal {D}, {\\textbf {v}} \\in \\Omega $ , where $g( {\\textbf {u}} )=\\lim _{\\alpha \\rightarrow 0}\\left\\lbrace d(\\alpha {\\textbf {u}} ) -k_1\\log \\alpha \\right\\rbrace -(k_1+k_2)\\log 2-d(\\tfrac{1}{2} {\\textbf {e}} )$ .", "Due to the property (REF ) equation (REF ) holds for any $ {\\textbf {u}} \\in \\Omega $ , so we arrive at the $ w $ -logarithmic Pexider equation.", "Lemma REF implies that there exist $ w $ -logarithmic function $e$ such that $\\bar{a}( {\\textbf {x}} )=e( {\\textbf {x}} )+C_1$ for any $ {\\textbf {x}} \\in \\Omega $ and a constant $C_1\\in \\mathbb {R}$ .", "Function $e$ is continuous, because $\\bar{a}$ is continuous.", "Coming back to the definition of $\\bar{a}$ , we obtain $a( {\\textbf {x}} )=\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle +e( {\\textbf {x}} )+C_1, \\quad {\\textbf {x}} \\in \\Omega .$ Analogously for function $b$ , considering equation (REF ) for $ {\\textbf {x}} = w ( {\\textbf {v}} )( {\\textbf {e}} -\\alpha {\\textbf {u}} )$ and $ {\\textbf {y}} =\\alpha w ( {\\textbf {v}} ) {\\textbf {u}} $ after passing to the limit as $\\alpha \\rightarrow 0$ , we show that there exist continuous $ w $ -logarithmic function $f$ such that $b( {\\textbf {x}} )=\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle +f( {\\textbf {x}} )+C_2, \\quad {\\textbf {x}} \\in \\Omega $ for a constant $C_2\\in \\mathbb {R}$ .", "The form of $c$ follows from (REF ).", "Taking $ {\\textbf {x}} =w( {\\textbf {e}} ) {\\textbf {u}} $ and $ {\\textbf {y}} = {\\textbf {e}} -w( {\\textbf {e}} ) {\\textbf {u}} $ in (REF ) for $ {\\textbf {u}} \\in \\mathcal {D}$ , we obtain the form of $d$ ." ], [ "The Lukacs-Olkin-Rubin theorem without invariance of the quotient", "In the following section we prove the density version of Lukacs-Olkin-Rubin theorem for any multiplication algorithm $w$ satisfying $ w (s {\\textbf {x}} )=s w ( {\\textbf {x}} )$ for $s>0$ and $ {\\textbf {x}} \\in \\Omega $ , differentiability of mapping $ \\Omega \\ni {\\textbf {x}} \\mapsto w ( {\\textbf {x}} )\\in G$ .", "We assume (ii) to ensure that Jacobian of the considered transformation exists.", "We start with the direct result, showing that the considered measures have desired property.", "The converse result is given in Theorem REF .", "For every generalized multiplication $ w $ , the family of these $ w $ -Wishart measures (as defined in (REF )) contains the Wishart laws.", "For $ w = w _1$ , there are no other distributions, while the $ w _2$ -Wishart measures consist of the Riesz distributions.", "It is an open question whether there is a generalized multiplication $ w $ that leads to other probability measures in this family.", "Theorem 4.1 Let $ w $ be a multiplication algorithm satisfying condition (ii) and define $ g = w ^{-1}$ .", "Suppose that $X$ and $Y$ are independent random variables with densities given by $\\begin{split}f_X( {\\textbf {x}} ) =C_X e( {\\textbf {x}} )\\exp \\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle I_ \\Omega ( {\\textbf {x}} ),\\\\f_Y( {\\textbf {x}} ) =C_Y f( {\\textbf {x}} )\\exp \\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle I_ \\Omega ( {\\textbf {x}} ),\\end{split}$ where $e$ and $f$ are $ w $ -multiplicative functions, $\\Lambda \\in \\mathbb {E}$ and $\\mathbb {E}$ is the Euclidean Jordan algebra associated with the irreducible symmetric cone $ \\Omega $ .", "Then vector $(U,V)=\\left( g (X+Y)X,X+Y\\right)$ have independent components.", "Note that if $ w ( {\\textbf {x}} )= w _1( {\\textbf {x}} )=\\mathbb {P}( {\\textbf {x}} ^{1/2})$ , then there exist positive constants $\\kappa _X$ and $\\kappa _Y$ such that $e( {\\textbf {x}} )=(\\det {\\textbf {x}} )^{\\kappa _X-\\dim \\Omega /r}$ and $f( {\\textbf {x}} )=(\\det {\\textbf {x}} )^{\\kappa _Y-\\dim \\Omega /r}$ .", "In this case $-\\Lambda =: {\\textbf {a}} \\in \\Omega $ and $(X,Y)\\sim \\gamma _{\\kappa _X, {\\textbf {a}} }\\otimes \\gamma _{\\kappa _Y, {\\textbf {a}} }$ .", "Similarly, if $ w ( {\\textbf {x}} )= w _2( {\\textbf {x}} )=t_ {\\textbf {x}} $ , $X$ and $Y$ follow Riesz distributions with the same scale parameter $-\\Lambda \\in \\Omega $ .", "In general we do not know whether $ {\\textbf {a}} =-\\Lambda $ should always belong to $ \\Omega $ .", "Let $\\psi \\colon \\Omega \\times \\Omega \\rightarrow \\mathcal {D}\\times \\Omega $ be a mapping defined through $\\psi ( {\\textbf {x}} , {\\textbf {y}} )=\\left( g ( {\\textbf {x}} + {\\textbf {y}} ) {\\textbf {x}} , {\\textbf {x}} + {\\textbf {y}} \\right)=( {\\textbf {u}} , {\\textbf {v}} ).$ Then $(U,V)=\\psi (X,Y)$ .", "The inverse mapping $\\psi ^{-1}\\colon \\mathcal {D}\\times \\Omega \\rightarrow \\Omega \\times \\Omega $ is given by $( {\\textbf {x}} , {\\textbf {y}} )=\\psi ^{-1}( {\\textbf {u}} , {\\textbf {v}} )=\\left( w ( {\\textbf {v}} ) {\\textbf {u}} , w ( {\\textbf {v}} )( {\\textbf {e}} - {\\textbf {u}} )\\right),$ hence $\\psi $ is a bijection.", "We are looking for the Jacobian of the map $\\psi ^{-1}$ , that is, the determinant of the linear map $\\begin{pmatrix}d {\\textbf {u}} \\\\d {\\textbf {v}} \\end{pmatrix}\\mapsto \\begin{pmatrix}d {\\textbf {x}} \\\\d {\\textbf {y}} \\end{pmatrix}=\\begin{pmatrix}d {\\textbf {x}} /d {\\textbf {u}} & d {\\textbf {x}} /d {\\textbf {v}} \\\\d {\\textbf {y}} /d {\\textbf {u}} & d {\\textbf {y}} /d {\\textbf {v}} \\end{pmatrix}\\begin{pmatrix}d {\\textbf {u}} \\\\d {\\textbf {v}} \\end{pmatrix}.$ We have $J=\\left\|\\begin{array}{cc} w ( {\\textbf {v}} ) & d {\\textbf {x}} /d {\\textbf {v}} \\\\- w ( {\\textbf {v}} ) & Id_ \\Omega -d {\\textbf {x}} /d {\\textbf {v}} \\end{array}\\right\|=\\left\|\\begin{array}{cc} w ( {\\textbf {v}} ) & d {\\textbf {x}} /d {\\textbf {v}} \\\\0 & Id_ \\Omega \\end{array}\\right\| = \\mathrm {Det}( w ( {\\textbf {v}} )).$ where $\\mathrm {Det}$ denotes the determinant in the space of endomorphisms on $ \\Omega $ .", "By (REF ) we get $\\mathrm {Det}\\left( w \\left( {\\textbf {v}} \\right)\\right)=(\\det {\\textbf {v}} )^{\\dim \\Omega /r}.$ Now we can find the joint density of $(U,V)$ .", "Since $(X,Y)$ have independent components, we obtain $f_{(U,V)}( {\\textbf {u}} , {\\textbf {v}} )=(\\det {\\textbf {v}} )^{\\dim \\Omega /r}f_X( w ( {\\textbf {v}} ) {\\textbf {u}} )f_Y( w ( {\\textbf {v}} )( {\\textbf {e}} - {\\textbf {u}} ))$ We assumed (REF ), thus there exist $\\Lambda \\in \\mathbb {E}$ , $C_X, C_Y \\in \\mathbb {R}$ and $ w $ -multiplicative functions $e$ , $f$ such that $f_{(U,V)}( {\\textbf {u}} , {\\textbf {v}} )= & (\\det {\\textbf {v}} )^{\\dim \\Omega /r}f_X( w ( {\\textbf {v}} ) {\\textbf {u}} )f_Y( w ( {\\textbf {v}} )( {\\textbf {e}} - {\\textbf {u}} )) \\\\= &C_1C_2\\, (\\det {\\textbf {v}} )^{\\dim \\Omega /r} e( w ( {\\textbf {v}} ) {\\textbf {u}} ) f( w ( {\\textbf {v}} )( {\\textbf {e}} - {\\textbf {u}} )) e^{\\left\\langle \\Lambda , {\\textbf {v}} \\right\\rangle }I_ \\Omega ( w ( {\\textbf {v}} ) {\\textbf {u}} )I_ \\Omega ( w ( {\\textbf {v}} )( {\\textbf {e}} - {\\textbf {u}} )) \\\\= &C_1C_2\\, (\\det {\\textbf {v}} )^{\\dim \\Omega /r} e( {\\textbf {v}} )f( {\\textbf {v}} ) e^{\\left\\langle \\Lambda , {\\textbf {v}} \\right\\rangle }I_ \\Omega ( {\\textbf {v}} ) \\,\\, e( w ( {\\textbf {e}} ) {\\textbf {u}} ) f( w ( {\\textbf {e}} )( {\\textbf {e}} - {\\textbf {u}} )) I_\\mathcal {D}( {\\textbf {u}} ), \\\\= & f_U( {\\textbf {u}} ) \\, f_V( {\\textbf {v}} ),$ what completes the proof.", "To prove the characterization of given measures, we need to show that the inverse implication is also valid.", "The following theorem generalizes results obtained in [2], [11], [13].", "We consider quotient $U$ for any multiplication algorithm $ w $ satisfying conditions (i) and (ii) given at the beginning of this section (note that multiplication algorithms $ w _1$ and $ w _2$ defined in (REF ) and (REF ), respectively, satisfy both of these conditions).", "Respective densities are then expressed in terms of $ w $ -multiplicative Cauchy functions.", "Theorem 4.2 (The Lukacs-Olkin-Rubin theorem with densities on symmetric cones) Let $X$ and $Y$ be independent rv's valued in irreducible symmetric cone $ \\Omega $ with strictly positive and continuous densities.", "Set $V=X+Y$ and $U= g \\left(X+Y\\right)X$ for any multiplication algorithm $ w = g ^{-1}$ satisfying conditions (i) and (ii).", "If $U$ and $V$ are independent then there exist $\\Lambda \\in \\mathbb {E}$ and $ w $ -multiplicative functions $e$ , $f$ such that (REF ) holds.", "In particular, if $ g ( {\\textbf {x}} )= g _1( {\\textbf {x}} )=\\mathbb {P}( {\\textbf {x}} ^{-1/2})$ , then there exist constants $p_i>\\dim \\Omega /r-1$ , $i=1,2$ , and $ {\\textbf {a}} \\in \\Omega $ such that $X\\sim \\gamma _{p_1, {\\textbf {a}} }$ and $Y\\sim \\gamma _{p_2, {\\textbf {a}} }$ , if $ g ( {\\textbf {x}} )= g _2( {\\textbf {x}} )=t_{ {\\textbf {x}} }^{-1}$ , then there exist constants $s_i=(s_{i,j})_{j=1}^r$ , $s_{i,j}>(j-1)d/2$ , $i=1,2$ , and $ {\\textbf {a}} \\in \\Omega $ such that $X\\sim R_{s_1, {\\textbf {a}} }$ and $Y\\sim R_{s_2, {\\textbf {a}} }$ .", "We start from (REF ).", "Since $(U,V)$ is assumed to have independent components, the following identity holds almost everywhere with respect to Lebesgue measure: $(\\det ( {\\textbf {x}} + {\\textbf {y}} ))^{\\dim \\Omega /r}f_X( {\\textbf {x}} )f_Y( {\\textbf {y}} )=f_U\\left( g \\left( {\\textbf {x}} + {\\textbf {y}} \\right) {\\textbf {x}} \\right)f_V( {\\textbf {x}} + {\\textbf {y}} ),$ where $f_X$ ,$f_Y$ ,$f_U$ and $f_V$ denote densities of $X$ , $Y$ , $U$ and $V$ , respectively.", "Since the respective densities are assumed to be continuous, the above equation holds for every $ {\\textbf {x}} , {\\textbf {y}} \\in \\Omega $ .", "Taking logarithms of both sides of the above equation (it is permitted since $f_X, f_Y>0$ on $ \\Omega $ ) we get $a( {\\textbf {x}} )+b( {\\textbf {y}} )=c( {\\textbf {x}} + {\\textbf {y}} )+d\\left( g \\left( {\\textbf {x}} + {\\textbf {y}} \\right) {\\textbf {x}} \\right),$ where $a( {\\textbf {x}} )&=\\log \\, f_X( {\\textbf {x}} ),\\\\b( {\\textbf {x}} )&=\\log \\, f_Y( {\\textbf {x}} ),\\\\c( {\\textbf {x}} )&=\\log \\, f_V( {\\textbf {x}} )-\\tfrac{\\dim \\Omega }{r}\\log \\det ( {\\textbf {x}} ),\\\\d( {\\textbf {u}} )&=\\log \\, f_U( {\\textbf {u}} ),$ for $ {\\textbf {x}} \\in \\Omega $ and $ {\\textbf {u}} \\in \\mathcal {D}$ .", "The first part of the conclusion follows now directly from Theorem REF .", "Thus there exist constants $\\Lambda \\in \\mathbb {E}$ , $C_i\\in \\mathbb {R}$ , $i\\in \\lbrace 1,2\\rbrace $ and $ w $ -logarithmic functions $e$ and $f$ such that $f_X( {\\textbf {x}} ) & =e^{a( {\\textbf {x}} )}=e^{C_1}e( {\\textbf {x}} )e^{\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle },\\\\f_Y( {\\textbf {x}} ) & =e^{b( {\\textbf {x}} )}=e^{C_2}f( {\\textbf {x}} )e^{\\left\\langle \\Lambda , {\\textbf {x}} \\right\\rangle },$ for any $ {\\textbf {x}} \\in \\Omega $ .", "Let us observe that if $ w ( {\\textbf {x}} )= w _1( {\\textbf {x}} )=\\mathbb {P}( {\\textbf {x}} ^{1/2})$ , then for Theorem REF there exist constants $\\kappa _i$ , $i=1,2,$ such that $e( {\\textbf {x}} )=(\\det {\\textbf {x}} )^{\\kappa _1}$ and $f( {\\textbf {x}} )=(\\det {\\textbf {x}} )^{\\kappa _2}$ .", "Since $f_X$ and $f_Y$ are densities it follows that $ {\\textbf {a}} =-\\Lambda \\in \\Omega $ , $k_i=p_i-(\\dim \\Omega )/r>-1$ and $e^{C_i}=(\\det ( {\\textbf {a}} ))^{p_i}/\\Gamma _ \\Omega (p_i)$ , $i=1,2$ .", "Analogously, if $ w ( {\\textbf {x}} )= w _2( {\\textbf {x}} )=t_ {\\textbf {x}} $ then Theorem REF and Remark REF imply that there exist constants $s_i=(s_{i,j})_{j=1}^r$ , $s_{i,j}>(j-1)d/2$ , $i=1,2$ , and $ {\\textbf {a}} =-\\Lambda \\in \\Omega $ such that $X\\sim R_{s_1, {\\textbf {a}} }$ i $Y\\sim R_{s_2, {\\textbf {a}} }$ .", "Remark 4.3 Fix $k\\in K$ and consider $w^{(k)}( {\\textbf {x}} )=w( {\\textbf {x}} ) k$ .", "The $w^{(k)}$ -multiplicative function $f$ satisfies equation $f( {\\textbf {x}} )f(w( {\\textbf {e}} )k {\\textbf {y}} )=f(w( {\\textbf {x}} ) k {\\textbf {y}} ).$ Substituting $ {\\textbf {y}} \\mapsto k^{-1} {\\textbf {y}} \\in \\Omega $ we obtain $f( {\\textbf {x}} )f(w( {\\textbf {e}} ) {\\textbf {y}} )=f(w( {\\textbf {x}} ) {\\textbf {y}} ),$ that is $w^{(k)}$ -multiplicative functions are the same as $w$ -multiplicative functions.", "This leads to the rather unsurprising observation that if we consider Theorem REF with $w( {\\textbf {x}} )=\\mathbb {P}( {\\textbf {x}} ^{1/2})k$ or $w( {\\textbf {x}} )=t_ {\\textbf {x}} k$ , regardless of $k\\in K$ , we will characterize the same distributions as in points $(1)$ and $(2)$ of Theorem REF .", "With Theorem REF one can easily re-prove original Lukacs-Olkin-Rubin theorem (version of [22] and [7]), when the distribution of $U$ is invariant under a group of automorphisms: Remark 4.4 Let us additionally assume in Theorem REF , that the quotient $U$ has distribution which is invariant under a group of automorphisms, that is $kU\\stackrel{d}{=}U$ for any $k\\in K$ .", "From the proof of Theorem REF it follows that there exist continuous $ w $ -multiplicative functions $e$ and $f$ and constant $C$ such that for $ {\\textbf {u}} \\in \\mathcal {D}$ , $ f_U( {\\textbf {u}} )=C e( w ( {\\textbf {e}} ) {\\textbf {u}} ) f( {\\textbf {e}} - w ( {\\textbf {e}} ) {\\textbf {u}} ).$ The distribution of $U$ is invariant under $K$ , thus density $f_U$ is a $K$ -invariant function, that is $f_U( {\\textbf {u}} )=f_U(k {\\textbf {u}} )$ for any $k\\in K$ .", "Note that $w( {\\textbf {e}} )\\in K$ , thus $e( {\\textbf {u}} )f( {\\textbf {e}} - {\\textbf {u}} )=e(k {\\textbf {u}} )f( {\\textbf {e}} -k {\\textbf {u}} ),\\quad (k, {\\textbf {u}} )\\in K\\times \\mathcal {D}.$ We will show that both functions $e$ and $f$ are $K$ -invariant.", "Recall that $e( {\\textbf {x}} )\\,\\,e( w ( {\\textbf {e}} ) {\\textbf {y}} )=e( w ( {\\textbf {x}} ) {\\textbf {y}} )$ , therefore after taking $ {\\textbf {y}} =\\alpha {\\textbf {e}} $ we obtain $e(\\alpha {\\textbf {x}} )=e( {\\textbf {x}} )e(\\alpha {\\textbf {e}} )$ for any $\\alpha >0$ and $ {\\textbf {x}} \\in \\Omega $ .", "Inserting $ {\\textbf {u}} =\\alpha {\\textbf {v}} $ into (REF ) we arrive at $e( {\\textbf {v}} ) e(\\alpha {\\textbf {e}} ) f( {\\textbf {e}} -\\alpha {\\textbf {v}} )=e(\\alpha {\\textbf {v}} ) f( {\\textbf {e}} -\\alpha {\\textbf {v}} )=e(\\alpha k {\\textbf {v}} ) f( {\\textbf {e}} -\\alpha k {\\textbf {v}} )= e(k {\\textbf {v}} ) e(\\alpha {\\textbf {e}} ) f( {\\textbf {e}} -k\\alpha {\\textbf {v}} ).$ Thus $e( {\\textbf {v}} ) f( {\\textbf {e}} -\\alpha {\\textbf {v}} )=e(k {\\textbf {v}} ) f( {\\textbf {e}} -k\\alpha {\\textbf {v}} )$ for any $\\alpha \\in (0,1]$ , $ {\\textbf {v}} \\in \\mathcal {D}$ .", "Since $f( {\\textbf {e}} )=1$ and $f$ is continuous on $ \\Omega $ , by passing to the limit as $\\alpha \\rightarrow 0$ we get that $e$ is $K$ -invariant and so is $f$ .", "By Theorem REF and continuity of $e$ and $f$ we get that there exist constants $\\kappa _1$ , $\\kappa _2$ such that $e( {\\textbf {x}} )=(\\det {\\textbf {x}} )^{\\kappa _1}$ and $f( {\\textbf {x}} )=(\\det {\\textbf {x}} )^{\\kappa _2}$ , hence $X$ and $Y$ have Wishart distributions." ], [ "Acknowledgement", "The author thanks J. Wesołowski for helpful comments and discussions.", "This research was partially supported by NCN grant No.", "2012/05/B/ST1/00554." ] ]	1403.0236
[ [ "Testing Composite Hypothesis based on the Density Power Divergence" ], [ "Abstract In any parametric inference problem, the robustness of the procedure is a real concern.", "A procedure which retains a high degree of efficiency under the model and simultaneously provides stable inference under data contamination is preferable in any practical situation over another procedure which achieves its efficiency at the cost of robustness or vice versa.", "The density power divergence family of Basu et al.", "(1998) provides a flexible class of divergences where the adjustment between efficiency and robustness is controlled by a single parameter $\\beta$.", "In this paper we consider general tests of parametric hypotheses based on the density power divergence.", "We establish the asymptotic null distribution of the test statistic and explore its asymptotic power function.", "Numerical results illustrate the performance of the theory developed." ], [ "Introduction", "Hypothesis testing is one of the fundamental paradigms of statistical inference.", "The likelihood ratio test is a key component of the classical theory of hypothesis testing; however, this test is known to be notoriously nonrobust under model misspecification and the presence of outliers.", "Many density based minimum distance procedures have been observed to have strong robustness properties in estimation and testing together with high efficiency, eg., [22] and [2].", "Among the available robust tests in the literature, those based on the class of disparities ([26] and [20]) are known to perform well in practical situations and have many theoretical advantages.", "However the effectiveness of these procedures in continuous models is tempered by the fact that it is necessary to construct a continuous density estimate of the data generating density as an intermediate step.", "The procedure thus becomes substantially more complicated and loses a part of its appeal.", "In contrast, none of the density power divergences require any density estimation to implement their minimization routines.", "[3] considered parametric hypothesis testing based on the density power divergence for simple null hypotheses.", "In this paper we extend, in a nontrivial way, the problem for composite null hypotheses in general populations.", "To do that we have introduced the minimum density power divergence estimator restricted to a general null hypothesis, i.e.", "the restricted minimum density power divergence estimator.", "In order to derive the asymptotic distribution of the new family of test statistics proposed in this paper for testing composite null hypotheses, we need the asymptotic distribution of the restricted minimum density power divergence estimator.", "Thus the theoretical results presented in this paper require a fresh approach and represent a non-trivial generalization of the [3] paper.", "Let $\\left\\lbrace P_{\\theta }:\\theta \\in \\Theta \\right\\rbrace $ be some identifiable parametric family of probability measures on a measurable space $(\\mathcal {X}$ ,$\\mathcal {A)}$ with an open parameter space $\\Theta \\subset {\\mathbb {R}}^{p},$ $p\\ge 1.$ Measures $P_{\\theta }$ are assumed to be described by densities $f_{\\theta }=dP_{\\theta }/d\\mu $ absolutely continuous with respect to a dominating $\\sigma $ -finite measure $\\mu $ on $\\mathcal {X}$ $.$ Let $X_{1},...,X_{n}$ be a random sample from a density belonging to the family $\\left\\lbrace f_{\\theta }:\\theta \\in \\Theta \\right\\rbrace $ , where the support of the random variables is independent of the parameter $\\theta $ .", "Consider a general null hypothesis of interest which restricts the parameter to a proper subset $\\Theta _{0}$ of $\\Theta $ , i.e.", "$H_{0}:\\theta \\in \\Theta _{0}~\\text{against}~H_{1}:\\theta \\notin \\Theta _{0}.", "$ In many practical hypothesis testing problems, the restricted parameter space $\\Theta _{0}$ is defined by a set of $r<p$ restrictions of the form $g(\\theta )=0_{r} $ on $\\Theta $ , where $g:\\mathbb {R}^{p}\\rightarrow \\mathbb {R}^{r}$ is a vector-valued function such that the $p\\times r$ matrix $\\mathbf {G}\\left( \\theta \\right) =\\frac{\\partial g^{T}(\\theta )}{\\partial \\theta } $ exists and is continuous in $\\theta $ and rank$\\left( \\mathbf {G}\\left( \\theta \\right) \\right) =r$ .", "Here $0_r$ denotes the null vector of dimension $r$ , and the superscript $T$ in the above represents the transpose of the matrix.", "In general, however, there are no uniformly most powerful tests for solving the class of problems formulated in (REF ).", "The canonical approaches for problems like these include the likelihood ratio test statistic, the Wald test statistic and the Rao test statistic; see, for instance, [25].", "The tests based on disparities (or divergences), already mentioned earlier, also provide attractive theoretical alternatives for performing the above tests.", "In this paper we will solve the hypothesis testing problem presented in (REF ) using the family of density power divergences.", "Let $\\mathcal {G}$ denote the set of all distributions having densities with respect to the dominating measure.", "Given any two densities $h$ and $f$ in $\\mathcal {G}$ , the density power divergence between them is defined, as the function of a nonnegative tuning parameter $\\beta $ , as $d_{\\beta }(h,f)=\\left\\lbrace \\begin{array}{ll}\\int \\left\\lbrace f^{1+\\beta }(x)-\\left( 1+\\frac{1}{\\beta }\\right) f^{\\beta }(x)h(x)+\\frac{1}{\\beta }h^{1+\\beta }(x)\\right\\rbrace dx, & \\text{for}\\mathrm {~}\\beta >0, \\\\[2ex]\\int h(x)\\log \\left( \\displaystyle \\frac{h(x)}{f(x)}\\right) dx, & \\text{for}\\mathrm {~}\\beta =0.\\end{array}\\right.", "$ The case corresponding to $\\beta = 0$ may be derived from the general case by taking the continuous limit as $\\beta \\rightarrow 0$ , and in this case $d_0(h, f)$ is the classical Kullback-Leibler divergence.", "The quantities defined in equation (REF ) are genuine divergences in the sense $d_{\\beta }(h,f)\\ge 0$ for all $h,f\\in \\mathcal {G}$ and all $\\beta \\ge 0$ , and $d_{\\beta }(h,f)$ is equal to zero if and only if the densities $h$ and $f $ are identically equal.", "In Section we introduce the restricted minimum density power divergence estimator (RMDPDE); we also study its asymptotic distribution and its relation with the minimum density power divergence estimator (MDPDE) in this section.", "The new family of test statistics and their asymptotic distributions are presented in Section .", "In Section we describe the relation of the proposed test with the likelihood ratio test for the normal model, and in Section we have considered testing hypotheses for the Weibull model.", "Numerical results including real data examples are presented in Section .", "The problem of tuning parameter selection is taken up in Section .", "Some concluding remarks are given in Section .", "In the rest of the paper, we will frequently use the standard assumptions of asymptotic inference as given by Assumptions A, B, C and D of [19].", "We will refer to them as the Lehmann conditions.", "Some of the proofs will also require the conditions D1–D5 of [2] which we will refer to as Basu et al.", "conditions.", "In order to avoid arresting the flow of the paper, these conditions have been presented in the Appendix." ], [ "Restricted Minimum Density Power Divergence Estimator", "We consider the parametric model of densities $\\lbrace f_{\\theta }:{\\theta }\\in \\Theta \\subset {\\mathbb {R}}^{p}\\rbrace $ ; suppose that we are interested in the estimation of ${{\\theta }}$ .", "Let $H$ represent the distribution function corresponding to the density $h$ .", "The minimum density power divergence functional $T_{\\beta }(H)$ at $H$ is defined by the requirement $d_{\\beta }(h,f_{T_{\\beta }(H)})=\\min _{{{\\theta }}\\in \\Theta }d_{\\beta }(h,f_{{\\theta }})$ .", "Clearly the term $\\int h^{1+\\beta }(x)dx$ in (REF ) has no role in the minimization of $d_{\\beta }(h,f_{{\\theta }})$ over ${{\\theta }}\\in \\Theta $ .", "Thus the essential objective function to be minimized in the computation of the minimum density power divergence functional $T_{\\beta }(H)$ reduces to $\\int \\left\\lbrace f_{{\\theta }}^{1+\\beta }(x)-\\left( 1+\\frac{1}{\\beta }\\right) f_{{\\theta }}^{\\beta }(x)h(x)\\right\\rbrace dx=\\int f_{{\\theta }}^{1+\\beta }(x)dx-\\left( 1+\\frac{1}{\\beta }\\right)\\int f_{{\\theta }}^{\\beta }(x)dH(x).", "$ Notice that in the above objective function the density $h$ appears only as a linear term (unlike, say, the computation of the of the minimum Hellinger distance functional where the square root of the density $h$ is the relevant quantity).", "Thus given a random sample $X_{1},\\ldots ,X_{n}$ from the distribution $H$ we can approximate the above objective function by replacing $H$ with its empirical distribution function $H_{n}$ .", "For a given tuning parameter $\\beta $ , therefore, the MDPDE $\\widehat{\\theta }_{\\beta }$ of ${\\theta }$ can be obtained by minimizing $\\int f_{{\\theta }}^{1+\\beta }(x)dx-\\left( 1+\\frac{1}{\\beta }\\right) \\int f_{{\\theta }}^{\\beta }(x)dH_{n}(x)& =\\int f_{{\\theta }}^{1+\\beta }(x)dx-\\left( 1+\\frac{1}{\\beta }\\right)\\frac{1}{n}\\sum _{i=1}^{n}f_{{\\theta }}^{\\beta }(X_{i})\\\\& =\\frac{1}{n}\\sum _{i=1}^{n}V_{\\theta }(X_{i}) $ over ${{\\theta }}\\in \\Theta $ , where $V_{\\theta }(x)=\\int f_{{\\theta }}^{1+\\beta }(y)dy-\\left( 1+\\frac{1}{\\beta }\\right) f_{{\\theta }}^{\\beta }(x)$ .", "In the special case $\\beta =0$ , the objective function reduces to $-\\frac{1}{n}\\sum _{i=1}^{n}\\log f_{\\theta }(X_{i})$ ; the corresponding minimizer turns out to be the maximum likelihood estimator (MLE) of $\\theta $ .", "The minimization of the expression in (REF ) over ${{\\theta }}$ does not require the use of a nonparametric density estimate of the true unknown distribution $H$ .", "Existing theory (e.g.", "[7]) shows that in general there is little or no advantage in introducing smoothing for such functionals which may be empirically estimated using the empirical distribution function alone, except in very special cases.", "Using $H_n$ as a substitute for $H$ , if possible, is therefore a natural step.", "Let $u_{{\\theta }}(x)=\\frac{\\partial }{\\partial \\theta }\\log f_{{\\theta }}(x)$ be the likelihood score function of the model.", "Under differentiability of the model the minimization of the objective function in equation (REF ) leads to an estimating equation of the form $\\frac{1}{n}\\sum _{i=1}^{n}u_{{\\theta }}(X_{i})f_{{\\theta }}^{\\beta }(X_{i})-\\int u_{{\\theta }}(x)f_{{\\theta }}^{1+\\beta }(x)dx=0_{p},$ which is an unbiased estimating equation under the model.", "Since the corresponding estimating equation weights the score $u_{\\theta }(X_{i})$ with the power of the density $f_{\\theta }^{\\beta }(X_{i})$ , the outlier resistant behavior of the estimator is intuitively apparent.", "See [1] and [18] for more details.", "The functional $T_{\\beta }(H)$ is Fisher consistent; it takes the value $\\theta _{0}$ when the true density $h=f_{\\theta _{0}}$ is in the model.", "When it is not, $\\theta _{\\beta }^{h}=T_{\\beta }(H)$ represents the best fitting parameter.", "For brevity we will suppress the $h$ superscript in the notation for $\\theta _{\\beta }^{h}$ ; $f_{\\theta _\\beta }$ is the model element closest to the density $h$ in the density power divergence sense corresponding to tuning parameter $\\beta $ .", "Let $h$ be the true data generating density, and ${\\theta }_\\beta =T_{\\beta }(H)$ be the best fitting parameter.", "To set up the notation we define the quantities $J_{\\beta }(\\theta )& =\\int u_{{\\theta }}(x)u_{{\\theta }}^{T}(x)f_{{\\theta }}^{1+\\beta }(x)dx+\\int \\lbrace I_{\\theta }(x)-\\beta u_{\\theta }(x)u_{\\theta }^{T}(x)\\rbrace \\lbrace h(x)-f_{\\theta }(x)\\rbrace f_{{\\theta }}^{\\beta }(x)dx, \\\\K_{\\beta }(\\theta )& =\\int u_{{\\theta }}(x)u_{{\\theta }}^{T}(x)f_{{\\theta }}^{2\\beta }(x)h(x)dx-\\xi _{\\beta }({{\\theta }})\\xi _{\\beta }^{T}({{\\theta }}), $ where $\\xi _{\\beta }({{\\theta }})=\\int u_{\\mathbf {\\theta }}(x)f_{\\theta }^{\\beta }(x)h(x)dx$ , and $I_{\\theta }(x)=-\\frac{\\partial }{\\partial \\theta }u_{\\theta }(x)$ is the so called information function of the model.", "The following results, proved in [2], form the basis of our subsequent developments.", "Theorem 1 We assume that the Basu et al.", "conditions are true.", "Then The minimum density power divergence estimating equation (REF ) has a consistent sequence of roots $\\widehat{\\theta }_{n,\\beta }$ (denoted, hereafter, as $\\widehat{\\theta }_{\\beta }$ ), i.e.", "$\\widehat{\\theta }_{\\beta }\\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {P}}{\\longrightarrow }}\\theta _\\beta $ .", "$n^{1/2}(\\widehat{\\theta }_{\\beta }-{\\theta }_\\beta )$ has an asymptotic multivariate normal distribution with (vector) mean zero and covariance matrix $J^{-1}KJ^{-1}$ , where $J=J_{\\beta }({{\\theta }}_\\beta )$ , $K=K_{\\beta }({{\\theta }}_\\beta )$ are as in (REF ) and () respectively.", "The above result is similar, in content and spirit, to those of [32].", "When the true distribution $H$ belongs to the model so that $H=F_{\\theta }$ for some $\\theta \\in \\Theta $ , the formula for $J$ , $K$ and $\\xi $ simplify to $J& =J_{\\beta }(\\theta )=\\int u_{\\theta }(x)u_{\\theta }^{T}(x)f_{\\theta }^{1+\\beta }(x)dx, \\\\K& =K_{\\beta }(\\theta )=\\int u_{\\theta }(x)u_{\\theta }^{T}(x)f_{\\theta }^{1+2\\beta }(x)dx-\\xi \\xi ^{T}, \\\\\\xi & =\\xi _{\\beta }(\\theta )=\\int u_{\\theta }(x)f_{\\theta }^{1+\\beta }(x)dx.", "$ The restricted minimum density power divergence functional $T_{\\beta }^{0}(H) $ at $H$ , on the other hand, is the value in the parameter space which satisfies $d_{\\beta }(h,f_{T_{\\beta }^{0}(H)})=\\min _{{{\\theta }}\\in \\Theta _{0}}d_{\\beta }(h,f_{{\\theta }}),$ provided such a minimizer exists.", "When a random sample $X_{1},\\ldots ,X_{n}$ is available from the distribution $H$ , the restricted minimum density power divergence estimator of $\\theta $ minimizes (REF ) subject to $g(\\theta )=0_{r}$ .", "Under this set up we will determine, in the next theorem, the asymptotic distribution of the restricted minimum density power divergence estimator (RMDPDE) $\\widetilde{\\theta }_{\\beta }$ of $\\theta $ .", "Theorem 2 Assume that the Lehmann and Basu et al.", "conditions hold.", "Suppose that the true distribution belongs to the model, and $\\theta _0 \\in \\Theta _0$ is the true parameter.", "Then the minimum density power divergence estimator $\\widetilde{\\theta }_{\\beta }$ of $\\theta $ obtained under the constraints $g(\\theta )=0_{r}$ of the null hypothesis has the distribution $n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})\\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {L}}{\\longrightarrow }}\\mathcal {N}(0_{p},\\Sigma _{\\beta }(\\theta _{0}))$ where $\\Sigma _{\\beta }(\\theta _{0})=P_{\\beta }(\\theta _{0})K_{\\beta }(\\theta _0)P_{\\beta }(\\theta _{0}),$ $P=P_{\\beta }(\\theta _{0})=J_{\\beta }^{-1}(\\theta _{0})-Q_{\\beta }(\\theta _{0})G^{T}(\\theta _{0}) J_{\\beta }^{-1}(\\theta _{0}), $ $Q=Q_{\\beta }(\\theta _{0})=J_{\\beta }^{-1}(\\theta _{0})G(\\theta _{0})\\left[ G^{T}(\\theta _{0})J_{\\beta }^{-1}(\\theta _{0})G(\\theta _{0})\\right] ^{-1}.", "$ and $J_{\\beta }(\\theta _{0})$ is as defined in (REF ), evaluated at $\\theta = \\theta _0$ .", "Proof.", "See the Appendix." ], [ "Testing Parametric Composite Hypotheses using Density Power\nDivergence", "Suppose $\\widehat{\\theta }_{\\beta }$ is the unconstrained estimator of $\\theta $ , whereas $\\widetilde{\\theta }_{\\beta }$ is the RMDPDE under the null hypothesis given in (REF ).", "In this section we will present the family of the density power divergence test statistics (DPDTS) for testing the composite null hypothesis in (REF ).", "This family of test statistics has the expression $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })=2nd_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }}) , $ where $d_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})$ is given in (REF ).", "In the following theorem we present the asymptotic distribution of the family of DPDTS defined in (REF ).", "Theorem 3 Assume that the Lehmann and Basu et al.", "conditions hold.", "The asymptotic distribution of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ defined in (REF ) coincides, under the null hypothesis $H_{0}$ given in (REF ), with the distribution of the random variable ${\\textstyle \\sum \\limits _{i=1}^{k}}\\lambda _{i}^{\\beta ,\\gamma }(\\theta _{0})Z_{i}^{2},$ where $Z_{1},\\ldots ,Z_{k}$ are independent standard normal variables, $\\lambda _{1}^{\\beta ,\\gamma }(\\theta _{0}),\\ldots ,\\lambda _{k}^{\\beta ,\\gamma }(\\theta _{0})$ are the nonzero eigenvalues of $A_{\\gamma }\\left( \\theta _{0}\\right) B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) $ and $k = rank\\left(B_{\\beta }\\left(\\theta _{0}\\right)K_{\\beta } (\\theta _{0})B_{\\beta }\\left(\\theta _{0}\\right)A_{\\gamma }\\left(\\theta _{0}\\right)B_{\\beta }\\left(\\theta _{0}\\right)K_{\\beta } (\\theta _{0}) B_{\\beta }\\left(\\theta _{0}\\right)\\right) .", "$ The matrices $A_{\\gamma }\\left( \\theta _{0}\\right)$ and $B_{\\beta }\\left( \\theta _{0}\\right) $ are defined by $A_{\\gamma }\\left( \\theta _{0}\\right) =\\left(a_{ij}^{\\gamma }\\left( \\theta _{0}\\right) \\right)_{i,j=1,...,p}=\\left( 1+\\gamma \\right) \\left( \\int \\nolimits _{\\mathcal {X}}f_{\\theta _{0}}^{\\gamma -1}\\left( x\\right) \\frac{\\partial f_{\\theta _{0}}\\left( x\\right) }{\\partial \\theta _{j}}\\frac{\\partial f_{\\theta _{0}}\\left( x\\right) }{\\partial \\theta _{i}}dx\\right) _{i,j=1,...,p} , $ and $B_{\\beta }\\left( \\theta _{0}\\right) =J_{\\beta }^{-1}(\\theta _{0})G(\\theta _{0})\\left[ G^{T}(\\theta _{0})J_{\\beta }^{-1}(\\theta _{0})G(\\theta _{0})\\right] ^{-1}G^{T}(\\theta _{0}) J_{\\beta }^{-1}(\\theta _{0}).", "$ In the above $\\theta _{0} \\in \\Theta _0$ represents the true unknown value of $\\theta $ .", "Proof.", "See the Appendix.", "Remark 4 The main point to note in the above proof is that it is by no means a trivial or simple extension of Theorem 1 of [3].", "The proof of the latter theorem simply requires the results involving the unrestricted MDPD estimator which has been very well studied in the literature.", "In the present scenario, one has to deal with both the restricted and unrestricted MDPD estimators.", "The random nature of the second argument of the DPD makes the derivations substantially more complicated and entirely different techniques have to be applied to the proof of Theorem REF in this paper.", "The restricted MDPD estimator which we have employed here only has a limited presence in the literature.", "In some sense Theorem REF may also be considered to be a part of Theorem REF , but here we have presented them separately for pedagogical reasons as well as to keep a clear focus in our presentations.", "Remark 5 We observe that the ranks of the matrices $B_{\\beta }\\left(\\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) $ and $B_{\\beta }\\left(\\theta _{0}\\right)K_{\\beta } (\\theta _{0})B_{\\beta }\\left(\\theta _{0}\\right)A_{\\gamma }\\left(\\theta _{0}\\right)B_{\\beta }\\left(\\theta _{0}\\right)K_{\\beta } (\\theta _{0}) B_{\\beta }\\left(\\theta _{0}\\right)$ are equal.", "Moreover, it can be easily shown that $rank\\left(B_{\\beta }\\left(\\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) \\right)=rank(G(\\theta _{0}))=r$ .", "So $k=r$ , i.e.", "there will be exactly $r$ non-zero eigenvalues.", "Corollary 6 For the special case when we test the null hypothesis $H_{0}:\\mu =\\mu _{0}$ against $H_{1}:\\mu \\ne \\mu _{0}$ under the $\\mathcal {N}(\\mu ,\\sigma ^{2})$ model with $\\sigma ^{2}$ unknown, the matrix $A_{\\gamma }\\left(\\theta _{0}\\right) B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) $ has the form $A_{\\gamma }\\left( \\theta _{0}\\right) B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) =\\left(\\begin{array}{cc}\\displaystyle \\frac{1}{\\sigma ^{\\gamma }}\\frac{(\\beta +1)^{3}}{\\sqrt{\\gamma +1}(2\\beta +1)^{3/2}(2\\pi )^{\\gamma /2}} & 0 \\\\0 & 0\\end{array}\\right) ,$ so that the DPDTS $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })=2nd_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})$ has the same asymptotic distribution as that of $\\lambda _{1}\\chi ^{2}(1)$ , where $\\lambda _{1}$ is the only nonzero eigenvalue of the above matrix (equal to its $(1,1)$ th element).", "In particular when $\\gamma =0$ and $\\beta =0$ , this eigenvalue becomes one, so that the DPDTS $T_{\\gamma =0}(\\hat{\\theta }_{\\beta =0},\\tilde{\\theta }_{\\beta =0}) = 2nd_{\\gamma =0}\\left( f_{\\hat{\\theta }_{\\beta =0}},f_{\\tilde{\\theta }_{\\beta =0}}\\right)$ has a simple asymptotic $\\chi ^{2}(1)$ distribution.", "We will revisit this problem again in Section .", "A simple approach to approximate the critical region of the DPDTS and perform the test could be the following.", "The $k$ eigenvalues described in Theorem REF can be expressed as a function of the parameter $\\theta _0$ .", "Under the null they can be consistently estimated by replacing $\\widetilde{\\theta }_{\\beta }$ in place of $\\theta _0$ .", "Let $\\hat{\\lambda }_1, \\hat{\\lambda }_2, \\cdots , \\hat{\\lambda }_k$ represent the corresponding estimated eigenvalues.", "Generating independent observations $Z_1, Z_2, \\cdots , Z_k$ from the $N(0,1)$ distribution repeatedly, one can estimate the quantiles of the distribution of $\\sum _{i=1}^k \\hat{\\lambda }_i Z_i^2$ , where $\\hat{\\lambda }_i$ 's are kept fixed during this exercise.", "The quantiles are then consistent approximations of the true quantiles of the asymptotic null distribution of the statistic in Theorem REF ; the experimenter can then perform the test based on the critical values thus obtained.", "In particular when $k=1$ one can perform the test by comparing $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })/\\hat{\\lambda }_1$ with the appropriate upper quantile of $\\chi ^2(1)$ distribution.", "Tables of the cumulative distribution of ${\\sum \\limits _{i=1}^{k}}c_{i}Z_{i}^{2}$ are also available in [27], [17], [10] and [16], which may be helpful in performing the test.", "[6] has proposed an algorithm to calculate the critical region corresponding to a linear combination of $\\chi ^2$ random variables.", "Several other conservative approximations of the critical value of the DPDTS are provided in [3]." ], [ "The Power Function", "By Theorem REF the null hypothesis should be rejected if $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })\\ge c_{\\alpha }^{\\beta ,\\gamma },$ where $c_{\\alpha }^{\\beta ,\\gamma ,}$ is the quantile of order $(1-\\alpha )$ of the asymptotic distribution of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ under $H_0$ .", "The following theorem can be used to approximate the power function.", "Theorem 7 Suppose Lehmann and Basu et al.", "conditions are satisfied.", "Assume that $\\theta ^{}\\notin \\Theta _0$ is the true value of the parameter such that $\\widehat{\\theta }_{\\beta }\\overset{p}{\\underset{n\\rightarrow \\infty }{\\longrightarrow }}\\theta ^{}$ under $H_1$ .", "Suppose there exists $\\theta ^\\in \\Theta _0$ such that the RMDPDE $\\widetilde{\\theta }_{\\beta }$ of $\\theta $ satisfies $\\widetilde{\\theta }_{\\beta }\\overset{p}{\\underset{n\\rightarrow \\infty }{\\longrightarrow }}\\theta ^$ .", "Further assume that $n^{1/2}\\left( (\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })-\\left( \\theta ^{},\\theta ^\\right) \\right)^T \\overset{\\mathcal {L}}{\\underset{n\\rightarrow \\infty }{\\longrightarrow }}\\mathcal {N}\\left( \\left(\\begin{array}{l}0_{p} \\\\0_{p}\\end{array}\\right) ,\\left(\\begin{array}{ll}J_{\\beta }^{-1}(\\theta ^{})K_{\\beta }(\\theta ^{})J_{\\beta }^{-1}(\\theta ^{}) & {A}_{12}\\left(\\theta ^{}, \\theta ^\\right) \\\\{A}_{12}\\left(\\theta ^{}, \\theta ^\\right)^T & \\Sigma \\left(\\theta ^{}, \\theta ^\\right)\\end{array}\\right) \\right) ,$ where ${A}_{12}\\left(\\theta ^{},\\theta ^\\right) $ and $\\Sigma \\left( \\theta ^, \\theta ^{}\\right) $ are appropriate $p\\times p$ matrices.", "Then, under $H_1$ , we have the following convergence $n^{1/2}\\left( d_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})-d_{\\gamma }(f_{\\theta ^{}},f_{\\theta ^})\\right) \\overset{L}{\\underset{n\\rightarrow \\infty }{\\longrightarrow }}\\mathcal {N}\\left( 0,\\sigma _{\\beta ,\\gamma }^{2}\\left( \\mathbf {\\theta }^{},\\theta ^\\right) \\right),$ where $\\sigma _{\\beta ,\\gamma }^{2}\\left( \\mathbf {\\theta }^{},\\theta _0\\right) =t^{T}J_{\\beta }^{-1}(\\theta ^{})K_{\\beta }(\\theta ^{})J_{\\beta }^{-1}(\\theta ^{})t+2t^{T}A_{12}\\left(\\theta ^{}, \\theta ^\\right)s+s^{T}\\Sigma \\left(\\theta ^{}, \\theta ^\\right) s, $ and $t=\\left( \\frac{\\partial d_{\\gamma }(f_{\\theta _{1}},f_{\\theta ^})}{\\partial \\theta _{1}}\\right) _{\\theta _{1}=\\mathbf {\\theta }^{}}\\text{ and }s=\\left( \\frac{\\partial d_{\\gamma }(f_{\\mathbf {\\theta }^{}},f_{\\theta _{2}})}{\\partial \\theta _{2}}\\right) _{\\theta _{2}=\\mathbf {\\theta }^}.$ Proof.", "The result follows in a straightforward manner by considering a first order Taylor expansion of $d_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})$ , which yields $d_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})=d_{\\gamma }(f_{\\theta ^{}},f_{\\theta ^})+t^{T}(\\widehat{\\theta }_{\\beta }-\\theta ^{})+s^{T}(\\widetilde{\\theta }_{\\beta }-\\theta ^)+o\\left(\\left\\Vert \\widehat{\\theta }_{\\beta }-\\theta ^{}\\right\\Vert +\\left\\Vert \\widetilde{\\theta }_{\\beta }-\\theta ^\\right\\Vert \\right) .$ Remark 8 On the basis of the previous theorem we get an approximation of the power function as $\\pi _{n,\\alpha }^{\\beta ,\\gamma }\\left( \\mathbf {\\theta }^{}\\right) &=&\\text{P}_{\\mathbf {\\theta }^{}}\\left( T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })\\ge c_{\\alpha }^{\\beta ,\\gamma }\\right) \\nonumber \\\\&=&1-\\Phi \\left( \\frac{n^{1/2}}{\\sigma _{\\beta ,\\gamma }\\left(\\mathbf {\\theta }^{},\\theta ^\\right) }\\left( \\frac{c_{\\alpha }^{\\beta ,\\gamma }}{2n}-d_{\\gamma }(f_{\\theta ^{}},f_{\\theta ^})\\right) \\right) , $ where $\\Phi (\\cdot )$ is the standard normal distribution function, $c_{\\alpha }^{\\beta ,\\gamma }$ is the quantile of order $1-\\alpha $ of the asymptotic distribution of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ under the null hypothesis, and $\\sigma _{\\beta ,\\gamma }^{2}\\left( \\mathbf {\\theta }^{},\\theta ^\\right) $ is as defined in (REF ).", "If some $\\mathbf {\\theta }^{}$ $\\ne \\theta ^$ is the true parameter, then the probability of rejecting $H_0$ for a fixed size $\\alpha $ tends to one as $n\\rightarrow \\infty $ .", "So the test statistic is consistent in the [12]'s ([12]) sense.", "Obtaining the approximate sample size $n$ to guarantee a power of $\\pi $ at a given alternative $\\mathbf {\\theta }^{}$ is an interesting application of formula (REF ).", "Let $n^{}$ be the positive root of the equation $\\pi =1-\\Phi \\left( \\frac{n^{1/2}}{\\sigma _{\\beta ,\\gamma }\\left( \\mathbf {\\theta }^{},\\theta ^\\right) }\\left( \\frac{c_{\\alpha }^{\\beta ,\\gamma }}{2n}-d_{\\gamma }(f_{\\theta ^{}},f_{\\theta ^})\\right) \\right),$ i.e.", "$n^{}=\\frac{A+B+\\sqrt{A(A+2B)}}{2d_{\\gamma }(f_{\\theta ^{}},f_{\\theta ^})^{2}},$ where $A=\\sigma _{\\beta ,\\gamma }^{2}\\left( \\mathbf {\\theta }^{},\\theta ^\\right) \\left( \\Phi ^{-1}\\left( 1-\\pi \\right)\\right) ^{2},$ and $B=c_{\\alpha }^{\\beta ,\\gamma }d_{\\gamma }(f_{\\theta ^{}},f_{\\theta ^}).$ Then the required sample size is $n=\\left[n^{}\\right] +1,$ where $\\left[ \\cdot \\right] $ is used to denote “integer part of”.", "We may also find an alternative approximation of the power of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ at an alternative close to the null hypothesis.", "Let $\\theta _{n}\\in \\Theta -\\Theta _{0}$ be a given sequence of alternatives, and let $\\theta _{0}$ be the element in $\\Theta _{0}$ closest to $\\theta _{n}$ in the Euclidean distance sense.", "One possibility to introduce contiguous alternative hypotheses is to consider a fixed $d\\in \\mathbb {R}^{p}$ and to permit $\\theta _{n}$ to move towards $\\theta _{0}$ as $n$ increases in the manner specified by the hypothesis $H_{1,n}:\\theta _{n}=\\theta _{0}+n^{-1/2}d. $ Theorem 9 Suppose that the model satisfies the Lehmann and Basu et al.", "conditions.", "Under the contiguous alternative hypotheses $H_{1,n}$ given in (REF ), the asymptotic distribution of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ coincides with the distribution of ${\\sum \\limits _{i=1}^{k}}\\lambda _{i}^{\\beta ,\\gamma }(\\theta _{0})\\left( Z_{i}+w_{i}\\right) ^{2}+\\eta ,$ where $Z_{1},\\ldots ,Z_{r}$ are independent standard normal variables, $\\lambda _{1}^{\\beta ,\\gamma }(\\theta _{0}),\\ldots ,\\lambda _{k}^{\\beta ,\\gamma }(\\theta _{0})$ are the positive eigenvalues of $A_{\\gamma }\\left( \\theta _{0}\\right) B_{\\beta }\\left( \\theta _{0}\\right)K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) $ , and the values of $w=\\left( w_{1},\\ldots ,w_{k}\\right) ^{T}$ and $\\eta $ are given by $w=\\Lambda _{k}^{-1}V^{T}S^{T}A_{\\gamma }(\\theta _{0})B(\\theta _{0})J_{\\beta }(\\theta _{0})d,~~\\eta =\\left( B(\\theta _{0})Jd\\right) ^{T}A_{\\gamma }(\\theta _{0})B(\\theta _{0})J_{\\beta }(\\theta _{0})d-w^{T}\\Lambda _{k}w.$ Also $S$ is any square root of $B_{\\beta }\\left(\\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) $ , $\\Lambda _{k}=diag(\\lambda _{1}^{\\beta ,\\gamma }(\\theta _{0}),\\ldots ,\\lambda _{k}^{\\beta ,\\gamma }(\\theta _{0}))$ and $V$ is the matrix of corresponding orthonormal eigenvectors.", "Proof.", "See the Appendix.", "From a practical point of view we will estimate the eigenvalues as well as $w$ and $\\eta $ by their consistent estimators." ], [ "Normal Case: Connection with the Likelihood Ratio Test", "Under the $\\mathcal {N}(\\mu ,\\sigma ^{2})$ model, consider the problem of testing $H_{0}:\\mu =\\mu _{0}\\text{ versus }H_{1}:\\mu \\ne \\mu _{0} , $ where $\\sigma $ is an unknown nuisance parameter.", "In this case the unrestricted and null parameter spaces are given by $\\Theta =\\lbrace (\\mu ,\\sigma )^T\\in {\\mathbb {R}}^{2}\|\\mu \\in {\\mathbb {R}},\\sigma \\in {\\mathbb {R}}^{+}\\rbrace $ and $\\Theta _{0}=\\lbrace (\\mu ,\\sigma )^T\\in {\\mathbb {R}}^{2}\|\\mu =\\mu _{0},\\sigma \\in {\\mathbb {R}}^{+}\\rbrace $ respectively.", "If we consider the function $g(\\theta )=\\mu -\\mu _{0},$ with $\\theta =\\left( \\mu ,\\sigma \\right) ^{T}$ , the null hypothesis $H_{0}$ can be written as $H_{0}:g(\\theta )=0,$ and we are in the situation considered in (REF ).", "We can observe that in our case $G\\left( \\theta \\right) =\\left( 1,0\\right) ^{T}.$ Based on (REF ) and taking into account the fact that $f_{\\theta }(x)$ is the normal density with mean $\\mu $ and variance $\\sigma ^{2}$ , the estimator $\\widehat{\\theta }_{\\beta }=(\\widehat{\\mu }_{\\beta },\\widehat{\\sigma }_{\\beta })^{T}$ of $\\theta =(\\mu ,\\sigma )^{T}$ is given by $(\\widehat{\\mu }_{\\beta },\\widehat{\\sigma }_{\\beta })^{T} =\\arg \\min _{(\\mu ,\\sigma )^{T} \\in \\mathbb {R} \\times \\mathbb {R}^+}\\frac{1}{\\sigma ^{\\beta }\\left( 2\\pi \\right)^{\\frac{\\beta }{2}}}\\left( \\frac{1}{\\left( 1+\\beta \\right) ^{3/2}} -\\frac{1}{n\\beta }\\sum _{i=1}^{n}\\exp \\left\\lbrace -\\frac{\\beta }{2}\\left( \\frac{X_{i}-\\mu }{{\\sigma }}\\right) ^{2}\\right\\rbrace \\right) ,$ where $\\beta >0$ .", "Similarly, the estimator $\\widetilde{\\theta }_{\\beta }=\\left(\\mu _{0},\\widetilde{\\sigma }_{\\beta }\\right) ^{T}$ , when $\\mu =\\mu _{0}$ , will be obtained from $\\widetilde{\\sigma }_{\\beta } = \\arg \\min _{\\sigma \\in \\mathbb {R}^+}\\frac{1}{\\sigma ^{\\beta }\\left( 2\\pi \\right) ^{\\frac{\\beta }{2}}}\\left( \\frac{1}{\\left( 1+\\beta \\right) ^{3/2}} -\\frac{1}{n\\beta }\\sum _{i=1}^{n}\\exp \\left\\lbrace -\\frac{1}{2}\\beta \\left( \\frac{X_{i}-\\mu _{0}}{\\sigma }\\right)^{2}\\right\\rbrace \\right).$ Simple calculations yield the expressions $J_{\\beta }(\\theta )=\\frac{1}{\\sqrt{1+\\beta }\\left( 2\\pi \\right) ^{\\beta /2}\\sigma ^{2+\\beta }}\\left(\\begin{array}{cc}\\frac{1}{1+\\beta } & 0 \\\\0 & \\frac{\\beta ^{2}+2}{\\left( 1+\\beta \\right) ^{2}}\\end{array}\\right),$ and $K_{\\beta }(\\theta )=\\frac{1}{\\sigma ^{2+2\\beta }\\left( 2\\pi \\right) ^{\\beta }}\\left( \\frac{1}{(1+2\\beta )^{3/2}}\\left(\\begin{array}{cc}1 & 0 \\\\0 & \\frac{4\\beta ^{2}+2}{1+2\\beta }\\end{array}\\right) -\\left(\\begin{array}{cc}0 & 0 \\\\0 & \\frac{\\beta ^{2}}{(1+\\beta )^{3}}\\end{array}\\right) \\right) .$ Based on these matrices we get $B_{\\beta }\\left( \\theta \\right) =\\left(\\begin{array}{cc}\\sigma ^{\\beta +2}\\left( \\beta +1\\right) ^{\\frac{3}{2}}\\left( 2\\pi \\right)^{\\beta /2} & 0 \\\\0 & 0\\end{array}\\right) .$ On the other hand $A_{\\gamma }\\left( \\theta \\right) =\\frac{1}{\\left( 2\\pi \\right)^{\\gamma /2}\\sigma ^{2+\\gamma }(1+\\gamma )^{1/2}}\\left(\\begin{array}{cc}1 & 0 \\\\0 & \\frac{\\gamma ^{2}+2}{\\left( 1+\\gamma \\right) }\\end{array}\\right),$ and $A_{\\gamma }\\left( \\theta \\right) B_{\\beta }\\left(\\theta \\right) K_{\\beta }(\\theta )B_{\\beta }\\left( \\theta \\right) =\\left(\\begin{array}{cc}\\frac{1}{\\sigma ^{\\gamma }}\\frac{\\left( \\beta +1\\right) ^{3}}{\\sqrt{\\gamma +1}\\left( 2\\beta +1\\right) ^{\\frac{3}{2}}}\\frac{1}{\\left( 2\\pi \\right) ^{\\frac{\\gamma }{2}}} & 0 \\\\0 & 0\\end{array}\\right) , $ which is identical to the matrix presented in Corollary REF .", "In order to apply the results of Theorem REF in this connection, we need to get the expression of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ .", "As in [3] we have $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })&=2nd_\\gamma (f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }}) \\\\& =\\frac{2n}{\\widetilde{\\sigma }_{\\beta }^{\\gamma }\\sqrt{1+\\gamma }\\left(2\\pi \\right) ^{\\gamma /2}}-\\left( 1+\\frac{1}{\\gamma }\\right) \\frac{1}{\\widetilde{\\sigma }^{\\gamma -1}\\left( \\gamma \\widehat{\\sigma }_{\\beta }^{2}+\\widetilde{\\sigma }_{\\beta }^{2}\\right) ^{1/2}\\left( 2\\pi \\right) ^{{\\gamma /2}}}\\exp \\left( -\\frac{1}{2}\\frac{\\mu _{0}^{2}}{\\left( \\frac{\\widehat{\\sigma }_{\\beta }}{\\sqrt{\\gamma }}\\right) ^{2}}+\\frac{\\widehat{\\mu }_{\\beta }^{2}}{\\widehat{\\sigma }_{\\beta }^{2}}\\right) \\\\& \\times \\exp \\left( \\frac{1}{2}\\frac{\\left( \\widehat{\\sigma }_{\\beta }^{2}\\mu _{0}+\\widehat{\\mu }_{\\beta }\\left( \\frac{\\widetilde{\\sigma }_{\\beta }}{\\sqrt{\\gamma }}\\right) ^{2}\\right) ^{2}}{\\left( \\widehat{\\sigma }_{\\beta }^{2}+\\left( \\frac{\\widehat{\\sigma }_{\\beta }}{\\sqrt{\\gamma }}\\right)^{2}\\right) \\left( \\frac{\\widetilde{\\sigma }_{\\beta }}{\\sqrt{\\gamma }}\\right) ^{2}\\widehat{\\sigma }_{\\beta }^{2}}\\right) +\\frac{1}{\\gamma \\widehat{\\sigma }_{\\beta }^{\\gamma }\\sqrt{1+\\gamma }\\left( 2\\pi \\right) ^{\\gamma /2}}.$ Using Corollary REF and the single nonzero eigenvalue of the matrix given in (REF ), we then get $\\frac{\\widetilde{\\sigma }_{\\beta } ^{\\gamma } \\sqrt{\\gamma +1}\\left( 2\\beta +1\\right) ^{3/2} \\left( 2\\pi \\right) ^{\\gamma /2}}{\\left( \\beta +1\\right) ^{3}}T_{\\gamma }\\left( \\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta }\\right) \\underset{n\\rightarrow \\infty }{\\overset{L}{\\longrightarrow }}\\chi ^{2}(1).", "$ A special case of interest is the situation where $\\beta =0$ and $\\gamma =0.$ The likelihood ratio test for the problem under study is equivalent to the ordinary $t$ -test and one can determine the exact small sample critical values for this test.", "On the other hand the standard asymptotic formulation of the likelihood ratio test leads to the rejection of the null hypothesis when $-2\\log \\lambda (X_{1},X_{2},\\ldots ,X_{n})>\\chi _\\alpha ^{2}(1)$ , where $\\lambda (X_{1},X_{2},\\ldots ,X_{n})=\\frac{\\sup _{\\theta \\in \\Theta _{0}}f_{\\theta }(X_{1},X_{2},\\ldots ,X_{n})}{\\sup _{\\theta \\in \\Theta }f_{\\theta }(X_{1},X_{2},\\ldots ,X_{n})}$ is the likelihood ratio, and $\\chi _\\alpha ^{2}(1)$ is the quantile of order $(1 - \\alpha )$ for the $\\chi ^2(1)$ distribution.", "The MLE of $\\theta $ under the parameter space $\\Theta $ is $\\widehat{\\theta }_{n}=\\left( \\bar{X},\\hat{\\sigma }_{n}^{2}=\\frac{1}{n}\\sum _{i=1}^{n}{(X_{i}-\\bar{X})^{2}}\\right) ^{T},$ while the MLE under $\\Theta _{0}$ is $\\widetilde{\\theta }_{n}=\\left( \\mu _{0},\\tilde{\\sigma }_{n}^{2}=\\frac{1}{n}\\sum _{i=1}^{n}{(X_{i}-\\mu _{0})^{2}}\\right) ^{T}.$ Straightforward calculations show that asymptotically we reject the null hypothesis when $-2\\log \\lambda (X_{1},X_{2},\\ldots ,X_{n})=n\\log \\left( \\frac{{\\tilde{\\sigma }_{n}}^{2}}{{\\hat{\\sigma }_{n}}^{2}}\\right) >\\chi _\\alpha ^{2}(1).$ This test may be looked upon as the asymptotic likelihood ratio test, as opposed to the usual $t$ -test which may be regarded as the exact version of the likelihood ratio test for the normal mean problem with unknown variance.", "What is the relation of the test statistic $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ given in (REF ) with the above test statistics?", "In the following we will demonstrate that for $\\gamma =0$ and $\\beta =0$ , our test statistic coincides with the asymptotic likelihood ratio test described in (REF ).", "Note that the density power divergence for the case $\\gamma =0$ between the densities of two normal distributions with different means and variances is given by $d_{\\gamma =0}(f_{{\\theta }_1},f_{{\\theta }_2})=\\log {\\frac{\\sigma _{2}}{\\sigma _{1}}}-\\frac{1}{2}+\\frac{1}{2}\\frac{\\sigma _{1}^{2}}{\\sigma _{2}^{2}}+\\frac{1}{2\\sigma _{2}^{2}}(\\mu _{1}-\\mu _{2})^{2}.$ Therefore for $\\gamma =0$ and $\\beta =0$ , we get $T_{\\gamma =0}(\\hat{\\theta }_{\\beta =0},\\tilde{\\theta }_{\\beta =0})=n\\left( \\log {\\frac{\\tilde{\\sigma }_{n}^{2}}{\\hat{\\sigma }_{n}^{2}}}-1+\\frac{\\hat{\\sigma }_{n}^{2}}{\\tilde{\\sigma }_{n}^{2}}+\\frac{(\\bar{X}-\\mu _{0})^{2}}{\\tilde{\\sigma }_{n}^{2}}\\right) .$ A routine calculation shows that $\\frac{\\hat{\\sigma }_{n}^{2}}{\\tilde{\\sigma }_{n}^{2}}+\\frac{(\\bar{X}-\\mu _{0})^{2}}{\\tilde{\\sigma }_{n}^{2}}=1,$ so that $T_{\\gamma =0}(\\hat{\\theta }_{\\beta =0},\\tilde{\\theta }_{\\beta =0})=n\\log \\left( {\\frac{\\tilde{\\sigma }_{n}^{2}}{\\hat{\\sigma }_{n}^{2}}}\\right) , $ and by equations (REF ) and (REF ), the asymptotic likelihood ratio test statistic is exactly same as the DPDTS for $\\gamma =0$ and $\\beta =0$ .", "Therefore when we are comparing the usual $t$ -test with the test statistic $T_{\\gamma =0}(\\widehat{\\theta }_{\\beta =0},\\widetilde{\\theta }_{\\beta =0}),$ we are comparing an exact likelihood ratio test with an asymptotic likelihood ratio test." ], [ "Testing for the Weibull Distribution", "While the normal model is the most important model where our methods are useful, it is also important to explore the applicability of the method in other models to demonstrate the general nature of the method.", "For this purpose we will include numerical results based on the Weibull distribution in our subsequent numerical study, together with the results on the normal model.", "Here we describe the statistic for the Weibull case.", "The probability density function of $\\mathcal {W}(\\sigma ,p)$ , a two parameter Weibull distribution, is given by $f_{\\theta }(x) = \\frac{p}{\\sigma } \\left(\\frac{x}{\\sigma }\\right)^{p-1} \\exp \\left\\lbrace -\\left(\\frac{x}{\\sigma }\\right)^p\\right\\rbrace , \\ x>0,$ where $\\theta =(\\sigma , p)^{T}$ , and the parameter space is given by $\\Theta =\\lbrace (\\sigma , p)\|\\sigma \\in {\\mathbb {R}}^{+}, p \\in {\\mathbb {R}}^{+}\\rbrace $ .", "We are interested in testing $H_{0}:\\sigma =\\sigma _{0}\\text{ versus }H_{1}:\\sigma \\ne \\sigma _{0} , $ where $p$ is a nuisance parameter.", "Let us consider the function $g(\\theta )=\\sigma -\\sigma _{0}$ .", "Then, as in the normal case which was considered in Section , the null hypothesis $H_{0}$ can be written as $H_{0}:g(\\theta )=0,$ and $G\\left( \\theta \\right) =\\left( 1,0\\right) ^{T}.$ Let us define $\\xi _{\\alpha , \\beta }( \\theta ) = \\int _0^\\infty \\left(\\frac{x}{\\sigma }\\right)^{\\alpha } f_{\\theta }^\\beta (x) dx ,$ and $\\eta _{\\alpha , \\beta , \\gamma }( \\theta ) = \\int _0^\\infty \\left(\\frac{x}{\\sigma }\\right)^{\\alpha } \\left[ \\log \\left(\\frac{x}{\\sigma }\\right)\\right]^{\\beta } f_{\\theta }^\\gamma (x) dx .$ It can be shown that $\\xi _{\\alpha , \\beta }( \\theta )=\\left(\\frac{p}{\\sigma }\\right)^{\\beta -1} \\beta ^{-\\frac{\\beta p - \\beta + \\alpha +1}{p}}\\Gamma \\left(\\frac{\\beta p - \\beta + \\alpha +1}{p}\\right) ,$ and $\\eta _{\\alpha , \\beta , \\gamma }( \\theta ) = \\sigma \\left(\\frac{p}{\\sigma }\\right)^\\gamma \\int _0^\\infty y^{\\alpha + \\gamma p - \\gamma }( \\log y)^\\beta \\exp ( -\\gamma y^p) dy ,$ where $\\Gamma (\\cdot )$ denotes the gamma function.", "Note that $\\xi _{\\alpha , \\gamma }( \\theta ) = \\eta _{\\alpha , 0, \\gamma }( \\theta )$ .", "For $\\beta \\ne 0$ the value of $\\eta _{\\alpha , \\beta , \\gamma }( \\theta )$ is calculated using numerical integration.", "Let us define $R_\\gamma ( \\theta ) = \\int _0^\\infty u_{\\theta }(x) u_{\\theta }^T(x) f_{\\theta }^\\gamma (x) dx = \\left(\\begin{array}{cc}r_{11} & r_{12} \\\\ r_{12} & r_{21}\\end{array}\\right),$ where $u_{\\theta }(x)$ , the score function of the Weibull distribution, is given by $u_{\\theta }(x) = \\frac{\\partial \\log f_{\\theta }(x)}{\\partial \\theta }= \\left(\\begin{array}{c}-\\frac{p}{\\sigma } + \\frac{p}{\\sigma }\\left(\\frac{x}{\\sigma }\\right)^{p} \\\\\\frac{1}{p} + \\log \\left(\\frac{x}{\\sigma }\\right) - \\left(\\frac{x}{\\sigma }\\right)^p \\log \\left(\\frac{x}{\\sigma }\\right)\\end{array}\\right).$ Then it can be shown that $r_{11} = \\left(\\frac{p}{\\sigma }\\right)^2 \\left\\lbrace \\xi _{0, \\gamma }( \\theta )- 2 \\xi _{p, \\gamma }( \\theta ) + \\xi _{2p, \\gamma }( \\theta ) \\right\\rbrace ,$ $r_{12} = \\frac{p}{\\sigma } \\left\\lbrace -\\frac{1}{p} \\xi _{0, \\gamma }( \\theta ) - \\eta _{0, 1, \\gamma }( \\theta ) + 2 \\eta _{p, 1, \\gamma }( \\theta )+ \\frac{1}{p} \\xi _{p, \\gamma }( \\theta ) - \\eta _{2p, 1, \\gamma }( \\theta ) \\right\\rbrace ,$ and $r_{22} &=& \\frac{1}{p^2} \\xi _{0, \\gamma }( \\theta ) + \\eta _{0, 2, \\gamma }( \\theta ) + \\eta _{2p, 2, \\gamma }( \\theta ) + \\frac{2}{p} \\eta _{0, 1, \\gamma }( \\theta ) - 2 \\eta _{p, 2, \\gamma }( \\theta ) - \\frac{2}{p} \\eta _{p, 1, \\gamma }( \\theta ) .$ Now $J_\\gamma ( \\theta ) = \\int _0^\\infty u_{\\theta }(x) u_{\\theta }^T(x) f_{\\theta }^{1+\\gamma }(x) dx = R_{1+\\gamma }( \\theta ),$ $K_\\gamma ( \\theta ) = \\int _0^\\infty u_{\\theta }(x) u_{\\theta }^T(x) f_{\\theta }^{1+2\\gamma }(x) dx = R_{1+2\\gamma }( \\theta ),$ and $A_\\gamma ( \\theta ) = (1+\\gamma ) \\int _0^\\infty u_{\\theta }(x) u_{\\theta }^T(x) f_{\\theta }^{1+\\gamma }(x) dx = (1+\\gamma ) R_{1+\\gamma }( \\theta ).$ Suppose we have two densities $f_{\\theta _1}$ and $f_{\\theta _2}$ from Weibull family, where $\\theta _1 =(\\sigma _1, p_1)^T$ and $\\theta _2 =(\\sigma _2, p_2)^T$ .", "If $\\gamma >0$ , then using (REF ) we get from equation (REF ) $d_{\\gamma }(f_{\\theta _1},f_{\\theta _2}) = \\xi _{0, 1+\\gamma }(\\theta _2) - \\left( 1+\\frac{1}{\\gamma }\\right) \\psi _\\gamma (\\theta _1, \\theta _2) + \\frac{1}{\\gamma } \\xi _{0, 1+\\gamma }(\\theta _1),$ where $\\psi _\\gamma (\\theta _1, \\theta _2) = \\int f_{\\theta _2}^{\\gamma }(x)f_{\\theta _1}(x) dx .$ The value of $\\psi _\\gamma (\\theta _1, \\theta _2)$ can also be calculated using numerical integration.", "For $\\gamma =0$ it can be shown that $d_{\\gamma =0}(f_{\\theta _1},& f_{\\theta _2}) = \\log p_1 - \\log \\sigma _1 + (p_1-1)\\eta _{0, 1,1}(\\theta _1) - \\xi _{p_1,1}(\\theta _1)\\nonumber \\\\&+ \\log p_2 - \\log \\sigma _2 +(p_2-1) \\log \\left(\\frac{\\sigma _1}{\\sigma _2}\\right) + (p_2-1)\\eta _{0, 1,1}(\\theta _1) - \\left(\\frac{\\sigma _1}{\\sigma _2}\\right)^{p_2} \\xi _{p_2,1}(\\theta _1).$ Using equations (REF )-(REF ) we calculate the test statistic as well as its asymptotic distribution.", "Suppose $\\widehat{\\sigma }_{\\beta }$ and $\\widehat{p }_{\\beta }$ are the unconstrained estimators of $\\sigma $ and $p$ respectively, and $\\widetilde{p }_{\\beta }$ is the RMDPDE of $p$ under the null hypothesis.", "For $\\gamma >0$ , the test statistic can be simplified as $& T_{\\gamma }(\\widehat{\\sigma }_{\\beta },\\widehat{p }_{\\beta },\\sigma _{0},\\widetilde{p }_{\\beta }) \\\\& =\\frac{2n\\widetilde{r}_{\\gamma +1}^{(2,2)}(\\widetilde{p }_{\\beta })\\left[\\widetilde{r}_{\\gamma +1}^{(1,1)}(\\widetilde{p }_{\\beta })\\widetilde{r}_{\\gamma +1}^{(2,2)}(\\widetilde{p }_{\\beta })-\\left( \\widetilde{r}_{\\gamma +1}^{(1,2)}(\\widetilde{p }_{\\beta })\\right) ^{2}\\right] }{(1+\\gamma )\\begin{pmatrix}-\\widetilde{r}_{\\gamma +1}^{(2,2)}(\\widetilde{p }_{\\beta }) & \\widetilde{r}_{\\gamma +1}^{(1,2)}(\\widetilde{p }_{\\beta })\\end{pmatrix}\\begin{pmatrix}\\widetilde{r}_{2\\gamma +1}^{(1,1)}(\\widetilde{p }_{\\beta }) & \\widetilde{r}_{2\\gamma +1}^{(1,2)}(\\widetilde{p }_{\\beta }) \\\\\\widetilde{r}_{2\\gamma +1}^{(1,2)}(\\widetilde{p }_{\\beta }) & \\widetilde{r}_{2\\gamma +1}^{(2,2)}(\\widetilde{p }_{\\beta })\\end{pmatrix}\\begin{pmatrix}-\\widetilde{r}_{\\gamma +1}^{(2,2)}(\\widetilde{p }_{\\beta }) \\\\\\widetilde{r}_{\\gamma +1}^{(1,2)}(\\widetilde{p }_{\\beta })\\end{pmatrix}} \\\\& \\times \\left\\lbrace \\frac{1}{\\gamma }\\left( \\frac{\\widehat{p }_{\\beta }\\sigma _{0}}{\\widetilde{p }_{\\beta }\\widehat{\\sigma }_{\\beta }}\\right) ^{\\gamma }\\varepsilon _{0,\\gamma +1}(\\widehat{p }_{\\beta })+\\varepsilon _{0,\\gamma +1}(\\widetilde{p }_{\\beta })-\\frac{\\gamma +1}{\\gamma }\\frac{1}{\\sigma _{0}^{\\gamma (\\widetilde{p }_{\\beta }-1)}}\\frac{\\widehat{p }_{\\beta }}{\\widehat{\\sigma }_{\\beta }^{\\widehat{p }_{\\beta }}}\\overline{I}_{\\gamma }(\\widehat{\\sigma }_{\\beta },\\widehat{p }_{\\beta },\\sigma _{0},\\widehat{p}_{\\beta })\\right\\rbrace , $ where $\\widetilde{r}_{\\gamma }^{(1,1)}(\\widetilde{p }_{\\beta })=\\varepsilon _{0,\\gamma }(\\widetilde{p }_{\\beta })-2\\varepsilon _{\\widetilde{p }_{\\beta },\\gamma }(\\widetilde{p }_{\\beta })+\\varepsilon _{2\\widetilde{p }_{\\beta },\\gamma }(\\widehat{p}_{\\beta }), $ $\\widetilde{r}_{\\gamma }^{(1,2)}(\\widetilde{p }_{\\beta })=-\\frac{1}{\\widehat{p}_{\\beta }}\\varepsilon _{0,\\gamma }(\\widetilde{p }_{\\beta })+\\left( \\log \\widetilde{p }_{\\beta }+\\frac{1}{\\widetilde{p }_{\\beta }}\\right) \\varepsilon _{\\widetilde{p }_{\\beta },\\gamma }(\\widetilde{p }_{\\beta })-\\log \\widehat{p}_{\\beta }\\varepsilon _{2\\widetilde{p }_{\\beta },\\gamma }(\\widetilde{p }_{\\beta })-\\kappa _{0,1,\\gamma }(\\widetilde{p }_{\\beta })+\\kappa _{\\widetilde{p }_{\\beta },1,\\gamma }(\\widetilde{p }_{\\beta }) , $ $\\widetilde{r}_{\\gamma }^{(2,2)}(\\widetilde{p }_{\\beta })& =\\frac{1}{\\widehat{p}_{\\beta }^{2}}\\varepsilon _{0,\\gamma }(\\widetilde{p }_{\\beta })-\\frac{2}{\\widetilde{p }_{\\beta }}\\log \\widetilde{p }_{\\beta }\\varepsilon _{\\widehat{p}_{\\beta },\\gamma }(\\widetilde{p }_{\\beta })+(\\log \\widehat{p}_{\\beta })^{2}\\varepsilon _{2\\widetilde{p }_{\\beta },\\gamma }(\\widetilde{p }_{\\beta })\\\\& +\\frac{2}{\\widetilde{p }_{\\beta }}\\kappa _{0,1,\\gamma }(\\widehat{p}_{\\beta })+\\kappa _{0,2,\\gamma }(\\widetilde{p }_{\\beta })-2\\log \\widehat{p}_{\\beta }\\kappa _{\\widetilde{p }_{\\beta },1,\\gamma }(\\widetilde{p }_{\\beta }) ,$ $\\overline{I}_{\\gamma }(\\widehat{\\sigma }_{\\beta },\\widetilde{p}_{\\beta },\\sigma _{0},\\widetilde{p }_{\\beta })=\\int _{0}^{\\infty }x^{\\gamma (\\widehat{p}_{\\beta }-1)+\\widehat{p }_{\\beta }-1}\\exp \\left\\lbrace -\\gamma \\left( \\frac{x}{\\sigma _{0}}\\right) ^{\\widetilde{p }_{\\beta }}-\\left( \\frac{x}{\\widetilde{\\sigma }_{\\beta }}\\right) ^{\\widehat{p }_{\\beta }}\\right\\rbrace dx ,$ $\\xi _{\\alpha ,\\gamma }(\\sigma ,p)& =\\left( \\frac{p}{\\sigma }\\right) ^{\\gamma -1}\\varepsilon _{\\alpha ,\\gamma }(p), \\\\\\varepsilon _{\\alpha ,\\gamma }(p)& =\\gamma ^{\\frac{(p-1)\\gamma +\\alpha +1}{p}}\\Gamma \\left( \\frac{(p-1)\\gamma +\\alpha +1}{p}\\right) ,$ $\\eta _{\\alpha ,\\delta ,\\gamma }(\\sigma _{0},\\widetilde{p }_{\\beta })& =\\widehat{p}_{\\beta }\\left( \\frac{\\widetilde{p }_{\\beta }}{\\sigma _{0}}\\right) ^{\\gamma -1}\\int _{0}^{\\infty }(\\log y)^{\\delta }y^{(\\widetilde{p }_{\\beta }-1)\\gamma +\\alpha }\\exp \\lbrace -\\gamma y^{\\widetilde{p }_{\\beta }}\\rbrace dy \\\\& =\\left( \\frac{\\widetilde{p }_{\\beta }}{\\sigma _{0}}\\right) ^{\\gamma -1}\\kappa _{\\alpha ,\\delta ,\\gamma }(\\widetilde{p }_{\\beta }), \\\\\\kappa _{\\alpha ,\\delta ,\\gamma }(\\widetilde{p }_{\\beta })& =\\widehat{p}_{\\beta }\\int _{0}^{\\infty }(\\log y)^{\\delta }y^{(\\widetilde{p }_{\\beta }-1)\\gamma +\\alpha }\\exp \\lbrace -\\gamma y^{\\widetilde{p }_{\\beta }}\\rbrace dy .$" ], [ "Numerical Studies", "In this section we provide some extensive numerical evidence of the performance of the proposed methods, demonstrating, in particular, their strong robustness properties.", "Notice that the test statistic depends on the data only through the value of the estimator (both unconstrained and constrained), so that the robustness of the test would appear to depend directly on the robustness of the estimator.", "However, it is still useful to develop actual theoretical robustness properties of the proposed tests.", "Fortunately there is a wealth of material available in this context which makes our work easy.", "[28] and [29] have, in general, touched upon the issue of theoretical robustness properties of tests.", "They have considered several theoretical measures of robustness in this context.", "In a more limited, but a more focused setting [15] have considered the robustness measures of test statistics based on the family of $S$ -divergences which include the DPD as a special case; in particular the influence functions of the tests and the so called level and power influence functions are derived.", "Taken together, the above references further reinforce the notion that the robustness of these tests are directly dependent on the robustness of the estimators as the influence function of the tests turn out to be directly related to the influence function of the estimators.", "The [15] paper relates only to the case of the simple null hypothesis; however it is not difficult to intuitively see how the robustness of these tests extend to the case of the composite hypothesis.", "The theoretical robustness properties of some similar tests have been considered in the Ph.D. dissertation of [13].", "On the whole, there is substantial overall indication and evidence of the theoretical robustness properties of the tests under study.", "For the sake of brevity we do not repeat these results here, but concentrate instead on the performance of the tests as observed in simulations and actual real data examples." ], [ "Telephone-Fault Data", "We consider the data on telephone line faults presented and analyzed by [31]; the data were also analyzed by [26].", "The data are given in Table REF , and consist of the ordered differences between the inverse test rates and the inverse control rates in 14 matched pairs of areas.", "A parametric approach to analyze this would be to model these data as a random sample from a normal distribution with mean $\\mu $ and standard deviation $\\sigma $ .", "It is obvious that the first observation of this dataset is a huge outlier with respect to the normal model, while the remaining 13 observations appear to be reasonable with respect to the same.", "[3] provided a limited analysis of these data by testing simple null hypotheses under the normal model.", "They tested null hypothesis about the mean by assuming the variance to be known, and also tested null hypothesis about the variance by assuming the mean to be known.", "These are contrived situations, and are less meaningful than the more realistic situation where both parameters are unknown.", "In this paper we consider tests for the normal mean without assuming the scale parameter to be known.", "For the full data, the $t$ -test for the null hypothesis $H_0: \\mu = 0$ against $H_1: \\mu \\ne 0$ fails to reject the null due to the presence of the large outlier (two sided $p$ -value is 0.6584); however the robust Hellinger deviance test [26] comfortably rejects the null (two sided $p$ -value based on the chi-square null distribution is 0.0061), as does the $t$ -test based on the cleaned data after the removal of the large outlier (two sided $p$ -value is 0.0076).", "Table: Telephone-Fault DataFigure: (a) Two sided pp-values of the density power divergence tests and(b) estimates of μ\\protect \\mu for different values of γ\\protect \\gamma in case of the telephone-fault data.Under the normal model, the maximum likelihood estimates of $\\mu $ (and $\\sigma $ ) are highly distorted due to the presence of the large outlier, and as a result the likelihood ratio test under the normal model fails to reject the null hypothesis.", "From the robustness perspective, this is precisely what we will like to avoid, and here we demonstrate that proper choices of the tuning parameter within the class of tests developed in this paper achieve this goal.", "Here we analyze the performance of the density power divergence tests with $\\beta = \\gamma $ .", "Figure REF (a) represents the $p$ -values of the test $H_0: \\mu = 0$ versus $H_1: \\mu \\ne 0$ for different values of $\\beta $ in a region of interest.", "While it is clearly seen that the tests fail to reject the null hypothesis for these data at very small values of $\\beta $ , the decision turns around sharply, as $\\beta $ crosses and goes beyond 0.1.", "On the other hand, the $p$ -values of the same test based on the outlier deleted data remain stable, supporting rejection, at all values of $\\beta $ (Figure REF (a)).", "The stable behavior of the test statistic based on the density power divergence for the full data approximately coincides with the stability of the density power divergence estimate of $\\mu $ itself, obtained under a two-parameter normal model, which is presented in Figure REF (b).", "The minimum density power divergence estimators of $\\mu $ for the full data and the outlier deleted data are practically identical for $\\beta > 0.12$ .", "At least in this example, the robustness of the test statistic is clearly linked to the robustness of the estimator.", "Figure: Two sided pp-values for the tests for the mean for thetelephone-fault data under the normal model against the first outlyingobservation.To further explore the robustness properties of the density power divergence tests we look at the two sided $p$ -values for different values of the outlier.", "For this purpose we vary the first outlying observation in the range from $-1000$ to 3000 by keeping the remaining 13 observations fixed.", "Figure REF shows the corresponding $p$ -values of the density power divergence tests with $\\beta = 0.15$ as well as $\\beta =0$ and the ordinary $t$ -test.", "It shows that initially the $p$ -value of the density power divergence test with $\\beta = 0.15$ increases as the first observation moves away from the center of the data set, but after a certain limit the test gradually nullifies the effect of the outlier.", "On the other hand, the $p$ -values of the $t$ -test and the density power divergence test with $\\beta = 0$ keep on increasing with the outlier on either tail.", "Indeed the $p$ -values of these two tests are remarkably close to each other." ], [ "Darwin's Plant Fertilization Data", "Charles Darwin had performed an experiment which may be used to determine whether self-fertilized plants and cross-fertilized plants have different growth rates.", "In this experiment pairs of Zea mays plants, one self and the other cross-fertilized, were planted in pots, and after a specific time period the height of each plant was measured.", "A particular sample of 15 such pairs of plants led to the paired differences (cross-fertilized minus self fertilized) presented in increasing order in Table REF (see [5]).", "Table: Darwin's Plant Fertilization DataAs in the previous example, we assume a normal model for the paired differences and test $H_0: \\mu = 0$ against $H_1: \\mu \\ne 0$ , i.e.", "we test whether the mean of the paired differences is different from zero.", "The unconstrained minimum DPD estimates of $\\mu $ under the normal model corresponding to different values of the tuning parameter $\\beta $ are presented in Figure REF (b).", "The two negative paired differences appear to be geometrically well separated from the rest of the data, though they are perhaps not as huge outliers as the first observation in the telephone-fault data.", "These two observations do have a substantial impact on the parameter estimates and the test statistic for testing $H_0$ using density power divergence tests with very small values of $\\gamma = \\beta $ , and it is instructive to compare to the case where these two outliers have been removed from the data.", "For small values of $\\beta $ , the two sided $p$ -values of the test statistics are drastically different for the full data and outlier deleted cases (Figure REF (b)), but they get closer with increasing $\\beta $ , and they essentially coincide for $\\beta \\ge 0.45$ .", "Once again this seems to be directly linked to the robustness of the parameter estimates; Figure REF (b), which also depicts the progression of the parameter estimates for the outlier deleted data, clearly demonstrates that.", "For comparison we note that the two sided $p$ -values for the ordinary $t$ -test in this case are 0.0497 (for full data) and $1.3119 \\times 10^{-4}$ (for the cleaned data with the two outliers removed).", "Figure: (a) Two sided pp-values of the density power divergence tests and(b) estimates of μ\\protect \\mu for different values of β\\protect \\beta in case of Darwin's fertilization data." ], [ "One Sided Tests", "In general, the default alternative hypotheses considered in our proposed tests are of the two sided type.", "Depending on the nature of the problem and the dimension of the parameter, one sided alternatives may sometimes be of interest.", "For the telephone fault data the primary interest could be in determining whether the mean fault rate is higher than zero (rather than simply whether it is different from zero).", "It is presumable that Darwin's interest in the fertilization problem was to determine whether cross fertilization leads to a higher growth rate compared to self fertilization; indeed the result of the test performed by R. A. Fisher (reported in [11]) on the plant fertilization data relates to the one sided alternative.", "In this subsection we consider appropriate one sided tests for these two real data examples presented earlier in this section.", "For this purpose we consider the signed divergence statistic (the signed square root of the statistic presented in (REF )) as was done in [26].", "The relevant one sided $p$ -values are determined using the normal approximation, or that based on the $t$ -distribution.", "In the following we will describe the problem of testing $H_0: \\mu = 0$ against $H_1: \\mu > 0$ under the normal model with unknown scale.", "The formal theory of constrained statistical inference (see [24]) established the expression of the asymptotic likelihood ratio test for the hypotheses $H_0: \\mu = 0$ against $H_1: \\mu > 0$ to be $T_{\\gamma =0,\\beta =0}^{(1)}=I(\\bar{X}>0)T_{\\gamma =0}(\\hat{\\theta }_{\\beta =0},\\tilde{\\theta }_{\\beta =0})=I(\\bar{X}>0)n\\log \\left( \\tfrac{\\tilde{\\sigma }_{n}^{2}}{\\hat{\\sigma }_{n}^{2}}\\right)$ with asymptotic distribution equal to $\\tfrac{1}{2}\\chi _{0}^{2}+\\tfrac{1}{2}\\chi _{1}^{2}$ under $H_0$ , where $\\chi _{0}^{2}=0$ a.s., $I(\\cdot )$ is the indicator function and $\\hat{\\theta }_{\\beta =0}$ , $\\tilde{\\theta }_{\\beta =0}$ are respectively the MDPDE and RMPDE for $\\theta =(\\mu ,\\sigma )^{T}$ when the parameter spaces are the unrestricted and restricted ones of the two sided test (REF ).", "This test is almost the same as the one provided by the signed divergence likelihood ratio test statistic, $\\widetilde{T}_{\\gamma =0,\\beta =0}^{(1)}=sign(\\bar{X})\\sqrt{T_{\\gamma =0}(\\hat{\\theta }_{\\beta =0},\\tilde{\\theta }_{\\beta =0})}=sign(\\bar{X})\\sqrt{n\\log \\left( \\tfrac{\\tilde{\\sigma }_{n}^{2}}{\\hat{\\sigma }_{n}^{2}}\\right) },$ since the corresponding $p$ -values at $t=n\\log \\left( \\frac{\\tilde{\\sigma }_{n}^{2}}{\\hat{\\sigma }_{n}^{2}}\\right) $ are given by $p\\mbox{-value}_{T_{\\gamma =0,\\beta =0}^{(1)}}(I(\\bar{x}>0)t)& =\\frac{1}{2}\\Pr (\\chi _{1}^{2}>I(\\bar{x}>0)t), \\\\p\\mbox{-value}_{\\widetilde{T}_{\\gamma =0,\\beta =0}^{(1)}}\\left( sign(\\bar{x})\\sqrt{t}\\right) & =\\Pr (Z>sign(\\bar{x})\\sqrt{t}),$ where $Z$ follows standard normal distribution.", "This means that if $\\bar{x}>0$ , both $p$ -values are equal, whereas for $\\bar{x}\\le 0$ , $ p\\mbox{-value}_{T_{\\gamma =0,\\beta =0}^{(1)}}(I(\\bar{x}>0)t)=1>p\\mbox{-value}_{\\widetilde{T}_{\\gamma =0,\\beta =0}^{(1)}}\\left( sign(\\bar{x})\\sqrt{t}\\right) >\\frac{1}{2}.$ Both tests are in practice equivalent, and such a difference for big $p$ -values comes from the fact that $\\widetilde{T}_{\\gamma =0,\\beta =0}^{(1)}$ is formally more appropriate for $H_{0}:\\mu \\le 0$ against $H_{1}:\\mu >0$ .", "We shall restrict ourselves, for simplicity, only to the signed divergence likelihood ratio test statistics and their DPD based analogues; the latter class of signed divergence DPDTS may be defined as $\\widetilde{T}_{\\gamma ,\\beta }^{(1)} = sign(\\hat{\\mu }_{\\beta })\\left\\lbrace \\frac{\\widetilde{\\sigma }_{\\beta } ^{\\gamma } \\sqrt{\\gamma +1}\\left( 2\\beta +1\\right) ^{3/2} \\left( 2\\pi \\right) ^{\\gamma /2}}{\\left( \\beta +1\\right) ^{3}} T_{\\gamma }(\\hat{\\theta }_{\\beta },\\tilde{\\theta }_{\\beta })\\right\\rbrace ^{-1/2}$ with asymptotic distribution equal to the standard normal under $H_0$ .", "In calculating the one sided $p$ -values based on the signed divergence Hellinger distance test in case of the telephone fault data, [26] used an approximation based on the $t$ -distribution, as the sample size was only 14.", "In large samples, the distribution of the statistic is approximately normal.", "The problem for a normal distribution with dimension bigger than one with inequality restrictions is more complicated and requires a specific theory based on [24].", "[21] illustrate the procedure of handing this problem when $\\phi $ -divergence based test statistics and MLEs are applied.", "Telephone-fault data The one sided $p$ -values for the signed divergence DPDTSs corresponding to $\\beta = 0.15$ and 0.3 are presented in Table REF for the full as well as outlier deleted data, using both the standard normal ($Z$ ) and $t$ (with suitable degrees of freedom) approximations.", "The result for the ordinary one-sided $t$ -test are also presented for comparison.", "The presence of the large outlier masks the significance in case of the $t$ -test, but the signed divergence DPDTSs provide consistent significant results with and without the outlier.", "Similar results were reported by [26] with the signed divergence Hellinger distance test.", "The mean of the ordered differences between the inverse test rates and inverse control rates does appear to be greater than zero.", "Table: pp-values of the one sided tests for the mean in case of the telephone-fault data.Darwin's plant fertilization data The results are presented in Table REF .", "The full data $p$ -value was reported by [11].", "In this case the one sided $p$ -values for the $t$ -test lead to a shift from marginal significance to solid rejection due to the deletion of the (two) outliers.", "This also seems to be the case for signed divergence DPDTSs for very small values of $\\beta $ .", "However, larger values of $\\beta $ lead to a more consistent behavior of the tests.", "This dataset requires stronger downweighting compared to the telephone-fault data, as the outliers here are less extreme, and therefore more difficult to identify.", "Under a suitable robust test, it appears that the mean growth of cross fertilized plants would be declared to be significantly higher than self fertilized plants.", "Table: pp-values of the one sided tests for the mean in case of the Darwin's plant fertilization data." ], [ "Normal Case", "To further explore the performance of our proposed test statistic in case of the $\\mathcal {N}(\\mu ,\\sigma ^{2})$ problem, we studied the behavior of the tests through simulation.", "We considered the hypothesis $H_{0}:\\mu =0$ against the alternative $H_{1}:\\mu \\ne 0$ with $\\sigma ^{2}$ unknown when data were generated from the $\\mathcal {N}(0,1)$ distribution.", "Subsequently, the same hypotheses were tested when the data were generated from the $\\mathcal {N}(1,1)$ distribution.", "In the first case our interest was in studying the observed level (measured as the proportion of test statistics exceeding the chi-square critical value in a large number – here 10000 – of replications) of the test under the correct null hypothesis, and in the second case we were interested in the observed power (obtained in a similar manner as above) of the test under the incorrect null hypothesis.", "The results are given in Figures REF (a) and REF (b).", "In either case the nominal level was $0.05$ .", "We have used the ordinary $t$ -test together with several DPD test statistics, corresponding to $\\beta = 0, 0.1, 0.15$ and $0.25$ , in this particular study.", "The horizontal lines in Figure REF (a), and later in Figure REF (c), represent the nominal level of 0.05.", "It may be noticed that all the tests excepting the exact likelihood ratio test (the $t$ -test) are slightly liberal for very small sample sizes and lead to somewhat inflated observed levels.", "However this discrepancy decreases rapidly, and by the time the sample size is 30 or more the observed levels have settled down reasonably around acceptable values.", "The observed powers of the tests as given in Figure REF (b) are, in fact, extremely close; in very small sample sizes the other tests have slightly higher power than the $t$ -test, but this must be a consequence of the observed levels of these tests being higher than the latter for such sample sizes.", "On the whole the proposed tests appear to be quite competitive to the ordinary $t$ -test for pure normal data.", "To evaluate the stability of the level of the tests under contamination, we repeated the tests for $H_0: \\mu = 0$ against $H_1: \\mu \\ne 0$ under data generated from the mixture of $N(0, 1)$ and $N(-10, 1)$ , where the mixing weight of the first component is 0.9.", "To illustrate the stability of power, the tests were performed with data generated under a mixture of $N(1, 1)$ and $N(-10, 1)$ , where the mixing weight of the first component is again 0.9.", "The results are given in REF (c) and REF (d) respectively.", "In this case there is a drastic and severe inflation in the observed level of the $t$ -test and that of the DPD(0) test.", "As $\\beta $ increases, however, the resistant nature of the tests are clearly apparent.", "By the time $\\beta = 0.25$ , the levels have already been reduced to acceptable values.", "The opposite behavior is seen in case of power.", "There appears to be a complete breakdown in power for small values of $\\beta $ , but the power remains quite stable for values of $\\beta $ equal to 0.25 or greater.", "On the whole it appears to be fair to claim that for sample sizes equal to or larger than 30 the efficiency of many of our DPDTSs are very close to the efficiency of the $t$ -test, but the robustness properties of our tests are often significantly better than the $t$ -test in terms of maintaining the stability of both the level and power.", "Figure: Simulated levels and powers of the DPDTSs for pure and contaminateddata in case of the normal distribution.Figure: Simulated levels and powers of the DPDTSs for pure and contaminateddata in case of the Weibull distribution." ], [ "Weibull Case", "As we have mentioned before, it is important to demonstrate the properties of the proposed method in models other than the normal so that one has a better idea about the scope of the method.", "Accordingly we performed tests of composite hypotheses under the Weibull model in the spirit of Section REF .", "Let us consider the hypothesis defined in (REF ), where $\\sigma _0$ is taken to be 1.5.", "In the first study we have generated data from the $\\mathcal {W}(1.5,1.5)$ distribution.", "The plot for the observed level for the hypothesis $H_0:\\sigma = 1.5$ against the two sided alternative is given in Figure REF (a), where we have used 1,000 replications.", "Next the same hypotheses were tested when the data were generated from the $\\mathcal {W}(1.1,1.5)$ distribution.", "The observed power function is plotted in Figure REF (b) for different values of $\\beta $ .", "The powers are remarkably close.", "In all cases the nominal level was $0.05$ .", "To evaluate the stability of the level and the power of the tests under contamination, we repeated the tests with data generated from the Weibull mixture consisting of 95% $\\mathcal {W}(1.5,1.5)$ and 5% $\\mathcal {W}(25,1.5)$ , and then from mixture of 95% $\\mathcal {W}(1.1,1.5)$ and 5% $\\mathcal {W}(25,1.5)$ .", "In either case the first larger component is our target.", "In Figures REF (c), the levels of the statistics under the contamination of first type are presented indicating the stability of levels for moderately large values of $\\beta $ .", "Figure REF (d) demonstrates the stability of powers under contaminated data of the second type for the same values of $\\beta $ ." ], [ "Comparison with Other Robust Tests", "Here we provide a comparison of our proposed tests with some other popular resistant tests available in the literature.", "In particular we have used a parametric test – the Winsorized test of [9] together with three nonparametric tests – the one sample Kolmogorov-Smirnov (KS) test, the two sided Wilcoxon signed rank test and the two sided sign test.", "The model, the hypotheses, the parameters chosen, the level of significance and other details of the set up of this simulation are the same as those in Section REF .", "We have Winsorized the 15% extreme observations on each tail of the data distribution in case of the Winsorized $t$ -test.", "Note that the null hypotheses are slightly different for the nonparametric tests.", "For the KS-test we first standardize the data using robust statistics, and then test whether the corresponding distribution is a standard normal.", "The data are standardized using the transformation $Z = (X - \\mu _0)/\\mbox{MAD}.$ Here $\\mu _0$ is the null value and ${\\rm MAD}$ is $1.4826 \\times $ (median absolute deviation about the median).", "In case of the Wilcoxon test and the sign test we perform tests for the population median without making any parametric model assumptions.", "For comparison just one DPDTS is used in these simulations, that corresponding to the tuning parameter $\\beta = 0.25$ .", "To emphasize the robustness properties of these tests we have also included the Student's $t$ -test in these investigations, so that the robust tests stand out in contrast.", "Our simulation results are presented in Figure REF .", "From Figure REF (a) it may be observed that the empirical levels of the Winsorized $t$ -test, the KS-test and the Wilcoxon test are very close to the nominal level for pure normal data.", "For small sample sizes the DPDTS is slightly liberal; however even at a sample size of 30, it is off by only one percent compared to the nominal level.", "On the other hand the sign test is a bit conservative, even at fairly large samples.", "The observed powers of all the tests in Figure REF (b) rapidly approach unity in fairly small samples.", "The results in Figure REF (c) demonstrate that for contaminated data all tests except the DPDTS fail to maintain the nominal level.", "The observed level of the sign test is close to the nominal level for small sample sizes but eventually as the sample size increases it also breaks down.", "The powers of the tests for the contaminated data, plotted in Figure REF (d) show that all the robust tests exhibit stable power.", "For small sample sizes the DPDTS exhibits the highest power.", "The overall observation on the basis of all the above appears to be that the DPD based test is superior to the classical Wald test under contamination, and is also competitive or better than several other standard resistant tests in terms of robustness, at least to the extent this particular simulation study is concerned.", "Figure: Simulated levels and powers of some robust tests for pure and contaminateddata in case of the normal distribution." ], [ "Choosing the Tuning Parameter", "By construction, the test statistic in (REF ) employs two different tuning parameters $\\beta $ and $\\gamma $ .", "These parameters have two different roles, in two different stages in the hypothesis testing process.", "The parameter $\\beta $ is used to evaluate the robust unconstrained and constrained (under the null hypothesis) estimators.", "In the next stage a density power divergence with parameter $\\gamma $ is constructed to quantify the disparity between the fitted unrestricted and the restricted models.", "As seen in Theorem REF , the null distribution of the statistic can be derived for all values of the parameters $\\beta , \\gamma > 0$ , and in practice one can choose them independently of one another.", "As the robustness of the test statistic depends primarily on the robustness of the estimators, the choice of the parameter $\\beta $ turns out to be more critical in our testing procedure.", "In repeated simulations (not presented here) our observation is that the parameter $\\gamma $ does not have a significant impact on the robustness of the procedure.", "Thus while the generality of the method allows us the choice of possibly different tuning parameters, for simplicity of implementation we will let $\\beta = \\gamma $ , so that the selection problem reduces to that of a single parameter.", "Throughout the paper, we have used $\\beta = \\gamma $ in our simulations and real data examples.", "In a real situation, the experimenter will require some guidance on the choice of this single tuning parameter $\\beta $ .", "[4] have reported that values of $\\beta \\in [0.1, 0.25]$ are often reasonable choices; we largely agree with this view, although tentative outliers and heavier contamination may require greater downweighting through a larger value of $\\beta $ ; this is the case, for example, in Darwin's plant fertilization data example.", "However, apart from fixed choices, other data driven and adaptive choices could also be useful, as one can then tune the parameter to make the procedure more robust as required.", "In this paper we follow the approach of [30] for this purpose, which minimizes an empirical measure of the mean square error of the estimator to determine the “optimal” tuning parameter.", "This requires the use of a robust pilot estimator of the parameter.", "[30] suggested the use of the MPDPDE corresponding to $\\beta = 1$ .", "The optimal parameter depends on the choice of the pilot estimator, however, and as larger values of $\\beta $ lead to a loss in efficiency, [14] suggested the choice of the MDPDE with $\\beta = 0.5$ as the pilot estimator, which appears to be reasonable in most cases.", "Our subsequent analysis is based on the [30] method, with the [14] modification.", "We do acknowledge that the criterion to be considered for the choice of the optimal $\\beta $ for the testing problem is not necessarily the same for the estimation problem.", "In hypothesis testing the appropriate criterion should involve a suitable linear combination of the inflation in the observed level under the null and the drop in power under contiguous alternatives in a contaminated scenario.", "However, an appropriate measure of this sort is not easy to construct.", "As it appears that the robustness of the proposed tests correspond almost exactly to the robustness of the MDPDEs, we feel that the optimal choice of $\\beta $ as described in the previous paragraph would generally work reasonably well in case of the hypothesis testing problem also.", "As of now, we recommend the choice of $\\beta $ according to the above recipe.", "The above criterion leads to estimated optimal choices of $\\beta $ to be 0.1919 for the telephone fault data, and 0.5657 for Darwin's plant fertilization data respectively.", "As the first observation in the telephone fault data is a massive outlier, it is easily recognized by the testing procedures even at fairly small values of $\\beta $ .", "However for Darwins' plant fertilization data the outliers are more tentative, and therefore require stronger downweighting to eliminate their effect." ], [ "Concluding Remarks", "This paper provides the appropriate theoretical machinery to perform general parametric tests of hypotheses based on the density power divergence.", "We demonstrate that one can construct a class of parametric tests of hypotheses based on the above measure which allows the experimenter to test for composite null hypotheses under the presence of nuisance parameters.", "The tests of this class have been shown to have excellent robustness properties in simulation studies and have a huge scope of application; for the purpose of numerical demonstration we have chosen the scenario of the usual $t$ -test and illustrated that for this situation the proposed test provides extremely satisfactory results.", "Similar improvements are also demonstrated outside the normal model, when data are generated from the Weibull distribution.", "When considered with the benefit of not requiring any intermediate smoothing technique as in the case of the Hellinger deviance test, our proposed techniques appear to prominently stand out among classes of robust tests for composite hypotheses.", "Our results also appropriately generalize the results of [3].", "Acknowledgments This work was partially supported by Grant MTM-2012-33740." ], [ "Appendix", "There is some overlap between the Lehmann and Basu et al.", "conditions.", "In the following we present the consolidated set of conditions which are the useful ones in our context.", "Lehmann and Basu et al.", "conditions (LB1) The model distributions $F_{\\theta }$ of $X$ have common support, so that the set $\\mathcal {X} =\\lbrace x\|f_{\\theta } (x) > 0\\rbrace $ is independent of $\\theta $ .", "The true distribution $H$ is also supported on $\\mathcal {X}$ , on which the corresponding density $h$ is greater than zero.", "(LB2) There is an open subset of $\\omega $ of the parameter space $\\Theta $ , containing the best fitting parameter $\\theta _0$ such that for almost all $x \\in \\mathcal {X}$ , and all $\\theta \\in \\omega $ , the density $f_{\\theta } (x)$ is three times differentiable with respect to $\\theta $ and the third partial derivatives are continuous with respect to $\\theta $ .", "(LB3) The integrals $\\int f_{\\theta }^{ 1+\\beta } (x)dx$ and $\\int f_{ \\theta }^\\beta (x)h(x)dx$ can be differentiated three times with respect to $\\theta $ , and the derivatives can be taken under the integral sign.", "(LB4) The $p \\times p$ matrix $J_\\beta (\\theta )$ , defined in (REF ), is positive definite.", "(LB5) There exists a function $M_{jkl} (x)$ such that $\|\\nabla _{jkl} V_{ \\theta } (x)\| \\le M_{jkl} (x)$ for all $\\theta \\in \\omega $ , where $E_h [M_{jkl} (X)] = m_{jkl} < \\infty $ for all $j$ , $k$ and $l$ , where $V_\\theta (x)$ is as defined in (REF ).", "Proof of Theorem REF This proof closely follows the approach of [23].", "Let $h_{n}(\\theta )=\\frac{1}{1+\\beta }\\left[ \\int f_{\\theta }^{1+\\beta }(x)dx-\\left( 1+\\frac{1}{\\beta }\\right) \\frac{1}{n}\\sum _{i=1}^{n}f_{\\theta }^{\\beta }(X_{i})\\right] $ be the function (REF ) divided by $1+\\beta $ .", "By differentiating both sides of equation (REF ) with respect to $\\theta $ we get $\\frac{\\partial }{\\partial \\theta }h_{n}(\\theta )=\\int u_{\\theta }(x)f_{\\theta }^{1+\\beta }(x)dx-\\frac{1}{n}\\sum _{i=1}^{n}u_{\\theta }(X_{i})f_{\\theta }^{\\beta }(X_{i}) $ and differentiating again with respect to $\\theta $ $\\frac{\\partial }{\\partial \\theta ^{T}}\\frac{\\partial }{\\partial \\theta }h_{n}(\\theta )& =(1+\\beta )\\int u_{\\theta }(x)u_{\\theta }^{T}(x)f_{\\theta }^{1+\\beta }(x)dx-\\int I_{\\theta }(x)f_{\\theta }^{1+\\beta }(x)dx\\\\& -\\frac{\\beta }{n}\\sum _{i=1}^{n}u_{\\theta }(X_{i})u_{\\theta }^{T}(X_{i})f_{\\theta }^{\\beta }(X_{i})+\\frac{1}{n}\\sum _{i=1}^{n}I_{\\theta }(X_{i})f_{\\theta }^{\\beta }(X_{i}).", "$ Here $u_{{\\theta }}(x)=\\frac{\\partial }{\\partial \\theta }\\log f_{{\\theta }}(x)$ and $I_{\\theta }(x)=-\\frac{\\partial }{\\partial \\theta }u_{\\theta }(x)$ .", "We assume that the null hypothesis is true, and $\\theta _{0} \\in \\Theta _0$ is the true value of the parameter.", "Since the model is correct $\\frac{\\partial }{\\partial \\theta ^{T}}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}$ converges in probability to $\\lim _{n\\rightarrow \\infty }\\frac{\\partial }{\\partial \\theta ^{T}}\\frac{\\partial }{\\partial \\theta }h_{n}(\\theta ) \\Big \\vert _{\\theta =\\theta _{0}}&=(1+\\beta )\\int u_{\\theta _{0}}(x)u_{\\theta _{0}}^{T}(x)f_{\\theta _{0}}^{1+\\beta }(x)dx-\\int I_{\\theta _{0}}(x)f_{\\theta _{0}}^{1+\\beta }(x)dx \\\\& -\\beta \\int u_{\\theta _{0}}(x)u_{\\theta _{0}}^{T}(x)f_{\\theta _{0}}^{1+\\beta }(x)dx+\\int I_{\\theta _{0}}(x)f_{\\theta _{0}}^{1+\\beta }(x)dx \\\\& =\\int u_{\\theta _{0}}(x)u_{\\theta _{0}}^{T}(x)f_{\\theta _{0}}^{1+\\beta }(x)dx.$ Notice that $\\lim _{n\\rightarrow \\infty }\\frac{\\partial }{\\partial \\theta ^{T}}\\frac{\\partial }{\\partial \\theta }h_{n}(\\theta )\\vert _{\\theta =\\theta _{0}}=J_\\beta (\\theta _{0})$ defined earlier in equation (REF ).", "Since $f_{\\theta _{0}}$ represents the true distribution, some simple algebra establishes that ${\\rm E}\\left[n^{1/2}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}\\right]=0_{p},~~\\mathrm {and ~~ Var} \\left[n^{1/2}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}\\right]=K_\\beta (\\theta _{0})$ where $K_\\beta (\\theta _{0})$ is as defined in equation ().", "Thus, asymptotically, $n^{1/2}\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}$ has a $\\mathcal {N}(0_{p},K_\\beta (\\theta _{0}))$ distribution.", "The restricted minimum density power divergence estimator of $\\theta $ , i.e.", "$\\widetilde{\\theta }_{\\beta }$ , will satisfy $\\left\\lbrace \\begin{array}{r}n\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\widetilde{\\theta }_{\\beta }}+G(\\widetilde{\\theta }_{\\beta })\\lambda _{n}=0_{p}, \\\\g(\\widetilde{\\theta }_{\\beta })=0_{r},\\end{array}\\right.", "$ where $\\lambda _{n}$ is a vector of Lagrangian multipliers.", "Now we consider the Taylor expansion of $\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\widetilde{\\theta }_{\\beta }}$ about the point $\\theta _{0}$ $\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\widetilde{\\theta }_{\\beta }}=\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}+\\frac{\\partial }{\\partial \\theta ^{T}}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{1}}(\\widetilde{\\theta }_{\\beta }-\\theta _0),$ where $\\theta _1$ belongs to the line segment joining $\\theta _0$ and $\\widetilde{\\theta }_{\\beta }$ .", "Now using the Khintchine's weak law of large numbers we have $\\frac{\\partial }{\\partial \\theta }\\frac{\\partial }{\\partial \\theta ^{T}}\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{1}} \\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {P}}{\\longrightarrow }}J_\\beta (\\theta _{0}).$ Therefore, from (REF ) we get $n^{1/2}\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\widetilde{\\theta }_{\\beta }}=n^{1/2}\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}+J_\\beta (\\theta _{0})n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1).", "$ On the other hand, the Taylor expansion of $g(\\widetilde{\\theta }_{\\beta })$ about the point $\\theta _{0}$ is $n^{1/2}g(\\widetilde{\\theta }_{\\beta })=G^{T}(\\theta _{0})n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1).", "$ Combining equations (REF ) and (REF ) we have $n^{1/2}\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}+J_\\beta (\\theta _{0})n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})+G(\\theta _{0})n^{-1/2}\\lambda _{n}+o_{p}(1)=0_{p}.$ The last expression also uses the fact that $G(\\widetilde{\\theta }_{\\beta }) - G(\\theta _{0})$ is an $o_p(1)$ term.", "Similarly from (REF ) and (REF ) it follows that $G^{T}(\\theta _{0})n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})+o_p(1)=0_{r}.", "$ Now we can express equations (REF ) and (REF ) in the matrix form as $\\left(\\begin{array}{cc}J_\\beta (\\theta _{0}) & G(\\theta _{0}) \\\\G^{T}(\\theta _{0}) & 0_{r\\times r}\\end{array}\\right) \\left(\\begin{array}{c}n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})\\\\n^{-1/2}\\lambda _{n}\\end{array}\\right) =\\left(\\begin{array}{c}-n^{1/2}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}} \\\\0_{r}\\end{array}\\right) +o_{p}(1).$ Therefore $\\left(\\begin{array}{c}n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})\\\\n^{-1/2}\\lambda _{n}\\end{array}\\right) =\\left(\\begin{array}{cc}J_\\beta (\\theta _{0}) & G(\\theta _{0}) \\\\G^{T}(\\theta _{0}) & 0_{r\\times r}\\end{array}\\right) ^{-1}\\left(\\begin{array}{c}-n^{1/2}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}\\\\0_{r}\\end{array}\\right) +o_{p}(1) .$ But $\\left(\\begin{array}{cc}J_\\beta (\\theta _{0}) & G(\\theta _{0}) \\\\G^{T}(\\theta _{0}) & 0\\end{array}\\right) ^{-1}={\\left(\\begin{array}{cc}P(\\theta _{0}) & Q(\\theta _{0}) \\\\Q(\\theta _{0})^{T} & R(\\theta _{0})\\end{array}\\right) } ,$ where $P(\\theta _{0})$ and $Q(\\theta _{0})$ are as given in (REF ) and (REF ) respectively.", "The matrix $R(\\theta _0)$ is the quantity needed to make the right hand side of the above equation equal to the indicated inverse.", "Then $n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})=-P(\\theta _{0})n^{1/2}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}+o_{p}(1) , $ and we know $n^{1/2}\\frac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}\\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {L}}{\\longrightarrow }}\\mathcal {N}(0,K_{\\beta }(\\theta _0)).", "$ Finally combining (REF ) and (REF ) we get the desired result.", "Proof of Theorem REF Consider the expression $d_{\\gamma }(f_{\\theta },f_{\\widetilde{\\theta }_{\\beta }})$ .", "A Taylor expansion for an arbitrary $\\theta \\in \\Theta $ , around $\\widetilde{\\theta }_{\\beta }$ leads to the relation $d_{\\gamma }(f_{\\theta },f_{\\widetilde{\\theta }_{\\beta }})& =d_{\\gamma }(f_{\\widetilde{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})+{\\textstyle \\sum \\limits _{i=1}^{p}}\\left( \\frac{\\partial d_{\\gamma }(f_{\\theta },f_{\\widetilde{\\theta }_{\\beta }})}{\\partial \\theta _{i}}\\right) _{\\theta =\\widetilde{\\theta }_{\\beta }}\\left( \\theta _{i}-\\widetilde{\\theta }_{i,{\\beta }}\\right) \\\\& +\\frac{1}{2}{\\textstyle \\sum \\limits _{i=1}^{p}}{\\textstyle \\sum \\limits _{j=1}^{p}}\\left( \\frac{\\partial ^{2}d_{\\gamma }(f_{\\theta },f_{\\widetilde{\\theta }_{\\beta }})}{\\partial \\theta _{i}\\partial \\theta _{j}}\\right) _{\\theta =\\widetilde{\\theta }_{\\beta }}\\left( \\theta _{i}-\\widetilde{\\theta }_{i,{\\beta }}\\right) \\left( \\theta _{j}-\\widetilde{\\theta }_{j,{\\beta }}\\right)+o\\left( \\left\\Vert \\theta -\\widetilde{\\theta }_{\\beta }\\right\\Vert ^{2}\\right) .$ It is clear that $d_{\\gamma }(f_{\\widetilde{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})=0$ , $\\left( \\frac{\\partial d_{\\gamma }(f_{\\theta },f_{\\widetilde{\\theta }_{\\beta }})}{\\partial \\theta _{i}}\\right) _{\\theta =\\widetilde{\\theta }_{\\beta }}=0$ for each $i$ , and $a_{ij}^{\\gamma }\\left( \\widetilde{\\theta }_{\\beta }\\right)=\\left( \\frac{\\partial ^{2}d_{\\gamma }(f_{\\theta },f_{\\widetilde{\\theta }_{\\beta }})}{\\partial \\theta _{i}\\partial \\theta _{j}}\\right) _{\\theta =\\widetilde{\\theta }_{\\beta }}=\\left( 1+\\gamma \\right) \\int \\nolimits _{\\mathcal {X}}f_{\\widetilde{\\theta }_{\\beta }}^{\\gamma -1}\\left( x\\right) \\frac{\\partial f_{\\widetilde{\\theta }_{\\beta }}\\left( x\\right) }{\\partial \\theta _{i}}\\frac{\\partial f_{\\widetilde{\\theta }_{\\beta }}\\left(x\\right) }{\\partial \\theta _{j}}dx.$ Therefore, $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })=2nd_{\\gamma }(f_{\\widehat{\\theta }},f_{\\widetilde{\\theta }_{\\beta }})=n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })^{T}A_{\\gamma }(\\widetilde{\\theta }_{\\beta })n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })+n\\times o\\left( \\left\\Vert \\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })\\right\\Vert ^{2}\\right) .$ Under $\\theta _{0} \\in \\Theta _0$ $A_{\\gamma }(\\widetilde{\\theta }_{\\beta })\\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {P}}{\\longrightarrow }}A_{\\gamma }\\left( \\theta _{0}\\right) .$ Using (REF ) and $n^{1/2}\\tfrac{\\partial }{\\partial \\theta }\\left.", "h_{n}(\\theta )\\right\|_{\\theta =\\theta _{0}}=-n^{1/2}J_\\beta \\left( \\theta _{0}\\right) (\\widehat{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1),$ we get $n^{1/2}(\\widetilde{\\theta }_{\\beta }-\\theta _{0})&=P(\\theta _{0})n^{1/2}J_\\beta \\left(\\theta _{0}\\right) (\\widehat{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1) \\\\& =J_\\beta ^{-1}\\left( \\theta _{0}\\right) n^{1/2}J_\\beta \\left( \\theta _{0}\\right) (\\widehat{\\theta }_{\\beta }-\\theta _{0})-Q\\left(\\theta _{0}\\right) G^{T}(\\theta _{0}) n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1) \\\\& =n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{0})-Q\\left( \\theta _{0}\\right) G^{T}(\\theta _{0}) n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1).$ Therefore $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })=Q(\\theta _{0}) G^{T}(\\theta _{0}) n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1).$ On the other hand, $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{0})\\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {L}}{\\longrightarrow }}\\mathcal {N}(0,J_{\\beta }^{-1}(\\theta _{0})K_\\beta (\\theta _{0})J_{\\beta }^{-1}(\\theta _{0}))$ .", "From equations (REF ) and (REF ) we have $B_{\\beta }\\left( \\theta _{0}\\right) = Q_{\\beta } \\left( \\theta _{0}\\right) G^{T}(\\theta _{0}) J_{\\beta }^{-1}(\\theta _{0})$ .", "Therefore it follows that $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })\\underset{n\\rightarrow \\infty }{\\overset{\\mathcal {L}}{\\longrightarrow }}\\mathcal {N}(0,B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) ).$ Now the asymptotic distribution of the random variables $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })=2nd_{\\gamma }(f_{\\widehat{\\theta }_{\\beta }},f_{\\widetilde{\\theta }_{\\beta }})$ and $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })^{T}A_{\\gamma }\\left( \\theta _{0}\\right) n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })$ are the same because $n\\times o\\left( \\left\\Vert \\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta }\\right\\Vert ^{2}\\right)=o_{p}\\left( 1\\right) .$ Now we apply Corollary 2.1 in [8], which essentially states the following.", "Let $X$ be a $q$ -variate normal random variable with mean vector 0 and variance-covariance matrix $\\Sigma $ .", "Let $M$ be a real symmetric matrix of order $q$ .", "Let $k=\\rm {rank}(\\Sigma M\\Sigma )$ , $k\\ge 1$ and let $\\lambda _{1},\\ldots ,\\lambda _{k},$ be the nonzero eigenvalues of $M\\Sigma .$ Then the distribution of the quadratic form $X^{T}MX$ coincides with the distribution of the random variable ${\\textstyle \\sum \\limits _{i=1}^{k}}\\lambda _{i}Z_{i}^{2},$ where $Z_{1},\\ldots ,Z_{k}$ are independent, each being a standard normal variable.", "In our case the asymptotic distribution of $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ coincides with the distribution of the random variable ${\\textstyle \\sum \\limits _{i=1}^{k}}\\lambda _{i}^{\\beta ,\\gamma }Z_{i}^{2}$ where $\\lambda _{1}^{\\beta ,\\gamma },\\ldots ,\\lambda _{k}^{\\beta ,\\gamma }$ , are the nonzero eigenvalues of $A_{\\gamma }\\left( \\theta _{0}\\right) B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) $ , where $k={\\rm rank}\\left( B_{\\beta }\\left( \\theta _{0}\\right)K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) A_{\\gamma }\\left(\\theta _{0}\\right) B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) \\right) .$ Proof of Theorem REF Notice that $Q_{\\beta } \\left( \\theta _{0}\\right) G^{T}(\\theta _{0}) = B_{\\beta }\\left( \\theta _{0}\\right) J_{\\beta }(\\theta _{0})$ .", "From equation (REF ) we have $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })=B_{\\beta }\\left( \\theta _{0}\\right) J_{\\beta }(\\theta _{0})n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{0})+o_{p}(1),$ then $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })& =B_{\\beta }\\left( \\theta _{0}\\right) J_{\\beta }(\\theta _{0})n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{n})+B_{\\beta }\\left( \\theta _{0}\\right) J_{\\beta }(\\theta _{0})n^{1/2}\\left( \\ \\theta _{n}-\\theta _{0}\\right) +o_{p}(1) \\\\& =B_{\\beta }\\left( \\theta _{0}\\right) J_{\\beta }(\\theta _{0})n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{n})+B_{\\beta }\\left( \\theta _{0}\\right) J_{\\beta }(\\theta _{0})d+o_{p}(1).$ Under $H_{1,n}$ one has $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\theta _{n})\\overset{\\mathcal {L}}{\\underset{n\\rightarrow \\infty }{\\longrightarrow }}\\mathcal {N}\\left( \\mathbf {0}_{p},J_{\\beta }^{-1}(\\theta _{0})K_{\\beta }(\\theta _{0})J_{\\beta }^{-1}(\\theta _{0})\\right),$ and $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })\\overset{\\mathcal {L}}{\\underset{n\\rightarrow \\infty }{\\longrightarrow }}\\mathcal {N}\\left( B_{\\beta }\\left(\\theta _{0}\\right) J_{\\beta }(\\theta _{0})d,B_{\\beta }\\left( \\theta _{0}\\right) K_{\\beta }(\\theta _{0})B_{\\beta }\\left( \\theta _{0}\\right) \\right) .$ We know that $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })=n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })^{T}A_{\\gamma }(\\theta _{0})n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })+o_{p}(1).$ Then, $T_{\\gamma }(\\widehat{\\theta }_{\\beta },\\widetilde{\\theta }_{\\beta })$ has the same asymptotic distribution as the quadratic form $n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta })^{T}A_{\\gamma }(\\theta _{0})n^{1/2}(\\widehat{\\theta }_{\\beta }-\\widetilde{\\theta }_{\\beta }).$ Now the result follows from Corollary 2.2 of [8]: Let $X\\sim \\mathcal {N}_{q}(\\mu ,\\Sigma )$ , a $q$ -variate normal distribution.", "Let $M$ be a real symmetric non-negative definite matrix of order $q$ .", "Let $k=\\rm {rank}(\\Sigma M\\Sigma )$ , $k\\ge 1$ , and let $\\lambda _{1},\\ldots ,\\lambda _{k}$ be the positive eigenvalues of $M\\Sigma $ .", "Then the quadratic form $X^{T}MX$ has the same distribution as the random variable $\\sum \\limits _{i=1}^{k}\\lambda _{i}\\left( Z_{i}+w_{i}\\right) ^{2}+\\eta ,$ where $Z_{1},\\ldots ,Z_{k}$ are independent, each having a standard normal distribution.", "Values of $w$ and $\\eta $ are given by $w=\\Lambda _{k}^{-1}V^{T}S^{T}M\\mu ,~~\\eta =\\mu ^{T}M\\mu -w^{T}\\Lambda _{k}w,$ where $S$ is any $q\\times k$ square root of $\\Sigma $ , $\\Lambda _{k}= \\rm {diag}\\left( \\lambda _{1},\\ldots ,\\lambda _{k}\\right) $ and $V$ is the matrix of corresponding orthonormal eigenvectors.", "We therefore have the desired result." ] ]	1403.0330
[ [ "The Structurally Smoothed Graphlet Kernel" ], [ "Abstract A commonly used paradigm for representing graphs is to use a vector that contains normalized frequencies of occurrence of certain motifs or sub-graphs.", "This vector representation can be used in a variety of applications, such as, for computing similarity between graphs.", "The graphlet kernel of Shervashidze et al.", "[32] uses induced sub-graphs of k nodes (christened as graphlets by Przulj [28]) as motifs in the vector representation, and computes the kernel via a dot product between these vectors.", "One can easily show that this is a valid kernel between graphs.", "However, such a vector representation suffers from a few drawbacks.", "As k becomes larger we encounter the sparsity problem; most higher order graphlets will not occur in a given graph.", "This leads to diagonal dominance, that is, a given graph is similar to itself but not to any other graph in the dataset.", "On the other hand, since lower order graphlets tend to be more numerous, using lower values of k does not provide enough discrimination ability.", "We propose a smoothing technique to tackle the above problems.", "Our method is based on a novel extension of Kneser-Ney and Pitman-Yor smoothing techniques from natural language processing to graphs.", "We use the relationships between lower order and higher order graphlets in order to derive our method.", "Consequently, our smoothing algorithm not only respects the dependency between sub-graphs but also tackles the diagonal dominance problem by distributing the probability mass across graphlets.", "In our experiments, the smoothed graphlet kernel outperforms graph kernels based on raw frequency counts." ], [ "Introduction", "In this paper, we are interested in comparing graphs by computing a kernel between graphs [39].", "Graph kernels are popular because many datasets from diverse domains such as bio-informatics [30], [3], chemo-informatics [2], and web data mining [41] naturally can be represented as graphs.", "Almost all graph kernels (implicitly or explicitly) represent a graph as a (normalized or un-normalized) vector which contains the frequency of occurrence of motifs or sub-graphsThe kernels proposed by [21] are a notable exception..", "The key idea here is that well chosen motifs can capture the semantics of the graph structure while being computationally tractable.", "For instance, counting walks in a graph leads to the random walk graph kernel of [3] (see [39] for an efficient algorithm for computing this kernel).", "Other popular motifs include subtrees [31], shortest paths [5], and cycles [16].", "Of particular interest to us are the graphlet kernels of [32].", "The motif used in this kernel is the set of unique sub-graphs of size $k$ , which were christened as graphlets by [28].", "Observation 1: Computing meaningful graphlet kernels that have high discriminative ability requires a careful selection of $k$ .", "If $k$ is small, then the number of unique graphlets is small (i.e., the length of the feature vector is small).", "See Figure REF .", "Consequently the feature vector does not provide meaningful discrimination between two graphs.", "On the other hand, if $k$ is large then a) the set of unique graphlets grows exponentially (i.e., the feature vector is very high dimensional) but b) only a small number of unique graphlets will be observed in a given graph (i.e., the feature vector is very sparse).", "Moreover, the probability that two graphs will contain a given large sub-graph is very small.", "Consequently, a graph is similar to itself but not to any other graph in the training data.", "This is well known as the diagonal dominance problem in the machine learning community [17], and the resulting kernel matrix is close to the identity matrix.", "In other words, the graphs are orthogonal to each other in the feature space.", "However, it is desirable to use large values of $k$ in order to gain better discriminative ability.", "One way to circumvent the diagonal dominance problem is to view the normalized graphlet-frequency vector as estimating a multinomial distribution, and use smoothing.", "Observation 2: The normalized graphlet-frequency vector exhibits power-law behavior, especially for large values of $k$ .", "In other words, a few popular graphlets occur very frequently while a vast majority of graphlets will occur very rarely.", "Put another way, a few graphlets dominate the distribution.", "To see this, we randomly sampled a graph from six benchmark datasets (details of the datasets can be found in Section ), and exhaustively computed occurrences of all graphlets of size $k=8$ and plotted the resulting histogram on a log-log scale in Figure REF .", "As can be seen, the frequencies are approximately linear in the log-log scale which indicates power law behavior.", "Therefore, any smoothing technique that we use on the normalized graphlet-frequency vector must respect this power-law behaviour.", "Observation 3: The space of graphlets is structured.", "What we mean by this is that graphlets of different sizes are related to each other.", "While many such relationships can be derived, we will work with perhaps the simplest one which is depicted in Figure REF .", "Here, we construct a directed acyclic graph (DAG) with the following property: a node at depth $k$ denotes a graphlet of size $k$ .", "Given a graphlet $g$ of size $k$ and other graphlet $g^{\\prime }$ of size $k+1$ we add an edge from $g$ to $g^{\\prime }$ if, and only if, $g$ can be obtained from $g^{\\prime }$ by deleting a node of $g^{\\prime }$ .", "This shows that graphlets of size $k$ have a strong relationship to graphlets of size $k+1$ and one must respect this relationship when deriving a smoothing technique.", "Figure: Number of unique graphlets increase exponentially withgraphlet size kk.Figure: We randomly select a graph from six benchmark graph datasetsand exhaustively searched for all graphlets of size k=8k=8 .", "Thehistogram is plotted in log-log scale in order to demonstrate thepower-law behaviour.Our contributions in this paper are as follows.", "First, we propose a new smoothing technique for graphlets which is inspired by Kneser-Ney smoothing [20] used for language models in natural language processing.", "Our model satisfies the desiderata that we outlined above, that is, it respects the power law behavior of the counts and yet takes into account the structure of the space of graphlets.", "Second, we provide a novel Bayesian version of our model that is extended from the Hierarchical Pitman-Yor process of [35].", "Unlike the traditional Hierarchical Pitman-Yor Process (HPYP) where the base distribution is given by another Pitman-Yor Process (PYP), in our case it is given by a transformation of a PYP that is guided by the structure of the space.", "Third, we perform experiments to validate and understand how smoothing affects the performance of graphlet kernels.", "The structure of the paper is as follows.", "In Section , we discuss background on graphlet kernels and smoothing techniques.", "In Section , we introduce our Kneser-Ney-inspired smoothing technique.", "In Section , we propose an alternate Bayesian version of our model.", "Related work is discussed in Section .", "In Section , we perform experiments and discuss our findings, and we conclude the paper with Section .", "A graph is a pair $G=(V,E)$ where $V = \\left\\lbrace v_1, v_2, \\ldots ,v_{\|V\|} \\right\\rbrace $ is an ordered set of vertices or nodes and $E \\subseteq V \\times V$ is a set of edges.", "Given $G = (V, E)$ and $H = (V_H , E_H )$ , $H$ is a sub-graph of $G$ iff there is an injective mapping $\\alpha : V_H \\rightarrow V$ such that $(v, w) \\in E_H$ iff $(\\alpha (v), \\alpha (w)) \\in E$ .", "Two graphs $G= (V, E)$ and $G^{\\prime } = (V^{\\prime }, E^{\\prime })$ are isomorphic if there exists a bijective mapping $g: V \\rightarrow V^{\\prime }$ such that $(v_i, v_j) \\in E$ iff $(g(v_i), g(v_j)) \\in E^{\\prime }$ .", "Graphlets are small, connected, non-isomorphic sub-graphs of a large network.", "They were introduced by [28] to design a new measure of local structural similarity between biological networks.", "Graphlets up to size five are shown in Figure REF .", "Figure: Connected, non-isomorphic induced sub-graphs of size k≤5k\\le 5.", "Plots are generated with NetworkX library ." ], [ "The Graphlet kernel", "Let $\\mathcal {G}_{k} = \\lbrace g_{1}, g_2, \\ldots , g_{n_k} \\rbrace $ be the set of size-$k$ graphlets where $n_k$ denotes the number of unique graphlets of size $k$ .", "Given a graph $G$ , we define $f_{G}$ as a normalized vector of length $n_k$ whose $i$ -th component corresponds to the frequency of occurrence of $g_{i}$ in $G$ : $f_G = (\\frac{c_1}{\\sum _{j}^{n_k} c_j}, \\cdots , \\frac{c_{n_k}}{\\sum _{j}^{n_k} c_j})^T.$ Here $c_i$ denotes number of times $g_i$ occurs as a sub-graph of $G$ .", "Given two graphs $G$ and $G^{\\prime }$ , the graphlet kernel $k_{g}$ is defined as: $k_{g}(G, G^{\\prime }):= f_{G}^{\\top } f_{G^{\\prime }},$ which is simply the dot product between the normalized graphlet-frequency vectors." ], [ "Smoothing multinomial distributions", "In this section we will briefly review smoothing techniques for multinomial distributions and show that graphlet kernels are indeed based on estimating a multinomial.", "Suppose we observe a sequence $e_1,e_2, \\ldots , e_n$ containing $n$ discrete events drawn from a ground set of size $M$ and we would like to estimate the probability $P(e_{i})$ of observing each event $e_{i}$ .", "Maximum likelihood estimation based on sequence counts obtained from the observations provides a way to compute $P(e_i)$ : $P_{MLE}(e_i) = \\frac{c_{i}}{\\sum _{j} c_{j}},$ where $c_{i}$ denotes the number of times the event $e_{i}$ appears in the observed sequence and $\\sum _{j} c_{j}$ denotes the total number of observed events.", "Therefore, one can easily see that the representation used in graphlet kernels in Section REF is actually an MLE estimate on the observed sequences of graphlets.", "However, MLE estimates of the multinomial distribution are spiky, that is, they assign zero probability to events that did not occur in the observed sequence.", "What this means is that an event with low probability is often estimated to have zero probability mass.", "This issue occurs in a number of different domains and therefore, unsurprisingly, has received significant research attention [42]; smoothing methods are typically used to address this problem.", "The general idea behind smoothing is to discount the probabilities of the observed events and to assign extra probability mass to unobserved events.", "Laplace smoothing is the simplest and one of the oldest smoothing methods, where only a fixed count of 1 is added to every event.", "This results in the estimate $P_{Laplace}(e_{i}) = \\frac{\\sum _{j} c_{j}}{\\sum _{j} c_{j} + M} P_{MLE} +\\frac{M}{\\sum _j c_j + M}\\frac{1}{M}$ or equivalently, $P_{Laplace}(e_{i}) = \\lambda P_{MLE}(e_{i}) + (1-\\lambda )\\frac{1}{M},$ where $\\lambda $ is a normalization factor which ensures that the distributions sum to one.", "The intuition behind Laplace smoothing is basically to interpolate a uniform distribution with the MLE distribution.", "Although Laplace smoothing resolves the zero-count problem, it does not produce a power law distribution which is a desirable feature in real-life models.", "Therefore, researchers have worked on finding smoothing techniques that respect power law behavior.", "The key idea behind these methods is to redistribute the probability mass using a so-called fallback model, where the fallback model is also recursively estimated.", "Kneser-Ney smoothing is a fallback based smoothing method which has been identified as the state-of-the-art smoothing in natural language processing by several studies [7].", "Kneser-Ney smoothing computes the probability of an event by using the raw counts that are discounted by using a fixed mass.", "Then, the discounted mass is re-added equally to all event probabilities by using a base distribution: $P_{KN}(e_i) = \\frac{max \\lbrace c_i -d, 0\\rbrace }{\\sum _{j} c_j} + \\sum _{j=1}^n\|\\lbrace e_j: c_j > d \\rbrace \|\\frac{d}{\\sum _{j} c_j} P_{0}(e_i),$ where $d \\ge 0 $ is the discounting parameter, $P_{0}(\\cdot )$ is the base distribution and $P_0(e_i)$ denotes the probability mass the base distribution assigned to event $e_i$ .", "The quantity $\\sum _{j=1}^n\|\\lbrace e_j: c_j > d \\rbrace \|$ is a normalization factor to ensure that the distribution sums to 1 and simply denotes the number of events the discount is applied.", "When discount parameter $d=0$ , we recover MLE estimation since no mass is taken away from any event.", "When $d$ is very large, then we recover the base distribution on the events since we discount all the available mass.", "One should interpolate between these two extremes in order to get a reasonable smoothed estimate.", "In order to propose a new Kneser-Ney-based smoothing framework, one needs to specify the discount parameter $d$ and the base distribution $P_{0}(\\cdot )$ .", "In the next section we will show how one can derive a meaningful base distribution for graphlets.", "The space of graphlets has an inherent structure.", "One can construct a directed acyclic graph (DAG) in order to show how different graphlets are related with each other.", "A node at depth $k$ denotes a graphlet of size $k$ .", "When it is clear from context, we will use the node of the DAG and a graphlet interchangeably.", "Given a graphlet $g_i$ of size $k$ and another graphlet $g_j$ of size $k+1$ we add an edge from $g_i$ to $g_j$ if, and only if, $g_i$ can be obtained from $g_j$ by deleting a node of $g_j$ .", "We first discuss how to construct this DAG and then discuss how we can use this DAG to define a base distribution.", "The first step towards constructing our DAG is to obtain all unique graphlet types of size $k$ .", "Therefore, we first exhaustively generate all possible graphs of size $k$ (this involves a one time $O(2^k)$ effort), and use Nauty [25] to obtain their canonically-labelled isomorphic representations.", "In order to obtain the edges of the DAG, we take a node at depth $k+1$ , which denotes a canonically-labeled isomorphic graphlet and delete a node to obtain a size $k$ graphlet.", "We use Nauty to canonically-label the size $k$ graph, which in turn allows us to link to a node at depth $k$ .", "By deleting each node of the $k+1$ sized graphlet we can therefore obtain $k+1$ possible links.", "We repeat this for all nodes at level $k+1$ before proceeding to level $k$ .", "Figure REF shows the constructed DAG for size $k=5$ graphlets.", "Since all descendants of a given graphlet at level $k$ are at level $k+1$ , a topological ordering of the vertices is possible, and hence it is easy to see that the resulting graph is a DAG.", "Figure: Graphlet g 15 g_{15} get the probability mass to itsparents g 7 ,g 6 ,g 5 g_{7}, g_{6}, g_{5} according to theweights w 1 ,w 2 ,w 3 w_1, w_2, w_3 respectively.Now, let us define the edge weight between an arbitrary graphlet $g_j$ of size $k+1$ and its parent $g_i$ of size $k$ .", "Let $s_{ij}$ denote the number of times $g_i$ occurs as a sub-graph of $g_j$ and $\\mathcal {C}_{g_{i}}$ denote all the children of graphlet $g_{i}$ in the DAG.", "Then, we define the edge weight between graphlet $g_i$ and $g_j$ as $w_{ij} = \\frac{s_{ij}}{\\sum _{g_{j^{\\prime }} \\in \\mathcal {C}_{g_i}} s_{ij^{\\prime }}}$ Next we show how the DAG can be used to define a base distribution.", "Suppose we have a distribution over graphlets of size $k$ .", "Then we can transform it into a distribution over size $k+1$ graphlets in a recursive way by exploiting edge connections in the DAG as follows: $P_{0}(g_j) = \\sum _{g_i \\in Pa(g_j)} w_{ij} P_0(g_i).$ Here $g_j$ denotes a graphlet of size $k+1$ and $Pa(g_{j})$ denotes the parents of graphlet $g_{j}$ in the DAG.", "Lemma 1 Given the set of child nodes of a graphlet $g_{i}$ , if the edge weights on the DAG are all non-negative and satisfy $\\sum _{g_j \\in C(g_i)} w_{ij} = 1,$ then (REF ) defines a valid probability distribution.", "For clarity, we introduce a few notations to facilitate the proof.", "Assume that there are in total $J$ child graphlets, which we denote by $g_1, g_2, \\cdots , g_J$ .", "Further assume that there are in total $I$ parent graphlets, which we denote by $g_1, \\cdots , g_I$ .", "Let $I_j$ denote the number of parents graphlet $g_j$ has, i.e.", "$I_j = \|Pa(g_j)\|$ .", "Clearly, we have that $\\sum _{j=1}^J I_j = I$ .", "Thus, we have $\\sum _{j=1}^J P_0(g_j) & = \\sum _{j=1}^J [\\sum _{i=1}^{I_j} w_{ij} P_0(g_i)] \\\\& = \\sum _{i=1}^I [\\sum _{\\lbrace j:g_j\\in C(g_i)\\rbrace } w_{ij} P_0(g_i)] \\\\& = \\sum _{i=1}^I P_0(g_i) [\\sum _{\\lbrace j:g_j\\in C(g_i)\\rbrace } w_{ij}]\\\\& = \\sum _{i=1}^I P_0(g_i) = 1,$ which completes the proof.", "The base distribution we defined above respects the structural space of the graphlets.", "Pretend for a moment that we are given only the frequencies of occurrences of size $k$ graphlets and are asked to infer the probability of occurrences of size $k+1$ graphlets.", "Without any additional information, one can infer the distribution as follows: each parent graphlet casts a vote for every child graphlet based on how many times the parent occurs in the child.", "The votes of all parents are accumulated and this provides a distribution over size $k+1$ graphlets.", "In other words, a natural way to infer the distribution at level $k+1$ is to use how likely we are to see its sub-graphs.", "Figure REF illustrates the relationship between a graphlet of size $k+1$ and its parent graphlets of size $k$ .", "Here, edge weights denote how many times each parent occurs as a sub-graph of $g_{15}$ .", "In the case that we do not observe graphlet $g_{15}$ , it still gets probability mass proportional to the edge weight from its parents $g_{7}, g_{6}, g_{5}$ , thus overcoming the sparsity problem of unseen data.", "Our model combines this base distribution with the observed real data to generates the final distribution.", "The way how the discounted mass is distributed is controlled by the edge weights between two graphlets.", "In Equation REF we defined edge weights according to the number of times a parent node occurs in its children.", "However, one can explore different weighting schemes between the nodes on the DAG based on domain knowledge.", "For example, in the case of structured graphs such as social networks, one might benefit from weighting the edges according to the PageRank [26] score of the nodes.", "Similarly, other link analysis algorithms such as Hubs or Authority given by HITS algorithm [19] can be used in order to exploit the domain knowledge." ], [ "Kneser-Ney Smoothing with a structural distribution", "We now have all the components needed to explain our Structural Kneser-Ney (SKN) framework.", "Given an arbitrary graphlet $g_j$ of size $k+1$ , we estimate the probability of observing that graphlet as follows: $P_{SKN}(g_j) = \\frac{max(c_j -d, 0)}{\\sum _{g_{j^{\\prime }} \\in \\mathcal {G}_{k+1}} c_{j^{\\prime }}} + \\frac{d}{\\sum _{g_{j^{\\prime }} \\in \\mathcal {G}_{k+1}} c_{j^{\\prime }}}\\\\\\sum _{g_{j^{\\prime }} \\in \\mathcal {G}_{k+1}}\|\\lbrace g_{j^{\\prime }}: c_{j^{\\prime }} > d \\rbrace \| \\sum _{i\\in \\mathcal {P}_{g_j}} P_{0}(g_i)\\frac{w_{ij}}{\\sum _{g_{j^{\\prime }} \\in \\mathcal {C}_{g_{i}}}w_{ij^{\\prime }}}$ As can be seen from the equation, we first discount the count of all graphlets by $d$ , and then redistribute this mass to all other graphlets.", "The amount of mass a graphlet receives is controlled by the base distribution.", "In order to automatically tune the discount parameter $d$ , we use the Pitman-Yor process (a Bayesian approximation of Kneser-Ney) in the next section." ], [ "Pitman-Yor Process", "We will only give a very high level overview of a Pitman-Yor process and refer the reader to the excellent papers by [35] and [12] for more details.", "A Pitman-Yor process $P$ on a ground set $\\mathcal {G}_{k+1}$ of size-$(k+1)$ graphlets is defined via $P_{k+1} \\sim PY(d_{k+1}, \\theta _{k+1}, P_{k}),$ where $d_{k+1}$ is a discount parameter $0 \\le d_{k+1} < 1$ , $\\theta >-d_{k+1}$ is a strength parameter, and $P_{k}$ is a base distribution.", "The most intuitive way to understand draws from the Pitman-Yor process is via the Chinese restaurant process (also see Figure REF ).", "Consider a restaurant with an infinite number of tables.", "Customers enter the restaurant one by one.", "The first customer sits at the first table, and is seated at the first table.", "Since this table is occupied for the first time, a graphlet is assigned to it by drawing a sample from the base distribution.", "The label of the first table is the first graphlet drawn from the Pitman-Yor process.", "Subsequent customers when they enter the restaurant decide to sit at an already occupied table with probability proportional to $c_{i}- d_{k+1}$ , where $c_{i}$ represents the number of customers already sitting at table $i$ .", "If they sit at an already occupied table, then the label of that table denotes the next graphlet drawn from the Pitman-Yor process.", "On the other hand, with probability $\\theta _{k+1} + d_{k+1} t$ , where $t$ is the current number of occupied tables, a new customer might decide to occupy a new table.", "In this case, the base distribution is invoked to label this table with a graphlet.", "Intuitively the reason this process generates power-law behavior is because popular graphlets which are served on tables with a large number of customers have a higher probability of attracting new customers and hence being generated again.", "This self reinforcing property produces power law behavior.", "In a hierarchical Pitman-Yor process, the base distribution $P_{k}$ is recursively defined via a Pitman-Yor process $P_{k} \\sim PY(d_{k},\\theta _{k}, P_{k-1})$ .", "In order to label a table, we need a draw from $P_{k}$ , which is obtained by inserting a customer into the corresponding restaurant.", "In our case $P_{k+1}$ is defined over $\\mathcal {G}_{k+1}$ of size $n_{k+1}$ while $P_{k}$ is defined over $\\mathcal {G}_{k}$ of size $n_{k}\\le n_{k+1}$ .", "Therefore, like we did in the case of Kneser-Ney smoothing we will use the DAG and Equation (REF ) to define a base distribution.", "This changes the Chinese Restaurant process as follows: When we need to label a table, we will first draw a size-$k$ graphlet $g_{i} \\sim P_{k}$ by inserting a customer into the corresponding restaurant.", "Given $g_{i}$ , we will draw a size-$(k+1)$ graphlet $g_{j}$ proportional to $w_{ij}$ , where $w_{ij}$ is obtained from the DAG.", "Deletion of a customer is handled similarly.", "Detailed pseudo-code can be found in Algorithms and .", "$d_{k+1}$ , $\\theta _{k+1}$ , $P_{k}$ $t \\leftarrow 0$ Occupied tables $c \\leftarrow ()$ Counts of customers $l \\leftarrow ()$ Labels of tables $t=0$ $t \\leftarrow 1$ append 1 to $c$ draw graphlet $g_{i} \\sim P_{k}$ Insert customer in parent draw $g_{j} \\sim w_{ij}$ append $g_{j}$ to $l$ $g_{j}$ with probability $\\propto \\max (0, c_j - d)$ $c_j \\leftarrow c_j+1$ $l_j$ with probability proportional to $\\theta + d t$ $t \\leftarrow t+1$ append 1 to $c$ draw graphlet $g_{i} \\sim P_{k}$ Insert customer in parent draw $g_{j} \\sim w_{ij}$ append $g_{j}$ to $l$ $g_{j}$ Insert a Customer $d$ , $\\theta $ , $P_{0}$ , $C$ , $L$ , $t$ with probability $\\propto c_l$ $c_l \\leftarrow c_l-1$ $g_j \\leftarrow l_j$ $c_l=0$ $P_k$ $\\propto $ $1/w_{ij}$ delete $c_l$ from $c$ delete $l_j$ from $l$ $t \\leftarrow t - 1$ $g$ Delete a Customer" ], [ "Related Work", "The problem of estimating multinomial distributions is a classic problem.", "In natural language processing they occur in the following context: suppose we are given a sequence of words $w_{1}, \\ldots , w_{k}$ and one is interested in asking what is the probability of observing word $w$ next.", "Estimating this probability lies at the heart of language models, and many sophisticated smoothing techniques have been proposed.", "This is a classic multinomial estimation problem that suffers from sparsity since the event space is unbounded.", "Moreover, natural language exhibits power law behavior since the distribution tends to be dominated by a small number of frequently occurring words.", "In extensive empirical evaluation it has been found the Kneser-Ney smoothing is very effective for language models [7], [24].", "Here, the base distribution is constructed using smaller context of $k-1$ words which naturally leads to a denser distribution.", "Even though language models and graphlets have some similarities, there is a significant fundamental difference between the two.", "In language models, one can derive the base distribution using a smaller context.", "However, in the case of graphlets there is no equivalent concept of a fallback model.", "Therefore, we need to derive the base distribution by using smaller size graphlets.", "However, this leads to a problem since the distribution is now defined on a smaller space.", "Therefore, we need to apply a transformation by using the DAG in order to convert the distribution back into to the original space.", "[13] and [35] independently showed that Kneser-Ney can be explained in a Bayesian setting by using the Pitman-Yor Process (PYP) [27].", "In the Bayesian interpretation, a hierarchical PYP where the Pitman-Yor prior comes from another PYP is used.", "Similar to Kneser-Ney, this interpretation is not directly applicable to our model since the previous PYP has a different space, thus we need to apply a transformation.", "Graph kernels can be considered as special cases of convolutional kernels proposed by [15].", "In general, graph kernels can be categorized into three classes: graph kernels based on walks and paths [11], [18],[5], graph kernels based on limited-size sub-graphs [16], [31], [34] and graph kernels based on subtree patterns [29].", "[33] performs a relaxation on the vertices and exploit labeling information embedded in the graphs to derive their so-called Weisfeiler-Lehman kernels.", "However, their kernel is applicable only to labeled graphs.", "The sparsity problem of graphlet kernels has been addressed before.", "Hash kernels proposed by [34] addresses the sparsity problem by applying a sparse projection into a lower dimensional space.", "The idea here is that many higher order graphlets will “collide” and therefore be mapped to the same lower dimensional representation, thus avoiding the diagonal dominance problem.", "Unfortunately, we find that in our experiments the hash kernel is very sensitive to the hash value used for embedding and rarely performed well as compared to the MLE estimate." ], [ "Experiments", "To compare the efficacy of our approach, we compare our Kneser-Ney and Pitman-Yor smoothed kernels with state-of-the-art graph kernels namel the graphlet kernel [32], the hash kernel [34], the random walk kernel [11], [18], [38], and the shortest path kernel [5].", "For random walk kernel, we uniformly set the decay factor $\\lambda = 10^{-4}$ , for shortest path we used the delta kernel to compare the shortest-path distances, and for the hash kernel we used a prime number of 11291.", "We adopted Markov chain Monte Carlo sampling based inference scheme for the hierarchical Pitman-Yor language model from [35] and modified the open source implementation of HPYP from https://github.com/redpony/cpyp.", "Due to lack of space we will only present a subset of our experimental results.", "Full results including the source code and experimental scripts will be made available for download from http://cs.purdue.edu/~ypinar/kdd." ], [ "Datasets", "In order to test the efficacy of our model, we applied smoothing to real-world benchmark datasets, namely MUTAG, PTC, NCI1, NCI109, ENZYMES and DD.", "MUTAG [9] is a binary data set of 188 mutagenic aromatic and heteroaromatic nitro compounds, labeled whether they have mutagenicity in Salmonella typhimurium.", "The Predictive Toxicology Challenge (PTC) [36] dataset is a chemical compound dataset that reports the carcinogenicity for male and female rats.", "NCI1 and NCI109 [40], (http://pubchem.ncbi.nlm.nih.gov) datasets, made publicly available by the National Cancer Institute (NCI), are two subsets of balanced data sets of chemical compounds screened for ability to suppress or inhibit the growth of a panel of human tumor cell lines.", "Enzymes is a data set of protein tertiary structures obtained from [4].", "DD [10] is a data set of protein structures where each protein is represented by a graph and nodes are amino acids that are connected by an edge if they are less than 6 Angstroms apart.", "Table REF shows summary statistics for these datasets.", "Note that we did not use edge or node labels in our experiments.", "Table: Properties of the datasetsAll data sets we work with consist of sparse graphs.", "However, counting all graphlets of size $k$ for a graph with $n$ nodes requires $O(n^k)$ effort which is intractable even for moderate values of $k$ .", "Therefore, we use random sampling, as advocated by [32], in order to obtain an empirical distribution of graphlet counts that is close to the actual distribution of graphlets in the graph.", "For each value of for each $k \\in \\lbrace 2,\\ldots ,8\\rbrace $ we randomly sampled 10,000 sub-graphs, and used Nauty [25] to get canonically-labeled isomorphic representations which are then used to construct the frequency representation.", "We performed 5-fold cross-validation with C-Support Vector Machine Classification using LibSVM [6], using 4 folds for training and 1 for testing.", "We used a linear kernel.", "Since we are interested in understanding the difference in performance between smoothed and un-smoothed kernels we did not tune the value of $C$ ; it was simply set to 1.", "In order to tune the discount parameter for Kneser-Ney based smoothed kernel, we tried different parameters vary from 0.01 to 10,000 and report results for the best one.", "Figure: Accuracy vs. graphlet size for PTC dataset with MLE, PYP and Kneser-Ney smoothed kernelsTable: Graphlet Kernel with exhaustive sampling vs. Smoothed Kernel with 10,000 samplesFirst, we investigate the effect of the discounting parameter on the classification performance.", "Since the trends are similar across different datasets, we pick PTC as a representative dataset to report results.", "Figure REF shows the classification accuracy on the PTC dataset with different discounting parameters for Kneser-Ney smoothing.", "As expected, applying very large discounts decreases the performance because the distribution (REF ) degrades to the base distribution.", "On the other hand, applying a very small discount also decreases the accuracy since the distribution degrades to the MLE estimate.", "From our experiments, we observe that the best performance in all datasets is achieved by using an intermediate discount value between these two extremes.", "However, the specific discount value is data dependent.", "Figure: Accuracy vs.", "Discounting for PTC dataset, k=5 graphletsNext we investigate how the value of $k$ affects performance.", "Again, we show results for a representative dataset namely PTC.", "Figure REF shows the classification accuracy on the PTC dataset as a function of graphlet size $k$ for MLE (the graphlet kernel), Pitman-Yor smoothed kernel and Kneser-Ney smoothed kernel.", "From the figure, we can see that small graphlet sizes such as $k=2,3$ do not perform well and are not informative since the number of unique graphlets is very small (see Table REF ).", "On the other hand, MLE does not perform well for large graphlet sizes such as $k=7,8$ because of the diagonal dominance problem.", "On the other hand, smoothed kernels in general obtain a balance between these two extreme situations and tend to yield better performance.", "Pitman-Yor smoothed kernel tends to achieve a better performance than MLE, but doesn't perform as good as Kneser-Ney.", "This is expected since we didn't tune the hyperparameters for Pitman-Yor process.", "[35] shows that Pitman-Yor yields a better performance if one tune the hyperparameters.", "Therefore, our Pitman-Yor kernel is open to improvement.", "We compare the proposed smoothed kernel with graphlet kernel and hash kernel on the benchmark data sets in Table REF .", "The results for the Shortest Path and Random Walk graph kernels are included mainly to show what is the state-of-the-art using other representations.", "We fixed the $k=5$ which is observed to be the best value for the MLE based graphlet kernel on most datasets.", "We randomly sample 10,000 graphlets from each graph and feed the same frequency vectors to graphlet kernel (GK), hash kernel (HK), Kneser-Ney smoothed kernel (KN), and Pitman-Yor smoothed kernel (PYP).", "Therefore, the differences in performance that we observe are solely due to the transformation of the frequency vectors that these kernels perform.", "We performed an unpaired $t$ -test and use bold numbers to indicate that the results were statistically significant at $p < 0.0001$ .", "We can see that KN kernel outperforms MLE and hash kernels on all of the benchmark data sets.", "The accuracy of the PYP kernel is usually lower than that of the KN kernel.", "We conjecture that this is because the Pitman-Yor process is sensitive to the hyper-parameters and we do not carefully tune the hyper-parameters in our experiments.", "On PTC, DD and Enzymes datasets, the KN kernel reached the highest accuracy.", "On MUTAG, NCI1 and NCI109 datasets, KN kernels also yield good results and got comparable classification accuracies to shortest path and random walk kernels.", "For the DD dataset, shortest path and random walk kernels were not able to finish in 24 hours, due to the fact that this dataset has a large maximum degree.", "To summarize, smoothed kernels turns out to be competitive in terms of classification accuracy on all datasets and are also applicable to very large graphs.", "Next, we investigate whether any of the difference in performance can be attributed to sampling a small number of graphlets.", "In other words, we ask do the results summarily change if we performed exhaustive sampling instead of using 10,000 samples.", "We give MLE an unfair advantage by performing exhaustive sampling on MUTAG, PTC, NCI and NCI109 datasets for $k=5$ by using a distributed memory implementation.", "Table REF shows mean, median and standard deviations of number of samples in bruteforce sampled datasets for $k=5$ .", "Here, we can see that the original frequencies of the graphlets are quite high in most of the datasets.", "Even though our algorithm only uses 10,000 samples, it outperforms the graphlet kernel with exhaustive sampling on MUTAG, PTC and NCI1 and achieves competitive performance on NCI109 dataset.", "Results are summarized in Table REF .", "Even though distribution with a small number of samples is close to the original distribution in the $L_{1}$ sense, bruteforce sampling reveals that the true underlying distribution of the datasets contains a larger number of unique graphlets comparing to random sampling.", "Since the graphlet kernel uses a MLE estimate its performance degrades.", "On the other hand, our smoothing technique uses structural information to redistribute the mass and hence is able to outperform MLE even with a small number of samples.", "Table: Mean, Std and Median number of size k=5k=5 graphlets per graph." ], [ "Discussion", "We presented a novel framework for smoothing normalized graphlet-frequency vectors inspired by smoothing techniques from natural language processing.", "Although our models are inspired by work done in language models, they are fundamentally different in the way they define a fallback base distribution.", "We believe that our framework has applicability beyond graph kernels, and can be used in any structural setting where one can naturally define a relationship such as the DAG that we defined in Figure REF .", "We are currently investigating the applicability of our framework to string kernels [37], [23], [22] and tree kernels [8].", "It is also interesting to investigate if our method can be extended to other graph kernels such as random walk kernels.", "Our framework is also applicable to node-labeled graphs since they also suffer from similar sparsity issues.", "We leave the application of our framework to labelled graphs to an extended version of this paper.", "We are also investigating better strategies for tuning the hyper-parameters of the Pitman-Yor kernels." ] ]	1403.0598
[ [ "Parameter estimation for the subcritical Heston model based on discrete\n time observations" ], [ "Abstract We study asymptotic properties of some (essentially conditional least squares) parameter estimators for the subcritical Heston model based on discrete time observations derived from conditional least squares estimators of some modified parameters." ], [ "Introduction", "The Heston model has been extensively used in financial mathematics since one can well-fit them to real financial data set, and they are well-tractable from the point of view of computability as well.", "Hence parameter estimation for the Heston model is an important task.", "In this paper we study the Heston model ${\\left\\lbrace \\begin{array}{ll}\\mathrm {d}Y_t = (a - b Y_t) \\, \\mathrm {d}t + \\sigma _1 \\sqrt{Y_t} \\, \\mathrm {d}W_t , \\\\\\mathrm {d}X_t = (\\alpha - \\beta Y_t) \\, \\mathrm {d}t+ \\sigma _2 \\sqrt{Y_t}\\bigl (\\varrho \\, \\mathrm {d}W_t+ \\sqrt{1 - \\varrho ^2} \\, \\mathrm {d}B_t\\bigr ) ,\\end{array}\\right.}", "\\qquad t \\geqslant 0 ,$ where $a > 0$ , $b, \\alpha , \\beta \\in \\mathbb {R}$ , $\\sigma _1 > 0$ , $\\sigma _2 > 0$ , $\\varrho \\in (-1, 1)$ , and $(W_t, B_t)_{t\\geqslant 0}$ is a 2-dimensional standard Wiener process, see Heston [7].", "We investigate only the so-called subcritical case, i.e., when $b > 0$ , see Definition REF , and we introduce some parameter estimator of $(a, b, \\alpha , \\beta )$ based on discrete time observations and derived from conditional least squares estimators (CLSEs) of some modified parameters starting the process $(Y,X)$ from some known non-random initial value $(y_0,x_0)\\in (0,\\infty )\\times \\mathbb {R}$ .", "We do not estimate the parameters $\\sigma _1$ , $\\sigma _2$ and $\\varrho $ , since these parameters could—in principle, at least—be determined (rather than estimated) using an arbitrarily short continuous time observation $(X_t)_{t\\in [0,T]}$ of $X$ , where $T>0$ , see, e.g., Barczy and Pap [1].", "In Overbeck and Rydén [15] one can find a strongly consistent and asymptotically normal estimator of $\\sigma _1$ based on discrete time observations for the process $Y$ , and for another estimator of $\\sigma _1$ , see Dokuchaev [5].", "Eventually, it turns out that for the calculation of the estimator of $(a, b, \\alpha , \\beta )$ , one does not need to know the values of the parameters $\\sigma _1, \\sigma _2$ and $\\varrho $ .", "For interpretations of $Y$ and $X$ in financial mathematics, see, e.g., Hurn et al.", "[8].", "CLS estimation has been considered for the Cox-Ingersoll-Ross (CIR) model, which satisfies the first equation of (REF ).", "For the CIR model, Overbeck and Rydén [15] derived the CLSEs and gave their asymptotic properties, however, they did not investigate the conditions of their existence.", "Specifically, Theorems 3.1 and 3.3 in Overbeck and Rydén [15] correspond to our Theorem REF , but they estimate the volatility coefficient $\\sigma _1$ as well, which we assume to be known.", "Li and Ma [14] extended the investigation to so-called stable CIR processes driven by an $\\alpha $ -stable process instead of a Brownian motion.", "For a more complete overview of parameter estimation for the Heston model see, e.g., the introduction in Barczy and Pap [1].", "It would be possible to calculate the discretized version of the maximum likelihood estimators derived in Barczy and Pap [1] using the same procedure as in Ben Alaya and Kebaier [3] valid for discrete time observations of high frequency.", "However, this would be basically different from the present line of investigation, therefore we will not discuss it further.", "The organization of the paper is the following.", "In Section 2 we recall some important results about the existence of a unique strong solution to (REF ), and study its asymptotic properties.", "In the subcritical case, i.e., when $b>0$ , we invoke a result due to Cox et al.", "[4] on the unique existence of a stationary distribution, and we slightly improve a result due to Li and Ma [14] and Jin et al.", "[10] and [11] on the ergodicity of the CIR process $(Y_t)_{t\\geqslant 0}$ , see Theorem REF .", "We also recall some convergence results for square-integrable martingales.", "In Section 3 we introduce the CLSE of a transformed parameter vector based on discrete time observations, and derive the asymptotic properties of the estimates – namely, strong consistency and asymptotic normality, see Theorem REF .", "Thereafter, we apply these results together with the so-called delta method to obtain the same asymptotic properties of the estimators for the original parameters, see Theorem REF .", "The point of the parameter transformation is to reduce the minimization in the CLS method to a linear problem, because our objective function depends on the original parameters through complicated functions.", "The covariance matrices of the limit normal distributions in Theorems REF and REF depend on the unknown parameters $a$ , $b$ and $\\beta $ , as well (but somewhat surprisingly not on $\\alpha $ ).", "They also depend on the volatility parameters $\\sigma _1$ , $\\sigma _2$ and $\\rho $ , but, again, we will assume these to be known.", "Since the considered estimators of $a$ , $b$ and $\\beta $ are proved to be strongly consistent, using random normalization, one may derive counterparts of Theorems REF and REF in a way that the limit distributions are four-dimensional standard normal distributions (having the identity matrix ${I}_4$ as covariance matrices)." ], [ "Preliminaries", "Let $\\mathbb {N}$ , $\\mathbb {Z}_+$ , $\\mathbb {R}$ , $\\mathbb {R}_+$ , $\\mathbb {R}_{++}$ , and $\\mathbb {R}_{--}$ denote the sets of positive integers, non-negative integers, real numbers, non-negative real numbers, positive real numbers, and negative real numbers, respectively.", "For $x , y \\in \\mathbb {R}$ , we will use the notation $x \\wedge y := \\min (x, y).$ By $\\Vert {x}\\Vert $ and $\\Vert {A}\\Vert $ , we denote the Euclidean norm of a vector ${x}\\in \\mathbb {R}^d$ and the induced matrix norm of a matrix ${A}\\in \\mathbb {R}^{d \\times d}$ , respectively.", "By ${I}_d \\in \\mathbb {R}^{d \\times d}$ , we denote the $d\\times d$ unit matrix.", "The Borel $\\sigma $ -algebra on $\\mathbb {R}$ is denoted by ${\\mathcal {B}}(\\mathbb {R})$ .", "Let $\\bigl (\\Omega , {\\mathcal {F}}, \\operatorname{\\mathbb {P}}\\bigr )$ be a probability space equipped with the augmented filtration $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ corresponding to $(W_t,B_t)_{t\\in \\mathbb {R}_+}$ and a given initial value $(\\eta _0,\\zeta _0)$ being independent of $(W_t,B_t)_{t\\in \\mathbb {R}_+}$ such that $\\operatorname{\\mathbb {P}}(\\eta _0\\in \\mathbb {R}_+)=1$ , constructed as in Karatzas and Shreve [12].", "Note that $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ satisfies the usual conditions, i.e., the filtration $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ is right-continuous and ${\\mathcal {F}}_0$ contains all the $\\operatorname{\\mathbb {P}}$ -null sets in ${\\mathcal {F}}$ .", "The next proposition is about the existence and uniqueness of a strong solution of the SDE (REF ), see, e.g., Barczy and Pap [1].", "Proposition.", "2.1 Let $(\\eta _0, \\zeta _0)$ be a random vector independent of $(W_t, B_t)_{t\\in \\mathbb {R}_+}$ satisfying $\\operatorname{\\mathbb {P}}(\\eta _0 \\in \\mathbb {R}_+) = 1$ .", "Then for all $a \\in \\mathbb {R}_{++}$ , $b, \\alpha , \\beta \\in \\mathbb {R}$ , $\\sigma _1, \\sigma _2 \\in \\mathbb {R}_{++}$ , and $\\varrho \\in (-1, 1)$ , there is a (pathwise) unique strong solution $(Y_t, X_t)_{t\\in \\mathbb {R}_+}$ of the SDE (REF ) such that $\\operatorname{\\mathbb {P}}((Y_0, X_0) = (\\eta _0, \\zeta _0)) = 1$ and $\\operatorname{\\mathbb {P}}(\\text{$Y_t \\in \\mathbb {R}_+$ \\ for all \\ $t \\in \\mathbb {R}_+$}) = 1$ .", "Further, for all $s, t \\in \\mathbb {R}_+$ with $s \\leqslant t$ , ${\\left\\lbrace \\begin{array}{ll}Y_t = \\mathrm {e}^{-b(t-s)} Y_s+ a \\int _s^t \\mathrm {e}^{-b(t-u)} \\, \\mathrm {d}u+ \\sigma _1 \\int _s^t \\mathrm {e}^{-b(t-u)} \\sqrt{Y_u} \\, \\mathrm {d}W_u, \\\\X_t = X_s + \\int _s^t (\\alpha - \\beta Y_u) \\, \\mathrm {d}u+ \\sigma _2\\int _s^t\\sqrt{Y_u} \\, \\mathrm {d}(\\varrho W_u + \\sqrt{1 - \\varrho ^2} B_u) .\\end{array}\\right.", "}$ Next we present a result about the first moment of $(Y_t, X_t)_{t\\in \\mathbb {R}_+}$ .", "For a proof, see, e.g., Barczy and Pap [1] together with (REF ) and Proposition 3.2.10 in Karatzas and Shreve [12].", "Proposition.", "2.2 Let $(Y_t, X_t)_{t\\in \\mathbb {R}_+}$ be the unique strong solution of the SDE (REF ) satisfying $\\operatorname{\\mathbb {P}}(Y_0 \\in \\mathbb {R}_+) = 1$ and $\\operatorname{\\mathbb {E}}(Y_0) < \\infty $ , $\\operatorname{\\mathbb {E}}(\|X_0\|) < \\infty $ .", "Then for all $s, t \\in \\mathbb {R}_+$ with $s \\leqslant t$ , we have $&\\operatorname{\\mathbb {E}}(Y_t \\,\|\\,{\\mathcal {F}}_s)= \\mathrm {e}^{-b(t-s)} Y_s + a \\int _s^t \\mathrm {e}^{-b(t-u)} \\, \\mathrm {d}u,\\\\ &\\operatorname{\\mathbb {E}}(X_t \\,\|\\,{\\mathcal {F}}_s)= X_s + \\int _s^t (\\alpha - \\beta \\operatorname{\\mathbb {E}}(Y_u \\,\|\\,{\\mathcal {F}}_s)) \\, \\mathrm {d}u \\\\\\nonumber &\\phantom{\\operatorname{\\mathbb {E}}(X_t \\,\|\\,{\\mathcal {F}}_s)\\,}= X_s + \\alpha (t - s)- \\beta Y_s \\int _s^t \\mathrm {e}^{-b(u-s)}\\,\\mathrm {d}u- a \\beta \\int _s^t \\left(\\int _s^u \\mathrm {e}^{-b(u-v)} \\, \\mathrm {d}v\\right) \\mathrm {d}u ,$ and hence $\\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_t) \\\\\\operatorname{\\mathbb {E}}(X_t) \\\\\\end{bmatrix}= \\begin{bmatrix}\\mathrm {e}^{-bt} & 0 \\\\- \\beta \\int _0^t \\mathrm {e}^{-bu} \\, \\mathrm {d}u & 1 \\\\\\end{bmatrix}\\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_0) \\\\\\operatorname{\\mathbb {E}}(X_0) \\\\\\end{bmatrix}+ \\begin{bmatrix}\\int _0^t \\mathrm {e}^{-bu} \\, \\mathrm {d}u & 0 \\\\- \\beta \\int _0^t \\left(\\int _0^u \\mathrm {e}^{-bv} \\, \\mathrm {d}v \\right) \\mathrm {d}u & t \\\\\\end{bmatrix}\\begin{bmatrix}a \\\\\\alpha \\\\\\end{bmatrix} .$ Consequently, if $b \\in \\mathbb {R}_{++}$ , then $\\lim _{t\\rightarrow \\infty } \\operatorname{\\mathbb {E}}(Y_t) = \\frac{a}{b} , \\qquad \\lim _{t\\rightarrow \\infty } t^{-1} \\operatorname{\\mathbb {E}}(X_t) = \\alpha - \\frac{\\beta a}{b} ,$ if $b = 0$ , then $\\lim _{t\\rightarrow \\infty } t^{-1} \\operatorname{\\mathbb {E}}(Y_t) = a , \\qquad \\lim _{t\\rightarrow \\infty } t^{-2} \\operatorname{\\mathbb {E}}(X_t) = - \\frac{1}{2} \\beta a ,$ if $b \\in \\mathbb {R}_{--}$ , then $\\lim _{t\\rightarrow \\infty } \\mathrm {e}^{bt} \\operatorname{\\mathbb {E}}(Y_t) = \\operatorname{\\mathbb {E}}(Y_0) - \\frac{a}{b} , \\qquad \\lim _{t\\rightarrow \\infty } \\mathrm {e}^{bt} \\operatorname{\\mathbb {E}}(X_t)= \\frac{\\beta }{b} \\operatorname{\\mathbb {E}}(Y_0) - \\frac{\\beta a}{b^2} .$ Based on the asymptotic behavior of the expectations $(\\operatorname{\\mathbb {E}}(Y_t), \\operatorname{\\mathbb {E}}(X_t))$ as $t \\rightarrow \\infty $ , we introduce a classification of the Heston model given by the SDE (REF ).", "Definition.", "2.3 Let $(Y_t, X_t)_{t\\in \\mathbb {R}_+}$ be the unique strong solution of the SDE (REF ) satisfying $\\operatorname{\\mathbb {P}}(Y_0 \\in \\mathbb {R}_+) = 1$ .", "We call $(Y_t, X_t)_{t\\in \\mathbb {R}_+}$ subcritical, critical or supercritical if $b \\in \\mathbb {R}_{++}$ , $b = 0$ or $b \\in \\mathbb {R}_{--}$ , respectively.", "In the sequel $\\stackrel{\\operatorname{\\mathbb {P}}}{\\longrightarrow }$ , $\\stackrel{{\\mathcal {L}}}{\\longrightarrow }$ and $\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }$ will denote convergence in probability, in distribution and almost surely, respectively.", "The following result states the existence of a unique stationary distribution and the ergodicity for the process $(Y_t)_{t\\in \\mathbb {R}_+}$ given by the first equation in (REF ) in the subcritical case, see, e.g., Cox et al.", "[4], Li and Ma [14], Theorem 3.1 with $\\alpha = 2$ and Theorem 4.1 in Barczy et al.", "[2], or Jin et al.", "[11].", "Only (REF ) of the following Theorem REF can be considered as a slight improvement of the existing results.", "Theorem.", "2.4 Let $a, b, \\sigma _1 \\in \\mathbb {R}_{++}$ .", "Let $(Y_t)_{t\\in \\mathbb {R}_+}$ be the unique strong solution of the first equation of the SDE (REF ) satisfying $\\operatorname{\\mathbb {P}}(Y_0 \\in \\mathbb {R}_+) = 1$ .", "Then $Y_t \\stackrel{{\\mathcal {L}}}{\\longrightarrow }Y_\\infty $ as $t \\rightarrow \\infty $ , and the distribution of $Y_\\infty $ is given by $\\operatorname{\\mathbb {E}}(\\mathrm {e}^{-\\lambda Y_\\infty })= \\left(1 + \\frac{\\sigma _1^2}{2b} \\lambda \\right)^{-2a/\\sigma _1^2} ,\\qquad \\lambda \\in \\mathbb {R}_+ ,$ i.e., $Y_\\infty $ has Gamma distribution with parameters $2a / \\sigma _1^2$ and $2b / \\sigma _1^2$ , hence $\\operatorname{\\mathbb {E}}(Y_\\infty ) = \\frac{a}{b}, \\qquad \\operatorname{\\mathbb {E}}(Y_\\infty ^2) = \\frac{(2a+\\sigma _1^2)a}{2b^2} , \\qquad \\operatorname{\\mathbb {E}}(Y_\\infty ^3) = \\frac{(2a+\\sigma _1^2)(a+\\sigma _1^2)a}{2b^3} .$ supposing that the random initial value $Y_0$ has the same distribution as $Y_\\infty $ , the process $(Y_t)_{t\\in \\mathbb {R}_+}$ is strictly stationary.", "for all Borel measurable functions $f : \\mathbb {R}\\rightarrow \\mathbb {R}$ such that $\\operatorname{\\mathbb {E}}(\|f(Y_\\infty )\|) < \\infty $ , we have $\\frac{1}{T} \\int _0^T f(Y_s) \\, \\mathrm {d}s \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\operatorname{\\mathbb {E}}(f(Y_\\infty )) \\qquad \\text{as \\ $T \\rightarrow \\infty $,}$ $\\frac{1}{n}\\sum _{i=0}^{n-1} f(Y_i) \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\operatorname{\\mathbb {E}}(f(Y_\\infty )) \\qquad \\text{as \\ $n\\rightarrow \\infty $.", "}$ Proof.", "Based on the references given before the theorem, we only need to show (REF ).", "By Corollary 2.7 in Jin et al.", "[10], the tail $\\sigma $ -field $\\bigcap _{t\\in \\mathbb {R}_+} \\sigma (Y_s,s\\geqslant t)$ of $(Y_t)_{t \\in \\mathbb {R}_+}$ is trivial for any initial distribution, i.e., the tail $\\sigma $ -field in question consists of events having probability 0 or 1 for any initial distribution on $\\mathbb {R}_+$ .", "But since the tail $\\sigma $ -field of $(Y_t)_{t \\in \\mathbb {R}_+}$ is richer than that of $(Y_i)_{i \\in \\mathbb {Z}_+}$ , the tail $\\sigma $ -field of $(Y_i)_{i\\in \\mathbb {Z}_+}$ is also trivial for any initial distribution.", "Denoting the distribution of $Y_0$ and $Y_\\infty $ by $\\nu $ and $\\mu $ , respectively, let us introduce the distribution $\\eta := (\\mu + \\nu )/2$ .", "Let us introduce the following processes: $(Z_t)_{t \\in \\mathbb {R}_+}$ , which is the pathwise unique strong solution of the first equation in (REF ) with initial condition $Z_0 = \\zeta _0$ , where $\\zeta _0$ has the distribution $\\mu $ ; and $(U_t)_{t \\in \\mathbb {R}_+}$ , which is the pathwise unique strong solution of the same SDE with initial condition $U_0 = \\xi _0$ , where $\\xi _0$ has the distribution $\\eta $ .", "We use Birkhoff's ergodic theorem (see, e.g., Theorem 8.4.1 in Dudley [6]) in the usual setting: the probability space is $(\\mathbb {R}^{\\mathbb {Z}_+}, {\\mathcal {B}}(\\mathbb {R}^{\\mathbb {Z}_+}), {\\mathcal {L}}((Z_i)_{i\\in \\mathbb {Z}_+}))$ , where ${\\mathcal {L}}((Z_i)_{i\\in \\mathbb {Z}_+})$ denotes the distribution of $(Z_i)_{i\\in \\mathbb {Z}_+}$ , and the measure-preserving transformation $T$ is the shift operator, i.e., $T((x_i)_{i\\in \\mathbb {Z}_+}):=(x_{i+1})_{i\\in \\mathbb {Z}_+}$ for $(x_i)_{i\\in \\mathbb {Z}_+}\\in \\mathbb {R}^{\\mathbb {Z}_+}$ (the measure preservability follows from (ii)).", "All invariant sets of $T$ are included in the tail $\\sigma $ -field of the coordinate mappings $\\pi _i$ , $i\\in \\mathbb {Z}_+$ , on $\\mathbb {R}^{\\mathbb {Z}_+}$ , since for any invariant set $A$ we have $A \\in \\sigma (\\pi _0, \\pi _1, \\ldots )$ , but as $T^k(A) = A$ for all $k \\in \\mathbb {N}$ , it is also true that $A \\in \\sigma (\\pi _k, \\pi _{k+1}, \\ldots )$ for all $k \\in \\mathbb {N}$ .", "This implies that $T$ is ergodic, since the tail $\\sigma $ -field is trivial.", "Hence we can apply the ergodic theorem for the function $g:\\mathbb {R}^{\\mathbb {Z}_+} \\rightarrow \\mathbb {R}, \\qquad g((x_i)_{i\\in \\mathbb {Z}_+}) := f(x_0) , \\qquad (x_i)_{i\\in \\mathbb {Z}_+}\\in \\mathbb {R}^{\\mathbb {Z}_+},$ where $f$ is given in (iii), to obtain $\\frac{1}{n}\\sum _{i=0}^{n-1} f(x_i) \\rightarrow \\int _{\\mathbb {R}_+} f(x_0)\\,\\mu (\\mathrm {d}x_0)\\qquad \\text{as \\ $n\\rightarrow \\infty $}$ for almost every $(x_i)_{i\\in \\mathbb {Z}_+}\\in \\mathbb {R}^{\\mathbb {Z}_+}$ with respect to the measure ${\\mathcal {L}}((Z_i)_{i\\in \\mathbb {Z}_+})$ , and consequently $\\frac{1}{n}\\sum _{i=0}^{n-1} f(Z_i) \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\operatorname{\\mathbb {E}}(f(Y_\\infty )) \\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ because, clearly, the distribution of $Y_\\infty $ does not depend on the initial distribution.", "We introduce the following event, which is clearly a tail event of $(Z_i)_{i \\in \\mathbb {Z}_+}$ and has probability 1 by (REF ): $C_Z:=\\left\\lbrace \\omega \\in \\Omega :\\text{$\\frac{1}{n}\\sum _{i=0}^{n-1} f(Z_i(\\omega )) \\rightarrow \\operatorname{\\mathbb {E}}(f(Y_\\infty ))$ \\ as \\ $n \\rightarrow \\infty $}\\right\\rbrace .$ The events $C_Y$ and $C_U$ are defined in a similar way and are clearly tail events of $(Y_i)_{i \\in \\mathbb {Z}_+}$ and $(U_i)_{i \\in \\mathbb {Z}_+}$ , respectively.", "Clearly, $\\operatorname{\\mathbb {P}}(C_U) = \\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U\\,\|\\,U_0=x) \\, \\mathrm {d}\\eta (x) &= \\frac{1}{2}\\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\, \\mathrm {d}\\mu (x)+ \\frac{1}{2}\\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\, \\mathrm {d}\\nu (x) \\\\& \\geqslant \\frac{1}{2}\\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\, \\mathrm {d}\\mu (x) = \\frac{1}{2}\\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_Z \\,\|\\,Z_0=x) \\, \\mathrm {d}\\mu (x) \\\\& = \\frac{1}{2} \\operatorname{\\mathbb {P}}(C_Z) = \\frac{1}{2}.$ Here we used that $\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x)= \\operatorname{\\mathbb {P}}(C_Z \\,\|\\,Z_0=x)$ $\\mu $ -a.e.", "$x\\in \\mathbb {R}_+$ , since the conditional probabilities on both sides depend only on the transition probability kernel of the CIR process given by the first SDE of (REF ) irrespective of the initial distribution.", "Further, we note that $\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x)$ is defined uniquely only $\\eta $ -a.e.", "$x\\in \\mathbb {R}_+$ , but, by the definition of $\\eta $ , this means both $\\mu $ -a.e.", "$x\\in \\mathbb {R}_+$ , and $\\nu $ -a.e.", "$x\\in \\mathbb {R}_+$ , and similarly $\\operatorname{\\mathbb {P}}(C_Z \\,\|\\,Z_0=x)$ is defined $\\mu $ -a.e.", "$x\\in \\mathbb {R}_+$ , so our equalities are valid.", "Thus, we have $\\operatorname{\\mathbb {P}}(C_U) \\geqslant \\frac{1}{2}$ .", "But since $C_U$ is a tail event of $(U_i)_{i \\in \\mathbb {Z}_+}$ , its probability must be either 0 or 1 (since the tail $\\sigma $ -field is trivial), hence $\\operatorname{\\mathbb {P}}(C_U)=1$ .", "Hence $2 = \\int _0^\\infty \\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\,\\mathrm {d}\\mu (x)+ \\int _0^\\infty \\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\,\\mathrm {d}\\nu (x)\\leqslant \\mu ([0,\\infty )) + \\nu ([0,\\infty ))=2,$ yielding that $\\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\, \\mathrm {d}\\mu (x) = \\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\, \\mathrm {d}\\nu (x) = 1,$ and the second equality is exactly (REF ) after we note that, by the same argument as above, $\\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_U \\,\|\\,U_0=x) \\, \\mathrm {d}\\nu (x)= \\int _{0}^{\\infty }\\operatorname{\\mathbb {P}}(C_Y \\,\|\\,Y_0=x) \\, \\mathrm {d}\\nu (x) = \\operatorname{\\mathbb {P}}(C_Y).$ With this our proof is complete.", "$\\Box $ In what follows we recall some limit theorems for (local) martingales.", "We will use these limit theorems later on for studying the asymptotic behaviour of (conditional) least squares estimators for $(a, b, \\alpha , \\beta )$ .", "First, we recall a strong law of large numbers for discrete time square-integrable martingales.", "Theorem.", "2.5 (Shiryaev [16]) Let $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_n)_{n\\in \\mathbb {N}}, \\operatorname{\\mathbb {P}}\\bigr )$ be a filtered probability space.", "Let $(M_n)_{n\\in \\mathbb {N}}$ be a square-integrable martingale with respect to the filtration $({\\mathcal {F}}_n)_{n\\in \\mathbb {N}}$ such that $\\operatorname{\\mathbb {P}}(M_0=0)=1$ and $\\operatorname{\\mathbb {P}}(\\lim _{n\\rightarrow \\infty } \\langle M\\rangle _n = \\infty )=1$ , where $(\\langle M\\rangle _n)_{n\\in \\mathbb {N}}$ denotes the predictable quadratic variation process of $M$ .", "Then $\\frac{M_n}{\\langle M \\rangle _n} \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }0 \\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Next, we recall a martingale central limit theorem in discrete time.", "Theorem.", "2.6 (Jacod and Shiryaev [9]) Let $\\lbrace ({M}_{n,k},{\\mathcal {F}}_{n,k}) : k=0,1,\\ldots ,k_n \\rbrace _{n\\in \\mathbb {N}}$ be a sequence of $d$ -dimensional square-integrable martingales with ${M}_{n,0} = {0}$ such that there exists some symmetric, positive semi-definite non-random matrix ${D}\\in \\mathbb {R}^{d\\times d}$ such that $\\sum _{k=1}^{k_n}\\operatorname{\\mathbb {E}}(({M}_{n,k} - {M}_{n,k-1})({M}_{n,k} - {M}_{n,k-1})^\\top \\,\|\\,{\\mathcal {F}}_{n,k-1})\\stackrel{\\operatorname{\\mathbb {P}}}{\\longrightarrow }{D}\\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ and for all $\\varepsilon \\in \\mathbb {R}_{++}$ , $\\sum _{k=1}^{k_n}\\operatorname{\\mathbb {E}}( \\Vert {M}_{n,k} - {M}_{n,k-1}\\Vert ^2\\mathbb {1}_{\\lbrace \\Vert {M}_{n,k}-{M}_{n,k-1}\\Vert \\geqslant \\varepsilon \\rbrace }\\,\|\\,{\\mathcal {F}}_{n,k-1})\\stackrel{\\operatorname{\\mathbb {P}}}{\\longrightarrow }0 \\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Then $\\sum _{k=1}^{k_n}({M}_{n,k} - {M}_{n,k-1}) = {M}_{n,k_n}\\stackrel{{\\mathcal {L}}}{\\longrightarrow }{\\mathcal {N}}_d({0}, {D}) \\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ where ${\\mathcal {N}}_d({0}, {D})$ denotes a $d$ -dimensional normal distribution with mean vector 0 and covariance matrix ${D}$ .", "In all the remaining sections, we will consider the subcritical Heston model (REF ) with a non-random initial value $(y_0, x_0) \\in \\mathbb {R}_+ \\times \\mathbb {R}$ .", "Note that the augmented filtration $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ corresponding to $(W_t,B_t)_{t\\in \\mathbb {R}_+}$ and the initial value $(y_0,x_0)\\in \\mathbb {R}_+\\times \\mathbb {R}$ , in fact, does not depend on $(y_0,x_0)$ ." ], [ "CLSE based on discrete time observations", "Using (REF ) and (), by an easy calculation, for all $i \\in \\mathbb {N}$ , $\\operatorname{\\mathbb {E}}\\left( \\begin{bmatrix}Y_i \\\\X_i\\end{bmatrix} \\bigg \| \\, {\\mathcal {F}}_{i-1} \\right)= \\begin{bmatrix}\\mathrm {e}^{-b} & 0 \\\\- \\beta \\int _0^1 \\mathrm {e}^{-bu} \\, \\mathrm {d}u & 1 \\\\\\end{bmatrix}\\begin{bmatrix}Y_{i-1} \\\\X_{i-1} \\\\\\end{bmatrix}+ \\begin{bmatrix}\\int _0^1 \\mathrm {e}^{-bu} \\, \\mathrm {d}u & 0 \\\\- \\beta \\int _0^1 \\left(\\int _0^u \\mathrm {e}^{-bv} \\, \\mathrm {d}v \\right) \\mathrm {d}u & 1 \\\\\\end{bmatrix}\\begin{bmatrix}a \\\\\\alpha \\\\\\end{bmatrix}.$ Using that $\\sigma (X_1,Y_1,\\ldots ,X_{i-1},Y_{i-1}) \\subseteq {\\mathcal {F}}_{i-1}$ , $i\\in \\mathbb {N}$ , by tower rule for conditional expectations, we have $&\\operatorname{\\mathbb {E}}\\left( \\begin{bmatrix}Y_i \\\\X_i\\end{bmatrix} \\Bigg \| \\, \\sigma (X_1,Y_1,\\ldots ,X_{i-1},Y_{i-1}) \\right)= \\operatorname{\\mathbb {E}}\\left(\\operatorname{\\mathbb {E}}\\left( \\begin{bmatrix}Y_i \\\\X_i\\end{bmatrix} \\Bigg \| \\, {\\mathcal {F}}_{i-1} \\right)\\Bigg \| \\, \\sigma (X_1,Y_1,\\ldots ,X_{i-1},Y_{i-1})\\right) \\\\&\\qquad = \\begin{bmatrix}\\mathrm {e}^{-b} & 0 \\\\- \\beta \\int _0^1 \\mathrm {e}^{-bu} \\, \\mathrm {d}u & 1 \\\\\\end{bmatrix}\\begin{bmatrix}Y_{i-1} \\\\X_{i-1} \\\\\\end{bmatrix}+ \\begin{bmatrix}\\int _0^1 \\mathrm {e}^{-bu} \\, \\mathrm {d}u & 0 \\\\- \\beta \\int _0^1 \\left(\\int _0^u \\mathrm {e}^{-bv} \\, \\mathrm {d}v \\right) \\mathrm {d}u & 1 \\\\\\end{bmatrix}\\begin{bmatrix}a \\\\\\alpha \\\\\\end{bmatrix}, \\qquad i\\in \\mathbb {N},$ and hence a CLSE of $(a, b, \\alpha , \\beta )$ based on discrete time observations $(Y_i, X_i)_{i\\in \\lbrace 1,\\ldots ,n\\rbrace }$ could be obtained by solving the extremum problem $\\operatornamewithlimits{arg\\,min}_{(a,b,\\alpha ,\\beta )\\in \\mathbb {R}^4}\\sum _{i=1}^n\\left[ (Y_i - d Y_{i-1} - c)^2+ (X_i - X_{i-1} - \\gamma - \\delta Y_{i-1})^2 \\right] ,$ where $\\begin{split}d &:= d(b) := \\mathrm {e}^{-b} , \\qquad c := c(a,b) := a \\int _0^1 \\mathrm {e}^{-bu} \\, \\mathrm {d}u , \\\\\\delta &:= \\delta (b, \\beta ) := -\\beta \\int _0^1 \\mathrm {e}^{-bu} \\, \\mathrm {d}u , \\qquad \\gamma := \\gamma (a, b, \\alpha , \\beta ):= \\alpha - a \\beta \\int _0^1 \\left(\\int _0^u \\mathrm {e}^{-bv} \\, \\mathrm {d}v \\right) \\mathrm {d}u.\\end{split}$ First, we determine the CLSE of $(c, d, \\gamma , \\delta )$ by minimizing the sum on the right hand side of (REF ) with respect to $(c, d, \\gamma , \\delta ) \\in \\mathbb {R}^4$ .", "We get $\\begin{bmatrix}\\widehat{c}_n^{\\mathrm {CLSE}} \\\\\\widehat{d}_n^{\\mathrm {CLSE}} \\\\\\widehat{\\gamma }_n^{\\mathrm {CLSE}} \\\\\\widehat{\\delta }_n^{\\mathrm {CLSE}}\\end{bmatrix}&= \\left( {I}_2 \\otimes \\begin{bmatrix}n & \\sum _{i=1}^n Y_{i-1} \\\\\\sum _{i=1}^n Y_{i-1} & \\sum _{i=1}^n Y_{i-1}^2\\end{bmatrix}^{-1} \\right)\\begin{bmatrix}\\sum _{i=1}^n Y_{i} \\\\\\sum _{i=1}^n Y_i Y_{i-1} \\\\X_n - x_0 \\\\\\sum _{i=1}^n (X_i - X_{i-1}) Y_{i-1}\\end{bmatrix}$ provided that $n \\sum _{i=1}^n Y_{i-1}^2 > \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2$ , where $\\otimes $ denotes Kronecker product of matrices.", "Indeed, with the notation $f(c, d, \\gamma , \\delta ):= \\sum _{i=1}^n\\left[ (Y_i - d Y_{i-1} - c)^2+ (X_i - X_{i-1} - \\gamma - \\delta Y_{i-1})^2 \\right] ,\\qquad (c, d, \\gamma , \\delta ) \\in \\mathbb {R}^4 ,$ we have $&\\frac{\\partial f}{\\partial c}(c, d, \\gamma , \\delta )= -2\\sum _{i=1}^n (Y_i - dY_{i-1} - c),\\\\&\\frac{\\partial f}{\\partial d}(c, d, \\gamma , \\delta )= -2\\sum _{i=1}^n Y_{i-1}(Y_i - dY_{i-1} - c),\\\\&\\frac{\\partial f}{\\partial \\gamma }(c, d, \\gamma , \\delta )= -2\\sum _{i=1}^n (X_i - X_{i-1} - \\gamma - \\delta Y_{i-1}),\\\\&\\frac{\\partial f}{\\partial \\delta }(c, d, \\gamma , \\delta )= -2\\sum _{i=1}^n Y_{i-1}(X_i - X_{i-1} - \\gamma - \\delta Y_{i-1}) .$ Hence the system of equations consisting of the first order partial derivates of $f$ being equal to 0 takes the form $\\left( {I}_2 \\otimes \\begin{bmatrix}n & \\sum _{i=1}^n Y_{i-1} \\\\\\sum _{i=1}^n Y_{i-1} & \\sum _{i=1}^n Y_{i-1}^2\\end{bmatrix} \\right)\\begin{bmatrix}c \\\\d \\\\\\gamma \\\\\\delta \\end{bmatrix}= \\begin{bmatrix}\\sum _{i=1}^n Y_i \\\\\\sum _{i=1}^n Y_{i-1} Y_i \\\\X_n - x_0 \\\\\\sum _{i=1}^n (X_i - X_{i-1}) Y_{i-1}\\end{bmatrix} .$ This implies (REF ), since the $4\\times 4$ -matrix consisting of the second order partial derivatives of $f$ having the form $2 {I}_2 \\otimes \\begin{bmatrix}n & \\sum _{i=1}^n Y_{i-1} \\\\\\sum _{i=1}^n Y_{i-1} & \\sum _{i=1}^n Y_{i-1}^2\\end{bmatrix}$ is positive definite provided that $n \\sum _{i=1}^n Y_{i-1}^2 > \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2$ .", "In fact, it turned out that for the calculation of the CLSE of $(c, d, \\gamma , \\delta )$ , one does not need to know the values of the parameters $\\sigma _1, \\sigma _2$ and $\\varrho $ .", "The next lemma assures the unique existence of the CLSE of $(c,d,\\gamma ,\\delta )$ based on discrete time observations.", "Lemma.", "3.1 If $a \\in \\mathbb {R}_{++}$ , $b \\in \\mathbb {R}$ , $\\sigma _1\\in \\mathbb {R}_{++}$ , and $Y_0 = y_0 \\in \\mathbb {R}_+$ , then for all $n\\geqslant 2$ , $n\\in \\mathbb {N}$ , we have $\\operatorname{\\mathbb {P}}\\left( n\\sum _{i=1}^n Y_{i-1}^2 > \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2\\right)=1,$ and hence, supposing also that $\\alpha ,\\beta \\in \\mathbb {R}$ , $\\sigma _2\\in \\mathbb {R}_{++}$ , $\\varrho \\in (-1,1)$ , there exists a unique CLSE $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}, \\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}})$ of $(c,d,\\gamma ,\\delta )$ which has the form given in (REF ).", "Proof.", "By an easy calculation, $n\\sum _{i=1}^n Y_{i-1}^2 - \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2= n \\sum _{i=1}^n \\left( Y_{i-1} - \\frac{1}{n}\\sum _{j=1}^n Y_{j-1} \\right)^2\\geqslant 0,$ and equality holds if and only if $Y_{i-1} = \\frac{1}{n}\\sum _{j=1}^n Y_{j-1},\\qquad i=1,\\ldots ,n\\qquad \\Longleftrightarrow \\qquad Y_0=Y_1=\\cdots =Y_{n-1}.$ Then, for all $n\\geqslant 2$ , $\\operatorname{\\mathbb {P}}(Y_0 = Y_1 = \\cdots = Y_{n-1})\\leqslant \\operatorname{\\mathbb {P}}(Y_0=Y_1) = \\operatorname{\\mathbb {P}}(Y_1=y_0)=0,$ since the law of $Y_1$ is absolutely continuous, see, e.g., Cox et al.", "[4].", "$\\Box $ Note that Lemma REF is valid for all $b\\in \\mathbb {R}$ , i.e., not only for the subcritical Heston model.", "Next, we describe the asymptotic behaviour of the CLSE of $(c,d,\\gamma ,\\delta )$ .", "Theorem.", "3.2 If $a, b \\in \\mathbb {R}_{++}$ , $\\alpha , \\beta \\in \\mathbb {R}$ , $\\sigma _1, \\sigma _2 \\in \\mathbb {R}_{++}$ , $\\varrho \\in (-1, 1)$ and $(Y_0, X_0) = (y_0, x_0) \\in \\mathbb {R}_{++} \\times \\mathbb {R}$ , then the CLSE $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}, \\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}})$ of $(c, d, \\gamma , \\delta )$ given in (REF ) is strongly consistent and asymptotically normal, i.e., $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}, \\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}})\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }(c, d, \\gamma , \\delta ) \\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ and $\\sqrt{n}\\begin{bmatrix}\\widehat{c}_n^{\\mathrm {CLSE}} - c \\\\\\widehat{d}_n^{\\mathrm {CLSE}} - d \\\\\\widehat{\\gamma }_n^{\\mathrm {CLSE}} - \\gamma \\\\\\widehat{\\delta }_n^{\\mathrm {CLSE}} - \\delta \\end{bmatrix}\\stackrel{{\\mathcal {L}}}{\\longrightarrow }{\\mathcal {N}}_4\\left({0}, {E}\\right)\\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ with some explicitly given symmetric, positive definite matrix ${E}\\in \\mathbb {R}^{2\\times 2}$ given in (REF ).", "Proof.", "By (REF ), we get $\\begin{split}\\begin{bmatrix}\\widehat{c}_n^{\\mathrm {CLSE}} \\\\\\widehat{d}_n^{\\mathrm {CLSE}}\\end{bmatrix}&=\\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\left( \\sum _{i=1}^{n}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix} Y_i \\right) \\\\&=\\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)\\begin{bmatrix} c \\\\ d \\end{bmatrix} \\\\&\\quad + \\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\sum _{i=1}^{n}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix} (Y_i - c - d Y_{i-1}) \\\\&= \\begin{bmatrix} c \\\\ d \\end{bmatrix}+ \\left( \\frac{1}{n}\\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\frac{1}{n}\\sum _{i=1}^{n} \\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix} \\varepsilon _i ,\\end{split}$ where $\\varepsilon _i := Y_i - c - d Y_{i-1}$ , $i \\in \\mathbb {N}$ , provided that $n \\sum _{i=1}^n Y_{i-1}^2 > \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2$ .", "By (REF ) and (REF ), $\\operatorname{\\mathbb {E}}(Y_i \\,\|\\,{\\mathcal {F}}_{i-1}) = d Y_{i-1} + c$ , $i \\in \\mathbb {N}$ , and hence $(\\varepsilon _i)_{i\\in \\mathbb {N}}$ is a sequence of martingale differences with respect to the filtration $({\\mathcal {F}}_i)_{i\\in \\mathbb {Z}_+}$ .", "By (REF ), we have $Y_i &= \\mathrm {e}^{-b} Y_{i-1} + a \\int _{i-1}^i \\mathrm {e}^{-b(i-u)} \\, \\mathrm {d}u+ \\sigma _1 \\int _{i-1}^i \\mathrm {e}^{-b(i-u)} \\sqrt{Y_u} \\, \\mathrm {d}W_u \\\\&= d Y_{i-1} + c+ \\sigma _1 \\int _{i-1}^i \\mathrm {e}^{-b(i-u)} \\sqrt{Y_u} \\, \\mathrm {d}W_u ,\\qquad i \\in \\mathbb {N},$ hence, by Proposition 3.2.10 in Karatzas and Shreve [12] and (REF ), we have $\\operatorname{\\mathbb {E}}(\\varepsilon _i^2\\,\|\\,{\\mathcal {F}}_{i-1})&= \\sigma _1^2\\operatorname{\\mathbb {E}}\\biggl (\\left( \\int _{i-1}^i \\mathrm {e}^{-b(i-u)} \\sqrt{Y_u} \\, \\mathrm {d}W_u \\right)^2\\, \\bigg \| \\, {\\mathcal {F}}_{i-1} \\biggr )= \\sigma _1^2 \\int _{i-1}^i \\mathrm {e}^{-2b(i-u)} \\operatorname{\\mathbb {E}}(Y_u \\,\|\\,{\\mathcal {F}}_{i-1}) \\, \\mathrm {d}u \\\\&= \\sigma _1^2 \\int _{i-1}^i \\mathrm {e}^{-2b(i-u)} \\mathrm {e}^{-b(u-i+1)} Y_{i-1} \\, \\mathrm {d}u+ \\sigma _1^2 \\int _{i-1}^{i}\\mathrm {e}^{-2b(i-u)}a \\int _{i-1}^{u} \\mathrm {e}^{-b(u-v)} \\, \\mathrm {d}v \\, \\mathrm {d}u \\\\&= \\sigma _1^2 Y_{i-1} \\int _0^1 \\mathrm {e}^{-b(2-v)} \\, \\mathrm {d}v+ \\sigma _1^2 a \\int _0^1 \\int _0^u \\mathrm {e}^{-b(2-v-u)} \\, \\mathrm {d}v \\, \\mathrm {d}u=: C_1 Y_{i-1} + C_2 .$ Now we apply Theorem REF to the square-integrable martingale $M_n^{(c)} := \\sum _{i=1}^{n} \\varepsilon _i$ , $n \\in \\mathbb {N}$ , which has predictable quadratic variation process $\\langle M^{(c)} \\rangle _n = \\sum _{i=1}^n \\operatorname{\\mathbb {E}}(\\varepsilon _i^2 \\,\|\\,{\\mathcal {F}}_{i-1})= C_1 \\sum _{i=1}^n Y_{i-1} + C_2 n$ , $n \\in \\mathbb {N}$ , see, e.g., Shiryaev [16].", "By (REF ) and (REF ), $\\frac{\\langle M^{(c)} \\rangle _n}{n} \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }C_1 \\operatorname{\\mathbb {E}}(Y_\\infty ) + C_2 \\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ and since $C_1,C_2\\in \\mathbb {R}_{++}$ , $\\langle M^{(c)}\\rangle _n \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\infty $ as $n \\rightarrow \\infty $ .", "Hence, by Theorem REF , $\\frac{1}{n} \\sum _{i=1}^n \\varepsilon _i= \\frac{M_n^{(c)}}{\\langle M^{(c)} \\rangle _n}\\frac{\\langle M^{(c)} \\rangle _n}{n}\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }0 \\cdot (C_1 \\operatorname{\\mathbb {E}}(Y_{\\infty }) + C_2) = 0\\qquad \\text{as \\ $n\\rightarrow \\infty $.", "}$ Similarly, $\\operatorname{\\mathbb {E}}(Y_{i-1}^2 \\varepsilon _i^2 \\,\|\\,{\\mathcal {F}}_{i-1})= Y_{i-1}^2 \\operatorname{\\mathbb {E}}(\\varepsilon _i^2\\,\|\\,{\\mathcal {F}}_{i-1})= C_1 Y_{i-1}^3 + C_2Y_{i-1}^2,\\qquad i \\in \\mathbb {N},$ and, by essentially the same reasoning as before, $\\frac{1}{n} \\sum _{i=1}^n Y_{i-1} \\varepsilon _i \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }0$ as $n \\rightarrow \\infty $ .", "By (REF ) and (REF ), $\\left( \\frac{1}{n}\\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}= \\begin{bmatrix}1 & \\frac{1}{n} \\sum _{i=1}^n Y_{i-1} \\\\\\frac{1}{n} \\sum _{i=1}^n Y_{i-1} & \\frac{1}{n} \\sum _{i=1}^n Y_{i-1}^2\\end{bmatrix}^{-1}\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}^{-1}$ as $n \\rightarrow \\infty $ , where we used that $\\operatorname{\\mathbb {E}}(Y_\\infty ^2) - (\\operatorname{\\mathbb {E}}(Y_\\infty ))^2 = \\frac{a\\sigma _1^2}{2b^2} \\in \\mathbb {R}_{++}$ , and consequently, the limit is indeed non-singular.", "Thus, by (REF ), $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}) \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }(c, d)$ as $n \\rightarrow \\infty $ .", "Further, by (REF ), $\\begin{split}\\begin{bmatrix} \\widehat{\\gamma }_n^{\\mathrm {CLSE}} \\\\ \\widehat{\\delta }_n^{\\mathrm {CLSE}} \\end{bmatrix}&= \\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}(X_i - X_{i-1}) \\right) \\\\&= \\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)\\begin{bmatrix}\\gamma \\\\ \\delta \\end{bmatrix} \\\\&\\quad + \\left( \\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}(X_i - X_{i-1} - \\gamma - \\delta Y_{i-1}) \\\\&= \\begin{bmatrix} \\gamma \\\\ \\delta \\end{bmatrix}+ \\left( \\frac{1}{n}\\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1}\\frac{1}{n}\\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\eta _i ,\\end{split}$ where $\\eta _i := X_i - X_{i-1} - \\gamma - \\delta Y_{i-1}$ , $i \\in \\mathbb {N}$ , provided that $n \\sum _{i=1}^n Y_{i-1}^2 > \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2$ .", "By (REF ) and (REF ), $\\operatorname{\\mathbb {E}}(X_i \\,\|\\,{\\mathcal {F}}_{i-1}) = X_{i-1} + \\delta Y_{i-1} + \\gamma $ , $i \\in \\mathbb {N}$ , and hence $(\\eta _i)_{i\\in \\mathbb {N}}$ is a sequence of martingale differences with respect to the filtration $({\\mathcal {F}}_i)_{i\\in \\mathbb {Z}_+}$ .", "By (REF ) and (REF ), with the notation $\\widetilde{W}_t := \\varrho W_t + \\sqrt{1 - \\varrho ^2} B_t$ , $t \\in \\mathbb {R}_+$ , we compute $X_i - X_{i-1} &= \\int _{i-1}^i (\\alpha - \\beta Y_u) \\, \\mathrm {d}u+ \\sigma _2 \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u= \\alpha - \\beta \\int _{i-1}^i Y_u \\, \\mathrm {d}u+ \\sigma _2 \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u \\\\&= \\alpha - \\beta \\int _{i-1}^i\\left( \\mathrm {e}^{-b(u-(i-1))} Y_{i-1}+ a \\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\, \\mathrm {d}v+ \\sigma _1\\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v \\right)\\mathrm {d}u \\\\&\\quad + \\sigma _2 \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u \\\\&= \\alpha - \\beta Y_{i-1} \\int _{i-1}^i \\mathrm {e}^{-b(u-i+1)} \\, \\mathrm {d}u- a \\beta \\int _{i-1}^i \\left( \\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\, \\mathrm {d}v \\right) \\mathrm {d}u \\\\&\\quad - \\beta \\sigma _1\\int _{i-1}^i\\left( \\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v \\right) \\mathrm {d}u+ \\sigma _2 \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u$ $&= \\alpha - \\beta Y_{i-1} \\int _{0}^1 \\mathrm {e}^{-bv} \\, \\mathrm {d}v- a \\beta \\int _{0}^1 \\left( \\int _{0}^u \\mathrm {e}^{-bv} \\,\\mathrm {d}v \\right) \\mathrm {d}u \\\\&\\quad - \\beta \\sigma _1\\int _{i-1}^i\\left( \\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v \\right) \\mathrm {d}u+ \\sigma _2 \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u \\\\&= \\delta Y_{i-1} + \\gamma - \\beta \\sigma _1\\int _{i-1}^i\\left( \\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v \\right) \\mathrm {d}u+ \\sigma _2 \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u ,$ and consequently, $\\operatorname{\\mathbb {E}}(\\eta _i^2 \\,\|\\,{\\mathcal {F}}_{i-1})&= \\beta ^2 \\sigma _1^2\\operatorname{\\mathbb {E}}\\biggl [ \\biggl ( \\int _{i-1}^i \\int _{i-1}^u\\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v\\, \\mathrm {d}u \\biggr )^2\\,\\bigg \|\\, {\\mathcal {F}}_{i-1} \\biggr ]+ \\sigma _2^2\\operatorname{\\mathbb {E}}\\biggl [ \\biggl ( \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u \\biggr )^2\\,\\bigg \|\\, {\\mathcal {F}}_{i-1} \\biggr ] \\\\&\\quad - 2 \\beta \\sigma _1 \\sigma _2\\operatorname{\\mathbb {E}}\\biggl [ \\bigg ( \\int _{i-1}^i \\int _{i-1}^u\\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v\\, \\mathrm {d}u \\biggr )\\biggl ( \\varrho \\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}W_u \\biggr )\\,\\bigg \|\\, {\\mathcal {F}}_{i-1} \\biggr ] \\\\&\\quad - 2 \\beta \\sigma _1 \\sigma _2\\operatorname{\\mathbb {E}}\\biggl [ \\biggl ( \\int _{i-1}^i \\int _{i-1}^u\\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v \\, \\mathrm {d}u \\biggr )\\biggl ( \\sqrt{1 - \\varrho ^2}\\int _{i-1}^i \\sqrt{Y_u} \\, \\mathrm {d}B_u \\biggr )\\,\\bigg \|\\, {\\mathcal {F}}_{i-1} \\biggr ] .$ We use Equation (3.2.23) from Karatzas and Shreve [12] to the first, second and third terms, and Proposition 3.2.17 from Karatzas and Shreve [12] to the fourth term (together with the independence of $W$ and $B$ ): $\\operatorname{\\mathbb {E}}(\\eta _i^2 \| {\\mathcal {F}}_{i-1})&= \\beta ^2 \\sigma _1^2 \\int _{i-1}^i \\int _{i-1}^i\\operatorname{\\mathbb {E}}\\left( \\int _{i-1}^u \\mathrm {e}^{-b(u-w)}\\sqrt{Y_w}\\,\\mathrm {d}W_w\\int _{i-1}^v \\mathrm {e}^{-b(v-w)}\\sqrt{Y_w}\\,\\mathrm {d}W_w\\;\\Big \\vert \\; {\\mathcal {F}}_{i-1}\\right)\\mathrm {d}v\\,\\mathrm {d}u\\\\&\\quad + \\sigma _2^2 \\int _{i-1}^i \\operatorname{\\mathbb {E}}(Y_u \\,\|\\,{\\mathcal {F}}_{i-1}) \\, \\mathrm {d}u \\\\&\\quad - 2 \\beta \\sigma _1 \\sigma _2 \\varrho \\int _{i-1}^i \\operatorname{\\mathbb {E}}\\left( \\int _{i-1}^u \\mathrm {e}^{-b(u-w)}\\sqrt{Y_w}\\,\\mathrm {d}W_w\\int _{i-1}^i \\sqrt{Y_w}\\,\\mathrm {d}W_w \\;\\Big \\vert \\; {\\mathcal {F}}_{i-1} \\right)\\mathrm {d}u - 0\\\\&= \\beta ^2 \\sigma _1^2\\int _{i-1}^i \\int _{i-1}^i \\int _{i-1}^{u \\wedge v}\\mathrm {e}^{-b(u+v-2w)} \\operatorname{\\mathbb {E}}(Y_w \\,\|\\,{\\mathcal {F}}_{i-1}) \\, \\mathrm {d}w \\, \\mathrm {d}u \\, \\mathrm {d}v+ \\sigma _2^2 \\int _{i-1}^i \\operatorname{\\mathbb {E}}(Y_u \\,\|\\,{\\mathcal {F}}_{i-1}) \\, \\mathrm {d}u \\\\&\\quad - 2 \\beta \\sigma _1 \\sigma _2 \\varrho \\int _{i-1}^i \\int _{i-1}^u \\mathrm {e}^{-b(u-v)}\\operatorname{\\mathbb {E}}(Y_v \\,\|\\,{\\mathcal {F}}_{i-1}) \\, \\mathrm {d}v \\, \\mathrm {d}u.$ Using again (REF ), we get $&\\operatorname{\\mathbb {E}}(\\eta _i^2 \| {\\mathcal {F}}_{i-1})= \\beta ^2 \\sigma _1^2 Y_{i-1}\\int _{i-1}^i \\int _{i-1}^i \\int _{i-1}^{u \\wedge v}\\mathrm {e}^{-b(u+v-w-(i-1))} \\, \\mathrm {d}w \\, \\mathrm {d}v \\, \\mathrm {d}u \\\\&\\phantom{\\operatorname{\\mathbb {E}}(\\eta _i^2 \| {\\mathcal {F}}_{i-1})\\quad }+ a \\beta ^2 \\sigma _1^2\\int _{i-1}^i \\int _{i-1}^i \\int _{i-1}^{u \\wedge v} \\int _{i-1}^w\\mathrm {e}^{-b(u+v-w-z)} \\, \\mathrm {d}z \\, \\mathrm {d}w \\, \\mathrm {d}v \\, \\mathrm {d}u+ \\sigma _2^2 Y_{i-1} \\int _{i-1}^i e^{-b(u-(i-1))} \\, \\mathrm {d}u \\\\&\\phantom{\\operatorname{\\mathbb {E}}(\\eta _i^2 \| {\\mathcal {F}}_{i-1})\\quad }+ a \\sigma _2^2 \\int _{i-1}^i \\int _{i-1}^u \\mathrm {e}^{-b(u-v)} \\, \\mathrm {d}v \\, \\mathrm {d}u- 2 \\beta \\sigma _1 \\sigma _2 \\varrho Y_{i-1}\\int _{i-1}^i \\int _{i-1}^u \\mathrm {e}^{-b(u-(i-1))} \\, \\mathrm {d}v \\, \\mathrm {d}u \\\\&\\phantom{\\operatorname{\\mathbb {E}}(\\eta _i^2 \| {\\mathcal {F}}_{i-1})\\quad }- 2 a \\beta \\sigma _1 \\sigma _2 \\varrho \\int _{i-1}^i \\int _{i-1}^u \\int _{i-1}^v\\mathrm {e}^{-b(u-w)} \\, \\mathrm {d}w \\, \\mathrm {d}v \\, \\mathrm {d}u$ $&= \\biggl ( \\beta ^2 \\sigma _1^2\\int _{0}^1 \\int _{0}^1 \\int _{0}^{u^{\\prime } \\wedge v^{\\prime }}\\!\\!\\mathrm {e}^{-b(u^{\\prime }+v^{\\prime }-w^{\\prime })} \\, \\mathrm {d}w^{\\prime } \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }- 2 \\beta \\sigma _1 \\sigma _2 \\varrho \\int _{0}^1 \\int _{0}^{u^{\\prime }} \\mathrm {e}^{-bu^{\\prime }} \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }+ \\sigma _2^2 \\int _{0}^1 \\!\\!\\mathrm {e}^{-bu^{\\prime }} \\, \\mathrm {d}u^{\\prime } \\biggr ) Y_{i-1} \\\\&\\quad + a \\beta ^2 \\sigma _1^2\\int _{0}^1 \\int _{0}^1 \\int _{0}^{u^{\\prime } \\wedge v^{\\prime }} \\int _{0}^{w^{\\prime }}\\!\\!", "\\mathrm {e}^{-b(u^{\\prime }+v^{\\prime }-w^{\\prime }-z^{\\prime })} \\, \\mathrm {d}z^{\\prime } \\, \\mathrm {d}w^{\\prime } \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime } \\\\&\\quad + a \\sigma _2^2 \\int _{0}^1 \\int _{0}^{u^{\\prime }} \\mathrm {e}^{-b(u^{\\prime }-v^{\\prime })} \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }- 2 a \\beta \\sigma _1 \\sigma _2 \\varrho \\int _{0}^1 \\int _{0}^{u^{\\prime }} \\int _{0}^{v^{\\prime }}\\mathrm {e}^{-b(u^{\\prime }-w^{\\prime })} \\, \\mathrm {d}w^{\\prime } \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }=: C_3 Y_{i-1} + C_4 .$ Now we apply Theorem REF to the square-integrable martingale $M_n^{(\\gamma )} := \\sum _{i=1}^n \\eta _i$ , $n \\in \\mathbb {N}$ , which has predictable quadratic variation process $\\langle M^{(\\gamma )} \\rangle _n = \\sum _{i=1}^n \\operatorname{\\mathbb {E}}(\\eta _i^2 \\,\|\\,{\\mathcal {F}}_{i-1})= C_3 \\sum _{i=1}^n Y_{i-1} + C_4 n$ , $n \\in \\mathbb {N}$ .", "By (REF ), $\\frac{\\langle M^{(\\gamma )} \\rangle _n}{n} \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }C_3 \\operatorname{\\mathbb {E}}(Y_\\infty ) + C_4 \\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Note that $C_3 \\geqslant 0$ and $C_4 \\geqslant 0$ , since $\\operatorname{\\mathbb {E}}(\\eta _1^2 \\,\|\\,{\\mathcal {F}}_0) = C_3 y_0 + C_4 \\geqslant 0$ for all $y_0 \\in \\mathbb {R}_+$ .", "By setting $y_0 = 0$ , we can see that $C_4 \\geqslant 0$ , and then, by taking the limit $y_0\\rightarrow \\infty $ on the right-hand side of the inequality $C_3\\geqslant -\\frac{C_4}{y_0}$ , $y_0>0$ , we get $C_3 \\geqslant 0$ as well.", "Note also that $\\langle M^{(\\gamma )} \\rangle _n \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\infty $ as $n \\rightarrow \\infty $ provided that $C_3 + C_4 > 0$ .", "If $C_3 = 0$ and $C_4 = 0$ , then $\\operatorname{\\mathbb {E}}(\\eta _i^2 \\,\|\\,{\\mathcal {F}}_{i-1}) = 0$ , $i \\in \\mathbb {N}$ , and consequently $\\operatorname{\\mathbb {E}}(\\eta _i^2)=0$ , $i\\in \\mathbb {N}$ , and, since $\\operatorname{\\mathbb {E}}(\\eta _i) = 0$ , $i \\in \\mathbb {N}$ , we have $\\operatorname{\\mathbb {P}}(\\eta _i = 0) = 1$ , $i \\in \\mathbb {N}$ , implying that $\\operatorname{\\mathbb {P}}(\\sum _{i=1}^n \\eta _i = 0) = 1$ and $\\operatorname{\\mathbb {P}}(\\sum _{i=1}^n Y_{i-1} \\eta _i = 0) = 1$ , $n \\in \\mathbb {N}$ , i.e., in this case, by (REF ), $(\\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}}) = (\\gamma , \\delta )$ , $n \\in \\mathbb {N}$ , almost surely.", "If $C_3 + C_4 > 0$ , then, by Theorem REF , $\\frac{1}{n} \\sum _{i=1}^n \\eta _i= \\frac{M_n^{(\\gamma )}}{\\langle M^{(\\gamma )} \\rangle _n}\\frac{\\langle M^{(\\gamma )} \\rangle _n}{n}\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }0 \\cdot (C_3 \\operatorname{\\mathbb {E}}(Y_{\\infty }) + C_4)= 0\\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Similarly, $\\operatorname{\\mathbb {E}}(Y_{i-1}^2 \\eta _i^2 \\,\|\\,{\\mathcal {F}}_{i-1})= Y_{i-1}^2 \\operatorname{\\mathbb {E}}(\\eta _i^2\\,\|\\,{\\mathcal {F}}_{i-1})= C_3 Y_{i-1}^3 + C_4Y_{i-1}^2 ,\\qquad i \\in \\mathbb {N},$ and, by essentially the same reasoning as before, $\\frac{1}{n} \\sum _{i=1}^n Y_{i-1} \\eta _i \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }0$ as $n \\rightarrow \\infty $ (in the case $C_3+C_4>0$ ).", "Using (REF ) and (REF ), we have $(\\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}}) \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }(\\gamma , \\delta )$ as $n \\rightarrow \\infty $ .", "Since the intersection of two events having probability 1 is an event having probability 1, we get $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}, \\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}})\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }(c, d, \\gamma , \\delta )$ as $n \\rightarrow \\infty $ , as desired.", "Next, we turn to prove that the CLSE of $(c, d, \\gamma , \\delta )$ is asymptotically normal.", "First, using (REF ) and (REF ), we can write $\\sqrt{n}\\begin{bmatrix}\\widehat{c}_n^{\\mathrm {CLSE}} - c \\\\\\widehat{d}_n^{\\mathrm {CLSE}} - d \\\\\\widehat{\\gamma }_n^{\\mathrm {CLSE}} - \\gamma \\\\\\widehat{\\delta }_n^{\\mathrm {CLSE}} - \\delta \\end{bmatrix}= \\left( {I}_2\\otimes \\left( n^{-1}\\sum _{i=1}^n\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}\\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix}^\\top \\right)^{-1} \\right)n^{-1/2}\\sum _{i=1}^n\\begin{bmatrix} \\varepsilon _i \\\\ \\eta _i \\end{bmatrix}\\otimes \\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix} ,$ provided that $n \\sum _{i=1}^n Y_{i-1}^2 > \\left(\\sum _{i=1}^n Y_{i-1}\\right)^2$ .", "By (REF ), the first factor converges almost surely to ${I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty )\\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}^{-1}\\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ For the second factor, we are going to apply the martingale central limit theorem (see Theorem REF ) with the following choices: $d = 4$ , $k_n = n$ , $n \\in \\mathbb {N}$ , ${\\mathcal {F}}_{n,k} = {\\mathcal {F}}_k$ , $n \\in \\mathbb {N}$ , $k \\in \\lbrace 1, \\ldots , n\\rbrace $ , and ${M}_{n,k} = n^{-\\frac{1}{2}}\\sum _{i=1}^k\\begin{bmatrix} \\varepsilon _i \\\\ \\eta _i \\end{bmatrix}\\otimes \\begin{bmatrix} 1 \\\\ Y_{i-1} \\end{bmatrix} ,\\qquad n \\in \\mathbb {N}, \\quad k \\in \\lbrace 1, \\ldots , n\\rbrace .$ Then, applying the identities $({A}_1\\otimes {A}_2)^\\top = {A}_1^\\top \\otimes {A}_2^\\top $ and $({A}_1\\otimes {A}_2)({A}_3\\otimes {A}_4)=({A}_1{A}_3)\\otimes ({A}_2{A}_4)$ , $&\\operatorname{\\mathbb {E}}\\big ( ({M}_{n,k} - {M}_{n,k-1})({M}_{n,k} - {M}_{n,k-1})^\\top \\,\\big \|\\, {\\mathcal {F}}_{n,k-1} \\big ) \\\\&\\qquad \\qquad = \\frac{1}{n}\\operatorname{\\mathbb {E}}\\left( \\left( \\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}\\otimes \\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix} \\right)\\left( \\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}\\otimes \\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix} \\right)^\\top \\,\\bigg \|\\, {\\mathcal {F}}_{k-1} \\right) \\\\&\\qquad \\qquad = \\frac{1}{n}\\operatorname{\\mathbb {E}}\\left( \\left( \\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}\\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}^\\top \\right)\\otimes \\left( \\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix}\\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix}^\\top \\right)\\,\\bigg \|\\, {\\mathcal {F}}_{k-1} \\right) \\\\&\\qquad \\qquad = \\frac{1}{n}\\operatorname{\\mathbb {E}}\\left( \\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}\\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}^\\top \\,\\bigg \|\\, {\\mathcal {F}}_{k-1} \\right)\\otimes \\left( \\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix}\\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix}^\\top \\right) ,\\qquad n\\in \\mathbb {N},\\quad k\\in \\lbrace 1,\\ldots ,n\\rbrace .$ Since $\\operatorname{\\mathbb {E}}(\\varepsilon _k^2 \\,\|\\,{\\mathcal {F}}_{k-1}) = C_1 Y_{k-1} + C_2$ , $k \\in \\mathbb {N}$ , and $\\operatorname{\\mathbb {E}}(\\eta _k^2 \\,\|\\,{\\mathcal {F}}_{k-1}) = C_3 Y_{k-1} + C_4$ , $k \\in \\mathbb {N}$ , it remains to calculate $&\\operatorname{\\mathbb {E}}(\\varepsilon _k\\eta _k \\,\|\\,{\\mathcal {F}}_{k-1})= \\operatorname{\\mathbb {E}}\\bigl ( (Y_k - c - dY_{k-1})(X_k - X_{k-1} - \\gamma - \\delta Y_{k-1})\\,\\big \|\\, {\\mathcal {F}}_{k-1} \\bigr ) \\\\&= \\operatorname{\\mathbb {E}}\\left( \\left.", "\\sigma _1\\int _{k-1}^k \\mathrm {e}^{-b(k-s)} \\sqrt{Y_s} \\, \\mathrm {d}W_s\\left( - \\beta \\sigma _1\\int _{k-1}^k \\int _{k-1}^u\\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v \\, \\mathrm {d}u+ \\sigma _2\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u \\right)\\right\| {\\mathcal {F}}_{k-1} \\right) \\\\&= - \\beta \\sigma _1^2\\int _{k-1}^k\\operatorname{\\mathbb {E}}\\left( \\left.", "\\int _{k-1}^k \\mathrm {e}^{-b(k-s)} \\sqrt{Y_s} \\, \\mathrm {d}W_s\\int _{k-1}^u \\mathrm {e}^{-b(u-v)} \\sqrt{Y_v} \\, \\mathrm {d}W_v\\right\| {\\mathcal {F}}_{k-1} \\right) \\mathrm {d}u \\\\&\\quad + \\sigma _1 \\sigma _2\\operatorname{\\mathbb {E}}\\left( \\left.", "\\int _{k-1}^k \\mathrm {e}^{-b(k-s)} \\sqrt{Y_s} \\, \\mathrm {d}W_s\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}\\widetilde{W}_u\\right\| {\\mathcal {F}}_{k-1} \\right) .$ Again, by Equation (3.2.23) and Proposition 3.2.17 from Karatzas and Shreve [12], we have $\\operatorname{\\mathbb {E}}(\\varepsilon _k \\eta _k \\,\|\\,{\\mathcal {F}}_{k-1})&= - \\beta \\sigma _1^2\\int _{k-1}^k \\int _{k-1}^u \\mathrm {e}^{-b(k+u-2v)}\\operatorname{\\mathbb {E}}(Y_v \\,\|\\,{\\mathcal {F}}_{k-1}) \\, \\mathrm {d}v \\, \\mathrm {d}u \\\\&\\quad + \\sigma _1 \\sigma _2 \\varrho \\int _{k-1}^k \\mathrm {e}^{-b(k-v)} \\operatorname{\\mathbb {E}}(Y_v \\,\|\\,{\\mathcal {F}}_{k-1}) \\, \\mathrm {d}v .$ Using (REF ), by an easy calculation, $&\\operatorname{\\mathbb {E}}(\\varepsilon _k \\eta _k \\,\|\\,{\\mathcal {F}}_{k-1}) \\\\&= - \\beta \\sigma _1^2\\int _{k-1}^k \\int _{k-1}^u\\mathrm {e}^{-b(k+u-2v)}\\left( \\mathrm {e}^{-b(v-k+1)} Y_{k-1}+ a \\int _{k-1}^v \\mathrm {e}^{-b(v-s)} \\, \\mathrm {d}s \\right)\\mathrm {d}v \\, \\mathrm {d}u \\\\&\\quad + \\sigma _1 \\sigma _2 \\varrho \\int _{k-1}^k\\mathrm {e}^{-b(k-v)}\\left( \\mathrm {e}^{-b(v-k+1)} Y_{k-1}+ a \\int _{k-1}^v \\mathrm {e}^{-b(v-s)} \\, \\mathrm {d}s \\right)\\mathrm {d}v \\\\&= \\left( - \\beta \\sigma _1^2\\int _{0}^1 \\int _{0}^{u^{\\prime }} \\mathrm {e}^{-b(u^{\\prime }-v^{\\prime }+1)} \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }+ \\sigma _1 \\sigma _2 \\varrho \\mathrm {e}^{-b} \\right)Y_{k-1}- a\\beta \\sigma _1^2\\int _{0}^1 \\int _{0}^{u^{\\prime }} \\int _{0}^{v^{\\prime }}\\mathrm {e}^{-b(u^{\\prime }-v^{\\prime }-s^{\\prime }+1)} \\, \\mathrm {d}s^{\\prime } \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime } \\\\&\\quad + a\\sigma _1 \\sigma _2 \\varrho \\int _{0}^1 \\int _{0}^{v^{\\prime }} \\mathrm {e}^{-b(1-s^{\\prime })} \\, \\mathrm {d}s^{\\prime } \\, \\mathrm {d}v^{\\prime }=: C_5 Y_{k-1} + C_6 , \\qquad k \\in \\mathbb {N}.$ Hence, by (REF ) and (REF ), $&\\sum _{k=1}^n\\operatorname{\\mathbb {E}}\\big ( ({M}_{n,k} - {M}_{n,k-1})({M}_{n,k} - {M}_{n,k-1})^\\top \\,\|\\,{\\mathcal {F}}_{n,k-1} \\big ) \\\\&\\qquad = \\frac{1}{n}\\sum _{k=1}^n\\begin{bmatrix}C_1 Y_{k-1} + C_2 & C_5 Y_{k-1} + C_6 \\\\C_5 Y_{k-1} + C_6 & C_3 Y_{k-1} + C_4\\end{bmatrix}\\otimes \\begin{bmatrix}1 & Y_{k-1} \\\\Y_{k-1} & Y_{k-1}^2\\end{bmatrix} \\\\&\\qquad = \\frac{1}{n}\\sum _{k=1}^n\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3\\end{bmatrix}\\otimes \\begin{bmatrix}Y_{k-1} & Y_{k-1}^2 \\\\Y_{k-1}^2 & Y_{k-1}^3\\end{bmatrix}+ \\frac{1}{n}\\sum _{k=1}^n\\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4\\end{bmatrix}\\otimes \\begin{bmatrix}1 & Y_{k-1} \\\\Y_{k-1} & Y_{k-1}^2\\end{bmatrix} \\\\&\\qquad \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3\\end{bmatrix}\\otimes \\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & \\operatorname{\\mathbb {E}}(Y_\\infty ^3)\\end{bmatrix}+ \\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4\\end{bmatrix}\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}=: {D}\\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ where the $4\\times 4$ limit matrix ${D}$ is necessarily symmetric and positive semi-definite (indeed, the limit of positive semi-definite matrices is positive semi-definite).", "Next, we check Lindeberg condition (REF ).", "Since $\\Vert {x}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {x}\\Vert \\geqslant \\varepsilon \\rbrace }\\leqslant \\frac{\\Vert {x}\\Vert ^4}{\\varepsilon ^2} \\mathbb {1}_{\\lbrace \\Vert {x}\\Vert \\geqslant \\varepsilon \\rbrace }\\leqslant \\frac{\\Vert {x}\\Vert ^4}{\\varepsilon ^2},\\qquad {x}\\in \\mathbb {R}^4 , \\quad \\varepsilon \\in \\mathbb {R}_{++} ,$ and $\\Vert {x}\\Vert ^4=(x_1^2 + x_2^2 + x_3^2 + x_4^2)^2 \\leqslant 4 (x_1^4 + x_2^4 + x_3^4 + x_4^4)$ , $x_1, x_2, x_3, x_4 \\in \\mathbb {R}$ , it is enough to check that $&\\frac{1}{n^2}\\sum _{k=1}^n\\bigl ( \\operatorname{\\mathbb {E}}(\\varepsilon _k^4 \\,\|\\,{\\mathcal {F}}_{k-1})+ Y_{k-1}^4 \\operatorname{\\mathbb {E}}(\\varepsilon _k^4 \\,\|\\,{\\mathcal {F}}_{k-1})+ \\operatorname{\\mathbb {E}}(\\eta _k^4 \\,\|\\,{\\mathcal {F}}_{k-1})+ Y_{k-1}^4 \\operatorname{\\mathbb {E}}(\\eta _k^4 \\,\|\\,{\\mathcal {F}}_{k-1}) \\bigr ) \\\\&\\qquad \\qquad = \\frac{1}{n^2}\\sum _{k=1}^n \\operatorname{\\mathbb {E}}( (1+Y_{k-1}^4)(\\varepsilon _k^4 + \\eta _k^4) \\,\|\\,{\\mathcal {F}}_{k-1})\\stackrel{\\operatorname{\\mathbb {P}}}{\\longrightarrow }0 \\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Instead of convergence in probability, we show convergence in $L^1$ , i.e., we check that $\\frac{1}{n^2} \\sum _{k=1}^n \\operatorname{\\mathbb {E}}((1+Y_{k-1}^4)(\\varepsilon _k^4 + \\eta _k^4)) \\rightarrow 0\\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Clearly, it is enough to show that $\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}((1+Y_{k-1}^4)(\\varepsilon _k^4 + \\eta _k^4)) < \\infty .$ By Cauchy–Schwarz inequality, $\\operatorname{\\mathbb {E}}((1+Y_{k-1}^4)(\\varepsilon _k^4 + \\eta _k^4))\\leqslant \\sqrt{\\operatorname{\\mathbb {E}}((1+Y_{k-1}^4)^2) \\operatorname{\\mathbb {E}}((\\varepsilon _k^4 + \\eta _k^4)^2)}\\leqslant \\sqrt{2} \\sqrt{\\operatorname{\\mathbb {E}}((1+Y_{k-1}^4)^2) \\operatorname{\\mathbb {E}}(\\varepsilon _k^8 + \\eta _k^8)}$ for all $k \\in \\mathbb {N}$ .", "Since, by Proposition 3 in Ben Alaya and Kebaier [3], $\\sup _{t\\in \\mathbb {R}_+} \\operatorname{\\mathbb {E}}( Y_t^\\kappa ) < \\infty , \\qquad \\kappa \\in \\mathbb {R}_+ ,$ it remains to check that $\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}(\\varepsilon _k^8 + \\eta _k^8) < \\infty $ .", "Since, by the power mean inequality, $\\operatorname{\\mathbb {E}}(\\varepsilon _k^8)= \\operatorname{\\mathbb {E}}(\\vert Y_k - d Y_{k-1} - c\\vert ^8)\\leqslant \\operatorname{\\mathbb {E}}((Y_k + d Y_{k-1} + c)^8)\\leqslant 3^7 \\operatorname{\\mathbb {E}}(Y_k^8 + d^8Y_{k-1}^8 + c^8) ,\\qquad k \\in \\mathbb {N},$ using (REF ), we have $\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}(\\varepsilon _k^8) < \\infty $ .", "Using (REF ) and again the power mean inequality, we have $\\operatorname{\\mathbb {E}}(\\eta _k^8)&= \\operatorname{\\mathbb {E}}((X_k - X_{k-1} - \\gamma - \\delta Y_{k-1})^8) \\\\&= \\operatorname{\\mathbb {E}}\\left( \\left( \\alpha - \\beta \\int _{k-1}^k Y_u \\, \\mathrm {d}u+ \\sigma _2 \\varrho \\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}W_u+ \\sigma _2 \\sqrt{1-\\varrho ^2}\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}B_u- \\gamma - \\delta Y_{k-1} \\right)^8 \\right) \\\\&\\leqslant 6^7 \\operatorname{\\mathbb {E}}\\Biggl ( \\alpha ^8+ \\beta ^8 \\left( \\int _{k-1}^k Y_u \\, \\mathrm {d}u \\right)^8+ \\sigma _2^8 \\varrho ^8\\left( \\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}W_u \\right)^8+ \\sigma _2^8 (1-\\varrho ^2)^4\\left( \\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}B_u \\right)^8 \\\\&\\phantom{ \\leqslant 6^7 \\operatorname{\\mathbb {E}}\\Biggl (}+ \\delta ^8 Y_{k-1}^8 + \\gamma ^8 \\Biggr ) ,\\qquad k \\in \\mathbb {N}.$ By Jensen's inequality and (REF ), $\\begin{split}\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}\\left( \\left( \\int _{k-1}^k Y_u \\, \\mathrm {d}u \\right)^8 \\right)&\\leqslant \\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}\\left( \\int _{k-1}^k Y_u^8 \\, \\mathrm {d}u \\right)= \\sup _{k\\in \\mathbb {N}} \\int _{k-1}^k \\operatorname{\\mathbb {E}}(Y_u^8) \\, \\mathrm {d}u\\\\&\\leqslant \\left(\\sup _{t\\in \\mathbb {R}_+} \\operatorname{\\mathbb {E}}(Y_t^8)\\right) \\left(\\sup _{k\\in \\mathbb {N}} \\int _{k-1}^k 1\\,\\mathrm {d}u\\right)= \\sup _{t\\in \\mathbb {R}_+} \\operatorname{\\mathbb {E}}(Y_t^8)< \\infty .\\end{split}$ By the SDE (REF ) and the power mean inequality, $\\operatorname{\\mathbb {E}}\\left(\\left( \\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}W_u \\right)^8\\right)& \\leqslant \\frac{1}{\\sigma _1^8} \\operatorname{\\mathbb {E}}\\left( \\left( Y_k - Y_{k-1} -a - b\\int _{k-1}^k Y_u\\,\\mathrm {d}u \\right)^8\\right) \\\\& \\leqslant \\frac{4^7}{\\sigma _1^8} \\operatorname{\\mathbb {E}}\\left( Y_k^8 + Y_{k-1}^8 + a^8 + b^8 \\left( \\int _{k-1}^k Y_u \\, \\mathrm {d}u \\right)^8 \\right),\\qquad k\\in \\mathbb {N},$ and hence, by (REF ), $\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}\\left( \\left( \\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}W_u \\right)^8 \\right)\\leqslant \\frac{4^7}{\\sigma _1^8} \\left(2\\sup _{t\\in \\mathbb {R}_+} \\operatorname{\\mathbb {E}}(Y_t^8) + a^8 + b^8\\sup _{t\\in \\mathbb {R}_+} \\operatorname{\\mathbb {E}}(Y_t^8)\\right)<\\infty .$ Further, using that the conditional distribution of $\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}B_u$ given $(Y_u)_{u\\in [0,k]}$ is normal with mean 0 and variance $\\int _{k-1}^k Y_u\\,\\mathrm {d}u$ for all $k\\in \\mathbb {N}$ , we have $\\operatorname{\\mathbb {E}}\\left( \\left(\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}B_u \\right)^8 \\, \\Big \\vert \\, (Y_u)_{u\\in [0,k]} \\right)= 105 \\left(\\int _{k-1}^k Y_u \\, \\mathrm {d}u \\right)^4,\\qquad k\\in \\mathbb {N},$ and consequently $\\operatorname{\\mathbb {E}}\\left( \\left(\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}B_u \\right)^8 \\right)= 105 \\operatorname{\\mathbb {E}}\\left( \\left(\\int _{k-1}^k Y_u \\, \\mathrm {d}u \\right)^4 \\right),\\qquad k\\in \\mathbb {N}.$ Hence, similarly to (REF ), we have $\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}\\left( \\left(\\int _{k-1}^k \\sqrt{Y_u} \\, \\mathrm {d}B_u \\right)^8 \\right)\\leqslant 105 \\sup _{t\\in \\mathbb {R}_+} \\operatorname{\\mathbb {E}}(Y_t^4)<\\infty ,$ which yields that $\\sup _{k\\in \\mathbb {N}} \\operatorname{\\mathbb {E}}(\\eta _k^8) < \\infty $ .", "All in all, by the martingale central limit theorem (see, Theorem REF ), ${M}_{n,n} =n^{-1/2}\\sum _{k=1}^n\\begin{bmatrix}\\varepsilon _k \\\\ \\eta _k\\end{bmatrix}\\otimes \\begin{bmatrix}1 \\\\ Y_{k-1}\\end{bmatrix}\\stackrel{{\\mathcal {L}}}{\\longrightarrow }{\\mathcal {N}}_4\\left( {0}, {D}\\right)\\qquad \\text{as \\ $n \\rightarrow \\infty $.", "}$ Consequently, by (REF ) and Slutsky's lemma, $\\sqrt{n}\\begin{bmatrix}\\widehat{c}_n^{\\mathrm {CLSE}} - c \\\\\\widehat{d}_n^{\\mathrm {CLSE}} - d \\\\\\widehat{\\gamma }_n^{\\mathrm {CLSE}} - \\gamma \\\\\\widehat{\\delta }_n^{\\mathrm {CLSE}} - \\delta \\end{bmatrix}\\stackrel{{\\mathcal {L}}}{\\longrightarrow }{\\mathcal {N}}_4\\left( {0},\\left({I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}\\right)^{-1} {D}\\left({I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}\\right)^{-1} \\right)$ as $n \\rightarrow \\infty $ , where the covariance matrix of the limit distribution takes the form $&\\left( {I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}\\right)^{-1}{D}\\left( {I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2)\\end{bmatrix}\\right)^{-1} \\nonumber \\\\&= \\left(\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}\\otimes \\left(\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1}\\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & \\operatorname{\\mathbb {E}}(Y_\\infty ^3) \\\\\\end{bmatrix}\\right)\\right)\\left({I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1} \\right) \\nonumber \\\\&\\quad +\\left(\\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4 \\\\\\end{bmatrix}\\otimes \\left(\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1}\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}\\right)\\right)\\left({I}_2\\otimes \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1} \\right)\\nonumber $ $& = \\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}\\otimes \\left( \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1}\\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & \\operatorname{\\mathbb {E}}(Y_\\infty ^3) \\\\\\end{bmatrix}\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1}\\right) \\nonumber \\\\& \\quad +\\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4 \\\\\\end{bmatrix}\\otimes \\left( \\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1}\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}\\begin{bmatrix}1 & \\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\end{bmatrix}^{-1}\\right) \\nonumber \\\\& =\\frac{1}{(\\operatorname{\\mathbb {E}}(Y_\\infty ^2) - (\\operatorname{\\mathbb {E}}(Y_\\infty ))^2)^2}\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}\\nonumber \\\\& \\phantom{=\\;}\\otimes \\left( \\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & -\\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\-\\operatorname{\\mathbb {E}}(Y_\\infty ) & 1 \\\\\\end{bmatrix}\\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ) & \\operatorname{\\mathbb {E}}(Y_\\infty ^2) \\\\\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & \\operatorname{\\mathbb {E}}(Y_\\infty ^3) \\\\\\end{bmatrix}\\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & -\\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\-\\operatorname{\\mathbb {E}}(Y_\\infty ) & 1 \\\\\\end{bmatrix}\\right) \\nonumber \\\\&\\quad + \\frac{1}{\\operatorname{\\mathbb {E}}(Y_\\infty ^2) - (\\operatorname{\\mathbb {E}}(Y_\\infty ))^2}\\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4 \\\\\\end{bmatrix}\\otimes \\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & -\\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\-\\operatorname{\\mathbb {E}}(Y_\\infty ) & 1 \\\\\\end{bmatrix} \\nonumber \\\\&= \\frac{1}{(\\operatorname{\\mathbb {E}}(Y_\\infty ^2) - (\\operatorname{\\mathbb {E}}(Y_\\infty ))^2)^2}\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix} \\nonumber \\\\& \\,\\quad \\otimes \\begin{bmatrix}-\\operatorname{\\mathbb {E}}(Y_\\infty )( (\\operatorname{\\mathbb {E}}(Y_\\infty ^2))^2 - \\operatorname{\\mathbb {E}}(Y_\\infty )\\operatorname{\\mathbb {E}}(Y_\\infty ^3) ) & (\\operatorname{\\mathbb {E}}(Y_\\infty ^2))^2 - \\operatorname{\\mathbb {E}}(Y_\\infty )\\operatorname{\\mathbb {E}}(Y_\\infty ^3) \\\\(\\operatorname{\\mathbb {E}}(Y_\\infty ^2))^2 - \\operatorname{\\mathbb {E}}(Y_\\infty )\\operatorname{\\mathbb {E}}(Y_\\infty ^3) & \\operatorname{\\mathbb {E}}(Y_\\infty ^3) - 2\\operatorname{\\mathbb {E}}(Y_\\infty )\\operatorname{\\mathbb {E}}(Y_\\infty ^2) + (\\operatorname{\\mathbb {E}}(Y_\\infty ))^3 \\\\\\end{bmatrix} \\nonumber \\\\& \\quad + \\frac{1}{\\operatorname{\\mathbb {E}}(Y_\\infty ^2) - (\\operatorname{\\mathbb {E}}(Y_\\infty ))^2}\\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4 \\\\\\end{bmatrix}\\otimes \\begin{bmatrix}\\operatorname{\\mathbb {E}}(Y_\\infty ^2) & -\\operatorname{\\mathbb {E}}(Y_\\infty ) \\\\-\\operatorname{\\mathbb {E}}(Y_\\infty ) & 1 \\\\\\end{bmatrix} \\nonumber $ $& = \\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}\\otimes \\begin{bmatrix}\\frac{a(2a+\\sigma _1^2)}{b\\sigma _1^2} & -\\frac{2a+\\sigma _1^2}{\\sigma _1^2} \\\\-\\frac{2a+\\sigma _1^2}{\\sigma _1^2} & \\frac{2b(a+\\sigma _1^2)}{a\\sigma _1^2} \\\\\\end{bmatrix}+ \\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4 \\\\\\end{bmatrix}\\otimes \\begin{bmatrix}\\frac{2a+\\sigma _1^2}{\\sigma _1^2} & -\\frac{2b}{\\sigma _1^2} \\\\-\\frac{2b}{\\sigma _1^2} & \\frac{2b^2}{a\\sigma _1^2} \\\\\\end{bmatrix} := {E}.$ Indeed, by (REF ), an easy calculation shows that $&\\left(\\operatorname{\\mathbb {E}}(Y_\\infty ) \\operatorname{\\mathbb {E}}(Y_\\infty ^3) - (\\operatorname{\\mathbb {E}}(Y_\\infty ^2))^2\\right) \\operatorname{\\mathbb {E}}(Y_\\infty )= \\frac{a^3\\sigma _1^2}{4b^5}(2a+\\sigma _1^2),\\\\&\\operatorname{\\mathbb {E}}(Y_\\infty ) \\operatorname{\\mathbb {E}}(Y_\\infty ^3) - (\\operatorname{\\mathbb {E}}(Y_\\infty ^2))^2= \\frac{a^2\\sigma _1^2}{4b^4}(2a+\\sigma _1^2),\\\\&\\operatorname{\\mathbb {E}}(Y_\\infty ^3) - 2 \\operatorname{\\mathbb {E}}(Y_\\infty ) \\operatorname{\\mathbb {E}}(Y_\\infty ^2) + (\\operatorname{\\mathbb {E}}(Y_\\infty ))^3= \\frac{a\\sigma _1^2}{2b^3}(a+\\sigma _1^2),\\\\&\\operatorname{\\mathbb {E}}(Y_\\infty ^2) - \\left(\\operatorname{\\mathbb {E}}(Y_\\infty )\\right)^2= \\frac{a\\sigma _1^2}{2b^2}.$ Finally, we show that ${E}$ is positive definite.", "To show this, it is enough to check that (i) the matrix $\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}$ is positive definite, (ii) the matrices $\\begin{bmatrix}C_2 & C_6 \\\\C_6 & C_4 \\\\\\end{bmatrix},\\qquad \\begin{bmatrix}\\frac{a(2a+\\sigma _1^2)}{b\\sigma _1^2} & -\\frac{2a+\\sigma _1^2}{\\sigma _1^2} \\\\-\\frac{2a+\\sigma _1^2}{\\sigma _1^2} & \\frac{2b(a+\\sigma _1^2)}{a\\sigma _1^2} \\\\\\end{bmatrix}\\qquad \\text{and} \\qquad \\begin{bmatrix}\\frac{2a+\\sigma _1^2}{\\sigma _1^2} & -\\frac{2b}{\\sigma _1^2} \\\\-\\frac{2b}{\\sigma _1^2} & \\frac{2b^2}{a\\sigma _1^2} \\\\\\end{bmatrix}$ are positive semi-definite.", "Indeed, the sum of a positive definite and a positive semi-definite square matrix is positive definite, the Kronecker product of positive semi-definite matrices is positive semi-definite and the Kronecker product of positive definite matrices is positive definite (as a consequence of the fact that the eigenvalues of the Kronecker product of two square matrices are the product of the eigenvalues of the two square matrices in question including multiplicities).", "The positive semi-definiteness of the matrices $\\begin{bmatrix}\\frac{a(2a+\\sigma _1^2)}{b\\sigma _1^2} & -\\frac{2a+\\sigma _1^2}{\\sigma _1^2} \\\\-\\frac{2a+\\sigma _1^2}{\\sigma _1^2} & \\frac{2b(a+\\sigma _1^2)}{a\\sigma _1^2} \\\\\\end{bmatrix}\\qquad \\text{and} \\qquad \\begin{bmatrix}\\frac{2a+\\sigma _1^2}{\\sigma _1^2} & -\\frac{2b}{\\sigma _1^2} \\\\-\\frac{2b}{\\sigma _1^2} & \\frac{2b^2}{a\\sigma _1^2} \\\\\\end{bmatrix}$ readily follows, since $\\frac{a(2a+\\sigma _1^2)}{b\\sigma _1^2}>0$ , $\\frac{2a+\\sigma _1^2}{\\sigma _1^2}>0$ , and the determinant of the matrices in question are $\\frac{2a+\\sigma _1^2}{\\sigma _1^2}>0$ and $\\frac{2b}{a\\sigma _1^2}>0$ , respectively.", "Next, we prove that the matrices $\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}\\qquad \\text{and} \\qquad \\begin{bmatrix}C_2 & C_4 \\\\C_4 & C_6 \\\\\\end{bmatrix}$ are positive semi-definite.", "Since $\\operatorname{\\mathbb {P}}(Y_0=y_0)=1$ , we have $\\operatorname{\\mathbb {E}}(\\varepsilon _1^2 \\,\|\\,{\\mathcal {F}}_0) = C_1 y_0 + C_2$ , $\\operatorname{\\mathbb {E}}( \\eta _1^2 \\,\|\\,{\\mathcal {F}}_0) = C_3 y_0 + C_4$ , and $\\operatorname{\\mathbb {E}}(\\varepsilon _1 \\eta _1 \\,\|\\,{\\mathcal {F}}_0) = C_5 y_0 + C_6$ $\\operatorname{\\mathbb {P}}$ -almost surely, hence $\\operatorname{\\mathbb {E}}( \\varepsilon _1^2 ) \\operatorname{\\mathbb {E}}(\\eta _1^2)- \\bigl ( \\operatorname{\\mathbb {E}}( \\varepsilon _1 \\eta _1 ) \\bigr )^2= (C_1 C_3 - C_5^2) y_0^2 + (C_1 C_4 + C_2 C_3 - 2 C_5 C_6) y_0+ C_2 C_4 - C_6^2 .$ Clearly, by Cauchy–Schwarz's inequality, $\\operatorname{\\mathbb {E}}( \\varepsilon _1^2 ) \\operatorname{\\mathbb {E}}( \\eta _1^2 )- \\bigl ( \\operatorname{\\mathbb {E}}( \\varepsilon _1 \\eta _1 ) \\bigr )^2\\geqslant 0 ,$ hence, by setting an arbitrary initial value $Y_0 = y_0 \\in \\mathbb {R}_+$ , we obtain $C_1 C_3 - C_5^2 \\geqslant 0$ and $C_2 C_4 - C_6^2 \\geqslant 0$ .", "Thus, both matrices $\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}\\qquad \\text{and}\\qquad \\begin{bmatrix}C_2 & C_4 \\\\C_4 & C_6 \\\\\\end{bmatrix}$ are positive semi-definite, since $C_1 > 0$ and $C_2 > 0$ .", "Now we turn to check that $\\begin{bmatrix}C_1 & C_5 \\\\C_5 & C_3 \\\\\\end{bmatrix}$ is positive definite.", "Since $C_1 > 0$ , this is equivalent to showing that $C_1 C_3 - C_5^2 > 0$ .", "Recalling the definition of the constants, we have $C_1 &= \\sigma _1^2 \\int _0^1 \\mathrm {e}^{-b(2-v)} \\, \\mathrm {d}v = \\sigma _1^2 \\mathrm {e}^{-2b}\\frac{\\mathrm {e}^b - 1}{b}, \\\\C_3 &=\\beta ^2 \\sigma _1^2\\int _{0}^1 \\int _{0}^1 \\int _{0}^{u^{\\prime } \\wedge v^{\\prime }}\\mathrm {e}^{-b(u^{\\prime }+v^{\\prime }-w^{\\prime })} \\, \\mathrm {d}w^{\\prime } \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }- 2 \\beta \\sigma _1 \\sigma _2 \\varrho \\int _{0}^1 \\int _{0}^{u^{\\prime }} \\mathrm {e}^{-bu^{\\prime }} \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }+ \\sigma _2^2 \\int _{0}^1 \\mathrm {e}^{-bu^{\\prime }} \\, \\mathrm {d}u^{\\prime } \\\\&= b^{-3}\\left(2 \\mathrm {e}^{-b} \\beta ^2 \\sigma _1^2 (\\sinh b -b) + 2 b \\beta \\varrho \\sigma _1 \\sigma _2((1 + b) \\mathrm {e}^{-b}-1) + b^2\\sigma _2^2(1-\\mathrm {e}^{-b}) \\right), \\\\C_5 &= - \\beta \\sigma _1^2\\int _{0}^1 \\int _{0}^{u^{\\prime }} \\mathrm {e}^{-b(u^{\\prime }-v^{\\prime }+1)} \\, \\mathrm {d}v^{\\prime } \\, \\mathrm {d}u^{\\prime }+ \\sigma _1 \\sigma _2 \\varrho \\mathrm {e}^{-b}= b^{-2} \\sigma _1 \\mathrm {e}^{-b}\\left(- \\mathrm {e}^{-b}\\beta \\sigma _1 (1+(b-1)\\mathrm {e}^{b}) + \\varrho \\sigma _2 b^2 \\right),$ thus we have $C_1 C_3 - C_5^2 &= b^{-4}\\mathrm {e}^{-2b} \\sigma _1^2\\Big (2b(2+b^2)\\beta \\varrho \\sigma _1 \\sigma _2+ 2 (\\beta ^2 \\sigma _1^2 - 2 b \\beta \\varrho \\sigma _1 \\sigma _2 + b^2 \\sigma _2^2) \\cosh b - (2+b^2)\\beta ^2 \\sigma _1^2 \\\\& \\qquad \\qquad \\qquad - b^2 (2+b^2 \\varrho ^2) \\sigma _2^2\\Big ).$ Consequently, using that $\\cosh b = \\sum _{k=0}^{\\infty } \\frac{b^{2k}}{(2k)!}", "> 1+\\frac{b^2}{2}$ and that $\\beta ^2 \\sigma _1^2 - 2 b \\beta \\varrho \\sigma _1 \\sigma _2 + b^2 \\sigma _2^2= (\\beta \\sigma _1 - b \\varrho \\sigma _2)^2 + b^2 (1-\\varrho ^2) \\sigma _2^2 >0,$ we have $C_1 C_3 - C_5^2 &> b^{-4} \\mathrm {e}^{-2b} \\sigma _1^2 \\Big ( 4b \\beta \\varrho \\sigma _1 \\sigma _2 + 2b^3 \\beta \\varrho \\sigma _1 \\sigma _2 + 2 \\beta ^2 \\sigma _1^2 + b^2 \\beta ^2 \\sigma _1^2 - 4 b \\beta \\varrho \\sigma _1 \\sigma _2 - 2 b^3 \\beta \\varrho \\sigma _1 \\sigma _2 \\\\& \\qquad \\qquad \\qquad + 2 b^2 \\sigma _2^2 + b^4 \\sigma _2^2 - 2\\beta ^2 \\sigma _1^2 - b^2 \\beta ^2 \\sigma _1^2 - 2b^2 \\sigma _2^2 - b^4 \\varrho ^2 \\sigma _2^2\\Big ) \\\\&= b^{-4} \\mathrm {e}^{-2b} \\sigma _1^2 ( b^4 (1-\\varrho ^2) \\sigma _2^2 ) > 0.$ With this our proof is finished.", "$\\Box $ So far we have obtained the limit distribution of the CLSE of the transformed parameters $(c,d,\\gamma ,\\delta )$ .", "A natural estimator of $(a,b,\\alpha ,\\beta )$ can be obtained from (REF ) using relation (REF ) detailed as follows.", "Calculating the integrals in (REF ) in the subcritical case, let us introduce the function $g: \\mathbb {R}_{++}^2 \\times \\mathbb {R}^2 \\rightarrow \\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ , $g(a,b,\\alpha ,\\beta ):=\\begin{bmatrix}a b^{-1} (1-\\mathrm {e}^{-b}) \\\\\\mathrm {e}^{-b} \\\\\\alpha - a \\beta b^{-2} (\\mathrm {e}^{-b}-1+b)\\\\-\\beta b^{-1} (1-\\mathrm {e}^{-b}) \\\\\\end{bmatrix}=\\begin{bmatrix}c \\\\ d \\\\ \\gamma \\\\ \\delta \\end{bmatrix},\\qquad (a,b,\\alpha ,\\beta )\\in \\mathbb {R}_{++}^2 \\times \\mathbb {R}^2.$ Note that $g$ is bijective having inverse $g^{-1}(c,d,\\gamma ,\\delta )=\\begin{bmatrix}- c \\frac{\\log d}{1-d} \\\\-\\log d \\\\\\gamma - c \\delta \\frac{d - 1 - \\log d}{(1-d)^2} \\\\\\delta \\frac{\\log d}{1-d}\\end{bmatrix}=\\begin{bmatrix}a \\\\ b \\\\ \\alpha \\\\ \\beta \\end{bmatrix},\\qquad (c,d,\\gamma ,\\delta )\\in \\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2.$ Indeed, for all $(c,d,\\gamma ,\\delta )\\in \\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ , we have $\\alpha &= \\gamma + a\\beta b^{-2}(\\mathrm {e}^{-b} -1+b)= \\gamma + (-c)\\frac{\\log d}{1-d}\\delta \\frac{\\log d}{1-d} (-\\log d)^{-2}(d -1-\\log d)\\\\&= \\gamma - c\\delta \\frac{d - 1 - \\log d}{(1-d)^2}.$ Under the conditions of Theorem REF the CLSE $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}, \\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}})$ of $(c,d,\\gamma ,\\delta )$ is strongly consistent, hence in the subcritical case $(\\widehat{c}_n^{\\mathrm {CLSE}}, \\widehat{d}_n^{\\mathrm {CLSE}}, \\widehat{\\gamma }_n^{\\mathrm {CLSE}}, \\widehat{\\delta }_n^{\\mathrm {CLSE}})$ fall into the set $\\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ for sufficiently large $n\\in \\mathbb {N}$ with probability one.", "Hence, in the subcritical case, one can introduce a natural estimator of $(a,b,\\alpha ,\\beta )$ based on discrete time observations $(Y_i, X_i)_{i\\in \\lbrace 1,\\ldots ,n\\rbrace }$ by applying the inverse of $g$ to the CLSE of $(c,d,\\gamma ,\\delta )$ , i.e., $(\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n):= g^{-1}(\\widehat{c}_n^{\\mathrm {CLSE}},\\widehat{d}_n^{\\mathrm {CLSE}},\\widehat{\\gamma }_n^{\\mathrm {CLSE}},\\widehat{\\delta }_n^{\\mathrm {CLSE}})$ for sufficiently large $n\\in \\mathbb {N}$ with probability one.", "Remark.", "3.3 We would like to stress the point that the estimator of $(a,b,\\alpha ,\\beta )$ introduced in (REF ) exists only for sufficiently large $n\\in \\mathbb {N}$ with probability of 1.", "However, as all our results are asymptotic, this will not cause a problem.", "From the considerations before this remark, we obtain $\\bigl (\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n\\bigr )= \\operatornamewithlimits{arg\\,min}_{(a,b,\\alpha ,\\beta )\\in \\mathbb {R}_{++}^2\\times \\mathbb {R}^2}\\sum _{i=1}^n\\left[ (Y_i - d Y_{i-1} - c)^2+ (X_i - X_{i-1} - \\gamma - \\delta Y_{i-1})^2 \\right]$ for sufficiently large $n\\in \\mathbb {N}$ with probability one.", "We call the attention that $\\bigl (\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n\\bigr )$ does not necessarily provides a CLSE of $(a,b,\\alpha ,\\beta )$ , since in (REF ) one takes the infimum only on the set $\\mathbb {R}_{++}^2\\times \\mathbb {R}^2$ instead of $\\mathbb {R}^4$ .", "Formula (REF ) serves as a motivation for calling $\\bigl (\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n\\bigr )$ essentially conditional least squares estimator in the Abstract.", "$\\Box $ Theorem.", "3.4 Under the conditions of Theorem REF the sequence $\\bigl (\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n\\bigr )$ , $n \\in \\mathbb {N}$ , is strongly consistent and asymptotically normal, i.e., $(\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n)\\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }(a, b, \\alpha , \\beta ) \\qquad \\text{as \\ $n \\rightarrow \\infty $,}$ and $\\sqrt{n}\\begin{bmatrix}\\widehat{a}_n - a \\\\\\widehat{b}_n - b \\\\\\widehat{\\alpha }_n - \\alpha \\\\\\widehat{\\beta }_n - \\beta \\end{bmatrix}\\stackrel{{\\mathcal {L}}}{\\longrightarrow }{\\mathcal {N}}_4\\left(0,{J}{E}{J}^\\top \\right)\\qquad \\text{as \\ $n\\rightarrow \\infty $,}$ where ${E}\\in \\mathbb {R}^{2\\times 2}$ is a symmetric, positive definite matrix given in (REF ) and ${J}:=\\begin{bmatrix}-\\frac{\\log d}{1-d} & -c \\frac{\\log d - 1 + d^{-1}}{(1-d)^2} & 0 & 0 \\\\0 & -\\frac{1}{d} & 0 & 0 \\\\\\delta \\frac{\\log d + 1 - d}{(1-d)^2} &c \\delta \\frac{2\\log d - d + d^{-1} }{(1-d)^3} & 1 & c \\frac{\\log d + 1 - d}{(1-d)^2} \\\\0 & \\delta \\frac{\\log d - 1 + d^{-1}}{(1-d)^2} & 0 & \\frac{\\log d}{1-d}\\end{bmatrix}$ with $c$ , $d$ and $\\delta $ given in (REF ).", "Proof.", "The strong consistency of $(\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n)$ , $n \\in \\mathbb {N}$ , follows from the strong consistency of the CLSE of $(c,d,\\gamma ,\\delta )$ proved in Theorem REF using also that the inverse function $g^{-1}$ given in (REF ) is continuous on $\\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ .", "For the second part of the theorem we use Theorem REF , and the so-called delta method (see, e.g., Theorem 11.2.14 in Lehmann and Romano [13]).", "Indeed, one can extend the function $g^{-1}$ to be defined on $\\mathbb {R}^4$ not only on $\\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ (e.g., let it be zero on the complement of $\\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ ), $(\\widehat{a}_n, \\widehat{b}_n, \\widehat{\\alpha }_n, \\widehat{\\beta }_n)$ takes the form given in (REF ) with this extension of $g^{-1}$ as well, and the Jacobian of $g^{-1}$ at $(c,d,\\gamma ,\\delta )\\in \\mathbb {R}_{++}\\times (0,1)\\times \\mathbb {R}^2$ is clearly ${J}$ .", "$\\Box $" ], [ "Acknowledgements", "We are undoubtedly grateful for the referee for pointing out mistakes in the proof of Theorem REF , and also for his/her several valuable comments that have led to an improvement of the manuscript." ] ]	1403.0527
[ [ "Real-time Topic-aware Influence Maximization Using Preprocessing" ], [ "Abstract Influence maximization is the task of finding a set of seed nodes in a social network such that the influence spread of these seed nodes based on certain influence diffusion model is maximized.", "Topic-aware influence diffusion models have been recently proposed to address the issue that influence between a pair of users are often topic-dependent and information, ideas, innovations etc.", "being propagated in networks (referred collectively as items in this paper) are typically mixtures of topics.", "In this paper, we focus on the topic-aware influence maximization task.", "In particular, we study preprocessing methods for these topics to avoid redoing influence maximization for each item from scratch.", "We explore two preprocessing algorithms with theoretical justifications.", "Our empirical results on data obtained in a couple of existing studies demonstrate that one of our algorithms stands out as a strong candidate providing microsecond online response time and competitive influence spread, with reasonable preprocessing effort." ], [ "Introduction", "In a social network, information, ideas, rumors, and innovations can be propagated to a large number of people because of the social influence between the connected peers in the network.", "Influence maximization is the task of finding a set of seed nodes in a social network such that the influence propagated from the seed nodes can reach the largest number of people in the network.", "More technically, a social network is modeled as a graph with nodes representing individuals and directed edges representing influence relationships.", "The network is associated with a stochastic diffusion model (such as independent cascade model and linear threshold model [13]) characterizing the influence propagation dynamics starting from the seed nodes.", "Influence maximization is to find a set of $k$ seed nodes in the network such that the influence spread, defined as the expected number of nodes influenced (or activated) through influence diffusion starting from the seed nodes, is maximized ([13], [6]).", "Influence maximization has a wide range of applications including viral marketing [8], [17], [13], information monitoring and outbreak detection [14], competitive viral marketing and rumor control [5], [12], or even text summarization [21] (by modeling a word influence network).", "As a result, influence maximization has been extensively studied in the past decade.", "Research directions include improvements in the efficiency and scalability of influence maximization algorithms (e.g., [7], [20], [11]), extensions to other diffusion models and optimization problems (e.g., [5], [3], [12]), and influence model learning from real-world data (e.g., [18], [19], [9]).", "Most of these works treat diffusions of all information, rumors, ideas, etc.", "(collectively referred as items in this paper) as following the same model with a single set of parameters.", "In reality, however, influence between a pair of friends may differ depending on the topic.", "For example, one may be more influential to the other on high-tech gadgets, while the other is more influential on fashion topics, or one researcher is more influential on data mining topics to her peers but less influential on algorithm and theory topics.", "Recently, Barbieri et al.", "[2] propose the topic-aware independent cascade (TIC) and linear threshold (TLT) models, in which a diffusion item is a mixture of topics and influence parameters for each item are also mixtures of parameters for individual topics.", "They provide learning methods to learn influence parameters in the topic-aware models from real-world data.", "Such topic-mixing models require new thinking in terms of the influence maximization task, which is what we address in this paper.", "In this paper, we adopt the models proposed in [2] and study efficient topic-aware influence maximization schemes.", "One can still apply topic-oblivious influence maximization algorithms in online processing of every diffusion item, but it may not be efficient when there are a large number of items with different topic mixtures or real-time responses are required.", "Thus, our focus is on preprocessing individual topic influence such that when a diffusion item with certain topic mixture comes, the online processing of finding the seed set is fast.", "To do so, our first step is to collect two datasets in the past studies with available topic-aware influence analysis results on real networks and investigate their properties pertaining to our preprocessing purpose (Section ).", "Our data analysis shows that in one network users and their relationships are largely separated by different topics while in the other network they have significant overlaps on different topics.", "Even with this difference, a common property we find is that in both datasets most top seeds for a topic mixture come from top seeds of the constituent topics, which matches our intuition that influential individuals for a mixed item are usually influential in at least one topic category.", "Motivated by our findings from the data analysis, we explore two preprocessing based algorithms (Section ).", "The first algorithm, Best Topic Selection (BTS), minimizes online processing by simply using a seed set for one of the constituent topics.", "Even for such a simple algorithm, we are able to provide a theoretical approximation ratio (when a certain property holds), and thus BTS serves as a baseline for preprocessing algorithms.", "The second algorithm, Marginal Influence Sort (MIS), further uses pre-computed marginal influence of seeds on each topic to avoid slow greedy computation.", "We provide a theoretical justification showing that MIS can be as good as the offline greedy algorithm when nodes are fully separated by topics.", "We then conduct experimental evaluations of these algorithms and comparing them with both the greedy algorithm and a state-of-the-art heuristic algorithm PMIA [20], on the two datasets used in data analysis as well as a third dataset for testing scalability (Section ).", "From our results, we see that MIS algorithm stands out as the best candidate for preprocessing based real-time influence maximization: it finishes online processing within a few microseconds and its influence spread either matches or is very close to that of the greedy algorithm.", "Our work, together with a recent independent work [1], is one of the first that study topic-aware influence maximization with focus on preprocessing.", "Comparing to [1], our contributions include: (a) we include data analysis on two real-world datasets with learned influence parameters, which shows different topical influence properties and motivates our algorithm design; (b) we provide theoretical justifications to our algorithms; (c) the use of marginal influence of seeds in individual topics in MIS is novel, and is complementary to the approach in [1]; (d) even though MIS is quite simple, it achieves competitive influence spread within microseconds of online processing time rather than milliseconds needed in [1]." ], [ "Preliminaries", "In this section, we introduce the background and problem definition on the topic-aware influence diffusion models.", "We focus on the independent cascade model [13] for ease of presentation, but our results also hold for other models parameterized with edge parameters such as the linear threshold model [13]." ], [ "Independent cascade model", "We consider a social network as a directed graph $G=(V,E)$ , where each node in $V$ represents a user, and each edge in $E$ represents the relationship between two users.", "For every edge $(u, v) \\in E$ , denote its influence probability as $p(u, v) \\in [0, 1]$ , and for all $(u, v) \\notin E$ or $u = v$ , we assume $p(u, v) = 0$ .", "The independent cascade (IC) model, defined in [13], captures the stochastic process of contagion in discrete time.", "Initially at time step $t=0$ , a set of nodes $S \\subseteq V$ called seed nodes are activated.", "At any time $t \\ge 1$ , if node $u$ is activated at time $t-1$ , it has one chance of activating each of its inactive outgoing neighbor $v$ with probability $p(u,v)$ .", "A node stays active after it is activated.", "This process stops when no more nodes are activated.", "We define influence spread of seed set $S$ under influence probability function $p$ , denoted $\\sigma (S, p)$ , as the expected number of active nodes after the diffusion process ends.", "As shown in [13], for any fixed $p$ , $\\sigma (S, p)$ is monotone (i.e., $\\sigma (S, p) \\le \\sigma (T, p)$ for any $S \\subseteq T$ ) and submodular (i.e., $\\sigma (S \\cup \\lbrace v \\rbrace , p) - \\sigma (S, p) \\ge \\sigma (T \\cup \\lbrace v \\rbrace , p) - \\sigma (T, p)$ for any $S \\subseteq T$ and $v \\in V$ ) on its seed set parameter.", "The next lemma further shows that for any fixed $S$ , $\\sigma (S, p)$ is monotone in $p$ .", "For two influence probability functions $p$ and $p^{\\prime }$ on graph $G=(V,E)$ , we denote $p \\le p^{\\prime }$ if for any $(u,v)\\in E$ , $p(u,v) \\le p^{\\prime }(u,v)$ .", "We say that influence spread function $\\sigma (S, p)$ is monotone in $p$ if for any $p\\le p^{\\prime }$ , we have $\\sigma (S,p) \\le \\sigma (S,p^{\\prime })$ .", "For any fixed seed set $S \\subseteq V$ , $\\sigma (S, p)$ is monotone in $p$ .", "[Proof sketch] We use the following coupling method.", "For any edge $(u,v)\\in E$ , we select a number $x(u,v)$ uniformly at random in $[0,1]$ .", "Then for any influence probability function $p$ , we select edge $(u,v)$ as a live edge if $x(u,v) \\le p(u,v)$ and otherwise it is a blocked edge.", "All live edges form a random live-edge graph $G_L(p)$ .", "One can verify that $\\sigma (S,p)$ is the expected value of the size of node set reachable from $S$ in random graph $G_L(p)$ .", "Moreover, for $p$ and $p^{\\prime }$ such that $p \\le p^{\\prime }$ , one can verify that after fixing the random numbers $x(u,v)^{\\prime }s$ , live-edge graph $G_L(p)$ is a subgraph of live-edge graph $G_L(p^{\\prime })$ , and thus nodes reachable from $S$ in $G_L(p)$ must be also reachable from $S$ in $G_L(p^{\\prime })$ .", "This implies that $\\sigma (S,p) \\le \\sigma (S,p^{\\prime })$ .", "$\\blacksquare $ We remark that using a similar idea as above we could show that influence spread in the linear threshold (LT) model [13] is also monotone in the edge weight parameter." ], [ "Influence maximization", "Given a graph $G=(V,E)$ , an influence probability function $p$ , and a budget $k$ , influence maximization is the task of selecting at most $k$ seed nodes such that the influence spread is maximized, i.e., finding set $S^{}=S^{}(k,p)$ such that $S^{}(k,p) = \\operatornamewithlimits{argmax}_{S \\subseteq V, \|S\| \\le k} \\sigma (S, p).$ In [13], Kempe et al.", "show that the influence maximization problem is NP-hard in both the IC model and the LT model.", "They propose the greedy approach for influence maximization, as shown in Algorithm REF .", "Given influence probability function $p$ , the marginal influence (MI) of a node $v$ under seed set $S$ is defined as ${\\it MI}(v \| S, p) = \\sigma (S \\cup \\lbrace v\\rbrace , p)-\\sigma (S, p)$ , for any $v\\in V$ .", "The greedy algorithm selects $k$ seeds in $k$ iterations, and in the $j$ -th iteration it selects a node $v_j$ with the largest marginal influence under the current seed set $S_{j-1}$ and adds $v_j$ into $S_{j-1}$ to obtain $S_j$ .", "Kempe et al.", "use Monte Carlo simulations to obtain accurate estimates on marginal influence ${\\it MI}(v \| S, p)$ , and later Chen et al.", "show that indeed exact computation of influence spread $\\sigma (S,p)$ or marginal influence ${\\it MI}(v \| S, p)$ is #P-hard [20].", "The monotonicity and submodularity of $\\sigma (S,p)$ in $S$ guarantees that the greedy algorithm selects a seed set with approximation ratio $1-\\frac{1}{e} - \\varepsilon $ , that is, it returns a seed set $S^{g}= S^{g}(k, p)$ such that $\\sigma (S^{g}, p) \\ge \\left(1-\\frac{1}{e} - \\varepsilon \\right) \\sigma (S^{},p),$ for any small $\\varepsilon > 0$ , where $\\varepsilon $ accommodates the inaccuracy in Monte Carlo estimations.", "[t] [1] $G=(V,E)$ , $p$ , $k$ .", "$S_0 = \\emptyset $ $j = 1,2,\\cdots ,k$ $v_j = \\operatornamewithlimits{argmax}_{v \\in V\\setminus S_{j-1}} {\\it MI}(v \| S_{j-1}, p)$ $S_j = S_{j-1} \\cup \\lbrace v_j\\rbrace $ $S_k$ Greedy algorithm." ], [ "Topic-aware independent cascade model and topic-aware influence maximization", "Topic-aware independent cascade (TIC) model [2] is an extension of the IC model to incorporate topic mixtures in any diffusion item.", "Suppose there are $d$ base topics, and we use set notation $[d] = \\lbrace 1,2,\\cdots ,d\\rbrace $ to denote topic $1,2, \\cdots , d$ .", "We regard each diffusion item as a distribution of these topics.", "Thus, any item can be expressed as a vector $I=(\\lambda _1, \\lambda _2, \\dots , \\lambda _{d})$ where $\\forall i \\in [d]$ , $\\lambda _i \\in [0,1]$ and $\\sum _{i \\in [d]} \\lambda _i=1$ .", "We also refer $(\\lambda _1, \\lambda _2, \\dots , \\lambda _{d})$ as a topic mixture.", "Given a directed social graph $G=(V,E)$ , for any topic $i \\in [d]$ , influence probability on that topic is $p_i: V \\times V \\rightarrow [0,1]$ , and for all $(u, v) \\notin E$ or $u = v$ , we assume $p_i(u, v) = 0$ .", "In the TIC model, the influence probability function $p$ for any diffusion item $I=(\\lambda _1,\\lambda _2,\\dots ,\\lambda _{d})$ is defined as $p(u,v) = \\sum _{i \\in [d]} \\lambda _i {p_i}(u,v)$ , for all $ u,v \\in V$ (or simply $p = \\sum _{i \\in [d]} \\lambda _i {p_i}$ ).", "Then, the stochastic diffusion process and influence spread $\\sigma (S, p)$ are exactly the same as defined in the IC model by using the influence probability $p$ on edges.", "Given a social graph $G$ , base topics $[d]$ , influence probability function $p_i$ for each base topic $i$ , a budget $k$ and an item $I=(\\lambda _1,\\lambda _2,\\dots ,\\lambda _{d})$ , the topic-aware influence maximization is the task of finding optimal seeds $S^{}= S^{}(k, p) \\subseteq V$ , where $p = \\sum _{i \\in [d]} \\lambda _i {p_i}$ , to maximize the influence spread, i.e., $S^{}= \\operatornamewithlimits{argmax}_{S \\subseteq V, \|S\| \\le k} \\sigma (S, p)$ ." ], [ "Data Observation", "There are relatively few studies on topic-aware influence analysis.", "For our study, we are able to obtain datasets from two prior studies, one is on social movie rating network Flixster [2] and the other is on academic collaboration network Arnetminer [19].", "In this section, we describe these two datasets, and present statistical observations on these datasets, which will help us in our algorithm design." ], [ "Data description", "We obtain two real-world datasets, Flixster and Arnetminer, which include influence analysis results from their respective raw data, from the authors of the prior studies [2], [19].", "Flixsterwww.flixster.com is an American social movie site for discovering new movies, learning about movies, and meeting others with similar tastes in movies.", "The raw data in Flixster dataset is the action traces of movie ratings of users.", "The Flixster network represents users as nodes, and two users $u$ and $v$ are connected by a directed edge $(u,v)$ if they are friends both rating the same movie and $v$ rates the movie shortly later after $u$ does so.", "The network contains 29357 nodes, 425228 directed edges and 10 topics [2].", "Barbieri et al.", "[2] use their proposed TIC model and apply maximum likelihood estimation method on the action traces to obtain influence probabilities on edges for all 10 topics.", "We found that there are a disproportionate number of edges with influence probabilities higher than $0.99$ , which is due to the lack of sufficient samplings of propagation events over these edges.", "We smoothen these influence probability values by changing all the probabilities larger than $0.99$ to random numbers according to the probability distribution of all the probabilities smaller than $0.99$ .", "We also obtain 11659 topic mixtures, and demonstrate the distribution of the number of topics in item mixtures in Table REF .", "We eliminate individual probabilities that are too weak ($\\forall i\\in [d], \\lambda _i < 0.01$ ).", "In general, most items are on a single topic only, with some two-topic mixtures.", "Mixtures with three or four topics are already rare and there are no items with five or more topics.", "Table: Distribution of topic numbers of mixture items in FlixsterArnetminerarnetminer.org is a free online service used to index and search academic social networks.", "The Arnetminer network represents authors as nodes and two authors have an edge if they coauthored a paper.", "The raw data in the Arnetminer dataset is not the action traces but the topic distributions of all nodes and the network structure [19].", "Tang et al.", "apply factor graph analysis to obtain influence probabilities on edges from node topic distributions and the network structure [19].", "The resulting network contains 5114 nodes, 34334 directed edges and 8 topics, and all 8 topics are related to computer science, such as data mining, machine learning, information retrieval, etc.", "Mixed items propagated in such academic networks could be ideas or papers from related topic mixtures, although there are no raw data of topic mixtures available in Arnetminer.", "Tables REF and REF provide statistics for the learned influence probabilities for every topic in Arnetminer and Flixster dataset.", "Column “nonzero” provides the number of edges having nonzero probabilities on the specific topic.", "Other columns are mean, standard deviation, 25-percentile, 50-percentile (median), and 75-percentile of the probabilities among the nonzero entries.", "The basic statistics show similar behavior between the two datasets, such as mean probabilities are mostly between $0.1$ and $0.2$ , standard deviations are mostly between $0.1$ to $0.3$ , etc.", "Comparing among different topics, even though the means and other statistics are similar to one another, the number of nonzero edges may have up to 10 fold difference.", "This indicates that some topics are more likely to propagate than others.", "Table: Influence probability statistics of ArnetminerTable: Influence probability statistics of Flixster" ], [ "Topic separation on edges and nodes", "For the two datasets, we would like to investigate how different topics overlap on edges and nodes.", "To do so, we define the following coefficients to characterize the properties of a social graph.", "Given threshold $\\theta \\ge 0$ , for every topic $i$ , denote edge set $\\tau _i(\\theta ) =\\lbrace (u,v) \\in E\\,\|\\, {p_i}(u,v) > \\theta \\rbrace $ , and node set $\\nu _i(\\theta ) = \\lbrace v\\in V \\,\|\\, \\sum _{u:(v,u)\\in E} p_i(v,u)+ \\sum _{u:(u,v)\\in E} p_i(u,v) > \\theta \\rbrace $ .", "For topics $i$ and $j$ , we define edge overlap coefficient as $\\Upsilon ^{E}_{ij}(\\theta ) =\\frac{\|\\tau _i(\\theta ) \\cap \\tau _j(\\theta )\|}{\\min \\lbrace \|\\tau _i(\\theta )\|, \|\\tau _j(\\theta )\|\\rbrace }$ , and node overlap coefficient as $ \\Upsilon ^{V}_{ij}(\\theta ) =\\frac{\|\\nu _i(\\theta ) \\cap \\nu _j(\\theta )\|}{\\min \\lbrace \|\\nu _i(\\theta )\|, \|\\nu _j(\\theta )\|\\rbrace }$ .", "If $\\theta $ is small and the overlap coefficient is small, it means that the two topics are fairly separated in the network.", "In particular, we say that the network is fully separable for topics $i$ and $j$ if $\\Upsilon ^{V}_{ij}(0) = 0$ , and it is fully separable for all topics if $\\Upsilon ^{V}_{ij}(0) = 0$ for any pair $i$ and $j$ with $i\\ne j$ .", "Then we apply the above coefficients to the Flixster and Arnetminer datasets.", "Table REF shows the edge and node overlap coefficients with threshold $\\theta =0.1$ for every pair of topics in the Arnetminer dataset.", "Correlating with Table REF , we see that $\\theta =0.1$ is around the mean value for all topics.", "Thus it is a reasonably small value especially for the node overlap coefficients, which is about aggregated probability of all edges incident to a node.", "A clear indication in Table REF is that topic overlap on both edges and nodes are very small in Arnetminer, with most node overlap coefficients less than $5\\%$ .", "We believe that this is because in academic collaboration network, most researchers work on one specific research area, and only a small number of researchers work across different research areas.", "Tables REF and REF show the edge and node overlap coefficients for the Flixster dataset.", "Different from the Arnetminer dataset, both edges and nodes have significant overlaps.", "For edge overlaps, even with threshold $\\theta =0.3$ , all topic pairs have edge overlap between $15\\%$ and $40\\%$ .", "For node overlap, we test the threshold for both $0.5$ to 5, but the overlap coefficients do not significantly change: at $\\theta =5$ , most pairs still have above $60\\%$ and up to $89\\%$ overlap.", "We think that this could be explained by the nature of Flixster, which is a movie rating site.", "Most users are interested in multiple categories of movies, and their influence to their friends are also likely to be across multiple categories.", "It is interesting to see that, even though the per-topic statistics between Arnetminer and Flixster are similar, they show quite different cross-topic overlap behaviors, which can be explained by the nature of the networks.", "This could be an independent research topic for further investigations on the influence behaviors among different topics.", "Table: Edge and Node overlap coefficients on Arnetminer.", "The upper black triangle represents edge overlap coefficient when θ=0.1\\theta =0.1.", "The entry on row ii, column jj represents Υ ij E (0.1)\\Upsilon ^{E}_{ij}(0.1); the lower blue triangle represents node overlap coefficient when θ=0.1\\theta =0.1.", "The entry on row ii, column jj represents Υ ij V (0.1)\\Upsilon ^{V}_{ij}(0.1).Table: Edge overlap coefficients on Flixster.", "The upper black triangle represents edge overlap coefficient when θ=0.1\\theta =0.1.", "The entry on row ii, column jj represents Υ ij E (0.1)\\Upsilon ^{E}_{ij}(0.1); the lower blue triangle represents edge overlap coefficient when θ=0.3\\theta =0.3.", "The entry on row ii, column jj represents Υ ij E (0.3)\\Upsilon ^{E}_{ij}(0.3).Table: Node overlap coefficients on Flixster.", "The upper black triangle represents node overlap coefficient when θ=0.5\\theta =0.5.", "The entry on row ii, column jj represents Υ ij V (0.5)\\Upsilon ^{V}_{ij}(0.5); the lower blue triangle represents node overlap coefficient when θ=5.0\\theta =5.0.", "The entry on row ii, column jj represents Υ ij V (5.0)\\Upsilon ^{V}_{ij}(5.0).Table: Overlap coefficient statistics for all topic pairsTable REF summarizes the edge and node overlap coefficient statistics among all pairs of topics for the two datasets.", "We can see that Arnetminer network has fairly separate topics on both nodes and edges, while Flixter network have significant topic overlaps.", "This may be explained by that in an academic network most researchers only work in one research area, but in a movie network many users are interested in more than one type of movies.", "Therefore, our first observation is: Observation 1 Topic separation in terms of influence probabilities is network dependent.", "In the Arnetminer network, topics are mostly separated among different edges and nodes in the network, while in the Flixster network there are significant overlaps on topics among nodes and edges." ], [ "Sources of seeds in the mixture", "Our second observation is more directly related to influence maximization.", "We would like to see if seeds selected by the greedy algorithm for a topic mixture are likely coming from top seeds for each individual topic.", "Intuitively, it seems reasonable to assume that top influencers for a topic mixture are likely from top influencers in their constituent topics.", "Table: Percentage of seeds in topic mixture that are also seeds of constituent topics.To check the source of seeds, we randomly generate 50 mixtures of two topics for both Arnetminer and Flixster, and use the greedy algorithm to select seeds for the mixture and the constituent topics.", "We then check the percentage of seeds in the mixture that is also in the constituent topics.", "Table REF shows our test results (Flixster (Dirhilect) is the result using a Dirichlet distribution to generate topic mixtures, see Section for more details).", "Our observation below matches our intuition: Observation 2 Most seeds for topic mixtures come from the seeds of constituent topics, in both Arnetminer and Flixster networks.", "For Arnetminer, it is likely due to the topic separation as observed in Table REF .", "For Flixster, even though topics have significant overlaps, these overlaps may result in many shared seeds between topics, which would also contribute as top seeds for topic mixtures." ], [ "Preprocessing Based Algorithms", "Topic-aware influence maximization can be solved by using existing influence maximization algorithms such as the ones in [13], [20]: when a query on an item $I = (\\lambda _1, \\lambda _2, \\cdots , \\lambda _d)$ comes, the algorithm first computes the mixed influence probability function $p = \\sum _j \\lambda _j p_j$ , and then applies existing algorithms using parameter $p$ .", "This, however, means that for each topic mixture influence maximization has to be carried out from scratch, which could be inefficient in large-scale networks.", "In this section, motivated by observations made in Section , we introduce two preprocessing based algorithms that cover different design choices.", "The first algorithm Best Topic Selection focuses on minimizing online processing time, and the second one MIS uses pre-computed marginal influence to achieve both fast online processing and competitive influence spread.", "For convenience, we consider the budget $k$ as fixed in our algorithms, but we could extend the algorithms to consider multiple $k$ values in preprocessing." ], [ "Best Topic Selection (BTS) algorithm", "The idea of our first algorithm is to minimize online processing by simply selecting a seed set for one of the constituent topics in the topic mixture that has the best influence performance, and thus we call it Best Topic Selection (BTS) algorithm.", "More specifically, given an item $I = (\\lambda _1, \\lambda _2, \\cdots , \\lambda _{d})$ , if we have pre-computed the seed set $S^{g}_i = S^{g}(k, \\lambda p_i)$ via the greedy algorithm for each topic $i$ , then we would simply use the seed set $S^{g}_{i^{\\prime }}$ that gives the best influence spread, i.e., $i^{\\prime } = \\operatornamewithlimits{argmax}_{i \\in [d]} \\sigma (S^{g}_i, \\lambda _i p_i)$ .", "However, in the preprocessing stage, the topic mixture $(\\lambda _1, \\lambda _2, \\cdots , \\lambda _{d})$ is not guaranteed to be pre-computed exactly.", "To deal with this issue, we pre-compute influence spread for a number of landmark points for each topic, and use rounding method in online processing to complete seed selection, as we explain in more detail now.", "Denote constant set $\\Lambda = \\lbrace \\lambda ^{c}_0, \\lambda ^{c}_1, \\lambda ^{c}_2, \\cdots , \\lambda ^{c}_m\\rbrace $ as a set of landmarks, where $0 = \\lambda ^{c}_0 < \\lambda ^{c}_1 < \\cdots < \\lambda ^{c}_m = 1$ .", "For each $\\lambda \\in \\Lambda $ and each topic $i \\in [d]$ , we pre-compute $S^{g}(k, \\lambda p_i)$ and $\\sigma (S^{g}(k, \\lambda p_i), \\lambda p_i)$ in the preprocessing stage, and store these values for online processing.", "In our experiments, we use uniformly selected landmarks and show that they are good enough for influence maximization.", "More sophisticated landmark selection method may be applied, such as the machine learning based method in [1].", "We define two rounding notations that return one of the neighboring landmarks in $\\Lambda = \\lbrace \\lambda ^{c}_0, \\lambda ^{c}_1, \\cdots , \\lambda ^{c}_m\\rbrace $ : for any $\\lambda \\in [0,1]$ , $\\underline{\\lambda }$ is denoted as rounding $\\lambda $ down to $\\lambda ^{c}_j$ where $\\lambda ^{c}_j \\le \\lambda < \\lambda ^{c}_{j+1}$ and $\\lambda ^{c}_j, \\lambda ^{c}_{j+1} \\in \\Lambda $ , and $\\overline{\\lambda }$ as rounding up to $\\lambda ^{c}_{j+1}$ where $\\lambda ^{c}_j < \\lambda \\le \\lambda ^{c}_{j+1}$ and $\\lambda ^{c}_j, \\lambda ^{c}_{j+1} \\in \\Lambda $ .", "Given $I = (\\lambda _1, \\lambda _2, \\cdots , \\lambda _{d})$ , let $D^+_I = \\lbrace i\\in [d] \\,\|\\, \\lambda _i > 0 \\rbrace $ .", "With the pre-computed $S^{g}(k, \\lambda p_i)$ and $\\sigma (S^{g}(k, \\lambda p_i), \\lambda p_i)$ for every $\\lambda \\in \\Lambda $ and every topic $i$ , the BTS algorithm is given in Algorithm REF .", "The algorithm basically rounds down the mixing coefficient on every topic to $(\\underline{\\lambda }_1, \\cdots , \\underline{\\lambda }_d)$ , and then returns the seed set $S^{g}(k, \\underline{\\lambda }_{i^{\\prime }} p_{i^{\\prime }})$ that gives the largest influence spread at the round-down landmarks: $i^{\\prime } = \\operatornamewithlimits{argmax}_{i \\in D^+_I} \\sigma (S^{g}(k, \\underline{\\lambda }_i p_{i}), \\underline{\\lambda }_i p_i)$ .", "[t] [1] $G=(V,E)$ , $k$ , $\\lbrace p_i\\,\|\\, i\\in [d]\\rbrace $ , $I = (\\lambda _1, \\cdots , \\lambda _{d})$ , $\\Lambda $ , $S^{g}(k, \\lambda p_i)$ and $\\sigma (S^{g}(k, \\lambda p_i), \\lambda p_i)$ , $\\forall \\lambda \\in \\Lambda , \\forall i\\in [d]$ .", "$I^{\\prime } = (\\underline{\\lambda }_1, \\cdots , \\underline{\\lambda }_d)$ $i^{\\prime } = \\operatornamewithlimits{argmax}_{i \\in D^+_I} \\sigma (S^{g}(k, \\underline{\\lambda }_i p_i), \\underline{\\lambda }_i p_i)$ $S^{g}(k, \\underline{\\lambda }_{i^{\\prime }} p_{i^{\\prime }})$ Best Topic Selection (BTS) Algorithm BTS is rather simple since it directly outputs a seed set for one of the constituent topics.", "However, we show below that even such a simple scheme could provide a theoretical approximation guarantee (if the influence spread function is sub-additive as defined below).", "Thus, we use BTS as a baseline for preprocessing based algorithms.", "We say that influence spread function $\\sigma (S,p)$ is $c$ -sub-additive in $p$ for some constant $c$ if for every set $S \\subseteq V$ with $\|S\| \\le k$ and every mixture $(\\lambda _1, \\lambda _2,\\ldots , \\lambda _d)$ , $\\sigma (S, \\sum _{i \\in D^+_I} \\lambda _i p_i)$ $ \\le $ $ c \\sum _{i \\in D^+_I} \\sigma (S, \\lambda _i p_i)$ .", "The sub-additivity property above means that the influence spread of any seed set $S$ in any topic mixture will not exceed constant times of the sum of the influence spread of the same seed set for each individual topic.", "It is easy to verify that, when the network is fully separable for all topic pairs, $\\sigma (S,p)$ is 1-sub-additive.", "The only counterexample to the sub-additivity assumption that we could find is a tree structure where even layer edges are for one topic and odd layer edges are for another topic.", "Such structures are rather artificial, and we believe that for real networks the influence spread is $c$ -sub-additive in $p$ with a reasonably small constant $c$ .", "We define $\\mu _{\\max } = \\max _{i \\in [d], \\lambda \\in [0,1]}\\frac{\\sigma (S^{g}(k, \\overline{\\lambda }p_i), \\overline{\\lambda }p_i)}{\\sigma (S^{g}(k, \\underline{\\lambda }p_i), \\underline{\\lambda }p_i)}$ , which is a value controlled by preprocessing.", "A fine-grained landmark set $\\Lambda $ could make $\\mu _{\\max }$ close to 1.", "The following Theorem REF guarantees the theoretical approximation ratio of Algorithm REF .", "If the influence spread function $\\sigma (S,p)$ is $c$ -sub-additive in $p$ , Algorithm REF achieves $\\frac{1-e^{-1}}{c \|D^+_I\| \\mu _{\\max }}$ approximation ratio for item $I = (\\lambda _1, \\lambda _2, \\cdots , \\lambda _{d})$ .", "Denote $S^{}= S^{}(k, p)$ , $\\overline{S}^{}_i = S^{}(k, \\overline{\\lambda }_i p_i)$ , $\\overline{S}^{g}_i = S^{g}(k, \\overline{\\lambda }_i p_i)$ and $\\underline{S}^{g}_i = S^{g}(k, \\underline{\\lambda }_i p_i)$ .", "Since $\\sigma (S, p)$ is monotone (Lemma REF ) and $c$ -sub-additive in $p$ , it implies $\\sigma (S^{}, p) = \\sigma (S^{}, \\sum _{i \\in D^+_I} \\lambda _i p_i)\\le c \\sum _{i \\in D^+_I} \\sigma (S^{}, \\lambda _i p_i)$ $\\le $ $c \\sum _{i \\in D^+_I} \\sigma (S^{}, \\overline{\\lambda }_i p_i)$ .", "From [13], we know $\\sigma (S^{}(k, p_0), p_0) \\le \\frac{1}{1-e^{-1}} \\sigma (S^{g}(k, p_0), p_0)$ holds for any $p_0$ in Algorithm REF .", "Thus we have, for each $i \\in D^+_I$ , $\\sigma (S^{}, \\overline{\\lambda }_i p_i)\\le \\sigma (\\overline{S}^{}_i, \\overline{\\lambda }_i p_i)\\le \\frac{\\sigma (\\overline{S}^{g}_i, \\overline{\\lambda }_i p_i)}{1-e^{-1}}\\le \\frac{\\mu _{\\max } \\cdot \\sigma (\\underline{S}^{g}_i, \\underline{\\lambda }_i p_i)}{1-e^{-1}}$ .", "According to line REF of Algorithm REF , $i^{\\prime }$ satisfies $\\sigma (\\underline{S}^{g}_{i^{\\prime }}, \\underline{\\lambda }_{i^{\\prime }} p_{i^{\\prime }}) = \\max _{i \\in D^+_I} \\sigma (\\underline{S}^{g}_i, \\underline{\\lambda }_i p_i)$ , and $\\sigma (\\underline{S}^{g}_{i^{\\prime }}, \\underline{\\lambda }_{i^{\\prime }} p_{i^{\\prime }}) \\le \\sigma (\\underline{S}^{g}_{i^{\\prime }}, \\lambda _{i^{\\prime }} p_{i^{\\prime }})$ .", "Thus, connecting all the inequalities, we have $\\sigma (S^{*}, p)$ $\\le $ $ \\frac{c \|D^+_I\| \\mu _{\\max }}{1-e^{-1}} \\sigma (\\underline{S}^{g}_{i^{\\prime }}, \\lambda _{i^{\\prime }} p_{i^{\\prime }})$ .", "Therefore, Algorithm REF achieves approximation ratio of $\\frac{1}{c \|D^+_I\| \\mu _{\\max }}(1-\\frac{1}{e})$ under the sub-additive assumption.", "$\\blacksquare $ The approximation ratio given in the theorem is a conservative bound for the worst case (e.g., a common setting may be $c=1$ , $\\mu _{\\max }=1.5$ , $\|D^+_I\|=2$ ).", "Tighter online bound in our experiment section based on [14] shows that Algorithm REF performs much better than the worst case scenario." ], [ "Marginal Influence Sort (MIS) algorithm", "Our second algorithm derives the seed set from pre-computed seed set of constituent topics, which is based on Observation REF .", "Moreover, it uses marginal influence information pre-computed to help select seeds from different seed sets.", "Our idea is partially motivated from Observation REF , especially the observation on Arnetminer dataset, which shows that in some cases the network could be well separated among different topics.", "Intuitively, if nodes are separable among different topics, and each node $v$ is only pertinent to one topic $i$ , the marginal influence of $v$ would not change much whether it is for a mixed item or the pure topic $i$ .", "The following lemma makes this intuition precise for the extreme case of fully separable networks.", "If a network is fully separable among all topics, then for any $v \\in V$ and topic $i \\in [d]$ such that $\\sigma (v, p_i) > 1$ , for any item $I = (\\lambda _1, \\lambda _2, \\dots , \\lambda _{d})$ , for any seed set $S \\subseteq V$ , we have ${\\it MI}(v \| S, \\lambda _i p_i) = {\\it MI}(v \| S, p)$ , where $p = \\sum _{j\\in [d]} \\lambda _j p_j$ .", "[Proof sketch] Let $G_i=(V_i,E_i)$ be the subgraph of $G$ generated by edges $(u,w)$ such that $p_i(u,w) > 0$ and their incident nodes.", "It is easy to verify that when the network is fully separable among all topics, $G_i$ and $G_j$ are disconnected for any $i\\ne j$ .", "In this case, we have (a) for any node $v$ and topic $i$ such that $\\sigma (v, p_i) > 1$ , $v \\in V_i$ ; (b) for any edge $(u,w)\\in E_i$ , $p(u,w) = \\lambda _i p_i(u,w)$ ; and (c) $\\sigma (S,p^{\\prime }) = \\sum _{j\\in [d]} \\sigma (S\\cap V_j, p^{\\prime })$ for any $p^{\\prime }$ .", "With the above property, a simple derivation following the definition of marginal influence will lead to ${\\it MI}(v \| S, \\lambda _i p_i) = {\\it MI}(v \| S, p)$ .", "$\\blacksquare $ The above lemma suggests that we can use the marginal influence of a node on each topic when dealing with a mixture of topics.", "Algorithm MIS is based on this idea.", "Recall the detail of Algorithm REF , given any fixed probability $p$ and budget $k$ , for each iteration $j = 1,2,\\cdots , k$ , it calculates $v_j$ to maximize marginal influence ${\\it MI}(v_j \| S_{j-1}, p)$ and let $S_j = S_{j-1} \\cup \\lbrace v_j \\rbrace $ every time, and output $S^{g}(k, p) = S_k$ as seeds.", "Let ${\\it MI}^{g}(v_j, p) = {\\it MI}(v_j \| S_{j-1}, p)$ , if $v_j \\in S^{g}(k, p)$ , and 0 otherwise.", "${\\it MI}^{g}(v_j, p) $ is the marginal influence of $v_j$ according to the greedy selection order.", "The preprocessing goes as follows.", "We also use the landmark set $\\Lambda = \\lbrace \\lambda ^{c}_0, \\lambda ^{c}_1, \\lambda ^{c}_2, \\cdots , \\lambda ^{c}_m\\rbrace $ .", "For every $\\lambda \\in \\Lambda $ , we pre-compute $S^{g}(k, \\lambda p_i)$ , for every single topic $i \\in [d]$ , and cache ${\\it MI}^{g}(v, \\lambda p_i)$ , $\\forall v \\in S^{g}(k, \\lambda p_i)$ in advance by Algorithm REF .", "With the above preparation, we can design Marginal Influence Sort (MIS) algorithm as described in Algorithm REF .", "Given an item $I = (\\lambda _1, \\cdots , \\lambda _{d})$ , the online processing stage first rounding down the mixture to $I^{\\prime }= (\\underline{\\lambda }_1, \\cdots , \\underline{\\lambda }_d)$ , and then use the union $V^g = \\cup _{i \\in [d], \\underline{\\lambda }_i > 0} S^{g}(k, \\underline{\\lambda }_i p_i)$ as seed candidates.", "If a node appears in multiple pre-computed seed sets, we add their marginal influence in each set together (line REF ).", "Then we simply sort all nodes in $V^g$ according to their computed marginal influence $f(v)$ and return the top $k$ nodes as seeds.", "[t] [1] $G=(V,E)$ , $k$ , $\\lbrace p_i\\,\|\\, i\\in [d]\\rbrace $ , $I = (\\lambda _1, \\cdots , \\lambda _{d})$ , $\\Lambda $ , $S^{g}(k, \\lambda p_i)$ and ${\\it MI}^{g}(v, \\lambda p_i)$ , $\\forall \\lambda \\in \\Lambda $ , $\\forall i \\in [d]$ .", "$I^{\\prime } = (\\underline{\\lambda }_1, \\cdots , \\underline{\\lambda }_d)$ $V^g = \\cup _{i \\in [d], \\underline{\\lambda }_i > 0} S^{g}(k, \\underline{\\lambda }_i p_i)$ $v \\in V^g$ $f(v) = \\sum _{i \\in [d], \\underline{\\lambda }_i > 0} {\\it MI}^{g}(v, \\underline{\\lambda }_i p_i)$ top $k$ nodes with the largest $f(v), \\forall v \\in V^g$ Marginal Influence Sort (MIS) Algorithm Although MIS is a heuristic algorithm, it does guarantee the same performance as the original greedy algorithm in fully separable networks when the topic mixtures is from the landmark set, as shown by the theorem below.", "Note that in a fully separable network, it is reasonable to assume that seeds for one topic comes from the subgraph for that topic, and thus seeds from different topics are disjoint.", "Suppose $I = (\\lambda _1, \\lambda _2, \\cdots , \\lambda _d)$ , where each $\\lambda _i \\in \\Lambda $ , and $S^{g}(k, \\lambda _1 p_1)$ , $\\cdots $ , $S^{g}(k, \\lambda _d p_d)$ are disjoint.", "If the network is fully separable for all topics, the seed set calculated by Algorithm REF is one of the possible sequences generated by Algorithm REF under the mixed influence probability $p = \\sum _{i \\in [d]} \\lambda _i p_i$ .", "[Proof sketch] Denote $v_1, v_2, \\cdots , v_k \\in V^g$ as the final seeds selected for the topic mixture in this order, and let $S_0 = \\emptyset $ and $S_\\ell = S_{\\ell - 1} \\cup \\lbrace v_\\ell \\rbrace $ , for $\\ell = 1,2,\\cdots ,k$ .", "Since the network is fully separable and topic-wise seed sets are disjoint, by Lemma 4.1 we can get that $v_1, v_2, \\cdots , v_k$ are selected from topic-wise seeds sets, and $\\forall v \\in V^g$ , $f(v) = {\\it MI}(v \| S_{\\ell - 1}, p)$ .", "We can prove that $v_{\\ell } = \\operatornamewithlimits{argmax}_{v \\in V \\setminus S_{\\ell -1}}$ ${\\it MI}(v \| S_{\\ell -1}, p)$ , $\\forall \\ell = 1,2,\\cdots ,k$ by induction.", "It is straightforward to see that $v_1 = \\operatornamewithlimits{argmax}_{v \\in V}$ $ {\\it MI}(v \| \\emptyset , p)$ .", "Assume it holds for $\\ell = j \\in \\lbrace 1,2,\\cdots ,k-1\\rbrace $ .", "Then, for $\\ell = j+1$ , for a contradiction we suppose that the $(j+1)$ -th seed $v^{\\prime }$ is chosen from $V \\setminus V^{g}$ other than $v_{j+1}$ , i.e., ${\\it MI}(v^{\\prime } \| S_{j}, p) > {\\it MI}(v_{j+1} \| S_{j}, p)$ .", "Denote $i^{\\prime }$ such that $\\sigma (v^{\\prime }, p_{i^{\\prime }}) > 1$ .", "Since budget $k > j$ , we can find a node $u \\in S^{g}(k, \\lambda _{i^{\\prime }} p_{i^{\\prime }}) \\setminus S_{j}$ , such that ${\\it MI}(u \| S_{j}, \\lambda _{i^{\\prime }} p_{i^{\\prime }})$ $\\ge $ ${\\it MI}(v^{\\prime } \| S_{j}, \\lambda _{i^{\\prime }} p_{i^{\\prime }})$ , and $u$ is selected before $v_{j+1}$ , which is a contradiction.", "Therefore, we will conclude that $v_1, v_2$ , $\\cdots $ , $v_k$ is one possible sequence from the greedy algorithm.", "$\\blacksquare $ The theorem suggests that MIS would work well for networks that are fairly separated among different topics, which are verified by our test results on the Arnetminer dataset.", "Moreover, even for networks that are not well separated, it is reasonable to assume that the marginal influence of nodes in the mixture is related to the sum of its marginal influence in individual topics, and thus we expect MIS to work also competitively in this case, which is verified by our test results on the Flixster dataset." ], [ "Experiments", "We test the effectiveness of our algorithms by using a number of real-world datasets, and compare them with state-of-the-art influence maximization algorithms." ], [ "Algorithms for comparison", "In our experiments, we test our topic-aware preprocessing based algorithms MIS and BTS comprehensively.", "We also select three classes of algorithms for comparison: (a) Topic-aware algorithms: The topic-aware greedy algorithm (TA-Greedy) and a state-of-the-art fast heuristic algorithm PMIA (TA-PMIA) from [20]; (b) Topic-oblivious algorithms: The topic-oblivious greedy algorithm (TO-Greedy), degree algorithm (TO-Degree) and random algorithm (Random); (c) Simple heuristic algorithms that do not need preprocessing: The topic-aware PageRank algorithm (TA-PageRank) from [4] and WeightedDegree algorithm (TA-WeightedDegree).", "The greedy algorithm we use employs lazy evaluation [14] to provide hundreds of time of speedup to the original Monte Carlo based greedy algorithm [13], and also provides the best theoretical guarantee.", "PMIA is a fast heuristic algorithm for the IC model based on trimming influence propagation to a tree structure and fast recursive computation on trees, and it achieves thousand fold speedup comparing to optimized greedy approximation algorithms with a small degradation on influence spread [20] (in this paper, we set a small threshold $\\theta = 1/1280$ to alleviate the degradation).", "Topic-oblivious algorithms work under previous IC model that does not identify topics, i.e., it takes the fixed mixture $\\forall j\\in [d], \\lambda _j=\\frac{1}{d}$ .", "TO-Greedy runs greedy algorithm for previous IC model and uses the top-$k$ nodes as its seeds.", "TO-Degree outputs the top-$k$ nodes with the largest degree based on the original graph.", "Random simply chooses $k$ nodes at random.", "We also carefully choose two simple heuristic algorithms that do not need preprocessing.", "TA-PageRank uses the probability of the topic mixture as its transfer probability, and runs PageRank algorithm to select $k$ nodes with top rankings.", "The damping factor is set to $0.85$ .", "TA-WeightedDegree uses the degrees weighted by the probability from topic mixtures, and selects top-$k$ nodes with the highest weighted degrees.", "Finally, we study the possibility of acceleration for large graphs by comparing PMIA with greedy algorithm in preprocessing stage.", "Therefore, we denote MIS and BTS algorithms, utilizing the seeds and marginal influence from greedy and PMIA, as MIS[Greedy], BTS[Greedy] and MIS[PMIA], BTS[PMIA], respectively." ], [ "Experiment setup", "We conduct all the experiments on a computer with 2.4GHz Intel(R) Xeon(R) E5530 CPU, 2 processors (16 cores), 48G memory, and an operating system of Windows Server 2008 R2 Enterprise (64 bits).", "The code is written in C++ and compiled by Visual Studio 2010.", "We test these algorithms on the Flixster and Arnetminer datasets as we described in Section , which have the advantage that the influence probabilities of all edges on all topics are learned from real action trace data or node topic distribution data.", "To further test the scalability of different algorithms, we use a larger network data DBLP, which is also used in [20].", "DBLP is an academic collaboration network extracted from the online service (www.DBLP.org), where nodes represent authors and edges represent coauthoring relationships.", "It contains 650K nodes and 2 million edges.", "As DBLP does not have influence probabilities from the real data, we simulate two topics according to the joint distribution of topics 1 and 2 in the Flixster and follow the practice of the TRIVALENCY model in [20] to rescale it into $0.1$ , $0.01$ , or $0.001$ , standing for strong, medium, and low influence, respectively.", "In terms of topic mixtures, in practice and also supported by our data, an item is usually a mixture of a small number of topics thus our tests focus on testing topic mixtures from two topics.", "First, we test random samples to cover most common mixtures as follows.", "For these three datasets, we uses 50 topic mixtures as testing samples.50 samples is mainly to fit for the slow greedy algorithm.", "Each topic mixture is uniformly selected from all possible two topic mixtures.", "Second, since we have the data of real topic mixtures in Flixster dataset, we also test additional cases following the same sampling technique described in Section 3.1 of [1].", "We estimate the Dirichlet distribution that maximizes the likelihood over topics learned from the data.", "After the distribution is learned, we re-sample 50 topic mixtures for testing.", "In the preprocessing stage, we use two algorithms, Greedy and PMIA, to pre-compute seed sets for MIS and BTS, except that for the DBLP dataset, which is too large to run the greedy algorithm, we only run PMIA.", "Algorithms MIS and BTS need to pre-select landmarks $\\Lambda $ .", "In our tests, we use 11 equally distant landmarks $\\lbrace 0, 0.1, 0.2, \\ldots , 0.9, 1\\rbrace $ .", "Each landmarks can be pre-computed independently, therefore we run them on 16 cores concurrently in different processes.", "We choose $k=50$ seeds in all our tests and compare the influence spread and running time of each algorithm.", "For the greedy algorithm, we use 10000 Monte Carlo simulations.", "We also use 10000 simulation runs and take the average to obtain the influence spread for each selected seed set.", "In addition, we apply offline bound and online bound to estimate influence spread of optimal solutions.", "Offline bound is the influence spread of any greedy seeds multiplied by factor $1/(1-e^{-1})$ .", "The online bound is based on Theorem 4 in [14]: for any seed set $S$ , its influence spread plus the sum of top $k$ marginal influence spread of $k$ other nodes is an upper bound on the optimal $k$ seed influence spread.", "We use the minimum of the upper bounds among the cases of $S=\\emptyset $ and $S$ being one of the greedy seed sets selected." ], [ "Experiment results", "Figure REF shows the total influence spread results on Arnetminer with random samples (a); Flixster with random and Dirichlet samples, (b) and (c), respectively; and DBLP with random samples (d).", "Table REF shows the preprocessing time based on greedy algorithm and PMIA algorithm on three datasets.", "Table REF shows the average online response time of various algorithms in finding 50 seeds (topic-oblivious algorithms always use the same seeds and thus are not reported).", "As is shown in Table REF , we run each landmark concurrently, and count both the total CPU time and the maximum time needed for one landmark.", "While the total time shows the cumulative preprocessing effort, the maximum time shows the latency when we use parallel preprocessing on multiple cores.", "The results indicate that the greedy algorithm is suitable for small graphs but infeasible for large graphs like DBLP, while PMIA is a scalable preprocessing solution on large graphs.", "For this reason, we test two preprocessing techniques and also compare their performance.", "For the Arnetminer dataset (Figure REF (a)), it clearly separates all algorithms into three tiers: the top tier is TA-Greedy, TA-PMIA, MIS[Greedy] and MIS[PMIA]; the middle tier is TA-WeightedDegree, BTS[Greedy], BTS[PMIA] and TA-PageRank; and the lower tier is topic-oblivious algorithms TO-Greedy, TO-Degree and Random.", "In particular, we measure the gaps of influence spread among different algorithms.", "We observe that the gap of top tiers are negligible, because TA-PMIA, MIS[Greedy] and MIS[PMIA] are only $0.61\\%$ , $0.32\\%$ and $1.08\\%$ smaller than TA-Greedy, respectively; the middle tier algorithms BTS[Greedy], BTS[PMIA], TA-WeightedDegree and TA-PageRank are $4.06\\%$ , $4.68\\%$ , $4.67\\%$ and $26.84\\%$ smaller, respectively; and the lower tier TO-Greedy, TO-Degree and Random have difference of $28.57\\%$ , $56.75\\%$ and $81.48\\%$ , respectively.", "(All percentages reported in this section are averages over influence spread from one seed to 50 seeds.)", "The detailed analyses are listed as follows: First, topic-oblivious algorithms does not perform well in topic-aware environment.", "Based on Observation REF , when topics are separated, algorithms ignoring topic mixtures cannot find influential seeds for all topics, and thus do not have good influence spread.", "Second, MIS[Greedy] and MIS[PMIA] almost match the influence spread of those of TA-Greedy and TA-PMIA.", "As is indicated from offline and online bounds, MIS[Greedy], BTS[Greedy] are 76.9% and 72.5% ($> 1-e^{-1}$ ) of the online bound, which demonstrates their effectiveness better than their conservative theoretical bounds could support.", "The MIS algorithm runs super fast in online processing, finishing 50 seeds selection in just a few microseconds (Table REF ), which is three orders of magnitude faster than the millisecond response time reported in [1], and at least three orders of magnitude faster than any other topic-aware algorithms.", "This is because it relies on pre-computed marginal influence and only a sorting process is needed online.", "Third, BTS[Greedy] and BTS[PMIA] are not expected to be better than MIS[Greedy] and MIS[PMIA], since BTS is a baseline algorithm only selecting a seed set from one topic.", "However, due to the preprocessing stage, we find that it can even perform better than other simple topic-aware heuristic algorithms that have short online response time.", "In addition, replacing the greedy algorithm with PMIA in the preprocessing stage, MIS and BTS only lose $0.76\\%$ and $0.62\\%$ in influence spread, indicating that PMIA is a viable choice for preprocessing, which greatly reduces the offline preprocessing time (Table REF ).", "What we can conclude from tests on Arnetminer is that, for networks where topics are well separated among nodes and edges such as in academic networks, utilizing preprocessing can greatly save the online processing time.", "In particular, MIS algorithm is well suited for this environment achieving microsecond response time with very small degradation in seed quality.", "For Flixster dataset (Figure REF (b) and (c)), we see that the influence spread of TA-PMIA, MIS[Greedy], MIS[PMIA], BTS[Greedy] and BTS[PMIA] are $1.78\\%$ , $3.04\\%$ , $4.58\\%$ , $3.89\\%$ and $5.29\\%$ smaller than TA-Greedy for random samples, and $1.41\\%$ , $1.94\\%$ , $3.37\\%$ , $2.31\\%$ and $3.59\\%$ smaller for Dirichlet samples, respectively.", "In Flixster, we can see that for networks where topics overlap with one another on nodes, our preprocessing based algorithms can still perform quite well.", "This is because most seeds of topic mixtures are from the constituent topics (Observation REF ).", "On the other hand, the influence of TA-WeightedDegree, TA-PageRank and TO-Greedy will suffer a noticeable degeneration demonstrated from two curves.", "In terms of online response time (Table REF ), the result is consistent with the result for Arnetminer: only MIS and BTS can achieve microsecond level online response, and all other topic-aware algorithms need at least milliseconds since they at least need a ranking computation among all nodes in the graph.", "In addition, TA-PMIA on Flixster is much slower than on Arnetminer, because both the network size and the computed MIA tree size are much larger, indicating that PMIA is not suitable in providing stable online response time.", "In contrast, the response time of MIS and BTS do not change significantly among different graphs.", "In DBLP (Figure REF (d)), the graph is too large to run greedy algorithm, thus we take TA-PMIA as the baseline algorithm to compare with other algorithms.", "For different algorithms, the influence spread is close to each other, and our results show that MIS[PMIA] has equal competitive influence spread with TA-PMIA ($0.44\\%$ slightly larger), while BTS[PMIA], TA-WeightedDegree, TO-Degree and TA-PageRank are $1.33\\%$ , $1.83\\%$ , $6.05\\%$ and $35.54\\%$ smaller than TA-PMIA, respectively.", "Combining Table REF and Table REF , we find that the greedy algorithm is not suitable for preprocessing for large graphs, while PMIA can be used in this case.", "To summarize, the greedy algorithm has the best influence spread performance, but is slow and not suitable for large-scale networks or fast response time requirements.", "PMIA as a fast heuristic can achieve reasonable performance in both influence spread and online processing time, but its response time varies significantly depending on graph size and influence probability parameters, and could take minutes or longer to complete.", "Our proposed MIS emerges as a strong candidate for fast real-time processing of topic-aware influence maximization task: it achieves microsecond response time, which does not depend on graph size or influence probability parameters, while its influence spread matches or is very close to the best greedy algorithm and outperforms other simple heuristics.", "Furthermore, in large graphs where greedy is too slow to finish, PMIA is a viable choice for preprocessing, and our MIS using PMIA as the preprocessing algorithm achieves almost the same influence spread as MIS using the greedy algorithm for preprocessing." ], [ "Related Work", "Domingos and Richardson [8], [17] are the first to study influence maximization in an algorithmic framework.", "Kempe et al.", "[13] first formulate the discrete influence diffusion models including the independent cascade model and linear threshold model, and provide the first batch of algorithmic results on influence maximization.", "A large body of work follows the framework of [13].", "One line of research improves on the efficiency and scalability of influence maximization algorithms [10], [7], [20], [11].", "Others extend the diffusion models and study other related optimization problems (e.g., [5], [3], [12]).", "A number of studies propose machine learning methods to learn influence models and parameters (e.g., [18], [19], [9]).", "A few studies look into the interplay of social influence and topic distributions [19], [16], [22], [15].", "They focus on inference of social influence from topic distributions or joint inference of influence diffusion and topic distributions.", "They do not provide a dynamic topic-aware influence diffusion model nor study the influence maximization problem.", "Barbieri et al.", "[2] introduce the topic-aware influence diffusion models TIC and TLT as extensions to the IC and LT models.", "They provide maximum-likelihood based learning method to learn influence parameters in these topic-aware models.", "We use the their proposed models and their dataset with the learned parameters.", "A recent independent work by Aslay et al.", "[1] is the closest one to our work.", "Their work focus on index building in the query space while we use pre-computed marginal influence to help guiding seed selection, and thus the two approaches are complementary.", "Other differences have been listed in the introduction and will not be repeated here." ], [ "Future Work", "One possible follow-up work is to combine the advantages of our approach and the approach in [1] to further improve the performance.", "Another direction is to study fast algorithms with stronger theoretical guarantee.", "An important work is to gather more real-world datasets and conduct a thorough investigation on the topic-wise influence properties of different networks, similar to our preliminary investigation on Arnetminer and Flixster datasets.", "This could bring more insights to the interplay between topic distributions and influence diffusion, which could guide future algorithm design." ], [ "Acknowledgments", "We would like to thank Nicola Barbieri and Jie Tang, the authors of [2], [19], respectively, for providing Flixster and Arnetminer datasets." ] ]	1403.0057
[ [ "Solving nth order fuzzy differential equation by fuzzy Laplace transform" ], [ "Abstract In this paper, we generalize the fuzzy Laplace transformation (FLT) for the nth derivative of a fuzzy-valued function named as nth derivative theorem and under the strongly generalized differentiability concept, we use it in an analytical solution method for the solution of an nth order fuzzy initial value problem (FIVP).", "This is a simple approach toward the solution of nth order fuzzy initial value problem (FIVP) by the nth generalized (FLT) form, and then we can use it to solve any order of FIVP.", "The related theorems and properties are proved.", "The method is illustrated with the help of some examples.", "We use MATLAB to evaluate the inverse Laplace transform." ], [ "Introduction", "The term fuzzy derivative has been introduced in 1972 by Chang and Zadeh [1], while the the term fuzzy differential equation (FDE) was first formulated by Kaleva [2] and Seikala [3].", "The theory of fuzzy differential equations (FDEs) has two branches, the one in which the Hukuhara derivative is the main tool and the other one is inclusion.", "This paper is based on the concept of generalized Hukuhara differentiability.", "The solution of FDE has a wide range of applications in the dynamic system of uncertainty.", "Moreover the field of FDEs is becoming a necessary part of science and real word problems [2], [3].", "In the last few decades, many method have been applied for the solution of FIVP as discussed in [4], [8], [21], [22] but every method has advantages and disadvantages.", "In the near past Allahviranloo, Kaini and Barkhordari [5] introduced an approach toward the existence and uniqueness to solve a second order FIVP.", "Here we will adopt FLT method in order to find an analytical solution of FIVP.", "Recently Salahshour and Allahviranloo [6] has found the analytical solution of second order FIVP, then the third order FIVP has been solved by Hawrra and Amal [7].", "In order to solve a FIVP Allahviranloo, Kiani and Barkhordari [5] stated under what condition FLT can be applied to solve a FIVP.", "In [5], they proposed two conditions for the existence of solution of FIVP using FLT and its inverse, and gives some useful results in the form of first order and second derivative theorem such as linearity, continuity, uniformity and convergency under the new definition of absolute value of fuzzy-valued functions etc.", "They also proposed two types of absolute value of fuzzy valued function which define the convergence and exponential order of a fuzzy-valued function to find an appropriate condition.", "In addition they have also proved that a large class of fuzzy-valued function can be solved with the help of FLT.", "In this paper, we generalize FLT for $nth$ order FIVP.", "This paper is arranged as follows: In section 2, we recall some basics definitions and theorems.", "In section 3, fuzzy Laplace Transform is defined.", "Then we prove $nth$ derivative theorem which is our contribution.", "In section 4, we solve FDEs by FLT.", "To illustrate the method, several examples are given in section 5.", "Conclusion is given in section 6." ], [ "Preliminaries", "In this section we will recall some basics definitions and theorems needed throughout the paper such as fuzzy number, fuzzy-valued function and the derivative of the fuzzy-valued functions as presented in [2], [3], [8], [10].", "Definition 2.1 A fuzzy number is defined as the mapping such that $u:R\\rightarrow [0,1]$ , which satisfies the following four properties $u$ is upper semi-continuous.", "$u$ is fuzzy convex that is $u(\\lambda x+(1-\\lambda )y) \\ge \\min {\\lbrace u(x), u(y)\\rbrace }.", "x, y\\in R$ and $\\lambda \\in [0,1]$ .", "$u$ is normal that is $\\exists $ $x_0\\in R$ , where $u(x_0)=1$ .", "$A=\\lbrace \\overline{x \\in \\mathbb {R}: u(x)>0}\\rbrace $ is compact that, where $\\overline{A}$ is closure of $A$ .", "The mapping $f:R\\rightarrow E$ , where $E$ denotes the set of all fuzzy numbers(what does it mean and why it is important to write it here).", "Definition 2.2 Obviously $R\\subset {E}$ for $0\\le r \\le 1$ set $[u]^{r}=\\lbrace x\\in {R}:u(x)\\ge r\\rbrace $ and $[u]^{0}=\\lbrace x\\in {R}: u(x)>0\\rbrace $ .", "So it must be well known that $r\\in [0,1]$ and $[u]^{r}$ is bounded closed interval.", "A fuzzy number can be defined in the parametric form as given in the following definition.", "(This definition does not make any sense to me) Definition 2.3 A fuzzy number in parametric form is an order pair of the form $u=(\\underline{u}(r), \\overline{u}(r))$ , where $0\\le r\\le 1$ satisfying the following conditions.", "$\\underline{u}(r)$ is a bounded left continuous increasing function in the interval $[0,1]$ $\\overline{u}(r)$ is a bounded left continuous decreasing function in the interval $[0,1]$ $\\underline{u}{(r)\\le \\overline{u}(r)}$ .", "If $\\underline{u}(r)=\\overline{u}(r)=r$ , then $r$ is called crisp number.", "Now we recall a triangular fuzzy number which must be in the form of $u=(l, c, r)$ where $l,c,r\\in R$ and $l\\le c\\le r$ , then $\\underline{u}(\\alpha )=l+(c-r)\\alpha $ and $\\overline{u}(\\alpha )=r-(r-c)\\alpha $ are the end points of the $\\alpha $ level set.", "Since each $y\\in R$ can be regarded as a fuzzy number if $\\widetilde{y}(t)={\\left\\lbrace \\begin{array}{ll}y, \\;\\;\\; if \\;\\; y=t,\\\\ 0, \\;\\;\\; if \\;\\; y\\ne t.\\end{array}\\right.", "}$ For arbitrary fuzzy numbers $u=(\\underline{u}(\\alpha ), \\overline{u}(\\alpha ))$ and $v=(\\underline{v}(\\alpha ), \\overline{v}(\\alpha ))$ and an arbitrary crisp number $j$ , we define addition and scalar multiplication as: $(\\underline{u+v})(\\alpha )=(\\underline{u}(\\alpha )+\\underline{v}(\\alpha ))$ .", "$(\\overline{u+v})(\\alpha )=(\\overline{u}(\\alpha )+\\overline{v}(\\alpha ))$ .", "$(j\\underline{u})(\\alpha )=j\\underline{u}(\\alpha )$ , $(j\\overline{u})(\\alpha )=j\\overline{u}(\\alpha )$ $j\\ge 0$ $(j\\underline{u})(\\alpha )=j\\overline{u}(\\alpha )\\alpha , (j\\overline{u})(\\alpha )=j\\underline{u}(\\alpha )\\alpha $ , $j<0$ Definition 2.4 Let us suppose that x, y $\\in E$ , if $\\exists $ $z\\in E$ such that $x=y+z$ , then $z$ is called the H-difference of $x$ and $y$ and is given by $x\\ominus y$ Remark 2.5 (see [11]) Let $X$ be a cartesian product of the universes, $X_1$ , $X_1, \\cdots , X_n$ , that is $X=X_1 \\times X_2 \\times \\cdots \\times X_n$ and $A_{1},\\cdots ,A_{n}$ be $n$ fuzzy numbers in $X_1, \\cdots , X_n$ respectively then $f$ is a mapping from $X$ to a universe $Y$ , and $y=f(x_{1},x_{2},\\cdots ,x_{n})$ , then the Zadeh extension principle allows us to define a fuzzy set $B$ in $Y$ as; $B=\\lbrace (y, u_B(y))\|y=f(x_1,\\cdots ,x_{n}),(x_{1},\\cdots ,x_{n})\\in X\\rbrace ,$ where $u_B(y)={\\left\\lbrace \\begin{array}{ll}\\sup _{{(x_1,\\cdots ,x_n)} \\in f^{-1}(y)} \\min \\lbrace u_{A_1}(x_1),\\cdots u_{A_n}(x_n)\\rbrace , \\;\\;\\; if \\;\\;\\; f^{-1}(y)\\ne 0,\\\\ 0, \\;\\;\\;\\; otherwise,\\end{array}\\right.", "}$ where $f^{-1}$ is the inverse of $f$ .", "The extension principle reduces in the case if $n=1$ and is given as follows: $B=\\lbrace (y, u_B(y)\|y=f(x), \\mbox{ } x \\in X)\\rbrace ,$ where $u_B(y)={\\left\\lbrace \\begin{array}{ll}\\sup _{x\\in f^{-1}(y)} \\min \\lbrace u_A(x)\\rbrace , \\mbox{ if } f^{-1}(y)\\ne 0,\\\\0, \\;\\;\\;\\; otherwise.", "\\end{array}\\right.}", "$ By Zadeh extension principle the approximation of addition of $E$ is defined by $(u\\oplus v)(x)=\\sup _{y\\in R} \\min (u(y), v(x-y))$ , $x \\in R$ and scalar multiplication of a fuzzy number is defined by $(k\\odot u)(x)=\\lbrace u(\\frac{x}{k}), k>0$ and $0, k=0\\rbrace $ , where $\\widetilde{0}\\in E$ .", "$(k\\odot u)(x)={\\left\\lbrace \\begin{array}{ll}u(\\frac{x}{k}), \\;\\;\\; k > 0,\\\\ 0 \\;\\;\\; \\mbox{ otherwise }.", "\\end{array}\\right.}", "$ The Housdorff distance between the fuzzy numbers defined by [6] $d:E\\times E\\longrightarrow R^{+}\\cup {0},$ $d(u,v)=\\sup _{r\\in [0,1]}\\max \\lbrace \|\\underline{u}(r)-\\underline{v}(r)\|, \|\\overline{u}(r)-\\overline{v}(r)\|\\rbrace ,$ where $u=(\\underline{u}(r), \\overline{u}(r))$ and $v=(\\underline{v}(r), \\overline{v}(r))\\subset R$ has been utilized by Bede and Gal [13].", "(need to rewrite it and check reference [13]) We know that if $d$ is a metric in $E$ , then it will satisfies the following properties, introduced by Puri and Ralescu [14]: $d(u+w,v+w)=d(u,v)$ , $\\forall $ u, v, w $\\in $ E. $(k \\odot u, k \\odot v)=\|k\|d(u, v)$ , $\\forall $ k $\\in $ R, and u, v $\\in $ E. $d(u \\oplus v, w \\oplus e)\\le d(u,w)+d(v,e)$ , $\\forall $ u, v, w, e $\\in $ E. Definition 2.6 (see Song and Wu [15] If $f:R\\times E \\longrightarrow E$ , then $f$ is continuous at point $(t_0,x_0) \\in R \\times E$ provided that for any fixed number $r \\in [0,1]$ and any $\\epsilon > 0$ , $\\exists $ $\\delta (\\epsilon ,r)$ such that $d([f(t,x)]^{r}, [f(t_{0},x_{0}]^{r}) < \\epsilon $ whenever $\|t-t_{0}\|<\\delta (\\epsilon , r)$ and $d([x]^{r}, [x_{0}]^{r})<\\delta (\\epsilon ,r)$ $\\forall $ t $\\in $ R, x $\\in E$ Theorem 2.7 (see Wu [16]) Let $f$ be a fuzzy-valued function $[a,\\infty )$ given in the parametric form as $(\\underline{f}(x,\\alpha ), \\overline{f}(x,\\alpha ))$ for any constant number $\\alpha \\in [0,1]$ .", "Here we assume that $\\underline{f}(x,\\alpha )$ and $\\overline{f}(x,\\alpha )$ are Reman-Integral on $[a,b]$ for every $b\\ge a$ .", "Also we assume that $\\underline{M}(\\alpha )$ and $\\overline{M}(\\alpha )$ are two positive function, such that $\\int _a^b\|\\underline{f}(x,\\alpha )\| dx \\le \\underline{M}(r)$ and $\\int _a^b \|\\overline{f}(x,\\alpha )\| dx \\le \\overline{M}(r)$ for every $b\\ge a$ , then $f(x)$ is improper integral on $[{a}, r)$ .", "Thus an improper integral will always a fuzzy number.", "In short $ \\int _a^r f(x) dx = ( \\int _a^b\|\\underline{f}(x,\\alpha )\| dx, \\int _a^b \|\\overline{f}(x,\\alpha )\| dx).$ It is will known that Hukuhare differentiability for fuzzy function was introduce by puried Ralescu in 1983.", "It is based on H-differentiable.", "Definition 2.8 (see [17]) Let $f:(a,b)\\rightarrow E$ where $x_{0}\\in (a,b)$ , then we say that $f$ is strongly generalized differentiable at $x_0$ (Beds and Gal differentiability).", "If $\\exists $ an element $f^{\\prime }(x_0)\\in E$ such that $\\forall $ $h>0$ sufficiently small $\\exists $ $f(x_0+h)\\ominus f(x_0)$ , $f(x_0)\\ominus f(x_0-h)$ , then the following limits hold (in the metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f(x_0+h)\\ominus f(x_0)}{h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0)\\ominus f(x_0-h)}{h}=f^{\\prime }(x_0)$ Or $\\forall h>0$ sufficiently small, $\\exists $ $f(x_0)\\ominus f(x_0+h)$ , $f(x_0-h)\\ominus f(x_0)$ , then the following limits holds (in the metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f(x_0)\\ominus f(x_0+h)}{-h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0-h)\\ominus f(x_0)}{-h}=f^{\\prime }(x_0)$ $\\forall $ $h>0$ sufficiently small $\\exists $ $f(x_0+h)\\ominus f(x_0)$ , $f(x_0-h)\\ominus f(x_0)$ and the following limits holds (in metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{(x_0+h)\\ominus f(x_0)}{h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0-h)\\ominus f(x_0)}{-h}=f^{\\prime }(x_0)$ $\\forall $ $h>0$ sufficiently small $\\exists $ $f(x_0)\\ominus f(x_0+h)$ , $f(x_0)\\ominus f(x_0-h)$ , then the following limits holds(in metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f(x_0)\\ominus f(x_0+h)}{-h}=\\lim _{h\\rightarrow 0}\\frac{f(x_0-h)\\ominus f(x_0)}{h}=f^{\\prime }(x_0)$ The denominators $h$ and $-h$ denotes multiplication by $\\frac{1}{h}$ $\\frac{-1}{h}$ respectively.", "Theorem 2.9 (See Chalco and Reman-Flores [18]) Let $F:R\\rightarrow E$ be a function and denoted by $[F(t)]^{\\alpha }=[f_{\\alpha }(t), g_{\\alpha }(t)]$ for each $\\alpha \\in [0,1]$ , then $f$ is (i)-differentiable, then $f_{\\alpha }(t)$ and $g_{\\alpha }(t)$ are differentiable functions.", "Lemma 2.10 (see Bede and Gal[19]) Let $x_0\\in R$ , then the FDE $y^{\\prime }=f(x,y)$ , $y(x_0)=y_0\\in R$ and $f:R\\times E\\rightarrow E$ is supposed to be a continuous and is equivalent to be one of the following integral equations.", "$y(x)=y_0+\\int _{x_0}^x f(t, y(t))dt \\;\\;\\; \\forall \\;\\;\\; x\\in [x_0, x_1],$ or $y(0)=y^1(x)+(-1)\\odot \\int _{x_0}^x f(t,y(t))dt\\;\\;\\; \\forall \\;\\;\\; x\\in [x_0, x_1],$ on some interval $(x_0, x_1)\\subset R$ depending on the strongly generalized differentiability.", "Integral equivalency shows that if one solution satisfies the given equation, then the other will also satisfies.", "Remark 2.11 (see Gal and Bede) In the case of strongly generalized differentiability to the FDE's $y^{\\prime }=f(x,y)$ we use two different integral equations.", "But in the case of differentiability as the definition of H-derivative, we use only one integral.", "The second integral equation as in Lemma $2.10$ will be in the form of $y^{\\prime }(t)=y^{\\prime }_0\\ominus (-1)\\int _{x_0}^x f(t,y(t))dt$ .The following theorem related to the existence of solution of FIVP under the generalized differentiability (see Bede and Gal) Theorem 2.12 Let us suppose that the following conditions are satisfied.", "Let $R_0=[x_0, x_0+s]\\times B(y_0, q), s,q>0, y\\in E, where B(y_0,q)=\\lbrace y\\in E: B(y,y_0)\\le q\\rbrace $ which denotes a closed ball in $E$ and let $f:R_0\\rightarrow E$ be continuous functions such that $D(0, f(x,y))\\le M$ $\\forall $ (x,y) $\\in $ $R_0$ and $0\\in $ E. Let $g:[x_0, x_0+s]\\times [0,q]\\rightarrow R$ such that $g(x, 0)\\equiv 0$ and $0\\le g(x,u)\\le M$ , $\\forall $ x $\\in [x_0, x_0+s], 0\\le u\\le q$ , such that $g(x,u)$ is increasing in u, and g is such that the FIVP $u^{\\prime }(x)=g(x, u(x))\\\\ u(x)\\equiv 0$ on $[x_0, x_0+s].$ We have $D[f(x,y),f(x,z)\\le g(x, D(y,z))]$ ,$\\forall $ (x,y), (x, z)$\\in R_0$ and $D(y,z)\\le q.$ $\\exists \\;\\; d>0$ such that for $x\\in [x_0, x_0+d]$ , the sequence $y^{\\prime }_n:[x_0, x_0+d]\\rightarrow E$ given by $y^{\\prime }_0(x)=y_0$ , $y^{\\prime }_n+1(x)=y_0\\ominus (-1)\\int _{x_0}^x f(t, y^{1}_n)dt$ defined for any $n\\in N$ .", "Then the FIVP $y^{\\prime }=f(x,y)$ , $y(x_0)=y_0$ has two solutions that is (i)-differentiable and the other one is (ii)-differentiable for $y$ .", "$y^{1}=[x_0, x_0+r]\\rightarrow B(y_0, q)$ , where $r=\\min \\lbrace s,\\frac{q}{M},\\frac{q}{M_1},d\\rbrace $ and the successive iteration is $y_n+1(x)=y_{0}+\\int (x-{0}^{x}f(t,y^{1}_{n}(t))dt$ converges to the two solutions respectively.", "Now according to Theorem (2.3), we restrict our attention to function which are (i) or (ii) differentiable on their domain except on a finite number of points(see also Bede and Chalco)." ], [ "Fuzzy Laplace Transform", "Suppose that $f$ is a fuzzy-valued function and $p$ is a real parameter, then according to [6], [19] FLT of the function $f$ is defined as follows: Definition 3.1 The FLT of fuzzy-valued function is [6] $\\widehat{F}(p)=L[f(t)]=\\int _{0}^{\\infty }e^{-pt}f(t)dt,$ $\\widehat{F}(p)=L[f(t)]=\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}f(t)dt,$ $\\widehat{F}(p)=[\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\\underline{f}(t)dt,\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\\overline{f}(t)dt],$ whenever the limits exist.", "Definition 3.2 Classical Fuzzy Laplace Transform: Now consider the fuzzy-valued function in which the lower and upper FLT of the function are represented by $\\widehat{F}(p;r)=L[f(t;r)]=[l(\\underline{f}(t;r)),l(\\overline{f}(t;r))]$ where $l[\\underline{f}(t;r)]=\\int _{0}^{\\infty }e^{-pt}\\underline{f}(t;r)dt=\\lim _{\\tau \\rightarrow \\infty } \\int _{0}^{\\tau }e^{-pt}\\underline{f}(t;r)dt,$ $l[\\overline{f}(t;r)]=\\int _{0}^{\\infty }e^{-pt}\\overline{f}(t;r)dt=\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\\overline{f}(t;r)dt.", "$" ], [ "$nth$ order fuzzy initial value problem", "In this section we are going to define an $nth$ order FIVP's under generalized H-differentiability, proposed in [19].", "We define $y^{(n)}(t)=f(t, y(t), y^{\\prime }(t),y^{\\prime \\prime }(t),\\cdots ,y^{(n-1)}(t)),$ $y(x_0)=y_0; y^{\\prime }(x_0)=y^{\\prime }_0; y^{\\prime \\prime }(x_0)=y^{\\prime \\prime }_0,\\cdots , y^{n-1}(x_0)=y^{n-1}_0,$ and $y(t)=(\\underline{y}{(t,r)}, \\overline{y}{(t,r)}),$ $y^{\\prime }(t)=(\\underline{y}^{\\prime }(t,r), \\overline{y}^{\\prime }(t,r)),$ $y^{\\prime \\prime }(t)=(\\underline{y}^{\\prime \\prime }(t,r), \\overline{y}^{\\prime \\prime }(t,r)),$ continuing the process we get for $(n-1)th$ order that is $y^{(n-1)}(t)=(\\underline{y}^{(n-1)}(t,r), \\overline{y}^{(n-1)}(t,r)),$ are fuzzy-valued functions for $t$ , where $f(t, y(t), y^{\\prime }(t),y^{\\prime \\prime }(t),\\cdots ,y^{(n-1)}(t))$ is continuous fuzzy-valued function.", "Definition 3.3 (see [8], [9]) Let $f:(a, b)\\rightarrow E$ and $x_0\\in (a, b)$ , then the $nth$ order derivative of the function is as follows: Let $f:(a,b)\\rightarrow E$ where $x_{0}\\in (a,b)$ , then we say that $f$ is strongly generalized differentiable of the $nth$ order at $x_0$ if $\\exists $ an element $f^k(x_0)\\in E$ such that $\\forall \\;\\; k=1,2\\cdots ,n$ one of the following holds.", "$\\forall $ $h>0$ sufficiently small $\\exists f^{k-1}(x_0+h)\\ominus f^{k-1}(x_0)$ , $f^{k-1}(x_0)\\ominus f^{k-1}(x_0-h)$ , then the following limits hold (in the metric $d$ ) $\\lim _{h}{\\rightarrow 0}\\frac{f^{k-1}(x_0+h)\\ominus f^{k-1}(x_0)}{h}=\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0)\\ominus f^{k-1}(x_0-h)}{h}=f^k(x_0)$ Or $\\forall $ $h>0$ sufficiently small, $\\exists $ $f^{k-1}(x_0)\\ominus f^{k-1}(x_0+h)$ , $f^{k-1}(x_0-h)\\ominus f^{k-1}(x_0)$ , then the following limits holds (in the metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0)\\ominus f^{k-1}(x_0+h)}{-h}=\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0-h)\\ominus f^{k-1}(x_0)}{-h}=f^k(x_0)$ $\\forall $ $h>0$ sufficiently small $\\exists $ $f^{k-1}(x_0+h)\\ominus f^{k-1}(x_0)$ , $f^{k-1}(x_0-h)\\ominus f^{k-1}(x_0)$ and the following limits holds (in metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0+h)\\ominus f^{k-1}(x_0)}{h}=\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0-h)\\ominus f^{k-1}(x_0)}{-h}=f^k(x_0)$ $\\forall $ $h>0$ sufficiently small $\\exists $ $f^{k-1}(x_0)\\ominus f^{k-1}(x_0+h)$ , $f^{k-1}(x_0)\\ominus f^{k-1}(x_0-h)$ , then the following limits holds(in metric $d$ ) $\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0)\\ominus f^{k-1}(x_0+h)}{-h}=\\lim _{h\\rightarrow 0}\\frac{f^{k-1}(x_0-h)\\ominus f^{k-1}(x_0)}{h}=f^k(x_0)$ Theorem 3.4 (see [7]) Let $F(t), F^{\\prime }(t), F^{\\prime \\prime }(t),\\cdots ,F^{(n)}(t)$ are $nth$ order differentiable fuzzy-valued functions and we denote $r$ -level set of a fuzzy-valued function $F(t)$ with $[F(t)]^{r}=[f_r(t), f_r(t)]$ , then $[F^{n}(t)]=[f^{n}_{r}(t), g^{n}_{r}(t)]$ Here $F(t)$ and $F^{\\prime }(t)$ are differentiable, then we can write as $[F^{\\prime \\prime }(t)]^{r}=[f^{\\prime \\prime }_{r}(t), g^{\\prime \\prime }_{r}(t)]$ .", "Since $F^{\\prime \\prime }(t)$ is differentiable, then by definition $2.6$ [7], a fuzzy-valued function $F:U\\rightarrow F_{0}(R^{n})$ is called Hukuhara differentiable at $t_0\\in U$ if $\\exists \\mbox{ } DF(t_0)=F^{\\prime }(t_0)\\in F_{0}\\times R^{n}$ such that the limits $\\lim _{h\\rightarrow 0}\\frac{F(t_0+h)\\ominus F(t_0)}{h}$ and $\\lim _{h\\rightarrow 0}\\frac{F(t_{0})\\ominus F(t_{0}-h}{h}$ exist and is equal to $DF(t_{0})$ .", "Similarly for $D^{2}F(t_0)$ we have $\\begin{split}[F^{\\prime }(t_0+h)\\ominus F^{\\prime }(t_0)]^{r}=[f^{\\prime }_{r}(t_0+h), g^{\\prime }_{r}(t_0+h)]\\ominus [f^{\\prime }_{r}(t_0), g^{\\prime }_{r}(t_0)]\\\\=[f^{\\prime }_{r}(t_0+h)\\ominus f^{\\prime }_{r}(t_0), g^{\\prime }_{r}(t_0+h)\\ominus g^{\\prime }_{r}(t_0)],\\end{split}$ and $\\begin{split}[F^{\\prime }(t_0)\\ominus F^{\\prime }(t_0-h)]^{r}=[f^{\\prime }_{r}(t_0),g^{\\prime }_{r}(t_0)]\\ominus [f^{\\prime }_{r}(t_0-h), g^{\\prime }_{r}(t_0-h)]\\\\=[f^{\\prime }_{r}(t_0)\\ominus f^{\\prime }_{r}(t_0-h), g^{\\prime }_{r}(t_0)\\ominus g^{\\prime }_{r}(t_0-h)].\\end{split}$ Similarly for third order, fourth order and continuing up to $nth$ order $i.e$ $D^{n}F(t_0)$ we have $\\begin{split}[F^{(n-1)}(t_0+h)\\ominus F^{(n-1)}(t_0)]^{r}=[f^{(n-1)}_{r}(t_0+h), g^{(n-1)}_{r}(t_0+h)]\\ominus [f^{(n-1)}_{r}(t_0), g^{(n-1)}_{r}(t_0)]\\\\=[f^{(n-1)}_{r}(t_0+h)\\ominus f^{(n-1)}_{r}(t_0), g^{(n-1)}_{r}(t_0+h)\\ominus g^{(n-1)}_{r}(t_0)],\\end{split}$ $\\begin{split}[F^{(n-1)}(t_0)\\ominus F^{(n-1)}(t_0-h)]^{r}=[f^{(n-1)}_{r}(t_0),g^{(n-1)}_{r}(t_0)]\\ominus [f^{(n-1)}_{r}(t_0-h), g^{(n-1)}_{r}(t_0-h)]\\\\=[f^{(n-1)}_{r}(t_0)\\ominus f^{(n-1)}_{r}(t_0-h), g^{(n-1)}_{r}(t_0)\\ominus g^{(n-1)}_{r}(t_0-h)].\\end{split}$ Now multiplying $\\frac{1}{h}$ to the second order, third order and so on up to $nth$ order and then applying limit as $h\\rightarrow 0$ on both sides we get the general form According to [20], if $n$ is a positive integer so in the case of (1) and (2)-differentiability we can write the $nth$ derivative of the functions $F,F^{\\prime },\\cdots ,F^{(n-1)}$ in the form of $D^{n}_{k_1\\cdots k-n}F(t_0)$ , where $k_i=1,2$ for $i=1,\\cdots ,n$ Now if we want to compute the $nth$ derivative of $F$ at $t_0$ Moreover $D^{(n-1)}_{1 1}F(t_0)$ is (1)-differentiable and $D^{(n-1)}_{2 2}F(t_0)$ is (2)-differentiable.", "Also $D^{(n-1)}_{1 2}F(t_0)$ is (1)and (2)-differentiable and $D^{(n-1)}_{2 1}F(t_0)$ is (2) and (1)-differentiable and hence proof is completed." ], [ "Convergence", "The FLT can be applied to a large number of fuzzy-valued functions [6], and in some of the examples FLT does not converge as explained below and reported in [6].", "Example 3.5 Let the fuzzy-valued function $f(t)=Ce^{t^2}$ , where $C\\in E$ , then we get $\\lim _{\\tau \\rightarrow \\infty } \\int _{0}^{\\tau }Ce^{-pt}e^{t^2}dt\\rightarrow {\\infty }$ for any choice of variable $p$ so the integral grows with out bounds as $\\tau \\rightarrow \\infty $ In the fuzzy Laplace theory we have to use absolute value of fuzzy-valued functions.", "Here we will define two types of absolute value of fuzzy-valued functions as discussed in [6] and is given in the following definition.", "Definition 3.6 Let us consider a fuzzy-valued function whose parametric form is given in the form $f(t;r)=[\\underline{f}(t;r),\\overline{f}(t;r)].$ Now if $f$ is (1)-absolute value function, then $\\forall \\mbox{ } r\\in [m_{1}, m_{2}]\\subset [0,1]$ $[f(t;r)]=[\|\\underline{f}(t;r)\|, \|\\overline{f}(t;r)\|].$ If $f$ is (2)-absolute value function, then $\\forall \\;r\\in [m_{1}, m_{2}]\\subseteq [0,1]$ $[f(t;r)]=[\|\\overline{f}(t;r)\|, \|\\underline{f}(t;r)\|],$ provided that the r-cut or r-level set is satisfied by the fuzzy-valued function $\|f(t;r)\|$ Theorem 3.7 According to [6], if a fuzzy-valued function $f$ defined as $[f(t;r)]=[\|\\underline{f}(t;r)\|, \|\\overline{f}(t;r)\|]$ , where $\\underline{f}(t;r)$ and $\\overline{f}(t;r)$ are lower and upper end points fuzzy-valued functions for $r\\in [0,1]$ respectively then If $\\underline{f}(t;r)\\ge 0$ $\\forall r$ then $f$ is (1)-absolute value fuzzy function.", "If $\\overline{f}(t;r)\\le 0$ $\\forall r$ then $f$ is (2)-absolute value fuzzy function.", "Example 3.8 Let us consider $f(t;r)=a(r)e^{t}$ , [6] where $a(r)=[1+r; 2-r]$ , then $f(t)$ is (1)-absolute and $\\forall \\; r\\in [0,1]$ , we have $\|f(t;r)\|=[\|(1+r)e^{t}\|, \|(2-r)e^{-t}\|]=[(1+r)e^{t}, (2-r)e^{t}].$ Definition 3.9 The integral (REF ) is absolute convergent if $\\lim _{\\tau \\rightarrow \\infty } \\int _{0}^{\\tau }\|e^{-pt}f(t)\|dt$ exists, that is $\\lim _{\\tau \\rightarrow \\infty }\\int _{0}^{\\tau }e^{-pt}\|\\underline{f}(t;r)\|dt, \\lim _{\\tau \\rightarrow \\infty } \\int _{0}^{\\tau }e^{-pt}\|\\overline{f}(t;r)\|dt$ exist.", "If $L[f(t)]$ does not converge absolutely and if $f(t)$ be (1)-absolute, then $\\mid \\int _{\\tau }^{\\acute{\\tau }}e^{-pt}f(t)dt\\mid =[\\mid \\int _{\\tau }^{\\acute{\\tau }}e^{-pt}\\underline{f}(t;r)dt\\mid , \|\\int _{\\tau }^{\\acute{\\tau }}e^{-pt}\\overline{f}(t;r)dt\|],$ $\|\\int _{\\tau }^{\\acute{\\tau }}e^{-pt}f(t)dt\|\\preceq [\\int _{\\tau }^{\\acute{\\tau }}e^{-pt}\|\\underline{f}(t;r)\|dt,\\int _{\\tau }^{\\acute{\\tau }}e^{-pt}\|\\overline{f}(t;r)\|dt],$ $\|\\int _{\\tau }^{\\acute{\\tau }}e^{-pt}f(t)dt\|=\\int _{\\tau }^{\\acute{\\tau }}e^{-pt}\|f(t)\|dt\\rightarrow \\widetilde{0},$ as $\\tau \\rightarrow {\\infty }$ , $\\forall $ $\\acute{\\tau } >{\\tau }$ .", "This implies that $L[f(t)]$ also converges.", "Similar case holds when $f$ is (2)-absolute.", "The symbol $\\le $ is an ordering relation defined as follows: For any two arbitrary fuzzy numbers $u$ and $v$ , $u\\le v$ $\\Leftrightarrow $ $\\underline{u}(r)\\le \\underline{v}(r)$ and $\\overline{u}(r)\\le \\overline{v}(r)$ , for all $r \\in [0,1]$ .", "Moreover in FIVP see [6] in the solution of second as in definition of the integral REF is uniformly convergent.", "Also in [6] the fuzzy-valued function has a jump i.e discontinuous at the point to $f$ .", "The left and right hand limit exist but not equal.", "Similar also in [6] the fuzzy-valued function is piece wise continuous in the interval $[0,{\\infty })$ and has exponential order of $p$ , then the LT of $\\widehat{F}(p)=L[f(t)]$ exist for $p>s$ and converges also discussed proceeding forward.", "If fuzzy-valued function $f$ is piece wise continuous in $[0,{\\infty })$ and has exponential order $p$ , then $\\widehat{F}(p)=L[f(t)]\\rightarrow 0$ as $t\\rightarrow {\\infty }$ .", "For the second order FIVP [6] present two theorem for translation of function as first translation and second translation theorem, Fuzzy Laplace Transform of Derivative.", "In [6] Derivative Theorem.", "If $f$ is continuous fuzzy-valued function on $[0,{\\infty })$ then $L[f^{\\prime }(t)]=pL[f(t)]\\ominus f(0)$ .", "If $f$ is (1)-differentiable.", "Also $L[f^{\\prime }(t)]=-f(0)\\ominus (-PL[f(t)])$ , if $f$ is (2)-differentiable Here,if we have to solve an $nth$ order derivative as in [5] under gH-differentiability, then we will prove the results in the following equation equation$1.1$ and equation $1.2$ for $nth$ order FIVP with $n$ number of initial values.As we have proved in the previous section.", "Theorem 3.10 Let suppose that $f,f^{\\prime },f^{\\prime \\prime },\\cdots , f^{n-1}$ are continuous fuzzy-valued functions on [0,${\\infty }$ ) and of exponential order and that $f^{(n-1)}$ is piecewise According to [6] suppose that $f$ and $f^{\\prime }$ are continuous fuzzy-valued functions on $[0,\\infty )$ and of exponential order and that $f^{\\prime \\prime }$ is piecewise continuous fuzzy-valued function on $[0,\\infty )$ , then $L(f^{\\prime \\prime }(t))=p^{2}L(f(t))\\ominus pf(0)\\ominus f^{\\prime }(0),$ if $f$ and $f^{\\prime }$ are (1)-differentiable $L(f^{\\prime \\prime }(t))=-f^{\\prime }(0)\\ominus (-p^{2})L(f(t))\\ominus pf(0),$ if $f$ is (1)-differentiable and $f^{\\prime }$ is (2)-differentiable $ L(f^{\\prime \\prime }(t))=-pf(0)\\ominus (-p^{2})L(f(t))\\ominus f^{\\prime }(0),$ if $f$ is (2)-differentiable and $f^{\\prime }$ is (1)-differentiable $L(f^{\\prime \\prime }(t))=p^{2}L(f(t))\\ominus pf(0)-f^{\\prime }(0),$ if $f$ and $f^{\\prime }$ are (2)-differentiable.", "Theorem 3.11 According to [7], suppose that $f(t),f^{\\prime }(t), f^{\\prime \\prime }(t)$ are the continuous fuzzy-valued function on [0,$\\infty $ ) and of exponential order while $f^{\\prime \\prime \\prime }(t)$ is piecewise continuous fuzzy-valued function on [0,$\\infty $ ) with $f(t)=(\\underline{f}(t,r),\\overline{f}(t,r))$ , then notations of the $nth$ derivative of the function is given by $L[f^{\\prime \\prime \\prime }(t,r)]=p^3L[f(t)]\\ominus p^2f(0)\\ominus pf^{\\prime }(t)\\ominus f^{\\prime \\prime }(0)$ .", "Here the notation we used $\\underline{f}^{\\prime },\\underline{f}^{\\prime \\prime },\\underline{f}^{\\prime \\prime \\prime }$ means the lower end points functions derivatives and $\\overline{f}^{\\prime }, \\overline{f}^{\\prime \\prime },{\\overline{f}}^{\\prime \\prime \\prime }$ are the upper end points functions derivatives, using theorem $3.2$ , we have $L[f^{\\prime \\prime \\prime }(t)]=L[\\underline{f}^{^{\\prime \\prime \\prime }}(t,r), \\overline{f}^{^{\\prime \\prime \\prime }}(t,r)]=[l\\underline{f}^{^{\\prime \\prime \\prime }}(t,r), l\\overline{f}^{^{\\prime \\prime \\prime }}(t,r)].", "$ Now for any arbitrary fixed number $r\\in [0,1]$ , using the definition of classical transform, we get $l[\\underline{f}^{\\prime \\prime \\prime }(t,r)]=p^3l[\\underline{f}(t,r)]-p^{2}\\underline{f}(0,r)-p\\underline{f}^{\\prime }(0,r)-\\underline{f}^{^{\\prime \\prime }}(0,r), $ $l[\\overline{f}^{n}(t,r)]=p^{3}l[\\overline{f}(t,r)]-p^{2}\\overline{f}(0,r)-p \\overline{f}^{\\prime }(0,r)-\\overline{f}^{^{\\prime \\prime }}(0,r).$ Now in order to solve the $nth$ order FIVP, we need the FLT of $nth$ derivative of the fuzzy-valued functions under the generalized H-differentiability.", "So we will prove the following theorem for the fuzzy Laplace Transform for $nth$ order FIVP as follows; Theorem 3.12 Suppose that $f, f^{\\prime }, \\cdots ,f^{(n-1)}$ are continuous fuzzy-valued functions on $[0, \\infty )$ and of exponential order and that $f^{(n)}$ is piecewise continuous fuzzy-valued function on $[0, \\infty )$ , then $\\begin{split} L(f^{(n)}(t))=p^{n}L(f(t))\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\ominus p^{n-3}f^{^{\\prime \\prime }}(0)\\\\ \\ominus \\cdots \\ominus f^{n-1}(0)\\end{split}$ If $f, f^{\\prime }\\cdots f^{(n-1)}$ are (1)-differentiable $\\begin{split} L(f^{(n)}(t))=-f^{(n-1)}(0)\\ominus (-p^{n})L(f(t))\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\\\ \\ominus \\cdots \\ominus p^{n-(n-1)}f^{(n-2)}(0) \\end{split}$ If $f, f^{\\prime }\\cdots f^{(n-2)}$ are (1)-differentiable and $f^{(n-1)}$ is (2)-differentiable $\\begin{split}L(f^{(n)}(t))=-p^{n-(n-1)}f^{(n-2)}\\ominus f^{(n-1)}(0)\\ominus (-p^{n})L(f(t))\\ominus p^{n-1}f(0)\\\\ \\ominus p^{n-2}f^{\\prime }(0)\\ominus \\cdots \\ominus p^{n-(n-2)}f^{(n-3)}(0)\\end{split} $ If $f, f^{\\prime }\\cdots f^{(n-3)}$ are (1)-differentiable and $f^{(n-1)}, f^{(n-2)}$ are (2)-differentiable Similarly $\\begin{split}L(f^{(n)}(t))=-p^{n-1}f(0)\\ominus (-p^{n})L(f(t))\\ominus p^{n-2}f^{\\prime }(0)\\\\ \\ominus \\cdots \\ominus f^{(n-1)}(0) \\end{split}$ If $f^{\\prime },\\cdots , f^{(n-1)}$ are (1)-differentiable and $f$ is (2)-differentiable Continuing the process until we obtain $2^{n}$ system of differential equations, hence the last equation is according to [19] $L(f^{(n)}(t))=p^{n}L(f(t))\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\ominus p^{n-3}f^{\\prime \\prime }(0)\\cdots -f^{n-1}(0)$ If $f, f^{\\prime }\\cdots f^{(n-1)}$ are (2)-differentiable REF $\\begin{split}p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\ominus \\cdots \\ominus f^{(n-1)}(0)=(p^{n}l[\\underline{f}(t,r)]- p^{n-1}\\underline{f}(0,r)\\\\-p^{n-2}\\underline{f}^{\\prime }(0,r)\\cdots -\\underline{f}^{(n-1)}(0,r), p^{n}l[\\overline{f}(t,r)]-p^{n-1}\\overline{f}(0,r)-p^{n-2}\\overline{f}^{\\prime }(0,r)\\\\\\cdots -\\overline{f}^{(n-1)}(0,r))\\end{split}$ since $l(\\underline{f}^{n}(t,r))=p^{n}l[\\underline{f}(t,r)]-p^{n-1}\\underline{f}(0,r)-p^{n-2}\\underline{f}^{\\prime }(0,r)\\cdots -\\underline{f}^{(n-1)}(0,r)$ $l(\\overline{f}^{(n)})=l(\\overline{f}^{n}(t,r))=p^{n}l[\\overline{f}(t,r)]-p^{n-1}\\overline{f}(0,r)-p^{n-2}\\overline{f}^{\\prime }(0,r)\\cdots -\\overline{f}^{(n-1)}(0,r)$ where $\\underline{f}^{(n-1)}(0,r)=\\underline{f}^{(n-1)}(0,r)$ and $\\overline{f}^{(n-1)}(0,r)=\\overline{f^{(n-1)}}(0,r)$ $\\begin{split}p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots \\ominus f^{(n-1)}(0)=(l(\\underline{f}^{(n)})(t,r), l(\\overline{f}^{(n)})(t,r))\\end{split}$ $p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots \\ominus f^{(n-1)}(0)=L(\\underline{f}^{(n)}(t,r), (\\overline{f}^{(n)}(t,r)))$ $p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots \\ominus f^{(n-1)}(0)=L(f^{(n)}(t))$ Hence proof is completed Now we are going to prove the final equation (REF ), while the middle equations are almost analogous to the proof of (REF ) and (REF ) REF $\\begin{split}p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots -f^{(n-1)}(0)=(p^{n}l[\\underline{f}(t,r)]-p^{n-1}\\underline{f}(0,r)-\\\\p^{n-2}\\underline{f}^{\\prime }(0,r)\\cdots -\\overline{f}^{(n-1)}(0,r), p^{n}l[\\overline{f}(t,r)]-p^{n-1}\\overline{f}(0,r)-p^{n-2}\\overline{f}^{\\prime }(0,r)\\cdots \\\\ -\\underline{f}^{(n-1)}(0,r))\\end{split}$ since $l(\\underline{f}^{(n)}(t,r))=l(\\underline{f^{n}(t,r)})=p^{n}l[\\underline{f}(t,r)]-p^{n-1}\\underline{f}(0,r)- p^{n-2}\\underline{f}^{\\prime }(0,r)\\cdots -\\underline{f}^{(n-1)}(0,r)$ $l(\\overline{f}^{(n)})=l(\\overline{f^{n}(t,r)})=p^{n}l[\\overline{f}(t,r)]-p^{n-1}\\overline{f}(0,r)-p^{n-2}\\overline{f}^{\\prime }(0,r)\\cdots -\\overline{f}^{(n-1)}(0,r)$ Here we have $\\overline{f}^{(n-1)}(0,r)=\\underline{f}^{(n-1)}(0,r)$ and $\\underline{f}^{(n-1)}(0,r)=\\overline{f}^{(n-1)}$ , so we know that $\\begin{split}p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots -f^{(n-1)}(0)=(l(\\underline{f}^{(n)})(t,r), l(\\overline{f}^{(n)})(t,r))\\end{split}$ $p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots \\ominus f^{(n-1)}(0)=L(\\underline{f}^{(n)}(t,r), (\\overline{f}^{(n)}(t,r)))$ $p^{n}L[f(t)]\\ominus p^{n-1}f(0)\\ominus p^{n-2}f^{\\prime }(0)\\cdots \\ominus f^{(n-1)}(0)=L(f^{(n)}(t))$ which is the required result." ], [ "Constucting Solutions Via FIVP", "Consider the following $nth$ order FIVP in general form $y^{(n)}(t)=f(t, y(t),y^{\\prime }(t),\\cdots , y^{(n-1)}(t)), $ subject to the $nth$ order initial conditions $y(0)=(\\underline{y}(0;r), \\overline{y}(0;r))$ , $y^{\\prime }(0)=(\\underline{{y}}^{\\prime }(0;r), \\overline{y}^{\\prime }(0;r))$ , $y^{\\prime \\prime }(0)=(\\underline{y}^{\\prime \\prime }(0;r), \\overline{y}^{\\prime \\prime }(0;r))$ .", "continuing for $nth$ initial conditions $y^{(n-1)}(0)=(\\underline{y}^{(n-1)}(0;r), \\overline{y}^{(n-1)}(0;r)).$ Now we use FLTM $L[y^{(n)}(t)]=L[f(t, y(t),y^{\\prime }(t),\\cdots , y^{(n-1)}(t))].", "$ Using the theorem $5.8$ and equation (REF ) $p^nL[y(t)]\\ominus p^{n-1}y(0)\\ominus p^{n-2}y^{\\prime }(0)\\ominus \\cdots \\ominus y^{(n-1)}(0)=L[f(t, y(t),y^{\\prime }(t),\\cdots ,y^{n-1}(t))].$ The classical form $\\begin{split}p^{n}l[\\underline{y}(t;r)]-p^{n-1}\\underline{y}(0;r)-p^{n-2}\\underline{y^{\\prime }}(0;r)-\\cdots - \\underline{y}^{(n-1)}(0;r)\\\\=l[\\underline{f}(t, y(0;r),y^{\\prime }(0;r),\\cdots ,y^{(n-1)}(0;r))]\\end{split}$ $\\begin{split}p^{n}l[\\overline{y}(t;r)]-p^{n-1}\\overline{y}(0;r)-p^{n-2}\\overline{y}^{^{\\prime }}(0;r)-\\cdots - \\overline{y}^{(n-1)}(0;r)\\\\=l[\\overline{f}(t, y(0;r),y^{\\prime }(0;r),\\cdots ,y^{(n-1)}(0;r))]\\end{split}$ In order to solve the system (REF ) and (REF ) we have assumed that $A(p;r)$ and $B(p;r)$ are the solutions of (REF ) and (REF ) respectively.", "So, we have $l[\\underline{y}(t;r)]=A(p;r),$ $l[\\overline{y}(t;r)]=B(p;r).$ Using Inverse Laplace Transform (ILT), we have $[\\underline{y}(t;r)]=l^{-1}[A(p;r)],$ $[\\overline{y}(t;r)]=l^{-1}[B(p;r)].$" ], [ "Examples", "Example 5.1 $y^{(iv)}=y^{^{\\prime \\prime \\prime }}(t)+y^{^{\\prime \\prime }}(t),$ with initial condition $y(0)=(3+r, 5+r)$ , $y^{\\prime }(0)=(-3+r, -1-r)$ , $y^{\\prime \\prime }(0)=(8+r, 10-r)$ , $y^{\\prime \\prime \\prime }(0)=(12+r, 14-r)$ , where $f(t,y(t), y^{\\prime }(t), y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))=y^{(iv)},\\\\$ The $r-level$ set of the FIVP is given in the following $f(t,y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t))=\\underline{y}^{\\prime \\prime \\prime }(t;r)+\\underline{y}^{\\prime \\prime }(t;r), \\overline{y}^{\\prime \\prime \\prime }(t;r)+\\overline{y}^{\\prime \\prime }(t;r),$ $\\underline{f}(t,y(t), y^{\\prime }(t),y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))=\\underline{y}^{\\prime \\prime \\prime }(t;r)+\\underline{y}^{\\prime \\prime }(t;r),$ $\\overline{f}(t,y(t), y^{\\prime }(t), y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))=\\overline{y}^{\\prime \\prime \\prime }(t;r)+\\overline{y}^{\\prime \\prime }(t;r).$ Applying Laplace Transform $l[\\underline{f}(t, y(t), y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t))]=l[\\underline{y}^{\\prime \\prime \\prime }(t;r)]+l[\\underline{y}^{\\prime \\prime }(t;r)],$ $l[\\overline{f}(t, y(t), y^{\\prime }(t), y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))]=l[\\overline{y}^{\\prime \\prime \\prime }(t;r)]+l[\\overline{y}^{\\prime \\prime }(t;r)],$ $\\begin{split}l[\\underline{f}(t,y(t), y^{\\prime }(t),y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))]=p^3l[\\underline{y}(t;r)]-p^2\\underline{y}(0;r)\\\\-p\\underline{y}^{\\prime }(0;r)- \\underline{y}^{^{\\prime \\prime }}(0;r)+p^{2}l[\\underline{y}(t;r)]-p\\underline{y}(0;r)-\\underline{y}^{\\prime }(0;r).\\end{split}$ Now for upper bound, we have $\\begin{split}l[\\overline{f}(t, y(t), y^{\\prime }(t), y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))]=p^3l[\\overline{y}(t;r)]-p^2\\overline{y}(0;r)\\\\-p\\overline{y}^{\\prime }(0;r)- \\overline{y}^{^{\\prime \\prime }}(0;r)+p^2l[\\overline{y}(t;r)]-p\\overline{y}(0;r)-\\overline{y}^{\\prime }(0;r).\\end{split}$ Now putting the initial conditions in (REF ), we get $\\begin{split}l[\\underline{f}(t,y(t), y^{\\prime }(t),y^{\\prime \\prime }(t), y^{\\prime \\prime \\prime }(t))]=p^3l[\\underline{y}(t;r)]-p^{2}(3+r)\\\\-p(-3+r)-(8+r)+p^{2}l[\\underline{y}(t;r)]- p(3+r)-(-3+r).\\end{split}$ Similarly putting the initial conditions in (REF ), we have $\\begin{split}l[\\overline{f}(t,y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t))]=p^3l[\\overline{y}(t;r)]-p^2(3+r)\\\\-p(-3+r)-(8+r)+p^2l[\\overline{y}(t;r)]-p(3+r)-(-3+r).\\end{split}$ In general, we have $L[\\underline{y}^{(iv)}(t)]=L[\\underline{f}(t,y(t), y^{\\prime }(t),y^{^{\\prime \\prime }}(t), y^{^{\\prime \\prime \\prime }}(t))],$ $\\begin{split}l[\\underline{f}(t, y(t), y^{\\prime }(t), y^{^{\\prime \\prime }}(t), y^{^{\\prime \\prime \\prime }}(t))]=p^4l[\\underline{y}(t;r)]-p^3\\underline{y}(0;r)\\\\-p^2\\underline{y}^{\\prime }(0;r)-p\\underline{y}^{^{\\prime \\prime }}(0;r)-\\underline{y}^{^{\\prime \\prime \\prime }}(0;r),\\end{split}$ $\\begin{split}[\\overline{f}(t,y(t), y^{\\prime }(t),y^{^{\\prime \\prime }}(t), y^{^{\\prime \\prime \\prime }}(t))]=p^4l[\\overline{y}(t;r)]-p^3\\overline{y}(0;r)\\\\-p^2\\overline{y}^{\\prime }(0;r)- p\\overline{y}^{^{\\prime \\prime }}(0;r)-\\overline{y}^{^{\\prime \\prime \\prime }}(0;r).\\end{split}$ Now comparing equations (REF ) and (REF ) $\\begin{split}p^{4}l[\\underline{y}(t;r)]-p^{3}(3+r)-p^{2}(-3+r)-p(8+r)-(12+r)=p^{3}l[\\underline{y}(t;r)]-\\\\p^2(3+r)-p(-3+r)- (8+r)+p^{2}l[\\underline{y}(t;r)]-p(3+r)-(-3+r),\\end{split}$ $\\begin{split}l[\\underline{y}(t;r)]=(3+r)\\frac{1}{p}+(-3+r)\\frac{1}{p^2}\\\\+(8+r)\\frac{p-1}{p^4-p^3-p^2}+(12+r)\\frac{1}{p^4-p^3-p^2},\\end{split}$ $\\begin{split}[\\underline{y}(t;r)]=(3+r)l^{-1}[\\frac{1}{p}]+(-3+r)l^{-1}[\\frac{1}{p^2}]\\\\+(8+r)l^{-1}[\\frac{p-1}{p^{4}-p^{3}-p^{2}}]+(12+r)l^{-1}[\\frac{1}{p^{4}-p^{3}}],\\end{split}$ $\\begin{split}\\underline{y}(t;r)=(3+r)+(-3+r)t+(8+r)[t+2e^\\frac{t}{2}\\cosh (\\frac{\\sqrt{5}t}{2})-\\frac{2}{\\sqrt{5}}\\sinh (\\frac{\\sqrt{5}t}{2})-2]\\\\+(12+r)[1-e^\\frac{t}{2}\\cos (\\frac{\\sqrt{5}t}{2})-\\frac{3}{\\sqrt{5}}\\sin (\\frac{\\sqrt{5}t}{2})-t].\\end{split}$ Similarly comparing (REF ) and (REF ) we get $\\begin{split}p^{4}l[\\overline{y}(t;r)]-p^{3}(5-r)-p^{2}(-1-r)-p(10-r)-(14-r)=(p^3l[\\overline{y}(t;r)]\\\\-p^2(5-r)-p(-1-r)- (10-r))+p^2l[\\overline{y}(t;r)]-p(5-r)-(-1-r).\\end{split}$ After simplification we get $l[\\overline{y}(t;r)]=(5-r)\\frac{1}{p}+(-1-r)\\frac{1}{p^2}+(10-r)\\frac{p-1}{p^4-p^3-p^2}+(14-r)\\frac{1}{p^4-p^3-p^2}.$ $\\begin{split}[\\overline{y}(t;r)]=(5-r)l^{-1}[\\frac{1}{p}]+(-1-r)l^{-1}[\\frac{1}{p^2}]+(10-r)l^{-1}[\\frac{p-1}{p^4-p^3-p^2}]+(14-r)l^{-1}[\\frac{1}{p^4-p^3-p^2}].\\end{split}$ $\\begin{split}\\overline{y}(t;r)=(15-r)+(-1-r)t+(10-r)[t+\\frac{1}{\\sqrt{2}}e^\\frac{t}{2}cosh(\\frac{\\sqrt{5}t}{2})-\\frac{2}{\\sqrt{5}}\\sinh (\\frac{\\sqrt{5}t}{2})-2]+\\\\(14-r)[1-e^\\frac{t}{2}\\cos (\\frac{\\sqrt{5}t}{2})-\\frac{3}{\\sqrt{5}}\\sinh (\\frac{\\sqrt{5}t}{2})-t].\\end{split}$ Example 5.2 Solve the FIVP $y^{(iv)}=-y^{^{\\prime \\prime }}(t)+2y^{\\prime }(t)+t,$ subject to the initial conditions $y(0)=(3-r, 1+r)$ , $y^{\\prime }(0)=(4-r, -2+r)$ , $y^{^{\\prime \\prime }}(0)=(7+r, 9-r)$ , $y^{^{\\prime \\prime \\prime }}(0)=(10-r, 8+r).$ $\\begin{split}l[\\underline{y}(t,r)]=p^4l[\\underline{y}(t,r)]-p^{3}\\underline{y}(0,r)-p^{2}\\underline{y}^{\\prime }(0,r)-p\\underline{y}^{^{\\prime \\prime }}(0,r)- \\\\\\underline{y}^{^{\\prime \\prime \\prime }}(0,r),\\end{split}$ $\\begin{split}L[\\underline{y}^{^{\\prime \\prime \\prime \\prime }}(t,r)]=p^4l[\\underline{y}(t,r)]\\ominus p^{3}\\underline{y}(0,r)\\ominus p^{2}\\underline{y}^{\\prime }(0,r)\\ominus \\\\p\\underline{y}^{^{\\prime \\prime }}(0,r)\\ominus \\underline{y}^{^{\\prime \\prime \\prime }}(0,r),\\end{split}$ $l[\\underline{y}(t,r)]=p^2l[\\underline{y}(t,r)]-p\\underline{y}(0,r)-\\underline{y}^{\\prime }(0,r),$ $L[\\underline{y}^{\\prime \\prime }(t,r)]=p^2l[\\underline{y}(t,r)]-p\\underline{y}(0,r)-\\underline{y}^{\\prime }(0,r),$ $l[\\underline{y}(t,r)]=pl[\\underline{y}(t,r)]-\\underline{y}(0,r),$ $L[\\underline{y}^{\\prime }(t,r)]=pl[\\underline{y}(t,r)]-\\underline{y}(0,r).$ Now putting the values, we get $\\begin{split}l[\\underline{y}(t,r)][p^{4}+p^{2}-2p]=p^{3}(3-r)+p^{2}(4-r)+p(7+r)\\\\+(10-r)+p(3-r)+(4-r)-2(3-r)+l(t),\\end{split}$ $\\begin{split}l[\\underline{y}(t,r)][p^{4}+p^{2}-2p]=(p^3+p-2)(3-r)+(p^2+1)(4-r)\\\\+p(7+r)+(10-r+4-r-6+2r)+l(t),\\end{split}$ $\\begin{split}l[\\underline{y}(t,r)][p^{4}+p^{2}-2p]=(p^3+p-2)(3-r)+(p^2+1)(4-r)+\\\\p(7+r)+8+l(t),\\end{split}$ $\\begin{split}\\underline{y}(t,r)=(3-r)l^{-1}[\\frac{p^3+p-2}{p^4+p^2-2p}]+(4-r)l^{-1}[\\frac{p^2+1}{p^4+p^2-2p}]+\\\\(7+r)l^{-1}[\\frac{p}{p^4+p^2-2p}]+8l^{-1}[\\frac{1}{p^4+p^2-2p}]+6,\\end{split}$ $\\begin{split}\\underline{y}(t,r)=(3-r)+(4-r)[e^\\frac{t}{2}+e^\\frac{t}{2}+\\frac{\\sqrt{7}\\sin \\frac{\\sqrt{7}t}{2}}{7e^\\frac{t}{2}}-1/2]+\\\\(7+r)[e^\\frac{t}{4}-\\cos (\\frac{\\sqrt{7} t}{2})+\\frac{(\\frac{3\\sqrt{7}\\sin \\frac{\\sqrt{7}t}{2}}{7})}{4e^\\frac{t}{2}}]+8[e^{\\frac{t}{4}}\\\\+\\cos \\frac{\\sqrt{7}t}{2}-\\frac{(\\frac{\\sqrt{7}\\sin \\frac{\\sqrt{7}t}{2}}{7})}{4e^\\frac{t}{2}}-1/2]+6.\\end{split}$ Example 5.3 Consider the following fourth order FIVP $y^{(iv)}(t)=y^{^{\\prime \\prime \\prime }}(t)-y^{^{\\prime \\prime }}(t),$ subject to the fuzzy initial conditions (FICs) $y(0)=(r-1, 1-r)$ , $y^{\\prime }(0)=(r+1, 3-r)$ , $y^{\\prime \\prime }(0)=(2+r, 4-r)$ , $y^{\\prime \\prime \\prime }(0)=(3+r, 5-r).$ In case of generalized H-differentiability, if we apply FLT we have a system of sixteen differential equations.", "It can be listed in the form of a differential operator as on the function $F(t_0)$ .The list is below Table: Differentiable OperatorNow here we will solve the two cases in the mentioned example that is (1)-differentiable and (2)-differentiable.", "First we consider that the FIVP is (1)-differentiable Let us consider that $y(t), y^{\\prime }(t), \\cdots ,y^{(4)}$ are (1)-differentiable As we know that $y^{(iv)}=y^{\\prime \\prime \\prime }(t)-y^{\\prime \\prime }(t).$ Now applying FLT on both sides of the above equation, we get $L[y^{(iv)}(t)]=L[y^{\\prime \\prime \\prime }(t)]-L[y^{\\prime \\prime }(t)],$ $L[f(t, y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t),y^{4}(t))]=L[y^{(iv)}],$ $L[y^{(iv)}]=p^4L[y(t)]\\ominus p^{3}y(0)\\ominus p^{2}y^{\\prime }(0)\\ominus py^{\\prime \\prime }(0)\\ominus y^{\\prime \\prime \\prime }(0).$ Now the classical FLT form of the above equation is $\\begin{split}l[\\underline{y}^{(iv)}(t,r)]=p^4l[\\underline{y}(t,r)]-p^{3}\\underline{y}(0,r)\\ominus p^{2}\\underline{y}^{\\prime }(0,r)- p\\underline{y}^{^{\\prime \\prime }}(0,r)-\\underline{y}^{^{\\prime \\prime \\prime }}(0,r)\\end{split}$ $\\begin{split}l[\\overline{y}^{(iv)}(t,r)]=p^4l[\\overline{y}(t,r)]-p^{3}\\overline{y}(0,r)-p^{2}\\overline{y}^{\\prime }(0,r)- p\\overline{y}^{^{\\prime \\prime }}(0,r)-\\overline{y}^{^{\\prime \\prime \\prime }}(0,r)\\end{split}$ $L[y^{\\prime \\prime \\prime }(t)]=p^{3}L[y(t)]\\ominus p^{2}y(0)\\ominus py^{\\prime }(0)\\ominus y^{\\prime \\prime }(0)$ The classical FLT of the above equation is $\\begin{split}l[\\underline{y}^{\\prime \\prime \\prime }(t,r)]=p^{3}l[\\underline{y}(t,r)]-p^{2}\\underline{y}(0,r)-p\\underline{y}^{\\prime }(0,r)-\\underline{y}^{\\prime \\prime }(0,r)\\end{split}$ $\\begin{split}l[\\overline{y}^{\\prime \\prime \\prime }(t,r)]=p^{3}l[\\overline{y}(t,r)]-p^{2}\\overline{y}(0,r)-\\overline{y}^{\\prime }(0,r)-\\overline{y}^{\\prime \\prime }(0,r)\\end{split}$ $L[y^{\\prime \\prime }(t)]=p^{2}L[y(t)]\\ominus py(0)\\ominus y^{\\prime }(0)$ The classical FLT form of the above equation is $l[\\underline{y}^{\\prime \\prime }(t,r)]=p^{2}l[\\underline{y}(t,r)]-p\\underline{y}(0,r)-\\underline{y}^{\\prime }(0,r)$ $l[\\overline{y}^{\\prime \\prime }(t,r)]=p^{2}l[\\overline{y}(t,r)]-p\\overline{y}(0,r)-\\overline{y}^{\\prime }(0,r)$ Now solve the above classical equations for lower and upper solutions, we have $\\begin{split}p^4l[\\underline{y}(t,r)]-p^{3}\\underline{y}(0,r)-p^{2}\\underline{y}^{\\prime }(0,r)-p\\underline{y}^{^{\\prime \\prime }}(0,r)- \\underline{y}^{^{\\prime \\prime \\prime }}(0,r)=p^{3}l[\\underline{y}(t,r)]-p^{2}\\underline{y}(0,r)-\\\\p\\underline{y}^{\\prime }(0,r)\\-\\underline{y}^{\\prime \\prime }(0,r)-(l[\\underline{y}^{\\prime \\prime }(t,r)]=p^{2}l[\\underline{y}(t,r)]\\\\-p\\underline{y}(0,r)-\\underline{y}^{\\prime }(0,r))\\end{split}$ Applying the initial conditions, we have $\\begin{split}p^4l[\\underline{y}(t,r)]-p^{3}(r-1)p^{2}(r+1)-p(2+r)-(3+r)=p^{3}l[\\underline{y}(t,r)]-p^{2}(r-1)\\\\-p(r+1)-(2+r)-(p^{2}l[\\underline{y}(t,r)] p(r-1)-(r+1))\\end{split}$ Solving the above equation for $\\underline{y}(t,r)$ , we get $\\begin{split}\\underline{y}(t,r)=(r-1)+(r+1)t+(2+r)[\\frac{2}{\\sqrt{3}}e^{\\frac{t}{2}}\\sin \\frac{\\sqrt{3}t}{2}-t]+(3+r)[t-e^{\\frac{t}{2}}(\\cos \\frac{\\sqrt{3}t}{2})\\\\+\\frac{1}{\\sqrt{3}}\\sin \\frac{\\sqrt{3}t}{2}+1]\\end{split}$ Now we will solve the classical FLT form for $\\overline{y}(t,r)$ , we have $\\begin{split}p^4l[\\overline{y}(t,r)]-p^{3}(1-r)-p^{2}(3-r)-p(4-r)-(5-r)=p^{3}l[\\overline{y}(t,r)]\\\\-p^{2}(1-r)-p(3-r)-(4-r)-(p^{2}l[\\overline{y}(t,r)]-p(1-r)(3-r))\\end{split}$ $\\begin{split}\\overline{y}(t,r)=(1-r)+(3-r)t+(4-r)[\\frac{2}{\\sqrt{3}}e^\\frac{t}{2}\\sin \\frac{\\sqrt{3}t}{2}-t]+(5-r)[t-e^\\frac{t}{2}\\cos \\frac{\\sqrt{3}t}{2}\\\\+\\frac{1}{\\sqrt{3}}\\sin \\frac{\\sqrt{3}t}{2}+1]\\end{split}$ Let us consider that $y(t), y^{\\prime }(t), \\cdots ,y^{(iv)}$ are (2)-differentiable Now applying FLT on both sides of the above FIVP, we get $L[y^{(iv)}(t)]=L[y^{\\prime \\prime \\prime }(t)]-L[y^{\\prime \\prime }(t)]$ $L[f(t, y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t),y^{iv}(t))]=L[y^{(iv)}]$ Also we know that $L[y^{(iv)}]=p^4L[y(t)]\\ominus p^{3}y(0)\\ominus p^{2}y^{\\prime }(0)\\ominus py^{\\prime \\prime }(0)\\ominus y^{\\prime \\prime \\prime }(0)$ Now comparing the above two equations, we get the classical FLT form of equation is $\\begin{split}l(\\underline{f}(t, y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t),y^{4}(t)))=p^4l[\\underline{y}(t,r)]-p^{3}\\underline{y}(0,r)- \\\\p^{2}\\underline{y}^{\\prime }(0,r)-p\\underline{y}^{^{\\prime \\prime }}(0,r)-\\underline{y}^{^{\\prime \\prime \\prime }}(0,r)\\end{split} $ $\\begin{split}l(\\overline{f}(t, y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t),y^{(iv)}(t)))=p^4l[\\overline{y}(t,r)]-p^{3}\\overline{y}(0,r)- \\\\p^{2}\\overline{y}^{\\prime }(0,r)-\\\\ p\\overline{y}^{^{\\prime \\prime }}(0,r)-\\overline{y}^{^{\\prime \\prime \\prime }}(0,r)\\end{split} $ In case of (2)-differentiability, we have the FIVP becomes $\\begin{split}l(\\underline{f}(t, y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t),y^{(iv)}(t)))=p^{3}l[\\overline{y}(t,r)]-p^{2}\\overline{y}(0,r)\\\\-p\\overline{y}^{\\prime }(0,r)-\\overline{y}^{\\prime \\prime }(0,r)-(p^{2}l[\\overline{y}(t,r)]\\ominus p\\overline{y}(0,r)-\\overline{y}^{\\prime }(0,r))\\end{split}$ $\\begin{split}l(\\overline{f}(t, y(t),y^{\\prime }(t),y^{\\prime \\prime }(t),y^{\\prime \\prime \\prime }(t),y^{(iv)}(t)))=p^{3}l[\\underline{y}(t,r)]-p^{2}\\underline{y}(0,r)\\\\-p\\underline{y}^{\\prime }(0,r)-\\underline{y}^{\\prime \\prime }(0,r)-(p^{2}l[\\underline{y}(t,r)]\\\\-p\\underline{y}(0,r)-\\underline{y}^{\\prime }(0,r))\\end{split}$ Now comparing (REF ), (REF ), (REF ) and (REF ), we get $\\begin{split}p^4l[\\underline{y}(t,r)]-p^{3}(r-1)-p^{2}(r+1)\\-p(2+r)-(3+r)=p^{3}l[\\overline{y}(t,r)]\\\\-p^{2}(1-r)-p(3-r)-(4-r)-(p^{2}l[\\overline{y}(t,r)]-p(1-r)-(3-r))\\end{split}$ $\\begin{split}p^4l[\\overline{y}(t,r)]-p^{3}(1-r)-p^{2}(3-r)-p(4-r)-(5-r)=p^{3}l[\\underline{y}(t,r)]\\\\-p^{2}(r-1)-p(r+1)-(2+r)-(p^{2}l[\\underline{y}(t,r)]-p(r-1)-(r+1)),\\end{split}$ First we will solve (REF ) and (REF ) for $ł[\\underline{y}(t,r)]$ and $ł[\\overline{y}(t,r)]$ $\\begin{split}l[\\underline{y}(t,r)]=r[\\frac{p^5+p^4+2p^2+1}{p^6-p^4+2p^3-P^2}]+[\\frac{3p^3+2p-4}{p^6-p^4+2p^3-P^2}],\\end{split}$ 7 $\\begin{split}l[\\overline{y}(t,r)]=r[\\frac{p^6-p^4+2p^3-2p^2+p-1}{p^8-p^6+2p^5-p^4}]+[\\frac{3p^4-3p^3+2p^2-6p-4}{p^8-p^6+2p^5-p^4}]\\end{split}.$ 8 Taking the inverse laplace transform, we get $\\begin{split}\\underline{y}(t,r)=rl^{-1}[\\frac{p^5+p^4+2p^2+1}{p^6-p^4+2p^3-P^2}]+l^{-1}[\\frac{3p^3+2p-4}{p^6-p^4+2p^3-P^2}]\\end{split}$ $\\begin{split}\\overline{y}(t,r)=rl^{-1}[\\frac{p^6-p^4+2p^3-2p^2+p-1}{p^8-p^6+2p^5-p^4}]+l^{-1}[\\frac{3p^4-3p^3+2p^2-6p-4}{p^8-p^6+2p^5-p^4}]\\end{split}$ $\\begin{split}\\underline{y}(t,r)=r[3\\cos (\\frac{\\sqrt{3}t}{2})+e^\\frac{-t}{2}(\\frac{7}{3\\sqrt{5}})\\sinh (\\frac{\\sqrt{5}t}{2})-t-2]+4t-7e^\\frac{t}{2}\\cos (\\frac{\\sqrt{3}t}{2})-\\\\\\frac{2\\sqrt{3}}{7}\\sin (\\frac{\\sqrt{3}t}{2})-5\\cosh ( \\frac{\\sqrt{5}t}{2})-2e^\\frac{t}{2}(\\frac{17\\sqrt{5}}{25})\\sinh (\\frac{\\sqrt{5}t}{2})+6\\end{split}$ $\\begin{split}\\overline{y}(t,r)=r[3t+\\frac{t^2}{2}+\\frac{t^3}{6}-3\\cosh (\\frac{\\sqrt{5}t}{2})+(\\frac{7}{3\\sqrt{5}})e^\\frac{-t}{2}\\sinh (\\frac{\\sqrt{5}t}{2})+3]\\\\+22t+7t^2+\\frac{2}{3}t^3-7e^\\frac{t}{2}\\cos (\\frac{\\sqrt{3}t}{2})+(\\frac{26}{7\\sqrt{3}})\\sin (\\frac{\\sqrt{3}t}{2})-59\\cosh (\\frac{\\sqrt{5}t}{2})+(\\frac{254\\sqrt{5}}{295})e^\\frac{t}{2}\\sinh (\\frac{\\sqrt{5}t}{2})+33\\end{split}$" ], [ "Conclusion", "we generalized the FLT for the $nth$ derivative of a fuzzy-valued function and provided method for the solution of an $nth$ order FIVP using the generalized differentiability concept.", "We have solved a number of different problems using this new approach.", "However some more research is needed to apply this method for the solution of system of FDEs which is in progress." ] ]	1403.0242
[ [ "General properties of the radiation spectra from relativistic electrons\n moving in a Langmuir turbulence" ], [ "Abstract We examine the radiation spectra from relativistic electrons moving in a Langmuir turbulence expected to exist in high energy astrophysical objects by using numerical method.", "The spectral shape is characterized by the spatial scale {\\lambda}, field strength {\\sigma}, and frequency of the Langmuir waves, and in term of frequency they are represented by {\\omega}_0 = 2{\\pi}c/{\\lambda}, {\\omega}_st = e{\\sigma}/mc, and {\\omega}_p, respectively.", "We normalize {\\omega}_st and {\\omega}_p by {\\omega}_0 as \\a \\equiv {\\omega}_st/{\\omega}_0 and \\b \\equiv{\\omega}_p/{\\omega}_0, and examine the spectral shape in the a-b plane.", "An earlier study based on Diffusive Radiation in Langmuir turbulence (DRL) theory by Fleishman and Toptygin showed that the typical frequency is {\\gamma}^2{\\omega}_p and that the low frequency spectrum behaves as F_{\\omega} pronto {\\omega}^1 for b > 1 irrespective of a.", "Here, we adopt the first principle numerical approach to obtain the radiation spectra in more detail.", "We generate Langmuir turbulence by superposing Fourier modes,inject monoenergetic electrons, solve the equation of motion, and calculate the radiation spectra using Lienard-Wiechert potential.", "We find different features from the DRL theory for a>b>1.", "The peak frequency turns out to be {\\gamma}^2{\\omega}_st which is higher than {\\gamma}^2{\\omega}_p predicted in the DRL theory, and the spectral index of low frequency region is not 1 but 1/3.", "It is because the typical deflection angle of electrons is larger than the angle of the beaming cone sim 1/{\\gamma}.", "We call the radiation for this case \"Wiggler Radiation in Langmuir turbulence\" (WRL)." ], [ "INTRODUCTION", "The radiation mechanisms of many high energy astrophysical objects are still an active issue since they often contain features which are not easily explained by the conventional synchrotron and inverse Compton emissions.", "Recently, much attention has been paid to the radiation signatures from the turbulent electromagnetic fields (e.g., Medvedev 2000, Fleishman 2006, Kelner, Aharonian, & Khangulyan 2013, Mao & Wang 2013, Teraki & Takahara 2013).", "The main scene of the emission regions of high energy astrophysical objects is collisionless shocks, and the turbulent electromagnetic fields would be generated in the shock region.", "Therefore, the electromagnetic turbulence should be taken into account when we consider the radiation.", "However, for the major emission mechanisms of synchrotron radiation and inverse Compton scattering, effects of small scale turbulence are not taken into account.", "There is a room for novel emission signatures in the consideration of turbulence, which may be of relevance to observations.", "By reproducing the observed spectra, we can extract physical parameters of astrophysical objects.", "Thus, researches of radiation signatures from the turbulent field would play a key role for the understanding of physical mechanisms of the high energy astrophysical objects.", "Radiation spectra from a small scale turbulent magnetic field have been well studied as the \"jitter radiation\" or \"Diffusive Synchrotron Radiation\" (Medvedev et al.", "2011, Fleishman & Urtiev 2010 and references therein).", "Differences from the synchrotron radiation become significant when the typical spatial scale of an eddy $1/k_\\mathrm {typ}$ is smaller than the Photon Formation Length (PFL) of the synchrotron photons $r_\\mathrm {L}/\\gamma $ , where $r_\\mathrm {L}$ is the Larmor radius and $\\gamma $ is the Lorentz factor of the electron.", "Such a small scale magnetic field is thought to be generated by Weibel instability around the shock front.", "When the strength of this small scale turbulent magnetic field is dominant, the radiation spectra are determined by the turbulence and reveal various signatures of the turbulent field.", "For example, when $2\\pi /k_\\mathrm {typ}\\ll r_\\mathrm {L}/\\gamma $ the peak frequency in $\\nu F_\\nu $ spectrum becomes $\\gamma ^2k_\\mathrm {typ}c$ , and the spectrum in the highest frequency region shows a power law $\\nu F_\\nu \\propto \\nu ^{-\\mu +1}$ when the turbulence exists up to the maximum wave number $k_{\\rm max}\\gg k_{\\rm typ}$ .", "The power law index $\\mu $ reflects that of the turbulent magnetic field $B^2(k)\\propto k^{-\\mu }$ .", "The spectra show more complex signatures when $2\\pi /k_\\mathrm {typ}\\sim r_\\mathrm {L}/\\gamma $ (Medvedev 2011, Reville & Kirk 2010, Teraki & Takahara 2011).", "We note that the anisotropy of turbulence also affects the radiation spectra (e.g.", "Kelner et al.", "2013, Reynolds & Medvedev 2012, Medvedev 2006).", "The radiation from an electron which moves in non-uniform magnetic field in a laboratory is well studied using the insertion device of synchrotron orbital radiation factory, where a series of magnets are line-upped to make the particle deflect periodically.", "It is called \"Wiggler\" or \"Undulator\" (Jackson 1999).", "For Undulator, the strength of magnets $B$ and gaps between them $\\lambda $ are chosen to satisfy the condition that the observer is always in the beaming cone.", "On the other hand, for Wiggler, the observer is periodically in and off the beaming cone.", "We estimate the critical distance $\\lambda _{\\rm c}$ which divides Wiggler and Undulator.", "The deflection angle in one deflection is $\\theta _{\\rm def}= \\lambda /r$ , where $r\\simeq \\gamma mc^2/eB$ is the typical curvature radius of the orbit.", "The radiation from a relativistic particle is concentrated into small cone with opening angle $\\sim 1/\\gamma $ .", "Therefore, the critical condition dividing Wiggler and Undulator is $\\theta _{\\rm def} = 1/\\gamma $ , which is rewritten as $\\lambda _{\\rm c} = r/\\gamma $ .", "Thus, the device is called Undulator when $\\lambda <\\lambda _{\\rm c}$ , while it is called Wiggler when $\\lambda >\\lambda _{\\rm c}$ .", "The radiation spectrum of Undulator shows a sharp peak at $\\gamma ^2 2\\pi c/\\lambda $ , while Wiggler shows a broad spectrum with peak frequency $\\sim \\gamma ^2 eB/mc$ .", "The relation between typical frequencies and deflection angle is a key point for understanding of the radiation spectra.", "Perturbative jitter radiation or perturbative DSR is recognized as extensions of the Undulator radiation, since the spatial scale of turbulence $\\lambda $ is assumed to be much smaller than $mc^2/eB$ .", "The fact that $\\theta _{\\rm def}$ determines the radiation feature was also noted in astrophysics in early seventies by, e.g., Rees, (1971), and Getmantsev & Tokarev (1972).", "Rees assumed that the strong electromagnetic wave is emitted from the Crab pulsar with frequency $\\Omega $ (30Hz) according to the \"oblique rotator\" model by Ostriker & Gunn (1969), and argued that the radiation from an electron moving in the strong wave is not inverse Compton scattering (with frequency $\\gamma ^2\\Omega $ ) but synchrotron-like (synchro-Compton) radiation, because the deflection angle $\\theta _{\\rm def}$ is estimated as $\\theta _{\\rm def}>10/\\gamma $ , where the factor 10 comes from the ratio of cyclotron frequency to the wave frequency $eB_{\\rm eq}/mc\\Omega $ .", "The magnetic field strength $B_{\\rm eq}$ is estimated by equating the spindown luminosity and power of magnetic dipole radiation of the Crab pulsar and by considering the distance from the pulsar.", "He argued that the radiation signature of synchro-Compton radiation resembles that of the synchrotron radiation.", "The typical frequency is determined by the field strength as $\\gamma ^2eB_{\\rm eq}/mc$ , and the low frequency spectral index is roughly $1/3$ .", "Getmantsev & Tokarev (1972) studied the radiation spectra under general electromagnetic fields and stated that the radiation signature from a single charged particle is determined by the frequency or wavelength of the field for $\\theta _{\\rm def}\\ll 1/\\gamma $ , while it is determined by the field strength for $\\theta _{\\rm def}\\gg 1/\\gamma $ in the same way as synchrotron radiation.", "Note that they did not discussed explicit expressions of the radiation spectra from a single electron for $\\theta _{\\rm def}\\sim 1/\\gamma $ , they presented a general expression of radiation spectrum from an ensemble of electrons with a power spectrum.", "Thirty years after these pioneering works, radiation spectra from a relativistic particle interacting with turbulent fields are gathering a renewed attention.", "For example, the radiation mechanism from Langmuir turbulence treated in the present paper has become an interesting topic.", "The Langmuir turbulence has been pointed out to be generated around the shock front of the relativistic shocks (Silva 2006, Dieckmann 2005, Bret, Dieckmann, & Deutch 2006), so it should be as important as the radiation from turbulent magnetic field.", "Fleishman & Toptygin (2007a,b) have made a systematic treatment of diffusive radiation in Langmuir turbulence (see references cited in Fleishman & Toptyigin 2007b for other earlier relevant works).", "Their method is the most sophisticated treatment for Langmuir turbulence, which is based on Toptygin & Fleishman (1987).", "For later discussions, we shortly review their treatment.", "They treat the electron motion by a statistical approach and use a perturbative treatment for calculation of the radiation.", "The calculation formula for the radiation spectra is based on the one written in Landau & Lifshitz (1971).", "For $\\theta _{\\rm def}\\ll 1/\\gamma $ , a rectilinear trajectory with constant velocity is assumed, but non-zero acceleration from the external field is taken into account.", "The wavenumber of the Langmuir waves is assumed to satisfy the condition $k_{\\rm typ} <\\omega _{\\rm p}/c$ , where $\\omega _\\mathrm {p}$ is the plasma frequency.", "They calculate the correlation between the acceleration and the Langmuir waves.", "The peak frequency is $\\gamma ^2\\omega _\\mathrm {p}$ .", "The spectrum shows an abrupt cutoff above the peak, and becomes $F_\\omega \\propto \\omega ^{-\\mu }$ in higher frequencies when the turbulence exists up to the maximum wavenumber $k_{\\rm max}\\gg c/\\omega _{\\rm p}$ for a power law turbulent spectrum $E^2(k)\\propto k^{-\\mu }$ .", "The spectrum just below the peak is $F_\\omega \\propto \\omega ^1$ , and becomes $F_\\omega \\propto \\omega ^0$ in lower frequencies and $F_\\omega \\propto \\omega ^{1/2}$ in even lower frequency region.$F_\\omega \\propto \\omega ^2$ spectrum is predicted in the lowest frequency region, which comes from the effect of wave dispersion in the plasma.", "We do not discuss such effects in this paper.", "$F_\\omega \\propto \\omega ^{1/2}$ spectrum comes from the effect of multiple scattering.", "The angle between the velocity and observer direction becomes larger than the beaming cone after many deflections even when the deflection angle in one deflection $\\theta _{\\rm def}$ is much smaller than $1/\\gamma $ , and the approximation of rectilinear trajectory is broken.", "Therefore, this treatment is beyond the perturbative treatment, and they call it \"non-perturbative treatment\".", "They treat the changing of direction of motion in many deflections by diffusion approximation.", "In consequence, a spectral break emerges in the low frequency region with a suppression of low frequency photons.", "The spectrum becomes $F_\\omega \\propto \\omega ^{1/2}$ from this effect and the index of $1/2$ comes from the diffusivity.", "They claimed that the break frequency approaches the peak frequency as $\\theta _{\\rm def}$ becomes large and the break and peak merge for $\\theta _{\\rm def}\\sim 1/\\gamma $ .", "They stated that even for $\\theta _{\\rm def}\\gg 1/\\gamma $ the spectrum in frequency region just below the peak of $\\gamma ^2\\omega _{\\rm p}$ is $F_\\omega \\propto \\omega ^{1/2}$ .", "This is inconsistent with the statement by Getmantsev & Tokarev (1972).", "This is one of the objectives of investigations in the present paper.", "The effect of large angle deflection would come into play in forming the radiation spectra for Langmuir turbulence as in the Wiggler radiation for $\\theta _{\\rm def}>1/\\gamma $ .", "This point has been discussed for magnetic turbulence in Kelner et al.", "(2013), Medevev et al.", "(2011), and Teraki & Takahara (2011).", "In this paper, we investigate general properties of the radiation spectra from relativistic electrons in a Langmuir turbulence for various cases including this regime.", "Before proceeding to the formulation of the calculation employed in the present paper, we point out the differences between the magnetic field generated by Weibel instability (entropy modes) and the Langmuir waves.", "The first is time variability.", "The entropy mode does not oscillate, so the typical timescale is determined by the turnover time of an eddy.", "It is longer than the crossing time of a relativistic electron if we assume that the background plasma is sub-relativistic.", "Therefore, we can treat the magnetic field as a static field when we calculate the radiation spectra for the zeroth-order approximation.", "On the contrary, we should not treat the Langmuir turbulence as a static field even for the zeroth-order approximation.", "The crossing time can be comparable to or longer than the period of the Langmuir waves, because the typical spatial scale is about inertial length $c/\\omega _{\\rm p}$ , governed by the plasma frequency $\\omega _\\mathrm {p}$ (Diekmann 2005).", "Therefore, the time variability of the electric field can not be ignored.", "This effect is well studied by Fleishman & Toptygin (2007a,b).", "The second is the energy change of the radiating electrons.", "For the case of turbulent magnetic field, the energy of electrons is conserved if we ignore the radiation back reaction.", "However, the energy change cannot be ignored for the Langmuir turbulence, because the electric field can accelerate the electrons parallel to their velocity.", "The Lorentz factor of the electron can change in a short time by strong Langmuir waves.", "This effect may play a role for calculation of the radiation spectra.", "The calculation of electron trajectory for the strong turbulence by analytical approach is hard to perform.", "Thus, we calculate the radiation spectra by numerical approach from first principle.", "We calculate radiation spectra for a wide range of the field parameters.", "We study about three factors, the scale length, time variability, and strength.", "By sweeping the parameter plane, we generally investigate the radiation spectra from a relativistic electron moving in Langmuir turbulence.", "In section 2, we describe calculation method, and we show the results in section 3.", "In section 4, we give the physical interpretations of the discovered spectral features using radiation for a spatially uniform plasma oscillation.", "In section 5, we make a summary and some discussions." ], [ "FORMULATION", "We use Lienard-Wiechert potential directly to calculate the radiation spectra.", "It is the same approach as employed in Teraki & Takahara 2011 for magnetic turbulence.", "This method is computationally expensive because the time step is restricted by observed frequency and the integration time has to be longer than the Photon Formation Time (PFT).", "PFT is defined as the coherence time for forming a photon with a given frequency (Reville & Kirk 2010, Akhiezer & Shul'ga 1987).", "Therefore, PFT is a function of the frequency.", "Since PFT is $\\gamma ^2$ times larger than the inverse of the observed frequency and since the time step should be shorter than PFT, the integration time is at least $\\gamma ^2$ times larger than the inverse of the observed frequency.", "As a result, large number of time steps are needed to calculate radiation spectra for highly relativistic case.", "Although less heavy approach is proposed by Reville & Kirk 2010 to overcome this problem by using appropriate approximations, we use the first principle method we show below, because it does not include any approximations in the calculation formula of the radiation spectra from prescribed trajectory.", "Therefore, this method is suitable for studying the cases for which any approximations are difficult to adopt.", "By using moderately relativistic particles, we reduce the computational expenses.", "We neglect radiation back reaction of electrons, because the strength of electric field is sufficiently weaker than $m^2c^4/e^3$ (cf.", "Jackson 1999).", "We use 3D isotropic Langmuir turbulence in this paper.", "If the Langmuir waves are generated near the shock front, they may be highly anisotropic on the spot.", "However, we assume the isotropic turbulence for two reasons.", "First, the Langmuir turbulence in large part of the emission region would be isotropic, since Langmuir waves interact with background ions and form the Kolmogorov type isotropic turbulence (Treumann & Baumjohan 1997).", "Second, the radiation spectra for 3D and 1D turbulence are not so different except for a particular case that the direction of radiating electrons and the wave vector are parallel.", "For this case, the radiation is linear acceleration emission.", "However, emission from the perpendicular acceleration dominates as long as the direction differs from the wave vector direction even slightly (Fleishman & Toptygin 2007a).", "If the shock is ultra-relativistic, the downstream plasma may be relativistically hot.", "For simplicity, we assume a mildly relativistic shock, so that the downstream plasma is sub-relativistic.", "We ignore the thermal velocity of background plasma in the dispersion relation of Langmuir waves $\\omega ^2 = \\omega _\\mathrm {p}^2+3/2k^2 v_\\mathrm {e,th}^2$ .", "Thus, we use the propagating Langmuir waves with the same frequency $\\omega _\\mathrm {p}$ .", "We generate 3D isotropic Langmuir turbulence by using Fourier transform description, which is slightly modified from the description for magnetic turbulence developed by Giacalone & Jokipii (1999).", "It is described by superpositions of $N$ Fourier modes, each with random phase, and direction $\\mathbf {E}(x) = \\sum _{n=1}^N A_n \\cos \\bigl \\lbrace (\\mathbf {k}_n \\cdot \\mathbf {x} -\\omega _\\mathrm {p}t + \\beta _n)\\bigr \\rbrace \\frac{\\mathbf {k_n}}{\|k_n\|}.$ Here, $A_n$ , $\\beta _n$ , $\\mathbf {k_n}$ , and $\\omega _\\mathrm {p}$ are the amplitude, phase, wave vector, frequency of the $n$ th mode, respectively.", "The amplitude $A_n$ of each mode is defined as $A_n^2 = \\sigma ^2 G_n \\Biggl [\\sum _{n=1}^N G_n\\Biggr ]^{-1},$ where the variable $\\sigma $ represents the amplitude of turbulent field.", "We use the following form for the power spectrum $G_n = \\frac{4\\pi k_n^2 \\Delta k_n}{1 + (k_n L_\\mathrm {c})^\\mu },$ where $L_\\mathrm {c}$ is the correlation length of the field.", "Here, $\\Delta k_n$ is chosen such that there is an equal spacing in logarithmic $k$ -space, over the finite interval $k_\\mathrm {min}\\le k\\le k_\\mathrm {max}$ and $N=10^3$ , where $k_\\mathrm {min} = 2\\pi /L_\\mathrm {c}$ , $\\mu =9/2$ and $k_\\mathrm {max}$ is chosen to be $10^3k_\\mathrm {min}$ or $10k_\\mathrm {min}$ .", "The spectrum has a peak at $k_\\mathrm {min}$ and the spectral index is for 3-dimensional isotropic Langmuir turbulence.", "Then we define two parameters which characterize radiation spectra.", "The first one is $a\\equiv \\frac{e\\sigma }{mc^2k_\\mathrm {min}}=\\frac{\\omega _\\mathrm {st}}{\\omega _0},$ where $\\omega _\\mathrm {st}\\equiv e\\sigma /mc$ and $\\omega _0 \\equiv k_\\mathrm {min}c$ .", "We call $\\omega _\\mathrm {st}$ \"strength omega\", and $\\omega _0$ \"spatial omega\".", "The strength omega $\\omega _\\mathrm {st}$ accounts for the effect of the field strength.", "The meaning of strength omega can be understood by considering the curvature of orbit and beaming effects for the relativistic particles as follows.", "For $\\gamma \\gg 1$ , the local curvature radius of the orbit suffering from perpendicular acceleration by the electric field is $\\sim \\gamma m c^2/(e\\sigma )$ .", "The radiation is concentrated in the beaming cone with an angle $\\sim 1/\\gamma $ , so the searchlight sweeps the observer in the timescale of $mc/e\\sigma =1/\\omega _\\mathrm {st}$ .", "It is an analogy of the cyclotron frequency in the mechanism of synchrotron radiation.", "Therefore, $\\omega _{\\rm st}$ represents the effect of the field strength on the radiation spectra.", "As for the spatial omega $\\omega _0$ , since the electron moves nearly at the light speed, the rate of change in the force direction for the electron is $2\\pi c/\\lambda =\\omega _0$ .", "The ratio $\\omega _\\mathrm {st}/\\omega _0=a$ parametrizes the field feature, which represents the ratio of the deflection angle $\\theta _{\\rm def}$ to $1/\\gamma $ .", "Therefore, the spectra from the turbulent magnetic field changes for $a\\lesssim 1$ to $a\\gtrsim 1$ drastically (Kelner et al.", "2013, Medvedev et al.", "2011, and Teraki & Takahara 2011).", "Note that $a$ is well known as the \"strength parameter\".", "It means the change of the Lorentz factor of an electron which is accelerated along the electric field on the spatial scale $l=1/k_\\mathrm {min}$ .", "We can understand it clearly by using the work by an electric field on an electron as $eEl = \\Delta \\gamma mc^2,$ where we assume the electric field is parallel to the velocity.", "We should note that it is different from the one we used in Teraki & Takahara 2011 by a factor of $2\\pi $ , and it agrees with the definition of Fleishman & Toptygin (2007a,b).", "The second one is the ratio of the frequency of the Langmuir waves to $\\omega _0$ , $b\\equiv \\frac{\\omega _\\mathrm {p}}{k_\\mathrm {min}c}=\\frac{\\omega _\\mathrm {p}}{\\omega _0}.$ For $b\\gg 1$ , the force direction changes with frequency $\\omega _\\mathrm {p}$ for all electrons.", "For $b\\ll 1$ , a change of the force direction is mainly from the spatial fluctuation.", "Summarizing above, although there are three parameters of the turbulent field ($\\omega _0$ , $\\omega _\\mathrm {st}$ , $\\omega _\\mathrm {p}$ ), we can reduce them to two parameters of ($a,b$ ) when our interest is in the spectral signature.", "We investigate the spectral features in the parameter plane of ($a,b$ ).", "We inject monoenergetic electrons with Lorentz factor $\\gamma _\\mathrm {init}=10$ into the generated turbulent field.", "The number of electrons used for each calculation is more than $10^2$ and is written in the caption of each figure.", "The initial velocity is randomly chosen to achieve an isotropic distribution.", "Next we solve the equation of motion $\\frac{d}{dt}(\\gamma m \\mathbf {v}) = -e \\mathbf {E},$ by using the method which has second order accuracy for each time step.", "The electrons get energy in the Langmuir turbulence from the parallel (to the velocity) component.", "If we pursue electrons much longer time than the PFT of a given frequency, the radiation spectrum in high frequency region corresponds to an \"integrated spectrum\".", "This spectrum would be understood by a superposition of \"instantaneous spectra\".", "Here, the term of \"instantaneous spectrum\" means a spectrum which does not contain the effect of secular energy change of the radiating electrons.", "If we do not obtain the instantaneous spectra, we can not discern either the intrinsic feature of instantaneous spectra or energy change of radiating electrons determines the spectral features.", "Therefore, in this paper we concentrate on \"instantaneous spectra\".", "On the other hand, spectral broadening due to finite integration time of particle orbit is inevitable.", "To compromise the accuracy and the instantaneousness, we choose the integration time as 100 times PFT of the peak or break frequency.", "Let us take an example of the spectrum for well known jitter radiationWe use the term \"jitter radiation\" when the orbit is rectilinear and acceleration is treated perturbatively whether acceleration is due to magnetic turbulence or electric turbulence., for $a\\ll 1, b=0$ .", "The break frequency of the jitter radiation is $2\\gamma ^2\\omega _0$ , and corresponding PFT is $T\\sim 1/\\omega _0$ .", "We can understand the origin of this frequency by using \"the method of virtual quanta\", (cf.", "Jackson 1999).", "The spatial fluctuation of the electric field with a spatial scale $\\lambda = 2\\pi c/\\omega _0$ .", "This fluctuation can be regarded as a photon which has the frequency with $\\gamma \\omega _0$ in the electron rest frame.", "The electron scatters the photon, and transformed back to the observer frame, the frequency of the scattered photon is written as $\\sim 2\\gamma ^2\\omega _0$ .", "In other words, we can understand this process as an analogy of the inverse Compton scattering for the photon with frequency $\\omega _0$ .", "We note that this analogy is based on the assumption that the observer is in almost the center of the beaming cone over the PFT.", "It is valid when $a\\ll 1$ .", "The deflection angle in the time scale of $1/\\omega _0$ is $eE\\lambda /\\gamma mc^2$ , and when we require that it is smaller than $1/\\gamma $ , we get the condition $eE\\lambda /mc^2<1$ .", "It can be transformed to $\\omega _\\mathrm {st}/\\omega _0=a<1$ .", "An alternative explanation can be done by using \"the photon chasing effect\" (Rybiki & Lightmani 1979).", "Since the radiation from an ultra-relativistic particle is concentrated into the beaming cone, the particle chases the photon.", "Therefore, the observer sees the emitted radiation in a time span $(1-v/c)T$ , so the frequency is $1/(1-v/c)T\\sim 2\\gamma ^2/T \\sim \\gamma ^2\\omega _0$ .", "As a consequence, the typical frequency is $2\\gamma ^2\\omega _0$ .", "As we see in the next section, PFT for the peak frequencies of the spectra is the one of the typical timescales of $1/\\omega _0$ , $1/\\omega _\\mathrm {st}$ , and $1/\\omega _\\mathrm {p}$ .", "We choose a suitable one for each case.", "The integration time 100 times the PFT of the peak frequency is sufficiently long to resolve the spectral shape in the low frequency regions.", "We calculate the radiation spectra using the acceleration, velocity, and position of electrons.", "The energy $dW$ emitted per unit solid angle $d\\Omega $ (around the direction $\\mathbf {n}$ ) and per unit frequency $d\\omega $ to the direction $\\mathbf {n}$ is computed as $\\frac{dW}{d\\omega d\\Omega } = \\frac{e^2}{4 \\pi c^2}\\Bigl \| \\int ^{\\infty }_{-\\infty } \\:dt^{\\prime } \\frac{ \\mathbf {n} \\times \\bigl [ (\\mathbf {n} - \\mathbf {\\beta }) \\times \\dot{\\mathbf {\\beta }} \\bigr ] }{(1 - \\mathbf {\\beta } \\cdot \\mathbf {n} )^2 }\\exp \\bigl \\lbrace {i\\omega ( t^{\\prime } - \\frac{\\mathbf {n} \\cdot \\mathbf {r}(t^{\\prime })}{c})}\\bigr \\rbrace \\Bigr \|^2,$ where $\\mathbf {r}(t^{\\prime })$ is the electron trajectory, $t^{\\prime }$ is retarded time (Jackson 1999).", "Since we have not any specific constraints for $a$ and $b$ which are realized in the high energy astrophysical object, we sweep wide parameter range of $a$ and $b$ ." ], [ "Short wavelength regime", "First, we show the radiation spectra for $b=\\omega _\\mathrm {p}/\\omega _0\\ll 1$ (Figure REF ), i.e., for the situation where typical spatial scale is shorter than the inertial length.", "We set $a= 0.1$ to 20, and fix $b=10^{-2}$ ; specifically we set $\\omega _0=1$ and $\\omega _\\mathrm {p}=10^{-2}$ , and change $\\omega _\\mathrm {st}$ from $0.1$ to 20, and take $k_\\mathrm {max}=10^3k_\\mathrm {min}$ .", "The inequality $b\\ll 1$ can be transformed to $\\lambda \\ll 2\\pi c/\\omega _\\mathrm {p}$ , which means that the fluctuation scale is much smaller than the inertial length.", "These fluctuations would be damped by Landau damping and may not be realized in high energy astrophysical objects (Treumann & Baumjohan 1997).", "However, we cannot reject the possibility that $b\\lesssim 1$ is realized for some time span in relativistic plasmas, so that we study the spectra for $b<1$ .", "To explore the regime for $b<1$ clearly, we set an extreme value $b=10^{-2}\\ll 1$ .", "An example of the temporal variation of the Lorentz factor of an electron is depicted in Figure REF .", "For $a=0.1$ , the low frequency spectrum is as flat as $F_\\omega \\propto \\omega ^0$ , and there is a break at $\\omega \\sim 200$ , and the spectrum declines with power law $F_\\omega \\propto \\omega ^{-5/2}$ in the high frequency region.", "For $a=5$ , the break frequency becomes higher than that for $a=0.1$ , and the high frequency spectrum deviates from a power law.", "For $a=10$ , the spectrum in low frequency region becomes hard with an index $\\sim 1/2$ , and the spectrum in higher frequencies reveals a cutoff feature.", "For $a=20$ , we see further different features.", "The spectrum in low frequency side of the peak becomes softer, with the spectral index of about $1/3$ , and we see a small deviation from a power law in the lowest frequency region.", "The features for these spectra can be understood by using the analogy to the radiation theory from the stochastic magnetic field (Medvedev et al.", "2011, Teraki & Takahara 2011).", "Since the wavelength of Langmuir waves for $b\\ll 1$ is very short, the oscillation of the electric field can be neglected in the particle crossing time of the wavelength.", "In fact, $b\\ll 1$ is also written by $\\lambda /c\\ll 2\\pi /\\omega _\\mathrm {p}$ .", "The crossing time $\\lambda /c$ corresponds to the PFT of the typical frequency.", "As explained above, we can use the straight analogy to the radiation theory of the stochastic magnetic fields for the radiation spectra from Langmuir turbulence for $b<1$ by substituting the electric field strength $E$ with magnetic field strength $B$ .", "For $a<1$ and $b<1$ , we can use the radiation theories of the DSR and jitter radiation theory (Fleishman 2006, Medvedev 2006).", "On the contrary, for $a>1$ and $b<1$ jitter radiation of strong deflection regime can be applicable (Medvedev et al.", "2011, Teraki & Takahara 2011).", "We call $a<1$ and $b<1$ regime as \"jitter radiation\" regime because the jitter radiation is basically perturbative theory for $a<1$ .", "We call the radiation for $a>1$ and $b<1$ regime as \"Wiggler Radiation in Langmuir turbulence\", or \"WRL\" in short.", "Although the Wiggler radiation is not the radiation mechanism from the stochastic field but that for a fixed field, it has a common picture that the observer is in and off the beaming cone in the course of time.", "First, we describe the signature of the radiation spectra of jitter radiation or Diffusive Synchrotron Radiation.", "The radiation signatures are determined by acceleration perpendicular to the motion, and the observer is always in the beaming cone, which is the same situation as the Langmuir turbulence for $a\\lesssim 1$ and $b<1$ .", "For $a\\ll 1$ , the spectrum is written by a broken power law $F_\\omega \\propto \\omega ^0$ in the low frequency region and $F_\\omega \\propto \\omega ^{-\\mu +2}$ in the high frequency region.", "The break frequency is $\\sim \\gamma ^2\\omega _0=\\gamma ^2k_\\mathrm {min}c$ .", "For $a\\sim 1$ , while the break frequency remains the same as $\\gamma ^2\\omega _0$ , the multiple deflection effect comes into play in the spectral features near the break frequency.", "The spectrum in the low frequency region becomes $F_\\omega \\propto \\omega ^{1/2}$ .", "Since the multiple deflection makes the angle between observer direction and velocity larger than the beaming cone angle $1/\\gamma $ , the observer sees the radiation over the timescale which is determined by the deflection condition.", "Fleishman supposed that the angle changes diffusively.", "The break frequency of $a\\gamma ^2\\omega _\\mathrm {st}$ is calculated from the angle diffusion (Fleishman 2006).", "The spectral index of $1/2$ comes from the diffusivity, too.", "Next, we describe the signatures for strong deflection regime of $a\\gg 1$ .", "For magnetic turbulence, the spectral shape resembles synchrotron radiation in the middle frequency region and deviations from it would be seen in low frequency region and highest frequency region (Teraki & Takahra 2011).", "The peak frequency of $\\gamma ^2\\omega _\\mathrm {st}$ is from the sweeping of the beaming cone on the observer.", "The picture is the same as the synchrotron radiation.", "For short wavelength regime $\\omega _0\\gg \\omega _{\\rm p}$ of Langmuir turbulence, the physical picture is the same as this, because the spatial fluctuation dominates the changing rate of the deflection angle.", "A single deflection angle is $\\sim eE\\lambda /\\gamma mc^2$ , which is larger than $1/\\gamma $ for $a=\\omega _\\mathrm {st}/\\omega _0=e\\sigma /mc^2k_\\mathrm {min}>1$ .", "As a result, the beaming cone sweeps out of the observer within one deflection.", "The intensity of the radiation off the beaming cone is weak.", "Therefore, the timescale which sweeps the observer $\\sim 1/\\omega _\\mathrm {st}$ corresponds to the PFT of the typical frequency.", "Considering the photon chasing effect, we get the peak frequency $\\sim \\gamma ^2 \\omega _\\mathrm {st}$ .", "The spectral break in the low frequency region would correspond to the deviation from local circular orbit, but the numerical error from finite integration time is also becoming large in the lower frequency region.", "We do not discuss this point further here, since it is not the main point of contents in this paper.", "The power law component in the highest frequency region comes from the smaller scale part of turbulence.", "It is the same as the spectra of jitter regime.", "We note here that the power law component in high frequency region cannot be seen for $a=10$ and $a=20$ .", "The reason may be as follows.", "In contrast to the magnetic turbulence, the energy of radiating electrons changes for $a\\gtrsim 1$ .", "Therefore, the high frequency region is determined by only the later part of integration time, because the peak frequency is $\\sim \\gamma ^2\\omega _\\mathrm {st}$ for $a>1$ , and the electrons get energy and radiate higher energy photon in later time.", "The power law component for the highest frequency region in our calculation is hidden by the component that small numbers of electrons with larger energy radiate by Wiggler mechanism.", "Finally, we consider effects of the energy change in WRL regime, which is the one of the different points from the magnetic case.", "Based on an example of the change of Lorentz factor is depicted in Figure REF , we estimate the energy change in PFT $1/\\omega _{\\rm st}$ for peak frequency.", "The PFT is $1/\\omega _\\mathrm {st}$ .", "The change of Lorentz factor is estimated as $\\Delta \\gamma =eE\\cdot \\frac{c}{\\omega _\\mathrm {st}}\\frac{1}{mc^2}\\sim 1.$ Therefore, the change of the Lorentz factor in this timescale is smaller than 1.", "Thus, we do not need to consider the energy change in PFT for peak frequency.", "However, for sufficiently large $a$ , we have to take it into account for lower frequency spectrum.", "We calculate the spectra for modest $a$ in this paper, so we omit this problem.", "It will be studied in future works." ], [ "Long wavelength and weak regime", "Next we show the radiation spectra for $a<1$ and $b>1$ (Figure REF ), i.e., a situation where long wavelength turbulence with weak amplitudes dominates; the spectra for $b=0.1$ and 1 are also depicted for comparison.", "An example of the temporal variation of the Lorentz factor of an electron is depicted in Figure REF .", "Firstly, we consider the meaning of the parameters of $a<1$ and $b>1$ , which correspond to $\\omega _\\mathrm {p}>\\omega _0>\\omega _\\mathrm {st}$ .", "In this regime, the changes of the direction of acceleration are mainly due to wave oscillation, rather than the spatial fluctuations because the crossing time is longer than the oscillation time.", "Moreover, $\\theta _{\\rm def}<1/\\gamma $ is derived from $b>a$ .", "Therefore, we can regard the orbit as straight in the time scale of plasma oscillation $1/\\omega _{\\rm p}$ .", "The condition $b>1$ can be transformed to $\\lambda >2\\pi c/\\omega _\\mathrm {p}$ , which means that the inertial length is shorter than the wavelength.", "Therefore, this regime is likely to occur in the astrophysical objects.", "We set $\\omega _0=1$ , $\\omega _\\mathrm {st}=10^{-2}$ and change $\\omega _\\mathrm {p}$ from $0.1$ to 10, so that $a=10^{-2}$ and $b=0.1$ to 10, and take $k_\\mathrm {max}=10^3k_\\mathrm {min}$ .", "As was discussed in the previous subsection for $b=0.1$ , the spectrum shows jitter radiation signature.", "The peak frequency is $\\gamma ^2\\omega _0$ and $F_\\omega \\propto \\omega ^0$ in the low frequency region, and $F_\\omega \\propto \\omega ^{-5/2}$ above the peak frequency reflecting the spectrum of the turbulent electric field $E^2(k)\\propto k^{-5/2}$ .", "As $b$ becomes larger, the peak shifts to higher frequency.", "For $b=10$ , the peak frequency is $\\sim 10^3$ , which is identified with $2\\gamma ^2\\omega _\\mathrm {p}$ .", "The spectral index of low frequency side is $\\sim 1$ .", "This feature coincides with the result of DRL theory (Fleishman 2006).", "We call this regime \"DRL regime\".", "The DRL theory predicts the spectral shape for $a<1$ and $b>1$ as follows.", "The peak frequency is $2\\gamma ^2\\omega _\\mathrm {p}$ with an abrupt cutoff in the higher frequency side and a power law component emerges in the highest frequency region for $k_\\mathrm {max}\\gg \\omega _\\mathrm {p}/c$ .", "In the low frequency side, the spectrum becomes $F_\\omega \\propto \\omega ^1$ .", "The peak frequency is determined by the timescale of plasma oscillation.", "The shortest timescale of the electron motion is $1/\\omega _{\\rm p}$ , and the observer located along the initial velocity direction can see this radiation, because the orbit is regarded as straight in this time scale as we showed above.", "We consider Doppler boosting, and we get the peak frequency $2\\gamma ^2 \\omega _{\\rm p}$ in the observer frame.", "The origin of the power law component in the highest frequency region is the same as jitter radiation.", "The hard spectral index 1 in the frequency region lower than the peak is from the photon chasing effect.", "The electrons oscillate by the same oscillating frequency $\\omega _\\mathrm {p}$ and move different angles to the observer in the observer frame.", "The photon chasing effect shifts the photon frequency from each electron and makes the $F_\\omega \\propto \\omega ^1$ spectrum (see Fleishman 2007a,b).", "It is regarded as the emission spectrum from an electron integrated over the solid angle, which can be understood in an analogy to the Undulator theory (Jackson 1999), although the force direction changes in this case not spatially but temporally.", "When the particle mean velocity and the wavenumber are fixed, that makes no difference for the radiation spectra.", "The peaky shape of the spectra is the most remarkable feature of the spectra for DRL case.", "A difference is that the wave number is a vector, while the frequency is a scalar.", "For DSR, the electron feels spatial fluctuation with wavenumber along the velocity $\\mathbf {k}\\cdot \\mathbf {v}/v$ , therefore the low frequency spectrum becomes flat.", "On the other hand, for DRL, all electrons feel the same frequency of $\\omega _\\mathrm {p}$ for the Langmuir turbulence (c.f.", "Fleishman & Toptygin 2007a,b and Medvedev 2006)." ], [ "Long wavelength and strong regime", "The remaining interesting regime of $a>1$ and $b>1$ is an open issue.", "As we explained in section , the radiation spectrum for Langmuir turbulence with $a>b>1$ predicted by Fleishman & Toptygin (2007b) is $F_\\omega \\propto \\omega ^{1/2}$ below the peak of $\\sim \\gamma ^2\\omega _{\\rm p}$ , which is inconsistent with the statements by Getamantsev and Tokarev (1972).", "Here we investigate this regime and clarify the radiation spectra.", "The inequations of $a>1$ and $b>1$ mean that the wavelength is longer than the inertial length and that the typical scale of PFT is $mc^2/e\\sigma $ .", "We set $a=10^2$ , $b=20$ to 800, so that $a/b=0.125$ to 5, and $\\omega _0=1$ .", "We set $\\omega _\\mathrm {st}=10^2$ , and we change $\\omega _\\mathrm {p}$ from 20 to 800, and $k_\\mathrm {max}$ is chosen to be $10k_\\mathrm {min}$ here.", "We show the interesting results for $a>1$ and $b>1$ (Figure REF ).", "An example of energy change for $a=100$ and $b=20$ is depicted in Figure REF .", "For clear discussion of the radiation spectra in this regime, we discuss the radiation spectra for $a>b$ and for $a<b$ separately.", "For $a<b$ , the peak frequency is $\\sim \\gamma ^2\\omega _\\mathrm {p}$ .", "The spectral index of low frequency region is 1, and cutoff feature can be seen above the peak.", "This region is regarded as the DRL regime from these signatures.", "For $a<b$ , i.e., $\\omega _\\mathrm {st}<\\omega _\\mathrm {p}$ , the particle is not deflected by large angle because the direction of the electric field changes in a time shorter than the time for which the beaming cone sweeps the observer.", "We set $k_\\mathrm {max}=10k_\\mathrm {min}$ , therefore, a power law component in high frequency region is not seen at all.", "On the contrary, different features emerge for $a>b$ .", "The peak frequency becomes larger than $\\gamma ^2\\omega _\\mathrm {p}$ .", "Moreover, the spectra below the peak frequency become softer, the index changes from 1 to $1/3$ .", "The energy change of electrons may cause the change of the peak frequency, but it cannot explain the soft spectrum.", "Rather, it would be naturally understood that the peak frequency is $\\gamma ^2\\omega _\\mathrm {st}$ and $F_\\omega \\propto \\omega ^{1/3}$ by using the analogy of the Wiggler radiation.", "We consider that we should use WRL theory not only for $a>1>b$ , but also for $a>b>1$ , because the deflection angle in one deflection is also larger than $1/\\gamma $ for this case.", "This is in contrast to the DRL theory, which predicts the same spectral signatures for the parameter range of $a>b>1$ as for $b>a>1$ .", "Thus, our numerical calculations reveal new features which have not been known previously for Langmuir turbulence.", "We ascribe that the parameter regime $a>b>1$ is in the WRL regime in $a-b$ plane.", "This radiation feature seems to be consistent with the statements by Getmantsev & Tokarev (1972).", "To clarify the spectral features for $a>b>1$ in more detail, and to confirm their statement and our consideration, we examine the radiation from a relativistic electron moving in pure plasma oscillation in the next section." ], [ "PURE PLASMA OSCILLATION", "In this section, we investigate the emission of a relativistic electron suffering from pure plasma oscillation in order to discuss the interpretation of the features of the radiation spectra for $a>b>1$ .", "To clarify the origin of the peak frequency $\\gamma ^2\\omega _{\\rm st}$ , we set a simple configuration of the electric field, where electron motion is deterministic compared to stochastic character in turbulent fields.", "We calculate the electron velocity analytically and the radiation spectra numerically.", "By comparing the motion and spectra, we interpret the mechanism which determines the peak frequency.", "Lastly we consider the radiation spectra from the turbulent field by using these results.", "We use a single Langmuir wave which has infinitely large wavelength $k=0$ , in other words, $\\omega _0=0$ .", "Therefore, it is a pure plasma oscillation.", "We set $\\mathbf {E}=(E_x,0,0)$ with $E_x=E_0 \\cos (\\omega _\\mathrm {p}t).$ We characterize the field by using a single parameter of $\\eta \\equiv \\omega _\\mathrm {st}/\\omega _\\mathrm {p}.$ We inject an electron along the $z$ -axis at $t=0$ with the initial Lorentz factor $\\gamma _\\mathrm {init}$ and solve the equation of motion (REF ).", "Therefore, the orbit is determined by $\\gamma _\\mathrm {init}$ and $\\eta $ .", "Solving the equation of motion, we get the momentum $\\begin{aligned}p_x =&& - \\frac{eE_0}{\\omega _\\mathrm {p}}\\sin {\\omega _\\mathrm {p}t}\\\\p_z =&& \\gamma _{\\rm init}\\beta _{\\rm init}mc = && \\mathrm {const}.\\end{aligned}$ Since the velocity is a periodic function, we can define the mean velocity by $\\bar{\\beta }=\\frac{\\omega _\\mathrm {p}}{2\\pi }\\int ^{2\\pi /\\omega _\\mathrm {p}}_0\\beta _zdz$ .", "The mean velocity $\\bar{\\beta }$ cannot be represented elementarily in a general form.", "Then, we take the parameter $\\eta \\ll \\gamma _{\\rm init}$ and approximate the motion hereafter.", "We expand the Lorentz factor and the velocity, and get the mean velocity and the mean Lorentz factor in the lowest order of $\\eta /\\gamma _{\\rm init}$ .", "$\\begin{aligned}\\bar{\\beta }=&&\\beta _\\mathrm {init}(1-\\frac{\\eta ^2}{4\\gamma _{\\rm init}^2})\\\\\\bar{\\gamma }=&&\\frac{\\gamma _\\mathrm {init}}{\\sqrt{1+\\frac{\\eta ^2}{2}}}\\end{aligned}$ For $\\eta \\ll 1$ , $\\bar{\\gamma }\\sim \\gamma _\\mathrm {init}$ , while for $\\eta \\gg 1$ , $\\bar{\\gamma }=\\sqrt{2}\\gamma _{\\rm init}/\\eta $ .", "We note that $\\bar{\\gamma }$ can be much smaller than $\\gamma _\\mathrm {init}$ for $\\eta \\gg 1$ .", "Using this approximated velocity, we calculate the maximum Lorentz factor in the mean velocity frame, to clarify the fact that the radiation signatures depend on $\\eta $ as $\\gamma ^{\\prime }_\\mathrm {max}=\\frac{\\gamma _\\mathrm {init}}{\\sqrt{1+\\frac{\\eta ^2}{2}}}\\left[\\sqrt{\\gamma _\\mathrm {init}^2+\\eta ^2} -\\beta _\\mathrm {init}(1-\\frac{\\eta ^2}{4\\gamma _{\\rm init}^2})\\sqrt{\\gamma _\\mathrm {init}^2-1}\\right].$ For $\\eta \\ll 1$ , $\\gamma _\\mathrm {max}^{\\prime } = 1+\\frac{\\eta ^2}{2}.$ The motion in this frame is non-relativistic, therefore, the radiation in this frame is dipole radiation.", "On the other hand, for $\\eta \\gg 1$ , the maximum Lorentz factor is $\\gamma _\\mathrm {max}^{\\prime } = \\frac{3\\sqrt{2}}{4}\\eta .$ Therefore, the motion is relativistic even in this frame and the radiation spectrum consists of higher harmonics, because $\\beta ^{\\prime }$ approaches 1.", "It should be noted that for $\\eta =1$ , the motion in the mean velocity frame is mildly relativistic with Lorentz factor $\\gamma ^{\\prime }_\\mathrm {max}=1.02$ , and $\\beta ^{\\prime }=1/5$ .", "We can see that the transition from non-relativistic harmonic motion to relativistic motion occurs around $\\eta \\sim $ a few from this fact.", "Next we show numerically calculated radiation spectra from the electron and their features are interpreted in terms of the properties of the orbit.", "We fix $\\omega _\\mathrm {st}=1$ , and change $\\omega _\\mathrm {p}$ to change the parameter $\\eta $ .", "The observer is on the $z$ -direction.", "We calculate radiation spectra using much longer integration time than the PFT, because the electron moves perfectly periodically.", "As a consequence, the spectra show very sharp features, which makes it easier to understand the relation between spectral features and orbit.", "First, we show the spectrum for $\\eta = 10^{-3}$ ($\\omega _\\mathrm {p}=10^3$ , Figure REF (a)).", "We see a sharp peak like a delta function at the frequency $2\\gamma _\\mathrm {init}^2 \\omega _\\mathrm {p} = 2\\times 10^5$ .", "This is understood in terms of the motion of the electron in the mean velocity frame.", "For $\\eta \\ll 1$ , in the mean velocity frame, electron motion is a non-relativistic simple harmonic motion with the frequency $\\sim \\bar{\\gamma }\\omega _\\mathrm {p}$ .", "Therefore, the radiation is the dipole radiation with the frequency of $\\bar{\\gamma }\\omega _\\mathrm {p}$ .", "Since $\\bar{\\gamma }\\sim \\gamma _\\mathrm {init}$ for $\\eta \\ll 1$ , the radiation frequency in the observer frame is $2\\gamma _\\mathrm {init}^2\\omega _\\mathrm {p}$ .", "Thus, we ascribe the frequency $2\\gamma _\\mathrm {init}^2\\omega _\\mathrm {p}$ in the radiation spectra of perturbative regime ($\\omega _\\mathrm {st}<\\omega _\\mathrm {p}$ ) to the Doppler shifted dipole radiation.", "Next we show the spectrum for $\\eta = 0.01$ ($\\omega _\\mathrm {p}=10^2$ , Figure REF (b)).", "We can see the higher harmonics of $2\\gamma _\\mathrm {init}^2 \\omega _\\mathrm {p}=2\\times 10^4$ .", "It is from effects of retarded time, as is clearly seen in the mean velocity frame.", "However, the effect is very weak, as the power of the second harmonics is about $10^{11}$ times smaller than the fundamental mode in the frequency resolution of this calculation.", "The ratio of the power of the second harmonics to the fundamental mode is proportional to $\\beta ^{\\prime 2}$ , so the second harmonics is much smaller than the fundamental mode in this case.", "For $\\eta \\sim 1$ , the spectral shape changes significantly.", "First, the higher harmonics stand more strongly, because $\\beta ^{\\prime }$ approaches 1.", "Many harmonics are as strong as the fundamental mode for $\\eta =1$ (Figure REF (c)), and the envelope of the peaks of the harmonics shows an exponential cutoff.", "We note that the spectrum in the frequency region higher than 5000 comes from numerical error.", "Second, the frequency of the fundamental mode deviates from $2\\gamma _\\mathrm {init}^2\\omega _\\mathrm {p}$ , because the difference between $\\bar{\\gamma }$ and $\\gamma _\\mathrm {init}$ becomes larger.", "The mean Lorentz factor $\\bar{\\gamma }$ is $\\sqrt{2/3}\\gamma _\\mathrm {init}$ for $\\eta =1$ , thus the frequency of the fundamental mode in the original frame is $2\\bar{\\gamma }^2\\omega _\\mathrm {p}=133$ .", "The difference between 133 and $2\\gamma _\\mathrm {init}^2\\omega _\\mathrm {p}=200$ is small, but we can discern it in Figure REF (c).", "Next we discuss the peak frequency (cutoff frequency) for $\\eta >1$ .", "We show the spectra for $\\eta =1,3$ , and 5 in Figure REF .", "The fundamental frequency is 133 for $\\eta =1$ , 12 for $\\eta =3$ , and 3 for $\\eta =5$ , but the cutoff frequency around a few hundreds does not change.", "We see the cutoff frequency is not of the fundamental mode, but it is determined by the higher harmonics for $\\eta >1$ .", "The radiation spectra in the observer frame also can be derived by regarding it as a Doppler boosted emission.", "However, since the mechanism which determines the peak frequency is the same as the Wiggler radiation we can understand the peak frequency more easily by considering the PFT in the observer frame.", "The condition $\\eta =\\omega _\\mathrm {st}/\\omega _\\mathrm {p}>1$ is equivalent to that PFT of the typical frequency in Wiggler mechanism, where $1/\\omega _\\mathrm {st}$ is shorter than the oscillating time $1/\\omega _\\mathrm {p}$ .", "On the other hand, the Lorentz factor relevant for the peak frequency is not $\\bar{\\gamma }$ , but $\\gamma _\\mathrm {init}$ , because the beaming cone sweeps the observer in the phase around $2n\\pi $ , where $n$ is a natural number.", "We note that the change of the Lorentz factor in the time scale of $1/\\omega _\\mathrm {st}$ is 1 at most, as seen in equation (REF ).", "As a result, the cutoff frequency is $\\sim \\gamma _\\mathrm {init}^2\\omega _\\mathrm {st}$ .", "In this way, we get a clear understanding of the mechanism of the peak frequency shift around $\\eta \\sim O(1)$ .", "Here, we compare the results in this section with the radiation spectra obtained in the preceding section.", "The case of Langmuir turbulence with $a>b>1$ ($\\omega _\\mathrm {st}>\\omega _\\mathrm {p}>\\omega _0$ ) corresponds to the case of pure plasma oscillation with $\\eta >1$ , since $\\eta =\\omega _\\mathrm {st}/\\omega _\\mathrm {p}$ and $\\omega _0=0$ for pure plasma oscillation.", "Moreover, the approximation we used in pure plasma oscillation of $\\gamma _\\mathrm {init}>\\eta $ is also applicable for the case of Langmuir turbulence, since $\\eta =a/b\\le 5$ and $\\gamma _\\mathrm {init}=10$ for the spectra shown in Figure REF .", "For pure plasma oscillation with $\\eta >1$ , we have shown that the peak frequency is $\\gamma _\\mathrm {init}^2\\omega _\\mathrm {st}$ , and it consists of the higher harmonics of $\\bar{\\gamma }^2\\omega _\\mathrm {p}$ .", "Therefore, the peak frequency for the Langmuir turbulence with $a/b>1$ in Figure REF is interpreted as $\\gamma ^2\\omega _\\mathrm {st}$ , and it naturally explains the fact that the peak frequency is larger than $\\gamma ^2\\omega _\\mathrm {p}$ .", "Lastly we consider the spectral index for the Langmuir turbulence with $a>b>1$ .", "As we see above, the spectral index for $a>b>1$ is neither 1 predicted by perturbative DRL nor $1/2$ predicted by the angle diffusion effect.", "We regard that the spectral index is around $1/3$ , because the radiation mechanism is identified as Wiggler mechanism, and the angle integrated spectral index is $1/3$ in Wiggler mechanism.", "The energy change in PFT for the frequency lower than peak can be larger than 1 for $a>b>1$ .", "This is a different point from the Wiggler, and it can affect forming the spectral index.", "However, we choose modest value of $a/b$ for Langmuir turbulence in the present paper, so that this effect does not stand out.", "Summarizing above, we confirm that the peak frequency is $\\sim \\gamma ^2\\omega _\\mathrm {st}$ and the spectral index in low frequency side of the peak $\\sim 1/3$ for the Langmuir turbulence with $a>b>1$ and $a/b=O(1)$ .", "The mechanism which determine these radiation features is Wiggler mechanism, which is consistent with the statements by Getamantsev and Tokarev (1972).", "Lastly we show an example of the extreme case of $\\eta \\gg \\gamma _\\mathrm {init}$ .", "For $\\eta \\gg \\gamma _\\mathrm {init}$ , the motion becomes strongly nonlinear and cannot be treated analytically.", "Thus, we show numerically calculated electron orbit.", "We show the radiation spectra and the orbit for $\\eta =500>\\gamma _\\mathrm {init}$ (Figure REF (d), Figure REF ).", "As we expected, the peak frequency is $\\sim \\gamma _\\mathrm {init}^2\\omega _\\mathrm {st}$ , because the observer sees this electron mainly in the phase that the electric field and the electron velocity is nearly perpendicular.", "We note that the spectral index of $2/3$ is the same as the Wiggler radiation when the observers located in a particular direction, i.e., the spectrum is not the angle integrated spectrum.", "This spectral index is an collateral evidence for the spectral index of $1/3$ for the case of turbulence with $a>b>1$ for which angle integrated spectrum is calculated.", "The Lorentz factor changes from 10 to $O(100)$ , but $\\gamma _\\mathrm {init}$ is relevant to the observer situated at $z$ -axis.", "In other words, the beaming cone sweeps the observer when $\\gamma \\sim \\gamma _\\mathrm {init}$ in the present geometry.", "The Lorentz factor relevant to an observer oriented in different angle is significantly different to each other.", "Moreover, in general the energy change in the PFT of peak frequency becomes larger than $mc^2$ , because the electric force in some phase of oscillation accelerates the electron linearly, and the curvature radius becomes larger.", "We have to consider the linear acceleration emission in this case.", "Thus, we draw the line on $a/b=\\gamma $ and $a=1$ , and divide the $a>b>1$ region.", "We call the radiation for this parameter range as \"non-linear trajectory\" radiation.", "This part of the spectra from an electron moving in the 3D turbulent electric field is determined by the chaotic trajectory.", "The generalization of the features of this regime is a future work.", "Summarizing this section, we have considered the motion and radiation in a single mode plasma oscillation.", "We clarify that the cutoff frequency for $\\eta >1$ ($\\omega _\\mathrm {st}>\\omega _\\mathrm {p}$ ) is $\\gamma _\\mathrm {init}^2\\omega _\\mathrm {st}$ , which consists of higher harmonics of the fundamental frequency of $\\bar{\\gamma }^2\\omega _\\mathrm {p}$ .", "It is from the effect that the beaming cone sweeps the observer, in the same way as the Wiggler radiation.", "Using this result, we interpret that the peak frequency for 3D Langmuir turbulence for $a>b>1$ ($\\omega _\\mathrm {st}>\\omega _\\mathrm {p}>\\omega _0$ ) is $\\gamma ^2\\omega _\\mathrm {st}$ , where $\\gamma $ is determined by the acceleration.", "The shallower spectrum for $a>b>1$ can be explained by WRL mechanism.", "Lastly, we show numerically calculated radiation spectra for the extreme case of $\\eta \\gg \\gamma _\\mathrm {init}$ .", "It shows Wiggler like spectra in the middle and high frequency region, while the flattening can be seen in the low frequency region.", "It is from the effect of elongated trajectory to the electric field direction.", "The radiation signatures are summarized as a chart in the $a-b$ plane in Figure REF ." ], [ "DISSCUSSION & SUMMARY", "We have calculated the radiation spectra from relativistic electrons moving in a Langmuir turbulence by using first principle numerical calculation.", "We characterize the radiation spectra by two parameters.", "The one is $a=\\omega _\\mathrm {st}/\\omega _0$ , where $\\omega _\\mathrm {st}=e\\sigma /mc$ is the strength omega, and $\\omega _0=2\\pi c/\\lambda $ is the spatial omega.", "The strength omega accounts for the effect of the field strength to the radiation spectra, and the spatial omega accounts for the effect of spatial fluctuation with a typical scale of $\\lambda $ .", "The other is $b = \\omega _\\mathrm {p}/\\omega _0$ , where $\\omega _\\mathrm {p}$ is the plasma frequency, which accounts for effects of the time variability of the waves.", "We investigate the spectral signatures in the $a-b$ plane (Figure REF ).", "For $a<1$ and $b<1$ , the spectral features are the same as those of jitter radiation or Diffusive Synchrotron Radiation.", "For $b>a>1$ and $b>1>a$ , the theory of the Diffusive Radiation in Langmuir turbulence is confirmed, where time variability plays a primary role.", "For $a>b>1$ , the spectra show novel features which are not predicted by DRL theory.", "In this regime, the peak frequency is $\\sim \\gamma ^2 \\omega _\\mathrm {st}$ , which is higher than the predicted frequency $\\gamma ^2\\omega _\\mathrm {p}$ from the DRL theory.", "The spectral index of the frequency region lower than the peak is $\\sim 1/3$ .", "These features are explained by the Wiggler mechanism.", "To clarify the radiation features in this regime, we calculate the radiation spectra from an electron moving in an oscillating electric field, i.e., for vanishing spatial omega.", "We analytically calculate the motion of the electron, and numerically calculate the radiation spectra form this electron.", "We show that for $\\eta =a/b\\gtrsim 1$ , the spectrum around the peak frequency consists of the higher harmonics of the fundamental mode, by considering the radiation in the mean velocity frame.", "The electron motion becomes relativistic for $\\eta >1$ even in this frame, so that strong higher harmonics photons are emitted because of the retarded time effect.", "As a result, the spectra in the observer frame consists of the higher harmonics of $\\gamma ^2\\omega _{\\rm p}$ .", "The peak frequency is characterized by $\\gamma ^2\\omega _{\\rm st}$ , which is understood by the analogy of the Wiggler radiation.", "The feature that the radiation spectra from Langmuir turbulence have a wide range of spectral indices can be important for high energy astrophysical objects, in particular gamma ray bursts.", "The emission mechanism of GRB is not settled for now.", "The spectral indices of low frequency side of the Band function are distributed as a Gaussian with the central value of 0.", "Non-negligible number of GRBs have spectral index harder than the theoretical limit for synchrotron radiation $1/3$ .", "The photospheric emission model can overcome this difficulty, but it also has a difficulty that intrinsic photosheric emission is too hard with a low energy spectral index of 2 and it is difficult to make it softer.", "On the other hand, the radiation mechanism from Langmuir turbulence in this paper has some advantages.", "Not only the spectral index is harder than the synchrotron radiation and it can reproduce very hard spectra of observed GRB (Fleishman 2007b), but also it may explain a wide range of spectral indices.", "Because the parameters of $a$ and $b$ are likely to have a value around 1 near the shock front (Silva 2006, Dieckmann 2005), the radiation spectra change drastically around these parameters.", "We are grateful to the referee for his/her constructive and helpful comments and for informing us of the interesting paper by Getmantsev & Tokarev (1972).", "We also thank K. Toma and S. Kimura for useful discussions.", "This work is supported by the JSPS Research Fellowships for Young Scientists (Y.T., 24593)." ] ]	1403.0369
[ [ "A note on doubly warped product spaces" ], [ "Abstract We present a coordinate free approach to derive curvature formulas for pseudo-Riemannian doubly warped product manifolds in terms of curvatures of their submanifolds.", "We also state the geodesics equation." ], [ "Introduction", "Singly warped products were first introduced by R. L. Bishop and B. O'Neil [1] in their attempt to construct a class of Riemannian manifolds with negative curvature.", "O'Neil also studied Robertson Walker, Schwarschild and Kruskal space-times as warped products in [2].", "Later Beem, Ehrlich and Powell pointed out that many exact solutions to Einstein's field equation can be expressed in terms of Lorentzian warped products in [3].", "Since then warped product spaces play a crucial role in a plethora of physical applications such as general relativity, string and supergravity theories [4].", "In the present work, for pseudo-Riemannian doubly warped product manifolds, we prove expressions that relate the Riemann, Ricci and scalar curvatures with those of their submanifolds.", "The derivation is coordinate independent and we also give the equation of geodesics.", "Finally, in the Appendix, we write the components of the curvatures in a local coordinate system." ], [ "Preliminaries", "In this section adopting the notations of [2] we recall briefly basic notions of product Riemannian manifolds and give the definition of the doubly warped product space.", "Let $(B,g_B)$ and $(F,g_F)$ be m and n-dimensional pseudo-Riemannian manifolds respectively.", "Then the $M=B\\times F$ is an $(m+n)$ -dimensional pseudo-Riemannian manifold with $\\pi :\\, M\\rightarrow B$ and $\\sigma : \\, M\\rightarrow F$ the usual smooth projection maps.", "We use the natural product coordinate systems on the product manifold $M$ .", "If $(p_0,q_0)\\in M$ and $(U_B,x)$ , $(U_F,y)$ are coordinate charts on $B$ and $F$ such that $p_0\\in B$ and $q_0\\in F$ , then we can define a coordinate chart $(U_M,z)$ on $M$ such that $U_M$ is an open subset in $M$ contained in $U_B\\times U_F$ , $(p_0,q_0)\\in U_M$ and $\\forall (p,q)\\in U_M$ , $z(p,q)=(x(p),y(q))$ where $x(p)=(x^1(p),\\cdots ,x^m(p))$ and $y(q)=(y^{m+1}(q), \\cdots , y^{m+n}(q))$ .", "The set of all smooth and positive valued functions $f: \\, B\\rightarrow R^{+}$ is denoted by $\\mathcal {F}(B)=C^{\\infty }(B)$ .", "The lift of $f$ to $M$ is defined by $\\tilde{f}=f\\circ \\pi \\in \\mathcal {F}(M)$ .", "If $x_p\\in T_p(B)$ and $q\\in F$ then the lift $\\tilde{x}_{(p,q)}$ of $x_p$ to $M$ is the unique tangent vector in $T_{(p,q)}(B\\times \\lbrace q\\rbrace )$ such that $d\\pi _{(p,q)}(\\tilde{x}_{(p,q)})=x_p$ and $d\\sigma _{(p,q)}(\\tilde{x}_{(p,q)})=0$ .", "The set of all such horizontal tangent vector lifts will be denoted by $L_{(p,q)}(B)$ .", "Moreover we can define lifts of vector fields.", "Let $X\\in \\Hr (B)$ where $\\Hr (B)$ is the set of smooth vector fields, then the lift $\\tilde{X}$ of $X$ to $M$ is the unique element of $\\Hr (M)$ whose value at each $(p,q)$ is the lift of $X_p$ to $(p,q)$ .", "The set of such lifts will be denoted by $\\mathcal {L}_{(p,q)}(B)$ .", "Definition 2.1 Let $(B,g_B)$ and $(F,g_F)$ be pseudo-Riemannian manifolds and $f:\\, B\\rightarrow \\mathbb {R}^+$ , $h:\\, F\\rightarrow \\mathbb {R}^+$ be smooth functions.", "The doubly warped product space is the product manifold furnished with the metric tensor defined by $g_M=(h\\circ \\sigma )^2 \\pi ^(g_B)+(f\\circ \\pi )^2 \\sigma ^(g_F).$ Explicitly if $v$ is tangent to $B\\times F$ at $(p,q)$ then $<v,v>=h^2(q)<d\\pi (v),d\\pi (v)>+f^2(p)<d\\sigma (v),d\\sigma (v)>.$ We will denote this structure by $M=B\\, _h\\!\\times _f F$ .", "The warped metric (REF ) is characterized by: $(\\alpha )$ For each $q\\in F$ the map $\\pi \\lceil (B\\times \\lbrace q\\rbrace )$ is a positive homothety onto $F$ with scale factor $1/h(q)$ .", "$(\\beta )$ For each $p\\in B$ the map $\\sigma \\lceil (\\lbrace p\\rbrace \\times F)$ is a positive homothety onto $B$ with scale factor $1/f(p)$ .", "$(\\gamma )$ For each $(p,q)\\in M$ , the leaf $B\\times \\lbrace q\\rbrace $ and the fiber $\\lbrace p\\rbrace \\times F$ are orthogonal at $(p,q)$ .", "If $h=1$ and $f\\ne 1$ then we obtain a singly warped product manifold.", "If $h=f=1$ then we have a product manifold." ], [ "Covariant derivatives", "As in the case of a pseudo-Riemannian product manifold we can define the orthogonal projections: $\\textrm {tan}: \\, T_{(p,q)}(M)\\rightarrow T_{(p,q)}(\\lbrace p\\rbrace \\times F), \\nonumber \\\\\\textrm {nor}: \\, T_{(p,q)}(M)\\rightarrow T_{(p,q)}(B\\times \\lbrace q\\rbrace )$ and thus vectors tangent to fibers $\\lbrace p\\rbrace \\times F=\\pi ^{-1}(p)$ are vertical while vectors tangent to leaves $B\\times \\lbrace q\\rbrace =\\sigma ^{-1}(q)$ are horizontal.", "The lifts of functions and vector fields will be denoted without the tilde for simplicity.", "Also all geometrical quantities with a superscript $B$ (or $F$ ) are the pullbacks by $\\pi $ (or $\\sigma $ ) of the corresponding ones on $B$ (or $F$ ).", "Proposition 3.1 On $M=B\\, _h\\!\\times _f F$ , if $X,Y \\in \\mathcal {L}(B)$ and $V,W \\in \\mathcal {L}(F)$ then $(1)$ $ \\textrm {nor} \\, {}^M \\!", "D_X Y= {}^B \\!", "D_X Y, \\quad \\textrm {nor} \\,{}^M \\!", "D_X Y\\in \\mathcal {L}(B) $ where $\\textrm {nor} \\,{}^M \\!", "D_X Y$ is the lift of the Levi-Civita connection ${}^B \\!", "D_X Y$ to $B$ .", "$(2)$ $ \\textrm {tan} \\, {}^M \\!", "D_X Y= II(X,Y)=-\\frac{<X,Y>}{h} \\textrm {grad} \\, h $ where the first fundamental form is defined by $g_M(X,Y)=<X,Y>$ and $II(X,Y)$ is the shape tensor (or second fundamental form) of $B$ to $M$ In this relation $h$ stands for $\\tilde{h}=h\\circ \\sigma $ and $grad\\, h$ for $grad\\, \\tilde{h}=\\widetilde{grad\\, h}$ .. $(3)$ $ {}^M D_X V={}^M D_V X, \\, \\, nor {}^M D_X V=\\frac{V h}{h} X, \\, \\, tan {}^M D_V X=\\frac{X f}{f} V $ $(4)$ $ \\textrm {nor} \\, {}^M \\!", "D_V W= II(V,W)=-\\frac{<V,W>}{f} \\textrm {grad} f $ where $II(V,W)$ is the shape tensor of $F$ .", "$(5)$ $ \\textrm {tan} \\, {}^M \\!", "D_V W= {}^F \\!", "D_V W, \\quad \\textrm {tan} \\, {}^M \\!", "D_V W\\in \\mathcal {L}(F).", "$ Proof 3.2 $(1)$ The vector fields $X, Y$ are tangent to the leaves.", "On a leaf, $\\textrm {nor} \\, {}^M \\!", "D_X Y$ is the leaf covariant derivative applied to the restrictions of $X$ and $Y$ to that leaf.", "Then $\\pi $ -relatedness follows since homotheties preserve Levi-Civita connections.", "$(2)$ The Koszul formula The Koszul formula for a pseudo-Riemannian manifold $M$ reads $ 2<D_X Y,Z>=X<Y,Z>-Z<X,Y>+Y<Z,X>-<X,[Y,Z]>+<Z,[X,Y]>+<Y,[Z,X]>$ for $X,Y,Z \\in \\Hr (M)$ .", "reduces to $2<{}^M D_X Y,V>=-V<X,Y>$ since $<X,V>=<Y,V>=0$ , $[X,V]=[Y,V]=0$ and $<V,[X,Y]>=0$ .", "Also $V<X,Y>=V[h^2(<X,Y>\\circ \\pi )]=2\\left(\\frac{Vh}{h}\\right)<X,Y>=2<\\frac{<X,Y> grad \\, h}{h},V>.$ From (REF ) and (REF ) we end up with the desired result.", "$(3)$ Since $[X,V]=0$ we have ${}^M D_X V={}^M D_V X$ .", "Using the Koszul formula twice for $<{}^M D_XV,Y>$ and $<{}^M D_X V,U>$ we obtain $2<{}^M D_XV,Y>&=& 2<\\textrm {nor}\\,{}^MD_X V,Y>=V<X,Y>=2<\\frac{V h}{h}X, Y> \\nonumber \\\\2<{}^M D_X V,W>&=& 2<\\textrm {tan}\\,{}^MD_X V,W>=X<V,W>=2<\\frac{Xf}{f}V,W>.$ $(4)$ The proof is identical to (2) for $V,W \\in \\mathcal {L}(F)$ .", "$(5)$ This proof is similar to (1) for vector fields tangent to fibers." ], [ "Riemann, Ricci and scalar curvatures", "The function $R: \\, \\Hr ^3(M)\\rightarrow \\Hr (M)$ given by $R_{XY}Z=D_{[X,Y]}Z-[D_X,D_Y]Z$ is a $(1,3)$ tensor field on $M$ the so called Riemann curvature of $M$ .", "Sometimes we use the symbols $R(X,Y)Z$ or $R(X,Y,Z)$ instead of $R_{XY}Z$ .", "Proposition 4.1 On $M=B\\, _h\\!\\times _f F$ , if $X,Y,Z \\in \\mathcal {L}(B)$ and $U,V,W \\in \\mathcal {L}(F)$ then $(1)$ $ {}^M R_{XY}Z={}^B R_{XY}Z- \\frac{\\parallel grad\\, h\\parallel ^2}{h^2}\\left(<X,Z>Y-<Y,Z>X\\right)$ where ${}^M R_{XY}Z$ is the lift of the Riemann curvature tensor ${}^B R_{XY}Z$ to $M$ .", "$(2)$ $ {}^M R_{VX}Y=\\frac{H^f(X,Y)}{f}V+\\frac{<X,Y>}{h}{}^FD_V (grad\\, h)={}^M R_{VY}X $ where $H^f$ is the Hessian of $f$ The Hessian $H^f$ of $f\\in \\mathcal {F}(M)$ is the symmetric $(0,2)$ tensor field such that $ H^f(X,Y)=XYf-(D_XY)f=<D_X(grad \\,f),Y>.$ .", "$(3)$ $ {}^M R_{XY}V=\\frac{Vh}{h}\\left[\\left(\\frac{Xf}{f}\\right)Y-\\left(\\frac{Yf}{f}\\right)X\\right], $ $ {}^M R_{VW}X=\\frac{Xf}{f} \\left[\\left(\\frac{Vh}{h}\\right)W-\\left(\\frac{Wh}{h}\\right)V\\right]$ $(4)$ $ {}^M R_{XV}W={}^M R_{XW}V=\\frac{H^h(V,W)}{h}X+\\frac{<V,W>}{f} {}^BD_X (grad\\, f) $ where $H^h$ is the Hessian of $h$ .", "$(5)$ $ {}^M R_{VW}U={}^F R_{VW} U-\\frac{\\parallel grad\\, f\\parallel ^2}{f^2}\\left( <V,U>W-<W,U>V\\right).", "$ Proof 4.2 $(1)$ For the vector fields $X,Y,Z,A\\in \\mathcal {L}(B)$ we have $&& <{}^M R_{XY}Z,A>= <{}^B R_{XY}Z,A>-<II(X,Z),II(Y,A)>+<II(X,A),II(Y,Z)> \\nonumber \\\\&=& <{}^B R_{XY}Z,A>-\\frac{\\parallel grad \\, h\\parallel ^2>}{h^2}\\left(<<X,Z>Y-<Y,Z>X,A>\\right)$ $(2)$ Since $[X,V]=0$ from definition (REF ) we find ${}^M R_{VX}Y=-{}^M D_V({}^M D_X Y)+{}^M D_X({}^M D_VY).$ The first term on the right hand side of (REF ), with the help of Proposition (REF ), gives $&& {}^MD_V({}^MD_X Y)= {}^MD_V(nor \\, {}^MD_XY+tan \\, {}^MD_XY)\\nonumber \\\\&=& \\frac{Vh}{h}{}^B D_XY+\\frac{({}^BD_XY)f}{f}V-\\frac{Vh}{h^2}<X,Y> grad\\, h -\\frac{<X,Y>}{h}{}^FD_V(grad \\, h)$ while the second term becomes ${}^MD_X({}^MD_VY)=\\frac{Vh}{h}\\left({}^BD_XY-\\frac{<X,Y>}{h}grad\\, h\\right)+\\frac{XYf}{f}V.$ Substituting (REF ) and (REF ) into (REF ) we recover the desired relation.", "$(3)$ Assuming that $[X,Y]=0$ (as, for example, for coordinate vector fields) we first prove that ${}^MD_X(D_YV)=\\frac{XYf}{f}V+\\left(\\frac{Yf}{f}\\right)\\left(\\frac{Vh}{h}\\right)X+\\frac{Vh}{h}{}^BD_YX$ and a similar expression for ${}^MD_Y(D_XV)$ with $X$ and $Y$ interchanged.", "The claim is then straightforward by (REF ).", "Also note that $<{}^MR_{XY}V,W>=<{}^MR_{VW}X,Y>=0$ .", "$(4)$ The proof is established following steps similar to those in (2).", "$(5)$ This relation is justified as in (1).", "The Ricci tensor relative to a frame field A frame field is a set $\\lbrace E_i\\rbrace , \\, i=1,\\cdots , dimM$ of orthonormal vector fields.", "is defined by $Ric(X,Y)=\\sum _{i}\\epsilon _i <R_{XE_i}Y,E_i>$ where $\\epsilon _i=<E_i,E_i>$ .", "Corollary 4.3 On $M=B\\, _h\\!\\times _f F$ , if $X,Y \\in \\mathcal {L}(B)$ , $V,W \\in \\mathcal {L}(F)$ , $m=dim B>1$ and $n=dim F>1$ then $(1)$ $ {}^MRic(X,Y)={}^B Ric(X,Y)-<X,Y>\\left[\\frac{{}^F\\Delta h}{h}+(m-1)\\frac{\\parallel grad\\,h\\parallel ^2}{h^2}\\right]-\\frac{n}{f}H^f(X,Y)$ $(2)$ $ {}^MRic(X,V)=(m+n-2)\\left(\\frac{Xf}{f}\\right)\\left(\\frac{Vh}{h}\\right) $ $(3)$ $ {}^MRic(V,W)={}^F Ric(V,W)-<V,W>\\left[\\frac{{}^B\\Delta f}{f}+(n-1)\\frac{\\parallel grad\\,f\\parallel ^2}{f^2}\\right]-\\frac{m}{h}H^h(V,W)$ Proof 4.4 Let $\\lbrace {}^BE^{i}\\rbrace , \\, i=1,\\cdots ,dim B=m$ be a frame field on an open set $U_B\\subseteq B$ and $\\lbrace {}^FE^{i}\\rbrace , \\, i=m+1,\\cdots ,m+dim F=m+n$ be a frame field on an open set $U_F\\subseteq F$ , then $\\lbrace {}^ME^{i}\\rbrace , \\, i=1,\\cdots ,dim M=m+n$ be a frame field on the open set $U_M\\subseteq U_B\\times U_F\\subseteq B\\times F$ .", "$(1)$ Using definition (REF ) we have $&& {}^MRic(X,Y)= \\sum _{i=1}^{m+n}{}^M\\epsilon _{i}<{}^MR_{X{}^ME_i}Y,{}^ME_i> \\nonumber \\\\&=&\\sum _{i=1}^{m}{}^B\\epsilon _{i}<{}^MR_{X{}^BE_i}Y,{}^BE_i>+\\sum _{i=m+1}^{m+n}{}^F\\epsilon _{i}<{}^MR_{X{}^FE_i}Y,{}^FE_i> \\nonumber \\\\&=& \\sum _{i=1}^{m}{}^B\\epsilon _{i}\\left[<{}^B R_{X{}^BE_i}Y,{}^BE_i>-\\frac{\\parallel grad \\, h\\parallel ^2}{h^2}\\left(<<X,Y>{}^BE_i-<{}^BE_i,Y>X,{}^BE_i>\\right)\\right]\\nonumber \\\\&-& \\sum _{i=m+1}^{m+n}{}^F\\epsilon _{i}\\left[\\frac{H^f(X,Y)}{f}<V,{}^FE_i>+\\frac{<X,Y>}{h}<{}^FD_{{}^FE_i} (grad\\, h),{}^FE_i>\\right]\\nonumber \\\\&=& {}^B Ric(X,Y)-\\frac{\\parallel grad \\, h\\parallel ^2}{h^2}<X,Y>\\left(\\sum _{i=1}^{m}{}^B\\epsilon _i^2-1\\right) \\nonumber \\\\&-& \\frac{H^f(X,Y)}{f}\\sum _{i=m+1}^{m+n}{}^F\\epsilon _i<{}^FE_i,{}^FE_i>-\\frac{<X,Y>}{h}\\sum _{i=m+1}^{m+n}{}^F\\epsilon _i <{}^FD_{{}^FE_i}(grad\\,h),{}^FE_i>$ from which taking into account that $\\sum _{i=m+1}^{m+n}{}^F\\epsilon _i <{}^FD_{E_i}(grad\\,h),{}^FE_i>={}^F\\Delta h$ we recover the known result.", "$(2)$ Using the definition of Ricci tensor we have $&& {}^MRic(X,V)=\\sum _{i=1}^{m}{}^B\\epsilon _i<{}^BR_{X{}^BE_i}V,{}^BE_i>+\\sum _{i=m+1}^{m+n}{}^F\\epsilon _i<{}^FR_{V{}^FE_i}X,{}^FE_i> \\nonumber \\\\&=& \\left(\\frac{Vh}{h}\\right)\\left[\\left(\\frac{Xf}{f}\\right)\\sum _{i=1}^{m}{}^B\\epsilon _i^2-\\frac{1}{f}\\left(\\sum _{i=1}^{m}{}^B\\epsilon <X,{}^BE_i>{}^BE_i\\right)f\\right]\\nonumber \\\\&+&\\left(\\frac{Xf}{f}\\right)\\left[\\left(\\frac{Vh}{h}\\right)\\sum _{i=m+1}^{m+n}{}^F\\epsilon _i^2-\\frac{1}{h}\\left(\\sum _{i=m+1}^{m+n}{}^F\\epsilon <V,{}^FE_i>{}^FE_i\\right)h\\right]\\nonumber \\\\&=& (m+n-2)\\left(\\frac{Xf}{f}\\right)\\left(\\frac{Vh}{h}\\right).$ $(3)$ The proof is similar to (1).", "The scalar curvature $R$ of $M$ is the contraction of its Ricci tensor which relative to a frame field yields $R=\\sum _{i\\ne j} K(E_i,E_j)=2\\sum _{i<j}K(E_i,E_j), \\quad \\textrm {where} \\quad K(X,Y)=\\frac{<R_{XY}U,V>}{<X,X><Y,Y>-<X,Y>^2}$ is the sectional curvature.", "Corollary 4.5 On $M=B\\, _h\\!\\times _f F$ , if $X,Y \\in \\mathcal {L}(B)$ , $V,W \\in \\mathcal {L}(F)$ , $m=dim B>1$ and $n=dim F>1$ then ${}^M R &=& \\frac{{}^B R}{h^2}-2m\\frac{{}^F\\Delta h}{h}-m(m-1)\\frac{\\parallel grad\\,h\\parallel ^2}{h^2}\\nonumber \\\\&+&\\frac{{}^F R}{f^2} - 2n\\frac{{}^B\\Delta f}{f}-n(n-1)\\frac{\\parallel grad\\,f\\parallel ^2}{f^2}$ If we change the sign in the definition of Riemann curvature then its components change sign but the Ricci tensor and scalar remain intact." ], [ "Geodesics", "A curve $\\gamma : \\, I \\rightarrow M$ can be written as $\\gamma (s)=(\\alpha (s),\\beta (s))$ with $\\alpha $ , $\\beta $ the projections of $\\gamma $ into $B$ and $F$ .", "Proposition 5.1 A curve $\\gamma =(\\alpha ,\\beta )$ in $M$ is geodesic iff $(1)$ $ \\beta ^{^{\\prime \\prime }}=<\\alpha ^{^{\\prime }},\\alpha ^{^{\\prime }}>h\\circ \\beta grad\\, h-\\frac{2}{(f\\circ \\alpha )}\\frac{d}{ds}(f\\circ \\alpha ) \\, \\beta ^{^{\\prime }}.", "$ $(2)$ $ \\alpha ^{^{\\prime \\prime }}=<\\beta ^{^{\\prime }},\\beta ^{^{\\prime }}>f\\circ \\alpha grad\\, f-\\frac{2}{h\\circ \\beta }\\frac{d}{ds}((h\\circ \\beta )) \\, \\alpha ^{^{\\prime }}.", "$ Proof 5.2 $(1)$ Consider the case when $\\gamma ^{^{\\prime }}(0)$ is neither horizontal nor vertical.", "Then $\\alpha $ , $\\beta $ are regular, namely $\\forall s\\in I, \\, \\alpha ^{^{\\prime }}(s), \\beta ^{^{\\prime }}(s)\\ne 0$ , and thus locally integral curves.", "This means that $\\alpha ^{^{\\prime }}(s)=X_{\\alpha (s)}$ and $\\beta ^{^{\\prime }}(s)=V_{\\beta (s)}$ for $X\\in \\mathcal {L}(B)$ and $V\\in \\mathcal {L}(F)$ .", "Moreover $\\gamma $ is an integral curve and $\\gamma ^{^{\\prime \\prime }}=D_{X+V}(X+V)=D_XX+D_XV+D_VX+D_VV.$ The curve $\\gamma $ is geodesic iff $\\gamma ^{^{\\prime \\prime }}=0$ or equivalently iff $\\tan \\, \\gamma ^{^{\\prime \\prime }}=\\rm {nor} \\, \\gamma ^{^{\\prime \\prime }}=0$ .", "Using Proposition (3.1) in (REF ) we obtain $\\textrm {tan}\\left(D_XX+D_XV+D_VX+D_VV \\right)&=& 0 \\quad \\Rightarrow \\nonumber \\\\-\\frac{<X,X>}{h}\\textrm {grad}\\, h+2\\frac{Xf}{f}V+{}^FD_VV&=& 0$ which proves (1).", "The second identity is reproduced by taking the $\\textrm {nor}$ component of (REF ).", "$(2)$ If $\\gamma ^{^{\\prime }}(0)$ is horizontal or vertical and nonzero then the geodesic $\\gamma $ does not remain in the leaves $\\sigma ^{-1}(q)$ or the fibers $\\pi ^{-1}(p)$ .", "Hence there is a sequence $\\lbrace s_i\\rbrace \\rightarrow 0$ such that $\\forall i$ , $\\gamma ^{^{\\prime }}(s_i)$ is neither horizontal nor vertical.", "Then $(1)$ and $(2)$ follow by continuity of Case 1. tocsubsectionAppendix" ], [ "Appendix", "Let $\\lbrace x^{\\mu }\\rbrace , \\, \\mu =1,\\cdots ,dim B=m$ be a local coordinate system on an open set $U_B\\subseteq B$ and $\\lbrace y^{\\alpha }\\rbrace , \\, \\alpha =m+1,\\cdots ,m+dim F=m+n$ be a local coordinate system on an open set $U_F\\subseteq F$ , then $\\lbrace x^{\\hat{\\mu }}\\rbrace , \\, \\hat{\\mu }=1,\\cdots ,dim M=m+n$ is a local coordinate system on the open set $U_M\\subseteq U_B\\times U_F\\subseteq B\\times F$ .", "The components of the Levi-Civita connections are $\\textrm {nor} \\, {}^M \\!", "D_{\\partial _{\\mu }} (\\partial _{\\nu })&=& {}^B \\!", "D_{\\partial _{\\mu }} (\\partial _{\\nu })={}^B\\Gamma ^{\\rho }_{\\mu \\nu } \\partial _{\\rho }, \\\\\\textrm {tan} \\, {}^M \\!", "D_{\\partial _{\\mu }} (\\partial _{\\nu })&=&-\\frac{<\\partial _{\\mu },\\partial _{\\nu }>}{h} grad \\, h=-\\frac{h}{f^2}g_{\\mu \\nu }g^{\\alpha \\beta } \\partial _{\\beta }h \\, \\partial _{\\alpha }=\\Gamma ^{\\alpha }_{\\mu \\nu }\\partial _{\\alpha }, \\\\\\textrm {nor} {}^MD_{\\partial _{\\mu }}( \\partial _{\\alpha })&=&\\frac{\\partial _{\\alpha }h}{h}\\delta ^{\\nu }_{\\mu }\\partial _{\\nu }=\\Gamma ^{\\nu }_{\\mu \\alpha } \\partial _{\\nu }, \\,\\, \\textrm {tan} {}^M D_{\\partial _{\\alpha }} (\\partial _{\\mu })=\\frac{\\partial _{\\mu }f}{f}\\delta ^{\\beta }_{\\alpha } \\partial _{\\beta }=\\Gamma ^{\\beta }_{\\alpha \\mu } \\partial _{\\beta }, \\\\\\textrm {nor} \\, {}^M \\!", "D_{\\partial _{\\alpha }} (\\partial _{\\beta })&=&-\\frac{f}{h^2}g_{\\alpha \\beta }g^{\\mu \\nu } \\partial _{\\nu }f \\, \\partial _{\\mu }=\\Gamma ^{\\mu }_{\\alpha \\beta } \\partial _{\\mu },\\\\\\textrm {tan} \\, {}^M \\!", "D_{\\partial _{\\alpha }} (\\partial _{\\beta })&=& {}^F\\Gamma ^{\\gamma }_{\\alpha \\beta } \\partial _{\\gamma }.$ For coordinate vector fields, $R_{\\partial _{\\hat{\\lambda }}\\partial _{\\hat{\\rho }}}(\\partial _{\\hat{\\nu }})=R^{\\hat{\\mu }}_{\\,\\,\\,\\hat{\\nu } \\hat{\\lambda } \\hat{\\rho }}\\partial _{\\hat{\\mu }}$ and the components of the Riemann curvature are given by ${}^MR^{\\mu }_{\\,\\,\\,\\nu \\lambda \\rho }&=&{}^BR^{\\mu }_{\\,\\,\\,\\nu \\lambda \\rho }-\\frac{\\parallel \\partial _{\\alpha }h\\parallel ^2}{f^2}\\left(\\delta ^{\\mu }_{\\lambda }g_{\\nu \\rho }-\\delta ^{\\mu }_{\\rho }g_{\\nu \\lambda }\\right), \\\\{}^MR^{\\alpha }_{\\,\\,\\, \\mu \\beta \\nu }&=& -\\frac{\\delta ^{\\alpha }_{\\beta }}{f}D_{\\nu }(\\partial _{\\mu }f) -\\frac{h}{f^2}g_{\\mu \\nu }D_{\\beta }(\\partial ^{\\alpha }h), \\\\{}^MR^{\\alpha }_{\\,\\,\\,\\mu \\beta \\gamma }&=& \\partial _{\\mu }(\\ln f)\\left[\\delta ^{\\alpha }_{\\beta }\\partial _{\\gamma }(\\ln h)-\\delta ^{\\alpha }_{\\gamma }\\partial _{\\beta }(\\ln h)\\right],\\\\{}^MR^{\\mu }_{\\,\\,\\,\\alpha \\nu \\lambda }&=& \\partial _{\\alpha }(\\ln h)\\left[\\delta ^{\\mu }_{\\nu }\\partial _{\\lambda }(\\ln f)-\\delta ^{\\mu }_{\\lambda }\\partial _{\\nu }(\\ln f)\\right],\\\\{}^MR^{\\alpha }_{\\,\\,\\,\\mu \\nu \\lambda }&=&\\frac{1}{2f^2}\\partial ^{\\alpha }(h^2)\\left[g_{\\mu \\lambda }\\partial _{\\nu }(\\ln f)-g_{\\mu \\nu }\\partial _{\\lambda }(\\ln f)\\right], \\\\{}^MR^{\\mu }_{\\,\\,\\,\\alpha \\beta \\gamma }&=&\\frac{1}{2h^2}\\partial ^{\\mu }(f^2)\\left[g_{\\alpha \\gamma }\\partial _{\\beta }(\\ln h)-g_{\\alpha \\beta }\\partial _{\\gamma }(\\ln h)\\right],\\\\{}^MR^{\\mu }_{\\,\\,\\, \\alpha \\nu \\beta }&=& -\\frac{\\delta ^{\\mu }_{\\nu }}{h}D_{\\beta }(\\partial ^{\\alpha }h) -\\frac{f}{h^2}g_{\\alpha \\beta }D_{\\nu }(\\partial ^{\\mu }f), \\\\{}^MR^{\\alpha }_{\\,\\,\\,\\beta \\gamma \\epsilon }&=&{}^FR^{\\alpha }_{\\,\\,\\,\\beta \\gamma \\epsilon }-\\frac{\\parallel \\partial _{\\mu }f\\parallel ^2}{h^2}\\left(\\delta ^{\\alpha }_{\\gamma }g_{\\beta \\epsilon }-\\delta ^{\\alpha }_{\\epsilon }g_{\\beta \\gamma }\\right).$ The components of the Ricci tensor are ${}^MR_{\\mu \\nu }&=& {}^BR_{\\mu \\nu }-\\frac{g_{\\mu \\nu }}{f^2}\\left[h {}^F\\Delta h+(m-1)\\parallel \\partial _{\\alpha } h\\parallel ^2\\right]-\\frac{n}{f}D_{\\mu }(\\partial _{\\nu }f),\\\\{}^MR_{\\mu \\alpha }&=&(m+n-2)\\partial _{\\mu }(\\ln f)\\partial _{\\alpha }(\\ln h),\\\\{}^MR_{\\alpha \\beta }&=& {}^BR_{\\alpha \\beta }-\\frac{g_{\\alpha \\beta }}{h^2}\\left[f{}^B\\Delta f+(n-1)\\parallel \\partial _{\\mu } f\\parallel ^2\\right]-\\frac{m}{h}D_{\\alpha }(\\partial _{\\beta }h)$ and the Ricci scalar is given by ${}^M R &=& \\frac{{}^B R}{h^2}-2m\\frac{{}^F\\Delta h}{hf^2}-m(m-1)\\frac{\\parallel \\partial _{\\alpha }(\\ln h)\\parallel ^2}{f^2}\\nonumber \\\\&+&\\frac{{}^F R}{f^2} - 2n\\frac{{}^B\\Delta f}{fh^2}-n(n-1)\\frac{\\parallel \\partial _{\\alpha }(\\ln f)\\parallel ^2}{h^2}.$ All the expressions of the Appendix have also been recorded in [5] and [6]." ] ]	1403.0204
[ [ "Regular Polygonal Complexes of Higher Ranks in E^3" ], [ "Abstract The paper establishes that the rank of a regular polygonal complex in 3-space E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in 3-space are the eight regular 4-apeirotopes." ], [ "Introduction", "A geometric polygonal complex $\\mathcal {K}$ of rank $n$ in Euclidean 3-space $\\mathbb {E}^3$ is a discrete incidence structure made up from finite or infinite, planar or skew polygons, assembled in a careful fashion into families of geometric polygonal complexes of smaller ranks.", "As combinatorial objects they are incidence complexes of rank $n$ with polygons as 2-faces, that is, abstract polygonal complexes of rank $n$ (see [3], [13]).", "A geometric polygonal complex $\\mathcal {K}$ is regular if $\\mathcal {K}$ has a flag-transitive geometric symmetry group.", "The regular polygonal complexes of rank 3 in $\\mathbb {E}^3$ are completely known.", "They comprise fourty-eight regular polyhedra as well as twenty-five regular complexes which are not polyhedra.", "The regular polyhedra were discovered by Grünbaum [6] and Dress [4], [5], and are described in detail in McMullen & Schulte [10] and McMullen [8].", "The regular complexes which are not polyhedra were recently classified in Pellicer & Schulte [11], [12].", "The present paper proves that a regular polygonal complex in $\\mathbb {E}^3$ cannot have rank larger than 4, and that the only regular polygonal complexes of rank 4 in $\\mathbb {E}^3$ are the eight regular 4-apeirotopes in $\\mathbb {E}^3$ described in McMullen & Schulte [10]." ], [ "Incidence complexes", "Following [3], [13], an incidence complex of rank $n$ , or simply an $n$ -complex, is a partially ordered set $\\mathcal {K}$ with a strictly monotone rank function with range $\\lbrace -1,0,\\ldots ,n\\rbrace $ satisfying the following conditions.", "The elements of rank $j$ are called the $j$ -faces of $\\mathcal {K}$ , or vertices, edges and facets of $\\mathcal {K}$ if $j=0$ , 1 or $n-1$ , respectively.", "Two faces $F$ and $G$ are said to be incident if $F\\le G$ or $G\\le F$ .", "Each flag (maximal totally ordered subset) of $\\mathcal {K}$ is required to contain exactly $n+2$ faces, including a unique minimal face $F_{-1}$ (of rank $-1$ ) and a unique maximal face $F_{n}$ (of rank $n$ ) as improper faces.", "We often find it convenient to suppress the improper faces in designating flags.", "Two flags $\\Phi $ and $\\Psi $ of $\\mathcal {K}$ are called adjacent if they differ in exactly one face; if this face is an $j$ -face for some $j$ (with $0\\le j\\le n-1$ ), then $\\Phi $ and $\\Psi $ of $\\mathcal {K}$ are $j$ -adjacent.", "Further, we ask that $\\mathcal {K}$ be strongly flag-connected, meaning that any two flags $\\Phi $ and $\\Psi $ of $\\mathcal {P}$ can be joined by a sequence of flags $\\Phi =\\Phi _{0},\\Phi _{1},\\ldots ,\\Phi _{l-1},\\Phi _{l}=\\Psi $ , all containing $\\Phi \\cap \\Psi $ , such that $\\Phi _{i-1}$ and $\\Phi _{i}$ are adjacent (differ by exactly one face) for each $i$ .", "Finally, $\\mathcal {K}$ has the property that if $F$ and $G$ are incident faces of ranks $j-1$ and $j+1$ for some $j$ , then there are at least two $j$ -faces $H$ such that $F<H<G$ .", "With regards to this latter condition, note that the corresponding condition required in [3], [13] is more restrictive, in that, for each $j$ , a fixed number $k_j$ of $j$ -faces $H$ is required to lie between $F$ and $G$ .", "This stronger condition is always satisfied for highly symmetric complexes like those studied in this paper.", "When $F$ and $G$ are two faces of a complex $\\mathcal {K}$ with $F \\le G$ , we call $ G/F := \\lbrace H \\mid F \\le H \\le G\\rbrace $ a section of $\\mathcal {K}$ .", "We usually identify a face $F$ with the section $F/F_{-1}$ .", "For a face $F$ we also call $F_{n}/F$ the co-face of $\\mathcal {K}$ at $F$ , or the vertex-figure at $F$ if $F$ is a vertex.", "An incidence complex $\\mathcal {K}$ is said to be regular if its (combinatorial) automorphism group $\\Gamma (\\mathcal {K})$ is transitive on the flags.", "A complex $\\mathcal {K}$ is an (abstract) polytope if, for all $j$ and all $(j-1)$ -faces $F$ and $(j+1)$ -faces $G$ with $F<G$ , there are exactly two $j$ -faces $H$ such that $F<H<G$ .", "This class of complexes has attracted a lot of attention (see [10]).", "More generally, an incidence complex $\\mathcal {K}$ is called an abstract polygonal complex if each 2-face of $\\mathcal {K}$ is isomorphic to (the face lattice of) a convex polygon or an (infinite) apeirogon.", "For $0\\le k\\le n-1$ , the $k$ -skeleton of a complex $\\mathcal {K}$ of rank $n$ is the incidence complex of rank $k+1$ , whose faces comprise the $n$ -face of $\\mathcal {K}$ and all faces of $\\mathcal {K}$ of rank less than or equal to $k$ , with the partial order inherited from $\\mathcal {K}$ ." ], [ "Geometric polygonal complexes", "Our definition of a “realization” of an abstract polygonal complex is inspired by the corresponding notion for abstract polytopes (see [7], [10]).", "A (faithful) realization of an abstract polygonal complex $\\mathcal {K}$ , again denoted by $\\mathcal {K}$ , is derived inductively from a bijection $\\beta $ of the vertex-set of $\\mathcal {K}$ into some Euclidean space $\\mathbb {E}$ .", "The vertices of $\\mathcal {K}$ are mapped by $\\beta $ to the vertices of the realization.", "Then each 1-face of $\\mathcal {K}$ can be viewed as being mapped under $\\beta $ to a line segment, called an edge of the realization, joining the images of the vertices of the 1-face under $\\beta $ .", "Moving up in rank, each 2-face of $\\mathcal {K}$ , which by assumption is isomorphic to a convex polygon or an apeirogon, is mapped to the finite or infinite polygon in $\\mathbb {E}$ , a 2-face of the realization, made up from the edges of the realization that are the images under $\\beta $ of the 1-faces of the given 2-face of $\\mathcal {K}$ .", "More generally, from the $j$ -faces of the realization we then obtain each $(j+1)$ -face of the realization as a family of $j$ -faces of the realization, namely those corresponding under $\\beta $ to the $j$ -faces of the underlying $(j+1)$ -face of $\\mathcal {K}$ .", "Finally, then, when $j=n-1$ we arrive at the desired realization of $\\mathcal {K}$ determined by $\\beta $ .", "We also call the resulting structure a geometric polygonal complex.", "Clearly, in order for this process to produce a faithful geometric copy of the given abstract polygonal complex $\\mathcal {K}$ we must assume that, for each $j$ , each $(j+1)$ -face of $\\mathcal {K}$ is uniquely determined by its $j$ -faces.", "Throughout we will be working under this assumption.", "Note, however, that an abstract complex may not satisfy this assumption; the regular map $\\lbrace 3,6\\rbrace _{(2,0)}$ on the 2-torus is an example.", "Alternatively we can proceed more directly and define geometric polygonal complexes with less explicit reference to realizations of abstract polygonal complexes, at least for small ranks, beginning with rank 2.", "Here we restrict ourselves to Euclidean 3-space $\\mathbb {E}^3$ , although similar notions carry over to realizations in higher-dimensional Euclidean spaces.", "Following [6], a finite polygon, or an $n$ -gon, in $\\mathbb {E}^3$ consists of a sequence $(v_1, v_2, \\dots , v_n)$ of $n$ distinct points, as well as of the line segments $(v_1, v_2), (v_2,v_3), \\ldots , (v_{n-1},v_n),(v_n, v_1)$ .", "A (discrete) infinite polygon, or apeirogon, in $\\mathbb {E}^3$ similarly consists of an infinite sequence of distinct points $(\\dots , v_{-2},v_{-1}, v_0, v_1, v_2, \\dots )$ , as well as of the line segments $(v_i, v_{i+1})$ for each $i$ , such that each compact subset of $\\mathbb {E}^3$ meets only finitely many line segments.", "In either case the points are the vertices and the line segments the edges of the polygon.", "Then following [11], a geometric polygonal complex $\\mathcal {K}$ of rank 3, or simply a geometric polygonal 3-complex, in $\\mathbb {E}^{3}$ is a triple $(\\mathcal {V},\\mathcal {E},\\mathcal {F}$ ) consisting of a set $\\mathcal {V}$ of points, called vertices, a set $\\mathcal {E}$ of line segments, called edges, and a set $\\mathcal {F}$ of (finite or infinite) polygons, called faces (here to mean 2-faces), satisfying the following properties.", "(a) The graph $(\\mathcal {V},\\mathcal {E})$ , the edge graph or net of $\\mathcal {K}$ , is connected.", "(b) The vertex-figure of $\\mathcal {K}$ at each vertex of $\\mathcal {K}$ is connected.", "By the vertex-figure of $\\mathcal {K}$ at a vertex $v$ we mean the graph, possibly with multiple edges, whose vertices are the vertices of $\\mathcal {K}$ adjacent to $v$ and whose edges are the line segments $(u,w)$ , where $(u, v)$ and $(v, w)$ are edges of a common face of $\\mathcal {K}$ .", "(There may be more than one such face in $\\mathcal {K}$ , in which case the corresponding edge $(u,w)$ of the vertex-figure at $v$ has multiplicity given by the number of such faces.)", "(c) Each edge of $\\mathcal {K}$ is contained in at least two faces of $\\mathcal {K}$ .", "(This requirement is less restrictive than the corresponding condition given in [11].", "For highly symmetric complexes like those discussed here, the two conditions are equivalent.)", "(d) $\\mathcal {K}$ is discrete, in the sense that each compact subset of $\\mathbb {E}^{3}$ meets only finitely many faces of $\\mathcal {K}$ .", "Each geometric polygonal complex of rank 3 in $\\mathbb {E}^3$ gives an incidence complex of the same rank.", "In fact, a quicker definition would consist of saying that $\\mathcal {K}$ as above consists of a triple $(\\mathcal {V},\\mathcal {E},\\mathcal {F}$ ), which, when ordered by inclusion, gives an abstract polygonal complex of rank 3.", "Proceeding with rank 4 structures, a geometric polygonal complex $\\mathcal {K}$ of rank 4, or simply a geometric polygonal 4-complex, in $\\mathbb {E}^{3}$ is a 4-tuple $(\\mathcal {F}^0,\\mathcal {F}^1,\\mathcal {F}^2,\\mathcal {F}^3)$ consisting of a set $\\mathcal {V}=\\mathcal {F}^0$ of points, called vertices, a set $\\mathcal {E}=\\mathcal {F}^1$ of line segments, called edges, a set $\\mathcal {F}=\\mathcal {F}^2$ of (finite or infinite) polygons, called 2-faces, and a set $\\mathcal {F}^3$ of geometric polygonal 3-complexes, called 3-faces, such that $\\mathcal {K}$ , partially ordered by inclusion, gives an abstract polygonal complex of rank 4 (with the same rank function).", "We can move up further in rank and similarly define higher rank geometric polygonal complexes in $\\mathbb {E}^3$ .", "The basic set-up is exactly the same.", "The $j$ -faces of an $n$ -complex are built from the $(j-1)$ -complexes representing the $(j-1)$ -faces of the $j$ -face.", "A geometric polygonal complex is a geometric polytope, or a geometric polyhedron when the rank is 3, if the underlying incidence complex is an abstract polytope.", "An apeirotope or apeirohedron is an infinite polytope or polyhedron, respectively.", "A geometric polygonal complex $\\mathcal {K}$ is called (geometrically) regular if its (geometric) symmetry group $G(\\mathcal {K})$ in $\\mathbb {E}^3$ is transitive on the flags of $\\mathcal {K}$ .", "The symmetry group $G(\\mathcal {K})$ of a polygonal complex $\\mathcal {K}$ is a (generally proper) subgroup of its combinatorial automorphism group $\\Gamma (\\mathcal {K})$ .", "In fact, $G(\\mathcal {K})$ is a flag-transitive subgroup of $\\Gamma (\\mathcal {K})$ if $\\mathcal {K}$ is regular.", "For a geometrically regular polytope, the symmetry group coincides with the full automorphism group and is necessarily simply flag-transitive.", "However, this is not true in general for geometrically regular polygonal complexes.", "We later require the classification of the regular apeirotopes of rank 4 in $\\mathbb {E}^3$ (see [10]).", "There are eight examples shown in the following display: $\\begin{matrix}\\lbrace 4,3,4\\rbrace = \\lbrace \\lbrace 4,3\\rbrace ,\\lbrace 3,4\\rbrace \\rbrace , & \\lbrace \\lbrace 4,6 \\mid 4\\rbrace ,\\lbrace 6,4\\rbrace _{3}\\rbrace ;\\\\[.02in]\\lbrace \\lbrace \\infty ,3\\rbrace _{6} \\# {\\lbrace \\,\\rbrace },\\lbrace 3,3\\rbrace \\rbrace , & \\lbrace \\lbrace \\infty ,4\\rbrace _{4} \\#\\lbrace \\infty \\rbrace ,\\lbrace 4,3\\rbrace _{3}\\rbrace ; \\\\[.02in]\\lbrace \\lbrace \\infty ,3\\rbrace _{6} \\# {\\lbrace \\,\\rbrace },\\lbrace 3,4\\rbrace \\rbrace , & \\lbrace \\lbrace \\infty ,6\\rbrace _{3} \\#\\lbrace \\infty \\rbrace ,\\lbrace 6,4\\rbrace _{3}\\rbrace ; \\\\[.02in]\\lbrace \\lbrace \\infty ,4\\rbrace _{4} \\# {\\lbrace \\,\\rbrace },\\lbrace 4,3\\rbrace \\rbrace , & \\lbrace \\lbrace \\infty ,6\\rbrace _{3} \\#\\lbrace \\infty \\rbrace ,\\lbrace 6,3\\rbrace _{4}\\rbrace .\\end{matrix} $ For notation we refer to [10].", "The eight examples fall into four pairs of “Petrie-duals\" (listed in the same row), where the apeirotopes in each pair share the same 2-skeleton.", "The familiar cubical tessellation $\\lbrace 4,3,4\\rbrace $ and its Petrie-dual listed in the first row both have (finite) square faces.", "All other apeirotopes have (infinite planar) zigzag faces." ], [ "Groups of regular incidence complexes", "The structure of a regular incidence complex $\\mathcal {K}$ can be completely described in terms of a well-behaved system of generating subgroups of any flag-transitive subgroup $\\Lambda $ of the full automorphism group $\\Gamma (\\mathcal {K})$ of $\\mathcal {K}$ (see [13]).", "Let $\\mathcal {K}$ be a regular incidence complex of positive rank $n$ , and let $\\Lambda $ be any flag-transitive subgroup of $\\Gamma (\\mathcal {K})$ .", "Let $\\Phi :=\\lbrace F_0,\\ldots ,F_{n-1}\\rbrace $ be a (fixed) base flag of $\\mathcal {K}$ , where $F_i$ denotes the $i$ -face in $\\Phi $ for each $i$ .", "For $\\Omega \\subseteq \\Phi $ , let $\\Lambda _\\Omega $ denote the stabilizer of $\\Omega $ in $\\Lambda $ , so in particular $\\Lambda _\\Phi $ is the stabilizer of $\\Phi $ itself, and $\\Lambda _\\emptyset = \\Lambda $ .", "For $i=-1,0,\\ldots ,n$ define the subgroup $ R_{i} := \\Lambda _{\\Phi \\setminus \\lbrace F_i\\rbrace } = \\langle \\varphi \\in \\Lambda \\mid F_j\\varphi =F_j \\mbox{ for all } j\\ne i\\rangle .$ Then $R_{-1}=R_{n}=\\Lambda _\\Phi \\le R_i$ for each $i=0,\\ldots ,n-1$ , and $\|R_{i}:R_{-1}\|-1$ is the number of flags $i$ -adjacent to $\\Phi $ .", "Moreover, $R_i \\cdot R_j = R_j \\cdot R_i \\qquad (-1\\le i <j-1\\le n-1), \\ $ as products of subgroups.", "The subgroup $\\Lambda $ is generated by the distinguished generating subgroups $R_0,\\ldots ,R_{n-1}$ , that is, $ \\Lambda = \\langle R_0,\\ldots ,R_{n-1} \\rangle .", "$ In fact, a stronger statement holds.", "When $\\emptyset \\ne I\\subseteq \\lbrace -1,0,\\ldots ,n\\rbrace $ set $\\Lambda _{I} := \\langle R_i \\mid i\\in I\\rangle $ ; and when $I=\\emptyset $ set $\\Lambda _I := R_{-1}=\\Lambda _\\Phi $ .", "Note that $\\Lambda _{I}=\\Lambda _{I\\setminus \\lbrace -1,n\\rbrace }$ for each subset $I$ .", "Then, for each $\\Omega \\subseteq \\Phi $ , $ \\Lambda _\\Omega = \\langle R_i \\mid F_{i}\\notin \\Omega \\rangle = \\Lambda _{\\lbrace i\\mid F_i \\notin \\Omega \\rbrace }.$ Here both sides coincide with $\\Lambda _\\Phi $ if $\\Omega =\\Phi $ .", "In addition, the following intersection property holds: $\\Lambda _I \\cap \\Lambda _J = \\Lambda _{I\\cap J}\\qquad (I,J\\subseteq \\lbrace -1,0,\\ldots ,n\\rbrace ) .$ The partial order on $\\mathcal {K}$ can be completely described in terms of the subgroups $R_{-1},R_0,\\ldots ,R_n$ of $\\Lambda $ .", "In fact, $ F_{i}\\varphi \\le F_{j}\\psi \\; \\leftrightarrow \\; \\psi ^{-1}\\varphi \\in \\Lambda _{\\lbrace i+1,\\ldots ,n\\rbrace }\\Lambda _{\\lbrace -1,0,\\ldots ,j-1\\rbrace }\\;\\quad (-1\\le i\\le j\\le n;\\, \\varphi ,\\psi \\in \\Lambda ) ,$ or equivalently, $ F_{i}\\varphi \\le F_{j}\\psi \\, \\leftrightarrow \\,\\Lambda _{\\lbrace -1,0,\\ldots ,n\\rbrace \\setminus \\lbrace i\\rbrace }\\varphi \\cap \\Lambda _{\\lbrace -1,0,\\ldots ,n\\rbrace \\setminus \\lbrace j\\rbrace }\\psi \\ne \\emptyset \\quad (-1\\le i\\le j\\le n;\\, \\varphi ,\\psi \\in \\Lambda ).", "$ Conversely, if a group $\\Lambda $ has a system of generating subgroups $R_{-1},R_0,\\ldots ,R_n$ with properties (REF ) and (REF ), then $\\Lambda $ is a flag-transitive subgroup of the full automorphism group of a regular incidence complex $\\mathcal {K}$ of rank $n$ (see [13]).", "For abstract regular polytopes, the groups $R_0,\\ldots ,R_{n-1}$ are generated by involutions $\\rho _0,\\ldots ,\\rho _{n-1}$ , where $\\rho _i$ maps the base flag to its unique $i$ -adjacent flag (see [10]).", "Also, $\\Lambda _\\Phi $ is trivial in this case." ], [ "Rank 3", "The regular polygonal complexes of rank 3 in $\\mathbb {E}^3$ have been completely enumerated.", "They comprise fourty-eight regular polyhedra and twenty-five regular complexes which are not polyhedra.", "The regular polyhedra in $\\mathbb {E}^3$ were discovered by Grünbaum [6] and Dress [4], [5] and are sometimes referred to as the Grünbaum-Dress polyhedra.", "For a quicker method of arriving at the classification see also McMullen & Schulte [9] and [10].", "The regular polyhedra include many well-known figures such as the Platonic solids, the Kepler-Poinsot polyhedra and the Petrie-Coxeter polyhedra (see [1], [2]).", "The regular polygonal complexes in $\\mathbb {E}^3$ which are not polyhedra were enumerated in Pellicer & Schulte [11], [12].", "The flag-stabilizers of their symmetry groups are either trivial or have order 2, and the methods of enumeration in these two cases are quite different.", "There are four regular polygonal complexes with a non-trivial flag-stabilizer, and each is the 2-skeleton of a regular 4-apeirotope in $\\mathbb {E}^3$ .", "The eight regular 4-apeirotopes in $\\mathbb {E}^3$ fall into four pairs of Petrie-duals, and the apeirotopes in each pair have the same 2-skeleton (see [10]).", "Thus each regular polygonal complex with a non-trivial flag-stabilizer is the common 2-skeleton of two regular 4-apeirotopes.", "There are also twenty-one regular polygonal complexes with a trivial flag-stabilizers, which are not polyhedra; accordingly, these complexes are referred to as the simply flag-transitive regular polygonal complexes." ], [ "Higher ranks", "In this section we bound the rank of a regular polygonal complex in $\\mathbb {E}^3$ by 4, and show that the eight regular 4-apeirotopes mentioned earlier are the only examples in rank 4.", "Suppose $\\mathcal {K}$ is a regular polygonal complex of rank $n\\ge 4$ .", "Then its geometric symmetry group $G(\\mathcal {K})$ is a flag-transitive subgroup of $\\Gamma (\\mathcal {K})$ to which the above structure results for flag-transitive subgroups of $\\Gamma (\\mathcal {K})$ apply with $\\Lambda =G(\\mathcal {K})$ .", "In particular, $G(\\mathcal {K})=\\langle R_0,\\ldots ,R_{n-1}\\rangle $ , where $R_0,\\ldots ,R_{n-1}$ are the distinguished generating subgroups of $G(\\mathcal {K})$ defined with respect to a base flag $\\Phi =\\lbrace F_0,\\ldots ,F_{n-1}\\rbrace $ of $\\mathcal {K}$ .", "Since $\\mathcal {K}$ is a polygonal complex, the base 2-face $F_2$ is a regular polygon in $\\mathbb {E}^3$ , planar or non-planar.", "Now each of the subgroups $R_{3},\\ldots ,R_{n-1}$ consists of isometries stabilizing $F_0,F_1,F_2$ and hence acting trivially (pointwise) on the entire affine hull of the polygon $F_2$ .", "This forces $F_2$ to be planar.", "In fact, otherwise the subgroups $R_{3},\\ldots ,R_{n-1}$ would have to be trivial; but this is impossible since $n\\ge 4$ .", "On the other hand, if $F_2$ is planar, then a nontrivial element from $R_{3},\\ldots ,R_{n-1}$ could only be the euclidean reflection, $\\rho _3$ (say), in the affine hull of $F_2$ .", "In this case we can immediately exclude the possibility that $n>4$ ; in fact, if $n>4$ then necessarily $R_{3}=R_{4}=\\langle \\rho _3\\rangle $ , which is impossible.", "This then only leaves the possibility that $n=4$ , the face $F_2$ is planar, $R_{3}=\\langle \\rho _3\\rangle $ , and there are just two facets of $\\mathcal {K}$ meeting at $F_2$ .", "Now suppose that $n=4$ and that $F_2$ and $\\rho _3$ are as described.", "Then the 2-skeleton $\\mathcal {L}$ of $\\mathcal {K}$ is a regular polygonal complex of rank 3 in $\\mathbb {E}^3$ whose symmetry group contains $G(\\mathcal {K})$ .", "In particular, $\\mathcal {L}$ has face mirrors, meaning that $\\mathcal {L}$ has planar 2-faces and that the affine hulls of the 2-faces are mirrors of plane symmetries of $\\mathcal {L}$ .", "In fact, $\\rho _3$ is a reflective symmetry of $\\mathcal {L}$ in the affine hull of the 2-face $F_2$ of $\\mathcal {L}$ , and its conjugates under $G(\\mathcal {K})$ (or $G(\\mathcal {L})$ ) provide all reflective symmetries in affine hulls of 2-faces of $\\mathcal {L}$ .", "In particular, $\\mathcal {L}$ is not simply flag-transitive, since $\\rho _3$ stabilizes the flag $\\lbrace F_0,F_1,F_2\\rbrace $ of $\\mathcal {L}$ .", "We now appeal to the classification in [11], [12] of the regular polygonal complexes of rank 3 in $\\mathbb {E}^3$ .", "In fact, it was shown in [11] that a regular polygonal 3-complex with face mirrors in $\\mathbb {E}^3$ is necessarily the 2-skeleton of a regular 4-apeirotope $\\mathcal {P}$ in $\\mathbb {E}^3$ with the same symmetry group, and that the fourth distinguished generator of the symmetry group $G(\\mathcal {P})$ of $\\mathcal {P}$ is given by the reflection in the planar base 2-face.", "Thus $\\mathcal {L}$ is the 2-skeleton of a regular 4-apeirotope $\\mathcal {P}$ with $G(\\mathcal {P})=G(\\mathcal {L})$ .", "It is also known that the number of 2-faces, $r$ , on an edge of $\\mathcal {P}$ and thus of $\\mathcal {L}$ must be 3 or 4.", "Now the facets of $\\mathcal {K}$ are also regular polygonal complexes of rank 3 in $\\mathbb {E}^3$ and have their 2-faces among the 2-faces of $\\mathcal {L}$ .", "In particular, the basic facet $F_3$ of $\\mathcal {K}$ gives rise to the complex $\\mathcal {K}_{3}:=F_{3}/F_{-1}$ whose full symmetry group $G(\\mathcal {K}_3)$ contains the stabilizer $\\langle R_{0},R_{1},R_{2}\\rangle $ of $F_3$ in $G(\\mathcal {K})$ as a flag-transitive subgroup.", "We must show that $\\mathcal {K}_3$ is a polyhedron.", "Then, since there are just two facets of $\\mathcal {K}$ meeting at $F_2$ , the complex $\\mathcal {K}$ itself would have to be a 4-apeirotope.", "Since $\\mathcal {K}_3$ is a subcomplex of $\\mathcal {L}$ , the number of 2-faces, $r_3$ , on an edge of $\\mathcal {K}_3$ must be 2, 3 or 4.", "We wish to show that $r_{3}=2$ .", "Clearly, we cannot have $r_{3}=r$ since $\\mathcal {L}$ is connected.", "In fact, otherwise, every 2-face of $\\mathcal {K}$ containing an edge of $\\mathcal {K}_3$ would also have to be a 2-face of $\\mathcal {K}_3$ ; but then connectedness would force all 2-faces of $\\mathcal {K}$ to be 2-faces of $\\mathcal {K}_3$ , which is impossible.", "Hence it remains to exclude the case when $r=4$ and $r_{3}=3$ .", "Now suppose $r=4$ and $r_{3}=3$ .", "Then we know from [11] that the pointwise stabilizer of the base edge $F_{1}$ of $\\mathcal {K}_3$ in $G(\\mathcal {K}_3)$ is a cyclic group $C_3$ or a dihedral group $D_3$ .", "Either way, the affine hulls of the (planar) faces of $\\mathcal {K}_3$ meeting at $F_1$ must be inclined at $120^\\circ $ .", "On the other hand, the affine hulls of the four 2-faces of $\\mathcal {P}$ (or $\\mathcal {K}$ or $\\mathcal {L}$ ) meeting at $F_1$ either coincide (when the faces are opposite relative to $F_1$ ) or are inclined at $90^\\circ $ .", "In any case, the two scenarios are incompatible and rule out the possibility that $r=4$ and $r_{3}=3$ .", "Thus $r_{3}=2$ .", "At this point we know that both $\\mathcal {K}$ and $\\mathcal {P}$ are regular 4-apeirotopes with a common 2-skeleton.", "In particular, $\\mathcal {K}$ must be among the eight regular 4-apeirotopes in $\\mathbb {E}^3$ .", "Another appeal to [11] then shows that $\\mathcal {K}$ must in fact coincide with either $\\mathcal {P}$ itself or with the Petrie-dual of $\\mathcal {P}$ .", "In summary we have established the following theorem.", "Theorem 1 There are no regular geometric polygonal complex in $\\mathbb {E}^3$ of rank $n\\ge 5$ .", "The only regular geometric polygonal complexes of rank 4 in $\\mathbb {E}^3$ are the eight regular 4-apeirotopes in $\\mathbb {E}^3$ .", "H.S.M.Coxeter.", "Regular Polytopes (3rd edition), Dover.", "H.S.M.Coxeter.", "Regular skew polyhedra in 3 and 4 dimensions and their topological analogues, Proc.", "London Math.", "Soc.", "(2) 43 (1937), 33–62.", "(Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press (Carbondale, 1968), 76–105.)", "L.Danzer, E.Schulte.", "Reguläre Inzidenzkomplexe, I, Geom.", "Dedicata 13 (1982), 295–308.", "A.W.M.Dress.", "A combinatorial theory of Grünbaum's new regular polyhedra, I: Grünbaum's new regular polyhedra and their automorphism group, Aequationes Math.", "23 (1981), 252–265.", "A.W.M.Dress.", "A combinatorial theory of Grünbaum's new regular polyhedra, II: complete enumeration, Aequationes Math.", "29 (1985), 222–243.", "B.Grünbaum.", "Regular polyhedra — old and new, Aequat.", "Math.", "16 (1977), 1–20.", "P.McMullen.", "Realizations of regular polytopes, Aequationes Math.", "37 (1989), 38–56.", "P.McMullen.", "Geometric regular polytopes, monograph in preparation.", "P.McMullen, E. Schulte.", "Regular polytopes in ordinary space, Discrete Comput.", "Geom.", "17 (1997), 449–478.", "P.McMullen, E. Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, Vol.", "92, Cambridge University Press, Cambridge, UK, 2002.", "D.Pellicer, E.Schulte.", "Regular polygonal complexes in space, I, Trans.", "Amer.", "Math.", "Soc.", "362 (2010), 6679–6714.", "D.Pellicer, E.Schulte.", "Regular polygonal complexes in space, II, Trans.", "Amer.", "Math.", "Soc.", "365 (2013), 2031–2061.", "E.Schulte.", "Reguläre Inzidenzkomplexe, II, Geom.", "Dedicata 14 (1983), 33–56." ] ]	1403.0251
[ [ "Chaos in two black holes with next-to-leading order spin-spin\n interactions" ], [ "Abstract We take into account the dynamics of a complete third post-Newtonian conservative Hamiltonian of two spinning black holes, where the orbital part arrives at the third post-Newtonian precision level and the spin-spin part with the spin-orbit part includes the leading-order and next-to-leading-order contributions.", "It is shown through numerical simulations that the next-to-leading order spin-spin couplings play an important role in chaos.", "A dynamical sensitivity to the variation of single parameter is also investigated.", "In particular, there are a number of \\textit{observable} orbits whose initial radii are large enough and which become chaotic before coalescence." ], [ "Introduction", "Massive binary black-hole systems are likely the most promising sources for future gravitational wave detectors.", "The successful detection of the waveforms means using matched-filtering techniques to best separate a faint signal from the noise and requires a very precise modelling of the expected waveforms.", "Post-Newtonian (PN) approximations can satisfy this requirement.", "Up to now, high-precision PN templates have already been known for the non-spin part up to 3.5PN order (i.e.", "the order $1/c^{7}$ in the formal expansion in powers of $1/c^{2}$ with $c$ being the speed of light) [1,2], the spin-orbit part up to 3.5PN order including the leading-order (LO, 1.5PN), next-to-leading-order (NLO, 2.5PN) and next-to-next-to-leading-order (NNLO, 3.5PN) interactions [3-5], and the spin-spin part up to 4PN order consisting of the LO (2PN), NLO (3PN) and NNLO (4PN) couplings [6-9].", "However, an extremely sensitive dependence on initial conditions as the basic feature of chaotic systems would pose a challenge to the implementation of such matched filters, since the number of filters required to detect these waveforms is exponentially large with increasing detection sensitivity.", "This has led to some authors focusing on research of chaos in the orbits of two spinning black holes.", "Chaos was firstly found and confirmed in the 2PN Lagrangian approximation of comparable mass binaries with the LO spin-orbit and LO spin-spin effects [10].", "Moreover, it was reported in [11] that the presence of chaos should be ruled out in these systems because no positive Lyapunov exponents could be found.", "As an answer to this claim, Refs.", "[12,13] obtained some positive Lyapunov exponents and pointed out these zero Lyapunov exponents of [11] due to the less rigorous calculation of the Lyapunov exponents of two nearby orbits with unapt rescaling.", "In fact, the conflicting results on Lyapunov exponents are because the two papers [11,12] used different methods to compute their Lyapunov exponents, as was mentioned in [14].", "Ref.", "[11] computed the stabilizing limit values of Lyapunov exponents, and Ref.", "[12] worked out the slopes of fit lines.", "This is the so-called doubt regarding different chaos indicators causing two distinct claims on the chaotic behavior.", "Besides this, there was second doubt on different dynamical approximations making the same physical system have distinct dynamical behaviors.", "The 2PN harmonic-coordinates Lagrangian formulation of the two-black hole system with the LO spin-orbit couplings of one spinning body allows chaos [15], but the 2PN ADM (Arnowitt-Deser-Misner) -coordinates Hamiltonian does not [16,17].", "Levin [18] thought that there is no formal conflict between them since the two approaches are not exactly but approximately equal, and different dynamical behaviors between the two approximately related systems are permitted according to dynamical systems theory.", "Seen from the canonical, conjugate spin coordinates [19], the former non-integrability and the latter integrability are clearer.", "As extensions, both any PN conservative Hamiltonian binary system with one spinning body and a conservative Hamiltonian of two spinning bodies without the constraint of equal mass or with the spin-orbit couplings not restricted to the leading order are still integrable.", "Recently, [20,21] argued the integrability of the 2PN Hamiltonian without the spin-spin couplings and with the NLO and/or NNLO spin-orbit contributions included.", "On the contrary, the corresponding Lagrangian counterpart with spin effects limited to the spin-orbit interactions up to the NLO terms exhibits the stronger chaoticity [22].", "Third doubt relates to different dependence of chaos on single dynamical parameter or initial condition.", "The description of the chaotic regions and chaotic parameter spaces in [15] are inconsistent with that in [23].", "The different claims are regarded to be correct according to the statement of [24] that chaos does not depend only on single physical parameter or initial condition but a complicated combination of all parameters and initial conditions.", "It is worth emphasizing that the spin-spin effects are the most important source for causing chaos in spinning compact binaries, but they were only restricted to the LO term in the published papers on research of the chaotic behavior.", "It should be significant to discuss the NLO spin-spin couplings included to a contribution of chaos.", "For the sake of this, we shall consider a complete 3PN conservative Hamiltonian of two spinning black holes, where the orbital part is up to the 3PN order and the spin-spin part as well as the spin-orbit part includes the LO and NLO interactions.", "In this way, we want to know whether the inclusion of the NLO spin-spin couplings have an effect on chaos, and whether there is chaos before coalescence of the binaries." ], [ "Third post-Newtonian order Hamiltonian approach", "It is too difficult to strictly describe the dynamics of a system of two mass comparable spinning black holes in general relativity.", "Instead, the PN approximation method is often used.", "Suppose that the two bodies have masses $m_1$ and $m_2$ with $m_1\\le m_2$ .", "Other mass parameters are the total mass $M=m_1+m_2$ , the reduced mass $\\mu =m_{1}m_{2}/M$ , the mass ratio $\\beta =m_1/m_2$ and the mass parameter $\\eta =\\mu /M=\\beta /(1+\\beta )^{2}$ .", "As to other specified notations, a 3-dimensional vector $\\mathbf {r}$ represents the relative position of body 1 to body 2, its unit radial vector is $\\mathbf {n}=\\mathbf {r}/r$ with the radius $r=\|\\mathbf {r}\|$ , and $\\mathbf {p}$ stands for the momenta of body 1 relative to the centre.", "The momenta, distances and time $t$ are respectively measured in terms of $\\mu $ , $M$ and $M$ .", "Additionally, geometric units $c=G=1$ are adopted.", "The two spin vectors are $\\mathbf {S}_{i}=S_{i}\\mathbf {\\hat{S}}_{i}$ ($i=1,2$ ) with unit vectors $\\mathbf {\\hat{S}}_{i} $ and the spin magnitudes $S_i=\\chi _{i}m_{i}^{2}/M^2$ ($0\\le \\chi _{i}\\le 1$ ).", "In ADM coordinates, the system can be expressed as the dimensionless conservative 3PN Hamiltonian $H(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2) &=&H_{o}(\\mathbf {r},\\mathbf {p})+H_{so}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2)\\nonumber \\\\ & & +H_{ss}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2).$ In the following, we write its detailed expressions although they are too long.", "For the conservative case, the orbital part $H_{o}$ does not include the dissipative 2.5PN term (which is the leading order radiation damping level) but the Newtonian term $H_{N}$ and the PN contributions $H_{1PN}$ , $H_{2PN}$ and $H_{3PN}$ , that is, $H_{o}=H_{N}+H_{1PN}+ H_{2PN}+H_{3PN}.$ As given in [25], they are $H_{N}&=&\\frac{\\mathbf {p}^2}{2}-\\frac{1}{r},$ $H_{1PN}&=&\\frac{1}{8}(3\\eta -1)\\mathbf {p}^4-\\frac{1}{2}[(3+\\eta )\\mathbf {p}^2+\\eta (\\mathbf {n}\\cdot \\mathbf {p})^2]\\frac{1}{r}\\nonumber \\\\& &+\\frac{1}{2r^{2}},$ $H_{2PN}&=&\\frac{1}{16}(1-5\\eta +5\\eta ^2)\\mathbf {p}^6+\\frac{1}{8}[(5-20\\eta -3\\eta ^2)\\mathbf {p}^4\\nonumber \\\\& &-2\\eta ^2{(\\mathbf {n}\\cdot \\mathbf {p})^2}\\mathbf {p}^2-3\\eta ^2{(\\mathbf {n}\\cdot \\mathbf {p})^4}]\\frac{1}{r}+\\frac{1}{2}[(5+8\\eta )\\mathbf {p}^2\\nonumber \\\\& & +3\\eta (\\mathbf {n}\\cdot \\mathbf {p})^2]\\frac{1}{r^2}-\\frac{1}{4}(1+3\\eta )\\frac{1}{r^3},$ $H_{3PN}&=&\\frac{1}{128}(-5+35\\eta -70\\eta ^2+35\\eta ^3)\\mathbf {p}^8+\\frac{1}{16}[(-7\\nonumber \\\\&&+42\\eta -53\\eta ^2-5\\eta ^3)\\mathbf {p}^6+(2-3\\eta )\\eta ^2(\\mathbf {n}\\cdot \\mathbf {p})^2\\nonumber \\\\& &\\times \\mathbf {p}^4+3(1-\\eta )\\eta ^2(\\mathbf {n}\\cdot \\mathbf {p})^4\\mathbf {p}^2-5\\eta ^3(\\mathbf {n}\\cdot \\mathbf {p})^6]\\frac{1}{r}\\nonumber \\\\& &+[\\frac{1}{16}(-27+136\\eta +109\\eta ^2) \\mathbf {p}^4+\\frac{1}{16}(17\\nonumber \\\\&&+30\\eta )\\eta (\\mathbf {n}\\cdot \\mathbf {p})^2\\mathbf {p}^2+\\frac{1}{12}(5+43\\eta )\\eta (\\mathbf {n}\\cdot \\mathbf {p})^4]\\frac{1}{r^2}\\nonumber \\\\&&+\\lbrace [-\\frac{25}{8}+(\\frac{1}{64}\\pi ^2-\\frac{335}{48})\\eta -\\frac{23}{8}\\eta ^2]\\mathbf {p}^2\\nonumber \\\\&&+(-\\frac{85}{16}-\\frac{3}{64}\\pi ^2-\\frac{7}{4}\\eta )\\eta (\\mathbf {n}\\cdot \\mathbf {p})^2\\rbrace \\frac{1}{r^3}\\nonumber \\\\&&+[\\frac{1}{8}+(\\frac{109}{12}-\\frac{21}{32}\\pi ^2)\\eta ]\\frac{1}{r^4}.$ The spin-orbit part $H_{so}$ is linear functions of the two spins.", "It is the sum of the LO spin-orbit term $H^{LO}_{so}$ and the NLO spin-orbit term $H^{NLO}_{so}$ , i.e.", "$H_{so}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2)=H^{LO}_{so}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2)+H^{NLO}_{so}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2).$ Ref.", "[5] gave their expressions $H_{so}&=&\\frac{1}{r^{3}}[g(\\mathbf {r},\\mathbf {p})\\mathbf {S}+g^{}(\\mathbf {r},\\mathbf {p})\\mathbf {S}^]\\cdot \\mathbf {L},$ where the related notations are $\\mathbf {S}=\\mathbf {S}_{1} + \\mathbf {S}_{2}, \\mathbf {S}^{}=\\frac{1}{\\beta }\\mathbf {S}_{1} + \\beta \\mathbf {S}_{2},$ $g(\\mathbf {r},\\mathbf {p})&=& 2+[\\frac{19}{8}\\eta \\mathbf {p}^{2}+\\frac{3}{2}\\eta (\\mathbf {n}\\cdot \\mathbf {p})^{2}-(6+2\\eta )\\frac{1}{r}],$ $g^{}(\\mathbf {r},\\mathbf {p})&=&\\frac{3}{2}+[-(\\frac{5}{8}+2\\eta )\\mathbf {p}^{2}+\\frac{3}{4}\\eta (\\mathbf {n}\\cdot \\mathbf {p})^{2}\\nonumber \\\\ & & -(5+2\\eta )\\frac{1}{r}],$ and the Newtonian-looking orbital angular momentum vector is $\\mathbf {L}=\\mathbf {r}\\times \\mathbf {p}.$ The constant terms in $g$ and $g^{}$ correspond to the LO part, and the others, the NLO part.", "Similarly, the spin-spin Hamiltonian $H_{ss}$ also consists of the LO spin-spin coupling term $H^{LO}_{ss}$ and the NLO spin-spin coupling term $H^{NLO}_{ss}$ , namely, $H_{ss}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2)=H^{LO}_{ss}(\\mathbf {r},\\mathbf {S}_1,\\mathbf {S}_2)+H^{NLO}_{ss}(\\mathbf {r},\\mathbf {p},\\mathbf {S}_1,\\mathbf {S}_2).$ The first sub-Hamiltonian reads [25] $H^{LO}_{ss} &=&\\frac{1}{2r^{3}}[3(\\mathbf {S}_0\\cdot \\mathbf {n})^{2}-\\mathbf {S}^{2}_0]$ with $\\mathbf {S}_0=\\mathbf {S}+\\mathbf {S}^{}$ .", "The second sub-Hamiltonian is made of three parts, $H^{NLO}_{ss}=H_{s^{2}_{1}p^{2}}+H_{s^{2}_{2}p^{2}}+H_{s_{1}s_{2}p^{2}}.$ They are written as [7,8] $H_{s^{2}_{1}p^{2}}&=&\\frac{\\eta ^{2}}{\\beta ^{2}r^{3}}[\\frac{1}{4}(\\mathbf {p}_{1}\\cdot \\mathbf {S}_{1})^{2}+\\frac{3}{8}(\\mathbf {p}_{1}\\cdot \\mathbf {n})^{2}\\mathbf {S}^{2}_{1} \\nonumber \\\\ && -\\frac{3}{8}\\mathbf {p}^{2}_{1}(\\mathbf {S}_{1}\\cdot \\mathbf {n})^{2}-\\frac{3}{4}(\\mathbf {p}_{1}\\cdot \\mathbf {n})(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{1}\\cdot \\mathbf {S}_{1})]\\nonumber \\\\ & &-\\frac{\\eta ^{2}}{r^{3}}[\\frac{3}{4}\\mathbf {p}^{2}_{2}\\mathbf {S}^{2}_{1}-\\frac{9}{4}\\mathbf {p}^{2}_{2}(\\mathbf {S}_{1}\\cdot \\mathbf {n})^{2}] \\nonumber \\\\& & +\\frac{\\eta ^{2}}{r^{3}\\beta }[\\frac{3}{4}(\\mathbf {p}_{1}\\cdot \\mathbf {p}_{2})\\mathbf {S^{2}_{1}}-\\frac{9}{4}(\\mathbf {p}_{1}\\cdot \\mathbf {p}_{2})(\\mathbf {S}_{1}\\cdot \\mathbf {n})^{2}\\nonumber \\\\ & &-\\frac{3}{2}(\\mathbf {p}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {S}_{1})(\\mathbf {S}_{1}\\cdot \\mathbf {n})\\nonumber \\\\ & &+3(\\mathbf {p}_{2}\\cdot \\mathbf {n})(\\mathbf {p}_{1}\\cdot \\mathbf {S}_{1})(\\mathbf {S}_{1}\\cdot \\mathbf {n})\\nonumber \\\\ & &+\\frac{3}{4}(\\mathbf {p}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {n})\\mathbf {S}^{2}_{1}\\nonumber \\\\ & &-\\frac{15}{4}(\\mathbf {p}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {n})(\\mathbf {S}_{1}\\cdot \\mathbf {n})^{2}],$ $H_{s^{2}_{2}p^{2}}=H_{S^{2}_{1}p^{2}} (1\\leftrightarrow 2),$ $H_{s_{1}s_{2}p^{2}}&=&\\frac{\\eta ^{2}}{2r^{3}}\\lbrace \\frac{3}{2}\\lbrace [(\\mathbf {p}_{1}\\times \\mathbf {S}_{1})\\cdot \\mathbf {n}][(\\mathbf {p}_{2}\\times \\mathbf {S}_{2})\\cdot \\mathbf {n}] \\nonumber \\\\ &&+6[(\\mathbf {p}_{2}\\times \\mathbf {S}_{1})\\cdot \\mathbf {n}][(\\mathbf {p}_{1}\\times \\mathbf {S}_{2})\\cdot \\mathbf {n}]\\nonumber \\\\ & & -15(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {S}_{2}\\cdot \\mathbf {n})(\\mathbf {p}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {n}) \\nonumber \\\\ & &-3(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {S}_{2}\\cdot \\mathbf {n})(\\mathbf {p}_{1}\\cdot \\mathbf {p}_{2})\\nonumber \\\\ & &+3(\\mathbf {S}_{1}\\cdot \\mathbf {p}_{2})(\\mathbf {S}_{2}\\cdot \\mathbf {n})(\\mathbf {p}_{1}\\cdot \\mathbf {n})\\nonumber \\\\ & &+3(\\mathbf {S}_{2}\\cdot \\mathbf {p}_{1})(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {n})\\nonumber \\\\ & &+3(\\mathbf {S}_{1}\\cdot \\mathbf {p}_{1})(\\mathbf {S}_{2}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {n})\\nonumber \\\\ & &+3(\\mathbf {S}_{2}\\cdot \\mathbf {p}_{2})(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{1}\\cdot \\mathbf {n})\\nonumber \\\\ & &-\\frac{1}{2}(\\mathbf {S}_{1}\\cdot \\mathbf {p}_{2})(\\mathbf {S}_{2}\\cdot \\mathbf {p}_{1})+(\\mathbf {S}_{1}\\cdot \\mathbf {p}_{1})(\\mathbf {S}_{2}\\cdot \\mathbf {p}_{2}) \\nonumber \\\\& &-3(\\mathbf {S}_{1}\\cdot \\mathbf {S}_{2})(\\mathbf {p}_{1}\\cdot \\mathbf {n})(\\mathbf {p}_{2}\\cdot \\mathbf {n})\\nonumber \\\\ & &+\\frac{1}{2}(\\mathbf {S}_{1}\\cdot \\mathbf {S}_{2})(\\mathbf {p}_{1}\\cdot \\mathbf {p}_{2})\\rbrace \\nonumber \\\\ & & +\\frac{3\\eta ^{2}}{2r^{3}\\beta }\\lbrace -[(\\mathbf {p}_{1}\\times \\mathbf {S}_{1})\\cdot \\mathbf {n}][(\\mathbf {p}_{1}\\times \\mathbf {S}_{2})\\cdot \\mathbf {n}]\\nonumber \\\\ & &+(\\mathbf {S}_{1}\\cdot \\mathbf {S}_{2})(\\mathbf {p}_{1}\\cdot \\mathbf {n})^{2}\\nonumber \\\\& &-(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {S}_{2}\\cdot \\mathbf {p}_{1})(\\mathbf {p}_{1}\\cdot \\mathbf {n})\\rbrace \\nonumber \\\\ & &+\\frac{3\\eta ^{2}\\beta }{2r^{3}}\\lbrace -[(\\mathbf {p}_{2}\\times \\mathbf {S}_{2})\\cdot \\mathbf {n}][(\\mathbf {p}_{2}\\times \\mathbf {S}_{1})\\cdot \\mathbf {n}] \\nonumber \\\\ & &+(\\mathbf {S}_{1}\\cdot \\mathbf {S}_{2})(\\mathbf {p}_{2}\\cdot \\mathbf {n})^{2}\\nonumber \\\\& &-(\\mathbf {S}_{2}\\cdot \\mathbf {n})(\\mathbf {S}_{1}\\cdot \\mathbf {p}_{2})(\\mathbf {p}_{2}\\cdot \\mathbf {n})\\rbrace \\nonumber \\\\ & &+\\frac{6\\eta }{r^{4}}[(\\mathbf {S}_{1}\\cdot \\mathbf {S}_{2})-2(\\mathbf {S}_{1}\\cdot \\mathbf {n})(\\mathbf {S}_{2}\\cdot \\mathbf {n})].$ Here, $\\mathbf {p}_{1}=-\\mathbf {p}_{2}=\\mathbf {p}$ .", "In a word, the conservative Hamiltonian (1) up to the 3PN order is not completely given until Eq.", "(15) appears.", "Clearly, Hamiltonian (1) does not depend on any mass but the mass ratio.", "The evolutions of position $\\mathbf {r}$ and momentum $\\mathbf {p}$ satisfy the canonical equations of the Hamiltonian (1): $\\frac{d\\mathbf {r}}{dt}=\\frac{\\partial H}{\\partial \\mathbf {p}}, \\quad \\quad \\frac{d\\mathbf {p}}{dt}=-\\frac{\\partial H}{\\partial \\mathbf {r}}.$ The spin variables vary with time according to the following relations $\\frac{d\\mathbf {S}_i}{dt}=\\frac{\\partial H}{\\partial \\mathbf {S}_i}\\times \\mathbf {S}_i.$ Besides the two spin magnitudes, there are four conserved quantities in the Hamiltonian (1), including the total energy $E=H$ and three components of the total angular momentum vector $\\mathbf {J}=\\mathbf {L}+\\mathbf {S}$ .", "A fifth constant of motion is absent, so the Hamiltonian (1) is non-integrable.Based on the idea of [19], the Hamiltonian (1) can be expressed as a completely canonical Hamiltonian with a 10-dimensional phase space when the canonical, conjugate spin coordinates are used instead of the original spin variables.", "If the system is integrable, at least five independent integrals of motion beyond the constant spin magnitudes are necessary.", "Its high nonlinearity seems to imply that it is a richer source for chaos.", "Next, we shall search for chaos, and particularly investigate the effect of the NLO spin-spin interactions on the dynamics of the system." ], [ "Detection of chaos before coalescence", "With numerical simulations, we use some chaos indicators to describe dynamical differences between the NLO spin-spin couplings excluded and included.", "The appropriate ones of the indicators are selected to study dependence of chaos on single parameter when the NLO spin-spin couplings are included.", "Finally, we expect to find chaos before coalescence by estimating the Lyapunov and inspiral decay times." ], [ "Comparisons", "Numerical methods are convenient to study nonlinear dynamics of the Hamiltonian (1).", "Symplectic integrators are efficient numerical tools since they have good geometric and physical properties, such as the symplectic structure conserved and energy errors without secular changes.", "However, they cannot provide high enough accuracies, and the computations are expensive when the mixed symplectic integration algorithms [21,26] with a composite of the second-order explicit leapfrog symplectic integrator and the second-order implicit midpoint rule are chosen.", "In this sense, we would prefer to adopt an 8(9) order Runge-Kutta-Fehlberg algorithm of variable time steps.", "In fact, it gives such high accuracy to the energy error in the magnitude of about order $10^{-13}\\sim 10^{-12}$ when integration time reaches $10^{6}$ , as shown in Fig.", "1.", "Here, orbit 1 we consider has initial conditions $(\\mathbf {p}(0);\\mathbf {r}(0))= (0,0.39, 0; 8.55, 0, 0)$ , which correspond to the initial eccentricity $e_0=0.30$ and the initial semi-major axis $a_0=12.2$ .", "Other parameters and initial spin angles are respectively $\\beta =0.79$ , $\\chi _1=\\chi _2=\\chi =1.0$ , $\\theta _{i}=78.46^{\\circ }$ and $\\phi _{i}=60^{\\circ }$ , where polar angles $\\theta _{i}$ and azimuthal angles $\\phi _{i}$ satisfy the relations $\\mathbf {\\hat{S}}_{i}=(\\cos \\phi _{i}\\sin \\theta _{i},\\sin \\phi _{i}\\sin \\theta _{i},\\cos \\theta _{i})$ , as commonly used in physics.", "The NLO spin-spin couplings are not included in Fig.", "1(a), but in Fig.", "1(b).", "It can be seen clearly that the inclusion of the NLO spin-spin couplings with a rather long expression decreases only slightly the numerical accuracy.", "Therefore, our numerical results are shown to be reliable although the energy errors have secular changes.", "We apply several chaos indicators to compare dynamical behaviors of orbit 1 according to the two cases without and with the NLO spin-spin couplings.", "The method of Poincaré surface of section can provide a clear description of the structure of phase space to a conservative system whose phase space is 4 dimensions.", "As a point to note, it is not suitable for such a higher dimensional system (1).", "Fortunately, power spectra, Lyapunov exponents and fast Lyapunov indicators would work well in finding chaos regardless of the dimensionality of phase space.", "Power spectrum analysis reveals a distribution of various frequencies $\\omega $ of a signal $x(t)$ .", "It is the Fourier transformation $X(\\omega )=\\int _{-\\infty }^{+\\infty }x(t)e^{-i\\omega t}dt,$ where $i$ is the imaginary unit.", "In general, the power spectra $X(\\omega )$ are discrete for periodic and quasi periodic orbits but continuous for chaotic orbits.", "That is to say, the classification of orbits can be distinguished in terms of different features of the spectra.", "On the basis of this, we know through Fig.", "2 that the orbit seems to be regular when the NLO spin-spin couplings are not included, but chaotic when the NLO spin-spin couplings are included.", "Notice that the method of power spectra is only a rough estimation of the regularity and chaoticity of orbits.", "More reliable chaos indicators are strongly desired." ], [ "Lyapunov exponents", "The maximum Lyapunov exponent is used to measure the average separation rate of two neighboring orbits in the phase space and gives quantitative analysis to the strength of chaos.", "Its calculations are usually based on the variational method and the two-particle method [27].", "The former needs solving the variational equations as well as the equations of motion, and the latter needs solving the equations of motion only.", "Considering the difficulty in deriving the variational equations of a complicated dynamical system, we pay attention to the application of the latter method.", "In the configuration space, it is defined as [28] $\\lambda =\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\ln \\frac{\|\\Delta \\mathbf {r}(t)\|}{\|\\Delta \\mathbf {r}(0)\|},$ where $\|\\Delta \\mathbf {r}(0)\|$ and $\|\\Delta \\mathbf {r}(t)\|$ are the separations between the two neighboring orbits at times 0 and $t$ , respectively.", "The initial distance cannot be too big or too small, and $10^{-8}$ is regarded as to its suitable choice in the double precision [27].", "For the sake of the overflow avoided, renormalizations from time to time are vital in the tangent space.", "A bounded orbit is chaotic if its Lyapunov exponent is positive, but regular when its Lyapunov exponent tends to zero.", "In this way, we can know from Fig.", "3 that orbit 1 is regular for the case without the NLO spin-spin couplings, but chaotic for the case with the NLO spin-spin couplings.", "Of course, it takes much computational cost to distinguish between the ordered and chaotic cases." ], [ "Fast Lyapunov indicators", "A quicker method to find chaos than the method of Lyapunov exponents is a fast Lyapunov indicator (FLI).", "This indicator that was originally considered to measure the expansion rate of a tangential vector [29] does not need any renormalization, while its modified version dealing with the use of the two-particle method [30] does.", "The modified version is of the form $\\textrm {FLI(t)}=\\log _{10}\\frac{\|\\Delta \\mathbf {r}(t)\|}{\|\\Delta \\mathbf {r}(0)\|}.$ Its computation is based on the following expression: $\\textrm {FLI}=-k(1+\\log _{10}\|\\Delta \\mathbf {r}(0)\|)+\\log _{10}\\frac{\|\\Delta \\mathbf {r}(t)\|}{\|\\Delta \\mathbf {r}(0)\|},$ where $k$ denotes the sequential number of renormalization.", "The FLI of Fig.", "4(a) corresponding to Fig.", "3(a) increases algebraically with logarithmic time $\\log _{10}t$ , and that of Fig.", "4(b) corresponding to Fig.", "3(b) does exponentially with logarithmic time.", "The former indicates the character of order, but the latter, the feature of chaos.", "Only when the integration time adds up to $1\\times 10^{5}$ , can the ordered and chaotic behaviors be identified clearly for the use of FLI unlike the application of Lyapunov exponent.", "There is a threshold value of the FLIs between order and chaos, 5.", "Orbits whose FLI are larger than 5 are chaotic, whereas those whose FLIs are less than 5 are regular.", "The above numerical comparisons seem to tell us that chaos becomes easier when the NLO spin-spin terms are included.", "This sounds reasonable.", "As claimed in [20,21], the system (1) is integrable and not at all chaotic when the spin-spin couplings are turned off.", "The occurrence of chaos is completely due to the spin-spin couplings, which include particularly the NLO spin-spin contributions leading to a sharp increase in the strength of nonlinearity.", "In fact, we employ FLIs to find that there are other orbits (such as orbits 2-5 in Table 1), which are not chaotic for the absence of the NLO spin-spin couplings but for the presence of the NLO spin-spin couplings.", "In addition, the strength of the chaoticity of orbits 6-8 increases.", "As a point to illustrate, the other initial conditions beyond Table 1 are those of orbit 1; the starting spin unit vectors of orbit 2 are those of orbit 1, and those of orbits 3-8 are $\\theta _{1}=84.26^{\\circ }$ , $\\phi _{1}=60^{\\circ }$ , $\\theta _{2}=84.26^{\\circ }$ and $\\phi _{2}=45^{\\circ }$ .", "Hereafter, only the dynamics of the complete Hamiltonian (1) with the NLO spin-spin effects included is focused on.", "Table: Values of FLIs and λT d \\lambda T_d for differentorbits.", "FLIa \\textrm {FLIa} corresponds to the NLO spin-spin couplingsturned off.", "FLIb \\textrm {FLIb}, λ\\lambda , λ d \\lambda _d and λT d \\lambda T_d correspond to the NLO spin-spin couplings included." ], [ "Lyapunov and inspiral decay times", "Taking $\\beta =0.5$ , the initial conditions and the initial unit spin vectors of orbit 1 as reference, we start with the spin parameter $\\chi $ at the value 0.2 that is increased in increments of 0.01 up to a final value of 1 and draw dependence of FLI on $\\chi $ in Fig.", "5(a).", "This makes it clear that chaos occurs when $\\chi \\ge 0.7$ .", "Precisely speaking, the larger the spin magnitudes get, the stronger the chaos gets.", "Note that this dependence of chaos on $\\chi $ relies typically on the choice of the initial conditions, the initial unit spin vectors and the other parameters.", "As claimed in [24], there is different dependence of chaos on $\\chi $ if the choice changes.", "On the other hand, taking the initial spin angles of orbit 3, fixing the spin parameter $\\chi =0.90$ and the initial conditions $(\\mathbf {p}(0);\\mathbf {r}(0))=(0, 0.39, 0; 8.4, 0, 0)$ , which correspond to the initial eccentricity $e_0=0.28$ and the initial semi-major axis $a_0=11.6$ , we study the range of the mass ratio $\\beta $ beginning at 0.5 and ending at 1 in increments of 0.01.", "At once, dependence of FLI on $\\beta $ can be described in Fig.", "5(b).", "There is chaos when $\\beta \\le 0.86$ and chaos seems easier for a smaller mass ratio.", "As in the panel (a), this result is given only under the present initial conditions, initial unit spin vectors and other parameters.", "Do the above-mentioned chaotic behaviors occur before the merger of the binaries?", "To answer it, we have to compare the Lyapunov time $T_{\\lambda }=1/\\lambda $ (i.e.", "the inverse of the Lyapunov exponent) with the inspiral decay time $T_d$ , estimated by [31] $T_{d}=\\frac{12}{19}\\frac{c_0^4}{\\gamma }\\int _{0}^{e_0}\\frac{e^{29/19}[1+(121/304)e^{2}]^{1181/2299}}{(1-e^2)^{3/2}} de,$ where the two parameters are $c_{0}=a_{0}(1-e^{2}_{0})e^{-12/19}_{0}(1+\\frac{121}{304}e^{2}_{0})^{-870/2299}$ and $\\gamma =64m_{1}m_{2}M/5$ .", "When $T_{\\lambda }$ is less than $T_d$ (or $\\lambda T_d>1$ ), chaos would be observed.", "Because $T_{\\lambda }=3.0\\times 10^{3}$ and $T_d=1.3\\times 10^{3}$ for orbit 1, the chaoticity can not be seen before the merger.", "Values of $\\lambda T_d$ for orbits 2-8 are listed in Table 1.", "Clearly, only chaotic orbit 8 is what we expect.", "Besides these, we plot two panels (a) and (b) of Fig.", "6 regarding dependence of Lyapunov exponent on single parameter, which correspond respectively to Figs 5(a) and 5(b).", "Here are two facts.", "First, the results in Fig.", "6 are the same as those in Fig.", "5.", "Second, lots of chaotic orbits whose Lyapunov times are many times greater than the inspiral times should be ruled out, and there are only a small quantity of desired chaotic orbits left.", "In order to make the accuracy of the PN approach better, we should choose orbits whose initial radii are larger enough than roughly $10M$ .", "All chaotic orbits in Table 2 are expected.", "Notice that the other initial conditions of these orbits beyond this table are $y=z=p_x=p_z=0$ , and the starting spin angles are still the same as those of orbit 3.", "Although an orbit has a large initial radius, it may still be chaotic when its initial eccentricity is high enough.", "This supports the result of [23] that a high eccentric orbit can easily yield chaos.", "Table: Values of λT d \\lambda T_d for chaotic orbits with biginitial radii when the NLO spin-spin contributions are included." ], [ "Conclusions", "This paper is devoted to studying the dynamics of the complete 3PN conservative Hamiltonian of spinning compact binaries in which the orbital part is accurate to the 3PN order and the spin-spin part as well as the spin-orbit part includes the LO and NLO contributions.", "Because of the high nonlinearity, the NLO spin-spin couplings included give rise to the occurrence of strong chaos in contrast with those excluded.", "By scanning single parameter with the FLIs, we obtained dependence of chaos on the parameter.", "It was shown sufficiently that chaos appears easier for larger spins or smaller mass ratios under the present considered initial conditions, starting unit spin vectors and other parameters.", "So does for a smaller initial radius.", "In spite of this, an orbit with a large initial radius is still possibly chaotic if its initial eccentricity is high enough.", "Above all, there are some observable chaotic orbits whose initial radii are suitably large and whose Lyapunov times are less than the corresponding inspiral times.", "This research is supported by the Natural Science Foundation of China under Grant Nos.", "11173012 and 11178002.", "Figure: Energy errors of orbit 1.", "The NLO spin-spin couplings arenot included in panel (a) but in Panel (b).Figure: Power spectra corresponding to Fig.", "1.Figure: The maximum Lyapunov exponents λ\\lambda corresponding toFig.", "1.Figure: The fast Lyapunov indicators (FLIs) corresponding to Fig.1.Figure: (color online) The FLIs as a function of χ\\chi or β\\beta when the NLO spin-spin interactions are included.", "All FLIs largerthan 5 mean chaos.Figure: (color online) The maximum Lyapunov exponents λ\\lambda corresponding to Fig.", "5.", "Note that λ>λ c \\lambda >\\lambda _c means chaos,and λ>λ d \\lambda >\\lambda _d with λ d =1/T d \\lambda _d=1/T_d indicates theoccurrence of chaos before coalescence." ] ]	1403.0378
[ [ "Warm-Intermediate Inflationary Universe Model with Viscous Pressure in\n High Dissipative Regime" ], [ "Abstract Warm inflation model with bulk viscous pressure in the context of \"intermediate inflation\" where the cosmological scale factor expands as $a(t)=a_0\\exp(At^f)$, is studied.", "The characteristics of this model in slow-roll approximation and in high dissipative regime are presented in two cases: 1- Dissipative parameter $\\Gamma$ as a function of scalar field $\\phi$ and bulk viscous coefficient $\\zeta$ as a function of energy density $\\rho$.", "2- $\\Gamma$ and $\\zeta$ are constant parameters.", "Scalar, tensor perturbations and spectral indices for this scenario are obtained.", "The cosmological parameters appearing in the present model are constrained by recent observational data (WMAP7)." ], [ "Introduction", "Big Bang model has many long-standing problems (horizon, flatness,...).", "These problems are solved in a framework of inflationary universe models [1].", "Scalar field as a source of inflation provides a causal interpretation of the origin of the distribution of large scale structure, and also observed anisotropy of cosmological microwave background (CMB) [2].", "Standard models for inflationary universe are divided into two regimes, slow-roll and reheating epochs.", "In slow-roll period, kinetic energy remains small compared to potential term.", "In this period, all interactions between scalar fields (inflatons) and other fields are neglected and as a result the universe inflates.", "Subsequently, in reheating poch, the kinetic energy is comparable to the potential energy that causes inflaton to begin an oscillation around minimum of the potential while losing their energy to other fields present in the theory.", "After this period, the universe is filled with radiation.", "In warm inflationary models radiation production occurs during inflationary period and reheating is avoided [3].", "Thermal fluctuations may be obtained during warm inflation.", "These fluctuations could play a dominant role to produce initial fluctuations which are necessary for Large-Scale Structure (LSS) formation.", "Density fluctuation arises from thermal rather than quantum fluctuation [4].", "Warm inflationary period ends when the universe stops inflating.", "After this period the universe enters in radiation phase smoothly [3].", "Finally, remaining inflatons or dominant radiation fields created the matter components of the universe.", "In warm inflation models, for simplicity, the particles which are created by the inflaton decay are considered as massless particles (or radiation).", "The existence of massive particles in the inflationary fluid model as a new model of inflation has been considered in Ref.", "[5], perturbation parameters of this model have been presented in Ref.[6].", "In this scenario the existence of massive particles may be altered the dynamics of the inflationary universe by modification the fluid pressure.", "Decay of the massive particles within the fluid is an entropy-producing scalar phenomenon, in other hand \"bulk viscous pressure\" has entropy-producing property.", "Therefore the decay of particles may be considered by a bulk viscous pressure $\\Pi =-3\\zeta H$ [7] where $H$ is Hubble parameter and $\\zeta $ is phenomenological coefficient of bulk viscosity.", "This coefficient is positive-definite by the second law of thermodynamics and depends on the energy density of the fluid.", "In this work we would like to consider the warm inflationary universe with bulk viscous pressure in the particular scenario \"intermediate inflation\" which is denoted by scale factor $a(t)=a_0\\exp (At^f), 0<f<1$ [8].", "The expansion of this model is faster than power-law inflation ($a=t^p; p>1$ ), but slower than standard de sitter inflation ($a=\\exp (Ht)$ ).� In term of string/M-theory [9], if the high order curvature corrections to Einstein-Hilbert action are proportional to the Gauss-Bonnet(GB) term, we obtain a free-ghost action.", "The GB term is the leading order of the $\\alpha $ (inverse string tension) expansion to the low-energy string effective action [10].", "This kind of theory is applied for study of initial singularity problem [11], black hole solutions [12] and the late time universe acceleration [13].", "The coupling of GB with dynamical dilatonic scalar field in $4D$ dark energy model leads to an intermediate form for scale factor, where $f=\\frac{1}{2}$ , $A=\\frac{2}{8\\pi G n}$ with a constant parameter \"n\" [9].", "Intermediate inflation model may be derived from an effective theory at low dimensions of a fundamental string theory.", "Therefore, the study of intermediate inflationary model is motivated by string/M-theory [9].", "In the other hand, It has been shown that, there are eight possible asymptotic solutions for cosmological dynamics [14].", "Three of these solutions have non-inflationary scale factor and another three one's of solutions give de Sitter (with scale factor $a(t)=a_0\\exp (H_0 t)$ ), power-low (with scale factor $a(t)=t^p, p>1$ ), inflationary expansions.", "Two cases of these solutions have asymptotic expansion with scale factor($a=a_0\\exp (A(\\ln t)^{\\lambda })$ , which is named \"logamediate inflation\" and finally intermediate inflation ($a(t)=a_0\\exp (At^f), 0<f<1$ ).", "The warm inflation with bulk viscous has been studied in Refs.", "[5] and [6].", "Intermediate scale factor has been used for non-viscous warm inflation models [15].", "To the best of our knowledge, the warm inflation model with viscous pressure in the context of intermediate inflation has not been yet studied.", "In this paper we will study warm inflationary universe model with bulk viscous pressure in the context of intermediate inflation.", "The paper is organized as: In section II, we give a brief review about warm inflationary universe model with bulk viscous pressure in high dissipative regime.", "In section III, we consider high dissipative warm-intermediate inflationary phase in two cases: 1- Dissipative parameter $\\Gamma $ as a function of field $\\phi $ and viscous coefficient $\\zeta $ as a function of energy density of the inflation fluid.", "2- Constant dissipative parameters $\\Gamma $ and constant viscous coefficient $\\zeta $ .", "In this section we also, investigate the cosmological perturbations for our model.", "Finally in section IV, we present a conclusion." ], [ "The model", "Warm inflation model in a spatially flat Friedmann Robertson Walker (FRW) universe which is filled with a scalar field $\\phi $ and an imperfect fluid is studied.", "Scalar field $\\phi $ or inflaton has energy density $\\rho _{\\phi }=\\frac{1}{2}\\dot{\\phi }^2+V(\\phi )$ .", "Imperfect fluid is a mixture of matter and radiation with adiabatic index $\\gamma $ , energy density $\\rho =Ts(\\phi ,T)$ ($T$ is the temperature and $s$ is the entropy density of the imperfect fluid [16].)", "and total pressure $P+\\Pi $ .", "$\\Pi =-3\\zeta H $ is viscous pressure[7], where $\\zeta $ is phenomenological coefficient of bulk viscosity.", "Friedmann equation of this model is $3H^2=\\frac{\\dot{\\phi }^2}{2}+V(\\phi )+\\rho $ where we choose $c=\\hbar =8\\pi G=1$ .", "Inflation field $\\phi $ decays into the imperfect fluid with rate $\\Gamma $ , so the conservation equation of fluid and inflaton field have these forms $\\dot{\\rho }+3H(\\rho +P+\\Pi )=\\dot{\\rho }+3H(\\gamma \\rho +\\Pi )=\\Gamma \\dot{\\phi }^2$ and $\\dot{\\rho }_{\\phi }+3H(\\rho _{\\phi }+P_{\\phi })=-\\Gamma \\dot{\\phi }^2\\Rightarrow \\ddot{\\phi }+(3H+\\Gamma )\\dot{\\phi }=-V^{\\prime }$ respectively.", "Where $V^{\\prime }=\\frac{dV}{d\\phi }$ , $P=(\\gamma -1)\\rho $ .", "Dissipation term denotes the inflaton decay into the imperfect fluid in the inflationary epoch.", "We would like to express the evolution equation (REF ) in terms of entropy density $s(\\phi ,T)$ .", "This parameter is defined by a thermodynamical relation [16] $s(\\phi ,T)=-\\frac{\\partial f}{\\partial T}=-\\frac{\\partial V}{\\partial T}.$ $f$ is Helmholtz free energy which is defined by $f=\\rho _{T}-Ts=\\frac{1}{2}\\dot{\\phi }^2+V(\\phi )+\\rho -Ts$ Free energy $f$ is dominated by the thermodynamical potential $V(\\phi , T)$ in slow-roll limit.", "The total energy density and total pressure are given by $\\rho _T=\\frac{1}{2}\\dot{\\phi }^2+V(\\phi )+Ts~~~~~~~~~~~~~~~~\\\\\\nonumber P_T=\\frac{1}{2}\\dot{\\phi }^2-V(\\phi )+(\\gamma -1)Ts+\\Pi $ The viscous pressure for an expanding universe is negative ($\\Pi =-3\\zeta H$ ), therefore this term acts to decrease the total pressure.", "Using Eq.", "(REF ), we can find the entropy density evolution for our model as $T\\dot{s}+3H(\\gamma Ts+\\Pi )=\\Gamma \\dot{\\phi }^2$ In the above equation, it is assumed that $\\dot{T}$ is negligible.", "For a quasi-equilibrium high temperature thermal bath as an inflationary fluid, we have $\\gamma =\\frac{4}{3}$ .", "The bulk viscosity effects may be read from above equation.", "Thus bulk viscous pressure $\\Pi $ as a negative quantity, enhances the source of entropy density on the RHS of the evolution equation (REF ).", "Therefore, energy density of radiation and entropy density increase by the bulk viscosity pressure $\\Pi $ (see FIG.1 and FIG.2).", "During the inflationary phase the energy density of inflation field $\\phi $ is the order of the potential, i.e.", "$\\rho _{\\phi }\\sim V(\\phi ),$ and this energy density dominates over the energy of imperfect fluid, i.e.", "$\\rho _{\\phi }>\\rho ,$ this limit is called stable regime [16].", "So the Friedmann equation (REF ) reduces to $3H^2=V(\\phi )$ In slow-roll limit, it is assumed that $\\dot{\\phi }^2\\ll V(\\phi ),$ and $\\ddot{\\phi }\\ll (3H+\\Gamma )\\dot{\\phi }$ [17].", "When the decay of the inflaton to imperfect fluid is quasi-stable, we have $\\dot{\\rho }\\ll 3H(\\gamma \\rho +\\Pi ),$ and $\\dot{\\rho }\\ll \\Gamma \\dot{\\phi }^2$ .", "Therefore the equations (REF ) and (REF ) are reduced to $3H(1+r)\\dot{\\phi }=-V^{\\prime }$ and $\\rho \\simeq \\frac{r\\dot{\\phi }^2-\\Pi }{\\gamma }$ where $r=\\frac{\\Gamma }{3H}$ .", "In the present work we will restrict our analysis in high dissipative regime, i.e.", "$r\\gg 1,$ where the dissipation coefficient $\\Gamma $ is much greater than $3H$ .", "The reason of this choice is as following.", "In weak dissipative, i.e.", "$r\\ll 1$ , the expansion of the universe in the inflationary era disperses the decay of the inflaton.", "There is a little chance for interaction between the sectors of the inflationary fluid, therefore we do not have non-negligible bulk viscosity.", "Warm inflation in high and weak dissipative regimes for a model without bulk viscous pressure have been studied in Refs.", "[3] and [17] respectively.", "Dissipation parameter $\\Gamma $ may be constant or a positive function of inflaton $\\phi $ and temperature $T$ by the second law of thermodynamics.", "There are some specific forms for the dissipative coefficient, with the most common which are found in the literatures being the $\\Gamma \\sim T^3$ form [16],[18],[19],[20].", "In some works $\\Gamma $ and potential of the inflaton have the same form [21].", "In Ref.", "[6], perturbation parameters for warm inflationary model with viscous pressure have obtained where $\\Gamma =\\Gamma (\\phi )=V(\\phi )$ and $\\Gamma =\\Gamma _0=const$ .", "In this work we will study the intermediate warm inflation with viscous pressure in high dissipative regime for these two cases.", "Slow-roll parameters $\\epsilon $ and $\\eta $ in high dissipative regime are given by [6] $\\epsilon \\equiv -\\frac{\\dot{H}}{H^2}=\\frac{1}{2r}[\\frac{V^{\\prime }}{V}]^2$ and $\\eta \\equiv -\\frac{\\ddot{H}}{H\\dot{H}}=\\frac{1}{r}[\\frac{V^{\\prime \\prime }}{V}-\\frac{1}{2}(\\frac{V^{\\prime }}{V})^2]$ respectively.", "We consider potentials of the form [22] $V=V_0\\phi ^n$ where $V_0$ is a constant.", "We restrict the model in the region $\\phi >0$ where the above potential is positive for all $n$ .", "The slow-roll condition ($\\eta ,\\epsilon \\ll 1$ ), are satisfied when $\\phi ^2$ is much greater than $\\frac{n^2}{r}$ .", "Therefore these potentials are classified as \"large field\" models [23].", "By using Eqs.", "(REF ), (REF ) and (REF ) in slow-roll limit, a relation between the energy densities $\\rho _{\\phi }$ and $\\rho $ is obtained as $\\rho =\\frac{1}{\\gamma }[\\frac{2}{3}\\epsilon \\rho _{\\phi }-\\Pi ]$ Using inflation condition, i.e.", "$\\ddot{a}>1,$ or equivalently $\\epsilon <1,$ and above equation, warm inflation epoch with viscose pressure could take place when $\\rho _{\\phi }>\\frac{3}{2}[\\gamma \\rho +\\Pi ]$ Our warm inflation model comes to close when $\\rho _{\\phi }\\simeq \\frac{3}{2}[\\gamma \\rho +\\Pi ]$ .", "The number of e-folds in high dissipative regime is given by $N(\\phi )=-\\int _{\\phi _i}^{\\phi _f} r\\frac{V}{V^{\\prime }}d\\phi $ where $\\phi _i$ and $\\phi _f$ are inflaton at the begining and end of inflation, respectively." ], [ "Intermediate inflation", "In this section we will study high dissipative warm inflation with viscous pressure in the context of intermediate inflation.", "The scale factor of intermediate inflation follows the law $a(t)=a_0\\exp (At^f),~~~~~0<f<1$ where $A$ is a positive constant with unit $m_{p}^f$ .", "We consider our model in two cases [6]: 1- $\\Gamma $ is a function of scalar field $\\phi $ and $\\zeta $ is a function of energy density $\\rho $ .", "2- $\\Gamma $ and $\\zeta $ are constant parameters." ], [ "$\\Gamma =\\Gamma (\\phi )=V(\\phi )$ , {{formula:619f6060-ac68-45ff-a75a-41b557dac1b5}} case", "By using Eqs.", "(REF ) and (REF ) in this case, we get the scalar field $\\phi $ , Hubble parameter $H(\\phi )$ and potential $V(\\phi )$ as $\\phi =2\\sqrt{2(1-f)t}$ $H(\\phi )=fA(\\frac{\\phi }{2\\sqrt{2(1-f)}})^{2f-2}$ and the potential in this case has form (REF ), where $n=4f-4~~~~~~~~~~~~~~V_0=3(fA)^2(2\\sqrt{2(1-f)})^{-n}$ Energy density $\\rho $ is obtained from Eq.", "(REF ) as $\\rho =\\frac{2fA(1-f)[\\frac{\\phi }{2\\sqrt{2(1-f)}}]^{2f-4}}{\\gamma -3\\zeta _1fA[\\frac{\\phi }{2\\sqrt{2(1-f)}}]^{2f-2}}$ Bulk viscous relation only holds for small deviations from equilibrium, so we consider this limitation as $\\mid \\Pi \\mid \\ll \\rho $ From two above equations the region of $\\phi $ is presented by $\\frac{\\phi }{2\\sqrt{2(1-f)}}\\gg (3fA\\zeta _1)^{\\frac{1}{2(1-f)}}$ The entropy density in terms of inflaton field $\\phi $ may be obtained from above equation $Ts=\\frac{2fA(1-f)[\\frac{\\phi }{2\\sqrt{2(1-f)}}]^{2f-4}}{\\gamma -3\\zeta _1fA[\\frac{\\phi }{2\\sqrt{2(1-f)}}]^{2f-2}}$ In FIG.1, we plot the entropy density in terms of scalar field.", "Figure: We plot the entropy density ss in terms of inflaton φ\\phi where, Π=0\\Pi =0 by dashed curve and Π=-3ζ 1 ρH\\Pi =-3\\zeta _1\\rho H by blue curve (T=5.47×10 -5 T=5.47\\times 10^{-5}, f=1 2,γ=1.5,A=5.01×10 8 ,ζ 1 =0.2×10 -8 f=\\frac{1}{2}, \\gamma =1.5, A=5.01\\times 10^{8}, \\zeta _1=0.2\\times 10^{-8})Using Eq.", "(REF ) slow-roll parameter $\\epsilon $ in terms of inflaton $\\phi $ is given by $\\epsilon =\\frac{1-f}{fA}(\\frac{\\phi }{2\\sqrt{2(1-f)}})^{-2f}$ Likewise, using (REF ) the slow-roll parameter $\\eta $ has this form $\\eta =\\frac{3-2f}{2fA}(\\frac{\\phi }{2\\sqrt{2(1-f)}})^{-2f}$ The inflation condition $\\ddot{a}>0$ (or equivalently $\\epsilon <1$ ) for this example is satisfied when $\\phi ^2>8(1-f)(\\frac{1-f}{fA})^{\\frac{1}{f}}$ From equation (REF ), the number of e-folds between initial and final fields $\\phi _i$ and $\\phi _f$ is $N(\\phi )=A([\\frac{\\phi _f}{2\\sqrt{2(1-f)}}]^{2f}-[\\frac{\\phi _i}{2\\sqrt{2(1-f)}}]^{2f})$ We find $\\phi _i$ at the begining of inflation (when $\\epsilon \\simeq 1$ ) $\\phi _i=2\\sqrt{2(1-f)}(\\frac{1-f}{fA})^{\\frac{1}{2f}}$ So, we could find the value of $\\phi _f$ in terms of $N$ , $A$ and $f$ as $\\phi _f=2\\sqrt{2(1-f)}[\\frac{N}{A}+\\frac{1-f}{fA}]^{\\frac{1}{2f}}$ Now, we study the scalar and tensor perturbation spectrums for our model in this case ($\\Gamma =V,\\zeta =\\zeta _1\\rho $ ).", "The power-spectrum of the curvature perturbation have the form [6] $P_R=\\frac{1}{2\\pi ^2}\\exp (-2\\Im (\\phi ))[\\frac{T_r}{\\epsilon \\sqrt{rV^3}}]$ The amplitude of tensor perturbation which could produce gravitational waves during inflation is given by $A_{g}^2=2(\\frac{H}{2\\pi })^2\\coth [\\frac{k}{2T}]\\simeq \\frac{f^2A^2}{2\\pi ^2}(\\frac{\\phi }{2\\sqrt{2(1-f)}})^{4f-4}\\coth [\\frac{k}{2T}]$ In the above equations the temperature of thermal background of gravitational wave has found in extra factor $\\coth [\\frac{k}{2T}]$ and $\\Im (\\phi )=-\\int {\\lbrace \\frac{\\Gamma ^{\\prime }}{3Hr}+\\frac{3}{8}[1-((\\gamma -1)+\\frac{\\Pi }{\\zeta }\\frac{d\\zeta }{d\\rho })\\frac{\\Gamma ^{\\prime }V^{\\prime }}{9\\gamma rH^2}]\\frac{V^{\\prime }}{V}}\\rbrace d\\phi \\\\\\nonumber =-\\frac{11}{2}(1-f)\\ln (\\phi )+\\alpha [\\frac{\\phi -\\phi _0}{2\\sqrt{2(1-f)}}]^{2f-4}\\\\\\nonumber +\\beta (\\frac{\\phi }{2\\sqrt{2(1-f)}})^{4f-6}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ where $\\alpha =\\frac{12(\\gamma -1)(1-f)^2fA}{(2-f)\\gamma }$ and $\\beta =-\\frac{18\\zeta _1(1-f)^2}{\\gamma (3-2f)}$ .", "From Eqs.", "(REF ) and (REF ) in high dissipative regime the tensor-to-scalar ratio is given by $R(k_0)=(\\frac{A_g^2}{P_R})\|_{k=k_0}=\\frac{2}{3}[\\frac{\\epsilon \\sqrt{rV^5}}{T_r}]\\exp (2\\Im (\\phi ))\\coth [\\frac{k}{2T}]\|_{k=k_0}$ Using seven-year Wilkinson microwave anisotropy probe (WMAP7) observational data we find an upper bound for $R=0.21<0.36$ [2].", "Spectral indices $n_g$ and $n_s$ in the present case are $n_g=\\frac{d}{d\\ln k}\\ln [\\frac{A_g^2}{\\coth [\\frac{k}{2T}]}]=-2\\epsilon =\\frac{2(f-1)}{fA} (\\frac{\\phi }{2\\sqrt{2(1-f)}})^{-2f}$ and $n_s=1-\\frac{4}{25}\\frac{d\\ln P_R}{d\\ln k}\\approx 1-[\\epsilon +2\\eta +\\sqrt{\\frac{2\\epsilon }{r}}[2\\Im ^{\\prime }(\\phi )-\\frac{r^{\\prime }}{2r}]]\\\\\\nonumber \\simeq 1-(2\\eta -5\\epsilon )=1-\\frac{3f-2}{fA}(\\frac{\\phi }{2\\sqrt{2(1-f)}})^{-2f}$ where $d\\ln k(\\phi )=-dN(\\phi )$ .", "Since $0<f<1$ we obviously see that the Harrison-Zeldovich spectrum (i.e.", "$n_s=1$ ) occurs for $f=\\frac{2}{3}$ which agrees with and non-viscous inflation models [8], [24], [25].", "$n_s>1$ is equivalent to $f<\\frac{2}{3}$ and $n_s<1$ is equivalent to $f>\\frac{2}{3}$ .", "Using the limitation of $\\phi $ (REF ) in this case, which is given by the condition (REF ), and using Eq.", "(REF ), we could find nearly scale invariant spectrum ($n\\simeq 1$ ) for all $f,$ which agrees with WMAP7 observational data [2].", "Running of the scalar spectral index is an important cosmological parameter which may be obtained by WMAP7 data $\\alpha _s=\\frac{dn_s}{d\\ln k}=-\\frac{dn_s}{d\\phi }\\frac{d\\phi }{dN (\\phi )}=-\\sqrt{\\frac{2\\epsilon }{r}}[\\epsilon ^{\\prime }+2\\eta ^{\\prime }]\\\\\\nonumber -\\frac{\\epsilon }{r}[(\\frac{\\epsilon ^{\\prime }}{\\epsilon }-\\frac{r^{\\prime }}{r})[2\\Im ^{\\prime }-\\frac{r^{\\prime }}{2r}]+[4\\Im ^{\\prime \\prime }-(\\ln r)^{\\prime \\prime }]]\\\\\\nonumber \\simeq \\frac{2-3f}{2\\sqrt{2(1-f)}}\\frac{V^{\\prime }}{rV}(\\frac{\\phi }{2\\sqrt{2(1-f)}})^{-2f-1}~~~~~~$ In term of WMAP7 results $\\alpha _s$ is approximately $-0.038$ [2].", "In the next subsection we will consider the specific case in which the dissipative parameter $\\Gamma $ and coefficient of bulk viscosity $\\zeta $ are constant parameters." ], [ "$\\Gamma =\\Gamma _0$ , {{formula:b80ff7f7-2d7f-43de-9aaf-21154f64a37e}} case ", "Where $\\Gamma =\\Gamma _0$ and $\\zeta =\\zeta _0$ and by using Eqs.", "(REF ) and (REF ) we get $\\phi =\\varpi t^{f-\\frac{1}{2}}$ where $\\varpi =\\frac{2fA}{2f-1}\\sqrt{\\frac{6(1-f)}{\\Gamma _0 }}$ .", "The effective potential and Hubble parameters in this case are obtained as $H=fA(\\frac{\\phi }{\\varpi })^{\\frac{2f-2}{2f-1}}$ and the potential in this case has form (REF ), where $n=\\frac{4f-4}{2f-1}~~~~~~~~~~~~~~V_0=3f^2A^2(\\varpi )^{-n}$ Viscous pressure $\\Pi $ and energy density $\\rho $ are obtained from Eqs.", "(REF ), (REF ) and (REF ) $\\Pi =-3\\zeta H=-3fA\\zeta _0(\\frac{\\phi }{\\varpi })^{\\frac{2f-2}{2f-1}}~~~~~~~~~\\\\\\nonumber \\rho =\\frac{fA}{\\gamma }[2(1-f)(\\frac{\\phi }{\\varpi })^{\\frac{2f-4}{2f-1}}+3\\zeta _0(\\frac{\\phi }{\\varpi })^{\\frac{2f-2}{2f-1}}]$ Using the above equation and Eq.", "(REF ) we could constrain the scalar field $\\phi $ , as: $f<\\frac{1}{2}~~\\Rightarrow ~~\\phi \\gg \\varpi (\\frac{\\zeta _0(\\gamma -1)}{2(1-f)})^{\\frac{1-2f}{2}}\\\\\\nonumber f>\\frac{1}{2}~~\\Rightarrow ~~\\phi \\ll \\varpi (\\frac{\\zeta _0(\\gamma -1)}{2(1-f)})^{\\frac{1-2f}{2}}$ We can find the entropy density $s$ in terms of scalar field $Ts=\\frac{fA}{\\gamma }[2(1-f)(\\frac{\\phi }{\\varpi })^{\\frac{2f-4}{2f-1}}+3\\zeta _0(\\frac{\\phi }{\\varpi })^{\\frac{2f-2}{2f-1}}]$ The entropy density and energy density in term of our model in this case increase by the bulk viscosity effect (see FIG.2).", "Figure: We plot the entropy density ss in terms of inflaton φ\\phi where, Π=0\\Pi =0 by dashed curve and Π=-3ζ 0 H\\Pi =-3\\zeta _0 H by blue curve (T=5.47×10 -5 T=5.47\\times 10^{-5}, f=3 4,γ=1.5,A=2×10 8 ,ζ 0 =1,Γ 0 =5.4×10 17 f=\\frac{3}{4}, \\gamma =1.5, A=2\\times 10^{8}, \\zeta _0=1, \\Gamma _0=5.4\\times 10^{17})Using Eqs.", "(REF ) and (REF ) the slow-roll parameters $\\epsilon $ and $\\eta $ in term of scalar field $\\phi $ are $\\epsilon \\simeq \\frac{1-f}{fA}(\\frac{\\phi }{\\varpi })^{\\frac{-2f}{2f-1}}\\\\\\nonumber \\eta \\simeq \\frac{1}{2fA}(\\frac{\\phi }{\\varpi })^{\\frac{-2f}{2f-1}}$ respectively.", "The inflation condition $\\ddot{a}>0$ (or equivalently $\\epsilon <1$ ) for this case is satisfied when $\\nonumber \\phi ^2>\\varpi (\\frac{1-f}{fA})^{\\frac{2f-1}{2f}}$ From Eq.", "(REF ) the number of e-folds is found $N=A[(\\frac{\\phi _f}{\\varpi })^{\\frac{2f}{2f-1}}-(\\frac{\\phi _i}{\\varpi })^{\\frac{2f}{2f-1}}]$ At the begining of the inflation period ($\\epsilon \\simeq 1$ ) $\\phi _i=\\varpi (\\frac{1-f}{fA})^{\\frac{2f-1}{2f}}$ , so the scalar field $\\phi _f$ at the end of inflation in term of the number of e-folds becomes $\\phi _f=\\varpi (\\frac{N}{A}+\\frac{1-f}{fA})^{\\frac{2f-1}{2f}}$ In the following we will find the perturbation parameters in terms of scalar field.", "From Eq.", "(REF ) we have $\\Im (\\phi )=-\\frac{3}{8}\\ln (V(\\phi ))$ The spectrum of the curvature perturbation in slow-roll limit in this case ($\\Gamma =\\Gamma _0$ , $\\zeta =\\zeta _0$ ), from above equation and equation (REF ), has the form $P_{R}\\approx B(\\frac{\\phi }{\\varpi })^{\\frac{2}{2f-1}}$ where $B=\\frac{T_r}{2\\pi ^2(1-f)\\sqrt{\\Gamma _0\\sqrt{3}}}$ .", "This parameter is found from WMAP7 results ($P_R=2.28\\times 10^{-9}$ )[2].", "Using Eq.", "(REF ) and (REF ) the amplitude of tensor perturbation becomes $A_g^{2}=\\frac{f^2A^2}{2\\pi ^2}(\\frac{\\phi }{\\varpi })^{\\frac{4f-4}{2f-1}}\\coth [\\frac{k}{2T}]$ From Eq.", "(REF ) the spectral index $n_s$ is given by $n_s=1-\\frac{2}{fA}(\\frac{\\phi }{\\varpi })^{\\frac{-2f}{2f-1}}$ For intermediate inflation where $0<f<1$ , the scalar index $n_s$ for this example becomes $n_s<1$ .", "By using the limitation (REF ) and above equation, we could obtain nearly scale invariant spectrum ($n\\simeq 1$ ) for $f>\\frac{1}{2}$ .", "Using Eqs.", "(REF ) and (REF ) we can re-express the above index in terms of number of e-folding $n_s=1-\\frac{2}{1+f(N-1)}$ and from above equation we find the value of $f$ in terms of $N$ and $n_s$ as $f=\\frac{1+n_s}{(1+n_s)(N-1)}$ Using WMAP7 observational data $n_s\\simeq 0.96$ and $N=60$ as a standard benchmark we obtain $f\\simeq 0.83$ .", "This amount of $f$ is also in the region $f>\\frac{1}{2}$ .", "From Eq.", "(REF ) the spectral index $n_g$ becomes $n_g=\\frac{2(f-1)}{fA}(\\frac{\\phi }{\\varpi })^{\\frac{-2f}{2f-1}}$ We could find the tensor-scalar ratio as $R=\\frac{(1-f)\\sqrt{\\sqrt{3}\\Gamma _0}f^2A^2}{T_r}(\\frac{\\phi }{\\varpi })^{\\frac{4f-6}{2f-1}}\\coth [\\frac{k}{2T}]\\\\\\nonumber =\\frac{(1-f)\\sqrt{\\Gamma _0\\sqrt{3}}f^2A^2}{T_r}\\coth [\\frac{k}{2T}][\\frac{2}{fA(1-n_s)}]^{\\frac{4f-6}{2f-1}}$ In FIG.3, the dependence of the tensor-to-scalar ratio on the spectrum index is shown.", "Three different values for parameter $\\Gamma _0$ have been used in this figure, when the value $f=\\frac{3}{4}$ is taken.", "We note that for different values of $\\Gamma _0$ which are bounded from below, $\\Gamma _0>0.275$ our model is well supported by WMAP data.", "Figure: We plot the evolution of the tensor-scalar ratio rr versus spectrum index n s n_s, for three cases: 1- Γ 0 =0.275\\Gamma _0=0.275 by green line,2- Γ=27.5\\Gamma =27.5 by red dashed line, 3- Γ 0 =2.75×10 4 \\Gamma _0=2.75\\times 10^4 by black dashed line.", "(f=3 4f=\\frac{3}{4}, A=1,A=1, T=T r =5.47×10 -5 ,T=T_r=5.47 \\times 10^{-5}, k=0.002Mpc -1 k=0.002 Mpc^{-1})Running of the scalar spectral index is obtained from Eq.", "(REF ) $\\alpha _s=\\frac{dn_s}{d\\ln k}=-\\frac{dn_s}{d\\phi }\\frac{d\\phi }{dN}=-\\frac{4f}{r(2f-1)\\varpi }\\frac{V^{\\prime }}{rV}(\\frac{\\phi }{\\varpi })^{\\frac{-4f+1}{2f-1}}$ This parameter may be found from WMAP7 observational data [2].", "Using WMAP7 data, $P_R(k_0)=\\simeq 2.28\\times 10^{-9}$ , $R(k_0)\\simeq 0.21$ and the characteristic of warm inflation $T>H$ [17], we may restrict the values of temperature to $T_r>5.47\\times 10^{-5}M_4$ using Eqs.", "(REF ), (REF ), (see FIG.4).", "We have chosen $k_0=0.002 Mpc^{-1}$ and $T\\simeq T_r$ .", "Note that, because of the bulk viscous pressure, the radiation energy density in our model increases.", "Therefore the minimum value of temperature for our model ($5.47\\times 10^{-5}M_4$ ) is bigger than the minimum value of temperature ($3.42\\times 10^{-6}M_4$ ) for the model without the viscous pressure effects [26].", "Figure: In this graph we plot the Hubble parameter HH in term of the temperature T r T_r.", "We can find the minimum amount of temperature T r =5.47×10 -5 T_r=5.47\\times 10^{-5} in order to have the necessary condition for warm inflation model (T r >HT_r>H)." ], [ "Conclusion", "In this article we have investigated the warm-intermediate inflationary model with viscous pressure.", "We have studied this scenario in two different cases of the dissipative coefficient $\\Gamma $ and bulk viscous coefficient $\\zeta $ .", "Our model have been described for $\\Gamma =\\Gamma _0=const$ , $\\zeta =\\zeta _0$ and for $\\Gamma $ as a function of field $\\phi $ , i.e.", "$\\Gamma =f(\\phi )=V(\\phi )$ , $\\zeta $ as a function of energy density $\\rho $ .", "For these two cases we have extracted the form of potential and Hubble parameters as a function of scalar field $\\phi $ .", "In $\\Gamma =f(\\phi )=V(\\phi )$ case, we introduced scalar field potential as $V(\\phi )\\propto \\phi ^{4(f-1)}, 0<f<1$ , but in non-viscous inflation potential has the form $\\phi ^{-4\\frac{1-f}{f}}$ .", "In this case, it is possible in the slow-roll approximation to have the Harrison-Zeldovich spectrum of density perturbation (i.e.", "$n_s=1$ ), provided $f$ takes the value of $\\frac{2}{3}$ which agrees with regular inflation model with a canonical scalar field characterized by a quasi-exponential expansion.", "Explicit expressions for tensor-scalar ratio $R$ , spectrum indices $n_g$ and $n_s$ , running of the scalar spectral index $\\alpha _s$ in slow-roll were obtained.", "We also have constrained these parameters by WMAP7 results." ] ]	1403.0186
[ [ "Teleportation of entanglement over 143 km" ], [ "Abstract As a direct consequence of the no-cloning theorem, the deterministic amplification as in classical communication is impossible for quantum states.", "This calls for more advanced techniques in a future global quantum network, e.g.", "for cloud quantum computing.", "A unique solution is the teleportation of an entangled state, i.e.", "entanglement swapping, representing the central resource to relay entanglement between distant nodes.", "Together with entanglement purification and a quantum memory it constitutes a so-called quantum repeater.", "Since the aforementioned building blocks have been individually demonstrated in laboratory setups only, the applicability of the required technology in real-world scenarios remained to be proven.", "Here we present a free-space entanglement-swapping experiment between the Canary Islands of La Palma and Tenerife, verifying the presence of quantum entanglement between two previously independent photons separated by 143 km.", "We obtained an expectation value for the entanglement-witness operator, more than 6 standard deviations beyond the classical limit.", "By consecutive generation of the two required photon pairs and space-like separation of the relevant measurement events, we also showed the feasibility of the swapping protocol in a long-distance scenario, where the independence of the nodes is highly demanded.", "Since our results already allow for efficient implementation of entanglement purification, we anticipate our assay to lay the ground for a fully-fledged quantum repeater over a realistic high-loss and even turbulent quantum channel." ], [ "1,2]Thomas Herbst 1]Thomas Scheidl 1]Matthias Fink 1]Johannes Handsteiner 1,2]Bernhard Wittmann 1,2]Rupert Ursin 1,2]Anton Zeilinger [1]Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria [2]Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria [ ] [ ](Dated: 2023/01/05 23:34:50) labelfont=bf As a direct consequence of the no-cloning theorem [1], the deterministic amplification as in classical communication is impossible for quantum states.", "This calls for more advanced techniques in a future global quantum network [2], [3], e.g.", "for cloud quantum computing [4], [5], [6].", "A unique solution is the teleportation of an entangled state, i.e.", "entanglement swapping [7], representing the central resource to relay entanglement between distant nodes.", "Together with entanglement purification [8], [9], [10], [11] and a quantum memory [12], [13] it constitutes a so-called quantum repeater [9], [14].", "Since the afore mentioned building blocks have been individually demonstrated in laboratory setups only, the applicability of the required technology in real-world scenarios remained to be proven.", "Here we present a free-space entanglement-swapping experiment between the Canary Islands of La Palma and Tenerife, verifying the presence of quantum entanglement between two previously independent photons separated by 143 km.", "We obtained an expectation value for the entanglement-witness operator, more than 6 standard deviations beyond the classical limit.", "By consecutive generation of the two required photon pairs and space-like separation of the relevant measurement events, we also showed the feasibility of the swapping protocol in a long-distance scenario, where the independence of the nodes is highly demanded.", "Since our results already allow for efficient implementation of entanglement purification, we anticipate our assay to lay the ground for a fully-fledged quantum repeater over a realistic high-loss and even turbulent quantum channel.", "The vision of a global quantum internet is to provide unconditionally secure communication [5], [13], blind cloud computing [15] and an exponential speedup in distributed quantum computation [5], [6].", "Teleportation of an entangled state, also known as entanglement swapping, plays a vital role in this vision.", "To date the entanglement-swapping protocol has been implemented in many different systems [16], [17], [18], [19], [20], [21], [22], owing to the fact that it represents a key resource for numerous quantum-information applications.", "Since an unknown single quantum state cannot be cloned nor amplified without destroying its essential quantum feature, the quantum repeater is the main resource for faithful entanglement distribution over long distances.", "The idea is to decompose the total distance into shorter elementary links, over which entanglement is shared, purified and eventually stored in quantum memories from which the entangled states can be retrieved on demand, once all the nodes are readily prepared.", "Finally the entanglement is swapped between adjacent nodes and faithfully extended over the whole distance.", "Entanglement purification and quantum memories serve solely to enhance the efficiency and the fidelity of the protocol, both of which are limited due to imperfection of the sources of entangled particles, of the involved quantum operations and of the interconnecting quantum channels.", "Entanglement swapping however provides the underlying non-classical correlations and constitutes the fundamental resource required for the implementation of a quantum repeater.", "Here we show that we were able to provide this resource via a realistic 143 km long-distance free-space (elementary) link under harsh atmospheric conditions, representing the largest geographical separation for this protocol to date.", "Furthermore, the simultaneous creation of two randomly generated photon pairs drastically reduces the signal-to-noise ratio, leading to technological requirements on the verge of practicability.", "Nonetheless, we ensured space-like separation of the remote measurement events, which is important for certain protocols e.g.", "quantum key distribution [23], [24].", "The entanglement swapping protocol is realized via the generation of two entangled pairs, photons \"0\" and \"1\" and photons \"2\" and \"3\", for example the maximally entangled singlet states $&\\left\| \\Psi ^- \\right\\rangle _{01}=\\frac{1}{\\sqrt{2}}(\\left\| H \\right\\rangle _0\\left\| V \\right\\rangle _1-\\left\| V \\right\\rangle _0\\left\| H \\right\\rangle _1)\\nonumber \\\\&\\left\| \\Psi ^- \\right\\rangle _{23}=\\frac{1}{\\sqrt{2}}(\\left\| H \\right\\rangle _2\\left\| V \\right\\rangle _3-\\left\| V \\right\\rangle _2\\left\| H \\right\\rangle _3)$ where $\\left\| H \\right\\rangle $ and $\\left\| V \\right\\rangle $ denote the horizontal and vertical polarization states, respectively.", "The product state $\\left\| \\Psi \\right\\rangle _{0123} =\\left\| \\Psi ^- \\right\\rangle _{01}\\otimes \\left\| \\Psi ^- \\right\\rangle _{23}$ may be written as $\\left\| \\Psi \\right\\rangle _{0123}=\\frac{1}{2}[&\\left\| \\Psi ^+ \\right\\rangle _{03}\\otimes \\left\| \\Psi ^+ \\right\\rangle _{12}-\\left\| \\Psi ^- \\right\\rangle _{03}\\otimes \\left\| \\Psi ^- \\right\\rangle _{12}\\nonumber \\\\-&\\left\| \\Phi ^+ \\right\\rangle _{03}\\otimes \\left\| \\Phi ^+ \\right\\rangle _{12}+\\left\| \\Phi ^- \\right\\rangle _{03}\\otimes \\left\| \\Phi ^- \\right\\rangle _{12}].$ Therefore a so-called Bell-state measurement (BSM) between photons \"1\" and \"2\" results randomly in one of the four maximally entangled Bell states $\\left\| \\Psi ^\\pm \\right\\rangle _{12}=\\frac{1}{\\sqrt{2}}(\\left\| H \\right\\rangle _1\\left\| V \\right\\rangle _2\\pm \\left\| V \\right\\rangle _1\\left\| H \\right\\rangle _2)$ and $\\left\| \\Phi ^\\pm \\right\\rangle _{12}=\\frac{1}{\\sqrt{2}}(\\left\| H \\right\\rangle _1\\left\| H \\right\\rangle _2\\pm \\left\| V \\right\\rangle _1\\left\| V \\right\\rangle _2)$ with an equal probability of 1/4.", "By that measurement, photons \"0\" and \"3\" are projected into the same entangled state as photons \"1\" and \"2\".", "Thus the entanglement is swapped from photons \"0-1\" and \"2-3\" to the photons \"1-2\" and \"0-3\" (Figure REF a).", "This procedure may also be seen as teleportation of the state of photon \"1\" onto photon \"3\" or photon \"2\" onto photon \"0\".", "Although the implementation of this protocol, solely based on linear optics, allows distinguishing between two out of four Bell states only [25], it provides a maximal fidelity of 1.", "Here we report successful entanglement swapping in an experiment performed on the Canary Islands, utilizing a 143 km horizontal free-space link between the Jacobus Kapteyn Telescope (JKT) building of the Isaac Newton Group of Telescopes (ING) on La Palma and the Optical Ground Station (OGS) of the European Space Agency (ESA) on Tenerife (Figure REF a).", "Both buildings are located at an altitude of 2400 m. The JKT served as the base station for the production of the two entangled photon pairs, for the BSM between photons \"1\" and \"2\" and for the polarization detection of photon \"0\" at Alice.", "The transmitter telescope, sending photon \"3\" to the receiving station on Tenerife, was installed on the rooftop of the JKT building.", "At the receiver the photons were collected by the 1 m diameter OGS reflector telescope and guided through the optical CoudÃ© path to the setup for polarization analysis and the final measurement by Bob.", "Figure: Entanglement swapping over a 143 km free-space channel between the Canary Islands La Palma and Tenerife.", "a, Experimental scheme.", "Both spontaneous parametric down-conversion (SPDC) sources, the Bell-state measurement (BSM) module and Alice were situated on La Palma and Bob on Tenerife.", "The two SPDC sources generated the entangled photon pairs \"0-1\" and \"2-3\".", "Photons \"1\" and \"2\" (photons are indicated by black numbers on red circles) were subjected to a BSM.", "A 100 m fibre delayed photon \"0\" with respect to photon \"3\", such that Alice's and Bob's measurements were space-like separated.", "Revealing entanglement of photons \"0\" and \"3\" between Alice and Bob verified successful entanglement swapping.", "b, Experimental setup.", "Two polarization-entangled photon pairs Ψ - 01 \\left\| \\Psi ^- \\right\\rangle _{01} and Ψ - 23 \\left\| \\Psi ^- \\right\\rangle _{23} were generated in two identical sources via spontaneous parametric down-conversion (SPDC) in a nonlinear β\\beta -barium borate (BBO) crystal.", "The photons were then coupled into single-mode (SM) fibres with fibre couplers (FC).", "Any polarization rotation in the SM fibres was compensated for by fibre polarization controllers (FPC).", "Photons \"1\" and \"2\" were spectrally filtered with interference filters (IF) with a full width at half maximum (FWHM) of 3 nm and overlapped in a fibre beam splitter (FBS).", "A subsequent polarization-dependent measurement was performed, utilizing a quarter wave plate (QWP), a half wave plate (HWP), a polarizing beam splitter (PBS) and four avalanche photodiodes (APD) a,b,c and d in the Bell-state measurement (BSM).", "Photon \"3\" was guided via a 50 m fibre to the transmitter (Tx) and sent to Bob in Tenerife, while photon \"0\" was delayed by a 100 m fibre before its polarization detection at Alice.", "The receiver (Rx) on Tenerife captured photon \"3\" where Bob performed his polarization-dependent measurement.", "Both Alice and Bob spectrally filtered their photons with IFs with 8 nm FWHM.", "All detection events were time stamped by time-tagging units (TTU) with a resolution of 156 ps and stored for subsequent analysis.", "See main text for detail.In our experimental setup (Figure REF b) a mode-locked femtosecond pulsed Ti:Saph laser emitted light with a central wavelength of 808 nm at a repetition rate of 80 MHz.", "Those near-infrared pulses were then frequency doubled to a central wavelength of 404 nm using second-harmonic generation in a type-I nonlinear $\\beta $ -barium borate (BBO) crystal.", "The individual polarization-entangled photon pairs employed in the protocol were generated via spontaneous parametric down-conversion (SPDC) in two subsequent type-II phase-matched BBOs [26] and coupled into single-mode (SM) optical fibres for spatial mode selection.", "The quality of entanglement was optimized by eliminating the spectral distinguishability [27], [28], [29], which is inherent to pulsed SPDC schemes.", "The first SPDC source provided the entangled state $\\left\| \\Psi ^- \\right\\rangle _{23}$ , with photon \"2\" as one input photon for the BSM and photon \"3\" being guided through a 50 m long SM fibre to the transmitter telescope.", "The second SPDC source prepared the state $\\left\| \\Psi ^- \\right\\rangle _{01}$ , where photon \"1\" was the second input photon for the BSM.", "Photon \"0\" was locally delayed in a 100 m fibre ($\\sim 500$ ns) and subsequently measured by Alice, thus ensuring space-like separation between Alice's and Bob's measurement events [24].", "In La Palma, the BSM was implemented using a tuneable fibre beam splitter (FBS) set to a 50:50 splitting ratio.", "While the spatial overlap of photons \"1\" and \"2\" is inherent to the FBS, a perfect temporal overlap is accomplished in the minimum of the Hong-Ou-Mandel [30] (HOM) interference dip.", "The latter was achieved by adjusting the optical path length for photon \"2\", by linearly moving the SM fibre coupler in the first SPDC source.", "Both output arms of the FBS were equipped with a quarter- and a half-wave plate followed by a polarizing beam splitter (PBS) in order to project on any desired polarization measurement basis.", "Intrinsic polarization rotations in the SM fibres were compensated for with in-fibre polarization controllers.", "The avalanche photodiodes (APDs) a, b, c and d, placed at the four outputs of the two PBSs, were connected to a home-made coincidence logic, providing the two valid outcomes of our BSM: simultaneous clicks of APDs (a $\\&$ d) $\\vee $ (b $\\&$ c) or (a $\\&$ b) $\\vee $ (c $\\&$ d) indicated that photons \"1\" and \"2\" were projected onto the maximally entangled $\\left\| \\Psi ^- \\right\\rangle _{12}$ singlet or $\\left\| \\Psi ^+ \\right\\rangle _{12}$ triplet Bell state, respectively.", "As can be seen from Eq.", "(2), conditioned on these BSM results, photons \"0\" and \"3\" were thus simultaneously projected onto the very same states $\\left\| \\Psi ^- \\right\\rangle _{03}$ and $\\left\| \\Psi ^+ \\right\\rangle _{03}$ , respectively.", "The projection onto the other Bell-states $\\left\| \\Phi ^\\pm \\right\\rangle _{12}$ does not result in a coincidence detection event by the BSM and thus cannot be resolved with a linear-optics scheme.", "Furthermore, the two valid BSM outcomes together with Alice's detection events of photon \"0\" (APDs e and f) were fed into a logic AND gate, providing four possible combinations.", "These local 3-fold coincidence events on La Palma as well as the remote detection events of photon \"3\" on Tenerife (APDs g and h) were then recorded by two separate time-tagging units (TTUs) with a temporal resolution of 156 ps.", "In order to retrieve the final 4-folds between Alice's events and those measured on Bob's side we calculated the cross-correlation between the remotely recorded individual measurement data - both synchronized to GPS standard time.", "To compensate for residual relative clock drifts between the distant TTU-clocks we employed entanglement assisted clock synchronization [31] between consecutive 30 s measurements, allowing for a coincidence-time window of down to 5 ns.", "Figure: A density plot of the entanglement-swapping visibility versus path-length difference and pump power measured locally on La Palma.", "The abscissa represents the relative optical path-length difference Δl\\Delta l between photon \"1\" and \"2\" in the Bell-state measurement and the ordinate represents the 2-fold count rate of the sources.", "All measured data points are indicated by black dots in the density plot.", "The entanglement-swapping visibility is illustrated by rainbow colours where the entanglement-witness bound and the bound for the violation of a Clauser-Horne-Shimony-Holt (CHSH) type Bell inequality are represented by a white and a black isoline at a visibility of 1/3 and 1/21/\\sqrt{2}, respectively.", "The hatched areas between visibilities of 1/3 and 1/21/\\sqrt{2} and above 1/21/\\sqrt{2} mark the regions where entanglement can be shown by an entanglement witness and a CHSH type Bell inequality, respectively.", "A plot of the individual measurement runs including the respective Gaussian fits is illustrated in the inset.", "The operating point (OP) for the local test of the CHSH inequality (see orange star in inset) was chosen at a 2-fold rate of 15 kHz and zero delay.", "At this set point a visibility of 0.87 was achieved.", "For the entanglement-swapping experiment via the 143 km and -32 dB free-space link we tuned the setup to a 2-fold rate of 130 kHz and again perfect overlap at zero delay, resulting in a visibility of 0.6 (see red star in inset).", "In the density plot white stars indicate both OPs.The strong average attenuation of -32 dB over the 143 km free-space quantum channel calls for high production rates of the SPDC sources in order to operate well above the noise level of the single-photon detectors on Tenerife.", "However, pumping the SPDC sources with high pump intensities reduces the achievable entanglement-swapping visibility due to increased multi-pair emissions.", "Hence, a reasonable trade-off between count rates and visibility was required.", "In order to find the optimal operating point, we locally characterized our setup for various 2-fold count rates of the SPDC sources (Figure REF ).", "The entanglement-swapping visibilities of our setup varied between 0.87 at lowest 2-fold count rate (15 kHz 2-folds, about 1 Hz 4-folds) and 0.49 at full pump power (240 kHz 2-folds, 370 Hz 4-folds).", "A traditional measure of entanglement is constituted by violation of a Clauser-Horne-Shimony-Holt [32] (CHSH) type Bell inequality.", "To accomplish this, a CHSH $S$ -value above the classical bound of $S\\le 2$ needs to be experimentally obtained, being equivalent to an entanglement visibility of $1/\\sqrt{2}\\approx 0.71$ .", "This was only achievable when operating at low pump powers and, given the resulting low count rates, therefore only feasible in the course of a measurement performed locally on La Palma.", "We accumulated data over 8000 s and measured the required S-value for both the singlet $\\left\| \\Psi ^- \\right\\rangle _{03}$ and triplet $\\left\| \\Psi ^+ \\right\\rangle _{03}$ state.", "In total we detected 5647 singlet and 5618 triplet swapping events and violated the inequality with $S_{singlet}=2.487\\pm 0.287$ and $S_{triplet}=2.469\\pm 0.287$ at a 15 kHz 2-fold rate, respectively (Figure REF ).", "This result clearly proves that photons \"0\" and \"3\" have been projected into an entangled state.", "Figure: Summary of the measurement results.", "Over the 143 km free-space link we obtained an expectation value for the entanglement-witness operator WW, more than 6σ6\\sigma below the classical bound of 0.", "This proved the presence of entanglement between the states Ψ - 03 \\left\| \\Psi ^- \\right\\rangle _{03} and Ψ + 03 \\left\| \\Psi ^+ \\right\\rangle _{03}.", "The violation of a Clauser-Horne-Shimony-Holt (CHSH) type Bell inequality was shown locally on La Palma underlining the quality of our setup.", "The bar chart illustrates the mean visibility V ¯\\overline{V} and the entanglement-witness operator WW over the 143 km link as well as the locally measured SS-value.", "All outcomes are given for the swapped states Ψ - 03 \\left\| \\Psi ^- \\right\\rangle _{03} and Ψ + 03 \\left\| \\Psi ^+ \\right\\rangle _{03} with error bars derived from Poissonian statistics, ±1σ\\pm 1\\sigma .", "The numerical values of the measurement results are given in the inset, including the individual visibilities V HV ,V PM V_{HV}, V_{PM} and V RL V_{RL} in the three mutually unbiased bases horizontal/vertical (HV), plus/minus (PM) and right/left (RL) as well as the mean visibility V ¯\\overline{V}, the entanglement-witness operator WW and the locally measured SS-value.In order to reduce the accumulation time in the remote measurement scenario, we set each SPDC source to a local 2-fold rate of 130 kHz, corresponding to a locally detected 4-fold count rate of 100 Hz and an average entanglement visibility of the swapped state of approximately 0.60.", "We measured the expectation value of an entanglement-witness operator $W$ , with $W < 0$ representing a sufficient condition for the presence of entanglement [33].", "Our entanglement-witness operator is given as $W=\\frac{1}{2}-\\frac{1}{4}(1+V_{HV} +V_{PM} +V_{RL}),$ with $V_{HV}, V_{PM}, V_{RL}$ being the correlation visibilities of state $\\left\| \\Psi \\right\\rangle _{03}$ in the three mutually unbiased bases horizontal/vertical (HV), plus/minus (PM) and right/left (RL), respectively.", "The visibility is given by $V=(CC_{max}-CC_{min})/(CC_{max}+CC_{min})$ with the max (min) coincidence counts $CC_{max}$ ($CC_{min}$ ).", "Inserting the measured visibilities into Eq.", "(3) yields a negative expectation value for the entanglement witness operator $W_{singlet}=-0.212\\pm 0.027$ and $W_{triplet}=-0.177\\pm 0.028$ with statistical significances of 7.99 and 6.37 standard deviations $\\sigma $ , respectively (assuming Poissonian photon statistics).", "Hence, we unambiguously demonstrated that the experimentally obtained states between photon \"0\" and \"3\" have become entangled over 143 km (Figure REF ).", "These results were obtained from subsequent 30 s data files, accumulated over a measurement time of 271 min during 4 consecutive nights.", "In total, 506 and 492 entanglement-swapping events have been recorded for the singlet and triplet state, respectively.", "Our data demonstrate successful entanglement swapping via a long-distance free-space link under the influence of highly demanding environmental conditions, in fact more challenging than expected for a satellite-to-ground link.", "This proves the feasibility of quantum repeaters in a future space- and ground-based worldwide quantum internet.", "In particular, in a quantum repeater scheme, a single step of the purification method realized in Ref.", "[11] would increase our obtained visibilities beyond the bound for the violation of a CHSH type Bell inequality even in the remote scenario.", "Together with a reliable quantum memory, our results set the benchmark for an efficient quantum repeater at the heart of a global quantum-communication network.", "The authors thank X.-S. Ma for fruitful discussions and the staff of IAC: F. Sanchez-Martinez, A. Alonso, C. Warden, M. Serra, J. Carlos and the staff of ING: M. Balcells, C. Benn, J. Rey, O. Vaduvescu, A. Chopping, D. GonzÃ¡lez, S. RodrÃguez, M. Abreu, L. GonzÃ¡lez, as well as J. Kuusela, E. Wille and Z. Sodnik from ESA for their support.", "This work was made possible by grants from the European Space Agency (Contract 4000105798/12/NL/CBi), the Austrian Science Foundation (FWF) under projects SFB F4008 and CoQuS, the FFG within the ASAP 7 (No.", "828316) program and the Federal Ministry of Science and Research (BMWF).", "T.H.", "conceived the research, designed, carried out the experiment and analyzed data with the help of T.S., M.F., J.H., B.W.", "and R.U.. A.Z.", "defined the scientific goals, conceived the research, designed and supervised the project and the experimental advancements.", "All authors contributed to the manuscript.", "References Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned.", "Nature 299, 802–803 (1982).", "Bose, S., Vedral, V. & Knight, P. L. Multiparticle Generalization of Entanglement Swapping.", "Physical Review A 57, 822–829 (1998).", "Kimble, H. J.", "The quantum internet.", "Nature 453, 1023–1030 (2008).", "Cirac, J. I., Ekert, A. K., Huelga, S. F. & Macchiavello, C. Distributed Quantum Computation over Noisy Channels.", "Physical Review A 59, 4249–4254 (1999).", "Nielsen, M. A.", "& Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).", "Ladd, T. D. et al.", "Quantum computers.", "Nature 464, 45–53 (2010).", "Zukowski, M., Zeilinger, A., Horne, M. A.", "& Ekert, A. K. \"Event-Ready-Detectors\" Bell experiment via entanglement swapping.", "Physical Review Letters 71, 4287–4290 (1993).", "Bennett, C. H. et al.", "Purification of noisy entanglement and faithful teleportation via noisy channels.", "Physical Review Letters 76, 722–725 (1996).", "Briegel, H. J., Dür, W., Cirac, J.", "& Zoller, P. Quantum repeaters: the role of imperfect local operations in quantum communication.", "Physical Review Letters 81, 5932–5935 (1998).", "Pan, J.-W., Simon, C., Brukner, C. & Zeilinger, A. Entanglement purification for quantum communication.", "Nature 410, 1067–1070 (2001).", "Pan, J.-W., Gasparoni, S., Ursin, R., Weihs, G. & Zeilinger, A.", "Experimental entanglement purification of arbitrary unknown states.", "Nature 423, 417–422 (2003).", "Clausen, C. et al.", "Quantum storage of photonic entanglement in a crystal.", "Nature 469, 508–511 (2011).", "Gisin, N. & Thew, R. Quantum communication.", "Nature Photonics 1, 165–171 (2007).", "Duan, L. M., Lukin, M. D. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics.", "Nature 413, 413–418 (2001).", "Barz, S. et al.", "Demonstration of blind quantum computing.", "Science 335, 303–308 (2012).", "Pan, J.-W., Bouwmeester, D., Weinfurter, H. & Zeilinger, A.", "Experimental Entanglement Swapping: Entangling Photons That Never Interacted.", "Physical Review Letters 80, 3891–3894 (1998).", "Jennewein, T., Weihs, G., Pan, J.-W. & Zeilinger, A.", "Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping.", "Physical Review Letters 88, 17903 (2001).", "Halder, M. et al.", "Entangling independent photons by time measurement.", "Nature Physics 3, 692–695 (2007).", "Yuan, Z.-S. et al.", "Experimental demonstration of a BDCZ quantum repeater node.", "Nature 454, 1098–1101 (2008).", "Kaltenbaek, R., Prevedel, R., Aspelmeyer, M. & Zeilinger, A. High-fidelity entanglement swapping with fully independent sources.", "Physical Review Letters 79, 040302 (2009).", "Ma, X.-S. et al.", "Experimental delayed-choice entanglement swapping.", "Nature Physics 8, 479–484 (2012).", "Hofmann, J. et al.", "Heralded Entanglement Between Widely Separated Atoms.", "Science 337, 72–75 (2012).", "Ekert, A. K. Quantum cryptography based on Bell's theorem.", "Physical Review Letters 67, 661–663 (1991).", "Scheidl, T. et al.", "Violation of local realism with freedom of choice.", "Proceedings of the National Academy of Sciences 107, 19708–19713 (2010).", "Calsamiglia, J.", "& Lütkenhaus, N. Maximum Efficiency of a Linear-Optical Bell-State Analyzer.", "Applied Physics B 72, 67–71 (2001).", "Kwiat, P. G. et al.", "New High-Intensity Source of Polarization-Entangled Photon Pairs.", "Physical Review Letters 75, 4337–4342 (1995).", "Kim, Y.-H. & Grice, W. P. Generation of pulsed polarization-entangled two-photon state via temporal and spectral engineering.", "Journal of Modern Optics 49, 2309–2323 (2002).", "Kim, Y.-H., Kulik, S. P., Chekhova, M. V., Grice, W. P. & Shih, Y.", "Experimental entanglement concentration and universal Bell-state synthesizer.", "Physical Review A 67, 010301 (2003).", "Poh, H., Lim, J., Marcikic, I., Lamas-Linares, A.", "& Kurtsiefer, C. Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion.", "Physical Review Letters 80, 043815 (2009).", "Hong, C. K., Ou, Z. Y.", "& Mandel, L. Measurement of subpicosecond time intervals between two photons by interference.", "Physical Review Letters 59, 2044–2046 (1987).", "Ma, X.-S. et al.", "Quantum teleportation over 143 kilometres using active feed-forward.", "Nature 489, 269–273 (2012).", "Clauser, J. F., Horne, M. A., Shimony, A.", "& Holt, R. A.", "Proposed Experiment to Test Local Hidden-Variable Theories.", "Physical Review Letters 23, 880–884 (1969).", "Gühne, O.", "& Tóth, G. Entanglement detection.", "Physics Reports 474, 1–75 (2009)." ] ]	1403.0009
[ [ "Fibonacci Oscillators in the Landau Diamagnetism problem" ], [ "Abstract We address the issue of the Landau diamagnetism problem via $q$-deformed algebra of Fibonacci oscillators through its generalized sequence of two real and independent deformation parameters $q_1$ and $q_2$.", "We obtain $q$-deformed thermodynamic quantities such as internal energy, number of particles, magnetization and magnetic susceptibility which recover their usual form in the degenerate limit $q_1^2 + q_2^2$=1." ], [ "Introduction", "The Landau diamagnetism problem continues to play a role in several issues of many physical systems and has strong relevance today [1], [2], [3], [4], [5].", "The diamagnetism can be used as an illustrative phenomenon that plays essential role in quantum mechanics on the surface, the perimeter, and the dissipation of statistical mechanics of non-equilibrium.", "In this paper, we are interested in investigating this phenomenon in $q$ -deformed algebra in order to understand impurities effects in, for example, magnetization and susceptibility.", "The magnetic susceptibility is an intrinsic characteristic of a material and its identity is related to the atomic and molecular structure.", "In Ref.", "[6] it was performed the calculation of susceptibility for electrons moving in a uniform external magnetic field, developing Landau diamagnetism, by applying the nonextensive Tsallis statistics [7], [8], [9], [10], which is a strong candidate for solving problems where the standard thermodynamics is not applicable — see also Ref.", "[11] for a similar study using another method.", "Of course, other noncommutative deformations can be applied, for example $q$ -deformation via Jackson derivative (JD) [12].", "The study of quantum groups and quantum algebras has attracted great interest in recent years, stimulated intense research in various fields of physics [13], [14], taking into account a range of applications, covering cosmology and condensed matter, e.g.", "black holes, fractional quantum Hall effect, high-temperature (high-T$_c$ ) superconductors [15], rational field theories, noncommutative geometry, quantum theory of super-algebras and so on [16].", "Furthermore, statistical and thermodynamic properties by studying $q$ -deformed physical systems have been intensively investigated in the literature [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31].", "Another important discussion is about the main reasons to consider two deformation parameters in some different physical applications.", "Starting from the generalization of the $q$ -algebra [32], in Ref.", "[33] it was generalized the Fibonacci sequence, which is a well-known linear combination where the third number is the sum of two predecessors and so on.", "Here, the numbers are in that sequence of generalized Fibonacci oscillators, where new parameters ($q_1,q_2$ ) are introduced [33], [34], [35].", "They provide a unification of quantum oscillators with quantum groups, keeping the degeneration property of the spectrum invariant under the symmetries of the quantum group.", "The quantum algebra with two deformation parameters may have a greater flexibility when it comes to application in the concrete phenomenological physical models [36], [37], and may increase interest in physical applications.", "The paper is organized as follows.", "In Sec.", "() we intrduce the $q$ -deformed algebra.", "In Sec.", "() we develop the ($q_1, q_2$ )-deformed Landau diamagnetism problem and in the Sec.", "() we make our final comments." ], [ "Fibonacci oscillators algebra", "We consider a system of generalized oscillators now entering two parameters in statistical distribution function, whose energy spectrum may be determined by Fibonacci's generalized sequence [33], [34], [35].", "This will establish a statistical system depending on the deformation parameters ($q_1,q_2$ ), allowing us to calculate the thermodynamic quantities in the limit of high temperatures.", "The $q$ -deformed quantum oscillator is now defined by the Heisenberg algebra in terms the annihilation and creation operators in $c$ , $c^{\\dagger }$ , respectively, and the number operator $N$ [19], [35], as follows $c_i c_{i}^\\dagger - Kq_1^{2}c_{i}^\\dagger c_i = q_2^{2n_i}\\qquad \\mbox{e}\\qquad c_i c_{i}^\\dagger -Kq_2^{2}c_{i}^\\dagger c_i = q_1^{2n_i}, $ $[N,c^{\\dagger }] = c^{\\dagger }, \\qquad \\qquad [N,c] = -c ,$ where $K = \\pm 1$ , stands for bosons and fermions, respectively.", "In addition, the operators also obey the relations $\\;\\;\\qquad c^\\dagger c=[N],\\;\\;\\qquad cc^{\\dagger } = [1+KN], $ $[1+Kn_{i,q_1,q_2}] = Kq_1^{2}[n_i]+q_2^{2n_i},\\;\\quad \\mbox{or}\\quad \\; [1+Kn_{i,q_1,q_2}] = Kq_2^{2}[n_i]+q_1^{2n_i}.", "$ The Fibonacci basic number is defined as [33] $[n_{i,q_1,q_2}] = c_{i}^\\dagger c_{i} = \\frac{q_2^{2n_i}-q_1^{2n_i}}{q_2^2-q_1^{2}}, $ The $q$ -Fock space spanned by the orthornormalized eigenstates $\|n \\rangle $ is constructed according to $ {\|n\\rangle } =\\frac{(c^{\\dagger })^{n}}{\\sqrt{[n]!", "}}{\|0\\rangle },\\qquad \\qquad c{\|0\\rangle }=0 ,$ The actions of $c$ e $c^{\\dagger }$ and $N$ on the states $\|n\\rangle $ in the $q$ -Fock space are known to be $ c^{\\dagger }{\|n\\rangle } = [n+1]^{1/2} {\|n+1\\rangle },$ $ c{\|n\\rangle } = [n]^{1/2} {\|n-1\\rangle },$ $ N{\|n\\rangle } = n{\|n\\rangle }.$ To calculate the $q$ -deformation statistical occupation number, we begin with the Hamiltonian of $q$ -deformed noninteracting oscillators (bosons or fermions) [16], ${\\cal H}_{q_1,q_2} = \\sum _{i}{(\\epsilon _i-\\mu _{q_1,q_2})}{N_i}, $ where $\\mu _{q_1,q_2}$ is the ($q_1,q_2$ )-deformed chemical potential.", "It should be noted that this Hamiltonian is a two-parameter deformed Hamiltonian and depends implicitly on the deformation parameters $q_1$ and $q_2$ , since the number operator is deformed via Eq.", "(REF ).", "The mean value of the ($q_1,q_2$ )-deformed occupation number can be calculated by $[n_i]\\equiv \\langle [n_i]\\rangle = \\frac{tr(\\exp (-\\beta {\\cal H})c_{i}^\\dagger c_{i})}{\\Xi }, $ $[n_{i,q_1,q_2}]=\\frac{z^{\\prime }\\left(\\exp (\\beta \\epsilon _i)-z^{\\prime }\\right)}{\\left(\\exp (\\beta \\epsilon _i)-q_2^2z^{\\prime }\\right)\\left(\\exp (\\beta \\epsilon _i)-q_1^2z^{\\prime }\\right)},$ where $z_{q_1,q_2}=\\exp (\\beta \\mu _{q_1,q_2})$ is the fugacity of the system, and we shall use the notation $z_{q_1,q_2}=z^{\\prime }$ .", "When $q_1=q_2=1$ , we find the usual form $n_i=\\frac{1}{z^{-1}\\exp {(\\beta \\epsilon _i)}-1}.$ In the present application of the Fibonacci oscillators, we are interested in obtain new $(q_1,q_2)$ -deformed thermodynamic quantities such as internal energy, magnetization, and magnetic susceptibility for the high-temperature case, i.e.", "the limit ($z\\ll 1$ )." ], [ "Fibonacci oscillators in the Landau diamagnetism", "To explain the phenomenon of diamagnetism, we have to take into account the interaction between the external magnetic field and the orbital motion of electrons.", "Disregarding the spin, the Hamiltonian of a particle of mass m and charge e in the presence of a magnetic field H is given by the expression [40] $ {\\cal H} = \\frac{1}{2m} \\left({p}-\\frac{e}{c}{\\bf A}\\right) ^2, $ where A is the vector potential associated with the magnetic field H and c is the speed of light in $CGS$ units.", "Let us start to formalize the statistical mechanical problem by using the grand partition function with the parameters $q_1$ and $q_2$ inserted through Eq.", "(REF ), in the form $ \\ln \\Xi &=& -K\\frac{2eHL^2}{hc}\\displaystyle \\sum _{n}^{\\infty }\\frac{L}{2\\pi }\\displaystyle \\int _{-\\infty }^{\\infty }dk_{z}\\frac{1}{(q_1^2-q_2^2)}\\Bigg \\lbrace \\ln {\\Big [1-Kz^{\\prime }q_1^2\\exp (-\\beta \\epsilon )\\Big ]}(q_1^{-2}-1)+\\nonumber \\\\&+&\\ln {\\Big [1-Kz^{\\prime }q_2^2\\exp (-\\beta \\epsilon )\\Big ](1-q_2^{-2})]}\\Bigg \\rbrace , $ where $k_{z} = -\\infty ,\\cdots ,\\infty $ , $\\epsilon =\\frac{\\hbar k_{z}^2}{2m}+\\hbar \\omega \\left(n+\\frac{1}{2}\\right)$ , $\\omega =\\frac{eH}{mc}$ .", "However, our study is focused on the analysis of diamagnetism in the limit of high temperatures $(z^{\\prime }\\ll 1)$ .", "Thus, performing the sum and integrals, we find the partition function is written as follows $\\ln \\Xi =\\frac{z^{\\prime }K^2HC_1}{\\sinh (\\gamma )}+\\frac{z^{\\prime 2}K^3HC_2Q}{2\\sinh (2\\gamma )},\\;\\;\\qquad \\mbox{where}\\qquad C_1=\\frac{eL^3}{2\\pi hc\\lambda }, C_2=\\frac{eL^3}{2\\pi \\sqrt{2}hc\\lambda }$ being $\\lambda =\\frac{\\hbar }{(2\\pi m\\kappa _B T)^{\\frac{1}{2}}}$ the thermal wavelength, $\\gamma =\\beta \\mu _{B}H$ and $Q=q_1^2+q_2^2-1$ .", "We note that Eq.", "(REF ) shows the ($q_1,q_2$ )-deformation in the second term.", "In the first order does not appear $q$ -deformation.", "It appears after considering at least the second order.", "Notice, however, the case ($q_1=q_2=1$ ), as expected, does recover the underformed thermodynamic quantities up to a second order correction which are usually disregarded.", "On the other hand, taking computation up to second order corrections is necessary to get the effects of the $q$ -deformation and as a consequence only in the unit circle on the ($q_1,q_2$ )-space, i.e., $q_1^2+q_2^2=1$ the deformation ceases.", "The case in Eq.", "(REF ) shows an interesting degeneration on the ($q_1,q_2$ )-space.", "Deformations show up as $q_1^2+q_2^2<1$ or $q_1^2+q_2^2>1$ .", "In former case appears the possibility of finding some unexpected negative thermodynamic quantities such as negative specific heat.", "In the following we shall consider the latter case to calculate the $q$ -deformed thermodynamic quantities of interest in the present study." ], [ "($q_1,q_2$ )-deformed thermodynamic quantities", "We obtain the number of particles $N$ by setting, $N=z^{\\prime }\\frac{\\partial }{\\partial z^{\\prime }}\\ln {\\Xi }=\\frac{z^{\\prime }K^2HC_1}{\\sinh (\\gamma )}+\\frac{z^{\\prime 2}K^3HC_2Q}{2\\sinh (2\\gamma )}.$ We determine the internal energy, and we can write it in terms of $N$ , in the form $ U=-\\frac{\\partial }{\\partial \\beta }\\ln \\Xi =\\frac{N\\mu _B H\\Big [C_1\\coth (\\gamma )\\sinh (2\\gamma )+z^{\\prime }KC_2\\sinh (\\gamma )\\coth (2\\gamma )Q\\Big ]}{C_1\\sinh (2\\gamma )+z^{\\prime }KC_2\\sinh (\\gamma )Q}.$ In Fig.", "REF we have the behavior of internal energy $U$ as a function of the magnetic field H and for some values of $q_1$ and $q_2$ - see caption.", "We note that all the curves have different maximum peaks (depending on the values adopted for $q_1$ and $q_2$ ), for small magnetic field $\\textbf {H}$ .", "The curves exhibit the same behavior asymptotically.", "We also have proven the symmetry between the oscillators, i.e.", "when $q_1=1$ and $q_2=2$ (black curve) and when $q_1=2$ and $q_2=1$ (red curve), they overlap.", "An expected effect due to the symmetry of $q_1$ and $q_2$ defined in $Q$ .", "Figure: (q 1 ,q 2 q_1,q_2)-deformed internal energy as a function of magnetic field H for several choices of q 1 q_1 and q 2 q_2.The grand potential $\\phi $ is determined as $\\phi =-\\frac{1}{\\beta }\\ln \\Xi =-\\left(\\frac{z^{\\prime }K^2HC_1}{\\beta \\sinh (\\gamma )}+\\frac{z^{\\prime 2}K^3HC_2Q}{2\\sinh (2\\gamma )}\\right).", "$ To determine the magnetization, we carried out the thermodynamic derivative by using Eq.", "(REF ), that gives $M=-\\frac{\\partial \\phi }{\\partial H}=\\frac{z^{\\prime }C_1K^2\\left(1-\\gamma \\coth (\\gamma )\\right)}{\\beta \\sinh (\\gamma )}-\\frac{z^{\\prime }C_2K^3Q\\left(1-\\gamma \\coth (2\\gamma )\\right)}{2\\beta \\sinh (2\\gamma )}.$ We can also eliminate the chemical potential through the number of particles $N$ and insert the Langevin functions ${\\cal L}(\\gamma ) = \\coth (\\gamma )-\\frac{1}{\\gamma },\\qquad {\\cal L}(2\\gamma ) = \\coth (2\\gamma )-\\frac{1}{2\\gamma }, $ to rewrite the magnetization as $ M= -\\frac{N\\mu _B\\Big [C_1\\sinh (2\\gamma ){\\cal L}(\\gamma )+z^{\\prime }KC_2\\gamma \\sinh (\\gamma ){\\cal L}(2\\gamma )\\Big ]}{C_{1}\\sinh (2\\gamma )+z^{\\prime }KC_{2}\\sinh (\\gamma )Q}.", "$ The results obtained for the deformed magnetization are very interesting, because we can compare it with experimental results obtained for superconducting materials (which are perfect diamagnetic materials) as a function of temperature variation [42], in order to strength the understanding of the $q$ -deformation as a factor of impurity.", "In these references [42] one was found that the minimum of magnetization deepens as temperature or pressure decreases.", "In Fig.", "REF we have the magnetization curves $(M)$ versus magnetic field (H) for some values of $q_1$ and $q_2$ , and we note that some observations made for internal energy such as oscillators symmetry are also valid for the magnetization, as expected.", "Notice that the minimum of magnetization deepens as $q$ -deformation increases.", "This means that increasing temperature or pressure we may diminish the effects of disorders or impurities of the system.", "This explains why we should reduce the deformation parameters until they assume the underformed degenerate case $q_1^2+q_2^2=1$ .", "Figure: (q 1 ,q 2 q_1,q_2)-deformed magnetization as a function of magnetic field H for several choices of q 1 q_1 and q 2 q_2.Now, computing the susceptibility reads, $\\chi &=&\\frac{\\partial M}{\\partial H}=\\frac{N\\beta \\mu _B^2}{C_1\\sinh (2\\gamma )+z^{\\prime }KC_2\\sinh (\\gamma )Q}\\Bigg [C_1\\sinh (2\\gamma )\\Big (2\\coth (\\gamma ){\\cal L}(\\gamma )-1\\Big )+\\nonumber \\\\&+& 2z^{\\prime }C_2\\sinh (\\gamma )Q\\Big (2\\coth (2\\gamma ){\\cal L}(2\\gamma )-1\\Big )\\Bigg ].$ In the limit of weak fields $\\gamma \\ll 1$ we have the leading term $M=-\\frac{2N\\mu _B\\sinh (\\gamma )\\cosh (\\gamma )(C_1+z^{\\prime }C_2KQ)}{3(2C_1\\cosh (\\gamma )+z^{\\prime }KC_2Q)}, $ and thus, we have the susceptibility in zero field $\\chi _0=-\\frac{2\\mu _B z^{\\prime }K^2(C_1+z^{\\prime }C_2KQ)}{3(2C_1+z^{\\prime }KC_2Q)}.", "$" ], [ "Conclusions", "As in our previous works [12], [38], [39], which we have shown that the $q$ -parameter is associated with impurities in a sample, in particular diamagnetic materials, as in the present study, we put forward new results to strength this interpretation of the $q$ -deformation.", "In this work, we expand the application of $q$ -calculation through two deformation parameters ($q_1,q_2$ ), known as Fibonacci oscillators.", "We work in the limit of high temperatures (`dilute gas' $z\\ll 1$ ), and a ($q_1,q_2$ )-deformed partition function.", "In first order the results reported in the literature [40], [41], are recovered.", "However, the $q$ -deformation takes place at second order for non-degenerate case $q_1^2+q_2^2>1$ .", "We note that the in the obtained results were found several interesting behaviors by just varying the values of $q_1$ and $q_2$ .", "Of course, we performed a theoretical application, and it allows various assumptions.", "By comparing these results with similar experimental curves, one could understand how impurities could be entering into a material that affects, e.g., superconductivity such its critical temperature increases, which would be of great interest to whom that works with high T$_c$ superconductors — see [43] for a recent alternative theoretical investigation on these type of superconductors whose structure can be extended via $q$ -deformation in order to introduce impurities.", "We would like to thank CNPq, CAPES, and PROCAD-CAPES, for partial financial support." ] ]	1403.0272
[ [ "Parametrized Positivity Preserving Flux Limiters for the High Order\n Finite Difference WENO Scheme Solving Compressible Euler Equations" ], [ "Abstract In this paper, we develop parametrized positivity satisfying flux limiters for the high order finite difference Runge-Kutta weighted essentially non-oscillatory (WENO) scheme solving compressible Euler equations to maintain positive density and pressure.", "Negative density and pressure, which often leads to simulation blow-ups or nonphysical solutions, emerges from many high resolution computations in some extreme cases.", "The methodology we propose in this paper is a nontrivial generalization of the parametrized maximum principle preserving flux limiters for high order finite difference schemes solving scalar hyperbolic conservation laws [22, 10, 20].", "To preserve the maximum principle, the high order flux is limited towards a first order monotone flux, where the limiting procedures are designed by decoupling linear maximum principle constraints.", "High order schemes with such flux limiters are shown to preserve the high order accuracy via local truncation error analysis and by extensive numerical experiments with mild CFL constraints.", "The parametrized flux limiting approach is generalized to the Euler system to preserve the positivity of density and pressure of numerical solutions via decoupling some nonlinear constraints.", "Compared with existing high order positivity preserving approaches [24, 26, 25], our proposed algorithm is positivity preserving by the design; it is computationally efficient and maintains high order spatial and temporal accuracy in our extensive numerical tests.", "Numerical tests are performed to demonstrate the efficiency and effectiveness of the proposed new algorithm." ], [ "Introduction", "The success of the high order essentially non-oscillatory (ENO) or weight ENO (WENO) methods solving hyperbolic conservation laws has been well documented in the literature [6], [15], [12], [9] and the references therein.", "At the heart of the high order ENO/WENO schemes solving hyperbolic problem is the robustness, namely stability in the sense of suppressing spurious oscillations around discontinuities.", "The application of the high order finite difference, finite volume ENO/WENO methods to hyperbolic systems [15], [9], such as the compressible Euler equations $\\left(\\begin{array}{l}\\rho \\\\\\rho u\\\\E\\end{array}\\right)_t+\\left(\\begin{array}{l}\\rho u \\\\\\rho u + P\\\\(E+ P)u\\end{array}\\right)_x=0,$ achieves the goal of suppressing oscillations when discontinuous solution emerges during the time evolution.", "However, in the extreme case, such as high Mach flow simulation, a slightly different (although equally important) problem is that the high order schemes that we are using might produce solutions with negative density and pressure, which leads to an ill-posed problem, often seen as blow-up of the numerical simulation.", "The failure of preserving positive density and pressure by the above mentioned schemes in such circumstance pose tremendous difficulty of applying high order schemes to some of the challenging simulations in practice.", "In the earlier work, see [3], [11], [13] and references included, much attention has been paid to the positivity preservation of schemes up to second order.", "It wasn't until the recent work by Zhang & Shu [23] that arbitrarily high order finite volume WENO and discontinuous Galerkin methods are designed to preserve positivity.", "The method proposed in [23] is a successful generalization of their earlier work on the maximum principle preserving (MPP) computations of scalar conservation laws, see [24].", "Their approach relies on limiting the reconstructed polynomials (finite volume WENO) or representing polynomials (discontinuous Galerkin) around cell averages to be MPP.", "The positivity preserving (PP) finite volume WENO scheme and DG scheme by Zhang & Shu can be proved to have the designed arbitrary high order accuracy when equipped with proper CFL number.", "In the later work by the authors [26], a PP finite difference WENO method is presented when the density and pressure is strictly greater than a fixed positive constant.", "In [8], a flux cut-off limiter method is applied to the high order finite difference WENO method to ensure positive density and pressure.", "In this paper, we continue along the line of research on the parametrized flux limiters proposed in [22], [10], [20] for high order ENO/WENO methods solving a scalar hyperbolic conservation law $ u_t+f(u)_x=0$ subject to the initial condition $u({x}, 0)= u_0 ({x})$ .", "For this particular family of equations, the solution satisfies a strict maximum principle $u_m\\le u(x, t) \\le u_M \\quad \\text{if} \\quad u_m\\le u_0(x)\\le u_M.$ The idea of the parametrized flux limiters for general conservative scheme solving scalar conservation laws is to modify high order numerical fluxes to enforce the discrete maximum principle for the updated solution.", "In general, a conservative high order scheme with explicit multi-stage Runge-Kutta (RK) time integration for (REF ) can be written as $u^{n+1}_j=u^{n}_j-\\frac{\\Delta t}{\\Delta x} (\\hat{H}^{rk}_{j+\\frac{1}{2}}-\\hat{H}^{rk}_{j-\\frac{1}{2}}),$ where $\\hat{H}^{rk}_{j\\pm \\frac{1}{2}}$ are the corresponding fluxes at the final stage of RK methods.", "The MPP properties of high order schemes are realized by taking a convex combination of a high order flux $\\hat{H}^{rk}_{j+\\frac{1}{2}}$ and a first order monotone flux $\\hat{h}_{j+\\frac{1}{2}}$ : $\\tilde{H}^{rk}_{j+\\frac{1}{2}}=\\hat{h}_{j+\\frac{1}{2}} + \\theta _{j+\\frac{1}{2}} (\\hat{H}^{rk}_{j+\\frac{1}{2}}-\\hat{h}_{j+\\frac{1}{2}})$ , with $\\theta _{j+\\frac{1}{2}} \\in [0, 1]$ .", "The limiting parameters $\\theta _{j+\\frac{1}{2}}$ , which measure the change of numerical fluxes, can be found out through decoupling the following MPP constraints that are linear with respect to $\\theta _{j\\pm \\frac{1}{2}}$ , $u_m\\le u^{n+1}_j=u^{n}_j-\\frac{\\Delta t}{\\Delta x} (\\tilde{H}^{rk}_{j+\\frac{1}{2}}-\\tilde{H}^{rk}_{j-\\frac{1}{2}})\\le u_M.$ The similar idea is utilized in this paper in the sense of making sufficient modification of the high order numerical fluxes to ensure that the updated density and pressure are positive.", "When such parametrized flux limiters are generalized to preserve the positivity of density and pressure of numerical solutions for Euler equations with source terms, there are several new challenges.", "One of the main difficulties is that the linear MPP constraint (REF ) becomes nonlinear for positivity preservation of pressure, which has nonlinear dependence on the density, momentum and energy.", "We address such challenges by decoupling the nonlinear PP constraint for a `convex set' of the limiting parameters.", "The proposed approach provides a sufficient condition for preserving positive pressure.", "The presence of the source term can also be conveniently handled in the parametrized flux limiting framework.", "Notice that we only require positivity preservation for the solutions at the final stage of RK method for the sake of preserving the designed high order temporal accuracy.", "If there are negative density and pressure in intermediate stages of the RK method, the speed of sound is computed by $c=\\sqrt{\\gamma \\frac{\|p\|}{\|\\rho \|}}$ .", "Our approach is similar to those very early discussions of the flux limiting approach [1], [2], [4], [17], [16] for the purpose of achieving a total variation diminishing (TVD) property, which is a much stronger stability requirement than the maximum principle.", "The schemes are expected to be TVD, therefore, most of the schemes are at most of second order accurate.", "To distinguish our work from others' in the context of designing arbitrarily high order schemes, we would like to point out that the method we are proposing only involves the modification of high order numerical fluxes.", "Another critical difference is that the parametrized flux limiters are only applied to the final stage of the multi-stage RK methods.", "These new features are designed to produce numerical solutions with positive density and pressure, while allowing for relatively large CFL numbers without sacrificing accuracy in our extensive numerical tests.", "The proposed method is essentially different from those by Zhang & Shu [26], in which the PP property is realized only with fine enough numerical meshes, when the density and the pressure is extremely close to 0.", "The flux limiting method we are proposing is also different from the flux cut-off method by Hu [8], whose approach demands significantly reduced CFL for accuracy as illustrated in their analysis and numerical tests.", "However, the proof of maintaining high order accuracy when the PP flux limiters are applied to the finite difference WENO method solving the Euler system is very difficult.", "In this paper, we rely on numerical observations to demonstrate the maintenance of high order accuracy.", "A rigorous proof of that the MPP flux limiters modify the original high order flux with up to third order accuracy for general nonlinear scalar cases is provided in [20] and that with up to fourth order accuracy for linear advection equations is provided in [21].", "The paper is organized as follows.", "In Section , we give a brief review of the parametrized MPP flux limiters for high order conservative schemes solving (REF ).", "We then generalize the MPP flux limiters to a scalar problem with source terms.", "In Section , we present the main algorithm of the parametrized PP finite difference WENO RK method for the compressible Euler equation in one and two dimensions.", "An implementation procedure is given in the presence of source terms.", "In Section , we perform extensive numerical tests to illustrate the effectiveness of the proposed method.", "We finally conclude in Section ." ], [ "Review of MPP flux limiters for scalar equations", "For simplicity, we consider a simple one-dimensional hyperbolic conservation equation $u_t+f(u)_x=0, \\quad x \\in [0, 1],$ with an initial condition $u(x,0) = u_0(x)$ and a periodic boundary condition.", "We adopt the following spatial discretization for the domain $[0, 1]$ $0 = x_\\frac{1}{2} < x_\\frac{3}{2} < \\cdots < x_{N+\\frac{1}{2}} = 1,$ where $I_j = [x_{j-\\frac{1}{2}}, x_{j+\\frac{1}{2}}]$ has the mesh size $\\Delta x = \\frac{1}{N}$ .", "Let $u_j(t)$ denote the solution at grid point $x_j = \\frac{1}{2}(x_{j-\\frac{1}{2}}+x_{j+\\frac{1}{2}})$ at continuous time $t$ .", "The finite difference scheme evolves the point values of the solution in a conservative form $\\frac{d}{dt}u_j(t)+ \\frac{1}{\\Delta x} (\\hat{H}_{j+1/2}-\\hat{H}_{j-1/2}) = 0.$ The numerical flux $\\hat{H}_{j+\\frac{1}{2}}$ in equation (REF ) can be reconstructed from neighboring flux functions $f(u(x_i, t)),$ $i=j-p, \\cdots , j+q$ with high order by WENO reconstructions [9], [14].", "By adaptively assigning nonlinear weights to neighboring candidate stencils, the WENO reconstruction preserves high order accuracy of the linear scheme around smooth regions of the solution, while producing a sharp and essentially non-oscillatory capture of discontinuities.", "Equation (REF ) can be further discretized in time by a high order time integrator via the method-of-line approach.", "For example, the scheme with a third order total variation diminishing (TVD) RK time discretization is $u_j^{(1)} &=& u_j^n+\\Delta t L(u_j^n), \\nonumber \\\\u_j^{(2)} &=& u_j^n+\\frac{1}{4}\\Delta t (L(u_j^n) + L(u_j^{(1)})), \\nonumber \\\\u_j^{n+1} &=& u_j^n+\\frac{1}{6} \\Delta t \\bigl ( L(u_j^{n})+ L(u_j^{(1)})+4 L(u_j^{(2)})\\bigr ).$ where $u^{(k)}_{j}$ and $u^n_j$ denotes the numerical solution at $x_j$ at $k^{th}$ RK stage and at time $t^n$ respectively.", "Let $\\Delta t$ be the time step size.", "$L(u^{(k)}) \\doteq -\\frac{1}{\\Delta x} (\\hat{H}^{(k)}_{j+2}-\\hat{H}^{(k)}_{j-2})$ with $\\hat{H}^{(k)}_{j+2}$ being the numerical flux from finite difference WENO reconstruction based on $\\lbrace u_j^{(k)}\\rbrace _{j=1}^N$ at intermedia RK stages.", "Equation (REF ) in the final stage of RK method can be re-written as $u^{n+1}_j=u^{n}_j-\\lambda (\\hat{H}^{rk}_{j+2}-\\hat{H}^{rk}_{j-2}),$ with $\\lambda =\\frac{\\Delta t}{\\Delta x}$ and $\\hat{H}^{rk}_{j+2} \\doteq \\frac{1}{6}\\left(\\hat{H}^n_{j+2}+\\hat{H}^{(1)}_{j+2}+4\\hat{H}^{(2)}_{j+2}\\right).$ The parametrized MPP flux limiters in [20] is based on the finite difference RK WENO scheme for equation (REF ) reviewed above.", "Let $u_{m}= \\underset{x}{\\text{min}}(u(x, 0))$ and $u_{M}=\\underset{x}{\\text{max}}(u(x, 0))$ .", "The idea of the parametrized MPP flux limiter is to modify the high order flux $\\hat{H}^{rk}_{j\\pm \\frac{1}{2}}$ in equation (REF ) towards a first order monotone flux denoted as $\\hat{h}_{j\\pm \\frac{1}{2}}$ by taking a linear combination of them, $\\tilde{H}^{rk}_{j\\pm 2} \\doteq \\hat{h}_{j\\pm \\frac{1}{2}} + \\theta _{j\\pm \\frac{1}{2}} (\\hat{H}^{rk}_{j\\pm \\frac{1}{2}}-\\hat{h}_{j\\pm \\frac{1}{2}}), \\quad \\theta _{j\\pm \\frac{1}{2}} \\in [0, 1].$ the original high order flux $\\hat{H}^{rk}_{j\\pm 2}$ in equation (REF ) is then replaced by the modified flux $\\tilde{H}^{rk}_{j\\pm 2}$ above.", "To preserve the MPP property, we wish to have $u_{m}\\le u^{n+1}_{j} \\le u_{M}$ at the final RK stage on each time step, i.e.", "$u_{m} \\le u^{n}_j-\\lambda (\\tilde{H}^{rk}_{j+2}-\\tilde{H}^{rk}_{j-2}) \\le u_{M}.$ For the parametrized MPP flux limiter, a pair $(\\Lambda _{-2, {I_j}}, \\Lambda _{+2, {I_j}})$ needs to be found such that any pair $(\\theta _{j-2}, \\theta _{j+2}) \\in [0, {\\Lambda _{-2, {I_j}}}]\\times [0, {\\Lambda _{+2, {I_j}}]}$ satisfies (REF ).", "Under such a constraint, $\\theta _{j\\pm 2}$ are chosen to be as close to 1 as possible for accuracy, which is done by the following three steps.", "Below $\\epsilon $ is a small positive number to avoid the denominator to be 0, e.g., $\\epsilon =10^{-13}$ .", "The right inequality of (REF ), that is the maximum value part, can be rewritten as $\\lambda \\theta _{j-2} (\\hat{H}^{rk}_{j-2}-\\hat{h}_{j-2}) - \\lambda \\theta _{j+2} (\\hat{H}^{rk}_{j+2}-\\hat{h}_{j+2})-\\Gamma ^M_j \\le 0,$ where $\\Gamma ^M_j=u_{M}-u_j+\\lambda (\\hat{h}_{j+2}-\\hat{h}_{j-2}) \\ge 0$ .", "Let $F_{j-2}=\\hat{H}^{rk}_{j-2}-\\hat{h}_{j-2}$ , the decoupling of (REF ) on cell $I_j$ gives: If $F_{j-2}\\le 0$ and $F_{j+2}\\ge 0$ , let $(\\Lambda ^M_{-2, I_j}, \\Lambda ^M_{+2, I_j})=(1, 1)$ .", "If $F_{j-2}\\le 0$ and $F_{j+2} < 0$ , let $(\\Lambda ^M_{-2, {I_j}}, \\Lambda ^M_{+2, {I_j}})=(1, \\min (1, \\frac{\\Gamma ^M_j}{-\\lambda F_{j+2}+\\epsilon }))$ .", "If $F_{j-2} > 0$ and $F_{j+2}\\ge 0$ , let $(\\Lambda ^M_{-2, {I_j}}, \\Lambda ^M_{+2, {I_j}})=(\\min (1, \\frac{\\Gamma ^M_j}{\\lambda F_{j-2}+\\epsilon }), 1)$ .", "If $F_{j-2} > 0$ and $F_{j+2} < 0$ , if $(\\theta _{j-2}, \\theta _{j+2})=(1, 1)$ satisfies (REF ), let $(\\Lambda ^M_{-2, {I_j}}, \\Lambda ^M_{+2, {I_j}})=(1, 1)$ ; otherwise, let $(\\Lambda ^M_{-2, {I_j}}, \\Lambda ^M_{+2, {I_j}})=(\\frac{\\Gamma ^M_j}{\\lambda F_{j-2}- \\lambda F_{j+2}+\\epsilon },\\frac{\\Gamma ^M_j}{\\lambda F_{j-2}- \\lambda F_{j+2}+\\epsilon } )$ .", "The left inequality of (REF ), that is the minimum value part, can be rewritten as $0\\le \\lambda \\theta _{j-2} (\\hat{H}^{rk}_{j-2}-\\hat{h}_{j-2}) - \\lambda \\theta _{j+2} (\\hat{H}^{rk}_{j+2}-\\hat{h}_{j+2})-\\Gamma ^m_j,$ where $\\Gamma ^m_j=u_{m}-u_j+\\lambda (\\hat{h}_{j+2}-\\hat{h}_{j-2}) \\le 0$ .", "Similar to the maximum value case, the decoupling of (REF ) on cell $I_j$ gives: If $F_{j-2}\\ge 0$ and $F_{j+2}\\le 0$ , let $(\\Lambda ^m_{-2, I_j}, \\Lambda ^m_{+2, I_j})=(1, 1)$ ; If $F_{j-2}\\ge 0$ and $F_{j+2}> 0$ , let $(\\Lambda ^m_{-2, {I_j}}, \\Lambda ^m_{+2, {I_j}})=(1, \\min (1, \\frac{\\Gamma ^m_j}{-\\lambda F_{j+2}-\\epsilon }))$ ; If $F_{j-2}< 0$ and $F_{j+2}\\le 0$ , let $(\\Lambda ^m_{-2, {I_j}}, \\Lambda ^m_{+2, {I_j}})=(\\min (1, \\frac{\\Gamma ^m_j}{\\lambda F_{j-2}-\\epsilon }), 1)$ ; If $F_{j-2}< 0$ and $F_{j+2}> 0$ , when $(\\theta _{j-2}, \\theta _{j+2})=(1, 1)$ satisfies (REF ), let $(\\Lambda ^m_{-2, {I_j}}, \\Lambda ^m_{+2, {I_j}})=(1, 1)$ ; otherwise, let $(\\Lambda ^m_{-2, {I_j}}, \\Lambda ^m_{+2, {I_j}})=(\\frac{\\Gamma ^m_j}{\\lambda F_{j-2}- \\lambda F_{j+2}-\\epsilon },\\frac{\\Gamma ^m_j}{\\lambda F_{j-2}- \\lambda F_{j+2}-\\epsilon } )$ .", "The locally defined limiting parameter is given as $\\Lambda _{j+2}=\\min (\\Lambda ^M_{+2, {I_j}}, \\Lambda ^M_{-2, {I_{j+1}}}, \\Lambda ^m_{+2, {I_j}}, \\Lambda ^m_{-2, {I_{j+1}}}), \\quad j = 0, \\cdots N.$ The flux limiting procedure above guarantees the MPP property of the numerical solution by the design.", "It is theoretically proved to preserve up to fourth order spatial and temporal accuracy for smooth solutions [20], [21]." ], [ "Scalar advection equations with source terms", "We consider scalar advection problems with a source term $u_t+f(u)_x=s(u).$ In particular, we consider the class of problems whose solutions enjoy the PP property, that is, the lower bound of the solution is 0 (such kind of problem might not preserve the MPP property).", "For example, when $s(u)=-k u$ with a positive $k$ , with positive initial values and periodic boundary conditions, the solution satisfies the PP property.", "The flux limiter is designed base on the PP property of a first order scheme $u^{n+1}_j=u^{n}_j-\\lambda (\\hat{h}_{j+2}-\\hat{h}_{j-2})+ \\Delta t s(u^n_j),$ under the time step constraint $\\Delta t \\le \\frac{\\text{CFL } \\Delta x}{\\lambda _{max}+ s_{max} \\Delta x},$ where $\\lambda _{max}=\\max \|f^{\\prime }(u)\|$ and $s_{max}=\\max \|s^{\\prime }(u)\|$ .", "We propose to first modify the source term such that $\\tilde{u}^{n+1}_j \\ge \\epsilon _s$ , with $\\epsilon _s=\\min _j(u^{n+1}_j, 10^{-13})$ , where $\\lbrace u^{n+1}_j\\rbrace $ are positive solutions computed from (REF ) and $10^{-13}$ is a small positive number related to machine precision.", "Here $\\tilde{u}^{n+1}_j$ is $\\tilde{u}^{n+1}_j=u^{n}_j-\\lambda (\\hat{h}_{j+2}-\\hat{h}_{j-2})+ \\Delta t \\tilde{s}^{rk}_j,$ with $\\tilde{s}^{rk}_j=r_j (\\hat{s}^{rk}_j-s(u^n_j))+s(u^n_j),$ and $\\hat{s}^{rk}_j \\doteq \\frac{1}{6}\\left(s(u^n_j)+ s(u^{(1)}_j) + 4 s(u^{(2)}_j)\\right), $ as in (REF ).", "$r_j$ is designed by the linear constraints to preserve the PP property of $\\lbrace \\tilde{u}^{n+1}_j\\rbrace _j$ .", "Specifically, $r_j={\\left\\lbrace \\begin{array}{ll}\\min (\\frac{\\epsilon _s-u^{n+1}_j}{\\Delta t \\Delta s_j}, 1),& \\quad \\text{if } \\tilde{\\tilde{u}}_j < \\epsilon _s \\\\1, &\\quad \\text{otherwise }\\end{array}\\right.", "},$ where $\\Delta s_j=\\hat{s}^{rk}_j-s(u^n_j)$ and $\\tilde{\\tilde{u}}_j=u^{n}_j-\\lambda (\\hat{h}_{j+2}-\\hat{h}_{j-2})+ \\Delta t \\hat{s}^{rk}_j$ .", "Next the parametrized MPP flux limiters are applied as in (REF ) to satisfy $\\epsilon _s \\le u^{n}_j-\\lambda (\\tilde{H}^{rk}_{j+2}-\\tilde{H}^{rk}_{j-2})+\\Delta t \\tilde{s}^{rk}_j .$ (REF ) leads to the same decomposed inequality (REF ) for the minimum value part, only with $\\Gamma ^m_j$ given by $\\Gamma ^m_j&=&\\epsilon _s-u_j+\\lambda (\\hat{h}_{j+2}-\\hat{h}_{j-2})-\\Delta t \\tilde{s}^{rk}_j \\le 0.$ The procedure proposed above for treating equations with a source term is PP by the design, and is shown to maintain high order accuracy by numerical tests in Section ." ], [ "Parametrized PP flux limiters for compressible Euler equations ", "In this section, we first extend the parametrized MPP flux limiters to PP flux limiters for the compressible Euler equations.", "We then describe how to generalize the proposed approach to systems with source terms and to high dimensional systems.", "In this section, we use letters in bold for vectors." ], [ "Parametrized positivity preserving flux limiters", "For compressible Euler equations in one dimension ${\\bf u}_t+{\\bf f}({\\bf u})_x=0,$ with ${\\bf u}=(\\rho , \\rho u, E)^T$ , ${\\bf f}({\\bf u})=(\\rho u, \\rho u^2+ p, (E+p)u )^T$ , where $\\rho $ is the density, $u$ is the velocity, $p$ is the pressure, $m=\\rho u$ is the momentum, $E=\\frac{1}{2}\\rho u^2+\\frac{p}{\\gamma -1}$ is the total energy from equation of state (EOS) and $\\gamma $ is the ratio of specific heat ($\\gamma =1.4$ for the air).", "Denote $\\hat{\\bf h}_{j+2}$ to be a first order monotone flux, and $\\hat{\\bf H}^{rk}_{j+2}$ to be the linear combinations of fluxes from multiple RK stages, similar to equation (REF ), but in a component-by-component fashion.", "For positivity preserving, we are seeking the flux limiters of the type $\\tilde{\\bf H}^{rk}_{j+2}=\\theta _{j+2} (\\hat{\\bf H}^{rk}_{j+2}-\\hat{\\bf h}_{j+2})+\\hat{\\bf h}_{j+2}$ such that ${\\left\\lbrace \\begin{array}{ll}\\rho ^{n+1}_{j}>0,\\\\p^{n+1}_j>0,\\end{array}\\right.", "}$ for the updated solution ${\\bf u}^{n+1}_j={\\bf u}^{n}_j-\\lambda (\\tilde{\\bf H}^{rk}_{j+2}-\\tilde{\\bf H}^{rk}_{j-2}).$ In the parametrized flux limiters' framework, a pair of $(\\Lambda _{-2, {I_j}}, \\Lambda _{+2, {I_j}})$ is found such that the updated solution satisfies (REF ) for any $(\\theta _{j-2}, \\theta _{j+2}) \\in [0, {\\Lambda _{-2, {I_j}}}]\\times [0, {\\Lambda _{+2, {I_j}}]}$ .", "The high order flux $\\hat{\\bf H}^{rk}_{j+2}$ is modified by (REF ) to preserve positive density and pressure.", "In simulations, preserving positivity is implemented by ${\\left\\lbrace \\begin{array}{ll}\\rho ^{n+1}_{j}\\ge \\epsilon _{\\rho },\\\\p^{n+1}_j\\ge \\epsilon _p.\\end{array}\\right.", "}$ where we introduce small positive numbers $\\epsilon _{\\rho }$ defined by $\\min _{j}(\\rho ^{n+1}_j, 10^{-13})$ and $\\epsilon _p$ defined by $\\min _{j}(p^{n+1}_j, 10^{-13})$ .", "$\\rho ^{n+1}_j$ and $p^{n+1}_j$ are positive density and pressure obtained by the first order monotone scheme and $10^{-13}$ is related to the machine precision.", "Let us denote the first order monotone flux by $\\hat{\\bf h}({\\bf u})=(f^\\rho , f^m, f^E)^T$ , similarly $\\hat{\\bf H}^{rk}=(\\hat{f}^\\rho , \\hat{f}^m, \\hat{f}^E)^T$ and $\\tilde{\\bf H}^{rk}=(\\tilde{f}^{\\rho }, \\tilde{f}^{m}, \\tilde{f}^{E})^T$ .", "The proposed process can be dissected into two steps.", "Find the limiting parameters $\\theta _{j\\pm 2}$ to preserve the positivity of the density, $\\rho ^{n+1}_j=\\rho ^n_j-\\lambda (\\tilde{f}^{\\rho }_{j+2}-\\tilde{f}^{\\rho }_{j-2}).$ Thus, the limiting parameters $\\theta _{j\\pm 2}$ are found to satisfy $\\epsilon _{\\rho } \\le \\Gamma _j-\\lambda (\\theta _{j+2} (\\hat{f}^{\\rho }_{j+2}-f^{\\rho }_{j+2}) -\\theta _{ j-2} (\\hat{f}^{\\rho }_{j-2}-f^{\\rho }_{j-2})),$ which is equivalent to $0 \\le \\Gamma _j-\\epsilon _{\\rho }-\\lambda (\\theta _{j+2} (\\hat{f}^{\\rho }_{j+2}-f^{\\rho }_{j+2}) -\\theta _{ j-2} (\\hat{f}^{\\rho }_{j-2}-f^{\\rho }_{j-2})),$ where $\\Gamma _j=\\rho ^n_j-\\lambda (f^{\\rho }_{j+2}- f^{\\rho }_{j-2}) \\ge \\epsilon _{\\rho }$ .", "A pair of limiting parameters $(\\Lambda ^{\\rho }_{-2, {I_j}}, \\Lambda ^{\\rho }_{+2, {I_j}})$ for the positive density of (REF ) can be identified by a similar procedure as described in Section REF .", "We can define a set for the positive density $\\rho ^{n+1}_j$ $S_{\\rho } =\\lbrace (\\theta _{j-2}, \\theta _{j+2}): 0\\le \\theta _{j-2} \\le \\Lambda ^{\\rho }_{-2, {I_j}}, 0\\le \\theta _{j+2} \\le \\Lambda ^{\\rho }_{+2, {I_j}} \\rbrace ,$ which is plotted as the rectangle bounded by the dash line in Figure REF .", "Find the limiting parameters $\\theta _{j\\pm 2}$ within the region $S_{\\rho }$ to preserve the positivity of the pressure.", "We seek a sufficient condition such that the pressure given by (REF ) satisfies $p^{n+1}_j (\\theta _{j-2}, \\theta _{j+2})=(\\gamma -1)\\left(E^{n+1}_j-\\frac{1}{2} \\frac{(m^{n+1}_j )^2}{\\rho ^{n+1}_j}\\right)\\ge \\epsilon _p.$ The decoupling of (REF ) for ($\\theta _{j-2}, \\theta _{j+2}$ ) is different from the scalar case since the principal variables are nonlinearly dependent on each other.", "However the idea is still to separate $\\theta _{j-2}$ and $\\theta _{j+2}$ .", "Since $\\rho ^{n+1}_j \\ge \\epsilon _{\\rho }$ is guaranteed by the previous step, we first put the concave property of pressure [23] in the following remark for future reference: Remark 3.1 The pressure as a function of $(\\rho , m, E)$ is concave, i.e., $p(\\alpha {\\bf U_1}+(1-\\alpha ) {\\bf U_2})\\ge \\alpha p({\\bf U_1})+(1-\\alpha ) p({\\bf U_2})$ for $0\\le \\alpha \\le 1$ if $\\rho _1, \\rho _2> 0$ .", "Therefore $p^{n+1}_j (\\theta _{j-2}, \\theta _{j+2})$ is a concave function of $(\\theta _{j-2}, \\theta _{j+2})$ on $S_\\rho $ due to the linear dependence of $(\\rho ^{n+1}_j , m^{n+1}_j, E^{n+1}_j)$ on $(\\theta _{j-2}, \\theta _{j+2})$ .", "Therefore, if $p^{n+1}_j (\\vec{\\theta }^l) \\ge \\epsilon _p$ , with $\\vec{\\theta }^l = (\\theta ^l_{j-2}, \\theta ^l_{j+2})$ for $l=1, 2$ , then $p^{n+1}_j (\\alpha \\vec{\\theta }^1 + (1-\\alpha ) \\vec{\\theta }^2) \\ge \\epsilon _p, \\quad 0\\le \\alpha \\le 1.$ We define an admissible set $S_\\theta =\\lbrace (\\theta _{j-2}, \\theta _{j+2})\\in S_\\rho : (\\theta _{j-2}, \\theta _{j+2}) \\text{ satisfies } (\\ref {pre})\\rbrace .$ $S_\\theta $ is a convex set thanks to Remark REF .", "Let the three vertices of the rectangle $S_\\rho $ other than $(0, 0)$ be denoted by $A^1=(0, \\Lambda ^{\\rho }_{+2, {I_j}}),\\quad A^2=(\\Lambda ^{\\rho }_{-2, {I_j}}, 0),\\quad A^3=(\\Lambda ^{\\rho }_{-2, {I_j}} , \\Lambda ^{\\rho }_{+2, {I_j}}),$ see Figure REF .", "Based on the concave property in Remark REF , we propose the following way of decoupling (REF ).", "For i=1, 2, 3, if $p(A^i)\\ge \\epsilon _p$ , let $B^i =A^i$ ; otherwise find $r$ such that $p(r A^i)\\ge \\epsilon _p$ and let $B^i= r A^i$ .", "The three $B^i$ 's and $(0, 0)$ form a convex polygonal region, denoted as $S_p$ , inside $S_\\theta $ .", "Such convex polygonal region $S_p$ is outlined by the dash dot line in Figure REF .", "We define the decoupling rectangle, as a subset of $S_p$ , to be $R_{\\rho , p}=[0, \\min (B^2_1, B^3_1)]\\times [0, \\min (B^1_2, B^3_2)],$ see the region outlined by the solid line in Figure REF .", "That is, within $S_p$ , we find the decoupling rectangle $R_{\\rho , p}$ with left-bottom node on $(0, 0)$ and right-top node $(\\Lambda _{-2, {I_j}}, \\Lambda _{+2, {I_j}})$ as close to $(1, 1)$ as possible to best preserve the accuracy while achieving the PP property of high order numerical schemes.", "Let $(\\Lambda _{-2, {I_j}}, \\Lambda _{+2, {I_j}})=(\\min (B^2_1, B^3_1), \\min (B^1_2, B^3_2)).$ Finally, similar to equation (REF ) for the MPP flux limiters, the locally defined limiting parameter is given as $\\theta _{j+2}=\\min (\\Lambda _{-2,{I_j}}, \\Lambda _{+2,{I_{j+1}}})$ .", "Figure: The decoupling rectangle R ρ,p R_{\\rho ,p} (bounded by the solid line) with the right-top node (Λ -2,I j ,Λ +2,I j )(\\Lambda _{-2, {I_j}}, \\Lambda _{+2, {I_j}}).", "S ρ S_{\\rho } is the rectangle bounded by the dash line.", "S p S_p is the polygonal bounded by the dash dot line.Remark 3.2 The limiter above can preserve positive density and pressure by its design due to the two sufficient conditions (REF ) and (REF ).", "For general equation of state, if $\\rho > 0$ , then $p>0 \\Leftrightarrow e>0$ , where the internal energy $e$ can always be written as a concave function of $(\\rho , m, E)^T$ similarly as (REF ) [26].", "Similar procedure can be followed for PP property of numerical solutions." ], [ "Extension to Euler system with source term", "The compressible Euler equations may come with source terms in the form of ${\\bf u}_t+{\\bf f}({\\bf u})_x={\\bf s}({\\bf u}),$ For example, four kinds of source terms were discussed in [25]: geometric, gravity, chemical reaction and radiative cooling.", "The PP flux limiters can be applied by the following three steps.", "Choose a time step, such that the first order scheme (REF ) is PP, ${\\bf u}^{n+1}_j={\\bf u}^{n}_j-\\lambda (\\hat{\\bf h}_{j+2}-\\hat{\\bf h}_{j-2})+\\Delta t{\\bf s}({\\bf u}^n_j).$ Find $r$ such that the scheme (REF ) with the modified source terms is PP ${\\bf u}^{n+1}_j={\\bf u}^{n}_j-\\lambda (\\hat{\\bf h}_{j+2}-\\hat{\\bf h}_{j-2})+\\Delta t\\tilde{\\bf s}^{rk}_j,$ with $\\tilde{\\bf s}^{rk}_j = r(\\hat{\\bf s}^{rk}_j-{\\bf s}({\\bf u}^n_j))+{\\bf s}({\\bf u}^n_j)$ , $\\hat{\\bf s}^{rk}_j$ is similarly defined as (REF ) component-by-component.", "Finally find $\\theta _{j\\pm 2}$ for the modified high order flux $\\tilde{\\bf H}^{rk}_{j+2}$ , such that (REF ) is PP ${\\bf u}^{n+1}_j={\\bf u}^{n}_j-\\lambda (\\tilde{\\bf H}^{rk}_{j+2}-\\tilde{\\bf H}^{rk}_{j-2})+\\Delta t\\tilde{\\bf s}^{rk}_j.$ The procedure is similar as in the previous subsection." ], [ "Extension to the multi-dimensional Euler system", "In this subsection, we extend the previously proposed PP flux limiters to Euler equations in two-dimensions ${\\bf u}_t+{\\bf f}({\\bf u})_x+{\\bf g}({\\bf u})_y=0,$ with ${\\bf u}=(\\rho , m_u, m_v, E)^T$ , ${\\bf f}({\\bf u})=(m_u, \\rho u^2+p, \\rho u v, (E+p)u)^T$ and ${\\bf g}({\\bf u})=(m_v, \\rho u v, \\rho v^2+p, (E+p)v)^T$ .", "$\\rho $ is the density, $u$ is the velocity in $x$ direction, $v$ is the velocity in $y$ direction, $p$ is the pressure, $m_u=\\rho u$ and $m_v=\\rho v$ are the momenta, $E=\\frac{1}{2}\\rho u^2+\\frac{1}{2}\\rho v^2+\\frac{p}{\\gamma -1}$ is the total energy and $\\gamma $ is the ratio of specific heat.", "The high order finite difference scheme with PP flux limiters at the final stage of a RK time discretization is given by ${\\bf u}^{n+1}_{i,j}={\\bf u}^{n}_{i,j}-\\lambda _x(\\tilde{\\bf H}^{rk}_{i+2,j}-\\tilde{\\bf H}^{rk}_{i-2,j})-\\lambda _y(\\tilde{\\bf G}^{rk}_{i,j+2}- \\tilde{\\bf G}^{rk}_{i,j-2}),$ with $\\tilde{\\bf H}^{rk}_{i+2,j}&=&\\theta _{i+2,j}(\\hat{\\bf H}^{rk}_{i+2,j}-\\hat{\\bf h}_{i+2,j})+\\hat{\\bf h}_{i+2,j}, \\\\\\tilde{\\bf G}^{rk}_{i,j+2}&=&\\theta _{i,j+2}(\\hat{\\bf G}^{rk}_{i,j+2}-\\hat{\\bf g}_{i,j+2})+\\hat{\\bf g}_{i,j+2},$ where $\\hat{\\bf H}^{rk}_{i+2,j}$ and $\\hat{\\bf G}^{rk}_{i,j+2}$ are linear combinations of fluxes from multiple RK stages similarly as (REF ) in the scalar case but in a component-wise fashion, $\\hat{\\bf h}_{i+2,j}$ and $\\hat{\\bf g}_{i,j+2}$ are first order monotone fluxes.", "Similar to the 1D case, we find the four parametrized limiters $\\Lambda ^{\\rho }_{L, {I_{ij}}}$ , $\\Lambda ^{\\rho }_{R,{I_{ij}}}$ , $\\Lambda ^{\\rho }_{U, {I_{ij}}}$ and $\\Lambda ^{\\rho }_{D,{I_{ij}}}$ , such that for all $\\theta _{i\\pm 2,j}$ and $\\theta _{i,j\\pm 2}$ in the set $S_{\\rho } = &\\lbrace (\\theta _{i-2,j}, \\theta _{i+2,j},\\theta _{i,j-2}, \\theta _{i,j+2}): 0\\le \\theta _{i-2,j} \\le \\Lambda ^{\\rho }_{L, {I_{ij}}}, \\nonumber \\\\&0\\le \\theta _{i+2,j} \\le \\Lambda ^{\\rho }_{R, {I_{ij}}},0\\le \\theta _{i,j-2} \\le \\Lambda ^{\\rho }_{D, {I_{ij}}},0\\le \\theta _{i,j+2} \\le \\Lambda ^{\\rho }_{U, {I_{ij}}} \\rbrace $ we have $\\rho ^{n+1}_{i,j}\\ge \\epsilon _\\rho $ .", "With the positive density $\\rho ^{n+1}_{i,j}$ , the pressure is updated by the constraint $p^{n+1}_{i,j}& (\\theta _{i-2,j}, \\theta _{i+2,j},\\theta _{i,j-2}, \\theta _{i,j+2})=\\nonumber \\\\&(\\gamma -1)(E^{n+1}_{i,j} -\\frac{1}{2} \\frac{((m_u)^{n+1}_{i,j} )^2+((m_v)^{n+1}_{i,j} )^2}{\\rho ^{n+1}_{i,j}})\\ge \\epsilon _p.$ Let the convex admissible set for positive pressure be $S_\\theta =\\lbrace (\\theta _{i-2,j}, \\theta _{i+2,j},\\theta _{i,j-2}, \\theta _{i,j+2})\\in S_\\rho :(\\theta _{i-2,j}, \\theta _{i+2,j},\\theta _{i,j-2}, \\theta _{i,j+2}) \\text{ satisfies } (\\ref {pre2d})\\rbrace $ Let the sixteen vertices of $S_\\rho $ denoted by $A^{k_1,k_2,k_3, k_4}=(k_1 \\Lambda ^{\\rho }_{L,{I_{ij}}}, k_2 \\Lambda ^{\\rho }_{R,{I_{ij}}},k_3 \\Lambda ^{\\rho }_{D,{I_{ij}}},k_4 \\Lambda ^{\\rho }_{U,{I_{ij}}}),$ with $k_1, k_2, k_3, k_4$ to be 0 or 1.", "We decouple (REF ) in the following way: For $(k_1,k_2,k_3,k_4)\\ne (0,0,0,0)$ , if $p(A^{k_1,k_2,k_3,k_4})\\ge \\epsilon _p$ , let $B^{k_1,k_2,k_3,k_4} =A^{k_1,k_2,k_3,k_4}$ ; otherwise find $r$ such that $P(r A^{k_1,k_2,k_3,k_4})\\ge \\epsilon _p$ and let $B^{k_1,k_2,k_3,k_4} =r A^{k_1,k_2,k_3,k_4}$ .", "The 15 $B^{k_1,k_2,k_3,k_4}$ 's with the origin $(0, 0,0,0)$ form a four dimensional polyhedra inside $S_{\\theta }$ ; The decoupling tesseract can be defined by $R_{\\rho , p}=&[0, \\min (B^{1,1,1,0}_1, B^{1,1,0,1}_1,B^{1,0,1,1}_1)] \\times [0, \\min (B^{1,1,1,0}_2, B^{1,1,0,1}_2,B^{0,1,1,1}_2)] \\nonumber \\\\\\times & [0, \\min (B^{1,1,1,0}_3, B^{1,0,1,1}_3, B^{0,1,1,1}_3)]\\times [0, \\min (B^{1,1,0,1}_4, B^{1,0,1,1}_4,B^{0,1,1,1}_4)].$ Let $(\\Lambda _{L, {I_{ij}}}&, \\Lambda _{R, {I_{ij}}},\\Lambda _{D, {I_{ij}}}, \\Lambda _{U, {I_{ij}}}) =(\\min (B^{1,1,1,0}_1, B^{1,1,0,1}_1,B^{1,0,1,1}_1), \\min (B^{1,1,1,0}_2, B^{1,1,0,1}_2, \\nonumber \\\\&B^{0,1,1,1}_2),\\min (B^{1,1,1,0}_3, B^{1,0,1,1}_3, B^{0,1,1,1}_3), \\min (B^{1,1,0,1}_4, B^{1,0,1,1}_4,B^{0,1,1,1}_4)).$ Finally, similar to equation (REF ) for the MPP flux limiters, the locally defined limiting parameter is given as $\\theta _{i+2,j}=\\min (\\Lambda _{L,{I_{ij}}}, \\Lambda _{R,{I_{i+1,j}}})$ and $\\theta _{i,j+2}=\\min (\\Lambda _{D,{I_{ij}}}, \\Lambda _{U,{I_{i,j+1}}})$ .", "Remark 3.3 For two dimensional compressible Euler equations with source terms, it can be done similarly as the one dimensional case." ], [ "Numerical simulations", "In this section, we will use the 5th order finite difference WENO scheme for space discretization [9] and a 4th order Runge-Kutta time discretization [15], denote as “WENO5RK4”, with the proposed PP flux limiters for simulating the compressible Euler equations.", "Here a 4th order RK time discretization is adopted for better observation of accuracy by taking the time step to be $\\Delta t=\\text{CFL } \\Delta x$ .", "Most of the tests are from [26].", "Below, $\\text{CFL }=0.6$ unless otherwise specified.", "Example 4.1 (Accuracy test for a scalar problem with a source term.)", "We consider $u_t+u_x=-u$ with the initial condition $u(x,0)=\\sin ^4(x),$ and the periodic boundary condition.", "The exact solution is given by $u(x,t)=e^{-t}\\sin ^4(x-t).$ The minimum value of the exact solution is $u_{m}=0$ .", "This example is used to test the PP property and accuracy of dealing with a source term.", "In Table REF , we can see the PP property is preserved and the 5th order accuracy has been maintained.", "Table: Example .", "A scalar advection problem with a source term at T=0.1T=0.1.", "v min v_{min} is the minimum value of the numerical solution.Example 4.2 (Accuracy test for the global Lax-Friedrichs flux.)", "We consider the Burgers' equation with the initial condition $u(x,0)=(1+\\sin (x))/2$ and a periodic boundary condition.", "We consider the WENO5RK4 scheme with the global Lax-Friedrichs (LxF) fluxes.", "Let $f^\\pm _i=2(f(u^n_i)\\pm \\alpha u^n_i),\\quad i=j-p, \\cdots , j+q,$ with $\\alpha \\ge \\max _{u} \|f^{\\prime }(u)\|$ .", "The numerical flux $\\hat{H}_{j+2}=f^-_{j+2}+f^+_{j+2}$ in (REF ), where $f^\\pm _{j+2}$ are reconstructed based on WENO schemes from (REF ) with the corresponding upwind mechanism.", "We numerically investigate the time step restriction for maintaining high order accuracy using the global Lax-Friedrichs flux, since it is frequently used in the computation of the Euler system.", "In [20], local truncation analysis is performed to prove that MPP flux limiters can maintain up to third order accuracy of the original scheme with no additional CFL constraint (i.e.", "$\\text{CFL}\\le 1$ ) when the upwind flux is used.", "However, when the global LxF flux with extra large $\\alpha $ in equation (REF ) is used, there is a mild time step restriction with $\\text{CFL}\\le 0.886$ .", "It is technically challenging to theoretically estimate such time step restriction for maintaining high order accuracy (e.g.", "fifth order) of the MPP flux limiters even for scalar equations, therefore we rely on extensive numerical tests.", "We consider the scheme with the global Lax-Friedrichs flux with extra large $\\alpha = 1.3$ (greater than $\\max _u\|f^{\\prime }(u)\|=1$ ).", "The time step is chosen to be $\\Delta t=\\text{CFL} \\Delta x / \\alpha $ .", "In Table REF , we show that for the 5th order linear scheme (linear weights instead of nonlinear weights in WENO5) with the 4th order Runge-Kutta time discretization, when $\\text{CFL}=0.886$ , the 5th order accuracy is maintained with the MPP flux limiters.", "In fact, $\\text{CFL}=0.886$ works for all other $\\alpha $ 's we tested, the results are not listed here to save space.", "Table: Example .", "Burgers' equation at T=0.2T=0.2.", "α=1.3\\alpha =1.3 for the global LxF flux ().", "Δt=0.886Δx/α\\Delta t=0.886 \\Delta x / \\alpha .", "u max -v max u_{max}-v_{max} is the difference of the maximum values between the numerical solution and the exact solution.Example 4.3 (Accuracy test for 2D vortex evolution problem.)", "We consider the vortex evolution problem [7] to test the accuracy.", "For this problem, the mean flow is $\\rho =p=u=v=1$ and is added by an isentropic vortex perturbation centered at $(x_0,y_0)$ in $(u,v)$ with $T=p/\\rho $ , no perturbation in entropy $S=p/\\rho ^\\gamma $ , $(\\delta u,\\delta v)=\\frac{\\varepsilon _{vortex}}{2\\pi }e^{0.5(1-r^2)}(-\\bar{y},\\bar{x}),\\quad \\delta T=-\\frac{(\\gamma -1)\\epsilon ^2}{8\\gamma \\pi ^2}e^{(1-r^2)},\\quad \\delta S=0,$ where $(\\bar{x},\\bar{y})=(x-x_0,y-y_0)$ , $r^2=\\bar{x}^2+\\bar{y}^2$ .", "The computational domain is taken to be $[-5, 15]\\times [-5, 15]$ and $(x_0,y_0)=(5,5)$ .", "The boundary condition is periodic.", "$\\gamma =1.4$ and the vortex strength is $\\varepsilon _{vortex}=10.0828$ as in [26].", "The exact solution is the passive convection of the vortex with the mean flow.", "The lowest density and pressure of the exact solution are $7.8\\times 10^{-15}$ and $1.7\\times 10^{-20}$ .", "$\\epsilon _{WENO}$ in the nonlinear WENO weights is chosen to be $10^{-5}$ , which is between $10^{-2}$ and $10^{-6}$ [7].", "In Table REF , we can clearly observe the 5th order accuracy with the PP flux limiters.", "Table: Example .", "Vortex evolution problem at T=0.01T=0.01.", "ϵ WENO =10 -5 \\epsilon _{WENO}=10^{-5}.", "ρ min \\rho _{min} andp min p_{min} are the minimum density and pressure of the numerical solution respectively.Example 4.4 1D low density and low pressure problems.", "We consider two 1D low density and low pressure problems for the ideal gas.", "The first one is a 1D Riemann problem, the initial condition is $\\rho _L=\\rho _R=7$ , $u_L=-1$ , $u_R=1$ , $p_L=p_R=0.2$ and $\\gamma =1.4$ , which is a double rarefaction problem.", "The exact solution contains vacuum.", "In Fig.", "REF (left), we show the results with the PP flux limiters at $T=0.6$ on a mesh size of $\\Delta x=1/200$ .", "The second one is the 1D Sedov blast wave.", "For the initial condition, the density is 1, the velocity is 0, the total energy is $10^{-12}$ everywhere except in the center cell, which is a constant $E_0/\\Delta x$ with $E_0=3200000$ .", "$\\gamma =1.4$ .", "In Fig.", "REF (right), we show the results with the PP flux limiters at $T=0.001$ on a mesh size of $\\Delta x=1/200$ .", "Figure: Example .", "Left: double rarefaction problem at T=0.6T=0.6.", "Right: 1D Sedov blast wave at T=0.001T=0.001.", "Δx=1 200\\Delta x=\\frac{1}{200}.", "The solid lines are the exact solutions.", "Symbols are the numerical solutions.Example 4.5 2D low density and low pressure problems.", "Now we consider two 2D low density and low pressure problems for the ideal gas.", "The first one is the 2D Sedov blast wave.", "The computational domain is a square of $[0, 1.1]\\times [0, 1.1]$ .", "For the initial condition, similar to the 1D case, the density is 1, the velocity is 0, the total energy is $10^{-12}$ everywhere except in the lower left corner is the constant $\\frac{0.244816}{\\Delta x \\Delta y}$ .", "$\\gamma =1.4$ .", "The numerical boundary on the left and bottom edges is reflective.", "In Fig.", "REF (left), we show the numerical density at the mesh sizes $\\Delta x=\\Delta y=\\frac{1.1}{160}$ with the PP flux limiters at $T=1$ .", "The numerical solution with cutting along the diagonal matches the exact solution very well in Fig.", "REF (right).", "Figure: Example .", "2D Sedov blast wave.", "T=1T=1.", "Δx=Δy=1.1 160\\Delta x=\\Delta y=\\frac{1.1}{160}.Left: contour of density.", "Right: cut along diagonal, the solid line is the exact solution, symbols arethe numerical solution.The second one is the shock diffraction problem.", "The computational domain is the union of $[0,1]\\times [6,11]$ and $[1,13]\\times [0,11]$ .", "The initial condition is a pure right-moving shock of $Mach=5.09$ , initially located at $x=0.5$ and $6\\le y \\le 11$ , moving into undisturbed air ahead of the shock.", "The undisturbed air has a density of $1.4$ and a pressure of 1.", "The boundary conditions are inflow at $x=0$ , $6\\le y \\le 11$ , outflow at $x=13$ , $0\\le y \\le 11$ , $1\\le x\\le 13$ , $y=0$ and $0\\le x \\le 13$ , $y=11$ , and reflective at the walls $0\\le x \\le 1$ , $y=6$ and $x=1$ , $0\\le y \\le 6$ .", "$\\gamma =1.4$ .", "The density and pressure at the mesh sizes $\\Delta x=\\Delta y=\\frac{1}{32}$ with the PP flux limiters at $T=2.3$ are presented in Fig.", "REF .", "Figure: Example .", "2D shock diffraction problem.", "T=2.3T=2.3.", "Δx=Δy=1 32\\Delta x=\\Delta y=\\frac{1}{32}.Left: density, 20 equally spaced contour lines from ρ=0.066227\\rho =0.066227 to ρ=7.0668\\rho =7.0668.", "Right: pressure,40 equally spaced contour lines from p=0.091p=0.091 to p=37p=37.Example 4.6 High Mach number astrophysical jets.", "We consider two high Mach number astrophysical jets without the radiative cooling [5], [26].", "The first one is a Mach 80 problem.", "$\\gamma =5/3$ .", "The computational domain is $[0,2]\\times [-0.5,0.5]$ , which is full of the ambient gas with $(\\rho , u, v, p)=(0.5,0,0,0.4127)$ initially.", "The boundary conditions for the right, top and bottom are outflows.", "For the left boundary, $(\\rho , u, v, p)=(5,30,0,0.4127)$ if $y\\in [-0.05, 0.05]$ and $(\\rho , u, v, p)=(5,0,0,0.4127)$ otherwise.", "The numerical density on a mesh of $448\\times 224$ grid points with the PP flux limiters at $T=0.07$ is shown in Fig.", "REF (left).", "Then a Mach 2000 problem is considered to show the robustness of the scheme with the PP flux limiters.", "The computational domain is taken as $[0,1]\\times [-0.25,0.25]$ , initially full of the ambient gas with $(\\rho , u, v, p)=(0.5,0,0,0.4127)$ .", "Similarly, the right, top and bottom boundary are outflows.", "For the left boundary, $(\\rho , u, v, p)=(5,800,0,0.4127)$ if $y\\in [-0.05, 0.05]$ and $(\\rho , u, v, p)=(5,0,0,0.4127)$ otherwise.", "The numerical density at a mesh of $800\\times 400$ grid points with the PP flux limiters at $T=0.001$ is shown in Fig.", "REF (right).", "Figure: Example .", "High Mach number astrophysical jet.", "T=2.3T=2.3.Left: density of Mach 80 at T=0.07T=0.07 with mesh 448×224448\\times 224.Right: density of Mach 2000 at T=0.001T=0.001 with mesh 800×400800\\times 400.Example 4.7 The reactive Euler equations.", "We consider the following two-dimensional Euler equations with a source term, which are often used to model the detonation waves [18], [26]: $& \\mathbf {u}_t+\\mathbf {f}(\\mathbf {u})_x+\\mathbf {g}(\\mathbf {u})=\\mathbf {s}(\\mathbf {u}),\\quad t\\ge 0,\\quad (x,y)\\in \\mathbb {R}^2,\\\\& \\mathbf {u}=\\begin{pmatrix}\\rho \\\\m_u \\\\m_v \\\\E \\\\\\rho Y\\end{pmatrix}, \\quad \\mathbf {f}(\\mathbf {u})=\\begin{pmatrix}m_u \\\\\\rho u^2+p \\\\\\rho u v \\\\(E+p)u \\\\\\rho u Y\\end{pmatrix} ,\\quad \\mathbf {g}(\\mathbf {u})=\\begin{pmatrix}m_v \\\\\\rho u v \\\\\\rho v^2+p\\\\(E+p)v \\\\\\rho v Y\\end{pmatrix},\\quad \\mathbf {s}(\\mathbf {u})=\\begin{pmatrix}0 \\\\0 \\\\0 \\\\0 \\\\\\omega \\end{pmatrix},$ with $m_u=\\rho u,\\quad m_v=\\rho v,\\quad E=\\frac{1}{2}\\rho u^2+\\frac{1}{2}\\rho v^2+\\frac{p}{\\gamma -1}+\\rho q Y,$ where $q$ is the heat release rate of reaction, $\\gamma $ is the specific heat ratio and $Y$ is the reactant mass fraction.", "The source term is assumed to be in an Arrhenius form $\\omega =-\\tilde{K}\\rho Y\\exp (-\\tilde{T}/T),$ where $T=\\frac{p}{\\rho }$ is the temperature, $\\tilde{T}$ is the activation temperature and $\\tilde{K}$ is a constant.", "The eigenvalues of the Jacobian $\\mathbf {f}^{\\prime }(\\mathbf {u})$ are $u-c, u, u, u, u+c$ and the eigenvalues of the Jacobian $\\mathbf {g}^{\\prime }(\\mathbf {u})$ are $v-c, v, v, v, v+c$ , where $c=\\sqrt{\\gamma \\frac{p}{\\rho }}$ .", "The computation domain for this problem is the union of $[0,1]\\times [2,5]$ and $[1,5]\\times [0,5]$ .", "The initial conditions are, if $x<0.5$ , $(\\rho ,u,v,E,Y)=(11,6.18,0,970,1)$ ; otherwise, $(\\rho ,u,v,E,Y)=(1,0,0,55,1)$ .", "The boundary conditions are reflective except at $x=0$ , $(\\rho ,u,v,E,Y)=(11,6.18,0,970,1)$ .", "Here the parameters are chosen to be $\\gamma =1.2$ , $q=50$ , $\\tilde{T}=50$ and $\\tilde{K}=2566.4$ .", "This problem is similar to the shock diffraction problem in Example REF , but this one has a source term.", "The time step is taken to be $\\Delta t=\\frac{\\text{CFL}}{ \\lambda _{max}(\\frac{1}{\\Delta x}+\\frac{1}{\\Delta y})+\\tilde{K} },$ where $\\lambda _{max}=\\max \\lbrace \\Vert \|u\|+c\\Vert _{\\infty }, \\Vert \|v\|+c\\Vert _{\\infty }\\rbrace $ on all grids, and $\\tilde{K}$ comes from the source term (REF ), such that the first order monotone scheme is PP.", "The numerical density and pressure at a mesh of $400\\times 400$ grid points with the PP flux limiters at $T=0.6$ are shown in Fig.", "REF , which are comparable to the results in [18], [26].", "Figure: Example .", "Detonation diffraction at a 90 ∘ 90^\\circ corner.", "T=0.6T=0.6.Mesh 400×400400\\times 400.", "Left: density; Right: pressure.Example 4.8 General equation of state.", "We consider the three species model of the one-dimensional Euler system with a more general equation of state in [19], [26].", "The model involves three species, $O_2$ , $O$ and $N_2$ ($\\rho _1=\\rho _O$ , $\\rho _2=\\rho _{O_2}$ and $\\rho _3=\\rho _{N_2}$ ) with the reaction $O_2 + N_2 \\rightleftharpoons O + O + N_2.$ The governing equations are $\\begin{pmatrix}\\rho _1 \\\\\\rho _2 \\\\\\rho _3 \\\\\\rho u \\\\E\\end{pmatrix}_t+\\begin{pmatrix}\\rho _1 u \\\\\\rho _2 u \\\\\\rho _3 u \\\\\\rho u^2 + p \\\\(E+p) u\\end{pmatrix}_x=\\begin{pmatrix}2M_1\\omega \\\\-M_2\\omega \\\\0 \\\\0 \\\\0\\end{pmatrix},$ and $\\rho =\\sum _{s=1}^3\\rho _s, \\quad p=RT\\sum _{s=1}^3\\frac{\\rho _s}{M_s}, \\quad E=\\sum _{s=1}^3 \\rho _s e_s(T)+\\rho _1 h_1^0+\\frac{1}{2}\\rho u^2,$ where the enthalpy $h_1^0$ is a constant, $R$ is the universal gas constant, $M_s$ is the molar mass of species $s$ , and the internal energy $e_s(T)=\\frac{3RT}{2M_s}$ and $\\frac{5RT}{2M_s}$ for monoatomic and diatomic species respectively.", "The rate of the chemical reaction is given by $\\omega =\\left(k_f(T)\\frac{\\rho _2}{M_2}-k_b(T)\\left(\\frac{\\rho _1}{M_1}\\right)^2\\right)\\sum _{s=1}^3\\frac{\\rho _s}{M_s}, \\quad k_f=C_0 T^{-2}\\exp (-E_0/T), \\\\k_b=k_f/\\exp (b_1+b_2\\log z+ b_3 z+b_4 z^2+b_5 z^3), \\quad z=10000/T.$ The parameters and constants are $h_1^0=1.558\\times 10^7$ , $R=8.31447215$ , $C_0=2.9\\times 10^{17} m^3$ , $E_0=59750 K$ , and $b_1=2.855$ , $b_2=0.988$ , $b_3=-6.181$ , $b_4=-0.023$ , $b_5=-0.001$ .", "The eigenvalues of the Jacobian are $(u,u,u,u+c,u-c)$ where $c=\\sqrt{\\gamma \\frac{p}{\\rho }}$ with $\\gamma =1+\\frac{p}{T\\sum _{s=1}^3\\rho _s e_s^{\\prime }(T)}$ .", "Similar to Example REF , the time step is chosen to be $\\Delta t=\\frac{\\text{CFL } \\Delta x}{\\lambda _{max}+s_{max} \\Delta x},$ here $\\lambda _{max}=\\max \\lbrace \\Vert \|u\|+c\\Vert _{\\infty }\\rbrace $ on all grids and $s_{max}$ is $s_{max}=\\max \\left\\lbrace \\left\|\\frac{M_2\\omega }{\\rho _2}\\right\|,\\left\|\\frac{2M_1\\omega }{\\rho _1}\\right\|\\right\\rbrace .$ A shock tube problem is considered for the reactive flows with high pressure on the left and low pressure on the right initially in the chemical equilibrium ($\\omega =0$ ).", "The initial conditions are: $(p_L, T_L)=(1000 N/m^2, 8000K), \\quad (p_R, T_R)=(1 N/m^2, 8000K),$ with zero velocity everywhere and the densities satisfying $\\frac{\\rho _1}{2M_1}+\\frac{\\rho _2}{M_2}=\\frac{21}{79}\\frac{\\rho _3}{M_3},$ where $M_1=0.016$ , $M_2=0.032$ and $M_3=0.028$ .", "The initial densities of $O$ , $O_2$ and $N_2$ are $5.251896311257204\\times 10^{-5}$ , $3.748071704863518\\times 10^{-5}$ and $2.962489471973072\\times 10^{-4}$ on the left respectively, and $8.341661837019181\\times 10^{-8}$ , $9.455418692098664\\times 10^{-11}$ and $2.748909430004963\\times 10^{-7}$ on the right respectively.", "The numerical solution with the PP flux limiter is computed on a mesh size of $\\Delta x=\\frac{2}{4000}$ up to $T=0.0001$ .", "$\\epsilon _{WENO}=10^{-20}$ is taken as in [26].", "In Fig.", "REF , the positivity of $\\rho _1$ , $\\rho _2$ , $\\rho _3$ and $p$ is preserved and converged solutions are observed.", "Figure: Example .", "Three species reaction problem at T=0.0001T=0.0001.", "Thesolid lines are the reference solutions at Δx=2 8000\\Delta x=\\frac{2}{8000}.", "Symbols are the numerical solutionsat Δx=2 4000\\Delta x=\\frac{2}{4000}." ], [ "Conclusion", "We addressed the potential negative density and pressure problem that emerges when the high order WENO schemes are applied to solve compressible Euler equations in some extreme situations.", "The approach that we propose is in the conservative high order finite difference WENO approximation framework.", "We generalized the MPP flux limiting technique for the high order finite difference WENO methods solving scalar conservation law to a class of PP flux limiters for compressible Euler equations.", "We also developed the parametrized flux limiters for equations with source terms.", "Extensive numerical tests show the capability of the proposed approach: without sacrificing accuracy and much of the efficiency, the new schemes produce solutions satisfying the PP property for scalar problems with a source term, and solutions with positive density and pressure for compressible Euler equations with or without source terms.", "Acknowledgement.", "We would like to thank Xiangxiong Zhang from MIT for helpful discussions." ] ]	1403.0594
[ [ "Complex Investigation of SBS Galaxies in Seven Selected Fields" ], [ "Abstract The main criterium for the selection of active objects in the First Byurakan Survey was the presence of uv-excess on low-dispersion spectra registered on photographic plates obtained with the 1m Shmidt type Byurakan telescope.", "Using the presence of emission lines as the second criterium became possible during the Second Byurakan Survey due to its improved technique.", "Through this criterium a majority of objects, extended by morphology, were selected into the separate \"sample of galaxies\".", "Certainly, there were cases of untrue selection, particularly, on faint magnitudes, when absorption lines were taken for emission ones and so on.", "Study of SBS galaxies, including evaluation of an effectivity of selection criteria, was undertaken by means of complex investigation of their very representative part, pooled in our basic sample.", "The completion of the follow-up slit spectroscopy of these about 500 objects formed the main stage of implementation of this program.", "Also, the scheme was developed to provide homogeneous classification, directed, in particular, to separate galaxies of AGN activity, of known types, and starforming, SfG, activity.", "For starforming galaxies, which constitute more than 80% of the basic sample, we provided two classes, SfGcontinual and SfGnebular.", "Averaged statistics of our SfG galaxies show, that every fifth of them is in more active, nebular phase of starforming activity, most of which are known as blue compact galaxies.", "However, it must be noted, that, by the analysis, namely for the latter objects, the effectiveness of the survey is the highest, so that BCGs represent the best product of SBS among extended objects.", "Aimed on further specifications in classification of SfG galaxies, other generalizations and statistics in frames of ongoing investigation, detailed studies of individual galaxies are currently beeing held, based on data of panoramic spectroscopy." ], [ "Introduction", "The Second Byurakan Spectral Sky Survey (SBS) ([14]) was undertaken by Markarian as a direct continuation of the First Byurakan Survey (FBS) ([13]) to achieve magnitudes fainter than in FBS in search of active objects.", "Both surveys were carried out with the 1-m Shmidt type telescope of Byurakan observatory with the use of objective prisms of the same size as the entrance port of the telescope.", "Main differences of conducting the two surveys are seen on Table 1.", "Besides, it must be noted that the objects in SBS have been selected not only by presence of uv-excess on their low dispersion spectra, as it was done in FBS, but also by the presence of emission lines, as the second selection criterium.", "More than 1500 objects, which have been included in the common sample of FBS, are more famous as \"Markarian galaxies\".", "About 3000 objects selected in the 65 fields, each of 16sq.", "deg., which compile the SBS area, are distributed in the three samples, two of which (the sample of quasars and stars) consist of starlike objects.", "In the third one, the sample of galaxies, initially about 1300 extended objects were included.", "Later, with the help of sources other than the SBS photographic plates, 200 more objects have been added to this list.", "As homogeneity of the object selection is important for the purposes of our program, we are working with the original list of SBS galaxies.", "Table: Main differences of conducting the two byurakan surveys.Figure: NO_CAPTION" ], [ "Accomplished Stages of the Program", "2.1 Construction of the basic sample.", "More than a third part of the whole initial sample of SBS galaxies, about 500, constituted our basic sample.", "Namely, it is composed of those selected as galaxies in the seven of 65 in all fields of the survey ([5]).", "The selected fields are numbered in order of ascending of right ascension of their centers, as shown on Fig.1, along with fields' boundaries given within the boundaries of the whole SBS area, in which the distribution of all SBS galaxies is depicted too.", "The second column of the table 2 gives initial numerical distribution of the galaxies by the seven fields (after excluding of the obviously wrong objects).", "These fields differ from many others, in particular by observations, completed with all the three objective prisms, high quality of the obtained photographic plates, providing the best for the survey limiting magnitudes (19.0 -19.5) and so on.", "The level of completeness, calculated for the corresponding seven subsamples of SBS galaxies, according to the test V/Vmax ([17]), was in the range 17.5 - 19.5 mag, i.e.", "higher than the value 16.5 - 17.0 mag, averaged for all 65 fields of the survey.", "The stated above is enough to claim, that our basic sample, being in addition statistically representative, must reflect productivity of the survey along the entire scale of apparent magnitudes, including the limiting ones ([4]).", "2.2.", "Completion of the follow-up slit spectroscopy.", "For the accomplishment of the follow-up slit spectroscopy of all the objects of the initial basic sample in frames of this program observations of more than 200 predominantly faint (17.5$^{\\begin{tiny}m\\end{tiny}}$ - 19.5$^{\\begin{tiny}m\\end{tiny}}$ ) objects have been undertaken for the first time and observations of about 30 objects have been repeated ([4], [6] and references therein, [8]).", "The observations were carried out from 1996 to 2002 with the 6-m telescope of Special Astrophysical Observatory of Russian Academy of Sciences (SAO RAS) (http://www.sao.ru), in combination with the Long Slit Spectrograph (LSS) in different modifications and Multi Pupil Fiber Spectrograph (MPFS) and with the 2.6-m Byurakan telescope in Armenia in combination with the focal cameras ByuFOSC and SCORPIO in its spectral mode.", "The best values for dispersion of spectral material (1.3 Å/pix) were obtained with the MPFS, the worst marks (5 Å/pix), with the LSS in 1996.", "2.3 Development of the classification scheme.", "On the basis of standard diagnostic criteria a scheme adapted to the spectral material has been developed to conduct preliminary and homogeneous classification of all objects of the basic sample.", "The first purpose was to confirm the presence of signs of activity on the slit spectra, then, to identify AGN or starforming activity, using the ratio of forbidden and allowed lines ([19]).", "Galaxies of starforming activity of any type (starburst, HII, BCG, etc.)", "have been combined under the designation SfG (Starforming Galaxy) with two subclasses, which are formed by analogy with the scheme of Terlevich ([18]).", "As seen from the structure of the scheme, shown on Fig.2, the AGN subclasses are widely known, with the exception of LINER 1, the description of which is given below, along with the description of the remaining ones.", "Figure: The structure of the adapted scheme used for objects' classification.2.3.1.Description of classes in the used scheme.", "AGN classes: Sy1 - galaxies of type Sy1.5 and lower ( [12], [15]); Sy2 - [NII]$\\lambda $ 6583/H$\\alpha $$>$ 0.7 and [OIII]$\\lambda $ 5007/H$\\alpha $$>$ 3; galaxies of type Sy1.6 and higher ([12], [15]); LINER - correspondence of at least two ratios of spectral lines intensities to the classical definition ( [11]); LINER1 - a one dimensional LINER - [NII]$\\lambda $ 6583/H$\\alpha $$>$ 0.7 (by analogy with [2]), with or without data on the [OIII]$\\lambda $ 5007 and H$\\alpha $ lines; Composite - galaxies with signs of both nuclear and starforming activity (by analogy with ).", "SfG classes: SfGneb - galaxies in the nebular starforming phase, referred to as the nebular phase in Terlevich's scheme ( [18]) and distinguished, in particular, by having EW(H$\\alpha $ )$>$ 100; SfGcont - galaxies in the continual starforming phase, referred to as one of the early continual or the later continual phases in Terlevich's scheme ( [18]), for which the equation EW(H$\\alpha $ )$<$ 100 stays common; Norm - a galaxy in emission spectrum of which only weak H$\\alpha $ is present, EW(H$\\alpha )$$<$ 5, with or without absorption lines (by analogy with [3]); Abs - a galaxy with spectrum including only absorption lines, when the spectral region comprising H$\\alpha $ line is available ([1]).", "Table: Common initial and final distribution of exploring galaxies by classes.2.4.Final samples and some of analytical results.", "Table 2 gives distribution of the objects of the basic sample in final subsamples, organized by classes.", "With the help of these data it is easy to find out that the percentage of galaxies with confirmed activity is about 70 %, relative to the initial number of objects in the basic sample, that the most of these, in their turn, are starforming galaxies and so on.", "The detailed analysis of the distribution of apparent and absolute magnitudes and, certainly, of the redshifts ([7], [4]) has been undertaken for the separate samples.", "The analysis of how different emission lines work in the spectral ranges obtained on photographic plates shows, that a sample can be complete within 250 Mpc (H$=$ 75kms$^{-1}$ Mpc$^{-1}$ ) ([4]).", "In a sphere of this radius 50% of the galaxies of nuclear activity of the basic sample (nearly all of the galaxies of type Sy2 and LINER and 6 out of 18 galaxies of type Sy1) are located and 80% of all galaxies of starforming activity, among which almost all of those in nebular phase, viz.", "53 of 55.", "As currently only two subclasses for SfG galaxies are provided by our scheme, it is easy to get, that among SfGs, of apparent magnitude up to 19-19.5 and of redshift z < 0.06, on an average, every fifth one is in the more active, nebular phase.", "In this statistics it must be accounted that an effectivity of selection of different types of galaxies in the survey is not the same.", "In particular, corresponding evaluations for the uv-excess, as a selection criterium, which works in the range up to 18,5 of apparent magnitude, was most effective in the selection of Sy1 type galaxies as well as for galaxies SfGneb, most of which are known as blue compact galaxies, BCG.", "The latter ones can be seen as the best result by productivity among SBS galaxies, since the second selection criterium used in SBS, i.e.", "emission lines, which works up to the surveys' limiting, 19,5 magnitude, is also mostly effective for their selection.", "3.Current Stage.", "In the current stage detailed investigation of separate objects of the basic sample is being carried out, the basis of which are the results of the panoramic spectroscopy with the coverage of spectral range, which includes the recombination H$\\alpha $ line of hydrogen and at least its nearest forbidden doublets of the nitrogen [NII]$\\lambda $$\\lambda $ 6548,6583 and sulfur [SII]$\\lambda $$\\lambda $ 6717,6731 lines.", "The observations were started from the most interesting by morphology galaxies with the usage of multipupil spectrographs MPFS and VAGR, with the 6-m telescope of SAO and the 2.6m telescope of BAO, respectively (e.g.,[9], [10]).", "The development of more detailed scheme of classification of starforming galaxies is within the goals connected to further explorations of our basic sample." ] ]	1403.0127
[ [ "Tropical optimization problems with application to project scheduling\n with minimum makespan" ], [ "Abstract We consider multidimensional optimization problems in the framework of tropical mathematics.", "The problems are formulated to minimize a nonlinear objective function that is defined on vectors over an idempotent semifield and calculated by means of multiplicative conjugate transposition.", "We start with an unconstrained problem and offer two complete direct solutions to demonstrate different practicable argumentation schemes.", "The first solution consists of the derivation of a sharp lower bound for the objective function and the solving of an equation to find all vectors that yield the bound.", "The second is based on extremal properties of the spectral radius of matrices and involves the evaluation of this radius for a certain matrix.", "This solution is then extended to problems with boundary constraints that specify the feasible solution set by a double inequality, and with a linear inequality constraint given by a matrix.", "To illustrate one application of the results obtained, we solve problems in project scheduling under the minimum makespan criterion subject to various precedence constraints on the time of initiation and completion of activities in the project.", "Simple numerical examples are given to show the computational technique used for solutions." ], [ "Introduction", "Tropical (idempotent) mathematics, which focuses on the theory and applications of semirings with idempotent addition, offers a useful framework for the formulation and solution of real-world optimization problems in various fields, including project scheduling.", "Even the early works by Cuninghame-Green [1] and Giffler [2] on tropical mathematics used optimization problems drawn from machine scheduling to motivate and illustrate the study.", "In the last few decades, the theory and methods of tropical mathematics have received much attention, which resulted in many published works, such as recent monographs by Golan [3], Heidergott et al.", "[4], Gondran and Minoux [5], Butkovič [6], McEneaney [7] and a great many contributed papers.", "Tropical optimization forms an important research domain within the field, which mainly concentrates on new solutions for problems in operations research.", "Applications to scheduling problems remain of great concern in a number of researches, such as the works by Zimmermann [8], [9], Butkovič et al.", "[10], [11], [12] and Krivulin [13], [14], [15], [16], [17].", "There are also applications in other areas, including applications in location analysis developed by Cuninghame-Green [18], [19] and Krivulin [20], [21], [22], [23], in decision making by Elsner and van den Driessche [24], [25], Akian et al.", "[26], Gaubert et al.", "[27] and Gursoy et al.", "[28], and in discrete event systems by Gaubert [29], De Schutter [30], and De Schutter and van den Boom [31], to name only a few.", "In this paper, we consider multidimensional optimization problems, which are formulated and solved in the tropical mathematics setting.", "The problems are to minimize a nonlinear objective function defined on vectors in a finite-dimensional semimodule over an idempotent semifield by means of a multiplicative conjugate transposition operator.", "We start with an unconstrained problem to propose two complete direct solutions to the problem, which offer different representations for the solution set.", "The first solution follows the approach developed in [13] to derive a sharp lower bound for the objective function and to solve an equation to find all vectors that yield the bound.", "The other one is based on extremal properties of the spectrum of matrices investigated in [14], [17], [16] and involves the evaluation of the spectral radius of a certain matrix.", "We show that, although these solutions are represented in different forms, they define the same solution set.", "The latter solution is then extended to solve the problem under constraints that specify the lower and upper boundaries for the feasible solution set, and the problem under a linear inequality constraint given by a matrix.", "To illustrate the application of the results obtained, we provide new exact solutions to problems in project scheduling under the minimum makespan.", "The problems are to minimize the overall duration of a project that consists of a number of activities to be performed in parallel subject to temporal precedence constraints, including start-finish, finish-start, early start and due date constraints.", "The problems under consideration are known to have, in the usual setting, polynomial-time solutions in the form of computational algorithms (see, eg, overviews in Demeulemeester and Herroelen [32], T’kindt and Billaut [33] and Vanhoucke [34]).", "In contrast to these algorithmic solutions, the new ones are given in a compact vector form of direct general solutions, which are immediately ready for further analysis and straightforward computations.", "The paper is organized as follows.", "In Section , we offer a brief overview of key definitions and notation that underlie the development of solutions to the optimization problem and their applications in the subsequent sections.", "Section includes preliminary results, which provide a necessary prerequisite for the solution of the problems.", "In Section , we first formulate an unconstrained optimization problem and then solve the problem in two different ways.", "Furthermore, the solution is extended in Section to problems with constraints added.", "Section presents application of the results to project scheduling.", "Numerical examples are given in Section ." ], [ "Definitions, Notation and General Remarks", "We start with a brief overview of main definitions and notation of tropical mathematics to provide a formal framework for the description of solutions to optimization problems in the next sections.", "The overview is mainly based on the results in [35].", "For additional details, insights and references, one can consult [3], [4], [36], [5], [37], [6], [7].", "Let $\\mathbb {X}$ be a set endowed with two associative and commutative operations, $\\oplus $ (addition) and $\\otimes $ (multiplication), and equipped with additive and multiplicative neutral elements, $\\mathbb {0}$ (zero) and $\\mathbb {1}$ (one).", "Addition is idempotent, which yields $x\\oplus x=x$ for every $x\\in \\mathbb {X}$ .", "Multiplication distributes over addition and is invertible to provide each nonzero $x\\in \\mathbb {X}$ with its inverse $x^{-1}$ such that $x^{-1}\\otimes x=\\mathbb {1}$ .", "The system $(\\mathbb {X},\\oplus ,\\otimes ,\\mathbb {0},\\mathbb {1})$ is commonly referred to as the idempotent semifield, and retains certain properties of the usual fields.", "We assume that the semifield is linearly ordered by an order that is consistent with the partial order induced by idempotent addition to define $x\\le y$ if and only if $x\\oplus y=y$ .", "From here on, we use the relation symbols as well as $\\max $ and $\\min $ operators in the sense of this definition.", "Specifically, it follows from the definition that $x\\oplus y=\\max (x,y)$ .", "Moreover, in terms of the above order, both operations $\\oplus $ and $\\otimes $ are monotone in each argument.", "As usual, the integer power specifies iterated multiplication, and is defined by $x^{p}=x^{p-1}\\otimes x$ , $x^{-p}=(x^{-1})^{p}$ , $x^{0}=\\mathbb {1}$ and $\\mathbb {0}^{p}=\\mathbb {0}$ for each nonzero $x\\ne \\mathbb {0}$ and integer $p\\ge 1$ .", "Moreover, the semifield is taken to be algebraically complete (radicable), which yields the existence of a solution of the equation $x^{m}=a$ for all $a\\ne \\mathbb {0}$ and integer $m$ , and hence the existence of the root $a^{1/m}$ .", "In the rest of the paper, we omit the multiplication sign $\\otimes $ for the sake of simplicity.", "Examples of the semifield include $\\mathbb {R}_{\\max ,+}=(\\mathbb {R}\\cup \\lbrace -\\infty \\rbrace ,\\max ,+,-\\infty ,0)$ , $\\mathbb {R}_{\\min ,+}=(\\mathbb {R}\\cup \\lbrace +\\infty \\rbrace ,\\min ,+,+\\infty ,0)$ , $\\mathbb {R}_{\\max ,\\times }=(\\mathbb {R}_{+}\\cup \\lbrace 0\\rbrace ,\\max ,\\times ,0,1)$ and $\\mathbb {R}_{\\min ,\\times }=(\\mathbb {R}_{+}\\cup \\lbrace +\\infty \\rbrace ,\\min ,\\times ,+\\infty ,1)$ , where $\\mathbb {R}$ is the set of real numbers and $\\mathbb {R}_{+}=\\lbrace x\\in \\mathbb {R}\|x>0\\rbrace $ .", "Let $\\mathbb {X}^{m\\times n}$ be the set of matrices with $m$ rows and $n$ columns over $\\mathbb {X}$ .", "A matrix with all entries equal to $\\mathbb {0}$ is the zero matrix denoted $\\mathbf {0}$ .", "If a matrix has no zero rows (columns), then it is called row-regular (column-regular).", "A matrix is regular if it is both row- and column-regular.", "Addition and multiplication of conforming matrices and scalar multiplication are defined by the standard rules with the scalar operations $\\oplus $ and $\\otimes $ used in place of the ordinary addition and multiplication.", "These operations are monotone with respect to the order relations defined component-wise.", "For any matrix $\\mathbf {A}\\in \\mathbb {X}^{m\\times n}$ , the transpose of $\\mathbf {A}$ is the matrix $\\mathbf {A}^{T}\\in \\mathbb {X}^{n\\times m}$ .", "Consider the set $\\mathbb {X}^{n\\times n}$ of square matrices of order $n$ .", "A matrix with $\\mathbb {1}$ on the diagonal and $\\mathbb {0}$ elsewhere is the identity matrix denoted $\\mathbf {I}$ .", "The integer powers of any matrix $\\mathbf {A}\\in \\mathbb {X}^{n\\times n}$ are defined as $\\mathbf {A}^{0}=\\mathbf {I}$ and $\\mathbf {A}^{p}=\\mathbf {A}^{p-1}\\mathbf {A}$ for any integer $p\\ge 1$ .", "Tropical analogues of the trace and the norm of a matrix $\\mathbf {A}=(a_{ij})$ are respectively given by $\\mathop \\mathrm {tr}\\mathbf {A}=\\bigoplus _{i=1}^{n}a_{ii},\\qquad \\Vert \\mathbf {A}\\Vert =\\bigoplus _{i=1}^{n}\\bigoplus _{j=1}^{n}a_{ij}.$ For any matrices $\\mathbf {A}$ and $\\mathbf {B}$ and a scalar $x\\in \\mathbb {X}$ , we obviously have $\\mathop \\mathrm {tr}(\\mathbf {A}\\mathbf {B})=\\mathop \\mathrm {tr}(\\mathbf {B}\\mathbf {A}),\\qquad \\mathop \\mathrm {tr}(x\\mathbf {A})=x\\mathop \\mathrm {tr}\\mathbf {A},\\qquad \\Vert \\mathbf {A}\\oplus \\mathbf {B}\\Vert =\\Vert \\mathbf {A}\\Vert \\oplus \\Vert \\mathbf {B}\\Vert .$ A scalar $\\lambda \\in \\mathbb {X}$ is an eigenvalue of the matrix $\\mathbf {A}$ if there exists a nonzero vector $\\mathbf {x}\\in \\mathbb {X}^{n}$ such that $\\mathbf {A}\\mathbf {x}=\\lambda \\mathbf {x}$ .", "The maximum eigenvalue is called the spectral radius of $\\mathbf {A}$ and calculated by the formula $\\lambda =\\bigoplus _{m=1}^{n}\\mathop \\mathrm {tr}\\nolimits ^{1/m}(\\mathbf {A}^{m}).$ Furthermore, we define a function that assigns to each matrix $\\mathbf {A}$ a scalar $\\mathop \\mathrm {Tr}(\\mathbf {A})=\\bigoplus _{m=1}^{n}\\mathop \\mathrm {tr}\\mathbf {A}^{m}.$ Provided the condition $\\mathop \\mathrm {Tr}(\\mathbf {A})\\le \\mathbb {1}$ holds, the asterate of $\\mathbf {A}$ (also known as the Kleene star) is the matrix given by $\\mathbf {A}^{\\ast }=\\bigoplus _{m=0}^{n-1}\\mathbf {A}^{m}.$ Under the above condition, the asterate possesses a useful property that takes the form of the inequality (the Carrè inequality) $\\mathbf {A}^{m}\\le \\mathbf {A}^{\\ast },\\qquad m\\ge 0.$ The set of column vectors of size $n$ over $\\mathbb {X}$ is denoted by $\\mathbb {X}^{n}$ .", "A vector with all entries equal to $\\mathbb {0}$ is the zero vector denoted $\\mathbf {0}$ .", "A vector is regular if it has no zero elements.", "For any nonzero vector $\\mathbf {x}=(x_{i})\\in \\mathbb {X}^{n}$ , the multiplicative conjugate transpose is the row vector $\\mathbf {x}^{-}=(x_{i}^{-})$ with elements $x_{i}^{-}=x_{i}^{-1}$ if $x_{i}\\ne \\mathbb {0}$ and $x_{i}^{-}=\\mathbb {0}$ otherwise.", "The conjugate transposition possesses certain useful properties, which are not difficult to verify.", "Specifically, for any nonzero vector $\\mathbf {x}$ , we have the obvious equality $\\mathbf {x}^{-}\\mathbf {x}=\\mathbb {1}$ .", "For any regular vectors $\\mathbf {x}$ and $\\mathbf {y}$ of the same order, the component-wise inequality $\\mathbf {x}\\mathbf {y}^{-}\\ge (\\mathbf {x}^{-}\\mathbf {y})^{-1}\\mathbf {I}$ is valid as well.", "Note that for any matrix $\\mathbf {A}$ we have $\\Vert \\mathbf {A}\\Vert =\\mathbf {1}^{T}\\mathbf {A}\\mathbf {1}$ , where $\\mathbf {1}=(\\mathbb {1},\\ldots ,\\mathbb {1})^{T}$ .", "If $\\mathbf {A}=\\mathbf {x}\\mathbf {y}^{T}$ , where $\\mathbf {x}$ and $\\mathbf {y}$ are vectors, then $\\Vert \\mathbf {A}\\Vert =\\Vert \\mathbf {x}\\Vert \\Vert \\mathbf {y}\\Vert $ ." ], [ "Preliminary Results", "We now present preliminary results concerning the solution of algebraic and optimization problems in the tropical mathematics setting to be used below.", "First, we assume that, given a vector $\\mathbf {a}\\in \\mathbb {X}^{n}$ and a scalar $d\\in \\mathbb {X}$ , we need to obtain vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ to satisfy the equation $\\mathbf {a}^{T}\\mathbf {x}=d.$ A complete solution to the problem can be described as follows [35].", "Lemma 1 Let $\\mathbf {a}=(a_{i})$ be a regular vector and $d\\ne \\mathbb {0}$ be a scalar.", "Then, the solution of equation (REF ) forms a family of solutions each defined for one of $k=1,\\ldots ,n$ as a set of vectors $\\mathbf {x}=(x_{i})$ with components $x_{k}=a_{k}^{-1}d,\\qquad x_{i}\\le a_{i}^{-1}d,\\quad i\\ne k.$ Given a matrix $\\mathbf {A}\\in \\mathbb {X}^{m\\times n}$ and a vector $\\mathbf {d}\\in \\mathbb {X}^{m}$ , consider the problem to find all regular vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ that satisfy the inequality $\\mathbf {A}\\mathbf {x}\\le \\mathbf {d}.$ The next statement offers a solution that is obtained as a consequence of the solution to the corresponding equation [35], and by independent proof [22].", "Lemma 2 Let $\\mathbf {A}$ be a column-regular matrix and $\\mathbf {d}$ a regular vector.", "Then, all regular solutions to inequality (REF ) are given by $\\mathbf {x}\\le (\\mathbf {d}^{-}\\mathbf {A})^{-}.$ Furthermore, assume that, for a given matrix $\\mathbf {A}\\in \\mathbb {X}^{n\\times n}$ , we need to find regular solutions $\\mathbf {x}\\in \\mathbb {X}^{n}$ to the problem $\\begin{aligned}&\\text{minimize}&&\\mathbf {x}^{-}\\mathbf {A}\\mathbf {x}.\\end{aligned}$ A complete solution to (REF ) is provided by the following result [14], [17], [16].", "Lemma 3 Let $\\mathbf {A}$ be a matrix with spectral radius $\\lambda >\\mathbb {0}$ .", "Then, the minimum value in problem (REF ) is equal to $\\lambda $ and all regular solutions are given by $\\mathbf {x}=(\\lambda ^{-1}\\mathbf {A})^{\\ast }\\mathbf {u},\\qquad \\mathbf {u}>\\mathbf {0}.$ We conclude with solutions obtained in [17] to constrained versions of problem (REF ).", "First, we offer a solution to the problem: given a matrix $\\mathbf {A}\\in \\mathbb {X}^{n\\times n}$ and vectors $\\mathbf {p},\\mathbf {q}\\in \\mathbb {X}^{n}$ , find all regular vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ that $\\begin{aligned}&\\text{minimize}&&\\mathbf {x}^{-}\\mathbf {A}\\mathbf {x},\\\\&\\text{subject to}&&\\mathbf {g}\\le \\mathbf {x}\\le \\mathbf {h}.\\end{aligned}$ Theorem 4 Let $\\mathbf {A}$ be a matrix with spectral radius $\\lambda >\\mathbb {0}$ and $\\mathbf {h}$ be a regular vector such that $\\mathbf {h}^{-}\\mathbf {g}\\le \\mathbb {1}$ .", "Then, the minimum in problem (REF ) is equal to $\\theta =\\lambda \\oplus \\bigoplus _{m=1}^{n}(\\mathbf {h}^{-}\\mathbf {A}^{m}\\mathbf {g})^{1/m},$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\theta ^{-1}\\mathbf {A})^{\\ast }\\mathbf {u},\\qquad \\mathbf {g}\\le \\mathbf {u}\\le (\\mathbf {h}^{-}(\\theta ^{-1}\\mathbf {A})^{\\ast })^{-}.$ Finally, we present a solution to the following problem.", "Given matrices $\\mathbf {A},\\mathbf {B}\\in \\mathbb {X}^{n\\times n}$ and a vector $\\mathbf {g}$ , we look for all regular vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ to $\\begin{aligned}&\\text{minimize}&&\\mathbf {x}^{-}\\mathbf {A}\\mathbf {x},\\\\&\\text{subject to}&&\\mathbf {B}\\mathbf {x}\\oplus \\mathbf {g}\\le \\mathbf {x}.\\end{aligned}$ Theorem 5 Let $\\mathbf {A}$ be a matrix with spectral radius $\\lambda >\\mathbb {0}$ and $\\mathbf {B}$ be a matrix such that $\\mathop \\mathrm {Tr}(\\mathbf {B})\\le \\mathbb {1}$ .", "Then, the minimum value in problem (REF ) is equal to $\\theta =\\bigoplus _{k=1}^{n}\\mathop {\\bigoplus \\hspace{11.00006pt}}_{0\\le i_{1}+\\cdots +i_{k}\\le n-k}\\mathop \\mathrm {tr}\\nolimits ^{1/k}(\\mathbf {A}\\mathbf {B}^{i_{1}}\\cdots \\mathbf {A}\\mathbf {B}^{i_{k}}),$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\theta ^{-1}\\mathbf {A}\\oplus \\mathbf {B})^{\\ast }\\mathbf {u},\\qquad \\mathbf {u}\\ge \\mathbf {g}.$" ], [ "Unconstrained Optimization Problem", "In this section we examine an unconstrained multidimensional optimization problem formulated in the tropical mathematics setting as follows.", "Given vectors $\\mathbf {p},\\mathbf {q}\\in \\mathbb {X}^{n}$ , the problem is to find regular vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ such that $\\begin{aligned}&\\text{minimize}&&\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p}.\\end{aligned}$ Below, we offer two direct complete solutions to the problem under fairly general assumptions.", "We show that, although these solutions have different forms, both forms determine the same solution set." ], [ "Straightforward Solution", "We start with a solution based on the derivation of a lower bound for the objective function and the solution of an equation to find all vectors that yield the bound.", "Theorem 6 Let $\\mathbf {p}$ be a nonzero vector and $\\mathbf {q}$ a regular vector.", "Then, the minimum value in problem (REF ) is equal to $\\Delta =\\mathbf {q}^{-}\\mathbf {p},$ and all regular solutions of the problem are given by $\\alpha \\mathbf {p}\\le \\mathbf {x}\\le \\alpha \\Delta \\mathbf {q},\\qquad \\alpha >\\mathbb {0}.$ First, we find the minimum of the objective function in the problem by using the properties of the conjugate transposition.", "With the inequality $\\mathbf {x}\\mathbf {x}^{-}\\ge \\mathbf {I}$ , which is valid for any regular vector $\\mathbf {x}$ , we derive a lower bound $\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p}\\ge \\mathbf {q}^{-}\\mathbf {p}=\\Delta .$ Note that, since $\\mathbf {p}$ is nonzero and $\\mathbf {q}$ is regular, we have $\\Delta >\\mathbb {0}$ .", "It remains to verify that this bound is attained at certain $\\mathbf {x}$ , say $\\mathbf {x}=\\Delta \\mathbf {q}$ .", "Indeed, substitution into the objective function and the equality $\\mathbf {q}^{-}\\mathbf {q}=\\mathbb {1}$ yield $\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p}=\\Delta (\\mathbf {q}^{-}\\mathbf {q})\\Delta ^{-1}(\\mathbf {q}^{-}\\mathbf {p})=\\mathbf {q}^{-}\\mathbf {p}=\\Delta .$ To obtain all regular vectors $\\mathbf {x}$ that solve the problem, we examine the equation $\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p}=\\Delta .$ It is clear that, if $\\mathbf {x}$ is a solution, then so is $\\alpha \\mathbf {x}$ for any $\\alpha >\\mathbb {0}$ , and thus all solutions of the equation are scale-invariant.", "Furthermore, we take an arbitrary $\\alpha >\\mathbb {0}$ and rewrite the equation in an equivalent form as the system of two equations $\\mathbf {q}^{-}\\mathbf {x}=\\alpha \\Delta ,\\qquad \\mathbf {x}^{-}\\mathbf {p}=\\alpha ^{-1}.$ Taking into account that all solutions are scale-invariant, we put $\\alpha =\\mathbb {1}$ to further reduce the system as $\\mathbf {q}^{-}\\mathbf {x}=\\Delta ,\\qquad \\mathbf {x}^{-}\\mathbf {p}=\\mathbb {1}.$ According to Lemma REF , the solutions to the first equation form a family of solutions $\\mathbf {x}=(x_{i})$ , each defined for one of $k=1,\\ldots ,n$ by the conditions $x_{k}=\\Delta q_{k},\\qquad x_{i}\\le \\Delta q_{i},\\quad i\\ne k.$ We now find those solutions from the family which satisfy the second equation at (REF ).", "Note that $\\mathbf {x}=\\Delta \\mathbf {q}$ solves the problem and thus this equation.", "Consider the minimum value of the problem and write $\\Delta =\\mathbf {q}^{-}\\mathbf {p}=\\bigoplus _{i=1}^{n}q_{i}^{-1}p_{i}=q_{k}^{-1}p_{k},$ where $k$ is an index that yields the maximum of $q_{i}^{-1}p_{i}$ over all $i=1,\\ldots ,n$ .", "We denote by $K$ the set of all such indices that produce $\\Delta $ , and then verify that all solutions to the second equation must have $x_{k}=\\Delta q_{k}$ for each $k\\in K$ .", "Assuming the contrary, let $k$ be an index in $K$ to satisfy the condition $x_{k}<\\Delta q_{k}=(\\mathbf {q}^{-}\\mathbf {p})q_{k}=q_{k}^{-1}p_{k}q_{k}=p_{k}.$ Then, for the left hand side of the second equation at (REF ), we have $\\mathbf {x}^{-}\\mathbf {p}=x_{1}^{-1}p_{1}\\oplus \\cdots \\oplus x_{n}^{-1}p_{n}\\ge x_{k}^{-1}p_{k}>p_{k}^{-1}p_{k}=\\mathbb {1},$ and thus the equation is not valid anymore and becomes a strict inequality.", "Furthermore, for all $i\\notin K$ if any, we can take $x_{i}\\le \\Delta q_{i}$ but not too small to keep the condition $\\mathbf {x}^{-}\\mathbf {p}\\le \\mathbb {1}$ .", "It follows from this condition that $\\mathbb {1}\\ge \\mathbf {x}^{-}\\mathbf {p}=x_{1}^{-1}p_{1}\\oplus \\cdots \\oplus x_{n}^{-1}p_{n}\\ge x_{i}^{-1}p_{i}.$ To satisfy the condition when $p_{i}\\ne \\mathbb {0}$ , we have to take $x_{i}\\ge p_{i}$ .", "With $p_{i}=\\mathbb {0}$ , the term $\\mathbf {x}^{-}\\mathbf {p}$ does not depend on $x_{i}$ , and we can write $x_{i}\\ge \\mathbb {0}=p_{i}$ .", "We can summarize the above consideration as follows.", "All solutions to the problem are vectors $\\mathbf {x}=(x_{i})$ that satisfy the conditions $x_{i}&=\\Delta q_{i},\\qquad i\\in K\\\\p_{i}\\le x_{i}&\\le \\Delta q_{i},\\qquad i\\notin K.$ Since we have $x_{i}=\\Delta q_{i}=q_{i}^{-1}p_{i}q_{i}=p_{i}$ for all $i\\in K$ , the solution can be written as one double inequality $p_{i}\\le x_{i}\\le \\Delta q_{i}$ for all $i=1,\\ldots ,n$ , or, in the vector form, as the inequality $\\mathbf {p}\\le \\mathbf {x}\\le \\Delta \\mathbf {q}.$ Considering that each solution is scale-invariant, we arrive at (REF ).", "$\\Box $" ], [ "Solution Using Spectral Radius", "To provide another solution to problem (REF ), we first put the objective function in the equivalent form $\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p}=\\mathbf {x}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {x},$ and then rewrite the problem as $\\begin{aligned}&\\text{minimize}&&\\mathbf {x}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {x}.\\end{aligned}$ The problem now becomes a special case of problem (REF ) with $\\mathbf {A}=\\mathbf {p}\\mathbf {q}^{-}$ , and can therefore be solved using Lemma REF .", "Theorem 7 Let $\\mathbf {p}$ be a nonzero vector and $\\mathbf {q}$ a regular vector.", "Then, the minimum value in problem (REF ) is equal to $\\Delta =\\mathbf {q}^{-}\\mathbf {p},$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-})\\mathbf {u},\\qquad \\mathbf {u}>\\mathbf {0}.$ We examine the problem in the form of (REF ).", "To apply Lemma REF , we take the matrix $\\mathbf {A}=\\mathbf {p}\\mathbf {q}^{-}$ and calculate $\\mathbf {A}^{m}=(\\mathbf {q}^{-}\\mathbf {p})^{m-1}\\mathbf {p}\\mathbf {q}^{-},\\qquad \\mathop \\mathrm {tr}\\mathbf {A}^{m}=(\\mathbf {q}^{-}\\mathbf {p})^{m},\\qquad m=1,\\ldots ,n.$ Let $\\Delta $ be the spectral radius of the matrix $\\mathbf {A}$ .", "Using formula (REF ), we obtain $\\Delta =\\mathbf {q}^{-}\\mathbf {p}$ , which, due to Lemma REF , presents the minimum value in the problem.", "To describe the solution set, we calculate $(\\Delta ^{-1}\\mathbf {A})^{m}=\\Delta ^{-m}\\mathbf {A}^{m}=\\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}$ , and then employ (REF ) to derive the matrix $(\\Delta ^{-1}\\mathbf {A})^{\\ast }=\\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}.$ Finally, the application of Lemma REF gives solution (REF ).", "$\\Box $ Note that, although the solution sets offered by Theorems REF and REF look different, it is not difficult to see that they are the same.", "Let us take any vector $\\mathbf {u}$ and verify that $\\mathbf {x}$ which is given by (REF ) satisfies inequality (REF ) for some $\\alpha $ .", "Indeed, if we put $\\alpha =\\Delta ^{-1}(\\mathbf {q}^{-}\\mathbf {u})$ , then we have $\\mathbf {x}=(\\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-})\\mathbf {u}\\ge \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {u}=\\Delta ^{-1}(\\mathbf {q}^{-}\\mathbf {u})\\mathbf {p}=\\alpha \\mathbf {p},$ which yields the left inequality at (REF ).", "Since $\\mathbf {q}\\mathbf {p}^{-}\\ge (\\mathbf {q}^{-}\\mathbf {p})^{-1}\\mathbf {I}=\\Delta ^{-1}\\mathbf {I}$ , we obtain the right inequality as follows $\\mathbf {x}=(\\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-})\\mathbf {u}\\le (\\mathbf {I}\\oplus \\mathbf {q}\\mathbf {p}^{-}\\mathbf {p}\\mathbf {q}^{-})\\mathbf {u}=(\\mathbf {I}\\oplus \\mathbf {q}\\mathbf {q}^{-})\\mathbf {u}=(\\mathbf {q}^{-}\\mathbf {u})\\mathbf {q}=\\alpha \\Delta \\mathbf {q}.$ Now assume that the vector $\\mathbf {x}$ satisfies (REF ) and then show that it can be written as (REF ).", "From the right inequality at (REF ), it follows that $\\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {x}\\le \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}(\\alpha \\Delta \\mathbf {q})=\\alpha \\mathbf {p}\\mathbf {q}^{-}\\mathbf {q}=\\alpha \\mathbf {p}.$ Considering the left inequality, we have $\\mathbf {x}\\ge \\alpha \\mathbf {p}\\ge \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {x}$ .", "Finally, by setting $\\mathbf {u}=\\mathbf {x}$ , we can write $\\mathbf {x}=\\mathbf {x}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {x}=\\mathbf {u}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {u}=(\\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {p}\\mathbf {q}^{-})\\mathbf {u},$ which gives a representation of the vector $\\mathbf {x}$ in the form of (REF )." ], [ "Constrained Optimization Problems", "We now add lower and upper boundary constraints on the feasible solutions.", "Given vectors $\\mathbf {p},\\mathbf {q},\\mathbf {g},\\mathbf {h}\\in \\mathbb {X}^{n}$ , consider the problem to find all regular vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ that $\\begin{aligned}&\\text{minimize}&&\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p},\\\\&\\text{subject to}&&\\mathbf {g}\\le \\mathbf {x}\\le \\mathbf {h}.\\end{aligned}$ The next theorem provides a complete direct solution to the problem.", "Theorem 8 Let $\\mathbf {p}$ be a nonzero vector, $\\mathbf {q}$ a regular vector, and $\\mathbf {h}$ a regular vector such that $\\mathbf {h}^{-}\\mathbf {g}\\le \\mathbb {1}$ .", "Then, the minimum in problem (REF ) is equal to $\\theta =\\mathbf {q}^{-}(\\mathbf {I}\\oplus \\mathbf {g}\\mathbf {h}^{-})\\mathbf {p},$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {p}\\mathbf {q}^{-})\\mathbf {u},\\qquad \\mathbf {g}\\le \\mathbf {u}\\le (\\mathbf {h}^{-}(\\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {p}\\mathbf {q}^{-}))^{-}.$ As in the previous proof, we rewrite the objective function in the form $\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p}=\\mathbf {x}^{-}\\mathbf {A}\\mathbf {x}$ , where $\\mathbf {A}=\\mathbf {p}\\mathbf {q}^{-}$ , and thus reduce the problem to (REF ).", "Furthermore, we obtain the spectral radius of $\\mathbf {A}$ in the form $\\Delta =\\mathbf {q}^{-}\\mathbf {p}$ , write $\\mathbf {A}^{m}=\\Delta ^{m-1}\\mathbf {p}\\mathbf {q}^{-}$ , and calculate $\\mathbf {h}^{-}\\mathbf {A}^{m}\\mathbf {g}=\\Delta ^{m-1}\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g}$ .", "Then, we apply Theorem REF to write the minimum value in the form $\\theta =\\Delta \\oplus \\bigoplus _{m=1}^{n}(\\Delta ^{m-1}\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g})^{1/m}=\\Delta \\left(\\mathbb {1}\\oplus \\bigoplus _{m=1}^{n}(\\Delta ^{-1}\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g})^{1/m}\\right).$ To simplify the last expression, consider two cases.", "First, suppose that $\\Delta \\ge \\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g}$ .", "It follows immediately from this condition that the inequality $(\\Delta ^{-1}\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g})^{1/m}\\le \\mathbb {1}$ holds for every $m$ , and therefore, $\\theta =\\Delta $ .", "Otherwise, if the opposite inequality $\\Delta <\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g}$ is valid, we see that $\\Delta ^{-1}\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g}\\ge (\\Delta ^{-1}\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g})^{1/m}>\\mathbb {1}$ , which gives $\\theta =\\mathbf {h}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {g}$ .", "By combining both results and considering $\\Delta =\\mathbf {q}^{-}\\mathbf {p}$ , we obtain the minimum value $\\theta =\\Delta \\oplus \\mathbf {q}^{-}\\mathbf {g}\\mathbf {h}^{-}\\mathbf {p}=\\mathbf {q}^{-}(\\mathbf {I}\\oplus \\mathbf {g}\\mathbf {h}^{-})\\mathbf {p}.$ We now define the solution set according to Theorem REF .", "We first calculate $(\\theta ^{-1}\\mathbf {A})^{m}=\\theta ^{-m}\\Delta ^{m-1}\\mathbf {p}\\mathbf {q}^{-}$ , and then apply (REF ) to write $(\\theta ^{-1}\\mathbf {A})^{\\ast }=\\mathbf {I}\\oplus \\theta ^{-1}\\left(\\bigoplus _{m=1}^{n-1}(\\theta ^{-1}\\Delta )^{m-1}\\right)\\mathbf {p}\\mathbf {q}^{-}.$ Since $\\theta \\ge \\Delta $ , the inequality $(\\theta ^{-1}\\Delta )^{m-1}\\le \\mathbb {1}$ is valid for all $m$ and becomes an equality if $m=1$ .", "Therefore, the term in the parenthesis on the right-hand side is equal to $\\mathbb {1}$ , and hence $(\\theta ^{-1}\\mathbf {A})^{\\ast }=\\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {p}\\mathbf {q}^{-}.$ Substitution into the solution provided by Theorem REF leads to (REF ).", "$\\Box $ Suppose that we replace the simple boundary constraints in the above problem by a linear inequality constraint given by a matrix $\\mathbf {B}\\in \\mathbb {X}^{n\\times n}$ and a vector $\\mathbf {g}\\in \\mathbb {X}^{n}$ .", "Consider the problem to find regular vectors $\\mathbf {x}\\in \\mathbb {X}^{n}$ that $\\begin{aligned}&\\text{minimize}&&\\mathbf {q}^{-}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {p},\\\\&\\text{subject to}&&\\mathbf {B}\\mathbf {x}\\oplus \\mathbf {g}\\le \\mathbf {x}.\\end{aligned}$ A solution to the problem can be obtained as follows.", "Theorem 9 Let $\\mathbf {p}$ be a nonzero vector, $\\mathbf {q}$ a regular vector, and $\\mathbf {B}$ be a matrix such that $\\mathop \\mathrm {Tr}(\\mathbf {B})\\le \\mathbb {1}$ .", "Then, the minimum value in problem (REF ) is equal to $\\theta =\\mathbf {q}^{-}\\mathbf {B}^{\\ast }\\mathbf {p},$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\theta ^{-1}\\mathbf {p}\\mathbf {q}^{-}\\oplus \\mathbf {B})^{\\ast }\\mathbf {u},\\qquad \\mathbf {u}\\ge \\mathbf {g}.$ To solve the problem, we again represent the objective function as $\\mathbf {x}^{-}\\mathbf {p}\\mathbf {q}^{-}\\mathbf {x}$ and then apply Theorem REF with $\\mathbf {A}=\\mathbf {p}\\mathbf {q}^{-}$ .", "First, we examine the minimum value provided by Theorem REF .", "This minimum now takes the form $\\theta =\\bigoplus _{k=1}^{n}\\mathop {\\bigoplus \\hspace{11.00006pt}}_{0\\le i_{1}+\\cdots +i_{k}\\le n-k}\\mathop \\mathrm {tr}\\nolimits ^{1/k}(\\mathbf {p}\\mathbf {q}^{-}\\mathbf {B}^{i_{1}}\\cdots \\mathbf {p}\\mathbf {q}^{-}\\mathbf {B}^{i_{k}}),$ where the properties of the trace allow us to write $\\mathop \\mathrm {tr}(\\mathbf {p}\\mathbf {q}^{-}\\mathbf {B}^{i_{1}}\\cdots \\mathbf {p}\\mathbf {q}^{-}\\mathbf {B}^{i_{k}})=\\mathbf {q}^{-}\\mathbf {B}^{i_{1}}\\mathbf {p}\\cdots \\mathbf {q}^{-}\\mathbf {B}^{i_{k}}\\mathbf {p}.$ By truncating the sum at $k=1$ , we bound from below the value of $\\theta $ as $\\theta \\ge \\bigoplus _{i=0}^{n-1}\\mathop \\mathrm {tr}(\\mathbf {p}\\mathbf {q}^{-}\\mathbf {B}^{i})=\\bigoplus _{i=0}^{n-1}\\mathbf {q}^{-}\\mathbf {B}^{i}\\mathbf {p}=\\mathbf {q}^{-}\\mathbf {B}^{\\ast }\\mathbf {p}.$ On the other hand, since $\\mathbf {B}^{m}\\le \\mathbf {B}^{\\ast }$ for any integer $m\\ge 0$ , we obtain $\\mathbf {q}^{-}\\mathbf {B}^{i_{1}}\\mathbf {p}\\cdots \\mathbf {q}^{-}\\mathbf {B}^{i_{k}}\\mathbf {p}\\le \\mathbf {q}^{-}\\mathbf {B}^{\\ast }\\mathbf {p}\\cdots \\mathbf {q}^{-}\\mathbf {B}^{\\ast }\\mathbf {p}=(\\mathbf {q}^{-}\\mathbf {B}^{\\ast }\\mathbf {p})^{k}.$ Consequently, we have the inequality $\\theta =\\bigoplus _{k=1}^{n}\\mathop {\\bigoplus \\hspace{11.00006pt}}_{0\\le i_{1}+\\cdots +i_{k}\\le n-k}(\\mathbf {q}^{-}\\mathbf {B}^{i_{1}}\\mathbf {p}\\cdots \\mathbf {q}^{-}\\mathbf {B}^{i_{k}}\\mathbf {p})^{1/k}\\le \\mathbf {q}^{-}\\mathbf {B}^{\\ast }\\mathbf {p},$ which together with the opposite inequality yields the desired result.", "Finally, we use Theorem REF to write the solution in the form of (REF ), and thus complete the proof.", "$\\Box $" ], [ "Application to Project Scheduling", "In this section, we show how the results obtained can be applied to solve real-world problems that are drawn from project scheduling under the minimum makespan criterion (see, e.g., [32], [33], [34] for further details).", "Consider a project that involves $n$ activities operating under start-finish, finish-start, early start, and late finish (due date) temporal constraints.", "The start-finish constraints define the lower limit for the allowed time lag between the initiation of one activity and the completion of another.", "The activities are assumed to be completed as early as possible within the start-finish constraints.", "The finish-start constraints determine the minimum time lag between the completion of one activity and the initiation of another.", "The early start and late finish constraints specify, respectively, the earliest possible initiation time and latest possible completion time for every activity.", "Below, we first examine a problem that has only start-finish constraints, and then extend the result obtained to problems with additional constraints.", "For each activity $i=1,\\ldots ,n$ , we denote the initiation time by $x_{i}$ and the completion time by $y_{i}$ .", "Let $c_{ij}$ be the minimum time lag between the initiation of activity $j=1,\\ldots ,n$ and the completion of $i$ .", "If $c_{ij}$ is not given for some $j$ , we put $c_{ij}=-\\infty $ .", "The completion time of activity $i$ must satisfy the start-finish relations written in terms of the usual operations as $x_{j}+c_{ij}\\le y_{i},\\qquad j=1,\\ldots ,n,$ where at least one inequality holds as equality.", "Combining the relations gives $y_{i}=\\max _{1\\le j\\le n}(x_{j}+c_{ij}).$ The makespan is defined as the duration between the earliest initiation time and the latest completion time in the project, and takes the form $\\max _{1\\le i\\le n}y_{i}-\\min _{1\\le i\\le n}x_{i}=\\max _{1\\le i\\le n}y_{i}+\\max _{1\\le i\\le n}(-x_{i}).$ After substitution of $y_{i}$ , the problem of scheduling under the start-finish constraints and the minimum makespan criterion can be formulated in the conventional form as follows: given $c_{ij}$ for $i,j=1,\\ldots ,n$ , find $x_{1},\\ldots ,x_{n}$ that $\\begin{aligned}&\\text{minimize}&&\\max _{1\\le i\\le n}\\max _{1\\le j\\le n}(x_{j}+c_{ij})+\\max _{1\\le i\\le n}(-x_{i}).\\end{aligned}$ In terms of the operations in the semifield $\\mathbb {R}_{\\max ,+}$ , the problem becomes $\\begin{aligned}&\\text{minimize}&&\\bigoplus _{i=1}^{n}\\bigoplus _{j=1}^{n}c_{ij}x_{j}\\bigoplus _{k=1}^{n}x_{k}^{-1}.\\end{aligned}$ Furthermore, we introduce the matrix $\\mathbf {C}=(c_{ij})$ and the vector $\\mathbf {x}=(x_{i})$ .", "Using this notation and considering that $\\mathbf {1}=(0,\\ldots ,0)^{T}$ in $\\mathbb {R}_{\\max ,+}$ , we put the problem in the vector form $\\begin{aligned}&\\text{minimize}&&\\mathbf {1}^{T}\\mathbf {C}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {1}.\\end{aligned}$ It is easy to see that the last problem is a special case of problem (REF ), where we take $\\mathbf {p}=\\mathbf {1}$ and $\\mathbf {q}^{-}=\\mathbf {1}^{T}\\mathbf {C}$ .", "As a consequence of Theorems REF and REF , we obtain the following result.", "Theorem 10 Let $\\mathbf {C}$ be a row-regular matrix.", "Then, the minimum value in problem (REF ) is equal to $\\Delta =\\mathbf {1}^{T}\\mathbf {C}\\mathbf {1}=\\Vert \\mathbf {C}\\Vert ,$ and all regular solutions of the problem are given by $\\alpha \\mathbf {1}\\le \\mathbf {x}\\le \\alpha \\Delta (\\mathbf {1}^{T}\\mathbf {C})^{-},\\qquad \\alpha \\in \\mathbb {R},$ or, equivalently, by $\\mathbf {x}=(\\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C})\\mathbf {u},\\qquad \\mathbf {u}\\in \\mathbb {R}^{n}.$ We now consider the problem with the early start and late finish constraints added.", "For each activity $i=1,\\ldots ,n$ , let $g_{i}$ be the earliest possible time to start, and $f_{i}$ the latest possible time to finish (the due date) for activity $i$ .", "The early start and late finish constraints imply the inequalities $g_{i}\\le x_{i},\\qquad y_{i}=\\max _{1\\le j\\le n}(x_{j}+c_{ij})\\le f_{i},$ which, combined with the objective function, yields a problem that is written in the usual form as $\\begin{aligned}&\\text{minimize}&&\\max _{1\\le i\\le n}\\max _{1\\le j\\le n}(x_{j}+c_{ij})+\\max _{1\\le i\\le n}(-x_{i}),\\\\&\\text{subject to}&&\\max _{1\\le j\\le n}(x_{j}+c_{ij})\\le f_{i},\\\\&&&g_{i}\\le x_{i},\\qquad i=1,\\ldots ,n.\\end{aligned}$ As before, we rewrite the problem in terms of the semifield $\\mathbb {R}_{\\max ,+}$ and use the vector notation $\\mathbf {g}=(g_{i})$ and $\\mathbf {f}=(f_{i})$ to extend the unconstrained problem at (REF ) to the problem $\\begin{aligned}&\\text{minimize}&&\\mathbf {1}^{T}\\mathbf {C}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {1},\\\\&\\text{subject to}&&\\mathbf {C}\\mathbf {x}\\le \\mathbf {f},\\quad \\mathbf {g}\\le \\mathbf {x}.\\end{aligned}$ It follows from Lemma REF that the first inequality constraint can be solved in the form $\\mathbf {x}\\le (\\mathbf {f}^{-}\\mathbf {C})^{-}$ .", "Then, the problem reduces to (REF ) with $\\mathbf {p}=\\mathbf {1}$ , $\\mathbf {q}^{-}=\\mathbf {1}^{T}\\mathbf {C}$ and $\\mathbf {h}=(\\mathbf {f}^{-}\\mathbf {C})^{-}$ .", "By applying Theorem REF and using properties of the norm to represent the minimum value, we come to the following result.", "Theorem 11 Let $\\mathbf {C}$ be a regular matrix, and $\\mathbf {f}$ a regular vector such that $\\mathbf {f}^{-}\\mathbf {C}\\mathbf {g}\\le \\mathbb {1}$ .", "Then, the minimum value in problem (REF ) is equal to $\\theta =\\mathbf {1}^{T}\\mathbf {C}(\\mathbf {I}\\oplus \\mathbf {g}\\mathbf {f}^{-}\\mathbf {C})\\mathbf {1}=\\Vert \\mathbf {C}\\Vert \\oplus \\Vert \\mathbf {C}\\mathbf {g}\\Vert \\Vert \\mathbf {f}^{-}\\mathbf {C})\\Vert ,$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C})\\mathbf {u},\\qquad \\mathbf {g}\\le \\mathbf {u}\\le (\\mathbf {f}^{-}\\mathbf {C}(\\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}))^{-}.$ Finally, suppose that, in the project under consideration, the late finish constraints are replaced by finish-start constraints.", "For each activity $i=1,\\ldots ,n$ , we denote by $d_{ij}$ the minimum allowed time lag between the completion of activity $j$ and initiation of $i$ .", "We take $d_{ij}=-\\infty $ if the time lag is not specified.", "The finish-start constraints are given in terms of the usual operations by the inequalities $y_{j}+d_{ij}\\le x_{i},\\qquad j=1,\\ldots ,n.$ Furthermore, we substitute $y_{j}$ from the start-finish constraints and combine the inequalities into one to write $\\max _{1\\le j\\le n}(\\max _{1\\le k\\le n}(x_{k}+c_{jk})+d_{ij})\\le x_{i}.$ The scheduling problem under finish-start and early start constraints can now be formulated as $\\begin{aligned}&\\text{minimize}&&\\max _{1\\le i\\le n}\\max _{1\\le j\\le n}(x_{j}+c_{ij})+\\max _{1\\le i\\le n}(-x_{i}),\\\\&\\text{subject to}&&\\max _{1\\le j\\le n}(\\max _{1\\le k\\le n}(x_{k}+c_{jk})+d_{ij})\\le x_{i},\\\\&&&g_{i}\\le x_{i},\\qquad i=1,\\ldots ,n.\\end{aligned}$ We introduce the matrix $\\mathbf {D}=(d_{ij})$ , and then represent the problem in terms of the semifield $\\mathbb {R}_{\\max ,+}$ to write $\\begin{aligned}&\\text{minimize}&&\\mathbf {1}^{T}\\mathbf {C}\\mathbf {x}\\mathbf {x}^{-}\\mathbf {1},\\\\&\\text{subject to}&&\\mathbf {D}\\mathbf {C}\\mathbf {x}\\oplus \\mathbf {g}\\le \\mathbf {x}.\\end{aligned}$ Application of Theorem REF with $\\mathbf {p}=\\mathbf {1}$ , $\\mathbf {q}^{-}=\\mathbf {1}^{T}\\mathbf {C}$ and $\\mathbf {B}=\\mathbf {D}\\mathbf {C}$ yields the next result.", "Theorem 12 Let $\\mathbf {C}$ and $\\mathbf {D}$ be matrices such that $\\mathop \\mathrm {Tr}(\\mathbf {D}\\mathbf {C})\\le \\mathbb {1}$ .", "Then, the minimum value in problem (REF ) is equal to $\\theta =\\mathbf {1}^{T}\\mathbf {C}(\\mathbf {D}\\mathbf {C})^{\\ast }\\mathbf {1}=\\Vert \\mathbf {C}(\\mathbf {D}\\mathbf {C})^{\\ast }\\Vert ,$ and all regular solutions of the problem are given by $\\mathbf {x}=(\\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}\\oplus \\mathbf {D}\\mathbf {C})^{\\ast }\\mathbf {u},\\qquad \\mathbf {u}\\ge \\mathbf {g}.$ Note that the solutions obtained are not unique.", "In the context of project scheduling, this leaves the freedom to account for additional temporal constraints." ], [ "Numerical Examples", "In this section, we show how the results obtained can be applied to solve simple example problems.", "The main aim of this section is to provide a transparent and detailed illustration of the computational technique in terms of the semifield $\\mathbb {R}_{\\max ,+}$ , which is used to obtain the solution.", "To this end, we examine relatively artificial low-dimensional problems which, however, clearly demonstrate the ability the approach to solve real-world problems of high dimensions.", "Consider an example project that involves $n=3$ activities and operates under the constraints given by $\\mathbf {C}=\\left(\\begin{array}{ccr}4 & 0 & \\mathbb {0}\\\\1 & 3 & -1\\\\0 & 2 & 2\\end{array}\\right),\\qquad \\mathbf {D}=\\left(\\begin{array}{ccc}\\mathbb {0} & \\mathbb {0} & \\mathbb {0}\\\\0 & \\mathbb {0} & \\mathbb {0}\\\\2 & 1 & \\mathbb {0}\\end{array}\\right),\\qquad \\mathbf {g}=\\left(\\begin{array}{c}3\\\\2\\\\1\\end{array}\\right),\\qquad \\mathbf {f}=\\left(\\begin{array}{c}8\\\\7\\\\4\\end{array}\\right),$ where the symbol $\\mathbb {0}=-\\infty $ is employed to save writing.", "We start with problem (REF ), where only the start-finish constraints are defined.", "Application of Theorem REF gives the minimum makespan $\\Delta =\\Vert \\mathbf {C}\\Vert =4.$ To represent the solution to the problem in the form of (REF ), we take the vector $\\mathbf {1}=(0,0,0)^{T}$ and calculate $\\mathbf {1}^{T}\\mathbf {C}=\\left(\\begin{array}{ccc}4 & 3 & 2\\end{array}\\right),\\qquad (\\mathbf {1}^{T}\\mathbf {C})^{-}=\\left(\\begin{array}{c}-4\\\\-3\\\\-2\\end{array}\\right),\\qquad \\Delta (\\mathbf {1}^{T}\\mathbf {C})^{-}=\\left(\\begin{array}{c}0\\\\1\\\\2\\end{array}\\right).$ Then, the solution vector $\\mathbf {x}=(x_{1},x_{2},x_{3})^{T}$ provided by the theorem is given by $\\alpha \\left(\\begin{array}{c}0\\\\0\\\\0\\end{array}\\right)\\le \\mathbf {x}\\le \\alpha \\left(\\begin{array}{c}0\\\\1\\\\2\\end{array}\\right),\\qquad \\alpha \\in \\mathbb {R},$ and can be written in terms of the usual operations as $x_{1}=\\alpha \\qquad \\alpha \\le x_{2}\\le \\alpha +1,\\qquad \\alpha \\le x_{3}\\le \\alpha +2,\\qquad \\alpha \\in \\mathbb {R}.$ Furthermore, we derive the solution represented as (REF ).", "After calculating the matrices $\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}=\\left(\\begin{array}{ccc}4 & 3 & 2\\\\4 & 3 & 2\\\\4 & 3 & 2\\end{array}\\right),\\qquad \\mathbf {I}\\oplus \\Delta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}=\\left(\\begin{array}{rrr}0 & -1 & -2\\\\0 & 0 & -2\\\\0 & -1 & 0\\end{array}\\right),$ we obtain the solution $\\mathbf {x}=\\left(\\begin{array}{rrr}0 & -1 & -2\\\\0 & 0 & -2\\\\0 & -1 & 0\\end{array}\\right)\\mathbf {u},\\qquad \\mathbf {u}\\in \\mathbb {R}^{3}.$ In the ordinary notation, assuming $u_{1},u_{2},u_{3}$ to be real numbers, we have $x_{1}&=\\max (u_{1},u_{2}-1,u_{3}-2),\\\\x_{2}&=\\max (u_{1},u_{2},u_{3}-2),\\\\x_{3}&=\\max (u_{1},u_{2}-1,u_{3}).$ Suppose that, in addition to the start-finish constraints, both early start and late finish constraints are also imposed.", "To check whether Theorem REF can be applied, we first obtain $\\mathbf {f}^{-}\\mathbf {C}=\\left(\\begin{array}{rrr}-4 & -2 & -2\\end{array}\\right),\\qquad \\mathbf {C}\\mathbf {g}=\\left(\\begin{array}{c}7\\\\5\\\\4\\end{array}\\right),\\qquad \\mathbf {f}^{-}\\mathbf {C}\\mathbf {g}=0.$ Since $\\mathbf {f}^{-}\\mathbf {C}\\mathbf {g}=0=\\mathbb {1}$ , we see that the conditions of Theorem REF are fulfilled.", "Considering that $\\Vert \\mathbf {C}\\Vert =4,\\qquad \\Vert \\mathbf {f}^{-}\\mathbf {C}\\Vert =-2,\\qquad \\Vert \\mathbf {C}\\mathbf {g}\\Vert =7,$ we evaluate the minimum makespan $\\theta =\\Vert \\mathbf {C}\\Vert \\oplus \\Vert \\mathbf {C}\\mathbf {g}\\Vert \\Vert \\mathbf {f}^{-}\\mathbf {C}\\Vert =5.$ Furthermore, we successively calculate the matrices $\\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}=\\left(\\begin{array}{rrr}-1 & -2 & -3\\\\-1 & -2 & -3\\\\-1 & -2 & -3\\end{array}\\right),\\qquad \\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}=\\left(\\begin{array}{rrr}0 & -2 & -3\\\\-1 & 0 & -3\\\\-1 & -2 & 0\\end{array}\\right),$ and the vector $\\mathbf {f}^{-}\\mathbf {C}(\\mathbf {I}\\oplus \\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C})=\\left(\\begin{array}{ccc}-3 & -2 & -2\\end{array}\\right).$ The solution given by Theorem REF at (REF ) takes the form $\\mathbf {x}=\\left(\\begin{array}{rrr}0 & -2 & -3\\\\-1 & 0 & -3\\\\-1 & -2 & 0\\end{array}\\right)\\mathbf {u},\\qquad \\left(\\begin{array}{c}3\\\\2\\\\1\\end{array}\\right)\\le \\mathbf {u}\\le \\left(\\begin{array}{c}3\\\\2\\\\2\\end{array}\\right).$ Scalar representation in the ordinary notation gives the equalities $x_{1}&=\\max (u_{1},u_{2}-2,u_{3}-3),\\\\x_{2}&=\\max (u_{1}-1,u_{2},u_{3}-3),\\\\x_{3}&=\\max (u_{1}-1,u_{2}-2,u_{3}),$ where the numbers $u_{1}$ , $u_{2}$ and $u_{3}$ satisfy the conditions $u_{1}=3,\\qquad u_{2}=2,\\qquad 1\\le u_{3}\\le 2.$ By combining these conditions with the above equalities, we obtain the single solution $x_{1}=3,\\qquad x_{2}=2,\\qquad x_{3}=2.$ Finally, we assume that the start-finish constraints given by the matrix $\\mathbf {D}$ are defined in the above project instead of the late finish constraints.", "To solve the problem, we use the result provided by Theorem REF .", "First, we calculate the matrices $\\mathbf {D}\\mathbf {C}=\\left(\\begin{array}{ccc}\\mathbb {0} & \\mathbb {0} & \\mathbb {0}\\\\4 & 0 & \\mathbb {0}\\\\6 & 4 & 0\\end{array}\\right),\\qquad (\\mathbf {D}\\mathbf {C})^{2}=\\left(\\begin{array}{ccc}\\mathbb {0} & \\mathbb {0} & \\mathbb {0}\\\\4 & 0 & \\mathbb {0}\\\\8 & 4 & 0\\end{array}\\right),\\qquad (\\mathbf {D}\\mathbf {C})^{3}=\\left(\\begin{array}{ccc}\\mathbb {0} & \\mathbb {0} & \\mathbb {0}\\\\4 & 0 & \\mathbb {0}\\\\8 & 4 & 0\\end{array}\\right).$ Application of formula (REF ) yields $\\mathop \\mathrm {Tr}(\\mathbf {D}\\mathbf {C})=0=\\mathbb {1},$ and hence the conditions of Theorem REF are satisfied.", "Furthermore, we have $(\\mathbf {D}\\mathbf {C})^{\\ast }=\\left(\\begin{array}{ccc}0 & \\mathbb {0} & \\mathbb {0}\\\\4 & 0 & \\mathbb {0}\\\\8 & 4 & 0\\end{array}\\right),\\qquad \\mathbf {C}(\\mathbf {D}\\mathbf {C})^{\\ast }=\\left(\\begin{array}{ccr}4 & 0 & \\mathbb {0}\\\\7 & 3 & -1\\\\10 & 6 & 2\\end{array}\\right),$ and then find the minimum makespan $\\theta =\\Vert \\mathbf {C}(\\mathbf {D}\\mathbf {C})^{\\ast }\\Vert =10.$ It remains to represent the solution according to Theorem REF .", "We successfully calculate the matrices $\\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}=\\left(\\begin{array}{rrr}-6 & -7 & -8\\\\-6 & -7 & -8\\\\-6 & -7 & -8\\end{array}\\right),\\qquad \\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}\\oplus \\mathbf {D}\\mathbf {C}=\\left(\\begin{array}{rrr}-6 & -7 & -8\\\\0 & -7 & -8\\\\2 & 1 & -8\\end{array}\\right),\\\\(\\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}\\oplus \\mathbf {D}\\mathbf {C})^{2}=\\left(\\begin{array}{rrr}-6 & -7 & -14\\\\-6 & -7 & -8\\\\1 & -5 & -6\\end{array}\\right),\\qquad (\\theta ^{-1}\\mathbf {1}\\mathbf {1}^{T}\\mathbf {C}\\oplus \\mathbf {D}\\mathbf {C})^{\\ast }=\\left(\\begin{array}{crr}0 & -7 & -8\\\\0 & 0 & -8\\\\2 & 1 & 0\\end{array}\\right).$ The solution defined by (REF ) becomes $\\mathbf {x}=\\left(\\begin{array}{crr}0 & -7 & -8\\\\0 & 0 & -8\\\\2 & 1 & 0\\end{array}\\right)\\mathbf {u},\\qquad \\mathbf {u}\\ge \\left(\\begin{array}{c}3\\\\2\\\\1\\end{array}\\right).$ By rewriting the solution in the usual notation, we have $x_{1}&=\\max (u_{1},u_{2}-7,u_{3}-8),\\\\x_{2}&=\\max (u_{1},u_{2},u_{3}-8),\\\\x_{3}&=\\max (u_{1}+2,u_{2}+1,u_{3}),$ under the conditions that $u_{1}\\ge 3,\\qquad u_{2}\\ge 2,\\qquad u_{3}\\ge 2.$ Specifically, with $u_{1}=3$ , $u_{2}=2$ and $u_{3}=2$ , we obtain the earliest optimal initiation times given by $x_{1}=3$ , $x_{2}=3$ and $x_{3}=5$ ." ] ]	1403.0268
[ [ "A Compilation Target for Probabilistic Programming Languages" ], [ "Abstract Forward inference techniques such as sequential Monte Carlo and particle Markov chain Monte Carlo for probabilistic programming can be implemented in any programming language by creative use of standardized operating system functionality including processes, forking, mutexes, and shared memory.", "Exploiting this we have defined, developed, and tested a probabilistic programming language intermediate representation language we call probabilistic C, which itself can be compiled to machine code by standard compilers and linked to operating system libraries yielding an efficient, scalable, portable probabilistic programming compilation target.", "This opens up a new hardware and systems research path for optimizing probabilistic programming systems." ], [ "Introduction", "Compilation is source to source transformation.", "We use the phrase intermediate representation to refer to a target language for a compiler.", "This paper introduces a C-language library that makes possible a C-language intermediate representation for probabilistic programming languages that can itself be compiled to executable machine code.", "We call this intermediate language probabilistic C. Probabilistic C can be compiled normally and uses only macros and POSIX operating system libraries [9] to implement general-purpose, scalable, parallel probabilistic programming inference.", "Note that in this paper we do not show how to compile any existing probabilistic programming languages (i.e.", "IBAL [11], BLOG [8], Church [5], Figaro [12], Venture [7], or Anglican [16]) to this intermediate representation; instead we leave this to future work noting there is a wealth of readily available resources on language to language compilation that could be leveraged to do this.", "We instead characterize the performance of the intermediate representation itself by writing programs directly in probabilistic C and then testing them on computer architectures and programs that illustrate the capacities and trade-offs of both the forward inference strategies probabilistic C employs and the operating system functionality on which it depends.", "Probabilistic C programs compile to machine executable meta-programs that perform inference over the original program via forward methods such as sequential Monte Carlo [3] and particle MCMC variants [1].", "Such inference methods can be implemented in a sufficiently general way so as to support inference over the space of probabilistic program execution traces using just POSIX operating system primitives." ], [ "Related work", "The characterization of probabilistic programming inference we consider here is the process of sampling from the a posteriori distribution of execution traces arising from stochastic programs constrained to reflect observed data.", "This is the view taken by the Church [5], Venture [7], and Anglican [16], programming languages among others.", "In such languages, models for observed data can be described purely in terms of a forward generative process.", "Markov chain Monte Carlo (MCMC) is used by these systems to sample from the posterior distribution of program execution traces.", "Single-site Metropolis Hastings (MH) [5] and particle MCMC (PMCMC) [16] are two such approaches.", "In the latter it was noted that a fork-like operation is a fundamental requirement of forward inference methods for probabilistic programming, where fork is the standard POSIX operating system primitive [10].", "[6] also noted that delimited continuations, a user-level generalization of fork could be used for inference, albeit in a restricted family of models." ], [ "Probabilistic Programming", "Any program that makes a random choice over the course of its execution implicitly defines a prior distribution over its random variables; running the program can be interpreted as drawing a sample from the prior.", "Inference in probabilistic programs involves conditioning on observed data, and characterizing the posterior distribution of the random variables given data.", "We introduce probabilistic programming capabilities into C by providing a library with two primary functions: observe which conditions the program execution trace given the log-likelihood of a data point, and predict which marks expressions for which we want posterior samples.", "Any random number generator and sampling library can be used for making random choices in the program, any numeric log likelihood value can be passed to an observe, and any C expression which can be printed can be reported using predict.", "The library includes a single macro which renames main and wraps it in another function that runs the original in an inner loop in the forward inference algorithms to be described.", "Although C is a comparatively low-level language, it can nonetheless represent many well-known generative models concisely and transparently.", "Figure REF shows a simple probabilistic C program for estimating the posterior distribution for the mean of a Gaussian, conditioned on two observed data points $y_1, y_2$ , corresponding to the model $\\mu &\\sim \\mathcal {N}(1, 5),&y_1, y_2 &\\stackrel{iid}{\\sim } \\mathcal {N}(\\mu , 2).$ We observe the data $y_1,y_2$ and predict the posterior distribution of $\\mu $ .", "The functions normal_rng and normal_lnp in Figure REF return (respectively) a normally-distributed random variate and the log probability density of a particular value, with mean and variance parameters mu and var.", "The observe statement requires only the log-probability of the data points 8 and 9 conditioned on the current program state; no other information about the likelihood function or the generative process.", "In this program we predict the posterior distribution of a single value mu.", "A hidden Markov model example is shown in Figure REF , in which $N = 10$ observed data points $y_{1:N}$ are drawn from an underlying Markov chain with $K$ latent states, each with Gaussian emission distributions with mean $\\mu _k$ , and a (known) $K\\times K$ state transition matrix $T$ , such that $z_0 &\\sim \\text{Discrete}([1/K, \\dots , 1/K]) \\\\z_n\|z_{n-1} &\\sim \\text{Discrete}(T_{z_{n-1}}) \\\\y_n\|z_n &\\sim \\mathcal {N}(\\mu _{z_n}, \\sigma ^2).$ Bayesian nonparametric models can also be represented concisely; in Figure REF we show a generative program for an infinite mixture of Gaussians.", "We use a Chinese restaurant process (CRP) to sequentially sample non-negative integer partition assignments $z_n$ for each data point $y_1, \\dots , y_N$ .", "For each partition, mean and variance parameters $\\mu _{z_n}, \\sigma _{z_n}^2$ are drawn from a normal-gamma prior; the data points $y_n$ themselves are drawn from a normal distribution with these parameters, defining a full generative model $z_n &\\sim \\mathrm {CRP}(\\alpha , z_1, \\dots , z_{n-1}) \\\\{1}/{\\sigma _{z_n}^2} &\\sim \\mathrm {Gamma}(1, 1) \\\\\\mu _{z_n}\|\\sigma _{z_n}^2 &\\sim \\mathcal {N}(0, \\sigma _{z_n}^2) \\\\y_n\|z_n, \\mu _{z_n}, \\sigma _{z_n}^2 &\\sim \\mathcal {N}(\\mu _{z_n}, \\sigma _{z_n}^2).$ This program also demonstrates the additional library function memoize, which can be used to implement stochastic memoization as described in [5].", "Figure: A infinite mixture of Gaussians on the real line.Class assignment variablesfor each of the 10 data pointsare drawn following a Blackwell-MacQueen urn scheme to sequentially sample from a Dirichlet process." ], [ "Operating system primitives", "Inference proceeds by drawing posterior samples from the space of program execution traces.", "We define an execution trace as the sequence of memory states (the entire virtual memory address space) that arises during the sequential step execution of machine instructions.", "The algorithms we propose for inference in probabilistic programs map directly onto standard computer operating system constructs, exposed in POSIX-compliant operating systems including Linux, BSD, and Mac OS X.", "The cornerstone of our approach is POSIX fork [10].", "When a process forks, it clones itself, creating a new process with an identical copy of the execution state of the original process, and identical source code; both processes then continue with normal program execution completely independently from the point where fork was called.", "While copying program execution state may naïvely sound like a costly operation, this actually can be rather efficient: when fork is called, a lazy copy-on-write procedure is used to avoid deep copying the entire program memory.", "Instead, initially only the pagetable is copied to the new process; when an existing variable is modified in the new program copy, then and only then are memory contents duplicated.", "The overall cost of forking a program is proportional to the fraction of memory which is rewritten by the child process [13].", "Using fork we can branch a single program execution state and explore many possible downstream execution paths.", "Each of these paths runs as its own process, and will run in parallel with other processes.", "In general, multiple processes run in their own memory address space, and do not communicate or share state.", "We handle inter-process communication via a small shared memory segment; the details of what global data must be stored are provided later.", "Synchronization between processes is handled via mutual exclusion locks (mutex objects).", "Mutexes become particularly useful for us when used in conjunction with a synchronized counter to create a barrier, a high-level blocking construct which prevents any process proceeding in execution state beyond the barrier until some fixed number of processes have arrived." ], [ "Probability of a program execution trace", "To notate the probability of a program execution trace, we enumerate all $N$ observe statements, and the associated observed data points $y_1, \\dots , y_N$ .", "During a single run of the program, some total number $N^{\\prime }$ random choices $\\mathbf {x}^{\\prime }_1, \\dots , \\mathbf {x}^{\\prime }_{N^{\\prime }}$ are made.", "While $N^{\\prime }$ may vary between individual executions of the program, we require that the number of observe directive calls $N$ is constant.", "The observations $y_n$ can appear at any point in the program source code and define a partition of the random choices $\\mathbf {x}^{\\prime }_{1:N^{\\prime }}$ into $N$ subsequences $\\mathbf {x}_{1:N}$ , where each $\\mathbf {x}_n$ contains all random choices made up to observing $y_n$ but excluding any random choices prior to observation $y_{n-1}$ .", "We can then define the probability of any single program execution trace $p(y_{1:N}, \\mathbf {x}_{1:N}) &= \\prod _{n=1}^N g(y_n\|\\mathbf {x}_{1:n}) f(\\mathbf {x}_n\|\\mathbf {x}_{1:n-1})$ In this manner, any model with a generative process that can be written in C code with stochastic choices can be represented in this sequential form in the space of program execution traces.", "Each observe statement takes as its argument $\\ln g(y_n\|\\mathbf {x}_{1:n})$ .", "Each quantity of interest in a predict statement corresponds to some function $h(\\cdot )$ of all random choices $\\mathbf {x}_{1:N}$ made during the execution of the program.", "Given a set of $S$ posterior samples $\\lbrace \\mathbf {x}_{1:N}^{(s)}\\rbrace $ , we can approximate the posterior distribution of the predict value as $h(\\mathbf {x}_{1:N}) \\approx \\frac{1}{S} \\sum _{s=1}^S h(\\mathbf {x}_{1:N}^{(s)}).$" ], [ "Sequential Monte Carlo", "Forward simulation-based algorithms are a natural fit for probabilistic programs: run the program and report executions that match the data.", "Sequential Monte Carlo (SMC, sequential importance resampling) forms the basic building block of other, more complex particle-based methods, and can itself be used as a simple approach to probabilistic programming inference.", "SMC approximates a target density $p(\\mathbf {x}_{1:N}\|y_{1:N})$ as a weighted set of $L$ realized trajectories $\\mathbf {x}_{1:N}^{\\ell }$ such that $p(\\mathbf {x}_{1:N}\|y_{1:N}) &\\approx \\sum _{\\ell = 1}^L w^\\ell _N \\delta _{\\mathbf {x}_{1:N}^\\ell }(\\mathbf {x}_{1:N}).$ For most probabilistic programs of interest, it will be intractable to sample from $p(\\mathbf {x}_{1:N}\|y_{1:N})$ directly.", "Instead, noting that (for $n > 1$ ) we have the recursive identity $p(&\\mathbf {x}_{1:n}\|y_{1:n}) \\\\ \\nonumber &= p(\\mathbf {x}_{1:n-1}\|y_{1:n-1}) g(y_n\|\\mathbf {x}_{1:n}) f(\\mathbf {x}_n\|\\mathbf {x}_{1:n-1}),$ we sample from $p(\\mathbf {x}_{1:N}\|y_{1:N})$ by iteratively sampling from each $p(\\mathbf {x}_{1:n}\|y_{1:n})$ , in turn, from 1 through $N$ .", "At each $n$ , we construct an importance sampling distribution by proposing from some distribution $q(\\mathbf {x}_n\|\\mathbf {x}_{1:n-1}, y_{1:n})$ ; in probabilistic programs we find it convenient to propose directly from the executions of the program, i.e.", "each sequence of random variates $\\mathbf {x}_n$ is jointly sampled from the program execution state dynamics $\\mathbf {x}_n^\\ell \\sim f(\\mathbf {x}_n\|\\mathbf {x}_{1:n-1}^{a_{n-1}^\\ell })$ where $a_{n-1}^\\ell $ is an “ancestor index,” the particle index $1, \\dots , L$ of the parent (at time $n-1$ ) of $\\mathbf {x}_n^\\ell $ .", "The unnormalized particle importance weights at each observation $y_n$ are simply the observe data likelihood $\\tilde{w}_n^\\ell = g( y_{1:n}, \\mathbf {x}_{1:n}^\\ell )$ which can be normalized as $w_n^\\ell &= \\frac{\\tilde{w}_n^\\ell }{\\sum _{\\ell =1}^L \\tilde{w}_n^\\ell }.$ After each step $n$ , we now have a weighted set of execution traces which approximate $p(\\mathbf {x}_{1:n}\|y_{1:n})$ .", "As the program continues, traces which do not correspond well with the data will have weights which become negligibly small, leading in the worst case to all weight concentrated on a single execution trace.", "To counteract this deficiency, we resample from our current set of $L$ execution traces after each observation $y_n$ , according to their weights $w_n^\\ell $ .", "This is achieved by sampling a count $O^\\ell _n$ for the number of “offspring” of a given execution trace $\\ell $ to be included at time $n+1$ ; any sampling scheme must ensure $\\mathbb {E}[O^\\ell _n] = w_n^\\ell $ .", "Sampling offspring counts $O^\\ell _n$ is equivalent to sampling ancestor indices $a_n^\\ell $ .", "Program execution traces with no offspring are killed; program execution traces with more than one offspring are forked multiple times.", "After resampling, all weights $w_n^\\ell = 1$ .", "[tb] Parallel SMC program execution $N$ observations, $L$ particles launch $L$ copies of the program (parallel) $n=1\\dots N$ wait until all $L$ reach observe $y_n$ (barrier) update unnormalized weights $\\tilde{w}^{1:L}_n$ (serial) $ESS < \\tau $ sample number of offspring $O_n^{1:L}$ (serial) set weight $\\tilde{w}^{1:L}_n = 1$ (serial) $\\ell =1\\dots L$ fork or exit (parallel) set all number of offspring $O_n^{\\ell } = 1$ (serial) continue program execution (parallel) wait until $L$ program traces terminate (barrier) predict from $L$ samples from $\\hat{p}(\\mathbf {x}_{1:N}^{1:L}\|y_{1:N})$ (serial) We only resample if the effective sample size $ESS\\approx \\frac{1}{\\sum _\\ell (w_n^{\\ell })^2}$ is less than some threshold value $\\tau $ ; we choose $\\tau = L/2$ .", "In probabilistic C, each observe statement forms a barrier: parallel execution cannot continue until all particles have arrived at the observe and have reported their current unnormalized weight.", "As execution traces arrive at the observe barrier, they take the number of particles which have already reached the current observe as a (temporary) unique identifier.", "Program execution is then blocked as the effective sample size is computed and the number of offspring are sampled.", "The number of offspring are stored in a shared memory block; when the number of offspring are computed, each particle uses the identifier assigned when reaching the observe barrier to retrieve (asynchronously) from shared memory the number of children to fork.", "Particles with no offspring wait for any child processes to complete execution, and terminate; particles with only one offspring do not fork any children but continue execution as normal.", "The SMC algorithm is outlined in Algorithm REF , with annotations for which steps are executed in parallel, serially, or form a barrier.", "After a single SMC sweep is complete, we sample values for each $\\texttt {\\footnotesize predict} $ , and then (if desired) repeat the process, running a new independent particle filter, to draw an additional batch of samples." ], [ "Particle Metropolis-Hastings", "Particle Markov chain Monte Carlo, introduced in [1], uses sequential Monte Carlo to generate high-dimensional proposal distributions for MCMC.", "The most simple formulation is the particle independent Metropolis-Hastings algorithm.", "After running a single particle filter sweep, we compute an estimate of the marginal likelihood, $\\hat{Z} \\equiv p(y_{1:N}) \\approx \\prod _{n=1}^N \\left[ \\frac{1}{L} \\sum _{\\ell = 1}^L w_n^\\ell \\right].$ We then run another iteration of sequential Monte Carlo which we use as a MH proposal; we estimate the marginal likelihood $\\hat{Z}^{\\prime }$ of the new proposed particle set, and then with probability $\\min (1, \\hat{Z}^{\\prime } / \\hat{Z})$ we accept the new particle set and output a new set of predict samples, otherwise outputting the same predict samples as in the previous iteration.", "The inner loop of Algorithm REF is otherwise substantially similar to SMC.", "[tb] Parallel PIMH program execution $M$ iterations, $N$ observations, $L$ particles $m=1 \\dots M$ launch $L$ copies of the program (parallel) $n=1\\dots N$ wait until all $L$ reach an observe (barrier) update unnormalized weights $\\tilde{w}_{1:L}$ (serial) $ESS < \\tau $ update proposal evidence estimate $\\hat{Z}^{\\prime }$ (serial) sample number of offspring $O_n^{1:L}$ (serial) set weight $\\tilde{w}^{1:L}_n = 1$ (serial) $\\ell =1\\dots L$ fork or exit (parallel) set all number of offspring $O_n^{\\ell } = 1$ (serial) continue program execution (parallel) wait until $L$ program traces terminate (barrier) accept or reject new particle set (serial) predict from $L$ samples from $\\hat{p}(\\mathbf {x}_{1:N}^{1:L}\|y_{1:N})$ (serial) store current particle set ${\\mathbf {x}}$ and evidence $\\hat{Z}$ (serial) continue to next iteration (parallel)" ], [ "Particle Gibbs", "[tb] Parallel Particle Gibbs program execution $M$ iterations, $N$ observations, $L$ particles $m=1 \\dots M$ $L^{\\prime } \\leftarrow L$ if $m = 1$ , otherwise $L-1$ launch $S^{\\prime }$ copies of the program (parallel) $n=1\\dots N$ wait until all $L^{\\prime }$ reach an observe (barrier) compute weights for all particles (serial) $m > 1$ signal num offspring to retained trace (serial) $\\ell =1\\dots L^{\\prime }$ spawn retain / branch process [Algo.", "REF ] (parallel) wait until $L$ particles finish branching (barrier) continue program execution (parallel) wait until $L$ program traces terminate (barrier) predict from $L$ samples from $\\hat{p}(\\mathbf {x}_{1:N}^{1:L}\|y_{1:N})$ (serial) select and signal particle to retain (serial) wait until $N$ processes are ready to branch (barrier) continue to next iteration (parallel) [tb] Retain and Branch inner loop input initial $C > 0$ children to spawn is_retained $\\leftarrow $ false true $ C = 0$ and not is_retained discard this execution trace, exit [$C \\ge 0$ ] spawn $C$ new children wait for signal which resets is_retained is_retained wait for signal which resets $C$ discard this execution trace, exit Particle Gibbs is a particle MCMC technique which also has SMC at its core, with better theoretical statistical convergence properties than PIMH but additional computational overhead.", "We initialize particle Gibbs by running a single sequential Monte Carlo sweep, and then alternate between (1) sampling a single execution trace $\\hat{\\mathbf {x}}_{1:M}$ from the set of $L$ weighted particles, and (2) running a “conditional” SMC sweep, in which we generate $L-1$ new particles in addition to the retained $\\hat{\\mathbf {x}}_{1:M}$ .", "The implementation based on operating system primitives is described in algorithms REF and REF .", "The challenge here is that we must “retain” an execution trace, which we can later revisit to resume and branch arbitrarily many times.", "This is achieved by spawning off a “control” process at every observation point, which from then on manages the future of that particular execution state.", "As before, processes arrive at an observe barrier, and when all particles have reached the observe we compute weights, and sample offspring counts $O_n^\\ell $ .", "Particles with $O_n^\\ell = 0$ terminate, but new child processes are no longer spawned right away.", "Instead, all remaining particles fork a new process whose execution path immediately diverges from the main codebase and enters the retain and branch loop in Algorithm REF .", "This new process takes responsibility for actually spawning the $O_n^\\ell $ new children.", "The spawned child processes (and the original process which arrived at the observe barrier) wait (albeit briefly) at a new barrier marking the end of observe $n$ , not continuing execution until all new child processes have been launched.", "Program execution continues to the next observe, during which the retain / branch process waits until a full particle set reaches the end of the program.", "Once final weights $\\tilde{w}_N^{1:L}$ are computed, we sample (according to weight) from the final particle set to select a single particle to retain during the next SMC iteration.", "When the particle is selected, a signal is broadcast to all retain / branch loops indicating which process ids correspond to the retained particle; all except the retained trace exit.", "The retain / branch loop now goes into waiting again (this time for a branch signal), and we begin SMC from the top of the program.", "As we arrive at each observe $n$ , we only sample $L-1$ new offspring to consider: we guarantee that at least one offspring is spawned from the retained particle at $n$ (namely, the retained execution state at $n+1$ ).", "However, depending on the weights, often sampling offspring will cause us to want more than a single child from the retained particle.", "So, we signal to the retained particle execution state at time $n$ the number of children to spawn; the retain / branch loop returns to its entry point and resumes waiting, to see if the previously retained execution state will be retained yet again.", "Note that in particle Gibbs, we must resample (select offspring and reset weights $w_n^\\ell = 1$ ) after every observation in order to be able to properly align the retained particle on the next iteration through the program." ], [ "Experiments", "We now turn to benchmarking probabilistic C against existing probabilistic programming engines, and evaluate the relative strengths of the inference algorithms in Section .", "We find that compilation improves performance by approximately 100 times over interpreted versions of the same inference algorithm.", "We also find evidence that suggests that optimizing operating systems to support probabilistic programming usage could yield significant performance improvements as well.", "The programs and models we use in our experiments are chosen to be sufficiently simplistic that we can compute the exact posterior distribution of interest analytically, allowing us to evaluate correctness of inference.", "Given the true posterior distribution $p$ , we measure sampler performance by the KL-divergence $KL(\\hat{p}\|\|p)$ , where $\\hat{p}$ is our Monte Carlo estimate.", "The first benchmark program we consider is a hidden Markov model (HMM) very similar to that of Figure REF , where we predict the marginal distributions of each latent state.", "The HMM used in our experiments here is larger; it has the same model structure, but with $K=10$ states and 50 sequential observations, and each state $k = 1, \\dots , 10$ has a Gaussian emission distribution with $\\mu _k = k-1$ and $\\sigma ^2 = 4$ .", "The second benchmark is the CRP mixture of Gaussians program in Figure REF , where we predict the total number of distinct classes." ], [ "Comparative performance of inference engines", "We begin by benchmarking against two existing probabilistic programming engines: Anglican, as described in [16], which also implements particle Gibbs but is an interpreted language based on Scheme, implemented in Clojure, and running on the JVM; and probabilistic-jshttps://github.com/dritchie/probabilistic-js, a compiled system implementing the inference approach in [15], which runs Metropolis-Hastings over each individual random choice in the program execution trace.", "The interpreted particle Gibbs engine is multithreaded, and we run it with 100 particles and 8 simultaneous threads; the Metropolis-Hastings engine only runs on a single core.", "In Figure REF we compare inference performance in both of these existing engines to our particle Gibbs backend, running with 100 and 1000 particles, in an 8 core cloud computing environment on Amazon EC2, running on Intel Xeon E5-2680 v2 processors.", "Our compiled probabilistic C implementation of particle Gibbs runs over 100 times faster that the existing interpreted engine, generating good sample sets in on the order of tens of seconds.", "The probabilistic C inference engine implements particle Gibbs, SMC, and PIMH sampling, which we compare in Figure REF using both 100 and 1000 particles.", "SMC is run indefinitely by simply repeatedly drawing independent sets of particles; in contrast to the PMCMC algorithms, this is known to be a biased estimator even as the number of iterations goes to infinity [14], and is not recommended as a general-purpose approach.", "Figures REF and REF plot wall clock time against KL-divergence.", "We use all generated samples as our empirical posterior distribution in order to produce as fair a comparison as possible.", "In all engines, results are reasonably stable across runs; the shaded band covers the 25th to 75th percentiles over multiple runs, with the median marked as a dark line.", "A sampler drawing from the target density will show approximately linear decrease in KL-divergence on these log-log plots; a steeper slope correspond to greater statistical efficiency.", "The methods based on sequential Monte Carlo do not provide any estimate of the posterior distribution until completing a single initial particle filter sweep; for large numbers of particles this may be a non-trivial amount of time.", "In contrast, the MH sampler begins drawing samples effectively immediately, although it may take a large number of individual steps before converging to the correct stationary distribution; individual Metropolis-Hastings samples are likely to be more autocorrelated, but producing each one is faster." ], [ "Performance characteristics across multiple cores", "As the probabilistic C inference engine offloads much of the computation to underlying operating system calls, we characterize the limitations of the OS implementation by comparing time to completion as we vary the number of cores.", "Tests for the hidden Markov model across core count (all on EC2, all with identical Intel Xeon E5-2680 v2 processors) are shown in Figure REF .", "Figure: Effect of system architecture on runtime performance.", "Here we run the HMM code on EC2 instances with identical processors (horizontal axis) with varying number of particles (individual bars) and report runtime to produce 10,000 samples.", "Despite adding more cores, after 16 cores performance begins to degrade.", "Similarly, adding more particles eventually degrades performance for any fixed number of cores.", "In combination these suggest the availability of operating system optimizations that could improve overall performance." ], [ "Discussion", "Probabilistic C is a method for performing inference in probabilistic programs.", "Methodologically it derives from the forward methods for performing inference in statistical models based on sequential Monte Carlo and particle Markov chain Monte Carlo.", "We have shown that it is possible to efficiently and scalably implement this particular kind of inference strategy using existing, standard compilers and POSIX compliant operating system primitives.", "What most distinguishes Probabilistic C from prior art is that it is highly compatible with modern computer architectures, all the way from operating systems to central processing units (in particular their virtual memory operations), and, further, that it delineates a future research program for scaling the performance of probabilistic programming systems to large scale problems by investigating systems optimizations of existing computer architectures.", "Note that this is distinct but compatible with approaches to optimizing probabilistic programming systems by compilation optimizations, stochastic hardware, and dependency tracking with efficient updating of local execution trace subgraphs.", "It may, in the future, be possible to delineate model complexity and hardware architecture regimes in which each approach is optimal; we assert that, for now, it is unclear what those regimes are or will be.", "This paper is but one step towards such a delineation.", "Several interesting research questions remain: (1) Is it more sensible to write custom memory management and use threads than fork and processes as we have done?", "The main contribution of this paper is to establish a probabilistic programming system implementation against a standardized, portable abstraction layer.", "It might be possible to eke out greater performance by capitalizing on the fact modern architectures are optimised for parallel threads more so than parallel processes; however, exploiting this would entail implementing memory management de facto equivalent in action to fork which may lead to lower portability.", "(2) Would shifting architectures to small page sizes help?", "There is a bias towards large page size support in modern computer architectures.", "It may be that the system use characteristics of probabilistic programming systems might provide a counterargument to that bias or inspire the creation of tuned end-to-end systems.", "Forking itself is lightweight until variable assignment which usually require manipulations of entire page tables.", "Large pages require large amounts of amortisation in order to absorb the cost of copying upon stochastic variable assignment.", "Smaller pages could potentially yield higher efficiencies.", "(3) What characteristics of process synchronisation can be improved specifically for probabilistic programming systems?", "This is both a systems and machine learning question.", "From a machine learning perspective we believe it may be possible to construct efficient sequential Monte Carlo algorithms that do not synchronize individual threads at observe barriers and instead synchronize in a queue.", "On a systems level it begs questions about what page replacement strategies to consider; perhaps entirely changing the page replacement schedule to reflect rapid process rather than thread multiplexing across cores.", "Probabilistic C does not disallow programmers from accidentally writing programs that are statistically incoherent.", "Many probabilistic programming languages (including Church, Anglican, and Venture) are nearly purely functional and so being disallow program variable value reassignment.", "This ensures a well-defined joint distribution on program variables.", "Probabilistic C offers no such guarantees.", "For this reason we are not, in this paper, making a claim about a new probabilistic programming language per se — rather we describe probabilistic C as an intermediate representation compilation target that implements a particular style of inference that is natively parallel and possible to optimize by system architecture choice.", "It is possible (as helpfully pointed out by Vikash Mansinghka) that compiler optimization techniques such as checking to see whether or not the program is natively in “static single assignment” form [2] can help avoid statistically incoherent programs being allowed (at compilation).", "This we leave as future work.", "Further (as helpfully pointed out by Noah Goodman), programming languages constructs such as delimited continuations [4] can be thought of as user level abstractions of fork, and might provide similar functionality at the user rather than system level in the context of statistically safer languages." ] ]	1403.0504
[ [ "The edge engineering of topological Bi(111) bilayer" ], [ "Abstract A topological insulator is a novel quantum state, characterized by symmetry-protected non-trivial edge/surface states.", "Our first-principle simulations show the significant effects of the chemical decoration on edge states of topological Bi(111) bilayer nanoribbon, which remove the trivial edge state and recover the Dirac linear dispersion of topological edge state.", "By comparing the edge states with and without chemical decoration, the Bi(111) bilayer nanoribbon offers a simple system for assessing conductance fluctuation of edge states.", "The chemical decoration can also modify the penetration depth and the spin texture of edge states.", "A low-energy effective model is proposed to explain the distinctive spin texture of Bi(111) bilayer nanoribbon, which breaks the spin-momentum orthogonality along the armchair edge." ], [ "bookmarks=true, unicode=false, pdftoolbar=true, pdfmenubar=true, pdffitwindow=false, pdfstartview=FitH, pdftitle=My title, pdfauthor=Author, pdfsubject=Subject, pdfcreator=Creator, pdfproducer=Producer, pdfkeywords=keyword1 key2 key3, pdfnewwindow=true, colorlinks=true, linkcolor=midnightblue, citecolor=magenta, filecolor=midnightblue, urlcolor=midnightblue, The edge engineering of topological Bi(111) bilayer Xiao Li Equal contribution.", "International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Center for Quantum Materials, Peking University, Beijing, China Hai-Wen Liu Equal contribution.", "International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Center for Quantum Materials, Peking University, Beijing, China Hua Jiang Department of Physics and Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China Fa Wang International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Center for Quantum Materials, Peking University, Beijing, China Ji Feng [email protected] International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Center for Quantum Materials, Peking University, Beijing, China A topological insulator is a novel quantum state, characterized by symmetry-protected non-trivial edge/surface states.", "Our first-principle simulations show the significant effects of the chemical decoration on edge states of topological Bi(111) bilayer nanoribbon, which remove the trivial edge state and recover the Dirac linear dispersion of topological edge state.", "By comparing the edge states with and without chemical decoration, the Bi(111) bilayer nanoribbon offers a simple system for assessing conductance fluctuation of edge states.", "The chemical decoration can also modify the penetration depth and the spin texture of edge states.", "A low-energy effective model is proposed to explain the distinctive spin texture of Bi(111) bilayer nanoribbon, which breaks the spin-momentum orthogonality along the armchair edge.", "71.15.Mb, 73.43.Nq, 73.20.Fz, 73.20.-r As an insulating state with symmetry-protected gapless interface electronic modes, the topological insulator (TI) has received considerable attention recently [1], [2], [3].", "The edge conduction channels of two-dimensional (2d) TI exhibits quantum spin Hall effect within bulk gap [4].", "A single bilayer Bi(111) film has been predicted to be a 2d TI with a large band gap of about 0.5 eV [5], [6], [7], while other 2d TIs, such as HgTe/CdTe quantum wells [8] and InAs/GaSb quantum wells [9], have gaps of only several tens of meV at best.", "Recently, Bi(111) bilayer has been readily grown on Bi$_{2}$ Te$_{3}$ or Bi$_{2}$ Se$_{3}$ substrates [10], [11], [12], [13], [14], [15].", "Therefore, it is very promising for room-temperature TI-based devices.", "However, the native edges of Bi bilayer suffer from the simultaneous presence of both trivial and non-trivial edge modes [6], [12], which complicate the fundamental investigation of its topological properties and eventual applications.", "Similar complication had perplexed the interpretation of surface states in three-dimensional (3d) TI Bi$_{1-x}$ Sb$_{x}$ [16], [17].", "Although localization in Anderson's sense will suppress trivial conducting channels, quantitively localizing trivial channels will still be an experimental challenge.", "A wide distribution of conductance induced by multiple edge states is not desirable for accurate transport measurement [18], [6], [19].", "The complicated edge or surface states may be a generic problem associated with dangling bond states at the termination of 2d or 3d TIs.", "In this Letter, we report a first-principle analysis of chemical decoration of the edge states of Bi(111) bilayer, which, as we show, is an effective route for precise engineering of conducting edge states.", "We demonstrate that chemical passivation quantitatively removes the trivial edge bands in Bi bilayer nanoribbons, restoring the desired Dirac dispersion of the non-trivial edges.", "We further compute transport and optical signatures of the chemical decoration of the Bi bilayer edges, which can be assessed experimentally.", "In particular, we suggest that the Bi bilayer nanoribbons, with and without chemical passivation, offer a simple system for assessing conductance fluctuation of edge states.", "Moreover, edge decoration has important consequences on the spatial distribution and spin texture of the edge states.", "A low-energy effective model is proposed to explain the distinctive spin texture of Bi bilayer nanoribbon, where the spin is no longer perpendicular to the momentum along the armchair edge.", "Figure: The geometric structure of (a) the zigzag and (b) the armchair Bi(111) nanoribbon.Up: The top view.", "Down: The side view.The primitive cell of single bilayer sheet,bounded by dashed lines, are also shown.The zigzag and armchair edges are perpendicular to eachother, which are along x- and y-axis of the Bi(111) sheet, respectively.The hydrogen-terminated edges shown in the figure are the most stablestructure of hydrogen adsorption.", "The big purple ball stands for bismuthatom, and the small pink ball for hydrogen atom.", "Idoine-terminatednanoribbons have the similar structures and they are not shown.We use density functional theory [20] calculations implemented in the Vienna Ab initio simulation package [21], [22] to investigate geometric and electronic structure of single Bi(111) bilayer and its nanoribbons.", "Computational details can be found in the Supplementary Information (S.I.)", "[23].", "Notice that spin-orbit coupling (SOC) is included in the calculation of electronic structure, unless otherwise specified.", "The single Bi(111) bilayer has the point group symmetry of $D_{3d}$ with spatial inversion included.", "As shown in Fig.", "REF , the top view of Bi(111) bilayer shows a bipartite honeycomb lattice with A and B sublattices.", "Two sublattices have different heights, forming bilayer structure.", "The calculated nearest-neighbor bond angles in single Bi bilayer is 91$^\\circ $ and the lattice constant $a=4.34$ Å.", "Based on the electronic structure of Bi bilayer (See S.I.", "[23]), the band inversion takes place between $p$ -like valence and conduction bands at $\\Gamma $ point, leading to a 2d topological insulator with a indirect band gap of 0.5 eV, agreeing with the previous results [5], [6], [7].", "To investigate the edge properties of the Bi(111) nanoribbons, we study two representative model systems: (1) a 40-atom (per unit cell) zigzag nanoribbon (about 7.3 nm wide), and (2) 50-atom (per unit cell) armchair nanoribbon (about 5.2 nm wide) (Fig.", "REF ).", "For nanoribbons with native edges, the band structures are shown in the left panels of Figs.", "REF (a) and (b).", "All bands remain spin-degenerate, owing to simultaneous time-reversal and inversion symmetry.", "Within the bulk band gap, the upper non-trivial and the lower trivial edge states are present simultaneously, which span the entire Brillouin zone (BZ).", "There are odd numbers of Kramers pairs of edge states at the Fermi level, showing that Bi(111) bilayer is indeed a 2d topological insulator.", "If we sweep the chemical potential across the gap by external gating, the number of conducting channel may change from 3 to 1 (or from 1 to 3).", "Here, the edge states do not show linear dispersion near $\\Gamma $ point, in contrast to Kane-Mele model of quantum spin Hall effect[4].", "Given that the atomic edge adsorption of graphene nanoribbon has been achieved via hydrogen plasma etching recently and the edge decoration has important effects on electronic properties of graphene nanoribbons [24], [25], [26], the edge states of Bi(111) nanoribbon may be modified by chemical adsorption.", "We study the adsorption of hydrogen and iodine atoms on Bi(111) zigzag and armchair nanoribbons.", "Considering different adsorption sites from the edge to the middle part of nanoribbons (Fig.", "S2 in S.I.", "[23]), the atom adsorption on the outmost bismuth atom is the most stable structure with the lowest adsorption energy [23], compared with atom adsorption on the basal plane.", "It will lead to selective edge decoration in experiment, similar with graphene nanoribbon [26].", "The adatoms restore three-fold coordination of the bismuths at two edges, indicating that the dangling bonds become saturated.", "Figure: The band structure of Bi(111) nanoribbons.", "(a) The zigzag nanoribbons with SOC.", "(b) The armchair nanoribbons with SOC.", "(c) The zigzag nanoribbons without SOC.", "The armchair nanoribbons withoutSOC are shown in Fig.", "S3.", "There are three panels in each figure.Left panel: The native edge.", "Middle panel:The hydrogen-terminated edge.", "Right panel: The iodine-terminatededge.The highest occupied energy level is set to zero energy.k ¯ x ≡k x a\\overline{k}_{x}\\equiv k_{x}aand k ¯ y ≡3k y a\\overline{k}_{y}\\equiv \\sqrt{3}k_{y}a.The middle and right panels in Figs.", "REF (a) and (b) show the band structure after the edge functionalization for zigzag and armchair nanoribbons, respectively.", "Compared with native nanoribbons, there are only linear dispersing non-trivial edge states in the center of BZ, while the trivial edge states are removed.", "The Fermi velocities are $8.5\\times 10^{5} m/s$ and $7.9\\times 10^{5} m/s$ for hydrogen- and iodine-terminated zigzag nanoribbon, respectively.", "And the values are $7.7\\times 10^{5} m/s$ and $7.3\\times 10^{5} m/s$ for hydrogen- and iodine-terminated armchair nanoribbon, respectively.", "The band structures Fig.", "REF (c) and Fig.", "S3 further show the corresponding band structure of Bi(111) nanoribbon without considering SOC.", "For the native edge, the edge states within the bulk gap are trivial ones in the absence of SOC.", "With atom adsorption at two edges, the trivial edge states are removed from the bulk gap.", "Therefore, taking into account SOC in our calculation, the emergent edge states (the middle and right panels in Figs.", "REF (a) and (b)) after chemical decoration are only non-trivial ones, resulting from the band inversion of TI.", "Figure: Conductance, GG, as the shift of the chemical potential for (a) the native and(b) hydrogen-terminated zigzag nanoribbon.Under Anderson disorders of differentstrengths WW, the conductances of the nanoribbons are shown in different colors and symbols.", "The baron every data point represents the conductance fluctuation, δG\\delta {G}.", "The insetof (a) shows the conductance fluctuation as a function of WW.", "The Fermi level is set to zero energy.", "The blue line stands for the edge states at 0.05eV.", "The red line stands for the ones at 0.25 eV, which keeps zero.Considering the significant modification of the edge bands by chemisorption, we suggest the effects of the edge engineering can be directly probed by transport and optical measurements.", "The key effect of the edge chemisorption is turning the number of edge conduction channels from three in the native nanoribbon to only one.", "This creates an interesting experimental apparatus to assess the effects of localization in the Anderson's paradigm.", "We expect that in the case of single non-trivial edge channel, the conduction will stay quantized and will not be affected by non-magnetic disordered Anderson scatterers.", "On the other hand, the simultaneous presence of both non-trivial and trivial edge channels will show rather different transport behavior.", "Sufficiently strong disorder will eventually localize the trivial channels.", "However, in the intermediate localization regime, we may have window to detect disorder-induced conductance fluctuation [18].", "The edge modification will also change the optical absorption of the material, which can also be measured experimentally.", "We therefore compute the transport and optical spectra of Bi(111) bilayer nanoribbons, based on a full-valence tight-binding (TB) model from DFT calculations [23].", "Taking Bi(111) zigzag nanoribbon as an example, the transport spectra are calculated with the non-equilibrium Green's function approach [27].", "For the native zigzag nanoribbon, the conductance is $6e^{2}/h$ at the Fermi level, as expected and also consistent with experimental measuration [19].", "The conductance of edge states changes from $6e^{2}/h$ to $2e^{2}/h$ as the chemical potential is gated up within the bulk band gap, leaving only the contribution from the non-trivial edge state.", "Upon introduction of Anderson disorder to the model, the conductance from trivial edge gradually decays with the increasing strength of disorder in the neighboring of the Fermi level (Fig.", "REF (a)), indicating localization.", "At the same time, we indeed observe significant conductance fluctuation, $\\delta {G}$ .", "We see that $\\delta {G}$ first increases with weak Anderson disorder, but eventually decreases to zero upon complete localization of trivial edge channels (the inset of Fig.", "REF (a)).", "For hydrogen-termination zigzag nanoribbon, the conductance stays at the quantized platform of $2e^{2}/h$ without any fluctuations within bulk band gap, as shown in Fig.", "REF (b).", "We also investigate the effect of less-than-full adsorption along two edges of nanoribbon with random occupancy of the adsorbed hydrogen atoms.", "It is found that 25% coverage of the edges with hydrogen adsorption can obtain a similar conductance plateau of $2e^{2}/h$ as the one of 100% hydrogen adsorption, showing that it is feasible to improve the transport properties by partial edge decoration.", "Figure: The imaginary part of the dielectric tensor component,ε xx (2) (ω)\\varepsilon _{xx}^{(2)}(\\omega ), as a function of the optical frequency ω\\omega for zigzag nanoribbon.", "The bare edge and the hydrogen-terminated edgeare shown in blue and red , respectively.Fig.", "REF shows the imaginary part of the dielectric tensor component for zigzag nanoribbon, $\\varepsilon _{xx}^{(2)}(\\omega )$ , under the $x$ -polarized irradiation field.", "Without any terminations, nonzero dielectric function shows that there are alway optical transitions between edge states or between edge states and bulk states.", "However, these transitions are inhibited to a great extent for hydrogen-terminated zigzag nanoribbon.", "These observable characteristics can be used as signals of the edge decoration of Bi nanoribbons.", "Alongside with the change of the band structure and the corresponding experimental signatures, the penetration depth and spin texture of edge states have been modified by edge engineering.", "Fig.", "REF (a) shows the change of the penetration depth with atomic adsorption for zigzag nanoribbon.", "For the zigzag nanoribbon with native edge, only one band of each group of the spin-degenerate edge bands are presented in the half of the BZ ($k_{x}a$ from $-\\pi $ to 0), while other states can be obtained by the inversion and time-reversal symmetry.", "For hydrogen-terminated nanoribbon, the edge states are presented in the smaller zone of the BZ ($-\\pi /5$ to 0) to zoom in on the linear dispersion.", "The penetration depth is very short for both the upper non-trivial (blue circle in the left panel of Fig.", "REF (a)) and lower trivial edge states (red square) of the 7.3 nm-width nanoribbon without any termination.", "For states away from the bulk states ($k_{x}a$ near $\\pi $ ), they are localized within 1nm closest to the edge.", "When the edge state approaches the bulk states ($k_{x}a$ near 0), the penetration depth is gradually getting longer.", "In contrast, the penetration depths of the hydrogen-terminated edge states are all more than 2 nm, much longer than the one of native edge, which agrees with the inverse relationship between the penetration depth and the momentum-space width of the edge state [28], [6].", "Besides, the penetration depths is also getting longer for hydrogen-terminated edge, as the edge state approaches the bulk states ($k_{x}a$ from 0 to $\\pi /5$ ).", "Figure: The electronic properties of Bi(111) zigzag edge states.", "(a) The penetration depth of the edge states as a function of the momentum k x k_{x}.", "Left: Thenative edge.", "Right: The hydrogen-terminated edge.", "The upper and loweredge states are shown by blue circle and red square, respectively.", "(b) The spin texture of edge states.", "The blue circle, red square and green trianglestand for three components of spin moment, m x m_{x}, m y m_{y} andm z m_{z}, respectively.", "k ¯ x ≡k x a\\overline{k}_{x}\\equiv k_{x}a.", "In (b), Only the more localized edge state (the upper edge) are shown,while the other one behaves in the same manner.The spin moments, $m_{x}$ , $m_{y}$ and $m_{z}$ , for Bi(111) zigzag nanoribbon are shown in Fig.", "REF (b).", "Compared with Kane-Mele model [4] and HgTe/CdTe quantum wells [8], the edge states of Bi(111) nanoribbons have more complicated spin textures.", "For native zigzag nanoribbon, the components, $m_{y}$ and $m_{z}$ , perpendicular to the momentum $k_{x}$ of zigzag edge, vary gradually and the spin direction rotates with the momentum.", "The component $m_{x}$ , parallel to the momentum, is zero.", "It is similar to the interface state of well-known topological insulators, such as the natural cleavage (111) surface of Bi$_{2}$ Se$_{3}$ , where the spins are locked to their momentums at right angles [1].", "For hydrogen-terminated zigzag edge, $m_{y}$ and $m_{z}$ have altered their trends and varied slowly in the momentum space near the Dirac point, with $m_{x}=0$ .", "The armchair edges have similar spin textures with zigzag edges, where $m_{x}$ is still zero, as shown in Fig.", "S4 [23].", "Given that the momentum direction is along y axis for armchair edges, the spin is no longer perpendicular to the momentum, in contrast with the spin-momentum orthogonality in well-known TIs [1], [4], [8].", "This departure can be explained by our effective model of single Bi(111) bilayer.", "We suggest a low-energy effective Hamiltonian of single Bi(111) bilayer based on the symmetry analysis.", "Considering the inversion symmetry of single bilayer, we combine the $p$ -orbitals near the Fermi level to form the bonding and anti-bonding states with definite parity, $\\left\|p_{\\lambda }^{\\pm }\\right\\rangle =\\frac{1}{\\sqrt{2}}\\left(\\left\|p_{A,\\lambda }\\right\\rangle \\mp \\left\|p_{B,\\lambda }\\right\\rangle \\right)$ , where $p_{\\lambda }=p_{x,y,z}$ stand for three $p$ -orbitals and A/B for A/B sublattice stands of the honeycomb lattice.", "The superscript $\\pm $ correspond to even and odd parity, respectively.", "Taking into account the band splitting from both crystal field and SOC, the band inversion mainly arises between degenerate states $\\left\|p_{z}^{-},\\pm \\frac{1}{2}\\right\\rangle $ and degenerate states $\\left\|p_{x,y}^{+},\\pm \\frac{3}{2}\\right\\rangle $ near $\\Gamma $ , where $\\pm \\frac{1}{2}$ and $\\pm \\frac{3}{2}$ denote the corresponding total azimuthal quantum numbers.", "We then construct four Wannier bases, $\|\\alpha ^{-}\\rangle ,$ $\|\\beta ^{+}\\rangle ,$ $\\hat{\\mathcal {T}}\|\\alpha ^{-}\\rangle ,$ and $\\hat{\\mathcal {T}}\|\\beta ^{+}\\rangle $ , to describe the low-energy excitations of single Bi(111) bilayer, where $\|\\alpha ^{-}\\rangle = \|p_{z}^{-},\\frac{1}{2}\\rangle $ and $\|\\beta ^{+}\\rangle =N_{0}\\left(\\left\|p_{x,y}^{+},\\frac{3}{2}\\right\\rangle +\\eta \\left\|p_{x,y}^{+},-\\frac{3}{2}\\right\\rangle \\right)$ .", "$\\hat{\\mathcal {T}}$ is time-reversal operator, $N_{0}$ is the normalization factor and $\\eta $ is the weight factor (See details in S.I.", "[23]).", "The effective Hamiltonian near $\\Gamma $ is separated into two subblocks as below, $\\mathcal {H}({\\bf k})=\\left[\\begin{array}{cc}H({\\bf k}) & 0\\\\0 & H^{*}(-{\\bf k})\\end{array}\\right],$ $H(\\mathbf {k})=m[\\sigma _{z}+k^{2}(\\lambda ^{2}\\sigma _{0}-\\xi {}^{2}\\sigma _{z})]+\\hbar v(k_{x}\\sigma _{x}-k_{y}\\sigma _{y}),$ where $\\sigma _{i}$ ($i=x,y,z,0$ ) is the Pauli matrices addressing the subspace spanned by $\|\\alpha ^{-}\\rangle $ and $\|\\beta ^{+}\\rangle .$ The parameters $m = 0.291$ eV, $\\lambda = 14.11$ Å, $\\xi = 15.51$ Å, and $v = 1.079\\times 10^{6}\\,\\text{m/s}$ , which are obtained by fitting the DFT band structure of Bi(111) bilayer near $\\Gamma $ point, as shown in Fig.", "S1 [23].", "The Hamiltonian leads to topological edge states with clean linear dispersion and a Fermi velocity of $6.0\\times 10^{5}\\,\\text{m/s}$ , agreeing with our first-principle results.", "Moreover, when spin Pauli matrix $s_{x}$ acts on the bases, we have $\\left\\langle \\varphi \\right\|s_{x}\\left\|\\varphi \\right\\rangle =0$ , where $\\varphi =\\alpha ^{-},\\beta ^{+}$ .", "That is, the low-energy bulk bands have vanishing $m_{x}$ , and so do the topological edge states of both zigzag and armchair edges, which arise from the bulk band inversion.", "In contrast, nonzero $m_{y}$ and $m_{z}$ can also be obtained by the Hamiltonian [23].", "This explains the distinctive spin-momentum relationship in Bi(111) bilayer.", "In conclusion, the edge chemical decoration of Bi(111) bilayer can significantly modify topological edge states.", "The experimental signatures and the low-energy effective Hamiltonian are also proposed.", "Clean edge state and model Hamiltonian will facilitate further investigations on topological properties of the Bi(111) bilayer, such as superconducting proximity effect [29] and topological Anderson Insulator [30].", "forestgreenAcknowledgements.— This work is supported by the National Science Foundation of China under Grants Nos.", "11174009 and 11374219, by China 973 Program Projects 2013CB921900." ] ]	1403.0147
[ [ "Applications of higher order QCD" ], [ "Abstract In this talk we summarize some recent developments in perturbative QCD and their application to particle physics phenomenology." ], [ "Introduction", "With the discovery of a new boson at the Large Hadron Collider (LHC), particle physics has entered a new era.", "Since this discovery, the field has quickly moved towards precision measurements on the new particle.", "In order to further improve these measurements and to find possible small deviations that may hint towards new physics, improved theoretical predictions, including higher-order perturbative QCD corrections for production rates and kinematics are urgently needed.", "The same is true for other reactions of interest at the LHC, like top quark production and $W/Z$ production.", "The toolkit used to this end ranges from fixed order calculations at the parton-level over resummation to parton showers and particle-level event generators.", "Tremendous progress has been made in the field during the past year.", "Some of the recent developments will be briefly summarized in this talk." ], [ "Higher-order calculations", "Fixed-order calculations are available for a large variety of processes.", "At the tree level, they have long been performed completely automatically using programs like ALPGEN [1], Amegic++ [2], Comix [3], CompHEP [4], HELAC [5], MadGraph [6] and Whizard [7].", "At the next-to-leading order (NLO), automation required two main ingredients: The implementation of known generic methods to perform the subtraction of infrared singularities [8], [9], [10], and the automated computation of one-loop amplitudes.", "As infrared subtraction terms consist of tree-level matrix elements joined by splitting operators, existing programs for leading order calculations are ideally suited to compute them.", "Correspondingly, Catani-Seymour dipole subtraction has been implemented in the existing generators Amegic++ [11], Comix, HELAC [12] and MadGraph [13], [14].", "FKS subtraction is realized in MadGraph only [15].", "Figure: Jet multiplicity distribution in pure jet events (right).Figure taken from .The automated computation of virtual corrections has received a boost from generalized unitarity [18], [19], [20], which can be used to determine one-loop amplitudes by decomposing them into known scalar one-loop integrals and rational coefficients determined from tree amplitudes, plus a rational piece [21], [22], [23], [24], [25].", "Programs like BlackHat [26], Gosam [27], HELACNLO [28], MadLoop [29], NJet [30], OpenLoops [31] and Rocket [32], [33] implement these techniques and supplement established programs like MCFM [34], [35] and dedicated codes based on improved tensor reduction approaches [36], [37].", "New techniques have also been proposed to accelerate the numerical calculation of the integrand of one-loop amplitudes, independent of the reduction scheme [31].", "Figures REF and REF show examples from recent NLO calculations for $W$ +5 jet production [16] and 5 jets production [17], both performed using unitarity based techniques.", "Other recently completed calculations include Higgs boson plus 3 jet production [38] and di-photon plus 2 jet production [39].", "The rapid progress in this field is reflected by the fact that all calculations from the experimenter's wishlist for the LHC have now been tackled [40].", "Most of the programs used to perform the calculations, or their results, are publicly available.", "Driven by the need for higher precision in some selected Standard-Model reactions, the field of next-to-next-to leading order (NNLO) calculations has significantly advanced in the past years.", "One of the most challenging problems is the regularization of infrared divergences at NNLO.", "Sector decomposition [41], [42], [43] has been used in the past to perform several $2\\rightarrow 1$ calculations [44], [45].", "Antenna subtraction [46], [47] was worked out and implemented for $e^+e^-\\rightarrow 3$ jets [48], [49].", "$q_T$ subtraction [50] was employed in several calculations, including Higgs production [51], $W/Z$ production [52], associated Higgs production [53] and di-photon production [54].", "More recently sector-improved subtraction methods were introduced [55], [56].", "They have been used to compute cross sections for $pp\\rightarrow t\\bar{t}$ [57], [58] and $pp\\rightarrow H+$ jet [59].", "At the same time, antenna subtraction was extended to initial states [60], [61], [62], [63] and employed to compute $pp\\rightarrow $ di-jets fully differentially at NNLO [64].", "Figures REF and REF show results from some of these calculations.", "The calculation of $pp\\rightarrow t\\bar{t}$ has also been combined with higher logarithmic resummation [65], [66], [67].", "Its theoretical uncertainty is such that uncertainties from scale choices, PDF, strong coupling measurements and top-quark mass measurements are all of the same order [68].", "Figure: Scale dependence of the pp→pp\\rightarrow di-jet cross section.Figure taken from ." ], [ "Resummation of jet vetoes", "The analysis of the Higgs-like particle discovered at the LHC places new demands on resummed calculations.", "Many of the Higgs analysis channels, most notably $H \\rightarrow WW^* \\rightarrow \\ell ^+ \\ell ^- \\nu \\bar{\\nu }$ , veto on the transverse momentum of final state jets to distinguish different Standard Model backgrounds and separate them from the signal.", "The leading systematic uncertainty is the theoretical uncertainty on the signal cross section in the jet bins.", "This uncertainty can be reduced by a proper resummation of the logarithms associated with the jet veto.", "Various groups have investigated this problem, in most cases up to next-to-next-to leading logarithmic accuracy matched to NNLO fixed order, relying either on more traditional resummation methods [69], [70], or on Soft Collinear Effective Theory [71], [72], [73], [74], [75].", "Higgs plus one jet production was studied at next-to-leading logarithmic order (NLL) and matched to NLO fixed order using SCET [76]." ], [ "Parton showers and matching to NLO calculations", "The interest in parton showers as a means to produce particle-level predictions fully differentially in the phase space of multi-jet events has increased significantly in recent years.", "New concepts for the construction of parton showers have been proposed, which are based on antenna subtraction [77], [78] and/or sectorizing the phase space [79], [80].", "Efforts were made to include subleading color corrections into showers as a means to improve their logarithmic accuracy [81], [82].", "However, the crucial development was the proposal of a method to match parton showers to NLO calculations [83], later extended to eliminate negative weights [84], [85].", "This matching has been partially or fully automated in several projects [86], [87], [88], [89], [90], such that particle-level predictions at NLO accuracy are now widely available.", "Figure: Jet multiplicity distribution in W+W+jets events.Figure taken from .The description of multi-jet final states with parton showers can be improved using so-called ME+PS merging methods [93], [94], [95], [96], [97], which, in contrast to matching methods, allow to correct the parton shower for an arbitrary number of emissions with higher-order tree-level calculations.", "These methods were recently refined and extended, leading to algorithms which can combine multiple NLO calculations of varying multiplicity (like $W+0$ jet, $W+1$ jet, $W+2$ jet, etc.)", "into a single, inclusive simulation (e.g.", "of $W+$ jets production) [98], [99], [92], [91], [100].", "Figures REF and REF show examples for the application of ME+PS merging to Higgs boson plus jets production and to $W+$ jets production.", "A particular scale choice is required for the evaluation of the strong coupling in ME+PS merging, which has also been adopted for the matching to higher-multiplicity NLO calculations on its own in the so-called MINLO approach [101].", "The MINLO method accounts for Sudakov suppression effects in higher-multiplicity final states and allows to extrapolate NLO calculations to zero jet transverse momentum, thus offering the opportunity to match to NNLO calculations for a limited class of processes and observables [102].", "A different proposal for a matching to NNLO parton-level calculations was made in [98], [91], which is based on a subtraction method similar to the one used in ME+PS merging at NLO.", "Both techniques are promising candidates to further increase the precision of event generators for collider physics." ], [ "Summary", "We have presented some of the recent developments in perturbative QCD and applications to particle physics phenomenology.", "NLO parton-level calculations can nowadays often be provided by fully automated tools.", "New techniques in event generation allow to also use them for particle-level predictions.", "NNLO calculations and higher-logarithmic resummation techniques are at the forefront of current research." ] ]	1403.0037
[ [ "Method of Fission Product Beta Spectra Measurements for Predicting\n Reactor Anti-neutrino Emission" ], [ "Abstract The nuclear fission process that occurs in the core of nuclear reactors results in unstable, neutron rich fission products that subsequently beta decay and emit electron anti-neutrinos.", "These reactor neutrinos have served neutrino physics research from the initial discovery of the neutrino to current precision measurements of neutrino mixing angles.", "The prediction of the absolute flux and energy spectrum of the emitted reactor neutrinos hinges upon a series of seminal papers based on measurements performed in the 1970s and 1980s.", "The steadily improving reactor neutrino measurement techniques and recent re-considerations of the agreement between the predicted and observed reactor neutrino flux motivates revisiting the underlying beta spectra measurements.", "A method is proposed to use an accelerator proton beam delivered to an engineered target to yield a neutron field tailored to reproduce the neutron energy spectrum present in the core of an operating nuclear reactor.", "Foils of the primary reactor fissionable isotopes placed in this tailored neutron flux will ultimately emit beta particles from the resultant fission products.", "Measurement of these beta particles in a time projection chamber with a perpendicular magnetic field provides a distinctive set of systematic considerations for comparison to the original seminal beta spectra measurements.", "Ancillary measurements such as gamma-ray emission and post-irradiation radiochemical analysis will further constrain the absolute normalization of beta emissions per fission.", "The requirements for unfolding the beta spectra measured with this method into a predicted reactor neutrino spectrum are explored." ], [ "Introduction", "Neutrino experiments at nuclear reactors have played a vital role in the study of neutrino properties and flavor oscillation phenomenon.", "The observed antineutrino rates at reactors are typically lower than model expectations [1], [2] .", "This observed deficit is called the “reactor neutrino anomaly”.", "Proposals exist for explaining this anomaly via non-standard neutrino physics models (sterile neutrinos, for example), and a new understanding of neutrino physics may again be required to account for this deficit.", "However, model estimation uncertainties may also play a role in the apparent discrepancy.", "An experimental technique is proposed to make precision measurements of the beta energy spectrum from neutron induced fission using a 30 MeV proton linear accelerator For example the Project X Injector Experiment at Fermilab -see http://www-bdnew.fnal.gov/pxie/ as a neutron generator [4].", "Each fission event produces fission products that decay and emit electrons (beta particles) and anti-neutrinos, and precise measurement of the beta energy spectrum is used to infer an associated anti-neutrino spectrum.", "The proposed new approach utilizes the flexibility of an accelerator-based neutron source with neutron spectral tailoring coupled with a careful design of an isotopic fission target and beta spectrometer.", "The inversion of the beta spectrum to the neutrino spectrum is intended to allow further reduction in the uncertainties associated with prediction of the reactor neutrino spectrum.", "Through the fission process, four isotopes, $^{235}$ U, $^{239}$ Pu, $^{241}$ Pu, and $^{238}$ U contribute more than 99% of all reactor neutrinos with energies above the inverse beta decay threshold (neutrino energy $\\ge $ 1.8 MeV).", "The resulting predicted reactor neutrino flux is an accumulation of thousands of beta decay branches of the fission fragments.", "Reactor neutrino fluxes from the thermal fission of $^{235}$ U, $^{239}$ Pu, and $^{241}$ Pu are currently obtained by inverting measured total beta spectra obtained in the 1980s at a beam port at the High Flux Reactor of the Institut Laue-Langevin (ILL) [1].", "Recent reevaluations of the 1980s data with a careful investigation and treatment of the various sources of correlated and uncorrelated uncertainties indicated an upward shift of about 3%, with uncertainties ranging from 2% to 29% across the neutrino spectrum [1].", "Clearly any limitations of the original ILL beta spectrum measurements in terms of energy resolution, absolute normalization, and statistical counting uncertainties will propagate into the predicted reactor antineutrino spectra.", "For a single beta decay branch, the neutrino energy spectrum is directly related to the beta energy spectrum by conservation of energy.", "However, there are hundreds of fission products and thousands of beta decay branches making measuring each branch individually practically impossible (especially for ultra-short half-life isotopes).", "Thus measuring the cumulative beta spectrum remains the most viable technique for producing a representative spectrum used as a basis for inversion.", "The beta spectrum from the fission target can be deconstructed into a set of individual beta decays modeled either as 'virtual branches' [1] or matched to expectations based on the information in nuclear decay databases.", "Likewise a parallel measurement of the gamma-ray emission from the irradiated fission foil (in situ and post-irradiation), provides a means to check the normalization of the beta emission per fission.", "These aspects of the proposed measurement seek to improve the confidence of the underlying reactor neutrino spectrum predictions." ], [ "Neutron production and spectra", "The neutron spectrum in a nuclear reactor core is composed of three different energy ranges.", "Neutrons from fission are emitted with an average energy of about 2 MeV and a most probable neutron energy of 0.73 MeV.", "Figure REF shows a representative neutron spectrum for a fuel pin in a pressurized water reactor (PWR).", "In such a reactor, the fast portion of the neutron spectrum, with energies greater than 0.1 MeV, has a shape similar to the primary fission neutrons.", "In an operating reactor, fine structures in the neutron spectrum are introduced by absorption resonances on the fuel, moderator, and structural materials.", "In the intermediate epi-thermal neutron energy range, from 0.1 MeV down to about 1 eV, the neutrons are slowing down with a characteristic 1/E dependence.", "This is due to elastic scatters in the moderator removing a constant fraction of the neutron energy per collision (on average).", "The thermal portion of the spectrum, below $\\sim $ 1 eV, is characterized by a thermal Maxwellian flux shape, where the neutrons are in thermal equilibrium with the moderator.", "The peak energy of the thermal flux depends upon the temperature of the moderator material.", "Higher temperatures will shift the peak to higher energies.", "At room temperature, the peak thermal flux is at 0.0265 eV, while for the PWR conditions in figure REF the coolant is around 320 $^{\\circ }\\mathrm {C}$ .", "The magnitude and shape of the thermal spectrum depends on the relative volume fractions of moderator and fuel, and on the presence of burnable poisons or neutron absorbers inside the fuel or mixed in the moderator.", "Figure: The neutron energy spectra from a PWR reactor, D 2 _{2}O thermalized neutrons (as at ILL), and the tailored spectrum from the 30 MeV proton source.The relative magnitudes of these three regions of the neutron spectrum depend a great deal on specific reactor conditions.", "Neutron spectra at the beginning and end of an operating cycle will differ because of changing fuel isotopics from burnup, buildup of fission products, burnout of burnable poison in the fuel, and (in the case of PWRs) deliberate changes in the boron concentration in the coolant through the cycle.", "The neutron spectra in various parts of the reactor core will vary because of increased leakage and/or reflection near the upper and lower surfaces and outer edges of the core compared to the interior of the core.", "For boiling water reactors (BWRs), the coolant/moderator water density varies axially from full density water near the bottom of the core to full steam at the top of the core.", "The primary fissioning isotopes in a typical commercial power reactor are $^{235}$ U, $^{239}$ Pu, $^{241}$ Pu, and $^{238}$ U.", "Figure REF shows the fission cross sections for these four isotopes.", "The cross sections for $^{235}$ U, $^{239}$ Pu, and $^{241}$ Pu are fairly flat at high energies, have a series of sharp resonances in the intermediate energy range, and a 1/v shape at thermal energies.", "Both $^{239}$ Pu and $^{241}$ Pu have broad low energy resonances that reside at the transition between the thermal neutrons and the epi-thermal neutrons.", "Uranium-238 has a threshold for fission at approximately 1 MeV, and therefore does not fission at lower energies.", "Figure: The fission cross-sections of key fuel isotopes.Where the spectral variability impacts the neutrino anomaly is through the fission product yields.", "There is a known dependence in fission product yields with the energy of the neutron causing fission.", "This is illustrated in figure REF , which compares the fission product mass yields for thermal (0.025 eV) and 0.5 MeV neutrons for $^{235}$ U fission.", "Large differences can be seen for fission product masses in the central valley and for the lower and upper mass ranges.", "Figure: Cumulative neutron-induced fission yields by mass number for 235 ^{235}U fission for 0.025 eV incident neutron energies and for 0.5 MeV neutron energies (top panel).", "Bottom panel shows the ratios of the cumulative yields for the two energies.", "These plots illustrate the differences in fission yields for different incident neutron energies.A major advantage of an accelerator neutron source over a neutron beam from a thermal reactor is that the fast neutrons can be slowed down or tailored to approximate various power reactor spectra.", "This provides an advantage for control in studying how changes in the neutron spectra (i.e.", "in the reactor core) affects the resulting fission product beta spectrum.", "Furthermore, the $^{238}$ U neutrino spectrum can be studied directly because of the enhanced 1 MeV fast neutron flux available at the accelerator source.", "Since $^{238}$ U contributes on the order of 10% of the fissions in a power reactor, measurement of the beta spectrum (and hence neutrino spectrum) should contribute to reducing the overall uncertainty in the reactor neutrino spectrum.", "Previous beta spectra measurements were conducted in the 1980’s by irradiating fission foils in a D$_{2}$ O-moderated thermal flux tube external to the ILL reactor.", "The neutron flux in that arrangement would be expected to be similar to that shown in figure REF , which does not have any intermediate or fast neutron components.", "In order to reproduce a PWR reactor spectrum, the temperature of the target moderator should be close to the temperature of the reactor moderator.", "This can be a problem for water moderators, since high pressures are required to keep the water from boiling at PWR temperatures.", "One way around this is to use metal hydride moderators that can maintain the hydrogen content at elevated temperatures, thereby matching the spectral shape of the in-core neutron flux.", "The objective of spectral tailoring is to make the generated neutron spectrum look more like the reactor spectrum through the use of moderators and reflectors.", "The major effort of the target studies was to see if an arrangement of proton beam target, moderator, and reflector could adequately simulate a representative PWR reactor neutron spectrum.", "A series of parametric calculations with a very simple model were done to evaluate various target, moderator, and reflector materials and dimensions.", "These results were then incorporated into more realistic models to further evaluate promising configurations.", "The results are shown in figure REF , which compares the spectrally tailored accelerator neutron spectrum with the PWR spectrum.", "Good agreement can be seen between the two spectra at fast, intermediate, and thermal energy ranges.", "When the $^{235}$ U fission cross section is folded with the entire neutron spectrum, the effective one group cross section for the tailored accelerator spectrum was 38 barns, compared to 39 barns for the PWR spectrum." ], [ "Proton beam and target design", "The beam of protons is produced by the linear accelerator and is characterized by proton energy, proton beam current, and beam profile at the target.", "The beam current defines the number of protons per second striking the target.", "The proton energy determines the reactions that produce neutrons and other particles in the target, and the depth of penetration.", "The beam profile determines the areal energy deposition rate in the target.", "The target converts the proton beam to neutrons through various (p,n) type reactions.", "The number of neutrons produced per incident proton and the energy distribution of the neutrons depends on the material of the target.", "The target also has to dissipate the energy deposited in it by the proton beam.", "Therefore, material properties such as heat transfer coefficients are important.", "A 30 MeV proton beam will deposit on the order of 20 kW of heat in the beam target.", "A means for removing the heat deposited in the target must also be supplied.", "The thickness and shape of the target must be designed to accommodate the heat deposited within the target.", "Since the 30 MeV protons penetrate only about 0.25 cm in the target, the heat and radiation damage is primarily in this thin surface layer exposed to the beam.", "Altering the shape of the target to distribute the heating and damage over a larger surface is advisable.", "This was investigated by using cone-shaped or wedge-shaped targets, and this appeared sufficient to limit target temperature to $<$ 1000 $^{\\circ }\\mathrm {C}$ and maintain target integrity for several materials.", "The neutrons generated in the target will generally have a distribution of energies up to the incident proton energy.", "This spectrum of neutrons must be modified to mimic a reactor spectrum.", "This can be done by including an adjustable length of moderator material for the neutrons to pass through.", "In order to reproduce a reactor spectrum, the temperature of the target moderator should be close to the temperature of the reactor moderator.", "A few centimeters of metal hydride moderator material at the temperature representative of PWR coolant conditions shows promise as the primary spectrum tailoring component.", "A neutron reflector is proposed to reduce neutron leakage from the system, and to scatter neutrons released from the proton target back to the moderator and fission foil regions.", "This reflector can also serve as a shield for the neutron, gamma, and other radiation generated in the proton target and fission foil.", "Lead was found to be a good reflector.", "Figure REF shows a diagram of the proposed design.", "Figure: Schematic diagram of the proposed experimental setup.", "The different labelled components are as follows: A) lead reflector 40 cm diameter ×\\times 40 cm in height, B) proton beam tube (0.5 cm in diameter), C) gamma window (see section ), D) beta tube (10cm in diameter), E) fission foil (see section ), F) moderator, G) proton target.", "The dimensions of some of the components may change in the final design." ], [ "Fission foil", "A fission foil will be placed in an area where the neutrons will have the desired spectrum.", "Fission foils of the primary fissioning isotopes will be used: $^{235}$ U, $^{239}$ Pu, $^{241}$ Pu, and $^{238}$ U.", "The neutron flux at the fission foil is predicted to be $>$ 10$^{11}$ n/cm$^{2}$ /s, with the tailored neutron spectrum.", "It is estimated that the beta rates from the foil will be $>$ 10$^{8}$ betas/s per mg of $^{235}$ U at an energy of $>$ 1 MeV.", "Aspects of foil design such as density and composition that impact the emitted beta spectrum were investigated.", "Two primary design constraints guide the fission foil design for this experiment.", "The encapsulating foils must first serve to retain the highly radioactive fission products and secondly the foil must not significantly alter the outgoing beta spectrum to a degree that measurement quality is degraded.", "The retention of fission products and the preservation of the energy spectrum of emitted beta particles are opposing requirements.", "Previous experiments like those conducted at ILL typically used nickel foils to encapsulate the fissionable material.", "Those experiments used nickel with an areal density of 7mg/cm$^{2}$ , which translates to a thickness of approximately 7.85 $\\mu $ m. The amount of fissionable material has varied between the past experiments.", "The $^{241}$ Pu foil used in [5] was 0.13 mg/cm$^{2}$ of 83% enriched PuO$_{2}$ on a 2 $\\times $ 6 cm$^{2}$ area.", "For $^{235}$ U, 0.15 mg/cm$^{2}$ [6] and 1 mg/cm$^{2}$ [7] of 93% enriched UO$_{2}$ on a 3 $\\times $ 6 cm$^{2}$ area have been used.", "Primarily nickel foil and sputtered graphite have been investigated as candidates for materials to encapsulate the fissionable isotope(s).", "These two materials were chosen because they provide unique benefits that will allow prioritization of certain design criteria later in time.", "For example, the graphite can be made thinner, with an effective lower-Z and density, but costs more to fabricate.", "Preliminary Monte Carlo studies have been done of the beta spectra for various foil thicknesses.", "A thickness of 7.85 $\\mu $ m can be tolerated without significantly degrading the energy resolution of the entire system.", "For the preliminary design, similar foil dimensions and densities as those used in the ILL measurements will be used." ], [ "Constraints on the beta spectra using gamma measurements", "Many beta decays are immediately followed by gamma radiation as the nucleus relaxes from an excited state to a ground or metastable state.", "This gamma radiation serves as a source of information with which to constrain the fission yields.", "The feasibility of measuring the gammas produced by the decay of many of the fission daughters using a germanium detector situated some distance from the fission foil is considered.", "The fission yields will then be obtained from the gamma measurements through a maximum likelihood fit in both energy and time." ], [ "Time dependent isotope populations", "Measuring the gammas produced during the decay of the fission daughter products can constrain fission yields.", "However, data that is in coincidence with the fission must be avoided since it will be dominated by prompt fission gammas and it will not be possible to resolve the necessary gamma energies required to determine fission yields.", "This will be easier using an accelerator source compared to a continuous reactor source of neutrons since the fissions will occur only when the proton beam is on target.", "The detector will be observing a population of isotopes within the uranium target which vary with time.", "The population of an isotope will grow when it is produced via fission or fed by the decay of a parent isotope, and it will decrease when it decays.", "If only a single isotope being produced by fission is considered, its population $N$ will be governed by the following rate equation $\\frac{dN}{dt} = f Y - \\lambda N$ where $f$ is the fission rate, $Y$ is the fission yield of the isotope, and $\\lambda $ is the decay constant of the isotope.", "It is expected that that there will be periods when the proton beam will not be incident on target.", "The measurement time is divided into \"windows\" of piecewise constant reaction rate.", "It may be chosen to have the detector not record events that occur during a given time window (for example, to ignore events while the beam is incident on the target in order to reduce the noise from neutron capture, neutron inelastic reactions, and so on).", "If time window $m$ starts at a time $t_m$ , then at any time $t$ within the window there is the solution $N(t) = \\frac{f Y}{\\lambda } \\left( 1-e^{-\\lambda (t-t_m)} \\right) + N(t_m) e^{-\\lambda (t-t_m)}.$ It is known that the population of all fission products is zero at the beginning of the experiment, so that the population at the beginning of all windows can be built up by calculating $N(t_{m+1})$ once $N(t_m)$ is known.", "It will also be important to know the time integral of the population $D(t_1,t_2) = \\int _{t_1}^{t_2} dt \\, N(t).$ If $t_1$ and $t_2$ are both in the same time window, then $D(t_1,t_2) & = \\frac{f Y}{\\lambda ^2}\\left[\\lambda (t_2-t_1) - e^{-\\lambda (t_1-t_m)} + e^{-\\lambda (t_2-t_m)}\\right]\\\\ \\nonumber &\\phantom{=}+\\frac{N(t_m)}{\\lambda }\\left[e^{-\\lambda (t_1-t_m)} - e^{-\\lambda (t_2-t_m)}\\right].$ Also, the overall integrated population is defined as the sum of integrated populations over time windows in which the detector is recording.", "$D = \\sum _m^{\\rm rec} D(t_m,t_{m+1})$ In practice, there is a system of isotopes which decay into each other $\\frac{dN_i}{dt} = f Y_i - \\sum _j \\lambda _{ij} N_j$ where $\\lambda _{ii}$ is the decay constant for isotope $i$ and $-\\lambda _{ij}/\\lambda _{ii}$ is the branching ratio for isotope $j$ to decay to isotope $i$ .", "It is convenient to express collections of quantities related to each isotope, such as yields or populations, as column vectors in the space of isotopes which will be denoted with bold symbols.", "Matrices in this space will be denoted with an underline and the inner product by the dot-product symbol.", "In this notation, Eq.", "REF reads $\\frac{d}{dt} \\mathbf {N}= f \\mathbf {Y}- \\underline{\\lambda }\\cdot \\mathbf {N}.$ This can be solved by diagonalizing the rate matrix $\\underline{\\lambda }= \\underline{U}\\cdot \\underline{\\Lambda }\\cdot \\underline{U}^{-1}$ where $\\underline{\\Lambda }$ is the diagonal matrix of eigenvalues and $\\underline{U}$ is the matrix of right eigenvectors.", "If a vector $\\mathbf {M}$ is defined such that $\\mathbf {N}= \\underline{U}\\cdot \\mathbf {M},$ then $\\frac{d}{dt} \\mathbf {M}= f \\underline{U}\\cdot \\mathbf {Y}- \\underline{\\Lambda }\\cdot \\mathbf {N}.$ These are a set of uncoupled equations for the scalars $M_i$ , each of which can be solved using Eq.", "$\\ref {rateeq1}$ and whose integrals can be found using Eq.", "$\\ref {drateeq1}$ .", "This allows the solution of the coupled rate equations via Eq.", "REF .", "The vector $\\mathbf {Y}$ is the quantity of interest.", "It will be fit to the data by selecting a model that best represents the data." ], [ "Fitting to data", "The data will be fit using a maximum likelihood minimization.", "To speed up the minimization, a binned maximum likelihood fit is performed over discrete energy and time bins.", "The following likelihood likelihood function is maximized $\\mathcal {F} = -n^{\\prime } \\ln N + \\sum _b^{\\rm nbins} n^{\\prime }_b \\ln W_b$ where the weight of a bin $b$ , $W_{b}$ , is the sum of expected gamma and background contributions in that particular energy-time bin.", "The parameter $n^{\\prime }$ is the total number of dead time corrected events, and $n^{\\prime }_b$ is the number of dead time corrected events in energy-time bin $b$ .", "Describing the methods of maximum likelihood estimation goes beyond the scope of this paper, as does the art of function minimization or maximization.", "Interested parties are directed to the literature [8], [9].", "Note that while grouping the data may be less accurate, evaluation can also be much faster.", "It may be worthwhile to pre-estimate the parameters using a binned analysis, and then finalize a solution using a maximum likelihood estimate on the individual events." ], [ "Notable gamma-active isotopes", "The relative contribution of various isotopes to the gamma spectra will depend on the length of the irradiation and measurement periods, and to a lesser extent on the neutron spectrum and irradiated material.", "Those lines with the strongest signals will be the most constrained by the data, allowing fits with lower uncertainty.", "Taking and analyzing data in list mode will also provide constraints on the fission yields of parents of the gamma-active isotopes.", "Table REF lists many of the isotopes with strong gamma lines between 100 and 6000 keV which are expected to be observable.", "Table: Prominent gamma-active isotopes from fission over time scales between several seconds and a day, with emissions between 100 and 6000 keV.", "Isotopes are arranged by isobars, with daughter products to the right." ], [ "Detector placement", "In order to estimate fission yields from gamma spectra, it is necessary to collect sufficient data to provide a good fit with statistically meaningful results.", "The effectiveness of a given instrument design and experimental irradiation and measurement schedule can be estimated by comparison to a previous analysis of beta-delayed gamma rays to determine fission yield.", "The analysis method described above was used in [10] to extract fission yields from a set of data taken at Oregon State University's TRIGA reactor [11].", "This data set was taken from exposure of an $^{235}$ U foil to a thermal beamline for 30 seconds and then measured for 150 seconds, repeated 100 times.", "As this data was primarily intended to demonstrate the analytical method, analysis was limited to the region between 3200 keV and 3650 keV.", "The measurements recorded approximately $2\\times 10^{6}$ gamma ray events in this energy range, of which approximately 350,000 were in the full energy peaks.", "This provided sufficient data to determine relative fission yields of some of the most prominent isotopes in the spectrum ($^{142}$ Cs, $^{137}$ I, $^{95}$ Y, $^{95}$ Sr, $^{91}$ Kr, $^{90}$ Kr) within 10% to 20%.", "It is expected that choosing measurement times more nearly equal to the irradiation times and performing the analysis over a wider spectral region would result in higher precision on the reported yields.", "A GEANT4 [12], [13] radiation transport simulation was set up to estimate a detector placement that would allow adequate statistics by comparison to the above experiment.", "An 8 cm diameter, 8 cm length HPGe detector was placed 100 cm from the fission foil.", "A 20 cm thickness of borated polyethylene was placed around the reflector box to reduce neutron exposure to the detector.", "A narrow wedge-shaped viewing port was modeled into the neutron shield and reflector to allow the foil gamma rays to be observed by the detector.", "An additional 10 cm slab of borated polyethylene was placed between the reflector and the detector, and 1 cm thickness of borated polyethylene was placed directly in front of the detector to reduce the dose from neutrons streaming down the viewing port.", "In this geometry, the absolute peak efficiency in the 3200 to 3650 keV region was estimated to be $3.5\\times 10^{-5}$ .", "To acquire comparable statistics $1\\times 10^{10}$ gamma rays emitted from the uranium foil in the spectral region of interest during the collection times is required.", "The model presented in section REF allow analysis of the gamma rays per fission with a suitable data set for the fission yields and gamma intensities.", "Using ENDF [14] and ENDSF [15] data sets, an estimate of a mean of $3.4\\times 10^{-2}$ gamma rays in the 3200 to 3650 keV spectral region per fission event emitted during the measured time windows is obtained.", "Given the expected mass of the fission foil, it is expected that the gamma flux will be low enough not to overwhelm the data acquisition while still being able to acquire the necessary statistics.", "This analysis suggests that a single high relative efficiency HPGe detector viewing gamma rays from the fission foil can achieve sufficient statistics for meaningful analysis without being overwhelmed by the rate of gamma interactions or destroyed by the neutrons." ], [ "Beta spectrometer", "A measurement of the fission foil beta spectrum needs to be made with good efficiency and energy resolution.", "Efficiency needs to be good, and precisely known, since uncertainty in the energy dependent efficiency leads directly to a systematic uncertainty in predicted anti-neutrino flux.", "Previous authors used magnetic beta spectrometry for the original fission beta specta measurements [6], [16], [7], [5], [17].", "Those measurements were made with a double focusing spectrometer named BILL [18].", "Electrons from the fission foil source were transported through a 14 m long 10 cm diameter beam tube to the spectrometer.", "The spectrometer formed an image of the aperture slit on a pair of multi-wire proportional counters in the focal plane.", "BILL was and still is an exquisite instrument, with relative momentum resolution of a few parts in $10^{4}$ for large targets and a momentum precision of one part in $10^5$ .", "A preliminary design is explored where betas emitted from a fission foil activated as described in Section are transported along a beam pipe to a simple dipole spectrometer with active tracking of betas performed by a time projection chamber (TPC) [19], [20] inserted between the pole faces.", "There are several qualitative reasons for pursuing this design and analysis.", "The active gaseous medium in the TPC is aninefficient detector of background gammas and neutrons.", "Furthermore, rare gamma and neutron interaction events can be rejected based on parameters of the TPC tracks.", "It will be similarly easy to identify betas originating somewhere other than the source, betas from the source that have scattered in the beam pipe, and any other charged backgrounds.", "Finally, this measurement method will provide a different set of systematic uncertainties to the measurements made with the BILL, ideally providing an \"independent\" test.", "This section describes preliminary simulations of the performance.", "Details of the TPC are given in Section REF .", "Track reconstruction and overall performance are described in Section REF .", "While the energy resolution of the TPC beta spectrometer will be worse than that of BILL, in the next section it is demonstrated that the resolution of the TPC beta spectrometer is sufficient for extracting the anti-neutrino flux with an uncertainty of $<$ 1%." ], [ "Tracking time projection chamber", "Most time projection chambers are large-volume devices, but smaller detectors do exist for specialized applications.", "To simulate the response of a time projection chamber of appropriate size, the NIFFTE Fission TPC [21] was used as the baseline.", "The NIFFTE TPC is designed to make precision cross section measurements of major actinides with an uncertainty of better than 1%, and its design characteristics make it a good candidate for a beta spectrometer baseline design.", "The NIFFTE TPC design consists of a cylinder 15 cm in diameter and 5.4 cm length (see figure REF ).", "In the NIFFTE experiment two such volumes are used with a target placed between them to identify fission fragments exiting in both directions.", "For the beta spectrometer application only one side was simulated.", "Each side is read out with a MICROMEGAS [22] gain region and 2976 hexagonal readout pads of 2mm pitch.", "The small drift volume allows for fast readout ($\\approx $ 1${\\mu }s$ in P10 gas) and minimizes electron cloud diffusion.", "The FPGA-based digital electronics are read out via Ethernet fiber to a central data acquisition computer.", "Figure: Field cage and target cathode for the NIFFTE TPC.", "The active volume is 5.4 cm deep and 15 cm across.The NIFFTE experiment design differs from the expected beta spectrometer design in several significant ways: only one TPC volume will be used; in NIFFTE the neutron beam enters the TPC axially, while in the beta spectrometer it will enter from the side; the NIFFTE experiment has no magnet, unlike the beta spectrometer application; and the signal gain in the NIFFTE MICROMEGAS is modest (10-40) since it is detecting fission fragments, while for the minimally ionizing betas in the spectrometer it will need to be significantly greater.", "A triple GEM structure is a likely candidate for the actual gain stage [23] in the spectrometer application.", "The simulation code for the NIFFTE TPC consists of a GEANT4 [12] particle transport and energy loss component, and a detector simulation component.", "The detector simulation code handles effects such as electron cloud drift and diffusion, charge sharing between pads, preamplifier noise, and signal crosstalk.", "For the beta spectrometer performance simulations, the output of the electron transport simulation was used as input to the TPC detector simulation.", "Total charge collected on each pad was output for each event and passed on to the track identification code." ], [ "Simulation of the Spectrometer", "The GEANT4 based simulation includes the entire path of the beta from the moment it exits the fission foil.", "A significant degradation in energy resolution occurs due to scattering along the pipe from the fission foil to the spectrometer.", "The final energy resolution reported at the end of this section includes any scattering that occurs during travel down the pipe.", "The simulation concludes with the beta going through the spectrometer.", "A track fitting algorithm is applied to determine its energy." ], [ "Track identification and reconstruction", "Track fitting and reconstruction is a two-step process.", "The first step is to perform a circular Hough transformation on the recorded track.", "That transformation is computationally intensive so the three-dimensional parameter space is divided into large bins representing values of the radius $r$ and the $(x,y)$ -coordinates of the center $(a,b)$ .", "The results of this initial step are used as initial guesses for the parameters of a maximum likelihood fit allowing continuous values of the parameters.", "A typical simulated track is shown in figure REF .", "Figure: A simulated TPC track for a 10 MeV beta.", "The color of each hexagonal pixel represent the collected charge above threshold in arbitrary uncalibrated units.", "The red circular arc represents the initial track parameter guesses from a Hough transform.", "The black circular arc represents the results of a continuous-parameter fit.The Hough transform uses recorded pixel charges to cast “votes” in a parameter space representing possible circular arcs of the form $r^2 = (x-a)^2 + (y-b)^2.$ For each pixel $i$ centered at $(x_i, y_i)$ with a charge $q_i$ above threshold, a family of circles $r^2 = (a - x_i)^2 + (b - y_i)^2$ with all possible values of $a, b$ and $r$ is drawn in the Hough space.", "A vote with weight $q_i$ is cast in every voxel intersected by each circle.", "The voxel in the $(a, b, r)$ Hough space with the most votes corresponds to the most likely circular arc as in Eqn.", "(REF ).", "A quantity $\\chi ^2$ is computed as the value to be minimized in the continuous fit that refines the guess made by the Hough transform: $\\chi ^2 = \\frac{1}{Q^2} \\sum _{i=0}^{N-1} \\left( \\frac{q_i d_i}{r_{\\rm rms}} \\right)^2,$ where the sum is over the $N$ pixels labeled $i = 0...N-1$ with charge above threshold, $q_i$ is the charge on the $i$ th pixel, $d_i$ is the shortest perpendicular distance from the track to the center of the $i$ th pixel, $r_{\\rm rms}$ is the radius of the circle which contains 2/3 of the total area of a pixel, and $Q = \\sum q_i$ is the total charge on all pixels above threshold.", "The distance $d_i$ is just the length of the segment perpendicular to the track and passing through the pixel center $(x_i, y_i)$ : $d_i = \\left\| r - \\sqrt{(x_i - a)^2 + (y_i - b)^2} \\right\|.$ Though the same notation is used, one should not interpret Eqn.", "(REF ) as the statistical parameter typically minimized in such a fit.", "The relationship between the reconstructed track radius and the kinetic energy $E$ of a beta follows straightforwardly from relativistic kinematics: $E = \\sqrt{(eBrc)^2 + m^2 c^4} - mc^2,$ where $B$ is the magnitude of the uniform magnetic flux density in the tracking region, $m$ and $e$ are the electron mass and charge, respectively, and $c$ is the speed of light." ], [ "Determining the overall performance of the spectrometer", "Spectrometer performance is characterized by the simulation of monoenergetic beta particles emitted isotropically from the foil surface.", "Scattering in the beam pipe is included but the effects of energy loss in the foil are not.", "Figure REF shows the response to 5, 8, and 10 MeV betas for a magnetic flux density of $B = 5000 G$ .", "Figure: The spectra of 5, 8, and 10 MeV betas as measured by the tracking algorithm after transport through the beam pipe.The spectra are all normalized to unit area.", "Only events with $\\chi ^2 < 0.03$ are included.", "Larger values of $\\chi ^2$ are indicative of multiply scattered events where the fitting algorithm fails since it is unable to distinguish more than one arc in an event.", "These spectra are fit to a function of the form $f(E) = A \\left( e^{\\frac{(E-E_0)^2}{\\sigma ^2}} + \\frac{m e^{\\frac{E - E_0}{c}}}{1 + e^{\\frac{E-E_0}{\\sigma }}} \\right).$ The first term in Eqn.", "(REF ) is Gaussian with mean $E_0$ and standard deviation $\\sigma $ .", "The second term is a low side tail attributed to scattering in the beam pipe characterized by the additional parameters $m$ and $c$ .", "The spectra in figure REF and similar spectra for lower energy betas are fit with the form of Eqn.", "(REF ).", "Figure REF shows the resolution $\\sigma $ versus the energy for each of three magnetic flux densities.", "Only fits with $p(\\chi ^2, {\\rm NDF}) > 0.05$ are included.", "Figure: The fitted resolution, σ\\sigma , versus true beta energy.", "The error bars on the points are taken from the uncertainties on the fits." ], [ "Implications for an Anti-neutrino Analysis", "The data from the beta spectrometer will ultimately be used to determine the corresponding anti-neutrino spectrum from the fission foil.", "This will be done through a maximum likelihood signal extraction on the beta spectrum to determine the yields of the various fission product beta branches.", "From the extracted yields the anti-neutrino spectrum is determined.", "To quantify how the beta resolution affects the anti-neutrino spectrum a maximum likelihood signal extraction is performed on Monte Carlo of the $^{235}$ U beta spectrum.", "The Monte Carlo includes the corresponding anti-neutrino spectrum, which is used to compare to the extracted anti-neutrino spectrum.", "Various energy resolutions were tested in the Monte Carlo signal extraction.", "As a worst case scenario, a 10% energy resolution is assumed, which is supported as a basis of estimation by figure REF .", "With this worst case assumption figure REF shows a comparison of the extracted anti-neutrino spectrum compared to the Monte Carlo's “true” anti-neutrino spectrum assuming a 10% energy resolution.", "With a 10% energy resolution the uncertainty on the integrated anti-neutrino flux is expected to be $<$ 1%.", "As shown in the previous sections, the resolution of the beta measurements, including the scattering in the pipe, will be less than 10%.", "Figure: Top panel shows a comparison of the extracted anti-neutrino spectrum and the “true” anti-neutrino spectrum given by the Monte Carlo for a 10% energy resolution.", "Bottom panel shows a histogram of the differences between the integrated true Monte Carlo and extracted anti-neutrino spectra for 50 signal extraction trials.The maximum likelihood signal extraction will also include measurements of the fission yields obtained from the proposed gamma analysis described in section .", "Also, if possible, radiochemical assays of the fission foil post irradiation will be used to obtain another set of measurements of the fission yields.", "These independent measurements will be used as constraints in the signal extraction.", "This will serve both to decrease the uncertainties on the extracted anti-neutrino spectrum as well as descrease the time required for the signal extraction to converge." ], [ "Summary", "The persistence of the “reactor neutrino anomaly” warrants a new approach for measuring the beta spectra from fissionable material found in common nuclear reactors.", "This paper outlines a plan for using an accelerator neutron source coupled with a fission foil and a beta spectrometer to provide an independent measurement of the fission beta spectra.", "The neutrons are produced through proton reactions on an appropriate target.", "This approach is advantageous since the neutron spectrum can be tailored to be similar to the neutron spectra from different reactor types.", "By careful study of target and moderator material a PWR neutron spectrum can be reproduced.", "Simulations of a beta spectrometer, which relies on active tracking of betas in a TPC, show that the beta energy resolution of the system will allow measurements of the beta spectrum with the necessary precision to produce valuable constraints on the reactor anti-neutrino spectrum.", "Furthermore, independent measurements of the fission yields using germanium gamma spectroscopy and subsequent radiochemistry are planned.", "These measurements will be used as external constraints in the maximum likelihood analysis to obtain the anti-neutrino spectrum.", "Details of an anti-neutrino spectrum extraction applied to the experimental setup described in the previous sections will be outlined in an upcoming paper.", "The research described in this paper was conducted under the Laboratory Directed Research and Development Program at Pacific Northwest National Laboratory, a multiprogram national laboratory operated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830.", "Detector simulations in this publication were based, in part, on the NIFFTE Fission Time Projection Chamber, and the authors would like to thank the NIFFTE Collaboration for use of their code." ] ]	1403.0107
[ [ "Origin of band gaps in graphene on hexagonal boron nitride" ], [ "Abstract Recent progress in preparing well controlled 2D van der Waals heterojunctions has opened up a new frontier in materials physics.", "In this paper we address the intriguing energy gaps that are sometimes observed when a graphene sheet is placed on a hexagonal boron nitride substrate, demonstrating that they are produced by an interesting interplay between structural and electronic properties, including electronic many-body exchange interactions.", "Our theory is able to explain the observed gap behavior by accounting first for the structural relaxation of graphene's carbon atoms when placed on a boron nitride substrate and then for the influence of the substrate on low-energy $\\pi$-electrons located at relaxed carbon atom sites.", "The methods we employ can be applied to many other van der Waals heterojunctions." ], [ "Introduction", "Recent progress in preparing vertical heterojunctions of graphene (G) and hexagonal boron nitride (BN) using either transfer [1] or growth techniques [2] has opened a new frontier for exploring both fundamental physics [3], [5], [4] and new device geometries [6].", "Experiments have made it clear that graphene on BN is very flat and that its low-energy electronic states are often very weakly perturbed by the substrate [1].", "However when the honeycomb lattices of graphene and BN are close to orientational alignment, the electronic coupling strengthens and is readily observed [7].", "The source of this variability in behavior is clearly related to variability in structure.", "For example, although ab initio theory [8], [9] predicts substantial gaps $\\sim 50$ meV when the two honeycomb lattices are identical, any incommensurability due to misorientation or lattice constant mismatch drastically reduces electronic coupling giving vanishingly small gaps [10].", "In this article we show that the large gaps observed [3] at the Fermi level of neutral graphene sheets that are nearly rotationally aligned with a BN substrate are not due solely in terms of the relative orientation-dependent moiré pattern, but require in addition both orientation-dependent structural relaxation of the carbon atoms, as suggested by recent experiments [11], and non-local many-body exchange interactions between electrons.", "Our theory involves two elements: i) structural relaxation due to interactions between G and the BN substrate and ii) an effective Hamiltonian for graphene's $\\pi $ -electrons which includes a substrate interaction term that is dependent on the local coordination between graphene and BN honeycombs.", "Our main results are summarized in Fig.", "REF where we show that atomic relaxation leads to substantially enhanced gap.", "The band gap for rotationally aligned layers is only $\\sim 1$ meV when the honeycomb lattices are held rigid, but increases to $\\sim $ 7 meV when relaxation is allowed.", "These gaps are further enhanced to $\\sim 20$ meV, in reasonable agreement with experiment, when we also account for electron-electron interactions.", "Moreover, unlike other proposed mechanisms for band gaps in graphene [12], ours does not degrade the mobility of graphene.", "Figure: Relaxation strains and band gaps of graphene on BN.a.", "Relaxation strain elastic and potential energies for orientation aligned graphene on BN as a function of ϵ\\epsilon therelative lattice constant difference.", "The black lines illustrates the case in which only carbon atom positions are allowed torelax (black) whereas the red curve is for the case in which both G and BN layer atoms are allowed to relax.The parabolic curve labelled FK (Frenkel-Kontorova) plots the energy difference between anundistorted graphene sheet and one with a lattice that has expanded to be commensurate with that of the substratethat is discussed in the text.", "E elastic E_{\\rm elastic} and E potential E_{\\rm potential} are respectively the elastic energy cost and the potentialenergy gained by straining both graphene and BN (black) layers, and the graphene (red) only while keeping the moirélattice constant fixed.We use ϵ=-0.017\\epsilon = -0.017 for graphene on BN in the absence of graphene lattice expansion.b.", "Energy gaps including strain effects vs. ϵ\\epsilon when graphene and BN layers are allowed to relax (black)and when only graphene atoms are allowed to relax (red), when the layers are held rigid at 3.4 ÅÅ separation (blue).and when electron-electron interactions are also included (inset).The interaction enhanced gaps are bracketed by Hartree-Fock calculations that usedielectric constants of 2.5 and 4 to account for screening effects.The θ\\theta label indicates the one-to-one relation with l M l_Mwhen we fix ε=0.017\\left\| \\varepsilon \\right\| = 0.017 and provides an approximate representationof the twist angle dependence." ], [ " Moiré patterns and strains", "The $\\pi $ -electron Hamiltonian of G/BN can be expressed as the sum of the continuum model Dirac Hamiltonian of an isolated flat graphene sheet, in which the honeycomb sublattice degree-of-freedom appears as a pseudo spin, and a correction from the interaction with the BN substrate [7], [9], [10], [13].", "We employ an approach in which the correction is given by a sub-lattice dependent but spatially local operator $H_{M}(\\vec{d})$ derived from ab initio theory [9] that depends on the local alignment between G and BN honeycomb lattices $\\vec{d}$ .", "This pseudo-spin dependent operator that gives rise to the moiré superlattice Hamiltonian is accurately parameterized in Ref. [moirebandtheory].", "(An alternate parameterization which allows spatial variation in the interlayer separation is discussed in the appendix.)", "When both G and BN form rigid honeycomb lattices $\\vec{d}(\\vec{r}) \\rightarrow \\vec{d}_{0}(\\vec{r}) \\equiv \\epsilon \\vec{r} + \\theta \\hat{z} \\times \\vec{r},$ where $\\epsilon $ is the difference between their lattice constants, $\\theta $ is the difference in their orientations, and $\\hat{z}$ is the direction normal to the G sheet.", "The two layers establish a moiré pattern in which equivalent alignments repeat periodically on a length scale that, when $\\epsilon $ and $\\theta $ are small, is long compared to the honeycomb lattice constant.", "(The moiré lattice vectors $\\vec{L}_M$ solve $\\vec{d}(\\vec{r}+ \\vec{L}_M) = \\vec{d}(\\vec{r}) +\\vec{L}$ where $\\vec{L}$ is a honeycomb lattice vector.)", "Since $H_{M}(\\vec{d})=H_{M}(\\vec{d}+\\vec{L})$ , the substrate interaction Hamiltonian has the periodicity of the moiré pattern.", "When the honeycomb lattices of the G and BN layers are allowed to relax, $\\vec{d}(\\vec{r})$ is no longer a simple linear function of position.", "We write $\\vec{d}(\\vec{r}) = \\vec{d}_0(\\vec{r}) + \\vec{u}(\\vec{r}) + (h_0+h(\\vec{r}) ) \\, \\hat{z}$ where $h_0$ is the mean separation between G and BN planes and the in-plane and vertical strains, $\\vec{u}(\\vec{r})$ and $h(\\vec{r})$ , also have the moiré pattern periodicity.", "If G/BN systems achieved thermal equilibrium $\\epsilon $ , $\\theta $ , $\\vec{u}(\\vec{r})$ and $h(\\vec{r})$ would be determined by minimizing free energy with respect to the positions of atoms in the G layer and in the BN layers close to the surface of the substrate.", "Evidently this is not the case since the observed value of $\\theta $ varies in an irreproducible fashion.", "In the following we take the view that because the thermodynamic bias favoring a particular value of $\\theta $ is weak, its observed value is fixed by transfer kinetics.", "Similarly the value of $\\epsilon $ , which can be adjusted only by atomic rearrangements on long length scales, is also likely determined by kinetics and not by equilibrium considerations.", "On the other hand, given values for $\\epsilon $ and $\\theta $ minimizing energy with respect to local strains $\\vec{u}(\\vec{r})$ and $h(\\vec{r})$ require only local atomic arrangements.", "We therefore view $\\epsilon $ and $\\theta $ as experimentally measurable system parameters.", "In practice $\\epsilon $ is close to the undisturbed relative lattice constant difference whereas $\\theta $ varies widely.", "The ratio of the honeycomb lattice constant to the moiré pattern lattice constant $l_M$ is $a/l_M = (\\varepsilon ^2 + \\theta ^2)^{1/2}$ .", "For given values of $\\theta $ and $\\epsilon $ , $H_{M}(\\vec{d}(\\vec{r}))$ is dependent on strains because of their contribution to Eq. .", "The strains must therefore be calculated first in order to fix the $\\theta ,\\epsilon $ -dependent $\\pi $ -band Hamiltonian of G/BN.", "As a side remark, we note that $\\vec{d}(\\vec{r}) = (a / ł_M) \\vec{r}$ is a convenient approximation for the coordination vector that can account for the twist angle dependence through the magnitude of $l_M$ but ignores the variations in the shape of the moire pattern.", "Like the $\\pi $ -electron Hamiltonian, the graphene sheet energy can be written as the sum of an isolated G layer contribution and a substrate interaction contribution that depends on the local band alignment $\\vec{d}(\\vec{r})$ .", "The substrate interaction $U(\\vec{d})$ is most attractive when half the carbon atoms are directly above boron atoms, and the centers of graphene's hexagonal plaquettes are directly above the nitrogen atoms (BA alignment).", "This alignment is energetically more stable than one in which half the carbon atoms sit on top of nitrogen (AB alignment), or one in which all carbon atoms sit on top of either boron and nitrogen atoms (AA stacking).", "By performing ab initio calculations for commensurate lattices we find that $U_{ BA} < U_{ AB} < U_{ AA}$ .", "The full dependence of $U$ on $\\vec{d}$ is plotted in Fig.", "REF .", "Figure: Relaxation strain and degree of commensuration as a function of moire pattern lattice constant.a.", "Substrate interaction energy U(d →)U(\\vec{d}) per unit cell area as a function of stacking coordination d →\\vec{d}.The arrows indicate the magnitudes and directions of substrate interaction forcesF →=-∇ → d → U\\vec{F} = -\\vec{\\nabla }_{\\vec{d}} U which drive atoms toward local BA coordination.The stacking arrangement cartoons use blue for boron, red for nitrogen, and black for carbon.b.", "Width of the distribution of carbon atom displacements (FWHM) as a function of moiré patternlattice constant at θ=0\\theta =0.The typical displacement varies from ∼5\\sim 5 to ∼8\\sim 8 nmwhen the moiré pattern lattice constant varies by a factor of four.c.", "Vertical strains for ε=0.017 \\left\| \\varepsilon \\right\| = 0.017 and d. for ε=0.0068\\left\| \\varepsilon \\right\| = 0.0068.Note that the vertical strain and the substrate interaction have similar spatial maps.The lower panels plot the height variation along the dashed lines of the upper panels for G&BN relaxed geometries.The maps for the elastic and substrate interaction energies are discussed in the appendix.When $\\epsilon $ or $\\theta $ are non-zero, the substrate interaction forces plotted in Fig.", "REF drive strains which attempt to match G and BN lattice constants locally and increase the sample area that is close to local BA coordination.", "For a given value of $\\epsilon $ , G sheet lattice constant expansion near BA points must be compensated by lattice compression elsewhere.", "This kind of local expansion and compression of the graphene lattice within the moiré unit cell was recently identified experimentally [11], [14].", "We determine the strains by minimizing the sum of the isolated graphene and substrate interaction energies.", "For the long-period moiré lattices the graphene sheet energy is accurately parameterized in terms of its elastic constants.", "The competition between isolated graphene and substrate interaction energies can then be understood by comparing the energies of the configurations in which the two terms are minimized separately.", "The substrate interaction energy is minimized by maintaining perfect BA alignment everywhere and therefore establishing commensurability between the BN and G lattices.", "Because the lattice constants of BN and G differ, this arrangement has an elastic energy cost in the graphene sheet.", "After an elementary calculation we find that the total energy per area is $e_{BA} = U_{BA}/A_0 + 2 (\\lambda + \\mu ) \\epsilon _{0}^2$ where $\\lambda $ and $\\mu $ are elastic constants, $\\epsilon _{0}$ is the relative difference between BN and G lattice constants, and $A_0$ is the unit cell area of graphene.", "The elastic energy, on the other hand, is minimized by keeping the graphene sheet lattice constant at its isolated value.", "In this configuration, because of the linear relationship between $\\vec{d}$ and $\\vec{r}$ the substrate interaction energy per unit area is equal to the average of $U(\\vec{d})$ over $\\vec{d}$ : $e_{iso} = \\overline{U}/A_0 > U_{ BA}/A_0.$ As indicated in Fig.", "REF , when our theoretical values for $U$ are combined with the elastic constants of a graphene sheet, the energy of the commensurate state is substantially lower.", "However, Eq.", "(REF ) overestimates the elastic energy cost of lattice matching between BN and G. For example in the extreme case of a single BN layer, lattice matching can be achieved by adjusting the lattice constants of each layer toward their mean value, approximately reducing the required strains by a factor of 2.", "In this case, the incommensurate structure still has lower energy, but the difference is smaller.", "We conclude that when they are orientationally aligned, the interaction between a G sheet and a BN sheet is nearly strong enough to favor lattice matching.", "G/BN is close enough to an incommensurate to commensurate transition that substantial strains can be driven by substrate interactions.", "Indeed we find by explicit energy minimization that both vertical and horizontal strains can assume values large enough to introduce changes in the electronic structure.", "We determined these strains numerically for the case of a single-layer BN substrate subject to a fixed periodic potential created by the layers underneath using methods explained in the appendix.", "We find that strains in the graphene sheet are comparable as those in the BN layer.", "Note that the atomic structure, and hence the $\\pi $ -band Hamiltonian, might therefore depend on the thickness of the BN and on other features that vary from one experimental study to another.", "Similarly the addition of encapsulating layers can lead to reductions in strains and hence gaps, as recently reported in Ref.", "[11], although the gap can in principle persist." ], [ "Strained moiré band Hamiltonian", "Given $H_{M}(\\vec{d})$ and $d(\\vec{r})$ , we obtain a sublattice-pseudospin dependent continuum Hamiltonian with the periodicity of the moiré pattern which is conveniently analyzed using a plane-wave expansion approach.", "We write the full Hamiltonian in the form $\\langle \\vec{k}^{\\prime },s^{\\prime }\| H \| \\vec{k},s \\rangle &=& \\delta _{\\vec{k},\\vec{k^{\\prime }}} \\langle s^{\\prime } \|H_{D}(\\vec{k}) \|s\\rangle + \\nonumber \\\\&+& \\sum _{\\vec{G}} \\langle s^{\\prime } \|H_{M, \\, \\vec{G}}\|s\\rangle \\; \\Delta \\left(\\vec{k^{\\prime }}-\\vec{k} -\\vec{G} \\right)$ where $H_{D}$ is the Dirac Hamiltonian and $H_{M, \\,\\vec{G}}$ is the Fourier transform over one period of the moiré pattern of $H_{M}(\\vec{d}(\\vec{r}))$ , and $\\vec{G}$ is a moiré pattern reciprocal lattice vector.", "In Eq.", "(), $\\Delta (\\vec{k})=1$ when $\\vec{k}$ is a moiré pattern reciprocal lattice vector and zero otherwise.", "The electronic structures implied by the Hamiltonian in Eq.", "(REF ) for rigid lattices, for graphene relaxation only, and for mutual G and BN relaxation are compared in Fig.", "(REF ).", "These results demonstrate that the electronic structure, and the gap at neutrality in particular, depend sensitively not only on $\\theta $ and $\\epsilon $ but also on the strains.", "Sizable band gaps appear at the neutral system Fermi level only when in-plane relaxation strains $\\vec{u}(\\vec{r})$ are allowed.", "Figure: Electronic structure of G/BN heterojunctions.a.", "Schematic representation of the moiré Brillouin zone and the moiré reciprocal lattice vectors.b.", "Three dimensional representation of the band structure in the moiré Brillouin zone showing superlattice Dirac point features.c.", "Local density of states (LDOS) maps near the charge neutrality Fermi energy for G&BN relaxed and rigid latticestructures at θ=0\\theta =0 that show contrasts for electrons and hole carrier doping.", "Lattice relaxation affects the LDOS maps.d.", "Band structure and density of states for three different values of ϵ\\epsilon at θ=0\\theta =0 allowing G&BN relaxation, G-relaxation only, and with no relaxation.In-plane lattice relaxation leads to sizeable band gaps in the limit of long moiré periods." ], [ "Physics of the Gaps", "Several potential mechanisms of gap formation in neutral graphene have been discussed in the literature including antidots [15], combinations of periodic scalar and vector fields [16], [17], and zero-line localization [18].", "Our approach allows for a simple classification based on the Fourier expansion of $H_{M}$ .", "We will discuss leading contributions to the gap at neutrality in terms of the expansion of each moiré pattern Fourier component of $H_{M}$ into four sublattice Pauli matrix components.", "We start with the $\\vec{G}_0=0$ Fourier component, i.e with the spatial average of $H_{M}$ .", "In the absence of relaxation, $H_{M, \\vec{G}_0=0}=0$ because the average of $H_{M}(\\vec{d})$ is zero [9] and $\\vec{d}$ in this case is a linear function of $\\vec{r}$ .", "(We neglect an irrelevant contribution proportional to the identity sublattice Pauli matrix $\\tau ^{0}$ .)", "When $\\vec{d}$ is a non-linear function of $\\vec{r}$ , however, the spatial average of Hamiltonian contributions which are sinusoidal functions of $\\vec{d}$ do not vanish.", "Among these, the term proportional to $\\tau ^{0}$ is an irrelevant constant, and the terms proportional to $\\tau ^{x}$ and $\\tau ^{y}$ , often interpreted in Dirac models as effective vector potentials, simply shift band crossings away from zero momentum.", "(In the continuum Dirac model of graphene, momentum is measured away from the Brillouin-zone corners.)", "However, the $\\vec{G}=0$ term proportional to $\\tau ^{z}$ produces a gap $\\Delta _0 = 2 H_{M,\\vec{G}_0=0}^{z} = H_{M,\\vec{G}_0=0}^{AA} - H_{M, \\vec{G}_0=0}^{BB}.$ Physically this gap appears simply because the average site energy is different on different honeycomb sublattices.", "The leading contributions to the gap from $\\vec{G} \\ne 0$ terms in $H_{M}$ are more subtle and appear at second order in perturbation theory.", "A perturbative treatment is in fact valid in practice because it turns out that $\\hbar \\upsilon \|\\vec{G}\|$ is substantially larger than $H_{M, \\vec{G}}$ .", "Applying degenerate state perturbation theory we obtain the following expression for their contribution to the effective $2 \\times 2$ sublattice Hamiltonian at the Dirac point: $H_{\\rm eff}=H_{M, \\vec{G}_0=0} -\\sum _{\\vec{G}\\ne 0} H_{M, \\vec{G}} \\, H_{\\vec{G}}^{-1}\\, H_{M, \\vec{G}}^{\\dagger },$ where $H_{\\vec{G}}=\\hbar \\upsilon \\tilde{G} \\cdot \\vec{\\tau }$ ($\\upsilon $ is the Dirac velocity and $\\vec{\\tau }$ is the vector of Pauli matrices,) ignoring the $\\vec{k}$ dependence close to the Dirac point which will be higher order.", "Note that $H_{M, \\vec{G}}$ connects the $\\vec{k}$ and $\\vec{k} + \\vec{G}$ blocks of the plane-wave expansion moiré band Hamiltonian.", "Because only the term proportional to $\\tau ^{z}$ can produce a gap at 2nd order, it is instructive to decompose $H_{\\rm eff}$ into Pauli matrix contributions.", "$H_{\\rm eff}= \\sum _{\\alpha =0,x,y,z} H_{\\rm eff}^{\\alpha } \\; \\tau ^{\\alpha }.$ Note that higher order terms proportional to $\\tau _{x}$ and $\\tau _{y}$ may in principle contribute to the gap, but we find them to be negligible.", "We have derived analytic expressions for $H_{\\rm eff}^{\\alpha }$ in terms of the Pauli matrix decomposition of $H_{M,\\vec{G}_i}$ which are discussed in detail in the appendix.", "We find that although the $\\vec{G}=0$ contribution to the gap is always larger, the $\\vec{G} \\ne 0$ contributions are not negligible.", "Both the difference in the spatial average of sub band energies and the detailed form of the full substrate interaction Hamiltonian play a role in determining the size of the gap at neutrality, and both are sensitive to the detailed structure of the lattice relaxation strains.", "In graphene non-local exchange interactions are expected to enhance gaps [21], [23], [22] at neutrality produced by sublattice-dependent potentials.", "We have performed plane-wave-expansion self-consistent Hartree-Fock calculations in which Coulomb interactions are added to the moiré band Hamiltonian we have discussed.", "The calculations were performed using effective dielectric constants bracketing the expected values between $\\varepsilon =2.5$ and $\\varepsilon _r = 4$ .", "When all effects are included we find band gaps $\\sim 20$ meV, as shown in the inset of Fig.", "REF .", "The values chosen for $\\varepsilon $ partly account for dielectric screening by the substrate and partly accounts for dynamic screening effects in the same spirit as in the screened exchange functionals used in density-functional theory.", "We have previously used a similar dielectric constant of $\\varepsilon _r = 4$ to successfully predict spontaneous band gaps $\\sim 50$ meV in ABC trilayer graphene [24].", "Further details of the Hartree-Fock theory in moire superlattice bands will be presented elsewhere [25]." ], [ "Discussion", "We have derived a $\\pi $ -band continuum model Hamiltonian intended to describe states near the Fermi level of G/BN and used it to address the energy gaps often observed in neutral graphene when it is nearly aligned with a BN substrate.", "In this theory the interaction of $\\pi $ -band electrons with the substrate is described by a local but sublattice dependent term $H_{M}$ that is dependent on the local relative displacement of the graphene sheet and substrate honeycomb lattices, $d(\\vec{r})$ .", "When neither the G sheet's carbon atoms nor the boron and nitrogen atoms in the substrate are allowed to relax, $d(\\vec{r})$ is a linear function of position because of the difference between the lattice constants $\\epsilon $ and because of difference in orientations specified by a relative angle $\\theta $ .", "The gap produced by substrate interactions in the absence of relaxation reaches its maximum at $\\theta =0$ , but is never larger than a few meV and too small to explain experimental observations.", "Only by allowing the carbon and substrate atoms to relax we can explain the much larger experimental gaps.", "The moiré pattern formed by graphene and a BN substrate is characterized in the first place by the lattice constant difference $\\epsilon $ and by the relative orientation angle $\\theta $ .", "These two quantities can be changed only by collective motion of many atoms.", "We take the view that because of large barriers and weak thermodynamic drivers these two macroscopic variables are not in practice relaxed to equilibrium values.", "We therefore view them as observables that characterize particular G/BN systems and calculate relaxation strains and $\\pi $ -band electronic structure as a function of $\\epsilon $ and $\\theta $ , and hence as a function of moiré pattern period.", "The explicit calculations reported on in this paper are for $\\theta =0$ , the orientation which leads to large experimental gaps.", "To account for relaxation strains, we minimize the total energy with respect to carbon and substrate atom positions.", "For this purpose we assume that the interaction energy $U$ between graphene and substrate is also a local function of $\\vec{d}$ and obtain $U(\\vec{d})$ from density functional calculations of commensurate structures.", "The strains minimize the total energy by increasing the number of carbon atoms that are on top of boron atoms and the number of hexagonal carbon atom plaquettes that are centered above nitrogen atoms.", "Our study emphasizes that atom relaxation in the BN sheets is as important for the electronic structure as atom relaxation in the graphene sheet.", "Although only atom positions in the top BN sheet are important for electronic structure, these will be affected by interactions with atoms in remote layers.", "We have performed calculations for two extreme cases, rigid BN atoms and a single-layer of BN in which atom positions relax to minimize total energy, finding that relaxation increases the energy gap substantially.", "The physical origin of these gaps can be revealed by expanding the continuum model $\\pi $ -band Hamiltonian in terms of Pauli-matrix pseudospin operators and in terms of moiré pattern reciprocal lattice vector components.", "Because of the wide $\\pi $ -band width and the relatively short moiré periods, the contribution of each term in the Hamiltonian to the gap can be analyzed using leading order perturbation theory.", "The $\\vec{G} \\ne 0$ terms which capture detailed spatial patterns contribute at second order and are not negligible.", "The largest contribution to the gap comes from the $\\vec{G}=0$ term, which vanishes in the absence of lattice relaxation has a very simple interpretation.", "Because of relaxation strains the average site energy in the carbon sheet is different for the two carbon atom sub lattices.", "It is well known that this type of perturbation produces a gap at the Fermi level of a neutral graphene sheet.", "Surprisingly the gap is a substantial fraction of the gap of the same origin present in the commensurate BA aligned graphene on BN.", "The gaps are therefore due to the contrast between the local classical physics of energy minimization with respect to atom position, and the wide $\\pi $ -bands and non-local quantum physics which forces the quantum wave functions to be smooth and sensitive mainly to spatial averages over the moiré period.", "When many-body interaction effects [21] are accounted for, these gaps are enhanced to values that are consistent with experiment.", "The approach described in this paper can be applied to other van der Waals materials which can form heterojunctions in which different layers have slightly different lattice constants or differ in orientation - such as transition metal dichalcogenide stacks." ], [ "Methods", "The elastic energy functional was modeled using the Born-von Karman plate theory [27].", "Neglecting the small bending rigidity of graphene $\\kappa = 1.6$ eV [28] the elasticity theory depends on the two Lamé parameters whose estimates for graphene from empirical potentials gives $\\lambda = 3.25$ eV/$Å^2$ and $\\mu = 9.57$ eV/$Å$ [29] in the low temperature limit, and for a single BN sheet we have used $\\lambda \\sim 3.5 \\,\\, {\\rm eV} \\, Å^{-2}$ and $\\mu \\sim 7.8 \\,\\, {\\rm eV} \\, Å^{-2}$ [26] obtained averaging the LDA and GGA values.", "The potential energy has been parametrized from the stacking-dependent and separation dependent energy curves in Ref.", "[sachs] calculated at the EXX+RPA level.", "The scalar functions used to obtain the moiré superlattice pattern for the height and the displacement vectors from their gradients have used $\\Phi $ written as a Fourier expansion in $\\vec{G}$ vectors as $\\Phi ( \\vec{d} ) &=& \\sum _{\\vec{G}} C_{\\vec{G}} \\exp \\left( - i \\vec{G} \\cdot \\vec{d} \\right) \\\\&\\simeq & C_0 + C^{\\prime }_1 g(\\vec{d}) + f_1(\\vec{d},C_1,\\varphi _1) + f_2(\\vec{d},C_2,\\varphi _2) \\nonumber $ where $C_{\\vec{G}}$ is in general a complex number and we retain up to three nearest $\\vec{G}$ vectors for the scalar field that preserves the symmetry of triangular lattices.", "The parameters $C_0$ , $C^{\\prime }_1$ , $C_1$ , $\\varphi _1$ , $C_2$ and $\\varphi _2$ are real valued constants and we defined auxiliary functions $f$ and $g$ in terms of the triangular lattice structure factors similar as those used in a general tight-binding model of graphene [30].", "The Fourier expansion coefficients within the first shell consisting of $C^{\\prime }_0$ and the first shell $ f_1(\\vec{d},C_1,\\varphi _1)$ are often good representation of the solutions.", "Further details on the calculation method and results of the elastostatic problem can be found in the appendix.", "The self-consistency Hartree-Fock calculations were calculated using an effective relative dielectric constant of $\\varepsilon _r = 4$ using 217 $k$ -points in the moiré Brillouin zone." ], [ "Acknowledgments", "The work in Singapore was supported by National Research Foundation of Singapore under its Fellowship program (NRF-NRFF2012-01).", "The work in Austin was supported by the Department of Energy, Office of Basic Energy Sciences under contract DE-FG02-ER45118, and by the Welch Foundation grant TBF1473.", "We acknowledge use of computational resources from the Texas Advanced Computing Center and the high performance computing center at the Graphene Research Centre at the National University of Singapore.", "We acknowledge helpful discussions with Andre Geim and Gareth Jones and the assistance from Miguel Dias Costa for troubleshooting the parallel implementation of the Hartree-Fock calculations." ], [ "Author contributions", "JJ and AMD executed research.", "All authors contributed in conceiving, discussing and preparing the manuscript." ], [ "Conflicts in financial interest", "The authors declare no conflicting financial interests.", "Appendix.", "In the following we supplement the information in the main text introducing: the parametrization of the Hamiltonian in real-space, the explicit form of the scalar fields with the moiré superlattice symmetry, the elastic energy functionals, the parametrization of the potential energy, the formulation and solutions to the elastostatic problem for the relaxed ground states, and the contributions of the different pseudospin terms in the Hamiltonian to the primary Dirac point band gap, both numerically and analytically through second-order perturbation theory." ], [ "Parametrization of the Hamiltonian", "The diagonal and off-diagonal elements of the Hamiltonian for a fixed interlayer separation distance can be written in the sublattice basis in a manner similar to the pseudospin representation used in Ref.", "[suppmoirebandtheory], $H_{ii}(\\vec{K} :\\vec{d}) &=& 2 C_{ii} {\\rm Re}[ f(\\vec{d}) \\exp [i \\varphi _{0}] ] \\\\H_{AB}(\\vec{K} :\\vec{d}) &=& 2 C_{AB} \\, \\cos ( \\frac{\\sqrt{3} }{2} G_1 d_x)\\left(\\cos \\left( \\frac{G_1 d_y}{2} - \\varphi _{AB} \\right)+\\sin \\left( \\frac{G_1 d_y}{2} - \\varphi _{AB} - \\frac{\\pi }{6} \\right)\\right)+ 2 C_{AB} \\, \\sin \\left( G_1 d_y + \\varphi _{AB} - \\frac{\\pi }{6} \\right) \\nonumber \\\\&+& i \\,2 C_{AB} \\, \\sin ( \\frac{\\sqrt{3} }{2} G_1 d_x )\\left(\\cos \\left( \\frac{G_1 d_y}{2} - \\varphi _{AB} \\right)- \\sin \\left( \\frac{G_1 d_y}{2} - \\varphi _{AB} - \\frac{\\pi }{6} \\right)\\right).\\nonumber $ The out-of-plane $z$ -axis layer separation dependence can be incorporated into the three main coefficients $C_{ii}(z)$ with an exponentially decaying behavior in the form $C(z) = C({z_0}) \\exp (-B\\cdot (z - z_0))$ where $z_0 = 3.35 \\, Å$ , and the three decay coefficients $B = 3.0, \\, 3.2, \\, 3.3 \\,\\, Å^{-1}$ for each one of the terms of the Hamiltonian in the sublattice basis were found fitting the $z$ -dependence between 2.8 $Å$ to 5 $Å$ , where we use the parameters obtained from ab initio calculations $C_{AA}(z_0) &=& -14.88 \\,\\, {\\rm meV}, \\quad \\varphi _{AA} = 50.19^{\\circ } \\\\C_{BB}(z_0) &=& 12.09 \\,\\, {\\rm meV}, \\quad \\varphi _{BB} = - 46.64^{\\circ } \\\\C_{AB}(z_0) &=& 11.34 \\,\\, {\\rm meV}, \\quad \\varphi _{AB} = 19.60^{\\circ }$ whose equivalent values in the pseudospin basis had been calculated previously [31].", "The variation of the phase with $z$ shows a weak linear dependence and we can approximate it as a constant value.", "The effects due to lattice relaxation can be conveniently incorporated when calculating the Fourier expansion of the above Hamiltonian by accounting for the in-plane displacement $\\vec{u}(\\vec{r}) = (u_x(\\vec{r}),u_y(\\vec{r}))$ in the stacking coordination vector $\\vec{d}(\\vec{r}) = \\vec{d}_0(\\vec{r}) + \\vec{u}(\\vec{r})$ , as explained in the main text, and the height $z = h(\\vec{r})$ that represents the local distance of graphene to BN, where $\\vec{r}=(x,y)$ is a two-dimensional vector.", "Both the displacement vectors $\\vec{u}(\\vec{r})$ and the height maps $h(\\vec{r})$ are assumed to respect the moiré periodicity and are therefore modeled from the scalar fields that we introduce in the following section." ], [ "Scalar fields for describing the moiré patterns", "In the main text we presented an approximation for a scalar field that varies smoothly in real space that respects the symmetry of the triangular superlattice to use in the variational trial functions.", "For brevity in notation here we use $(x,y)$ to indicate the $(d_x, d_y)$ .", "The specific form of the trial functions we use are given by $\\Phi \\left( \\vec{r} \\right) &=& \\sum _{\\vec{G}} C_{\\vec{G}} \\exp \\left( - i \\vec{G} \\cdot \\vec{r} \\right) \\\\&\\simeq & C_0 + C^{\\prime }_1 g(\\vec{r}) + f_1(\\vec{r},C_1,\\varphi _1) + f_2(\\vec{r},C_2,\\varphi _2) \\nonumber $ where the constants $C_{\\vec{G}}$ are complex numbers.", "The $f$ function $f_j (\\vec{r},C_j,\\varphi _j) &=& C_j \\exp (i \\varphi _j) \\widetilde{f}_j (\\vec{r}) + {\\rm c.c.", "}$ is defined in terms of the structure factors $\\widetilde{f}_j (\\vec{r}) &=& \\exp (- i j G_1 y ) \\nonumber \\\\&+& 2 \\exp ( i j G_1 y / 2 ) \\cos ( j \\sqrt{3} G_1 x / 2 )$ where $G_1 = 4\\pi /3a$ , where $a$ is the real-space periodicity of the moiré superlattice and $j = 1, 2$ .", "These are momentum space analogues of the real space inter-sublattice hopping structure factors in a honeycomb lattice [32].", "The explicit form of the functions defined along the symmetry lines $x = 0$ or $y = 0$ can be obtained from sums of $f_j(\\vec{r}, y=0, C, \\varphi ) &=& 2 C \\cos \\varphi \\left( 1 + 2 \\cos ( j \\sqrt{3} G_1 x / 2 )\\right) \\nonumber \\\\f_j(\\vec{r}, x=0, C, \\varphi ) &=& 2 C \\cos \\left( \\varphi - jG_1 y \\right) \\nonumber \\\\&+& 4 C \\cos \\left( \\varphi + jG_1 y / 2 \\right).$ The analytical expression for the $g$ function shell contribution reduces to a simpler form $g (\\vec{r}) = 2 \\cos \\left( G_2 x \\right) + 4 \\cos \\left( \\sqrt{3} G_2 y / 2 \\right) \\cos \\left( G_2 x / 2 \\right),$ where $G_2 = 4 \\pi $ , that for the symmetry lines reduce to $g(\\vec{r}, y = 0 ) &=& 2 \\cos \\left( G_2 x \\right) + 4 \\cos \\left( G_2 x / 2 \\right) \\nonumber \\\\g(\\vec{r}, x = 0 ) &=& 2 + 4 \\cos \\left( \\sqrt{3} G_2 y / 2 \\right).$ The vector fields such as in-plane forces, displacement vectors, and stresses can be obtained as gradients of the scalar potentials given by the above forms that can preserve the symmetry of the triangular moiré superlattice.", "The vector field that can be obtained from the gradient of the scalar field is $\\vec{\\nabla } \\Phi = \\vec{\\nabla } f_1 + \\vec{\\nabla } f_2 + \\vec{\\nabla } g$ and can be obtained taking the respective partial derivatives.", "Thus we have $\\vec{\\nabla } f_j = C_j \\exp (i \\varphi _j) \\vec{\\nabla } \\widetilde{f}_j + c.c.$ where the partial derivatives of the constituent functions are given by $\\partial _x \\widetilde{f}_j(\\vec{r}) &=& - j \\sqrt{3}G_1 \\exp (ij G_1 y / 2 ) \\sin (j \\sqrt{3}G_1 x / 2) \\nonumber \\\\\\partial _y \\widetilde{f}_j(\\vec{r}) &=& ijG_1 \\left( - \\exp (-ij G_1 y ) \\right.", "\\nonumber \\\\&+& \\left.", "\\exp (ij G_1 y / 2 ) \\cos (j \\sqrt{3} G_1 x/2) \\right).$ For the $g$ terms we have $\\partial _x g(\\vec{r}) &=& - 2 G_2 \\left( \\sin (G_2 x) + \\cos (\\sqrt{3}G_2 y/2) \\sin (G_2 x/2) \\right) \\nonumber \\\\\\partial _y g(\\vec{r}) &=& - 2 \\sqrt{3} G_2 \\sin (\\sqrt{3} G_2 y/2) \\cos ( G_2 x/2).$ Likewise higher order derivatives used in the stress tensors or the gauge fields can be evaluated analytically.", "The pair of parameters $C$ and $\\varphi $ for each $f_j$ function and the single parameter accompanying the $g$ function specify the variational space we used to minimize the energy functionals.", "Because the $f_1$ term captures the first harmonic contribution, the different variables such as $\\vec{u}$ , $\\vec{h}$ , $E_{pot}$ can be characterized in terms of just two parameters $C_1$ and $\\varphi _1$ , or up to three when the average value of the origin $C_0$ is required." ], [ "Elastic energy functional", "The elastic energy for the Born-von Karman plate theory can be obtained in terms of the Lamé parameters since the small bending stiffness $\\kappa $ for graphene or BN plays a negligible role [28].", "For graphene we use $\\lambda = 3.25 \\,\\, {\\rm eV} \\, Å^{-2}$ , $\\mu = 9.57 \\,\\, {\\rm eV} \\, Å^{-2}$ valid close to zero 0 K have been estimated from empirical potentials [29] and the for a single sheet of BN use the DFT estimates $\\lambda = 3.5 \\,\\, {\\rm eV} \\, Å^{-2}$ , $\\mu = 7.8 \\,\\, {\\rm eV} \\, Å^{-2}$ obtained averaging the LDA and GGA values.", "The total elastic energy per superlattice area is given by [27], [33] $E_{\\rm elastic} &= & \\frac{\\kappa }{2 A_M} \\int _{A_M} d^2 \\vec{r} \\left[\\nabla ^2 h ( \\vec{r} )\\right]^2 + \\\\ &+ &\\frac{1}{2 A_M}\\int _{A_M} d^2 \\vec{r} \\left\\lbrace \\lambda \\left[u_{11} ( \\vec{r} ) + u_{22} ( \\vec{r} ) \\right]^2 \\right.", "\\nonumber \\\\&+& \\left.2 \\mu \\left[ u_{11}^2 ( \\vec{r} ) + u_{22}^2 ( \\vec{r} ) + u_{12}^2 ( \\vec{r} ) \\right] \\right\\rbrace \\nonumber $ The strain tensors $u_{ij} ( \\vec{r} )$ associated to the deformation of the graphene layer depend both on the in-plane displacements and heights in Monge's representation: $u_{11} &= &\\frac{\\partial u_x}{\\partial x} + \\frac{1}{2} \\left(\\frac{\\partial h}{\\partial x} \\right)^2 \\\\u_{22} &= &\\frac{\\partial u_y}{\\partial y} + \\frac{1}{2}\\left(\\frac{\\partial h}{\\partial y} \\right)^2 \\nonumber \\\\u_{12} &= & \\frac{1}{2} \\left( \\frac{\\partial u_x}{\\partial y} +\\frac{\\partial u_y}{\\partial x} \\right) + \\frac{1}{2}\\frac{\\partial h}{\\partial x}\\frac{\\partial h}{\\partial y} \\nonumber $ In a practical calculation it is convenient to use an integration domain that remains fixed for every moiré period.", "For this purpose we use rescaled coordinates to operate in the coordination vector $\\vec{d}$ defined in the unit cell of graphene.", "Using the chain rule to relate the reduced vector $\\vec{d}$ in graphene's unit cell and the real-space $\\vec{r}$ coordinates for zero twist angle and assuming variable lattice constant mismatch $\\varepsilon $ we have $\\nabla _{\\vec{r}}= \\varepsilon \\nabla _{ \\vec{d} }.$ When we neglect the contributions from the height variation the elastic energy can be written as ${E}_{\\rm elastic} = \\frac{\\varepsilon ^2}{ A_{M}} \\int _{A_M} d^2 \\vec{r} \\,\\, S_{el} (\\vec{d}(\\vec{r}), \\vec{u})$ where $S_{el}$ represents the integrand of Eq.", "(REF ) in rescaled coordinates $\\vec{d}$ .", "This form shows more explicitly a $\\varepsilon ^2$ weakening of the elastic energy as the lattice constant mismatch becomes smaller." ], [ "Parametrization of the potential energy", "Likewise it is convenient to use the parametrization of the potential energy in the coordination vector $\\vec{d}(\\vec{r})$ and the interlayer separation height.", "The potential energy term has been parametrized from EXX+RPA calculations binding energy curves for different stacking configurations [26] as a starting point to extract the potential energy curves needed for the formulation of the Frenkel-Kontorova (FK) model for this two-dimensional bipartite lattice.", "We can neglect the van der Waals tail corrections from the bulk that bring the equilibrium distances closer because their influence in distinguishing different stacking energies are small.", "We make use of the property that the energy landscape for a fixed $z$ -axis separation is given by a simple expansion in the first shell of G-vectors in Fourier space [31] to represent the energy map with three parameters.", "As noted previously the simplest approximation for a scalar field that varies smoothly in real space with the triangular lattice symmetry is given by $\\Phi \\left( \\vec{r} \\right) &=& \\sum _{\\vec{G}} C_{\\vec{G}} \\exp \\left( - i \\vec{G} \\cdot \\vec{r} \\right) \\simeq C_0 + f_1(\\vec{r},C_1,\\varphi _1) \\nonumber $ where the constants $C_{\\vec{G}}$ are complex numbers.", "Thanks to this simple form it is possible to parametrize the whole energy landscape from the values of the potentials at three inequivalent stacking configurations, for example the three symmetric stacking configurations AA, AB and BA.", "Its explicit expression $\\Phi (x,y, C_0, C_1, \\varphi ) &=& C_0 + 2 C_1 \\cos (\\varphi - G_1 y) \\\\ & +& 4 C_1 \\cos (G_1 y / 2 + \\varphi ) \\cos (\\sqrt{3} G_1 x / 2) \\nonumber $ repeats with the periodicity of a triangular lattice.", "The scalar function at the three distinct symmetry points in units of graphene's lattice constant $A &=& \\Phi (0,0)= C_0 + 6 C_1 \\cos \\varphi \\\\B &=& \\Phi (0, \\frac{1}{\\sqrt{3}}) \\\\&=& C_0 + 2 C_1 \\cos \\left( \\varphi - 4 \\pi / 3 \\right) + 4 C_1 \\cos \\left( 2 \\pi /3 + \\varphi \\right) \\nonumber \\\\C &=& \\Phi (0, \\frac{2}{\\sqrt{3}}) \\\\&=& C_0 + 2 C_1 \\cos \\left( \\varphi - 8 \\pi /3 \\right) + 4C_1 \\cos \\left( 4 \\pi /3 + \\varphi \\right).", "\\nonumber $ These equations lead to the explicit values of the parameters $\\varphi &=& \\arctan \\left[ - \\frac{\\sqrt{3}}{2(D + 1/2)} \\right] \\\\C_1 &=& \\frac{C - B}{6 \\sqrt{3} \\sin \\varphi } \\\\C_0 &=& - 6C_1 \\cos \\varphi + A$ where we have used the relation $D = (A - B) / (B - C)$ .", "From the layer separation $z$ -dependence of these three coefficients $C_0(z)$ , $C_1(z)$ and $\\varphi (z)$ we can obtain the complete potential landscape $U(x,y,z)$ that we need for our model.", "We note that the $C_0(z) = (A(z) + B(z) + C(z))/3$ term is the average value of $\\Phi (x,y)$ in the periodic domain for every value of $z$ and that the remaining $C_1(z)$ and $\\varphi (z)$ terms accounts for the landscape of the energy in the first harmonic approximation, which is often an accurate approximation for functions varying smoothly with the moiré pattern [31].", "The difference between this average and the minimum $U_{\\rm dif} = U_{\\rm av} - U_{\\rm min}$ gives a measure of the in-plane forces associated to the energy gradient in a Frenkel-Kontorova problem [34].", "The numerical values for $A(z)$ , $B(z)$ and $C(z)$ for the binding energy curves as a function of separation distance $z$ can be obtained from the calculations provided in Ref.", "[26].", "They can be interpolated numerically or alternatively we can use analytic fitting expressions similar to that in Ref.", "[35] used in the G/G case.", "We define the auxiliary functions $M(x) &=& -M_0 ( 1 + \\tau x ) \\exp ( -\\tau x ) \\\\T(x) &=& T_0 / (x^4 + T_1 ) \\\\W(x) &=& (1 + \\exp ( -16 (x - 4 ) ))^{-1}$ with the parameters $M_0 = 0.06975$ , $\\tau = 7$ , $D_0 = 3.46$ , $T_0 = - 10.44 $ , $ T_1 = -58.87$ to define the fitting function for the average value of $C_0(z)$ for all the stacking configurations through $C_0(z) &=& M\\left( z/ D_0 - 1 \\right) \\\\&+& \\left( T( z ) - M\\left( z / D_0 - 1 \\right) \\right) W( z)$ where $z$ is given in angstroms.", "We used a rather simple model for $W(x)$ which is fairly accurate but can still be improved through additional parameters to better capture the behavior away from the equilibrium point.", "The $z$ -dependence of the $C_1(z)$ term is easily captured through an exponentially decaying form $C_1(z) = a \\exp (- b (z/a_0 - z_0))$ where $a = 2.226$ , $b= -3.295$ and $z_0 = 1.295$ , $a_0 = 2.46 \\, Å$ is the lattice constant of graphene.", "The $\\varphi = -50.4^{\\circ }$ term shows a weak linear dependence with respect to $z$ so we use a constant value.", "When necessary, the long-ranged van der Waals tails originating from the bulk BN layers can be added through $T_{\\rm tail}(z) = \\sum _{n} T(z + n c)$ where $c$ is the separation lattice constant between the layers and whose sum saturates quickly.", "However, this correction term has a small influence for the differences in energy for different stacking arrangements and we neglect this term.", "The energy landscape plots for a fixed separation distance $z_0 = 3.4\\, Å$ presented in Fig.", "REF Figure: Left Panel: The total energy per unit cell area as a function of sliding in the yy axisfor x=0x=0 shows a minimum when one of the carbon atoms sitsin the middle of the hexagon and another sits on top of boron.Right top panel:Potential energy of graphene's carbon atoms per unit cell area.Right bottom panel: potential energy experienced by the individual carbon atom per unit cell areaobtained assuming additivity of the energies.allows to estimate the average in-plane traction force being applied on the two inequivalent carbon atoms in the unit cell.", "Even though the LDA binding energies are substantially smaller than in an EXX+RPA calculation, we find that this in-plane energy map obtained through parametrization in Fig.", "3 of the main text is closely similar to the LDA energy map obtained in Ref.", "[31], whose agreement is attributable to the dominance of short-range character of the interactions near equilibrium distances that is captured reasonably well by the LDA approximation [35].", "From this potential landscape per two carbon atom unit cell we can infer the potential experienced by the individual carbon atoms that can be useful for lattice force-field calculations where the higher energy optical modes are treated explicitly.", "This is done assuming that the total potential energy consists of the sum of the potentials experienced by each carbon atom which is separated by a distance $\\tau = a/\\sqrt{3}$ $U(x,y) = U_{C}(x,y) + U_{ C}(x,y+\\tau ).$ Solving the above equation we get $U_{ C}(x,y) = \\Phi (x,y,C_0^{\\prime },C_1,\\varphi ^{\\prime }_1)$ where $C_0^{\\prime } = C_0/2$ and $\\varphi _1^{\\prime } = \\varphi _1 - \\pi /3$ .", "Likewise if the long-range van der Waals tails are used they would need to be reduced to one half of its value.", "In Fig.", "REF we show the potential energy repeated over several periods as well as the energy landscape seen by each carbon atom, derived assuming additivity in the total potential energy." ], [ "Relaxation of the graphene and substrate atoms in the G/BN heterojunction", "The relaxed geometries can be readily obtained minimizing the elastic and potential energy functionals using the trial functions that we introduced earlier.", "We distinguish two different scenarios in our elasticity problem.", "In the first case we solve for the elastostatic solutions where we relax the atoms of graphene subject to a periodic moiré potential of a rigid substrate.", "A second scenario allows the coupled relaxation of the BN substrate atoms.", "For simplicity we consider only the zero twist angle case and variable lattice constant mismatch $\\varepsilon $ ." ], [ "Graphene relaxation only model", "The resolution of the elastostatic problem of graphene subject to a superlattice potential requires the minimization of the total energy functional ${E}_{\\rm total} = {E}_{\\rm elastic} + { E}_{\\rm potential}$ where ${E}_{\\rm elastic}$ is given in Eq.", "(REF ) and the potential energy is given by the integral in the moire supercell of area $A_M$ of the potential energy kernel $U (\\vec{r},\\vec{d},h)$ given in Eq.", "(REF ) ${E}_{\\rm potential} = \\frac{1}{A_M} \\int _{A_M} d \\vec{r} \\,\\, U(\\vec{r},\\vec{d},h).$ The stacking coordination vector is modeled as $\\vec{d}_G(\\vec{r}) = \\vec{d}_0 (\\vec{r}) + \\vec{u}_{G}(\\vec{r})$ assuming that the substrate produces a rigid periodic potential pattern.", "We use the gradients of the scalar field in Eq.", "(REF ) to model the displacement vectors $\\vec{u}_G$ .", "The local elastic and potential energy maps corresponding to the small and large strain limits are represented in Fig.", "REF .", "Figure: Map of local elastic and potential energies per unit cell area (see Eqs.", ", ) correspondingto small and large strains using constant hh model, see also , corresponding tolattice constant differences of of ε=0.017,0.0082\\left\| \\varepsilon \\right\| = 0.017, 0.0082 repectively.The large strain configuration we represent here is just before the poitnt of steep transitionas shown in Fig.", ", which happens for longer moiré periods than whenthe zz-axis relaxation is allowed." ], [ "Coupled relaxation of the BN lattice", "Here we explore the influence in the elastic of energy of graphene when the BN atoms of the topmost layer in the substrate are allowed to relax in response to the stacking rearrangement of the graphene sheet.", "The coupled motion of the substrate atoms contribute in decreasing the total elastic energy of the graphene BN heterojunction because a smaller displacement in the graphene sheet is needed than if the substrate remains rigid.", "For solving the coupled G/BN elasticity problem we will assume that the topmost BN sheet is subject to a potential stemming from the graphene sheet itself and the BN layers underneath, assuming that the BN atoms below the topmost layer remain fixed.", "The potential energy for fixed interlayer separation of $c = 3.4\\, Å$ for G/BN and BN/NB along the $y$ direction of stacking arrangement vector is shown in Fig.", "REF .", "The interaction potential between the two topmost BN layers are defined by $C_1 = -2.47$ meV and $\\varphi _1 = -57.75^{\\circ }$ through the funciton $U_{\\rm BN/NB}(\\vec{r}; \\vec{d}_{BN}) = f_{1}(\\vec{d}_{BN}, C_1,\\varphi _1)$ .", "Figure: In (a) we show the total energy per unit cell area for sliding along the vertical yy-axisfor different stacking configurations for G/BN within RPA and BN/NB heterojunctionswithin LDA near the equilibrium interlayer separation for fixed c=3.4Åc = 3.4 \\, Å.The energy curves were obtainedusing the information at threedifferent symmetric stacking configurations for AA, AB and BA for interlayer sliding vectorsτ AA =(0,0)\\tau _{AA} = (0,0), τ AB =(0,a/3)\\tau _{AB} = (0, a / \\sqrt{3}) and τ BA =(0,2a/3)\\tau _{BA} = (0, 2 a / \\sqrt{3}).The energy minimum for G/BN stacking happens at τ BA \\tau _{BA} whereasfor BN/NB the energies are smallest near τ AA \\tau _{AA} and τ AB \\tau _{AB}.In (b) we show the stacking configurations for G/BN and BN/NB and minimize the total energywhich shows that deformation of the topmost BN layer is easier when it preserves the BA stackingof the G/BN hererojunction.", "The minimum energy configurations are indicated with labels 1, 2 whereasthe maximum energy ones are labeled with 3, 4.Even though the binding energies and forces predicted by the LDA typically underestimate the values obtained from higher level RPA calculations [37], [38], [39], [40] we assume that the energy landscape for different strackings We used LDA energies for BN/NB coupling as a function of sliding, assuming that their sliding energy maps are comparable to EXX+RPA as we found for the G/BN case.", "The total energy of G/BN/NB where both sheets are allowed to relax is given by the sum of the elastic and potential energies of graphene and the topmost BN sheet.", "The total potential energy term can be obtained from the interaction energies between the neighboring layers through ${ E}_{\\rm potential} &=& E_{\\rm potential, \\, G/BN} + E_{\\rm potential, \\, BN/NB}, $ and can be calculated from the parametrized potential energies evaluating the integrals in the moiré supercell ${ E}_{\\rm potential, \\, G/BN} &=& \\frac{1}{A_M} \\int _{A_M} d \\vec{r} \\,\\, \\, U_{\\rm G/BN}(\\vec{r} \\,; \\, \\vec{d}_G , h_G), \\\\{ E}_{\\rm potential, \\, BN/NB} &=& \\frac{1}{A_M} \\int _{A_M} d \\vec{r} \\,\\, \\, U_{\\rm BN/NB}(\\vec{r} \\,; \\, \\vec{d}_{BN} ),$ where the kernels are functionals of the local stacking coordination functions $\\vec{d}_G$ and $\\vec{d}_{BN}$ that depend on the displacements relative to the neighboring layers $\\vec{d}_{G}(\\vec{r}) &=& \\vec{d}_0(\\vec{r}) + \\vec{u}_{G}(\\vec{r}) - \\vec{u}_{BN}(\\vec{r}), \\\\\\vec{d}_{BN}(\\vec{r}) &=& \\vec{u}_{BN}(\\vec{r}).", "\\\\ \\nonumber $ We used explicit labels G/BN and BN/NB to distinguish the interaction potentials.", "For the graphene sheet the only relevant reference frame is the topmost BN layer whereas the latter interacts both with the graphene sheet and the BN layers underneath whose coordinates are assumed to remain fixed.", "Figure: Potential energy maps U G/ BN U_{\\rm G/BN} and U BN / NB U_{\\rm BN/NB} in real space.The left column represents the interaction potentials of rigid graphene and BN sheets.The right column shows the potential energy maps for U G/ BN U_{\\rm G/BN} and U BN / NB U_{\\rm BN/NB}corresponding to the relaxed geometry configurations determined bythe strains in the graphene and boron nitride sheet given by u → G \\vec{u}_{G} and u → BN \\vec{u}_{BN} respectively." ], [ "Strains in relaxed ground-states", "As noted earlier, the strained geometries can be characterized by the magnitude and phases that define the scalar fields in Eq.", "(REF ) and vector fields in Eq.", "(REF ) within a restricted variational space that preserves the triangular moiré periodicity dictated by the lattice constant mismatch $\\varepsilon $ and twist angle $\\theta $ .", "The solutions for the strains in the graphene layer can be largely characterized by two parameters $C_{u,1}$ and $\\varphi _{u, 1}$ whereas the height profiles require three $C_{h,0}$ , $C_{h,1}$ and $\\varphi _{h,1}$ for the additional average interlayer separation.", "For the topmost BN sheet we only consider in-plane strains assuming that its separation from the additional BN layer underneath takes a constant average value.", "Because the relative magnitudes of the elastic and potential energies scale with the lattice constant mismatch $\\varepsilon $ a correction in the potential profiles or the average value of the elasticity constants would have an overall effect of shifting the solutions in the abscissa.", "Within our approximation, the solutions are completely characterized by the $\\varepsilon $ -dependent values of the parameters that define the scalar field.", "In Fig.", "REF we show the values of the relaxed solution parameters where only the graphene sheet is allowed to relax both in-plane and out of plane.", "We also show a comparison with the solutions where only in-plane relaxation is permitted and we fix the interlayer separation to a constant value.", "Figure: Left panel:Elastostatic solutions for the strains in the graphene sheet relaxation onlythat is subject to the potential of a rigid BN substrate.For smaller ε\\left\| \\varepsilon \\right\|the potential energy dominates and the deformation becomes larger.The increase of the deformation is steady until it reaches a tipping point where the solutions become unstable.Comparison of in-plane relaxation only and that allowing out-of-plane relaxation shows that the both approximationsgive similar in-plane displacements but allowing the full relaxation makes the transition easier.Right panel:Elastostatic solutions of the coupled graphene and topmost BN layer subject to the potentials of a rigid BN layer potential underneath.We notice that the magnitude of the in-plane deformation of the graphene and BN sheets are comparableto the strains in the BN sheet as the latter can relax along the easy sliding axis directions.When we consider the coupled graphene and BN layer relaxation we notice an interesting behavior where the largest strain magnitudes are for the topmost BN sheet rather than graphene itself.", "The results for the relaxed strains of the coupled G/BN/NB heterojunction where both graphene and the topomost BN layer are relaxed is shown in Fig.", "REF .", "This is possible thanks to a special total energy landscape with easy sliding path in BN [41].", "The BN sheets in the crystal substrate follows an AA$^{\\prime }$ stacking order and for this stacking configuration they have a minimum energy sliding path when going from AA$^{\\prime }$ to AB$^{\\prime }$ with a small barrier of about $\\sim $ 3 meV and even smaller total energy differences of about $\\sim $ 1 meV within the LDA.", "A more elaborate GGA + vdW functional calculation [41] predicted similar barrier magnitudes but with the minimum of energy happening for the AB$^{\\prime }$ stacking configuration with a total energy lower by $\\sim $ 1.5 meV.", "These minute differences are unimportant for the solutions we discuss." ], [ "Band gaps and the Fourier components of the strained Hamiltonian", "The Hamiltonian of graphene is modified in a G/BN heterojunction by moiré patterns [31] that can be described in a transparent manner when represented in a pseudospin basis.", "The different contributions consist of a site potential $H^{0}_M$ , the mass or sublattice staggering potential $H^z_{M}$ and an in-plane pseudospin inter-sublattice coherence term $H^{xy}_{M} = H^{AB}_{M}$ as shown in Fig.", "REF .", "The latter is closely related with a pseudomagnetic field derived from the straining of the graphene sheet represented in Fig.", "REF .", "Figure: Real space representation of the pseudospin Hamiltonianfor unrelaxed (top row) and relaxed (bottom row) geometries near zero twist angle.The H M 0 H^0_{M} term accounts for the site potential fluctuations normally seen in scanning probestudies, the H M z H^z_{M} term is the mass term dictating the local band gap in real space,and H M xy H^{xy}_{M} reflects the anisotropic strains.Figure: Real space map of the pseudomagnetic field magnitude A →(r →)\\left\| \\vec{A}(\\vec{r}) \\right\|for relaxed solutions.", "The left panel represents graphene relaxation only allowingonly in-plane strains with a long moiré period corresponding to ε=0.0082\\left\| \\varepsilon \\right\| = 0.0082.On the right hand we present the magnitude of the vector potential due to the relaxation of the graphene sheetwhen both graphene and BN sheets are simultaneously allowed to relax near zero twist angle moiré period.As mentioned in the main text and explained in more detail in Ref.", "[31] it is possible to obtain the full band structure from the Fourier components represented in the moiré reciprocal lattice vectors $\\vec{G}$ .", "The term that most directly influences the band gap comes from $H^z_{M}$ , in particular the $\\vec{G}_0 = (0,0)$ contribution which is the average mass in the a moiré supercell.", "This term normally vanishes to zero in a rigid crystal [31] but here we showed that they generally average to a nonzero value in the presence of in-plane strains.", "Farther shell contributions in $\\vec{G}$ do also contribute to the band gap through higher order corrections.", "The contributions from the first shell in G-vectors through second order perturbation theory play the most relevant role.", "In Fig.", "REF we show a comparison of the nonzero average mass term and the total band gap as a function of moiré period.", "It is noteworthy that when out of plane relaxations are absent the values of the band gaps are generally smaller and the relative cancellation between the average mass $\\Delta _0$ and the second order contributions from the first shell are more substantial.", "The pseudomagnetic field term from $H^{AB}_{M}$ , due to the anisotropic strains generated by the coupling with the BN substrate, plays a minor role in configuring the band gap at the primary Dirac point as will be made clearer in the perturbative analysis.", "Figure: Breakdown of different contributions to the single-particle band gap.Left panel: Comparison of the band gap Δ\\Delta represented with connectedfilled circles and the non-zero average contribution Δ 0 \\Delta _0 represented withempty circles for graphene relaxed, graphene and boron nitride relaxed, and restricted in-plane only relaxationof graphene.We notice that the presence of out-of-plane relaxation prevents the complete cancellationof the average mass in the presence of small in-plane strains.Top right panel:Perturbation theory analysis of the gap in the configuration where graphene relaxes due to in-plane strains.", "The non-perturbative gap is shown in a black solid line, which is closely approximated by the 2nd order perturbation theory (dashed black line).The dash-dotted line is the gap due to the average mass.The decomposition of the Hamiltonian into H eff 0 H^{0}_{\\rm eff} (grey circles) andH eff α H^{\\alpha }_{\\rm eff} for α=x,y,z\\alpha = x,y,z (blue, green, and red circles, respectively)indicates that the primary source of the gap is H eff z H^{z}_{\\rm eff}.Bottom right panel:Perturbation theory analysis of the gap in the configuration where graphene and boron nitride both relax due to in-plane strains.The non-perturbative gap is shown in a black solid line, which is closely approximated by the 2nd order perturbation theory (dashed black line).The dash-dotted line is the G → 0 =0\\vec{G}_0 = 0 contribution to the gap.The decomposition of the Hamiltonian into H eff 0 H^{0}_{\\rm eff} (grey circles) and H eff α H^{\\alpha }_{\\rm eff} forα=x,y,z\\alpha = x,y,z (blue, green, and red circles, respectively) indicates that the primary source of the gap is H eff z H^{z}_{\\rm eff}.Figure: A representation of the real and imaginary parts of the Fourier components of the H M z H^{z}_{M} localmass term distribution in real space of the Hamiltonian for substantially strained configurations for in-plane onlyrelaxation of the graphene sheet.The band gap is determined mainly by the average mass term from the G →=(0,0)\\vec{G}=(0,0) contribution and modifiedby the first hexagonal shell in G →\\vec{G} vectors contributing to second order in perturbation theory.The other two components of the Hamiltonian, potential fluctuations $H^0_{M}$ and in-plane pseudospin terms $H^{xy}_{M}$ represented in Figs.", "(REF , REF ), also see modifications due to straining, typically acquiring contributions beyond the first shell in G-vectors with the most important contributions ranging up to three nearest neighbor hoppings, and their Fourier components having magnitudes in the order of $\\sim $ 10 meV.", "Figure: Fourier expansion of the Hamiltonian for the site potential fluctuations H 0 H_0 and the anisotropic strainH AB H_{AB} terms.", "The contributions to the band gap of these two terms are much smaller than those fromH z H_{z}.The pseudomagnetic term generated by the strains in the graphene sheet itself can in principle have a contribution comparable to the contribution due to the electron virtual hopping to and back from the BN sheet.", "Using the expressions for pseudomagnetic fields in graphene provided in Refs.", "[42], [20] $A_x(\\vec{r}) &=& g [u_{11}(\\vec{r}) - u_{22}(\\vec{r})] \\\\A_y(\\vec{r}) &=& - 2 g u_{12}(\\vec{r})$ and using $g \\sim 1.5/a$ a typical map of its magnitude in real space is shown in Fig.", "REF .", "We note that the pseudomagnetic vector potentials follow a moiré period scaling relation given by $\\left\| \\vec{A}(\\vec{r})\\right\| \\propto \\left( {a}/{l_M} \\right) \\left\| \\widetilde{A}(\\vec{d}(\\vec{r}))\\right\|$ when represented in rescaled coordinates of the moiré superlattice $\\widetilde{A}(\\vec{d}(\\vec{r}))$ , in turn defined by the parameters that determine the displacement vectors $\\vec{u}(\\vec{r})$ ." ], [ "Second order perturbation theory", "Further insight on the contributions to the band gaps can be achieved from second order perturbation theory from the first shell approximation.", "We distinguish two scenarios, one for rigid unrelaxed lattices and another where strains are allowed to modify the stacking coordination.", "Formally it is possible to show that for rigid unrelaxed lattices the in-plane $H^{AB}_{M, \\vec{G}}$ gives a zero contribution to the band gap to second order in perturbation theory.", "When in-plane strains are allowed, band gaps develop thanks primarily to a nonzero average mass and all three pseudospin components make a nonzero contribution to the gap to second order.", "Among these, in our calculations the in-plane pseudospin terms contribute to the gap with a smaller magnitude than the Fourier expansion of the mass terms $H^{z}_{M, \\vec{G}}$ ." ], [ "Unrelaxed configuration", "Here we discuss the effective 2x2 Hamiltonian obtained from perturbation theory around the Dirac point.", "Our initial Hamiltonian is a $2N\\times 2N$ matrix, where $N$ is two times the number of Moiré reciprocal lattice vectors in the Fourier transform.", "Treating the $2\\times 2$ diagonal blocks as the unperturbed Hamiltonian, second order degenerate perturbation theory gives an effective Hamiltonian for the low energy states, $H_{\\rm eff}=H_{M, \\vec{G}=0}-\\sum _{\\vec{G}\\ne 0}H_{M, \\, \\vec{G}} \\, H_{G}^{-1}H_{M, \\, \\vec{G}}^{\\dagger }$ where $H_{G}$ are the $2\\times 2$ blocks in the Hamiltonian associated with the moiré vector $\\vec{G}$ , and $H_{M, \\vec{G}}$ connect the $\\vec{k}$ and $\\vec{k}+\\vec{G}$ blocks of the Hamiltonian.", "If we ignore for a moment the relaxation due to in-plane strains, the diagonal blocks are $H_{M, \\vec{G}}=\\hbar \\upsilon \\vec{G}\\cdot \\vec{\\tau }$ , which has an inverse $(H_{\\vec{G}})^{-1}=\\hbar \\upsilon \\vec{G}\\cdot \\vec{\\tau }/(\\hbar \\upsilon \\vec{G})^{2}$ .", "We can decompose both the effective Hamiltonian and the $H_{M, \\,\\vec{G}}$ into terms proportional to Pauli matrices, $H_{\\rm eff}=&\\sum _{\\alpha =0,x,y,z}H_{\\rm eff}^{\\alpha }\\tau ^{\\alpha }\\\\H_{M, \\vec{G}_{j}}=&\\sum _{\\alpha =0,x,y,z}M_{j}^{\\alpha }\\tau ^{\\alpha }$ Since $H_{\\rm eff}$ is hermitian, the parameters $H_{\\rm eff}^{\\alpha }$ must be real numbers.", "However, each block $H_{M, \\vec{G}_{j}}$ is not necessarily hermitian, so $M_{j}^{\\alpha }$ are complex numbers.", "Plugging in the decomposed forms, and restricting to just the nearest shell of reciprocal lattice vectors $j=1,...,6$ (we use the index $j=0$ for $\\vec{G}=0$ ), we get $H_{\\rm eff}^{0}= & \\frac{4\\hbar \\upsilon }{(\\hbar \\upsilon \\vec{G})^{2}}\\sum _{j=1}^{3}\\left[\\operatorname{Re}M_{j}^{z}\\left(\\operatorname{Im}\\vec{M}_{j}\\times \\vec{G}_{j}\\right)\\cdot \\hat{z}-\\right.\\nonumber \\\\& \\qquad \\left.-\\operatorname{Im}M_{j}^{z}\\left(\\operatorname{Re}\\vec{M}_{j}\\times \\vec{G}_{j}\\right)\\cdot \\hat{z}\\right]\\\\H_{\\rm eff}^{x}=&-\\frac{4\\hbar \\upsilon }{(\\hbar \\upsilon \\vec{G})^{2}}\\sum _{j=1}^{3}\\vec{G}_{j,y}\\operatorname{Im}\\left\\lbrace M_{j}^{0}M_{j}^{z}\\right\\rbrace \\\\H_{\\rm eff}^{y}=& \\frac{4\\hbar \\upsilon }{(\\hbar \\upsilon \\vec{G})^{2}}\\sum _{j=1}^{3}\\vec{G}_{j,x}\\operatorname{Im}\\left\\lbrace M_{j}^{0}M_{j}^{z}\\right\\rbrace \\\\H_{\\rm eff}^{z}=& \\frac{4\\hbar \\upsilon }{(\\hbar \\upsilon \\vec{G})^{2}}\\sum _{j=1}^{3}\\left[\\operatorname{Re}M_{j,0}\\left(\\operatorname{Im}\\vec{M}_{j}\\times \\vec{G}_{j}\\right)\\cdot \\hat{z}-\\right.\\nonumber \\\\& \\qquad \\left.-\\operatorname{Im}M_{j}^{0}\\left(\\operatorname{Re}\\vec{M}_{j}\\times \\vec{G}_{j}\\right)\\cdot \\hat{z}\\right]$ The sums in the above equations are restricted to $j=1,2,3$ due to the relation $\\vec{G}_{j+3}=-\\vec{G}_{j}$ and the property of the corresponding matrices, $M_{j+3}=M_{j}^{\\dagger }$ .", "We will now prove that $h_{x}=h_{y}=0$ .", "The moiré Hamiltonian has the property $M_{1}^{0}=M_{3}^{0}=M_{2}^{0}$ and $M_{1}^{z}=M_{3}^{z}=M_{2}^{z}$ .", "Therefore $\\operatorname{Im}M_{3}^{0}M_{3}^{z}=-\\operatorname{Im}M_{1}^{0}M_{1}^{z}$ .", "However, $\\vec{G}_{1}-\\vec{G}_{2}+\\vec{G}_{3}=0$ .", "Examining the above equations, we see that $H_{\\rm eff}^{x}=H_{\\rm eff}^{y}=0$ due to the symmetry properties of the Hamiltonian, and the gap for the unrelaxed configuration arises entirely from a mass term $H_{\\rm eff}^{z}$ .", "We have numerically calculated the low energy eigenvalues as a function of the parameters $M_{j}^{\\alpha }$ to verify the second order perturbation theory result and show which terms contribute at third order and higher.", "Our numerical calculations are performed by multiplying each of the $M_{i}^{\\alpha }$ by interpolation parameters $\\lambda _{\\alpha }$ which range from 0 to 1, thus keeping the same relationship (magnitude and phase) between the different $\\vec{G}_{i}$ terms while allowing us to see explicitly the power law behavior of the gap due to each term.", "Figure: Top left: Gap vs λ\\lambda for individual contributions to H M,G → H_{M, \\vec{G}} from M 0 M^{0} (black dots), M x M^{x} (blue dots), M y M^{y} (green dots) and M z M^{z} (red solid line).", "The labels in the legend correspond to (λ 0 ,λ x ,λ y ,λ z )(\\lambda _{0},\\,\\lambda _{x},\\,\\lambda _{y},\\,\\lambda _{z}).", "The dashed line is a fit to λ 3 \\lambda ^{3}.Top right:Gap vs λ\\lambda for contributions with both M z M^{z} and either M x M^{x} (blue) or M y M^{y} (green).", "The two terms are equal.", "Such terms contribute to H eff 0 H_{\\rm eff}^{0} at second order (see main text), but H eff 0 H_{\\rm eff}^{0} does not contribute to a gap opening.", "Dashed line (black) is a fit to λ 3 \\lambda ^{3}, showing that indeed, no second order contribution is evident.Bottom left:Gap vs λ\\lambda for contributions with both M 0 M^{0} and M z M^{z} (solid red line).", "Such terms contribute to H eff x H_{\\rm eff}^{x} and H eff y H_{\\rm eff}^{y} at second order (see main text), which is zero due to the symmetry of the Hamiltonian.", "Dashed line (black) is a fit to λ 3 \\lambda ^{3}, showing that indeed, no second order contribution is evident.", "Also shown (blue circles) is the contribution with both M x M^{x} and M y M^{y}, which is zero.Bottom right:Gap vs λ\\lambda for contributions with both M 0 M^{0} and either M x M^{x} (blue circles) or M y M^{y} (green circles).", "The two contributions are equal.", "Such terms contribute to h z h^{z} in the second order perturbation theory (see main text), and therefore contribute to the gap.", "Solid black line is a fit to λ 2 \\lambda ^{2}, confirming the perturbation theory result.First we set all $\\lambda _{\\alpha }=0$ except for one.", "Power law fits show that there is no 2nd order contribution from any of the terms individually (Fig.", "REF ).", "The $\\lambda _{z}\\ne 0$ term contributes at the 3rd order, while all others are 5th order or higher.", "Next we look at the interplay between the different matrix elements $M_{j}^{\\alpha }$ which are found to contribute to the perturbation theory results for $H_{\\rm eff}=H_{\\rm eff}^{0}+\\vec{H}_{\\rm eff}\\cdot \\vec{\\tau }$ as described in the main text.", "Figure REF confirm that to second order, the gap is not opened by terms proportional to $M^{x}M^{z}$ , $M^{y}M^{z}$ , $M^{0}M^{z}$ , and $M^{x}M^{y}$ .", "The first two, $M^{x}M^{z}$ and $M^{y}M^{z}$ do contribute to the energy levels at second order: they lead to a nonzero $H_{\\rm eff}^{0}$ which does not open a gap.", "The $M^{0}M^{z}$ term we found to be zero in the second order perturbation theory due to the symmetry of the Hamiltonian, which is verified here.", "Finally, the term $M^{x}M^{y}$ does not appear in the second order perturbation theory at all, which is again confirmed by our numerical results.", "The only terms which contribute to the gap at second order, and are therefore most efficient at opening a gap, are $M^{0}M^{x}$ and $M^{0}M^{y}$ ." ], [ "Relaxed configuration", "We showed in the main text that the in-plane relaxation of the graphene and boron nitride lattices has a large effect on the size of the gap.", "This is due primarily to the emergence of nonzero mass in the $2\\times 2$ block $H^{z}_{M, \\vec{G}_0=0}$ .", "This term alone slightly overestimates the gap.", "We again calculate a second order perturbation theory, Eqn.", "(REF ).", "However, it is no longer a good approximation to restrict to the nearest six reciprocal lattice vectors.", "This means that although the decompositions given in Equations (REF $-$) remain valid, the symmetry properties that cause $H_{\\rm eff}^{x}$ and $H_{\\rm eff}^{y}$ to vanish do not strictly hold.", "We do, however, find that these terms are small.", "The primary contribution to the gap comes from the $H^{z}_{M, \\vec{G}_0=0}$ term which overshoots the gap.", "Including the second order terms produces an excellent approximation to the calculated gap.", "Thus we see explicitly that the relaxation is a key source of the gap opening in graphene/boron nitride bilayer systems." ] ]	1403.0496
[ [ "A new approach to Steiner symmetrization of coercive convex functions" ], [ "Abstract In this paper, a new approach of defining Steiner symmetrization of coercive convex functions is proposed and some fundamental properties of the new Steiner symmetrization are proved.", "Further, using the new Steiner symmetrization, we give a different approach to prove a functional version of the Blaschke-Santalo inequality due to Ball." ], [ "Introduction", "The purpose of this paper is to introduce a new way of defining Steiner symmetrization for coercive convex functions, and to explore its applications.", "Our new definition is motivated by and can be regarded as an improvement of a functional Steiner symmetrization of [1].", "In particular, our new definition has a key property: the invariance of integral, which is not true for the definition of [1].", "Moreover, our definition provides a new approach to the familiar functional Steiner symmetrization (see [7], [8]), but we do not use geometric Steiner symmetrization and our approach is more suitable for certain functional problems.", "Steiner symmetrization was invented by Steiner [32] to prove the isoperimetric inequality.", "For over 160 years Steiner symmetrization has been a fundamental tool for attacking problems regarding isoperimetry and related geometric inequalities [17], [18], [32], [33].", "Steiner symmetrization appears in the titles of dozens of papers (see e.g.", "[4], [5], [6], [8], [10], [13], [16], [20], [21], [24], [26], [27], [31]) and plays a key role in recent work such as [19], [25], [34], [35].", "Steiner symmetrization is a type of rearrangement.", "In the 1970s, interest in rearrangements was renewed, as mathematicians began to look for geometric proofs of functional inequalities.", "Rearrangements were generalized from smooth or convex bodies to measurable sets and to functions in Sobolev spaces.", "Functional Steiner symmetrization, as a kind of important rearrangement of functions, has been studied in [1], [7], [8], [9], [11], [12], [14].", "In [7], Brascamp, Lieb, and Luttinger established that the spherical symmetrization of a nonnegative function can be approximated in $L^p(\\mathbb {R}^{n+1})$ by a sequence of Steiner symmetrizations and rotations.", "In [8], Burchard proved that Steiner symmetrization is continuous in $W^{1,p}(\\mathbb {R}^{n+1})$ , $1\\le p<\\infty $ , for every dimension $n\\ge 1$ , in the sense that $f_k\\rightarrow f$ in $W^{1,p}$ implies $S f_k\\rightarrow S f$ in $W^{1,p}$ .", "In [14], Fortier gave a thorough review and exposition of results regarding approximating the symmetric decreasing rearrangement by polarizations and Steiner symmetrizations.", "For a nonnegative measurable function $f$ , the familiar definition of its Steiner symmetrization (see [7], [8], [9], [14]) is defined as following: Definition 1 For a measurable function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}^+$ , let $m$ denote the Lebesgue measure, if $m([f>t])<+\\infty $ for all $t>0$ , then its Steiner symmetrization is defined as $ \\bar{S}_u f(x)=\\int _{0}^{\\infty }\\mathcal {X}_{S_u E(t)}(x)dt,$ where $S_u E(t)$ is the Steiner symmetrization of the level set $E(t):=\\lbrace x\\in \\mathbb {R}^n: f(x)>t\\rbrace $ about $u^{\\perp }$ and $\\mathcal {X}_{A}$ denotes the characteristic function of set $A$ .", "During the study of the analogy between convex bodies and log-concave functions, Artstein-Klartag-Milman in [1] defined another functional Steiner transformation as follows: Definition 2 For a coercive convex function $f: \\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ and a hyperplane $H=u^{\\bot }$ ($u\\in S^{n-1}$ ) in $\\mathbb {R}^n$ , for any $x=x^{\\prime }+tu$ , where $x^{\\prime }\\in H$ and $t\\in \\mathbb {R}$ , we define the Steiner symmetrization $\\widetilde{S}_u f$ of $f$ about $H$ by $(\\widetilde{S}_u f)(x)=\\inf _{t_1+t_2=t}[\\frac{1}{2}f(x^{\\prime }+2t_1u)+\\frac{1}{2}f(x^{\\prime }-2t_2u)].$ In this paper, we introduce a new way of defining the functional Steiner symmetrization for coercive convex functions.", "A function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ , not identically $+\\infty $ , is called convex if $f(\\alpha x+(1-\\alpha )y)\\le \\alpha f(x)+(1-\\alpha )f(y)$ for all $x$ , $y\\in \\mathbb {R}^n$ and for $0\\le \\lambda \\le 1$ .", "A convex function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is called coercive if $\\lim _{\|x\|\\rightarrow +\\infty }f(x)=+\\infty $ .", "Definition 3 For a coercive convex function $f: \\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ and a hyperplane $H=u^{\\bot }$ ($u\\in S^{n-1}$ ) in $\\mathbb {R}^n$ , for any $x=x^{\\prime }+tu$ , where $x^{\\prime }\\in H$ and $t\\in \\mathbb {R}$ , we define the Steiner symmetrization $S_u f$ of $f$ about $H$ by $(S_u f)(x)=\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=t}[\\lambda f(x^{\\prime }+2t_1u)+(1-\\lambda )f(x^{\\prime }-2t_2u)].$ Our definition $S_u f$ is motivated by and can be regarded as an improvement of $\\widetilde{S}_u f$ in Definition REF .", "When compared with $\\bar{S}_uf$ in Definition REF , our definition symmetrizes a parabola-like (one-dimension) cure once at a time instead of symmetrizing the level set as in $\\bar{S}_uf$ .", "The rest of the paper is organized as follows.", "In Section 2, we explore the analogy between convex bodies and coercive convex functions using our new definition (see Table REF ).", "Table: NO_CAPTIONIn Section 3, we will elaborate on the relations between Definition REF and Definitions REF , REF .", "In Section 4, we give a completely different approach to prove a functional version of the Blaschke-Santaló inequality due to Ball [2]." ], [ " The functional Steiner symmetrization", "We first study the one-dimensional case.", "In Definition REF , when $n=1$ , $S^0=\\lbrace -1, 1\\rbrace $ and $H=\\lbrace 0\\rbrace $ , it is clear that $(S_1f)(x)=(S_{-1}f)(x)$ for any $x\\in \\mathbb {R}$ .", "Let $Sf$ denote Steiner symmetrization of one-dimensional function, then $Sf(x)=\\sup _{\\lambda \\in [0,1]}\\inf _{x_1+x_2=x}[\\lambda f(2x_1)+(1-\\lambda )f(-2x_2)].$ Theorem 1 If $f: \\mathbb {R}\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is a coercive convex function, then $Sf(x)$ is a coercive even convex function and for any $s\\in \\mathbb {R}$ , $Vol_1 ([f\\le s])=Vol_1([Sf\\le s]),$ where $[f\\le s]=\\lbrace x\\in \\mathbb {R}: f(x)\\le s\\rbrace $ denotes the sublevel set of $f$ .", "The following lemma is straightforward, and we omit its proof.", "Lemma 1 Let $f: \\mathbb {R}\\rightarrow \\mathbb {R}$ be a coercive convex function, then we have (i) If $a=\\inf f(t)$ , then $a\\in (-\\infty ,+\\infty )$ and $f^{-1}(a)=\\lbrace x\\in \\mathbb {R}: f(x)=a\\rbrace $ is a nonempty finite closed interval $[\\mu ,\\nu ]$ , where $\\mu $ may equal to $\\nu $ .", "(ii) $f(t)$ is strictly decreasing on the interval $(-\\infty ,\\mu ]$ and strictly increasing on the interval $[\\nu ,+\\infty )$ .", "(iii) If $f(c)=f(d)$ and $c<d$ , then $\\mu <d$ and $c<\\nu $ .", "(iv) For $c$ and $d$ given in (iii), we have the right derivative $f^{\\prime }_{r}(d)\\ge 0$ for $f$ is increasing on $[\\mu ,+\\infty )$ , we also have $f^{\\prime }_{r}(c)\\le 0$ for $f$ is decreasing on $(-\\infty ,\\nu ]$ .", "(v) For two intervals $[a, a+t_0]$ and $[b, b+t_0]$ with the same length $t_0>0$ , if $f(a)=f(a+t_0)$ , then either $f(b)\\ge f(a)$ or $f(b+t_0)\\ge f(a+t_0)$ .", "Proof of Theroem REF .", "First, we show that $Sf$ is even.", "For any $x\\in \\mathbb {R}$ , by (REF ), we have $Sf(-x)&=&\\sup _{\\lambda \\in [0,1]}\\inf _{x_2\\in \\mathbb {R}}[\\lambda f(-2x_2-2x)+(1-\\lambda )f(-2x_2)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{x_2\\in \\mathbb {R}}[\\lambda f(2x_2-2x)+(1-\\lambda )f(2x_2)]\\nonumber \\\\&=&\\sup _{\\lambda ^{\\prime }\\in [0,1]}\\inf _{x_2\\in \\mathbb {R}}[\\lambda ^{\\prime }f(2x_2)+(1-\\lambda ^{\\prime })f(2x_2-2x)]\\nonumber \\\\&=&Sf(x),$ which implies that $Sf$ is even.", "Let ${\\rm dom}f:=\\lbrace x\\in \\mathbb {R}^n: f(x)<+\\infty \\rbrace $ denote the effective domain of $f$ .", "To prove the remaining part of the theorem, we shall consider two cases: ${\\rm dom} f =\\mathbb {R}$ and ${\\rm dom}f\\ne \\mathbb {R}$ .", "Case (1) ${\\rm dom} f =\\mathbb {R}$ .", "There are two steps.", "First Step.", "We shall prove that $Sf(0)=\\inf f$ and for any $x>0$ , there exists some $x^{\\prime }\\in \\mathbb {R}$ such that $Sf(x)=f(x^{\\prime })=f(x^{\\prime }-2x).$ Let $x=0$ , by (REF ), we have $Sf(0)&=&\\sup _{\\lambda \\in [0,1]}\\inf _{x_1+x_2=0}[\\lambda f(2x_1)+(1-\\lambda ) f(-2x_2)]\\nonumber \\\\&=&\\inf _{x_1\\in \\mathbb {R}}f(2x_1)=\\inf _{x\\in \\mathbb {R}} f(x).$ For $x>0$ , since $f$ is coercive and convex, there exists some $x^{\\prime }\\in \\mathbb {R}$ satisfying $f(x^{\\prime })=f(x^{\\prime }-2x).$ Indeed, let $f_x(x_1):=f(x_1)-f(x_1-2x)$ , $a=\\inf f$ and $f^{-1}(a)=[\\mu ,\\nu ]$ , by Lemma REF (ii), $f_x(x_1)<0$ if $x_1<\\mu $ and $f_x(x_1)>0$ if $x_1>\\nu $ .", "Since $f(x_1)$ and $f(x_1-2x)$ are convex functions about $x_1\\in \\mathbb {R}$ and any convex function is continuous on the interior of its effective domain, thus $f_x(x_1)$ is continuous in $\\mathbb {R}$ .", "Therefore, there exists some $x^{\\prime }$ such that $f_x(x^{\\prime })=0$ .", "Now we prove $Sf(x)=f(x^{\\prime })$ , where $x>0$ and $x^{\\prime }$ satisfies equality (REF ).", "Let $G_x(\\lambda )$ be a function about $\\lambda \\in [0,1]$ defined as $G_x(\\lambda ):=\\inf _{x_1\\in \\mathbb {R}}[\\lambda f(2x_1)+(1-\\lambda )f(2x_1-2x)].$ For any $\\lambda \\in [0,1]$ , choose $x_1=\\frac{x^{\\prime }}{2}$ , we have $G_x(\\lambda )\\le \\lambda f(x^{\\prime })+(1-\\lambda )f(x^{\\prime }-2x)=f(x^{\\prime }).$ Thus, $Sf(x)=\\sup _{\\lambda \\in [0,1]}G_x(\\lambda )\\le f(x^{\\prime })$ .", "On the other hand, we prove that there exists some $\\lambda _0\\in [0,1]$ such that $G_x(\\lambda _0)=f(x^{\\prime })$ .", "Since $f$ is a convex function defined in $\\mathbb {R}$ and by Theorem 1.5.2 in [30], both the right derivative $f^{\\prime }_{r}$ and the left derivative $f^{\\prime }_{l}$ exist and $f^{\\prime }_{l}\\le f^{\\prime }_{r}$ .", "Claim 1 There exists some $\\lambda _0\\in [0,1]$ satisfying $\\lambda _0f^{\\prime }_{r}(x^{\\prime })+(1-\\lambda _0)f^{\\prime }_{r}(x^{\\prime }-2x)=0.$ Proof of Claim REF .", "Since $f(x^{\\prime })=f(x^{\\prime }-2x)$ and $x>0$ , by Lemma REF (iv), we have $f_r^{\\prime }(x^{\\prime })\\ge 0$ and $f_r^{\\prime }(x^{\\prime }-2x)\\le 0$ , thus $f_r^{\\prime }(x^{\\prime })-f_r^{\\prime }(x^{\\prime }-2x)\\ge 0$ .", "(i) If $f^{\\prime }_{r}(x^{\\prime })-f^{\\prime }_{r}(x^{\\prime }-2x)>0$ , then (REF ) can be obtained by choosing $\\lambda _0=\\frac{-f^{\\prime }_{r}(x^{\\prime }-2x)}{f^{\\prime }_{r}(x^{\\prime })-f^{\\prime }_{r}(x^{\\prime }-2x)}.$ (ii) If $f^{\\prime }_{r}(x^{\\prime })-f^{\\prime }_{r}(x^{\\prime }-2x)=0$ , then $f_r^{\\prime }(x^{\\prime })=f_r^{\\prime }(x^{\\prime }-2x)= 0$ , thus, for any $\\lambda _0\\in [0,1]$ , we can get (REF ).", "$\\Box $ Choose a $\\lambda _0$ satisfying (REF ), we define $\\Phi _{\\lambda _0}(x_1)=\\lambda _0f(2x_1)+(1-\\lambda _0)f(2x_1-2x).$ Since $f$ is a convex function, then $\\Phi _{\\lambda _0}$ is a convex function about $x_1$ .", "By (REF ), we have that the right derivative and the left derivative of $\\Phi _{\\lambda _0}$ at $x_1=\\frac{x^{\\prime }}{2}$ satisfy $\\Phi _{\\lambda _0r}^{\\prime }(x_1)\|_{x_1=\\frac{x^{\\prime }}{2}}=2\\lambda _0f_r^{\\prime }(x^{\\prime })+2(1-\\lambda _0)f_r^{\\prime }(x^{\\prime }-2x)=0,$ and $\\Phi _{\\lambda _0l}^{\\prime }(x_1)\|_{x_1=\\frac{x^{\\prime }}{2}}\\le \\Phi _{\\lambda _0r}^{\\prime }(x_1)\|_{x_1=\\frac{x^{\\prime }}{2}}=0$ .", "By (REF ), (REF ) and the fact that if a convex function $f:\\mathbb {R}\\rightarrow \\mathbb {R}$ satisfies $f^{\\prime }_r(x_0)\\ge 0$ and $f^{\\prime }_l(x_0)\\le 0$ then $f(x_0)=\\min \\lbrace f(x):x\\in \\mathbb {R}\\rbrace $ , we have $\\inf _{x_1\\in \\mathbb {R}}\\Phi _{\\lambda _0}(x_1)=\\Phi _{\\lambda _0}(\\frac{x^{\\prime }}{2})=f(x^{\\prime }).$ By (REF ) and (REF ), we have $Sf(x)=\\sup _{\\lambda \\in [0,1]}G_x(\\lambda )\\ge G_x(\\lambda _0)=\\inf _{x_1\\in \\mathbb {R}}\\Phi _{\\lambda _0}(x_1)=f(x^{\\prime }).$ Thus, we have $Sf(x)=f(x^{\\prime })=f(x^{\\prime }-2x)$ .", "Second Step.", "We shall prove that $Sf$ is coercive and convex, and for any $s\\in \\mathbb {R}$ , $Vol_1 ([Sf\\le s])=Vol_1([f\\le s])$ .", "First, we prove that $Sf$ is coercive.", "Suppose that there exists $M_0>0$ and a sequence $\\lbrace x_n\\rbrace $ satisfying $\|x_n\|>n$ and $Sf(x_n)<M_0$ for any positive integer $n$ , then by (REF ), there exists $x_n^{\\prime }$ such that $Sf(x_n)=f(x_n^{\\prime })=f(x_n^{\\prime }-2x_n)<M_0.$ Since $2\\max \\lbrace \|x_n^{\\prime }\|,\|x_n^{\\prime }-2x_n\|\\rbrace \\ge \|x_n^{\\prime }\|+\|x_n^{\\prime }-2x_n\|\\ge 2\|x_n\|>2n$ , there is a sequence $\\lbrace y_n\\rbrace $ , where $y_n=x_n^{\\prime }$ if $\|x_n^{\\prime }\|\\ge \|x_n^{\\prime }-2x_n\|$ and $y_n=x_n^{\\prime }-2x_n$ if $\|x_n^{\\prime }\|\\le \|x_n^{\\prime }-2x_n\|$ , satisfying $\\lim _{n\\rightarrow +\\infty }\|y_n\|=+\\infty $ and $f(y_n)<M_0$ , which is contradictory with $f$ is coercive.", "Next, we prove that $Sf$ is a convex function on $\\mathbb {R}$ .", "First, we prove that $Sf(x)$ is increasing on $[0,+\\infty )$ .", "In fact, by (REF ), for any $0< x_1<x_2$ , there exist $x_1^{\\prime }$ and $x_2^{\\prime }$ such that $Sf(x_i)=f(x_i^{\\prime })=f(x_i^{\\prime }-2x_i)$ $(i=1,2)$ .", "By Lemma REF (iii), for $\\mu $ and $\\nu $ given in Lemma REF , we have $x_i^{\\prime }>\\mu $ $(i=1,2)$ and $x_i^{\\prime }-2x_i<\\nu $ $(i=1,2)$ .", "If $f(x_1^{\\prime })> f(x_2^{\\prime })$ , since $f$ is increasing on the interval $[\\mu ,+\\infty )$ , then $x_1^{\\prime }>x_2^{\\prime }$ .", "By $0<x_1<x_2$ , we have $x_1^{\\prime }-2x_1>x_2^{\\prime }-2x_2$ .", "Since $f$ is decreasing on the interval $(-\\infty ,\\nu ]$ , we have $f(x_1^{\\prime }-2x_1)\\le f(x_2^{\\prime }-2x_2)$ , which is a contradiction.", "The contradiction means that $f(x_1^{\\prime })\\le f(x_2^{\\prime })$ , thus $Sf$ is increasing on $[0,+\\infty )$ .", "Since $Sf$ is even, to prove $Sf$ is convex on $\\mathbb {R}$ , it suffices to prove that $Sf$ is convex on $[0,+\\infty )$ .", "For any $0\\le x_1<x_2$ and $0<\\alpha <1$ , by (REF ), let $x_1^{\\prime }$ , $x_2^{\\prime }$ and $x_0\\triangleq (\\alpha x_1+(1-\\alpha ) x_2)^{\\prime }$ be three real numbers satisfying $Sf(x_1)=f(x^{\\prime }_1)=f(x_1^{\\prime }-2x_1),$ $Sf(x_2)=f(x^{\\prime }_2)=f(x_2^{\\prime }-2x_2),$ $Sf(\\alpha x_1+(1-\\alpha )x_2)=f(x_0)=f(x_0-2(\\alpha x_1+(1-\\alpha ) x_2)).$ Since $f$ is a convex function, we have $\\alpha f(x_1^{\\prime })+(1-\\alpha )f(x_2^{\\prime })\\ge f(\\alpha x_1^{\\prime }+(1-\\alpha )x_2^{\\prime }),$ $&&\\alpha f(x_1^{\\prime }-2x_1)+(1-\\alpha )f(x_2^{\\prime }-2x_2)\\nonumber \\\\&\\ge & f(\\alpha x_1^{\\prime }+(1-\\alpha )x_2^{\\prime }-2(\\alpha x_1+(1-\\alpha )x_2)).$ Since $f(x_0)=f(x_0-2(\\alpha x_1+(1-\\alpha ) x_2))$ and both $[x_0-2(\\alpha x_1+(1-\\alpha )x_2),x_0]$ and $[\\alpha x^{\\prime }_1+(1-\\alpha )x^{\\prime }_2-2(\\alpha x_1+(1-\\alpha )x_2),\\alpha x^{\\prime }_1+(1-\\alpha )x^{\\prime }_2]$ have the same length $2(\\alpha x_1+(1-\\alpha )x_2)>0$ , by Lemma REF (v), either $f(\\alpha x_1^{\\prime }+(1-\\alpha )x_2^{\\prime })\\ge f(x_0)$ or $&&f(\\alpha x_1^{\\prime }+(1-\\alpha )x_2^{\\prime }-2(\\alpha x_1+(1-\\alpha )x_2))\\nonumber \\\\&\\ge & f(x_0-2(\\alpha x_1+(1-\\alpha )x_2)).$ If (REF ) holds, then we use (REF ) and if (REF ) holds, then we use (REF ), by (REF )-(REF ), $Sf$ is a convex function.", "Finally, we prove that $Vol_1 ([f\\le s])=Vol_1([Sf\\le s])$ for any $s\\in \\mathbb {R}$ .", "Since $Sf(x)$ is an even convex function, $Sf(0)=\\inf Sf$ .", "Since $Sf(0)=\\inf f$ by (REF ), thus $\\inf Sf=\\inf f$ .", "Let $a=\\inf Sf=\\inf f$ , $(Sf)^{-1}(a)=[-\\delta ,\\delta ]$ , and $f^{-1}(a)=[\\mu ,\\nu ]$ .", "If $s=a$ , then $Vol_1 ([f\\le s])=\\nu -\\mu $ and $Vol_1([Sf\\le s])=2\\delta $ .", "Next, we prove $\\nu -\\mu =2\\delta $ .", "By Lemma REF , $Sf$ is strictly decreasing on $(-\\infty ,-\\delta )$ and strictly increasing on $(\\delta ,+\\infty )$ , and $f$ is strictly decreasing on $(-\\infty ,\\mu )$ and strictly increasing on $(\\nu ,+\\infty )$ .", "For $\\delta \\ge 0$ , if $\\nu -\\mu >2\\delta $ , then $x_0:=\\delta +\\frac{\\nu -\\mu -2\\delta }{2}>\\delta $ , thus $Sf(x_0)>Sf(\\delta )$ , which is contradictory with $Sf(x_0)&=&\\sup _{\\lambda \\in [0,1]}\\inf _{x_1\\in \\mathbb {R}}[\\lambda f(2x_1)+(1-\\lambda )f(2x_1-2x_0)]\\nonumber \\\\&\\le &\\sup _{\\lambda \\in [0,1]}[\\lambda f(\\nu )+(1-\\lambda )f(\\nu -2x_0)]=a,$ where inequality is by choosing $x_1=\\frac{\\nu }{2}$ and last equality is by $\\nu -2x_0=\\mu $ .", "Thus, $\\nu -\\mu \\le 2\\delta $ .", "Thus if $\\delta =0$ , then $\\mu =\\nu $ .", "For $\\delta >0$ , by (REF ), there exists $\\delta ^{\\prime }$ such that $Sf(\\delta )=f(\\delta ^{\\prime })=f(\\delta ^{\\prime }-2\\delta )=a$ , which implies that $\\nu -\\mu \\ge 2\\delta $ .", "Thus, $\\nu -\\mu = 2\\delta $ .", "If $s>a$ , by Lemma REF , equality (REF ), and $Sf$ is even, there is a unique $x>0$ and a unique $x^{\\prime }\\in \\mathbb {R}$ such that $Sf(-x)=Sf(x)=s=f(x^{\\prime })=f(x^{\\prime }-2x)$ , thus we have $Vol_1([f\\le s])=Vol_1([Sf\\le s])=2x$ .", "If $s<a$ , then $[Sf\\le s]=[f\\le s]=\\emptyset $ , thus $Vol_1 ([f\\le s])=Vol_1([Sf\\le s])=0$ .", "Case (2) ${\\rm dom}f\\ne \\mathbb {R}$ .", "There exist eight cases for ${\\rm dom}f$ : 1) $[\\alpha ,\\beta ]$ ; 2) $(\\alpha ,\\beta )$ ; 3) $(\\alpha ,\\beta ]$ ; 4) $[\\alpha ,\\beta )$ ; 5) $(-\\infty ,\\beta ]$ ; 6) $(-\\infty ,\\beta )$ ; 7) $[\\alpha ,+\\infty )$ ; 8) $(\\alpha ,+\\infty )$ .", "We need only prove our conclusion for ${\\rm dom}f=(\\alpha ,\\beta )$ .", "By the same method we can prove our conclusion for other cases.", "For ${\\rm dom}f=(\\alpha ,\\beta )$ , there exist three cases: (i) $f$ is decreasing on $(\\alpha ,\\beta )$ ; (ii) $f$ is increasing on $(\\alpha ,\\beta )$ ; (iii) $f$ is decreasing on $(\\alpha ,\\gamma ]$ and increasing on $[\\gamma ,\\beta )$ for some $\\gamma \\in (\\alpha ,\\beta )$ .", "Cases (i) and (ii) are corresponding to the cases of $\\lim _{\\gamma \\rightarrow \\beta ,\\gamma <\\beta }\\gamma $ and $\\lim _{\\gamma \\rightarrow \\alpha ,\\gamma >\\alpha }\\gamma $ in case (iii), respectively, thus we need only prove our conclusion for case (iii).", "If $\\lim _{x\\rightarrow \\alpha ,x>\\alpha }f(x)=\\lim _{x\\rightarrow \\beta ,x<\\beta }f(x)$ , following the proof of Case (1) (i.e., ${\\rm dom}f=\\mathbb {R}$ ), we have that $Sf$ is convex on $(-\\frac{\\beta -\\alpha }{2},\\frac{\\beta -\\alpha }{2})$ and $Vol_1([Sf\\le s])=Vol_1([f\\le s])$ for any $s<\\lim _{x\\rightarrow \\alpha ,x>\\alpha }f(x)$ .", "If $\\lim _{x\\rightarrow \\alpha ,x>\\alpha }f(x)\\ne \\lim _{x\\rightarrow \\beta ,x<\\beta }f(x)$ , we may assume that $\\lim _{x\\rightarrow \\alpha ,x>\\alpha }f(x)=b>\\lim _{x\\rightarrow \\beta ,x<\\beta }f(x)=c.$ If $c=a=\\inf f$ , then $f$ is decreasing on $(\\alpha , \\beta )$ .", "Thus we may suppose that $c>a$ .", "Let $\\gamma \\in (\\alpha ,\\beta )$ satisfy $f(\\gamma )=c$ .", "If $\|x\|<\\frac{\\beta -\\gamma }{2}$ , by the proof of Case (1), there exists $x^{\\prime }\\in (\\gamma ,\\beta )$ such that $Sf(x)=f(x^{\\prime })=f(x^{\\prime }-2x)$ .", "Step 1.", "We shall prove that for $\|x\|\\ge \\frac{\\beta -\\gamma }{2}$ and $\|x\|<\\frac{\\beta -\\alpha }{2}$ , $Sf(x)=f(\\beta -2\|x\|).$ Since $Sf$ is even, we may assume $\\frac{\\beta -\\gamma }{2}\\le x<\\frac{\\beta -\\alpha }{2}$ .", "For any $\\lambda \\in [0,1]$ , $&&\\inf _{x_1\\in \\mathbb {R}}[\\lambda f(2x_1)+(1-\\lambda )f(2x_1-2x)]\\nonumber \\\\&\\le &\\lambda \\lim _{t\\rightarrow \\beta ,t<\\beta }f(t)+(1-\\lambda )f(\\beta -2x)\\nonumber \\\\&=&\\lambda c+(1-\\lambda )f(\\beta -2x).$ Since $\\frac{\\beta -\\gamma }{2}\\le x<\\frac{\\beta -\\alpha }{2}$ , then $\\alpha <\\beta -2x\\le \\gamma $ .", "Since $f$ is decreasing on $(\\alpha ,\\gamma ]$ , thus $f(\\beta -2x)\\ge f(\\gamma )=c$ .", "Thus, by (REF ), we have $Sf(x)&=&\\sup _{\\lambda \\in [0,1]}\\inf _{x_1\\in \\mathbb {R}^n}[\\lambda f(2x_1)+(1-\\lambda )f(2x_1-2x)]\\nonumber \\\\&\\le &\\sup _{\\lambda \\in [0,1]}[\\lambda c+(1-\\lambda )f(\\beta -2x)]=f(\\beta -2x).$ On the other hand, we prove that $Sf(x)\\ge f(\\beta -2x)$ .", "Since ${\\rm dom}f=(\\alpha ,\\beta )$ and $\\inf _{x_1\\in \\mathbb {R}}[\\lambda f(x_1)+(1-\\lambda )f(x_1-2x)]=\\inf f$ for $\\lambda =0$ or $\\lambda =1$ , we have $Sf(x)=\\sup _{\\lambda \\in (0,1)}\\inf _{x_1\\in (\\alpha +2x,\\beta )}[\\lambda f(x_1)+(1-\\lambda )f(x_1-2x)].$ By $b>c>a$ , if $f^{-1}(a)=[\\mu ,\\nu ]$ , then $\\alpha <\\mu \\le \\nu <\\beta $ , thus $f$ is strictly decreasing on $(\\alpha ,\\mu ]$ and strictly increasing on $[\\nu ,\\beta )$ .", "Claim 2 For a fixed $\\beta ^{\\prime }\\in (\\nu ,\\beta )\\cap (\\alpha +2x,\\beta )$ , there exists $\\delta >0$ such that function $G_x(x_1):=\\lambda f(x_1)+(1-\\lambda )f(x_1-2x)$ is decreasing on $(\\alpha +2x,\\beta ^{\\prime }]$ for any $0<\\lambda <\\delta $ .", "Proof of Claim REF .", "For $x_1\\in (\\alpha +2x,\\beta ^{\\prime }]$ , the right derivative of $G_x(x_1)$ $G_{xr}^{\\prime }(x_1)&=&\\lambda f_r^{\\prime }(x_1)+(1-\\lambda )f_r^{\\prime }(x_1-2x)\\nonumber \\\\&\\le &\\lambda f_r^{\\prime }(\\beta ^{\\prime })+(1-\\lambda )f_r^{\\prime }(\\beta ^{\\prime }-2x),$ where the inequality is by the right derivative of a convex function is increasing on the interior of its effective domain.", "Since $\\beta ^{\\prime }\\in (\\nu ,\\beta )\\cap (\\alpha +2x,\\beta )$ and $x\\in [\\frac{\\beta -\\gamma }{2},\\frac{\\beta -\\alpha }{2})$ , then $\\beta ^{\\prime }-2x\\in (\\alpha ,\\gamma +\\beta ^{\\prime }-\\beta )$ , thus $f_r^{\\prime }(\\beta ^{\\prime })>0$ and $f_r^{\\prime }(\\beta ^{\\prime }-2x)<0$ for $f$ is strictly increasing on $(\\nu ,\\beta )$ and strictly decreasing on $(\\alpha ,\\gamma ]$ .", "Thus, by (REF ), we choose $\\delta =\\frac{-f_r^{\\prime }(\\beta ^{\\prime }-2x)}{f_r^{\\prime }(\\beta ^{\\prime })-f_r^{\\prime }(\\beta ^{\\prime }-2x)},$ then $G_{xr}^{\\prime }(x_1)<0$ on $(\\alpha +2x,\\beta ^{\\prime }]$ for any $\\lambda \\in (0,\\delta )$ .", "Therefore, $G_x(x_1)$ is decreasing on $(\\alpha +2x,\\beta ^{\\prime }]$ for any $\\lambda \\in (0,\\delta )$ .", "$\\Box $ By (REF ) and Claim REF , we have that $Sf(x)&=&\\sup _{\\lambda \\in (0,1)}\\inf _{x_1\\in (\\alpha +2x,\\beta )}[\\lambda f(x_1)+(1-\\lambda )f(x_1-2x)]\\nonumber \\\\&\\ge &\\sup _{\\lambda \\in (0,\\delta )}\\inf _{x_1\\in (\\alpha +2x,\\beta )}[\\lambda f(x_1)+(1-\\lambda )f(x_1-2x)]\\nonumber \\\\&=&\\sup _{\\lambda \\in (0,\\delta )}\\inf _{x_1\\in [\\beta ^{\\prime },\\beta )}[\\lambda f(x_1)+(1-\\lambda )f(x_1-2x)]\\nonumber \\\\&\\ge &\\sup _{\\lambda \\in (0,\\delta )}[\\lambda f(\\beta ^{\\prime })+(1-\\lambda )f(\\beta -2x)]\\nonumber \\\\&=&f(\\beta -2x),$ where the second inequality is by $x_1\\in [\\beta ^{\\prime },\\beta )\\subset (\\nu ,\\beta )$ and $\\beta ^{\\prime }-2x\\le x_1-2x<\\beta -2x\\le \\gamma $ and $f$ is strictly increasing on $(\\nu ,\\beta )$ and strictly decreasing on $(\\alpha ,\\gamma ]$ , and the last equality is by $f(\\beta -2x)\\ge f(\\beta ^{\\prime })$ .", "Step 2.", "We shall prove that $Sf$ is convex in $\\mathbb {R}$ .", "Since $Sf$ is increasing on $[0,\\frac{\\beta -\\alpha }{2})$ and $Sf$ is even on $(-\\frac{\\beta -\\alpha }{2},\\frac{\\beta -\\alpha }{2})$ .", "Thus, it suffices to prove $Sf$ is convex in $[0,\\frac{\\beta -\\alpha }{2})$ .", "For any $x_1, x_2\\in [\\frac{\\beta -\\gamma }{2},\\frac{\\beta -\\alpha }{2})$ and $\\lambda \\in (0,1)$ , by (REF ) and $f$ is convex function, then $\\lambda Sf(x_1)+(1-\\lambda )Sf(x_2)&=&\\lambda f(\\beta -2x_1)+(1-\\lambda )f(\\beta -2x_2)\\nonumber \\\\&\\ge &f(\\beta -2(\\lambda x_1+(1-\\lambda )x_2))\\nonumber \\\\&=&Sf(\\lambda x_1+(1-\\lambda )x_2),$ where the last equality is by $\\lambda x_1+(1-\\lambda )x_2\\in [\\frac{\\beta -\\gamma }{2},\\frac{\\beta -\\alpha }{2})$ .", "By (REF ), $Sf$ is convex on $[\\frac{\\beta -\\gamma }{2},\\frac{\\beta -\\alpha }{2})$ .", "Because that $Sf$ is convex in $[0,\\frac{\\beta -\\gamma }{2}]$ by the proof in Case (1), it suffices to prove that the left derivative of $Sf$ at $x=\\frac{\\beta -\\gamma }{2}$ is less than its right derivative at $x=\\frac{\\beta -\\gamma }{2}$ .", "By (REF ), we have $Sf_r^{\\prime }(\\frac{\\beta -\\gamma }{2})&=&\\lim _{t\\rightarrow 0,t>0}\\frac{Sf(\\frac{\\beta -\\gamma }{2}+t)-Sf(\\frac{\\beta -\\gamma }{2})}{t}\\nonumber \\\\&=&\\lim _{t\\rightarrow 0,t>0}\\frac{f(\\gamma -2t)-f(\\gamma )}{t}=-2f_l^{\\prime }(\\gamma ).$ For any $t\\in (-\\frac{\\beta -\\gamma }{2},0)$ , we have $\\frac{\\beta -\\gamma }{2}+t\\in (0,\\frac{\\beta -\\gamma }{2})$ .", "Thus there exist $x^{\\prime },\\;x^{\\prime \\prime }\\in (\\gamma ,\\beta )$ such that $x^{\\prime \\prime }-x^{\\prime }=2(\\frac{\\beta -\\gamma }{2}+t)$ and $Sf(\\frac{\\beta -\\gamma }{2}+t)=f(x^{\\prime })=f(x^{\\prime \\prime })$ .", "Since $(x^{\\prime }-\\gamma )+2(\\frac{\\beta -\\gamma }{2}+t)=(x^{\\prime }-\\gamma )+(x^{\\prime \\prime }-x^{\\prime })=x^{\\prime \\prime }-\\gamma <\\beta -\\gamma ,$ $x^{\\prime }<\\gamma -2t$ .", "Let $\|t\|$ be sufficiently small such that $\\gamma +2\|t\|<\\mu $ , where $\\mu $ satisfies $f^{-1}(a)=[\\mu ,\\nu ]$ , then $f(x^{\\prime })>f(\\gamma -2t)$ for $f$ is strictly decreasing on $(\\gamma ,\\mu )$ .", "Then $Sf_l^{\\prime }(\\frac{\\beta -\\gamma }{2})&=&\\lim _{t\\rightarrow 0,t<0}\\frac{Sf(\\frac{\\beta -\\gamma }{2}+t)-Sf(\\frac{\\beta -\\gamma }{2})}{t}=\\lim _{t\\rightarrow 0,t<0}\\frac{f(x^{\\prime })-f(\\gamma )}{t}\\nonumber \\\\&\\le &\\lim _{t\\rightarrow 0,t<0}\\frac{f(\\gamma -2t)-f(\\gamma )}{t}=-2f_r^{\\prime }(\\gamma ).$ Since $f$ is a convex function, then $f_l^{\\prime }(\\gamma )\\le f_r^{\\prime }(\\gamma )$ , by (REF ) and (REF ), we have $Sf_l^{\\prime }(\\frac{\\beta -\\gamma }{2})\\le Sf_r^{\\prime }(\\frac{\\beta -\\gamma }{2})$ .", "Step 3.", "Proof of $Vol_1([Sf\\le a])=Vol([f\\le s])$ for any $s\\in \\mathbb {R}$ .", "If $s<c$ , the proof is the same as in Case (1).", "If $c\\le s<b$ , since $f$ is strictly decreasing on $(\\alpha ,\\gamma )$ , there is a unique $x^{\\prime }\\in (\\alpha ,\\gamma )$ such that $f(x^{\\prime })=s$ , thus $[f<s]=[x^{\\prime },\\beta )$ .", "By (REF ), we have $Sf(\\frac{\\beta -x^{\\prime }}{2})=f(x^{\\prime })=s$ , thus $[Sf<s]=[-\\frac{\\beta -x^{\\prime }}{2},\\frac{\\beta -x^{\\prime }}{2}]$ .", "Therefore, $Vol_1([Sf<s])=Vol_1([f<s])=\\beta -x^{\\prime }$ .", "If $s\\ge b$ , then $b<+\\infty $ for $s\\in \\mathbb {R}$ , we have $Vol_1([Sf<s])=Vol_1([f<s])=\\beta -\\alpha $ .$\\Box $ Remark 1) By Theorem REF , for any $x\\in \\mathbb {R}$ , if $x=0$ , then $Sf(0)=\\inf f$ ; if $x\\ne 0$ , then there exist three cases: i) $Sf(x)=f(x^{\\prime })=f(x^{\\prime }-2\|x\|)$ for some $x^{\\prime }\\in \\mathbb {R}$ ; ii) $Sf(x)=f(x_0-2\|x\|)$ for some $x_0\\in \\mathbb {R}$ ; iii) $S_u f(x)=f(x_0+2\|x\|)$ for some $x_0\\in \\mathbb {R}$ .", "2) In Theorem REF , there exist three cases for ${\\rm dom}Sf$ : i) ${\\rm dom}Sf=(-\\delta ,\\delta )$ ; ii) ${\\rm dom}Sf=[-\\delta ,\\delta ]$ ; iii) ${\\rm dom}Sf=\\mathbb {R}$ .", "${\\rm dom}Sf=(-\\delta ,\\delta )$ is corresponding to ${\\rm dom}f=(\\alpha ,\\beta )$ , ${\\rm dom}f=(\\alpha ,\\beta ]$ or ${\\rm dom}f=[\\alpha ,\\beta )$ , where $\\delta =\\frac{\\beta -\\alpha }{2}$ .", "${\\rm dom}Sf=[-\\delta ,\\delta ]$ is corresponding to ${\\rm dom}f=[\\alpha ,\\beta ]$ .", "${\\rm dom}Sf=\\mathbb {R}$ is corresponding to ${\\rm dom}f=(-\\infty ,\\beta )$ , ${\\rm dom}f=(-\\infty ,\\beta ]$ , ${\\rm dom}f=(\\alpha ,+\\infty )$ , ${\\rm dom}f=[\\alpha ,+\\infty )$ or ${\\rm dom}f=\\mathbb {R}$ .", "For a non-empty convex set $K\\subset \\mathbb {R}^n$ and a hyperplane $H=u^{\\perp }$ , where $u\\in S^{n-1}$ , the Steiner symmetrization $S_H K$ of $K$ about $H$ is defined as $S_{H}K=\\lbrace x^{\\prime }+\\frac{1}{2}(t_1-t_2)u:\\;x^{\\prime }\\in P_H(K),\\;t_i\\in I_K(x^{\\prime })\\;{\\rm for}\\;i=1,2\\rbrace ,$ where $P_H(K)=\\lbrace x^{\\prime }\\in H:\\;x^{\\prime }+tu\\in K\\; {\\rm for\\;some}\\;t\\in \\mathbb {R}\\rbrace $ is the projection of $K$ onto $H$ and $I_K(x^{\\prime })=\\lbrace t\\in \\mathbb {R}:\\;x^{\\prime }+tu\\in K\\rbrace $ .", "By the above definition and Definition REF , for coercive convex function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ and its Steiner symmetrization $S_u f$ , we have ${\\rm dom}(S_{u^{\\perp }} f)=S_{u^{\\perp }} ({\\rm dom}f).$ We know that ${\\rm dom}f$ is convex if $f$ is convex and the Steiner symmetrization of a non-empty convex set is still a convex set, thus by (REF ), ${\\rm dom}(S_{u^{\\perp }} f)$ is a convex set.", "3) For a convex function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ , the epigraph of $f$ is defined as ${\\rm epi}f:=\\lbrace (x,y)\\in \\mathbb {R}^{n+1}: x\\in {\\rm dom}f,\\;y\\ge f(x)\\rbrace $ .", "By the definition of epigraph and Theorem REF , for one-dimensional coercive convex function $f:\\mathbb {R}\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ , we have ${\\rm cl}({\\rm epi}Sf)=S_{e^{\\perp }}({\\rm cl}({\\rm epi}f))$ , where $e$ is a unit vector along the $x$ -axis and ${\\rm cl} A$ denotes the closure of a subset $A\\subset \\mathbb {R}^n$ .", "Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ be a coercive and convex function and $u\\in S^{n-1}$ .", "For any $x^{\\prime }\\in u^{\\perp }$ and $t\\in \\mathbb {R}$ , if $\\tilde{f}(t)=f(x^{\\prime }+tu)$ is considered as a one-dimensional function about $t$ , then $S\\tilde{f}(t)=S_uf(x^{\\prime }+tu)$ .", "By Theorem REF , ${\\rm cl}({\\rm epi}(S\\tilde{f}))=S_{e^{\\perp }}({\\rm cl}({\\rm epi}\\tilde{f}))$ .", "Since $x^{\\prime }\\in u^{\\perp }$ is arbitrary, thus we have $ {\\rm cl}({\\rm epi}(S_u f))=S_{\\tilde{u}^{\\perp }}({\\rm cl}({\\rm epi}f)),$ where $\\tilde{u}^{\\perp }\\subset \\mathbb {R}^{n+1}$ denotes the hyperplane through the origin and orthogonal to the unit vector $\\tilde{u}=(u,0)\\in \\mathbb {R}^{n+1}$ .", "Next, by Definition REF and Theorem REF , we shall prove five propositions which are corresponding to properties 1-5 in Table 1.", "The following lemma is an obvious fact, and we omit its proof.", "Lemma 2 For $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ , let $u\\in S^{n-1}$ and $H=u^{\\perp }$ , if i) $f$ is symmetric with respect to hyperplane $H$ , i.e., for any $x^{\\prime }\\in H$ and $t\\in \\mathbb {R}$ , $f(x^{\\prime }+tu)=f(x^{\\prime }-tu)$ ; ii) for any $x^{\\prime }\\in H$ and $t_1$ , $t_2\\in \\mathbb {R}$ , if $\|t_1\|\\le \|t_2\|$ , then $f(x^{\\prime }+t_1 u)\\le f(x^{\\prime }+t_2u)$ ; iii) $f$ is convex on half-space $H^{+}:=\\lbrace x^{\\prime }+tu: x^{\\prime }\\in u^{\\perp },t\\ge 0\\rbrace $ .", "Then $f$ is a convex function on $\\mathbb {R}^n$ .", "Proposition 2.1 If $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is a coercive convex function and $u\\in S^{n-1}$ , then $S_uf$ is a coercive convex function and symmetric about $u^{\\perp }$ .", "It is clear that $S_uf$ is symmetric about $u^{\\perp }$ .", "Indeed, for any $x^{\\prime }\\in u^{\\perp }$ and $t\\in \\mathbb {R}$ , if we consider $S_uf(x^{\\prime }+tu)$ as a one-dimensional function about $t$ , then by Theorem REF and Definition REF , we have $S_uf(x^{\\prime }+tu)=S_uf(x^{\\prime }-tu)$ .", "Step 1.", "We shall prove that $S_u f$ is coercive.", "Suppose that there exist $M_0>0$ and a sequence $\\lbrace x_n\\rbrace _{n=1}^{\\infty }\\subset \\mathbb {R}^n$ satisfying that $\|x_n\|>n$ and $S_u f(x_n)<M_0$ .", "Next, we shall construct a sequence $\\lbrace y_n\\rbrace $ satisfying $\|y_n\|>n$ and $f(y_n)<M_0$ , which is contradictory with $f$ is coercive.", "For any positive integer $n\\ge 1$ , let $x_n=x_n^{\\prime }+t_n u$ and $x_n^{\\prime }\\in u^{\\perp }$ .", "There exist two cases of $t_n\\ne 0$ and $t_n=0$ .", "(1) If $t_n\\ne 0$ , then by Theorem REF , there exist three cases: i) $S_uf(x_n)=f(x_n^{\\prime }+t_n^{\\prime }u)=f(x_n^{\\prime }+(t_n^{\\prime }-2t_n)u)$ for some $t_n^{\\prime }\\in \\mathbb {R}$ ; ii) $S_u f(x_n)=f(x_n^{\\prime }+(t_0-2t_n)u)$ for some $t_0\\in \\mathbb {R}$ ; iii) $S_u f(x_n)=f(x_n^{\\prime }+(t_0+2t_n)u)$ for some $t_0\\in \\mathbb {R}$ .", "For case i), since $\|t_n^{\\prime }\|+\|t_n^{\\prime }-2t_n\|\\ge 2\|t_n\|$ , then either $\|t_n^{\\prime }\|\\ge \|t_n\|$ or $\|t_n^{\\prime }-2t_n\|\\ge \|t_n\|$ .", "If $\|t_n^{\\prime }\|\\ge \|t_n\|$ , let $y_n=x_n^{\\prime }+t_n^{\\prime }u$ , then $S_uf(x_n)=f(y_n)$ and $\|y_n\|=\|x_n^{\\prime }\|+\|t_n^{\\prime }\|\\ge \|x_n^{\\prime }\|+\|t_n\|=\|x_n\|$ .", "If $\|t_n^{\\prime }-2t_n\|\\ge \|t_n\|$ , let $y_n=x_n^{\\prime }+(t_n^{\\prime }-2t_n)u$ , then $S_uf(x_n)=f(y_n)$ and $\|y_n\|=\|x_n^{\\prime }\|+\|t_n^{\\prime }-2t_n\|\\ge \|x_n^{\\prime }\|+\|t_n\|=\|x_n\|$ .", "Since $\|x_n\|>n$ , we have $\|y_n\|> n$ and $f(y_n)=S_uf(x_n)<M_0$ .", "For case ii), since $\|t_0\|+\|t_0-2t_n\|\\ge 2\|t_n\|$ , we have either $\|t_0\|\\ge \|t_n\|$ or $\|t_0-2t_n\|\\ge \|t_n\|$ .", "If $\|t_0-2t_n\|\\ge \|t_n\|$ , let $y_n=x_n^{\\prime }+(t_0-2t_n)u$ , then $S_uf(x_n)=f(y_n)$ and $\|y_n\|\\ge \|x_n\|$ .", "If $\|t_0\|\\ge \|t_n\|$ , let $y_n=x_n^{\\prime }+t_0u$ if $x_n^{\\prime }+t_0u\\in {\\rm dom} f$ , otherwise let $y_n=x_n^{\\prime }+t_0^{\\prime }u$ , where $t_0^{\\prime }$ satisfies $x_n^{\\prime }+t_0^{\\prime }u\\in {\\rm dom}f$ , $\|x_n^{\\prime }+t_0^{\\prime }u\|>n$ and $f(x_n^{\\prime }+t_0^{\\prime }u)<f(x_n^{\\prime }+(t_0-2t_n)u)$ , which can be satisfied for $\\lim _{t\\rightarrow t_0,\\;t<t_0}f(x_n^{\\prime }+t u)\\le f(x_n^{\\prime }+(t_0-2t_n)u)$ by Theorem REF .", "Thus, we have $\|y_n\|> n$ and $f(y_n)< M_0$ .", "For case iii), we can construct $\\lbrace y_n\\rbrace $ with the same method as in case (ii).", "(2) If $t_n=0$ , by Definition REF , we have $Sf(x_n)=\\inf _{t\\in \\mathbb {R}}f(x_n^{\\prime }+tu)$ .", "Since $S_uf(x_n)<M_0$ , there exists $y_n=x_n^{\\prime }+t^{\\prime }u$ such that $f(y_n)<M_0$ .", "Since $\|y_n\|=\|x_n^{\\prime }\|+\|t^{\\prime }\|\\ge \|x_n^{\\prime }\|=\|x_n\|$ , we have $\|y_n\|>n$ and $f(y_n)<M_0$ .", "Step 2.", "We shall prove that $S_uf$ is convex.", "Claim 3 $S_uf$ is proper, i.e., $[S_uf=+\\infty ]\\ne \\mathbb {R}^n$ and $[S_uf=-\\infty ]=\\emptyset $ .", "Proof of Claim REF .", "For any $x\\in \\mathbb {R}^n$ , let $x=x^{\\prime }+tu$ , where $x^{\\prime }\\in u^{\\perp }$ .", "Since $f$ is a coercive convex function defined on $\\mathbb {R}^n$ , one dimensional function $f(x^{\\prime }+tu)$ about $t\\in \\mathbb {R}$ either is a coercive convex function or is identically $+\\infty $ .", "If $f(x^{\\prime }+tu)$ is a coercive convex function, then there exists $s\\in \\mathbb {R}$ such that $s=\\inf \\lbrace f(x^{\\prime }+tu):t\\in \\mathbb {R}\\rbrace $ .", "Thus, we have $Sf(x)=\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=t}[\\lambda f(x^{\\prime }+2t_1u)+(1-\\lambda )f(x^{\\prime }-2t_2u)]\\ge s,$ which implies that $Sf(x)>-\\infty $ .", "If $f(x^{\\prime }+tu)$ is identically $+\\infty $ , then $S_uf(x)=+\\infty >-\\infty $ .", "By the definition of convex functions, $f$ is not identically $+\\infty $ , there exists $x\\in \\mathbb {R}^n$ such that $f(x)<+\\infty $ .", "Let $x=x_0+tu$ , where $x_0\\in u^{\\perp }$ , then $S_uf(x_0)=\\inf _{t_1\\in \\mathbb {R}}f(x_0+t_1u)\\le f(x)<+\\infty ,$ which implies that $S_uf$ is not identically $+\\infty $ .$\\Box $ By Definition REF and Theorem REF , for any $x^{\\prime }\\in u^{\\perp }$ , one-dimensional function $S_uf(x^{\\prime }+tu)$ is either an even and coercive convex function about $t\\in \\mathbb {R}$ or identically $+\\infty $ .", "Thus, $S_uf$ satisfies conditions i) and ii) in Lemma REF .", "Therefore, to prove that $S_uf$ is convex, it suffices to prove that $S_uf$ satisfies condition iii) of Lemma REF .", "For any $x$ , $y\\in \\lbrace x^{\\prime }+tu:x^{\\prime }\\in u^{\\perp },t\\ge 0\\rbrace $ and $\\lambda \\in (0,1)$ , if $x\\notin {\\rm dom} (S_uf)$ or $y\\notin {\\rm dom} (S_uf)$ , then $S_uf(x)=+\\infty $ or $S_uf(y)=+\\infty $ , thus $S_uf(\\lambda x+(1-\\lambda )y)\\le \\lambda S_uf(x)+(1-\\lambda )S_uf(y).$ By Remark 2), ${\\rm dom}(S_uf)$ is convex.", "Therefore, if $x\\in {\\rm dom}(S_uf)$ and $y\\in {\\rm dom} (S_uf)$ , then $\\lambda x+(1-\\lambda )y\\in {\\rm dom}(S_uf)$ .", "Let $x=x^{\\prime }+tu$ and $y=y^{\\prime }+su$ , where $x^{\\prime }$ , $y^{\\prime }\\in u^{\\perp }$ and $t\\ge 0$ and $s\\ge 0$ , then $\\lambda x+(1-\\lambda )y=[\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }]+[\\lambda t+(1-\\lambda )s]u$ .", "Case 3.1.", "The case of $t=0$ and $s=0$ .", "For the case we have $x$ , $y\\in u^{\\perp }$ , thus $\\lambda x+(1-\\lambda )y\\in u^{\\perp }$ .", "By Definition REF and $f$ is convex, we have $&&\\lambda S_uf(x)+(1-\\lambda )S_uf(y)\\nonumber \\\\&=&\\lambda \\inf _{t\\in \\mathbb {R}}f(x+tu)+(1-\\lambda )\\inf _{s\\in \\mathbb {R}}f(y+su)\\nonumber \\\\&=&\\inf _{(t,s)\\in \\mathbb {R}^2}[\\lambda f(x+tu)+(1-\\lambda )f(y+su)]\\nonumber \\\\&\\ge &\\inf _{(t,s)\\in \\mathbb {R}^2}f(\\lambda x+(1-\\lambda )y+(\\lambda t+(1-\\lambda )s)u)\\nonumber \\\\&=&S_uf(\\lambda x+(1-\\lambda )y).$ Case 3.2.", "The case of $t>0$ and $s>0$ .", "For $x=x^{\\prime }+tu\\in {\\rm dom}(S_uf)$ , by Theorem REF , there exist three cases: $a_1$ ) There exists some $t^{\\prime }\\in \\mathbb {R}$ such that $S_uf(x)=f(x^{\\prime }+t^{\\prime }u)=f(x^{\\prime }+(t^{\\prime }-2t)u);$ $a_2$ ) There exists some $t_0\\in \\mathbb {R}$ such that $S_uf(x)=f(x^{\\prime }+(t_0-2t)u)\\ge \\lim _{t_0^{\\prime }\\rightarrow t_0,t_0^{\\prime }<t_0}f(x^{\\prime }+t_0^{\\prime }u);$ $a_3$ ) There exists some $t_0\\in \\mathbb {R}$ such that $S_uf(x)=f(x^{\\prime }+(t_0+2t)u)\\ge \\lim _{t_0^{\\prime }\\rightarrow t_0,t_0^{\\prime }>t_0}f(x^{\\prime }+t_0^{\\prime }u).$ For $y=y^{\\prime }+su\\in {\\rm dom}(S_uf)$ , by Theorem REF , there exist three cases: $b_1$ ) There exists some $s^{\\prime }\\in \\mathbb {R}$ such that $S_uf(y)=f(y^{\\prime }+s^{\\prime }u)=f(y^{\\prime }+(s^{\\prime }-2s)u);$ $b_2$ ) There exists some $s_0\\in \\mathbb {R}$ such that $S_uf(y)=f(y^{\\prime }+(s_0-2s)u)\\ge \\lim _{s_0^{\\prime }\\rightarrow s_0, s_0^{\\prime }<s_0}f(y^{\\prime }+s_0^{\\prime }u);$ $b_3$ ) There exists some $s_0\\in \\mathbb {R}$ such that $S_uf(y)=f(y^{\\prime }+(s_0+2s)u)\\ge \\lim _{s_0^{\\prime }\\rightarrow s_0,s_0^{\\prime }>s_0}f(y^{\\prime }+s_0^{\\prime }u).$ We may assume that $&&f(x^{\\prime }+t_0 u)=\\lim _{t_0^{\\prime }\\rightarrow t_0,t_0^{\\prime }<t_0}f(x^{\\prime }+t_0^{\\prime }u)\\;\\;\\;{\\rm for\\;\\;case}\\;\\; a_2),\\nonumber \\\\&&f(x^{\\prime }+t_0 u)=\\lim _{t_0^{\\prime }\\rightarrow t_0,t_0^{\\prime }>t_0}f(x^{\\prime }+t_0^{\\prime }u)\\;\\;\\;{\\rm for\\;\\;case}\\;\\; a_3),\\nonumber \\\\&&f(y^{\\prime }+s_0 u)=\\lim _{s_0^{\\prime }\\rightarrow s_0,s_0^{\\prime }<s_0}f(y^{\\prime }+s_0^{\\prime }u)\\;\\;\\;{\\rm for\\;\\;case}\\;\\; b_2),\\nonumber \\\\&&f(y^{\\prime }+s_0 u)=\\lim _{s_0^{\\prime }\\rightarrow s_0,s_0^{\\prime }>s_0}f(y^{\\prime }+s_0^{\\prime }u)\\;\\;\\;{\\rm for\\;\\;case}\\;\\; b_3).$ Let $(\\tilde{t}_1,\\tilde{t}_2)$ be a pair of real numbers satisfying $ (\\tilde{t}_1,\\tilde{t}_2)=\\left\\lbrace \\begin{aligned}(t^{\\prime }-2t,t^{\\prime })\\;\\;\\;{\\rm for\\;\\;case}\\;\\; a_1)\\\\(t_0-2t,t_0)\\;\\;\\;{\\rm for\\;\\;case}\\;\\; a_2)\\\\(t_0,t_0+2t)\\;\\;\\;{\\rm for\\;\\; case}\\;\\; a_3).\\end{aligned} \\right.$ Let $(\\tilde{s}_1,\\tilde{s}_2)$ be a pair of real numbers satisfying $ (\\tilde{s}_1,\\tilde{s}_2)=\\left\\lbrace \\begin{aligned}(s^{\\prime }-2s,s^{\\prime })\\;\\;\\;{\\rm for\\;\\;case}\\;\\; b_1)\\\\(s_0-2s,s_0)\\;\\;\\;{\\rm for\\;\\;case}\\;\\; b_2)\\\\(s_0,s_0+2s)\\;\\;\\;{\\rm for\\;\\; case}\\;\\; b_3).\\end{aligned} \\right.$ Since $f$ is convex and by (REF -REF ), for $i=1,2$ , we have $&&\\lambda S_u f(x)+(1-\\lambda )S_u f(y)\\nonumber \\\\&\\ge &\\lambda f(x^{\\prime }+\\tilde{t}_iu)+(1-\\lambda )f(y^{\\prime }+\\tilde{s}_iu)\\nonumber \\\\&\\ge &f(\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+(\\lambda \\tilde{t}_i+(1-\\lambda )\\tilde{s}_i)u).$ By (REF ) and (REF ), we have $&&[\\lambda \\tilde{t}_2+(1-\\lambda )\\tilde{s}_2]-[\\lambda \\tilde{t}_1+(1-\\lambda )\\tilde{s}_1]\\nonumber \\\\&=&\\lambda (\\tilde{t}_2-\\tilde{t}_1)+(1-\\lambda )(\\tilde{s}_2-\\tilde{s}_1)=2[\\lambda t+(1-\\lambda )s].$ By $\\lambda x+(1-\\lambda )y=\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+(\\lambda t+(1-\\lambda )s)u$ and Definition REF , we have $&&S_uf(\\lambda x+(1-\\lambda )y)\\nonumber \\\\&=&\\sup _{\\delta \\in [0,1]}\\inf _{\\omega \\in \\mathbb {R}}\\left[\\delta f(\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+\\omega u)\\right.\\nonumber \\\\&&\\left.+(1-\\delta )f(\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+(\\omega -2(\\lambda t+(1-\\lambda )s)u)\\right]\\nonumber \\\\&\\le &\\sup _{\\delta \\in [0,1]}\\left[\\delta f(\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+(\\lambda \\tilde{t}_2+(1-\\lambda )\\tilde{s}_2)u)\\right.\\nonumber \\\\&&\\left.+(1-\\delta )f(\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+(\\lambda \\tilde{t}_1+(1-\\lambda )\\tilde{s}_1)u)\\right]\\nonumber \\\\&\\le & \\max _{i=1,2}f(\\lambda x^{\\prime }+(1-\\lambda )y^{\\prime }+(\\lambda \\tilde{t}_i+(1-\\lambda )\\tilde{s}_i)u)\\nonumber \\\\&\\le &\\lambda S_u f(x)+(1-\\lambda )S_u f(y),$ where the first inequality is by choosing $\\omega =\\lambda \\tilde{t}_2+(1-\\lambda )\\tilde{s}_2$ and (REF ), and the last inequality is by (REF ).", "Case 3.3.", "The case of $t=0$ and $s>0$ (or $t>0$ and $s=0$ ).", "In this case, there exists $t_0$ such that $S_uf(x)=\\lim _{ t\\rightarrow t_0,\\;x+tu\\in {\\rm dom}f}f(x+tu).$ We may assume that $f(x+t_0u)=\\lim _{t\\rightarrow t_0,\\; x+tu\\in {\\rm dom}f}f(x+tu).$ In the proof of Case 3.2, let $\\tilde{t}_1=\\tilde{t}_2=t_0$ , we can get the required inequality.", "Proposition 2.2 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ be a coercive convex function and $u\\in S^{n-1}$ , then $\\int _{\\mathbb {R}^n}e^{-(S_u f)(x)}dx= \\int _{\\mathbb {R}^n}e^{-f(x)}dx.$ By (REF ), for any $t\\in \\mathbb {R}$ , we have ${\\rm cl}[S_uf<t]=S_u ({\\rm cl}[f<t])$ .", "Since Steiner symmetrization of convex sets preserves volume, $Vol([S_uf<t])=Vol([f<t])$ .", "By Fubini's theorem, we have $\\int _{\\mathbb {R}^n}e^{-(S_uf)(x)}dx&=&\\int _{\\mathbb {R}}Vol([S_uf<t])e^{-t}dt\\nonumber \\\\&=&\\int _{\\mathbb {R}}Vol([f<t])e^{-t}dt=\\int _{\\mathbb {R}^n}e^{-f(x)}dx.$ Lemma 3 Let $u_1,u_2\\in S^{n-1}$ and $\\langle u_1, u_2\\rangle =0$ .", "If $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is a coercive convex function and $f$ is symmetric about $u_1^{\\perp }$ , then $S_{u_2} f$ is symmetric about both $u_1^{\\perp }$ and $u_2^{\\perp }$ .", "By Proposition REF , $S_{u_2}f$ is symmetric about $u_2^{\\perp }$ .", "Next, we prove that $S_{u_2} f$ is symmetric about $u_1^{\\perp }$ .", "Since $\\langle u_1,u_2\\rangle =0$ , then $u_1\\in u_2^{\\perp }$ and $u_2\\in u_1^{\\perp }$ .", "For any $x^{\\prime }\\in u_1^{\\perp }$ , let $x^{\\prime }=x^{\\prime \\prime }+t_{x^{\\prime }}u_2$ , where $x^{\\prime \\prime }=x^{\\prime }\|u_2^{\\perp }$ .", "Then $x^{\\prime \\prime }=x^{\\prime }-t_{x^{\\prime }}u_2\\in u_1^{\\perp }$ , thus $x^{\\prime \\prime }+tu_2\\in u_1^{\\perp }$ .", "Because that $x^{\\prime \\prime }\\in u_2^{\\perp }$ and $u_1\\in u_2^{\\perp }$ , thus $x^{\\prime \\prime }+tu_1\\in u_2^{\\perp }$ .", "Thus, for any $x^{\\prime }\\in u_1^{\\perp }$ and $t\\in \\mathbb {R}$ , we have $&~&(S_{u_2}f)(x^{\\prime }+tu_1)=(S_{u_2}f)(x^{\\prime \\prime }+tu_1+t_{x^{\\prime }}u_2)\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=t_{x^{\\prime }}}[\\lambda f(x^{\\prime \\prime }+tu_1+2t_1u_2)+(1-\\lambda )f(x^{\\prime \\prime }+tu_1-2t_2u_2)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=t_{x^{\\prime }}}[\\lambda f(x^{\\prime \\prime }-tu_1+2t_1u_2)+(1-\\lambda )f(x^{\\prime \\prime }-tu_1-2t_2u_2)]\\nonumber \\\\&=&(S_{u_2}f)(x^{\\prime \\prime }-tu_1+t_{x^{\\prime }}u_2)\\nonumber \\\\&=&(S_{u_2}f)(x^{\\prime }-tu_1),$ where the second equality is by $f$ is symmetric about $u_1^{\\perp }$ and $x^{\\prime \\prime }+tu_2\\in u_1^{\\perp }$ .", "This completes the proof.", "We say that a function $f:\\mathbb {R}^n\\mapsto \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is unconditional if $f(x_1,\\dots , x_n)=f(\|x_1\|,\\dots ,\|x_n\|)$ for every $(x_1,\\dots ,x_n)\\in \\mathbb {R}^n$ .", "Proposition 2.3 Any coercive convex function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ can be transformed into an unconditional function $\\bar{f}$ using $n$ Steiner symmetrizations.", "Let $\\lbrace u_1,\\dots , u_n\\rbrace $ be an orthonormal basis of $\\mathbb {R}^n$ .", "By Proposition REF and Lemma REF , $S_{u_n}\\cdots S_{u_1}f$ is symmetric about $u_i^{\\bot }$ , $i=1,\\cdots , n$ , which implies that $f$ can be transformed into an unconditional function $\\bar{f}=S_{u_n}\\cdots S_{u_1} f$ using $n$ Steiner symmetrizations.", "Proposition 2.4 Let $f_1:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ and $f_2:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ be coercive convex functions and $u\\in S^{n-1}$ .", "If $f_1\\le f_2$ (which implies that $f_1(x)\\le f_2(x)$ for any $x\\in \\mathbb {R}^n$ ), then $S_u f_1\\le S_u f_2$ .", "By Definition REF and $f_1\\le f_2$ , for $x=x^{\\prime }+tu$ , where $x^{\\prime }\\in u^{\\perp }$ , we have $S_u f_1(x)&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=t}[\\lambda f_1(x^{\\prime }+2t_1u)+(1-\\lambda ) f_1(x^{\\prime }-2t_2u)]\\nonumber \\\\&\\le &\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=t}[\\lambda f_2(x^{\\prime }+2t_1u)+(1-\\lambda )f_2(x^{\\prime }-2t_2u)]\\nonumber \\\\&=&S_u f_2(x).$ We say a function $f$ is even about point $z\\in \\mathbb {R}^n$ if $f(z+x)=f(z-x)$ for any $x\\in \\mathbb {R}^n$ .", "Let $z\|H$ denote the projection of $z$ onto hyperplane $H$ .", "Proposition 2.5 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ be a coercive convex function and $u\\in S^{n-1}$ , if $f$ is even about $z$ , then $S_u f$ is even about $z\|u^{\\perp }$ .", "For any $x\\in \\mathbb {R}^n$ , let $x=x^{\\prime }+tu$ , where $x^{\\prime }=x\|u^{\\perp }$ .", "Let $z=z^{\\prime }-t_0u$ , where $z^{\\prime }=z\|u^{\\perp }$ .", "By Definition REF , we have $&~&(S_u f)(z^{\\prime }+x)=(S_u f)(z^{\\prime }+x^{\\prime }+tu)=(S_u f)(z^{\\prime }+x^{\\prime }-tu)\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=-t}[\\lambda f(z^{\\prime }+x^{\\prime }+2t_1u)+(1-\\lambda )f(z^{\\prime }+x^{\\prime }-2t_2u)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_2\\in \\mathbb {R}}[\\lambda f(z+t_0u+x^{\\prime }-2t_2u-2tu)+(1-\\lambda )f(z+t_0u+x^{\\prime }-2t_2u)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_2\\in \\mathbb {R}}[\\lambda f(z+x^{\\prime }-2t_2u-2tu)+(1-\\lambda )f(z+x^{\\prime }-2t_2u)]\\nonumber \\\\&=&\\sup _{\\lambda ^{\\prime }\\in [0,1]}\\inf _{t_2\\in \\mathbb {R}}[\\lambda ^{\\prime }f(z+x^{\\prime }-2t_2u)+(1-\\lambda ^{\\prime })f(z+x^{\\prime }-2t_2u-2tu)],$ where the second equality is by $S_uf$ is symmetric about $u^{\\perp }$ and the fifth equality is by replacing $t_0-2t_2$ by $-2t_2$ .", "On the other hand, since $f$ is even about $z$ , we have $&~&(S_u f)(z^{\\prime }-x)=(S_u f)(z^{\\prime }-x^{\\prime }-tu)\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1+t_2=-t}[\\lambda f(z^{\\prime }-x^{\\prime }+2t_1u)+(1-\\lambda )f(z^{\\prime }-x^{\\prime }-2t_2u)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1\\in \\mathbb {R}}[\\lambda f(z+t_0u-x^{\\prime }+2t_1u)+(1-\\lambda )f(z+t_0u-x^{\\prime }+2t_1u+2tu)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1\\in \\mathbb {R}}[\\lambda f(z-x^{\\prime }+2t_1u)+(1-\\lambda )f(z-x^{\\prime }+2t_1u+2tu)]\\nonumber \\\\&=&\\sup _{\\lambda \\in [0,1]}\\inf _{t_1\\in \\mathbb {R}}[\\lambda f(z+x^{\\prime }-2t_1u)+(1-\\lambda )f(z+x^{\\prime }-2t_1u-2tu)],$ where the last equality is by $f$ is even about $z$ .", "By (REF ) and (REF ), we have $(S_uf)(z^{\\prime }+x)=(S_uf)(z^{\\prime }-x)$ for any $x\\in \\mathbb {R}^n$ ." ], [ " The relation between Definition ", "The relation can be generalized as follows: (i) $S_uf$ is in general larger than $\\widetilde{S}_u f$ (look at Example REF ).", "(ii) For one-dimensional coercive convex function $f:\\mathbb {R}\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ , if $f$ is symmetric about an axes $x=x_0$ , i.e., $f(x_0-x)=f(x_0+x)$ for any $x\\in \\mathbb {R}$ , then $S f=\\widetilde{S}f$ .", "(iii) For $n$ -dimensional coercive convex function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ and $u\\in S^{n-1}$ , if for any $x^{\\prime }\\in u^{\\perp }$ , one-dimensional function $f(x^{\\prime }+tu)$ about $t\\in \\mathbb {R}$ is symmetric about an axes $t=t_0$ , then $S_uf=\\widetilde{S}_uf$ .", "Example 1 For one-dimensional coercive convex function $ f(x)=\\left\\lbrace \\begin{aligned}x^3\\;\\;{\\rm if}\\;\\;x\\ge 0,\\\\x^2\\;\\;{\\rm if}\\;\\;x\\le 0.\\end{aligned} \\right.$ We compare $Sf$ with $\\widetilde{S} f$ , where $Sf(x)=\\sup _{\\lambda \\in [0,1]}\\inf _{x_1+x_2=x}[\\lambda f(2x_1)+(1-\\lambda )f(-2x_2)]$ and $\\widetilde{S} f(x)=\\inf _{x_1+x_2=x}[\\frac{1}{2}f(2x_1)+\\frac{1}{2}f(-2x_2)].$ By calculation, we can get that $ \\widetilde{S} f(x)=\\left\\lbrace \\begin{aligned}\\frac{(-12x-1)\\sqrt{1+12x}+18x+1}{27}+2x^2\\;\\;{\\rm if}\\;\\;x\\ge 0,\\\\\\frac{(12x-1)\\sqrt{1-12x}-18x+1}{27}+2x^2\\;\\;\\;{\\rm if}\\;\\;x\\le 0.\\end{aligned} \\right.$ and $S f(x)=g^{-1}(\|x\|),$ where $g^{-1}$ is the inverse function of $g(x)=\\frac{1}{2}(\\@root 3 \\of {x}+\\sqrt{x}),\\;\\;x\\in [0,\\infty ).$ By Matlab, we can draw their figures (see Figure 1).", "In the figure, we can find that the level sets of $Sf$ and $f$ have the same size and $Sf>\\widetilde{S}f$ .", "Figure: NO_CAPTION" ], [ " The relation between Definition ", "In this section, we show that the two definitions are same for log-concave functions (Theorem REF ).", "Lemma 3.1 Let $F=e^{-f}$ be a log-concave function, where $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ is a coercive convex function, then $[\\bar{S}_u F>t]=S_u ([F>t])$ .", "By Definition REF , if $\\bar{S}_u F(x)>t$ , then $x\\in S_u([F>t])$ .", "On the other hand, if $x\\in S_u ([F>t])$ , since $S_u([F>t])$ is an open set and $F$ is continuous, then there exists $t^{\\prime }>t$ such that $x\\in S_u ([F>t^{\\prime }])$ , by (REF ), we have $\\bar{S}_u F(x)>t$ .", "Theorem 3.2 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ be a coercive convex function and $u\\in S^{n-1}$ , then $e^{(-S_u f)}=\\bar{S}_u(e^{-f})$ , where $S_uf$ and $\\bar{S}_u(e^{-f})$ are given in (REF ) and (REF ), respectively.", "For $t>0$ , we have $[e^{(-S_u f)}>t]=[S_uf<-\\ln t]=S_u([f<-\\ln t])=S_u([e^{-f}>t]),$ where the second equality holds by (REF ).", "By Lemma REF , we have $[\\bar{S}_u (e^{-f})>t]=S_u ([e^{-f}>t])$ , thus $[e^{(-S_u f)}>t]=[\\bar{S}_u (e^{-f})>t]$ .", "Using the “layer-cake representation\", we have $e^{(-S_u f)}=\\int _{0}^{\\infty }\\mathcal {X}_{[e^{(-S_u f)}>t]}(x)dt=\\int _{0}^{\\infty }\\mathcal {X}_{[\\bar{S}_u (e^{-f})>t]}(x)dt=\\bar{S}_u (e^{-f}).$ The continuity and convergence of Steiner symmetrization in $L^p$ space have been proved in many papers [7], [8], [9], [14], especially Proposition 3 and Theorem 2 in [14] are corresponding to the properties 6-7 in Table 1." ], [ "Application to functional Blaschke-Santaló inequality", "We can use the new definition to prove some important inequalities, such as functional Blaschke-Santaló inequality, Prékopa-Leindler inequality for log-concave functions, Hardy-Littlewood inequality for log-concave functions, etc.", "As an illustration, here we only use it to prove the functional Blaschke-Santaló inequality for even convex functions.", "For a convex body $K\\subset \\mathbb {R}^n$ , its polar about $z$ is defined by $K^{z}=\\lbrace x\\in \\mathbb {R}^n: \\sup _{y\\in K}\\langle x-z,y-z\\rangle \\le 1\\rbrace $ .", "For a log-concave function $f:\\mathbb {R}^n\\rightarrow [0,\\infty )$ , its polar about $z$ is defined by $f^{z}(x)=\\inf _{y\\in \\mathbb {R}^n}\\frac{e^{-\\langle x-z,y-z\\rangle }}{f(y)}.$ To better understand this definition recall the classical Legendre transform: For a function $\\phi : \\mathbb {R}^n\\rightarrow \\mathbb {R}$ , its Legendre transform about $z$ is defined by $\\mathcal {L}^{z}\\phi (x)=\\sup _{y\\in \\mathbb {R}^n}[\\langle x-z,y-z\\rangle -\\phi (y)]$ .", "From above definition of polarity, if $f(x)=e^{-\\phi (x)}$ , where $\\phi (x)$ is a convex function, then $f^{z}(x)=e^{-\\mathcal {L}^{z}\\phi (x)}$ .", "Since $\\mathcal {L}^z(\\mathcal {L}^z\\phi )=\\phi $ for a convex function $\\phi $ , $(f^z)^z=f$ .", "For $z=0$ , we denote $\\mathcal {L}^{0}\\phi =\\mathcal {L}\\phi $ .", "For a convex body $K$ , its Santaló point $s(K)$ satisfies $Vol(K^{s(K)})=\\min _{z}Vol(K^{z})$ .", "The Blaschke-Santaló inequality [3], [29] states that $Vol(K)Vol(K^{s(K)})\\le Vol(B_2^n)^2$ , where $B_2^n=\\lbrace x\\in \\mathbb {R}^n: \|x\|\\le 1\\rbrace $ is the Euclidean ball ($\|\\cdot \|$ denote the Euclidean norm).", "The functional Blaschke-Santaló inequality of log-concave functions is the analogue of Blaschke-Santaló inequality of convex bodies.", "If $f$ is a nonnegative integrable function on $\\mathbb {R}^n$ such that $f^{0}$ has its barycenter at 0, then $\\int _{\\mathbb {R}^n}f(x)dx\\int _{\\mathbb {R}^n}f^{0}(y)dy\\le \\left(\\int _{\\mathbb {R}^n}e^{-\\frac{1}{2}\|x\|^2}dx\\right)^2=(2\\pi )^n.$ In the special case where the function $f$ is even, this result follows from an earlier inequality of Ball [2]; and in [15], Fradelizi and Meyer prove something more general (see also [22]).", "Recently, Lehec [23] gave a direct proof of the functional Blaschke-Santaló inequality.", "In this paper, inspired by the proof of K. Ball [2] for Santaló inequality for centrally symmetric convex bodies, we prove functional Blaschke-Santaló inequality for even convex functions.", "For the non-even case, we can prove the inequality by the similar method, but we don't prove it here.", "Theorem 4.1 (K. Ball, [2]) Let $f:\\mathbb {R}^n\\rightarrow [0,\\infty )$ be an even convex function.", "Assume that $0<\\int e^{-f}<\\infty $ .", "Then $\\int e^{-f}\\int e^{-\\mathcal {L} f}\\le (2\\pi )^n.$ First, we give the following lemmas.", "Lemma 4.2 Let $f:\\mathbb {R}^n\\rightarrow [0,\\infty )$ be an even convex function and $u\\in S^{n-1}$ .", "Assume that $0<\\int e^{-f}<\\infty $ .", "Then $\\int e^{-\\mathcal {L} f}\\le \\int e^{-\\mathcal {L} (S_uf)}.$ After a linear transformation, it may be supposed that $H=u^{\\perp }=\\lbrace (x_i)_{i=1}^{i=n}:x_n=0\\rbrace $ .", "For $f$ and $t\\in \\mathbb {R}$ , we define a new function $f_{(t)}(x^{\\prime }):=f(x^{\\prime }+tu)$ , where $x^{\\prime }\\in H$ .", "By the definition of Steiner symmetrization, for $x^{\\prime }=x_1^{\\prime }+x_2^{\\prime }$ , where $x^{\\prime }$ , $x_1^{\\prime }$ and $x_2^{\\prime }\\in H$ , let $(x^{\\prime },t)$ denote $x^{\\prime }+tu$ , we have $&&(\\mathcal {L}(S_u f))_{(t)}(x^{\\prime })=(\\mathcal {L}(S_u f))(x^{\\prime }+tu)\\nonumber \\\\&=&\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}\\left[\\langle (x^{\\prime },t),(y^{\\prime },s)\\rangle -(S_uf)(y^{\\prime }+su)\\right]\\nonumber \\\\&=&\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}[\\langle (x^{\\prime },t),(y^{\\prime },s)\\rangle -\\sup _{\\lambda \\in [0,1]}\\inf _{s_1+s_2=s}(\\lambda f(y^{\\prime }+2s_1u)+(1-\\lambda )f(y^{\\prime }-2s_2u))]\\nonumber \\\\&=&\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}\\inf _{\\lambda \\in [0,1]}\\sup _{s_1+s_2=s}[\\langle (x^{\\prime },t),(y^{\\prime },s)\\rangle -(\\lambda f(y^{\\prime }+2s_1u)+(1-\\lambda )f(y^{\\prime }-2s_2u))]\\nonumber \\\\&\\le &\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}\\sup _{s_1+s_2=s}[\\langle (x^{\\prime },t),(y^{\\prime },s)\\rangle -(\\frac{1}{2} f(y^{\\prime }+2s_1u)+\\frac{1}{2}f(y^{\\prime }-2s_2u))]\\nonumber \\\\&=&\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}\\sup _{s_1\\in \\mathbb {R}}[\\langle (x^{\\prime },t),(y^{\\prime },s)\\rangle -(\\frac{1}{2} f(y^{\\prime }+2s_1u)+\\frac{1}{2}f(y^{\\prime }+2(s_1-s)u))]\\nonumber \\\\&\\le &\\frac{1}{2}\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}\\sup _{s_1\\in \\mathbb {R}}[\\langle (2x_1^{\\prime },t),(y^{\\prime },2s_1)\\rangle -f(y^{\\prime }+2s_1u)]\\nonumber \\\\&~&+\\frac{1}{2}\\sup _{(y^{\\prime },s)\\in H\\times \\mathbb {R}}\\sup _{s_1\\in \\mathbb {R}}[\\langle (2x_2^{\\prime },-t),(y^{\\prime },2s_1-2s)\\rangle -f(y^{\\prime }+2(s_1-s)u)]\\nonumber \\\\&=&\\frac{1}{2}[(\\mathcal {L}f)(2x_1^{\\prime }+tu)+(\\mathcal {L}f)(2x_2^{\\prime }-tu)],$ where the first inequality is by choosing $\\lambda =\\frac{1}{2}$ and the second inequality is by $\\sup \\sup (A+B)\\le \\sup \\sup A+\\sup \\sup B$ .", "Since $x_1^{\\prime }$ and $x_2^{\\prime }$ are arbitrary, by (REF ), we can get $\\left(e^{-(\\mathcal {L}(S_u f))_{(t)}}\\right)(x^{\\prime })\\ge \\sup _{x_1^{\\prime }+x_2^{\\prime }=x^{\\prime }}\\left(e^{-\\frac{1}{2}(\\mathcal {L}f)_{(t)}(2x_1^{\\prime })}\\times e^{-\\frac{1}{2}(\\mathcal {L}f)_{(-t)}(2x_2^{\\prime })}\\right).$ By (REF ) and Prékopa-Leindler inequality, we have $\\int _{H}e^{-(\\mathcal {L}(S_uf))_{(t)}}(x^{\\prime })dx^{\\prime }&\\ge &\\left(\\int _{H}e^{-(\\mathcal {L}f)_{(t)}(x^{\\prime })}dx^{\\prime }\\right)^{\\frac{1}{2}}\\left(\\int _{H}e^{-(\\mathcal {L}f)_{(-t)}(x^{\\prime })}dx^{\\prime }\\right)^{\\frac{1}{2}}\\nonumber \\\\&=&\\int _{H}e^{-(\\mathcal {L}f)_{(t)}(x^{\\prime })}dx^{\\prime },$ where the last equality is by $\\mathcal {L}f$ is even (since $f$ is even).", "Thus, by Fubini's theorem, we can get the desired inequality.", "Lemma 4.3 Let $h(t)$ be an increasing convex function defined on $[0,+\\infty )$ and $\\int _{0}^{+\\infty }e^{-h(t)}dt<\\infty $ .", "Let $\\mathcal {L}(h(\|\\cdot \|))$ denote the Legendre transform of function $h(\|x\|)$ defined on $\\mathbb {R}^n$ .", "Then $\\int _{\\mathbb {R}^n}e^{-h(\|x\|)}dx\\int _{\\mathbb {R}^n}e^{-(\\mathcal {L}(h(\|\\cdot \|)))(x)}dx\\le (2\\pi )^n.$ By spherical coordinate transformation, we have $\\int _{\\mathbb {R}^n}e^{-h(\|x\|)}dx=\\int _{S^{n-1}}\\left[\\int _{0}^{+\\infty }e^{-h(r)}r^{n-1}dr\\right]d\\omega .$ For any $x\\in \\mathbb {R}^n$ , let $x=t_x\\theta _x$ , where $\\theta _x=\\frac{x}{\|x\|}\\in S^{n-1}$ for $\|x\|\\ne 0$ and $\\theta _x$ is any unit vector for $\|x\|=0$ , and $t_x=\|x\|$ .", "Then, we have $\\mathcal {L}(h(\|\\cdot \|))(x)&=&\\sup _{y\\in \\mathbb {R}^n}(\\langle x,y\\rangle -h(\|y\|))\\nonumber \\\\&=&\\sup _{\\theta _y\\in S^{n-1},t_y\\ge 0}(\\langle t_x\\theta _x,t_y\\theta _y\\rangle -h(t_y))=\\sup _{t_y\\ge 0}(t_xt_y-h(t_y)).$ Thus, we have $\\int _{\\mathbb {R}^n}e^{-(\\mathcal {L}(h(\|\\cdot \|)))(x)}dx=\\int _{S^{n-1}}\\left[\\int _{0}^{+\\infty }\\left(e^{-\\sup _{t\\ge 0}(rt-h(t))}\\right)r^{n-1}dr\\right]d\\omega .$ For $r\\in [0,+\\infty )$ , let $f_1(r)=\\left(e^{-h(r)}\\right)r^{n-1}$ , $f_2(r)=\\left(e^{-\\sup _{t\\ge 0}(rt-h(t))}\\right)r^{n-1}$ and $f_3(r)=\\left(e^{-\\frac{r^2}{2}}\\right)r^{n-1}$ .", "Next, we shall prove that $\\int _{0}^{+\\infty }f_1(r)dr\\int _{0}^{+\\infty }f_2(r)dr\\le \\left(\\int _{0}^{+\\infty }f_3(r)dr\\right)^2.$ Let $g_i(t)=f_i(e^t)e^t$ for $i=1,2,3,$ then $\\int _{0}^{+\\infty }f_i(r)dr=\\int _{\\mathbb {R}}g_i(t)dt$ and for every $s,t\\in \\mathbb {R}$ , $g_1(s)g_2(t)\\le \\left(g_3(\\frac{s+t}{2})\\right)^2$ .", "Hence inequality (REF ) follows from Prékopa-Leindler inequality.", "By (REF ), (REF ) and (REF ), we have $&&\\int _{\\mathbb {R}^n}e^{-h(\|x\|)}dx\\int _{\\mathbb {R}^n}e^{-(\\mathcal {L}(h(\|\\cdot \|)))(x)}dx\\nonumber \\\\&=&\\omega _n^2\\int _{0}^{+\\infty }f_1(r)dr\\int _{0}^{+\\infty }f_2(r)dr\\le \\omega _n^2\\left(\\int _{0}^{+\\infty }e^{-\\frac{r^2}{2}}r^{n-1}dr\\right)^2=(2\\pi )^n,$ where $\\omega _n=n\\pi ^{n/2}/\\Gamma (1+\\frac{n}{2})$ is the surface area of Euclidean unit ball.", "Proof of Theorem REF .", "By the integral invariance under Steiner symmetrization (Proposition REF ), for any $u\\in S^{n-1}$ , we have $\\int _{\\mathbb {R}^n}e^{-(S_uf)(x)}dx=\\int _{\\mathbb {R}^n}e^{-f(x)}dx.$ By (REF ) and Lemma REF , we have $\\int e^{-f}\\int e^{-\\mathcal {L} f}\\le \\int e^{-S_uf}\\int e^{-\\mathcal {L}(S_uf)}.$ By property 7 in Table REF , for log-concave function $e^{-f}\\in L^1(\\mathbb {R}^n)$ , there exists a sequence of directions $\\lbrace u_i\\rbrace _{i=1}^{\\infty }\\subset S^{n-1}$ such that $e^{-S_{u_1,\\dots ,u_i}f}$ converges to a radial function $e^{-h(\|\\cdot \|)}$ , where $h(t)$ is a one-dimensional increasing convex function defined on $[0,+\\infty )$ .", "By (REF ) and Lemma REF and the continuity of integral in $L^1(\\mathbb {R}^n)$ , we have $\\int e^{-f}\\int e^{-\\mathcal {L}f} &\\le & \\lim _{i\\rightarrow +\\infty } \\int e^{-S_{u_1,\\dots ,u_i}f}\\int e^{-\\mathcal {L}(S_{u_1,\\dots ,u_i}f)}\\nonumber \\\\&=&\\int e^{-h(\|\\cdot \|)}\\int e^{-\\mathcal {L}(h(\|\\cdot \|))}\\le (2\\pi )^n.$ This completes the proof.$\\Box $" ] ]	1403.0319
[ [ "A Scalarization Proximal Point Method for Quasiconvex Multiobjective\n Minimization" ], [ "Abstract In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions.", "We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point.", "Our method may be seen as an extension, for the non convex case, of the inexact proximal method for multiobjective convex minimization problems studied by Bonnel et al.", "(SIAM Journal on Optimization 15, 4, 953-970, 2005)." ], [ "Introduction", "In this work we consider the unconstrained multiobjective minimization problem: $\\textrm {min}\\lbrace F(x): x \\in \\mathbb {R}^n\\rbrace $ where $F: \\mathbb {R}^n\\longrightarrow \\mathbb {R}^m$ is a locally Lipschitz and quasiconvex vector function on the Euclidean space $ \\mathbb {R}^n.$ A motivation to study this problem are the consumer demand theory in economy, where the quasiconvexity of the objective vector function is a natural condition associated to diversification of the consumption, see Mas Colell et al.", "[21], and the quasiconvex optimization models in location Theory, see [12].", "Another motivation are the extensions of well known methods in convex optimization to quasiconvex one, we mentioned the following works: Bello Cruz et al.", "[3], considered the projected gradient method for solving the problem of finding a Pareto optimum of a quasiconvex multiobjective function.", "They proved the convergence of the sequence generated by the algorithm to a stationary point and when the components of the multiobjective function are pseudoconvex, they obtained the convergence to a weak Pareto solution.", "da Cruz Neto et al.", "[10], extended the classical subgradient method for real-valued mi-nimization to multiobjective optimization.", "Assuming the basically componentwise quasiconvexity of the objective components they obtained the full convergence of the sequence to a Pareto solution.", "Papa Quiroz and Oliveira [24], [25], [27], have been extended the convergence of the proximal point method for quasiconvex minimization problems on general riemannian manifolds wich includes the euclidean space.", "Furthermore, in [26] the authors extended the convergence of the proximal point method for the nonnegative orthant.", "Kiwiel [16], extended the convergence of the subgradient method to solve quasiconvex minimization problems in Hilbert spaces.", "Brito et al.", "[6], proposed an interior proximal algorithm inspired by the logarithmic-quadratic proximal method for linearly constrained quasiconvex minimization problems.", "For that method, they proved the global convergence when the proximal parameters go to zero.", "The latter assumption could be dropped when the function is assumed to be pseudoconvex.", "Langenberg and Tichatschke [17] studied the proximal method when the objective func-tion is quasiconvex and the problem is constrained to an arbitrary closed convex set and the regularization is a Bregman distance.", "Assuming that the function is locally Lipschitz and using the Clarke subdifferential, the authors proved the global convergence of the method to a critical point.", "In this paper we are interested in extending the convergence properties of the proximal point method to solve the quasiconvex multiobjective problem (REF ).", "The proximal point method, introduced by Martinet [20], to solve the problem $\\min \\lbrace f(x) : x \\in \\mathbb {R}^n\\rbrace $ where $f$ is a escalar function, generates a sequence $\\lbrace x^k\\rbrace _{k \\in \\mathbb {N}}\\subset \\mathbb {R}^n$ , from an iterative process starting with a point $x^0 \\in \\mathbb {R}^n$ , arbitrary, and $x^{k+1} \\in \\textrm {argmin}\\lbrace f(x) + \\frac{\\lambda _k}{2}\\Vert x - x^k \\Vert ^2: x \\in \\mathbb {R}^n\\rbrace $ , where $\\lambda _k > 0,$ is a regularization parameter.", "It is well known, see Guler [13], that if $f$ is convex and $\\lbrace \\lambda _k\\rbrace $ satisfies $\\sum \\limits _{k=1}^{+\\infty }(1/\\lambda _k)=+\\infty ,$ then $\\lim _{k\\rightarrow \\infty }f(x^k)=\\inf \\lbrace f(x):x\\in \\mathbb {R}^n \\rbrace .$ Furthermore, if the optimal set is nonempty, we obtain that $\\lbrace x^k\\rbrace $ converges to an optimal solution of the problem.", "When $F$ is convex in (REF ), Bonnel at al.", "[5] have been proved the convergence of the proximal point method for a weak Pareto solution of the problem (REF ) in a general context, see also Villacorta and Oliveira [31] using proximal distances and Gregório and Oliveira [11] using a logarithmic quadratic proximal scalarization method.", "In this work we introduce a scalarization proximal point method to solve the quasiconvex multiobjective minimization problem (REF ).", "The iteration is the following: given $x^{k} \\in \\mathbb {R}^n$ , find $x^{k+1}\\in \\Omega _k$ such that: $0 \\in \\partial ^o\\left( \\left\\langle F(.", "), z_k\\right\\rangle + \\dfrac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\Vert \\ .\\ - x^k \\Vert ^2 \\right) (x^{k+1}) + \\mathcal {N}_{\\Omega _k}(x^{k+1})$ where ${\\partial }^o$ is the Clarke subdifferential, $\\Omega _k= \\left\\lbrace x\\in \\mathbb {R}^n: F(x) \\preceq F(x^k)\\right\\rbrace $ , $\\alpha _k > 0 $ , $\\left\\lbrace e_k\\right\\rbrace \\subset \\mathbb {R}^m_{++}$ , $\\left\\Vert e_k\\right\\Vert = 1$ , $\\left\\lbrace z_k\\right\\rbrace \\subset \\mathbb {R}^m_+\\backslash \\left\\lbrace 0\\right\\rbrace $ , $\\left\\Vert z_k\\right\\Vert = 1$ and $\\mathcal {N}_{\\Omega _k}(x^{k+1})$ the normal cone to $\\Omega _k$ at $x^{k+1}$ .", "We prove the well definition of the sequence generated by the method and we obtain the global convergence to a Pareto-Clarke critical point and when $F$ is convex we obtain the convergence to a weak Pareto solution of the problem.", "The paper is organized as follows: In Section 2 we recall some concepts and results basic on multiobjective optimization, quasiconvex and convex functions, Fréchet, Limiting and Clarke subdiferential, descent direction and Fejér convergence theory.", "In Section 3 we introduce our method and analyze the convergence of the iterations.", "In Section 4, we present some quasiconvex optimization models and in Section 5 we give our conclusion and some ideas for future researchers." ], [ "Preliminaries", "In this section, we present some basic concepts and results that are of fundamental importance for the development of our work.", "These facts can be found, for example, in Hadjisavvas [14], Mordukhovich [23] and, Rockafellar and Wets [29]." ], [ "Definitions, notations and some basic results", "Along this paper $ \\mathbb {R}^n$ denotes an euclidean space, that is, a real vectorial space with the canonical inner product $\\langle x,y\\rangle =\\sum \\limits _{i=1}^{n} x_iy_i$ and the norm given by $\|\|x\|\|=\\sqrt{\\langle x, x\\rangle }$ .", "Given a function $f :\\mathbb {R}^n\\longrightarrow \\mathbb {R}\\cup \\left\\lbrace +\\infty \\right\\rbrace $, we denote by dom $(f)= \\left\\lbrace x \\in \\mathbb {R}^n: f(x) < + \\infty \\right\\rbrace $, the effective domain of $f$ .", "If dom $(f)\\ne \\emptyset $, $f $ is called proper.", "If $\\lim \\limits _{\\left\\Vert x\\right\\Vert \\rightarrow +\\infty }f(x) = +\\infty $, $f$ is called coercive.", "We denote by arg min $\\left\\lbrace f(x): x \\in \\mathbb {R}^n\\right\\rbrace $ the set of minimizer of the function $f$ and by $f * $ , the optimal value of problem: $\\min \\left\\lbrace f(x): x \\in \\mathbb {R}^n\\right\\rbrace ,$ if it exists.", "The function $f$ is lower semicontinuous at $\\bar{x}$ if for all sequence $\\left\\lbrace x_k\\right\\rbrace _{k \\in \\mathbb {N}} $ such that $\\lim \\limits _{k \\rightarrow +\\infty }x_k = \\bar{x}$ we obtain that $f(\\bar{x}) \\le \\liminf \\limits _{k \\rightarrow +\\infty }f(x_k)$ .", "Definition 2.1.1 Let $f:\\mathbb {R}^n\\longrightarrow \\mathbb {R}\\cup \\left\\lbrace +\\infty \\right\\rbrace $ be a proper function.", "We say that $f$ is locally Lipschitz at $x\\in $ dom $(f)$ if there exists $\\varepsilon _x>0$ such that $\\vert f(z) - f(y)\\vert \\le L_{x}\\Vert z - y\\Vert , \\ \\forall z, y \\in B(x,\\varepsilon _x)\\cap {\\small dom (f)},$ where $B(x,\\varepsilon _x) = \\lbrace y \\in \\mathbb {R}^n: \\Vert y - x\\Vert < \\varepsilon _{x}\\rbrace $ and $L_x$ is some positive number.", "$f$ is locally Lipschitz on $\\mathbb {R}^n$ if $f$ is locally Lipschitz for each $x\\in $ dom $(f)$ The next result ensures that the set of minimizers of a function, under some assumptions, is nonempty.", "Proposition 2.1.1 (Rockafellar and Wets [29], Theorem 1.9) Suppose that $f:\\mathbb {R}^n\\longrightarrow \\mathbb {R}\\cup \\left\\lbrace +\\infty \\right\\rbrace $ is proper, lower semicontinuous and coercive, then the optimal value $ f^$ is finite and the set $\\textnormal {arg min}$ $\\left\\lbrace f(x): x \\in \\mathbb {R}^n\\right\\rbrace $ is nonempty and compact.", "Definition 2.1.2 Let $D \\subset \\mathbb {R}^n$ and $\\bar{x} \\in D$ .", "The normal cone at the point $\\bar{x}$ related to the set $D$ is given by $\\mathcal {N}_{D}(\\bar{x}) = \\left\\lbrace v \\in \\mathbb {R}^n: \\langle v, x - \\bar{x}\\rangle \\le 0, \\forall \\ x \\in D\\right\\rbrace $ ." ], [ "Multiobjective optimization", "In this subsection we present some properties and notation on multiobjective optimization.", "Those basic facts can be seen, for example, in Miettinen [22] and Luc [18].", "Throughout this paper we consider the cone $\\mathbb {R}^m_+ = \\lbrace y\\in \\mathbb {R}^m : y_i\\ge 0, \\forall \\ i = 1, ... , m \\rbrace $ , which induce a partial order $\\preceq $ in $\\mathbb {R}^m$ given by, for $y,y^{\\prime }\\in \\mathbb {R}^m$ , $y\\ \\preceq \\ y^{\\prime }$ if, and only if, $ y^{\\prime }\\ - \\ y$ $ \\in \\mathbb {R}^m_+$ , this means that $ y_i \\le \\ y^{\\prime }_i$ for all $ i= 1,2,...,m $ .", "Given $ \\mathbb {R}^m_{++}$ the above relation induce the following one $\\prec $ , induced by the interior of this cone, given by, $y\\ \\prec \\ y^{\\prime }$ , if, and only if, $ y^{\\prime }\\ - \\ y$ $ \\in \\mathbb {R}^m_{++}$ , this means that $ y_i < \\ y^{\\prime }_i$ for all $ i= 1,2,...,m$ .", "Those partial orders establish a class of problems known in the literature as Multiobjective Optimization.", "Let us consider the unconstrained multiobjective optimization problem (MOP) : $\\textrm {min} \\left\\lbrace G(x): x \\in \\mathbb {R}^n \\right\\rbrace $ where $G:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m$ , with $G = \\left(G_1, G_2, ... , G_m\\right)^T$ .", "Definition 2.2.1 (Miettinen [22], Definition 2.2.1) A point $x^ \\in \\mathbb {R}^n$ is a Pareto solution of the problem $\\left(\\ref {POM}\\right)$ , if there does not exist $x \\in \\mathbb {R}^n $ such that $ G_{i}(x) \\le G_{i}(x^)$ , for all $i \\in \\left\\lbrace 1,...,m\\right\\rbrace $ and $ G_{j}(x) < G_{j}(x^)$ , for at least one index $ j \\in \\left\\lbrace 1,...,m\\right\\rbrace $ .", "Definition 2.2.2 (Miettinen [22],Definition 2.5.1) A point $x^* \\in \\mathbb {R}^n$ is a weak Pareto solution of the problem $\\left(\\ref {POM}\\right)$ , if there does not exist $x \\in \\mathbb {R}^n $ such that $ G_{i}(x) < G_{i}(x^)$ , for all $i \\in \\left\\lbrace 1,...,m\\right\\rbrace $ .", "We denote by arg min$\\left\\lbrace G(x):x\\in \\mathbb {R}^n \\right\\rbrace $ and by arg min$_w$ $\\left\\lbrace G(x):x\\in \\mathbb {R}^n \\right\\rbrace $ the set of Pareto solutions and weak Pareto solutions to the problem $\\left(\\ref {POM}\\right)$ , respectively.", "It is easy to check that arg min$\\left\\lbrace G(x):x\\in \\mathbb {R}^n \\right\\rbrace \\subset $ arg min$_w$ $\\left\\lbrace G(x):x\\in \\mathbb {R}^n \\right\\rbrace $ ." ], [ "Quasiconvex and Convex Functions", "In this subsection we present the concept and characterization of quasiconvex functions and quasiconvex multiobjective function.", "This theory can be found in Bazaraa et al.", "[2], Luc [18], Mangasarian [19], and their references.", "Definition 2.3.1 Let $f:\\mathbb {R}^n\\longrightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ be a proper function.", "Then, f is called quasiconvex if for all $x,y\\in \\mathbb {R}^n$ , and for all $ t \\in \\left[0,1\\right]$ , it holds that $f(tx + (1-t) y)\\le \\textnormal {max}\\left\\lbrace f(x),f(y)\\right\\rbrace $ .", "Definition 2.3.2 Let $f:\\mathbb {R}^n\\longrightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ be a proper function.", "Then, f is called convex if for all $x,y\\in \\mathbb {R}^n$ , and for all $ t \\in \\left[0,1\\right]$ , it holds that $f(tx + (1-t) y)\\le tf(x) + (1 - t)f(y)$ .", "Observe that if $f$ is a quasiconvex function then dom$(f)$ is a convex set.", "On the other hand, while a convex function can be characterized by the convexity of its epigraph, a quasiconvex function can be characterized by the convexity of the lower level sets: Definition 2.3.3 (Luc [18], Corollary $6.6$ ) Let $F= (F_1,...,F_m)^T:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m$ be a function, then $F$ is $\\mathbb {R}^m_+$ - quasiconvex if and only if every component function of $F$ , $F_i: \\mathbb {R}^n\\longrightarrow \\mathbb {R}$ , is quasiconvex.", "Definition 2.3.4 Let $F= (F_1,...,F_m)^T:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m$ be a function, then $F$ is $\\mathbb {R}^m_+$ - convex if and only if every component function of $F$ , $F_i: \\mathbb {R}^n\\longrightarrow \\mathbb {R}$ , is convex.", "Definition 2.3.5 Let $F= (F_1,...,F_m)^T:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m$ be a function, then $F$ is locally Lipschitz on $\\mathbb {R}^n$ if and only if every component function of $F$ , $F_i: \\mathbb {R}^n\\longrightarrow \\mathbb {R}$ , is locally Lipschitz on $\\mathbb {R}^n$ ." ], [ "Fréchet and Limiting Subdifferentials", "Definition 2.4.1 Let $f: \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace +\\infty \\rbrace $ be a proper function.", "For each $x \\in \\textnormal {dom}(f)$ , the set of regular subgradients (also called Fréchet subdifferential) of $f$ at $x$ , denoted by $\\hat{\\partial }f(x)$ , is the set of vectors $v \\in \\mathbb {R}^n$ such that $f(y) \\ge f(x) + \\left\\langle v,y-x\\right\\rangle + o(\\left\\Vert y - x\\right\\Vert )$ , where $\\lim \\limits _{y \\rightarrow x}\\frac{o(\\left\\Vert y - x\\right\\Vert )}{\\left\\Vert y - x\\right\\Vert } =0$ .", "Or equivalently, $\\hat{\\partial }f (x) := \\left\\lbrace v \\in \\mathbb {R}^n : \\liminf \\limits _{y\\ne x,\\ y \\rightarrow x} \\dfrac{f(y)- f(x)- \\langle v , y - x\\rangle }{\\Vert y - x \\Vert } \\ge 0 \\right\\rbrace $ .", "If $x \\notin \\textnormal {dom}(f)$ then $\\hat{\\partial }f(x) = \\emptyset $ .", "The set of general subgradients (also called limiting subdifferential) $f$ at $x \\in \\mathbb {R}^n$ , denoted by $\\partial f(x)$ , is defined as follows: $\\partial f(x) := \\left\\lbrace v \\in \\mathbb {R}^n : \\exists \\ x_n \\rightarrow x, \\ \\ f(x_n) \\rightarrow f(x), \\ \\ v_n \\in \\hat{\\partial } f(x_n)\\ \\textnormal {and}\\ v_n \\rightarrow v \\right\\rbrace $ .", "Proposition 2.4.1 For a function $f: \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ and a point $\\bar{x}\\in \\textnormal {dom}(f)$ , the subgradient sets $\\partial f(\\bar{x})$ and $\\hat{\\partial } f(\\bar{x})$ are closed, with $\\hat{\\partial } f(\\bar{x})$ convex and $\\hat{\\partial }f\\left(\\bar{x}\\right)$ $\\subset $ $\\partial f\\left(\\bar{x}\\right)$ .", "Proof.", "See Rockafellar and Wets [29], Theorem 8.6.", "Proposition 2.4.2 (Fermat’s rule generalized) If a proper function $f: \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ has a local minimum at $\\bar{x} \\in dom{(f)}$ , then $0\\in \\hat{\\partial } f\\left(\\bar{x}\\right)$ .", "Proof.", "See Rockafellar and Wets [29], Theorem 10.1.", "Proposition 2.4.3 Let $f, g : \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ proper functions such that $f$ is locally Lipschitz at $\\bar{x} \\in $ dom$(f)\\ \\cap \\ $ dom$(g)$ and $g$ is lower semicontinuous function at this point.", "Then, $\\partial (f + g)(\\bar{x})\\subset \\partial f(\\bar{x}) + \\partial g(\\bar{x})$ Proof.", "See Mordukhovich [23] Theorem 2.33." ], [ "Definition 2.5.1 Let $f: \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ be a proper locally Lipschitz function at $x \\in \\textnormal {dom}(f)$ and $d \\in \\mathbb {R}^n$ .", "The Clarke directional derivative of $f$ at $x$ in the direction $d$ , denoted by $f^o(x,d)$ , is defined as $f^o (x,d) = \\limsup \\limits _{t\\downarrow 0 \\ \\ y \\rightarrow x} \\dfrac{f(y + td)- f(y)}{t}$ and the Clarke subdifferential of $f$ at $x$ , denoted by $\\partial ^of(x)$ , is defined as $\\partial ^of(x)= \\lbrace w \\in \\mathbb {R}^n:\\langle w , d \\rangle \\le f^o (x,d), \\forall \\ d \\in \\mathbb {R}^n \\rbrace $ .", "Remark 2.5.1 From the above definitions it follows directly that for all $x \\in \\mathbb {R}^n$ , one has $\\hat{\\partial }f(x) \\subset \\partial f(x) \\subset \\partial ^o f(x)$ (see Bolte et al.", "[4], Inclusion (7)).", "Lemma 2.5.1 Let $f, g : \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ be locally Lipschitz functions at $x \\in \\mathbb {R}^n $ .", "Then, $\\forall d \\in \\mathbb {R}^n$ : $\\left( f + g \\right) ^ o \\left( x , d\\right) \\le f^o \\left( x , d\\right) + g^o \\left( x, d\\right)$ ; $ \\left(\\lambda f \\right)^ o \\left( x , d\\right) = \\lambda \\left( f^o (x , d) \\right),\\ \\forall \\lambda \\ge 0 $ ; $ f ^ o \\left( x , \\lambda d\\right) = \\lambda f^o (x , d),\\ \\forall \\lambda \\ge 0 $ .", "Proof.", "It is immediate from Clarke directional derivative.", "Lemma 2.5.2 Let $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be locally Lipschitz function at $x$ and any scalar $\\lambda $ , then $\\partial ^o \\left(\\lambda f\\right)(x)\\subset \\lambda \\partial ^o f(x)$ Proof.", "See Clarke [8], Proposition 2.3.1.", "Lemma 2.5.3 Let $f_i: \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace , i=1,2,...,m,$ be locally Lipschitz functions at $x$ , then $\\partial ^o \\left(\\displaystyle \\sum _{i=1}^{m} f_i\\right)(x)\\subset \\displaystyle \\sum _{i=1}^{m}\\partial ^o f_i (x)$ Proof.", "See Clarke [8], Proposition 2.3.3.", "Proposition 2.5.1 Let $f: \\mathbb {R}^n \\rightarrow \\mathbb {R} \\cup \\lbrace + \\infty \\rbrace $ be a proper locally Lipschitz function on $\\mathbb {R}^n$ .", "Then, $f^ o $ is upper semicontinuous, i.e, if $ \\lbrace (x^k, d^k)\\rbrace $ is a sequence in $\\mathbb {R}^n\\times \\mathbb {R}^n$ such that $\\lim \\limits _{k \\rightarrow + \\infty }(x^k, d^k) = (x , d) $ then $\\limsup \\limits _{k \\rightarrow + \\infty }f^o (x^k, d^k) \\le f^o (x , d)$ .", "Proof.", "See Clarke [8], Proposition 2.1.1, (b).", "Proposition 2.5.2 Let $f:\\mathbb {R}^n\\longrightarrow \\mathbb {R}$ be a quasiconvex locally Lipschitz function on $\\mathbb {R}^n.$ If $g \\in \\partial ^o f(x),$ such that $\\left\\langle g ,\\tilde{x} - x \\right\\rangle > 0$ then, $f(x)\\le f(\\tilde{x}).$ Proof.", "See Aussel [1], Theorem 2.1.", "Proposition 2.5.3 Let $f : \\mathbb {R}^n \\longrightarrow \\mathbb {R}$ be a convex function.", "Then $\\partial ^ o f(x)$ coincides with the subdifferential at $x$ in the sense of convex analysis, and $f^ o (x,d)$ coincides with the directional derivative $f^{\\prime }(x,d)$ for each $d$ .", "Proof.", "See Clarke [8], Proposition 2.2.7" ], [ "Descent direction", "We are now able to introduce the definition of Pareto-Clarke critical point for locally Lipschitz functions on $\\mathbb {R}^n$ , which will play a key role in our paper.", "Definition 2.6.1 (Custódio et al.", "[9], Definition $4.6$ ) Let $F= (F_1,...,F_m)^T:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m $ be locally Lipschitz on $\\mathbb {R}^n$ .", "We say that $x^ \\in \\mathbb {R}^n$ is a Pareto-Clarke critical point of $F$ if, for all directions $d \\in \\mathbb {R}^n$ , there exists $i_0 = i_{0} (d) \\in \\lbrace 1,...,m\\rbrace $ such that $F^o_{i_o}(x^, d)\\ge 0$ .", "Definition $\\ref {paretoclarke}$ says essentially that there is no direction in $\\mathbb {R}^n$ that is descent for all the objective functions (see, for instance, (Custódio et al.", "[9]).", "If a point is a Pareto minimizer (local or global), then it is necessarily a Pareto-Clarke critical point .", "Remark 2.6.1 Follows from the previous definition that, if a point $x$ is not Pareto-Clarke critical, there exists a direction $d \\in \\mathbb {R}^n$ satisfying $F_{i}^o(x, d) < 0, \\forall \\ i \\in \\left\\lbrace 1,...,m\\right\\rbrace $ This implies that, for each $i \\in \\lbrace 1,..., m \\rbrace $ , $d$ is a descent direction, for each function $F_i$ , i.e, there exists $\\varepsilon > 0 $ , such that $F_i(x + td) < F_i(x), \\forall \\ t \\in (0 , \\varepsilon ], \\forall \\ i \\in \\lbrace 1,..., m \\rbrace $ .", "It is a well known fact that such $d$ is a descent direction for the multiobjective function $F$ at $x$ , i.e, $\\exists \\ \\ \\varepsilon > 0 $ such that $ F(x + td) \\prec F(x), \\ \\forall \\ t \\in (0 , \\varepsilon ]$ .", "Proposition 2.6.1 Let $\\bar{x}$ be a Pareto-Clarke critical point of a locally Lipschitz $G:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m.$ If $G$ is $\\mathbb {R}^m_{+}$ -convex, then $\\bar{x}$ is weak Pareto solution of the problem (REF ).", "Proof.", "As $\\bar{x}$ is a Pareto-Clarke critical point of $G$ then for all directions $d$ there exists $i_0 = i_{0} (d) \\in \\lbrace 1,...,m\\rbrace $ such that $G^o_{i_o}(\\bar{x}, d)\\ge 0.$ Now, due that $G$ is $\\mathbb {R}^m_{+}-$ convex then the last is equivalent, see Proposition REF , to $G^{\\prime }_{i_o}(\\bar{x}, d)\\ge 0,$ where $G^{\\prime }_{i_0}(\\bar{x}, d)$ is the directional derivative of the convex function $G_{i_0}$ at $\\bar{x}$ in the direction $d$ .", "On the other hand, suppose by contradiction that $\\bar{x}$ is not a weak Pareto solution of the problem (REF ), then exists $x^ \\in \\mathbb {R}^n$ such that $G(x^) \\prec G(\\bar{x}),\\ \\textnormal {i.e},\\ G_i(x^) < G_i(\\bar{x}), \\forall i \\in {1,...,m}.$ Thus, for all $i,$ there exists $\\alpha =\\alpha (i) > 0$ such that $G_{i}(x^) =G_{i}(\\bar{x}) - \\alpha $ .", "Define $x_\\lambda = \\lambda x^ + (1-\\lambda )\\bar{x}$ , $\\lambda \\in (0,1)$ .", "From the $\\mathbb {R}^m_+$ -convexity of $G$ we have $G_{i}(x_\\lambda )=G_{i}(\\lambda x^* + (1-\\lambda )\\bar{x})\\le \\lambda G_{i}(x^) + (1-\\lambda )G_{i}(\\bar{x})= -\\alpha \\lambda + G_{i}(\\bar{x})$ Its follows that $\\dfrac{G_{i}(\\bar{x} + \\lambda (x^ -\\bar{x})) - G_{i}(\\bar{x})}{\\lambda } \\le - \\alpha < 0$ , $\\forall \\lambda \\in (0,1)$ .", "Taking $\\bar{d} = x^* - \\bar{x} \\in \\mathbb {R}^n$ and limit when $\\lambda $ converges to zero in the above inequality we obtain a contradiction with (REF ).", "Therefore $\\bar{x}$ is a weak Pareto solution of the problem (REF )." ], [ "Fejér convergence", "Definition 2.7.1 A seguence $\\left\\lbrace y_k\\right\\rbrace \\subset \\mathbb {R}^n$ is said to be Fejér convergent to a set $U\\subseteq \\mathbb {R}^n$ if, $\\left\\Vert y_{k+1} - u \\right\\Vert \\le \\left\\Vert y_k - u\\right\\Vert , \\forall \\ k \\in \\mathbb {N},\\ \\forall \\ u \\in U$ .", "The following result on Fejér convergence is well known.", "Lemma 2.7.1 If $\\left\\lbrace y_k\\right\\rbrace \\subset \\mathbb {R}^n$ is Fejér convergent to some set $U\\ne \\emptyset $ , then: The sequence $\\left\\lbrace y_k\\right\\rbrace $ is bounded.", "If an accumulation point $y$ of $\\left\\lbrace y_k\\right\\rbrace $ belongs to $ U$ , then $\\lim \\limits _{k\\rightarrow +\\infty }y_k = y$ .", "Proof.", "See Schott [30], Theorem $2.7$ ." ], [ "Scalarization proximal point method (SPPM)", "We are interested in solving the unconstrained multiobjective optimization problem (MOP): $\\textrm {min}\\lbrace F(x): x \\in \\mathbb {R}^n\\rbrace $ where $F: \\mathbb {R}^n \\rightarrow \\mathbb {R}^m$ is a vector function satisfying the following assumptions: $\\bf (H_1)$ $F$ is locally Lipschitz on $\\mathbb {R}^n$ .", "$\\bf (H_2)$ $F$ is $\\mathbb {R}^m_+$ -quasiconvex." ], [ "The algorithm", "In this subsection, we propose a Scalarization Proximal Point Method with quadratic regula-rization, denoted by SPPM, to solve the problem $(\\ref {pom})$ .", "SPPM Algorithm Initialization: Choose an arbitrary initial point $x^0\\in \\mathbb {R}^n$ Main Steps: Given $x^k,$ find $x^{k+1}\\in \\Omega _k$ such that $0 \\in \\partial ^o\\left( \\left\\langle F(.", "), z_k\\right\\rangle + \\dfrac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\Vert \\ .\\ - x^k \\Vert ^2 \\right) (x^{k+1}) + \\mathcal {N}_{\\Omega _k}(x^{k+1})$ where $\\Omega _k= \\left\\lbrace x\\in \\mathbb {R}^n: F(x) \\preceq F(x^k)\\right\\rbrace $ , $\\alpha _k > 0 $ , $\\left\\lbrace e_k\\right\\rbrace \\subset \\mathbb {R}^m_{++}$ , $\\left\\Vert e_k\\right\\Vert = 1$ , $\\left\\lbrace z_k\\right\\rbrace \\subset \\mathbb {R}^m_+\\backslash \\left\\lbrace 0\\right\\rbrace $ and $\\left\\Vert z_k\\right\\Vert = 1$ .", "Stop Criterion: If $x^{k+1}=x^{k}$ or $x^{k+1}$ is a Pareto-Clarke critical point, then stop.", "Otherwise, to do $k \\leftarrow k+1$ and return to Main Steps.", "Remark 3.1.1 If $F$ is $\\mathbb {R}^{n}_{+}-$ convex the main step (REF ) is equivalent to: $x^{k+1}=\\textrm {argmin} \\left\\lbrace \\left\\langle F(x), z_k\\right\\rangle +\\frac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\left\\Vert x - x^k\\right\\Vert ^2 : x\\in \\Omega _k\\right\\rbrace $ This iteration has been studied by Bonnel et al.", "[5], so we can say that, in certain sense, our iteration is an extension for the nonconvex case of that work .", "On the other hand, when $F$ is $\\mathbb {R}^{n}_{+}-$ quasiconvex, the regularized function $F_k=\\left\\langle F(x), z_k\\right\\rangle + \\frac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\left\\Vert x - x^k\\right\\Vert ^2$ is not necessarily quasiconvex and so (REF ) is a global optimization problem, it is the reason for which we consider the more weak iteration (REF )." ], [ "Existence of the iterates", "Theorem 3.2.1 Let $F:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m $ be a function satisfying $\\bf (H_1), \\bf (H_2)$ and $ 0 \\prec F$ .", "Then the sequence $\\left\\lbrace x^k\\right\\rbrace $ , generated by the SPPM algorithm, given by $(\\ref {inicio})$ and $(\\ref {subdiferencial}),$ is well defined.", "Proof.", "We proceed by induction.", "It holds for $k=0,$ due to (REF ).", "Assume that $x^k$ exists and define $\\varphi _k(x)=\\left\\langle F(x), z_k\\right\\rangle + \\frac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\left\\Vert x - x^k\\right\\Vert ^2 +\\delta _{\\Omega _k}(x)$ , where $\\delta _{\\Omega _k}(.", ")$ is the indicator function of ${\\Omega _k}$ .", "Then we have that min$\\lbrace \\varphi _k(x): x \\in \\mathbb {R}^n\\rbrace $ is equivalent to min$\\lbrace \\left\\langle F(x), z_k\\right\\rangle + \\frac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\left\\Vert x - x^k\\right\\Vert ^2: x \\in \\Omega _k\\rbrace $ .", "Due that $ 0 \\prec F$ and $z_k \\in \\mathbb {R}^m_+\\backslash \\left\\lbrace 0\\right\\rbrace $ the function $\\left\\langle F(.", "), z_k\\right\\rangle $ is bounded from below.", "Then, by the lower boundedness and continuity of the function $\\left\\langle F(.", "), z_k\\right\\rangle $ , as also, by the continuity and coercivity of $\|\|.-x^k\|\|^2,$ and using Proposition REF , we obtain that there exists $x^{k+1} \\in \\Omega _k$ which is a global minimum of $\\varphi _k(.", ").$ From Proposition $\\ref {otimo}$ , $x^{k+1}$ satisfies $ 0 \\in \\hat{ \\partial }\\left( \\left\\langle F(.", "), z_k\\right\\rangle + \\dfrac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\Vert \\ .\\ - x^k \\Vert ^2 + \\delta _{\\Omega _k}(.", ")\\right) (x^{k+1})$ and by Proposition $\\ref {Rockwets1}$ and Proposition $\\ref {mordukhovich}$ , we have that $0 \\in \\partial \\left( \\left\\langle F(.", "), z_k\\right\\rangle + \\dfrac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\Vert \\ .\\ - x^k \\Vert ^2\\right) (x^{k+1}) + \\mathcal {N}_{\\Omega _k}(x^{k+1}).$ From Remark $\\ref {frechetclarke}$ , the iteration $(\\ref {subdiferencial})$ is obtained of $(\\ref {cone})$ .", "Remark 3.2.1 (Huang and Yang [15]) Without loss of generality, always we can assume that the function $F:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m$ satisfies $0 \\prec F.$ Of fact, consider the following multiobjective optimization problem $(P^{^{\\prime }})\\ \\ \\textnormal {min}\\left\\lbrace e^{F(x)}:x\\in \\mathbb {R}^n\\right\\rbrace $ Observe that both, (REF ) and $(P^{^{\\prime }})$ , have the same set of Pareto solutions, weak Pareto solutions and Pareto-Clarke critical points.", "Furthermore, if $F$ is $\\mathbb {R}^m_+$ - quasiconvex and locally Lipschitz on $\\mathbb {R}^n,$ then $e^{F(x)}$ is also $\\mathbb {R}^m_+$ - quasiconvex and locally Lipschitz on $\\mathbb {R}^n$ .", "Therefore, along this paper and from now on we implicitly assume that $0 \\prec F.$ Remark 3.2.2 We are interest in the asymptotic convergence of the (SPPM) algorithm, so we also assume along this paper that in each iteration $x^k$ is not a Pareto-Clarke critical point and $x^{k+1}\\ne x^k.$ This implies, from Remark REF that the interior of $\\Omega _{k+1},$ denoted by $\\Omega _{k+1}^0,$ is nonempty.", "When the condition $x^{k+1}=x^k$ is not satisfied, that is, if there exists $k_0$ such that $x^{k_0+1}=x^{k_0}$ then it is easy to prove that this point is a Pareto-Clarke critical point of $F.$" ], [ "Weak Convergence", "In this subsection we prove, under the assumption that the consecutive iterations converges to zero, that any cluster point is a Pareto-Clarke critical point of the problem $(\\ref {pom}).$ Proposition 3.3.1 Let $F:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m $ be a function satisfying $\\bf (H_1)$ and $\\bf (H_2).$ If $0 < \\alpha _k < \\tilde{\\alpha }$ , with $\\tilde{\\alpha }>0,$ and the sequence $\\lbrace x^k\\rbrace $ generated by the SPPM algorithm, $(\\ref {inicio})$ and $(\\ref {subdiferencial}),$ satisfies $\\lim _{k\\rightarrow +\\infty }\|\|x^{k+1}-x^{k}\|\|=0,$ and has a cluster point, then it is a Pareto-Clarke critical point of the problem $(\\ref {pom})$ .", "Proof.", "By assumption, there exists a convergent subsequence $\\left\\lbrace x^{k_j}\\right\\rbrace $ of $\\left\\lbrace x^{k}\\right\\rbrace $ whose limit is some $\\widehat{x} \\in \\mathbb {R}^n$ .", "Since $F$ is locally Lipschitz on $\\mathbb {R}^n$ , then the function $\\left\\langle F(.", "), z\\right\\rangle $ is also locally Lipschitz on $\\mathbb {R}^n$ and so, continuos for all $z \\in \\mathbb {R}^m$ , in particular, for all $z \\in \\mathbb {R}^m_+ \\backslash \\left\\lbrace 0\\right\\rbrace $ , and $\\lim \\limits _{j\\rightarrow +\\infty }\\left\\langle F(x^{k_j}) , z\\right\\rangle = \\left\\langle F(\\widehat{x}),z\\right\\rangle $ .", "On the other hand, as $x^{k+1} \\in \\Omega _k$ , we have $F(x^{k+1}) \\preceq F(x^{k})$ and since $z \\in \\mathbb {R}^m_+ \\backslash \\left\\lbrace 0\\right\\rbrace $ , we conclude that the sequence $\\left\\lbrace \\left\\langle F(x^k),z\\right\\rangle \\right\\rbrace $ is convergent to $\\left\\langle F(\\widehat{x}),z\\right\\rangle $ because it is nonincreasing and admits a subsequence converging to $\\left\\langle F(\\widehat{x}),z\\right\\rangle $ .", "So $\\lim \\limits _{k\\rightarrow +\\infty }\\left\\langle F(x^{k}) , z\\right\\rangle = \\left\\langle F(\\widehat{x}),z\\right\\rangle = inf_{k\\in \\mathbb {N}}\\left\\lbrace \\left\\langle F(x^k),z\\right\\rangle \\right\\rbrace \\le \\left\\langle F(x^k),z\\right\\rangle $.", "Thus, $\\left\\langle F(x^k)-F(\\widehat{x}),z\\right\\rangle \\ge 0, \\forall \\ k \\in \\mathbb {N}, \\forall \\ z \\in \\mathbb {R}^m_+ \\backslash \\left\\lbrace 0\\right\\rbrace $ .", "We conclude that $F(x^k) - F(\\widehat{x}) \\in \\mathbb {R}^m_+$ , i.e, $F(\\widehat{x})\\preceq F(x^k), \\forall \\ k \\in \\mathbb {N}$ .", "This implies that $\\widehat{x} \\in \\Omega _{k}$ .", "Assume, to arrive at a contradiction, that $\\widehat{x}$ is not Pareto-Clarke critical point in $\\mathbb {R}^n$ , then there exists a direction $d \\in \\mathbb {R}^n$ such that $F_{i}^o(\\widehat{x}, d) < 0, \\forall \\ i \\in \\left\\lbrace 1,...,m\\right\\rbrace $ Therefore $d$ is a descent direction for the multiobjective function $F$ in $\\widehat{x}$ , so, $\\exists \\ \\varepsilon > 0$ such that $F(\\widehat{x} + \\lambda d) \\prec F(\\widehat{x}),\\ \\forall \\ \\lambda \\in (0, \\varepsilon ].$ Thus, $ \\widehat{x} + \\lambda d \\in \\Omega _{k}$ .", "On the other hand, as $\\left\\lbrace x^{k}\\right\\rbrace $ is generated by SPPM algorithm, from Theorem $\\ref {existe}$ , $(\\ref {subdiferencial})$ , Lemma $\\ref {subcontido}$ and from Lemma $\\ref {escalar}$ , this implies that there exists $\\beta _k (x^{k} - x^{k+1}) - v_k\\in \\partial ^ o \\left( \\left\\langle F(.", "), z_k\\right\\rangle \\right) (x^{k+1})$ , with $v_k \\in \\mathcal {N}_{\\Omega _k}(x^{k+1})$ and $\\beta _k =\\alpha _k\\left\\langle e_k , z_k\\right\\rangle > 0$ , such that $\\beta _k \\langle x^k - x^{k+1} , p \\rangle - \\langle v_k , p \\rangle \\le \\langle F(.", "), z_k \\rangle ^ o (x^{k+1},p), \\forall p\\in \\mathbb {R}^n$ Consider $p = (\\widehat{x} + \\lambda d) - x^{k+1}$ and as $v_k \\in \\mathcal {N}_{\\Omega _k}(x^{k+1})$ , from $(\\ref {subclarke4})$ we have $\\beta _k \\langle x^k - x^{k+1} , \\widehat{x} + \\lambda d - x^{k+1} \\rangle \\le \\langle F(.", "), z_k \\rangle ^ o (x^{k+1},\\widehat{x} + \\lambda d - x^{k+1})$ As $\\left\\lbrace z_k\\right\\rbrace $ is bounded, then there exists a subsequence denoted also, without loss of generality, by $\\left\\lbrace z^{k_j}\\right\\rbrace $ such that $\\lim \\limits _{j\\rightarrow +\\infty }z^{k_j}=\\bar{z}$ , with $\\bar{z} \\in \\mathbb {R}^m_+\\backslash \\left\\lbrace 0\\right\\rbrace $ .", "From $(\\ref {subclarke3})$ , we have: $\\beta _{k_j} \\langle x^{k_j} - x^{{k_j}+1} , \\widehat{x} + \\lambda d - x^{{k_j}+1} \\rangle \\le \\langle F(.", "), z_{k_j} \\rangle ^ o (x^{{k_j}+1}, \\widehat{x} + \\lambda d - x^{{k_j}+1})$ Lemma $\\ref {algebra}$ , $(i)$ and $(ii)$ , we have: $\\beta _{k_j} \\langle x^{k_j} - x^{{k_j}+1} , \\widehat{x} + \\lambda d - x^{{k_j}+1} \\rangle \\le \\sum \\limits _{i=1}^m z_{k_j}^iF_i^0(x^{{k_j}+1}, \\widehat{x} + \\lambda d - x^{{k_j}+1}),$ where $z_{k_j}^i$ are the components of the vector $z_{k_j}.$ Then using Lemma $\\ref {algebra}$ , $(iii),$ we obtain: $\\beta _{k_j} \\langle x^{k_j} - x^{{k_j}+1} , \\widehat{x} + \\lambda d - x^{{k_j}+1} \\rangle \\le \\sum \\limits _{i=1}^m F_i^0\\left(x^{{k_j}+1}, z_{k_j}^i(\\widehat{x} + \\lambda d - x^{{k_j}+1})\\right),$ Taking lim sup in the above inequality, using the condition (REF ), Proposition $\\ref {limsup}$ and as $\\lambda > 0$ , we conclude that $0\\le F_1^o(\\widehat{x}, d)\\bar{z}_1 + ... + F_m^o(\\widehat{x}, d)\\bar{z}_m$ Without loss of generality, consider the set $J = \\left\\lbrace i \\in I: \\bar{z}_i > 0 \\right\\rbrace $ , where $I = \\left\\lbrace 1,...,m\\right\\rbrace $ .", "Thus, from $(\\ref {somatorio1})$ , there exists $i_0 \\in J$ such that $F_{i_0}^o(\\widehat{x}, d)\\bar{z}_{i_0} \\ge 0$ contradicting $(\\ref {desce})$ ." ], [ "Global Convergence", "For this subsection we make the following assumption on the function $F$ and the initial point $x^0$ : The set $\\left(F(x^0) - \\mathbb {R}^m_+\\right)\\cap F(\\mathbb {R}^n)$ is $\\mathbb {R}^m_+$ - complete, meaning that for all sequences $\\left\\lbrace a_k\\right\\rbrace \\subset \\mathbb {R}^n$ , with $a_0 = x^0$ , such that $F(a_{k+1}) \\preceq F(a_k)$ , there exists $ a \\in \\mathbb {R}^n$ such that $F(a)\\preceq F(a_k), \\ \\forall \\ k \\in \\mathbb {N}$ .", "Remark 3.4.1 The assumption ${\\bf (H_3)}$ is cited in various works on proximal point method for convex functions, see Bonnel et al.", "[5], Ceng and Yao [7] and Villacorta and Oliveira [31].", "As the sequence $\\left\\lbrace x^k\\right\\rbrace $ generated by SPPM algorithm, satisfies the assumption ${\\bf (H_3)}$ and from assumptions ${\\bf (H_1)}$ and ${\\bf (H_2)}$ then $E = \\left\\lbrace x \\in \\mathbb {R}^n: F\\left(x\\right)\\preceq F\\left(x^k\\right),\\ \\ \\forall \\ k \\in \\mathbb {N}\\right\\rbrace $ is a nonempty closed convex set.", "Proposition 3.4.1 (Fejér convergence) Under assumptions ${\\bf (H_1)}$ , ${\\bf (H_2)}$ and ${\\bf (H_3)}$ , the sequence $\\left\\lbrace x^k\\right\\rbrace $ generated by the SPPM algorithm, $(\\ref {inicio})$ and $(\\ref {subdiferencial})$ , is Fejér convergent to $E$ .", "Proof.", "From Theorem $\\ref {existe}$ , $(\\ref {subdiferencial})$ , Lemma $\\ref {subcontido}$ and from Lemma $\\ref {escalar}$ we obtain that there exist $g_i^k\\in \\partial ^ oF_i (x^{k+1}),i=1,...,m$ such that $0 \\in \\sum \\limits _{i=1}^{m}z_k^i g_i^k + \\alpha _k\\left\\langle e_k , z_k\\right\\rangle ( x^{k+1} \\ - \\ x^k )+ \\mathcal {N}_{\\Omega _k}(x^{k+1})$ where $z_k^i$ are the components of $z_k.$ Thus there exist vectors $g_i^k\\in \\partial ^ oF_i (x^{k+1}),i=1,...,m,$ and $v_k \\in \\mathcal {N}_{\\Omega _k}(x^{k+1})$ such that $\\sum \\limits _{i=1}^{m}z_k^i g_i^k = \\beta _k(x^k - x^{k+1}) - v_k$ where $\\beta _k = \\alpha _k\\left\\langle e_k , z_k\\right\\rangle $ , $ \\forall \\ k \\in \\mathbb {N}$ .", "Note that $\\beta _k > 0$ , because $\\alpha _k > 0$ , $e_k$ belongs to $\\mathbb {R}^m_{++}$ , and $z_k$ belongs to $\\mathbb {R}^m_+\\backslash \\left\\lbrace 0\\right\\rbrace $ .", "From $(\\ref {gk})$ we have $x^k - x^{k+1} = \\dfrac{1}{\\beta _k}\\left( \\sum \\limits _{i=1}^{m}z_k^i g_i^k + v_k \\right)$ Now take $x^* \\in E,$ then by definition of $E$ , $x^* \\in \\Omega _{k+1}$ for all $k,$ and from Remark REF , there exists $\\lbrace x^l\\rbrace \\in \\Omega _{k+1}^0$ such that $x^l\\rightarrow x^.$ Observe that, $\\forall \\ x \\in \\mathbb {R}^n$ : $\\left\\Vert x^k - x\\right\\Vert ^2 = \\left\\Vert x^k - x^{k+1}\\right\\Vert ^2 + \\left\\Vert x^{k+1} - x\\right\\Vert ^2 +2\\left\\langle x^k - x^{k+1}, x^{k+1} - x\\right\\rangle .$ Now,combining $(\\ref {norma2})$ , with $x = x^l$ , and $(\\ref {xk})$ , we have: $\\left\\Vert x^k - x^l\\right\\Vert ^2 & = & \\left\\Vert x^k - x^{k+1}\\right\\Vert ^2 + \\left\\Vert x^{k+1} - x^l \\right\\Vert ^2 +\\frac{2}{\\beta _k} \\left(\\sum \\limits _{i=1}^{m}z_k^i\\left\\langle g_i^k ,\\ x^{k+1} -x^l\\right\\rangle + \\left\\langle v_k\\ ,\\ x^{k+1} -x^l\\right\\rangle \\right)$ As $F(x^l)\\prec F(x^{k+1}),$ then $F_i(x^l)<F_i(x^{k+1}), \\forall i=1,...,m$ .", "Furthermore, $g^k_i \\in \\partial ^ o F_i(x^{k+1})$ and as $F_i$ is quasiconvex, using Proposition $\\ref {t11}$ we have $\\left\\langle g_i^k ,\\ x^{k+1} -x^l\\right\\rangle \\ge 0, \\forall i=1,...,m.$ Now, as $v_k \\in \\mathcal {N}_{\\Omega _k}(x^{k+1})$ , the inequality $(\\ref {desigualdade2})$ and $(\\ref {conclusao})$ , imply, taking $l\\rightarrow \\infty $ $0\\le \\left\\Vert x^{k+1} - x^k\\right\\Vert ^2 \\le \\left\\Vert x^k - x^\\right\\Vert ^2 - \\left\\Vert x^{k+1} - x^\\right\\Vert ^2, \\forall k \\in \\mathbb {N}$ Thus, $\\left\\Vert x^{k+1} - x^\\right\\Vert \\le \\left\\Vert x^k - x^\\right\\Vert $ Proposition 3.4.2 Under assumptions ${\\bf (H_1)}$ , ${\\bf (H_2)}$ and ${\\bf (H_3)},$ the sequence $\\left\\lbrace x^k\\right\\rbrace $ generates by the SPPM algorithm, $(\\ref {inicio})$ and $(\\ref {subdiferencial}),$ satisfies $\\lim \\limits _{k\\rightarrow +\\infty }\\left\\Vert x^{k+1} - x^k\\right\\Vert = 0$ .", "Proof.", "It follows from $(\\ref {fejer3})$ , that $ \\forall x^ \\in E$ , $\\left\\lbrace \\left\\Vert x^k - x^*\\right\\Vert \\right\\rbrace $ is a nonnegative and nonincreasing sequence, and hence is convergent.", "Thus, the right-hand side of $(\\ref {desigual})$ converges to 0 as $k \\rightarrow +\\infty $ , and the result is obtained.", "Proposition 3.4.3 Under assumptions ${\\bf (H_1)}$ , ${\\bf (H_2)}$ and ${\\bf (H_3)},$ the sequence $\\left\\lbrace x^k\\right\\rbrace $ generated by the SPPM algorithm converges some point of $E$ .", "Proof.", "From Proposition $\\ref {fejer2}$ and Lemma $\\ref {fejerlim1}$ , $(i)$ , $\\left\\lbrace x^k\\right\\rbrace $ is bounded, then exists a subsequence $\\left\\lbrace x^{k_j}\\right\\rbrace $ such that $\\lim \\limits _{j\\rightarrow +\\infty }x^{k_j} = \\widehat{x}$ .", "Since $F$ is locally Lipschitz on $\\mathbb {R}^n$ , then the function $\\left\\langle F(.", "), z\\right\\rangle $ is also locally Lipschitz on $\\mathbb {R}^n$ and so, continuous for all $z \\in \\mathbb {R}^m$ , in particular, for all $z \\in \\mathbb {R}^m_+ \\backslash \\left\\lbrace 0\\right\\rbrace $ , and $\\lim \\limits _{j\\rightarrow +\\infty }\\left\\langle F(x^{k_j}) , z\\right\\rangle = \\left\\langle F(\\widehat{x}),z\\right\\rangle $ .", "On the other hand, as $x^{k+1} \\in \\Omega _k$ , we have $F(x^{k+1}) \\preceq F(x^{k})$ and since $z \\in \\mathbb {R}^m_+ \\backslash \\left\\lbrace 0\\right\\rbrace $ , we conclude $\\left\\langle F(x^{k+1}) , z\\right\\rangle \\le \\left\\langle F(x^{k}) , z\\right\\rangle $ .", "Furthermore, from Remark $\\ref {limitada}$ , we can assume that the function $\\left\\langle F(.", "), z\\right\\rangle $ is bounded below, for each $z \\in \\mathbb {R}^m_+\\backslash \\left\\lbrace 0\\right\\rbrace $ .", "Then the sequence $\\left\\lbrace \\left\\langle F(x^k),z\\right\\rangle \\right\\rbrace $ is nonincreasing and bounded below, hence convergent.", "So $\\lim \\limits _{k\\rightarrow +\\infty }\\left\\langle F(x^{k}) , z\\right\\rangle = \\left\\langle F(\\widehat{x}),z\\right\\rangle = inf_{k\\in \\mathbb {N}}\\left\\lbrace \\left\\langle F(x^k),z\\right\\rangle \\right\\rbrace \\le \\left\\langle F(x^k),z\\right\\rangle $.", "Thus, $\\left\\langle F(x^k)-F(\\widehat{x}),z\\right\\rangle \\ge 0, \\forall \\ k \\in \\mathbb {N}, \\forall \\ z \\in \\mathbb {R}^m_+ \\backslash \\left\\lbrace 0\\right\\rbrace $ .", "We conclude that $F(x^k) - F(\\widehat{x}) \\in \\mathbb {R}^m_+$ , i.e, $F(\\widehat{x})\\preceq F(x^k), \\forall \\ k \\in \\mathbb {N}$ .", "Thus $\\widehat{x}\\in E,$ then using Lemma $\\ref {fejerlim1}$ , $(ii)$ , we obtain the result.", "Finally, we prove that the sequence of the iterations converges to a Pareto-Clarke critical point when the sequence of regularization parameters $\\lbrace \\alpha _k\\rbrace $ is bounded.", "Theorem 3.4.1 Consider $F:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m $ a function satisfying the assumptions $\\bf (H_1)$ , $\\bf (H_2)$ and $\\bf (H_3)$ .", "If $0 < \\alpha _k < \\tilde{\\alpha }$ , then the sequence $\\lbrace x_k\\rbrace $ generated by the SPPM algorithm, $(\\ref {inicio})$ and $(\\ref {subdiferencial})$ , converges to a Pareto-Clarke critical point of the problem $(\\ref {pom})$ .", "Proof.", "From Proposition REF , $\\lbrace x^k\\rbrace $ converges, then this sequence has a unique cluster point $\\bar{x}$ and from Proposition REF and Proposition REF we obtain the result.", "Corollary 3.4.1 If $F:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m $ is $\\mathbb {R}^m_{+}$ -convex and $\\bar{x}$ the point of convergence given by the SPPM algorithm, given by $(\\ref {inicio})$ and $(\\ref {subdiferencial})$ , then $\\bar{x}$ is weak Pareto solution of the problem $(\\ref {pom})$ .", "Proof.", "It is inmediate from Proposition REF .", "Corollary 3.4.2 If $F:\\mathbb {R}^n\\longrightarrow \\mathbb {R}^m $ is continuously differentiable on $\\mathbb {R}^n$ and satisfies the assumptions $\\bf (H_2),$ $\\bf (H_3),$ then the point of convergence given by the SPPM algorithm $\\bar{x}$ , given by $(\\ref {inicio})$ and $(\\ref {subdiferencial})$ , is a Pareto critical point of the problem $(\\ref {pom}),$ that is, there exists $i_0\\in \\lbrace 1,2,...,m\\rbrace $ such that $\\left\\langle \\nabla F_{i_0}(\\bar{x}),d\\right\\rangle \\ge 0, \\forall d\\in \\mathbb {R}^n.$ Proof.", "It is immediate since continuously differentiable on $\\mathbb {R}^n$ implies the assumption $\\bf (H_1),$ and $=F_i^0(x,d)=F_i^{\\prime }(x,d)=\\left\\langle \\nabla F_i(x),d\\right\\rangle ,$ where $F_i^{\\prime }$ is the directional derivative of $F_i.$" ], [ "Optimization models with quasiconvex multivalued functions", "In this section we present some general quasiconvex multiobjective problems where the proposed algorithm may be applied." ], [ "A quasiconvex model in demand theory", "Let n be a finite number of consumer goods.", "A consumer is an agent who must choose how much to consume of each good.", "An ordered set of numbers representing the amounts consumed of each good set is called vector of consumption, and denoted by $ x = (x_1, x_2, ..., x_n) $ where $ x_i $ with $ i = 1,2, ..., n $ , is the quantity consumed of good i. Denote by $ X $ , the feasible set of these vectors which will be called the set of consumption, usually in economic applications we have $ X \\subset \\mathbb {R} ^ n_ + $ .", "In the classical approach of demand theory, the analysis of consumer behavior starts specifying a preference relation over the set $X,$ denoted by $\\succeq $ .", "The notation: $ \"x \\succeq y \" $ means that \"$ x $ is at least as good as $ y $ \" or \"$ y $ is not preferred to $x$ \".", "This preference relation $ \\succeq $ is assumed rational, i.e, is complete because the consumer is able to order all possible combinations of goods, and transitive, because consumer preferences are consistent, which means if the consumer prefers $\\bar{x}$ to $\\bar{y} $ and $\\bar{y}$ to $\\bar{z}$ , then he prefers $\\bar{x}$ to $\\bar{z} $ (see Definition 3.B.1 of Mas-Colell et al.", "[21]).", "A function $ \\mu :X \\longrightarrow \\mathbb {R} $ is said to be an utility function representing a preference relation $\\succeq $ on $X$ , if the following condition is satisfied: $x \\succeq y , \\textrm {if and only if,}\\ \\mu (x) \\ge \\mu (y)$ for all $x, y \\in X$ .", "The utility function is a way to represent preferences between two vectors of consumption.", "If they have the same value of the utility function, then the consumer is indifferent.", "Moreover, if we have several preferences relations $\\succeq _i, i=1,2,...,m,$ (multiple criteria), which satisfy the condition $(\\ref {prefe})$ , then we have a utility function $\\mu _i$ for each one of these preferences $\\succeq _i$ .", "Observe that the utility function not always exist.", "In fact, define in $ X = \\mathbb {R} ^ 2 $ a lexicographic relation, given by: for $ x, y \\in \\mathbb {R}^2 $ , $ x \\succeq y $ if and only if $ \"x_1 > y_1\"$ or $\"x_1 = y_1 \\ \\textnormal {e} \\ x_2 \\ge y_2 \"$ .", "Fortunately, a very general class of preference relations can be represented by utility functions, see for example 3.C.1 Proposition of Mas-Colell et al.", "[21].", "If a preference relation $ \\succeq $ is represented by a utility function $ \\mu $ , then the problem of maximizer the consumer preference on $X$ is equivalent to solve the optimization problem $\\textnormal {(P)\\ max}\\lbrace \\mu (x) :x \\in X\\rbrace .$ Now consider a multiple criteria, that is, consider $ m $ preference relations denoted by $\\succeq _i, i=1,2,...,m.$ Suppose that for each preference $\\succeq _i,$ there exists an utility function, $ \\mu _i,$ respectively, then the problem of maximizer the consumer preference on $ X $ is equivalent to solve the multiobjective optimization problem $\\textnormal {(P^{\\prime })\\ max}\\lbrace (\\mu _{1}(x), \\mu _{2}(x), ..., \\mu _{m}(x)) \\in \\mathbb {R}^m :x \\in X\\rbrace .$ Since there is not a single point which maximize all the functions simultaneously the concept of optimality is established in terms of Pareto optimality or efficiency.", "On the other hand, a natural psychological assumption in economy is that the consumer tends to diversify his consumption among all goods, that is, the preference $ \\succeq $ satisfies the following convexity property: $ X$ is convex and if $x \\succeq z $ and $ y \\succeq z $ then $ \\lambda x + (1 - \\lambda ) y \\succeq z $ , $ \\forall \\lambda \\in [0,1] $ .", "It can be proved that if there is a utility function representing the preference relation $\\succeq ,$ then the convexity property of $ \\succeq $ is equivalent to the quasiconcavity of the utility function $ \\mu $ .", "Therefore ${\\rm (P^{\\prime })} $ becomes a maximization problem with quasiconcave multiobjective function, since each component function is quasiconcave.", "Taking F = $ (- \\mu _1, - \\mu _2, ..., - \\mu _m) $ , we obtain a minimization problem with quasiconvex multiobjective function, since each component function is quasiconvex one.", "There are various class of utilities functions which are frequently used to generate demand functions.", "One of the most common is the Cobb-Douglas utility function, which is defined on $ \\mathbb {R} ^ 2$ by $ \\mu (x_1, x_2) = k x_1 ^\\alpha x_2 ^ {\\beta } $ , with $ \\alpha ,\\beta >0$ and $ k > 0 $ .", "Another utility function CES (Constant Elasticity of Substitution), defined on $ \\mathbb {R} ^ 2$ by $ \\mu (x_1, x_2) = (\\lambda _1x_1 ^ \\rho + \\lambda _2x_2 ^ \\rho ) ^ {1 / \\rho }$ , where $ \\lambda _1, \\lambda _2 \\ge 0$ , $ \\lambda _1 + \\lambda _2 = 1 $ , and $ \\rho $ is a constant." ], [ "A quasiconvex model in location theory", "Location problems are related to determining the location for one or more facilities, considering a given set of demand points, with which interactions should be established.", "These terms are not part of a standard terminology, are sometimes replaced by: clients, existing facilities, businesses or users.", "The following problem of locating a facility is motivated from the Chapter IV of Gromicho, [12].", "For each $ i = 1 , ... , m,$ let the cluster set $ d^i = \\lbrace d ^{i}_1 , d^{i}_2,..., d ^{i}_{p(i)} \\rbrace \\subset \\mathbb { R} ^n $ , $ n \\ge $ 2 (there exist $m$ cluster).", "We need to find a location $ x \\in \\mathbb {R} ^ n $ for an installation so that this location minimizes some real function involving the distance between the new location and each cluster set of demand points.", "For each $ i = 1 , ... , m $ , if $ C ^ i_j $ , $ j = 1 , ... , p(i),$ are compact convex sets with $ 0 \\in \\textnormal {int } ( C ^ i_j ) $ and $ \\textnormal { int } ( C ^ i_j ) $ denotes the interior of $ C ^ i_j $ then, for each $ i = 1 , ... , m,$ we define the distance between $ x $ and $ d ^ i_j$ by $ \\gamma _{C ^{i}_j} ( x - d ^ i_j ) $ with $ \\gamma _ {C^ i_j}$ the gauge or Minkowsky functional of the set $C^i_j$ , i.e.", "$ \\gamma _ {C^i_j}(x) = \\textnormal {inf} \\lbrace t > 0: x \\in tC ^i_j \\rbrace $ .", "Note that if $ { C ^ i_j } $ is the unit ball in $ \\mathbb {R} ^ n,$ then $ \\gamma _ { C ^ i_j } ( x ) $ is the Euclidean distance from $ x $ to 0.", "To introduce the model, consider, for each $ i = 1, ..., m $ , the function $\\gamma _i: \\mathbb {R} ^ n \\longrightarrow \\mathbb {R}^p_+$ , given by $ \\gamma _i (x) = (\\gamma _{C^ i_1}(x - d ^i_1), ...,\\gamma _{C ^i_{p(i)}} (x - d^i_{p(i)})) $ .", "And suppose, for each $i,$ that the functions $ f^i_j: \\mathbb {R} ^{p(i)}_ + \\longrightarrow \\mathbb {R}_ +$ , with $ j = 1, ..., p(i) $ is nondecreasing in $ \\mathbb {R}^ {p(i)}_ + $ , that is, if $ x, y \\in \\mathbb {R} ^{p(i)}_+$ , satisfying for each $ j = 1, ..., p(i)$ , $ x_j \\le y_j $ , then $f^i_j (x) \\le f^i_j (y) $ .", "The localization model is given by $min \\lbrace (\\phi _1 (x), \\phi _2 (x), ..., \\phi _m (x)): x \\in \\mathbb {R} ^ n \\rbrace $ , where, for each $ i = 1, ..., m$ , $\\phi _i (x) = max_{1 \\le j \\le p(i)} f^i_j (\\gamma _i (x)) $ .", "If for each $i=1,...,m,$ the functions $ f_j^i: \\mathbb {R} ^{p(i)}_+ \\longrightarrow \\mathbb {R} ^ + $ are quasiconvex in $ \\mathbb {R} ^{p(i)}_ + $ , then it can proved that for every $ i = 1, ..., m $ , each function $ \\phi _i (.)", "$ is quasiconvex in $ \\mathbb {R} ^ n $ ." ], [ "Conclusion and future works", "In this paper we introduced a scalarization proximal point method to solve unconstrained (possibly nonconvex and non-differentiable) multiobjective minimization problems with locally Lipschitz functions.", "Then, for quasiconvex objective functions we show a strong convergence (global convergence) to a Pareto-Clarke critical point satisfying the completeness assumption $\\bf (H_3)$ .", "Note this assumption has been considered in the convergence analysis of the proximal point method for the convex case, see [3].", "We also present, in Section , two optimization models where the quasiconvexity of the multiobjective functions appear naturally.", "We present quasiconvex models in demand theory and location theory.", "The (SPPM) algorithm, introduced in this paper, is the first attempt to construct efficient proximal point methods to solve quasiconvex multiobjective minimization problems and in its actual version may be considered as a based algorithm to develop other methods that consider computational errors, lower computational costs, lower complexity order and improves the convergence rate.", "Observe that in this paper we do not present an inexact version because, according to our knowledge, the theory of $\\epsilon -$ subdifferencial Clarke has not yet been developed.", "To reduce considerably the computational cost in each iteration of the (SPPM) algorithm it is need to consider the unconstrained iteration $0 \\in \\partial ^o\\left( \\left\\langle F(.", "), z_k\\right\\rangle + \\dfrac{\\alpha _k}{2}\\left\\langle e_k , z_k\\right\\rangle \\Vert \\ .\\ - x^k \\Vert ^2 \\right) (x^{k+1})$ which is more practical that (REF ).", "One natural condition to obtain (REF ) is that $x ^{k +1} \\in (\\Omega _k)^0$ (interior of $\\Omega _k$ ).", "So we believe that a variant of the (SPPM) algorithm may be an interior variable metric proximal point method.", "Observe also that in practice the iteration (REF ) or (REF ) should be solve using a local algorithm, which only provides an approximate solution.", "Therefore, we consider that in a future work it is important to analyze the convergence of the proposed algorithm considering now inexact iterations, see [28].", "Also the introduction of bundle methods are welcome.", "Acknowledgements The research of H.C.F.Apolinário was partially supported by CAPES/Brazil.", "The research of P.R.Oliveira was partially supported by CNPQ/Brazil.", "The research of E.A.Papa Quiroz was partially supported by the Postdoctoral Scholarship CAPES-FAPERJ Edital PAPD-2011." ] ]	1403.0150
[ [ "Microwave spectroscopy of Lambda-doublet transitions in the ground state\n of CH" ], [ "Abstract The Lambda-doublet transitions in CH at 3.3 and 0.7 GHz are unusually sensitive to variations in the fine-structure constant and the electron-to-proton mass ratio.", "We describe methods used to measure the frequencies of these transitions with Hz-level accuracy.", "We produce a pulsed supersonic beam of cold CH by photodissociation of CHBr3, and we measure the microwave transition frequencies as the molecules propagate through a parallel-plate transmission line resonator.", "We use the molecules to map out the amplitude and phase of the standing wave field inside the transmission line.", "We investigate velocity-dependent frequency shifts, showing that they can be strongly suppressed through careful timing of the microwave pulses.", "We measure the Zeeman and Stark effects of the microwave transitions, and reduce systematic shifts due to magnetic and electric fields to below 1 Hz.", "We also investigate other sources of systematic uncertainty in the experiment." ], [ "Introduction", "Many extensions to the standard model of particle physics make the intriguing prediction that quantities we normally consider to be fundamental constants - such as the fine-structure constant $\\alpha $ or the electron-to-proton mass ratio $\\mu $ - may in fact vary with time, position or with the local density of matter [1].", "These theories aim to unify gravity with the other forces, or to explain the nature of dark energy.", "The lowest $\\Lambda $ -doublet and millimetre-wave transitions in the CH molecule are particularly sensitive to variation of $\\alpha $ and $\\mu $ [2], [3].", "Recently, we tested the hypothesis that these constants may differ between the high density environment of the Earth and the vastly lower density of the interstellar medium, by comparing microwave frequencies of CH observed in cold interstellar gas clouds in our own galaxy to those measured in the laboratory [4].", "Using this method, we were able to constrain variations in these fundamental constants at the 0.1 parts-per-million level.", "To measure the laboratory frequencies with Hz-level accuracy, we developed a source which produces short pulses of CH molecules at low temperature, and we developed a variation on the method of Ramsey spectroscopy [5].", "In our method, the pulse of molecules propagate along a parallel-plate transmission line which supports a standing wave of the microwave field.", "In this geometry, the amplitude, polarization, and phase of the field are exceptionally well controlled.", "The microwaves can be pulsed on when the molecules are at any chosen position, and because the molecular pulse is short and the velocity spread is low, they can be used to map out the amplitude and phase of the field as a function of position [6].", "Doppler shifts are cancelled because the field is (nearly) a standing wave, and any residual velocity-dependent shifts are easily measured by changing the beam velocity.", "Here, we describe the source of CH, the spectroscopic technique, and the methods we use to control systematic frequency shifts." ], [ "The CH molecule", "The methylidyne radical, CH, plays an essential role in both chemistry and physics.", "Early investigations of optical emission spectra of CH helped to explain the spectra of diatomic molecules (see [7] and references therein).", "CH is a major participant in most combustion processes.", "It was one of the first molecules to be detected in the interstellar medium, in stellar atmospheres, and in comets, by means of optical absorption spectroscopy [8], [9].", "It is an essential building block in the formation of complex carbon-chain molecules in interstellar gas clouds [10] and it is commonly used as a tracer for atomic carbon and molecular hydrogen [11].", "In 1937 Dunham and Adams recorded the first optical spectrum of CH in the interstellar medium [12].", "Over forty years later the frequencies of the two lowest-lying $\\Lambda $ -doublet transitions, at 3.3 and 0.7, were determined by radio-astronomy [13], [14], [15].", "Laboratory measurements followed [16], [17], but the radio-astronomy measurements remained the most precise until our own recent measurements [4] using the methods described here.", "Figure: Relevant energy levels and transitions between the X and A states of CH.", "Each state labelled by JJ is split into two Λ\\Lambda -doublet levels of opposite parity (±\\pm , also labelled by their e/f parity).", "Each of these is further split by the hyperfine interaction into a pair of states with total angular momentum quantum numbers F=J±1/2F=J\\pm 1/2.", "Dotted lines show the optical transitions used for detecting molecules in the J=3/2J=3/2 and J=1/2J=1/2 states, labelled R 11ff (3/2)_{11ff}(3/2) and R 22ff (1/2)_{22ff}(1/2) respectively.", "The hyperfine components of the latter transition are labelled as a, b and c. Transition frequencies are given in MHz unless stated otherwise.Figure REF shows the low-lying energy level structure in the ground electronic state of CH, $\\text{X} ^2\\Pi (v=0)$ , and the first electronically excited state, $\\text{A} ^2\\Delta (v=0)$ , both described using the Hund's case (b) coupling scheme.", "Here, the electronic orbital angular momentum $\\mathbf {L}$ and the rotational angular momentum $\\mathbf {R}$ are coupled to a resultant $\\mathbf {N}=\\mathbf {L}+\\mathbf {R}$ .", "The spin angular momentum $\\mathbf {S}$ adds to $\\mathbf {N}$ to give the total electronic angular momentum $\\mathbf {J}=\\mathbf {N}+\\mathbf {S}$ .", "Each $J$ -level is split into a $\\Lambda $ -doublet which is composed of two closely-spaced states of opposite parity $\\left\|p=\\pm 1\\right\\rangle =\\left\|+\|\\Lambda \|\\right\\rangle \\pm (-1)^{J-S}\\left\|-\|\\Lambda \|\\right\\rangle $ , where $\\Lambda $ is the quantum number for the projection of $\\mathbf {L}$ onto the internuclear axis.", "The interaction with the $I=1/2$ hydrogen nuclear spin splits each $\\Lambda $ -doublet component into a pair of hyperfine levels, labelled by the total angular momentum quantum number $F=J\\pm 1/2$ .", "We use the short-hand notation $(J^p,F)$ to label the energy levels of the X state." ], [ "Production and detection of CH", "Figure REF shows the apparatus.", "We produce a pulsed, supersonic beam of CH molecules by photo-dissociation of bromoform (CHBr$_3$ ), following some of the methods described in [18], [19].", "A carrier gas, with a backing pressure of 4bar, is bubbled through liquid bromoform (Sigma Aldrich, 96% purity, stabilized in ethanol) which is held at room temperature in a stainless steel container.", "The mixture expands through the 1 orifice of a solenoid valve (General Valve Series 99) into a vacuum chamber at a repetition rate of 10.", "To dissociate the bromoform, we use pulses of light from an excimer laser, with an energy of 220, a wavelength of 248, and a duration of 20.", "This light propagates along the $x$ -axis, and is focussed to a rectangular spot, 1 high (along $y$ ) and 4 wide (along $z$ ), in the region immediately beyond the nozzle of the valve.", "The excimer pulse sets our origin of time.", "Further details of the source are given in [20].", "Figure: Sketch of the experiment to measure the Λ\\Lambda -doublet transitions of CH (not to scale).", "The molecular beam is produced via photodissociation of bromoform.", "The beam passes through a skimmer and a state selector, and then travels between the plates of a magnetically-shielded parallel-plate transmission line where the microwave transition is driven.", "Finally, the molecules are detected by laser induced fluorescence.", "For the measurement of the J=1/2J=1/2 Λ\\Lambda -doublet transition the outer magnetic shield and the inner magnetic field coils were absent.The molecules pass through a skimmer with a 2 diameter orifice, situated at $z={78}{}$ , then through the apparatus used to measure the microwave transition frequencies (described below), and are finally detected at $z=D={780}{}$ by laser-induced fluorescence.", "The probe laser used for detection is a frequency-doubled continuous-wave titanium-sapphire laser, tuned to the $A^2\\Delta (v=0)\\leftarrow X^2\\Pi (v=0)$ transition near 430.15.", "This probe is linearly polarized along $z$ , propagates along $x$ , has a power of 5, and is shaped to a rectangular cross-section, 4 in the $y$ -direction and 1.4 in the $z$ direction.", "The induced fluorescence is imaged onto a photomultiplier tube, and the signal is recorded with a time resolution of approximately 5.", "To vary the beam velocity, $v_0$ , we use He, Ne, Ar and Kr carrier gases.", "Figure REF (a) shows the time-of-flight profiles of CH molecules arriving at the detector when He and Ar are used.", "From the arrival times we measure the mean speeds to be $v_0=1710$ , 800, 570 and 420 for He, Ne, Ar and Kr respectively.", "The duration of the pulse of CH produced in the source is determined by the 4 width of the excimer beam and is always very short compared with the width of the time-of-flight profile measured at the detector.", "This width therefore measures the translational temperature of the CH beam.", "From the data in figure REF (a) we measure a CH translational temperature of 2 and 0.4 for He and Ar carrier gases respectively.", "We selectively detect molecules in either the $(1/2^{-},F)$ or the $(3/2^{+},F)$ levels, by driving one of the two transitions labelled R$_{22ff}(1/2)$ and R$_{11ff}(3/2)$ in figure REF .", "Figure REF (b) shows the spectrum recorded as the probe laser is scanned over the three well resolved hyperfine components of the R$_{22ff}(1/2)$ transition.", "Because the detected molecules have a range of transverse speeds, the spectral lines are Doppler broadened to a full width at half maximum of 23.", "For the microwave spectroscopy, the laser frequency is locked to one of the hyperfine components of the relevant transition using an optical cavity which is itself locked to a stabilized He-Ne laser.", "Figure: (a) Time of flight profiles of CH molecules using (separately) He and Ar carrier gases.", "Points: data, line: Gaussian fits.", "(b) Spectrum showing the three hyperfine components of the R 22ff (1/2)_{22ff}(1/2) line of the A 2 Δ(v=0,N=2)←X 2 Π(v=0,N=1)A^2\\Delta (v=0,N=2)\\leftarrow X^2\\Pi (v=0, N=1) transition.", "The labels a, b, c correspond to those in figure .", "Points: data.", "Line: Fit to a sum of three Gaussians." ], [ "Microwave spectroscopy setup", "At $z={241}{}$ , the molecules pass through the state-selector, which selectively populates one of the two parity eigenstates of a $\\Lambda $ -doublet.", "For the $J=1/2$ measurements, the state selection is done by optically pumping molecules out of the relevant $(1/2^-,F)$ state, using 40 of laser light at the same frequency as the probe.", "This light is reflected almost back on itself to increase the interaction time with the molecules.", "The optical pumping removes about 70% of the initial population.", "Over 95% of the CH molecules produced in our source are in the ground $J=1/2$ state, so for the $J=3/2$ measurements the state selection is done by driving population into the $(3/2^+,1)$ or $(3/2^+,2)$ states using 10 of radiation near 533.", "This millimetre-wave radiation is generated by an amplifier-multiplier chain that produces the 54th harmonic of a frequency synthesizer.", "The transfer efficiency is about 40%.", "A description of the measurement of this lowest millimeter-wave transition is given in reference [21].", "Following the state-selector the molecules enter the transmission line resonator where we drive the $\\Lambda $ -doublet transition.", "The transmission line is formed by a pair of parallel copper plates of length $L$ and width $w$ , separated vertically by a distance $d$ .", "It is fed from a semi-rigid, non-magnetic coaxial cable, with the inner conductor connected to one plate and the outer conductor connected to the other.", "The other end of the transmission line is open, and the wave reflects from this end to form a standing wave.", "The quality factor of this resonator is determined by the reflectivity of the open end and the transmission between the coaxial cable and the transmission line.", "We cut the plates to a length such that a resonance frequency of the transmission line matches the approximate $\\Lambda $ -doublet transition frequency.", "This length is approximately $L=n/(2 \\lambda )$ where $n$ is the number of electric field antinodes in the resonator, but this is insufficiently accurate because the position where the wave reflects from the ends is not well defined.", "Instead, we measure the spectrum of the resonator with a vector network analyzer and then reduce the length to obtain the desired resonance frequency.", "The lengths were approximately 480 for the $J=1/2$ measurements and 440 for the $J=3/2$ measurements.", "Figure REF shows the spectrum of the resonator set up for measuring the $(1/2^+,1)-(1/2^-,0)$ transition.", "The free spectral range of the resonator is 300, and the full width at half maximum (FWHM) of the peaks is 35, corresponding to a round-trip loss of 50%.", "Figure: Spectrum of the transmission line resonator.", "The plate length is chosen so that the resonator has a peak near the resonance frequency of the molecules.", "The width of the transmission line resonance is about 35.", "The dashed line indicates the approximate transition frequency of the (1/2 + ,1)-(1/2 - ,0)(1/2^+,1)-(1/2^-,0) line at 3.264.We aim to propagate only the TEM$_{00}$ mode so that the electric field is uniform between the plates and accurately polarized along $y$ .", "If the plate spacing is large enough, the transmission line can support the higher-order TE and TM modes that have components of the k-vector along $y$ , but these modes are cut off if the plate spacing is smaller than half the wavelength, $d < \\lambda / 2$ .", "The shortest wavelength we use in this experiment is 90, whereas the plate spacing is only 5, so these higher-order modes are very strongly attenuated.", "There can also be modes that have components of the k-vector along $x$ .", "These lesser-known modes zig-zag horizontally in the space between the two plates, reflecting from the open edges of the structure.", "Because they can radiate into the open space beyond the plates, they are referred to as leaky modes.", "The procedure for calculating the k-vectors of these modes is described in [22].", "In an early version of the experiment, we used plates of width $w={50}{}$ spaced by $d={10}{}$ .", "In this case, we find theoretically that there is a leaky mode which has a propagation wavelength of 130 and a $1/e$ attenuation distance of 220.", "Using the characterization methods described below, we indeed observed a mode with this wavelength.", "To cut-off the leaky mode, we reduced the width of the plates to $w={30}{}$ and also reduced the plate spacing to $d={5}{}$ .", "This increases the leaky mode propagation wavelength to 600 and reduces the attenuation distance to just 20.", "Since the latter is far smaller than the length of the interaction region, the mode is very effectively eliminated from the experiment.", "The spectrum shown in figure REF shows that there are no significant higher-order modes.", "The microwave radiation is generated by a frequency synthesizer which is phase locked to a GPS-disciplined frequency reference to a fractional uncertainty better than $10^{-13}$ .", "The synthesizer is connected to the transmission line resonator through a fast, high isolation switch." ], [ "Characterizing the microwave field", "The electric field inside the transmission line resonator can be described by $E=\\frac{A}{2} (1+\\Delta )\\cos (k (z-z_0) -\\omega t)+\\frac{A}{2}(1-\\Delta )\\cos (k(z-z_0) +\\omega t)\\, ,$ where $k=2\\pi /\\lambda $ is the wave vector, $A$ is the amplitude of the electric field, $z_{0}$ is the position of an antinode of the standing wave, and $\\Delta $ is an amplitude imbalance to account for the fact that the wave is not perfectly reflected at the end of the transmission line.", "This equation can be rewritten as $E=A\\sqrt{\\cos ^2\\left(k (z-z_0)\\right) + \\Delta ^2\\sin ^2\\left(k (z-z_0)\\right)}\\cos (\\omega t-\\phi (z))\\, ,$ where $\\phi (z)=\\tan ^{-1}\\left[\\Delta \\tan \\left(k (z-z_0)\\right)\\right] +\\phi _{0}\\,.$ Here, $\\phi _{0}=0$ when $\\cos (k (z-z_0))>0$ and $\\pi $ otherwise.", "For a single microwave pulse, the interaction of the molecules with the radiation transfers population from the initial to the final state with a probability of $P_{\\text{1-pulse}}=\\frac{\\Omega ^2}{\\Omega ^2+\\delta ^2}\\sin ^2\\left(\\frac{\\sqrt{\\Omega ^2+\\delta ^2}\\tau }{2}\\right)\\, ,$ where, $\\Omega =d_{12} E/\\hbar $ is the Rabi frequency, $d_{12}$ is the transition dipole moment, $\\delta =\\omega -\\omega _0$ is the detuning of the microwave angular frequency $\\omega $ from the resonance angular frequency $\\omega _0$ , and $\\tau $ is the interaction time.", "When $\\Omega \\tau =\\pi $ and $\\delta =0$ ($\\pi $ -pulse condition) the entire population is transferred from the initial to the final state.", "To begin an experiment, we first pulse the microwaves on for just a short period, $\\tau ={15}{}$ , at the time when the molecular pulse is at the centre of the transmission line, near an antinode of the electric field.", "Scanning the microwave frequency over the resulting broad resonance gives a first estimate of the transition frequency.", "The frequency is then fixed at the transition frequency and the microwave power is scanned.", "We observe Rabi oscillations in the measured population, an example of which is shown in figure REF (a).", "Since $\\Omega $ is proportional to the electric field of the radiation, we plot the signal versus the square root of the microwave power, and then fit equation (REF ) to the data.", "This identifies the exact power needed to drive a $\\pi $ -pulse for this pulse duration and for this particular position of the molecules.", "Figure REF (b) shows the power needed for a $\\pi $ -pulse with the molecules centred on each of the antinodes.", "The variation along the the transmission line is small.", "Figure: (a) Rabi oscillations.", "The detuning is set to δ=0\\delta =0, and the interaction time to τ=15\\tau ={15}{}.", "By scanning the microwave power we observe oscillations in the (1/2 - ,1)(1/2^-,1) population.", "This allows us to find the right power for a π\\pi -pulse.", "We fit S=S 0 +Asin 2 (αx)S=S_0+A\\sin ^2(\\alpha x) (red solid line) to the data (blue dots), where xx is proportional to the square root of the applied microwave power, and S 0 S_0, AA and α\\alpha are fitting parameters.", "(b) Power needed to drive a π\\pi -pulse for each antinode of the standing wave.", "Solid and dashed lines are the mean and standard deviation of the set.Next, we use the molecules to make a map of the microwave electric field amplitude inside the transmission line.", "We do this by measuring the population as a function of the time when a short, resonant microwave pulse is applied.", "Figure REF shows an example of the data obtained.", "Here, we have set $\\delta =0$ , $\\tau ={15}{}$ , and the microwave power such that $\\Omega \\tau = \\pi $ when the molecules are at the central antinode.", "When $\\delta =0$ the transition probability becomes $P=\\sin ^2\\left(\\Omega \\tau /2\\right)$ .", "The Rabi frequency $\\Omega $ is proportional to the electric field which varies along the transmission line according to equation (REF ).", "For this measurement, we take $\\Delta =0$ and so $\\Omega =\\Omega _{\\text{max}}\\cos \\left[2\\pi \\left(z-z_0\\right)/\\lambda \\right]$ .", "To account for the finite spread of the molecules we introduce an averaged Rabi frequency such that $\\Omega _{\\text{max}}\\tau =q\\pi $ with $q<1$ .", "The line in figure REF is a fit to the data using the expected model $S=S_0+A\\sin ^2\\left(q\\frac{\\pi }{2}\\cos \\left(\\frac{2\\pi \\left(v t-v t_0\\right)}{\\lambda }\\right)\\right)\\, ,$ where $t=z/v$ and $v={570}{}$ .", "The parameter $q$ , the offset $S_0$ , the amplitude $A$ , the initial time $t_0$ , and the wavelength $\\lambda $ are all fitting parameters.", "The fit yields $\\lambda =8.99\\pm 0.01{}{}$ , in agreement with the expected wavelength at 3.335.", "This field map is essential for the Ramsey experiments described below, where we need to know exactly when the molecular pulse passes each antinode.", "Figure: Mapping the amplitude of the microwave field by recording the fluorescence as a function of the time when a resonant 15 microwave pulse is applied.", "The power corresponds to a π\\pi -pulse when the molecules are at the central antinode.", "Blue dots: data.", "Red line: fit using the model of equation ()." ], [ "Frequency measurements", "Next, we increase the interaction time, $\\tau $ , and decrease the microwave power accordingly to maintain $\\Omega \\tau = \\pi $ .", "Figure REF shows the signal as a function of microwave frequency for the $(1/2^+,1)-(1/2^-,0)$ transition, when $\\tau ={326}{}$ .", "The timing is chosen so that the molecules pass through the central antinode half way through this interaction time.", "Because the field consists of two counter-propagating waves, parallel and anti-parallel to the molecular beam direction, we see two resonances separated by twice the Doppler shift $\\Delta \\omega _D=\\pm \\omega _{0} \\frac{v}{c}$ , where $v$ is the velocity of the molecules.", "To each resonance we fit the function $S=S_0+\\frac{\\Omega ^2}{\\Omega ^{2}+\\delta ^{2}}\\sin ^2\\left(\\frac{\\sqrt{\\Omega ^{2}+\\delta ^{2}}\\tau }{2}\\right)$ , with $\\tau =~{326}{}$ , $\\Omega \\tau =q\\pi $ , and $\\delta =\\omega -\\omega _0$ .", "Here, $S_0$ , $\\omega _0$ and $q$ are fitting parameters.", "The amplitudes of the two peaks are equal within their uncertainties of 3%.", "We find that the power needed for a $\\pi $ -pulse is the same for the two peaks to within 2% showing that $\\Delta $ is less than 0.005 at this frequency.", "The mean of the two centre frequencies gives the Doppler-free resonance frequency, which we find to be $3263793456 \\pm {17}{}$ for the $(1/2^+,1)-(1/2^-,0)$ transition and $3335479349\\pm {7}{}$ for the $(1/2^+,1)-(1/2^-,1)$ transition.", "With transitions produced by a single pulse of radiation, it is well known that splitting the line to such high accuracy may suffer from systematic errors.", "Specifically, inhomogeneities in either the static field or the ac field can produce an asymmetric lineshape and a corresponding shift of the line centre.", "Remarkably, however, these resonance frequencies agree at the level of 7 Hz with the measurements below, using the Ramsey method.", "This agreement indicates that there is no such lineshape distortion at this level.", "Figure: Single pulse measurement of the (1/2 + ,1)-(1/2 - ,0)(1/2^+,1)-(1/2^-,0) transition with τ=326\\tau =326 .", "The red line is a fit using the model discussed in the text.For the most precise measurement of the resonance frequency we use Ramsey's method of separated oscillatory fields [5].", "This method is less sensitive to the field inhomogeneities noted above, and it reduces the resonance linewidth by approximately 40% compared with a single-pulse measurement of the same over-all duration.", "The pulse sequence consists of two short $\\pi /2$ pulses of duration $\\tau $ and angular frequency $\\omega $ separated by a period of free evolution $T$ .", "The first $\\pi /2$ pulse is applied when the molecular pulse is at antinode $m_1$ and creates a superposition of the two $\\Lambda $ -doublet components.", "The coherence evolves freely for a time $T$ at the transition angular frequency $\\omega _0$ and develops a phase difference $\\delta T$ relative to the microwave oscillator, where $\\delta =\\omega -\\omega _0$ .", "A second $\\pi /2$ pulse completes the population transfer with a probability of $P_{\\text{2-pulse}}\\left(\\delta \\right)=&\\,\\frac{4\\pi ^2\\sin ^2\\left(\\frac{X}{4}\\right)}{X^4}\\times \\\\&\\,\\left[X\\cos \\left(\\frac{X}{4}\\right)\\cos \\left(\\frac{\\delta T+\\beta }{2}\\right)-2\\delta \\tau \\sin \\left(\\frac{X}{4}\\right)\\sin \\left(\\frac{\\delta T+\\beta }{2}\\right)\\right]^2\\, ,$ where $X=\\sqrt{\\pi ^2+4\\delta ^2\\tau ^2}$ and $\\beta $ is any change in the phase of the microwave field between one pulse and the next (here is no such phase shift if the field is a perfect standing wave).", "We set $\\tau ={15}{}$ , and choose the free evolution time $T=m\\lambda /(2v_0)-\\tau $ , where $m$ is an integer, so that the molecules travel an integer number of half wavelengths between the start of one pulse and the start of the next, making $\\beta =0$ (modulo $\\pi $ ) even for a travelling wave.", "Figure REF shows data taken this way to measure the $(1/2^-,1)-(1/2^+,1)$ and $(3/2^-,2)-(3/2^+,1)$ transition frequencies.", "The figure shows data for several different free evolution times $T$ .", "The lines are fits using the model $b+a P_{\\text{2-pulse}}(\\delta )$ with $\\beta =0$ , and with $\\tau $ and $T$ set to the values used in the experiment.", "This leaves only the offset $b$ , the amplitude $a$ (negative if $m$ is odd) and the resonance angular frequency $\\omega _0$ as fitting parameters.", "The frequencies measured for different values of $m$ are all in agreement.", "Figure: Frequency measurements using the method of separated oscillatory fields.", "Top: Population in the (1/2 - ,1)(1/2^-,1) state as a function of the microwave frequency for three different free evolution times, 458 (green), 380 (blue), 302 (red).", "Bottom: Population in the (3/2 + ,1)(3/2^+,1) state as a function of the microwave frequency for two different free evolution times, 650 (green), 330 (blue).", "Points: data, Lines: fits using equation ()" ], [ "Velocity-dependent frequency shifts", "If $\\beta $ is not zero, there will be a systematic frequency shift, $\\delta f=\\beta /\\left(2\\pi T\\right)$ .", "An obvious contribution to $\\beta $ comes from the amplitude imbalance between the co-propagating and counter-propagating waves, $\\beta =\\phi (z_2)-\\phi (z_1)$ , where $\\phi (z)$ is given by equation (REF ) and $z_{1,2}$ are the positions of the molecules at the start of the first and second pulses.", "As mentioned above, we aim to place $z_{1}$ and $z_{2}$ at antinodes of the standing wave.", "This can be achieved with high accuracy using the field map that we make for each frequency measurement, such as the one shown in figure REF , but there will always be some error, and there is a spread in the positions of the molecules.", "In fact, this spread in positions can usefully be used to map out the position dependence of the phase.", "Consider a molecule whose arrival time at the detector is $t_d$ .", "It is at position $z_{1}=v t_0$ at the time $t_0$ when the first pulse starts, and at position $z_{2}=v(t_0+\\tau + T)$ when the second pulse starts, where $v=D/t_d$ is its speed.", "For this molecule, the expected systematic shift is $\\delta f = \\frac{1}{2\\pi T}\\left\\lbrace \\tan ^{-1}\\left[\\Delta \\tan \\left(k v(t_0+\\tau + T)- k z_0\\right)\\right] - \\tan ^{-1}\\left[\\Delta \\tan \\left(k v t_0- k z_0\\right)\\right] \\right\\rbrace .$ We divide the time-of-flight profile into slices 5 wide.", "For each slice we find the resonance frequency using the Ramsey method, and plot this against the arrival time.", "Figure REF shows the data obtained this way for the $(1/2^+,1)-(1/2^-,1)$ transition.", "We see that the frequency shift is small for molecules near the central arrival time ($\\approx 1.35$ ms) because, when the pulses were applied, these molecules were close to the antinodes where the phase changes very slowly.", "The molecules that arrive later are moving more slowly, and they are closer to the nodes when the pulses are applied.", "Molecules arriving at 1.4 are near the node when the second pulse is applied, and here we see a sudden change in the frequency shift.", "Those arriving even later, at about 1.44, are at a node when the first pulse is applied, and we see another sudden change in the frequency shift.", "The line in figure REF is a fit to the model $f_0 + \\delta f$ where $\\delta f$ is given by equation (REF ) and $f_{0}$ is the resonance frequency for molecules in the centre of the pulse.", "We fix $t_0$ , $\\tau $ , $T$ and $D$ to the values used in the experiment, while $f_0$ , $z_0$ and $\\Delta $ are fitting parameters.", "From the fit, we find $\\Delta =-0.079\\pm 0.002$ .", "This model agrees remarkably well with the data, showing that we have excellent control over position-dependent phases in the experiment.", "For frequency measurements, we use only those molecules that arrive during the period indicated by the shaded region in figure REF , where the frequency changes by less than 20 Hz.", "We note that for the $J=3/2$ measurements, a plot similar to figure REF showed no frequency dependence at the 20 Hz level over the entire range of arrival times.", "Figure: Systematic frequency shift of the (1/2 + ,1)-(1/2 - ,1)(1/2^+,1)-(1/2^-,1) transition as a function of the arrival time of the molecules.", "For each 5 interval of the time-of-flight profile, the resonance frequency is measured by fitting equation () to Ramsey data.", "The red solid line is a fit to equation () where the fitting parameters are z 0 z_0, Δ\\Delta , and an overall frequency offset.", "All other parameters are fixed to the values they have in the experiment.", "The shaded regions indicates the range of arrival times used for the frequency measurements.For molecules near the centre of the pulse, we can find a simple expression for the systematic frequency shift discussed above.", "Let $z_1=z_0+\\delta z_0$ and $z_2=z_1 + (1+\\epsilon )m\\lambda /2$ , where $z_0$ is now the position of any antinode, while $\\delta z_0/\\lambda \\ll 1$ and $\\epsilon \\ll 1$ account for the imperfect timing of the microwave pulses due to the uncertainty in the velocity of the molecule.", "Expanding the trigonometric functions in equation (REF ) to first order in these small quantities, we find that the systematic frequency shift is $\\delta f \\simeq \\epsilon \\,\\Delta \\left(\\frac{v}{\\lambda }\\right).$ This result shows that the frequency shift is independent of $\\delta z_0$ to first order and that the first-order Doppler shift ($v/\\lambda $ ) is suppressed by the product of two small quantities - $\\Delta $ , which is the imbalance factor, and $\\epsilon $ which is the fractional error in setting the interaction length to an integer number of half-wavelengths.", "We can set upper limits to $\\Delta $ in three ways: the finesse of the transmission line resonator (figure REF ), the power needed to drive $\\pi $ -pulses for each of the resolved Doppler-shifted components (figure REF ), and fitting to data such as that in figure REF .", "The first method gives $\\Delta < 0.09$ but does not give a tight constraint because the finesse of the resonator is only partly determined by the reflection at the open end, the other part being the transmission at the input end.", "The other two methods give consistent results as follows: $\\Delta < 0.08$ for the $J=1/2$ components at 3335 and 3349, $\\Delta < 0.005$ for the $J=1/2$ component at 3264After measuring the first two frequencies the plates were cut to a new length to measure the 3264 line.", "This must be the reason for the change in $\\Delta $ ., and $\\Delta < 0.08$ for all the $J=3/2$ components.", "An upper limit to $\\epsilon $ comes from the maximum fractional uncertainty in determining the central speed of the pulse, which we estimate to be 0.03.", "In addition, molecules with different speeds have different values of $\\epsilon $ .", "We select molecules from the time-of-flight profile with arrival times in the range $t_0\\pm \\delta t$ , where $t_0$ is the most probable arrival time and $\\delta t\\simeq 0.02 t_0$ , and so the spread in $\\epsilon $ values is $\\pm 0.02$ .", "Thus, the maximum possible value of $\\epsilon $ in the experiment is 0.05.", "For the $J=1/2$ measurement the upper limit to this systematic shift is 0.04().", "Note that for $J=3/2$ , the shift is about 5 times smaller for the same $\\epsilon $ and $\\delta $ , because of the longer wavelength.", "There is also a second contribution to the velocity-dependent frequency shift which stems from the motion of the molecules during the two short $\\pi /2$ pulses.", "Consider first the interaction with a travelling wave tuned to the resonant angular frequency $\\omega _0$ .", "For a stationary molecule, the phase difference between the microwave oscillator and the oscillating dipole is $\\pi /2$ , but for a moving molecule there is an additional contribution to this phase difference due to the Doppler shift $\\delta _D=2\\pi v/\\lambda $ .", "To second order in $\\delta _D$ , this phase difference is $[1-\\tan (\\Omega \\tau /2)/(\\Omega \\tau )]\\delta _D\\tau $ .", "Due to this phase shift, the population transfer is maximized by slightly detuning the microwave oscillator.", "To find the resulting systematic frequency shift, we include the Doppler shift in the expression for the Ramsey lineshape, expand this to second order in both the detuning and the phase shift, $\\beta $ , between the two pulses, and then find the value of $\\beta $ that maximizes the population transfer.", "When $T\\gg \\tau $ the frequency shift is $\\delta f=\\left(1+\\frac{2(\\tan (\\Omega \\tau )-\\csc (\\Omega \\tau ))}{\\Omega \\tau }\\right)\\frac{v_0}{\\lambda }\\frac{\\tau }{T}.$ For our case, where $\\Omega \\tau =\\pi /2$ , the bracketed quantity is $(1-4/\\pi )$ .", "We see that the Doppler shift is suppressed by the small quantity $\\tau /T$ , which we may have expected since the microwave field is only applied for this fraction of the time.", "When there are two counter-propagating waves, we might expect the above expression to be further suppressed by the imbalance factor $\\Delta $ , and our numerical modelling shows that this is indeed the case.", "With $\\Delta <0.08$ and typical values of $\\tau $ and $T$ , we find the shift to be less than 0.01().", "There are other possible velocity-dependent frequency shifts in addition to those discussed above.", "For example, a position-dependence of the polarization can produce a frequency shift proportional to the velocity.", "To control these shifts, we measure the transition frequency for at least three different velocities.", "Figure REF (a) shows the measured frequency of the $(1/2^+,1)-(1/2^-,0)$ transition as a function of $v_0$ for three different values of $m_1$ (the antinode used for the first microwave pulse).", "For each data point in the figure, we average together at least three measurements with different values of $m$ , since we find no dependence on $m$ .", "We see that the measured frequency depends linearly on the velocity of the molecules and that the gradient $df/dv_0$ differs for different values of $m_1$ .", "The largest gradient observed is $0.05\\pm 0.01$ ().", "After extrapolating to zero velocity, the measurements using various $m_1$ are all in agreement, as shown in figure REF (b).", "We average together these zero-velocity results to obtain the final transition frequency, and we do this for all seven frequencies measured.", "For the $J=3/2$ measurements, the largest velocity-dependence we observed was $0.03\\pm 0.01$ (), and we observed no dependence on $m_1$ .", "Figure: (a) Resonance frequency of the (1/2 + ,1)-(1/2 - ,0)(1/2^+,1)-(1/2^-,0) transition as a function of the most probable velocity v 0 v_0, for m 1 =3,4,5m_1=3,4,5 (blue, red, green).", "(b) Extrapolated zero-velocity frequencies for the three values of m 1 m_1.", "They agree within the uncertainty of the linear fits and we take the weighted mean (solid line).", "The 1-σ\\sigma standard error of the weighted mean is shown by the dashed lines." ], [ "Zeeman shifts", "Next, we consider systematic frequency shifts due to magnetic fields in the interaction region.", "For small magnetic fields, the Zeeman splitting is linear and symmetric about the line centre.", "The amplitudes of the components are also symmetric if the microwave field is linearly polarized.", "Frequency shifts arise due to circular polarization components and/or higher-order Zeeman shifts.", "To minimize these shifts the interaction region is magnetically shielded (see figure REF ).", "In the $J=1/2$ state the magnetic moments arising from the orbital and spin angular momenta are very nearly equal and opposite and the $g$ -factor is of order $10^{-3}$ .", "For the $J=1/2$ measurements, we used just a single layer mu-metal shield to reduce the background magnetic field to acceptable levels.", "We applied fields as large as 50 in each direction and observed no shift of the frequencies measured using the Ramsey method, at the 1 level.", "Since the residual magnetic field is far smaller than this, Zeeman shifts are negligible in the $J=1/2$ measurements.", "The g-factor is much larger in the $J=3/2$ state - $g=1.081$ for $F=1$ and $g=0.648$ for $F=2$ - and so for the $J=3/2$ measurements we improved the shielding by using a two-layer shield.", "Inside the shields we create homogeneous magnetic fields along $x$ and $y$ using Helmholtz coils and along $z$ using a solenoid.", "These coils are calibrated with a fluxgate magnetometer.", "To measure the Zeeman splitting of the hyperfine components, we apply large enough magnetic fields to resolve the splitting of the spectral line obtained using a single microwave pulse of 780 duration.", "Figure REF (a) shows the Zeeman shift of the $(3/2^+,2)-(3/2^-,2)$ hyperfine component as a function of the magnetic field applied in the $y$ -direction, parallel to the polarization of the microwaves.", "In this case, only the $\\Delta M_F=0$ components are driven, and the shift is quadratic and negative.", "This shift is due to mixing of the $M_F=0,\\pm 1$ levels of $F=2$ with those same components in $F=1$ .", "In the f state, $F=1$ lies lower (see figure REF ) and the mixing raises the energy of $F=2$ .", "The opposite is true for the e state, where the shift is also far smaller because of the larger hyperfine splitting.", "Therefore, the shift of this transition is negative and is determined mainly by the shift of the lower $F=2$ level.", "We measure a quadratic Zeeman shift of $-12.6\\pm 1.1$ Hz/($\\mu $ T)$^2$ , which is consistent with our calculation.", "Figure REF (b) shows the Zeeman shift of the same hyperfine component as a function of the magnetic field applied in the $x$ -direction.", "In this case, we drive $\\Delta M_F=\\pm 1$ transitions and observe a linear Zeeman shift of $9.34\\pm 0.28$ Hz/nT, again consistent with our calculation.", "Here, the error is dominated by the uncertainty in the calibration of the magnetic field coils.", "We determine the residual magnetic field averaged over the interaction region by measuring the change in Zeeman shift upon reversal of the applied magnetic fields.", "We also look for any broadening of the single-pulse lineshape due to residual Zeeman splittings.", "Together, these set upper limits to the background magnetic field of 3, 56, and 25 along $x$ , $y$ and $z$ respectively.", "Figure: Zeeman shifts of the (3/2 + ,2)-(3/2 - ,2)(3/2^+,2)-(3/2^-,2) transition, measured using single microwave pulses of 780 duration.", "(a) Quadratic Zeeman shift of the ΔM F =0\\Delta M_F=0 transitions versus field applied along yy.", "Line: fit to a quadratic model.", "(b) Linear Zeeman shift of the ΔM F =±1\\Delta M_F=\\pm 1 transitions versus field along xx.", "We plot the component that shifts to lower frequency, which is the ΔM F =+1(-1)\\Delta M_F=+1 (-1) component for negative (positive) values of the field.", "Line: fit to a linear model.In the frequency measurements made using the Ramsey method, we look for a shift in the resonance frequency as a function of applied magnetic fields.", "For the $(3/2^+,2)-(3/2^-,2)$ transition we measure a maximum gradient of $-0.017\\pm 0.013$ Hz/nT for fields applied along $x$ , $-0.036\\pm 0.005$ Hz/nT along $y$ and $-0.005\\pm 0.003$ Hz/nT along $z$ .", "Using these, and the upper limits for background magnetic fields, we get upper limits for systematic frequency shifts of 0.1, 2 and 0.2 for residual fields along $x$ , $y$ and $z$ respectively.", "Adding these in quadrature gives a total systematic uncertainty due to uncontrolled magnetic fields of 2.", "We assume the same systematic uncertainty for the $(3/2^+,1)-(3/2^-,1)$ transition due to the similar Zeeman structure.", "For the $(3/2^+,1)-(3/2^-,2)$ transition we could not rule out gradients as large as 0.1 Hz/nT along x, 0.2 Hz/nT along y and 0.05 Hz/nT along z. Multiplying these by the upper limits to the residual field gives a systematic uncertainty of 11.", "We assume the same uncertainty for the $(3/2^+,2)-(3/2^-,1)$ transition due to its similar Zeeman structure." ], [ "Stark shifts", "Figure REF (a) shows the calculated Stark shift of the two $J=1/2$ $\\Lambda $ -doublet levels in low electric fields.", "The two levels shift oppositely and quadratically.", "Using a dipole moment of 1.46 D [23] we calculate a shift of $\\mp 17.8$ Hz/(V/cm)$^2$ .", "For $J=1/2$ the shift has virtually no dependence on $F$ or $M_F$ .", "To measure the Stark shift of the microwave transition we use single, long microwave pulses, apply a DC voltage to one of the plates of the transmission line via a biased tee, and record the transition frequency as a function of the electric field.", "Figure REF (b) shows our data for the $(1/2^+,1)-(1/2^-,1)$ transition.", "The line is a fit to a parabola, $\\delta f = a (E-E_b)^{2}$ , where $E$ is the applied field and $E_b$ the background electric field.", "This fit gives a Stark shift of $33 \\pm 3$ Hz/(V/cm)$^2$ , where the error is dominated by the systematic uncertainty in the plate spacing.", "The background field is consistent with zero, and from the uncertainty in this field we obtain an upper limit to uncontrolled Stark shifts of 0.1.", "For $J=3/2$ , the Stark shift depends on $F$ and $M_F$ .", "Using the same procedure as before we obtain a systematic uncertainty of 0.2 for the $J=3/2$ measurements.", "Figure: (a) Calculated Stark shift of the Λ\\Lambda -doublet states for low electric fields.", "There is no dependence on FF or M F M_F.", "(b) Measured frequency shift of the (1/2 + ,1)-(1/2 - ,1)(1/2^+,1)-(1/2^-,1) transition." ], [ "Other systematic uncertainties", "We test for systematic frequency shifts that depend on the microwave power by using shorter $\\pi /2$ pulses in a Ramsey experiment.", "We reduce the pulse length from 15 to 4, increasing the power by a factor of 14, and do not find any frequency shift.", "We also do not find any dependence on the probe laser detuning.", "Scanning the microwave frequency can change the power in the resonator and this can lead to a systematic frequency shift.", "Consider a worst-case model where the molecular signal depends linearly on the microwave power and the resonator is badly tuned so that the molecular frequency is half way down the side of a resonance, where the gradient is steepest.", "Suppose the molecular signal is a Gaussian with a standard deviation $w$ , and the transmission line resonances are Lorentzian with FWHM $W$ .", "We find that there is a systematic shift of $\|\\Delta f\|=2w^2/W$ .", "In the experiment, typical values are $w\\simeq 1.5$ kHz and $W \\simeq 30$ MHz, leading to a shift of only 0.15.", "In reality, the shift is much smaller for two reasons.", "First, we tune the transmission line to be on resonance at the molecular frequency.", "Second, we choose the power that maximizes the molecular signal, i.e.", "a $\\pi $ -pulse in a single-pulse measurement or $\\pi /2$ -pulses in a Ramsey measurement, and so the signal has no first derivative with respect to power.", "Unwanted frequency sidebands may be produced, for example by the microwave oscillator or by the switching electronics, and these can lead to systematic shifts if there is an asymmetry in the amplitudes of the sidebands.", "We have measured the frequency spectrum and find no sidebands down to -40 dB within 1 of the oscillator frequency.", "For our parameters, we estimate that the systematic frequency shift will be largest for an asymmetric sideband with an offset of 40 [24].", "Then, if the amplitude of this single sideband is -40 dB, the resulting frequency shift is only 10.", "Systematic frequency shifts due to blackbody radiation, the motional Stark effect, the second-order Doppler shift, and collisional shifts, are also all negligible at the current accuracy level." ], [ "Conclusions", "Table REF gives our final transition frequencies, reproduced from [4].", "We add the statistical and systematic uncertainties in quadrature to give the total uncertainty.", "Table: The measured Λ\\Lambda -doublet transition frequencies with their 1σ1\\sigma uncertainties.Our method of microwave spectroscopy is exceptionally versatile and accurate.", "The method can be used for any molecule that can be produced in a pulsed supersonic beam, and the same apparatus can be used over a very wide frequency range, including low frequencies where a conventional microwave cavity would be too large.", "Ramsey spectroscopy is more commonly done using two separate cavities with their axes perpendicular to the molecular beam direction.", "The phase difference between the two cavities then needs to be accurately controlled, which can be difficult to achieve.", "In the transmission line resonator, control over the relative phase of the two pulses is straightforward because the field is supported by a single structure.", "With high-Q cavities it is necessary to scan the cavity in synchronism with the microwave frequency.", "By contrast, the transmission line resonances are broad enough that it is not necessary to tune the line as the frequency is scanned.", "We emphasize the importance of choosing the plate width and spacing so that higher-order modes, include the `leaky modes', are strongly attenuated.", "With only the TEM mode able to propagate, the field is very well controlled.", "In our setup, molecular resonance frequencies can be measured either using a single, long microwave pulse, or using the Ramsey method of two short pulses separated in time.", "These pulses can be applied when the molecules are at any position along the transmission line and so the molecules can be used to map out the amplitude and phase of the microwave field, providing an exceptional degree of control.", "For this mapping method to work well it is important to produce short, cold pulses of molecules, and to use a detector with adequate time resolution.", "Our CH source produces pulses that are just a few millimetres in length and with a translational temperature as low as 400 mK.", "Because the field is a standing wave, Doppler shifts are very strongly suppressed in the experiment.", "As described in section , the residual Doppler shift in the Ramsey measurements is proportional to two small factors, one being the imbalance in amplitude between the two counter-propagating waves, the other being the fractional error in setting the interaction length to an integer number of half-wavelengths.", "In this experiment, we observed velocity-dependent shifts at the level of 0.05() or less.", "We measured and eliminated these shifts by varying the beam velocity.", "The experiment reached a precision of 3, limited mainly by the statistical uncertainty in extrapolating to zero velocity.", "The velocity-dependent shifts could be reduced by improving the reflection at the end of the transmission line to reduce the amplitude imbalance between the counter-propagating waves, and by improving the way the microwaves are launched into the transmission line to eliminate field non-uniformities in this region.", "A Stark decelerator [25] could also be used to reduce the velocity of the beam by a factor of 10, giving both longer interaction times and improved velocity control [26].", "Individual frequency measurements reached a statistical uncertainty of 1 within about 1 hour.", "This was partly limited by a background of scattered laser light that reaches the detector, and partly by shot-to-shot fluctuations of the source.", "The background could be reduced by improving the shape of the probe laser mode, and the source noise could be reduced by using a second laser-induced-fluorescence detector upstream of the experiment to record the number of molecules produced in each shot.", "The photon shot noise limit could then be reached, giving an uncertainty of 1 in just a few minutes of integration." ], [ "Acknowledgements", "We thank Ben Sauer, Jony Hudson and Heather Lewandowski for their help and advice.", "We are indebted to Jon Dyne, Steve Maine and Valerijus Gerulis for their expert technical assistance.", "This work was supported by the EPSRC and the Royal Society." ] ]	1403.0195
[ [ "Large Scale Structure Formation in Eddington-inspired Born-Infeld\n Gravity" ], [ "Abstract We study the large scale structure formation in Eddington-inspired Born-Infeld (EiBI) gravity.", "It is found that the linear growth of scalar perturbations in EiBI gravity deviates from that in general relativity for modes with large wave numbers ($k$), but the deviation is largely suppressed with the expansion of the Universe.", "We investigate the integrated Sachs-Wolfe effect in EiBI gravity, and find that its effect on the angular power spectrum of the anisotropy of the cosmic microwave background (CMB) is almost the same as that in the Lambda-cold dark matter ($\\Lambda$CDM) model.", "We further calculate the linear matter power spectrum in EiBI gravity and compare it with that in the $\\Lambda$CDM model.", "Deviation is found on small scales ($k\\gtrsim 0.1 h$ Mpc$^{-1}$), which can be tested in the future by observations from galaxy surveys." ], [ "Introduction", "Purely affine theory of gravity has drawn a lot of attention since it was first proposed by Eddington [1].", "Schrödinger generalized Eddington's theory to a nonsymmetric metric [2].", "One of the advantages of Eddington affine theory is that it can automatically generate a cosmological term.", "But in these early papers, matter fields are not included.", "Attempts to add matter fields in this theory have been an interesting topic [3], [4].", "Recently, a new alternative theory called Eddington-inspired Born-Infeld (EiBI) gravity was proposed by Banados and Ferreira [5].", "EiBI gravity is equivalent to general relativity in vacuum; but when matter fields are included, it presents many interesting properties.", "It is claimed to be singularity free both at the beginning of the Universe [5], [6] and during the gravitational collapse of dust [7].", "In Ref.", "[8], EiBI gravity as an alternative to inflation was discussed.", "Despite the good properties EiBI gravity exhibits, the validity of this theory has also been an important topic.", "It was found that the tensor perturbation and the nonzero wave number modes of scalar perturbations in EiBI gravity are unstable deep in the Eddington regime [8], [9], [10], [11], while it was shown that the vector perturbations and the zero wave number modes of scalar perturbations are stable for positive $\\kappa $ (an extra parameter in EiBI gravity) in Ref.", "[10].", "In Ref.", "[12], the authors argued that there exist curvature singularities at the surface of polytropic stars and unacceptable Newtonian limit in EiBI gravity.", "On the other hand, researchers try to find out how we can remove these pathologies.", "In Ref.", "[13], Liu et al.", "investigated a thick brane model in EiBI gravity.", "They found that the instability of tensor perturbation does not exit in their model.", "In Ref.", "[8], Avelino and Ferreira found another solution to the instability problem of tensor perturbation by considering matter sources with a time-dependent state parameter.", "Recently, Kim argued that the problem of singularity at the surface of a star can be cured by taking into account the gravitational backreaction [14].", "These extensions make EiBI gravity a more consistent theory and a prospective alternative to general relativity.", "Other papers have also been done to constrain the parameter $\\kappa $ from compact stars [7], [15], [16], tests in solar system [17], astrophysical and cosmological observations [18], and nuclear physics [19], [20].", "The strongest constraint on the parameter $\\kappa $ implies $\|\\kappa \|<10^{-3}$ kg$^{-1}$ m$^5$ s$^{-2}$[19].", "More relevant studies can be found in Refs.", "[21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39].", "It was shown in Refs.", "[5], [6], [24] that, in the low density and curvature limit, EiBI gravity recoveries the conventional Friedman cosmology.", "But these studies only considered a homogeneous and isotropic Universe.", "It is worthwhile to examine the cosmological consequences of perturbations in EiBI gravity.", "In Ref.", "[11], the authors found that a nearly scale-invariant power spectrum for both scalar and tensor primordial quantum perturbations can be obtained in EiBI gravity without introducing the inflation.", "However, it remains to be seen whether these primordial quantum perturbations can lead to proper cosmic microwave background (CMB) and large scale structure of the Universe consistent with observations.", "In this paper, we investigate the evolution of cosmological perturbations after the last scattering and the large scale structure formation in EiBI gravity.", "First, we use the linear perturbed equations derived in [10] to obtain the approximate equations governing the scalar perturbations.", "Then we discuss these equations in subhorizon and superhorizon regimes and compare them with those in the $\\Lambda $ CDM model.", "Finally, we solve the perturbed equations by numerical methods for all wave numbers in the range we are concerned with and calculate the integrated Sachs-Wolfe effect and linear matter power spectrum.", "We find that the linear matter power spectrum in EiBI gravity deviates from that in the $\\Lambda $ CDM model when $k\\gtrsim 0.1 h$ Mpc$^{-1}$ , which can be further tested in the future by observations from galaxy surveys.", "Arrangement for this paper is as follows.", "In section , we briefly review the framework of EiBI gravity and its application to cosmology.", "In section , we discuss the linear scalar perturbed equations in EiBI gravity.", "In section , we solve the perturbed equations and compare the results with those in the $\\Lambda $ CDM model.", "In section , we discuss the integrated Sachs-Wolfe effect and the linear matter power spectrum in EiBI gravity.", "Finally, conclusions and discussions are presented in Section ." ], [ "Field Equations and Cosmological Background", "The action for EiBI gravity is given by [5] $S=\\frac{2}{\\kappa }\\int {d^4 x\\left[\\sqrt{-\|g_{\\mu \\nu }+\\kappa R_{\\mu \\nu }(\\Gamma )\|}-\\lambda \\sqrt{-\|g_{\\mu \\nu }\|} \\right]}+S_{M},$ where $R_{\\mu \\nu }(\\Gamma )$ represents the symmetric part of the Ricci tensor built with the connection $\\Gamma $ and $\\lambda $ is a dimensionless constant which is different from 0 (here we work in Planck units $c=8 \\pi G=1$ ).", "In the following part we will take $\\lambda =1+\\kappa \\Lambda $ , as it is well known that $\\Lambda $ here acts as an effective cosmological constant when $\|\\kappa R\|$ is small [5].", "Varying the action (REF ) independently with respect to the metric and the connection, respectively, yields $\\sqrt{q}q^{\\mu \\nu }&=&\\lambda \\sqrt{g}g^{\\mu \\nu }-\\kappa \\sqrt{g}T^{\\mu \\nu },\\\\q_{\\mu \\nu }&=&g_{\\mu \\nu }+\\kappa R_{\\mu \\nu },$ where $q_{\\mu \\nu }$ is the auxiliary metric compatible with the connection.", "Now we consider the case of a homogeneous and isotropic Universe which can be described by the Friedmann-Robertson-Walker (FRW) metric $ds^2=-dt^2+a^2(t)\\delta _{i j}dx^i dx^j.$ The corresponding auxiliary metric is taken to be $q_{\\mu \\nu }dx^{\\mu }dx^{\\nu }=-X(t)^2dt^2+a^2(t)Y(t)^2\\delta _{ij}dx^i dx^j.$ For simplicity, we have assumed the spacetime to be spatial flat.", "Furthermore, we assume that the matter field is dominated by pressureless cold dark matter and the effect of radiation can be neglected.", "So the energy-momentum tensor of matter field can be written as $T_{\\mu \\nu }=\\rho u_{\\mu } u_{\\nu }$ .", "Then solving Eqs.", "(REF ) and () yields [5], [9], [6] $H^2=\\frac{G}{6 F^2},$ with $G&=&\\frac{1}{\\kappa }\\left(1+2 X^2-3\\frac{X^2}{Y^2}\\right),\\\\F&=&1-\\frac{3\\kappa (1+\\kappa \\Lambda )\\rho }{4[1+\\kappa (\\rho +\\Lambda )](1+\\kappa \\Lambda )},\\\\X^2&=&\\sqrt{\\frac{(1+\\kappa \\Lambda )^3}{1+\\kappa (\\rho +\\Lambda )}},\\\\Y^2&=&\\sqrt{[1+\\kappa (\\rho +\\Lambda )](1+\\kappa \\Lambda )}.$ Here $\\rho $ is the energy density of dark matter.", "If $\|\\kappa \|$ is sufficiently small so that $\\lbrace \|\\kappa \\rho \|,\|\\kappa \\Lambda \|\\rbrace \\ll 1$ , Eq.", "(REF ) can be expended in terms of $\\kappa \\rho $ and $\\kappa \\Lambda $ : $H^2=\\frac{1}{3}(\\rho +\\Lambda )+\\frac{1}{8}\\kappa \\rho ^2+\\mathcal {O}((\\kappa \\rho )^2).$ Note that, since $\\rho $ is larger than $\\Lambda $ or at least has the same order as $\\Lambda $ up to now, we have used $\\mathcal {O}((\\kappa \\rho )^2)$ to stand for the higher-order terms such as $(\\kappa \\rho )^2$ , $\\kappa ^2\\rho \\Lambda $ , $(\\kappa \\Lambda )^2$ , etc.", "When $\\kappa \\rightarrow 0$ , Eq.", "(REF ) reduces to the standard Friedmann equation with a cosmological constant.", "Taking the derivative with respect to $t$ of Eq.", "(REF ) and considering the continuity equation for cold dark matter $\\dot{\\rho }+3 H \\rho =0$ , we can obtain a useful equation $\\dot{H}=-\\frac{1}{2}\\rho -\\frac{3}{8}\\kappa \\rho ^2+\\mathcal {O}((\\kappa \\rho )^2).$ It will be used to simplify the perturbed equations in the following sections.", "To parametrize Eqs.", "(REF ) and (REF ) for later calculations, we take $\\rho &=&3 H_0^2 \\Omega _m a^{-3},\\\\\\Lambda &=&3 H_0^2 \\Omega _{\\Lambda },\\\\\\gamma &=&\\kappa H_0^2.$ Here $\\Omega _m$ , $\\Omega _{\\Lambda }$ , and $H_0$ are the matter density parameter, the vacuum energy density parameter, and the Hubble constant at present, respectively ($a$ is normalized so that $a=1$ at present), and $\\gamma $ characterizes the deviation from general relativity.", "These parameters satisfy $\\Omega _m+\\Omega _{\\Lambda }+\\frac{9}{8}\\gamma \\Omega _m^2 = 1.$ So only two of them are independent.", "Furthermore, it is also useful to define a dimensionless Hubble parameter $h$ , so that $H_0=100h$ km s$^{-1}$ Mpc$^{-1}$ ." ], [ "Linear Scalar Perturbations", "Now we consider a perturbed FRW metric in the Newtonian gauge (here we are only concerned with the scalar perturbations): $ds^2=-(1+2\\Phi )dt^2+a^2(t)(1-2\\Psi )\\delta _{i j}dx^i dx^j.$ Noting that the auxiliary metric is related to the physical metric and matter fields according to Eq.", "(REF ), the corresponding perturbed auxiliary metric is taken to be $q_{\\mu \\nu }dx^{\\mu }dx^{\\nu }=-X(t)^2(1+2\\alpha )dt^2+a^2(t)Y(t)^2(1-2\\beta )\\delta _{ij}dx^i dx^j.$ The relations between the perturbations of the auxiliary metric and the physical metric are given in Refs.", "[6], [13], [10] as $\\alpha &=&\\Phi -\\frac{1}{4}\\,\\frac{\\kappa \\delta \\rho }{1+\\kappa (\\rho +\\Lambda )},\\\\\\beta &=&\\Psi -\\frac{1}{4}\\,\\frac{\\kappa \\delta \\rho }{1+\\kappa (\\rho +\\Lambda )}.$ Then the equations for the growth of perturbations in the linear regime have been obtained in Ref.", "[10].", "The perturbed conservation equations in Fourier space lead to $\\dot{\\delta }&=&3\\dot{\\Psi }+\\frac{k^2}{a^2}\\delta u,\\\\\\dot{\\delta u}\\!&=&-\\Phi .$ Here $\\delta =\\frac{\\delta \\rho }{\\rho }$ is the relative energy density perturbation, and $\\delta u$ is related to the longitudinal part of the spatial velocity perturbation: $\\delta u_i^L=\\partial _i\\delta u$ .", "The perturbed Eq.", "() gives $\\!&\\!\\!&\\!\\ \\frac{X^2}{Y^2}a^{-2}\\nabla ^2\\Phi +6\\Big (\\frac{\\ddot{a}}{a}+\\frac{\\ddot{Y}}{Y}-H\\frac{\\dot{X}}{X}+2H\\frac{\\dot{Y}}{Y}-\\frac{\\dot{X}}{X}\\frac{\\dot{Y}}{Y}\\Big )\\Phi +3\\Big (H+\\frac{\\dot{Y}}{Y}\\Big )\\dot{\\Phi }+3\\ddot{\\Psi }+6\\Big (H-\\frac{1}{2}\\frac{\\dot{X}}{X}+\\frac{\\dot{Y}}{Y}\\Big )\\dot{\\Psi }\\nonumber \\\\\\!&\\!\\!&\\!\\ -\\frac{1}{4}\\kappa a^{-2}\\frac{X^2}{Y^2}\\frac{\\nabla ^2\\delta \\rho }{1+\\kappa (\\rho +\\Lambda )}-\\frac{3}{4}\\kappa \\partial _{t}^2\\Big [\\frac{\\delta \\rho }{1+(\\rho +\\Lambda )}\\Big ]-\\frac{3}{4}\\kappa \\Big (3H+3\\frac{\\dot{Y}}{Y}-\\frac{\\dot{X}}{X}\\Big )\\partial _t\\Big [\\frac{\\delta \\rho }{1+\\kappa (\\rho +\\Lambda )}\\Big ]\\nonumber \\\\\\!&\\!\\!&\\!\\ -\\frac{1}{2}\\Big [1+3\\kappa \\Big (\\frac{\\ddot{a}}{a}+\\frac{\\ddot{Y}}{Y}-H\\frac{\\dot{X}}{X}+2H\\frac{\\dot{Y}}{Y}-\\frac{\\dot{X}}{X}\\frac{\\dot{Y}}{Y}\\Big )\\Big ]\\frac{\\delta \\rho }{1+\\kappa (\\rho +\\Lambda )}-\\kappa a^{-2}\\partial _t \\Big [\\frac{\\rho }{1+\\kappa (\\rho +\\Lambda )}\\nabla ^2\\delta u\\Big ]\\nonumber \\\\\\!&\\!\\!&\\!\\ -\\kappa a^{-2}\\Big (2\\frac{\\dot{Y}}{Y}-\\frac{\\dot{X}}{X}\\Big )\\frac{\\rho }{1+\\kappa (\\rho +\\Lambda )}\\nabla ^2\\delta u=0,$ $\\!&\\!\\!&\\!\\Big (H+\\frac{{\\dot{Y}}}{Y}\\Big )\\Phi +\\dot{\\Psi }-\\frac{1}{4}\\kappa \\partial _t \\Big [\\frac{\\delta \\rho }{1+\\kappa (\\Lambda +\\rho )}\\Big ]-\\frac{1}{4}\\kappa \\Big (H+\\frac{\\dot{Y}}{Y}\\Big )\\frac{\\delta \\rho }{1+\\kappa (\\Lambda +\\rho )}+\\frac{1}{2}\\frac{\\rho }{1+\\kappa (\\Lambda +\\rho )}\\delta u = 0,$ $\\!&\\!\\!&\\!\\ \\Phi -\\Psi =\\kappa \\frac{Y^2}{X^2}\\Big [\\partial _t\\Big (\\frac{\\rho }{1+\\kappa (\\rho +\\Lambda )}\\delta u\\Big )+\\Big (H-\\frac{\\dot{X}}{X}+3\\frac{\\dot{Y}}{Y}\\Big )\\frac{\\rho }{1+\\kappa (\\rho +\\Lambda )}\\delta u \\Big ].$ Substituting Eqs.", "(), () into Eqs.", "(REF ), (REF ), (REF ) and expanding them with respect to $\\kappa \\rho $ , we obtain $\\!&\\!\\!&\\!\\ 3H(1-\\frac{3}{4}\\kappa \\rho )\\dot{\\Phi }+6\\Big [\\dot{H}(1-\\frac{3}{4}\\kappa \\rho )+H^2\\Big ]\\Phi -\\frac{k^2}{a^2}\\Phi +3\\ddot{\\Psi }+6H\\Big (1+\\frac{3}{8}\\kappa \\rho \\Big )\\dot{\\Psi }\\nonumber \\\\\\!&\\!\\!&\\!\\ -\\frac{3}{4}\\kappa \\rho \\ddot{\\delta }-\\frac{3}{4}\\kappa \\rho H\\dot{\\delta }+\\frac{3}{4}\\kappa \\rho (\\dot{H}-2H^2)\\delta -\\frac{1}{2}\\rho [1-\\kappa (\\rho +\\Lambda )]\\delta +\\frac{1}{4}\\kappa \\rho \\frac{k^2}{a^2}\\delta =\\mathcal {O}((\\kappa \\rho )^2),$ $H\\Big (1-\\frac{3}{4}\\kappa \\rho \\Big )\\Phi +\\dot{\\Psi }-\\frac{1}{4}\\kappa \\rho \\dot{\\delta }+\\frac{1}{2}\\kappa \\rho H\\delta +\\frac{1}{2}\\rho [1-\\kappa (\\rho +\\Lambda )]\\delta u=\\mathcal {O}((\\kappa \\rho )^2),$ $\\!&\\!\\!&\\!\\ \\Phi -\\Psi =\\kappa \\rho (\\dot{\\delta u}-2 H\\delta u)+\\mathcal {O}((\\kappa \\rho )^2).$ Here we have used the continuity equation $\\dot{\\rho }+3 H \\rho =0$ to eliminate the term $\\dot{\\rho }$ and written the equations in Fourier space.", "Equations.", "(REF ), (), (REF ), (REF ), and (REF ) govern the cosmological scalar perturbations $\\Phi $ , $\\Psi $ , $\\delta $ and $\\delta u$ .", "But it should be noted that only four of theses equations are actually independent.", "Given appropriate initial conditions and the expansion history governed by Eqs.", "(REF ) and (REF ), we can solve these differential equations and calculate related observational quantities to compare with observations.", "However, these equations are too complicated for an analytic treatment, so we first look at two wavelength regimes: wavelengths much smaller than the Hubble horizon (subhorizon), and wavelengths much larger than the Hubble horizon (superhorizon).", "It will give us a first impression how EiBI gravity deviates from general relativity.", "Then we will show the numerical results in the next section." ], [ "Subhorizon regime", "First, we look at the perturbations which are deep inside the Hubble horizon, i.e., $\\frac{k}{a}\\gg H$ .", "To get the equation governing the perturbation $\\delta $ deep inside the Hubble horizon, we also apply the following approximations [40], [41], [42]: $\\Big \\lbrace \\frac{k^2}{a^2}\|\\Phi \|,\\frac{k^2}{a^2}\|\\Psi \|\\Big \\rbrace \\gg \\lbrace H^2\|\\Phi \|,H^2\|\\Psi \|,H\|\\dot{A}\|,\|\\ddot{A}\|\\rbrace ,$ where $A=\\Phi , \\Psi $ .", "Then from Eqs.", "(REF ), (), (REF ), and (REF ), we finally arrive at $\\Big (1-\\frac{3}{4}\\kappa \\rho \\Big )\\ddot{\\delta }+2H\\Big (1-\\frac{3}{8}\\kappa \\rho \\Big )\\dot{\\delta }-\\frac{1}{2}\\Big (\\rho -\\frac{1}{2}\\kappa \\rho \\frac{k^2}{a^2}\\Big )\\delta =0.$ Note that if $\|\\kappa \\rho \|$ is sufficiently small so that the terms associating with it can be neglected except the term $-\\frac{1}{2}\\kappa \\rho \\frac{k^2}{a^2}$ due to large value of $k$ , Eq.", "(REF ) is exactly the same as that derived by a different method in the nonrelativistic regime in Ref.", "[18].", "Unlike in general relativity, the pressureless cold dark matter has a nonzero effective sound speed $c_{seff}=\\frac{1}{2}\\sqrt{\\kappa \\rho }$ in EiBI gravity.", "For positive $\\kappa $ , the perturbation of dark matter density exhibits an oscillating behavior when $\\frac{1}{2}\\kappa \\frac{k^2}{a^2}>1$ , which was first pointed out by Avelino [18].", "However, in the case we consider in this paper, we will choose $\|\\kappa \|$ sufficiently small so that $\|\\frac{1}{2}\\kappa \\frac{k^2}{a^2}\|<1$ for all wave numbers in the range we are concerned with.", "An important observational quantity is the growth rate of clustering defined as $f\\equiv \\frac{d\\ln {\\Delta }}{d\\ln {a}},$ where $\\Delta $ is the the relative density perturbation in the comoving gauge, $\\Delta =\\delta -3H\\delta u.$ Here we do not use the relative density perturbation in the Newtonian gauge because it depends on the specific gauge we choose, while the observational quantities should be gauge invariant.", "It is easy to check that the combination $\\delta -3H\\delta u$ is invariant under gauge transformations." ], [ "Superhorizon Regime", "Now we go on with the superhorizon regime, i.e., $\\frac{k}{a} \\ll H$ .", "It is well known that the quantity $\\mathcal {R}\\equiv -\\Psi +H\\delta u$ defined in the Newtonian gauge is conserved outside the Hubble horizon in general relativity [43], [44].", "In Ref.", "[45], Bertschinger had proven that the constancy of $\\mathcal {R}$ also holds for modified gravity theories that obey the energy-momentum conservation $\\nabla _\\mu T^{\\mu \\nu }=0$ (see also in Ref.", "[46]).", "Thus we have $-\\dot{\\Psi }+H\\dot{\\delta u}+\\dot{H}\\delta u=0.$ Along with Eq.", "(), we can get $\\ddot{\\Psi }+H\\dot{\\Phi }-\\frac{\\ddot{H}}{\\dot{H}}\\dot{\\Psi }+H\\Big (2\\frac{\\dot{H}}{H}-\\frac{\\ddot{H}}{\\dot{H}}\\Big )\\Psi =0.$ It is the same as the case in general relativity.", "However, from Eqs.", "(), (REF ), and (REF ), we can obtain $\\Psi -\\Phi =\\kappa \\rho \\,\\Big [2\\frac{H}{\\dot{H}}\\dot{\\Psi }+\\Big (1+2\\frac{\\,\\,H^2}{\\dot{H}}\\Big )\\Phi \\Big ],$ which implies that $\\Phi $ does not equal to $\\Psi $ as in general relativity when the matter fields carry no anisotropic stress.", "But the deviation will be suppressed by the expansion of the Universe because $\\rho \\propto a^{-3}$ .", "An important quantity is the metric combination $\\Psi _{+}\\equiv \\frac{\\Psi +\\Phi }{2},$ which affects the CMB power spectrum through the integrated Sachs-Wolfe effect and weak gravitational lensing.", "We will discuss the integrated Sachs-Wolfe effect in detail later." ], [ "Numerical Evolution", "In this section, we present the numerical solutions for the scalar perturbations.", "We choose Eqs.", "(REF ), (), (REF ), and (REF ) to be a complete set of differential equations, because they are only first order and easier to solve.", "As mentioned before, we need to give appropriate initial conditions.", "It requires a knowledge of the evolution of the perturbations at early time before the photon decoupled with matter (last scattering).", "However, to give an accurate prescription of this era, we need to solve the multispecies Boltzmann equations [47], [49], [48], which is beyond the scope of this paper.", "We will focus on the evolution of linear perturbations from the time when photon decoupled with matter to the present.", "At the time of decoupling, the Universe is already dominated by matter fields.", "Neglecting the effects of radiation, baryonic matter, and cosmological constant $\\Lambda $ , we can solve the perturbed equations analytically.", "From the analytic solutions, we assume a set of initial conditions.", "Using these initial conditions, we solve Eqs.", "(REF ), (),(REF ), and (REF ) numerically." ], [ "Initial Conditions", "In section , we have expanded the perturbed equations with respect to $\\kappa \\rho $ .", "The zeroth-order equations are just the same as those in general relativity.", "So we assume the solutions can be expanded as $\\delta &=&\\delta ^{(0)}+\\gamma \\,\\delta ^{(1)}+\\mathcal {O}(\\gamma ^2),\\\\\\Phi &=&\\Phi ^{(0)}+\\gamma \\,\\Phi ^{(1)}+\\mathcal {O}(\\gamma ^2),\\\\\\Psi &=&\\Psi ^{(0)}+\\gamma \\,\\Psi ^{(1)}+\\mathcal {O}(\\gamma ^2),\\\\\\delta u\\!&=&\\delta u^{(0)}+\\gamma \\,\\delta u^{(1)}+\\mathcal {O}(\\gamma ^2),$ where $\\delta ^{(0)}$ , $\\Phi ^{(0)}$ , $\\Psi ^{(0)}$ , and $\\delta u^{(0)}$ are the same as the solutions in general relativity ($\\kappa \\rightarrow 0$ ): $\\delta ^{(0)}&=&2 c_1\\Big (1+\\frac{k^2}{3\\Omega _m H_0^2}a\\Big )+c_2\\Big (3 a^{-\\frac{5}{2}}-\\frac{2 k^2}{3 \\Omega _m H_0^2}a^{-\\frac{3}{2}}\\Big ),\\\\\\Phi ^{(0)}&=&-c_1+c_2 a^{-\\frac{5}{2}},\\\\\\Psi ^{(0)}&=&-c_1+c_2 a^{-\\frac{5}{2}},\\\\\\delta u^{(0)}\\!&=&\\frac{1}{\\Omega _m^{\\frac{1}{2}}H_0}\\Big (\\frac{2}{3}c_1 a^{\\frac{3}{2}}+c_2 a^{-1}\\Big ).$ We are only interested in the growth modes of $\\delta $ which are important for structure formation, so we take $c_2=0$ .", "Then substituting Eqs.", "(REF ), (), (), and () into Eqs.", "(REF ), (), (REF ), and (REF ), we can obtain the first-order solutions $\\delta ^{(1)}&=&c_1\\Big ( \\frac{3}{4}\\Omega _m a^{-3}-\\frac{k^2}{2 H_0^2}a^{-2}+\\frac{k^4}{2\\Omega _m H_0^4}a^{-1}\\Big )+2 c_3\\Big (1+\\frac{k^2}{3\\Omega _m H_0^2}a\\Big )+c_4\\Big (3 a^{-\\frac{5}{2}}-\\frac{2 k^2}{3\\Omega _m H_0^2}a^{-\\frac{3}{2}}\\Big ), \\\\\\Phi ^{(1)}&=&-c_1\\Big (\\frac{3\\Omega _m}{4}a^{-3}+\\frac{k^2}{4 H_0^2}a^{-2}\\Big )-c_3+c_4 a^{-\\frac{5}{2}},\\\\\\Psi ^{(1)}&=&-c_1 \\Big (-\\frac{\\Omega _m}{4}a^{-3}+\\frac{k^2}{4 H_0^2}a^{-2} \\Big )-c_3+c_4 a^{-\\frac{5}{2}},\\\\\\delta u^{(1)}\\!&=&\\frac{1}{\\Omega _m^{\\frac{1}{2}}H_0}\\Big [-c_1\\Big (\\frac{1}{8}\\Omega _m a^{-\\frac{3}{2}}+\\frac{k^2}{2 H_0^2}a^{-\\frac{1}{2}}\\Big )+\\frac{2}{3} c_3 a^{\\frac{3}{2}}+c_4 a^{-1}\\Big ].$ Is it easy to find that the contributions of the terms associating with $c_3$ and $c_4$ in the above equations are just modifications to the integral constants $c_1$ and $c_2$ in the zeroth-order solutions, while the terms associating with $c_1$ present much more interesting properties.", "So we assume $c_3=c_4=0$ .", "Actually, if $c_1$ , $c_3$ , and $c_4$ all have the same order, the terms associating with $c_1$ will be the dominated parts of the first-order solutions at the initial time.", "Under the above assumptions, we use the analytic forms of the approximate solutions to determine the initial values of $\\delta $ , $\\Phi $ , and $\\delta u$ .", "Note that $\\Psi $ can be expressed by $\\Phi $ and $\\delta u$ algebraically using Eqs.", "() and (REF )." ], [ "Numerical Solutions", "From Eqs.", "(REF ), (), (), and (), we can acquire some important information about the modifications of EiBI gravity to general relativity.", "It can be seen that the deviations grow with $k$ at the beginning, but then will be largely suppressed by the expansion of the Universe.", "It is also confirmed by our numerical calculations.", "As is known that when $k$ is larger, the nonlinear effects will become more significant and finally make the linear analysis unsuitable.", "So we restrict $k$ in the range $(0,0.5 h)$ Mpc$^{-1}$ .", "As mentioned in Section REF , we require $\|\\frac{1}{2}\\kappa \\frac{k^2}{a_i^2}\|<1$ , where $a_i=\\frac{1}{1+z^{}}$ is the scale factor at the time of decoupling.", "The redshift $z^{}$ at that time is $1090.43\\pm 0.54$ according to the results of Planck 2013 [50].", "So the parameter $\|\\gamma \|=\|\\kappa H_0^2\|$ should be the order of $10^{-13}$ or smaller.", "Figure REF shows the growth rate $f$ with respect to the scale factor $a$ for $k=0.5 h$ Mpc$^{-1}$ and $k=0.2 h$ Mpc$^{-1}$ .", "The parameter $\\gamma $ is taken to be $\\pm 10^{-13}$ and $\\pm 10^{-14}$ .", "We have also taken $\\Omega _m=0.315$ suggested by Ref.", "[50].", "It can be seen that as $\|\\gamma \|$ becomes smaller, the growth rate in EiBI gravity approaches to that in $\\Lambda $ CDM model.", "The deviation is larger for bigger $k$ , but due to the expansion of the Universe, this deviation is largely suppressed at present.", "For positive $\\gamma $ , the growth rate is smaller than that in the $\\Lambda $ CDM model, so the growth of structure is suppressed by the effect of the modification to $\\Lambda $ CDM model at an early time, which will affect the later process of the formation of the large scale structure.", "The case with negative $\\gamma $ is just the opposite.", "With the observations from galaxy surveys, we may find some constraints on EiBI gravity.", "Figure: Evolution of the growth rate ff for k=0.5hk=0.5 hMpc -1 ^{-1} (left) and k=0.2hk=0.2 hMpc -1 ^{-1} (right)in EiBI gravity with different values of γ\\gamma .Figure REF shows the evolution of $\\Psi _{+}$ for $k=0.5 h$ Mpc$^{-1}$ and $k=0.2 h$ Mpc$^{-1}$ .", "Similarly, as $\|\\gamma \|$ becomes smaller, the evolution of $\\Psi _{+}$ in EiBI gravity approaches to that in $\\Lambda $ CDM model.", "The deviation is larger for bigger $k$ .", "The overall change of $\\Psi _{+}$ from the time of decoupling to the present is larger than that in the $\\Lambda $ CDM model for the case with positive $\\gamma $ , while it is the opposite for negative $\\gamma $ .", "This deviation from $\\Lambda $ CDM model will affect the angular power spectrum of CMB at low multipoles through the integrated Sachs-Wolfe effect.", "Figure: Evolution of the metric combination Ψ + \\Psi _{+} for k=0.5hk=0.5 hMpc -1 ^{-1} (left) and k=0.2hk=0.2 hMpc -1 ^{-1} (right)in EiBI gravity with different values of γ\\gamma .", "Here Ψ +i \\Psi _{+i} is the initial value of Ψ + \\Psi _{+}." ], [ "Observations", "Galaxy surveys, such as the PSCz, 2dF, VVDS, SDSS, 6dF, 2MASS, BOSS, and WiggleZ, provide us plenty of information about the large scale structure formation.", "They report data of the growth rate $f$ at low redshift or its combination with the rms matter fluctuations at $8h^{-1}$ Mpc ($\\sigma _8$ ) and the matter power spectrum for a large range of $k$ .", "We can use them to test the concordance cosmology model $\\Lambda $ CDM and modified gravity theories such as $f(R)$ [51], [52], [53], [54], [55], [56].", "Furthermore, WMAP and Planck spacecraft provide us accurate data of the anisotropy of CMB, which can also be used to constrain different gravity models.", "But in our case, according to the results presented in the last section, the growth rate in EiBI gravity is nearly indistinguishable from that in the $\\Lambda $ CDM at late time (low redshift), so we will focus on the effects on CMB and matter power spectrum below." ], [ "Integrated Sachs-Wolfe Effect", "In Ref.", "[11], the authors showed that we can obtain a nearly scale-invariant power spectrum for scalar perturbations in EiBI gravity.", "So we assume the curvature power spectrum in EiBI gravity can be written as $\\frac{k^3 P_{\\mathcal {R}}}{2\\pi ^2}=A_s\\left(\\frac{k}{k_0}\\right)^{n_s-1}T(k)^2,$ where $A_s$ is the amplitude of curvature power spectrum on the scale $k_0=0.05$ Mpc$^{-1}$ , $n_s$ is the scalar spectrum power-law index, and $T(k)$ is the matter-radiation transfer function.", "The integrated Sachs-Wolfe effect contributes to the angular power spectrum of the temperature anisotropies as [51] $C_l^{II}=4\\pi \\int \\frac{dk}{k}[I_l^I]^2\\frac{9}{25}\\frac{k^3 P_\\mathcal {R}}{2\\pi ^2},$ where $I_l^I(k)=2\\int dz G^{\\prime }(z) j_l(k D).$ Here $G(z)=\\frac{\\Psi _{+}(a,k)}{\\Psi _{+}(a_i,k)}$ , $j_l$ is the spherical Bessel function, and $D=\\int dz/H(z)$ is the comoving distance.", "It can be seen that the integrated Sachs-Wolfe effect depends on the variation of $\\Psi _{+}$ with time.", "For a matter dominated Universe in general relativity, $\\Psi _{+}$ is time independent, so there is no integrated Sachs-Wolfe effect.", "But in EiBI gravity, even at the matter-dominated era, $\\Psi _{+}$ is not time independent.", "So there will be a difference between these two theories.", "As is shown in the last section, the overall change of $\\Psi _{+}$ in EiBI gravity is larger than that in the $\\Lambda $ CDM model for positive $\\gamma $ , while the case with negative $\\gamma $ is just the opposite.", "These deviations will cause an elevation or reduction of the angular power spectrum at low multipoles.", "However, a further analysis shows that the differences caused by the modifications to general relativity are extremely small.", "It is not difficult to understand, considering that the deviations appear for large $k$ , but the transfer function $T(k)$ decreases significantly for these wave numbers, making the contributions from these modes very small.", "In fact, if we use the fitting formulas for $T(k)$ proposed in Ref.", "[57], we can calculate the contributions of the integrated Sachs-Wolfe effect.", "The results show that the CMB quadrupole power $6 C_2^{II}/2\\pi $ contributed by the integrated Sachs-Wolfe effect is nearly indistinguishable between EiBI gravity and $\\Lambda $ CDM model, they all give a value about $362.3$ .", "Here we have also taken $A_s=2.196\\times 10^{-9}$ , $n_s=0.9603$ , and $\\Omega _b h^2=0.02205$ [50].", "Note that the transfer function is actually different in EiBI gravity.", "So for a more specific analysis, we need to calculate it numerically by modifying the numerical codes, such as CMBFAST [47], [49], [48] and CAMB [58], [59].", "And a slower but easier to modify program called CMBquick is also available [60].", "But it will be a complicated work and needs a knowledge of the early evolution of the perturbations at radiation-dominated era and matter-radiation transition in EiBI gravity." ], [ "Linear Matter Power Spectrum", "As in Ref.", "[51], we can define the density growth: $D_G(a,k)=\\frac{\\Delta (a,k)}{\\Delta (a_i,k)} a_i.$ Then the linear matter power spectrum takes the form of $\\frac{k^3 P_L}{2\\pi ^2}=\\frac{4}{25}D_G^2(a,k)\\frac{k^4}{\\Omega _m^2 H_0^4}\\frac{k^3 P_{\\mathcal {R}}}{2\\pi ^2}.$ As mentioned in the last subsection, $T(k)$ is different in EiBI gravity.", "But in this paper, we focus on the growth of structure after the time of decoupling, thus to compare with $\\Lambda $ CDM model we use the same fitting formulate for transfer function as before.", "Figure REF shows the linear matter power spectrum at present in EiBI gravity and $\\Lambda $ CDM model.", "The parameter $\\gamma $ is taken to be $\\pm 10^{-13}$ .", "It can be found that when $k\\gtrsim 0.1 h$ Mpc$^{-1}$ , the power spectrum is lower in EiBI gravity with positive $\\gamma $ , while the case with negative $\\gamma $ is just the opposite.", "It is not a surprising result, if we consider the discussions in section REF .", "For positive $\\gamma $ , at early time the growth of density perturbation is suppressed by the modifications of EiBI gravity to $\\Lambda $ CDM model.", "This effect is especially significant for large $k$ .", "On the contrary, for negative $\\gamma $ , the growth of density perturbation is strengthened, so the power spectrum is higher than $\\Lambda $ CDM at high $k$ .", "Deviations will be much more significant for even larger $k$ .", "However, with the analysis in this paper, we cannot yet compare the above results with observations directly.", "One reason is that we don't know the accurate form of the transfer function in EiBI gravity.", "The other is that for large $k$ , the nonlinear effects will become important.", "So nonlinear analysis is needed in future works.", "But from the results of the present work, we can see a trend of increasing departure from $\\Lambda $ CDM model with increasing wave number $k$ .", "So there should be a more significant departure for even larger $k$ , when nonlinear regime become important.", "Nonlinear measurements of the mass power spectrum through the cluster abundance, Lyman-$\\alpha $ forest, and cosmic shear will provide us a way to test EiBI gravity at high $k$ and give constraint on the only extra parameter $\\kappa $ .", "Figure: Linear matter power spectrum at present in EiBI gravity (γ=±10 -13 \\gamma =\\pm 10^{-13})comparing with Λ\\Lambda CDM model." ], [ "Conclusions and Discussions", "The EiBI gravity has been one of the prospective candidates for modified gravity theories, in which the singularity at the beginning of the Universe can be avoided.", "A lot of works have been done to analyze the stability of cosmological perturbations in this theory and constrain it using different astrophysical and cosmological observations.", "In this paper, we have discussed the evolution of linear scalar perturbations since the matter-dominated era and the large scale structure formation in EiBI gravity.", "The growth rate of clustering in EiBI gravity is found to deviate from that in the $\\Lambda $ CDM model at an early time of the Universe.", "The departure increases with wave number $k$ .", "But at relative low redshift, the growth rate in EiBI gravity approaches to that in the $\\Lambda $ CDM model.", "The suppression (for positive $\\gamma $ ) or enhancement (for negative $\\gamma $ ) on the growth of density perturbation at early time for large $k$ affects the linear matter power spectrum on small scales (large $k$ ).", "For $k\\gtrsim 0.1h$ Mpc$^{-1}$ , the matter power spectrum in EiBI gravity with positive $\\gamma $ is lower than that in the $\\Lambda $ CDM.", "And for negative $\\gamma $ , it is just the opposite.", "So it is prospective to use the observational matter power spectrum to test EiBI gravity and constrain the parameter $\\kappa =\\gamma /H_0^2$ .", "If we require that the linear matter power spectrum in EiBI gravity does not deviate significantly from that in the $\\Lambda $ CDM model, the parameter $\|\\gamma \|$ should be the order of $10^{-14}$ or smaller.", "At present it is still not a very strong constraint.", "In Ref.", "[19], Avelino obtained $\|\\kappa \|<10^{-3}$ kg$^{-1}$ m$^5$ s$^{-2}$ by requiring that gravity plays a subdominant role inside atomic nuclei, which leads to $\|\\gamma \|<10^{-47}$ .", "If we consider this constraint, there will be no distinguishable deviation.", "Besides, we also calculate the integrated Sachs-Wolfe effect in EiBI gravity, and find that its effect on the angular power spectrum of CMB is almost the same as that in the $\\Lambda $ CDM model.", "However, some work still needs to be done before we can compare the predictions of EiBI gravity directly with observations.", "First, we must also analyze in detail the evolution of scalar perturbations at a much earlier time when the Universe is dominated by radiation, and the transition from a radiation-dominated Universe to a matter-dominated one.", "Along with these analyses, we can obtain more accurate transfer function by modifying corresponding numerical codes such as CAMB, CMBFAST and CMBquick.", "Secondly, to distinguish EiBI gravity with $\\Lambda $ CDM model, we need to compare the evolution of perturbations with large $k$ , at which wave number nonlinear effect cannot be neglected.", "This can be solved by following the halo-based description of nonlinear gravitational clustering [61] or using numerical simulations.", "Although within this paper, we cannot yet give a strong constraint on the parameter $\\kappa $ comparable with that derived from other methods such as considering the compact objects or structure of nucleon, we have shown some interesting behaviors of EiBI gravity on large scale structure formation, especially the linear matter power spectrum.", "An analysis of nonlinear matter power spectrum in future papers will give a stronger constraint on the deviations from general gravity.", "Furthermore, despite the success of $\\Lambda $ CDM model in predicting the large scale structure, it may have some problems on small scales, such as the missing satellites problem or the cuspy halo problem.", "So it is motivated to consider some modifications on these scales." ], [ "Acknowledgement", "The authors thank Shruti Thakur for her kind reply about the calculation of some observational quantities.", "XLD would like to thank Shao-Wen Wei for his help on figure processing.", "This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No.", "11375075, and the Fundamental Research Funds for the Central Universities under Grant No.", "lzujbky-2013-18.", "K. Yang was supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.", "X.H.", "Meng was partially supported by the Natural Science Foundation of China (NSFC) under Grant No.", "11075078." ] ]	1403.0083
[ [ "Evidence for a gas-rich major merger in a proto-cluster at z=2.5" ], [ "Abstract Gas-rich major mergers in high-redshift proto-clusters are important events, perhaps leading to the creation of the slowly rotating remnants seen in the cores of clusters in the present day.", "Here, we present a deep Jansky Very Large Array observation of CO J = 1-0 emission line in a proto-cluster at z = 2.5, USS1558-003.", "The target field is an extremely dense region, where 20 H-alpha emitters (HAEs) are clustering.", "We have successfully detected the CO emission line from three HAEs and discovered a close pair of red and blue CO-emitting HAEs.", "Given their close proximity (~30 kpc), small velocity offset (~300 km/s), and similar stellar masses, they could be in the early phase of a gas-rich major merger.", "For the red HAE, we derive a total infrared luminosity of L(IR)=5.1e12 Lsun using MIPS 24 um and radio continuum images.", "The L(IR)/L'(CO) ratio is significantly enhanced compared to local spirals and high-redshift disks with a similar CO luminosity, which is indicative of a starburst mode.", "We find the gas depletion timescale is shorter than that of normal star-forming galaxies regardless of adopted CO-H2 conversion factors.", "The identification of such a rare event suggests that gas-rich major mergers frequently take place in proto-clusters at z > 2 and may involve the formation processes of slow rotators seen in local massive clusters." ], [ "Introduction", "Properties of galaxies seen in the local universe strongly depend on their surrounding environments, as is well known from the morphology–density relation.", "Early-type galaxies are frequently observed in high-density regions such as clusters and late-type galaxies are common in low-density regions .", "The ATLAS$^\\mathrm {3D}$ survey further demonstrates that early-type galaxies are divided into two kinds of populations based on the kinematics: fast rotators and slow rotators .", "This categorization makes the trend in the morphology–density relation more prominent.", "Whereas fast rotators, which form the majority of early-type galaxies, appear in a wide range of environments, slow rotators reside exclusively in dense cores of mature clusters .", "Such spatial segregation and the difference in their kinematics could be related to formation processes and subsequent quenching mechanisms of star formation, which probably take place at high redshift.", "In theoretical models, gas-rich major mergers leading to a spin-down of remnants successfully produce simulated slow rotators and are considered to be one of the possible formation processes of slow rotators .", "Given high number densities of star-forming galaxies in proto-clusters at $z>2$ , we naturally expect a high frequency of major-merger events.", "What is important here is whether they are interactions between gas-dominated systems.", "While stellar components within galaxies are collisionless, systems comprising gas are dissipational.", "Therefore, gas-rich major mergers trigger an intense, dusty star formation due to shocks and an inflow of gas that has lost its angular momentum, as well as establishment of the outer profile through violent relaxation .", "CO observations are critical for measuring the molecular gas mass within galaxies, $M_\\mathrm {gas}$ , and investigating the star-formation mode characterized by the star-formation efficiency, SFE=SFR/$M_\\mathrm {gas}$ , and the gas depletion timescale, $\\tau _\\mathrm {depl}=M_\\mathrm {gas}$ /SFR.", "CO studies at high-redshift have rapidly developed over the past years not only for very bright galaxies in the dust emission such as submillimeter galaxies (SMGs; , , , , ) but for optical/near-infrared selected galaxies such as $BzK$ galaxies , .", "However, most of them observe high-excitation CO lines mainly using the Plateau de Bure Interferometer.", "This significantly affects the estimates of molecular gas mass because high-$J$ lines trace dense gas regions rather than total gas reservoirs probed by CO $J=1-0$ emission.", "In a cluster field at $z=0.4$ , detect the CO $J=1-0$ line from five dusty star-forming galaxies and find the environmental dependence of SFE is not seen.", "also find that the SFEs of two galaxies in a proto-cluster at $z=1.5$ are comparable to that of field galaxies at similar redshift.", "On the other hand, discover four CO luminous galaxies across an $\\sim $ 100 kpc region at $z=2.4$ and find that two of them have a high SFE.", "No conclusive result could be obtained due to a small sample size.", "In this paper, we report results from a deep CO $J=1-0$ observation of a proto-cluster at $z=2.5$ to search for gas-rich galaxies and see if there is any environmental effect of the star-formation mode in the formation phase of the progenitors of cluster early-type galaxies seen today.", "We assume the Salpeter initial mass function and cosmological parameters of H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _\\mathrm {M}$ = 0.3, and $\\Omega _\\Lambda $ = 0.7.", "Figure: A 2-D map of the USS1558 proto-cluster at z=2.5z = 2.5.Red and blue squares represent CO-detected and non-detected Hα\\alpha emitters (HAEs), respectively.Coordinates are relative to the radio galaxy (a red star).The primary beam is shown by a magenta circle.", "Contours denote the local number densities (Σ 5 th \\Sigma _\\mathrm {5th}) of all HAEs, in a step of 2 Mpc -2 ^{-2}.Our target is a proto-cluster at $z=2.5$ , USS1558-003, where an over-density of massive, red-sequence galaxies has been discovered around a radio galaxy .", "This region has also been observed as part of a systematic H$\\alpha $ narrow-band imaging campaign with MOIRCS on Subaru Telescope called “$MAHALO-Subaru$ ” project (MApping HAlpha and Lines of Oxygen with Subaru; ), and it is found to host numerous H$\\alpha $ emitters (HAEs; ).", "Membership of about half of the 68 HAEs has been spectroscopically confirmed with a success rate of 70% .", "An extremely dense clump lies about three arc-minutes away from a radio galaxy in the southwest (Figure REF ).", "Because the dynamical mass of this clump is estimated to be $\\sim 10^{14}~M_\\odot $ from the H$\\alpha $ spectroscopy, it is expected to evolve into a single massive system with $>10^{15}~M_\\odot $ similar to the Coma cluster .", "Therefore, this proto-cluster is very likely the site where slow rotators, which will eventually dominate a rich cluster by the present-day, are just in their formation phase." ], [ "JVLA observations", "We have conducted CO $J=1-0$ emission line observations with the Jansky Very Large Array (JVLA) during February-April 2013.", "The target field includes 20 HAEs at $z=2.5$ (Figure REF ), of which 12 have spectroscopic redshifts based on their H$\\alpha $ line.", "The observations were made in the compact D array configuration to securely detect emission lines from entire galaxies.", "Because the narrow-band survey samples star-forming galaxies at $z=2.53\\pm 0.02$ , the CO $J=1-0$ emission line ($\\nu _\\mathrm {rest}=115.271$ GHz) can be observed with the Ka-band receiver ($\\nu _\\mathrm {obs}=33$ GHz), providing the primary beam size of 82.", "The WIDAR correlator was set up to cover 32.078–34.082 GHz, corresponding to the CO line at $z$ =2.382–2.593.", "We observed the standard calibrators 3C286 (1.9 Jy) for bandpass and flux calibration.", "Phase calibration is performed with observations of J1557-0001 (0.7 Jy).", "The data are processed through the VLA CASA Calibration Pipeline .", "Channels at the edges of each spectral window are flagged, which brings 14 spectral gaps of 14 MHz.", "The total integration time is 14 hours on source.", "Two kinds of CO maps are reconstructed with the CASA task CLEAN, using “briggs” weighting with a robustness parameter of -0.5 and natural weighting with 5.0, which provide a synthesized beam of 2.2$\\times $ 1.9 arcsec$^2$ /3.0$\\times $ 2.5 arcsec$^2$ and a rms level of 80–90 $\\mu $ Jy beam$^{-1}$ /40–50 $\\mu $ Jy beam$^{-1}$ per 2 MHz channel (18 km s$^{-1}$ ) before the primary beam correction, respectively.", "Figure: (Top) Velocity-averaged maps of the CO emission for three HAEs.The visibility data are averaged over the spectral bins where positive signals are detected (yellow-shaded regions in the bottom panel).The images are obtained using briggs weighting for ID 191/ID 193 and natural weighting for ID 213.The beamsize is at bottom left .Magenta and green contours present MIPS 24 μ\\mu m and radio continuum (ν rest =116\\nu _\\mathrm {rest}=116 GHz) map, respectively.", "(Bottom) The CO spectra extracted from the peak position in the top images.The horizontal axis shows the relativistic velocity in the rest frame at z=2.515z=2.515.The velocity resolution is 18 km s -1 ^{-1}, but the spectrum of ID 213 is binned over the velocity range of 92 km s -1 ^{-1} to detect a faint emission line.The red and blue lines show the best-fitting profile with single and double gaussian model, respectively.Dashed lines indicate the redshift expected from the Hα\\alpha spectroscopy.The channels flagged at the edges of each spectral window are shown by gray-shaded regions." ], [ "Spitzer MIPS 24 $\\mu $ m data", "We use a $Spitzer$ /MIPS 24 $\\mu $ m image to identify dusty star formation and estimate total infrared (IR) luminosities of HAEs.", "The data is retrieved from the Spitzer Heritage Archive.", "Data reduction is performed in a standard manner (flat fielding, background subtraction and mosaic) using MOPEX software .", "Source extraction and PSF-fitted photometry are performed using APEX module in MOPEX.", "The limiting flux reaches to $5\\sigma \\sim 150~\\mu $ Jy in the final combined image." ], [ "Detections of CO J=1-0 emission line", "We search for the CO $J=1-0$ emission line within a 0.5 arcsec radius from the position of 20 HAEs by using the 10 MHz ($\\sim $ 92 km s$^{-1}$ ) binned data cube.", "To avoid spurious detections, we identify a $>3\\sigma $ peak associated with two additional bins of $>1.5\\sigma $ significance.", "The probability that such signals are detected, by chance, in the frequency range (32.471–32.841 GHz, 37 bins) of the narrow-band redshift is estimated to be 0.5% in Gaussian noise.", "Actually, negative peaks are not seen except for one object located at the edge of the primary beam.", "The significance of the detection is defined by the signal-noise ratio of peak flux in the 10MHz binned cube.", "ID 191, ID 193, and ID213 are eventually detected in 6.9, 6.3, and $3.8\\sigma $ , respectively.", "The separation between ID 191 and ID 193 is about 4 arcsec corresponding to 32 kpc in the physical scale and the velocity offset is 300 km s$^{-1}$ (Figure REF ).", "We also look into the relation between the detection rate of CO emission and the rest-frame optical color.", "Our sample consists of four red HAEs with $J-K_s>1.38$ and 16 blue HAEs.", "The two (ID 193 and ID 213) out of four red HAEs are actually detected in the CO emission line while only one (ID 191) out of 16 blue HAEs is detected.", "Red and massive galaxies tend to be relatively bright in CO emission compared to blue and less massive ones.", "Table: IR vs. CO J=1-0J=1-0 luminosities of the HAEs with CO detection along with SMGs at z>1z>1 , , , ULIRGs at z<1z<1 , optical/near-IR selected star-forming galaxies at z>1z>1 , and local spirals , , .Dashed black and red lines show the best-fitting relations for normal star-forming galaxies and luminous mergers ." ] ]	1403.0040
[ [ "CLASH: Extending galaxy strong lensing to small physical scales with\n distant sources highly-magnified by galaxy cluster members" ], [ "Abstract We present a strong lensing system in which a double source is imaged 5 times by 2 early-type galaxies.", "We take advantage in this target of the multi-band photometry obtained as part of the CLASH program, complemented by the spectroscopic data of the VLT/VIMOS and FORS2 follow-up campaign.", "We use a photometric redshift of 3.7 for the source and confirm spectroscopically the membership of the 2 lenses to the galaxy cluster MACS J1206.2-0847 at redshift 0.44.", "We exploit the excellent angular resolution of the HST/ACS images to model the 2 lenses in terms of singular isothermal sphere profiles and derive robust effective velocity dispersions of (97 +/- 3) and (240 +/- 6) km/s.", "The total mass distribution of the cluster is also well characterized by using only the local information contained in this lensing system, that is located at a projected distance of more than 300 kpc from the cluster luminosity center.", "According to our best-fitting lensing and composite stellar population models, the source is magnified by a total factor of 50 and has a luminous mass of about (1.0 +/- 0.5) x 10^{9} M_{Sun}.", "By combining the total and luminous mass estimates of the 2 lenses, we measure luminous over total mass fractions projected within the effective radii of 0.51 +/- 0.21 and 0.80 +/- 0.32.", "With these lenses we can extend the analysis of the mass properties of lens early-type galaxies by factors that are about 2 and 3 times smaller than previously done with regard to, respectively, velocity dispersion and luminous mass.", "The comparison of the total and luminous quantities of our lenses with those of astrophysical objects with different physical scales reveals the potential of studies of this kind for investigating the internal structure of galaxies.", "These studies, made possible thanks to the CLASH survey, will allow us to go beyond the current limits posed by the available lens samples in the field." ], [ "Introduction", "Gravitational lensing studies have radically improved our understanding of the internal structure of galaxies and clusters of galaxies (e.g., [85]; [10]).", "In particular, the combination of strong lensing with stellar dynamics or stellar population synthesis models has allowed to characterize some properties, previously almost unexplored, of the galaxy dark-matter haloes and sub-haloes.", "For example, it has become possible to measure the dark over total mass fraction and dark-matter halo density slope in the inner regions of galaxies (e.g., [44], [42], [39]; [3]; [8], [7]; [77]; [29]), and to estimate the mass function of dark satellites (also called substructure) (e.g., [53]; [91], [92]; [32]; [94]) and the spatial extent (e.g., [48]; [83]; [69]; [28]; [30]) of galaxy dark-matter haloes.", "Moreover, the same combinations of mass diagnostics have enabled to investigate the total mass density profile (e.g., [72], [71]; [57], [56]; [70]; [78]; [1]), the stellar Initial Mass Function (IMF; [45], [44]; [86]; [79], [80]; [9]), and the origin of the tilt of the Fundamental Plane (e.g., [44], [43]; [4]) of massive early-type galaxies and to use single lenses or statistical samples of them to infer cosmologically relevant quantities (e.g., [46]; [75]; [84], [82]; [33]).", "Taking advantage of the excellent data collected by the Lenses Structure and Dynamics (LSD; [88]), Sloan Lens ACS Survey (SLACS; [16]; [89]; [4]), CFHT Strong Lensing in the Legacy Survey (SL2S; [65]; [36]; [76]), and the BOSS Emission-Line Lens Survey (BELLS; [19]; [15]), the analyses conducted so far have mainly examined the physical properties of isolated, massive early-type galaxies, acting as strong lenses on background sources.", "Only more recently, thanks also to the Cambridge And Sloan Survey Of Wide ARcs in the skY (CASSOWARY; [11]; [81]), growing interest has been shown in the study of early-type lens galaxies residing in galaxy groups and clusters (e.g., [47], [40], [41]; [61]; [27]).", "Despite the large amount of results published to date, detailed strong lensing studies in “small” lens galaxies are still lacking, mostly because these systems are not observed frequently.", "One possibility to find more such systems is to look at clusters of galaxies, where, owing to the increase in the strong lensing cross section of a cluster member due to the presence of the extended mass distribution of the cluster, low-mass galaxies are more likely to produce strong lensing features than in less dense environments.", "Investigations of these objects are particularly useful, as they can provide the necessary piece of information to elucidate what is the amount and distribution of dark matter in astrophysical objects extending from the lowest to the highest ends of the galaxy mass function.", "The comparison of these observational measurements over a wide range of physical scales with the outcomes of cosmological simulations can give fundamental clues about the precise nature of dark matter and the role played by the interaction of baryons and dark matter during the mass assembly of cosmological structures.", "The Cluster Lensing And Supernova survey with Hubble (CLASH; GO 12065, PI Postman) was awarded 524 orbits of Hubble Space Telescope (HST) time to observe 25 massive (virial mass $M_{\\mathrm {vir}} \\approx 5$ -$30 \\times 10^{14} M_{\\odot }$ , X-ray temperature $T_{X} \\ge 5$ keV) galaxy clusters in 16 broadband filters, ranging from approximately 2000 to 17000 Å$\\,$ with the Wide Field Camera 3 (WFC3; [50]) and the Advanced Camera for Surveys (ACS; [35]).", "The sample, spanning a wide redshift range ($z$ = 0.18-0.90), was carefully chosen to be largely free of lensing bias and representative of relaxed clusters, on the basis of their symmetric and smooth X-ray emission profiles (for a thorough overview, see [68]).", "CLASH has four main scientific goals: 1) measure the cluster total mass profiles over a wide radial range, by means of strong and weak lensing analyses (e.g., [96]; [25]; [63]); 2) detect new Type Ia supernovae out to redshift z $\\sim $ 2.5, to improve the constraints on the dark energy equation of state (e.g., [37]; [66]); 3) discover and study some of the first galaxies that formed after the Big Bang (z $>$ 7) (e.g., [95]; [26]; [17]); 4) perform galaxy evolution analyses on cluster members and background galaxies.", "Ancillary science that can surely be carried out with the superb data set of CLASH is the analysis of several new strong lensing systems on galaxy scale.", "A Large Programme (186.A-0798, PI Rosati) of 225 hours with the VIMOS instrument at the Very Large Telescope (VLT) has also been approved to perform a panoramic spectroscopic survey of the 14 CLASH clusters that are visible from ESO-Paranal (Rosati et al.", "2014, in preparation).", "This observational campaign aims at measuring in each cluster the redshifts of 1) approximately 500 cluster members within a radius of more than 3 Mpc; 2) 10-30 lensed multiple images inside the HST field of view, including possible highly-magnified candidates out to z $\\approx $ 7 (e.g., [64]; [6]); 3) possible supernova hosts.", "In one of the CLASH clusters (i.e., MACS J1206.2$-$ 0847, hereafter MACS 1206), the first spectroscopic redshifts have already been exploited to build robust strong lensing models ([97]; [90]), to obtain an independent total mass estimate from the spatial distribution and kinematics of the cluster members ([14]; [59]), and to confirm a source at $z = 5.703$ ([18]).", "Strong lensing (with spectroscopically confirmed systems) and cluster dynamics analyses are planned for all 14 southern clusters (e.g., in MACS J0416.1$-$ 2403, Grillo et al.", "2014, in preparation; Balestra et al.", "2014, in preparation).", "Here, we focus on a rare strong lensing system in which two angularly close early-type galaxies, members of the galaxy cluster MACS 1206 at $z=0.44$ , produce in total ten multiple images of a double source located at $z \\approx 3.7$ .", "This is the first example of the kind of strong lensing studies that can be conducted on galaxy cluster members, capitalizing on the extraordinary multi-band photometric observations obtained as part of the CLASH program and spectroscopic measurements of the VLT/VIMOS follow-up campaign.", "This work is organized as follows.", "In Sect.", "2, we introduce the photometric and spectroscopic observations used for this analysis.", "In Sect.", "3, we present the strong lensing modeling performed to measure principally the total mass values of the lens galaxies and the magnification factors of the multiple images.", "In Sect.", "4, we estimate the luminous mass values of the lens and lensed galaxies by means of stellar population synthesis models.", "In Sect.", "5, we compare some physical quantities, related to the luminous and total masses, of the two lens galaxies with those of lower and higher-mass galaxies.", "In Sect.", "6, we summarize our conclusions.", "All quoted errors are 68.3% confidence limits (CL) unless otherwise stated.", "Throughout this work we assume $H_{0}=70$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _{m}=0.3$ , and $\\Omega _{\\Lambda }=0.7$ .", "In this model, 1$\\,$ corresponds to a linear size of 5.69 kpc at the cluster redshift of $z=0.44$ ." ], [ "Observations", "MACS 1206 was observed as part of the CLASH program in HST Cycle 18 between April 3 and July 20 2011 to a total depth of 20 orbits in 16 broadband filters.", "The images were processed for debias, flats, superflats, and darks using standard techniques, and then co-aligned and combined using drizzle algorithms to a pixel scale of 0.065$\\,$ (for details, see [54], [55]).", "By fitting the full UV to near-IR isophotal aperture magnitudes, photometric redshift estimates of all detected sources were measured through the BPZ ([12], [13]; [24]) and LePhare ([2]; [58]) codes.", "A color composite image of the lensing system analyzed in this work is shown in Figure REF .", "The coordinates and photometric redshift measurements, $z_{\\mathrm {ph}}$ , of the two lenses G1 and G2 and of the multiple images are listed in Tables REF and REF .", "For the lensed source, we adopt the photometric redshift estimate of 3.7 presented by [97] (see system number 7 in the cited paper; for more details on the photometric redshifts measured with the CLASH data, and particularly in MACS 1206, see [49]).", "This value was determined as an average estimate of the BPZ measurements of the three images (1, 3, and 5; see Figure REF ) for which the photometry is less contaminated by the light distribution of the two lens galaxies.", "In passing, we mention that the BPZ probability distribution functions of the redshift of the three images were all unimodal and that the combined 95% CL interval of the redshifts extended from 3.4 to 4.2 (this redshift uncertainty is not significant for the lensing analysis performed below; for example, it introduces a percentage error smaller than 1.5% on the values of the lens effective velocity dispersions plotted in Figure REF ).", "We used the public code Galfithttp://users.obs.carnegiescience.edu/peng/work/galfit/galfit.html ([67]) to derive the luminosity structural parameters of the two lenses in the F160W band (see Table REF ): the axis ratio, $q_{L}$ , the position angle of the major axis, $\\theta _{q_{L}}$ , the half-light angle, $\\theta _{e}$ , and the Sersic index, $n$ .", "Spectroscopic follow-up observations were taken as part of the VLT/VIMOS Large Programme 186.A-0798.", "Four VIMOS pointings were used, keeping one of the four quadrants fixed to the cluster core in order to allow for long integrations on the strong lensing features.", "In each pointing, we took 45 to 60 minute exposure times.", "We used 1-wide slits with either the low-resolution LR-Blue grism or the intermediate-resolution MR grism, covering a layout of $20^{\\prime }$ -$25^{\\prime }$ across.", "MACS 1206 is the first of the 14 southern galaxy clusters in the CLASH sample targeted by this spectroscopic program for which the observational campaign was concluded.", "The measurements have resulted in a total integration time of approximately 11 hours, providing about 600 secure cluster members (see [14]) and 4 confirmed multiple image systems (see [97]).", "Additional spectroscopic measurements on some cluster members of MACS 1206 were obtained with the VLT/FORS2 in April 17 2012 (Programme ID 089.A-0879, PI Gobat).", "The observations were taken in good seeing conditions, with the medium resolution grism 600RI and 1-wide slits, with a total exposure time of 60 minutes.", "The flux-calibrated VIMOS MR and LR-Blue spectra of G1 and G2, respectively, are shown in Figure REF .", "The identification of the most prominent absorption lines, like CaII K and H and G-band, have provided spectroscopic redshift estimates, $z_{\\mathrm {sp}}$ , of 0.436 and 0.439 for these two galaxies.", "These values are included in the 95% CL intervals of the estimated photometric redshifts, ranging from 0.36 to 0.46 and from 0.42 to 0.52 for, respectively, G1 and G2.", "In Figure REF , we also display several template emission and absorption features redshifted to the best-fitting values of the galaxy spectroscopic redshifts.", "The FORS2 spectrum of G2 is shown in Figure REF and will be discussed in the following section.", "Table: Astrometric and photometric measurements for the multiple images.Figure: Flux-calibrated VLT/VIMOS spectra of G1 and G2 obtained with the MR and LR-Blue grisms, respectively.", "The exposure times are 45 and 6 minutes.", "The 1D spectra with several template emission and absorption lines shifted to the best-fitting redshift values and the 2D spectra are shown.", "On the top: Spectra of G1 which provide a redshift value of 0.436.", "On the bottom: Spectra of G2 which provide a redshift value of 0.439." ], [ "Strong lensing modeling", "In this section we present two different models of the strong gravitational lensing system.", "We use the public code gravlenshttp://redfive.rutgers.edu/$\\sim $ keeton/gravlens/ ([51]) to reconstruct the total mass distribution of the lenses and to estimate the position and magnification factor of the sources.", "We stress the fact that the analysis presented below has a different perspective compared to the previous strong lensing models of MACS 1206 ([97]; [30]).", "We concentrate here on radial scales of a few kpc, where the total mass of the galaxies G1 and G2 is the main source of the gravitational potential and the mass of the cluster (extended over a typical radial scale of 100 kpc) is instead approximated and treated as a second-order term.", "We keep the model complexity to a minimum and describe the three main lenses, i.e.", "the two galaxies G1 and G2 and the cluster, in terms of either three singular isothermal spheres (3SISs) or two singular isothermal spheres and a singular isothermal ellipsoid (2SISs+SIE).", "The mass components are fixed to the luminosity centroids of the galaxies G1 and G2 and of the BCG (the hypothesis of the center of mass of the cluster coinciding with that of the BCG is supported by the results of the studies cited in the previous paragraph) and parametrized by angular scales (labeled as $b_{\\mathrm {G1}}$ , $b_{\\mathrm {G2}}$ , and $b_{\\mathrm {H}}$ ), which represent the strength of the lenses and are equal to the values of the Einstein angles in the spherical case.", "The SIE model requires two additional parameters: the values of the axis ratio, $q$ , and of the major axis position angle $\\theta _{q}$ .", "The ten multiple images are approximated to point-like objects and associated to two close sources (A and B), each of which is lensed five times.", "The multiple images are identified with indices running from 1 to 5 (see Table REF ).", "For each image we assume an observational error $\\delta _{x,y}$ on the determination of its position of one image pixel (i.e., 0.065).", "Although in each HST filter the luminosity centroids of the multiple images can be measured with positional errors of some fractions of a pixel, we have decided to consider a conservative uncertainty value of one pixel to take into account the point-like approximation, the centroid differences in the individual HST bands, and the contamination from the light of the lens galaxies in estimating the multiple image positions.", "To quantify the goodness of a model, we use a standard chi-square function, $\\chi ^{2}$ , defined as the sum over all the sources $i$ and their multiple images $j$ of the squared ratios of the differences between the observed (${x}^{i,j}_{\\mathrm {obs}}$ ) and model-predicted (${x}^{i,j}_{\\mathrm {mod}}$ ) positions divided by the adopted positional uncertainties ($\\sigma _{{x}^{i,j}}$ ): $\\chi ^{2}({p}) := \\sum _{i,j} \\frac{\|\| {x}^{i,j}_{\\mathrm {obs}} - {x}^{i,j}_{\\mathrm {mod}} \|\|^{2}}{\\sigma _{{x}^{i,j}}^{2}}\\, .$ We minimize this $\\chi ^{2}$ estimator by varying the model parameters (${p}$ ) and compare the minimum value of the chi-square with the number of degrees of freedom (d.o.f.).", "These are the number of observables (twenty coordinates of the ten images) minus the number of the parameters of a model (four coordinates for the two sources, three lens angular scales, and, when present, two lens ellipticity parameters).", "The best-fitting (minimum chi-square) parameters are shown in Table REF .", "There, we also present the median ($\\tilde{\\Delta }$ ) and root mean square ($\\Delta _{\\mathrm {rms}}$ ) values of the Euclidean distances ($\\Delta $ ) between the observed and model-predicted angular positions of the multiple images.", "Table: The parameters of the best-fitting models.The values of the best-fitting chi-square are very close to the number of degrees of freedom.", "This fact and the small values of $\\tilde{\\Delta }$ and $\\Delta _{\\mathrm {rms}}$ confirm that our relatively simple mass modeling choices are adequate to describe the lenses.", "More quantitatively, the probability that the value of a random variable extracted from the chi-square distribution with 11(13) degrees of freedom is greater than 13.3(19.1) is 0.27(0.12).", "It is not surprising that the spherical approximation used for describing the total mass distributions of the galaxies G1 and G2 is proved suitable, given the large values of the minor-to-major axis ratio $q_{L}$ of their luminous components (see Table REF ).", "Looking at Table REF , we conclude that the values of $b_{\\mathrm {G1}}$ and $b_{\\mathrm {G2}}$ , which characterize the total mass distribution of the two main lenses, are robust and not sensitive to the details of the cluster total mass modeling.", "In Figure REF , we plot the best-fitting 2SISs+SIE model with the observed and model-predicted multiple images and the critical curves.", "We show also the values of the magnification factors in proximity to the reconstructed positions of each multiple image.", "Figure: The best-fitting 2SISs+SIE strong lensing model.", "On the left: The model-predicted critical curves and multiple image positions (diamond and square symbols) of the two sources, AA and BB, around the two main lenses, G1 and G2.", "The observed positions of the multiple images (cross and plus symbols) are shown for comparison.", "On the right: Reconstructed values of the magnification factor at the positions of the model-predicted multiple images.", "Positive and negative values on the contour levels indicate, respectively, if the images have conserved or inverted parity with respect to the source.", "North is top and East is left.Interestingly, starting from the only lensing system analyzed in this work we measure that for the cluster component, when modeled as an SIE, the values of axis ratio and position angle are aligned with the prominent intracluster light (for more details on the properties of the intracluster light in MACS 1206, we refer to Presotto et al.", "2014, in preparation).", "The values of these parameters are consistent with those obtained from the thorough lensing analyses performed on the cluster scale by [97], [30], and [90].", "The first two and the last studies exploit, respectively, the full strong lensing and strong plus weak lensing information in MACS 1206.", "Moreover, the best-fitting parameters for the SIE model associated to the cluster component provide a total mass estimate of about $4\\times 10^{14} M_{\\odot }$ projected within a cylinder of radius equal to 320 kpc (the approximate average distance of the multiple images from the BCG luminosity center).", "Given our simplified assumptions on the cluster total mass distribution, it is remarkable that this estimate is only approximately $10\\%$ higher than those obtained in the previously cited lensing works and in the cluster dynamical analysis by [14].", "If one used otherwise the very crude approximation of the Einstein radius of the cluster (for a source at redshift 3.7) given by the average projected distance between the strong lensing system and the BCG luminosity centre, this would translate into a total projected mass that is more than 1.5 times larger than what obtained from the other cluster total mass diagnostics.", "Several previous studies of strong lensing systems around galaxy cluster members (e.g., [47]; [61]) have demonstrated that although these systems contain enough information to characterize reasonably well the cluster mass distribution, this last term is not the main focus of such studies and it can be safely modeled with an approximated convergence plus shear contribution.", "In Table REF , we consider the two best-fitting models and list the values of the magnification factor for images A$_{5}$ and B$_{5}$ , $\\mu (\\mathrm {A}_{5})$ and $\\mu (\\mathrm {B}_{5})$ , and of the total magnification factor for the sources A and B, i.e., the sum of the magnification values over all the multiple images of each source: $\\mu _{\\mathrm {tot}}(\\mathrm {A}) &:=& \\sum _{i=1}^{5} \\mu (\\mathrm {A}_{i}) \\, ; \\nonumber \\\\\\mu _{\\mathrm {tot}}(\\mathrm {B}) &:=& \\sum _{i=1}^{5} \\mu (\\mathrm {B}_{i}) \\, .$ We decide to concentrate on images A$_{5}$ and B$_{5}$ because they are the most distant objects from the luminosity (and mass) centers of the two lenses G1 and G2.", "For this reason, their photometry is less contaminated by the light distribution of the much brighter lens galaxies.", "In addition, their relatively large distance from the critical curves (see Figure REF ) makes the measurements of their magnification factors less dependent on the modeling details.", "We conclude that A$_{5}$ and B$_{5}$ are magnified by a factor of approximately 9 and that each of the two sources is magnified in total by a factor of approximately 50.", "Table: Values of the magnification factor for the best-fitting models.To estimate the statistical uncertainties on the model parameters $b_{\\mathrm {G1}}$ and $b_{\\mathrm {G2}}$ , we perform a bootstrapping analysis in the 2SISs+SIE case.", "We resample the position of the 10 multiple images by extracting random values from Gaussian distributions with average and standard deviation values equal to, respectively, the positions and positional uncertainties listed in Table REF .", "We simulate in this way $10^{4}$ data samples, minimize the positional $\\chi ^{2}$ shown in Eq.", "(REF ), and consider the best-fitting values of the lens strength.", "We recall that the value of the effective velocity dispersion $\\sigma $ of an SIS model is related to that of the lens strength $b$ in the following way: $b = 4 \\pi \\bigg ( \\frac{\\sigma }{c} \\bigg )^2 \\frac{D_{ls}}{D_{os}} \\, ,$ where $c$ is the speed of light and $D_{ls}$ and $D_{os}$ are, respectively, the angular diameter distances between the lens and the source and the observer and the source.", "We show the results of this analysis in Table REF and Figure REF .", "Table: Median and statistical 1σ\\sigma error values of the SIS effective velocity dispersions of the two main lenses.Figure: Values of the effective velocity dispersion of the two main lenses, G1 and G2, for the 2SISs+SIE model.", "The best-fitting values are represented by a cross.", "The 68% and 95% confidence regions (contour levels) and the 68% confidence intervals (thick lines on the axes) are obtained from the χ 2 \\chi ^{2} minimization of 10 4 10^{4} resampled multiple image positions.Figure: VLT/FORS2 rest-frame spectrum of G2 from an aperture of 6 pixels (i.e., 1.5 arcsec) around the peak of emission.", "The total exposure time is 60 minutes.", "The data have been slightly smoothed with a box of 3 pixels for display purpose.", "Black and red solid lines show the data and the best-fitting model, respectively, and green dots display the difference of these last two spectra.", "The grey shaded areas correspond to the sky line regions which have been masked while performing the fit due to their high residuals.", "The main absorption lines and possible emission lines are labelled.The values of $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ are positively correlated and measured with a small statistical uncertainty.", "The median values with 1$\\sigma $ errors of $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ are $97 \\pm 3$ and $240 \\pm 6$ km s$^{-1}$ , respectively.", "As already found in several strong lensing studies (e.g., [47], [40]), the total mass projected within the average distance of the multiple images from the lens center can be measured precisely.", "In this specific system, we confirm that an increase of the total mass component associated to the two lens galaxies (i.e., larger values of $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ ) is correlated to a decrease of the total mass contribution related to the cluster (i.e., smaller values of $\\sigma _{\\mathrm {H}}$ ), and viceversa, in order to keep their sum approximately constant.", "Then, we repeat two more times the bootstrapping analysis in the 2SISs+SIE case, first allowing the center of the SIE component (i.e., the cluster dark-matter halo) to vary and then including the two candidate (based on the photometric redshift values) cluster members nearest in projection to G1 and G2.", "In the first case, we obtain $94 \\pm 3$ and $227 \\pm 8$ km s$^{-1}$ for the median and 1$\\sigma $ error values of the SIS effective velocity dispersions of the two main lenses, $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ , respectively.", "Compared to the previous estimates (see the last row in Table REF ), we remark that our assumption on the center of mass of the cluster dark-matter halo does not affect significantly the measurements of the most relevant quantities of our lensing model, i.e.", "$\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ .", "Furthermore, the estimates of the lens strength of the halo, $b_{\\mathrm {H}}$ , change from $37 \\pm 1$$\\,$ (in the previous analysis with the total mass center fixed) to $38 \\pm 2$$\\,$ (in this new analysis with the total mass center free).", "We remind that the squared value of the strength of a lens is proportional to the mass of that lens projected within its Einstein radius.", "From this consideration and from the cited estimates of $b_{\\mathrm {H}}$ , we notice that the mass measurements of the cluster dark-matter halo with and without its mass center fixed are consistent, given their uncertainties.", "In the second case, we add two galaxies at a projected distance from G1 of approximately 6$\\,$ (to the North), with luminosity and size values not larger than those of G1.", "We include these two lenses in the model, fixing their total mass center and strength values (the former to the galaxy luminosity centroids and the latter to the upper limit given by the strength value of G1), and find $97 \\pm 3$ and $249 \\pm 6$ km s$^{-1}$ , respectively, for $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ .", "As from the previous test, comparing these new values with those of Table REF , we can exclude a significant effect of the two nearest candidate cluster members on our estimates of the total mass distributions of G1 and G2.", "Given their larger projected distances, the possible influence of other cluster members is expected to be even smaller that that of the two neighboring galaxies considered above.", "From these tests, we can confirm that our measurements of $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ are robust and that the possible systematic uncertainties, due to our specific modeling assumptions, are approximately on the same order of the statistical uncertainties.", "This means that, even considering both statistical and systematic errors, the values of $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ can be measured with relative errors of less than 10% and, therefore, that the errors on these quantities are not dominating the error budget of the galaxy luminous over total mass fractions presented in Sect.", "5.", "The errors on these last quantities are in fact mainly driven by the errors on the luminous mass values, estimated in the next section from the galaxy spectral energy distribution fitting.", "We remark that our estimates of $\\sigma _{\\mathrm {G1}}$ and $\\sigma _{\\mathrm {G2}}$ are consistent, given the errors, with the values of 101 and 236 km s$^{-1}$ , respectively, obtained by [30].", "They performed a strong lensing study of this cluster using the multiple images of 13 background sources and modeling the cluster total mass distribution with a combination of an extended NFW profile and several, smaller, truncated isothermal profiles (representing the candidate cluster member mass contribution), scaled according to the Faber-Jackson relation ([31]).", "We emphasize that the adoption of scaling relations to model the total mass distributions of candidate cluster members, necessary in order to reduce the number of parameters of a cluster strong lensing model, provides interesting results on the statistical ensemble of galaxies, but these results should be interpreted very carefully if the main focus is on the study of the mass properties of individual cluster members.", "We caution that the choice of particular scaling relations can drive the results on possible variations in the amount of dark matter present in the inner regions of different cluster members.", "For this reason, tailored strong lensing models, like the one presented above, are needed.", "Furthermore, we measure the stellar velocity dispersion of G2, $\\sigma _{,\\mathrm {G2}}$ , within an aperture with diameter equal to 1.5$\\,$ , from a fit of the VLT/FORS2 spectrum shown in Figure REF .", "We use the pixel-fitting method of [22] and adopt the template stellar spectra of the MILES library ([74], [34]).", "We estimate a redshift of 0.4402, consistent with the VIMOS measurement, and a value of $\\sigma _{,\\mathrm {G2}}$ of $(250 \\pm 30)$ km s$^{-1}$ , also consistent, given the uncertainties, with our strong lensing estimate.", "We notice that a good agreement between the values of the effective velocity dispersion of an isothermal model and of the central stellar velocity dispersion is common in galaxy-scale strong lensing systems where the multiple image geometry can be reconstructed well (e.g., [89], [46]).", "By exploiting the optimized lens mass models obtained from our bootstrapping analysis, we can also estimate the statistical uncertainties on the values of the different magnification factors.", "We show the 1$\\sigma $ errors (in parentheses) in Table REF .", "The values of only a few per cent for the magnification relative errors are not surprising, because of the large number of multiple images that provide detailed information about the lens total mass distributions.", "We remark that the small errors on the total magnification factors are a minor source of uncertainty on the unlensed luminous mass of the source presented in the next section.", "Finally, assuming that an isothermal profile is a good description of the total mass distribution of the two lenses out to their effective radius (for these values in angular units, see $\\theta _{e}$ in Table REF ) and using the results of the bootstrapping analysis (see Table REF ), we measure total mass values $M_{T}$ projected within $R_{e}$ , $M_{T}(<R_{e}) = \\frac{\\pi \\sigma ^{2} R_{e}}{G} \\, ,$ (being $G$ the value of the gravitational constant) of $1.7_{-0.1}^{+0.1}\\times 10^{10}$ and $2.8_{-0.1}^{+0.2}\\times 10^{11}$ $M_{\\odot }$ for $M_{T,\\mathrm {G1}}(<2.4\\, \\mathrm {kpc})$ and $M_{T,\\mathrm {G2}}(<6.7\\, \\mathrm {kpc})$ , respectively.", "Table: Details of the composite stellar population models adopted to measure the luminous mass values of G1, G2, and A 5 +B 5 \\mathrm {A}_{5}+\\mathrm {B}_{5} (after correcting for the lensing magnification effect)." ], [ "Luminous mass estimates", "Here we model the multicolor photometry, composed of 16 HST bands, of the two lens galaxies G1 and G2 and of the sum of the multiple images $\\mathrm {A}_{5}$ and $\\mathrm {B}_{5}$ .", "We concentrate on the measurement of the luminous mass of these objects, leaving the study of the physical properties of the source to a future work.", "We use composite stellar population (CSP) models based on [20] templates at solar metallicity and with a [73] stellar IMF.", "We consider constant and delayed exponential (with a possible cut) Star Formation Histories (SFHs).", "We allow for the presence of dust, according to [21], and take into account the flux contribution of emission lines.", "For the two early-type galaxies and high-redshift source we choose, respectively, truncated delayed exponential and constant SFHs, which we believe are the most suitable SFHs for these classes of objects.", "We summarize our modeling prescriptions and final results in Table REF .", "The best-fitting models for the two lens cluster members G1 and G2 are shown in Figure REF and for the lensed objects $A_{5}$ and $B_{5}$ in Figure REF .", "We have decided to exclude from the fitting and plots the 4 bluest bands because mostly affected by relevant contamination from very close objects.", "We have checked that removing these bands from the SED fitting does not change appreciably the results on the values of the luminous masses.", "In fact, photometric mass estimates are known to be more sensitive to the fluxes measured in the redder filters (e.g., [44]).", "The best-fitting values of the luminous masses of G1, G2, and $\\mathrm {A}_{5}+\\mathrm {B}_{5}$ ($M_{L,G1}$ , $M_{L,G2}$ , and $M_{L,\\mathrm {A}_{5}+\\mathrm {B}_{5}}$ ), are, respectively, $1.7 \\times 10^{10}$ , $4.5 \\times 10^{11}$ , and $8.8 \\times 10^{9}$ $M_{\\odot }$ .", "From the ranges of results obtained by considering the different photometric uncertainties, systematic errors associated to the several possible stellar population modeling assumptions (i.e., SFH, dust, emission lines), and rest-frame wavelength range covered by the HST observations, we estimate relative errors of 40% on $M_{L,G1}$ and $M_{L,G2}$ and of 50% on $M_{L,\\mathrm {A}_{5}+\\mathrm {B}_{5}}$ .", "Furthermore, taking into account the value of the average magnification factor of approximately 9 at the positions where $\\mathrm {A}_{5}$ and $\\mathrm {B}_{5}$ are observed (see Figure REF and Table REF ), we conclude that the measured luminous mass values and errors are $(1.7 \\pm 0.7)\\times 10^{10}$ for G1, $(4.5 \\pm 1.8)\\times 10^{11}$ for G2, and $(1.0 \\pm 0.5)\\times 10^{9}$ $M_{\\odot }$ for $\\mathrm {A}_{5}+\\mathrm {B}_{5}$ .", "We remark that to avoid possible artifacts in our following investigation we have explicitly omitted the recent results (e.g., [5]; [86]; [23]; [9]) suggesting systematic variations in the stellar IMF of a galaxy as a function of its luminous mass or stellar velocity dispersion.", "Taking these results into account would probably result in luminous mass estimates approximately 2 times smaller for G1 and $\\mathrm {A}_{5}+\\mathrm {B}_{5}$ ." ], [ "Discussion", "In this section we compare the values of the luminous mass, effective velocity dispersion, and luminous over total mass fraction projected within the effective radius of G1 and G2 with those of three samples of SLACS lens galaxies, massive early-type galaxies from the SDSS, and dwarf spheroidals.", "First, starting from the total and luminous mass estimates derived in the previous two sections, we measure for G1 and G2 the values of the fraction of luminous over total mass projected inside the effective radius, $f_{L}(<R_{e})$ , in the following way: $f_{L}(<R_{e}) := \\frac{M_{L}/2}{M_{T}(<R_{e})} \\, .$ We obtain $0.51 \\pm 0.21$ and $0.80 \\pm 0.32$ for G1 and G2, respectively.", "Then, we consider early-type galaxies with physical properties similar to those of the lenses selected by the SLACS survey.", "For the SLACS galaxies, we use the luminous mass estimates by [44], that are obtained by fitting the galaxy SEDs with CSP models built on [20] templates at solar metallicity and with a Salpeter stellar IMF, and the effective velocity dispersion measurements $\\sigma _{\\mathrm {SIE}}$ , presented in [87], that are derived by modeling the total mass distribution of the lenses with SIE profiles.", "Several studies have shown that the SLACS lens galaxies are an unbiased subsample of the family of SDSS massive early-type galaxies (e.g., [16]; [38]; [4]), as far as their luminous and mass properties are concerned.", "For this reason, we also use here the results on the luminous over total mass fractions of approximately 2$\\times 10^{5}$ SDSS early-type galaxies selected by [38].", "In this last study, the values of the galaxy luminous and total mass were measured under the same hypotheses adopted in this work.", "We show the results in Table REF and Figures REF and REF .", "Looking at Table REF and Figure REF , we notice that G2 has values of $M_{L}$ and $\\sigma _{\\mathrm {SIE}}$ that are consistent with those of the galaxies in the SLACS sample.", "Interestingly, G1 has values of luminous mass and effective stellar velocity dispersion lower by approximately factors of 30 and 3, respectively, than the average SLACS lens galaxy, but these values are in good agreement with the extrapolation of the scaling relation based on the SLACS galaxies only.", "From the same Table and Figure REF , we observe that G2 has a value of luminous over total mass fraction that is typical of SDSS massive early-type galaxies.", "The value of $f_{L}(<R_{e})$ of G1 instead is smaller, but still consistent with the lower end of the distribution of the SDSS sample.", "Considering the example provided by G1, we can conclude that in clusters of galaxies it is possible to study galaxy strong lensing on physical scales that are different from (i.e., smaller than) those characterizing isolated early-type galaxies (e.g., the SLACS lenses).", "This possibility is offered by the increase with the overdensity of the environment in the probability of one source to be strongly lensed by a (small) galaxy.", "In different words, the lensing cross section of a single (small) galaxy can be significantly enhanced by the presence of the mass distributions primarily on the cluster scale and secondarily on the scale of the neighboring cluster members.", "This opens the way to studies on the internal structure of lens galaxies over a more extended range of physical properties than done so far.", "We remark that several studies (e.g., [48]; [60]; [30]) have found evidence for the truncation of the total mass profiles of early-type galaxies residing in galaxy clusters.", "Nonetheless, for a given galaxy, the value of its truncation radius is estimated to be significantly larger that that of its effective radius.", "This allows us to disregard, to a first approximation, the possible differences in the values of the stellar over total mass ratios projected within the effective radii of cluster and field early-type galaxies because of their different truncation radii.", "Moreover, we observe that the results of [89] and [43] have shown that the SLACS (i.e., mainly field) and Coma (i.e., cluster) galaxies, with comparable stellar masses, do not differ appreciably as far as their inner total mass structure and stellar IMF are concerned.", "For these reasons, we consider appropriate to plot the values of the central luminous over total mass ratios of cluster and field galaxies in the same plots, as done in Figures REF and REF .", "Figure: Same quantities and symbols plotted in Figures and over ranges of values that include also 11 dwarf spheroidals (grey triangles).Following the previous results, we expand the intervals of physical scales plotted in Figures REF and REF to include a sample of 11 pressure-supported dwarf spheroidals (dSph) for which all the relevant quantities are available in the literature.", "We take the values of effective radius and luminous mass from [62] and those of stellar velocity dispersion averaged along the line of sight $\\sigma _{*}$ from [93].", "Simplistically, we decide to use the same stellar IMF (i.e., Salpeter) adopted to estimate the luminous mass of the previous galaxies and the expression given in Eq.", "(REF ) to measure the total mass projected within the effective radius.", "The results are plotted in Fig.", "REF .", "The observed values of luminous mass and velocity dispersion of the dwarf spheroidals do not differ dramatically from the expected values obtained by extrapolating the SLACS scaling relation at several orders of magnitude difference.", "The projected fractions of luminous over total mass inside the effective radius show instead a clear variation from centrally luminous to dark-matter-dominated systems, moving from massive early-type galaxies to dwarf spheroidals.", "We speculate that the similarities and differences between these two classes of astrophysical objects might be explored effectively by extending strong lensing analyses to lenses with diverse physical scales, as started here with G1.", "The CLASH survey seems to be particularly well suited to this aim, as several other interesting systems of strong lensing on galaxy scale have already been discovered and are currently under investigation." ], [ "Conclusions", "The combination of unprecedented HST multi-wavelength observations and VLT spectra has allowed us to perform a detailed strong lensing and stellar population analysis of an unusual system composed in total of ten multiple images of a double source, lensed by two early-type galaxies in the field of the CLASH galaxy cluster MACS 1206.", "Our main results can be summarized in the following points.", "$\\bullet $ Based on our 16-band photometry and low-resolution spectroscopy, we measure a photometric redshift of 3.7 for the source and spectroscopic redshifts of 0.436 and 0.439 for the two lens galaxies G1 and G2, respectively, thus confirming their membership to MACS 1206.", "$\\bullet $ By modeling the total mass distribution of the cluster members and cluster in terms of singular isothermal profiles, we can reconstruct well the observed positions of the multiple images and predict a total magnification factor of approximately 50 for the source.", "$\\bullet $ From the lensing modeling statistics, we estimate effective velocity dispersion values of $97 \\pm 3$ and $240 \\pm 6$ km s$^{-1}$ , corresponding to total mass values projected within the effective radii of $1.7_{-0.1}^{+0.1}\\times 10^{10}$ and $2.8_{-0.1}^{+0.2}\\times 10^{11}$ $M_{\\odot }$ for G1 and G2, respectively.", "Moreover, we obtain reasonable values for the distribution and amount of projected total mass in the galaxy cluster component.", "$\\bullet $ Through composite stellar populations synthesis models (adopting a Salpeter stellar IMF), we infer luminous mass values of $(1.7 \\pm 0.7)\\times 10^{10}$ and $(4.5 \\pm 1.8)\\times 10^{11}$ $M_{\\odot }$ for, respectively, G1 and G2, and $(1.0 \\pm 0.5)\\times 10^{9}$ $M_{\\odot }$ for the source, taking into account the estimated lensing magnification factor.", "$\\bullet $ In G1 and G2, respectively, we derive luminous over total mass fractions of $0.51 \\pm 0.21$ and $0.80 \\pm 0.32$ .", "We compare these values with those typical of massive early-type galaxies and dwarf spheroidals and conclude that more analyses in the CLASH fields of systems similar to that presented here will enable us to extend the investigation of the internal structure of galaxies in an important and still relatively unexplored region of the physical parameter space.", "The CLASH Multi-Cycle Treasury Program is based on observations made with the NASA/ESA Hubble Space Telescope.", "The Space Telescope Science Institute is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.", "ACS was developed under NASA Contract NAS 5-32864.", "This research is supported in part by NASA Grant HST-GO-12065.01-A.", "We thank ESO for the continuous support of the Large Programme 186.A-0798.", "The Dark Cosmology Centre is funded by the DNRF.", "We acknowledge partial support by the DFG Cluster of Excellence Origin Structure of the Universe.", "V.P.", "acknowledges the grant PRIN INAF 2010 and “Cofinanziamento di Ateneo 2010”.", "The work of L.A.M.", "was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.", "Support for A.Z.", "is provided by NASA through Hubble Fellowship grant HST-HF-51334.01-A awarded by STScI.", "Part of this work was also supported by contract research “Internationale Spitzenforschung II/2-6” of the Baden Württemberg Stiftung." ] ]	1403.0573
[ [ "A Back-to-Basics Empirical Study of Priority Queues" ], [ "Abstract The theory community has proposed several new heap variants in the recent past which have remained largely untested experimentally.", "We take the field back to the drawing board, with straightforward implementations of both classic and novel structures using only standard, well-known optimizations.", "We study the behavior of each structure on a variety of inputs, including artificial workloads, workloads generated by running algorithms on real map data, and workloads from a discrete event simulator used in recent systems networking research.", "We provide observations about which characteristics are most correlated to performance.", "For example, we find that the L1 cache miss rate appears to be strongly correlated with wallclock time.", "We also provide observations about how the input sequence affects the relative performance of the different heap variants.", "For example, we show (both theoretically and in practice) that certain random insertion-deletion sequences are degenerate and can lead to misleading results.", "Overall, our findings suggest that while the conventional wisdom holds in some cases, it is sorely mistaken in others." ], [ "Introduction", "The priority queue is a widely used abstract data structure.", "Many theoretical variants and implementations exist, supporting a varied set of operations with differing guarantees.", "We restrict our attention to the following base set of commonly used operations: [leftmargin=*] $\\operatorname{\\textsc {Insert}}\\left({Q,x,k}\\right)$ — insert item $x$ with key $k$ into heap $Q$ and return a handle $\\bar{x}$ $\\operatorname{\\textsc {DeleteMin}}\\left({Q}\\right)$ — remove the item of minimum key from heap $Q$ and return its corresponding key $k$ $\\operatorname{\\textsc {DecreaseKey}}\\left({Q,\\bar{x},k^{\\prime }}\\right)$ — given a handle $\\bar{x}$ , change the key of item $x$ belonging to heap $Q$ to be $k^{\\prime }$ , where $k^{\\prime }$ is guaranteed to be less than the original key $k$ It has long been known that either $\\operatorname{\\textsc {Insert}}$ or $\\operatorname{\\textsc {DeleteMin}}$ must take ${\\log n}$ time due to the classic lower bound for sorting [24], but that the other operations can be done in 1 time.", "In practice, the worst-case of $\\log n$ is often not encountered or can be treated as a constant, and for this reason simpler structures with logarithmic bounds have traditionally been favored over more complicated, constant-time alternatives.", "In light of recent developments in the theory community [12], [11], [19], [5], [2] and the outdated nature of the most widely cited experimental studies on priority queues [26], [31], [25], we aim to revisit this area and reevaluate the state of the art.", "More recent studies [3], [13], [11] have been narrow in focus with respect to the implementations considered (e.g., comparing a single new heap to a few classical ones), the workloads tested (e.g., using a few synthetic tests), or the metrics collected (e.g., measuring wallclock time and element comparisons).", "In addition to the normal metric of wallclock time, we have collected additional metrics such as branching and caching statistics.", "Our goal is to identify experimentally verified trends which can provide guidance to future experimentalists and theorists alike.", "We stress that this is not the final word on the subject, but merely another line in the continuing dialogue.", "In implementing the various heap structures, we take a different approach from the existing algorithm engineering literature, in that we do not perform any algorithm engineering.", "That is, our implementations are intentionally straightforward from their respective descriptions in the original papers.", "The lack of considerable tweaking and algorithm engineering in this study is, we believe, an example of naïveté as a virtue.", "We expect that this would accurately reflect the strategy of a practitioner seeking to make initial comparisons between different heap variants.", "As a sanity check, we also compare our implementations with a state-of-the-art, well-engineered implementation often cited in the literature.", "Our high-level findings can be summarized as follows.", "We find that wallclock time is highly correlated with the cache miss rate, especially in the L1 cache.", "High-level theoretical design decisions—such as whether to use an array-based structure or a pointer-based one—have a significant impact on caching, and which design fares best is dependent on the specific workload.", "For example, Fibonacci heaps sometimes outperform implicit $d$ -ary heaps, in contradiction to conventional wisdom.", "Even a well-engineered implementation like Sanders' sequence heap [29] can be bested by our untuned implementations if the workload favors a different method.", "Beyond caching behavior, those heaps with the simplest implementations tend to perform very well.", "It is not always the case that a theoretically superior or simpler structure lends itself to simpler code in practice.", "Pairing heaps dominate Fibonacci heaps across the board, but interestingly, recent theoretical simplifications to Fibonacci heaps tend to do worse than the original structure.", "Furthermore we found that a widely-used benchmarking workload is degenerate in a certain sense.", "As the sequence of operations progresses, the distribution of keys in the heap becomes very skewed towards large keys, contradicting the premise that the heap contains a uniform distribution of keys.", "This can be shown both theoretically and in practice.", "Our complete results are detailed in Sections and .", "We first describe the heap variants we implemented in Section , and then discuss our experimental methodology and the various workloads we tested in Section .", "We conclude in Section with some remarks." ], [ "Heap Variants", "Aiming to be broad, but not necessarily comprehensive, this study includes both traditional heap variants and new variants which have not previously undergone much experimental scrutiny.", "We have implemented the following structures, listed here in order of program length: implicit $d$ -ary heaps, pairing heaps, Fibonacci heaps, binomial queues, explicit $d$ -ary heaps, rank-pairing heaps, quake heaps, violation heaps, rank-relaxed weak queues, and strict Fibonacci heaps.", "Table REF lists the logical lines of code; in our experience, this order corresponded exactly to perceived programming difficulty.", "There are several other heap variants which may be worth investigating, but which have not been included in this study.", "Among those not included are the 2-3 heap [32], thin/thick heaps [23], and the buffer heap [6].", "Williams' binary heap [34] is the textbook example of a priority queue.", "Lauded for its simplicity and taught in undergraduate computer science courses across the world, it is likely the most widely used variant today.", "Storing a complete binary tree whose nodes obey the heap order gives a very rigid structure; indeed, the heap supports all operations in worst-case ${\\log n}$ time.", "The tree can be stored explicitly using heap-allocated nodes and pointers, or it can be encoded implicitly as a level-order traversal in an array.", "We refer to these variations as explicit and implicit heaps respectively.", "The implicit heap carries a small caveat, such that in order to support $\\operatorname{\\textsc {DecreaseKey}}$ efficiently, we must rely on a level of indirection: encoding the tree's structure as an array of node pointers and storing the current index of a node's pointer in the node itself (allowing us to return the node pointer to the client as $\\bar{x}$ ).", "This study includes two versions of implicit heaps—one that supports $\\operatorname{\\textsc {DecreaseKey}}$ through this indirection, and one that doesn't.", "We refer to the latter as the implicit-simple heap.", "Explicit and implicit heaps can be generalized beyond the binary case to have any fixed branching factor $d$ .", "We refer to these heaps collectively as $d$ -ary heaps; this study examines the cases where $d=2,4,8,16$ .", "To distinguish between versions with different branching factors, we label the heaps in this fashion: implicit-2, explicit-4, implicit-simple-16, and so forth.", "Table: Programming effortBeyond the $d$ -ary heaps, all other heap variants are primarily pointer-based structures, though some make use of small auxiliary arrays.", "All are conceptual successors to Vuillemin's binomial queue [33].", "Originally developed to support efficient melding (which takes linear time in $d$ -ary heaps), we have included it in our study due to its simplicity.", "The binomial queue stores a forest of perfect, heap-ordered binomial trees of unique rank.", "This uniqueness is maintained by linking trees of equal rank such that the root with lesser key becomes the new parent.", "To support deletion of a node, each of its children is made into a new root, and the resulting forest is then processed to restore the unique-rank invariant.", "This can lead to a fair amount of structural rearrangement, but the code to do so is rather simple.", "Key decreases are handled as in $d$ -ary heaps by sifting upwards.", "Like $d$ -ary heaps, binomial queues support all operations in worst-case ${\\log n}$ time.", "Most other heap variants can be viewed as some sort of relaxation of the binomial queue, with the chronologically first one being the Fibonacci heap [17].", "The Fibonacci heap achieves amortized 1-time $\\operatorname{\\textsc {Insert}}$ and $\\operatorname{\\textsc {DecreaseKey}}$ by only linking after deletions and allowing some imperfections in the binomial trees.", "The imperfections are generated by key decreases: instead of sifting, a node is cut from its parent and made into a new root.", "To prevent the trees from becoming too malformed, a node is also cut from its parent as soon as it loses a second child.", "This can lead to a series of upwardly cascading cuts.", "The violation heap [12] and rank-pairing heaps [19] can be viewed as further relaxations of the Fibonacci heap.", "The rank-pairing heaps allow rank differences greater than one, and propagate ranks instead of cascading cuts so that at most one cut is made per $\\operatorname{\\textsc {DecreaseKey}}$ .", "Two rank rules were proposed by the authors, leading to our implementations being labeled rank-pairing-t1 and rank-pairing-t2.", "The violation heap also propagates ranks instead, only considering rank differences in the two most significant children of a node.", "It allows two trees of each rank and utilizes a three-way linking method.", "The pairing heap [16] is essentially a self-adjusting, single-tree version of the Fibonacci heap, where ranks are not stored explicitly, and linking is done eagerly.", "Its amortized complexity is still an open question, though it has been shown that $\\operatorname{\\textsc {DecreaseKey}}$ requires ${\\log \\log n}$ time if all other operations are ${\\log n}$ [15].", "Two different amortization arguments can be used to prove either 1 and ${\\log n}$ bounds for $\\operatorname{\\textsc {Insert}}$ and $\\operatorname{\\textsc {DecreaseKey}}$ respectively [21] or ${2^{2\\sqrt{\\log \\log n}}}$ for both operations [27].", "It remains an open question to prove an ${\\log n}$ bound for $\\operatorname{\\textsc {DecreaseKey}}$ simultaneously with 1-time $\\operatorname{\\textsc {Insert}}$ .", "All three relaxations are intended to be in some way simpler than Fibonacci heaps, with the hope that this makes them faster in practice.", "The strict Fibonacci heap [2] on the other hand, intends to match the Fibonacci time bounds in the worst case, rather than in an amortized sense.", "This leads to a fair amount of extra code to manage structural imperfections somewhat lazily.", "Rank-relaxed weak queues [11] are essentially a tweaked version of rank-relaxed heaps, with an emphasis on minimizing key comparisons.", "They mark nodes as potentially violating after a $\\operatorname{\\textsc {DecreaseKey}}$ operation and clean them up lazily.", "Quake heaps [5] are a departure from the Fibonacci model, but are still vaguely reminiscent of binomial queues.", "A forest of uniquely-ranked tournament trees is maintained.", "Subtrees may be missing, but the number of nodes at a given height decays exponentially in the height, a property guaranteed through a set of global counters and a global rebuilding process triggered after deletions.", "There are multiple implementation strategies mentioned in the original paper, but only the one that was fully detailed (the full tournament representation) has been implemented here.", "It is possible that the other implementations would be more efficient." ], [ "Experimental Design and Workloads", "Our codebase is written primarily in C99 and is available online for inspection, modification, and further development [1].", "As we stated earlier, our implementations are intentionally straightforward from their respective descriptions in the original papers, or use only the most basic, well-known optimizations for the more studied structures.", "Further optimization is left to the compiler (gcc -O4) so as not to unfairly bias toward one variant or another.", "Keys are 64-bit unsigned integers, while the items themselves are 32-bit unsigned integers.", "In most cases, the key actually consists of a 32-bit key in the high-order bits and the item identifier in the low-order bits, in order to break ties during comparisons.", "We experimented with different memory allocation schemes using our own simple fixed-size memory pool implementation.", "This abstraction layer allowed us to allocate all memory eagerly using a single malloc, allocate lazily by doubling when space fills, or allocate completely on the fly using a malloc for each $\\operatorname{\\textsc {Insert}}$ .", "In our experiments, the memory allocation scheme made very little difference regardless of heap variant, indicating that this layer of optimization was superfluous.", "Thus, all the results in this paper use the eager strategy.", "The workloads we tested are described in the subsections below.", "These include workloads generated by code sourced (with modifications) from DIMACS implementation challenges [9], [10], as well as workloads generated by a packet-level network simulator [7].", "All experiments use trace-based simulation.", "More specifically, a workload is generated once using a reference heap and the sequence of operations and values is recorded in a trace file.", "This trace file can then be executed against each of many drivers—one for each heap variant included in the study—as well as a dummy driver that simply parses the trace file but does not execute any heap operations.", "The dummy driver captures the overhead of the simulation and its collected metrics are subtracted from those of the other drivers before any comparisons are done.", "Wallclock time is measured by the driver itself.", "For purposes of timing, each execution of a trace file is run for a minimum of five iterations and two seconds of wallclock time (whichever takes longer), and the time is averaged over all iterations.", "Other metrics are collected over the course of a single iteration using cachegrind [4], a cache and branch-prediction profiler.", "The profiler simulates actual machine parameters and does not vary between executions, providing accurate measurements that are isolated from other system processes.", "We have used it to collect dynamic instruction and branching counts as well as reads, writes and misses for both the L1 and L2 caches.", "Additionally, cachegrind allows for simulating branch prediction in a basic model (that does not correspond exactly to the real machines); we have collected this misprediction count as well.", "All experiments were run on a high-performance computing cluster in Princeton consisting of Dell PowerEdge SC1435 nodes with dual AMD Opteron 2212 processors (dual-core, 2.0GHz, 64KB L1 cache and 1MB L2 cache per core) and 8GB of RAM (DDR2-667).", "The machines ran Springdale/PUIAS Linux (a Red-Hat Enterprise clone) with kernel version 2.6.32.", "All executions remained in-core." ], [ "Artificial randomized workloads.", "The first workload we consider is a standard and ubiquitous one: sorting sequences of $n$ uniformly random integers.", "This translates to $n$ random insertions followed by $n$ minimum deletions in the trace files.", "The next type of sequence intermixes insertions and deletions, but in a very structured way which turns out to be degenerate.", "It is a very natural sequence to test, and due to its presence in the DIMACS test set, we worry that its use in benchmarks may be more widespread than one might hope of a broken test.", "The sequence begins with $n$ random insertions as in the sorting case.", "It is then followed by $cn$ repetitions of the following: one random insertion followed by one minimum deletion.", "It is not hard to show that the evolving distribution of keys remaining in the heap is far from uniform.", "Lemma 3.1 After the initial $n$ insertions and $cn$ iterations of insert-delete, the items remaining in the heap consist of the $n$ largest keys inserted thus far.", "The next item inserted has roughly a ${c}{c+1}$ probability of becoming the new minimum.", "For the purpose of this analysis, we consider the inserted keys to be reals distributed uniformly at random in the range $\\left[{0,1}\\right]$ , rather than 32-bit integers.", "The pattern of operations leaves the $n$ largest keys inserted thus far in the heap, as the following simple inductive argument shows.", "Initially, $n$ keys are inserted; being the only keys thus far, they are trivially the largest.", "Then, each iteration consists of a single insertion followed by a minimum deletion.", "Since there are $n+1$ keys in the heap after the insertion, and the minimum is deleted, the remaining $n$ keys are the largest thus far.", "We can view the random variables of all keys inserted thus far to be the collection $X_1, \\dots , X_{\\left({c+1}\\right)n}$ , and the current minimum in the heap to be the $\\left({cn + 1}\\right)^{th}$ order statistic, $X_{\\left({cn+1}\\right)}$ .", "The expectation of this variable is well-known: ${}{X_{\\left({cn+1}\\right)}} ={cn+1}{\\left({c+1}\\right)n} \\approx {c}{c+1}$ .", "From this we deduce that the probability $p$ of the next inserted key becoming the new minimum is roughly ${c}{c+1}$ .", "As $c$ grows, the most recent insertion becomes exceedingly likely to be the next deleted item.", "In other words, the behavior of the queue becomes increasingly stack-like as the sequence lengthens.", "On the other hand, if we introduce $\\operatorname{\\textsc {DecreaseKey}}$ operations to the sequence, we can ameliorate the degeneracy.", "This brings us to our third type of artificial sequence.", "We again build an initial heap of size $n$ with random insertions.", "We then perform $cn$ repetitions of the following: one random insertion, $k$ key decreases on random nodes, and one minimum deletion.", "We also consider two cases for the $k$ key decreases.", "In the first, we decrease the key to some random number between its current value and the minimum.", "In the second, we decrease it so that it becomes the new minimum.", "We refer to these options as “middle” and “min”, respectively.", "In both the insertion-deletion workloads and the key-decrease workloads we consider $c \\in \\left\\lbrace {1,32,1024}\\right\\rbrace $ , while in the key-decrease workloads we also consider $k \\in \\left\\lbrace {1,32,1024}\\right\\rbrace $ ." ], [ "More realistic workloads.", "Of our remaining workloads, some are still artificial in the sense that they are generated by running real algorithms on artificial inputs, but others make use of real inputs.", "The first two of these are Dijkstra's algorithm for single source shortest paths and the Nagamochi-Ibaraki algorithm for the min-cut problem.", "We run both algorithms against well-structured or randomly generated graphs.", "Dijkstra's algorithm in particular is run on several classes of graphs, including some which guarantee a $\\operatorname{\\textsc {DecreaseKey}}$ operation for each edge.", "Additionally, we run Dijkstra's algorithm on real road networks of different portions of the United States.", "Our final set of trace files is generated from the htsim packet-level network simulator [7], written by the authors of the multipath TCP (MPTCP) protocol.", "The simulator models arbitrary networks using pipes (that add delays) and queues (with fixed processing capacity and finite buffers), and implements both TCP and MPTCP.", "One of these workloads is based on real traffic traces from the VL2 network [18]." ], [ "Results", "The results reveal a more nuanced truth than that which has been traditionally accepted.", "It is not true that implicit-4 heaps are optimal for all workloads, nor is it true that Fibonacci heaps are always exceptionally slow.", "We focus on the most interesting cases here, and include the remaining results in the appendix.", "We present most of our data in tables sorted in ascending order of wallclock time.", "Each table is for a single, large input file.", "The tables represent raw metrics divided by the minimum value attained by any heap, such that a highlighted value of 1.00 is the minimum, while a value $c$ is $c$ times the minimum.", "These ratios make it easier to interpret relative performance instead of the full counts.", "In order to keep the tables compact, the column titles have been abbreviated: time is wallclock time, inst is the dynamic instruction count, l1_rd and l1_wr are the number of L1 reads and writes respectively, l2_rd and l2_wr are the L2 reads and writes respectively, br is the number of dynamic branches, and l1_m, l2_m and br_m are the number of L1 misses, L2 misses, and branch mispredictions.", "Figure: Dijkstra on the full USA road map.", "All operation counts are scaled bylogn\\log n.Figure: Dijkstra on the full USA road map.", "The error\\operatorname{\\textsc {DeleteMin}} count is scaledby logn\\log n.We initially ran each experiment on many problem sizes.", "We found that in most cases the relative performance stabilized very quickly, so from here on we only present data for the largest problem size.", "See Figures 1 and 2 for some evidence of this stabilization.", "The heaps are separated into two classes so as to unclutter the plots and give a consistent axis.", "The operation counts are the sum of the counts of $\\operatorname{\\textsc {Insert}}$ , $\\operatorname{\\textsc {DecreaseKey}}$ , and $\\operatorname{\\textsc {DeleteMin}}$ operations.", "In Figure 1, all operation counts are scaled by $\\log n$ , where $n$ is the average size of the heap.", "In Figure 2, only the $\\operatorname{\\textsc {DeleteMin}}$ count is scaled by $\\log n$ .", "This scaling approximately reflects the amortized bounds for each heap.", "Before diving into the results, we first make a high-level observation.", "The number of L1 cache misses appears to be the metric most strongly correlated with wallclock time.", "It is not a perfect predictor, and inversions in ordering certainly exist.", "Some of these inversions can be explained by L2 cache misses, write counts, or branch misprediction.", "Others appear to be outliers or are otherwise yet unexplained.", "Table: SortingTable: Dijkstra – full USA road map" ], [ "We first examine two cases where the conventional wisdom holds.", "As seen in Table REF , the implicit-simple heaps handle sorting workloads very well.", "The best performance is achieved by the implicit-simple-4 heap.", "The Fibonacci heap is almost seven times as slow as the fastest, which does indeed echo old complaints about its speed.", "The pairing heap and binomial queue fare better here, but still poorly at at least four times as slow as the fastest.", "Without any key decreases, the rank-relaxed weak queue is essentially just an alternate implementation of a binomial queue, so it is not terribly surprising that it does better than the Fibonacci heap.", "Similarly with Dijkstra's algorithm on the full USA road map (Table REF ), we see implicit-4 heaps performing quite well, while Fibonacci heaps are roughly three times as slow.", "The explicit heaps are noticeably slower even than Fibonacci heaps, and the only Fibonacci relaxation to perform well here is the pairing heap.", "The others are in fact slower than their conceptual ancestor.", "Although they exhibit similar caching behavior, their code is somewhat more complicated, which may be contributing to the slowdown.", "Both of the above workloads are very well-studied, and as such the relative performance of the older heap variants should not be very surprising.", "Table: Randomized error\\operatorname{\\textsc {Insert}}–error\\operatorname{\\textsc {DeleteMin}} (Degenerate) – c=1024c=1024" ], [ "We now turn to our randomized insertion-deletion workload.", "The results here are more surprising.", "Recall from Lemma REF that this workload is degenerate, in that as the sequence goes on, the most recently inserted item is very likely to be the next item deleted.", "Nevertheless, this sequence is commonly used in empirical studies.", "The shortest sequence we tested, $c=1$ (Table REF ), remains rather close to the sorting workload.", "On the other hand, when the sequence is very long ($c=1024$ ), as shown in Table REF , we see a very different picture.", "The queue-based structures outperform the implicit heaps by a factor of at least two.", "Under these assumptions about the distribution, an $\\operatorname{\\textsc {Insert}}$ operation in a $d$ -ary heap results in the node being sifted all the way to the top, and the subsequent $\\operatorname{\\textsc {DeleteMin}}$ on average results in another long sifting sequence.", "In a queue structure with lazy insertion, the $\\operatorname{\\textsc {Insert}}$ commonly results in a singleton node which is simply removed afterwards with little to no restructuring.", "Although degenerate in the above case, a generalization of this sequence becomes a natural sequence for which efficient structures have been designed.", "Consider workloads which frequently insert new items near the minimum rather than toward the bottom of the heap.", "Let $r({x})$ denote the rank of $x$ among the items in the heap, such that the rank of the minimum is 1 and the maximum is $n$ .", "Similarly let $m({x})$ be the maximum value of $r({x})$ over the lifetime of $x$ in the heap.", "Then there are structures which are optimized for both the case of frequently deleting small-rank items and the case of frequently deleting large-rank items.", "The fishspear data structure achieves an ${\\log m({x})}$ bound for deletion, while rank-sensitive priority queues achieve an ${\\log \\left({{n}{r({x})}}\\right)}$ bound [14], [8].", "Additonally, pairing heaps have been shown to support $\\operatorname{\\textsc {DeleteMin}}$ in ${\\log k}$ time where $k$ is the number of heap operations since the minimum item was inserted [21].", "The event simulation literature, largely orthogonal to the theory literature, includes more sophisticated random models for generating insertion-deletion workloads.", "One in particular to note is the so-called “classic hold” model which is essentially the same as the degenerate model, except that instead of inserting a completely random key in each iteration, the new key is equal to the most recently deleted key plus a positive random value.", "This avoids the degeneracy.", "This and other models were explored in a previous experimental study [28].", "That study also considers several special-case priority queues with poor theoretical bounds (e.g., $\\omega (\\log n)$ ) which nonetheless perform quite well for event simulation workloads.", "Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1c=1, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1c=1, k=1k=1" ], [ "As noted in our discussion of the workloads, adding even a single key decrease per iteration to the random sequences lessens the degeneracy.", "Furthermore in Table REF we see that if the key decreases do not always generate a new minimum, as would be the case in many applications such as graph search, then implicit heaps with large branching factors continue to perform well.", "When the key decreases always produce new minima, the amortized structures come out ahead, while worst-case structures (implicit heaps and binomial queues included) fare poorly, as shown in Table REF .", "As these sequences get longer, e.g.", "$c=1024$ and $k=1$ (Table REF ), the Fibonacci relaxations gain ground, with rank-pairing-t1 heaps surpassing Fibonacci heaps.", "We note that the change in performance coincides with a large gap in L2 cache misses, and is likely due to the long sifting process in $d$ -ary heaps.", "Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1c=1, k=1024k=1024If we increase the density of key decreases in the sequence, then we see something strange (Table REF ).", "Suddenly the $d$ -ary heaps are doing well, and in particular the explicit heaps outperform the implicit ones.", "One possible explanation for this is that the level of indirection in the implicit heap implementations requires them to not only touch the same allocated nodes that the explicit heaps touch, but also to jump around in the structural array while doing path traversals.", "Noting that implicit and explicit heaps have a similar number of L1 misses, this is one of the few other workloads for which L2 behavior is a better performance predictor.", "As to why the $d$ -ary heaps outperform the amortized structures, consider the overall pattern.", "If many nodes have their keys decreased in a pairing heap, for instance, then their subtrees are simply reattached underneath the root.", "Each node access is likely to trigger a cache miss, as there will be little revisiting of nodes other than the root, and the subsequent minimum deletion will have to examine each of these nodes again in order to restructure the tree.", "On the other hand, in the $d$ -ary heaps, all the sifting is along ancestral paths which share many nodes between operations, and hence the caching effects are more favorable." ], [ "Among the other workloads we tested, those that generate relatively small heap sizes (e.g.", "Tables REF , REF ) favor the implicit heaps, while those operating on larger heap sizes (e.g.", "Tables REF , REF , REF , REF , REF ) favor amortized structures, especially the pairing heap.", "Very dense key-decrease workloads, including some “bad” inputs for Dijkstra's algorithm (Tables REF and REF ) and the Nagamochi-Ibaraki workloads, (Table REF ) favor implicit and explicit heaps.", "The network simulation workloads, (Tables REF –REF ) which produce relatively small heap sizes and have no key decreases, all favor implicit-simple heaps." ], [ "Sanity Checks", "We performed a few auxiliary experiments to verify our findings in the previous section." ], [ "Testing the caching hypothesis.", "In order to lend some credence to our claim that caching is the primary predictor of performance in many of these test cases, we ran a few additional tests, tweaking the parameters of our implementations.", "We added an extra padding field to the node in our pairing heap and implicit-4 heap implementations.", "The extra field does not generate additional instructions in the code other than in the original memory allocation process (not included in the timing procedures) and thus the only change should be in the memory address allocated to the nodes.", "This can affect both caching and branch prediction.", "Through repeated doubling of node size, we find that even though the dynamic instruction count does not grow, the wall-clock time does—in fact, it grows roughly in proportion to the cache miss rate.", "A less-pronounced effect also accompanies the growth of the misprediction rate.", "Table: Tweaking node size to test caching effects.One potentially interesting observation from these experiments is this: the instruction patterns of pairing heaps is write-first, while that of implicit heaps is read-first.", "By this we mean that typically, whenever an implicit heap touches a node, it does so first via a read, while a pairing heap quite often simply overwrites data in the node without reading it.", "This means that the cache behavior for pairing heaps is skewed toward write misses, while implicit heaps are skewed toward read misses.", "Table REF shows the read and write miss rates for both heaps." ], [ "Comparison to an existing implementation.", "We ran a few experiments against Sanders' implementation of the sequence heap [29], which has a reputation of being hard to beat in practice.", "This gives us an easy way to benchmark our own untuned implementations to see how they compare against a well-engineered one.", "The results were encouraging.", "Of the four workloads we tested, the sequence heap was faster than any of our implementations on two of them, while it was slower on the other two.", "More specifically, the sequence heap was $1.97$ times faster than the implicit-simple-4 heap on the sorting workload, and a significant $3.69$ times faster than the pairing heap on the randomized insertion-deletion workload with $c=32$ .", "Our pairing heap implementation performed $1.36$ times faster than the sequence heap on the insertion-deletion workload with $c=1024$ , and the implicit-simple-2 heap was $1.15$ times faster on one of the network simulator workloads." ], [ "Remarks", "As declared in the introduction, this is by no means a final study.", "The push in the past decade for better-performing Fibonacci-like heaps, while it may have led to theoretical simplifications, does not seem to have yielded obvious practical benefits.", "The results show that the optimal choice of implementation is strongly input-dependent.", "Furthermore, it shows that care must be taken to optimize for cache performance, primarily at the L1-L2 barrier.", "This suggests that complicated, cache-oblivious structures are unlikely to perform well compared to simpler, cache-aware structures.", "Some obvious candidates for renewed testing are sequence heaps and B-heaps [22].", "Another obvious direction for future work is to explore other classes of workloads.", "We hope that our study gives future theorists and practitioners a new outlook on the state of affairs.", "Unfortunately, there is no simple answer to which heap should be used when.", "Picking the best tool for the job will likely require experimentation between existing implementations or careful analysis of the expected workload's caching behavior against each heap.", "To this end, we hope that our simple implementations of various heap structures will serve as a useful resource." ], [ "Acknowledgments.", "We would like to thank Jeff Erickson for the considerable guidance he provided the first author in the initial stages of this project.", "Some of the compute cluster resources at Princeton were donated by Yahoo!.", "5pt Appendix Below we have included the data tables for the rest of our results.", "Due to space constraints, we were not able to discuss all of them in detail, but we offer a brief description of the workloads involved.", "Dijkstra on contrived graphs.", "In Tables REF –REF , we see the results of running Dijkstra's shortest paths algorithm on graphs from the DIMACS generators.", "In general these are well-structured graphs, though some of them do contain randomness.", "A more complete description of the generators is in the DIMACS codebase [9], [10].", "Nagamochi-Ibaraki.", "In Table REF we see the results of running the Nagamochi-Ibaraki algorithm for the minimum-cut problem on random graphs (again from the DIMACS generators).", "Further artificial workloads.", "In Tables REF –REF we see the results of the rest of our artificial workloads.", "These include extra settings of the $c$ and $k$ parameters described in Section REF .", "Network event simulation.", "In tables REF –REF we see the results of running different protocols and workloads on the htsim network simulator.", "They vary between the use of the Equal-Cost Multi-Path (ECMP) [20] or MPTCP protocols, a permutation traffic matrix or traffic from VL2 network traces [18], and a 128-node fat-tree or a 512-node, 4-to-1 oversubscribed fat-tree network topology.", "See [30] for further discussion of these inputs.", "Table: Dijkstra – acyc_pos graphsTable: Dijkstra – grid_phard graphsTable: Dijkstra – grid_slong graphsTable: Dijkstra – grid_ssquare graphsTable: Dijkstra – grid_ssquare_s graphsTable: Dijkstra – rand_1_4 graphsTable: Dijkstra – rand_4 graphsTable: Dijkstra – spbad_dense graphsTable: Dijkstra – spbad_sparse graphsTable: Nagamochi-IbarakiTable: Randomized error\\operatorname{\\textsc {Insert}}–error\\operatorname{\\textsc {DeleteMin}} (Broken) – c=1c=1Table: Randomized error\\operatorname{\\textsc {Insert}}–error\\operatorname{\\textsc {DeleteMin}} (Broken) – c=32c=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=32c=32, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1024c=1024, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1c=1, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=32c=32, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1024c=1024, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1c=1, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=32c=32, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1024c=1024, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=32c=32, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1024c=1024, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1c=1, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=32c=32, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1024c=1024, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=32c=32, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1024c=1024, k=1024k=1024Table: Network Simulation – ECMP, PERM, fat-treeTable: Network Simulation – ECMP, PERM, over fat-treeTable: Network Simulation – ECMP, VL2, over fat-treeTable: Network Simulation – MPTCP, PERM, fat-treeTable: Network Simulation – MPTCP, PERM, over fat-treeTable: Network Simulation – MPTCP, VL2, over fat-treeBelow we have included the data tables for the rest of our results.", "Due to space constraints, we were not able to discuss all of them in detail, but we offer a brief description of the workloads involved." ], [ "In Tables REF –REF , we see the results of running Dijkstra's shortest paths algorithm on graphs from the DIMACS generators.", "In general these are well-structured graphs, though some of them do contain randomness.", "A more complete description of the generators is in the DIMACS codebase [9], [10]." ], [ "In Table REF we see the results of running the Nagamochi-Ibaraki algorithm for the minimum-cut problem on random graphs (again from the DIMACS generators)." ], [ "In Tables REF –REF we see the results of the rest of our artificial workloads.", "These include extra settings of the $c$ and $k$ parameters described in Section REF ." ], [ "In tables REF –REF we see the results of running different protocols and workloads on the htsim network simulator.", "They vary between the use of the Equal-Cost Multi-Path (ECMP) [20] or MPTCP protocols, a permutation traffic matrix or traffic from VL2 network traces [18], and a 128-node fat-tree or a 512-node, 4-to-1 oversubscribed fat-tree network topology.", "See [30] for further discussion of these inputs.", "Table: Dijkstra – acyc_pos graphsTable: Dijkstra – grid_phard graphsTable: Dijkstra – grid_slong graphsTable: Dijkstra – grid_ssquare graphsTable: Dijkstra – grid_ssquare_s graphsTable: Dijkstra – rand_1_4 graphsTable: Dijkstra – rand_4 graphsTable: Dijkstra – spbad_dense graphsTable: Dijkstra – spbad_sparse graphsTable: Nagamochi-IbarakiTable: Randomized error\\operatorname{\\textsc {Insert}}–error\\operatorname{\\textsc {DeleteMin}} (Broken) – c=1c=1Table: Randomized error\\operatorname{\\textsc {Insert}}–error\\operatorname{\\textsc {DeleteMin}} (Broken) – c=32c=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=32c=32, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1024c=1024, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1c=1, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=32c=32, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1024c=1024, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1c=1, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=32c=32, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Middle, c=1024c=1024, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=32c=32, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1024c=1024, k=1k=1Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1c=1, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=32c=32, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1024c=1024, k=32k=32Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=32c=32, k=1024k=1024Table: Randomized error\\operatorname{\\textsc {DecreaseKey}} – Min, c=1024c=1024, k=1024k=1024Table: Network Simulation – ECMP, PERM, fat-treeTable: Network Simulation – ECMP, PERM, over fat-treeTable: Network Simulation – ECMP, VL2, over fat-treeTable: Network Simulation – MPTCP, PERM, fat-treeTable: Network Simulation – MPTCP, PERM, over fat-treeTable: Network Simulation – MPTCP, VL2, over fat-tree" ] ]	1403.0252
[ [ "An alternative to Riemann-Siegel type formulas" ], [ "Abstract Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series).", "The formulas enable square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control.", "The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor.", "Practical parameter choices are discussed." ], [ "Introduction", "The Riemann zeta function is defined for $s=\\sigma +it$ by $\\zeta (s) = \\sum _{n=1}^{\\infty } n^{-s}$ , $\\sigma >1$ .", "It can be analytically continued everywhere except for a simple pole at $s=1$ .", "The zeta function satisfies the functional equation $\\zeta (s) = \\chi (s) \\zeta (1-s)$ where $\\chi (s) := \\pi ^{s-1/2}\\Gamma ((1-s)/2)/\\Gamma (s/2)$ .", "One is usually interested in numerically evaluating $\\zeta (\\sigma +it)$ on the critical line $\\sigma = 1/2$ (e.g.", "to verify the Riemann hypothesis).", "However, one cannot use the Dirichlet series $\\sum _{n=1}^{\\infty } n^{-s}$ to numerically evaluate zeta when $\\sigma <1$ because the series diverges.", "Rather, one can use partial summation and integration by parts to analytically continue the series to $\\sigma >0$ , obtaining $ \\zeta (s) = \\sum _{1\\le n<M} \\frac{1}{n^s} +\\frac{M^{-s}}{2}+ \\frac{M^{1-s}}{s-1}+\\mathcal {R}_M(s), \\qquad \|\\mathcal {R}_M(s)\| \\le \\frac{\\mathfrak {q}(s)}{\\sigma M^{\\sigma }},$ where $\\mathfrak {q}(s):=\|s\|+3$ is the analytic conductor of zeta; see [11].", "The analytic conductor terminology was introduced by Iwaniec and Sarnak; see [10] for example.", "This terminology will be useful when we generalize our formulas to Dirichlet $L$ -functions, and it ensures that $\\log \\mathfrak {q}(s)>0$ .", "We remark, though, that the precise definition of the analytic conductor does not affect the asymptotic content of the results, since $\\mathfrak {q}(s)$ needs only be of a comparable size to $\|s\|$ .", "Formula (REF ) can be viewed as consisting of a main sum $\\sum _{n<M} n^{-s}$ , an extra term $M^{-s}/2+M^{1-s}/(s-1)$ , and a remainder $\\mathcal {R}_M(s)$ .", "The main sum accounts for the bulk of the computational effort, the extra term can be computed easily, and the remainder can be controlled by choosing $M$ accordingly.", "For example, one can ensure that $\|\\mathcal {R}_M(s)\|<\\epsilon $ on taking $M > (\\mathfrak {q}(s)/(\\sigma \\epsilon ))^{1/\\sigma }$ .", "So when $\\sigma = 1/2$ , the main sum consists of $\\gg \\mathfrak {q}(s)^2$ terms, even if $\\epsilon = 1$ say.", "Using a more careful analysis, however, one can show that $\\mathcal {R}_M(s) \\ll M^{-\\sigma }$ if $M \\gg \\mathfrak {q}(s)$ .", "Alternatively, one can use the Euler-Maclaurin summation (see §) which allows for far more accuracy.", "In either case, though, the resulting main sum is of length $\\gtrsim \\mathfrak {q}(s)$ .", "So these formulas are rather impractical for numerical computations on single processor when $t \\gtrsim 10^{10}$ , say, especially if high precision is sought.", "This is unfortunate since they are simple to derive and analyze, and have explicit error bounds.", "So, instead, one typically uses the Riemann-Siegel asymptotic formula which has a much shorter main sum of length $\\lfloor \\sqrt{t/(2\\pi )}\\rfloor $ (see §).", "The Riemann-Siegel formula was discovered around 1932 in Riemann's unpublished papers by C.L.", "Siegel.", "Some of its history is narrated in [6].", "In lieu of the Riemann-Siegel formula, one can use the efficient smoothed formulas in [14].", "We propose a new method for computing zeta based on slowly converging Dirichlet series such as (REF ).", "Then we generalize our method to Dirichlet $L$ -functions to a power-full modulus.", "Interestingly, our results can be derived without knowing about the functional equation of the associated $L$ -function, nor using analysis of similar strength, such as the Poisson summation.", "To state the results, we introduce some notation.", "Let $\\begin{split}f_s(z) := \\frac{e^{sz}}{(1+z)^s},\\,\\,\\, f_s(0)=1,\\quad g_K(z) := \\sum _{k=0}^{K-1} e^{kz} =\\frac{e^{Kz}-1}{e^z-1},\\,\\, z\\notin 2\\pi i \\mathbb {Z},\\end{split}$ where $f_s^{(j)}(z)$ and $g_K^{(j)}(z)$ denote the $j$ -th derivative in $z$ .", "We choose integers $u_0\\ge 1$ , $v_0\\ge u_0$ , and $M\\ge v_0$ , and construct sequences $K_r= \\lceil v_r/u_0\\rceil $ and $v_{r+1}= v_r + K_r$ for $0\\le r< R$ , where $R:=R(v_0,u_0,M)$ is the largest integer such that $v_R < M$ .", "We define $K_R :=\\min \\lbrace \\lceil v_R/u_0\\rceil ,M-v_R\\rbrace $ , so that $v_{R+1} = M$ .", "Then we divide the main sum in (REF ) into an initial sum of length $v_0$ , followed by $R+1$ consecutive blocks where the $r$ -th block starts at $v_r$ and has length $K_r$ .", "The sequences $K_r$ and $v_r$ are so defined in order to implement a more efficient version of dyadic subdivision of the main sum.", "There will be substantial flexibility in choosing them (need only $K_r-1\\le v_r/u_0$ , $u_0\\ge \\sqrt{\\mathfrak {q}(s)}$ ), but we do not exploit this here.", "We plan to approximate the $r$ -th block $\\sum _{v_r\\le n<v_r+K_r} n^{-s}$ by $v_r^{-s}B_r(s,m)$ where $B_r(s,m):=\\sum _{j=0}^m \\frac{f_s^{(j)}(0)}{j!", "},\\frac{g_{K_r}^{(j)}(-s/v_r)}{v_r^j},$ which is a linear combination of a geometric sum and its derivatives.", "Also, we let $\\mathcal {B}_M(s,u_0,v_0):=\\sum _{r=0}^R v_r^{-\\sigma }\\min \\lbrace g_{K_r}(-\\sigma /v_r),\|\\csc (t/(2v_r))\|\\rbrace ,$ $ \\epsilon _m(s,u):=\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{3.5\\, e^{0.78(m+1)}}{(m+1)^{(m+1)/2}}\\frac{\|s\|^{(m+1)/2}}{u^{m+1}},\\,\\, &m\\le \|s\|/4,\\\\\\\\\\displaystyle \\frac{2^m e^{0.194\|s\|}}{u^m},\\,\\, &m> \|s\|/4.\\end{array}\\right.$ We prove the following theorem in §.", "Theorem 1.1 Given $s=\\sigma +it$ with $\\sigma >0$ , let $u_0$ and $v_0$ be any integers satisfying $v_0\\ge u_0\\ge 2\\max \\lbrace 6,\\sqrt{\\mathfrak {q}(s)},\\sigma \\rbrace $ .", "Then for any integers $M\\ge v_0$ and $m\\ge 0$ we have $\\zeta (s) = \\sum _{n=1}^{v_0-1} \\frac{1}{n^s}+ \\sum _{r=0}^R \\frac{B_r(s,m)}{v_r^s}+\\frac{M^{-s}}{2}+ \\frac{M^{1-s}}{s-1}+ \\mathcal {T}_{M,m}(s,u_0,v_0) + \\mathcal {R}_M(s),$ where $\\displaystyle \|\\mathcal {T}_{M,m}(s,u_0,v_0)\|\\le \\epsilon _m(s,u_0) \\mathcal {B}_M(s,u_0,v_0)$ .", "We have $R < 2u_0 \\log (M/v_0)+1$ .", "We could have used the main sum from the Euler-Maclaurin formula, instead of the main sum in (REF ), to derive Theorem REF .", "This permits one to choose $M$ smaller.", "Indeed, replacing $\\mathcal {R}_M(s)$ by the Euler-Maclaurin correction terms, one can restrict $M\\ll \\mathfrak {q}(s)$ while retaining high accuracy.", "In this case, Theorem REF , applied with $m=0$ , leads to a simple proof of the bound $\\zeta (1/2+it)\\ll \\mathfrak {q}(1/2+it)^{1/4}$ ; see corollary REF in §.", "The truncation error $\\mathcal {T}_{M,m}$ in Theorem REF is bounded by $\\epsilon _m\\mathcal {B}_M$ , where, by lemma REF , we have $\\mathcal {B}_M(s,u_0,v_0)\\le v_0^{-\\sigma } +(M^{1-\\sigma }-v_0^{1-\\sigma })(1-\\sigma )^{-1}$ if $\\sigma \\ne 1$ , and $\\mathcal {B}_M(s,u_0,v_0) \\le v_0^{-\\sigma }+\\log (M/v_0)$ if $\\sigma = 1$ .", "This estimate is quite generous, however.", "It can be improved by computing $\\mathcal {B}_M(s,u_0,v_0)$ directly, which should yield a bound like $u_0/(\\sigma v_0^{\\sigma })$ .", "The said computation can be done in about $R$ steps, and so it is subsumed by the computational effort for the main sum.", "In either case, the remainder term is clearly easy to control when $u_0\\ge \\sqrt{\\mathfrak {q}(s)}$ , due to the rapid decay of $\\epsilon _m(s,u_0)$ with $m$ (decays like $1/\\lfloor (m+1)/2\\rfloor !$ ).", "The main sum in Theorem REF has $v_0 + (m+1)(R+1)$ terms, where each term is, basically, a geometric sum.", "To ensure that $\|\\mathcal {T}_{M,m}(s)\| + \|\\mathcal {R}_M(s)\| <\\epsilon $ for $\\sigma =1/2$ , it suffices to take $M \\ll (\\mathfrak {q}(s)/\\epsilon )^2$ and $m \\ll \\log (\\mathfrak {q}(s)/\\epsilon )$ .", "Since $R\\le 2u_0\\log (M/v_0)+1$ , this is of length $\\ll v_0+u_0\\log ^2 (\\mathfrak {q}(s)/\\epsilon )$ terms.", "Choosing $u_0=v_0 = 2\\lceil \\sqrt{\\mathfrak {q}(s)}\\rceil $ , which is a typical choice, the main sum thus consists of $\\ll \\sqrt{\\mathfrak {q}(s)} \\log ^2 (\\mathfrak {q}(s)/\\epsilon )$ terms.", "We show how to compute these terms (geometric sums) efficiently in §, using $\\ll \\log (\\mathfrak {q}(s)/\\epsilon )$ precision.", "So, put together, the complexity of the formula in Theorem REF depends only logarithmically on $M$ and the error tolerance $\\epsilon $ .", "The formula enables square-root of the analytic conductor complexity, up to logarithmic loss, without using the functional equation, or the approximate functional equation.", "Also, the usual factor $\\chi (s)$ does not appear, and the conditions on $v_0$ and $u_0$ imply that $v_0u_0 \\gg \\mathfrak {q}(s)$ .", "Nevertheless, the idea behind the theorem is fairly simple.", "Writing $n^{-s}=e^{-s\\log n}$ , we have $\\sum _{v\\le n<v+K}n^{-s}= v^{-s}\\sum _{0\\le k<K} e^{-s\\log (1+k/v)}$ .", "So if $K/v \\ll 1/\\sqrt{\\mathfrak {q}(s)}$ , as we will have, then $s\\log (1+k/v) = sk/v + O(1)$ .", "In particular, using Taylor expansions, we can approximate $\\sum _{0\\le k<K} e^{-s\\log (1+k/v)}$ by a linear combination of the geometric sum $g_K(-s/v)$ and several of its derivatives.", "These geometric sums are easy to compute, which is the reason for the savings.", "One can shorten the length of the main sum in Theorem REF to be roughly $\\mathfrak {q}(s)^{1/3}$ .", "But then instead of obtaining linear exponential sums, one obtains quadratic exponential sums.", "The length can be further shortened, leading to cubic and higher degree exponential sums.", "In view of this, Theorem REF belongs to the family of methods for computing zeta that were derived in [9].", "And like these methods (see [8]), Theorem REF can be generalized to Dirichlet $L$ -functions $L(s,\\chi )$ , $\\chi \\bmod {q}$ , when $q$ is power-full.", "To this end, define the analytic conductor for $L(s,\\chi )$ by $\\mathfrak {q}(s,\\chi ) := q (\|s\|+3)$ .", "If $\\chi \\bmod {q}$ is non-principal, then we have the trivial bound $\|\\sum _n \\chi (n)\|< q$ .", "Combined with partial summation we obtain, for $\\sigma >0$ , thatTo estimate $\\mathcal {R}_M(s,\\chi )$ , we used the following partial summation formula (see [14]): Let $f:\\mathbb {Z}^+\\rightarrow \\mathbb {C}$ and $g:\\mathbb {R}\\rightarrow \\mathbb {C}$ such that $g^{\\prime }$ exists on $[1,x]$ .", "Then for $y\\in [1,x]$ we have $\\sum _{y< n\\le x} f(n)g(n) =\\left(\\sum _{y< n \\le x} f(n)\\right)g(x) +\\left(\\sum _{1\\le n \\le y}f(n)\\right)(g(x)- g(y)) -\\int _y^x \\left(\\sum _{1\\le n \\le \\tau } f(n)\\right) g^{\\prime }(\\tau ) \\,d\\tau .$ $L(s,\\chi ) = \\sum _{1\\le n<M} \\frac{\\chi (n)}{n^s} + \\mathcal {R}_M(s,\\chi ),\\qquad \|\\mathcal {R}_M(s,\\chi )\| \\le \\frac{2\\mathfrak {q}(s,\\chi )}{\\sigma M^{\\sigma }}.$ We will only consider the case $q=p^a$ for $p$ prime.", "As in Theorem REF , we divide the main sum in (REF ) into an initial sum of length $v_0$ , followed by $R+1$ consecutive blocks, where the $r$ -th block starts at $v_r$ and has length $K_r$ .", "Let $g_K(z,\\chi ,v) := \\sum _{0\\le k<K} \\chi (v+k) e^{kz}$ .", "Then, in analogy with zeta, we approximate the $r$ -th block $\\sum _{v_r\\le n<v_r+K_r} \\chi (n) n^{-s}$ by $v_r^{-s}B_r(s,\\chi ,m)$ where $B_r(s,\\chi ,m):=\\sum _{j=0}^m \\frac{f_s^{(j)}(0)}{j!", "}\\frac{g_{K_r}^{(j)}(-s/v_r,\\chi ,v_r)}{v_r^j},$ and $g_K^{(j)}(z,\\chi ,v)$ denotes the $j$ -th derivative in $z$ .", "The analogue of $\\mathcal {B}_M$ from Theorem REF is going to be more complicated to define.", "To this end, let $b:=\\lceil a/2\\rceil $ and, for $0\\le d<p^b$ , let $H_{r,d}:= \\lceil (K_r-d)/p^b\\rceil $ and $w_{r,d}:=2\\pi \\overline{v_r+d}L/p^{a-b}$ where $L$ is as in lemma REF and $(\\overline{v_r+d})(v_r+d)\\equiv 1\\bmod {p^a}$ if $\\gcd (v_r+d,p)=1$ .", "Then let $\\begin{split}\\mathcal {B}_M(s,\\chi ,u_0,v_0):=\\sum _{r=0}^R\\sum _{d=0}^{p^b-1}\\delta _{\\gcd (v_r+d,p)=1}&\\min \\lbrace e^{-\\sigma d/v_r}g_{H_{r,d}}(-p^b\\sigma /v_r),\\\\&\|\\csc (w_{r,d}/2-p^bt/(2v_r))\|\\rbrace v_r^{-\\sigma }.\\end{split}$ In §, we prove the following.", "Theorem 1.2 Given $s=\\sigma +it$ with $\\sigma >0$ , a non-principal Dirichlet character $\\chi \\bmod {p^a}$ with $p$ a prime, let $b=\\lceil a/2\\rceil $ , and let $u_0$ and $v_0$ be any integers satisfying $v_0\\ge u_0\\ge 2\\max \\lbrace 6,\\sqrt{\\mathfrak {q}(s)},\\sigma \\rbrace $ .", "Then for any integers $M\\ge v_0$ and $m\\ge 0$ we have $\\begin{split}L(s,\\chi ) =& \\sum _{n=1}^{v_0-1} \\frac{\\chi (n)}{n^s}+ \\sum _{r=0}^{R} \\frac{B_r(s,\\chi ,m)}{v_r^s}+ \\mathcal {T}_{M,m}(s,\\chi ) + \\mathcal {R}_M(s,\\chi ),\\end{split}$ where $\\displaystyle \|\\mathcal {T}_{M,m}(s,\\chi )\|\\le \\epsilon _m(s,u_0) \\mathcal {B}_M(s,\\chi ,u_0,v_0)$ .", "We have $R < 2u_0 \\log (M/v_0)+1$ .", "We use the Postnikov character formula in § to show that $g_K(z,\\chi ,v)$ can be written as a sum of $p^b$ geometric sums.", "Lemma 1.3 Given a Dirichlet character $\\chi \\bmod {p^a}$ with $p$ a prime, let $b=\\lceil a/2\\rceil $ , and $H_d:= \\lceil (K-d)/p^b\\rceil $ .", "Then $g_K(z,\\chi ,v) = \\sum _{d=0}^{p^b-1} \\chi (v+d) e^{zd} g_{H_d}(p^bz+iw_d)$ , where $w_d:=2\\pi \\overline{v+d}L/p^{a-b}$ if $(v+d,p)=1$ , with $(\\overline{v+d})(v+d)\\equiv 1\\bmod {p^a}$ , otherwise $w_d:=0$ .", "Here, $L\\in [0,p^{a-b})$ is the integer determined by the equation $\\chi (1+p^b)=e^{2\\pi i L/p^{a-b}}$ .", "The main sum in Theorem REF has $\\le v_0 + (m+1)(R+1)p^b$ terms, where the extra $p^b$ is from the formula for $g_K(z,\\chi ,v)$ in lemma REF .", "One can easily deduce from the proof of lemma REF that $\\mathcal {B}_M(s,\\chi ,u_0,v_0) \\le v_0^{-\\sigma }+ (M^{1-\\sigma }-v_0^{1-\\sigma })(1-\\sigma )^{-1}$ if $\\sigma \\ne 1$ , and $\\mathcal {B}_M(s,\\chi ,u_0,v_0)\\le v_0^{-\\sigma } +\\log (M/v_0)$ if $\\sigma = 1$ .", "This bound is generous, of course, and can be improved by computing $\\mathcal {B}_M(s,\\chi ,u_0,v_0)$ directly, as was pointed out earlier for zeta.", "In any case, we can ensure that $\|\\mathcal {T}_{M,m}(s,\\chi )\| + \|\\mathcal {R}_M(s,\\chi )\| <\\epsilon $ for $\\sigma =1/2$ , by taking $M \\ll (\\mathfrak {q}(s,\\chi )/\\epsilon )^2$ and $m \\ll \\log (\\mathfrak {q}(s,\\chi )/\\epsilon )$ .", "So, choosing $u_0 = 2\\lceil \\sqrt{\\mathfrak {q}(s)}\\rceil $ and $v_0 = p^b u_0$ , we see that the main sum on the critical line can be made of length $\\ll p^b\\sqrt{\\mathfrak {q}(s)} \\log ^2 (\\mathfrak {q}(s,\\chi )/\\epsilon )$ terms.", "If $a$ is an even integer, or a large integer, then $p^b\\approx \\sqrt{q}$ , and so the length of the main sum is about $\\sqrt{\\mathfrak {q}(s,\\chi )}\\log ^2 (\\mathfrak {q}(s,\\chi )/\\epsilon )$ .", "We remark that one can apply the Euler-Maclaurin formula along arithmetic progressions to the main sum in (REF ) (for each residue class of $p^a$ ).", "This way, one can restrict $M\\ll \\mathfrak {q}(s,\\chi )$ , replacing $\\mathcal {R}_M(s,\\chi )$ by the correction terms resulting from the Euler-Maclaurin formula.", "These correction terms will involve sums over the residue classes of $p^a$ .", "But it will not be too hard to see that these sums can be tackled using the same methods presented here.", "Remark If $\\sigma > 0$ , then one has the exact expression $L(s,\\chi ) = \\sum _{n=1}^{v_0-1} \\frac{\\chi (n)}{n^s}+ \\sum _{r=0}^{\\infty }\\frac{1}{v_r^s}\\sum _{j=0}^{\\infty } \\frac{f_s^{(j)}(0)}{j!", "}\\frac{g_{K_r}^{(j)}(-s/v_r,\\chi ,v_r)}{v_r^j}.$ The order of the double sum can be switched if $\\sigma >1$ ." ], [ "Previous methods and motivation", "In the case of the Riemann zeta function, one can use the Euler-Maclaurin summation to obtain a main sum of length about $\\mathfrak {q}(s)$ .", "One notes that $n^{-s}$ changes slowly with $n$ when $n\\gg \\mathfrak {q}(s)$ , and so $n^{-s}$ becomes approximable by the integral $\\int _n^{n+1}x^{-s}\\,dx$ .", "This gives an efficient way to compute the tail $\\sum _{n\\gg \\mathfrak {q}(s)} n^{-s}$ .", "Specifically, following [13], [14], we have, for any positive integers $N$ and $L_1$ , $\\zeta (s) = \\sum _{n=1}^{N-1} n^{-s} + \\frac{N^{-s}}{2} +\\frac{N^{1-s}}{s-1} +\\sum _{\\ell =1}^{L_1} T_{\\ell ,N}(s) + E_{N,L_1}(s),$ where $T_{\\ell ,N}(s)=\\frac{B_{2\\ell }}{(2\\ell )!", "}N^{-s}\\prod _{l=0}^{2\\ell -2}(s+l)/N$ , $B_2=1/6$ , $B_4=-1/30,\\ldots ,$ are the Bernoulli numbers, and, by the estimate in [14], we have, for any $\\sigma >-(2L_1+1)$ , $\|E_{N,L_1}(s)\| \\le \\frac{\\zeta (2L_1)}{\\pi N^{\\sigma }}\\frac{\|s+2L_1-1\|}{\\sigma +2L_1-2}\\prod _{l=0}^{2L_1-2} \\frac{\|s+l\|}{2\\pi N}.$ It follows from (REF ) that, for $\\sigma \\ge 1/2$ say, one can ensure that $\|E_{N,L_1}(s)\|<\\epsilon $ by taking $2\\pi N \\ge e \|s+2L_1-1\|$ and $2L_1-1 > 0.5 \\log \|s+2L_1-1\| -\\log \\epsilon $ .", "Therefore, the remainder term in the Euler-Maclaurin summation is easy to control, enabling very accurate computations of zeta.", "Rubinstein showed [14] that one could reduce the length of the main sum in the Euler-Maclaurin formula to $\\ll \\mathfrak {q}(s)^{1/2}$ terms, but requiring $\\approx \\log (\\mathfrak {q}(s)/\\epsilon )\\log (\\mathfrak {q}(s))$ precision due to substantial cancellation that occurs, and with each term involving an incomplete Gamma function.", "The Riemann-Siegel formula offers good control over the required precision, and is often used in zeta computations.", "The derivation of the Riemann-Siegel formula is quite involved.", "One begins by expressing $\\zeta (s)$ as a contour integral, then moves the contour of integration suitably.", "This leads to a remainder term that requires careful saddle-point analysis; see [17] and [6] for example.", "One version of the Riemann-Siegel formula on the critical line is the following.", "For $t>2\\pi $ , let $a:=\\sqrt{t/(2\\pi )}$ , $n_1:=\\lfloor a\\rfloor $ the integer part of $a$ , and $z:= 1-2(a-\\lfloor a\\rfloor )$ .", "Then $ e^{i\\theta (t)}\\zeta (1/2+it) = 2\\, \\Re \\left(e^{-i\\theta (t)}\\sum _{n=1}^{n_1}\\frac{e^{it \\log n}}{\\sqrt{n}}\\right) - \\frac{(-1)^{n_1}}{\\sqrt{a}}\\sum _{r=0}^m \\frac{C_r(z)}{a^r} + R_m(t).$ The $C_r(z)$ can be written as a linear combination of derivatives of the function $F(z):=\\cos ((\\pi /2)(z^2+3/4))(\\cos (\\pi z))^{-1}$ (up to the $3r$ -th derivative).", "For example, $C_0(z) = F(z)$ and $C_1(z) = F^{(3)}(z)/(12\\pi ^2)$ , where $F^{(3)}(z)$ is the third derivative of $F(z)$ with respect to $z$ .", "(Note that $F(z)$ is not periodic in $z$ .)", "The general form of $C_r(z)$ can be found in Gabcke's thesis [7].", "Using formal manipulations of Dirichlet series, Berry showed [2] (see also [3]) that the series of the correction terms $\\sum _{r\\ge 0} C_r(z) a^{-r}$ is divergent, and, therefore, improvement from adding more correction terms in (REF ) is not to continue indefinitely, instead, the series should be stopped at the least term for a given $t$ .", "The phase $\\theta (t)$ is defined by $\\theta (t):=\\arg [\\pi ^{-it/2}\\Gamma (1/4+it/2)]$ .", "We can also define $\\theta (t)$ by a continuous variation of $s$ in $\\pi ^{-s/2}\\Gamma (s/2)$ , starting at $s=1/2$ and going up vertically, which gives the formula $\\theta (t)=(t/2)\\log (t/(2\\pi e))-\\pi /8 + 1/(48t)+O(t^{-3})$ for large $t$ .", "We note that the rotation factor $e^{i\\theta (t)}$ is chosen so that $e^{i\\theta (t)}\\zeta (1/2+it)$ is real.", "Thus, one may locate non-trivial zeros of zeta by looking for sign changes in the r.h.s.", "of (REF ).", "As for the remainder term $R_m(t)$ , we have $R_m(t) \\ll t^{-(2m+3)/4}$ .", "Gabcke derived explicit bounds for $R_m(t)$ , for $m=0,\\ldots ,10$ in his thesis [7].", "For example, for $t\\ge 200$ , we have $\|R_1(t)\| < .053 t^{-5/4}$ , $\|R_4(t)\| < 0.017t^{-11/4}$ , and $\|R_{10}(t)\| < 25966 t^{-23/4}$ .", "While Gabcke's estimates are sufficient for most applications, they do not allow for very high accuracy for relatively small $t$ , such as required when computing zeta zeros to many digits in order to test their linear independence.", "(Recently, very good bounds have been derived in [1].)", "A source of the difficulty towards explicit estimates of $R_m(t)$ is that the main sum of the Riemann-Siegel formula has a sharp cut-off (dictated by the location of the saddle-point), which complicates the analysis of the remainder term significantly.", "The analysis is much simplified by using a smoothing function.", "Indeed, Turing had proposed [18] a type of smoothed formula for computing zeta in the intermediate range where $t$ is neither so small that the Euler-Maclaurin summation can be used nor large enough for the Riemann-Siegel asymptotic formula.It is worth mentioning that Theorem REF is useful in such a range, in order to carry out high precision computations.", "Rubinstein provides [14] the following smoothed formula, which has a main sum of length $\\mathfrak {q}(s)^{1/2+o_{\\epsilon }(1)}$ , and which can be generalized to a fairly large class of $L$ -functions.", "$ \\pi ^{-s/2}\\Gamma (s/2)\\zeta (s)\\delta ^{-s}= -\\frac{1}{s}-\\frac{\\delta ^{-1}}{1-s}+\\sum _{n=1}^{\\infty } G(s/2,\\pi n^2\\delta ^2)+\\delta ^{-1}\\sum _{n=1}^{\\infty } G((1-s)/2,\\pi n^2/\\delta ^2),$ where $G(z,w)$ is a smoothing function that can be expressed in terms of the incomplete Gamma function $\\Gamma (z,w)$ , $G(z,w) := w^{-z}\\Gamma (z,w) = \\int _1^{\\infty } e^{-wx}x^{z-1}\\,dx$ , $\\Re (w)> 0$ , and $\\delta $ is a complex parameter of modulus one, with a simple dependence on $t$ , such that $\|\\Im (\\log \\delta )\| \\in (-\\pi ,\\pi ]$ and $\\Im (\\log \\delta )$ tends to $\\operatorname{sgn}(t) \\pi /4$ for large $t$ .", "In explicit form, $\\delta = \\exp (i\\operatorname{sgn}(t) (\\pi /4 - \\theta ))$ , where $\\theta = \\pi /4$ if $\|t\| \\le 2c/\\pi $ , $\\theta = c/\|2t\|$ if $\|t\|>2c/\\pi $ , and $c>0$ is a free parameter that we can optimize.", "In particular, $\\delta ^{-s}$ is chosen to cancel out the exponential decay in $\\Gamma (s/2)$ as $t$ gets large on the l.h.s of (), ensuring that the l.h.s.", "is $\\gg \|s\|^{(\\sigma -1)/2}\|\\zeta (s)\| e^{-c}$ for large $t$ .", "Although the series in (REF ) are infinite, the weights $G(z,w)$ decay exponentially fast when $\\Re (w) \\gg 1$ .", "Specifically, following [14], we have for $\\Re (w)>0$ and $\\Re (z)\\le 1$ that $\|G(z,w)\| < e^{-\\Re (w)}/\\Re (w)$ .", "So, for $\|t\| > 2c/\\pi $ and $\\sigma \\in [0,1]$ say, we have $\\Re (\\pi n^2\\delta ^2)=\\Re (\\pi n^2/\\delta ^2) = \\pi n^2 \\cos (\\pi /2 -c/\|t\|) > \\pi n^2 c/\|2t\|$ , where we used the inequality $\\cos (\\pi /2-x)\\ge x/2$ for $0\\le x\\le 1$ .", "Therefore, the series can be truncated after $M$ terms with truncation error $< 4\|t\|/(\\pi c) \\sum _{n\\ge M} n^{-2} e^{-\\pi n^2c/\|2t\|}$ .", "So to ensure that the truncation error is $<\\epsilon $ , it certainly suffices to take $M > \\sqrt{\|2t\|/(\\pi c) \\log (\|4t\|/(\\epsilon c))}$ .", "Once the series is truncated, it can be evaluated term by term to give a numerical approximation of $\\zeta (\\sigma +it)$ for $\|t\|>2c/\\pi $ .", "The number of terms in the resulting main sum (i.e.", "truncated series) is roughly equal to $\\sqrt{\\mathfrak {q}(s)\\log (\\mathfrak {q}(s)/\\epsilon )}$ .", "The terms in the main sum are more complicated than in the Riemann-Siegel formula since each term involves the smoothing function $G(z,w)$ .", "In the case of Dirichlet $L$ -functions, Davies [4], Deuring [5], Lavrik [12], and others had developed Riemann-Siegel type formulas for $L(1/2+it,\\chi )$ , where $\\chi $ is a primitive character mod $q$ and $t\\gg 1$ .", "Such formulas, whose general form was already considered by Siegel [16], require the numerical evaluation of a main sum of length $q\\lfloor \\sqrt{t/(2\\pi q)}\\rfloor \\approx \\mathfrak {q}(\\chi ,s)^{1/2}$ terms, where each term is of the form $\\chi (n) n^{-1/2}\\exp (it\\log n)$ .", "Unfortunately, however, it does not seem that we have an analogue of Gabcke's explicit estimate for the remainder terms in such formulas.", "And it is not clear how to obtain a posteriori error estimate either.", "Therefore, we are not prepared to find the accuracy of the numerics resulting from these formulas explicitly.", "Still, if one is willing to live with a much longer main sum, consisting of about $\\mathfrak {q}(\\chi ,s)$ terms, then one can keep the simplicity of an unsmoothed main sum while having an explicit estimate for the remainder term.", "The basic idea is well-known, and was implemented carefully by Rumely [15].", "Essentially, one uses the periodicity of $\\chi $ to write $L(1/2+it,\\chi )$ as a linear combination of about $\\mathfrak {q}(\\chi ,0)$ Hurwitz zeta functions, then one approximates each Hurwitz zeta function using the Euler-Maclaurin summation formula.", "However, since the Euler-Maclaurin formula requires a main sum of length about $\\mathfrak {q}(s)$ , the cost of this method is prohibitive in comparison with a Riemann-Siegel approach with explicit remainder.", "In view of this, one typically uses a smoothing function to accelerate the convergence.", "Such formulae (see [14]) are applicable even for small $t$ and have a main sum of length $\\mathfrak {q}(\\chi ,s)^{1/2+o_{\\epsilon }(1)}$ terms, where each term involves the computation of a smoothing function." ], [ "Proofs of Theorems ", "We first prove Theorem REF .", "The proof of Theorem REF will be similar, but will additionally require a specialization of the Postnikov character formula, lemma REF .", "Recall that we choose integers $u_0\\ge 1$ , $v_0\\ge u_0$ , $M\\ge v_0$ , and we construct the sequences $K_r= \\min \\lbrace \\lceil v_r/u_0\\rceil , M-v_r\\rbrace $ and $v_{r+1}= v_r + K_r$ for $0\\le r\\le R$ , where $R:=R(v_0,u_0,M)$ is the smallest integer such that $v_{R+1} = M$ .", "Lemma 3.1 $R = R(v_0,u_0,M) < 2u_0 \\log (M/v_0)+1$ .", "For $r<R$ , we have $v_{r+1} = v_r + K_r \\ge v_r(1+1/u_0)$ , and so by induction $v_{r+1} \\ge v_0(1+1/u_0)^r$ .", "If $R>0$ , then taking $r=R-1$ and noting that $v_R < M$ , we obtain $R < \\log (M/v_0)/\\log (1+1/u_0) +1 \\le 2u_0 \\log (M/v_0)+1$ , where we used the inequality $\\log (1+x) \\ge x/2$ for $0\\le x<1$ .", "If $R=0$ , then clearly the last bound still holds.", "Lemma 3.2 Let $s=\\sigma +it$ , $\\sigma \\ge 0$ .", "Using the same notation for $K_r$ , $v_r$ , and $R$ , we have $\\sum _{0\\le r\\le R} g_{K_r}(-\\sigma /v_r) v_r^{-\\sigma }\\le v_0^{-\\sigma }+ \\left\\lbrace \\begin{array}{ll}\\frac{M^{1-\\sigma }-v_0^{1-\\sigma }}{1-\\sigma }, &\\sigma \\ne 1, \\\\\\log (M/v_0), &\\sigma = 1.\\\\\\end{array}\\right.$ For $k<v$ , we have $\\log (1+k/v) = k/v - k^2/(2v^2) +\\cdots \\le k/v$ .", "Thus, $e^{-\\sigma k/v} \\le (1+k/v)^{-\\sigma }$ .", "Hence, $g_K(-\\sigma /v) v^{-\\sigma }\\le v^{-\\sigma }\\sum _{0\\le k<K} (1+k/v)^{-\\sigma }= \\sum _{0\\le k<K} (v+k)^{-\\sigma }$ .", "So $\\sum _{0\\le r\\le R} g_{K_r}(-\\sigma /v_r) v_r^{-\\sigma }\\le \\sum _{v_0\\le n<M} n^{-\\sigma } \\le v_0^{-\\sigma } + \\int _{v_0}^M x^{-\\sigma }\\,dx$ .", "The lemma follows on evaluating the integral.", "Lemma 3.3 Let $s=\\sigma +it$ , $\\sigma \\ge 0$ .", "For any integers $v\\ge u\\ge 2\\max \\lbrace 6,\\sqrt{\|s\|},\\sigma \\rbrace $ , $K\\ge 1$ , and $m\\ge 0$ , such that $(K-1)/v \\le 1/u$ , we have $\\sum _{0\\le k<K} e^{-s\\log (1+k/v)} =\\sum _{j=0}^m c_j(s) \\frac{g_K^{(j)}(-s/v)}{v^j} + \\mathcal {E}_m(s,v,K),$ $c_j(s) = \\frac{f_s^{(j)}(0)}{j!", "}$ , $\|\\mathcal {E}_m(s,v,K)\| \\le \\epsilon _m(s,u)\\min \\lbrace g_K(-\\sigma /v),\|\\csc (t/(2v))\|e^{-\\sigma (K-1)/v}\\rbrace $ , and $\\epsilon _m(s,u)$ is defined in (REF ).", "We have $e^{-s\\log (1+k/v)} = e^{-sk/v} f_s(k/v)$ .", "The function $f_s(z)$ is analytic in $\|z\| <1$ .", "Taking the branch of the logarithm determined by $f_s(0)=1$ , we have $f_s(z) = e^{-s\\log (1+z)+sz} =e^{sz^2/2 - sz^3/3+\\cdots }$ for $\|z\|<1$ .", "We expand $f_s(z)$ into a power series $1+\\cdots +c_m(s) z^m + \\cdots $ .", "By definition, we have $\\mathcal {E}_m(s,v,K) = \\sum _{0\\le k < K} e^{-sk/v}\\sum _{j > m} c_j(s) (k/v)^j$ .", "So, interchanging the order of summation in $j$ and $k$ , we obtain $\|\\mathcal {E}_m(s,v,K)\| \\le \\sum _{j>m} \|c_j(s)\|\|\\sum _{0\\le k<K} (k/v)^j e^{-sk/v} \|$ .", "We note that the function $x^j e^{-\\sigma x}$ is increasing with $x$ if $0\\le x< j/\\sigma $ .", "So, if $0\\le k < jv/\\sigma $ , then $(k/v)^je^{-\\sigma k/v}$ increases with $k$ .", "This last condition, $k<jv/\\sigma $ , is satisfied because, by hypothesis, $j>m\\ge 0$ , so $j\\ge 1$ , and $k/v\\le 1/u< 1/\\sigma $ .", "Thus, it follows by partial summation that $\|\\mathcal {E}_m(s,v,K)\| \\le e^{-\\sigma (K-1)/v} \\max _{x\\in [0,K]} \|\\sum _{x\\le k<K} e^{-itk/v}\|\\sum _{j>m} \|c_j(s)\| (K-1)^j/v^j.$ Executing the summation in the geometric sum, we see that it is bounded by $\|\\csc (t/(2v))\|$ .", "Also, by a trivial estimate, $\|\\sum _{0\\le k<K}e^{-sk/v} \|\\le g_K(-\\sigma /v)$ .", "Thus, $\|\\mathcal {E}_m(s,v,K)\|\\le \\min \\lbrace g_K(-\\sigma /v),\|\\csc (t/(2v))\|e^{-\\sigma (K-1)/v}\\rbrace \\sum _{j>m} \|c_j(s)\| \\frac{(K-1)^j}{v^j}.$ We bound $c_j(s)$ by a standard application of Cauchy's theorem using a circle around the origin.", "We have $2\\pi \|c_j(s)\| \\le \|\\int _{\|z\|=c}f_s(z)/z^{j+1}\\,dz \| \\le 2\\pi c^{-j}e^{\|s\|c^2/2+\\cdots }$ , $c\\in (0,1)$ .", "If $0< j\\le \|s\|/4$ , let $c =\\sqrt{j/\|s\|} \\le 1/2$ .", "So $\|s\|c^2/2+\\cdots \\le \|s\|c^2\\sum _{r=2}^{\\infty }c^{r-2}/r \\le \\alpha j$ , where $\\alpha :=\\sum _{r=2}^{\\infty }(1/2)^{r-2}/r= -2+4\\log 2<0.78$ .", "We conclude that $\|c_j(s)\| \\le \|s\|^{j/2} j^{-j/2} e^{\\alpha j}$ for $0< j\\le \|s\|/4$ .", "Also, for any $j\\ge 0$ , we may choose $c=1/2$ .", "So we have $\|c_j(s)\| \\le 2^j e^{\\alpha \|s\|/4}$ for each $j\\ge 0$ .", "Since $(K-1)/v\\le 1/u$ , by hypothesis, we have by the estimate for $c_j(s)$ , and assuming that $m\\le \|s\|/4$ , that $\\sum _{j>m} \|c_j(s)\| \\frac{(K-1)^j}{v^j} \\le \\sum _{m<j \\le \|s\|/4}\|s\|^{j/2} u^{-j}j^{-j/2} e^{\\alpha j} +\\sum _{j > \|s\|/4} u^{-j} 2^j e^{\\alpha \|s\|/4}.$ If $\|s\|/4$ is not an integer, then $\\sum _{j > \|s\|/4} u^{-j}2^je^{\\alpha \|s\|/4}\\le 0.2e^{\\alpha } (u/2)^{-\\lfloor \|s\|/4\\rfloor }e^{\\alpha \\lfloor \|s\|/4\\rfloor }$ , where we used $\\sum _{\\ell >0} (2/u)^{\\ell }\\le 0.2$ and $u\\ge 12$ .", "Since this is at most $0.2e^{\\alpha }<0.44$ times the last term in first sum on the r.h.s.", "above, we obtain the estimate $\\sum _{j>m} \|c_j(s)\| \\frac{(K-1)^j}{v^j}\\le 1.44\\sum _{j>m} \|s\|^{j/2} u^{-j} j^{-j/2} e^{\\alpha j}.$ Now, for $\\ell \\ge 0$ , $(m+1+\\ell )^{-(m+1+\\ell )/2}\\le (m+1)^{-(m+1)/2} (1+\\ell )^{-\\ell /2}$ .", "Therefore, $\\sum _{j>m} \|c_j(s)\| \\frac{(K-1)^j}{v^j} \\le \\frac{3.5\\, e^{\\alpha (m+1)}}{(m+1)^{(m+1)/2}}\\frac{\|s\|^{(m+1)/2}}{u^{m+1}}< \\epsilon _m(s,u),$ where we used $u\\ge 2\\sqrt{\|s\|}$ and $\\sum _{\\ell =0}^{\\infty } \\frac{\|s\|^{\\ell /2}u^{-\\ell }e^{\\alpha \\ell }}{(1+\\ell )^{\\ell /2}}\\le 2.42$ , so $(2.42)(1.44)<3.5$ .", "If $\|s\|/4$ is an integer, on the other hand, then the same bound holds (with an even better constant).", "It remains to consider the case when $m>\|s\|/4$ .", "Here, we have $\\sum _{j>m} \|c_j(s)\| (K-1)^j/v^j \\le \\sum _{j > m} u^{-j} 2^j e^{\\alpha \|s\|/4}\\le 2^m e^{\\alpha \|s\|/4}/u^m$ .", "Therefore, $\\sum _{j>m} \|c_j(s)\|(K-1)^j/v^j \\le \\epsilon _m(s,u)$ .", "Put together, we arrive at the claimed bound on $\\mathcal {E}_m(s,v,K)$ .", "To complete the proof of the lemma, notice that $\\sum _{0\\le k<K} e^{-s\\log (1+k/v)}= \\sum _{0\\le k <K} \\sum _{j=0}^m c_j(s) (k/v)^j e^{-sk/v} +\\mathcal {E}_m(s,v,K).$ So the formula (REF ) follows on interchanging the order of the double sum.", "Lemma 3.4 Let $\\chi \\bmod {p^a}$ be a Dirichlet character, where $p$ is a prime, and let $b=\\lceil a/2\\rceil $ .", "Then there exists an integer $L\\bmod {p^{a-b}}$ , depending on $\\chi $ , $p$ , $a$ , and $b$ only (so independent of $x$ ), such that $\\chi (1+p^b x)=e^{2\\pi i L x/p^{a-b}}$ for all $x\\in \\mathbb {Z}$ .", "The proof is similar to that of [8], but we still give it here for completeness.", "Let $H$ be the subgroup in $\\left(\\mathbb {Z}/p^a\\mathbb {Z}\\right)^$ consisting of the residue classes congruent to $1\\bmod {p^b}$ , so $H$ has size $\|H\|=p^{a-b}$ .", "We identify the elements of $H$ with the set of integers $\\lbrace 1+p^b x\\, \|\\, 0\\le x <p^{a-b}\\rbrace $ .", "Consider the function $\\psi : H\\rightarrow \\mathbb {C}$ , defined by $\\psi (1+p^b x):=e^{2\\pi i x/p^{a-b}}$ By our choice of $b=\\lceil a/2\\rceil $ , we have $p^{b}\\equiv 0\\bmod {p^{a-b}}$ .", "Therefore, $\\psi ((1+p^b x)(1+p^b y))=\\psi (1+p^b x)\\psi (1+p^b y)$ for all $x,y,\\in \\mathbb {Z}$ , meaning that $\\psi $ is multiplicative.", "Also, $\\psi $ is not identically zero; e.g.", "$\\psi (1) = 1$ .", "Therefore, $\\psi $ is a character of $H$ .", "Moreover, the values $\\psi (1+p^b)^u = e^{2\\pi i u /p^{a-b}}$ , $0\\le u < p^{a-b}$ , are all distinct.", "In particular, $\\psi $ has order $p^{a-b}$ , which is the same as the order of $H$ .", "So $\\psi $ generates the full character group of $H$ .", "Since $\\left.\\chi \\right\|_H$ is a character of $H$ , then $\\left.\\chi \\right\|_{H} \\equiv \\psi ^L$ for some $L \\bmod {p^{a-b}}$ .", "To find $L$ , we calculate $\\chi (1+p^b)$ , then use the relation $\\chi (1+p^b)=e^{2\\pi i L /p^{a-b}}$ .", "We divide the main sum in (REF ) according to the positions of $v_r$ as follows: $\\sum _{1\\le n<v_0} n^{-s} + \\sum _{0\\le r\\le R} v_r^{-s} \\sum _{0\\le k<K_r}e^{-s\\log (1+k/v_r)}$ .", "Note that $K_r = \\lceil v_r/u_0\\rceil \\le v_r/u_0+1$ for $r<R$ , and $K_R \\le v_R/u_0+1$ .", "So $(K_r-1)/v_r \\le 1/u_0$ throughout $0\\le r\\le R$ .", "Thus, the conditions for lemma REF are satisfied and we can apply it to each block $\\sum _{0\\le k<K_r} e^{-s\\log (1+k/v_r)}$ .", "This yields $\\mathcal {T}_{M,m}(s,u_0,v_0)=\\sum _{0\\le r\\le R} v_r^{-s} \\mathcal {E}_m(s,v_r,K_r)$ .", "And using the estimate for $\\mathcal {E}_m(s,v,K)$ in lemma REF yields the required bound on $\\mathcal {T}_{M,m}(s,u_0,v_0)$ .", "This follows from from the definitions and lemma REF : $\\begin{split}g_K(z,\\chi ,v) &= \\sum _{d=0}^{p^b-1} \\sum _{0\\le k<H_d}e^{z(d+p^bk)}\\chi (v+d+p^bk)\\\\&= \\sum _{d=0}^{p^b-1} \\delta _{\\gcd (v+d,p)=1} \\chi (v+d) e^{zd}\\sum _{0\\le k<H_d}e^{p^bzk}\\chi (1+p^b\\overline{v+d}k)\\\\&= \\sum _{d=0}^{p^b-1} \\delta _{\\gcd (v+d,p)=1} \\chi (v+d) e^{zd}\\sum _{0\\le k<H_d}e^{p^bzk + 2\\pi iL \\overline{v+d}k/p^{a-b}}\\\\&= \\sum _{d=0}^{p^b-1} \\chi (v+d) e^{zd} g_{H_d}(p^bz+iw_d).\\end{split}$ Lemma 3.5 Given $s=\\sigma +it$ , $\\sigma \\ge 0$ , and a Dirichlet character $\\chi \\bmod {p^a}$ with $p$ a prime, let $b=\\lceil a/2\\rceil $ .", "Then for any integers $v\\ge u\\ge 2\\max \\lbrace 6,\\sqrt{\|s\|},\\sigma \\rbrace $ , $K\\ge 1$ , and $m\\ge 0$ , such that $(K-1)/v \\le 1/u$ , we have $\\sum _{0\\le k<K} \\chi (v+k) e^{-s\\log (1+k/v)} =\\sum _{j=0}^m c_j(s) \\frac{g_K^{(j)}(-s/v,\\chi ,v)}{v^j} + \\mathcal {E}_m(s,\\chi ,v,K),$ where $c_j(s) = \\frac{f_s^{(j)}(0)}{j!", "}$ , and, with $H_d=\\lceil (K-d)/p^b\\rceil $ , we have $\|\\mathcal {E}_m(s,\\chi ,v,K)\| \\le \\epsilon _m(s,u)\\sum _{d=0}^{p^b-1}\\delta _{\\gcd (v+d,p)=1}\\min \\lbrace g_{H_d}(-p^b\\sigma /v),\|\\csc (w_d/2-p^bt/(2v))\|\\rbrace .$ The $\\epsilon _m(s,u)$ is defined in (REF ).", "Proceeding in the same way as in Theorem REF and lemma REF , we arrive at $\\begin{split}&\\mathcal {E}_m(s,\\chi ,v,K) =\\sum _{0\\le k<K} \\chi (v+k) e^{-sk/v} \\sum _{j>m} c_j(s) (k/v)^j\\\\&= \\sum _{j>m}c_j(s) \\sum _{0\\le k<K} \\chi (v+k) e^{-sk/v} (k/v)^j\\\\&= \\sum _{j>m}c_j(s) \\sum _{d=0}^{p^b-1}\\sum _{0\\le k<H_d} \\chi (v+d+p^bk) e^{-s(d+p^bk)/v} ((d+p^bk)/v)^j\\\\&= \\sum _{j>m}c_j(s) \\sum _{d=0}^{p^b-1} \\chi (v+d)\\sum _{\\ell =0}^j \\binom{j}{\\ell } d^{j-\\ell } v^{\\ell -j}e^{-sd/v} \\sum _{0\\le k<H_d} (p^bk/v)^{\\ell } e^{(-p^bs/v+iw_d)k}.\\end{split}$ Therefore, using partial summation, as in the proof of lemma REF , we obtain $\\begin{split}&\|\\mathcal {E}_m(s,\\chi ,v,K)\| \\\\&\\le \\sum _{j>m} \|c_j(s)\| \\sum _{d=0}^{p^b-1}\\delta _{\\gcd (v+d,p)=1}\\sum _{\\ell =0}^j \\binom{j}{\\ell } (d/v)^{j-\\ell } e^{-\\sigma d/v}\|\\sum _{0\\le k<H_d} (p^bk/v)^{\\ell } e^{(-p^bs/v+iw_d)k}\|\\\\&\\le \\sum _{j>m} \|c_j(s)\| \\sum _{d=0}^{p^b-1} \\delta _{\\gcd (v+d,p)=1}((d+p^b(H_d-1))/v)^j\\max _{x\\in [0,H_d]} \|\\sum _{x\\le k<H_d} e^{i(-p^bt/v+w_d)k}\|\\\\&\\le \\sum _{j>m} \|c_j(s)\|(K-1)^j/v^j\\sum _{d=0}^{p^b-1}\\delta _{\\gcd (v_r+d,p)=1}\\max _{x\\in [0,H_d]} \|\\sum _{x\\le k<H_d} e^{i(-p^bt/v+w_d)k}\|\\\\&\\le \\epsilon _m(s,u) \\sum _{d=0}^{p^b-1}\\delta _{\\gcd (v_r+d,p)=1}\|\\csc (w_d/2-p^bt/(2v))\|.\\end{split}$ Combined with the trivial estimate, this yields the lemma.", "We divide the main sum in (REF ) according to the positions of $v_r$ as before: $\\sum _{1\\le n<v_0} \\chi (n)n^{-s} + \\sum _{0\\le r\\le R} v_r^{-s} \\sum _{0\\le k<K_r}\\chi (v+r)e^{-s\\log (1+k/v_r)}$ .", "We apply lemmas REF and REF to the sum over $k$ .", "This yields the result with $\\mathcal {T}_{M,m}(s,\\chi ,u_0,v_0)=\\sum _{0\\le r\\le R} v_r^{-s}\\mathcal {E}_m(s,\\chi ,v_r,K_r)$ .", "By the estimate for $\\mathcal {E}_m(s,\\chi ,v,K)$ in lemma REF we obtain the desired bound on $\\mathcal {T}_{M,m}(s,\\chi , u_0,v_0)$ ." ], [ "Computing $\\displaystyle \\frac{f_s^{(j)}(0)}{j!}\\frac{g_K^{(j)}(z)}{v^j}$ \nfor {{formula:c127c662-4020-4860-ba4e-d2c127932f3c}}", "One can choose the parameters in Theorems REF & REF so that one can achieve moderate accuracy with $m\\le 8$ , say.", "So, in general, computing $(f_s^{(j)}(0)/j!", ")(g_K^{(j)}(z)/v^j)$ will be quite easy, and can be done using closed-form formulas to evaluate the geometric sum.", "The methods that we present below are intended for when $j$ is large, but they can be used for any $j\\ge 0$ .", "In our application (Theorems REF & REF ), we have $(K-1)/v \\le 1/u_0\\le 1/(2\\sqrt{\\mathfrak {q}(s)})$ , and $-1/2\\le \\Re (z)\\le 0$ .", "So we will assume that this holds throughout.", "We recall that $f_s(z) = e^{sz^2/2-sz^3/3+\\cdots } = \\sum _{j\\ge 0} \\frac{f^{(j)}_s(0)}{j!}", "z^j$ for $\|z\|<1$ .", "For example, $\\begin{split}&f_s(0) = 1,\\qquad f^{(1)}_s(0) = 0,\\qquad f^{(2)}_s(0) = s,\\qquad f^{(3)}_s(0)= -2s,\\\\&f^{(4)}_s(0) = 3s(2+s),\\qquad f^{(5)}_s(0) = -4s(6+5s),\\\\& f^{(6)}_s(0) = 5s(24+26s+3s^2),\\qquad f^{(7)}_s(0) = -6s(120+154s+35s^2),\\\\&f^{(8)}_s(0) = 7s(720+1044s+340s^2+15s^3),\\quad \\ldots .\\end{split}$ To find $f^{(j)}_s(0)$ in general, let $q(z):=\\sum _{\\alpha \\ge 2} (-1)^{\\alpha }z^{\\alpha }/\\alpha $ , so $f_s^{(j)}(z) = Q_{s,j}(z) e^{sq(z)}$ for some $Q_{s,j}(z) = \\sum _{l\\ge 0} w_{j,l}(s) z^l$ that satisfies the recursion $Q_{s,0}(z) := 1$ and $Q_{s,j+1}(z) = \\frac{d}{dz} Q_{s,j}(z) +s Q_{s,j}(z)\\frac{d}{dz}q(z)$ .", "Therefore, $w_{j,0}(s) = f^{(j)}_s(0)$ , $w_{0,0}(s) = 1$ , $w_{0,l}(s)=0$ for $l>0$ , and $w_{j+1,l}(s) = (l+1)w_{j,l+1}(s) -s\\sum _{\\alpha =1}^l (-1)^{\\alpha } w_{j,l-\\alpha }(s)$ .", "Using this recursion, one can find all of $f^{(j)}_s(0)=w_{j,0}(s)$ for $0\\le j\\le m$ in about $(m+1)^2$ steps.", "In carrying out the recursion, one may treat $s$ symbolically, so $w_{j,0}(s)$ is viewed as a polynomial in $s$ and the recursion is finding the coefficients of this polynomial.", "In fact, it follows from the recursion that, more generally, $w_{j,l}(s)$ is a polynomial in $s$ of degree $\\le \\min \\lbrace (j+l)/2,j\\rbrace $ .", "So we may write $w_{j,l}(s)=\\sum _{0\\le \\eta \\le j/2} \\beta _{j,l,\\eta } s^{\\eta }$ .", "Also, $\\beta _{0,0,0} = 1$ , $\\beta _{0,0,\\eta }=0$ for $\\eta >0$ , $\\beta _{0,l,\\eta }=0$ for $l>0$ , and we have $\\beta _{j+1,l,\\eta } = (l+1)\\beta _{j,l+1,\\eta } -\\sum _{\\alpha =1}^l (-1)^{\\alpha } \\beta _{j,l-\\alpha ,\\eta -1}$ .", "Therefore, using induction, we obtain the bound $\|\\beta _{j,l,\\eta }\|\\le (j+l)!2^{l+1}/l!$ .", "In particular, $\|\\beta _{j,0,\\eta }\|/j!\\le 2$ .", "Thus, the number of bits needed to represent $\|\\beta _{j,0,\\eta }\|/j!$ , and hence to compute $f_s^{(j)}(0)/j!$ as a polynomial in $s$ , to a given precision, is also well-controlled.", "As for computing $g_K^{(j)}(z)$ , one can use the formula $g_K^{(j)}(z) = \\sum _{\\ell =0}^j \\binom{j}{\\ell } w^{(j-\\ell )}(z)y^{(\\ell )}(z)$ , where $w(z) := e^{Kz}-1$ and $y(z):=(e^z-1)^{-1}$ .", "So for $z\\notin 2\\pi i \\mathbb {Z}$ we have $\\begin{split}\\frac{f_s^{(j)}(0)}{j!}", "\\frac{g_K^{(j)}(z)}{v^j} &=\\frac{f_s^{(j)}(0)}{j!", "}\\frac{2^j(K-1)^j}{v^j} \\frac{g_K^{(j)}(z)}{2^j(K-1)^j} \\\\&=\\frac{f_s^{(j)}(0)}{j!}", "\\frac{2^j(K-1)^j}{v^j}\\left( e^{Kz}\\sum _{\\ell =0}^j \\frac{1}{2^j}\\binom{j}{\\ell }\\frac{y^{(\\ell )}(z)}{(K-1)^{\\ell }} - \\frac{y^{(j)}(z)}{(K-1)^j}\\right).\\end{split}$ The factor $2^{-j}$ is inserted inside the sum in (REF ) in order to to control the size of the binomial coefficient $\\binom{j}{l}\\le 2^j$ .", "By hypothesis, $(K-1)/v\\le 1/u_0\\le 1/(2\\sqrt{\\mathfrak {q}(s)})$ .", "So, recalling that $f_s^{(j)}(0) = \\sum _{0\\le \\eta \\le j/2} \\beta _{j,0,\\eta }s^{\\eta }$ , $\|\\beta _{j,0,\\eta }\|\\le 2(j!", ")$ , and $\\mathfrak {q}(s)\\ge 3$ , we obtain $\|f_s^{(j)}(0) 2^j(K-1)^j\|/(j!v^j)\\le 5$ .", "In particular, the number of bits needed to represent the outside factor in (REF ) is well-controlled, and we may focus on computing the sum enclosed in parentheses.", "To that end, we consider the computation of $y^{(\\ell )}(z)/(K-1)^{\\ell }$ in (REF ).", "If $\\ell $ is small, this can be done by directly differentiating $y(z)$ , but this is not a practical method if $\\ell $ is large.", "Instead, we note that $zy(z)=z/(e^z-1)$ is the exponential generating function for the Bernoulli numbers, specifically, $y(z)= \\frac{1}{z} -\\frac{1}{2} + \\sum _{l=1}^{\\infty } \\frac{B_{2l}}{(2l)!", "}z^{2l-1},\\quad 0<\|z\|<2\\pi .$ Therefore, for $\\ell >0$ , $y^{(\\ell )}(z) = \\frac{(-1)^{\\ell } \\ell !", "}{z^{\\ell +1}} +\\sum _{l=\\lceil (\\ell +1)/2\\rceil }^{\\infty }\\frac{B_{2l}}{2l} \\frac{z^{2l-\\ell -1}}{(2l-\\ell -1)!", "},\\quad 0<\|z\|<2\\pi .$ Using the periodicity of $e^z$ , and our assumption on $z$ , we can ensure that the argument given to $g_K^{(j)}(z)$ satisfies $\|z\| < 3\\pi /2$ .", "Thus, the above formulas will suffice to compute $y^{(\\ell )}(z)/(K-1)^{\\ell }$ provided that $\|z\|$ is sufficiently bounded away from 0, say $\|z\|> (m+1)/(K-1)$ .", "For such $z$ , and assuming that $K> 2\\pi (m+1)$ (otherwise, we may compute $g_K^{(j)}(z)$ by direct summation in $\\ll m+1$ steps), we obtain that $y^{(\\ell )}(z)/(K-1)^{\\ell }$ is bounded by a constant, and so its size is well-controlled.", "Thus, the only remaining case is when $\|z\|<(m+1)/(K-1)$ , with $K>2\\pi (m+1)$ .", "In this case, we use the Euler-Maclaurin summation.", "To this end, let $h_{j,z}(x) :=x^je^{z x}$ .", "Then $g_K^{(j)}(z) = \\sum _{0\\le k < K} h_{j,z}(k)$ .", "Note that, using the periodicity of $e^{zk}$ and conjugating if necessary, we may assume that $0\\le \\Im (z) \\le \\pi $ .", "By the Euler-Maclaurin formula (see [14]), we have $\\begin{split}g_K^{(j)}(z) &= \\int _0^{K-1} h_{j,z}(x)\\,dx+ \\sum _{\\ell =1}^L \\frac{B_{2\\ell }}{(2\\ell ) !", "}(h_{j,z}^{(2\\ell -1)}(K-1)-h_{j,z}^{(2\\ell -1)}(0)) \\\\&+ \\frac{1}{2}(h_{j,z}(K-1)+ h_{j,z}(0))+\\mathcal {E}_{K,j,z,L},\\end{split}$ where $h_{j,z}^{(2\\ell -1)}(x)$ is the $(2\\ell -1)$ -st derivative of $h_{j,z}(x)$ with respect to $x$ , and the remainder term $\\mathcal {E}_{K,j,z,L} = (-1/(2L)!", ")\\int _0^{K-1}B_{2L}(\\lbrace x\\rbrace )h_{j,z}^{(2L)}(x)\\,dx$ , where $B_{2L}(x)$ is the $2L$ -th Bernoulli polynomial (e.g.", "$B_2(x) = x^2-x+1/6$ ), and $\\lbrace x\\rbrace $ is the factional part of $x$ .", "Now, $h_{j,z}^{(2\\ell -1)}(x)= \\sum _{l=0}^{2\\ell -1} \\binom{2\\ell -1}{l}(\\frac{d^l}{dx^l} x^j) (\\frac{d^{2\\ell -l-1}}{dx^{2\\ell -l-1}} e^{zx})$ .", "Thus, we have $h_{j,z}^{(2\\ell -1)}(x) = e^{zx} \\sum _{l=0}^{\\min \\lbrace 2\\ell -1,j\\rbrace } \\binom{2\\ell -1}{l}\\frac{j!}{(j-l)!}", "x^{j-l} z^{2\\ell -l-1}.$ Also, from the Fourier expansion for $B_{2L}(\\lbrace x\\rbrace )$ (see [14]), $\|B_{2L}(\\lbrace x\\rbrace )\|\\le 4\\frac{(2L)!", "}{(2\\pi )^{2L}}$ .", "Therefore, since $\\Re (z)\\le 0$ , we deduce that $\|\\mathcal {E}_{K,j,z,L}\|/(K-1)^j \\le 4(K-1)(2\\pi )^{-2L}$ , which decays exponentially with $L$ .", "As for the main term $\\int _0^{K-1} h_{j,z}(x)\\,dx$ in formula (REF ), its computation does not present any difficulty since $\|z\|<m/K$ (so $z$ is small).", "For example, one can split the interval of integration into $m+1$ consecutive subintervals of equal length, then, after a suitable change of variable, apply Taylor expansions to the integrand in each subinterval, which reduces the problem to integrating polynomials.", "Alternatively, one can use a numerical quadrature rule." ], [ "A convexity bound", "We will use the following well-spacing lemma to prove corollary REF .", "Lemma 5.1 Let $\\lbrace x_n, n=0,1,\\ldots \\rbrace $ be a set of real numbers.", "Suppose there exists a positive integer $Q$ such that $\\min _{n\\ne n^{\\prime }}\|x_n-x_{n^{\\prime }}\| \\ge 1/(2Q)$ .", "Then, for any $y\\ge x$ and any $P\\ge 1$ , we have $\\sum _{x_n \\in [x,y]}\\min \\lbrace P,\|\\csc (\\pi x_n)\|\\rbrace \\le (1+ \\lfloor y-x\\rfloor )(2(A+1)P+2Q\\log (Q/A)),$ where $A$ is any positive integer that satisfies $A \\le Q/P$ .", "Since $\|x_n-x_{n^{\\prime }}\| \\ge 1/(2Q)$ for $n\\ne n^{\\prime }$ , then for any integer $k$ we have $\\sum _{x_n\\in [k-1/2,k+1/2]} \\min \\lbrace P,\|\\csc (\\pi x_n)\|\\rbrace \\le P+\\sum _{\|l\|\\le Q} \\min \\lbrace P,\|\\csc (\\pi l/(2Q))\|\\rbrace =:$ .", "Using the inequality $\|\\sin (\\pi \\alpha )\|\\ge 2\|\\alpha \|$ , $-1/2\\le \\alpha \\le 1/2$ , we obtain that $\\le 2(A+1)P+ \\sum _{A< \|l\|\\le Q} Q/\|l\|$ .", "Combined with the inequality $\\sum _{A<l\\le Q} 1/l \\le \\log (Q/A)$ , this gives $\\le 2(A+1)P+2Q\\log (Q/A)$ .", "Since the interval $[x,y]$ contains $\\le 1+ \\lfloor y-x\\rfloor $ integers, the lemma follows.", "The bound that we obtain in corollary REF for zeta is, of course, superseded by the bound that one can obtain from the Riemann-Siegel formula.", "Nevertheless, it illustrates that Theorem REF yields a convexity bound of similar strength to the Riemann-Siegel formula, up to a constant factor, even though it is quite elementary.", "Corollary 5.2 $\|\\zeta (1/2+it)\|\\ll \\mathfrak {q}(1/2+it)^{1/4}$ .", "We will use Theorem REF , but replacing $\\mathcal {R}_M(s)$ by the correction terms from the Euler-Maclaurin formula for $\\zeta (s)$ (see the paragraph following the statement of the theorem).", "We take $s=1/2+it$ , $m=0$ , $v_0 = u_0 = 4\\lceil \\sqrt{t}\\rceil $ , $M=10 \\lceil t\\rceil $ , and assume that $t\\ge 36$ , as we may.", "Given our choice of $M$ , it is not hard to show that the Euler-Maclaurin correction terms contribute $\\ll 1$ .", "And given our choice of $u_0$ , we have $\\epsilon _0(s,u_0)\\ll 1$ .", "By routine calculations, $\\sum _{n=1}^{v_0-1} n^{-1/2} \\le 2\\sqrt{v_0}$ and $\|\\sum _{r=0}^R g_{K_r}(-s/v_r)v_r^{-s}\| \\le \\mathcal {B}_M(s,u_0,v_0)$ .", "Thus, $\|\\zeta (1/2+it)\|\\ll \\sqrt{v_0}+\\mathcal {B}_M(1/2+it,u_0,v_0).$ It is helpful to recall that $K_r=\\lceil v_r/u_0\\rceil $ for $r<R$ , $v_{r+1}=v_r+K_r$ , and $\\mathcal {B}_M(s,u_0,v_0)\\le \\sum _{r=0}^R\\min \\lbrace K_r,\|\\csc (t/(2v_r))\|\\rbrace v_r^{-\\sigma }$ .", "So, letting $I_{\\ell } := (2^{\\ell }u_0, 2^{\\ell +1}u_0]$ , we see that if $v_r\\in I_{\\ell }$ , then $2^{\\ell }<K_r \\le 2^{\\ell +1}$ .", "We let $I_{\\ell _0}$ denote the interval containing $M$ , so $M\\le 2^{\\ell _0+1}u_0$ .", "Then, using simple estimates, we obtain $\\mathcal {B}_M(s,u_0,v_0)\\le u_0^{-1/2}\\sum _{\\ell = 0}^{\\ell _0} 2^{-\\ell /2}\\sum _{v_r\\in I_{\\ell }} \\min \\lbrace 2^{\\ell +1},\|\\csc (t/(2v_r)\|\\rbrace .$ Now, consider that for $v_r,v_{r+1}\\in I_{\\ell }$ , with $r<R$ , we have $\\begin{split}\\frac{t}{2 v_r}-\\frac{t}{2 v_{r+1}} =\\frac{tK_r}{2 v_rv_{r+1}} \\ge \\frac{t2^{\\ell }}{2^{2\\ell +3} u_0^2} \\ge \\frac{t}{2^{\\ell +3}u_0^2}\\ge \\frac{\\pi }{B2^{\\ell +2}},\\end{split}$ where $B$ is the smallest positive integer such that $t/(2\\pi u_0^2)\\ge 1/B$ .", "Note that, since $u_0=4\\lceil \\sqrt{t}\\rceil $ , then $B\\ll 1$ .", "Also, as $v_r$ ranges over $I_{\\ell }$ , the argument $t/(2\\pi v_r)$ moves by increments $\\ge 1/(B2^{\\ell +2})$ , and it spans an interval of length $\\le t/(2\\pi 2^{\\ell } u_0) - t/(2\\pi 2^{\\ell +1}u_0)= t/(\\pi 2^{\\ell +2}u_0)$ .", "Therefore, applying lemma REF to the set $\\lbrace v_r\\in I_{\\ell }\\rbrace $ with $Q=B2^{\\ell +1}$ , $A=B$ , and $P=2^{\\ell +1}$ , we obtain $\\begin{split}\\sum _{v_r\\in I_{\\ell }} \\min \\lbrace 2^{\\ell +1},\|\\csc (t/(2(v_r))\|\\rbrace \\le & (1+ t/(\\pi 2^{\\ell +2}u_0))(2(B+1)2^{\\ell +1}\\\\&+ B2^{\\ell +2}\\log (2^{\\ell +1}))\\ll t(\\ell +1)/u_0.\\end{split}$ It follows that $\\mathcal {B}_M(s,u_0,v_0)\\le (t/(u_0\\sqrt{v_0})\\sum _{\\ell =0}^{\\ell _0} (\\ell +1)/2^{-\\ell /2} \\ll t/(u_0\\sqrt{v_0})$ .", "So we conclude, $\\zeta (1/2+it) \\ll \\sqrt{v_0} + t/(u_0\\sqrt{v_0}) \\ll t^{1/4}$ ." ], [ "Parameter choices", "Theorem REF offers a simple method for computing $\\zeta (\\sigma +it)$ with an explicit error bound.", "The control over the error term in the theorem goes beyond what the Riemann-Siegel asymptotic formula enables.", "Theorem REF achieves the same for $L(\\sigma +it,\\chi )$ when $\\chi $ is power-full.", "We implemented a basic version of Theorem REF in Mathematica 9, which is an application for computation, see http://www.wolfram.com/mathematica/.", "This was sufficient for our purposes as we were mainly interested in learning about reasonable choices of the parameters.", "This way, we could appraise the accuracy and running time in practice.", "The Mathematica notebook containing the implementation is available at https://people.math.osu.edu/hiary.1/.", "Our computation relies on finite precision arithmetic, which introduces round-off errors.", "Such errors become significant for large $t$ .", "This is primarily because the computation of $t\\log n\\bmod {2\\pi }$ will contain only a few correct digits for large $t$ .", "In general, one cannot expect more than $\\pm \\epsilon _{mach}\\, t\\log n$ accuracy when computing $e^{it\\log n}$ , where $\\epsilon _{mach}$ is the machine epsilon.", "So if $t>1/\\epsilon _{mach}$ say, then, certainly, numerical results will not be meaningful.", "To overcome this problem, one could switch to an arithmetic system with a smaller machine epsilon (but having a slower performance).", "Assuming that round-off errors behave like independent random variables, which is a reasonable model, the accumulated round-off error in computing $\\sum _{n<M} n^{-1/2-it}$ will be typically like $\\pm \\epsilon _{mach}\\,t(\\sum _{n<M} (\\log n)^2/n)^{1/2}$ .", "For double-precision arithmetic, $\\epsilon _{mach} = 2^{-52} \\approx 2 \\times 10^{-16}$ .", "So, if we use double-precision arithmetic with $t=10^d$ and $M\\approx 10t$ , the accumulated round-off error will be like $\\pm 10^{d-16} \\log 10^{3(d+1)/2}$ .", "With this in mind, we obtained marginally better control over the round-off errors by using the main sum from the Euler-Maclaurin formula with 6 correction terms, and with $M=10\\lceil \\mathfrak {q}(s)\\rceil $ , in particular we did not need to take $M$ very large.", "We computed $g_K(z)$ using the formula $(e^{Kz}-1)/(e^z-1)$ when $\|z\|>10(m+1)/(K-1)$ (as is typically the case), and using the Mathematica built-in Euler-Maclaurin summation routine when $\|z\|<10(m+1)/(K-1)$ .", "To check the accuracy of the results, we compared them with the outputs from lcalc and the Mathematica built-in zeta routine, leading to Table REF .", "We attempted to increase the accuracy by inputting $t$ in Mathematica using a higher precision.", "However, it is likely that Mathematica still uses double-precision arithmetic in intermediate steps and some built-in routines.", "So the accuracy of many stages of the computation will be limited by the machine epsilon for double-precision numbers.", "The error entries in Table REF are significantly smaller than the explicit bound for $\\mathcal {T}_{M,m}(s,u_0,v_0)$ given in Theorem REF .", "For example, when $t=10^{10}$ and $m=6$ , the explicit bound gives $\|\\mathcal {T}_{M,m}(s,u_0,v_0)\|\\le 2.9\\times 10^{-3}$ (here, we calculated $\\mathcal {B}_M(s,u0,v0)$ directly).", "This is significantly larger than the observed error $1.9\\times 10^{-10}$ in Table REF .", "This is not surprising, and is due to the pseudo-random nature of round-off errors.", "Table: Error for various tt and mm, and using σ=1/2\\sigma =1/2,u 0 =6⌈𝔮(s)⌉u_0 = 6\\lceil \\sqrt{\\mathfrak {q}(s)}\\rceil , and v 0 =10(m+1)u 0 v_0=10(m+1)u_0.There was no attempt to optimize our implementation since, in any case, it is not competitive with an implementation directly in C/C++.", "With our parameter choices, and for large $t$ , the implementation was slower by factor of about $2(m+1)^2\\log t$ compared to computing the main sum in a Riemann-Siegel formula directly (in both cases we input $t$ in higher precision than double-precision).", "The implementation was faster by a factor of about $10 \\sqrt{t}/((m+1)^2\\log t)$ than computing $\\sum _{n\\le M} n^{-s}$ directly (this is essentially the main sum in the Euler-Maclaurin formula).", "It might be possible to speed up the implementation by a factor of $m+1$ if the derivatives $g_K^{(j)}(z)$ , $0\\le j\\le m$ , are computed simultaneously via a recursion.", "One can also save a factor of 2 by choosing $u_0=3\\lceil \\sqrt{\\mathfrak {q}(s)}\\rceil $ instead of $u_0=6\\lceil \\sqrt{\\mathfrak {q}(s)}\\rceil $ , at the expense of a larger truncation error $\\mathcal {T}_{M,m}(s,u_0,v_0)$ ." ] ]	1403.0317
[ [ "Parameterized Algorithms for Graph Partitioning Problems" ], [ "Abstract We study a broad class of graph partitioning problems, where each problem is specified by a graph $G=(V,E)$, and parameters $k$ and $p$.", "We seek a subset $U\\subseteq V$ of size $k$, such that $\\alpha_1m_1 + \\alpha_2m_2$ is at most (or at least) $p$, where $\\alpha_1,\\alpha_2\\in\\mathbb{R}$ are constants defining the problem, and $m_1, m_2$ are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in $U$, respectively.", "This class of fixed cardinality graph partitioning problems (FGPP) encompasses Max $(k,n-k)$-Cut, Min $k$-Vertex Cover, $k$-Densest Subgraph, and $k$-Sparsest Subgraph.", "Our main result is an $O^(4^{k+o(k)}\\Delta^k)$ algorithm for any problem in this class, where $\\Delta \\geq 1$ is the maximum degree in the input graph.", "This resolves an open question posed by Bonnet et al.", "[IPEC 2013].", "We obtain faster algorithms for certain subclasses of FGPPs, parameterized by $p$, or by $(k+p)$.", "In particular, we give an $O^(4^{p+o(p)})$ time algorithm for Max $(k,n-k)$-Cut, thus improving significantly the best known $O^(p^p)$ time algorithm." ], [ "Introduction", "Graph partitioning problems arise in many areas including VLSI design, data mining, parallel computing, and sparse matrix factorizations (see, e.g., [1], [12], [7]).", "We study a broad class of graph partitioning problems, where each problem is specified by a graph $G\\!=\\!", "(V,\\!E)$ , and parameters $k$ and $p$ .", "We seek a subset $U\\!\\subseteq \\!V$ of size $k$ , such that $\\alpha _1m_1 \\!+\\!", "\\alpha _2m_2$ is at most (or at least) $p$ , where $\\alpha _1,\\!\\alpha _2\\!\\in \\!\\mathbb {R}$ are constants defining the problem, and $m_1, m_2$ are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in $U$ , respectively.", "This class encompasses such fundamental problems as Max and Min $(k,\\!n\\!-\\!k)$ -Cut, Max and Min $k$ -Vertex Cover, $k$ -Densest Subgraph, and $k$ -Sparsest Subgraph.", "For example, Max $(k,\\!n\\!-\\!k)$ -Cut is a max-FGPP (i.e., maximization FGPP) satisfying $\\alpha _1\\!=\\!0$ and $\\alpha _2\\!=\\!1$ , Min $(k,\\!n\\!-\\!k)$ -Cut is a min-FGPP (i.e., minimization FGPP) satisfying $\\alpha _1\\!=\\!0$ and $\\alpha _2\\!=\\!1$ , and Min $k$ -Vertex Cover is a min-FGPP satisfying $\\alpha _1\\!=\\!\\alpha _2\\!=\\!1$ .", "A parameterized algorithm with parameter $k$ has running time $O^\\!", "(\\!f(k))$ for some function $f$ , where $O^$ hides factors polynomial in the input size.", "In this paper, we develop a parameterized algorithm with parameter $(k+\\Delta )$ for the class of all FGPPs, where $\\Delta \\!", "\\ge \\!", "1$ is the maximum degree in the graph $G$ .", "For certain subclasses of FGPPs, we develop algorithms parameterized by $p$ , or by $(k+p)$ .", "Related Work: Parameterized by $k$ , Max and Min $(\\!k,\\!n\\!-\\!k\\!", ")$ -Cut, and Max and Min $k$ -Vertex Cover are W[1]-hard [8], [4], [11].", "Moreover, $k$ -Clique and $k$ -Independent Set, two well-known W[1]-hard problems [9], are special cases of $k$ -Densest Subgraph where $p\\!=\\!k(k\\!-\\!1)$ , and $k$ -Sparsest Subgraph where $p\\!=\\!0$ , respectively.", "Therefore, parameterized by $(k\\!+\\!p)$ , $k$ -Densest Subgraph and $k$ -Sparsest Subgraph are W[1]-hard.", "Cai et al.", "[5] and Bonnet et al.", "[2] studied the parameterized complexity of FGPPs with respect to $(k\\!+\\!\\Delta )$ .", "Cai et al.", "[5] gave $O^\\!", "(2^{(k\\!+\\!1)\\Delta })$ time algorithms for $k$ -Densest Subgraph and $k$ -Sparsest Subgraph.", "Recently, Bonnet et al.", "[2] presented an $O^\\!", "(\\Delta ^k)$ time algorithm for degrading FGPPs.", "This subclass includes max-FGPPs in which $\\alpha _1\\!/2\\!\\le \\!", "\\alpha _2$ , and min-FGPPs in which $\\alpha _1\\!/2\\!\\ge \\!", "\\alpha _2$ .A max-FGPP (min-FGPP) is non-degrading if $\\alpha _1/2\\ge \\alpha _2$ ($\\alpha _1/2\\le \\alpha _2$ ).", "They also proposed an $O^\\!", "(k^{2k}\\Delta ^{2k})$ time algorithm for all FGPPs, and posed as an open question the existence of constants $a$ and $b$ such that any FGPP can be solved in time $O^\\!", "(a^k\\!\\Delta ^{bk})$ .", "In this paper we answer this question affirmatively, by developing an $O^\\!", "(4^{k+o(k)}\\!\\Delta ^k\\!", ")$ time algorithm for any FGPP.", "Parameterized by $p$ , Max and Min $k$ -Vertex Cover can be solved in times $O^\\!", "(1.396^p)$ and $O^\\!", "(4^p)$ , respectively, and in randomized times $O^\\!", "(1.2993^p)$ and $O^\\!", "(3^p)$ , respectively [14].", "Moreover, Max $(k,\\!n\\!-\\!k)$ Cut can be solved in time $O^\\!", "(p^p)$ [2], and Min $(k,\\!n\\!-\\!k)$ Cut can be solved in time $O(2^{O(p^3)})$ [6].", "Parameterized by $(k\\!+\\!p)$ , Min $(k,\\!n\\!-\\!k)$ Cut can be solved in time $O^\\!(k^{2k}(k\\!+\\!p)^{2k}\\!", ")$ [2].", "We note that the parameterized complexity of FGPPs has also been studied with respect to other parameters, such as the treewidth and the vertex cover number of $G$ (see, e.g., [13], [3], [2]).", "Contribution: Our main result is an $O^\\!", "(4^{k+o(k)}\\!\\Delta ^k\\!", ")$ time algorithm for the class of all FGPPs, answering affirmatively the question posed by Bonnet et al.", "[2] (see Section ).", "In Section , we develop an $O^\\!", "(4^{p+o(p)})$ time algorithm for Max $(k,\\!n\\!-\\!k)$ -Cut, that significantly improves the $O^\\!", "(p^p)$ running time obtained in [2].", "We also obtain (in Section ) an $O^\\!", "(2^{k+\\frac{p}{\\alpha _2}+o(k+p)})$ time algorithm for the subclass of positive min-FGPPs, in which $\\alpha _1\\!\\ge \\!0$ and $\\alpha _2\\!>\\!0$ .", "Finally, we develop (in Section ) a faster algorithm for non-degarding positive min-FGPPs (i.e., min-FGPPs satisfying $\\alpha _2\\!\\ge \\!\\frac{\\alpha _1}{2}\\!>\\!0$ ).", "In particular, we thus solve Min $k$ -Vertex Cover in time $O^\\!", "(2^{p+o(p)})$ , improving the previous randomized $O^\\!", "(3^p)$ time algorithm.", "Techniques: We obtain our main result by establishing an interesting reduction from non-degrading FGPPs to the Weighted $k^{\\prime }$ -Exact Cover ($k^{\\prime }$ -WEC) problem (see Section ).", "Building on this reduction, combined with an algorithm for degrading FGPPs given in [2], and an algorithm for $k^{\\prime }$ -WEC given in [18], we develop an algorithm for any FGPP.", "To improve the running time of our algorithm, we use a fast construction of representative families [10], [17].", "In designing algorithms for FGPPs, parameterized by $p$ or $(k+p)$ , we use as a key tool randomized separation [5] (see Sections –).", "Roughly speaking, randomized separation finds a `good' partition of the nodes in the input graph $G$ via randomized coloring of the nodes in red or blue.", "If a solution exists, then, with some positive probability, there is a set $X$ of only red nodes that is a solution, such that all the neighbors of nodes in $X$ that are outside $X$ are blue.", "Our algorithm for Max $(k,\\!n\\!-\\!k)$ -Cut makes non-standard use of randomized separation, in requiring that only some of the neighbors outside $X$ of nodes in $X$ are blue.", "This yields the desired improvement in the running time of our algorithm.", "Our algorithm for non-degrading positive FGPPs is based on a somewhat different application of randomized separation, in which we randomly color edges rather than the nodes.", "If a solution exists, then, with some positive probability, there is a node-set $X$ that is a solution, such that some edges between nodes in $X$ are red, and all edges between nodes in $X$ and nodes outside $X$ are blue.", "In particular, we require that the subgraph induced by $X$ , and the subgraph induced by $X$ from which we delete all blue edges, contain the same connected components.", "We derandomize our algorithms using universal sets [16].", "Notation: Given a graph $G\\!=\\!", "(V,\\!E)$ and a subset $X\\!\\subseteq \\!", "V$ , let $E(X)$ denote the set of edges in $E$ having both endpoints in $X$ , and let $E(X,\\!V\\!\\setminus \\!", "X)$ denote the set of edges in $E$ having exactly one endpoint in $X$ .", "Moreover, given a subset $X\\!\\subseteq \\!V$ , let $\\mathrm {val}(X)\\!=\\!", "\\alpha _1\|E(X)\|\\!+\\!\\alpha _2\|E(X,\\!V\\!\\setminus \\!X)\|$ ." ], [ "Solving FGPPs in Time $O^(4^{k+o(k)}\\Delta ^k)$", "In this section we develop an $O^\\!", "(4^{k+o(k)}\\Delta ^k)$ time algorithm for the class of all FGPPs.", "We use the following steps.", "In Section REF we show that any non-degrading FGPP can be reduced to the Weighted $k^{\\prime }$ -Exact Cover ($k^{\\prime }$ -WEC) problem, where $k^{\\prime }=k$ .", "Applying this reduction, we then show (in Section REF ) how to decrease the size of instances of $k^{\\prime }$ -WEC, by using representative families.", "Finally, we show (in Section REF ) how to solve any FGPP by using the results in Sections REF and REF , an algorithm for $k^{\\prime }$ -WEC, and an algorithm for degrading FGPPs given in [2]." ], [ "From Non-Degrading FGPPs to ", "We show below that any non-degrading max-FGPP can be reduced to the maximization version of $k^{\\prime }$ -WEC.", "Given a universe $U$ , a family ${\\cal S}$ of nonempty subsets of $U$ , a function $w\\!", ": {\\cal S}\\!\\rightarrow \\!\\mathbb {R}$ , and parameters $k^{\\prime }\\!\\in \\!\\mathbb {N}$ and $p^{\\prime }\\!\\in \\!\\mathbb {R}$ , we seek a subfamily ${\\cal S}^{\\prime }$ of disjoint sets from ${\\cal S}$ satisfying $\|\\bigcup {\\cal S}^{\\prime }\|=k^{\\prime }$ whose value, given by $\\sum _{S\\in {\\cal S}^{\\prime }}w(S)$ , is at least $p^{\\prime }$ .", "Any non-degrading min-FGPP can be similarly reduced to the minimization version of $k^{\\prime }$ -WEC.", "Let $\\Pi $ be a max-FGPP satisfying $\\frac{\\alpha _1}{2}\\!\\ge \\!\\alpha _2$ .", "Given an instance ${\\cal I}\\!=\\!(G\\!=\\!", "(V,\\!E),\\!k,\\!p)$ of $\\Pi $ , we define an instance $f({\\cal I})\\!=\\!(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ of the maximization version of $k^{\\prime }$ -WEC as follows.", "$U\\!=\\!V$ .", "${\\cal S}\\!=\\!\\bigcup _{i=1}^k{\\cal S}_i$ , where ${\\cal S}_i$ contains the node-set of any connected subgraph of $G$ on exactly $i$ nodes.", "$\\forall S\\!\\in \\!", "{\\cal S}: w(S) = \\mathrm {val}(S)$ .", "$k^{\\prime }\\!=\\!k$ , and $p^{\\prime }\\!=\\!p$ .", "We illustrate the reduction in Figure REF (see Appendix ).", "We first prove that our reduction is valid.", "${\\cal I}$ is a yes-instance iff $f({\\cal I})$ is a yes-instance.", "First, assume there is a subset $X\\!\\subseteq \\!", "V$ of size $k$ satisfying $\\mathrm {val}(X)\\!\\ge \\!p$ .", "Let $G_1\\!=\\!", "(V_1,\\!E_1),\\ldots ,G_t\\!=\\!", "(V_t,\\!E_t)$ , for some $1\\!\\le \\!", "t\\!\\le \\!", "k$ , be the maximal connected components in the subgraph of $G$ induced by $X$ .", "Then, for all $1\\!\\le \\!", "\\ell \\!\\le \\!", "t$ , $V_{\\ell }\\!\\in \\!", "{\\cal S}$ .", "Moreover, $\\displaystyle {\\sum _{\\ell =1}^t\|V_{\\ell }\|\\!=\\!\|X\|\\!=\\!k^{\\prime }}$ , and $\\displaystyle {\\sum _{\\ell =1}^tw(V_{\\ell })\\!=\\!\\mathrm {val}(X)\\!\\ge \\!p^{\\prime }}$ .", "Now, assume there is a subfamily of disjoint sets $\\lbrace S_1,\\ldots ,S_t\\rbrace \\!\\subseteq \\!", "{\\cal S}$ , for some $1\\!\\le \\!", "t\\!\\le \\!", "k$ , such that $\\displaystyle {\\sum _{\\ell =1}^t\|S_{\\ell }\|\\!=\\!k^{\\prime }}$ and $\\displaystyle {\\sum _{\\ell =1}^tw(S_{\\ell })\\!\\ge \\!p^{\\prime }}$ .", "Thus, there are connected subgraphs $G_1\\!=\\!", "(V_1,\\!E_1),\\ldots ,G_t\\!=\\!", "(V_t,\\!E_t)$ of $G$ , such that $V_{\\ell }\\!=\\!S_{\\ell }$ , for all $1\\!\\le \\!", "\\ell \\!\\le \\!", "t$ .", "Let $X_{\\ell }\\!=\\!\\bigcup _{j=\\ell }^tV_j$ , for all $1\\!\\le \\ell \\!\\le \\!", "t$ .", "Clearly, $\|X_1\|\\!=\\!k$ .", "Since $\\frac{\\alpha _1}{2}\\!\\ge \\!\\alpha _2$ , we get that $\\begin{array}{ll}\\end{array}\\mathrm {val}(X_1) &= \\mathrm {val}(V_1) \\!+\\!", "\\mathrm {val}(X_2) \\!+\\!", "\\alpha _1\|E(V_1,X_2)\| \\!-\\!", "2\\alpha _2\|E(V_1,X_2)\|\\\\$ val(V1) + val(X2) = val(V1) + val(V2) + val(X3) + 1\|E(V2,X3)\| - 22\|E(V2,X3)\| val(V1) + val(V2) + val(X3) ... =1tval(V).", "$Thus, $ val(X1)=1tw(V)p$.$ We now bound the number of connected subgraphs in $G$ .", "[[15]] There are at most $4^i(\\Delta \\!-\\!1)^i\|V\|$ connected subgraphs of $G$ on at most $i$ nodes, which can be enumerated in time $O(4^i(\\Delta \\!-\\!1)^i(\|V\|\\!+\\!\|E\|)\|V\|)$ .", "Thus, we have the next result.", "The instance $f({\\cal I})$ can be constructed in time $O(4^k(\\Delta \\!-\\!1)^k(\|V\|\\!+\\!\|E\|)\|V\|)$ .", "Moreover, for any $1\\!\\le \\!i\\!\\le \\!k$ , $\|{\\cal S}_i\|\\!\\le \\!", "4^i(\\Delta \\!-\\!1)^i\|V\|$ ." ], [ "Decreasing the Size of Inputs for ", "In this section we develop a procedure, called Decrease, which decreases the size of an instance $\\!(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ of $k^{\\prime }$ -WEC.", "To this end, we find a subfamily $\\widehat{\\cal S}\\!\\subseteq \\!", "{\\cal S}$ that contains \"enough\" sets from ${\\cal S}$ , and thus enables to replace ${\\cal S}$ by $\\widehat{\\cal S}$ without turning a yes-instance to a no-instance.", "The following definition captures such a subfamily $\\widehat{\\cal S}$ .", "Given a universe $E$ , nonnegative integers $k$ and $p$ , a family ${\\cal S}$ of subsets of size $p$ of $E$ , and a function $w\\!:\\!", "{\\cal S}\\!\\rightarrow \\!\\mathbb {R}$ , we say that a subfamily $\\widehat{\\cal S}\\!\\subseteq \\!", "{\\cal S}$ max (min) represents $\\cal S$ if for any pair of sets $X\\!\\in \\!", "{\\cal S}$ , and $Y\\!\\subseteq \\!E\\!\\setminus \\!X$ such that $\|Y\|\\!\\le \\!k\\!-\\!p$ , there is a set $\\widehat{X}\\!\\in \\!\\widehat{\\cal S}$ disjoint from $Y$ such that $w(\\!\\widehat{X}\\!", ")\\!\\ge \\!w(\\!X\\!", ")$ ($w(\\!\\widehat{X}\\!", ")\\!\\le \\!w(\\!X\\!", ")$ ).", "The following result states that small representative families can be computed efficiently.This result builds on a powerful construction technique for representative families presented in [10].", "[[17]] Given a constant $c\\!\\ge \\!1$ , a universe $E$ , nonnegative integers $k$ and $p$ , a family ${\\cal S}$ of subsets of size $p$ of $E$ , and a function $w\\!:\\!", "{\\cal S}\\!\\rightarrow \\!\\mathbb {R}$ , a subfamily $\\widehat{\\cal S}\\!\\subseteq \\!", "{\\cal S}$ of size at most $\\displaystyle {\\frac{(ck)^{k}}{p^p(ck\\!-\\!p)^{k\\!-\\!p}}2^{o(k)}\\!\\log \\!\|E\|}$ that max (min) represents $\\cal S$ can be computed in time $\\displaystyle {O(\\!\|{\\cal S}\|(ck/(ck\\!-\\!p))^{k\\!-\\!p}2^{o(k)}\\!\\log \\!\|E\|\\!", "+\\!", "\|{\\cal S}\|\\!\\log \\!\|{\\cal S}\|)}$ .", "We next consider the maximization version of $k^{\\prime }$ -WEC and max representative families.", "The minimization version of $k^{\\prime }$ -WEC can be similarly handled by using min representative families.", "Let RepAlg$(E,\\!k,\\!p,\\!", "{\\cal S},\\!w)$ denote the algorithm in Theorem REF where $c\\!=\\!2$ , and let ${\\cal S}_i\\!=\\!\\lbrace S\\!\\in \\!", "{\\cal S}\\!", ": \|S\|\\!=\\!i\\rbrace $ , for all $1\\!\\le \\!i\\!\\le \\!k^{\\prime }$ .", "We now present procedure Decrease (see the pseudocode below), which replaces each family ${\\cal S}_i$ by a family $\\widehat{\\cal S}_i\\!\\subseteq \\!", "{\\cal S}_i$ that represents ${\\cal S}_i$ .", "First, we state that procedure Decrease is correct (the proof is given in Appendix ).", "algorithmProcedure [!ht] Decrease($U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime }$ ) [1] for $i=1,2,\\ldots ,k^{\\prime }$ do $\\widehat{\\cal S}_i\\Leftarrow $ RepAlg($U,\\!k^{\\prime },\\!i,\\!", "{\\cal S}_i,\\!w$ ).", "end for $\\widehat{\\cal S}\\Leftarrow \\bigcup _{i=1}^k\\widehat{\\cal S}_i$ .", "return ($U,\\!\\widehat{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime }$ ).", "$(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ is a yes-instance iff $(U,\\!\\widehat{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ is a yes-instance.", "Theorem REF immediately implies the following result.", "Procedure Decrease runs in time $\\displaystyle {O(\\sum _{i=1}^{k^{\\prime }}(\|{\\cal S}_i\|(\\frac{2k^{\\prime }}{2k^{\\prime }\\!-\\!i})^{k^{\\prime }-i}2^{o(k^{\\prime })}\\!\\log \\!\|U\|}$ $+ \|{\\cal S}_i\|\\!\\log \\!\|{\\cal S}_i\|))$ .", "Moreover, $\\displaystyle {\|\\widehat{\\cal S}\| \\le \\sum _{i=1}^{k^{\\prime }}\\frac{(2k^{\\prime })^{k^{\\prime }}}{i^i(2k^{\\prime }\\!-\\!i)^{k^{\\prime }-i}}2^{o(k^{\\prime })}\\!\\log \\!\|U\| \\le 2.5^{k^{\\prime }+o(k^{\\prime })}\\!\\log \\!\|U\|}$ ." ], [ "An Algorithm for Any FGPP", "We now present FGPPAlg, which solves any FGPP in time $O^(4^{k+o(k)}\\Delta ^k)$ .", "Assume w.l.o.g that $\\Delta \\!\\ge \\!2$ , and let DegAlg($G,\\!k,\\!p$ ) denote the algorithm solving any degrading FGPP in time $O((\\Delta \\!+\\!1)^{k+1}\|V\|)$ , given in [2].", "The algorithm given in Section 5 of [18] solves a problem closely related to $k^{\\prime }$ -WEC, and can be easily modified to solve $k^{\\prime }$ -WEC in time $O(2.851^{k^{\\prime }}\|{\\cal S}\|\|U\|\\cdot $ $\\log ^2\|U\|)$ .", "We call this algorithm WECAlg($U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime }$ ).", "Let $\\Pi $ be an FGPP having parameters $\\alpha _1$ and $\\alpha _2$ .", "We now describe algorithm FGPPAlg (see the pseudocode below).", "First, if $\\Pi $ is a degrading FGPP, then FGPPAlg solves $\\Pi $ by calling DegAlg.", "Otherwise, by using the reduction $f$ , FGPPAlg transforms the input into an instance of $k^{\\prime }$ -WEC.", "Then, FGPPAlg decreases the size of the resulting instance by calling the procedure Decrease.", "Finally, FGPPAlg solves $\\Pi $ by calling WECAlg.", "algorithmAlgorithm [!ht] FGPPAlg($G\\!=\\!", "(V,\\!E),\\!k,\\!p$ ) [1] ($\\Pi $ is a max-FGPP and $\\frac{\\alpha _1}{2}\\!\\le \\!", "\\alpha _2$ ) or ($\\Pi $ is a min-FGPP and $\\frac{\\alpha _1}{2}\\!\\ge \\!", "\\alpha _2$ ) accept iff DegAlg$(G,\\!k,\\!p)$ accepts.", "$(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })\\Leftarrow f(G,\\!k,\\!p)$ .", "$(U,\\!\\widehat{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })\\Leftarrow $ Decrease$(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ .", "accept iff WECAlg($U,\\!\\widehat{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime }$ ) accepts.", "Algorithm FGPPAlg solves $\\Pi $ in time $O(4^{k+o(k)}\\Delta ^k(\|V\|\\!+\\!\|E\|)\|V\|)$ .", "The correctness of the algorithm follows immediately from Lemmas REF and REF , and the correctness of DegAlg and WECAlg.", "By Lemmas REF and REF , and the running times of DegAlg and WECAlg, algorithm FGPPAlg runs in time $\\begin{array}{lll}\\end{array}& &\\displaystyle {O(4^k(\\Delta \\!-\\!1)^k(\|V\|\\!+\\!\|E\|)\|V\| + \\sum _{i=1}^k(4^i(\\Delta \\!-\\!1)^i\|V\|(\\frac{2k}{2k-i})^{k-i}2^{o(k)}\\!\\log \\!\|V\|)}\\\\& & \\displaystyle {+\\ 2.851^k2.5^{k+o(k)}\|V\|\\log ^3\|V\|)}\\\\$ = O(4kk(\|V\|+\|E\|)\|V\| + 2o(k)\|V\|\|V\|[01{4(22-)1-}]k) = O(4kk(\|V\|+\|E\|)\|V\| + 4k+o(k)k\|V\|\|V\|) = O(4k+o(k)k(\|V\|+\|E\|)\|V\|).", "$\\vspace{-5.0pt}$ We give below an $O^\\!", "(4^{p+o(p)})$ time algorithm for Max $(k,n\\!-\\!k)$ Cut.", "In Section REF we show that it suffices to consider an easier variant of Max $(k,n\\!-\\!k)$ Cut, that we call NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "We solve this variant in Section REF .", "Finally, our algorithm for Max $(k,n\\!-\\!k)$ Cut is given in Section REF ." ], [ "Simplifying ", "We first define an easier variant of Max $(k,\\!n\\!-\\!k)$ Cut.", "Given a graph $G\\!=\\!", "(V,\\!E)$ in which each node is either red or blue, and positive integers $k$ and $p$ , NC-Max $(k,\\!n\\!-\\!k)$ -Cut asks if there is a subset $X\\!\\subseteq \\!", "V$ of exactly $k$ red nodes and no blue nodes, such that at least $p$ edges in $E(X,\\!V\\!\\setminus \\!", "X)$ have a blue endpoint.", "Given an instance $(G,\\!k,\\!p)$ of Max $(k,\\!n\\!-\\!k)$ Cut, we perform several iterations of coloring the nodes in $G$ ; thus, if $(G,\\!k,\\!p)$ is a yes-instance, we generate at least one yes-instance of NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "To determine how to color the nodes in $G$ , we need the following definition of universal sets.", "Let ${\\cal F}$ be a set of functions $f\\!:\\!", "\\lbrace 1,\\!2,\\!\\ldots ,\\!n\\rbrace \\rightarrow \\lbrace 0,\\!1\\rbrace $ .", "We say that ${\\cal F}$ is an $(n,t)$ -universal set if for every subset $I\\!\\subseteq \\!\\lbrace 1,\\!2,\\!\\ldots ,\\!n\\rbrace $ of size $t$ and a function $f^{\\prime }\\!", ":\\!I\\!\\rightarrow \\!\\lbrace 0,\\!1\\rbrace $ , there is a function $f\\!\\in \\!", "{\\cal F}$ such that for all $i\\!\\in \\!I$ , $f(i)\\!=\\!f^{\\prime }(i)$ .", "The following result asserts that small universal sets can be computed efficiently.", "[[16]] There is an algorithm, UniSetAlg, that given a pair of integers $(n,\\!t)$ , computes an $(n,\\!t)$ -universal set ${\\cal F}$ of size $2^{t\\!+\\!o(t)}\\!\\log \\!", "n$ in time $O\\!", "(2^{t\\!+\\!o(t)}n\\!\\log \\!n\\!", ")$ .", "We now present ColorNodes (see the pseudocode below), a procedure that given an input ($G,\\!k,\\!p,\\!q$ ), where ($G,\\!k,\\!p$ ) is an instance of Max $(k,\\!n\\!-\\!k)$ Cut and $q\\!=\\!k+p$ , returns a set of instances of NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "Procedure ColorNodes first constructs a $(\|V\|,\\!k\\!+\\!p)$ -universal set $\\cal F$ .", "For each $f\\!\\in \\!", "{\\cal F}$ , ColorNodes generates a colored copy $V^f$ of $V$ .", "Then, ColorNodes returns a set $\\cal I$ , including the resulting instances of NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "algorithmProcedure [!ht] ColorNodes($G\\!=\\!", "(V,E),k,p,q$ ) [1] let $V\\!=\\!\\lbrace v_1,v_2,\\ldots ,v_{\|V\|}\\rbrace $ .", "${\\cal F}\\Leftarrow $ UniSetAlg$(\|V\|,q)$ .", "$f\\!\\in \\!", "{\\cal F}$ let $V^f\\!=\\!\\lbrace v^f_1,v^f_2,\\ldots ,v^f_{\|V\|}\\rbrace $ , where $v^f_i$ is a copy of $v_i$ .", "$i=1,2,\\ldots ,\|V\|$ if $f(i)\\!=\\!0$ then color $v^f_i$ red.", "else color $v^f_i$ blue.", "end if return ${\\cal I}=\\lbrace (G_f\\!=\\!", "(V_f,E),k,p): f\\!\\in \\!", "{\\cal F}\\rbrace $ .", "The next lemma states the correctness of procedure ColorNodes.", "An instance $(G,\\!k,\\!p)$ of Max $(k,\\!n\\!-\\!k)$ -Cut is a yes-instance iff ColorNodes($G,\\!k,\\!p,\\!k\\!+\\!p$ ) returns a set ${\\cal I}$ containing at least one yes-instance of NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "If $(G,\\!k,\\!p)$ is a no-instance of Max $(k,\\!n\\!-\\!k)$ -Cut, then clearly, for any coloring of the nodes in $V$ , we get a no-instance of NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "Next suppose that $(G,\\!k,\\!p)$ is a yes-instance, and let $X$ be a set of $k$ nodes in $V$ such that $\|E(X,\\!V\\!\\setminus \\!X)\|\\!\\ge \\!p$ .", "Note that there is a set $Y$ of at most $p$ nodes in $V\\!\\setminus \\!X$ such that $\|E(X,\\!Y)\|\\!\\ge \\!p$ .", "Let $X^{\\prime }$ and $Y^{\\prime }$ denote the indices of the nodes in $X$ and $Y$ , respectively.", "Since ${\\cal F}$ is a $(\|V\|,\\!k\\!+\\!p)$ -universal set, there is $f\\!\\in \\!", "{\\cal F}$ such that: (1) for all $i\\!\\in \\!X^{\\prime }$ , $f(i)\\!=\\!0$ , and (2) for all $i\\!\\in \\!Y^{\\prime }$ , $f(i)\\!=\\!1$ .", "Thus, in $G_f$ , the copies of the nodes in $X$ are red, and the copies of the nodes in $Y$ are blue.", "We get that $(G_f,\\!k,\\!p)$ is a yes-instance of NC-Max $(k,\\!n\\!-\\!k)$ -Cut.", "Furthermore, Lemma REF immediately implies the following result.", "Procedure ColorNodes runs in time $O(2^{q+o(q)}\|V\|\\log \\!\|V\|)$ , and returns a set ${\\cal I}$ of size $O(2^{q+o(q)}\\log \\!\|V\|)$ ." ], [ "A Procedure for ", "We now present SolveNCMaxCut, a procedure for solving NC-Max $(k,\\!n\\!-\\!k)$ -Cut (see the pseudocode below).", "Procedure SolveNCMaxCut orders the red nodes in $V$ according to the number of their blue neighbors in a non-increasing manner.", "If there are at least $k$ red nodes, and the number of edges between the first $k$ red nodes and blue nodes is at least $p$ , procedure SolveNCMaxCut accepts; otherwise, procedure SolveNCMaxCut rejects.", "algorithmProcedure [!ht] SolveNCMaxCut($G\\!=\\!", "(V,E),k,p$ ) [1] for all red $v\\!\\in \\!", "V$ do compute the number $n_b(v)$ of blue neighbors of $v$ in $G$ .", "end for let $v_1,\\!v_2,\\!\\ldots ,\\!v_r$ , for some $0\\le r\\le \|V\|$ , denote the red nodes in $V$ , such that $n_b(v_i)\\ge n_b(v_{i+1})$ for all $1\\!\\le \\!", "i\\!\\le \\!", "r\\!-\\!1$ .", "accept iff ($r\\!\\ge \\!k$ and $\\sum _{i=1}^kn_b(v_i)\\!\\ge \\!", "p$ ).", "Clearly, the following result concerning SolveNCMaxCut is correct.", "Procedure SolveNCMaxCut solves NC-Max $(k,\\!n\\!-\\!k)$ -Cut in time $O(\|V\|\\!\\log \|V\|\\!+\\!\|E\|)$ ." ], [ "An Algorithm for ", "Assume w.l.o.g that $G$ has no isolated nodes.", "Our algorithm, MaxCutAlg, for Max $(k,n\\!-\\!k)$ Cut, proceeds as follows.", "First, if $p\\!<\\!\\min \\lbrace k,\|V\|\\!-\\!k\\rbrace $ , then MaxCutAlg accepts, and if $\|V\|\\!-\\!k < k$ , then MaxCutAlg calls itself with $\|V\|\\!-\\!k$ instead of $k$ .", "Then, MaxCutAlg calls ColorNodes to compute a set of instances of NC-Max $(k,\\!n\\!-\\!k)$ -Cut, and accepts iff SolveNCMaxCut accepts at least one of them.", "algorithmAlgorithm [!ht] MaxCutAlg($G\\!=\\!", "(V,E),k,p$ ) [1] if $p < \\min \\lbrace k,\|V\|\\!-\\!k\\rbrace $ then accept.", "end if if $\|V\|\\!-\\!k < k$ then accept iff MaxCutAlg($G,\|V\|\\!-\\!k,p$ ) accepts.", "end if ${\\cal I}\\Leftarrow $ ColorNodes($G,k,p,k\\!+\\!p$ ).", "$(G^{\\prime },k^{\\prime },p^{\\prime })\\in {\\cal I}$ if SolveNCMaxCut$(G^{\\prime },k^{\\prime },p^{\\prime })$ accepts then accept.", "end if reject.", "The next lemma implies the correctness of Step REF in MaxCutAlg.", "[[2]] In a graph $G\\!=\\!", "(V,\\!E)$ having no isolated nodes, there is a subset $X\\!\\subseteq \\!V$ of size $k$ such that $\|E(X,\\!V\\!\\setminus \\!", "X)\|\\ge \\min \\lbrace k,\\!\|V\|\\!-\\!k\\rbrace $ .", "Our main result is the following.", "Algorithm MaxCutAlg solves Max $(k,\\!n\\!-\\!k)$ Cut in time $O(4^{p\\!+\\!o(p)}\\cdot $ $(\|V\|+\\!\|E\|)\\log ^2\|V\|)$ .", "Clearly, ($G,k,p$ ) is a yes-instance iff ($G,\|V\|\\!-\\!k,p$ ) is a yes-instance.", "Thus, Lemmas REF , REF and REF immediately imply the correctness of MaxCutAlg.", "Denote $m\\!=\\!\\min \\lbrace k,\\!\|V\|\\!-\\!k\\rbrace $ .", "If $p \\!<\\!", "m$ , then MaxCutAlg runs in time $O(1)$ .", "Next suppose that $p \\!\\ge \\!", "m$ .", "Then, by Lemmas REF and REF , MaxCutAlg runs in time $O(2^{m+p+o(m+p)}(\|V\|\\!+\\!\|E\|)\\log ^2\|V\|)=O(4^{p+o(p)}(\|V\|\\!+\\!\|E\|)\\log ^2\|V\|)$ ." ], [ "Solving Positive Min-FGPPs in Time $O^\\!(2^{k+\\frac{p}{\\alpha _2}+o(k+p)})$", "Let $\\Pi $ be a min-FGPP satisfying $\\alpha _1\\!\\ge \\!0$ and $\\alpha _2\\!>\\!0$ .", "In this section we develop an $O^\\!", "(2^{k+\\frac{p}{\\alpha _2}+o(k+p)})$ time algorithm for $\\Pi $ .", "Using randomized separation, we show in Section REF that we can focus on an easier version of $\\Pi $ .", "We solve this version in Section REF , using dynamic programming.", "Then, Section REF gives our algorithm." ], [ "Simplifying the Positive Min-FGPP $\\Pi $", "We first define an easier variant of $\\Pi $ .", "Given a graph $G\\!=\\!", "(V,\\!E)$ in which each node is either red or blue, and parameters $k\\!\\in \\!\\mathbb {N}$ and $p\\!\\in \\!\\mathbb {R}$ , NC-$\\Pi $ asks if there is a subset $X\\!\\subseteq \\!", "V$ of exactly $k$ red nodes and no blue nodes, whose neighborhood outside $X$ includes only blue nodes, such that val$(X)\\!\\le \\!p$ .", "The simplification process is similar to that performed in Section REF .", "However, we now use the randomized separation procedure ColorNodes, defined in Section REF , with instances of $\\Pi $ , and consider the set ${\\cal I}$ returned by ColorNodes as a set of instances of NC-$\\Pi $.", "We next prove that ColorNodes is correct.", "An instance $(G,\\!k,\\!p)$ of $\\Pi $ is a yes-instance iff ColorNodes($G,\\!k,\\!p,\\!k\\!+\\!\\frac{p}{\\alpha _2}$ ) returns a set ${\\cal I}$ containing at least one yes-instance of NC-$\\Pi $.", "If $(G,\\!k,\\!p)$ is a no-instance of $\\Pi $, then clearly, for any coloring of the nodes in $V$ , we get a no-instance of NC-$\\Pi $.", "Next suppose that $(G,\\!k,\\!p)$ is a yes-instance, and let $X$ be a set of $k$ nodes in $V$ such that val$(X)\\!\\le \\!p$ .", "Let $Y$ denote the neighborhood of $X$ outside $X$ .", "Note that $\|Y\|\\!\\le \\!", "\\frac{p}{\\alpha _2}$ .", "Let $X^{\\prime }$ and $Y^{\\prime }$ denote the indices of the nodes in $X$ and $Y$ , respectively.", "Since ${\\cal F}$ is a $(\|V\|,\\!k\\!+\\!\\frac{p}{\\alpha _2})$ -universal set, there is $f\\!\\in \\!", "{\\cal F}$ such that: (1) for all $i\\!\\in \\!X^{\\prime }$ , $f(i)\\!=\\!0$ , and (2) for all $i\\!\\in \\!Y^{\\prime }$ , $f(i)\\!=\\!1$ .", "Thus, in $G_f$ , the copies of the nodes in $X$ are red, and the copies of the nodes in $Y$ are blue.", "We get that $(G_f,\\!k,\\!p)$ is a yes-instance of NC-$\\Pi $." ], [ "A Procedure for ", "We now present SolveNCP, a dynamic programming-based procedure for solving NC-$\\Pi $ (see the pseudocode below).", "Procedure SolveNCP first computes the node-sets of the maximal connected red components in $G$ .", "Then, procedure SolveNCP generates a matrix M, where each entry $[i,j]$ holds the minimum value val$(X)$ of a subset $X\\!\\subseteq \\!", "V$ in $Sol_{i,j}$ , the family containing every set of exactly $j$ nodes in $V$ obtained by choosing a union of sets in $\\lbrace C_1,C_2\\ldots ,C_i\\rbrace $ , i.e., $Sol_{i,j}\\!=\\!\\lbrace (\\bigcup {\\cal C}^{\\prime }): {\\cal C}^{\\prime }\\subseteq \\lbrace C_1,C_2,\\ldots ,C_i\\rbrace ,\|\\bigcup {\\cal C}^{\\prime }\|=j\\rbrace $ .", "Procedure SolveNCP computes M by using dynamic programming, assuming an access to a non-existing entry returns $\\infty $ , and accepts iff M$[t,k]\\!\\le \\!p$ .", "algorithmProcedure [!ht] SolveNCP($G\\!=\\!", "(V,E),k,p$ ) [1] use DFS to compute the family ${\\cal C}\\!=\\!\\lbrace C_1,C_2,\\ldots ,C_t\\rbrace $ , for some $0\\!\\le \\!", "t\\!\\le \\!", "\|V\|$ , of the node-sets of the maximal connected red components in $G$ .", "let M be a matrix containing an entry $[i,j]$ for all $0\\!\\le \\!", "i\\!\\le \\!", "t$ and $0\\!\\le \\!", "j\\!\\le \\!", "k$ .", "initialize M$[i,0]\\Leftarrow 0$ for all $0\\!\\le \\!i\\!\\le \\!", "t$ , and M$[0,j]\\Leftarrow \\infty $ for all $1\\!\\le \\!j\\!\\le \\!k$ .", "$i\\!=\\!1,2,\\ldots ,t,$ and $j\\!=\\!1,2,\\ldots ,k$ M$[i,j]\\Leftarrow \\min \\lbrace \\mathrm {M}[i\\!-\\!1,j],\\mathrm {M}[i\\!-\\!1,j\\!-\\!\|C_i\|]+\\mathrm {val}(C_i)\\rbrace $ .", "accept iff M$[t,k]\\!\\le \\!p$ .", "The following lemma states the correctness and running time of SolveNCP.", "Procedure SolveNCP solves NC-$\\Pi $ in time $O(\|V\|k\\!+\\!\|E\|)$ .", "For all $0\\!\\le \\!", "i\\!\\le \\!", "t$ and $0\\!\\le \\!", "j\\!\\le \\!", "k$ , denote val$(i,\\!j)\\!=\\!\\min _{X\\in Sol_{i,j}}\\lbrace \\mathrm {val}(X)\\rbrace $ .", "Using a simple induction on the computation of M, we get that M$[i,\\!j]\\!=\\!\\mathrm {val}(i,\\!j)$ .", "Since $(G,\\!k,\\!p)$ is a yes-instance of NC-$\\Pi $ iff val$(t,\\!k)\\!\\le \\!", "p$ , we have that SolveNCP is correct.", "Step REF , and the computation of val$(C)$ for all $C\\!\\in \\!", "{\\cal C}$ , are performed in time $O(\|V\|\\!+\\!\|E\|)$ .", "Since M is computed in time $O(\|V\|k)$ , we have that SolveNCP runs in time $O(\|V\|k\\!+\\!\|E\|)$ ." ], [ "An Algorithm for $\\Pi $", "We now conclude PAlg, our algorithm for $\\Pi $ (see the pseudocode below).", "Algorithm PAlg calls ColorNodes to compute several instances of NC-$\\Pi $, and accepts iff SolveNCP accepts at least one of them.", "algorithmAlgorithm [!ht] PAlg($G\\!=\\!", "(V,E),k,p$ ) [1] ${\\cal I}\\Leftarrow $ ColorNodes($G,k,p,k+\\frac{p}{\\alpha _2}$ ).", "$(G^{\\prime },k^{\\prime },p^{\\prime })\\in {\\cal I}$ if SolveNCP$(G^{\\prime },k^{\\prime },p^{\\prime })$ accepts then accept.", "end if reject.", "By Lemmas REF , REF and REF , we have the following result.", "Algorithm PAlg solves $\\Pi $ in time $O(2^{k+\\frac{p}{\\alpha _2}\\!+\\!o(k+p)}(\|V\|\\!+\\!\|E\|)\\!\\log \\!\|V\|)$ ." ], [ "Solving a Subclass of Positive Min-LGPPs Faster", "Let $\\Pi $ be a min-FGPP satisfying $\\alpha _2\\!\\ge \\!\\frac{\\alpha _1}{2}\\!>\\!0$ .", "Denote $x\\!=\\!\\max \\lbrace \\frac{p}{\\alpha _2},\\min \\lbrace \\frac{p}{\\alpha _1},\\frac{p}{\\alpha _2}\\!+\\!", "(1\\!-\\!\\frac{\\alpha _1}{\\alpha _2})k\\rbrace \\rbrace $ .", "In this section we develop an $O^(2^{x+o(x)})$ time algorithm for $\\Pi $, that is faster than the algorithm in Section .", "Applying a divide-and-conquer step to the edges in the input graph $G$ , Section REF shows that we can focus on an easier version of $\\Pi $.", "This version is solved in Section REF by using dynamic programming.", "We give the algorithm in Section REF ." ], [ "Simplifying the Non-Degrading Positive Min-FGPP $\\Pi $", "We first define an easier variant of $\\Pi $.", "Suppose we are given a graph $G\\!=\\!", "(V,\\!E)$ in which each edge is either red or blue, and parameters $k\\in \\mathbb {N}$ and $p\\in \\mathbb {R}$ .", "For any subset $X\\!\\subseteq \\!V$ , let C$(X)$ denote the family containing the node-sets of the maximal connected components in the graph $G_r\\!=\\!", "(X,E_r)$ , where $E_r$ is the set of red edges in $E$ having both endpoints in $X$ .", "Also, let val$^(X)\\!=\\!\\sum _{C\\in \\mathrm {C}(X)}\\mathrm {val}(C)$ .", "The variant EC-$\\Pi $ asks if there is a subset $X\\!\\subseteq \\!", "V$ of exactly $k$ nodes, such that all the edges in $E(X,V\\!\\setminus \\!", "X)$ are blue, and val$^(X)\\!\\le \\!p$ .", "We now present a procedure, called ColorEdges (see the pseudocode below), whose input is an instance ($G,\\!k,\\!p$ ) of $\\Pi $.", "Procedure ColorEdges uses a universal set to perform several iterations coloring the edges in $G$ , and then returns the resulting set of instances of EC-$\\Pi $.", "algorithmProcedure [!ht] ColorEdges($G\\!=\\!", "(V,E),k,p$ ) [1] let $E\\!=\\!\\lbrace e_1,e_2,\\ldots ,e_{\|E\|}\\rbrace $ .", "${\\cal F}\\Leftarrow $ UniSetAlg$(\|E\|,x)$ .", "$f\\!\\in \\!", "{\\cal F}$ let $E^f\\!=\\!\\lbrace e^f_1,e^f_2,\\ldots ,e^f_{\|E\|}\\rbrace $ , where $e^f_i$ is a copy of $e_i$ .", "$i=1,2,\\ldots ,\|E\|$ if $f(i)\\!=\\!0$ then color $e^f_i$ red.", "else color $e^f_i$ blue.", "end if return ${\\cal I}=\\lbrace (G_f\\!=\\!", "(V,E_f),k,p): f\\!\\in \\!", "{\\cal F}\\rbrace $ .", "The following lemma states the correctness of ColorEdges.", "An instance $(G,\\!k,\\!p)$ of $\\Pi $ is a yes-instance iff ColorEdges($G,\\!k,\\!p$ ) returns a set ${\\cal I}$ containing at least one yes-instance of EC-$\\Pi $.", "Since $\\alpha _2\\!\\ge \\!\\frac{\\alpha _1}{2}$ , val$^(X)\\!\\ge \\!\\mathrm {val}(X)$ for any set $X\\!\\subseteq \\!V$ and coloring of edges in $E$ .", "Thus, if $(G,\\!k,\\!p)$ is a no-instance of $\\Pi $, then clearly, for any coloring of edges in $E$ , we get a no-instance of EC-$\\Pi $.", "Next suppose that $(G,\\!k,\\!p)$ is a yes-instance, and let $X$ be a set of $k$ nodes in $V$ such that val$(X)\\!\\le \\!p$ .", "Let $\\widetilde{E}_r\\!=\\!E(X)$ , and $E_b\\!=\\!E(X,\\!V\\!\\setminus \\!", "X)$ .", "Also, choose a minimum-size subset $E_r\\!\\subseteq \\!\\widetilde{E}_r$ such that the graphs $G_r^{\\prime }\\!=\\!", "(X,\\widetilde{E}_r)$ and $G_r\\!=\\!", "(X,E_r)$ contain the same set of maximal connected components.", "Let $E_r^{\\prime }$ and $E_b^{\\prime }$ denote the indices of the edges in $E_r$ and $E_b$ , respectively.", "Note that $\|E_r^{\\prime }\|\\!+\\!\|E_b^{\\prime }\|\\!\\le \\!x$ .", "Since ${\\cal F}$ is an $(\|E\|,x)$ -universal set, there is $f\\!\\in \\!", "{\\cal F}$ such that: (1) for all $i\\!\\in \\!E_r^{\\prime }$ , $f(i)\\!=\\!0$ , and (2) for all $i\\!\\in \\!E_b^{\\prime }$ , $f(i)\\!=\\!1$ .", "Thus, in $G_f$ , the copies of the edges in $E_r$ are red, and the copies of the edges in $E_b$ are blue.", "Then, val$^(X)\\!=\\!\\mathrm {val}(X)$ .", "We get that $(G_f,\\!k,\\!p)$ is a yes-instance of EC-$\\Pi $.", "Furthermore, Lemma REF immediately implies the following result.", "Procedure ColorEdges runs in time $O(2^{x+o(x)}\|E\|\\!\\log \\!\|E\|)$ , and returns a set ${\\cal I}$ of size $O(2^{x+o(x)}\\!\\log \\!\|E\|)$ ." ], [ "A Procedure for ", "By modifying the procedure given in Section REF , we get a procedure, called SolveECP, satisfying the following result (see Appendix ).", "Procedure SolveECP solves EC-$\\Pi $ in time $O(\|V\|k\\!+\\!\|E\|)$ ." ], [ "A Faster Algorithm for $\\Pi $", "Our faster algorithm for $\\Pi $ , FastPAlg, calls ColorEdges to compute several instances of EC-$\\Pi $, and accepts iff SolveECP accepts at least one of them (see the pseudocode below).", "algorithmAlgorithm [!ht] FastPAlg($G\\!=\\!", "(V,E),k,p$ ) [1] ${\\cal I}\\Leftarrow $ ColorEdges($G,k,p$ ).", "$(G^{\\prime },k^{\\prime },p^{\\prime })\\in {\\cal I}$ if SolveECP$(G^{\\prime },k^{\\prime },p^{\\prime })$ accepts then accept.", "end if reject.", "By Lemmas REF , REF and REF , we have the following result.", "Algorithm FastPAlg solves $\\Pi $ in time $O(2^{x+o(x)}(\|V\|k\\!+\\!\|E\|)\\!\\log \\!\|E\|)$ .", "Since Min $k$ -Vertex Cover satisfies $\\alpha _1\\!=\\!\\alpha _2\\!=\\!1$ , we have the following result.", "Algorithm FastPAlg solves Min $k$ -Vertex Cover in time $O(2^{p+o(p)}(\|V\|k\\!+\\!\|E\|)\\!\\log \\!\|E\|)$ ." ], [ "An Illustration of the Reduction $f$", " Figure: An illustration of the reduction ff, given in Section ." ], [ "A Procedure for ", "We now present the details of procedure SolveECP (see the pseudocode below).", "Procedure SolveECP first computes the node-sets of the maximal connected components in the graph obtained by removing all the blue edges from $G$ .", "Then, procedure SolveECP generates a matrix M, where each entry $[i,j]$ holds the minimum value val$^(X)$ of a subset $X\\!\\subseteq \\!", "V$ in $Sol_{i,j}$ , the family containing every set of exactly $j$ nodes in $V$ obtained by choosing a union of sets in $\\lbrace C_1,C_2\\ldots ,C_i\\rbrace $ , i.e., $Sol_{i,j}\\!=\\!\\lbrace (\\bigcup {\\cal C}^{\\prime }): {\\cal C}^{\\prime }\\subseteq \\lbrace C_1,C_2,\\ldots ,C_i\\rbrace ,\|\\bigcup {\\cal C}^{\\prime }\|=j\\rbrace $ .", "Procedure SolveNCP computes M by using dynamic programming, assuming an access to a non-existing entry returns $\\infty $ , and accepts iff M$[t,k]\\!\\le \\!p$ .", "algorithmProcedure [!ht] SolveECP($G\\!=\\!", "(V,E),k,p$ ) [1] use DFS to compute the family ${\\cal C}\\!=\\!\\lbrace C_1,C_2,\\ldots ,C_t\\rbrace $ , for some $0\\!\\le \\!", "t\\!\\le \\!", "\|V\|$ , of the node-sets of the maximal connected components in the graph obtained by removing all the blue edges from $G$ .", "let M be a matrix containing an entry $[i,j]$ for all $0\\!\\le \\!", "i\\!\\le \\!", "t$ and $0\\!\\le \\!", "j\\!\\le \\!", "k$ .", "initialize M$[i,0]\\Leftarrow 0$ for all $0\\!\\le \\!i\\!\\le \\!", "t$ , and M$[0,j]\\Leftarrow \\infty $ for all $1\\!\\le \\!j\\!\\le \\!k$ .", "$i\\!=\\!1,2,\\ldots ,t,$ and $j\\!=\\!1,2,\\ldots ,k$ M$[i,j]\\Leftarrow \\min \\lbrace \\mathrm {M}[i\\!-\\!1,j],\\mathrm {M}[i\\!-\\!1,j\\!-\\!\|C_i\|]+\\mathrm {val}^(C_i)\\rbrace $ .", "accept iff M$[t,k]\\!\\le \\!p$ .", "We next prove the correctness of Lemma REF .", "For all $0\\!\\le \\!", "i\\!\\le \\!", "t$ and $0\\!\\le \\!", "j\\!\\le \\!", "k$ , denote val$(i,\\!j)\\!=\\!\\min _{X\\in Sol_{i,j}}\\lbrace \\mathrm {val}^(X)\\rbrace $ .", "Using a simple induction on the computation of M, we get that M$[i,\\!j]\\!=\\!\\mathrm {val}(i,\\!j)$ .", "Since $(G,\\!k,\\!p)$ is a yes-instance of EC-$\\Pi $ iff val$(t,\\!k)\\!\\le \\!", "p$ , we have that SolveECP is correct.", "Step , and the computation of val$^*(C)$ for all $C\\!\\in \\!", "{\\cal C}$ , are performed in time $O(\|V\|\\!+\\!\|E\|)$ .", "Since M is computed in time $O(\|V\|k)$ , we have that SolveECP runs in time $O(\|V\|k\\!+\\!\|E\|)$ ." ], [ "Some Proofs", "Proof of lemma REF : First, assume that $(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ is a yes-instance.", "Let ${\\cal S}^{\\prime }$ be a subfamily of disjoint sets from ${\\cal S}$ , such that $\|\\bigcup {\\cal S}^{\\prime }\|\\!=\\!k^{\\prime }$ , $\\sum _{S\\in {\\cal S}^{\\prime }}w(S)\\!\\ge \\!p^{\\prime }$ , and there is no subfamily ${\\cal S}^{\\prime \\prime }$ satisfying these conditions, and $\|{\\cal S}^{\\prime }\\!\\cap \\!\\widehat{\\cal S}\|\\!<\\!\|{\\cal S}^{\\prime \\prime }\\!\\cap \\!\\widehat{\\cal S}\|$ .", "Suppose, by way of contradiction, that there is a set $S\\!\\in \\!", "({\\cal S}_i\\cap {\\cal S}^{\\prime })\\!\\setminus \\!\\widehat{\\cal S}$ , for some $1\\!\\le \\!i\\!\\le \\!k^{\\prime }$ .", "By Theorem REF , there is a set $\\widehat{S}\\!\\in \\!\\widehat{\\cal S}_i$ such that $w(\\widehat{S})\\!\\ge \\!w(S)$ , and $\\widehat{S}\\!\\cap \\!S^{\\prime }\\!=\\!\\emptyset $ , for all $S^{\\prime }\\!\\in \\!", "{\\cal S}^{\\prime }\\!\\setminus \\!\\lbrace S\\rbrace $ .", "Thus, ${\\cal S}^{\\prime \\prime }\\!=\\!", "({\\cal S}^{\\prime }\\!\\setminus \\!\\lbrace S\\rbrace )\\!\\cup \\!\\lbrace \\widehat{S}\\rbrace $ is a solution to $(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ .", "Since $\|{\\cal S}^{\\prime }\\!\\cap \\!\\widehat{\\cal S}\|\\!<\\!\|{\\cal S}^{\\prime \\prime }\\!\\cap \\!\\widehat{\\cal S}\|$ , this is a contradiction.", "Now, assume that $(U,\\!\\widehat{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ is a yes-instance.", "Since $\\widehat{\\cal S}\\!\\subseteq \\!", "{\\cal S}$ , we immediately get that $(U,\\!", "{\\cal S},\\!w,\\!k^{\\prime },\\!p^{\\prime })$ is also a yes-instance." ] ]	1403.0099
[ [ "Asymptotic Properties of the Misclassification Errors for Euclidean\n Distance Discriminant Rule in High-Dimensional Data" ], [ "Abstract Performance accuracy of the Euclidean Distance Discriminant rule (EDDR) is studied in the high-dimensional asymptotic framework which allows the dimensionality to exceed sample size.", "Under mild assumptions on the traces of the covariance matrix, our new results provide the asymptotic distribution of the conditional misclassification error and the explicit expression for the consistent and asymptotically unbiased estimator of the expected misclassification error.", "To get these properties, new results on the asymptotic normality of the quadratic forms and traces of the higher power of Wishart matrix, are established.", "Using our asymptotic results, we further develop two generic methods of determining a cut-off point for EDDR to adjust the misclassification errors.", "Finally, we numerically justify the high accuracy of our asymptotic findings along with the cut-off determination methods in finite sample applications, inclusive of the large sample and high-dimensional scenarios." ], [ "Introduction", "In this paper, we focus on the discrimination problem which is concerned with the allocation of a given object, ${\\text{$x$}}$ , a random vector represented by a set of features $(x_{1},\\dots ,x_{p})$ , to one ot two populations, $\\Pi _1$ and $\\Pi _2$ given by ${\\cal N}_p({\\text{$\\mu $}}_1, \\Sigma )$ and ${\\cal N}_p({\\text{$\\mu $}}_2, \\Sigma )$ , respectively, where ${\\text{$\\mu $}}_1\\ne {\\text{$\\mu $}}_2$ and common covariance matrix $\\Sigma $ is non-singular.", "Let $\\lbrace {\\text{$x$}}_{gj}\\rbrace ^{N_g}_{j=1}$ be a random sample of independent observations drawn from $g$ th population ${\\cal N}_p({\\text{$\\mu $}}_g, \\Sigma )$ , $g=1, 2$ .", "Let also $N=N_{1}+N_{2}$ denote the total sample size and set $n=N-2$ .", "We are interested to explore the discrimination procedure that can accomodate $p>n$ cases, with the main focus on the performance accuracy in the asymptototic framework that allows $p$ to grow together with $n$ .", "Clearly, the classical discriminant procedures, like Fisher linear discriminant rule, cannot be used when $p>n$ since the sample covariance matrix is singular and hence cannot be inverted.", "An intuitively appealing alternative considered in this study focuses on geometrical properties of the sample space and re-formulates the classification problem in terms of the Euclidian distance discriminant rule (EDDR): assign a new observation ${\\text{$x$}}$ to the \"nearest\" population $\\Pi _{g}$ , i.e.", "assign to $\\Pi _{g}$ if it is on average closer to the data from $\\Pi _{g}$ than to the data from the other population.", "Matusita's papers (see Matusita (1955), and Matusita and Motoo (1956)) are perhaps the oldest references dealing with the discriminant rule based on distance measures, including the case when the multivariate distributions underlying the data are not specified.", "Recently, Aoshima and Yata (2013) have been considered the EDDR for the high-dimensional multi-class problem with different class covariance matrices.", "In particular, they derived asymptotic conditions which ensure that the expected misclassification error converges to zero.", "Recent paper by Srivastava (2006) used the Moore-Penrose inverse of the estimated covariance matrix and suggested a second-order approximation of the expected error rate in high-dimensional data.", "We, in this study, focus on the asymptotic behavior of the misclassification errors of EDDR.", "Continuing with the normality assumption, with ${\\text{$\\mu $}}_g$ acting as the centered of the $\\Pi _{g}$ 's distribution we define $T_0(x)=\\parallel x-{\\text{$\\mu $}}_2\\parallel ^2-\\parallel x-{\\text{$\\mu $}}_1\\parallel ^2,$ and its sample based version as $\\widetilde{T}({\\text{$x$}}) = \\parallel x-\\overline{x}_2\\parallel ^2-\\parallel x-\\overline{x}_1\\parallel ^2$ where $\\Vert \\cdot \\Vert $ denotes the Euclidean norm and $\\overline{{\\text{$x$}}}_g$ 's denote the sample mean vectors, $g=1,2$ .", "Hence, each term in $(1.1)$ and $(1.2)$ represents the distance between the observed vector ${\\text{$x$}}$ and the centroid of $\\Pi _{g}$ 's or its sample based counterpart.", "The natural advantage of using $\\widetilde{T}({\\text{$x$}})$ for classifying high-dimensional data is its ability to mitigate the effect of dimensionality on the performance accuracy.", "Indeed, as it is seen from $(\\ref {eqn:ttil})$ , $\\widetilde{T}({\\text{$x$}})$ utilizes only the marginal distribution of the $p$ variables, thereby naturally reducing the effect of large $p$ in implementations.", "But the dimensionality has impact on the classification accuracy.", "To show this, we first point out that classifier $\\widetilde{T}({\\text{$x$}})$ has a bias.", "In fact, ${\\mathrm {E}}[\\widetilde{T}({\\text{$x$}}) \| {\\text{$x$}}\\in \\Pi _g] &=&(-1)^{g-1}\\Vert {\\text{$\\mu $}}_1-{\\text{$\\mu $}}_2\\Vert ^2+\\frac{N_1 - N_2}{N_1 N_2} {\\rm tr\\,}{\\Sigma },~g=1,2,$ and thus the impact of dimensionality is implied by the quantity $(N_1 - N_2) {\\rm tr\\,}\\Sigma /(N_1N_2)$ .", "In this study, we introduce the bias-corrected version $\\widetilde{T}({\\text{$x$}})$ defined as $T({\\text{$x$}}) = \\Vert {\\text{$x$}}- \\overline{{\\text{$x$}}}_2\\Vert ^2 -\\Vert {\\text{$x$}}- \\overline{{\\text{$x$}}}_1\\Vert ^2- \\frac{N_1-N_2}{N_1 N_2} {\\rm tr\\,}S,$ where the subtraction of $(N_1-N_2)/(N_1 N_2){\\rm tr\\,}S$ in $(\\ref {eqn:tt})$ is to guarantee that ${\\mathrm {E}}[T({\\text{$x$}})\| {\\text{$x$}}\\in \\Pi _g]=(-1)^{g-1}\\Vert {\\text{$\\mu $}}_1-{\\text{$\\mu $}}_2 \\Vert ^2,~g=1,2.$ Here, $S= (1/n) \\sum _{g=1}^2 \\sum _{j=1}^{N_g}({\\text{$x$}}_{gj}-\\overline{{\\text{$x$}}}_g)({\\text{$x$}}_{gj}-\\overline{{\\text{$x$}}}_g)^\\prime .$ Now, the EDDR given by $T({\\text{$x$}})$ places a new observation ${\\text{$x$}}$ to $\\Pi _1$ if $T({\\text{$x$}}) > \\tilde{c}$ , and to $\\Pi _2$ otherwise, where $\\tilde{c}$ is an appropriate cut-off point.", "Then, for a specific $\\tilde{c}$ , the performance accuracy of EDDR will be represented by the pair of misclassification error rates that result.", "Precisely, we define the conditional misclassification error of EDDR by $ce(2\|1) =\\Pr (T({\\text{$x$}}) \\le \\tilde{c} \|{\\text{$x$}}\\in \\Pi _1,~\\overline{{\\text{$x$}}}_1,~\\overline{{\\text{$x$}}}_2,~S)$ and its expected version by $e(2\|1)={\\mathrm {E}}[ce(2\|1)]$ , where the expectation is taken with respect to $\\overline{{\\text{$x$}}}_1$ , $\\overline{{\\text{$x$}}}_2$ and $S$ .", "Our main objective is to derive characteristic properties of both conditional and expected misclassification error in high-dimensional data.", "In many practical problems one type of misclassification error is generally regarded as more serious than the other, examples include e.g.", "medical applications associated with the diagnosis of diseases.", "In such a case, it might be desired to determine the cut-off $\\tilde{c}$ to obtain a specified probability of the error, or at least to approximate a specified probability.", "Then, one might base the choice of $\\tilde{c}$ on the expected misclassification error.", "This method, denoted in what follows by $\\mathbf {M1}$ , suggests to set a cut-off point $\\tilde{c}$ such that $\\mathbf {M1}: \\quad e(2\|1)={\\mathrm {E}}[ce(2\|1)] = \\alpha ,$ where $\\alpha $ is a value given by experimenters.", "On the other hand, one may exploit the confidence of the conditional error rate when determining $\\tilde{c};$ we denote this method by $\\mathbf {M2}: \\quad \\Pr (ce(2\|1) <eu)=1-\\beta ,$ where $1-\\beta $ is the desired level of confidence and $eu$ is an upper bound.", "Both determination methods $\\mathbf {M1}$ and $\\mathbf {M2}$ have been established by using large sample approximation, see Anderson (1973), McLachlan (1977) and Shutoh et al.", "(2012).", "In this study, we extend the consideration to the high-dimensional case.", "Our main theoretical results provide the asymptotically unbiased and consistent estimator of $e(2\|1)$ and the limit distribution of $ce(2\|1)$ under general assumptions covering the case when $p > n$ .", "In fact, ${\\bf M1}$ and ${\\bf M2}$ procedures can be considered as specific examples of using our generic results in the theory of EDDR in high-dimensions.", "The remaining part of the paper is organized as follows.", "In Section 2, we derived the asymptotically unbiased and consistent estimator of $e(2\|1)$ .", "Further, the limiting approximations of the cut-off point defined by $\\mathbf {M1}$ are established by using this estimator.", "In Section 3, two estimators of the confidence-based cut-off point defined by $\\mathbf {M2}$ are proposed, for which the asymptotic normality of the conditional error rate is shown.", "Section 4 summaries the results of numerical experiments justifying the validity of the suggested cut-off estimators for various strength of dependence underlying the data along with a number of high-dimensional scenarios where $p$ far exceeds the sample size.", "We conclude in Section 5, and give a through proofs of newly established asymptotic results together with some auxiliary lemmas in Appendix A." ], [ "Evaluation of the expected misclassification error", "Getting the closed-form expression for the expected error is too demanding, therefore we first shall derive its asymptotic approximation, and then based on this result, propose the consistent and asymptotically unbiased estimator of $e(2\|1)$ in high dimensions.", "We further show how these results can be used to provide the cut-off by the determination procedure $\\mathbf {M1}$ .", "Let ${\\text{$\\delta $}}={\\text{$\\mu $}}_1-{\\text{$\\mu $}}_2, a_i={\\rm tr\\,}\\Sigma ^i/p,i=1,\\dots ,8,\\Delta _i={\\text{$\\delta $}}^{\\prime }\\Sigma ^i{\\text{$\\delta $}},i=1,\\dots ,7$ and $\\Delta _0={\\text{$\\delta $}}^{\\prime }{\\text{$\\delta $}}.$ We make the following assumptions for the consistency and unbiasedness of the estimator of $e(2\|1)$ : $\\displaystyle ({\\rm A1}):0<\\lim _{(n,p) \\rightarrow \\infty }\\frac{p}{n}<\\infty , ~~~~0<\\lim _{(n,N_i) \\rightarrow \\infty }\\frac{N_i}{n+2}<1,~i=1,2.$ $\\displaystyle ({\\rm A2}):0<\\lim _{(n,p) \\rightarrow \\infty }\\Delta _0, \\lim _{(n,p) \\rightarrow \\infty }\\Delta _1<\\infty , ~~~~0<\\lim _{(n,p) \\rightarrow \\infty }a_1, \\lim _{(n,p) \\rightarrow \\infty }a_2<\\infty .$ $\\displaystyle ({\\rm A3}):\\lim _{(n,p) \\rightarrow \\infty }\\frac{\\Delta _3}{n} \\rightarrow 0, \\lim _{(n,p) \\rightarrow \\infty }\\frac{a_4}{n} \\rightarrow 0.$ Assume henceforth ${\\text{$x$}}\\in \\Pi _1$ .", "The symmetry of our classification rule makes the probability of error if the mean of ${\\text{$x$}}$ is ${\\text{$\\mu $}}_1$ the same as that under ${\\text{$\\mu $}}_2$ .", "Then for the conditional distribution of $T({\\text{$x$}})$ given $({\\overline{{\\text{$x$}}}}_1,{\\overline{{\\text{$x$}}}}_2,S)$ it holds that $T({\\text{$x$}})\| ({\\overline{{\\text{$x$}}}}_1,{\\overline{{\\text{$x$}}}}_2,S) \\sim {\\cal N}\\left( -2U-\\frac{N_1-N_2}{N_1N_2}{\\rm tr\\,}S, 4V\\right),$ where $U=& ({\\overline{{\\text{$x$}}}}_1-{\\overline{{\\text{$x$}}}}_2)^{\\prime }({\\overline{{\\text{$x$}}}}_1-{\\text{$\\mu $}}_1)-\\frac{1}{2}({\\overline{{\\text{$x$}}}}_1-{\\overline{{\\text{$x$}}}}_2)^{\\prime }({\\overline{{\\text{$x$}}}}_1-{\\overline{{\\text{$x$}}}}_2),\\\\V=& ({\\overline{{\\text{$x$}}}}_1-{\\overline{{\\text{$x$}}}}_2)^{\\prime }\\Sigma ({\\overline{{\\text{$x$}}}}_1-{\\overline{{\\text{$x$}}}}_2).$ Now the expected error rate $e(2\|1)$ of $T({\\text{$x$}})$ can be expressed in terms of $U$ and $V$ as $e(2\|1)={\\mathrm {E}}[ce(2\|1)]={\\mathrm {E}}\\left[\\Phi \\left(\\frac{U+(N_2^{-1}-N_1^{-1})p\\hat{a}_1/2+c}{\\sqrt{V}}\\right)\\right],$ where the expectation is with respect to $U$ and $V$ , $c=\\tilde{c}/2,$ ${\\hat{a}}_1={\\rm tr\\,}S/p$ and $\\Phi (\\cdot )$ is the cumulative distribution function of the standard normal distribution.", "In order to proceed to asymptotic approximation of $e(2\|1)$ , we need some preparatory stochastic evaluation of $U$ and $V$ .", "We introduce the auxiliary random variables $\\textrm {z}_1 &=&N^{- \\frac{1}{2}}\\Gamma ^\\prime \\Sigma ^{- \\frac{1}{2}}(N_1\\overline{\\textrm {x}}_1+ N_2 \\overline{\\textrm {x}}_2- N_1 \\textrm {\\mu }_1- N_2 \\textrm {\\mu }_2),\\\\\\textrm {z}_2 &=&\\left( \\frac{N}{N_1 N_2} \\right)^{- \\frac{1}{2}}\\Gamma ^\\prime \\Sigma ^{- \\frac{1}{2}}(\\overline{\\textrm {x}}_1- \\overline{\\textrm {x}}_2- \\textrm {\\mu }_1+ \\textrm {\\mu }_2),$ and observe that $\\textrm {$ z$}_1$ and $\\textrm {$ z$}_2$ are independent and identically distributed as $\\mathcal {N}_p(\\textrm {$ 0$},I_p)$ , where $\\Gamma $ is an orthogonal matrix such that $\\Sigma =\\Gamma \\Lambda \\Gamma ^{\\prime }$ and $\\Lambda $ is a diagonal matrix of eigenvalues of $\\Sigma $ .", "By means of ${\\text{$z$}}_1$ and ${\\text{$z$}}_2$ , we further define $U_0&=&-\\Delta _0/2, \\\\U_1&=&\\frac{1}{\\sqrt{N}}\\textrm {\\delta }^\\prime \\Gamma \\Lambda ^{\\frac{1}{2}}\\textrm {z}_1-\\left(\\frac{N_1}{N N_2} \\right)^{\\frac{1}{2}}\\textrm {\\delta }^\\prime \\Gamma \\Lambda ^{\\frac{1}{2}}\\textrm {z}_2+\\frac{1}{(N_1 N_2)^{\\frac{1}{2}}}\\textrm {z}_1^\\prime \\Lambda \\textrm {z}_2-\\frac{N_1 -N_2}{2 N_1 N_2}(\\textrm {z}_2^\\prime \\Lambda \\textrm {z}_2-pa_1), \\nonumber \\\\& &\\\\U_2&=&\\frac{(N_1-N_2)p}{2N_1N_2}(\\hat{a}_1-a_1),$ and observe that by using (2.2)-(2.4) the numerator in (2.1) can be decomposed as $U+\\frac{(N_1-N_2)p\\hat{a}_1}{2N_1N_2}=U_0+U_1+U_2.$ By analogy with $U$ , $V$ can also be decomposed by first defining $V_0$ and $V_1$ as $V_0 &=&\\Delta _1+\\frac{Npa_2}{N_1N_2},~V_1=2 \\left(\\frac{N}{N_1 N_2} \\right)^{\\frac{1}{2}} \\textrm {\\delta }^\\prime \\Gamma \\Lambda ^{\\frac{3}{2}}\\textrm {z}_2+ \\frac{N}{N_1 N_2}(\\textrm {z}_2^\\prime \\Lambda ^2\\textrm {z}_2-pa_2)$ and then observing that $V=V_0+V_1$ .", "Now for the first moments, we have by (2.5) and (2.6) ${\\mathrm {E}}\\left[U + \\frac{(N_1-N_2)p{\\hat{a}}_1}{2N_1N_2}\\right]=U_0,~{\\mathrm {E}}[V]=V_0.$ To evaluate the second moments, we apply Lemma A.3 (see Appendix) and obtain ${\\mathrm {E}}\\left[ \\left(U+\\frac{(N_1-N_2)p{\\hat{a}}_1}{2N_1N_2}-U_0 \\right)^2\\right] =H_U(\\Delta _1, a_2)+o(n^{-1}),~{\\mathrm {E}}[(V-V_0)^2] =H_V(\\Delta _3, a_4),$ where $H_U({\\Delta }_1,a_2)=\\frac{1}{N_2}\\Delta _1+\\frac{(N_1^2 + N_2^2) pa_2}{2 N_1^2 N_2^2},~H_V({\\Delta }_3,a_4)=\\frac{4N}{N_1 N_2} \\Delta _3+ \\frac{2 N^2pa_4}{(N_1 N_2)^2}.$ Under the assumptions (A1)-(A3), it holds that ${\\mathrm {E}}\\left[ \\left(U+\\frac{(N_1-N_2)p{\\hat{a}}_1}{2N_1N_2}-U_0 \\right)^2\\right] \\rightarrow 0,~{\\mathrm {E}}[(V-V_0)^2] \\rightarrow 0,$ and by Chebyshev's inequality, (2.7) implies that $U+\\frac{(N_1-N_2)p{\\hat{a}}_1}{2N_1N_2} \\xrightarrow{} U_0,~~ V \\xrightarrow{} V_0,$ where $\\xrightarrow{}$ denotes convergence in probability.", "Since $\\Phi (\\cdot )$ in (2.1) is a continuous function of $U$ and $V$ , it follows from (2.8), by the continuous mapping theorem, that $&&\\left\|\\Phi \\left(\\frac{U+(N_1-N_2)p{\\hat{a}}_1/(2N_1N_2)+c}{\\sqrt{V}}\\right) - \\Phi \\left(\\frac{U_0+c}{\\sqrt{V_0}}\\right)\\right\| \\xrightarrow{} 0.", "\\nonumber $ On the other hand, it naturally holds that $\\left\|\\Phi \\left(\\frac{U+(N_1-N_2)p{\\hat{a}}_1/(2N_1N_2)+c}{\\sqrt{V}}\\right) - \\Phi \\left(\\frac{U_0+c}{\\sqrt{V_0}}\\right)\\right\| < 1.$ Hence, by the dominated convergence theorem we have ${\\mathrm {E}}\\left[\\left\|\\Phi \\left(\\frac{U+(N_1-N_2)p{\\hat{a}}_1/(2N_1N_2)+c}{\\sqrt{V}}\\right) - \\Phi \\left(\\frac{U_0+c}{\\sqrt{V_0}}\\right)\\right\|\\right] \\rightarrow 0.$ Further, by applying the Jensen's inequality to (2.9) we get $&&\\left\|{\\mathrm {E}}\\left[\\Phi \\left(\\frac{U+(N_1-N_2)p{\\hat{a}}_1/(2N_1N_2)+c}{\\sqrt{V}}\\right)\\right] - \\Phi \\left(\\frac{U_0+c}{\\sqrt{V_0}}\\right)\\right\|\\\\&&\\le {\\mathrm {E}}\\left[\\left\|\\Phi \\left(\\frac{U+(N_1-N_2)p{\\hat{a}}_1/(2N_1N_2)+c}{\\sqrt{V}}\\right) - \\Phi \\left(\\frac{U_0+c}{\\sqrt{V_0}}\\right)\\right\|\\right]\\rightarrow 0.$ The above results are summarized in the following lemma.", "Lemma 2.1 Under assumptions (A1)-(A3) $e(2\|1) \\rightarrow \\Phi \\left(\\frac{U_0+c}{\\sqrt{V_0}}\\right),$ where $U_0$ and $V_0$ are defined in (2.2) and (2.6), respectively.", "In words, Lemma 2.1 provides a closed form expression for the limiting term of $e(2\|1)$ .", "Hence, to identify the cut-off point for $T({\\text{$x$}})$ , we derive a consistent and unbiased estimator of $e(2\|1)$ by plugging-in consistent estimators of $U_0$ and $V_0$ into the right hand side of (2.10).", "As $U_0$ and $V_0$ are functions of ${\\Delta }_0$ , ${\\Delta }_1$ and $a_2$ , we begin by obtaining their consistent estimators.", "Lemma 2.2 Let estimators of $\\Delta _0, \\Delta _1, a_2$ be defined as ${\\widehat{{\\Delta }}}_0&=&{\\widehat{{\\text{$\\delta $}}}}^\\prime {\\widehat{{\\text{$\\delta $}}}}-\\frac{Np}{N_1N_2}{\\hat{a}}_1,\\\\{\\widehat{{\\Delta }}}_1&=&{\\widehat{{\\text{$\\delta $}}}}^\\prime S {\\widehat{{\\text{$\\delta $}}}}-\\frac{Np}{N_1N_2}{\\hat{a}}_2,\\\\\\hat{a}_2&=&\\frac{n^2}{p(n+2)(n-1)} \\left( {\\rm tr\\,}{S^2} -\\frac{({\\rm tr\\,}{S})^2}{n} \\right),$ respectively, where ${\\widehat{{\\text{$\\delta $}}}}=\\overline{\\textrm {x}}_1 - \\overline{\\textrm {x}}_2.$ Then under assumptions (A1)-(A3) ${\\widehat{{\\Delta }}}_0 \\xrightarrow{} {\\Delta }_0, ~~{\\widehat{{\\Delta }}}_1 \\xrightarrow{} {\\Delta }_1,~~ {\\hat{a}}_2 \\xrightarrow{} a_2.$ (Proof) To show consistency of $a_1$ and $a_2$ , we use exact expressions for the variances of these estimators derived in Srivastava (2005) as ${\\mathrm {E}}[({\\hat{a}}_1 - a_1)^2] &=& \\frac{2a_2}{np}, \\\\{\\mathrm {E}}[({\\hat{a}}_2 - a_2)^2] &=& \\frac{8(n+2)(n+3)(n-1)^2}{p n^5} a_4 +\\frac{4(n+2)(n-1)}{n^4}(a_2^2 - p^{-1}a_4).$ Then by applying Chebyshev's inequality to (2.14) and (2.15) it can be seen that ${\\hat{a}}_1 \\xrightarrow{} a_1, {\\hat{a}}_2 \\xrightarrow{} a_2.$ To show consistency of ${\\widehat{{\\Delta }}}_0$ and ${\\widehat{{\\Delta }}}_1$ , we first consider the following random variables $\\widetilde{\\Delta }_0={\\widehat{{\\text{$\\delta $}}}}^\\prime {\\widehat{{\\text{$\\delta $}}}}-\\frac{Np}{N_1N_2}a_1,~\\widetilde{\\Delta }_1={\\widehat{{\\text{$\\delta $}}}}^\\prime S {\\widehat{{\\text{$\\delta $}}}}-\\frac{Np}{N_1N_2}a_2$ and evaluate the first two moments of ${\\widehat{{\\text{$\\delta $}}}}^\\prime {\\widehat{{\\text{$\\delta $}}}}$ and ${\\widehat{{\\text{$\\delta $}}}}^\\prime S {\\widehat{{\\text{$\\delta $}}}}$ .", "We rewrite ${\\widehat{{\\text{$\\delta $}}}}^\\prime {\\widehat{{\\text{$\\delta $}}}}&=&{\\text{$\\delta $}}^{\\prime }{\\text{$\\delta $}}+2\\left(\\frac{N}{N_1N_2}\\right)^{1/2}{\\text{$\\delta $}}^{\\prime }\\Sigma ^{1/2}{\\text{$z$}}+\\frac{N}{N_1N_2}{\\text{$z$}}^{\\prime }\\Sigma {\\text{$z$}},$ and ${\\widehat{{\\text{$\\delta $}}}}^\\prime S{\\widehat{{\\text{$\\delta $}}}}&=&{\\text{$\\delta $}}^{\\prime }S{\\text{$\\delta $}}+2\\left(\\frac{N}{N_1N_2}\\right)^{1/2}{\\text{$\\delta $}}^{\\prime }S\\Sigma ^{1/2}{\\text{$z$}}+\\frac{N}{N_1N_2}{\\text{$z$}}^{\\prime }\\Sigma ^{1/2}S\\Sigma ^{1/2}{\\text{$z$}},$ where ${\\text{$z$}}\\sim {\\cal N}(0,I_p)$ .", "Then it easily follows that ${\\mathrm {E}}[\\widetilde{\\Delta }_0]=\\Delta _0,~{\\mathrm {E}}[\\widetilde{\\Delta }_1]=\\Delta _1$ and ${\\rm Var}[\\widetilde{\\Delta }_0]&=&\\frac{4N}{N_1N_2}{\\Delta }_1+\\frac{2N^2p}{N_1^2N_2^2}a_2, \\\\{\\rm Var}[\\widetilde{\\Delta }_1]&=&\\frac{2 a_2^2 N^2 p^2}{n N_1^2 N_2^2}+\\frac{4 a_2 {\\Delta }_1 N p}{n N_1 N_2}+\\frac{2 a_4 N^3p}{n N_1^2 N_2^2}+\\frac{2 {\\Delta }_1^2}{n}+\\frac{4 {\\Delta }_3 N^2}{n N_1 N_2}.$ By applying Chebyshev's inequality to (2.18)-(2.20), we obtain $\\widetilde{{\\Delta }}_0 \\xrightarrow{} {\\Delta }_0,~\\widetilde{{\\Delta }}_1 \\xrightarrow{} {\\Delta }_1.$ Finally, from (2.16) and (2.21), we see that consistency of $\\widetilde{{\\Delta }}_0$ and $\\widetilde{{\\Delta }}_1$ imply consistency of ${\\widehat{{\\Delta }}}_0$ and ${\\widehat{{\\Delta }}}_1$ .", "$\\hfill \\square $ Now by substituting the estimators of ${\\Delta }_0, {\\Delta }_1, a_2$ into the limiting term in Lemma 2.1. the consistent estimator of $e(2\|1)$ is given by $\\Phi (({\\widehat{U}}_0+c){\\widehat{V}}_0^{-1/2})$ , where ${\\widehat{U}}_0 = - {\\widehat{{\\Delta }}}_0/2$ and ${\\widehat{V}}_0 = {\\widehat{{\\Delta }}}_1 + Np\\hat{a}_2/(N_1N_2)$ .", "The following theorem is provided by the consistency of estimators ${\\widehat{{\\Delta }}}_0, {\\widehat{{\\Delta }}}_1$ and ${\\hat{a}}_2$ , continuous mapping theorem and dominated convergence theorem.", "Theorem 2.1 Under assumptions (A1) - (A3) $\\Phi \\left(({\\widehat{U}}_0+c){\\widehat{V}}_0^{-1/2}\\right)\\xrightarrow{} e(2\|1)\\hspace{14.22636pt} and \\hspace{14.22636pt}{\\mathrm {E}}\\left[\\Phi \\left(({\\widehat{U}}_0+c){\\widehat{V}}_0^{-1/2}\\right)\\right]\\rightarrow e(2\|1).$ By the results of Theorem 2.1 and Lemma 2.1, the M1-based cut-off point for EDDR using $T({\\text{$x$}})$ is provided by $&{\\hat{c}}_1 = {\\widehat{V}}^{1/2}_0 z_\\alpha - {\\widehat{U}}_0,$ where $z_\\alpha $ is the $\\alpha $ -percentile of $\\mathcal {N}(0,1)$ and $\\alpha \\in (0,1)$ ." ], [ "Asymptotic distribution of the conditional misclassification error", "Our objective in this section is to establish the asymptotic distribution of $ce(2\|1)$ , for which we need some auxiliary notations and assumptions.", "We begin by modifying the high-dimensional asymptotic framework from Section 2 by replacing the Assumption (A3) with (B3) as follows: $\\displaystyle ({\\rm B3}):\\lim _{(n,p) \\rightarrow \\infty }\\frac{\\Delta _i}{n} \\rightarrow 0, i=2, \\cdots , 5,~~\\lim _{(n,p) \\rightarrow \\infty }\\frac{a_i}{n} \\rightarrow 0, i=3, \\cdots , 6.$ As $ce(2\|1)$ is a function of the variable set of $(U,V)$ , we first obtain the joint asymptotic distribution of $(U,V)$ .", "Lemma 3.1 Let $\\widetilde{U}=U+(N_1-N_2)p{\\hat{a}}_1/(N_1N_2).$ Then under assumptions $\\mathrm {(A1)}$ , $\\mathrm {(A2)}$ and $\\mathrm {(B3)}$ the following holds $\\sqrt{n}\\left\\lbrace \\left(\\begin{array}{c}\\widetilde{U}\\\\V\\end{array}\\right)-\\left(\\begin{array}{c}U_0\\\\V_0\\end{array}\\right)\\right\\rbrace \\xrightarrow{} \\mathcal {N}_2(0,\\Theta ),$ where $\\Theta =n\\left(\\begin{array}{cc}H_U(\\Delta _1,a_2)&H_{UV}(\\Delta _2,a_3)\\\\H_{UV}(\\Delta _2,a_3)&H_V(\\Delta _3,a_4)\\end{array}\\right),$ $H_{UV}({\\Delta }_2,a_3)=&-\\frac{2}{N_2} \\Delta _2-\\frac{N(N_1-N_2)pa_3}{(N_1 N_2)^2},$ and $\\xrightarrow{}$ denotes convergence in distribution.", "(Proof) Let $d_1$ and $d_2$ denote two non-random values which satisfy $\\displaystyle 0 < \\lim _{(n, p) \\rightarrow \\infty } \|d_1\| < \\infty $ and $\\displaystyle 0 < \\lim _{(n, p) \\rightarrow \\infty } \|d_2\| < \\infty $ , and introduce the statistic $Q$ which is defined as the linear combination of $\\widetilde{U}$ and $V$ .", "$Q=\\sqrt{n}\\left\\lbrace d_1\\left(\\widetilde{U}+\\frac{1}{2} \\Delta _0\\right)+d_2\\left(V-\\Delta _1-\\frac{Npa_2}{N_1N_2}\\right)\\right\\rbrace .$ The asymptotic normality of $Q$ would imply that the joint distribution of $\\widetilde{U}$ and $V$ is asymptotically normal.", "Thus, Lemma 3.1 will be proven if we show the normal convergence of $Q$ under (A1), (A2) and (B3).", "We introduced the following notations $\\omega _1&=&\\frac{d_1\\sqrt{n}}{\\sqrt{N}}\\Lambda ^{\\frac{1}{2}}\\Gamma ^{\\prime }{\\text{$\\delta $}}, \\\\\\omega _2&=&\\frac{2d_2\\sqrt{n N}}{\\sqrt{N_1 N_2}} \\Lambda ^{3/2}\\Gamma ^{\\prime }{\\text{$\\delta $}}-\\frac{d_1\\sqrt{n N_1}}{\\sqrt{N N_2}} \\Lambda ^{1/2}\\Gamma ^{\\prime }{\\text{$\\delta $}}, \\\\\\Omega _3&=&\\frac{d_1\\sqrt{n}}{\\sqrt{N_1N_2}} \\Lambda , \\\\\\Omega _4&=&\\frac{d_2 \\sqrt{n} N}{N_1N_2} \\Lambda ^2- \\frac{d_1 \\sqrt{n} (N_1-N_2)}{2N_1N_2}\\Lambda .$ Now, since ${\\hat{a}}_1 -a_1= O_p(n^{-1})$ by (2.14), the statistic $Q$ can be expressed as $Q&=&\\omega _1^\\prime \\textrm {z_1}+\\omega _2^\\prime \\textrm {z_2}+\\textrm {z_1}^\\prime \\Omega _3\\textrm {z_2}+\\textrm {z_2}^\\prime \\Omega _4\\textrm {z_2}+o_p(1).$ Note also that $\\omega _1^{\\prime }\\omega _1&=&\\frac{d_1^2n}{N}{\\text{$\\delta $}}^{\\prime }\\Sigma {\\text{$\\delta $}}, \\\\\\omega _2^{\\prime }\\omega _2&=&\\frac{4d_2^2nN}{N_1N_2}{\\text{$\\delta $}}^{\\prime }\\Sigma ^3{\\text{$\\delta $}}+\\frac{d_1^2nN_1}{NN_2}{\\text{$\\delta $}}^{\\prime }\\Sigma {\\text{$\\delta $}}-\\frac{4d_1d_2n}{N_2}{\\text{$\\delta $}}^{\\prime }\\Sigma ^2{\\text{$\\delta $}},\\\\{\\rm tr\\,}\\Omega _3^2&=&\\frac{d_1^2n}{N_1N_2}{\\rm tr\\,}\\Sigma ^2, \\\\{\\rm tr\\,}\\Omega _4^2&=&\\frac{d_2^2nN^2}{N_1^2N_2^2}{\\rm tr\\,}\\Sigma ^4+\\frac{d_1^2n(N_1-N_2)^2}{4N_1^2N_2^2}{\\rm tr\\,}\\Sigma ^2-\\frac{d_1d_2n(N_1^2-N_2^2)}{N_1^2N_2^2}{\\rm tr\\,}\\Sigma ^3.$ By combining these terms, we now obtain the asymptotic variance of $Q$ as $\\sigma _Q^2=\\lim _{(n,p)\\rightarrow \\infty }n\\lbrace d_1^2H_U(\\Delta _1,a_2)-2d_1d_2H_{UV}(\\Delta _2,a_3)+d_2^2H_V(\\Delta _3,a_4)\\rbrace $ and observe that (A1), (A2) and (B3) $0<\\sigma _Q^2<\\infty .$ Furthermore, the following convergence results hold $\\omega _1^{\\prime }\\Omega _3\\omega _2\\rightarrow 0,~\\omega _2^{\\prime }\\Omega _4\\omega _2\\rightarrow 0,~{\\rm tr\\,}\\Omega _3^2\\Omega _4\\rightarrow 0~ {\\rm and} ~{\\rm tr\\,}\\Omega _4^3\\rightarrow 0.$ Now by using (3.1) and (3.2), and by applying (A.1) from Lemma A.1 (see Appendix), we obtain $\\frac{\\omega _1^{\\prime }\\Omega _3\\omega _2}{\\sigma _Q^3}\\rightarrow 0,~\\frac{\\omega _2^{\\prime }\\Omega _4\\omega _2}{\\sigma _Q^3}\\rightarrow 0,~\\frac{{\\rm tr\\,}\\Omega _3^2\\Omega _4}{\\sigma _Q^3}\\rightarrow 0~ {\\rm and} ~\\frac{{\\rm tr\\,}\\Omega _4^3}{\\sigma _Q^3}\\rightarrow 0.$ (3.3) in combination with Lemma A.1 show that the asymptotic normality of $Q$ holds, which completes the proof.", "$\\hfill \\square $ Now we are ready to state our main results on the distribution of $ce(2\|1)$ .", "Besides the distribution of the latter we also find the asymptotic distribution of the logit transform of $ce(2\|1)$ .", "Our motivation to make this particular type of transform will be clear below.", "Theorem 3.1 Let the logit transform of $ce(2\|1)$ be defined by $\\ell (2\|1)= \\log {\\frac{ce(2\|1)}{1-ce(2\|1)}}$ and let the operator $\\nabla _{(u,v)} (\\cdot )$ for a function $f(u,v)$ be defined as $\\nabla _{(u,v)}f(u,v) = \\left( \\frac{\\partial f}{\\partial u} , \\frac{\\partial f}{\\partial v} \\right)^\\prime .$ Then in the framework $\\mathrm {(A1)}$ , $\\mathrm {(A2)}$ and $\\mathrm {(B3)}$ $ce(2\|1)$ and $\\ell (2\|1)$ are asymptotically normal, i.e.", "$&{\\rm (i)}&ce(2\\mid 1)\\xrightarrow{} \\mathcal {N} \\left(e_0 ,\\tau ^2 \\right),\\\\&{\\rm (ii)}& \\ell (2\\mid 1) \\xrightarrow{}\\mathcal {N}\\left(\\ell _{0},\\tau _{\\ell }^2\\right)$ with $e_0= \\Phi \\left(\\frac{U_0 + c}{V_0^{1/2}}\\right),~\\ell _{0}= \\log {\\frac{e_0}{1-e_0}}, ~\\tau ^2=\\nabla _{(U_0,V_0)}^{\\prime } \\Theta \\nabla _{(U_0,V_0)}, ~\\tau ^2_{\\ell }=\\frac{\\tau ^2}{(1-e_0)e_0},$ where $\\nabla _{(U_0,V_0)}$ is defined as $\\nabla _{(U_0,V_0)}= \\left(V_0^{-1/2}\\phi \\left(\\frac{U_0 +c }{\\sqrt{V_0}} \\right),- \\frac{(U_0 + c)}{2 V_0^{3/2}} \\phi \\left(\\frac{U_0 + c}{\\sqrt{V_0}} \\right)\\right)^\\prime .$ (Proof) By using asymptotic normality of $(\\widetilde{U},V)$ and by applying Lemma A.4 (see Appendix) to the function $g(\\widetilde{U},V)=\\Phi \\left(\\frac{\\widetilde{U}+c}{V^{1/2}}\\right)$ it easily follows that $\\nabla _{(\\tilde{u},v)}g(\\tilde{u},v)=\\left(\\frac{\\partial g}{\\partial \\tilde{u}}, \\frac{\\partial g}{\\partial v}\\right)^\\prime = \\left(v^{-1/2}\\phi \\left(\\frac{\\tilde{u} + c}{\\sqrt{v}} \\right),- \\frac{(\\tilde{u} + c)}{2 v^{3/2}} \\phi \\left(\\frac{\\tilde{u} + c}{\\sqrt{v}} \\right)\\right)^\\prime .$ Then we obtain $ce(2\|1)=g(\\widetilde{U},V)\\xrightarrow{}{\\cal N}(\\Phi ((U_0 + c)V_0^{-1/2}),\\nabla _{(U_0,V_0)}^{\\prime } \\Theta \\nabla _{(U_0,V_0)}).$ The statement (ii) can be proven similarly.", "$\\hfill \\square $ Now we are ready to explore the determination method ${\\bf M2}$ which chooses the cut-off point $c$ to get the desired level of confidence $1-\\beta $ of a pre-specified upper bound $eu$ .", "By the asymptotic normality of $ce(2\|1)$ and $\\ell (2\|1)$ , we propose to set the cut-off points for the EDDR using $T({\\text{$x$}})$ as $&{\\rm (i)}&c_{2,1}~~s.t.~~c_{2,1}=\\frac{-U_0+V_0^{1/2}z_{\\gamma }}{a_1},\\\\&{\\rm (ii)}&c_{2,2}~~s.t.~~c_{2,2}=\\frac{-U_0+V_0^{1/2}z_{\\gamma _{\\ell }}}{a_1},\\\\$ where $\\gamma =eu-\\tau z_{1-\\beta },~\\gamma _{\\ell }=\\frac{eu}{(1-eu)\\exp (\\tau _{\\ell } z_{1-\\beta })+eu}.$ Remark 3.1 If $\\gamma \\notin [0,1]$ then (i) is not defined.", "This motivates our logit trance form of $ce(2\|1)$ which yields the result (ii) where $\\gamma _\\ell \\in [0,1]$ always.", "For practical use, the unknown parameters ${\\Delta }_0$ , ${\\Delta }_1$ , ${\\Delta }_2$ , ${\\Delta }_3$ , $a_1$ , $a_2$ , $a_3$ and $a_4$ in (i)-(ii) should be replaced by their consistent estimators.", "To ensure consistency, the asymptotic framework (A1)-(A3) is modified by replacing (A3) with $\\displaystyle ({\\rm B^{\\prime }3}):0 < \\lim _{(n,p) \\rightarrow \\infty }a_i < \\infty , ~i=3, \\dots , 8, ~~0 < \\lim _{(n,p) \\rightarrow \\infty }{\\Delta }_i < \\infty , ~i=2, \\dots , 7.$ By the consistency results of Lemma A.5 and A.6 (see Appendix), obtained under the assumptions (A1), (A2) and (B$^{\\prime }$ 3), we now propose the M2-based cut-off point estimator as, $&{\\rm (i)}&\\hat{c}_{2,1}~~s.t.~~\\hat{c}_{2,1}=\\frac{-{\\widehat{U}}_0+{\\widehat{V}}_0^{1/2}z_{\\hat{\\gamma }}}{{\\hat{a}}_1},\\\\&{\\rm (ii)}&\\hat{c}_{2,2}~~s.t.~~\\hat{c}_{2,2}=\\frac{-{\\widehat{U}}_0+{\\widehat{V}}_0^{1/2}z_{\\hat{\\gamma }_{\\ell }}}{\\hat{a}_1},\\\\$ where $\\hat{\\gamma }=eu-\\hat{\\tau } z_{1-\\beta },~\\hat{\\gamma }_\\ell =\\frac{eu}{(1-eu)\\exp (\\hat{\\tau }_{\\ell } z_{1-\\beta })+eu}.$ Remark 3.2 The problem described in Remark 3.1 remains for $\\hat{\\gamma }$ .", "Therefore for practical use we recommend to replace $\\hat{\\gamma }$ with $\\hat{\\gamma _\\ell }$ when the observed value of $\\hat{\\gamma }\\notin [0,1]$ ." ], [ "Simulation study", "We now turn to numerical evaluation of the asymptotic results and the suggested cut-off points.", "The goal of the simulation experiment is threefold: to investigate the finite sample behaviour of newly derived asymptotic approximations, to compare the performance of our approach under independence with that for dependent data with various dependence strength, and to investigate the effect of choice of the confidence level in combination with the upper bound specification.", "The data sets for each $\\Pi _{g}$ , $g=1,2$ are independently generated as $&x_{11},x_{12},\\ldots ,x_{1N_1} \\stackrel{i.i.d.", "}{\\sim }\\mathcal {N}_p(\\mu _1,\\Sigma ), \\hspace{-42.67912pt}&x_{21},x_{22},\\ldots ,x_{2N_2} \\stackrel{i.i.d.", "}{\\sim }\\mathcal {N}_p(\\mu _2,\\Sigma ),$ respectively.", "To assess the performance for dependent data, $\\Sigma $ will be assumed to have band correlation $\\Sigma =\\left( \\sigma _{ij} \\right)$ , $\\sigma _{ij} ={\\left\\lbrace \\begin{array}{ll}{}\\rho ^{\|i-j\|}, ~~~\|i-j\| \\le 50,\\\\0,~~~~~~~~\|i-j\|>50,\\end{array}\\right.", "}$ with $\\rho $ ranging from 0 to $0.5$ , which is chosen to fulfill the condition $(\\textrm {A}2)$ .", "To constrain the classification complexity, we set $\\Sigma ^{-1/2}\\mu _1=(p)^{-1/2}(5^{1/2},5^{1/2},\\ldots ,5^{1/2})^{\\prime }~~~ {\\textrm {and}} ~~~\\mu _2=(0,0,\\ldots ,0)^{\\prime },$ through the whole simulation experiment.", "To evaluate the effect of high-dimensionality and sample size, we let $p=64,128,256,$ $512,1024$ and $N_1=N_2$ , $N=64,128,256$ for each choice of $\\rho $ .", "First, as in the previous sections, we focus without loss of generality on evaluation of $ce(2\|1)$ .", "For each triple $(p,N,\\rho )$ , we generate data according to $(\\ref {datasim})$ , apply EDDR given by $T({\\text{$x$}})$ in (1.3) with both $\\mathbf {M1}$ -based cut-offs, $\\hat{c}_1$ established in Section 2, and repeat the whole process independently $100 ~000$ times.", "As a result, we get $100 ~000$ conditional classification errors of $T({\\text{$x$}})$ : $C^{(i)}=\\Phi \\left(\\frac{U^{(i)}+(N_2^{-1}-N_1^{-1})p\\hat{a}^{(i)}_1/2+\\hat{c}^{(i)}_1}{\\sqrt{V^{(i)}}}\\right),~~~i=1,\\dots ,100~000,$ which after averaging provides attained error rate $ae(\\hat{c}_1)=\\frac{1}{100~000}\\sum _{i=1}^{100~000}C^{(i)}.$ This result, being summarized in Table 1 through Table 9, suggest that the EDDR based on $\\hat{c}_1$ is optimally adaptive in a sense that its performance accuracy is closely approaching the actual value of the misclassification, $\\alpha $ .", "Stably good result is obtained when varying the dependence strength $\\rho $ and the value of the actual error $\\alpha $ , in both large sample and high-dimensional cases.", "To evaluate the performance of the $\\mathbf {M2}$ -based cut-offs we use the simulation setting $(\\ref {datasim})$ , with the same variety of covariance strength, a range of $\\beta $ varying between $0.01$ to $0.1$ representing higher respective lower confidence levels, and two values of $eu$ , $0.1$ and $0.2$ representing the upper bound on the actual misclassification probability.", "We summarize the combination of the values of $1-\\beta $ and $eu$ in Table 10.", "Then for each setting, the classification procedure by $T({\\text{$x$}})$ with cut-offs $\\hat{c}_{2,1}$ and $\\hat{c}_{2,2}$ in section 3, respectively.", "Proceeding with the same simulation strategy as above for each cut-off choice, we consider the attained confidence level $acl(\\hat{c}_{2,i}) = \\frac{\\# \\left\\lbrace \\Phi \\left(\\lbrace U + (N_2^{-1}-N_1^{-1})\\hat{a}_1/2 + \\hat{c}_{2,i}\\rbrace /\\sqrt{V} \\right) \\le eu \\right\\rbrace }{100~000},~ i=1,2,$ which is obtained by averaging the observed confidence level of $ce(2\|1)$ of $T({\\text{$x$}})$ with $\\hat{c}_{2,i}$ for each, $i$ , over $100 ~000$ independent replicates of the data generation step, estimation of parameters and classification.", "This result, being summarized in Table 11 through Table 28.", "In most tables, the case in using $\\hat{c}_{2,2}$ is better accuracy than the case in using $\\hat{c}_{2,1}$ , and conservative." ], [ "Conclusion", "This paper contributes to the asymptotic analyses of the EDDR performance in high-dimensional data, with particular focus on determining a cut-off point to adjust the probabilities of misclassification.", "Two generic cut-off determination approaches, $\\mathbf {M1}$ based on the expected error and $\\mathbf {M2}$ based on the upper bound of the actual misclassification probability, $eu$ with the specified confidence level $1-\\beta $ , are proposed.", "To establish the cut-off by $\\mathbf {M1}$ , an approximation of the expected misclassification error along with its asymptotic unbiased estimator, is derived; our result extends the approach of Anderson (1973) by considering a more general asymptotic set-up that allows $p > N$ .", "Subsequently, the cut-off based on the main term of the asymptotic expression is suggested.", "To set up the cut-off based on $\\mathbf {M2}$ , the asymptotic normality of the conditional misclassification error and its logit transform are established for a given $\\beta $ and $eu$ in high-dimensions.", "Based on the asymptotic results, two types of cut-offs are also established.", "Our newly derived results extend the asymptotic consideration by McLachlan (1977) to a high-dimensional case.", "For both $\\mathbf {M1}$ and $\\mathbf {M2}$ approaches, the practically workable expressions of the theoretical cut-offs are established, for which we obtain consistent and asymptotic unbiased estimators of a set of unknown parameters.", "The validity of the new asymptotic results in a finite sample case is numerically shown by applying the cut-offs in the suggested EDDR classifier $T({\\text{$x$}})$ for a range of confidence levels, various strength of correlation and a set of $p$ and $N$ values.", "As the both suggested cut-off determination procedures demonstrate stably good accuracy in high dimensions, they can generally be recommended for practical applications in distance-based classifiers, with EDDR as special case, when it is desired to set a cut-off point to achieve a specified misclassification error.", "Acknowledgments.", "The authors thank Professor Makoto Aoshima and Professor Yasunori Fujikoshi for extensive discussions, references and encouragements.", "The research of Tatjana Pavlenko is in part supported by the grant 2013-45266 VR of Sweden.", "The research of Takashi Seo was supported in part by Grant-in-Aid for Scientific Research (C) (23500360).", "The research of Masashi Hyodo is in part supported by the Stiftelsen G.S.", "Magnuson travel grant (2013), the Royal Swedish Academy of Sciences." ], [ "Appendix", "Lemma A.", "1 (The central limit theorem for quadratic forms) Let ${\\text{$z$}}_1$ and ${\\text{$z$}}_2$ be independent, ${\\cal N}_p(0,I_p)$ distributed random variables, $\\omega _i~(i=1,2)$ be arbitrary non-random $p$ -dimensional vectors and $\\Omega _i~(i=3,4)$ be arbitrary non-random $p\\times p$ diagonal matrices.", "Define $K=\\omega _1^{\\prime }{\\text{$z$}}_1+\\omega _2^{\\prime }{\\text{$z$}}_2+{\\text{$z$}}_1^\\prime \\Omega _3{\\text{$z$}}_2+({\\text{$z$}}_2^{\\prime }\\Omega _4{\\text{$z$}}_2-{\\rm tr\\,}\\Omega _4)$ with $\\sigma _K^2=\\omega _1^{\\prime }\\omega _1+\\omega _2^{\\prime }\\omega _2+{\\rm tr\\,}\\Omega _3^2+2{\\rm tr\\,}\\Omega _4^2$ .", "If the following limiting conditions are fulfilled $1\\endcsname \\frac{\\omega _1^{\\prime }\\Omega _3\\omega _2}{\\sigma _K^3}\\rightarrow 0,~\\frac{\\omega _2^{\\prime }\\Omega _4\\omega _2}{\\sigma _K^3}\\rightarrow 0,~\\frac{{\\rm tr\\,}\\Omega _3^2\\Omega _4}{\\sigma _K^3}\\rightarrow 0~and ~\\frac{{\\rm tr\\,}\\Omega _4^3}{\\sigma _K^3}\\rightarrow 0,$ then $K/\\sigma _K\\xrightarrow{}{\\cal N}(0,1)$ as $p\\rightarrow \\infty $ .", "(Proof) Let $\\omega _{ij} (i=1,2)$ be the j-th element of $\\omega _i, \\omega _{ij} (i=3,4)$ be the j-th diagonal element of $\\Omega _j$ and $z_{ij} (i=1,2)$ be the j-th element of $z_i$ .", "$K$ can be expressed as $K&=&\\omega _1^{\\prime }{\\text{$z$}}_1+\\omega _2^{\\prime }{\\text{$z$}}_2+{\\text{$z$}}_1^\\prime \\Omega _3{\\text{$z$}}_2+({\\text{$z$}}_2^\\prime \\Omega _4{\\text{$z$}}_2-{\\rm tr\\,}\\Omega _4)\\\\&=&\\sum _{i=1}^p\\omega _{1i}z_{1i}+\\sum _{i=1}^p\\omega _{2i}z_{2i}+\\sum _{i=1}^p\\omega _{3i}z_{1i}z_{2i}+\\sum _{i=1}^p(\\omega _{4i}z_{2i}^2-\\omega _{4i}).$ Consider $\\varepsilon _i=\\omega _{1i}z_{1i}+\\omega _{2i}z_{2i}+\\omega _{3i}z_{1i}z_{2i}+\\omega _{4i}z_{2i}^2-\\omega _{4i},~(i=1,2,\\ldots ,p)$ and note that $\\lbrace \\varepsilon _i \\rbrace _{i=1}^p$ is a sequence of i.i.d.", "random variables such that $K=\\sum _{i=1}^p\\varepsilon _i$ and the third moment of $\\varepsilon _i$ is given by ${\\mathrm {E}}[\\varepsilon _i^3]=2(3\\omega _{1i}\\omega _{2i}\\omega _{3i} + 3\\omega _{2i}^2\\omega _{4i} + 3\\omega _{3i}^2\\omega _{4i} + 4\\omega _{4i}^3).$ Then to ensure that $K/\\sigma _K\\xrightarrow{}\\mathcal {N}(0,1)$ , we consider the Lyapunov-based sufficient condition for the sequences $\\lbrace \\varepsilon _i \\rbrace _{i=1}^p$ which states that there exists such $\\eta >0$ that $2\\endcsname \\frac{\\sum _{i=1}^p{\\mathrm {E}}[\\varepsilon _i^{2+\\eta }]}{\\sigma _K^{2+\\eta }}\\rightarrow 0~{\\rm as}~p\\rightarrow \\infty .$ For now, we check (A.", "2) with $\\eta =1$ .", "Based on the third moment of $\\varepsilon _i$ , we obtain $3\\endcsname \\sum _{i=1}^p{\\mathrm {E}}[\\varepsilon _i^3]=2( 3\\omega _1^{\\prime }\\Omega _3\\omega _2 + 3\\omega _2^{\\prime }\\Omega _4\\omega _2 + 3{\\rm tr\\,}\\Omega _3^2\\Omega _4 + 4{\\rm tr\\,}\\Omega _4^3).$ From (REF ) and the condition (REF ), $\\frac{\\sum _{i=1}^p{\\mathrm {E}}[\\varepsilon _i^3]}{\\sigma _K^3} \\rightarrow 0$ as $p\\rightarrow \\infty $ , from which the convergence $K/\\sigma _K\\xrightarrow{}{\\cal N}(0,1)$ follows.", "$\\hfill \\square $ Lemma A.", "2 (Higher order moments of the traces of Wishart matrices) Let $W$ be distributed as $\\mathcal {W}_p(n,\\Sigma )$ , where $\\mathcal {W}_p$ denoted Wishart distribution with freedom parameter $n$ and scale parameter $\\Sigma $ .", "Let $A$ and $B$ denote $p \\times p$ symmetric non-random matrices.", "Then the following assertions hold: $&&\\mathrm {(i)}~E[({\\rm tr\\,}A W)({\\rm tr\\,}BW)]=n^2 {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B+2 n {\\rm tr\\,}\\Sigma A \\Sigma B,\\\\&&\\mathrm {(ii)}~E[{\\rm tr\\,}AWBW]=(n^2+n){\\rm tr\\,}\\Sigma A \\Sigma B+n {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B,\\\\&&\\mathrm {(iii)}~E[({\\rm tr\\,}A W^3)]=(n^3+3n^2+4n) {\\rm tr\\,}\\Sigma ^3 A+(n^2+n) {\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma A\\\\&&~~~~~~~~~~~~~~~~~~~~~~~+2 n(n+1) {\\rm tr\\,}\\Sigma {\\rm tr\\,}\\Sigma ^2 A+n ({\\rm tr\\,}\\Sigma )^2 {\\rm tr\\,}\\Sigma A,\\\\&&\\mathrm {(iv)}~{\\mathrm {E}}[({\\rm tr\\,}A W^2)({\\rm tr\\,}BW^2)]=n(n^2+n+2)(n+1){\\rm tr\\,}\\Sigma ^2 A {\\rm tr\\,}\\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+({\\rm tr\\,}\\Sigma )^2 \\lbrace n^2 {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B+2 n {\\rm tr\\,}\\Sigma A \\Sigma B\\rbrace \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+{\\rm tr\\,}(\\Sigma )\\lbrace n (n^2+n+2) {\\rm tr\\,}\\Sigma ^2 A {\\rm tr\\,}\\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^2+n+2) {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^2 B+8 n (n+1) {\\rm tr\\,}\\Sigma ^2A \\Sigma B\\rbrace \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+{\\rm tr\\,}\\Sigma ^2 (2 n {\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma B+2 n (n+1) {\\rm tr\\,}\\Sigma A \\Sigma B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n+1)^2 {\\rm tr\\,}\\Sigma ^2 A \\Sigma ^2 B+4 n(n^2+3n+4){\\rm tr\\,}\\Sigma ^3A\\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n+1)({\\rm tr\\,}\\Sigma ^3A {\\rm tr\\,}\\Sigma B+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^3 B),\\\\&&\\mathrm {(v)}~{\\mathrm {E}}[{\\rm tr\\,}AW^2BW^2]=2n(n+1)^2{\\rm tr\\,}\\Sigma ^2 A{\\rm tr\\,}\\Sigma ^2 B+n(n^2+3n+4)(n+1){\\rm tr\\,}\\Sigma ^2 A \\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~+({\\rm tr\\,}\\Sigma )^2(n{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B+n(n+1){\\rm tr\\,}\\Sigma A \\Sigma B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~+{\\rm tr\\,}(\\Sigma ) \\lbrace 2 n (n+1)({\\rm tr\\,}\\Sigma ^2 A{\\rm tr\\,}\\Sigma B+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^2 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~+2n(n^2+3n+4){\\rm tr\\,}\\Sigma ^2 A \\Sigma B\\rbrace \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~+{\\rm tr\\,}(\\Sigma ^2)\\lbrace n(n+1) {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B+n (n+3) {\\rm tr\\,}\\Sigma A \\Sigma B\\rbrace \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^2+3n+4)({\\rm tr\\,}\\Sigma ^3 A{\\rm tr\\,}\\Sigma B+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^3 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^2+7n+8) {\\rm tr\\,}\\Sigma ^3 A \\Sigma B, \\\\&&\\mathrm {(vi)}~E[({\\rm tr\\,}A W^3)({\\rm tr\\,}BW^3)]= (n^2{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma B+2n{\\rm tr\\,}\\Sigma A \\Sigma B ) ({\\rm tr\\,}\\Sigma )^4\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \\lbrace 2 n (n^2+n+2 )({\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^2 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^2 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+16 n (n+1) {\\rm tr\\,}\\Sigma ^2 A \\Sigma B \\rbrace ({\\rm tr\\,}\\Sigma )^3\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \\lbrace 2 n (n^2+n+4 ) {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+12 n (n+1) {\\rm tr\\,}\\Sigma A \\Sigma B {\\rm tr\\,}\\Sigma ^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n+1) (n^2+n+4 ){\\rm tr\\,}\\Sigma ^2 A {\\rm tr\\,}\\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^3+3n+24n+20) ({\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^3 A+{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma ^3 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (5 n^2+11n+8) {\\rm tr\\,}\\Sigma ^2 A \\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+24 n (n^2+3n+4) {\\rm tr\\,}\\Sigma ^3 A \\Sigma B\\rbrace ({\\rm tr\\,}\\Sigma )^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \\lbrace 2 n (n+1) (n^2+n+10 )\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times ({\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^2 {\\rm tr\\,}\\Sigma ^2 A+{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma ^2 {\\rm tr\\,}\\Sigma ^2 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^4+4n^3+21n^2+38n+32) {\\rm tr\\,}\\Sigma ^3 B{\\rm tr\\,}\\Sigma ^2 A\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+16 n (n+1){\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^3\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+8 n (n^2+3n+4) {\\rm tr\\,}\\Sigma ^3 {\\rm tr\\,}\\Sigma A \\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^4+4n^3+21n^2+38n+32) {\\rm tr\\,}\\Sigma ^2 B {\\rm tr\\,}\\Sigma ^3 A\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+16 n (2 n^2+5n+5) {\\rm tr\\,}\\Sigma ^2 {\\rm tr\\,}\\Sigma ^2 A \\Sigma B$ $&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (7 n^2+19n+22) ({\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^4 A+{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma ^4 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+16 n (2 n^3+9n^2+21n+16) {\\rm tr\\,}(\\Sigma ^3 A \\Sigma ^2 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+16 n (n^3+6n^2+21n+20) {\\rm tr\\,}\\Sigma ^4 A \\Sigma B\\rbrace {\\rm tr\\,}\\Sigma \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n+1)(n^2+n+4) {\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma B ({\\rm tr\\,}\\Sigma ^2)^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (5 n^2+11n+8) {\\rm tr\\,}\\Sigma ^2 {\\rm tr\\,}\\Sigma ^2 A {\\rm tr\\,}\\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (3 n^2+7n+6) ({\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^2 A +{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma ^2 B) {\\rm tr\\,}\\Sigma ^3\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (2 n^2+5n+5) ({\\rm tr\\,}\\Sigma ^2)^2{\\rm tr\\,}(\\Sigma A \\Sigma B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^4+4n^3+19n^2+36n+36)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times ({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^3 A +{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma ^3 B){\\rm tr\\,}\\Sigma ^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^5+6n^4+27n^3+74n^2+156n+120) {\\rm tr\\,}\\Sigma ^3 A {\\rm tr\\,}\\Sigma ^3 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (2 n^2+5n+5) {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^4\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^3+6n^2+21n+20) {\\rm tr\\,}\\Sigma A \\Sigma B {\\rm tr\\,}\\Sigma ^4\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+8 n (n^3+5n^2+14n+12){\\rm tr\\,}\\Sigma ^3 {\\rm tr\\,}\\Sigma ^2 A \\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (2 n^3+9n^2+21n+16)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times (2{\\rm tr\\,}\\Sigma ^2 B {\\rm tr\\,}\\Sigma ^4 A+2{\\rm tr\\,}\\Sigma ^2 A {\\rm tr\\,}\\Sigma ^4 B+{\\rm tr\\,}\\Sigma ^2 {\\rm tr\\,}\\Sigma ^2 A \\Sigma ^2 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+12 n (n^3+5n^2+14n+12)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times ({\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^3 A \\Sigma B+{\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^5 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^5 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (3 n^4+20n^3+77n^2+152n+132) {\\rm tr\\,}\\Sigma ^3 A \\Sigma ^3 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+8 n (n^4+8n^3+39n^2+80n+64) {\\rm tr\\,}\\Sigma ^4 A \\Sigma ^2B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^4+10n^3+65n^2+160n+148) {\\rm tr\\,}\\Sigma ^5 A \\Sigma B,\\\\&&\\mathrm {(vii)}~E[{\\rm tr\\,}AW^3BW^3]=\\lbrace (n^2+n){\\rm tr\\,}\\Sigma A \\Sigma B+n{\\rm tr\\,}\\Sigma A \\Sigma B\\rbrace ({\\rm tr\\,}\\Sigma )^4\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\lbrace 4n(n+1)({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^2 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^2 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4n(n^2+3n+4){\\rm tr\\,}\\Sigma ^2 A \\Sigma B\\rbrace ({\\rm tr\\,}\\Sigma )^3\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\lbrace 6n(n+1){\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2n(n^2+4n+7){\\rm tr\\,}\\Sigma A \\Sigma B{\\rm tr\\,}\\Sigma ^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2n(5n^2+11n+8){\\rm tr\\,}\\Sigma ^2 A {\\rm tr\\,}\\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+6n(n^2+3n+4)({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^3 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^3 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2n(2 n^3+9n^2+21n+16) {\\rm tr\\,}\\Sigma ^2 A \\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n(n^3+9n^2+42n+44) {\\rm tr\\,}\\Sigma ^3 A \\Sigma B\\rbrace ({\\rm tr\\,}\\Sigma )^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\lbrace 4 n (2 n^2+5n+5)({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^2 A+{\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma ^2 B){\\rm tr\\,}\\Sigma ^2\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4n((2 n^3+9n^2+21n+16)({\\rm tr\\,}\\Sigma ^3 B{\\rm tr\\,}\\Sigma ^2 A+{\\rm tr\\,}\\Sigma ^2 B{\\rm tr\\,}\\Sigma ^3 A)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^2+3n+4) {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^3\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^2+7n+8) {\\rm tr\\,}\\Sigma ^3{\\rm tr\\,}\\Sigma A \\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^3+6n^2+21n+20)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times ({\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^2 A\\Sigma B+{\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^4 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^4 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^4+8n^3+39n^2+80n+64) {\\rm tr\\,}\\Sigma ^3 A \\Sigma ^2B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+8 n (n^3+13n^2+40n+42) {\\rm tr\\,}\\Sigma ^4 A \\Sigma B\\rbrace {\\rm tr\\,}\\Sigma \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (2 n^2+5n+5) {\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma B ({\\rm tr\\,}\\Sigma ^2)^2$ $&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (2 n^3+9n^2+21n+16) {\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^2 A{\\rm tr\\,}\\Sigma ^2B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n(n^3+5n^2+14n+12)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\lbrace 2({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^2 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^2 B){\\rm tr\\,}\\Sigma ^3\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+3({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^3 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^3 B)\\rbrace \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^3+4n^2+10n+9) ({\\rm tr\\,}\\Sigma ^2)^2 {\\rm tr\\,}\\Sigma A \\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (3 n^4+20n^3+77n^2+152n+132) {\\rm tr\\,}\\Sigma ^3 A {\\rm tr\\,}\\Sigma ^3 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^3+6n^2+21n+20) {\\rm tr\\,}\\Sigma A {\\rm tr\\,}\\Sigma B {\\rm tr\\,}\\Sigma ^4\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^3+14n^2+41n+40) {\\rm tr\\,}\\Sigma A \\Sigma B{\\rm tr\\,}\\Sigma ^4\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^3+11n^2+28n+24) {\\rm tr\\,}\\Sigma ^3{\\rm tr\\,}\\Sigma ^2 A \\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^4+8n^3+39n^2+80n+64)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times ({\\rm tr\\,}\\Sigma ^2 B{\\rm tr\\,}\\Sigma ^4 A+{\\rm tr\\,}\\Sigma ^2 A{\\rm tr\\,}\\Sigma ^4 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (2n^3+19n^2+43n+32) {\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^2 A \\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^5+7n^4+34n^3+78n^2+72) {\\rm tr\\,}\\Sigma ^2{\\rm tr\\,}\\Sigma ^3 A \\Sigma B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^4+10n^3+65n^2+160n+148)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times ({\\rm tr\\,}\\Sigma B{\\rm tr\\,}\\Sigma ^5 A+{\\rm tr\\,}\\Sigma A{\\rm tr\\,}\\Sigma ^5 B)\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n (n^5+9n^4+47n^3+151n^2+308n+252) {\\rm tr\\,}\\Sigma ^3 A \\Sigma ^3 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+4 n (n^4+16n^3+75n^2+164n+128) {\\rm tr\\,}\\Sigma ^4 A \\Sigma ^2 B\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2 n (n^4+22n^3+125n^2+328n+292) {\\rm tr\\,}\\Sigma ^5 A \\Sigma B.\\\\$ (Proof) The proof of assertions (i)-(vii) follows directly by applying the technique derived in Lemma A.2, in Hyodo et al.", "(2012).", "$\\hfill \\square $ Lemma A.", "3 (Moments of quadratic form) Let $x$ be distributed $\\mathcal {N}_p(0,I_p)$ .", "Then the following assertions hold: $&{\\rm (i)}&~{\\mathrm {E}}[{\\text{$x$}}^{\\prime }A{\\text{$x$}}]={\\rm tr\\,}A,\\\\&{\\rm (ii)}&~{\\mathrm {E}}[{\\text{$x$}}^{\\prime } A{\\text{$x$}}{\\text{$x$}}^{\\prime } B{\\text{$x$}}]=2{\\rm tr\\,}AB+{\\rm tr\\,}A{\\rm tr\\,}B,$ where $A$ and $B$ are $p\\times p$ non-random symmetric matrices.", "(Proof) See, Gupta and Nagar (1999).", "Lemma A.", "4 (Multivariate Delta Method) Suppose that ${\\text{$y$}}_n=(y_{n1},\\ldots ,y_{nk})^{\\prime }$ is a sequence of the random vectors such that $\\sqrt{n}({\\text{$y$}}_n-{\\text{$\\mu $}})\\xrightarrow{}{\\cal N}_k(0,\\Theta )~{as}~n\\rightarrow \\infty ,$ where ${\\text{$\\mu $}}=(\\mu _1,\\cdots ,\\mu _k)^{\\prime }$ is the asymptotic mean vector and $\\Theta $ is the asymptotic covariance matrix which is assumed to be positive definite.", "Let $g:\\mathbb {R}^k\\rightarrow \\mathbb {R}$ and g is continuously differentiable.", "Let $\\nabla g({\\text{$y$}})=\\left(\\frac{\\partial g}{\\partial y_1},\\cdots ,\\frac{\\partial g}{\\partial y_k}\\right)^{\\prime }.$ Let $\\nabla _\\mu $ denote $\\nabla g({\\text{$y$}})$ evaluated at ${\\text{$y$}}={\\text{$\\mu $}}$ and assume that the elements of $\\nabla _\\mu $ are nonzero.", "Then it holds that $\\sqrt{n}(g({\\text{$y$}}_n)-g({\\text{$\\mu $}}))\\rightarrow {\\cal N}(0,\\nabla _\\mu ^{\\prime }\\Theta \\nabla _\\mu )~{as}~n\\rightarrow \\infty .$ (Proof) See, Rao (1973).", "Lemma A.", "5 (The consistent estimators of $a_3$ and $a_4$ ) The consistent estimators of $a_3$ and $a_4$ are $\\hat{a}_3&=&\\frac{n^2}{(n+4)(n+2)(n-1)(n-2)p} \\lbrace n^2{\\rm tr\\,}{S^3} - 3n{\\rm tr\\,}{S^2}{\\rm tr\\,}{S}+ 2({\\rm tr\\,}{S})^3 \\rbrace ,\\\\\\hat{a}_4&=&\\frac{1}{p} \\lbrace b_1 {\\rm tr\\,}{S^4}+ b_2 {\\rm tr\\,}{S^3}{\\rm tr\\,}{S}+ b_3 ({\\rm tr\\,}{S^2})^2+ b_4 ({\\rm tr\\,}{S})^2{\\rm tr\\,}{S^2}+b_5 ({\\rm tr\\,}{S})^4 \\rbrace ,$ where $b_1&=&\\frac{n^5(n^2 +n +2)}{(n+6)(n+4)(n+2)(n+1)(n-1)(n-2)(n-3)},\\\\b_2&=&-\\frac{4n^4(n^2+n+2)}{(n+6)(n+4)(n+2)(n+1)(n-1)(n-2)(n-3)},\\\\b_3&=&-\\frac{n^4(2n^2+3n-6)}{(n+6)(n+4)(n+2)(n+1)(n-1)(n-2)(n-3)},\\\\b_4&=&\\frac{2n^4(5n+6)}{(n+6)(n+4)(n+2)(n+1)(n-1)(n-2)(n-3)},\\\\b_5&=&-\\frac{n^3(5n+6)}{(n+6)(n+4)(n+2)(n+1)(n-1)(n-2)(n-3)}.$ (Proof) See, Hyodo et al.", "(2012).", "Lemma A.", "6 (The consistent estimators of ${\\Delta }_2$ and ${\\Delta }_3$ ) The consistent estimators of ${\\Delta }_2$ and ${\\Delta }_3$ are $\\widehat{\\Delta }_2&=&\\left(1+\\frac{1}{n}\\right)^{-1}\\left\\lbrace {\\widehat{{\\text{$\\delta $}}}}^\\prime S^2{\\widehat{{\\text{$\\delta $}}}}-\\frac{p}{n}{\\hat{a}}_1{\\widehat{{\\Delta }}}_1-\\frac{Np}{N_1N_2}\\left(\\frac{n+1}{n}{\\hat{a}}_3+\\frac{p}{n}{\\hat{a}}_1{\\hat{a}}_2\\right)\\right\\rbrace ,\\\\\\widehat{\\Delta }_3&=&\\left(\\frac{n(n+3)+4}{n^2}\\right)^{-1}\\left\\lbrace {\\widehat{{\\text{$\\delta $}}}}^\\prime S^3{\\widehat{{\\text{$\\delta $}}}}-\\frac{(n+1)p}{n^2}{\\hat{a}}_2{\\widehat{{\\Delta }}}_1^2-\\frac{2(n+1)p}{n^2}{\\hat{a}}_1{\\widehat{{\\Delta }}}_2^2-\\frac{p^2}{n^2}{\\hat{a}}_1^2{\\widehat{{\\Delta }}}_1^2\\right.\\\\& &-\\left.\\frac{Np}{N_1N_2}\\left(\\frac{n(n+3)+4}{n^2}{\\hat{a}}_4+\\frac{(n+1)p}{n^2}{\\hat{a}}_2^2+\\frac{2(n+1)p}{n^2}{\\hat{a}}_1{\\hat{a}}_3+\\frac{p^2}{n^2}{\\hat{a}}_1^2{\\hat{a}}_2\\right)\\right\\rbrace .$ (Proof)We consider following random variables $\\widetilde{\\Delta }_2&=&\\left(1+\\frac{1}{n}\\right)^{-1}\\left\\lbrace {\\widehat{{\\text{$\\delta $}}}}^\\prime S^2{\\widehat{{\\text{$\\delta $}}}}-\\frac{p}{n}a_1{\\Delta }_1-\\frac{Np}{N_1N_2}\\left(\\frac{n+1}{n}a_3+\\frac{p}{n}a_1a_2\\right)\\right\\rbrace ,\\\\\\widetilde{\\Delta }_3&=&\\left(\\frac{n(n+3)+4}{n^2}\\right)^{-1}\\left\\lbrace {\\widehat{{\\text{$\\delta $}}}}^\\prime S^3{\\widehat{{\\text{$\\delta $}}}}-\\frac{(n+1)p}{n^2}a_2{\\Delta }_1-\\frac{2(n+1)p}{n^2}a_1{\\Delta }_2-\\frac{p^2}{n^2}a_1^2{\\Delta }_1\\right.\\\\& &-\\left.\\frac{Np}{N_1N_2}\\left(\\frac{n(n+3)+4}{n^2}a_4+\\frac{(n+1)p}{n^2}a_2^2+\\frac{2(n+1)p}{n^2}a_1a_3+\\frac{p^2}{n^2}a_1^2a_2\\right)\\right\\rbrace .$ Then the conditional moments of ${\\widehat{{\\text{$\\delta $}}}}^\\prime S^i{\\widehat{{\\text{$\\delta $}}}}$ are given by $4\\endcsname E[{\\widehat{{\\text{$\\delta $}}}}^\\prime S^i{\\widehat{{\\text{$\\delta $}}}}\|S]&=&{\\text{$\\delta $}}^{\\prime }S^i{\\text{$\\delta $}}+\\frac{N}{N_1N_2}{\\rm tr\\,}\\Sigma S^i,\\\\E[({\\widehat{{\\text{$\\delta $}}}}^\\prime S^i{\\widehat{{\\text{$\\delta $}}}})^2\|S]&=&({\\text{$\\delta $}}^{\\prime }S^i{\\text{$\\delta $}})^2+\\frac{4N}{N_1N_2}{\\text{$\\delta $}}^{\\prime }S^i\\Sigma S^i{\\text{$\\delta $}}+\\frac{N^2}{N_1^2N_2^2}\\lbrace 2({\\rm tr\\,}\\Sigma S^i\\Sigma S^i)+({\\rm tr\\,}\\Sigma S^i)^2\\rbrace \\nonumber \\\\& &+\\frac{2N}{N_1N_2}{\\rm tr\\,}\\Sigma S^i{\\text{$\\delta $}}^{\\prime }S^i{\\text{$\\delta $}}.$ By using (A.4), (A.5) and Lemma A.2, we can calculate the expectations of $\\widetilde{\\Delta }_2,~\\widetilde{\\Delta }_3$ and these variances $E[\\widetilde{\\Delta }_2]&=&{\\text{$\\delta $}}^\\prime \\Sigma ^2 {\\text{$\\delta $}},~E[\\widetilde{\\Delta }_3]={\\text{$\\delta $}}^\\prime \\Sigma ^3 {\\text{$\\delta $}},\\\\{\\rm Var}[\\widetilde{\\Delta }_2]&=&\\frac{n^2}{(n+1)^2} \\left\\lbrace \\frac{4N}{N_1 N_2} {\\Delta }_5 + \\frac{8 N p}{n N_1 N_2} {\\Delta }_4 a_1 + \\frac{4}{n} {\\Delta }_1 {\\Delta }_3 + \\frac{4 N p}{n N_1 N_2} {\\Delta }_3 a_2 \\right.", "\\\\&& + \\frac{4 N p^2}{n^2 N_1 N_2} {\\Delta }_3 a_1^2 + \\frac{4}{n} {\\Delta }_2^2 + \\frac{8 p}{n^2} {\\Delta }_1 {\\Delta }_2 a_1 + \\frac{8 N p}{n N_1 N_2} {\\Delta }_2 a_3 + \\frac{8 N p^2}{n^2 N_1 N_2} {\\Delta }_2 a_1 a_2 \\\\&& + \\frac{2 p}{n^2} {\\Delta }_1^2 a_2 + \\frac{2 p^2}{n^3} {\\Delta }_1^2 a_1^2 + \\frac{4 N p}{n N_1 N_2} {\\Delta }_1 a_4 + \\frac{8 N p^2}{n^2 N_1 N_2} {\\Delta }_1 a_1 a_3 + \\frac{4 N p^2}{n^2 N_1 N_2} {\\Delta }_1 a_2^2 \\\\&& + \\frac{4 N p^3}{n^3 N_1 N_2} {\\Delta }_1 a_1^2 a_2 + \\frac{2 N^2 p}{N_1^2 N_2^2} a_6 + \\frac{4 N^2 p^2}{n N_1^2 N_2^2} a_1 a_5 + \\frac{4 N^2 p^2}{n N_1^2 N_2^2} a_2 a_4 \\\\&& + \\frac{2 N^2 p^3}{n^2 N_1^2 N_2^2} a_1^2 a_4 + \\frac{4 N^2 p^2}{n N_1^2 N_2^2} a_3^2 + \\frac{8 N^2 p^3}{n^2 N_1^2 N_2^2} a_1 a_2 a_3 + \\frac{2 N^2 p^3}{n^2 N_1^2 N_2^2} a_2^3 \\\\&& \\left.", "+ \\frac{2 N^2 p^4}{n^3 N_1^2 N_2^2} a_1^2 a_2^2 \\right\\rbrace + o(n^{-1}), \\\\{\\rm Var}[\\widetilde{\\Delta }_3]&=&\\frac{n^4}{(n^2 + 3 n + 4)^2} \\left\\lbrace \\frac{4 N}{N_1 N_2} {\\Delta }_7 + \\frac{16 N p}{n N_1 N_2} {\\Delta }_6 a_1 + \\frac{4}{n} {\\Delta }_1 {\\Delta }_5 + \\frac{12 N p}{n N_1 N_2} {\\Delta }_5 a_2 \\right.\\\\&& + \\frac{24 N p^2}{n^2 N_1 N_2} {\\Delta }_5 a_1^2 + \\frac{8}{n} {\\Delta }_2 {\\Delta }_4 + \\frac{16 p}{n^2} {\\Delta }_1 {\\Delta }_4 a_1 + \\frac{8 N p}{n N_1 N_2} {\\Delta }_4 a_3 \\\\&& + \\frac{32 N p^2}{n^2 N_1 N_2} {\\Delta }_4 a_1 a_2 + \\frac{16 N p^3}{n^3 N_1 N_2} {\\Delta }_4 a_1^3 + \\frac{6}{n} {\\Delta }_3^2 + \\frac{32 p}{n^2} {\\Delta }_2 {\\Delta }_3 a_1 \\\\&& + \\frac{12 p}{n^2} {\\Delta }_1 {\\Delta }_3 a_2 + \\frac{24 p^2}{n^3} {\\Delta }_1 {\\Delta }_3 a_1^2 + \\frac{12 N p}{n N_1 N_2} {\\Delta }_3 a_4 + \\frac{32 N p^2}{n^2 N_1 N_2} {\\Delta }_3 a_1 a_3 \\\\&& + \\frac{16 N p^2}{n^2 N_1 N_2} {\\Delta }_3 a_2^2 + \\frac{32 N p^3}{n^3 N_1 N_2} {\\Delta }_3 a_1^2 a_2 + \\frac{4 N p^4}{n^4 N_1 N_2} {\\Delta }_3 a_1^4 + \\frac{8 p}{n^2} {\\Delta }_2^2 a_2 \\\\&& + \\frac{20 p^2}{n^3} {\\Delta }_2^2 a_1^2+ \\frac{8 p}{n^2} {\\Delta }_1 {\\Delta }_2 a_3 + \\frac{32 p^2}{n^3} {\\Delta }_1 {\\Delta }_2 a_1 a_2 + \\frac{16 p^3}{n^4} {\\Delta }_1 {\\Delta }_2 a_1^3 \\\\&& + \\frac{8 N p}{n N_1 N_2} {\\Delta }_2 a_5 + \\frac{32 N p^2}{n^2 N_1 N_2} {\\Delta }_2 a_1 a_4 + \\frac{24 N p^2}{n^2 N_1 N_2} {\\Delta }_2 a_2 a_3 + \\frac{40 N p^3}{n^3 N_1 N_2} {\\Delta }_2 a_1^2 a_3 \\\\&& + \\frac{32 N p^3}{n^3 N_1 N_2} {\\Delta }_2 a_1 a_2^2 + \\frac{16 N p^4}{n^4 N_1 N_2} {\\Delta }_2 a_1^3 a_2 + \\frac{2 p}{n^2} {\\Delta }_1^2 a_4 + \\frac{8 p^2}{n^3} {\\Delta }_1^2 a_1 a_3$ $&& + \\frac{4 p^2}{n^3} {\\Delta }_1^2 a_2^2 + \\frac{12 p^3}{n^4} {\\Delta }_1^2 a_1^2 a_2 + \\frac{2 p^4}{n^5} {\\Delta }_1^2 a_1^4 + \\frac{4 N p}{n N_1 N_2} {\\Delta }_1 a_6 + \\frac{16 N p^2}{n^2 N_1 N_2} {\\Delta }_1 a_1 a_5 \\\\&& + \\frac{16 N p^2}{n^2 N_1 N_2} {\\Delta }_1 a_2 a_4 + \\frac{24 N p^3}{n^3 N_1 N_2} {\\Delta }_1 a_1^2 a_4 + \\frac{8 N p^2}{n^2 N_1 N_2} {\\Delta }_1 a_3^2 \\\\&& + \\frac{48 N p^3}{n^3 N_1 N_2} {\\Delta }_1 a_1 a_2 a_3 + \\frac{16 N p^4}{n^4 N_1 N_2} {\\Delta }_1 a_1^3 a_3 + \\frac{8 N p^3}{n^3 N_1 N_2} {\\Delta }_1 a_2^3 \\\\&& + \\frac{24 N p^4}{n^4 N_1 N_2} {\\Delta }_1 a_1^2 a_2^2 + \\frac{4 N p^5}{n^5 N_1 N_2} {\\Delta }_1 a_1^4 a_2 + \\frac{2 N^2 p}{N_1^2 N_2^2} a_8 + \\frac{8 N^2 p^2}{n N_1^2 N_2^2} a_1 a_7 \\\\&& + \\frac{8 N^2 p^2}{n N_1^2 N_2^2} a_2 a_6 + \\frac{12 N^2 p^3}{n^2 N_1^2 N_2^2} a_1^2 a_6 + \\frac{8 N^2 p^2}{n N_1^2 N_2^2} a_3 a_5 + \\frac{24 N^2 p^3}{n^2 N_1^2 N_2^2} a_1 a_2 a_5 \\\\&& + \\frac{8 N^2 p^4}{n^3 N_1^2 N_2^2} a_1^3 a_5 + \\frac{6 N^2 p^2}{n N_1^2 N_2^2} a_4^2 + \\frac{32 N^2 p^3}{n^2 N_1^2 N_2^2} a_1 a_3 a_4 + \\frac{16 N^2 p^3}{n^2 N_1^2 N_2^2} a_2^2 a_4 \\\\&& + \\frac{28 N^2 p^4}{n^3 N_1^2 N_2^2} a_1^2 a_2 a_4 + \\frac{2 N^2 p^5}{n^4 N_1^2 N_2^2} a_1^4 a_4 + \\frac{16 N^2 p^3}{n^2 N_1^2 N_2^2} a_2 a_3^2 \\\\&& + \\frac{20 N^2 p^4}{n^3 N_1^2 N_2^2} a_1^2 a_3^2 + \\frac{40 N^2 p^4}{n^3 N_1^2 N_2^2} a_1 a_2^2 a_3 + \\frac{16 N^2 p^5}{n^4 N_1^2 N_2^2} + \\frac{4 N^2 p^4}{n^3 N_1^2 N_2^2} a_2^4 \\\\&&\\left.", "+ \\frac{12 N^2 p^5}{n^4 N_1^2 N_2^2} a_1^2 a_2^3 + \\frac{2 N^2 p^6}{n^5 N_1^2 N_2^2} a_1^4 a_2^2 \\right\\rbrace + o(n^{-1}).$ Using the Chebyshev's inequality, We get $\\widetilde{\\Delta }_i\\xrightarrow{}{\\Delta }_i,~i=2,3$ .", "Replacing the unknown values in $\\widetilde{{\\Delta }}_2$ with their consistent estimator of ${\\Delta }_2$ , we have $\\widehat{\\Delta }_2&=&\\left(1+\\frac{1}{n}\\right)^{-1}\\left\\lbrace {\\widehat{{\\text{$\\delta $}}}}^\\prime S^2{\\widehat{{\\text{$\\delta $}}}}-\\frac{p}{n}{\\hat{a}}_1{\\widehat{{\\Delta }}}_1-\\frac{Np}{N_1N_2}\\left(\\frac{n+1}{n}{\\hat{a}}_3+\\frac{p}{n}{\\hat{a}}_1{\\hat{a}}_2\\right)\\right\\rbrace .$ Using consistency of ${\\hat{a}}_i$ , i=1,2,3 and ${\\widehat{{\\Delta }}}_1$ , we can prove the consistency of ${\\widehat{{\\Delta }}}_2$ .", "The consistency of ${\\widehat{{\\Delta }}}_3$ can be proven similarly.", "$\\hfill \\square $" ] ]	1403.0329
[ [ "Difference equations of q-Appell polynomials" ], [ "Abstract In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known calssical (q=1) results.", "We also provide the recurrence relations and the q-difference equations for q-Bernoulli polynomials, q-Euler polynomials, q-Genocchi polynomials and for newly defined q-Hermite polynomials, as special cases of q-Appell polynomials" ], [ "Introduction", "He and Ricci [4] obtained the differential equations of the Appell polynomials via the factorization method.", "Moreover, they found differential equations satisfied by Bernoulli and Euler polynomials as a special case.", "Afterward, Da-Qian Lu found differential equations for generalized Bernoulli polynomials in [6].", "Recently, several interesting properties and relationships involving the classical Appell type polynomials were investigated [5]-[13].", "The proof given by He and Ricci used the factorization method, which based on raising and lowering operators techniques.", "Note that the raising operators are not available for general polynomials, although lowering operators always exist.", "The proof of the main results given here for $q$ -Appell polynomials does not use raising operators.", "In this paper, we derive $q$ -difference equations for $q$ -Appell polynomials $A_{n,q}\\left( x\\right) $ defined in Al-Salam [1].", "As special cases of $q$ -Appell polynomials, we also provide the $q$ -difference equations for $q$ -Bernoulli polynomials $B_{n,q}\\left( x\\right) $ , $q$ -Euler polynomials $E_{n,q}\\left( x\\right) $ , $q$ -Genocchi polynomials and for newly defined $q$ -Hermite polynomials $H_{n,q}\\left( x\\right) $ .", "We briefly recall some of the properties of these polynomials.", "The Appell polynomials can be defined by considering the following generating function: $A\\left( x,t\\right) :=A_{q}\\left( t\\right) e_{q}\\left( tx\\right)=\\sum _{n=0}^{\\infty }A_{n,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[ n\\right]_{q}!", "},\\ \\ 0<q<1, $ where $A_{q}\\left( t\\right) :=\\sum _{n=0}^{\\infty }A_{n,q}\\dfrac{t^{n}}{\\left[n\\right] _{q}!", "},\\ \\ \\ A\\left( 0\\right) \\ne 0,\\ \\ \\ $ is analytic function at $t=0$ , and $A_{n,q}:=A_{n,q}\\left( 0\\right) ,$ and $e_{q}\\left( t\\right) =\\sum _{n=0}^{\\infty }\\dfrac{t^{n}}{\\left[ n\\right]_{q}!", "}.$ Differentiating generation equation (REF ) with respect to $x$ and equating coefficients of $t^{n},$ we obtain $D_{q,x}A_{n,q}\\left( x\\right) =\\left[ n\\right] _{q}A_{n-1,q}\\left(x\\right) .$ Then the lowering operator $\\Phi _{n}=\\dfrac{1}{\\left[ n\\right] _{q}}D_{q,x}$ satisfies the following operational relation: $\\Phi _{n}A_{n,q}\\left( x\\right) =A_{n-1,q}\\left( x\\right) .$ It follows that $A_{n-k,q}\\left( x\\right) =\\left( \\Phi _{n-k}\\circ ...\\circ \\Phi _{n}\\right)A_{n,q}\\left( x\\right) =\\frac{\\left[ n-k\\right] _{q}!", "}{\\left[ n\\right]_{q}!", "}D_{q,x}^{k}A_{n,q}\\left( x\\right) .", "$" ], [ " Recursion formulas and $q$ -difference equations", "In this section, we derive a difference equation for the $q$ -Appell polynomials $A_{n,q}\\left( x\\right) $ and give the recurrence relations and difference equations for the $q$ -Appell polynomials.", "Theorem 1 The following linear homogeneous recurrence relation for the $q$ -Appell polynomials holds true: $A_{n,q}\\left( qx\\right) & =\\frac{1}{\\left[ n\\right] _{q}}\\sum _{k=0}^{n}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}\\alpha _{n-k}q^{k}A_{k,q}\\left( x\\right) +xq^{n}A_{n-1,q}\\left(x\\right) \\\\& =\\frac{1}{\\left[ n\\right] _{q}}\\alpha _{0}q^{n}A_{n,q}\\left( x\\right)+q^{n}\\left( x+\\alpha _{1}q^{-1}\\right) A_{n-1,q}\\left( x\\right) +\\frac{1}{\\left[ n\\right] _{q}}\\sum _{k=0}^{n-2}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}\\alpha _{n-k}q^{k}A_{k,q}\\left( x\\right) .$ Proof.", "See formula (REF ) in the proof of Theorem REF .", "Theorem 2 Assume that $t\\dfrac{D_{q,t}A_{q}\\left( t\\right) }{A_{q}\\left( qt\\right) }=\\sum _{n=0}^{\\infty }\\alpha _{n}\\dfrac{t^{n}}{\\left[ n\\right] _{q}!}.", "$ The $q$ -Appell polynomials $A_{n,q}\\left( x\\right) $ satisfy the $q$ -difference equation $\\dfrac{\\alpha _{n}}{\\left[ n\\right] _{q}!", "}D_{q,x}^{n}A_{n,q}\\left( x\\right)+\\dfrac{q\\alpha _{n-1}}{\\left[ n-1\\right] _{q}!", "}D_{q,x}^{n-1}A_{n,q}\\left(x\\right) +...+\\dfrac{q^{n-2}\\alpha _{2}}{\\left[ 2\\right] _{q}!", "}D_{q,x}^{2}A_{n,q}\\left( x\\right) \\\\+\\dfrac{q^{n-1}\\alpha _{1}}{\\left[ 1\\right] _{q}!", "}D_{q,x}A_{n,q}\\left(x\\right) +\\dfrac{q^{n}\\alpha _{0}}{\\left[ 0\\right] _{q}!", "}A_{n,q}\\left(x\\right) +xq^{n}D_{q,x}A_{n,q}\\left( x\\right) -\\left[ n\\right]_{q}A_{n,q}\\left( qx\\right) =0.$ Proof.", "Differentiating generating equation $A_{q}\\left( qx,t\\right) =A_{q}\\left( t\\right) e_{q}\\left( tqx\\right)=\\sum _{n=0}^{\\infty }A_{n,q}\\left( qx\\right) \\dfrac{t^{n}}{\\left[ n\\right]_{q}!}", "$ with respect to $t$ and multiplying the obtained equality by $t,$ we get the following two equations $tD_{q,t}A_{q}\\left( qx,t\\right) & =D_{q,t}\\left( A_{q}\\left( t\\right)e_{q}\\left( tqx\\right) \\right) =\\left( D_{q,t}A_{q}\\left( t\\right)\\right) e_{q}\\left( tqx\\right) +qxA_{q}\\left( qt\\right) e_{q}\\left(tqx\\right) \\\\& =A_{q}\\left( x,qt\\right) \\left[ t\\dfrac{D_{q,t}A_{q}\\left( t\\right)}{A_{q}\\left( qt\\right) }+tqx\\right] \\\\tD_{q,t}A_{q}\\left( qx,t\\right) & =t\\sum _{n=0}^{\\infty }\\left[ n\\right]_{q}A_{n,q}\\left( qx\\right) \\dfrac{t^{n-1}}{\\left[ n\\right] _{q}!", "}=\\sum _{n=0}^{\\infty }\\left[ n\\right] _{q}A_{n,q}\\left( qx\\right)\\dfrac{t^{n}}{\\left[ n\\right] _{q}!", "}.$ Now from the assumption (REF ) it follows that $\\sum _{n=0}^{\\infty }\\left[ n\\right] _{q}A_{n,q}\\left( qx\\right)\\dfrac{t^{n}}{\\left[ n\\right] _{q}!}", "& =A_{q}\\left( x,qt\\right) \\left[t\\dfrac{D_{q,t}A_{q}\\left( t\\right) }{A_{q}\\left( qt\\right) }+tqx\\right]\\nonumber \\\\& =\\sum _{n=0}^{\\infty }q^{n}A_{n,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[n\\right] _{q}!", "}\\left[ \\sum _{n=0}^{\\infty }\\alpha _{n}\\dfrac{t^{n}}{\\left[n\\right] _{q}!", "}+tqx\\right] \\nonumber \\\\& =\\sum _{n=0}^{\\infty }\\sum _{k=0}^{n}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}\\alpha _{k}q^{n-k}A_{n-k,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[n\\right] _{q}!", "}+x\\sum _{n=0}^{\\infty }q^{n+1}A_{n,q}\\left( x\\right)\\dfrac{t^{n+1}}{\\left[ n\\right] _{q}!}.", "$ Equating coefficients of $t^{n}$ in equation (REF ), we obtain $\\left[ n\\right] _{q}A_{n,q}\\left( qx\\right) =\\sum _{k=0}^{n}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}\\alpha _{k}q^{n-k}A_{n-k,q}\\left( x\\right) +x\\left[ n\\right]_{q}q^{n}A_{n-1,q}\\left( x\\right) .", "$ Inserting (REF ) into (REF ) we get $\\left[ n\\right] _{q}A_{n,q}\\left( qx\\right) & =\\sum _{k=0}^{n}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}\\alpha _{k}q^{n-k}\\frac{\\left[ n-k\\right] _{q}!", "}{\\left[n\\right] _{q}!", "}D_{q,x}^{k}A_{n,q}\\left( x\\right) +x\\left[ n\\right]_{q}q^{n}\\frac{\\left[ n-1\\right] _{q}!", "}{\\left[ n\\right] _{q}!", "}D_{q,x}A_{n,q}\\left( x\\right) \\\\& =\\sum _{k=0}^{n}\\frac{q^{n-k}}{\\left[ k\\right] _{q}!", "}\\alpha _{k}D_{q,x}^{k}A_{n,q}\\left( x\\right) +xq^{n}D_{q,x}A_{n,q}\\left( x\\right) \\\\& =\\left( \\sum _{k=0}^{n}\\frac{q^{n-k}}{\\left[ k\\right] _{q}!", "}\\alpha _{k}D_{q,x}^{k}+xq^{n}D_{q,x}\\right) A_{n,q}\\left( x\\right) .$" ], [ "$q$ -Bernoulli polynomials", "The Bernoulli polynomials $B_{n,q}\\left( x\\right) $ are defined (see [2], [3]) starting from the generating function: $B_{q}\\left( x,t\\right) :=\\frac{t}{e_{q}\\left( t\\right) -1}e_{q}\\left(tx\\right) =\\sum _{n=0}^{\\infty }B_{n,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[n\\right] _{q}!", "},\\ \\ \\ \\left\|t\\right\|<2\\pi ,$ and consequently, the Bernoulli numbers $b_{n,q}:=B_{n,q}\\left( 0\\right) $ can be obtained by the generating function: $B_{q}\\left( t\\right) :=\\frac{t}{e_{q}\\left( t\\right) -1}=\\sum _{n=0}^{\\infty }b_{n,q}\\dfrac{t^{n}}{\\left[ n\\right] _{q}!", "}.$ Theorem 3 The following linear homogeneous recurrence relation for the $q$ -Bernoulli polynomials holds true: $B_{n,q}\\left( qx\\right) =q^{n}\\left( x-\\frac{1}{q\\left[ 2\\right] _{q}}\\right) B_{n-1,q}\\left( x\\right) -\\frac{1}{\\left[ n\\right] _{q}}\\sum _{k=0}^{n-2}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}q^{k-1}b_{n-k,q}B_{k,q}\\left( x\\right)$ Theorem 4 The $q$ -Bernoulli polynomials $B_{k,q}\\left( x\\right) $ satisfy the $q$ -difference equation $\\dfrac{b_{n,q}}{q\\left[ n\\right] _{q}!", "}D_{q,x}^{n}B_{n,q}\\left( x\\right)+\\dfrac{b_{n-1,q}}{\\left[ n-1\\right] _{q}!", "}D_{q,x}^{n-1}B_{n,q}\\left(x\\right) +...+q^{n-3}\\dfrac{b_{2,q}}{\\left[ 2\\right] _{q}!", "}D_{q,x}^{2}B_{n,q}\\left( x\\right) \\\\-q^{n}\\left( x-\\frac{1}{q\\left[ 2\\right] _{q}}\\right) D_{q,x}B_{n,q}\\left( x\\right) +\\left[ n\\right] _{q}B_{n,q}\\left( qx\\right) =0.$" ], [ "$q$ -Euler polynomials", "The Euler numbers $e_{n,q}$ can be defined by the generating function $E_{q}\\left( t\\right) :=\\frac{te_{q}\\left( t\\right) }{e_{q}\\left(2t\\right) -1}=\\sum _{n=0}^{\\infty }e_{n,q}\\dfrac{t^{n}}{\\left[ n\\right]_{q}!", "}.$ The Euler polynomials $E_{n,q}\\left( x\\right) $ (see [3]) can be defined by the generating function $E_{q}\\left( x,t\\right) :=\\frac{2}{e_{q}\\left( t\\right) +1}e_{q}\\left(tx\\right) =\\sum _{n=0}^{\\infty }E_{n,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[n\\right] _{q}!", "},\\ \\ \\ \\left\|t\\right\|<\\pi .$ The connection to the Euler numbers is given by $e_{n,q}=2^{n}E_{n,q}\\left( \\frac{1}{2}\\right) .$ Theorem 5 The following linear homogeneous recurrence relation for the $q$ -Euler polynomials holds true: $E_{n,q}\\left( qx\\right) =\\frac{1}{2}\\sum _{k=0}^{n-1}\\left[\\begin{array}[c]{c}n-1\\\\k\\end{array}\\right] _{q}q^{k}E_{n-k-1}E_{k,q}\\left( x\\right) +xq^{n}E_{n-1,q}\\left(x\\right) .$ Theorem 6 The $q$ -Euler polynomials $B_{k,q}\\left( x\\right) $ satisfy the $q$ -difference equation $\\frac{1}{2}\\dfrac{e_{n-1,q}}{\\left[ n-1\\right] _{q}!", "}D_{q,x}^{n}E_{n,q}\\left( x\\right) +\\frac{1}{2}\\dfrac{qe_{n-2,q}}{\\left[ n-2\\right]_{q}!", "}D_{q,x}^{n-1}E_{n,q}\\left( x\\right) +...+\\frac{1}{2}\\dfrac{q^{n-2}e_{1,q}}{\\left[ 2\\right] _{q}!", "}D_{q,x}^{2}E_{n,q}\\left( x\\right) \\\\-\\dfrac{1}{2}q^{n-1}D_{q,x}E_{n,q}\\left( x\\right) +xq^{n}D_{q,x}E_{n,q}\\left( x\\right) -\\left[ n\\right] _{q}E_{n,q}\\left( qx\\right) =0.$" ], [ "$q$ -Genocchi polynomials", "The $q$ -Genocchi numbers $g_{n,q}$ can be defined by the generating function $G_{q}\\left( t\\right) :=\\frac{2t}{e_{q}\\left( t\\right) +1}=\\sum _{n=0}^{\\infty }g_{n,q}\\dfrac{t^{n}}{\\left[ n\\right] _{q}!", "}.$ The $q$ -Genocchi polynomials $G_{n,q}\\left( x\\right) $ (see [3]) can be defined by the generating function $G_{q}\\left( x,t\\right) :=\\frac{2t}{e_{q}\\left( t\\right) +1}e_{q}\\left(tx\\right) =\\sum _{n=0}^{\\infty }G_{n,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[n\\right] _{q}!", "},\\ \\ \\ \\left\|t\\right\|<\\pi .$ Theorem 7 The following linear homogeneous recurrence relation for the $q$ -Genocchi polynomials holds true: $\\frac{1}{2q}\\sum _{k=0}^{n-2}\\left[\\begin{array}[c]{c}n\\\\k\\end{array}\\right] _{q}g_{n-k,q}q^{k}G_{k,q}\\left( x\\right) +\\left[ n\\right]_{q}\\left( xq-\\frac{1}{2q}\\right) q^{n-1}G_{n-1,q}\\left( x\\right)+q^{n-1}G_{n,q}\\left( x\\right) -\\left[ n\\right] _{q}G_{n,q}\\left(qx\\right) =0.$ Theorem 8 The $q$ -Genocchi polynomials $G_{n,q}\\left( x\\right) $ satisfy the $q$ -difference equation $\\frac{1}{2q}\\dfrac{g_{n,q}}{\\left[ n\\right] _{q}!", "}D_{q,x}^{n}G_{n,q}\\left(x\\right) +\\dfrac{g_{n-1,q}}{2\\left[ n-1\\right] _{q}!", "}D_{q,x}^{n-1}G_{n,q}\\left( x\\right) +...+\\dfrac{q^{n-3}g_{2,q}}{2\\left[ 2\\right] _{q}!", "}D_{q,x}^{2}G_{n,q}\\left( x\\right) \\\\-\\dfrac{q^{n-2}}{2}D_{q,x}G_{n,q}\\left( x\\right) +q^{n-1}G_{n,q}\\left(x\\right) +xq^{n}D_{q,x}G_{n,q}\\left( x\\right) -\\left[ n\\right]_{q}G_{n,q}\\left( qx\\right) =0.$" ], [ " $q$ -Hermite polynomials", "In this section we construct a $q$ -Hermite polynomials and give of their some properties.", "Also, we derive the three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials.", "We define new $q$ -Hermite polynomials $H_{n,q}\\left( x\\right) $ by means of the generating function $H_{q}\\left( x,t\\right) & :=H_{q}\\left( t\\right) e_{q}\\left( tx\\right)=\\sum _{n=0}^{\\infty }H_{n,q}\\left( x\\right) \\dfrac{t^{n}}{\\left[ n\\right]_{q}!", "},\\\\H_{q}\\left( t\\right) & :=\\sum _{n=0}^{\\infty }\\left( -1\\right)^{n}q^{n\\left( n-1\\right) }\\dfrac{t^{2n}}{\\left[ 2n\\right] _{q}!!", "},\\ \\ \\ \\ \\left[ 2n\\right] _{q}!", "!=\\left[ 2n\\right] _{q}\\left[2n-2\\right] _{q}...\\left[ 2\\right] _{q}.$ It is clear that $\\lim \\limits _{q\\rightarrow 1^{-}}H_{q}\\left( x,t\\right) & =\\lim \\limits _{q\\rightarrow 1^{-}}H_{q}\\left( t\\right) e_{q}\\left( tx\\right)=e^{tx}\\lim \\limits _{q\\rightarrow 1^{-}}\\sum _{n=0}^{\\infty }\\left( -1\\right)^{n}q^{n\\left( n-1\\right) }\\dfrac{t^{2n}}{\\left[ 2n\\right] _{q}!!", "}\\\\& =e^{tx}\\lim \\limits _{q\\rightarrow 1^{-}}\\sum _{n=0}^{\\infty }\\left( -1\\right)^{n}\\dfrac{t^{2n}}{\\left( 2n\\right) \\left( 2n-2\\right) ....2}=e^{tx}\\lim \\limits _{q\\rightarrow 1^{-}}\\sum _{n=0}^{\\infty }\\left( -1\\right)^{n}\\dfrac{t^{2n}}{2^{n}n!", "}\\\\& =\\exp \\left( tx-\\frac{t^{2}}{2}\\right) .$ Moreover $\\dfrac{D_{q,t}H_{q}\\left( t\\right) }{H_{q}\\left( qt\\right) }=-t\\ \\ \\ \\ \\ \\text{and\\ \\ \\ }D_{q,x}H_{n,q}\\left( x\\right) =\\left[n\\right] _{q}H_{n-1,q}\\left( x\\right) .$ Theorem 9 The series form of the $q$ -Hermite polynomial is given by $H_{n,q}\\left( x\\right) =\\sum _{k=0}^{\\left[ \\frac{n}{2}\\right] }\\frac{\\left( -1\\right) ^{k}q^{k\\left( k-1\\right) }x^{n-2k}}{\\left[2k\\right] _{q}!", "!\\left[ n-2k\\right] _{q}!", "}$ Proof.", "Indeed, expanding the generation function $H_{n,q}\\left( x,t\\right) $ , we have $H_{q}\\left( x,t\\right) & =\\sum _{k=0}^{\\infty }\\left( -1\\right)^{k}q^{k\\left( k-1\\right) }\\dfrac{t^{2k}}{\\left[ 2k\\right] _{q}!!", "}\\sum _{l=0}^{\\infty }x^{l}\\dfrac{t^{l}}{\\left[ l\\right] _{q}!", "}\\\\& =\\sum _{n=0}^{\\infty }\\sum _{l=0}^{\\infty }\\frac{\\left( -1\\right)^{k}q^{k\\left( k-1\\right) }x^{l}}{\\left[ 2k\\right] _{q}!", "!\\left[ l\\right]_{q}!", "}t^{2k+l}\\ \\ \\ (2k+l=n)\\\\& =\\sum _{n=0}^{\\infty }\\sum _{k=0}^{\\left[ \\frac{n}{2}\\right] }\\frac{\\left(-1\\right) ^{k}q^{k\\left( k-1\\right) }x^{n-2k}}{\\left[ 2k\\right]_{q}!", "!\\left[ n-2k\\right] _{q}!", "}t^{n}.$ Theorem 10 The following linear homogeneous recurrence relation for the $q$ -Hermite polynomials holds true: $H_{n,q}\\left( qx\\right) =xq^{n}H_{n-1,q}\\left( x\\right) -\\left[n-1\\right] _{q}q^{n-2}H_{n-2,q}\\left( x\\right) ,\\ \\ \\ n\\ge 2.", "$ Using the recurrence relation (REF ), we get $H_{0,q}\\left( x\\right) & =1,\\ \\text{(by definition)}\\\\H_{1,q}\\left( x\\right) & =x,\\\\H_{2,q}\\left( x\\right) & =x^{2}-1,\\\\H_{3,q}\\left( x\\right) & =x^{3}-\\left[ 3\\right] _{q}x,\\\\H_{4,q}\\left( x\\right) & =x^{4}-\\left( 1+q^{2}\\right) \\left[ 3\\right]_{q}x^{2}+\\left[ 3\\right] _{q}q^{2}.$ Theorem 11 The $q$ -Hermite polynomials $G_{n,q}\\left( x\\right) $ satisfy the $q$ -difference equation $q^{n-2}D_{q,x}^{2}H_{n,q}\\left( x\\right) -xq^{n}D_{q,x}H_{n,q}\\left(x\\right) +\\left[ n\\right] _{q}H_{n,q}\\left( qx\\right) =0.", "$ In the limit when $q\\rightarrow 1^{-}$ , the equation (REF ) is reduced to the second order differential equation satisfied by the Hermite polynomials." ] ]	1403.0189
[ [ "Relativistic nuclear magnetic resonance J-coupling with ultrasoft\n pseudopotentials and the zeroth-order regular approximation" ], [ "Abstract We present a method for the first-principles calculation of nuclear magnetic resonance (NMR) J-coupling in extended systems using state-of-the-art ultrasoft pseudopotentials and including scalar-relativistic effects.", "The use of ultrasoft pseudopotentials is allowed by extending the projector augmented wave (PAW) method of Joyce et.", "al [J. Chem.", "Phys.", "127, 204107 (2007)].", "We benchmark it against existing local-orbital quantum chemical calculations and experiments for small molecules containing light elements, with good agreement.", "Scalar-relativistic effects are included at the zeroth-order regular approximation (ZORA) level of theory and benchmarked against existing local-orbital quantum chemical calculations and experiments for a number of small molecules containing the heavy row six elements W, Pt, Hg, Tl, and Pb, with good agreement.", "Finally, 1J(P-Ag) and 2J(P-Ag-P) couplings are calculated in some larger molecular crystals and compared against solid-state NMR experiments.", "Some remarks are also made as to improving the numerical stability of dipole perturbations using PAW." ], [ "Introduction", "Modern high resolution solid-state NMR experiments [1], [2] are a valuable tool for materials characterisation due to their sensitivity to the local atomic environment.", "Importantly, solid-state NMR can provide information on materials with compositional, positional or dynamic disorder[3].", "However, there is no straight-forward analytic technique to obtain atomic-level structure directly from an NMR spectrum; a `Bragg's law' for NMR.", "Instead, one must pursue computational-theoretical prediction of the NMR parameters that influence a spectrum in order to fully take advantage of the information present to interpret and assign spectra.", "First principles predictions of NMR parameters can also assist in the design of NMR experiments, such as determining observability and orientation of tensors.", "Overall, first-principles calculations offer the ability to fully exploit the information in experimental NMR data.", "The sensitivity of NMR experiments to molecular geometry and electronic structure is a `double-edged sword,' being both an important chemical probe and a challenge to the computational theorist.", "For small, finite, systems, NMR parameters such as magnetic shielding, electric field gradients and J-coupling can be routinely calculated with quantum chemical methods based on local orbitals and have demonstrated value in assigning solution-state spectra [4].", "Treatment of solid-state NMR systems with these methods requires the creation of finite-clusters, which need careful convergence with respect to the size of the cluster to ensure that the appropriate electronic environment is reconstructed.", "They also require careful selection of the basis set used to represent the wave function to ensure numerical convergence.", "A planewave approach with pseudopotentials is appealing for its algorithmic efficiency, automatic inclusion of periodic boundary conditions and easy systematic convergence of basis sets via the maximum kinetic energy of the waves used.", "However, since such calculations require the use of pseudopotentials, the calculated pseudo-wave function is non-physical near the nucleus, the very region that is so influential to NMR parameters.", "The development of the gauge-including projector augmented wave (GIPAW) method[5] has enabled calculations of magnetic shielding in extended systems using pseudopotentials by reconstructing the form of the all-electron wavefunction near the nucleus.", "Extensive reviews are available in Refs.", "bonhommefirst-principles2012 and charpentierpaw/gipaw2011.", "This paper concerns itself with the theoretical prediction of NMR J-coupling, or indirect spin-spin coupling, particularly in solid-state systems with heavy ions.", "J-coupling is the indirect magnetic coupling between two nuclei mediated via the bonding electrons.", "It manifests in NMR spectra as fine structure splitting of resonant peaks, providing information on bonds such as strength, angles and the connectivity network.", "J-coupling has been well-studied in the gas and solution state for many decades, as the multiplet splitting in peaks is well resolved due to molecular tumbling decoupling anisotropic interactions.", "In contrast, solid-state J-coupling studies are more challenging due to anisotropic broadening.", "Recent advances in solid-state NMR experiments, such as higher MAS spinning rates (up to 90KHz) and ultra-high magnetic field strengths (up to 23.3T), have resulted in increased experimental and theoretical interest[8] in measurements of J. Joyce et al.", "[9] developed a method to calculate J-coupling constants from first-principles in extended systems within a planewave-pseudopotential density-functional theory (DFT) framework, using PAW to reconstruct the all-electron properties of the system.", "This method has been validated for a small number of systems containing light atoms against quantum chemical calculations and against experimental data [10], [11], [12], [13], [14].", "There is great interest in making this a `full periodic table' method, i.e.", "being able to reliably treat systems containing any elements.", "However, it is known[15], [16] that J-coupling in systems containing heavy ions is extremely sensitive to the effects of special relativity.", "This is because both core states and valence states near the nucleus attain high kinetic energy and so should be treated using the Dirac equation, leading to contraction in the wave function and corrections to the operators representing electromagnetic (EM) interactions.", "Scalar relativistic effects (i.e.", "ignoring spin-orbit terms) in particular have been found to be the dominant correction in full four-component Dirac equation calculations[16], at least for one-bond couplings.", "Autschbach and Ziegler[17], [18] developed the application of the zeroth-order regular approximation (ZORA), an approximation to the Dirac equation, using DFT to the prediction of J-coupling in an all-electron, local orbital framework.", "Yates et al.", "[19] developed the use of ZORA with pseudopotentials and PAW for the calculation of NMR chemical shifts in systems containing heavy ions.", "In this paper we will incorporate the scalar-relativistic terms of Autschbach and Ziegler's ZORA approach within a planewave pseudopotential DFT[20] framework to give a highly efficient method for predicting J-coupling within extended systems containing heavy ions at negligible extra computational cost as compared to the non-relativistic method.", "We also provide some improvements to the non-relativistic method of Joyce et al.", "[9], removing some of the numerical difficulties present in that approach, and generalising the method to use state-of-the-art ultrasoft pseudopotentials[21].", "We carefully benchmark our planewave-pseudopotential implementation against both experiment and existing quantum chemical calculations, and conclude by examining the effects of relativity on J-couplings in two Ag-containing molecular crystals.", "We will proceed by first reviewing the derivation of the zeroth-order regular approximation from the Dirac equation.", "Then we will show how a scalar-relativistic theory of NMR J-coupling can be derived from the ZORA Hamiltonian.", "We then discuss Blöchl's PAW as a general formalism for the calculation of all-electron properties from pseudopotential calculations and derive a form of the scalar-relativistic theory that is suitable for efficient pseudopotentials calculations." ], [ "Zeroth-order regular approximation", "We start with the time-independent single particle Dirac equation[22] for an electron in Hartree atomic units, $[c\\mathbf {\\alpha }\\cdot \\hat{\\mathbf {p}} + \\beta c^2 + V]\\psi = E \\psi ,$ where $\\alpha $ and $\\beta $ are the Dirac matrices and in the case of density-functional theory[23] $V$ represents the nuclear, Hartree and exchange-correlation potentials.", "The wave function $\\psi $ is a complex four-component spinor, alternatively expressed in terms of the small component $\\chi $ and the large component $\\phi $ .", "We can eliminate the small component $\\chi $ by substitution, $\\chi = \\hat{X} \\phi = \\frac{c \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}}}{2c^2 + E - V} \\phi ,$ and retrieve an energy-dependent Hamiltonian in the large component $\\phi $ only: $\\hat{H}^{\\textit {esc}}\\phi = E\\phi = V\\phi + \\frac{1}{2}\\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}} (1 - \\frac{E-V}{2c^2})^{-1} \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}}.$ To normalize $\\phi $ we introduce a normalization operator $\\hat{O} = \\sqrt{1 + \\hat{X}^\\dagger \\hat{X}}$ and so we find the transformed Hamiltonian: H = (1 + XX)12 [V + cp X] (1 + XX)-12 The standard expansion of $\\hat{X}$ and $\\hat{H}^{\\textit {esc}}$ in $(E-V)/(2c^2)$ to give the relativistic Pauli approximation is appropriate when the classical velocity of the electrons is small compared to the speed of light.", "This breaks down for a nuclear Coulomb potential.", "Instead, following van Lenthe [24], [25], we expand in $1/(2c^2 - V)$ , which is justified even near the singularity of a nuclear Coulomb potential.", "This gives $\\hat{H}^{\\textit {esc}} \\approx V + \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}} \\frac{c^2}{2c^2 - V} \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}} - \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}} \\frac{c^2}{2c^2 - V} \\frac{E}{2c^2 - V} \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}} + ... .$ To lowest order the expansion of $\\hat{O}$ in $1/(2c^2 - V)$ gives the identity, so we find that the first two terms of Eqn.", "REF are the ZORA Hamiltonian: $\\hat{H}^{\\textit {ZORA}} = V + \\frac{1}{2}\\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}} \\mathcal {K} \\mathbf {\\sigma }\\cdot {\\hat{\\mathbf {p}}},$ where $\\mathcal {K}$ effectively determines the local influence of relativity on the system (Fig.", "REF ), $\\mathcal {K} = \\frac{2c^2}{2c^2 - V}.$ It is known that ZORA describes valence states in many-electron systems well [25], [26], while describing core states less well.", "However, as valence states are the main contributors to J-coupling it should provide an appropriate level of theory for the present work.", "Substituting the canonical momentum for a magnetic vector potential $\\mathbf {A}$ , $\\hat{\\mathbf {p}} \\rightarrow \\pi = \\hat{\\mathbf {p}} + \\mathbf {A}$ , and expanding we obtain the ZORA Hamiltonian in a magnetic field: HZORA = V + 12(pKp + i (pK)p + pKA + AKp + i[p(KA) + A(Kp)] + KAA) It can be observed that for $\\mathcal {K}=1$ the ZORA Hamiltonian reduces to the nonrelativistic Levy-Leblond Hamiltonian [27] plus spin-orbit coupling.", "The right hand side of REF corresponds to the EM-free ZORA Hamiltonian.", "We concentrate on the scalar-relativistic terms, parts REF and REF , so we neglect the third term of REF , representing spin-orbit coupling.", "However, we note that the effect of spin-orbit coupling on J-coupling can be significant in some compounds [18], [28]." ], [ "NMR J-coupling", "In NMR, indirect spin-spin coupling or J-coupling is an interaction between two nuclear moments due to indirect coupling mediated by the electrons in the system.", "The first analysis of this interaction came with Ramsey and Purcel[29] and Ramsey[30], who decomposed the interaction into four mechanisms: two due to interactions of the electron spins with the nuclear moments and two due to electron currents induced by the nuclear moments.", "When spin-orbit coupling is neglected these can be treated separately.", "Following Ramsey's second-order perturbation analysis, the reduced spin coupling tensor, $K^{AB}$ between nuclei A and B, can be expressed as a second derivative of the system energy with respect to the two interacting nuclear moments, $\\mu = \\gamma \\hbar \\mathbf {I}$ , where $\\gamma $ is the nucleus' gyromagnetic ratio and $\\mathbf {I}$ is the nucleus' spin: $K^{AB} = \\left.", "\\frac{\\partial ^2E}{\\partial \\mu _A\\partial \\mu _B} \\right\|_{\\mu _A=0,\\mu _B=0}$ We can then express the observed $J$ tensor in terms of the reduced spin coupling tensor, $J^{AB} = \\frac{\\hbar }{2\\pi } \\gamma _A \\gamma _B K^{AB}$ .", "For our system of nuclear dipole moments the magnetic vector potential is, in the symmetric gauge with ($\\mathbf {r}_{N} = \\mathbf {r} - \\mathbf {R}_N$ ), $\\mathbf {A} = \\sum _N \\alpha ^2 \\frac{\\mu _{N} \\times \\mathbf {r}_{N}}{\|\\mathbf {r}_{N}\|^3}$ , where $\\alpha $ is the fine-structure constant.", "Determining the derivatives of the ZORA Hamiltonian in this $\\mathbf {A}$ -field with respect to the interacting nuclear magnetic moments will allow us to use second-order perturbation theory to calculate $K^{AB}$ .", "We will use superscripts to represent order of perturbation with respect to the perturbation parameters $\\mu _A$ and $\\mu _B$ : $\\hat{H}^{(n,m)} = \\left.", "\\left(\\frac{\\partial }{\\partial \\mu _A}\\right)^n\\left(\\frac{\\partial }{\\partial \\mu _B}\\right)^m\\hat{H}\\right\|_{\\mu _A=0,\\mu _B=0}$ Autschbach[17] obtained the following derivatives of the ZORA Hamiltonian, along with their equivalent $\\hat{H}^{(0,1)}$ derivatives, as follows: H(1,1)Z-dia;i,j = K4 ij(rA rB) - rA;i rB;j\|rA\|3\|rB\|3 = K H(1,1)dia H(1,0)Z-para;i = 22i [K\|rA\|3 (rA )i + (rA )i K\|rA\|3] = 2i K\|rA\|3 (rA )i = K H(1,0)para H(1,0)Z-spin;i = 826 K i(rA) + K22(3(rA)rA;i\|rA\|5 - i\|rA\|3) + 221\|rA\|3 [{(K)rA}i - (i K)(rA)] We can choose to split the spin term into a spin-dipole analogue, $\\mathcal {K}\\hat{H}^{(1,0)}_{\\textrm {SD}}$ , plus a Fermi-contact analogue: H(1,0)Z-FC;i = 826 K i(rA) + dKdr22((rA)rA;i\|rA\|4- i\|rA\|2).", "All the terms reduce to the non-relativistic versions [9] in the limit $c \\rightarrow \\infty $ , that is, $\\mathcal {K}=1$ and $\\nabla \\mathcal {K}=0$ .", "We see changes for the relativistic case, $\\mathcal {K} \\ne 1$ , especially in the case of the Eqn.", "REF where term REF is non-zero and the term REF (equivalent to the Fermi-contact term) is zero for a nuclear Coulomb potential, neglecting finite-size nucleus effects.", "Finite-size nucleus effects have been found [31] to be somewhat significant in certain circumstances, contributing a 10 to 15% reduction in isotropic J-couplings between heavy elements in the sixth row of the periodic table and light elements, and larger between two heavy elements.", "We can now proceed to calculate $K$ by separating it into contributions from the interactions of the nuclear magnetic moments mediated by the electron spin, $K^{\\textrm {Z-spin}}$ , Eqn.", "REF , and from the interactions of the nuclear magnetic moments mediated by the electron charge current, $K^{\\textrm {Z-orb}}$ , Eqn.", "REF and REF ." ], [ "Spin magnetization density", "To calculate the contribution from the electron spin density we first find an expression for the magnetization density induced by a perturbing nuclear moment.", "From this we can compute the magnetic field induced at all of the other nuclei in the system.", "We take the wave function to be a product of spin-restricted independent electron orbitals $\\psi $ , with a spin index $\\sigma $ .", "The magnetization density induced by a nuclear magnetic moment aligned along the $\\mathbf {u}_j$ direction, $\\mathbf {m}_j$ , is calculated by summing the magnetization density induced when applying the perturbation with the spin quantized along each direction $\\mathbf {u}_k$ .", "$\\mathbf {m}_j(\\mathbf {r}) = \\sum _k m_{jk}(\\mathbf {r}) \\mathbf {u}_k$ where $m_{jk}(\\mathbf {r})$ is given by $m_{jk}(\\mathbf {r}) = 4g\\beta \\sum _o \\langle \\psi ^{(1)}_{o\\uparrow jk} \| \\mathbf {r} \\rangle \\langle \\mathbf {r} \| \\psi ^{(0)}_{o\\uparrow } \\rangle ,$ and the first order response in the wave function, $\| \\psi ^{(1)}_{o\\sigma jk} \\rangle $ , is given by $\| \\psi ^{(1)}_{o\\sigma jk} \\rangle = \\mathcal {G}(\\epsilon ) \| \\psi ^{(0)}_{o\\sigma } \\rangle .$ $\\mathcal {G}(\\epsilon )$ is Green's function, $\\mathcal {G}(\\epsilon ) = \\sum _e \\frac{\| \\psi ^{(0)}_{e} \\rangle \\langle \\psi ^{(0)}_{e} \| }{\\epsilon _o - \\epsilon _e} (\\hat{H}^{(0,1)}_{\\textrm {Z-spin};j} + \\hat{H}^{(1)}_{\\textrm {xc}}),$ where $\\epsilon _o$ and $\\epsilon _e$ are the eigenvalues of the occupied and empty bands respectively, $\\sum _e$ is a sum over empty bands and $\\hat{H}^{(1)}_{\\textrm {xc}}$ is the self-consistent first order variation in the Kohn-Sham exchange-correlation potential due to the first order change in the spin density: $\\hat{H}^{(1)}_{\\textrm {xc}} = V^{(1)}_{\\rm xc} [m^{(1)}],$ where $m^{(1)}$ is the first order magnetization density.", "As there is a first-order change in the Hamiltonian, we solve Eqn.", "REF self-consistently by iteration.", "Given the first order variation in the wave functions, the full expression for the spin term contribution to $K$ is, with an implicit rotation over the spin axes, $K^{\\textrm {Z-spin}}_{ij} = 2\\textrm {Re} \\sum _{o,\\sigma } \\int d\\mathbf {r} \\: \\psi ^{(0)}_{o}(\\mathbf {r})^\\dagger \\hat{H}^{(1,0)}_{\\textrm {Z-spin};i} \\psi ^{(1)}_{o\\sigma j}(\\mathbf {r}).$ We can rearrange this expression to get the effective spin coupling in terms of the induced magnetisation, including an implicit rotation over the spin axes, $K^{\\textrm {Z-spin}}_{ij} = \\alpha ^2 \\int d\\mathbf {r} \\: \\left[ \\mathcal {K} \\left(\\frac{3\\mathbf {r}_{A;i} \\mathbf {r}_{A;j} - \|\\mathbf {r}_A\|^2 \\delta _{ij}}{\|\\mathbf {r}_{A}\|^5}\\right) - \\frac{d\\mathcal {K}}{dr} \\frac{\\mathbf {r}_{A;i} \\mathbf {r}_{A;j} - \|\\mathbf {r}_{A}\|^2\\delta _{ij}}{\|\\mathbf {r}_A\|^4} \\right] \\mathbf {m}_j(\\mathbf {r}).$" ], [ "Current density", "To calculate the contribution from the electron charge current density to $K$ we first find an expression for the current density induced by a perturbing nuclear moment and subsequently the magnetic field induced at the receiving magnetic moment.", "As with the induced spin-magnetization density, we can calculate the induced magnetic field at all the receiving nuclei from this induced current density.", "To first order the current perturbation term, $\\hat{H}^{(0,1)}_{\\textrm {Z-para},j}$ , does not modify the magnetization density or the charge density so there is no first order change in the self-consistent potential.", "The first order variation in the orbitals is therefore $\| \\psi ^{(1)}_{o j} \\rangle = \\mathcal {G}(\\epsilon ) \| \\psi ^{(0)}_{o} \\rangle ,$ where $\\mathcal {G}(\\epsilon )$ is Green's function, $\\mathcal {G}(\\epsilon ) = \\sum _e \\frac{\| \\psi ^{(0)}_{e} \\rangle \\langle \\psi ^{(0)}_{e} \| }{\\epsilon _o - \\epsilon _e} \\hat{H}^{(0,1)}_{\\textrm {Z-para};j},$ in which $\\epsilon _o$ and $\\epsilon _e$ are the eigenvalues of the occupied and empty bands respectively and $\\sum _e$ is a sum over empty bands.", "From second order perturbation theory we can write down $K^{\\textrm {Z-orb}}$ : $K^{\\textrm {Z-orb}}_{ij} = \\sum _{o,\\sigma } \\int d\\mathbf {r} \\: & \\psi ^{(0)}_{o}(\\mathbf {r})^\\dagger \\hat{H}^{(1,0)}_{\\textrm {Z-para};i} \\psi ^{(1)}_{o j}(\\mathbf {r}) + c.c.", "\\\\+ & \\psi ^{(0)}_{o}(\\mathbf {r})^\\dagger \\hat{H}^{(1,1)}_{\\textrm {Z-dia},ij} \\psi ^{(0)}(\\mathbf {r}).$ From the first order variation in the orbitals we can re-arrange Eqn.", "REF to find the induced current: j(1)j(r) = 2o [ 2 Re(0)o \| JZ-P(r) \| (1)oj + (0)o \| JjZ-D(r) \| (0)o ], where our modified paramagnetic and diamagnetic current operators have the form $\\mathbf {J}^{\\textrm {Z-P}}(\\mathbf {r}) &= -\\mathcal {K} (\\hat{\\mathbf {p}}\|\\mathbf {r}\\rangle \\langle \\mathbf {r}\| + \|\\mathbf {r}\\rangle \\langle \\mathbf {r}\|\\hat{\\mathbf {p}}) / 2 \\\\&= \\mathcal {K} \\mathbf {J}^{\\textrm {P}}(\\mathbf {r})$ $\\mathbf {J}_j^{\\textrm {Z-D}}(\\mathbf {r}) &= -\\mathcal {K} \\alpha ^2 \\frac{\\mathbf {r}_{B} \\times \\hat{\\mu }^j}{\|\\mathbf {r}_{kB}\|^3} \\\\&= \\mathcal {K} \\mathbf {J}^{\\textrm {D}}(\\mathbf {r}),$ and so the orbital current contribution to $K$ is simply the Biot-Savart law acting on the induced current: $K^{\\textrm {Z-orb}}_{ij} = \\alpha ^2 \\int d\\mathbf {r} \\: \\left( \\mathbf {j}^{(1)}_{j}(\\mathbf {r}) \\times \\frac{\\mathbf {r}_A}{\|\\mathbf {r}_A\|^3}\\right)_i.$" ], [ "Projector augmented wave", "The expressions derived in Sections REF and REF cannot be directly applied in a pseudopotential based formalism.", "The use of pseudopotential implies that the valence wave functions have a non-physical form in the region close to the nucleus; the all-electron operators used in Sections REF and REF are sensitive to the precise form of the wavefunctions near to the nuclei.", "The standard approach to deal with this problem is the projector augmented wave (PAW) formalism introduced by Van de Walle and Blöchl [32], [33].", "PAW provides a practical way to transform an operator acting on the all-electron wavefunction (AE) $\| \\psi \\rangle $ into an operator acting on the pseudo-wavefunction (PS) $\| \\tilde{\\psi } \\rangle $ , allowing pseudopotentials to be used in calculations of properties that are sensitive to the form of the wavefunction near the nucleus.", "In particular, PAW proposes a linear transformation $\\mathcal {T}$ , $\\mathcal {T}= 1 + \\sum _i(\| \\phi _i \\rangle - \| \\tilde{\\phi _i} \\rangle )\\langle p_i \|,$ such that $\\mathcal {T}\| \\tilde{\\psi } \\rangle = \| \\psi \\rangle $ where $\\phi _i$ and $\\tilde{\\phi _i}$ are atomic-like AE and PS partial waves at each atomic site for the spherically symmetric atom and pseudo-atom.", "The AE and PS partial waves form a complete basis within the augmentation region ($r<r_c$ ) and are equal outside.", "The functions $p_i$ are the corresponding projectors to $\\tilde{\\phi _i}$ , defined such that $\\langle p_i \| \\tilde{\\phi }_j \\rangle = \\delta _{ij}$ and that they vanish outside the augmentation region.", "So, for an AE operator $\\hat{O}$ , $\\langle \\psi \| \\hat{O} \| \\psi \\rangle = \\langle \\tilde{\\psi } \| \\mathcal {T}^{\\dagger } \\hat{O} \\mathcal {T}\| \\tilde{\\psi } \\rangle ,$ we can derive the equivalent pseudo-operator $\\widetilde{O} = \\mathcal {T}^{\\dagger } \\hat{O} \\mathcal {T}$ .", "The explicit form of $\\widetilde{O}$ is $\\widetilde{O} = \\hat{O} + \\sum _{ij} \| p_i \\rangle (\\langle \\phi _i \| \\hat{O} \| \\phi _j \\rangle - \\langle \\tilde{\\phi _i} \| \\hat{O} \| \\tilde{\\phi _j} \\rangle ) \\langle p_j \|.$ This allows us to calculate all-electron properties from pseudopotential calculations.", "Blöchl notes that there is a degree of freedom in Eqn.", "REF to add a term of the form $\\hat{B} - \\sum _{ij} \| p_i \\rangle \\langle \\tilde{\\phi _i} \| \\hat{B} \| \\tilde{\\phi _j} \\rangle \\langle p_j \|,$ where $\\hat{B}$ is an arbitrary local operator acting solely within the augmentation region.", "We might describe an all electron operator $\\hat{O}$ in terms of a `fictitious' operator $\\hat{O}_{\\textrm {fict}}$ and a `real' operator $\\hat{O}_{\\textrm {real}}$ , which are equal outside the augmentation region: $\\hat{O} = \\hat{O}_{\\textrm {fict}} + f(r) ( \\hat{O}_{\\textrm {real}} - \\hat{O}_{\\textrm {fict}}),$ where $f(r)$ is a cutoff function which is unity for $r<r_c$ and zero otherwise and so $\\hat{B} =\\hat{O}_{\\textrm {real}} - \\hat{O}_{\\textrm {fict}}$ .", "If we apply the PAW operator transform to $\\hat{O}$ (Eqn.", "REF ) and add Eqn.", "REF we find: $\\widetilde{O} = \\hat{O}_{\\textrm {fict}} + \\sum _{ij} \| p_i \\rangle (\\langle \\phi _i \| \\hat{O}_{\\textrm {real}} \| \\phi _j \\rangle - \\langle \\tilde{\\phi _i} \| \\hat{O}_{\\textrm {fict}} \| \\tilde{\\phi _j} \\rangle ) \\langle p_j \|.$ This allows us to substitute a new operator $\\hat{O}_{\\textrm {fict}}$ , which is easier to numerically represent, for the pseudo calculation, so long as it is equal to the real operator, $\\hat{O}_{\\textrm {real}}$ , outside the augmentation region, and perform the correction in the PAW augmentation.", "Blöchl goes on to use this freedom to set pseudopotential theory within the PAW formalism by substituting the real nuclear Coulomb potential of an atom for a smoothed fictitious potential that is more amenable for evaluation in a plane wave basis set.", "We can use this technique to better represent the radially divergent part of the magnetic dipole perturbation, $\\frac{1}{r^3}$ , which when naïvely applied in real space is ill-represented on the cartesian grid, leading to varying numerical predictions depending on the relative position of the perturbing nucleus to the real space grid.", "We replace it with a smoothed term, $\\hat{O}_{\\textrm {fict}} = \\frac{1 - \\exp (-(r/r_0)^3)}{r^3},$ which is well represented on a coarser real-space grid and make the correction using Eqn.", "REF during PAW augmentation.", "Figure: Plot of 𝒦(r)\\mathcal {K}(r), the 6s and 6p all-electron partial waves φ\\phi , and d𝒦 dr\\frac{d\\mathcal {K}}{dr} for a scalar-relativistic lead atom.", "Note that 𝒦(r)\\mathcal {K}(r) quickly reaches unity and d𝒦 dr\\frac{d\\mathcal {K}}{dr} quickly reaches zero for r≪r c =2.36a 0 r \\ll r_c = 2.36 a_0, the pseudopotential cut off radius.", "Also note that only the 6s orbital has significant character in the region where d𝒦 dr≫0\\frac{d\\mathcal {K}}{dr} \\gg 0 while the 6p has none, meaning the behaviour of the Z-FC term will be similar to the non-relativistic FC term." ], [ "J-coupling with PAW", "Unlike the calculation of NMR magnetic shielding, which requires the use of GIPAW to preserve translational invariance in a magnetic field [5], the calculation of J-coupling can use a gauge fixed on the perturbing atom and so only standard PAW is required.", "We will now apply PAW to the derived ZORA-level scalar-relativistic theory of J-coupling to get pseudo-operators in terms of modifications to the non-relativistic operators.", "The diamagnetic and paramagnetic current operators are simple multiples of $\\mathcal {K}$ , while the Fermi-contact term disappears for a nuclear Coulomb potential and the spin-dipole term turns into a simple multiple of $\\mathcal {K}$ plus a Fermi-contact-like term which is sharply localised by the gradient of $\\mathcal {K}$ (Figure REF ).", "As the region where $\\mathcal {K}\\ne 1$ is localised inside the augmentation region, the operators are equal to their non-relativistic equivalents outside, and so we can apply all these corrections accurately in PAW augmentation for no extra computational cost [19] using Eqn.", "REF .", "In the case of the spin contribution, the sharply localised FC/Z-FC perturbation operator can be written as H(1,0)Z-FC = R,n,m \| pR,n R,n \|H(1,0)Z-FC \| R,m pR,m \|, where we have used the on-site approximation.", "The SD/Z-SD PAW augmentation is also evaluated on a real radial grid, while the un-augmented contribution is calculated in Fourier space on the augmented magnetization density.", "$\\widetilde{H}^{(1,0)}_{\\textrm {Z-SD}} = \\hat{H}^{(1,0)}_{\\textrm {SD}} + \\Delta \\hat{H}^{(1,0)}_{\\textrm {Z-SD}},$ where, using the on-site approximation again, H(1,0)Z-SD = R,n,m \| pR,n (R,n \| KH(1,0)SD \| R,m - R,n\| H(1,0)SD \| R,m ) pR,m \|.", "The total spin contribution to the reduced coupling tensor is then KZ-spin = KZ-spin + KZ-spin KZ-spin = -03 d G [ 3(m(1)(G) G)G - m(1)(G) G2G2 ] eiGrA KZ-spin = oo (0)o \| H(1,0)Z-FC + H(1,0)SD \| (1)o , where $\\mathbf {\\tilde{m}}^{(1)}(\\mathbf {G})$ is the un-augmented spin density in reciprocal space, to which we apply the spin dipole operator and slow Fourier transform at the position of the receiving nucleus The diamagnetic current operators do not receive PAW augmentation, as we cannot make the off-site approximation due to the complexity of calculating the relevant matrix elements, and so only contributes to the bare current.", "This has the effect of ignoring the diamagnetic augmented current for off-site nuclei, though the diamagnetic contribution is the smallest J-coupling component and may only be really relevant for the calculation of anisotropic couplings [34].", "The PAW augmented paramagnetic operator is H(1,0)Z-para = H(1,0)Z-para + H(1,0)Z-para H(1,0)Z-para = R,n,m \| pR,n (R,n \| K H(1,0)para \| R,m - R,n \| H(1,0)para \| R,m ) pR,m \|.", "The bare contribution of the paramagnetic current is calculated in Fourier space on the pseudo-current density and is PAW augmented on a real radial grid with an on-site approximation, justified by the short-rangeness of the interaction.", "The total orbital current contribution is then KZ-orb = KZ-orb + KZ-orb KZ-orb = 0 d G i G j(1)(G)G2 eiGrA KZ-orb = 2 Reoo (0)o \| H(1,0)Z-para \| (1)o , where $\\mathbf {\\tilde{j}}^{(1)}(\\mathbf {G})$ is the un-augmented induced current density, paramagnetic and diamagnetic, in Fourier space, to which we apply the Biot-Savart law in reciprocal space and slow Fourier transform at the position of the receiving nucleus.", "The total indirect coupling tensor between the two nuclei is then the sum of $K^{\\textrm {Z-spin}}$ and $K^{\\textrm {Z-orb}}$ ." ], [ "Additional considerations for ultrasoft pseudopotentials", "The so-called `ultrasoft' pseudopotential formalism introduced by Vanderbilt[21] is the most computationally efficient form of pseudopotential generally providing numerically converged results with significantly smaller basis-sets.", "This is particularly important for elements which require semi-core states to be treated as valence for accurate results e.g.", "`3p' states in the 3d transition metal series.", "While ultrasoft potentials are efficient from a user's point of view, there is some additional complexity when implemented in an electronic-structure code.", "The key ingredient of the ultrasoft scheme is that the norm of the the pseudo partial-waves in the augmentation region is different from that of the corresponding all-electron partial-waves.", "We can thus define a non-zero charge augmentation term ${\\rm Q}_{{\\bf R},nm}({\\mathbf {r}})$ : ${\\rm Q}_{{\\bf R},nm}({\\mathbf {r}}) = \\langle \\phi _{{\\bf R},n} \|{\\bf r}\\rangle \\langle {\\bf r}\| \\phi _{{\\bf R},m} \\rangle - \\langle \\tilde{\\phi }_{{\\bf R},n} \|{\\bf r}\\rangle \\langle {\\bf r}\| \\tilde{\\phi }_{{\\bf R},m} \\rangle .$ The norm of a pseudo-wave function can be computed as the expectation value of the pseudo operator $\\tilde{1}=S$ .", "Using Eqn.", "REF , $S = 1 + \\sum _{{\\bf R},n,m} \|p_{{\\bf R},n} \\rangle {\\rm q}_{{\\bf R},nm} \\langle p_{{\\bf R},m} \|$ where ${\\rm q}_{{\\bf R},nm} = \\langle \\phi _{{\\bf R},n} \| \\phi _{{\\bf R},m} \\rangle -\\langle \\tilde{\\phi }_{{\\bf R},n} \| \\tilde{\\phi }_{{\\bf R},m} \\rangle .$ As a result, a normalized eigenstate of the pseudo Hamiltonian obeys the generalized equations: $\\tilde{H}{\|\\tilde{\\psi }_{o} \\rangle } = \\varepsilon _{o}S{\|\\tilde{\\psi }_{o}\\rangle },$ and ${\\langle \\tilde{\\psi }_{o}\|}S{\|\\tilde{\\psi }_{o^\\prime }\\rangle }=\\delta _{o,o^\\prime }.$ The pseudo-Hamiltonian $\\tilde{H}$ can be derived using Eqn.", "REF as $\\tilde{H} = -\\nabla ^2 + V_{\\textrm {eff}} + \\sum _{{\\bf R},n,m} \|p_{{\\bf R},n} \\rangle {\\rm D}_{{\\bf R},nm} \\langle p_{{\\bf R},m} \|$ where $ {\\rm D}_{{\\bf R},nm}$ is given by ${\\rm D}_{{\\bf R},nm}= {\\rm D}_{{\\bf R},nm}^{0} + \\int d{\\mathbf {r} \\:} V_{\\textrm {eff}}({\\bf r}){\\rm Q}_{{\\bf R},nm}({\\bf r}).$ ${\\rm D}_{{\\bf R},nm}^{0}$ is obtained from the construction of the pseudopotential[35], and $V_{\\textrm {eff}}$ is the screened local potential.", "We note that the norm-conserving pseudopotential scheme can be regarded as special case in which ${\\rm q}_{{\\bf R},nm}=0$ by definition, and by convention the terms ${\\rm Q}_{{\\bf R},nm}$ are assumed to vanish.", "The charge density in the ultrasoft scheme is given by n(r) = o [ (0)o(r) (0)o(r) + R,n,m QR,nm(r) pR,n \| (0)o (0)o \| pR,m ] In practice $Q_{\\mathbf {R},nm}$ and hence $n(\\mathbf {r})$ would be prohibitively expensive to represent in a planewave basis, and so $Q_{\\mathbf {R},nm}$ is replaced by a pseudized augmentation charge $\\tilde{Q}_{\\mathbf {R},nm}$ , where the pseudization conserves the electrostatic moments of the charge[35].", "The grid-representable charge density $\\bar{n}(\\mathbf {r})$ is given by n(r) = o [ (0)o(r) (0)o(r) + R,n,m QR,nm(r) pR,n \| (0)o (0)o \| pR,m ] The ZORA J-coupling operators derived in Section REF are still valid for ultrasoft potentials, as the PAW formulation made no assumption regarding the norm of the partial waves used in the PAW transformation.", "However, changes are required when computing the first order induced current and magnetization density.", "Firstly, we note that to avoid a costly sum over unoccupied states the first-order change in the wave functions is computed using a conjugate-gradient minimization[9].", "For ultrasoft pseudopotentials the minimization routines must be adapted to allow for the generalised orthonormality condition, Eqn.", "REF .", "This can be done straight-forwardly following the method outlined in Appendix B of Ref.", "yatescalculation2007.", "Secondly, care must be taken in computing the induced magnetization density.", "Following the ultrasoft charge density in Eqn.", "REF we introduce the magnetization density $\\bar{m}^{(1)}$ m(1)(r) = 2o [ *(1)o(r) (0)o(r) + c.c.", "+ R,n,m QR,nm(r) (pR,n \| (1)o (0)o \| pR,m + c.c.)", "] When using ultrasoft potentials we compute Eqn.", "REF using $\\bar{m}^{(1)}(\\mathbf {r})$ in order to capture off-site contributions to the spin-dipolar contribution.", "However, this requires that care is taken to subtract from every receiving atom the on-site contribution from the augmentation charge so as not to double count with the PAW on-site augmentation.", "In addition, the self-consistent exchange-correlation term, Eqn.", "REF , should also be augmented to reflect the first order change in the $D$ matrix [37]: H(1)xc = V(1)xc [m(1)] + R,n,m \|pR,n [ dr V(1)xc [m(1)] (r) QR,nm(r) ] pR,m \|." ], [ "Calculations", "The PAW scalar-relativistic theory as proposed has been implemented in the CASTEP planewave DFT code[38] using norm-conserving pseudopotentials [39] and ultrasoft pseudopotentials [21], [35].", "In the case of norm-conserving pseudopotentials this involved modifying only the preparatory calculation of PAW matrix elements, as the pseudo-Hamiltonian remains the same as for the existing non-relativistic theory except for PAW augmentation.", "In the case of ultrasoft pseudopotentials modifications were needed to address the issues mentioned in Section REF .", "In addition, PAW is also used to smooth the dipole operator and reconstruct it in real space to provide greater numerical stability with respect to real grid spacing, as described in Eqn.", "REF .", "We will first seek to validate our implementation against all-electron, local orbital basis set quantum chemical predictions and experimental measurements of J-coupling for both molecules containing light atoms, to be treated non-relativistically, and molecules containing heavy atoms, to be treated relativistically.", "We will then proceed to use the implemented code to predict J-coupling in two large novel molecular crystals containing silver atoms." ], [ "Validation against existing quantum chemistry: Non-relativistic", "Lantto, Vaara and Helgaker [40] provide a benchmark on 34 small molecules containing light atoms with a total of 60 couplings calculated using localised Gaussian orbital basis sets at the LDA, BLYP, B3LYP and MCSCF levels of theory along with some experimental values.", "Included in the light atoms is fluorine, which is notoriously difficult to treat well with density functional theory[41].", "We have run the same calculations at a non-relativistic level of theory using both norm-conserving and ultrasoft pseudopotentials with the LDA functional and present the results in Tables REF , REF , REF and REF and Figures REF , REF and REF .", "As expected, the fluorine couplings perform poorly compared to experiment, particularly $^1$ J(F-C).", "In addition, they compare poorly to the Gaussian orbital basis set calculations.", "The majority of the difference between the pseudopotential calculations and the Gaussian orbital basis set calculations is from the Fermi-contact term contribution, suggesting that the origin of the disagreement might lie in the difficulty of constructing Gaussian basis sets with sufficient flexibility in the core to represent Fermi-contact coupling[42], a problem that is circumvented with pseudopotentials when using the PAW transformed operator.", "The root mean squared deviation (RMSD) agreements of, where available, experimental couplings with LDA Gaussian orbital, norm-conserving and ultrasoft calculations was 31.3, 23.5 and 22.7 Hz respectively.", "Excluding systems including fluorine gives RMSD agreements with experiment of 25.6, 20.9 and 17.6 Hz.", "Excluding systems involving sodium, the other significantly poorly performing element, gives RMSD agreements with experiment of 25.0, 19.2 and 20.0 Hz respectively.", "Excluding systems including both fluorine and sodium gives RMSD agreements of 11.5, 13.0 and 11.6 Hz respectively.", "The RMSD agreements of Gaussian orbital LDA with norm-conserving and ultrasoft calculations was 42.2 and 36.4 Hz respectively, 42.4 and 39.4 Hz excluding fluorine systems, 23.5 and 14.9 Hz excluding sodium systems and 5.4 and 4.4 Hz excluding both fluorine and sodium systems.", "This indicates that, by and large, norm-conserving and ultrasoft pseudopotential calculations give couplings favourably comparable to equivalent localised Gaussian orbital calculations and to experiment.", "Table: Non-relativistic benchmark on fluorine containing molecules.", "Gaussian orbital LDA, BLYP, MCSCF calculations and experimental numbers from Ref. lanttospinspin2002.", "NCP LDA and USP LDA values calculated with the implementation in CASTEP.", "All values are isotropic J-coupling in Hz.Table: Non-relativistic benchmark on nitrogen containing molecules.", "Gaussian orbital LDA, BLYP, MCSCF calculations and experimental numbers from Ref. lanttospinspin2002.", "NCP LDA and USP LDA values calculated with the implementation in CASTEP.", "All values are isotropic J-coupling in Hz.Table: Non-relativistic benchmark on lithium, sodium and potassium containing molecules.", "Gaussian orbital LDA, BLYP, MCSCF calculations and experimental numbers from Ref. lanttospinspin2002.", "NCP LDA and USP LDA values calculated with the implementation in CASTEP.", "All values are isotropic J-coupling in Hz.Table: Non-relativistic benchmark on remaining molecules.", "Gaussian orbital LDA, BLYP, MCSCF calculations and experimental numbers from Ref. lanttospinspin2002.", "NCP LDA and USP LDA values calculated with the implementation in CASTEP.", "All values are isotropic J-coupling in Hz." ], [ "Validation against existing quantum chemistry: Relativistic", "We make a comparison against a comprehensive benchmark performed by Moncho and Autschbach[28] on a set of 45 molecules containing sixth row elements.", "For this study they used the Amsterdam Density Functional code (ADF) at the scalar-relativistic and spin-orbit levels of ZORA theory and with both the PBE[43] (more precisely a mix of VWN[44] and PBE) and PBE0[45] functionals.", "As neither PBE0 nor spin-orbit effects have yet been implemented for the calculation of NMR parameters in periodic DFT calculations we compare only to their PBE scalar-relativistic ZORA calculations.", "We now highlight some of the physical and numerical differences between the scalar relativistic approach used in Ref.", "monchorelativistic2010 and our planewave-pseudopotential formalism.", "In Ref.", "monchorelativistic2010 the SD term is neglected due to computational expense, however, we include it in our calculation.", "It is generally less than $1\\%$ of the final isotropic value, though in the case of WF$_6$ it accounts for $24\\%$ of the coupling.", "The ADF calculations took $V$ from four-component numerical Dirac equation calculations on isolated, neutral atoms.", "This potential was then used to determine $\\mathcal {K}$ for a ZORA Hamiltonian.", "Similarly, in our implementation, $\\mathcal {K}$ for each species of atom is determined by the all-electron potential of a four-component calculation on an isolated, neutral atom.", "This was then used to construct a scalar ZORA isolated atom and so generate an ultrasoft pseudopotential.", "The norm-conserving relativistic potentials were generated with the atom code as maintained by José Luís Martins from four-component calculations, with the all-electron partial waves taking the spin-up electron state of the valence orbitals.", "By using pseudopotentials we are implicitly using the frozen core approximation.", "While Moncho and Autschbach do not use the frozen core approximation, Autschbach[17] notes that the frozen core approximation yields almost the same couplings as the respective all-electron computations as long as a sufficiently complete basis set is used.", "We neglect finite nucleus effects, which can be relevant for row-six elements, as noted in Section REF .", "Finally, we also neglect solvent effects, which are included in the ADF calculations with the conductor-like screening model (COSMO) [46].", "However, Moncho and Autschbach find that the COSMO model does not significantly improve the median relative deviations of couplings with experiment over the gas phase calculations and suggest that explicit solvent models are necessary.", "We also note that the main goal of our methodology is to compute J-couplings in solid materials, where solvent effects are not relevant.", "The PBE functional was used with a converged planewave basis set cut-off of 80 rydberg and a single k-point in each case.", "As we are implicitly using periodic boundary conditions, we must place the molecules in a supercell of sufficient size to reduce interactions between periodic images.", "A cubic cell size of $(15$ Å$)^3$ is found to be sufficient in most cases.", "The calculated isotropic reduced couplings for tungsten, lead, mercury, platinum and thallium are shown in Tables REF , REF , REF , REF and REF respectively.", "Fig.", "REF shows CASTEP and ADF calculations against experiment and Fig.", "REF shows the CASTEP implementation against ADF.", "To be consitent with Ref.", "monchorelativistic2010 in this section we report the reduced coupling constants in S.I.", "units ($10^{19}\\,\\cdot \\,$ T$^2\\,\\cdot \\,$ J$^{-1}$ ).", "Considering the full set of compounds we find that, as expected, the non-relativitistic pseudopotential calculations show large deviation from experiment, with a RMSD of 4529.2, mean absolute deviation of 1352.3 and median absolute deviation of 224.1 $\\times $ $10^{19}\\,\\cdot \\,$ T$^2\\,\\cdot \\,$ J$^{-1}$ .", "Using relativistic ultrasoft pseudopotentials and the ZORA formalism of Section REF significantly improves the agreement with experiment, giving a RMSD of 1215.2, mean absolute deviation of 511.6 and median absolute deviation of 87.6.", "These results are comparable to the equivalent results of Moncho and Autschbach [28] (1558.7, 517.8, 82.1).", "We also note that there is good agreement between the norm-conserving and ultrasoft pseudopotential results.", "For tungsten containing molecules (Table REF ), the performance of ultrasoft pseudopotentials was equivalent to the all-electron calculations, with marginally larger RMSD and smaller mean and median absolute deviations from experiment.", "Lead-containing molecules (Table REF ) also gave equivalent performance for ultrasoft pseudopotentials, with marginally smaller RMSD, mean and median absolute deviations.", "Mercury-containing molecules (Table REF ) have a significantly smaller RMSD, dominated by the size of the $^2$ J(Hg-Sn) coupling in IrCl(SnCl$_3$ )(HgCl)(CO)(PH$_3$ )$_2$ , and smaller mean and median absolute deviations.", "Platinum-containing molecules (Table REF ) have a larger RMSD, mean and median deviations, largely due to the significant error in the $^1$ J(Pt-Pt) coupling in the charged molecules.", "We believe this error arises from the difficulty in treating charged systems with periodic boundary conditions and the influence of the solvation model on the all-electron couplings.", "For thallium containing molecules (Table REF ), the statistics are poor compared to experiment and show some deviations from the all-electron results.", "It was previously found [28] that greater accuracy for couplings in thallium-containing molecules requires going to PBE0 and spin-orbit levels of theory.", "The reported couplings in molecules TlF, TlCl, TlBr and TI are taken from gas phase experiments and, except for TlI, are in good agreement with all-electron calculations.", "The $^1$ J(Tl-I) coupling exhibits a very large orbital current contribution, an order of magnitude larger than the total value, and a large Fermi-contact-like contribution with opposite sign.", "The couplings in the remaining thallium molecules are measured in solution; Tl$_4$ (OCH$_3$ )$_4$ was in a toluene solvent and the rest in a water solvent.", "Calculations presented at the PBE0 and spin-orbit level of theory in Ref.", "monchorelativistic2010 show that, in contrast to other elements, couplings in thallium containing molecules are significantly affected by the inclusion of solvent, with the mean absolute deviation going from 1163.0 to 737.1 when moving to the COSMO model.", "Neglect of solvation effects is hence likely to be the main source of disagreement between our results and all-electron couplings.", "Table: Relativistic benchmark on tungsten containing molecules.", "Local orbital PBE ZORA calculations using ADF and experimental numbers from Ref. monchorelativistic2010.", "Ultrasoft PBE ZORA values calculated with the implementation in CASTEP.", "All values are isotropic reduced J-coupling in 10 19 ·10^{19}\\,\\cdot \\,T 2 ·^2\\,\\cdot \\,J -1 ^{-1}.Table: Relativistic benchmark on lead containing molecules.", "Local orbital PBE ZORA calculations using ADF and experimental numbers from Ref. monchorelativistic2010.", "Ultrasoft PBE ZORA values calculated with the implementation in CASTEP.", "All values are isotropic reduced J-coupling in 10 19 ·10^{19}\\,\\cdot \\,T 2 ·^2\\,\\cdot \\,J -1 ^{-1}.Table: Relativistic benchmark on mercury containing molecules.", "Local orbital PBE ZORA calculations using ADF and experimental numbers from Ref. monchorelativistic2010.", "Ultrasoft PBE ZORA values calculated with the implementation in CASTEP.", "All values are isotropic reduced J-coupling in 10 19 ·10^{19}\\,\\cdot \\,T 2 ·^2\\,\\cdot \\,J -1 ^{-1}.Table: Relativistic benchmark on platinum containing molecules.", "Local orbital PBE ZORA calculations using ADF and experimental numbers from Ref. monchorelativistic2010.", "Ultrasoft PBE ZORA values calculated with the implementation in CASTEP.", "All values are isotropic reduced J-coupling in 10 19 ·10^{19}\\,\\cdot \\,T 2 ·^2\\,\\cdot \\,J -1 ^{-1}.Table: Relativistic benchmark on thallium containing molecules.", "Local orbital PBE ZORA calculations using ADF and experimental numbers from Ref. monchorelativistic2010.", "Ultrasoft PBE ZORA values calculated with the implementation in CASTEP.", "All values are isotropic reduced J-coupling in 10 19 ·10^{19}\\,\\cdot \\,T 2 ·^2\\,\\cdot \\,J -1 ^{-1}.Figure: Comparison of reduced isotropic J-couplings in a number of small molecules containing row six elements calculated using ADF's implementation of scalar-relativistic ZORA J-coupling (red crosses), from Ref.", "monchorelativistic2010, and the authors' implementation in CASTEP of the same (blue stars), both using the PBE exchange-correlation functional, against experiment.", "Two points are excluded due to scale: 2 ^2J(Hg-Sn) in Hg-Ir-SnCl 3 _3; and 1 ^1J(Pt-C) in Pt(SnCl 3 _3).Figure: Comparison of reduced isotropic J-couplings in a number of small molecules containing row six elements calculated using ADF's implementation of scalar-relativistic ZORA J-coupling, from Ref.", "monchorelativistic2010, against the authors' implementation in CASTEP of the same, both using the PBE exchange-correlation functional.", "Two points are excluded due to scale: 2 ^2J(Hg-Sn) in Hg-Ir-SnCl 3 _3; and 1 ^1J(Pt-C) in Pt(SnCl 3 _3)." ], [ "Example: Prediction of J-coupling in silver triphenylphosphine systems", "Two molecular crystals of interest are (Ph$_3$ P)$_2$ Ag(O$_2$ COH) and [{(Ph$_3$ P)$_2$ Ag}$_2$ (CO$_3$ )]$\\cdot $ 2H$_2$ O, shown in Figure REF , part of a family of compounds incorporating phosphorous atoms bonded via a silver atom, which is in turn bonded to a varying complex.", "These are useful model systems for studying J-coupling between a heavy atom and a lighter atom.", "The J-couplings were experimentally measured with $^{31}$ P CPMAS NMR by Bowmaker et al.", "[47] and are quoted for both $^{107}$ Ag and $^{109}$ Ag when the two splittings were distinguishable.", "When converted to reduced couplings this gives a range of values, which could be considered as an indication of the error in the measurements.", "We have calculated J-couplings using both non-relativistic theory and ZORA scalar-relativistic theory with ultrasoft pseudopotentials for both structures with a 80 rydberg kinetic energy cutoff and a single k-point with primitive cells containing 148 atoms and 296 atoms respectively.", "Table REF shows the calculated couplings.", "Two sets of calculations for each structure are given: one for which only the hydrogen atomic positions have been optimized; and the other for which all atomic positions have been optimized, both using ultrasoft pseudopotentials with an energy cut-off of 50 rydberg.", "In both cases the lattice vectors were kept fixed to their experimental values.", "To illustrate the cost of treating such a system, each calculation at a single perturbing atom took approximately 4,500 seconds for (Ph$_3$ P)$_2$ Ag(O$_2$ COH) and 25,000 seconds for [{(Ph$_3$ P)$_2$ Ag}$_2$ (CO$_3$ )]$\\cdot $ 2H$_2$ O on a cluster containing four dual-quad core Intel Xeon E5620 processors, giving a total of 32 cores.", "When a relativistic pseudopotential is used on the silver atom the $^2$ J($^{31}$ P-$^{31}$ P) couplings are in good agreement with experiment.", "The difference in $^2$ J($^{31}$ P-$^{31}$ P) coupling between the relativistic and non-relativistic cases demonstrate indirect effects in the J-coupling caused by changes in the ground state electronic structure due to use of a relativistic pseudopotential on the silver atom, as noted in previous studies involving two-bond couplings through a heavy ion[16].", "The $^1$ J($^{107/109}$ Ag-$^{31}$ P) couplings are in reasonable agreement with experiment when ZORA-level corrections to the operators are applied.", "The computed couplings agree better with the lower end of the quoted experimental ranges.", "The $^1$ J($^{109}$ Ag-$^{31}$ P) couplings demonstrate direct effects due to use of ZORA operators when performing the perturbation as well as indirect ground state electronic structure effects.", "Performing a full optimization of all the atomic positions gives only a small improvement in the calculated couplings.", "For [{(Ph$_3$ P)$_2$ Ag}$_2$ (CO$_3$ )]$\\cdot $ 2H$_2$ O, the calculations give a spread of $^1$ J(Ag-P) couplings corresponding to the four crystallographically distinct phosporous sites - experimentally only a single coupling was determined, presumably an average value.", "Table: Calculated and experimental isotropic reduced coupling constants (K), in 10 19 ·10^{19}\\,\\cdot \\,T 2 ·^2\\,\\cdot \\,J -1 ^{-1}, for (Ph 3 _3P) 2 _2Ag(O 2 _2COH) and [{(Ph 3 _3P) 2 _2Ag} 2 _2(CO 3 _3)]·\\cdot 2H 2 _2O.", "Their structures are shown in Figure .", "For K Calc 𝑍𝑂𝑅𝐴 K_{\\textrm {Calc}}^{\\textit {ZORA}} silver atoms were treated with relativistic pseudopotentials and ZORA modified operators." ], [ "Conclusions", "We have presented a method for the calculation of J-coupling tensors for systems containing heavy elements using state-of-the-art ultrasoft pseudopotentials which gives accuracy comparable to existing quantum chemistry methods at the same level of theory.", "The use of pseudopotentials allows fewer electrons to be explicitly treated, and we have shown that the treatment of relativity using the ZORA approach allows cheap corrections to be made using PAW.", "This allows calculations of J-coupling tensors between heavy ions in periodic and molecular systems containing hundreds of atoms." ], [ "Supplementary materials", "Compressed archives of calculation input and output files, including structures and magnetic resonance tensors, are available online for the non-relativistic benchmark[48], relativistic benchmark[49] and silver-containing molecular crystals[50].", "The authors acknowledge financial support from EPSRC (TFGG, JRY) and the Royal Society (JRY)." ] ]	1403.0524
[ [ "Sharp weighted bounds for one-sided and multiple integral operators" ], [ "Abstract In this paper we establish sharp weighted bounds (Buckley type theorems) for one{sided maximal and fractional integral operators in terms of one{sided $A_p$ characteristics.", "Appropriate sharp bounds for strong maximal functions, multiple potentials and singular integrals are derived." ], [ "Introduction", "One of the main problem in Harmonic analysis is to characterize a weight $w$ for which a given integral operator is bounded in $L^p_{w}$ .", "An important class of such weights is the well-known $A_{p}$ class.", "It is known that $A_p$ condition is necessary and sufficient for the boundedness of Hardy–Littlewood and singular integral operators see, e.g., ([5], [7], [24]).", "However, the sharp dependence of the corresponding $L^p_{w}$ norms in terms of $A_{p}$ characteristic of $w$ is known only for some operators.", "The interest in the sharp weighted norm for singular integral operators is motivated by applications in partial differential equations (see e.g [2], [27], [25], [26]).", "In this paper, new sharp weighted estimates for one–sided maximal functions, one–sided fractional integrals, strong maximal functions, singular and fractional integrals with product kernels are derived.", "Let $w$ be an almost everywhere positive locally integrable function on a subset $\\Omega $ of $\\mathbb {R}^n$ .", "We denote by $L_{w}^{p}(\\Omega )$ , $1<p<\\infty $ , the set of all measurable functions $f:\\Omega \\rightarrow \\mathbb {R}$ for which the norm $\\Vert f\\Vert _{L_{w}^{p}(\\Omega )}=\\bigg (\\int \\limits _{\\Omega }\|f(x)\|^{p}w(x)dx\\bigg )^{\\frac{1}{p}}$ is finite.", "If $w\\equiv $ const, then we denote $L_{w}^{p}(\\Omega )=L^{p}(\\Omega )$ .", "Suppose that $L^{p,\\infty }_{w}(\\Omega )$ be the weighted weak Lebesgue space defined by the quasi norm: $\\Vert f\\Vert _{L^{p,\\infty }_{w}(\\Omega )}=\\sup \\limits _{\\lambda >0}\\lambda [w(\\lbrace x\\in \\Omega : \|f(x)\|>\\lambda \\rbrace )]^{1/p}.$ Let $X$ and $Y$ be two Banach spaces.", "Given a bounded operator $T:X\\rightarrow Y$ , we denote the operator norm by $\\Vert T\\Vert _{X\\rightarrow Y}$ which is defined in the standard way i.e.", "$\\sup \\limits _{\\Vert f\\Vert _{X}\\le 1}\\Vert Tf\\Vert _{Y}$ .", "If $X=Y$ we use the symbol $\\Vert T\\Vert _{X}$ .", "A non-negative locally integrable function $w$ define on ${R}^{n}$ is said to satisfy $A_p({{R}}^{n})$ condition ($w\\in A_p({{R}}^{n})$ ) for $1<p<\\infty $ if $\\Vert w\\Vert _{A_{p}({{R}^{n}})}:=\\sup \\limits _{Q}\\bigg (\\frac{1}{\|Q\|}\\int \\limits _{Q}w(x)dx\\bigg )\\bigg (\\frac{1}{\|Q\|}\\int \\limits _{Q}w(x)^{1-p^{\\prime }}dx\\bigg )^{p-1}<\\infty ,$ where $p^{\\prime }=\\frac{p}{p-1}$ and supremum is taken over all cubes $Q$ in ${R}^{n}$ with sides parallel to the co-ordinate axes.", "We call $\\Vert w\\Vert _{A_p({{R}}^n)}$ the $A_p$ characteristic of $w$ .", "In 1972 B. Muckhenhoupt [24] showed that if $w\\in A_p({R}^n)$ , where $1<p<\\infty $ , then the Hardy–Littlewood maximal operator $Mf(x)=\\sup \\limits _{x\\in Q}\\frac{1}{\|Q\|}\\int \\limits _{Q}\|f(y)\|dy$ is bounded in $L^p_w({{R}}^n)$ .", "Later R. Hunt, B. Muckhenhoupt and R. L. Wheeden [7] proved that the Hilbert transform $\\mathcal {H}f(x)=p.v\\int \\limits _{{R}}\\frac{f(y)}{x-y}dy, \\;\\;\\; x\\in {{R}}, $ is also bounded in $L^p_w({{R}})$ if $w\\in A_p({{R}})$ .", "S. Buckley [3] investigated the sharp $A_p$ bound for the operator $M$ and established the inequality $\\Vert M\\Vert _{L^{p}_w({{R}}^n)}\\le C\\Vert w\\Vert _{A_p({{R}}^n)}^{\\frac{1}{p-1}},\\;\\;\\;\\; 1<p<\\infty ,$ Moreover, he showed that the power $\\frac{1}{p-1}$ is best possible in the sense that we can not replace $\\Vert w\\Vert _{A_p}^{\\frac{1}{p-1}}$ by $\\psi (\\Vert w\\Vert _{A_p})$ for any positive non-decreasing function $\\psi $ growing slowly than $x^{\\frac{1}{p-1}}$ .", "From here it follows that for any $\\lambda >0$ , $\\sup \\limits _{w\\in A_p}\\frac{\\Vert M\\Vert _{L^{p}_w}}{\\Vert w\\Vert _{A_p}^{\\frac{1}{p-1}-\\lambda }}=\\infty .$ It was also shown by S. Buckley that for $1<p<\\infty $ convolution Calderón Zygmund singular operator satisfies $\\Vert T\\Vert _{L^p_w({R}^n)}\\le c\\Vert w\\Vert _{A_p({R}^n)}^{\\frac{p}{p-1}}$ and the best possible exponent is at least $\\max \\lbrace 1,\\frac{1}{p-1}\\rbrace $ .", "S. Petermichl [25], [26] proved that the estimate $\\Vert S\\Vert _{L^p_w({R}^n)}\\le c\\Vert w\\Vert _{A_p({R}^n)}^{\\max \\lbrace 1,\\frac{1}{p-1}\\rbrace }$ is sharp, where $S$ is either the Hilbert transform or one of the Riesz transforms in ${R}^n$ $R_jf(x)=c_{n} p.v\\int \\limits _{{{R}}^{n}}\\frac{x_{j}-y_j}{\|x-y\|^{n+1}}f(y)dy.$ S. Petermichl obtained the results for $p=2$ .", "The general case $p\\ne 2$ then follows by the sharp version of he Rubio de Francia extrapolation theorem given by O. Dragičević, L. Grafakos, C. Pereyra and S. Petermichl [6] (see also, T. Hytönen [13] regarding the $A_2$ conjecture for Calderón-Zygmund operators which, in fact,implies appropriate estimate for all exponents $1<p<\\infty $ by applying a sharp version of Rubio de Francia's extrapolation theorem.).", "In 1974 B. Muckenhoupt and R. Wheeden [21] found necessary and sufficient condition for the one-weight inequality; namely, they proved that the Riesz potential $I_{\\alpha }$ (resp the fractional maximal operator $M_{\\alpha }$ ) is bounded from $L^p_{w^p}({{R}}^n)$ to $L^q_{w^q}({{R}}^n)$ , where $1<p<\\infty $ , $0<\\alpha <n/p$ , $q= \\frac{np}{n-\\alpha p}$ if and only if $w$ satisfies the so called $A_{p,q}({{R}}^n)$ condition (see the definition below).", "Moreover, from their result it follows that there is a positive constant $c$ depending only on $p$ and $\\alpha $ such that $\\Vert T_{\\alpha } \\Vert _{L^p_{w^p} \\rightarrow L^q_{w^q}} \\le c \\Vert w\\Vert _{A_{p,q}}^{\\beta },$ for some positive exponent $\\beta $ , where $T_{\\alpha }$ is either $I_{\\alpha }$ or $M_{\\alpha }$ , and $\\Vert w\\Vert _{A_{p,q}} $ is the $A_{p,q}$ characteristic of $w$ .", "M. T. Lacey, K. Moen, C. Perez and R. H. Torres [17] proved that the best possible value of $\\beta $ in (REF ) is $p^{\\prime }/q(1-\\alpha /n)$ (resp.", "$(1-\\alpha /n)\\max \\lbrace 1, p^{\\prime }/q \\rbrace $ ) for $M_{\\alpha }$ (resp.", "for $I_{\\alpha }$ ) (see also [4] for this and other sharp results).", "In 1986 E. Sawyer proved the following inequality for the right maximal operator $M^+$ : $ \\Vert M^+ f\\Vert _{L^p_w({{R}})} \\le C_{p} \\Vert w\\Vert _{A_{p}^+({{R}})}^{\\beta } \\Vert f\\Vert _{L^p_w({{R}})}, \\;\\;\\; f\\in L^p_{w}({{R}}),$ with some positive exponent $\\beta $ , where $\\Vert w\\Vert _{A_{p}^+({{R}})}$ is $A_p^+$ characteristic of a weight $w$ (see the appropriate definitions below).", "Later K. Andersen and E. Sawyer, in their celebrated work [1] completely characterized the one-weight boundedness for one–sided fractional operators.", "In particular, they proved that if $1<p<\\infty $ , $0<\\alpha <1/p$ , $q=\\frac{p}{1-\\alpha p}$ , then $ \\Vert w {\\mathcal {N}}^{+}_{\\alpha } f\\Vert _{L^q({{R}})} \\le C_{p,\\alpha } \\Vert w\\Vert _{A_{p,q}^+({{R}})}^{\\beta } \\Vert w f\\Vert _{L^p({{R}})}, \\;\\;\\; f\\in L^p_{w^p}({{R}}),$ for some positive $\\beta $ , where ${\\mathcal {N}}^{+}_{\\alpha }$ is either the Weyl transform ${\\mathcal {W}}_{\\alpha }$ or the right fractional maximal operator $M^+_{\\alpha }$ , and $\\Vert w\\Vert _{A_{p,q}^+}$ is the right $A_{p,q}^+$ characteristic of a weight $w$ (see the appropriate definitions below).", "Our aim is to obtain sharp bounds in inequalities (REF ), (REF ) (as well as to investigate their ”left” analogs), in particular, to establish best possible value for $\\beta $ .", "From the obtained results for one–sided potentials, for example, can be easily obtained the sharp estimates for two–sided fractional integrals $I_{\\alpha }$ in the case $n=1$ established in [17].", "Known results and the derived statements are applied to give analogous sharp estimates for multiple operators (strong maximal functions, multiple singular integrals and potentials with product kernels).", "In some cases, appropriate examples of weighted bounds are given.", "To explain better the point of sharp estimates for multiple operators, let us discuss, for example, the strong Hardy–Littlewood maximal operator $M^{(s)}$ defined on ${{R}}^2$ .", "Denote by $A_p^{(s)}({{R}}^2)$ the Muckenhoupt class taken with respect to the rectangles with sides parallel to the co-ordinate axes (see Section for the definitions).", "Let $\\Vert w\\Vert _{A_p^{(s)}({{R}}^2)}$ be $A_p^{(s)}$ characteristic of $w$ .", "There arises a natural questions regarding the sharp bound in the inequality $\\Vert M^{(s)} \\Vert _{L^p_w({{R}}^2)} \\le c \\Vert w\\Vert _{A_p^{(s)}({{R}}^2)}^{\\beta }.$ We show that the following estimate is sharp $\\Vert M^{(s)}\\Vert _{L^{p}_{w}({{R}}^2)}\\le c\\Bigg ( \\Vert w\\Vert _{A_p(x_{1})} \\Vert w\\Vert _{A_p(x_{2})}\\Bigg )^{1/(p-1)},$ where $\\Vert w\\Vert _{A_p(x_{i})}$ be the characteristic of the weight $w$ defined with respect to the $i$ - th variable uniformly to another one $i=1,2$ (see e.g., [10], [14], [11], Ch.", "IV for the one–weight theory for multiple integral operators).", "Inequality (REF ) together with the Lebesgue differentiation theorem implies that (REF ) holds for $\\beta = \\frac{n}{p-1}$ ; however, unfortunately we do not know whether it is or not sharp.", "The proofs of the main results are based on the two-weight theory for one–sided fractional integrals and the ideas of [3]; [17]; [25], [26].", "Under the symbol $A\\approx B$ we mean that there are positive constants $c_1$ and $c_2$ (depending on appropriate parameters) such that $ c_1A \\le B \\le c_2 A$ ; $A \\ll B$ means that there is a positive constant $c$ such that $A \\le c B$ Finally we mention that constants (often different constants in one and the same lines of inequalities) will be denoted by $c$ or $C$ .", "The symbol $p^{\\prime }$ stands for the conjugate number of $p$ : $p^{\\prime }=p/(p-1)$ , where $1<p<\\infty $ ." ], [ "One-sided Hardy–Littlewood maximal operator", "Let $f$ be a locally integrable function on ${R}$ .", "Then we define one–sided Hardy–Littlewood maximal functions as: $M^{+}f(x)=\\sup \\limits _{h>0} \\frac{1}{ h } \\int \\limits _{x}^{x+h} \|f(t)\|dt, \\;\\;\\;\\; M^{-}f(x)=\\sup \\limits _{h>0}\\frac{1}{h}\\int \\limits _{x-h}^h \|f(t)\|dt, \\;\\;\\;\\;\\; x\\in {R}.$ E. Sawyer [28] characterized the one-weighted inequality for $M^{+}$ and $M^{-}$ under the so-called one–sided Muckenhoupt condition; namely he proved that if $1<p<\\infty $ , then (i) the inequality $ \\Vert M^{+} f\\Vert _{L^p_w({{R}})} \\le C \\Vert f\\Vert _{L^p_w({{R}})}, \\;\\;\\; f\\in L^p_w({{R}})$ holds if and only if $w\\in A_p^{+}({R})$ , i.e., $\\Vert w\\Vert _{A_{p}^{+}({R})}:=\\sup \\limits _{x\\in {R},h>0}\\bigg (\\frac{1}{h}\\int \\limits _{x-h}^{x}w(t)dt\\bigg )\\bigg (\\frac{1}{h}\\int \\limits ^{x+h}_{x}w^{1-p^{\\prime }}(t)dt\\bigg )^{p-1}<\\infty .$ (ii) the inequality $ \\Vert M^{-} f\\Vert _{L^p_w({{R}})} \\le C \\Vert f\\Vert _{L^p_w({{R}})}, \\;\\;\\; f\\in L^p_w({{R}})$ holds if and only if $w\\in A_p^{-}({R})$ , i.e., $\\Vert w\\Vert _{A_{p}^{-}({R})}:= \\sup \\limits _{x\\in {R},h>0}\\bigg (\\frac{1}{h}\\int \\limits _{x}^{x+h}w(t)dt\\bigg )\\bigg (\\frac{1}{h}\\int \\limits ^{x}_{x-h}w^{1-p^{\\prime }}(t)dt\\bigg )^{p-1}<\\infty ,$ Our main result of this section reads as follows: Theorem 2.1 Let $1<p<\\infty $ .", "Then (i) $\\Vert M^{+}\\Vert _{L^p_w({R})}\\le c\\Vert w\\Vert _{A_{p}^{+}({R})}^{\\frac{1}{p-1}}$ holds and the exponent $\\frac{1}{p-1}$ is best possible.", "(ii) $\\Vert M^{-}\\Vert _{L^p_w({R})}\\le c\\Vert w\\Vert _{A_{p}^{-}({R})}^{\\frac{1}{p-1}}$ holds and the exponent $\\frac{1}{p-1}$ is best possible.", "To prove Theorem REF we need some auxiliary statements.", "The fact that $A_p^+$ (resp $A_p^-$ ) condition implies $A_{p-\\varepsilon }^+$ (resp.", "$A_{p-\\varepsilon }^-$ ) is well-known (see e.g., the papers [22] and [28]) but we formulate and prove the next lemma because of sharp estimates.", "Lemma A If $w\\in A_p^{+}({R})$ $($ resp.", "$w\\in A_p^{-}({R})$$)$ , then $w\\in A_{p-\\epsilon }^+({R})$ $($ resp.", "$w\\in A_{p-\\epsilon }^{-}({R})$$)$ , where $\\epsilon \\approx \\Vert w\\Vert ^{1-p^{\\prime }}_{A_p^+({R})}$ $($ resp.", "$\\epsilon \\approx \\Vert w\\Vert ^{1-p^{\\prime }}_{A_p^-({R})}$ $)$ and $\\Vert w\\Vert _{A_{p-\\epsilon }^{+}({R})}\\le c\\Vert w\\Vert _{A_p^+({R})}$ , $($ resp.", "$\\Vert w\\Vert _{A_{p-\\epsilon }^{-}({R})}\\le c\\Vert w\\Vert _{A_p^-({R})}$ $)$ .", "We only give the proof of this Lemma for $w\\in A_p^+({R})$ the case $w\\in A_p^{-}({R}) $ follows analogously.", "It should be noted that if $w\\in A_p^{+}({R})$ then $w$ in general does not satisfy reverse Hölder inequality (see [28]), but nevertheless it satisfies the so called weak reverse Hölder inequality which is stated as follows: Lemma B ([22]) Let $1<p<\\infty $ .", "If $w$ satisfies $A_p^+({R})$ then there exist positive constants $C$ and $c$ independent of $w$ , $a$ and $b$ such that $\\int _{a}^{b}w^{1+\\delta }(x) dx \\le C (M^{-}(w\\chi _{(a,b)}(b)))^\\delta \\int _a^b w(x) dx, $ where $\\delta =\\frac{1}{c\\Vert w\\Vert _{A_p^{+}({R})}}$ , and therefore $M^{-}(w^{1+\\delta }\\chi _{(a,b)})(b)\\le C (M^{-}(w_{\\chi _{(a,b)}})(b))^{1+\\delta }$ with the same constants $C$ and $c$ .", "The proof of this lemma is given in [22], but here we present the proof only for estimate of $\\delta $ .", "Following the proof of Lemma 5 in that paper we have the following estimate $\\bigg (1-\\frac{\\delta }{(1+\\delta )\\alpha \\beta ^{1+\\delta }}\\bigg )\\int \\limits _a^b w^{1+\\delta }(x) dx \\le (M^{-}(w\\chi _{(a,b)}(b)))^\\delta \\int \\limits _a^b w(x) dx ,$ where $\\alpha = 1- 2 \\big ( 4 \\Vert w\\Vert _{A_p^+({{R}})}\\beta )^{1/(p-1)}$ and $\\beta $ can be taken appropriately later.", "Finally, by assuming that $\\beta = 1/ (2^{2p} \\Vert w\\Vert _{A_p^+({{R}})})$ (in this case $\\alpha =1/2$ ) we have that $\\frac{\\delta }{(1+\\delta )\\alpha \\beta ^{1+\\delta }} \\le 1 $ taking $\\delta =\\frac{1}{c\\Vert w\\Vert _{A_p^+({R})}}$ with the constant $c$ large enough independently $w$ , $a$ and $b$ .", "The lemma is proved.", "The following Theorem is in [28].", "Theorem A Suppose that $1<p<\\infty $ .", "Then (i) there is a positive constant $c_p$ depending only on $p$ such that the weak type inequalities $w(\\lbrace M^{+}f>\\alpha \\rbrace )\\le c_p \\Vert w\\Vert _{A_p^{+}({R})}\\bigg (\\frac{\\Vert f\\Vert _{L^p_w({R})}}{\\alpha }\\bigg )^p$ (ii) $w(\\lbrace M^{-}f>\\alpha \\rbrace )\\le c_p \\Vert w\\Vert _{A_p^{-}({R})}\\bigg (\\frac{\\Vert f\\Vert _{L^p_w({R})}}{\\alpha }\\bigg )^p$ holds.", "Now we recall the Marcinkiewicz interpolation theorem.", "Theorem B Suppose $1\\le p_0<p_1<\\infty $ and that $T$ is a sublinear operator of weak-type $p_0$ and $p_1$ , with respect to the measure $d\\mu =wdx$ , with norms $R_0$ and $R_1$ respectively, then $T$ is bounded on $L^p_w({R})$ for all $p_0<p<p_1$ .", "In fact, for any $0<t<1$ , $\\Vert Tf\\Vert _{L^{p_t}_w({R})}\\le C_tR_0^{1-t}R_1^{t}\\Vert f\\Vert _{L^{p_t}_w({R})}$ where $\\frac{1}{p_t}=\\frac{1-t}{p_0}+\\frac{t}{p_1}\\textrm {and} \\;\\;C_t^p=\\frac{2^{p_t}}{p_t}\\bigg (\\frac{p_1}{p_1-p_t}+\\frac{p_0}{p_t-p_0}\\bigg ).$ Proof of Lemma REF .", "We follow the proof of Proposition 3 in [22].", "For simplicity let us denote $s=p-\\epsilon $ .", "To show $w\\in A_{s}^{+}({R})$ , $1<s<p$ , it is enough to show that ${\\mathcal {M}}_s^+(w) := \\sup \\limits _{a,b}\\sup \\limits _{a<x<b}\\frac{1}{(b-a)^s}\\bigg ( \\int \\limits _{a}^{x}w(t) dt \\bigg ) \\bigg (\\int \\limits _{x}^{b} w^{-1/(s-1)(t) dt}\\bigg )^{s-1}\\ll \\Vert w\\Vert _{A_+^p({{R}})} $ because (see [22]) $ {\\mathcal {M}}_r^+(w) \\approx \\Vert w\\Vert _{A_r^+({{R}})}, \\;\\;\\;\\; 1<r<\\infty .", "$ Observe that $w\\in A_{p}^{+}({R})$ if and only if $\\sigma \\in A_{p^{\\prime }}^{-}({R})$ .", "Then by analogue Lemma REF for $A_{p^{\\prime }}^{-}({R})$ classes, we have that there is a constant $C$ independent of $w$ , $x$ and $b$ such that $M^{+}(\\sigma ^{1+\\delta }\\chi _{(x,b)})(x)\\le C (M^{+}(\\sigma _{\\chi _{(x,b)}})(x))^{1+\\delta },$ where $\\delta =\\frac{1}{c\\Vert \\sigma \\Vert _{A_{p^{\\prime }}^{-}({R})}}=\\frac{1}{c\\Vert w\\Vert _{A_{p}^{+}({R})}^{p^{\\prime }-1}}$ .", "Now we claim that $w\\in A_s^{+}({{R}})$ , where $s=\\frac{p+\\delta }{1+\\delta }$ .", "Fix $a<x<b$ .", "Since $\\sigma $ is locally integrable, it follows from (REF ) that same holds for $\\sigma ^{1+\\delta }$ .", "Hence, there exists a finite deceasing sequence $x_0=x>x_1>\\cdots >x_{N}\\ge a=x_{N+1}$ such that $\\int \\limits _{a}^{b}\\sigma ^{1+\\delta }=2^k\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }\\textrm {if}\\;\\; k=0,\\cdots ,N \\;\\textrm {and}\\;\\int \\limits _{a}^{x_N}\\sigma ^{1+\\delta }<2^N\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }.$ From (REF ) it follows the fact that for every $k=0,\\cdots ,N$ $\\int \\limits _{x_{k+1}}^{b}\\sigma ^{1+\\delta }\\le 2^{k+1}\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }.$ On the other hand, (REF ) and (REF ) yield $\\int \\limits _a^x w\\bigg (\\frac{1}{b-a}\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }\\bigg )^s&=&\\sum \\limits _{k=0}^{N}\\frac{1}{2^{ks}}\\int \\limits _{x_{k+1}}^{x_k}w\\bigg (\\frac{1}{b-a}\\int \\limits _{x_k}^{b}\\sigma ^{1+\\delta }\\bigg )^s\\\\&\\le &\\sum \\limits _{k=0}^{N}\\frac{1}{2^{ks}}\\int \\limits _{x_{k+1}}^{x_k}w(y)(M^{+}(\\sigma ^{1+\\delta }\\chi _{(y,b)})(y))^s dy\\\\&\\le &\\sum \\limits _{k=0}^{N}\\frac{2}{2^{ks}}\\int \\limits _{x_{k+1}}^{x_k}w(y)(M^{+}(\\sigma \\chi _{(x_{k+1},b)})(y))^{p+\\delta } dy.$ Since $w\\in A_p^{+}({R})$ , we know by Theorem REF that $M^{+}$ maps $L^{p}_{w}({R})$ into weak $L^{p}_{w}({R})$ with the norm less than or equal to $\\Vert w\\Vert _{A_p^{+}({R})}^{1/p}$ .", "Then, by the Marcinkiewicz interpolation theorem, $M^+$ maps $L^{p+\\delta }_{w}({R})$ into $L^{p+\\delta }_{w}({R})$ with the norm $\\le \\Vert w\\Vert ^{1/(p+\\delta )}_{A_p^+({{R}})}$ .", "This together with (REF ) and the fact that $s>1$ gives $\\int \\limits _a^x w\\bigg (\\frac{1}{b-a}\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }\\bigg )^s&\\le &2\\Vert w\\Vert _{A_{p}^{+}({R})}\\sum \\limits _{k=0}^{N}\\frac{1}{2^{ks}}\\int \\limits _{x_{k+1}}^{b}\\sigma ^{1+\\delta }\\\\&\\le &2\\Vert w\\Vert _{A_{p}^{+}({R})}\\sum \\limits _{k=0}^{N}\\frac{2^{k+1}}{2^{ks}}\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }\\le 2\\Vert w\\Vert _{A_{p}^{+}}\\int \\limits _{x}^{b}\\sigma ^{1+\\delta }<\\infty .$ Now the lemma follows.", "Further, notice that $\\epsilon =\\frac{p-1}{1+c\\Vert w\\Vert ^{p^{\\prime }-1}_{A_p^{+}({R})}}\\approx c_p\\Vert w\\Vert ^{1-p^{\\prime }}_{A_{p}^{+}({R})}$ .", "$\\;\\;\\;\\; \\Box $ Proof of Theorem REF.", "(i) We follow [3].", "Let $w\\in A_{p}^{+}({R})$ .", "Then by Lemma REF , $w\\in A_{p-\\epsilon }^{+}({R})$ , and trivially, $w$ is also an $A_{p+\\epsilon }^{+}({R})$ weight.", "Moreover, the following inequalities $\\Vert w\\Vert _{A_{p-\\epsilon }^{+}({R})}\\le c\\Vert w\\Vert _{A_p^+({R})},$ $ \\Vert w\\Vert _{A_{p+\\epsilon }^{+}({R})}\\le \\Vert w\\Vert _{A_p^+({R})}.$ hold.", "Applying the Marcinkiewicz interpolation theorem corresponding to weak type results at $p-\\epsilon $ and $p+\\epsilon $ , we get estimate $\\Vert M^{+}f\\Vert _{L^{p}_{w}({R})}\\le c\\Vert w\\Vert _{A_{p}^{+}({R})}^{\\frac{1}{p-1}}\\Vert f\\Vert _{L^{p}_{w}({{R}})}.$ Sharpness.", "Let us take $0<\\epsilon <1$ .", "Let $w(x)=\|1-x\|^{(1-\\epsilon )(p-1)}$ .", "Then it is easy to check that $\\Vert w\\Vert _{A_{p}^{+}({R})}^{1/(p-1)}\\approx \\frac{1}{\\epsilon }.$ Observe also that for $f(t)=(1-t)^{\\epsilon (p-1)-1}\\chi _{(0,1)}(t),$ we have $f\\in {L^p_w}$ .", "Now let $0<x<1$ .", "Then we find that the following estimate $M^{+}f(x)\\ge \\frac{1}{1-x}\\int \\limits _{x}^{1}f(t)dt=c\\frac{1}{\\epsilon }f(x)$ holds.", "Finally we conclude $\\Vert M^{+}f\\Vert _{L^{p}_w({R})}\\ge c\\frac{1}{\\epsilon }\\Vert f\\Vert _{L^{p}_{w}({{R}})}.$ Thus the exponent $1/(p-1)$ is sharp.", "Proof of (ii) is similar.", "We show only Sharpness.", "For that let us take $0<\\epsilon <1$ .", "Suppose that $w(x)=\|x\|^{(1-\\epsilon )(p-1)}$ .", "Then it is easy to check that $\\Vert w\\Vert _{A_{p}^{-}({R})}^{1/(p-1)}\\approx \\frac{1}{\\epsilon }.$ Observe also that for $f(t)=t^{\\epsilon (p-1)-1}\\chi _{(0,1)}(t)$ , $f\\in {L^p_w}({{R}})$ .", "Now let $0<x<1$ .", "Then we have the following estimate $M^{-}f(x)\\ge \\frac{1}{x}\\int \\limits _{0}^{x}f(t)dt=c\\frac{1}{\\epsilon }f(x).$ Finally $\\Vert M^{-}f\\Vert _{L^{p}_w({R})}\\ge c\\frac{1}{\\epsilon }\\Vert f\\Vert _{L^{p}_{w}({{R}})}.$ Thus the exponent $1/(p-1)$ is sharp.", "$\\;\\;\\;\\; \\Box $" ], [ "One-sided fractional integrals", "Let $f$ be a locally integrable function on ${R}$ and let $0<\\alpha <1$ .", "Then we define one–sided fractional integrals $\\mathcal {W}_{\\alpha }f(x)=\\int \\limits _{x}^{\\infty }\\frac{f(t)}{(t-x)^{1-\\alpha }}dt,\\mathcal {R}_{\\alpha }f(x)=\\int \\limits _{-\\infty }^{x}\\frac{f(t)}{(x-t)^{1-\\alpha }}dtx\\in {{R}},$ and corresponding fractional maximal functions: $M_{\\alpha }^{+}f(x)=\\sup \\limits _{h>0} \\frac{1}{ h^{1-\\alpha } } \\int \\limits _{x}^{x+h} \|f(t)\| dt, \\;\\;\\;\\; M_{\\alpha }^{-}f(x)=\\sup \\limits _{h>0}\\frac{1}{h^{1-\\alpha }}\\int \\limits _{x-h}^h \|f(t)\|dt, \\;\\;\\;\\;\\; x\\in {R}.$ Taking formally $\\alpha =0$ , we have one–sided Hardy–Littlewood maximal operators.", "As it was mentioned above (see Introduction) necessary and sufficient conditions governing the one-weight inequality for one–sided fractional integrals under the $A_{p,q}^{\\pm }$ conditions were established in [1].", "Now we give the definition of $A_{p,q}^{\\pm }$ classes: Definition 3.1 Let $1<p,q<\\infty $ .", "We say that a weight function $w$ defined on ${R}$ satisfies the $A_{p,q}^{+}({R})$ condition $(w\\in A_{p,q}^{+}({R}))$ , if $\\Vert w\\Vert _{A_{p,q}^{+}({R})}:=\\sup \\limits _{\\begin{array}{c}x\\in {R}\\\\h>0\\end{array}}\\bigg (\\frac{1}{h}\\int \\limits _{x-h}^{x}w^q(t)dt\\bigg )\\bigg (\\frac{1}{h}\\int \\limits ^{x+h}_{x}w^{-p^{\\prime }}(t)dt\\bigg )^{q/p^{\\prime }}<\\infty ;$ a weight function $w$ satisfies the $A_{p,q}^{-}({R})$ condition $(w\\in A_{p,q}^{-}({R}))$ , if $\\Vert w\\Vert _{A_{p,q}^{-}({R})}:=\\sup \\limits _{\\begin{array}{c}x\\in {R}\\\\h>0\\end{array}}\\bigg (\\frac{1}{h}\\int \\limits _{x}^{x+h}w^q(t)dt\\bigg )\\bigg (\\frac{1}{h}\\int \\limits ^{x}_{x-h}w^{-p^{\\prime }}(t)dt\\bigg )^{q/p^{\\prime }}<\\infty .$" ], [ "The two-weight problem", "It is well-known two-weight criteria of various type for one-sided fractional integrals.", "The Sawyer type two-weight characterization for the operators $M^+_{\\alpha }$ , $M^-_{\\alpha }$ reads as follows (see [23]): Theorem C Let $1<p\\le q<\\infty $ and let $0<\\alpha <1$ .", "Suppose that $v$ and $w$ be weight functions on ${R}$ .", "Then (i) $M^+_{\\alpha }$ is bounded from $L^p_w({R})$ to $L^{q}_v({R})$ if and only if $(v,w)\\in S^+_{p,q}$ , i.e.", "$A^+_{MT}(v,w, p,q):= \\sup _{I} \\frac{\\Big \\Vert M^{+}_{\\alpha }(\\sigma \\chi _I)\\Vert _{L^q_v({{R}})}}{\\big (\\sigma (I)\\big )^{1/p}}<\\infty , $ where the supremum is taken over all intervals $I$ with $\\sigma (I):= w^{1-p^{\\prime }}(I)<\\infty $ .", "Moreover, $\\Vert M^+_{\\alpha }\\Vert _{L^p_w \\rightarrow L^{q}_v}\\approx A^+_{MT}(v,w,p,q)$ .", "(ii) $M^-_{\\alpha }$ is bounded from $L^p_w({R})$ to $L^{q}_v({R})$ if and only if $(v,w)\\in S^-_{p,q}$ , i.e.", "$A^-_{MT}(v,w, p,q):= \\sup _{I} \\frac{\\Big \\Vert M^{-}_{\\alpha }(\\sigma \\chi _I)\\Vert _{L^q_v({{R}})}}{\\big (\\sigma (I)\\big )^{1/p}}<\\infty , $ where the supremum is taken over all intervals $I$ with $\\sigma (I):= w^{1-p^{\\prime }}(I)<\\infty $ .", "Moreover, $\\Vert M^-_{\\alpha }\\Vert _{L^p_w \\rightarrow L^{q}_v}\\approx A^-_{MT}(v,w,p,q)$ .", "M. Lorente and A.", "De la Torre ([20]) gave necessary and sufficient conditions for the boundness of one–sided potentials from $L^p_w({R})$ to $L^q_v({R})$ (resp.", "to $L^{q, \\infty }({R})$ ) under the Sawyer type conditions.", "In what follows, ${\\mathcal {N}}_{\\alpha }$ is one of the operators ${\\mathcal {W}}_{\\alpha }$ or ${\\mathcal {R}}_{\\alpha }$ .", "Observe that if ${\\mathcal {N}}^_{\\alpha }$ denotes the adjoint of ${\\mathcal {N}}_{\\alpha }$ , then it is easy to see that ${\\mathcal {N}}^_{\\alpha }= {\\mathcal {R}}_{\\alpha }$ if ${\\mathcal {N}}_{\\alpha }= {\\mathcal {W}}_{\\alpha }$ , and viceversa.", "Theorem D Let $1<p\\le q<\\infty $ and let $0<\\alpha <1$ .", "Suppose that $v$ and $w$ be weight functions on ${R}$ .", "Then (i) ${\\mathcal {N}}_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^{q, \\infty }_v({{R}})$ if and only if there is a positive constant $C$ such that $ A^_{LT}(v,w, p,q):= \\sup _{I} \\frac{ \\Big \\Vert {\\mathcal {N}}^_{\\alpha }(v\\chi _I) \\Big \\Vert _{ L^{p^{\\prime }}_{ w^{1-p^{\\prime }} } ({{R}})} }{\\big (v(I)\\big )^{1/q^{\\prime }}}<\\infty , $ where, the supremum is taken over all intervals $I$ with $v(I)<\\infty $ .", "Moreover, $\\Vert {\\mathcal {N}}_{\\alpha }\\Vert _{L^p_w \\rightarrow L^{q, \\infty }_v} \\approx A^_{LT}(v,w,p,q)$ .", "(ii) ${\\mathcal {N}}_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^{q}_v({{R}})$ if and only if $\\sup _{I} \\frac{\\Vert {\\mathcal {N}}_{\\alpha }(\\sigma \\chi _I)\\Vert _{L^q_v({{R}})}}{\\big (\\sigma (I)\\big )^{1/p}}<\\infty , $ where the supremum is taken over all intervals $I$ with $\\sigma (I)<\\infty $ ; $\\sup _{I} \\frac{\\Vert {\\mathcal {N}}^_{\\alpha }(\\sigma \\chi _I)\\Vert _{ L^{p^{\\prime }}_\\sigma ({{R}})}}{\\big (v(I)\\big )^{1/p}}<\\infty , $ where $\\sigma :=w^{1-p^{\\prime }}$ and $v(I)<\\infty $ .", "The following statements are from [8] (see also $§2.2$ of [9]) and give Gabidzashvili–Kokilashvili type two-weight criteria for one-sided potentials (we refer to [15], pp.", "189-190 for the latter conditions in the case of Riesz potentials).", "Theorem E Let $1<p<q<\\infty $ and let $0<\\alpha <1$ .", "Suppose that $v$ and $w$ be weight functions on ${R}$ .", "Then $\\mathcal {W}_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^{q,\\infty }_v({{R}})$ if and only if $[v,w]^{+}_{Glo}(p,q):=\\sup \\limits _{\\begin{array}{c}a\\in {R}\\\\h>0\\end{array}}\\Bigg (\\int \\limits _{a-h}^{a+h}v(t)dt\\Bigg )^{1/q}\\Bigg (\\int \\limits ^{\\infty }_{a+h}(t-a)^{(\\alpha -1)p^{\\prime }}w^{1-p^{\\prime }}(t)dt\\Bigg )^{1/p^{\\prime }}<\\infty .$ Moreover, $\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q,\\infty }_v}\\approx [v,w]_{Glo}^{+}(p,q)$ .", "Theorem F Let $1<p<q<\\infty $ and let $0<\\alpha <1$ .", "Suppose that $v$ and $w$ be weight functions on ${R}$ .", "Then $\\mathcal {R}_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^{q,\\infty }_v({{R}})$ if and only if $[v,w]^{-}_{Glo}(p,q):=\\sup \\limits _{\\begin{array}{c}a\\in {R}\\\\h>0\\end{array}}\\Bigg (\\int \\limits _{a-h}^{a+h}v(t)dt\\Bigg )^{1/q}\\Bigg (\\int \\limits _{-\\infty }^{a-h}(a-t)^{(\\alpha -1)p^{\\prime }}w^{1-p^{\\prime }}(t)dt\\Bigg )^{1/p^{\\prime }}<\\infty .$ Moreover, $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q,\\infty }_v}\\approx [v,w]_{Glo}^{-}(p,q)$ .", "Theorem G Let $1<p<q<\\infty $ and let $0<\\alpha <1$ .", "Suppose that $v$ and $w$ be weight functions on ${R}$ .", "Then $\\mathcal {W}_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^{q}_v({{R}})$ if and only if (i) $[v,w]^{+}_{Glo}(p,q)<\\infty $ .", "(ii) $A^{+}_{GK}(v,w,p,q):=\\sup \\limits _{\\begin{array}{c}a\\in {R}\\\\h>0\\end{array}}\\Bigg (\\int \\limits _{a-h}^{a+h}w^{1-p^{\\prime }}(t)dt\\Bigg )^{1/p^{\\prime }}\\Bigg (\\int \\limits _{-\\infty }^{a-h}\\frac{v(y)}{(a-y)^{(1-\\alpha )q}}dy\\Bigg )^{1/q}<\\infty .$ Moreover, $\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q}_v}\\approx [v,w]_{Glo}^{+}(p,q)+A^{+}_{GK}(v,w,p,q)$ .", "Theorem H Let $1<p<q<\\infty $ and let $0<\\alpha <1$ .", "Suppose that $v$ and $w$ be weight functions on ${R}$ .", "Then $\\mathcal {R}_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^{q}_v({{R}})$ if and only if (i) $[v,w]^{-}_{Glo}(p,q)<\\infty $ .", "(ii) $A^{-}_{GK}(v,w,p,q):=\\sup \\limits _{\\begin{array}{c}a\\in {R}\\\\h>0\\end{array}}\\Bigg (\\int \\limits _{a-h}^{a+h}w^{1-p^{\\prime }}(t)dt\\Bigg )^{1/p^{\\prime }}\\Bigg (\\int \\limits ^{\\infty }_{a+h}\\frac{v(y)}{(y-a)^{(1-\\alpha )q}}dy\\Bigg )^{1/q}<\\infty .$ Moreover, $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q}_v}\\approx [v,w]_{Glo}^{-}(p,q)+ A^{-}_{GK}(v,w,p,q)$ .", "Remark 3.2 The strong type statements (see Theorems REF and REF ) follow from appropriate weak type results (see Theorems REF and REF ) and the Sawyer type two-weight criteria (see Theorem REF ).", "We refer to $§2.2$ of [9] for details.", "Remark 3.3 It is easy to see that, (i) $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q}_v}\\approx \\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q,\\infty }_v}+\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{q^{\\prime }}_{v^{1-q^{\\prime }}}\\rightarrow L^{p^{\\prime },\\infty }_{w^{1-p^{\\prime }}}};$ (ii) $\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q}_v}\\approx \\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q,\\infty }_v}+\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{q^{\\prime }}_{v^{1-q^{\\prime }}}\\rightarrow L^{p^{\\prime },\\infty }_{w^{1-p^{\\prime }}}}.$ We need similar criteria for one–sided fractional maximal operators defined on ${{R}}$ (cf.", "[29]).", "Proposition 3.4 Suppose that $1<p<q<\\infty $ and that $0<\\alpha <1$ .", "Then (i) $M^+_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^q_v({{R}})$ if and only if $A^{+}_{GK}(v,w,p,q)<\\infty $ (see Theorem REF ).", "Moreover, $\\Vert M_{\\alpha }^+\\Vert _{L^{p}_w\\rightarrow L^{q}_v}\\approx A^+_{GK}(v,w,p,q)$ .", "(ii) $M^-_{\\alpha }$ is bounded from $L^p_w({{R}})$ to $L^q_v({{R}})$ if and only if $A^{-}_{GK}(v,w,p,q)<\\infty $ (see Theorem REF ).", "Moreover, $ \\Vert M^-_{\\alpha } \\Vert _{ L^{p}_w \\rightarrow L^{q}_v }\\approx A^-_{GK}(v,w,p,q)$ .", "We prove (i).", "Proof for (ii) is similar.", "Let $A^+_{GK}(v,w,p,q)<\\infty $ .", "Observe that $A^+_{GK}(v,w,p,q)= [w^{1-p^{\\prime }},v^{1-q^{\\prime }}]_{Glo}^{-}(q^{\\prime },p^{\\prime }).$ Hence, by Theorem REF we have that $\\Vert {\\mathcal {R}}_{\\alpha }\\Vert _{ L^{q^{\\prime }}_{v^{1-q^{\\prime }}} \\rightarrow L^{p^{\\prime },\\infty }_{w^{1-p^{\\prime }}}} \\approx A^+_{GK}(v,w,p,q).$ Applying now Theorem REF (part (i)) for ${\\mathcal {N}}_{\\alpha }= {\\mathcal {R}}_{\\alpha }$ (in that case ${\\mathcal {N}}^_{\\alpha }= {\\mathcal {W}}_{\\alpha }$ ) we conclude that $A^_{LT}(w^{1-p^{\\prime }}, v^{1-q^{\\prime }}, q^{\\prime },p^{\\prime }) \\ll \\Vert \\mathcal {R}_{\\alpha }\\Vert _{ L^{q^{\\prime }}_{v^{1-q^{\\prime }}} \\rightarrow L^{p^{\\prime },\\infty }_{w^{1-p^{\\prime }}}} \\ll A^+_{GK}(v,w,p,q) < \\infty .$ Hence, by Theorem REF we have that $\\Vert M^+_{\\alpha }\\Vert _{L^p_w \\rightarrow L^q_v}\\ll A^+_{MT}(v,w,p,q) \\ll A^*_{LT}(w^{1-p^{\\prime }}, v^{1-q^{\\prime }}, q^{\\prime },p^{\\prime }) \\ll A^+_{GK}(v,w,p,q) <\\infty .$ Necessity, and consequently, the inequality $A^+_{GK}(v,w,p,q) \\ll \\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_w\\rightarrow L^{q}_v} $ follows by the standard way choosing appropriate test functions." ], [ "The main results", "Now we are in the position to formulate our main results of this section.", "Theorem 3.5 Suppose that $0<\\alpha <1$ , $1<p<1/\\alpha $ and that $q$ is such that $1/p-1/q-\\alpha =0$ .", "Then (i) there exists a positive constant $c$ depending only on $p$ and $\\alpha $ such that $\\Vert M^{+}_{\\alpha }\\Vert _{L^p_{w^p}\\rightarrow L^q_{w^q}}\\le c \\Vert w\\Vert _{A_{p,q}^{+}({R})}^{\\frac{p^{\\prime }}{q}(1-\\alpha )}.$ Moreover, the exponent ${\\frac{p^{\\prime }}{q}(1-\\alpha )}$ is best possible.", "(ii) there exists a positive constant $c$ depending only on $p$ and $\\alpha $ such that $\\Vert M^{-}_{\\alpha } \\Vert _{ L^p_{w^p}\\rightarrow L^q_{w^q} } \\le c \\Vert w \\Vert _{ A_{p,q}^{-}({R}) }^{\\frac{p^{\\prime }}{q}(1-\\alpha )}.$ Moreover, the exponent ${\\frac{p^{\\prime }}{q}(1-\\alpha )}$ is best possible.", "Theorem 3.6 Let $1< p<\\frac{1}{\\alpha }$ , where $0<\\alpha <1$ .", "We set $q=\\frac{p}{1-\\alpha p}$ .", "Then (a) $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_{w^p}\\rightarrow L^{q,\\infty }_{w^q}}\\le c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1-\\alpha },$ where the positive constant $c$ depends only on $p$ and $\\alpha $ .", "(b) $\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{p}_{w^p}\\rightarrow L^{q,\\infty }_{w^q}}\\le c\\Vert w\\Vert _{A_{p,q}^{+}({R})}^{1-\\alpha },$ where the positive constant $c$ depends only on $p$ and $\\alpha $ .", "Theorem 3.7 Let $0<\\alpha <1$ , $1 < p<1/\\alpha $ and let $q$ satisfy $q=\\frac{p}{1-\\alpha p}$ .", "Then (a) there is a positive constant $c$ depending only on $p$ and $\\alpha $ such that $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_{w^p}\\rightarrow L^{q}_{w^q}}\\le c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{(1-\\alpha )\\max \\lbrace 1,p^{\\prime }/q\\rbrace }.$ Furthermore, this estimate is sharp; (b) there is a positive constant $c$ depending only on $p$ and $\\alpha $ such that $\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{p}_{w^p}\\rightarrow L^{q}_{w^q}}\\le c\\Vert w\\Vert _{A_{p,q}^{+}({R})}^{(1-\\alpha )\\max \\lbrace 1,p^{\\prime }/q\\rbrace }.$ Moreover, this estimate is sharp." ], [ "Main lemmas", "In this subsection we prove some lemmas regarding the weights satisfying one–sided conditions.", "Lemma 3.8 Let $u\\in A_{p,q}^{-}({R})$ .", "Then for every $a\\in {R}$ and $h>0$ , $\\int \\limits _{a}^{a+h}u^q(x)dx\\le \\Vert u\\Vert _{A_{p,q}^{-}({R})}\\int \\limits _{a-h}^{a}u^q(x)dx.$ Further, let $u\\in A_{p,q}^{+}({R})$ .", "Then for every $a\\in {R}$ and $h>0$ , $\\int \\limits _{a-h}^{a}u^q(x)dx\\le \\Vert u\\Vert _{A_{p,q}^{+}{(R)}}\\int \\limits _{a}^{a+h}u^q(x)dx.$ Let us prove the first part.", "The second one follows analogously.", "Let $u\\in A_{p,q}^{-}({R})$ .", "Using Hölder's inequality we have $\\int \\limits _{a}^{a+h}u^{q}(x)dx&\\le &\\Bigg (\\int \\limits _{a}^{a+h}u^q(x)dx\\Bigg )h^{-q}\\Bigg (\\int \\limits _{a-h}^{a}u^q(x)dx\\Bigg )\\Bigg (\\int \\limits _{a-h}^{a}u^{-q^{\\prime }}(x)dx\\Bigg )^{q/q^{\\prime }}\\\\&\\le &h^{-1}\\Bigg (\\int \\limits _{a}^{a+h}u^q(x)dx\\Bigg )\\Bigg (\\int \\limits _{a-h}^{a}u^q(x)dx\\Bigg )\\Bigg (\\frac{1}{h}\\int \\limits _{a-h}^{a}u^{-p^{\\prime }}(x)dx\\Bigg )^{q/p^{\\prime }}\\\\&\\le &\\Vert u\\Vert _{A_{p,q}^{-}({R})}\\int \\limits _{a-h}^{a}u^{q}(x)dx.$ Lemma 3.9 Let $u\\in A_{p,q}^{-}({R})$ .", "Then for all $a\\in {R}$ and $r>0$ we have, $\\frac{\\int \\limits _{a-r}^{a}u^q(x)dx}{\\int \\limits _{a-2r}^{a}u^q(x)dx}\\le \\frac{\\Vert u\\Vert _{A_{p,q}^{-}{(R)}}}{\\Vert u\\Vert _{A_{p,q}^{-}{(R)}}+1}.$ Further, let $u\\in A_{p,q}^{+}({R})$ .", "Then for all $a\\in {R}$ and $r>0$ we have, $\\frac{\\int \\limits _{a}^{a+r}u^q(x)dx}{\\int \\limits _{a}^{a+2r}u^q(x)dx}\\le \\frac{\\Vert u\\Vert _{A_{p,q}^{+}({R})}}{\\Vert u\\Vert _{A_{p,q}^{+}({R})}+1}.$ Let $u\\in A_{p,q}^{-}({R})$ .", "By Lemma REF we find that $\\int \\limits _{a-2r}^{a}u^{q}(x)dx&=&\\int \\limits _{a-2r}^{a-r}u^{q}(x)dx+\\int \\limits _{a-r}^{a}u^{q}(x)dx\\\\&\\ge &\\Bigg (\\frac{1}{\\Vert u\\Vert _{A_{p,q}^{-}({R})}}+1\\Bigg )\\int \\limits _{a-r}^{a}u^q(x)dx.$ The remaining part of the lemma follows analogously.", "To prove Theorems REF and REF we need to prove also the following Lemma: Lemma 3.10 Let $0<\\alpha <1$ , $1<p<1/\\alpha $ and let $q$ be such that $\\frac{1}{q}=\\frac{1}{p}-\\alpha $ .", "Then $[w^q, w^p]^{+}_{Glo}\\ll c\\Vert w\\Vert ^{1-\\alpha }_{A_{p,q}^{+}({R})}.$ $[w^q, w^p]^{-}_{Glo}\\ll c\\Vert w\\Vert ^{1-\\alpha }_{A_{p,q}^{-}({R})}.$ We prove (REF ).", "Proof for (REF ) follows in the same manner.", "First observe that $p^{\\prime }(1-\\alpha )=1+\\frac{p^{\\prime }}{q}$ .", "Further, let us take $a\\in {R}$ and $h>0$ .", "Then by Lemma REF , $&&\\Bigg (\\int \\limits _{a-2h}^{a}w^{q}(x)dx\\Bigg )^{1/q}\\Bigg (\\int \\limits _{-\\infty }^{a-2h}(a-x-h)^{(\\alpha -1)p^{\\prime }}w^{-p^{\\prime }}(x)dx\\Bigg )^{1/p^{\\prime }}\\\\&\\le &c\\Bigg (\\int \\limits _{a-2h}^{a}w^{q}(x)dx\\Bigg )^{1/q}\\Bigg (\\sum \\limits _{j=1}^{\\infty }(2^{j}h)^{(\\alpha -1)p^{\\prime }}\\int \\limits _{a-2^{j+1}h}^{a-2^{j}h}w^{-p^{\\prime }}(x)dx\\Bigg )^{1/p^{\\prime }}\\\\&=&c\\Bigg [\\sum \\limits _{j=1}^{\\infty }(2^{j}h)^{(\\alpha -1)p^{\\prime }}\\Bigg (\\int \\limits _{a-2h}^{a}w^{q}(x)dx\\Bigg )^{p^{\\prime }/q}\\Bigg (\\int \\limits _{a-2^{j+1}h}^{a-2^{j}h}w^{-p^{\\prime }}(x)dx\\Bigg )\\Bigg ]^{1/p^{\\prime }}\\\\&=&c\\Bigg [\\sum \\limits _{j=1}^{\\infty }\\Bigg (\\frac{\\int \\limits _{a-2h}^{a}w^{q}(x)dx}{\\int \\limits _{a-2^jh}^{a}w^{q}(x)dx}\\Bigg )^{p^{\\prime }/q}\\Bigg (\\frac{1}{2^jh}\\int \\limits _{a-2^jh}^{a}w^{q}(x)dx\\Bigg )^{p^{\\prime }/q}\\Bigg (\\frac{1}{2^jh}\\int \\limits _{a-2^{j+1}h}^{a-2^jh}w^{q}(x)dx\\Bigg )\\Bigg ]^{1/p^{\\prime }}\\\\&\\le & c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1/q}\\Bigg (\\sum \\limits _{j=1}^{\\infty }\\Bigg (\\frac{\\int \\limits _{a-2h}^{a}w^{q}(x)dx}{\\int \\limits _{a-2^jh}^{a}w^{q}(x)dx}\\Bigg )^{p^{\\prime }/q}\\Bigg )^{1/p^{\\prime }}\\\\&\\le & c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1/q}\\Bigg (\\sum \\limits _{j=0}^{\\infty }\\Bigg (\\frac{\\Vert w\\Vert _{A_{p,q}^{-}({R})}}{1+\\Vert w\\Vert _{A_{p,q}^{-}({R})}}\\Bigg )^{p^{\\prime }j/q}\\Bigg )^{1/p^{\\prime }}\\\\&=&c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1/q}\\Bigg (\\frac{1}{1-(\\frac{\\Vert w\\Vert _{A_{p,q}^{-}({R})}}{1+\\Vert w\\Vert _{A_{p,q}^{-}({R})}})^{p^{\\prime }/q}}\\Bigg )^{1/p^{\\prime }}\\\\&\\le &c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1/q}\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1/p^{\\prime }}=c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1-\\alpha }.$ In the latter inequality we used the fact that $\\Vert w\\Vert _{A^{-}_{p,q}({{R}})}\\ge 1$ and the relation $ \\Big (1- \\big ( \\frac{s}{1+s}\\big )^{p^{\\prime }/q}\\Big )^{-1} = O(s)$ as $s\\rightarrow \\infty $ ." ], [ "Proofs of the main results", "Proof of Theorem REF .", "(i) Sharpness.", "Let us take $0<\\epsilon <1$ .", "Suppose that $w(x)=\|1-x\|^{(1-\\epsilon )/p^{\\prime }}$ .", "Then it is easy to check that $\\Vert w\\Vert _{A_{p,q}^{+}({R})}=\\Vert w^q\\Vert _{A^{+}_{1+q/p^{\\prime }}({R})}\\approx \\frac{1}{\\epsilon ^{q/p^{\\prime }}}.$ Observe also that for $f(t)=(1-t)^{\\epsilon -1}\\chi _{(0,1)}(t)$ , $\\Vert wf\\Vert _{L^p}\\approx c\\frac{1}{\\epsilon ^{1/p}}$ .", "Now let $0<x<1$ .", "Then we find that $M^{+}_{\\alpha }f(x)\\ge \\frac{1}{(1-x)^{1-\\alpha }}\\int \\limits _{x}^{1}f(t)dt=c\\frac{1}{\\epsilon }(1-x)^{\\epsilon -1+\\alpha }.$ Finally, $\\Vert wM^{+}_{\\alpha }f\\Vert _{L^{q}({R})}\\ge \\epsilon ^{-1-1/q}.$ Thus letting $\\epsilon \\rightarrow 0$ we conclude that the exponent $p^{\\prime }/q(1-\\alpha )$ is sharp.", "Inequality (REF ) follows from Proposition REF , (part (i)) Lemma REF and the following observations: $ A^+_{GK}(w^q, w^p, p,q)= \\big [ w^{-p^{\\prime }}, w^{-q^{\\prime }}\\big ]_{Glo} \\ll \\Vert w^{-1}\\Vert _{A^-_{q^{\\prime },p^{\\prime }}}^{1-\\alpha }= \\Vert w\\Vert _{A^{+}_{p,q}}^{(1-\\alpha )p^{\\prime }/q}.$ Notice also that $\\frac{1}{q^{\\prime }}-\\frac{1}{p^{\\prime }}=\\alpha $ .", "For part (ii), we show only the sharpness.", "Inequality (REF ) follows in the same way as (REF ) was proved.", "Let $0<\\epsilon <1$ .", "Let $w(x)=\|x\|^{(1-\\epsilon )/p^{\\prime }}$ .", "Then $\\Vert w\\Vert _{A_{p,q}^{-}({R})}=\\Vert w^q\\Vert _{A^{-}_{1+q/p^{\\prime }}({R})}\\approx \\frac{1}{\\epsilon ^{q/p^{\\prime }}}.$ Suppose that $f(t)=t^{\\epsilon -1}\\chi _{(0,1)}(t)$ .", "Suppose that $\\Vert wf\\Vert _{L^p}\\approx c\\frac{1}{\\epsilon ^{1/p}}$ .", "If $0<x<1$ , then we find that $M^{-}_{\\alpha }f(x)\\ge \\frac{1}{x^{1-\\alpha }}\\int \\limits _{0}^{x}f(t)dt=c\\frac{1}{\\epsilon }x^{\\epsilon -1+\\alpha }.$ Finally, $\\Vert wM^{-}_{\\alpha }f\\Vert _{L^{q}({R})}\\ge \\epsilon ^{-1-1/q}.$ Letting $\\epsilon \\rightarrow 0$ we conclude that the exponent $p^{\\prime }/q(1-\\alpha )$ is sharp.", "$\\;\\;\\; \\Box $ Proof of Theorem REF follows immediately from Lemma REF , Theorems REF and REF .", "Proof of Theorem REF .", "We prove only (REF ); the proof of the estimate (REF ) follows analogously.", "For estimate (REF ), observe that Theorem REF and Theorem REF yield $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^p_{w^p}\\rightarrow L^q_{v^q}}&\\ll & \\Big ( \\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1-\\alpha }+ A^-_{GK}(w^q, w^p, p,q) \\Big ) \\ll \\Big ( \\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1-\\alpha }+ \\Vert w\\Vert _{A_{p,q}^{-}({R})}^{p^{\\prime }/q(1-\\alpha )}\\Big ) \\\\&\\ll & \\Vert w\\Vert _{A_{p,q}^{-}({R})}^{(1-\\alpha )\\max \\lbrace 1,p^{\\prime }/q\\rbrace }.$ Here we used the relations: $A^-_{GK}(w^q, w^p, p,q) \\ll \\Vert w^{-1}\\Vert ^{1-\\alpha }_{A_{q^{\\prime },p^{\\prime }}^{+}({R})}=c\\Vert w\\Vert ^{p^{\\prime }/q(1-\\alpha )}_{A_{p,q}^{-}({R})}.$ The first inequality is due to Lemma REF , while the latter equality is easy to check.", "For sharpness first we assume that $p^{\\prime }/q\\ge 1$ .", "Observe that the following pointwise estimate $M_{\\alpha }^{-}f\\le \\mathcal {R}_{\\alpha }f,f\\ge 0$ holds.", "Then by using same $w$ and $f$ as in the proof of Theorem REF we have estimate (REF ) with $M_{\\alpha }^{-}$ replaced by $\\mathcal {R}_{\\alpha }$ showing sharpness.", "Similarly for $\\mathcal {W}_{\\alpha }$ observe that the following pointwise estimate: $M_{\\alpha }^{+}f\\le \\mathcal {W}_{\\alpha }f,f\\ge 0$ holds.", "Then by applying the same $w$ and $f$ as in the proof of (ii) of Theorem REF , we have estimate (REF ) with $M_{\\alpha }^{+}$ replaced by $\\mathcal {W}_{\\alpha }$ shows sharpness.", "The case when $p^{\\prime }/q<1$ follows from duality argument.", "Indeed, $\\Vert \\mathcal {R}_{\\alpha }\\Vert _{L^{p}_{w^p}\\rightarrow L^{q}_{w^q}}=\\Vert \\mathcal {W}_{\\alpha }\\Vert _{L^{q^{\\prime }}_{w^{-q^{\\prime }}}\\rightarrow L^{p^{\\prime }}_{w^{-p^{\\prime }}}}\\le c\\Vert w^{-1}\\Vert ^{(1-\\alpha )q/p^{\\prime }}_{A^{+}_{q^{\\prime },p^{\\prime }}({R})}=c\\Vert w\\Vert _{A_{p,q}^{-}({R})}^{1-\\alpha }.$ $\\Box $" ], [ "Strong Maximal and Multiple Integral Operators", "Let $T$ be an operator acting on a class of functions defined on ${R}^n$ .", "We denote by $T^k$ , $k=1\\cdots n$ , the operator defined on functions in ${R}$ by letting $T$ act on the $k$ -th variable while keeping the remaining variable fixed.", "Formally, $(T^k f)(x)=(Tf(x_1,x_2\\cdots ,x_{k-1},\\cdot ,x_{k+1},\\cdots ,x_n)(x_k).$ Definition 4.1 A weight function $w$ satisfies $A_p^{(s)}({R}^n)$ condition $(w\\in A_{p}^{(s)}({R}^n))$ , $1<p<\\infty $ , if $\\Vert w\\Vert _{A_{p}^{(s)}({R}^n)}:=\\sup \\limits _{P}\\bigg (\\frac{1}{\|P\|}\\int \\limits _{P}w(x)dx\\bigg )\\bigg (\\frac{1}{\|P\|}\\int \\limits _{P}w(x)^{-1/p-1}dx\\bigg )^{p-1}<\\infty ,$ where the supremum is taken over all parallelepipeds $P$ in ${R}^{n}$ with sides parallel to the co-ordinate axes.", "Definition 4.2 Let $1<p<\\infty $ .", "A weight function $w=w(x_1,\\cdots ,x_n)$ defined on ${{R}}^n$ is said to satisfy $A_p$ condition in $x_{i}$ uniformly with respect to other variables $(w\\in A_{p}(x_i))$ if $\\Vert w\\Vert _{A_{p}(x_i)}:= \\operatornamewithlimits{ess\\,sup}\\limits _{\\begin{array}{c}(x_{1},\\cdots x_{i-1},\\\\x_{i+1}\\cdots ,x_{n})\\in {{R}}^{n-1}\\end{array}}\\sup _{I} &&\\bigg (\\frac{1}{\|I\|}\\int \\limits _{Q}w(x_{1},\\cdots ,x_{n})dx_{i}\\bigg )\\times \\\\&&\\times \\bigg (\\frac{1}{\|I\|}\\int \\limits _{I}w^{1-p^{\\prime }}(x_{1},\\cdots ,x_{n}) dx_{i}\\bigg )^{p-1}<\\infty ,$ where by $I$ we denote a bounded interval in ${R}$ .", "Remark 4.3 $w(x_1,\\cdots ,x_n)\\in A_{p}^{(s)}({R}^n)\\Leftrightarrow w\\in \\bigcap \\limits _{i=1}^{n} A_{p}(x_i)$ (see e.g., pp.", "453-454 of [11], [14]).", "Proposition 4.4 Let $T$ be an operator defined on ${R}^{n}$ such that $\\Vert T^k\\Vert _{L^{p}_{w}({R})}\\le c\\Vert w\\Vert _{A_p(x_k)}^{\\gamma (p)}k=1,\\cdots ,n,$ holds, where $\\gamma (p)$ is a constant depending only on $p$ .", "Then the following estimate $\\Vert T\\Vert _{L^{p}_{w}({R}^{n})}\\le c(\\Vert w\\Vert _{A_p(x_{1})}\\cdots \\Vert w\\Vert _{A_p(x_{n})} )^{\\gamma (p)}.$ holds.", "For simplicity we give proof for $n=2$ the proof general case is the same.", "Suppose that $f\\ge 0$ .", "Using (REF ) two times and Fubini's theorem we have, $\\Vert Tf\\Vert ^p_{L^p_w({R}^2)}&=&\\int \\limits _{{R}}\\int \\limits _{{R}}(Tf(x_{1},x_{2}))^{p}w(x_{1},x_{2})dx_{1}dx_{2}\\\\&=&\\int \\limits _{{R}}\\Bigg (\\int \\limits _{{R}}(T^{1}(T^{2}f(\\cdot ,x_{2})))(x_{1})^{p}w(x_{1},x_{2})dx_{1}\\Bigg )dx_{2}\\\\&\\le &c\\Vert w\\Vert _{A_{p}(x_{1})}^{p\\gamma (p)}\\int \\limits _{{R}}\\Bigg (\\int \\limits _{{R}}(T^{2}f(x_{1},x_{2}))^{p}w(x_{1},x_{2})dx_{1}\\Bigg )dx_{2}\\\\&=&c\\Vert w\\Vert _{A_{p}(x_{1})}^{p\\gamma (p)}\\int \\limits _{{R}}\\Bigg (\\int \\limits _{{R}}(T^{2}f(x_{1},x_{2}))^{p}w(x_{1},x_{2})dx_{2}\\Bigg )dx_{1}\\\\&=&c(\\Vert w\\Vert _{A_{p}(x_{1})}\\Vert w\\Vert _{A_{p}(x_{2})})^{p\\gamma (p)}\\Vert f\\Vert ^p_{L^{p}_{w}({R}^{2})}$" ], [ "Strong Hardy–Littlewood Maximal Functions and Multiple Singular Integrals", "The following theorem is due to S. M. Buckley [3].", "Theorem I If $w\\in A_p({{R}}^n)$ , then $\\Vert Mf\\Vert _{L^{p}_{w}({{R}}^n)}\\le c_{n,p}\\Vert w\\Vert _{A_{p}({{R}}^n)}^{1/(p-1)}\\Vert f\\Vert _{L^{p}_{w}({{R}}^n)}$ .", "The power $1/(p-1)$ is best possible.", "Let $f$ be a locally integrable function on ${R}^{n}$ .", "Then we define strong Hardy–Littlewood maximal operator as $\\big (M^{(s)}f\\big )(x)=\\sup \\limits _{P\\ni x}\\frac{1}{\|P\|}\\int \\limits _{P}\|f(y)\|dy,x\\in {R}^{n},$ where the supremum is taken over all parallelepipeds $P\\ni x$ in ${R}^{n}$ with sides parallel to the co-ordinate axes.", "Theorem 4.5 Let $1<p<\\infty $ and $w$ be a weight function on ${R}^{n}$ such that $w\\in A^{(s)}_{p}({{R}}^n)$ .", "Then there exists a constant $c$ depending only on $n$ and $p$ such that the following inequality $\\Vert M^{(s)}f\\Vert _{L^{p}_{w}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_p(x_{i})}\\Bigg )^{1/(p-1)}\\Vert f\\Vert _{L^{p}_{w}({{R}}^n)}$ holds, for all $f\\in L^{p}_w({{R}}^n)$ .", "Further, the power $1/(p-1)$ in estimate (REF ) is sharp.", "We can estimate $M^{(s)}$ as follows $\\big ( M^{(s)}f\\big ) (x)\\le \\big ( M^{1}\\circ M^{2}\\circ \\dots \\circ M^{n}\\big ) f(x),$ where $\\big ( M^{k}f\\big ) (x_{1},\\cdots x_{n})=\\sup \\limits _{ I_{k}\\ni x_{k}}\\frac{1}{\|I_k\|}\\int \\limits _{I_k}\|f(x_{1},\\cdots ,x_{k-1},t,x_{k+1},\\cdots ,x_{n})\|dt.$ Now by Theorem REF (for $n=2$ ) we have $\\gamma (p)=\\frac{1}{p-1}$ and consequently, by Proposition REF we find that $\\Vert M^{(s)}f\\Vert _{L^{p}_{w}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_p(x_{i})}\\Bigg )^{1/(p-1)}\\Vert f\\Vert _{L^{p}_{w}({{R}}^n)}.$ For sharpness we consider the case for $n=2$ .", "Observe that when $w$ is of product type, i.e.", "$w(x_{1},x_{2})=w_{1}(x_{1})w_{2}(x_{2})$ , then $ \\Vert w\\Vert _{A_{p}(x_{1})}=\\Vert w_1\\Vert _{A_{p}({R})},\\Vert w\\Vert _{A_{p}(x_{2})}=\\Vert w_2\\Vert _{A_{p}({R})}.$ Let us take $0<\\epsilon <1$ .", "Suppose that $w(x_{1},x_{2})=\|x_{1}\|^{(1-\\epsilon )(p-1)}\|x_{2}\|^{(1-\\epsilon )(p-1)}$ .", "Then it is easy to check that $(\\Vert w\\Vert _{A_{p(x_1)}}\\Vert w\\Vert _{A_{p(x_2)}})^{1/(p-1)}\\approx \\frac{1}{\\epsilon ^2}.$ Observe also that for $f(x_{1},x_{2})={x_{1}}^{\\epsilon (p-1)-1}\\chi _{(0,1)}(x_{1}){x_{2}}^{\\epsilon (p-1)-1}\\chi _{(0,1)}(x_{2}),$ we have $\\Vert f\\Vert ^p_{L^p_w}\\approx \\frac{1}{\\epsilon ^2}$ .", "Now let $0<x_{1},x_{2}<1$ .", "Then we find that the following estimate $\\big ( M^{(s)}f\\big ) (x_{1},x_{2})\\ge \\frac{1}{x_{1}x_{2}}\\int \\limits _{0}^{x_{1}}\\int \\limits _{0}^{x_{2}}f(t,\\tau )dtd\\tau =c\\frac{1}{\\epsilon ^{2}}f(x,y)$ holds.", "Finally $\\Vert M^{(s)}f\\Vert _{L^{p}_w}\\ge c\\frac{1}{\\epsilon ^2}\\Vert f\\Vert _{L^{p}_{w}}$ Thus we have the sharpness in (REF ).", "Let us denote by ${\\mathcal {H}}^{(n)}$ the Hilbert transform with product kernels (or $n$ -dimensional Hilbert transform) defined by $\\big ( {\\mathcal {H}}^{(n)}f\\big ) (x)=\\lim \\limits _{\\begin{array}{c}\\epsilon _{1}\\rightarrow 0\\\\\\cdots \\\\ \\epsilon _{n}\\rightarrow 0\\end{array}}\\int \\limits _{\|x_{1}-t_{1}\|>\\epsilon _{1}}\\cdots \\int \\limits _{\|x_{n}-t_{n}\|>\\epsilon _{n}}\\frac{f(t_{1},\\cdots ,t_{n})}{(x_{1}-t_{1})\\cdots (x_{n}-t_{n})}dt_{1}\\cdots dt_{n}.$ We denote ${\\mathcal {H}}^{(1)}=: { {\\mathcal {H}} }$ .", "Notice that for each $x\\in {R}^{n}$ , we can write $\\big ( {\\mathcal {H}}^{(n)}f\\big ) (x)=\\big (\\mathcal {H}^{1}\\circ \\cdots \\circ \\mathcal {H}^{n}\\big )f(x)$ where, $\\big ( \\mathcal {H}^{k}f\\big ) (x)=\\lim \\limits _{\\epsilon _{k}\\rightarrow 0} \\int \\limits _{\|x_{k}-y_{k}\|>\\varepsilon _k} \\frac{f(x_1, \\cdots , y_k, \\cdots , x_n)}{x_{k}-y_{k}}dy.$ The following theorem is due to S. Petermichl [25].", "Theorem J Let $1<p<\\infty $ .", "Then there exist positive constant $c$ depending only on $p$ such that for all weights $w\\in A_p({{R}})$ we have $\\Vert \\mathcal {H}f\\Vert _{L^{p}_w({{R}})}\\le c\\Vert w\\Vert _{A_{p}({{R}})}^{\\beta }\\Vert f\\Vert _{L^p_w({{R}})}, \\;\\;\\; f\\in L^p_w({{R}}),$ where $\\beta =\\max \\lbrace 1,p^{\\prime }/p\\rbrace $ .", "Moreover, the exponent $\\beta $ in this estimate is sharp.", "Theorem 4.6 Let $1<p<\\infty $ and $w$ be a weight function on ${R}^{n}$ such that $w\\in {A^{(s)}_{p}}({R}^n)$ .", "Then there exists a constant $c$ depending only on $n$ and $p$ such that the following inequality $\\Vert {\\mathcal {H}}^{(n)}f\\Vert _{L^{p}_{w}({{R}}^n)}\\le c(\\Vert w\\Vert _{A_p(x_{1})}\\cdots \\Vert w\\Vert _{A_p(x_{n})})^{\\max \\lbrace 1,p^{\\prime }/p\\rbrace }\\Vert f\\Vert _{L^{p}_{w}({{R}}^n)}$ holds for all $f\\in L^{p}_w$ .", "Further the power ${\\max \\lbrace 1, p^{\\prime }/p\\rbrace }$ in estimate (REF ) is sharp.", "Using representation (REF ), Proposition REF and Theorem REF , we have that $\\Vert {\\mathcal {H}}^{(n)}f\\Vert _{L^{p}_{w}({{R}}^n)} \\ll \\big (\\Vert w\\Vert _{A_p(x_{1})}\\cdots \\Vert w\\Vert _{A_p(x_{n})}\\big )^{\\max \\lbrace 1, p^{\\prime }/p\\rbrace }\\Vert f\\Vert _{L^{p}_{w}({{R}}^n)}.$ Let $n=2$ .", "For sharpness we observe that when $w$ is of product type i.e.", "$w(x_{1},x_{2})=w_{1}(x_{1})w_{2}(x_{2})$ , then inequality (REF ) holds.", "Let us first derive sharpness for $p=2$ .", "Let us take $0<\\epsilon <1$ and let $w(x_{1},x_{2})=w_1(x_1)w_2(x_2)$ , where $w_1(x_1)= \|x_{1}\|^{1-\\epsilon }$ and $w_2(x_2)=\|x_{2}\|^{1-\\epsilon }$ .", "Then it is easy to check that (REF ) holds.", "Observe also that for $f(x_{1},x_{2})={x_{1}}^{\\epsilon -1}\\chi _{(0,1)}(x_{1}){x_{2}}^{\\epsilon -1}\\chi _{(0,1)}(x_{2}),$ $\\Vert f\\Vert ^2_{L^2_w}\\approx \\frac{1}{\\epsilon }.$ Now let $0<x_{1},x_{2}<1$ .", "Then we find that $\\Vert {\\mathcal {H}}^{(2)}f\\Vert _{L^{2}_w({{R}}^2)}\\ge 4\\epsilon ^{-3}.$ Letting $\\epsilon \\rightarrow 0$ we have sharpness in (REF ) for $p=2$ i.e., the estimate $\\Vert {\\mathcal {H}}^{(2)}\\Vert _{L^{2}_{w}({{R}}^2)}\\ll \\Vert w\\Vert _{A_{2}(x_1)}\\Vert w\\Vert _{A_{2}(x_2)}$ is sharp.", "Let $1<p<2$ .", "Suppose that $0<\\epsilon <1$ and that $w(x_{1},x_{2})=\|x_{1}\|^{(1-\\epsilon )}\|x_{2}\|^{(1-\\epsilon )}$ .", "Then it is easy to check that $(\\Vert w\\Vert _{A_{p(x_1)}}\\Vert w\\Vert _{A_{p(x_2)}})^{1/(p-1)}\\approx \\frac{1}{\\epsilon ^2}.$ Observe also that for the function defined by (REF ) the relation $\\Vert f\\Vert _{L^p_w}\\approx (\\frac{1}{\\epsilon ^2})^{\\frac{1}{p}}$ holds.", "Now let $0<x_{1},x_{2}<1$ .", "Then we find that following estimates $\\Vert {\\mathcal {H}}^{(2)}f\\Vert _{L^{p}_w({{R}}^2)}\\ge \\frac{1}{\\epsilon ^2} \\Vert f\\Vert _{L^p_w({{R}}^2)} \\approx \\big ( \\Vert w\\Vert _{A_{p}(x_1)} \\Vert w\\Vert _{A_{p}(x_2)}\\big )^{p^{\\prime }/p}\\Vert f\\Vert _{L^p_{w}({{R}}^2)}$ are fulfilled.", "Thus we have sharpness in (REF ) for $1<p<2$ .", "Using the fact that $n$ -dimensional Hilbert transform is essentially self-adjoint and applying duality argument together with the obvious equality $\\Vert u^{1-p^{\\prime }}\\Vert _{A_{p^{\\prime }}}=\\Vert u\\Vert ^{1/(p-1)}_{A_p}, \\;\\;\\; u\\in A_p, $ we have sharpness for $p>2$ .", "This completes the proof.", "Let $x=(x^{(1)},\\cdots ,x^{(n)})\\in {R}^{d_{1}}\\times \\cdots \\times {R}^{d_{n}}$ , where $d_{1},d_{2},\\cdots , d_{n}\\in {N}$ .", "Suppose that $x^{(k)}_{j_k}$ are components of $x^{(k)}$ , $k=1, \\cdots , n$ , $1\\le j_k \\le d_k$ .", "Then we define $n$ -fold Riesz transform $\\big ( R_{(j_{1},\\cdots ,j_{n})}^{(n)}f\\big ) (x)=p.v.\\int \\limits _{{R}^{d_{1}}}\\!\\!\\!\\cdots \\!\\!\\!\\int \\limits _{{R}^{d_{n}}}\\prod _{k=1}^{n}\\frac{(x_{j_{k}}^{(k)}-y_{j_{k}}^{(k)})}{\|x^{(k)}-y^{(k)}\|^{d_{k}+1}}f(y^{(1)},\\cdots ,y^{(n)})dy^{(1)}\\cdots dy^{(n)},$ where $1\\le j_{k}\\le d_k$ , $k=1,\\cdots ,n$ .", "It can be noticed that $\\big ( R_{(j_{1},\\cdots ,j_{n})}^{(n)}f\\big ) (x)=\\big ( R^{1}_{j_{1}}\\circ \\dots \\circ R^{n}_{j_{n}}f\\big ) (x)$ where $\\big ( R_{(j_{1},\\cdots ,j_{n})}^{k}f\\big ) (x)=p.v\\int \\limits _{{R}^{d_{k}}}\\!\\!\\!\\frac{x_{j_{k}}^{(k)}-y_{j_{k}}^{(k)}}{\|x^{(k)}-y^{(k)}\|^{d_{k}+1}}f(x^{(1)},\\cdots , x^{(k-1)}, y^{(k)}, x^{(k+1)}\\cdots , x^{(n)})dy^{(k)}.$ Theorem 4.7 Let $1<p<\\infty $ and $w$ be a weight function on ${R}^{d_{1}}\\times \\cdots \\times {R}^{d_{n}}$ satisfy the condition $w\\in {A^{(s)}_{p}({{R}}^{d_1}\\times \\cdots \\times {{R}}^{d_n})}$ .", "Then there exist a constant $c$ depending only on $n$ and $p$ such that the following inequalities $\\Vert R^{(n)}_{(j_{1},\\cdots ,j_{n})}f\\Vert _{L^{p}_{w}({{R}}^{d})}\\le c(\\Vert w\\Vert _{A_p(x^{(1)})}\\cdots \\Vert w\\Vert _{A_p(x^{(n)})})^{\\max \\lbrace 1,p^{\\prime }/p\\rbrace }\\Vert f\\Vert _{L^{p}_{w}({{R}}^{d})}$ hold for all $f\\in L^{p}_w({{R}}^{d})$ and $1\\le j_{k}\\le d_{k}$ , $k=1\\cdots n$ , where $d= d_1+ \\cdots , +d_n$ .", "Further, the power ${\\max \\lbrace 1, p^{\\prime }/p\\rbrace }$ in estimate (REF ) is sharp.", "Proof of this statement is similar to that of the previous one; we need to apply Proposition REF and the results of [26].", "$\\;\\;\\; \\Box $ .", "Example 4.8 Let $-1<\\gamma <p-1$ .", "It is known that $w(x)=\|x\|^{\\gamma }$ belongs to $A_p^{(s)}({R}^n)$ .", "Let $\\overline{w}(t)=\|t\|^\\gamma $ , $t\\in {R}$ .", "We set $b_\\gamma :=\\max \\lbrace 2^{\\frac{\\gamma }{2}},1\\rbrace $ and $d_\\gamma :=\\max \\lbrace 2^{\\frac{-\\gamma }{2}-p+1},1\\rbrace $ .", "(i) Then it follows from Theorem REF that $\\Vert M^{(s)}\\Vert _{L^p_w({R}^n)}\\le C^n {C_{\\gamma }}^{\\frac{n}{p-1}} \\bigg (1+\\Vert \\overline{w}\\Vert _{A_p({R})}\\bigg )^{\\frac{n}{p-1}}$ where $C$ is the constant from the Buckley's estimate (see (REF )) and $C_\\gamma = \\left\\lbrace \\begin{array}{l}b_\\gamma , \\;\\; 0\\le \\gamma < p-1, \\\\d_\\gamma , \\;\\; -1<\\gamma <0;\\end{array}\\right.$ It is known (see [12] pp 287–289) that $C$ in (REF ) can be taken as $C=3^{p+p^{\\prime }}2^{p^{\\prime }-p}p^{\\prime }24^{\\frac{2}{p}}p^{\\frac{1}{p-1}}$ .", "(ii) From Theorem REF it follows that $\\Vert \\mathcal {H}^{(n)}\\Vert _{L^p_w({R}^n)}\\le c^n C_{\\gamma }^{{n}\\max \\lbrace 1, p^{\\prime }/p \\rbrace } \\bigg (1+\\Vert \\overline{w}\\Vert _{A_p({R})}\\bigg )^{n \\max \\lbrace 1,\\frac{p^{\\prime }}{p}\\rbrace }$ holds, where $c$ is the constant from (REF ) and $C_\\gamma $ is defined in (REF ).", "Following [12], pp 285–286, it can be verified that $\\Vert \\overline{w}\\Vert _{A_p({R})}\\le \\max \\bigg \\lbrace 2^{\|\\gamma \|},\\frac{4^p}{(\\gamma +1)(\\gamma (1-p^{\\prime })+1)^{p-1}}\\bigg \\rbrace .$ We can get also another type of estimate of the norms in $L^p_w$ .", "By using the same arguments as in [12], pp 285–286, we find that $\\Vert w\\Vert _{A_p(x_i)}\\le \\left\\lbrace \\begin{array}{l}\\Gamma _\\gamma , \\;\\; 0\\le \\gamma < p-1, \\\\G_\\gamma , \\;\\; -1<\\gamma <0;\\end{array}\\right.i=1,2,$ where $\\Gamma _\\gamma =\\max \\bigg \\lbrace ((4/3)^2+1)^{\\gamma /2}(2/3)^{\\gamma }, b_\\gamma 4^p\\bigg ((\\gamma +1)^{-1}(\\gamma (1-p^{\\prime })+1)^{1-p}+1\\bigg )\\bigg \\rbrace ,$ $G_\\gamma =\\max \\bigg \\lbrace ((4/3)^2+1)^{-\\gamma /2}(2/3)^{\\gamma },d_0 d_\\gamma 4^p\\bigg ((\\gamma +1)^{-1}(\\gamma (1-p^{\\prime })+1)^{1-p}+1\\bigg )\\bigg \\rbrace .$ Consequently, using directly Theorems REF and REF we have the following estimate $\\Vert M^{(s)}\\Vert _{L^{p}_w({R}^n)}\\le C^n\\left\\lbrace \\begin{array}{l}\\Gamma _\\gamma ^{n/(p-1)}, \\;\\; 0\\le \\gamma < p-1, \\\\G_\\gamma ^{n/(p-1)}, \\;\\; -1<\\gamma <0;\\end{array}\\right.$ $\\Vert \\mathcal {H}^{(n)}\\Vert _{L^{p}_w({R}^n)}\\le c^n\\left\\lbrace \\begin{array}{l}\\Gamma _\\gamma ^{n\\max \\lbrace 1, p^{\\prime }/p \\rbrace }, \\;\\; 0\\le \\gamma < p-1, \\\\G_\\gamma ^{n\\max \\lbrace 1, p^{\\prime }/p \\rbrace }, \\;\\; -1<\\gamma <0;\\end{array}\\right.$ where $C$ and $c$ are constants in (REF ) and (REF ) respectively." ], [ "Strong Fractional Maximal Functions and Reisz Potentials with Product Kernels", "In this subsection we state and prove sharp weighted norm estimates for strong fractional maximal and Reisz potential with product kernels.", "To get the main results we use the ideas of the previous subsection.", "Definition 4.9 A weight function $w$ satisfies $A_{p,q}^{(s)}$ condition $(w\\in A_{p,q}^{(s)})$ , $1<p<\\infty $ if $\\Vert w\\Vert _{A_{p,q}^{(s)}}:=\\sup \\limits _{P\\ni x}\\bigg (\\frac{1}{\|P\|}\\int \\limits _{P}w^{q}(x)dx\\bigg )^{1/q}\\bigg (\\frac{1}{\|P\|}\\int \\limits _{P}w(x)^{-p^{\\prime }}dx\\bigg )^{1/p^{\\prime }}<\\infty ,$ where the supremum is taken over all parallelepipeds $P$ in ${R}^{n}$ with sides parallel to the co-ordinate axes.", "Definition 4.10 Let $1<p\\le q<\\infty $ .", "A weight function $w=w(x_1,\\cdots ,x_n)$ defined on ${{R}}^n$ is said to satisfy $A_{p,q}$ condition in $x_{i}$ uniformly with respect to other variables $(w\\in A_{p,q}(x_i))$ if $\\Vert w\\Vert _{A_{p,q}(x_i)}:= \\operatornamewithlimits{ess\\,sup}\\limits _{\\begin{array}{c}(x_{1},\\cdots , x_{i-1},\\\\x_{i+1}\\cdots ,x_{n}) \\in {{R}}^{n-1}\\end{array}}\\sup _{I\\ni x_{i}} &&\\bigg (\\frac{1}{\|I\|}\\int \\limits _{I}w^{q}(x_{1},\\cdots ,x_{n})dx_{i}\\bigg )^{1/q}\\times \\\\&&\\times \\bigg (\\frac{1}{\|I\|}\\int \\limits _{I}w^{-p^{\\prime }}(x_{1},\\cdots ,x_{n})dx_{i}\\bigg )^{1/p^{\\prime }}<\\infty ,$ where $I$ is a bounded interval.", "Remark 4.11 Like $A^{(s)}_p({R}^n)$ weights for given $w(x_1,\\cdots ,x_n)\\in A_{p,q}^{(s)}\\Leftrightarrow w\\in \\bigcap \\limits _{i=1}^{n} A_{p,q}(x_i)$ .", "Proposition 4.12 Let $1<p\\le q<\\infty $ and let $T$ be an operator defined on ${R}^{n}$ .", "Suppose that $w$ is weight, $w\\in A_{p,q}^{(s)}$ .", "Let $\\Vert T^k\\Vert _{L^{p}_{w^p}({R})\\rightarrow L^{q}_{w^q}({R})}\\le c\\Vert w\\Vert _{A_{p,q}(x_k)}^{\\gamma (p,q)}k=1,\\cdots ,n,$ hold, where $\\gamma (p,q)$ is a constant depending only on $p$ and $q$ .", "Then $\\Vert T\\Vert _{L^{p}_{w^p}({R}^{n})\\rightarrow L^{q}_{w^q}({R}^{n})}\\le c(\\Vert w\\Vert _{A_p(x_{1})}\\cdots \\Vert w\\Vert _{A_p(x_{n})} )^{\\gamma (p,q)}.$ Proof is similar to that of Proposition REF ; therefore it is omitted.", "The following Theorem is from [17].", "Theorem K Suppose that $0<\\alpha <n$ , $1<p<n/\\alpha $ and $q$ is defined by the relationship $1/q=1/p-\\alpha /n$ .", "If $w\\in A_{p,q}({{R}}^n)$ , then $\\Vert wM_{\\alpha }f\\Vert _{L^q({{R}}^n)}\\le c\\Vert w\\Vert _{A_{p,q}({{R}}^n)}^{\\frac{p^{\\prime }}{q}(1-\\alpha /n)}\\Vert wf\\Vert _{L^p({{R}}^n)}.$ Furthermore, the exponent ${\\frac{p^{\\prime }}{q}(1-\\alpha /n)}$ is sharp.", "Let $f$ be a locally integrable function and let $0<\\alpha <1$ .", "The strong fractional maximal operator is defined by $\\big ( M_{\\alpha }^{(s)}f\\big ) (x)=\\sup \\limits _{ P\\ni x }\\frac{1}{\|P\|^{1-\\alpha }}\\int \\limits _{P}\|f(y)\|dy,$ where the supremum is taken over all parallelepipeds $P$ in ${R}^{n}$ with sides parallel to the co-ordinate axes.", "It is easy to see that $\\big ( M_{\\alpha }^{(s)}f\\big ) (x)\\le \\big ( M_{\\alpha }^{1}\\circ M_{\\alpha }^{2}\\circ \\dots \\circ M_{\\alpha }^{n}f\\big ) (x),$ where $\\big ( M_{\\alpha }^{k}f\\big ) (x_{1},\\cdots x_{n})=\\sup \\limits _{I_{k}\\ni x_{k}}\\frac{1}{\|I_k\|^{1-\\alpha }}\\int \\limits _{I_k}\|f(x_{1},\\cdots ,x_{k-1},t,x_{k+1},\\cdots ,x_{n})\|dt,$ where $I_k$ are intervals in ${R}$ such that $P=I_{1}\\times \\cdots \\times I_{k}$ .", "Theorem 4.13 Let $0<\\alpha <1$ , $1<p<\\frac{1}{\\alpha }$ , $q=\\frac{p}{1-\\alpha p}$ and $w$ be a weight function on ${R}^{n}$ such that $w\\in {A^{(s)}_{p,q}({{R}}^n)}$ .", "Then there exists a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that the following inequality $\\Vert wM_{\\alpha }^{(s)}f\\Vert _{L^{q}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_{p,q}(x_{i})}\\Bigg )^{\\frac{p^{\\prime }}{q}(1-\\alpha )}\\Vert wf\\Vert _{L^{p}({{R}}^n)}$ holds, for all $f\\in L^{p}_{w^p}({{R}}^n)$ .", "Further, the power ${\\frac{p^{\\prime }}{q}(1-\\alpha )}$ in estimate (REF ) is sharp.", "Using estimate (REF ), Theorem REF and Proposition REF we get easily (REF ).", "The main ”difficulty” here is to derive sharpness.", "Let, for simplicity, $n=2$ .", "Let us take $0<\\epsilon <1$ .", "suppose that $w$ is of product type $w(x_{1},x_{2})=w_1(x_1) w_2(x_2)$ , where $w_1(x_1)= \|x_{1}\|^{(1-\\epsilon )/p^{\\prime }}$ and $w_2(x_2)= \|x_{2}\|^{(1-\\epsilon )/p^{\\prime }}$ .", "Then it is easy to see that $\\Vert w\\Vert _{A_{p,q}(x_1)}=\\Vert w_1\\Vert _{A_{1+q/p^{\\prime }}({{R}})}\\approx \\epsilon ^{-q/p^{\\prime }}; \\;\\; \\Vert w\\Vert _{A_{p,q}(x_2)}=\\Vert w_2\\Vert _{A_{1+q/p^{\\prime }}({{R}})}\\approx \\epsilon ^{-q/p^{\\prime }}.", "$ Further, if $f(t_{1},t_{2})=\|t_{1}\|^{\\epsilon -1}\\chi _{(0,1)}(t_{1})\|t_{2}\|^{\\epsilon -1}\\chi _{(0,1)}(t_{2}),$ then $\\Vert wf\\Vert _{L^p({{R}}^2)} \\approx \\frac{1}{\\epsilon ^{2/p}}$ .", "Let $0<x_1,x_2<1$ .", "Then we find that $M^{(s)}_{\\alpha }f(x_{1},x_{2})\\ge \\frac{1}{\|x_{1}\|^{1-\\alpha }\|x_{2}\|^{1-\\alpha }}\\int \\limits _{0}^{x_{1}}\\int \\limits _{0}^{x_{2}}f(t_{1},t_{2})dt_{1}dt_{2}\\approx c\\frac{\|x_{1}\|^{\\epsilon -1+\\alpha }\|x_{2}\|^{\\epsilon -1+\\alpha }}{\\epsilon ^{2}}.$ Finally we conclude that, $\\Vert wM^{(s)}_{\\alpha }f\\Vert _{L^{q}({{R}}^2)}\\ge \\epsilon ^{-2-2/q}.$ Thus letting $\\epsilon \\rightarrow 0$ we have sharpness.", "Let $0<\\alpha <1$ .", "We define Riesz potential with product kernels on ${R}^{n}$ as follows: $(I_{\\alpha }^{(n)}f)(x)=\\int \\limits _{{R}^{n}}\\frac{f(t_1,\\cdots ,t_n)}{\\prod \\limits _{i=1}^{n}\|x_{i}-t_{i}\|^{1-\\alpha }}dt_{1}\\cdots dt_n, \\;\\;\\; x=(x_{1},\\cdots ,x_{n}) \\in {{R}}^n.$ When $n=1$ we use the symbol $I_{\\alpha }$ for $I_{\\alpha }^{(1)}$ .", "The following Theorem is from [17].", "Theorem L Let $0<\\alpha <n$ , $1<p<n/\\alpha $ .", "We put $q=\\frac{np}{n-\\alpha p}$ .", "Suppose that $w\\in A_{p,q}({{R}}^n)$ .", "Then $\\Vert wI_{\\alpha }f\\Vert _{L^{q}({{R}}^n)}\\le c\\Vert w\\Vert _{A_{p,q}({{R}}^n)}^{(1-\\alpha /n)\\max \\lbrace 1,p^{\\prime }/q\\rbrace }\\Vert wf\\Vert _{L^p({{R}}^n)}.$ Furthermore, this estimate is sharp.", "Our result regarding $I_{\\alpha }^{(n)}$ reads as follows: Theorem 4.14 Let $0<\\alpha <1$ , $1<p<1/\\alpha $ .", "We put $q= \\frac{p}{1-\\alpha p}$ .", "Let $w$ be a weight function on ${R}^{n}$ such that $w\\in {A^{(s)}_{p,q}({{R}}^n)}$ .", "Then there exists a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that the following inequality $\\Vert wI^{(n)}_{\\alpha }f\\Vert _{L^{q}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_{p,q}(x_{i})}\\Bigg )^{\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}\\Vert wf\\Vert _{L^{p}({{R}}^n)}$ holds for all $f\\in L^{p}_{w^p}({{R}}^n)$ .", "Further, the power ${\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}$ in estimate (REF ) is sharp.", "Proof of this statement follows using the same arguments as in the proof of Theorem REF together with Theorem REF ." ], [ "Strong one–sided maximal operators", "Based on the results derived in this paper we give sharp bounds for one–sided strong maximal operators.", "Let $f$ be locally integrable function on ${R}^{n}$ .", "We define one–sided strong fractional maximal operators as $M^{+(s)}f(x_{1},\\cdots ,x_{n})=\\frac{1}{\\prod \\limits _{i=1}^{n}{h_i}}\\int \\limits _{x_{1}}^{x_{1}+h_{1}}\\cdots \\int \\limits _{x_{n}}^{x_{n}+h_{n}}\|f(y_1,\\cdots y_n)\|dy_1\\cdots dy_{n},$ $M^{-(s)}_{\\alpha }f(x_{1},\\cdots ,x_{n})=\\frac{1}{\\prod \\limits _{i=1}^{n}{h_i}}\\int \\limits _{x_{1}-h_{1}}^{x_{1}}\\cdots \\int \\limits _{x_{n}-h_n}^{x_{n}}\|f(y_1,\\cdots y_n)\|dy_1\\cdots dy_{n}.$ Let $1<p<\\infty $ .", "We say that a weight function $w$ belongs to the class $A_{p}^{-(s)}({R}^{n})$ if $\\Vert w\\Vert _{A_{p}^{-(s)}({R}^{n})}&:=&\\sup \\limits _{\\begin{array}{c}h_1,\\cdots ,h_n>0\\\\x_{1},\\cdots x_{n}\\in {R}\\end{array}}\\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits _{x_{1}}^{x_{1}+h_{1}}\\cdots \\int \\limits _{x_{n}}^{x_{n}+h_{n}}w(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )\\\\&&\\times \\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits ^{x_{1}}_{x_{1}-h_{1}}\\cdots \\int \\limits ^{x_{n}}_{x_{n}-h_{n}}w^{1-p^{\\prime }}(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )^{p-1}<\\infty ;$ further, $w\\in A_{p}^{-(s)}({R}^{n})$ if $\\Vert w\\Vert _{A_{p}^{+(s)}({R}^{n})}&:=&\\sup \\limits _{\\begin{array}{c}h_1,\\cdots ,h_n>0\\\\x_{1},\\cdots x_{n}\\in {R}\\end{array}}\\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits ^{x_{1}}_{x_{1}-h_{1}}\\cdots \\int \\limits ^{x_{n}}_{x_{n}-h_{n}}w(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )\\\\&&\\times \\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits _{x_{1}}^{x_{1}+h_{1}}\\cdots \\int \\limits _{x_{n}}^{x_{n}+h_{n}}w^{1-p^{\\prime }}(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )^{p-1}<\\infty .$ Definition 4.15 Let $1<p<\\infty $ .", "A weight function $w=w(x_1,\\cdots ,x_n)$ defined on ${{R}}^n$ is said to satisfy $A_p^{-}$ condition in $x_{i}$ uniformly with respect to other variables $(w\\in A_{p}^{-}(x_i))$ if $\\Vert w\\Vert _{A^{-}_{p}(x_i)}\\equiv \\operatornamewithlimits{ess\\,sup}\\limits _{\\begin{array}{c}(x_{1},\\cdots x_{i-1},\\\\x_{i+1}\\cdots ,x_{n})\\in {{R}}^{n-1}\\end{array}}\\sup _{h_{i}>0} \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}}^{x_{i}+h_i}w(x_1,\\cdots ,x_{i-1}, t, x_{i-1}, \\cdots , x_n)dt\\bigg )\\\\\\times \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}-h_i}^{x_i}w(x_1,\\cdots ,x_{i-1}, t, x_{i-1}, \\cdots , x_n)^{-1/(p-1)}dt \\bigg )^{p-1}<\\infty .$ further, $w\\in A_{p}^{+}(x_i)$ if $\\Vert w\\Vert _{A^{+}_{p}(x_i)}\\equiv \\operatornamewithlimits{ess\\,sup}\\limits _{\\begin{array}{c}(x_{1},\\cdots x_{i-1},\\\\x_{i+1}\\cdots ,x_{n})\\in {{R}^{n-1}}\\end{array}}\\sup _{h_{i}>0} \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}-h_{i}}^{x_{i}}w(x_1,\\cdots ,x_{i-1}, t, x_{i-1}, \\cdots , x_n)dt \\bigg )\\\\\\times \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}}^{x_i+h_i}w(x_1,\\cdots ,x_{i-1}, t, x_{i-1}, \\cdots , x_n)^{-1/(p-1)}dt\\bigg )^{p-1}<\\infty .$ Remark.", "It is known that (see [16], Ch.", "5) that $w(x_1,\\cdots ,x_n)\\in A_{p}^{\\pm (s)}\\Leftrightarrow w\\in \\bigcap \\limits _{i=1}^{n} A_{p}^{\\pm }(x_i)$ .", "Theorem 4.16 Let $1<p<\\infty $ .", "(i) Suppose that a weight function $w$ on ${{R}}^n$ belongs to the class $A^{+(s)}_p({{R}}^n)$ .", "Then there exists a constant $c$ depending only on $n$ and $p$ such that the following inequality $\\Vert M^{+(s)}f\\Vert _{L^{p}_{w}({R}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_p^{+}(x_{i})}\\Bigg )^{1/(p-1)}\\Vert f\\Vert _{L^{p}_{w}({R}^n)}$ holds for all $f\\in L^{p}_w({R}^n)$ .", "Further, the power $1/(p-1)$ in estimate (REF ) is sharp.", "(ii) Let $w \\in A^{-(s)}_p({{R}}^n)$ .", "Then there exists a constant $c$ depending only on $n$ and $p$ such that the following inequality $\\Vert M^{-(s)}f\\Vert _{L^{p}_{w}({R}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_p^{-}(x_{i})}\\Bigg )^{1/(p-1)}\\Vert f\\Vert _{L^{p}_{w}({R}^n)}$ holds for all $f\\in L^{p}_w({R}^n)$ .", "Further, the power $1/(p-1)$ in estimate (REF ) is sharp.", "We show (i).", "The proof of (ii) is similar.", "Since the proof of inequality (REF ) follows in the same way as in the case of $M^{(s)}$ (see Theorem REF ), we show only sharpness.", "Let $n=2$ .", "We take $0<\\epsilon <1$ .", "Let $w(x_{1},x_{2})=\|1-x_{1}\|^{(1-\\epsilon )(p-1)}\|1-x_{2}\|^{(1-\\epsilon )(p-1)}$ .", "Then it is easy to check that $(\\Vert w\\Vert _{A_{p}^{+}(x_1)}\\Vert w\\Vert _{A_{p}^{+}(x_2)})^{1/(p-1)}\\approx \\frac{1}{\\epsilon ^2}.$ Observe also that for $f(x_{1},x_{2})={(1-x_{1})}^{\\epsilon (p-1)-1}\\chi _{(0,1)}(x_{1}){(1-x_{2})}^{\\epsilon (p-1)-1}\\chi _{(0,1)}(x_{2}),$ we have $\\Vert f\\Vert _{L^p_w}\\approx \\frac{1}{\\epsilon ^2}$ .", "Now let $0<x_{1},x_{2}<1$ .", "Then $M^{+(s)}f(x_{1},x_{2})\\ge \\frac{1}{(1-x_{1})(1-x_{2})}\\int \\limits ^{1}_{x_{1}}\\int \\limits ^{1}_{x_{2}}f(t,\\tau )dtd\\tau =c\\frac{1}{\\epsilon ^{2}}f(x_{1},x_{2}).$ Finally $\\Vert M^{+(s)}f\\Vert _{L^{p}_w({R}^2)}\\ge c\\frac{1}{\\epsilon ^2}\\Vert f\\Vert _{L^{p}_{w}}.$ Thus we have the sharpness in (REF )." ], [ "One–sided Multiple Fractional Integrals", "Now we discuss sharp bounds for one–sided strong maximal potential operators with product kernels.", "Let $f$ be a locally integrable function on ${R}^{n}$ and let $0<\\alpha <1$ .", "We define one–sided strong fractional maximal operators as $M^{+(s)}_{\\alpha }f(x_{1},\\cdots ,x_{n})=\\sup _{h_1,\\cdots ,h_n>0} \\frac{1}{\\prod \\limits _{i=1}^{n}{h_{i}^{1-\\alpha }}}\\int \\limits _{x_{1}}^{x_{1}+h_{1}}\\cdots \\int \\limits _{x_{n}}^{x_{n}+h_{n}}\|f(y_1,\\cdots y_n)\|dy_1\\cdots dy_{n},$ $M^{-(s)}_{\\alpha }f(x_{1},\\cdots ,x_{n})=\\sup _{h_1,\\cdots ,h_n>0} \\frac{1}{\\prod \\limits _{i=1}^{n}{h_{i}^{1-\\alpha }}}\\int \\limits _{x_{1}-h_{1}}^{x_{1}}\\cdots \\int \\limits _{x_{n}-h_n}^{x_{n}}\|f(y_1,\\cdots y_n)\|dy_1\\cdots dy_{n}.$ Let $1<p\\le q<\\infty $ .", "We say that a weight function $w$ belongs to the class $A_{p,q}^{-(s)}({R}^{n})$ if $\\Vert w\\Vert _{A_{p,q}^{-(s)}({R}^{n})}&:=&\\sup \\limits _{\\begin{array}{c}h_1,\\cdots ,h_n>0\\\\x_{1},\\cdots x_{n}\\in {R}\\end{array}}\\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits _{x_{1}}^{x_{1}+h_{1}}\\cdots \\int \\limits _{x_{n}}^{x_{n}+h_{n}}w^{q}(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )^{1/q}\\\\&&\\times \\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits ^{x_{1}}_{x_{1}-h_{1}}\\!\\!\\!\\cdots \\!\\!\\!\\int \\limits ^{x_{n}}_{x_{n}-h_{n}}w^{-p^{\\prime }}(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )^{1/p^{\\prime }}<\\infty ;$ further, $w\\in A_{p,q}^{+(s)}({R}^{n})$ if $\\Vert w\\Vert _{A_{p,q}^{+(s)}({R}^{n})}&:=&\\sup \\limits _{\\begin{array}{c}h_1,\\cdots ,h_n>0\\\\x_{1},\\cdots x_{n}\\in {R}\\end{array}}\\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits ^{x_{1}}_{x_{1}-h_{1}}\\cdots \\int \\limits ^{x_{n}}_{x_{n}-h_{n}}w^q(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )^{1/q}\\\\&&\\times \\bigg (\\frac{1}{h_{1}\\cdots h_{n}}\\int \\limits _{x_{1}}^{x_{1}+h_{1}}\\!\\!\\!\\cdots \\!\\!\\!\\int \\limits _{x_{n}}^{x_{n}+h_{n}}w^{-p^{\\prime }}(t_{1},\\cdots ,t_{n})dt_{1}\\cdots dt_{n}\\bigg )^{1/p^{\\prime }}<\\infty .$ Definition 4.17 Let $1<p\\le q<\\infty $ .", "A weight function $w=w(x_1,\\cdots ,x_n)$ defined on ${{R}}^n$ is said to satisfy $A_{p,q}^{-}$ condition in $x_{i}$ uniformly with respect to other variables $(w\\in A_{p,q}^{+}(x_i))$ if $\\Vert w\\Vert _{A^{+}_{p,q}(x_i)}:= \\operatornamewithlimits{ess\\,sup}\\limits _{\\begin{array}{c}(x_{1},\\cdots x_{i-1},\\\\x_{i+1}\\cdots ,x_{n})\\in {{R}}^{n-1}\\end{array}}\\sup _{h_{i}>0} \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}}^{x_{i}+h_i}w^{q}(x_1,\\cdots , x_{i-1}, t, x_{i+1} \\cdots , x_n)dt\\bigg )^{1/q}\\times $ $\\times \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}-h_i}^{x_i}w^{-p^{\\prime }}(x_1,\\cdots , x_{i-1}, t, x_{i+1} \\cdots , x_n) dt \\bigg )^{1/p^{\\prime }}<\\infty ,$ further, $w\\in A_{p,q}^{-}(x_i)$ if $\\Vert w\\Vert _{A^{-}_{p,q}(x_i)}\\equiv \\sup \\limits _{\\begin{array}{c}(x_{1},\\cdots x_{i-1},\\\\x_{i+1}\\cdots ,x_{n})\\in {{R}}^{n-1}\\end{array}}\\sup _{h_{i}>0} \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}-h_{i}}^{x_{i}}w^{q}(x_1,\\cdots , x_{i-1}, t, x_{i+1} \\cdots , x_n)dt\\bigg )^{1/q}\\times $ $\\times \\bigg (\\frac{1}{h_{i}}\\int \\limits _{x_{i}}^{x_i+h_i}w^{-p^{\\prime }} (x_1,\\cdots , x_{i-1}, t, x_{i+1} \\cdots , x_n)dt\\bigg )^{1/p^{\\prime }}<\\infty .$ Remark 4.18 It is easy to check that $w(x_1,\\cdots ,x_n)\\in A_{p,q}^{\\pm (s)}\\Leftrightarrow w\\in \\bigcap \\limits _{i=1}^{n}A_{p,q}^{\\pm }(x_i)$ .", "Theorem 4.19 Let $0<\\alpha <1$ , $1<p<1/\\alpha $ .", "We put $q=\\frac{p}{1-\\alpha p}$ .", "Suppose that $w$ is a weight function defined on ${R}^{n}$ such that $w\\in A^{+(s)}_{p,q}({{R}}^n)$ .", "Then there exists a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that the following inequality $\\Vert wM_{\\alpha }^{+(s)}f\\Vert _{L^{q}({{R}}^n)}\\le c\\big (\\Vert w\\Vert _{A_{p,q}^{+}(x_{1})}\\cdots \\Vert w\\Vert _{A_{p,q}^{+}(x_{n})}\\big )^{\\frac{p^{\\prime }}{q}(1-\\alpha )}\\Vert wf\\Vert _{L^{p}({{R}}^n)}$ holds for all $f\\in L^{p}_{w^p}({{R}}^n)$ .", "Further, the power ${\\frac{p^{\\prime }}{q}(1-\\alpha )}$ in estimate (REF ) is sharp.", "Estimate (REF ) follows in the same way as in the previous cases.", "For sharpness we take $n=2$ and $w(x_{1},x_{2})=w_1(x_1) w_2(x_2)$ , where $w_1(x_1)= \|1-x_{1}\|^{(1-\\epsilon )p^{\\prime }}$ ; $w_2(x_2)= \|1-x_{2}\|^{(1-\\epsilon )p^{\\prime }}$ , $0<\\epsilon <1$ .", "Then $ \\Vert w\\Vert _{A^+_{p,q}(x_1)} \\Vert w\\Vert _{A^+_{p,q}(x_2)} = \\prod \\limits _{i=1}^2\\Vert w_i\\Vert _{A^+_{p,q}({{R}})} =\\prod \\limits _{i=1}^{2}\\Vert w_i^q\\Vert _{A^+_{1+q/p^{\\prime }}({{R}})} \\approx \\varepsilon ^{2q/p^{\\prime }}.$ If $f(t_{1},t_{2})=(1-t_{1})^{\\epsilon -1}\\chi _{(0,1)}(t_{1})(1-t_{2})^{\\epsilon -1}\\chi _{(0,1)}(t_{2}),$ then $\\Vert wf\\Vert _{L^p({{R}})^2}\\approx \\frac{1}{\\epsilon ^{2/p}}$ .", "Now let $0<x<1$ .", "Then we find that the following estimate $M^{+(s)}_{\\alpha }f(x_{1},x_{2})\\ge \\frac{1}{\\prod \\limits _{i=1}^{2}\|1-x_{i}\|^{1-\\alpha }}\\int \\limits ^{1}_{x_{1}}\\int \\limits ^{1}_{x_{2}}f(t_{1},t_{2})dt_{1}dt_{2}\\approx \\frac{\\prod \\limits _{i=1}^{2}\|1-x_{i}\|^{\\epsilon -1+\\alpha }}{\\epsilon ^{2}}$ holds.", "Finally $\\Vert wM^{+(s)}_{\\alpha }f\\Vert _{L^{q}({{R}}^2)}\\ge \\epsilon ^{-2-2/q}.$ Thus, letting $\\epsilon \\rightarrow 0$ we are done.", "The next statement can be proved analogously.", "Details are omitted.", "Theorem 4.20 Let $\\alpha $ , $p$ and $q$ satisfy the condition of Theorem REF .", "Let $w$ be a weight function on ${R}^{n}$ such that $w\\in {A^{-(s)}_{p,q}({R}^{n})}$ .", "Then there exist a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that the following inequality $\\Vert wM_{\\alpha }^{-(s)}f\\Vert _{L^{q}({R}^{n})}\\le c(\\Vert w\\Vert _{A_{p,q}^{-}(x_{1})}\\cdots \\Vert w\\Vert _{A_{p,q}^{+}(x_{n})})^{\\frac{p^{\\prime }}{q}(1-\\alpha )}\\Vert wf\\Vert _{L^{p}({R}^{n})}$ holds for all $f\\in L^{p}_{w^p}({R}^{n})$ .", "Further, the power ${\\frac{p^{\\prime }}{q}(1-\\alpha )}$ in estimate (REF ) is sharp.", "Let $f$ be a measurable function on ${R}^{n}$ and let $0<\\alpha <1$ .", "We define one–sided potentials $\\mathcal {R}_{\\alpha }^{(n)}$ and $\\mathcal {W}_{\\alpha }^{(n)}$ with product kernels $\\mathcal {R}_{\\alpha }^{(n)}f(x_{1},\\cdots ,x_n)=\\int \\limits _{-\\infty }^{x_1}\\cdots \\int \\limits _{-\\infty }^{x_n}\\frac{f(t_{1},\\cdots ,t_n)}{(x_{1}-t_{1})^{1-\\alpha }\\cdots (x_{n}-t_{n})^{1-\\alpha }}dt_1\\cdots dt_n,$ $\\mathcal {W}_{\\alpha }^{(n)}f(x_{1},\\cdots ,x_n)=\\int \\limits ^{\\infty }_{x_1}\\cdots \\int \\limits ^{\\infty }_{x_n}\\frac{f(t_{1},\\cdots ,t_n)}{(t_{1}-x_{1})^{1-\\alpha }\\cdots (t_{n}-x_{n})^{1-\\alpha }}dt_1\\cdots dt_n,$ where $x_i\\in {R}$ , $i=1,\\cdots , n$ .", "Finally we formulate the ”sharp result” for one–sided potentials with product kernels.", "We do not repeat the arguments using above, and therefore omit the proof of the next statement.", "Theorem 4.21 Let $\\alpha $ , $p$ and $q$ satisfy the conditions of Theorem REF .", "Suppose that $w$ be a weight function on ${R}^{n}$ such that $w\\in {A^{-(s)}_{p,q}({{R}}^n)}$ .", "Then (i) there exists a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that the following inequality $\\Vert w\\mathcal {R}^{(n)}_{\\alpha }f\\Vert _{L^{q}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_{p,q}^{-}(x_{i})}\\Bigg )^{\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}\\Vert wf\\Vert _{L^{p}({{R}}^n)}$ holds for all $f\\in L^{p}_{w^p}({{R}}^n)$ .", "Further, the power ${\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}$ in estimate (REF ) is sharp.", "(ii) There is a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that $\\Vert w\\mathcal {W}^{(n)}_{\\alpha }f\\Vert _{L^{q}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_{p,q}^{+}(x_{i})}\\Bigg )^{\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}\\Vert wf\\Vert _{L^{p}({{R}}^n)}$ for all $f\\in L^{p}_{w^p}({{R}}^n)$ .", "Further, the power ${\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}$ in estimate (REF ) is sharp.", "(ii) There is a constant $c$ depending only on $n$ , $p$ and $\\alpha $ such that $\\Vert w\\mathcal {W}^{(n)}_{\\alpha }f\\Vert _{L^{q}({{R}}^n)}\\le c\\Bigg (\\prod \\limits _{i=1}^{n}\\Vert w\\Vert _{A_{p,q}^{+}(x_{i})}\\Bigg )^{\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}\\Vert wf\\Vert _{L^{p}({{R}}^n)}$ for all $f\\in L^{p}_{w^p}({{R}}^n)$ .", "Further, the power ${\\max \\lbrace 1,\\frac{p^{\\prime }}{q}\\rbrace (1-\\alpha )}$ in estimate (REF ) is sharp." ], [ "Acknowledgements", "The first and second authors were partially supported by the Shota Rustaveli National Science Foundation Grant (Contract Numbers: D/13-23 and 31/47).", "The third author is thankful to the Higher Education Commission, Pakistan for the financial support." ] ]	1403.0372
[ [ "The unstable set of a periodic orbit for delayed positive feedback" ], [ "Abstract In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no.", "4, 727--790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback.", "We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures.", "In this paper we focus on a large-amplitude periodic orbit $\\mathcal{O}_{p}$ with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set $\\mathcal{W}^{u}\\left(\\mathcal{O}_{p}\\right)$.", "We prove that $\\mathcal{W}^{u}\\left(\\mathcal{O}_{p}\\right)$ is a three-dimensional $C^{1}$-submanifold of the phase space and admits a smooth global graph representation.", "Within $\\mathcal{W}^{u}\\left(\\mathcal{O}_{p}\\right)$, there exist heteroclinic connections from $\\mathcal{O}_{p}$ to three different periodic orbits.", "These connecting sets are two-dimensional $C^{1}$-submanifolds of $\\mathcal{W}^{u}\\left(\\mathcal{O}_{p}\\right)$ and homeomorphic to the two-dimensional open annulus.", "They form $C^{1}$-smooth separatrices in the sense that they divide the points of $\\mathcal{W}^{u}\\left(\\mathcal{O}_{p}\\right)$ into three subsets according to their $\\omega$-limit sets." ], [ "Introduction", "Consider the delay differential equation $\\dot{x}\\left(t\\right)=-\\mu x\\left(t\\right)+f\\left(x\\left(t-1\\right)\\right),$ where $\\mu $ is a positive constant and $f:\\mathbb {R}\\rightarrow \\mathbb {R}$ is a smooth monotone nonlinearity.", "The natural phase space for Eq.", "(REF ) is $C=C\\left(\\left[-1,0\\right],\\mathbb {R}\\right)$ equipped with the supremum norm.", "For any $\\varphi \\in C$ , there is a unique solution $x^{\\varphi }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ of (REF ).", "For each $t\\ge 0$ , $x_{t}^{\\varphi }\\in C$ is defined by $x_{t}^{\\varphi }\\left(s\\right)=x^{\\varphi }\\left(t+s\\right)$ , $-1\\le s\\le 0$ .", "Then the map $\\Phi :\\left[-1,\\infty \\right)\\times C\\ni \\left(t,\\varphi \\right)\\mapsto x_{t}^{\\varphi }\\in C$ is a continuous semiflow.", "In [8], the authors of this paper have studied Eq.", "(REF ) under the subsequent hypothesis: 00.00.0000 (H1) $\\mu >0$ , $f\\in C^{1}\\left(\\mathbb {R},\\mathbb {R}\\right)$ with $f^{\\prime }\\left(\\xi \\right)>0$ for all $\\xi \\in \\mathbb {R}$ , and $\\xi _{-2}<\\xi _{-1}<\\xi _{0}=0<\\xi _{1}<\\xi _{2}$ are five consecutive zeros of $\\mathbb {R}\\ni \\xi \\mapsto -\\mu \\xi +f\\left(\\xi \\right)\\in \\mathbb {R}$ with $f^{\\prime }\\left(\\xi _{j}\\right)<\\mu <f^{\\prime }\\left(\\xi _{k}\\right)$ for $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ and $k\\in \\left\\lbrace -1,1\\right\\rbrace $ (see Fig. 1).", "Figure: A feedback function satisfying condition (H1).Under hypothesis (H1), $\\hat{\\xi }_{j}\\in C$ , defined by $\\hat{\\xi }_{j}\\left(s\\right)=\\xi _{j}$ , $-1\\le s\\le 0$ , is an equilibrium point of $\\Phi $ for all $j\\in \\left\\lbrace -2,-1,0,1,2\\right\\rbrace $ , furthermore $\\hat{\\xi }_{-2},$ $\\hat{\\xi }_{0}$ and $\\hat{\\xi }_{2}$ are stable, and $\\hat{\\xi }_{-1}$ and $\\hat{\\xi }_{1}$ are unstable.", "By the monotone property of $f$ , the subsets $C_{-2,2}=\\left\\lbrace \\varphi \\in C:\\,\\xi _{-2}\\le \\varphi \\left(s\\right)\\le \\xi _{2}\\mbox{ for all }s\\in \\left[-1,0\\right]\\right\\rbrace ,$ $C_{-2,0}=\\left\\lbrace \\varphi \\in C:\\,\\xi _{-2}\\le \\varphi \\left(s\\right)\\le 0\\mbox{ for all }s\\in \\left[-1,0\\right]\\right\\rbrace ,$ $C_{0,2}=\\left\\lbrace \\varphi \\in C:\\,0\\le \\varphi \\left(s\\right)\\le \\xi _{2}\\mbox{ for all }s\\in \\left[-1,0\\right]\\right\\rbrace $ of the phase space $C$ are positively invariant under the semiflow $\\Phi $ (see Proposition REF in Section ).", "Let $\\mathcal {A}$ , $\\mathcal {A}_{-2,0}$ and $\\mathcal {A}_{0,2}$ denote the global attractors of the restrictions $\\Phi \|_{\\left[0,\\infty \\right)\\times C_{-2,2}}$ , $\\Phi \|_{\\left[0,\\infty \\right)\\times C_{-2,0}}$ and $\\Phi \|_{\\left[0,\\infty \\right)\\times C_{0,2}}$ , respectively.", "If (H1) holds and $\\xi _{-2},\\xi _{-1},0,\\xi _{1},\\xi _{2}$ are the only zeros of $-\\mu \\xi +f\\left(\\xi \\right)$ , then $\\mathcal {A}$ is the global attractor of $\\Phi $ .", "The structures of $\\mathcal {A}_{-2,0}$ and $\\mathcal {A}_{0,2}$ are (at least partially) well understood, see e.g.", "[5], [6], [7], [9], [10], [11].", "$\\mathcal {A}_{-2,0}$ and $\\mathcal {A}_{0,2}$ admit Morse decompositions [18].", "Further technical conditions regarding $f$ ensure that $\\mathcal {A}_{-2,0}$ and $\\mathcal {A}_{0,2}$ have spindle-like structures [5], [9], [10], [11]: $\\mathcal {A}_{0,2}$ is the closure of the unstable set of $\\hat{\\xi }_{1}$ containing the equilibrium points $\\hat{\\xi }_{0}$ , $\\hat{\\xi }_{1}$ , $\\hat{\\xi }_{2}$ , periodic orbits in $C_{0,2}$ and heteroclinic orbits among them.", "In other cases $\\mathcal {A}_{0,2}$ is larger than the the closure of the unstable set of $\\hat{\\xi }_{1}$ .", "The structure of $\\mathcal {A}_{-2,0}$ is similar.", "See Fig.", "2 for a simple situation.", "Figure: A spindle-like structureThe monograph [10] of Krisztin, Walther and Wu has addressed the question whether the equality $\\mathcal {A}=\\mathcal {A}_{-2,0}\\cup \\mathcal {A}_{0,2}$ holds under hypothesis (H1).", "The authors of this paper have constructed an example in [8] so that (H1) holds, and Eq.", "(REF ) admits periodic orbits in $\\mathcal {A}\\setminus \\left(\\mathcal {A}_{-2,0}\\cup \\mathcal {A}_{0,2}\\right)$ , that is, besides the spindle-like structures.", "The periodic solutions defining these periodic orbits oscillate slowly around 0 and have large amplitudes in the following sense.", "A periodic solution $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) is called a large amplitude periodic solution if $r(\\mathbb {R})\\supset (\\xi _{-1},\\xi _{1})$ .", "A solution $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is slowly oscillatory if for each $t$ , the restriction $r\|_{[t-1,t]}$ has one or two sign changes.", "Note that here slow oscillation is different from the usual one used for equations with negative feedback condition [2], [21].", "A large-amplitude slowly oscillatory periodic solution $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is abbreviated as an LSOP solution.", "We say that an LSOP solution $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is normalized if $r(-1)=0$ , and for some $\\eta >0$ , $r(s)>0$ for all $s\\in (-1,-1+\\eta )$ .", "The first main result of [8] is as follows.", "Theorem A There exist $\\mu $ and $f$ satisfying (H1) such that Eq.", "(REF ) has exactly two normalized LSOP solutions $p:\\mathbb {R}\\rightarrow \\mathbb {R}$ and $q:\\mathbb {R}\\rightarrow \\mathbb {R}$ .", "For the ranges of $p$ and $q$ , $(\\xi _{-1},\\xi _{1})\\subset p(\\mathbb {R})\\subset q(\\mathbb {R})\\subset \\left(\\xi _{-2},\\xi _{2}\\right)$ holds.", "The corresponding periodic orbits $\\mathcal {O}_{p}=\\left\\lbrace p_{t}:\\, t\\in \\mathbb {R}\\right\\rbrace \\ \\mathit {and}\\ \\mathcal {O}_{q}=\\left\\lbrace q_{t}:\\, t\\in \\mathbb {R}\\right\\rbrace $ are hyperbolic and unstable.", "$\\mathcal {O}_{p}$ admits two different Floquet multipliers outside the unit circle, which are real and simple.", "$\\mathcal {O}_{q}$ has one real simple Floquet multiplier outside the unit circle.", "Note that although Theorem 1.1 in [8] does not mention that the Floquet multipliers found outside the unit circle are simple and real, these properties are verified in Section 4 of the same paper.", "In the proof of the theorem, $\\mu =1$ and $f$ is close to the step function $f^{K,0}\\left(x\\right)={\\left\\lbrace \\begin{array}{ll}-K & \\mbox{if }x<-1,\\\\0 & \\mbox{if }\\left\|x\\right\|\\le 1,\\\\K & \\mbox{if }x>1,\\end{array}\\right.", "}$ where $K>0$ is chosen large enough.", "In their paper [3], Fiedler, Rocha and Wolfrum considered a special class of one-dimensional parabolic partial differential equations and obtained a catalogue listing the possible structures of the global attractor.", "In particular, the result of Theorem A motivated Fiedler, Rocha and Wolfrum to find an analogous configuration for their equation.", "It is an interesting question whether all the structures found by them have counterparts in the theory of Eq.", "(REF ).", "Let $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and $\\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)$ denote the unstable sets of $\\mathcal {O}_{p}$ and $\\mathcal {O}_{q}$ , respectively.", "A solution $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is called slowly oscillatory around $\\xi _{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , if $\\mathbb {R}\\ni t\\mapsto r(t)-\\xi _{k}\\in \\mathbb {R}$ admits one or two sign changes on each interval of length 1.", "As it is described by Proposition 2.7 in [8], $f$ and $\\mu $ in Theorem A are set so that there exist at least one periodic solution oscillating slowly around $\\xi _{1}$ with range in $(0,\\xi _{2})$ , furthermore there is a solution $x^{1}:\\mathbb {R}\\rightarrow \\mathbb {R}$ among such periodic solutions that has maximal range $x^{1}(\\mathbb {R})$ in the sense that $x^{1}(\\mathbb {R})\\supset x(\\mathbb {R})$ for all periodic solutions $x$ oscillating slowly around $\\xi _{1}$ with range in $(0,\\xi _{2})$ .", "Similarly, there exists a maximal periodic solution $x^{-1}$ oscillating slowly around $\\xi _{-1}$ with range in $(\\xi _{-2},0)$ .", "Set $\\mathcal {O}_{1}=\\left\\lbrace x_{t}^{1}:t\\in \\mathbb {R}\\right\\rbrace \\mbox{ and }\\mathcal {O}_{-1}=\\left\\lbrace x_{t}^{-1}:t\\in \\mathbb {R}\\right\\rbrace .$ Let $\\omega \\left(\\varphi \\right)$ denote the $\\omega $ -limit set of any $\\varphi \\in C$ .", "Introduce the connecting sets $C_{j}^{p}= & \\left\\lbrace \\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right):\\,\\omega \\left(\\varphi \\right)=\\hat{\\xi }_{j}\\right\\rbrace ,\\qquad j\\in \\left\\lbrace -2,0,2\\right\\rbrace ,$ $C_{k}^{p}= & \\left\\lbrace \\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right):\\,\\omega \\left(\\varphi \\right)=\\mathcal {O}_{k}\\right\\rbrace ,\\qquad k\\in \\left\\lbrace -1,1\\right\\rbrace ,$ and $C_{q}^{p}= & \\left\\lbrace \\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right):\\,\\omega \\left(\\varphi \\right)=\\mathcal {O}_{q}\\right\\rbrace .$ Sets $C_{j}^{q}$ , $j\\in \\left\\lbrace -2,2\\right\\rbrace $ , are defined analogously.", "The next theorem has also been given in [8] and describes the dynamics in $\\mathcal {A}\\setminus (\\mathcal {A}_{-2,0}\\cup \\mathcal {A}_{0,2})$ .", "Theorem B One may set $\\mu $ and $f$ satisfying (H1) such that the statement of Theorem A holds, and for the global attractor $\\mathcal {A}$ we have the equality $\\mathcal {A}=\\mathcal {A}_{-2,0}\\cup \\mathcal {A}_{0,2}\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right).$ Moreover, the dynamics on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and $\\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)$ is as follows.", "The connecting sets $C_{j}^{p}$ , $C_{q}^{p}$ , $C_{k}^{p}$ , $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , are nonempty, and $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)=\\mathcal {O}_{p}\\cup C_{-2}^{p}\\cup C_{-1}^{p}\\cup C_{0}^{p}\\cup C_{1}^{p}\\cup C_{2}^{p}\\cup C_{q}^{p}.$ The connecting sets $C_{-2}^{q}$ and $C_{2}^{q}$ are nonempty, and $\\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)=\\mathcal {O}_{q}\\cup C_{-2}^{q}\\cup C_{2}^{q}.$ The system of heteroclinic connections is represented in Fig. 3.", "Figure: Connecting orbits: the dashed arrows represent heteroclinic connectionsin 𝒜 -2,0 \\mathcal {A}_{-2,0} and in 𝒜 0,2 \\mathcal {A}_{0,2}, while the solidones represent connecting orbits given by Theorem B.Hereinafter we fix $\\mu =1$ and set $f$ in Eq.", "(REF ) so that Theorems A and B hold.", "The purpose of this paper is to characterize the geometrical properties of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and the connecting sets within $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "We say that a subset $W$ of $C$ admits global graph representation, if there exists a splitting $C=G\\oplus E$ with closed subspaces $G$ and $E$ of $C$ , a subset $U$ of $G$ and a map $w:U\\rightarrow E$ such that $W=\\left\\lbrace \\chi +w\\left(\\chi \\right):\\,\\chi \\in U\\right\\rbrace .$ $W$ is said to have a smooth global graph representation if in the above definition $U$ is open in $G$ and $w$ is $C^{1}$ -smooth on $U$ .", "Note that in this case $W$ is a $C^{1}$ -submanifold of $C$ in the usual sense with dimension $\\dim G$ , see e.g.", "the definition of Lang in [12].", "$W$ is said to admit a smooth global graph representation with boundary if $G$ is $n$ dimensional with some integer $n\\ge 1$ , $U$ is the closure of an open set $U^{0}$ , $w$ is $C^{1}$ -smooth on $U^{0}$ , the boundary $\\mbox{bd}U$ of $U$ in $G$ is an $(n-1)$ -dimensional $C^{1}$ -submanifold of $G$ , and all points of $\\mbox{bd}U$ have an open neighborhood in $G$ on which $w$ can be extended to a $C^{1}$ -smooth function.", "In this case $W$ is an $n$ -dimensional $C^{1}$ -submanifold of $C$ with boundary in the usual sense [12].", "The first result of this paper is the following.", "Theorem 1.1 $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , $C_{-2}^{p},\\, C_{0}^{p}$ and $C_{2}^{p}$ are three-dimensional $C^{1}$ -submanifolds of $C$ admitting smooth global graph representations.", "The next objects of our study are the connecting sets $C_{q}^{p}$ , $C_{-1}^{p}$ , $C_{1}^{p}$ containing the heteroclinic orbits from $\\mathcal {O}_{p}$ to $\\mathcal {O}_{q}$ , $\\mathcal {O}_{-1}$ , $\\mathcal {O}_{1}$ , respectively.", "We actually get a detailed picture of the structure of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ by characterizing the unions $S_{-1}=C_{-1}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\qquad \\mbox{and}\\qquad S_{1}=C_{1}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}.$ A solution $x:\\mathbb {R}\\rightarrow \\mathbb {R}$ is said to oscillate around $\\xi _{i}$ , $i\\in \\left\\lbrace -2,-1,0,1,2\\right\\rbrace $ , if the set $x^{-1}\\left(\\xi _{i}\\right)\\subset \\mathbb {R}$ is not bounded from above.", "It is a direct consequence of Theorem B that for $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , $S_{k}=\\left\\lbrace \\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right):\\, x^{\\varphi }\\mbox{ oscillates around }\\xi _{k}\\right\\rbrace .$ We say that a subset $W$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is above $S_{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , if to each $\\varphi \\in W$ there corresponds an element $\\psi $ of $S_{k}$ with $\\psi \\ll \\varphi $ (that is, $\\psi \\left(s\\right)<\\varphi \\left(s\\right)$ for all $s\\in \\left[-1,0\\right]$ ).", "Similarly, a subset $W$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is below $S_{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , if for all $\\varphi \\in W$ there exists $\\psi \\in S_{k}$ with $\\varphi \\ll \\psi $ .", "$W$ is between $S_{-1}$ and $S_{1}$ if it is below $S_{1}$ and above $S_{-1}$ .", "Our main result offers geometrical and topological descriptions of $C_{q}^{p}$ , $C_{-1}^{p}$ , $C_{1}^{p}$ , $S_{-1}$ and $S_{1}$ , and their closures in $C$ .", "It shows that $S_{-1}$ and $S_{1}$ separate the points of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ into three groups according to their $\\omega $ -limit sets.", "Thereby, $S_{-1}$ and $S_{1}$ play a key role in the dynamics of the equation.", "Theorem 1.2 (i) The sets $C_{q}^{p}$ , $C_{-1}^{p}$ , $C_{1}^{p}$ , $S_{-1}$ and $S_{1}$ are two-dimensional $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ with smooth global graph representations.", "They are homeomorphic to the open annulus $A^{\\left(1,2\\right)}=\\left\\lbrace u\\in \\mathbb {R}^{2}:\\,1<\\left\|u\\right\|<2\\right\\rbrace .$ (ii) The equalities $\\overline{C_{q}^{p}}=\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q},\\qquad \\overline{C_{k}^{p}}=\\mathcal {O}_{p}\\cup C_{k}^{p}\\cup \\mathcal {O}_{k}$ and $\\overline{S_{k}}=\\mathcal {O}_{k}\\cup S_{k}\\cup \\mathcal {O}_{q}=\\mathcal {O}_{k}\\cup C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ hold for both $k\\in \\left\\lbrace -1,1\\right\\rbrace $ .", "The sets $\\overline{C_{q}^{p}}$ , $\\overline{C_{-1}^{p}}$ , $\\overline{C_{1}^{p}}$ , $\\overline{S_{-1}}$ and $\\overline{S_{1}}$ admit smooth global graph representations with boundary, and thereby they are two-dimensional $C^{1}$ -submanifolds of $C$ with boundary.", "In addition, they are homeomorphic to the closed annulus $A^{\\left[1,2\\right]}=\\left\\lbrace u\\in \\mathbb {R}^{2}:\\,1\\le \\left\|u\\right\|\\le 2\\right\\rbrace .$ (iii) $S_{_{-1}}$ and $S_{1}$ are separatrices in the sense that $C_{2}^{p}$ is above $S_{1}$ , $C_{0}^{p}$ is between $S_{-1}$ and $S_{1}$ , furthermore $C_{-2}^{p}$ is below $S_{-1}$ .", "Fig.", "4 visualizes the structure of the closure $\\overline{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ in $C$ .", "To get an overview of the above results regarding $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , see the inner part of Fig.", "4, drawn in black.", "We emphasize a particular consequence of Theorem REF : the tangent spaces of $S_{-1}$ and $S_{1}$ coincide along $\\mathcal {O}_{p}$ , see Fig. 5.", "Figure: 𝒲 u 𝒪 p ¯\\overline{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)} can be visualizedas a “tulip” rotated around the vertical axis: the dots correspondto equilibria and periodic orbits, the thick arrows symbolize two-dimensionalheteroclinic connecting sets, and the three groups of thin arrowsrepresent three-dimensional connecting sets.", "The elements of 𝒲 u 𝒪 p \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)are drawn in black.", "Grey is used for the boundary of 𝒲 u 𝒪 p \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).Figure: The tangent spaces of S -1 S_{-1} and S 1 S_{1} coincide along 𝒪 p \\mathcal {O}_{p}.Let $\\mathcal {W}^{u}\\left(\\mathcal {O}_{1}\\right)$ and $\\mathcal {W}^{u}\\left(\\mathcal {O}_{-1}\\right)$ denote the unstable sets of $\\mathcal {O}_{1}$ and $\\mathcal {O}_{-1}$ , respectively, defined as the forward extension of a one-dimensional local unstable manifold of a return map (corresponding to the only Floquet multiplier outside the unit circle which is real and simple), see (REF ).", "We expect $\\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)$ , $\\mathcal {W}^{u}\\left(\\mathcal {O}_{-1}\\right)$ and $\\mathcal {W}^{u}\\left(\\mathcal {O}_{1}\\right)$ to be two-dimensional $C^{1}$ -submanifolds of $C$ .", "We conjecture that for the closure $\\overline{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ in $C$ , the equality $\\overline{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}=\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{1}\\right)\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{-1}\\right)\\cup \\left\\lbrace \\hat{\\xi }_{-2},\\hat{0},\\hat{\\xi }_{2}\\right\\rbrace $ holds, as it represented in Fig.", "4.", "Moreover, all points of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{1}\\right)\\cup \\mathcal {W}^{u}\\left(\\mathcal {O}_{-1}\\right)$ have an open neighborhood on which the $C^{1}$ -map in the graph representation of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ can be smoothly extended.", "It also remains an open question whether $\\mathcal {A}\\setminus (\\mathcal {A}_{-2,0}\\cup \\mathcal {A}_{0,2})$ is homeomorphic to the three-dimensional body $\\mathcal {B}_{3}\\left(\\left(0,0,0\\right),2\\right)\\backslash \\left\\lbrace \\mathcal {B}_{3}\\left(\\left(0,0,1\\right),1\\right)\\cup \\mathcal {B}_{3}\\left(\\left(0,0,-1\\right),1\\right)\\right\\rbrace \\subset \\mathbb {R}^{3},$ where $\\mathcal {B}_{3}\\left(\\left(a_{1},a_{2},a_{3}\\right),r\\right)$ denotes the three-dimensional closed ball with center $\\left(a_{1},a_{2},a_{3}\\right)$ and radius $r$ .", "The proofs of Theorems REF –REF apply general results on delay differential equations, the Floquet theory (Appendix VII of [10], [14]), results on local invariant manifolds for maps in Banach spaces (Appendices I-II of [10]), correspondences between different return maps (Appendices I and V of [10]), a result from transversality theory [1] and also a discrete Lyapunov functional of Mallet-Paret and Sell counting the sign changes of the elements of $C$ (Appendix VI of [10], [16]).", "This paper is organized as follows.", "Section offers a general overview of the theoretical background and introduces the discrete Lyapunov functional.", "As the Floquet theory and certain results on local invariant manifolds of return maps play essential role in this work, Section is devoted to the discussion of these concepts.", "Sections 4 and 5 contain the proofs of Theorems REF and REF , respectively.", "The proof of Theorem REF in Section 4 takes advantage of the fact that the unstable set of a hyperbolic periodic orbit is the forward continuation of a local unstable manifold of a Poincaré map by the semiflow.", "In consequence, by using the smoothness of the local unstable manifold and the injectivity of the derivative of the solution operator, we prove that all points $\\varphi $ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ belong to a subset $W_{\\varphi }$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ that is a three-dimensional $C^{1}$ -submanifold of $C$ .", "This means that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is an immersed submanifold of $C$ .", "In general, an immersed submanifold is not necessarily an embedded submanifold of the phase space.", "In order to prove that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is embedded in $C$ , we have to show that for any $\\varphi $ in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , there is no sequence in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\backslash W_{\\varphi }$ converging to $\\varphi $ .", "We define a projection $\\pi _{3}$ from $C$ into $\\mathbb {R}^{3}$ .", "Using well-known properties of the discrete Lyapunov functional, we show that $\\pi _{3}$ is injective on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and on the tangent spaces of $W_{\\varphi }$ .", "This implies that $\\pi _{3}W_{\\varphi }$ is open in $\\mathbb {R}^{3}$ .", "If a sequence $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\backslash W_{\\varphi }$ converges to $\\varphi $ as $n\\rightarrow \\infty $ , then $\\pi _{3}\\varphi ^{n}\\rightarrow \\pi _{3}\\varphi $ as $n\\rightarrow \\infty $ , and $\\pi _{3}\\varphi ^{n}\\in \\pi _{3}W_{\\varphi }$ for all $n$ large enough.", "The injectivity of $\\pi _{3}$ on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ then implies that $\\varphi ^{n}\\in W_{\\varphi }$ , which is a contradiction.", "So $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a three-dimensional embedded $C^{1}$ -submanifold of the phase space.", "The description of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is rounded up by giving a graph representation for $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ in order to present the simplicity of it structure.", "The smoothness of the sets $C_{-2}^{p},\\, C_{0}^{p}$ and $C_{2}^{p}$ then follows at once because they are open subsets of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "We also obtain as an important consequence that the semiflow defined by the solution operator extends to a $C^{1}$ -flow on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ with injective derivatives.", "The proof of Theorem REF in Section 5 is built from several steps, and it is organized into five subsections.", "In Subsection 5.1 we list preliminary results regarding the closure $\\overline{S_{k}}$ of $S_{k}$ in $C$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ .", "We introduce in particular a projection $\\pi _{2}$ from $C$ into $\\mathbb {R}^{2}$ , and – using the special properties of the discrete Lyapunov functional – we show that $\\pi _{2}$ is injective on $\\overline{S_{k}}$ .", "The injectivity of $\\pi _{2}\|_{\\overline{S_{k}}}$ is already sufficient to give a two-dimensional graph representation for any subset $W$ of $\\overline{S_{k}}$ (without smoothness properties): there is a linear isomorphism $J_{2}:\\mathbb {R}^{2}\\rightarrow C$ such that $P_{2}=J_{2}\\circ \\pi _{2}:C\\rightarrow C$ is a projection onto a two-dimensional subspace $G_{2}$ of $C$ , and there exists a map $w_{k}$ defined on the image set $P_{2}\\overline{S_{k}}$ with range in $P_{2}^{-1}\\left(0\\right)$ such that for any subset $W\\subseteq S_{k}$ , $W=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}W\\right\\rbrace .$ The smoothness of $w_{k}$ and the properties of its domain $P_{2}\\overline{S_{k}}\\subset G_{2}$ are investigated later.", "Subsection 5.1 is closed with showing that $\\pi _{2}\|_{\\overline{S_{k}}}$ is a homeomorphism onto its image, furthermore $\\pi _{2}$ is injective on the tangent spaces of $\\overline{S_{k}}$ .", "It is clear that $\\left(\\mathcal {O}_{k}\\cup S_{k}\\cup \\mathcal {O}_{q}\\right)\\subset \\overline{S_{k}}$ for both $k\\in \\left\\lbrace -1,1\\right\\rbrace $ .", "The converse inclusion is proved in Subsection 5.2 based on the previously obtained result that $\\overline{S_{k}}$ is mapped injectively into $\\mathbb {R}^{2}$ .", "Then it follows easily that $\\overline{C_{k}^{p}}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace ,$ and $\\overline{C_{q}^{p}}$ are not larger than the unions $\\mathcal {O}_{p}\\cup C_{k}^{p}\\cup \\mathcal {O}_{k}$ and $\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ , respectively.", "It is a more challenging task to show that $C_{q}^{p}$ and $C_{k}^{p}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace ,$ are $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ (as stated by Theorem REF .", "(i)).", "The proof of this assertion is contained in Subsection 5.3.", "It is partly based on transversality [1]; we verify that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ intersects transversally a local center-stable manifold of a Poincaré return map at a point of $\\mathcal {O}_{k}$ and a local stable manifold of a Poincaré return map at a point of $\\mathcal {O}_{q}$ , and thereby the intersections – subsets of $C_{q}^{p}$ and $C_{k}^{p}$ – are one-dimensional submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "The main difficulty in this task is that the hyperbolicity of $\\mathcal {O}_{k}$ is not known.", "Krisztin, Walther and Wu have proved transversality in a similar situation [10].", "Then we apply techniques that already appeared in Section 4.", "The injectivity of the derivative of the flow induced by the solution operator on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ guarantees that each point $\\varphi $ in $C_{q}^{p}$ or $C_{k}^{p}$ belongs to a “small” subset of $C_{q}^{p}$ or $C_{k}^{p}$ , respectively, that is a two-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Therefore, $C_{q}^{p}$ and $C_{k}^{p}$ are immersed $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "In order to prove that $C_{q}^{p}$ and $C_{k}^{p}$ are embedded in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , we repeat an argument from the proof of Theorem REF with $\\pi _{2}$ in the role of $\\pi _{3}$ .", "Based on the property that $C_{q}^{p}$ and $C_{k}^{p}$ are $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , we prove at the end of Subsection 5.3 that $w_{k}$ is continuously differentiable on the open sets $P_{2}C_{q}^{p}$ and $P_{2}C_{k}^{p}$ , i.e., the representations $C_{q}^{p}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}C_{q}^{p}\\right\\rbrace \\quad \\mbox{and}\\quad C_{k}^{p}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}C_{k}^{p}\\right\\rbrace .$ are smooth.", "Next we verify in Subsection 5.4 that the images of $C_{q}^{p}$ , $C_{k}^{p}$ and $S_{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace ,$ under $\\pi _{2}$ are topologically equivalent to the open annulus, and the images of their closures are topologically equivalent to the closed annulus.", "As $S_{k}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}S_{k}\\right\\rbrace \\quad \\mbox{and}\\quad P_{2}S_{k}=P_{2}C_{k}^{p}\\cup P_{2}\\mathcal {O}_{p}\\cup P_{2}C_{q}^{p},$ we have a smooth representation for $S_{k}$ if we show that $P_{2}S_{k}$ is open in $G_{2}$ and $w_{k}$ is smooth at the points of $P_{2}\\mathcal {O}_{p}$ .", "This is done in Subsection 5.5.", "It follows immediately that $S_{k}$ is a $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Simultaneously, we verify that all points of $P_{2}\\mathcal {O}_{k}\\cup P_{2}\\mathcal {O}_{q}$ have open neighborhoods on which $w_{k}$ can be extended to $C^{1}$ -functions.", "As $P_{2}\\mathcal {O}_{k}\\cup P_{2}\\mathcal {O}_{q}$ is the boundary of $P_{2}\\overline{S_{k}}$, this step guarantees that $\\overline{S_{k}}$ has a smooth representation with boundary, and thereby $\\overline{S_{k}}$ is a $C^{1}$ -submanifold of $C$ with boundary.", "The same reasonings yield the analogous results for $\\overline{C_{q}^{p}}$ and $\\overline{C_{k}^{p}}$ .", "Summing up, the proofs of Theorem REF .", "(i) and (ii) are completed in Subsection 5.5.", "It remains to show that $S_{-1}$ and $S_{1}$ are indeed separatrices in the sense described by Theorem REF .", "(iii).", "It is easy to see that the assertion restricted to a local unstable manifold of $\\mathcal {O}_{p}$ holds.", "Then we use the monotonicity of the semiflow to extend the statement for $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).$ Several techniques applied here have already appeared in the monograph [10] of Krisztin, canadianWalther and Wu.", "The novelty of this paper compared to [10] is that here we describe the unstable set of a periodic orbit, while [10] considers the unstable set of an equlibrium point.", "Acknowledgments.", "Both authors were supported by the Hungarian Scientific Research Fund, Grant No.", "K109782.", "The research of Gabriella Vas was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÃMOP-4.2.4.A/ 2-11/1-2012-0001 ‘National Excellence Program’.", "The research of Tibor Krisztin was also supported by the European Union and co-funded by the European Social Fund.", "Project title: “Telemedicine-focused research activities on the field of Matematics, Informatics and Medical sciences” Project number: TÃMOP-4.2.2.A-11/1/KONV-2012-0073." ], [ "Preliminaries", "We fix $\\mu =1$ and set $f$ in Eq.", "(REF ) so that Theorems A and B hold.", "In this section we give a summary of the theoretical background.", "In particular, we discuss the differentiability of the semiflow, the basic properties of the global attractor, the discrete Lyapunov functional of Mallet-Paret and Sell, and we list some technical results.", "The discussion of the Floquet theory and the Poincaré return maps is left to the next section." ], [ "Phase space, solution, segment.", "The natural phase space for Eq.", "(REF ) is the Banach space $C=C\\left(\\left[-1,0\\right],\\mathbb {R}\\right)$ of continuous real functions defined on $\\left[-1,0\\right]$ equipped with the supremum norm $\\left\\Vert \\varphi \\right\\Vert =\\sup _{-1\\le s\\le 0}\\left\|\\varphi \\left(s\\right)\\right\|.$ If $J$ is an interval, $u:J\\rightarrow \\mathbb {R}$ is continuous and $\\left[t-1,t\\right]\\subseteq J$ , then the segment $u_{t}\\in C$ is defined by $u_{t}\\left(s\\right)=u\\left(t+s\\right)$ , $-1\\le s\\le 0$ .", "Let $C^{1}$ denote the subspace of $C$ containing the continuously differentiable functions.", "Then $C^{1}$ is also a Banach space with the norm $\\left\\Vert \\varphi \\right\\Vert _{C^{1}}=\\left\\Vert \\varphi \\right\\Vert +\\left\\Vert \\varphi ^{\\prime }\\right\\Vert .$ For all $\\xi \\in \\mathbb {R}$ , $\\hat{\\xi }\\in C$ is defined by $\\hat{\\xi }\\left(s\\right)=\\xi $ for all $s\\in \\left[-1,0\\right]$ .", "A solution of Eq.", "(REF ) is either a continuous function on $\\left[t_{0}-1,\\infty \\right)$ , $t_{0}\\in \\mathbb {R}$ , which is differentiable for $t>t_{0}$ and satisfies equation Eq.", "(REF ) on $\\left(t_{0},\\infty \\right)$ , or a continuously differentiable function on $\\mathbb {R}$ satisfying the equation for all $t\\in \\mathbb {R}$ .", "To all $\\varphi \\in C$ , there corresponds a unique solution $x^{\\varphi }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) with $x_{0}^{\\varphi }=\\varphi $ .", "On $\\left(0,\\infty \\right)$ , $x^{\\varphi }$ is given by the variation-of-constants formula for ordinary differential equations repeated on successive intervals of length 1: $x^{\\varphi }\\left(t\\right)=e^{n-t}x^{\\varphi }\\left(n\\right)+\\int _{n}^{t}e^{s-t}f\\left(x^{\\varphi }\\left(s-1\\right)\\right)\\mbox{d}s\\quad \\mbox{for all }n\\in \\mathbb {N},\\, n\\le t\\le n+1.$ The solutions of Eq.", "(REF ) define the continuous semiflow $\\Phi :\\mathbb {R}^{+}\\times C\\ni \\left(t,\\varphi \\right)\\mapsto x_{t}^{\\varphi }\\in C.$ All maps $\\Phi \\left(t,\\cdot \\right):C\\rightarrow C$ , $t\\ge 1$ , are compact [4].", "As $f^{\\prime }>0$ on $\\mathbb {R}$ , all maps $\\Phi \\left(t,\\cdot \\right):C\\rightarrow C$ , $t\\ge 0$ , are injective [10].", "It follows that for every $\\varphi \\in C$ there is at most one solution $x:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) with $x_{0}=\\varphi .$ Whenever such solution exists, we denote it also by $x^{\\varphi }$ .", "For fixed $\\varphi \\in C$ , the map $\\left(1,\\infty \\right)\\ni t\\mapsto \\Phi (t,\\varphi )\\in C$ is continuously differentiable with $D_{1}\\Phi \\left(t,\\varphi \\right)1=\\dot{x_{t}}^{\\varphi }$ for all $t>1$ .", "For all $t\\ge 0$ fixed, $C\\ni \\varphi \\mapsto \\Phi (t,\\varphi )\\in C$ is continuously differentiable, and $D_{2}\\Phi (t,\\varphi )\\eta =v_{t}^{\\eta }$ , where $v^{\\eta }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ is the solution of the linear variational equation $\\dot{v}(t) & =-v(t)+f^{\\prime }\\left(x^{\\varphi }\\left(t-1\\right)\\right)v\\left(t-1\\right)$ with $v_{0}^{\\eta }=\\eta $ .", "So the restriction of $\\Phi $ to the open set $\\left(1,\\infty \\right)\\times C$ is continuously differentiable.", "Proposition 2.1 Suppose that $\\eta \\in C$ , $b:\\mathbb {R}\\rightarrow \\mathbb {R}$ is positive, and the problem ${\\left\\lbrace \\begin{array}{ll}\\dot{v}(t) & =-v(t)+b(t)v\\left(t-1\\right)\\\\v_{0} & =\\eta \\end{array}\\right.", "}$ has a solution $v^{\\eta }$ either on $\\left[t_{0}-1,\\infty \\right)$ with $t_{0}\\le 0$ or on $\\mathbb {R}$ (i.e., there is a continuous function $v^{\\eta }:\\left[t_{0}-1,\\infty \\right)\\rightarrow \\mathbb {R}$ with $v_{0}^{\\eta }=\\eta $ that is differentiable and satisfies the equation for $t>t_{0}$ , or there exists a differentiable function $v^{\\eta }:\\mathbb {R}\\rightarrow \\mathbb {R}$ with $v_{0}^{\\eta }=\\eta $ satisfying the equation for all real $t$ , respectively).", "Then $v^{\\eta }$ is unique.", "As the solution on $\\left[0,\\infty \\right)$ is determined by a variation-of-constants formula analogous to (REF ), the uniqueness in forward time is clear.", "For $t<0$ , the uniqueness follows from $v\\left(t-1\\right)=\\left(\\dot{v}\\left(t\\right)+v\\left(t\\right)\\right)/b\\left(t\\right)$ .", "In particular, the solution operator $D_{2}\\Phi (t,\\varphi )$ corresponding to the variational equation (REF ) is injective for all $\\varphi \\in C$ and $t\\ge 0$ .", "A function $\\hat{\\xi }\\in C$ is an equilibrium point (or stationary point) of $\\Phi $ if and only if $\\hat{\\xi }\\left(s\\right)=\\xi $ for all $-1\\le s\\le 0$ with $\\xi \\in \\mathbb {R}$ satisfying $-\\xi +f\\left(\\xi \\right)=0$ .", "Then $x^{\\hat{\\xi }}\\left(t\\right)=\\xi $ for all $t\\in \\mathbb {R}$ .", "As it is described in Chapter 2 of [10], condition $f^{\\prime }\\left(\\xi \\right)<1$ implies that $\\hat{\\xi }$ is stable and locally attractive.", "If $f^{\\prime }\\left(\\xi \\right)>1$ , then $\\hat{\\xi }$ is unstable.", "So hypothesis (H1) with $\\mu =1$ implies that $\\hat{\\xi }_{-2},$ $\\hat{\\xi }_{0}$ and $\\hat{\\xi }_{2}$ are stable, and $\\hat{\\xi }_{-1}$ and $\\hat{\\xi }_{1}$ are unstable.", "If $\\varphi \\in C$ and $x^{\\varphi }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ is a bounded solution of Eq.", "(REF ), then the $\\omega $ -limit set $\\omega \\left(\\varphi \\right)= & \\left\\lbrace \\psi \\in C:\\,\\mbox{there exists a sequence }\\left(t_{n}\\right)_{0}^{\\infty }\\mbox{ in }\\left[0,\\infty \\right)\\right.\\\\& \\left.\\mbox{ with }t_{n}\\rightarrow \\infty \\mbox{ and }\\Phi \\left(t_{n},\\varphi \\right)\\rightarrow \\psi \\mbox{ as }n\\rightarrow \\infty \\right\\rbrace $ is nonempty, compact, connected and invariant.", "For a solution $x:\\mathbb {R}\\rightarrow \\mathbb {R}$ such that $x\|_{\\left(-\\infty ,0\\right]}$ is bounded, the $\\alpha $-limit set $\\alpha \\left(x\\right)= & \\left\\lbrace \\psi \\in C:\\,\\mbox{there exists a sequence }\\left(t_{n}\\right)_{0}^{\\infty }\\mbox{ in }\\mathbb {R}\\right.\\\\& \\left.\\mbox{ with }t_{n}\\rightarrow -\\infty \\mbox{ and }x_{t_{n}}\\rightarrow \\psi \\mbox{ as }n\\rightarrow \\infty \\right\\rbrace $ is also nonempty, compact, connected and invariant.", "According to the Poincaré–Bendixson theorem of Mallet-Paret and Sell [17], for all $\\varphi \\in C_{-2,2}=\\left\\lbrace \\varphi \\in C:\\,\\xi _{-2}\\le \\varphi \\left(s\\right)\\le \\xi _{2}\\mbox{ for all }s\\in \\left[-1,0\\right]\\right\\rbrace ,$ the set $\\omega \\left(\\varphi \\right)$ is either a single nonconstant periodic orbit, or for each $\\psi \\in \\omega \\left(\\varphi \\right)$ , $\\alpha \\left(x^{\\psi }\\right)\\cup \\omega \\left(\\psi \\right)\\subseteq \\left\\lbrace \\hat{\\xi }_{-2},\\hat{\\xi }_{-1},\\hat{\\xi }_{0},\\hat{\\xi }_{1},\\hat{\\xi }_{2}\\right\\rbrace .$ An analogous result holds for $\\alpha \\left(x\\right)$ in case $x$ is defined on $\\mathbb {R}$ and $\\left\\lbrace x_{t}:\\ t\\le 0\\right\\rbrace \\subset C_{-2,2}$ .", "By Theorem 4.1 in Chapter 5 of [20], there is an open and dense set of initial functions in $C_{-2,2}$ so that the corresponding solutions converge to equilibria.", "Note that there is no homoclinic orbit to $\\hat{\\xi }_{j}$ , $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ , as these equilibria are stable.", "It follows from Proposition 3.1 in [7] that there exists no homoclinic orbits to the unstable equilibria $\\hat{\\xi }_{-1}$ and $\\hat{\\xi }_{1}$ .", "The global attractor $\\mathcal {A}$ of the restriction $\\Phi \|_{\\left[0,\\infty \\right)\\times C_{-2,2}}$ is a nonempty, compact set in $C$ , that is invariant in the sense that $\\Phi \\left(t,\\mathcal {A}\\right)=\\mathcal {A}$ for all $t\\ge 0$ , and that attracts bounded sets in the sense that for every bounded set $B\\subset C_{-2,2}$ and for every open set $U\\supset \\mathcal {A}$ , there exists $t\\ge 0$ with $\\Phi \\left(\\left[t,\\infty \\right)\\times B\\right)\\subset U$ .", "Global attractors are uniquely determined [4].", "It can be shown that $\\mathcal {A}= & \\left\\lbrace \\varphi \\in C_{-2,2}:\\mbox{ there is a bounded solution }x:\\mathbb {R}\\rightarrow \\mathbb {R}\\right.\\\\& \\left.\\mbox{ of Eq.\\,}(\\ref {eq:eq_general})\\mbox{ so that }\\varphi =x_{0}\\right\\rbrace ,$ see [9], [14], [18].", "The compactness of $\\mathcal {A}$ , its invariance property and the injectivity of the maps $\\Phi \\left(t,\\cdot \\right):C\\rightarrow C$ , $t\\ge 0$ , combined permit to verify that the map $\\left[0,\\infty \\right)\\times \\mathcal {A}\\ni \\left(t,\\varphi \\right)\\mapsto \\Phi \\left(t,\\varphi \\right)\\in \\mathcal {A}$ extends to a continuous flow $\\Phi _{\\mathcal {A}}:\\mathbb {R}\\times \\mathcal {A}\\rightarrow \\mathcal {A}$ ; for every $\\varphi \\in \\mathcal {A}$ and for all $t\\in \\mathbb {R}$ we have $\\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)=x_{t}^{\\varphi }$ with the uniquely determined solution $x^{\\varphi }:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) satisfying $x_{0}^{\\varphi }=\\varphi $ .", "Note that we have $\\mathcal {A}=\\Phi \\left(1,\\mathcal {A}\\right)\\subset C^{1}$ ; $\\mathcal {A}$ is a closed subset of $C^{1}$ .", "Using the flow $\\Phi _{\\mathcal {A}}$ and the continuity of the map $C\\ni \\varphi \\mapsto \\Phi \\left(1,\\varphi \\right)\\in C^{1},$ one obtains that $C$ and $C^{1}$ define the same topology on $\\mathcal {A}$ .", "Following Mallet-Paret and Sell in [16], we use a discrete Lyapunov functional $V:C\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace \\rightarrow 2\\mathbb {N}\\cup \\left\\lbrace \\infty \\right\\rbrace $ .", "For $\\varphi \\in C\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace ,$ set $sc\\left(\\varphi \\right)=0$ if $\\varphi \\ge \\hat{0}$ or $\\varphi \\le \\hat{0}$ (i.e., $\\varphi \\left(s\\right)\\ge 0$ for all $s\\in \\left[-1,0\\right]$ or $\\varphi \\left(s\\right)\\le 0$ for all $s\\in \\left[-1,0\\right]$ , respectively), otherwise define $sc\\left(\\varphi \\right)=\\textrm {sup}\\Bigl \\lbrace k\\in \\mathbb {N}\\setminus \\left\\lbrace 0\\right\\rbrace :\\mbox{ there exist a strictly increasing sequence}$ $\\left.\\left(s_{i}\\right)_{0}^{k}\\subseteq \\left[-1,0\\right]\\textrm { with }\\varphi \\left(s_{i-1}\\right)\\varphi \\left(s_{i}\\right)<0\\textrm { for }i\\in \\left\\lbrace 1,2,..,k\\right\\rbrace \\right\\rbrace .$ Then set $V\\left(\\varphi \\right)=\\left\\lbrace \\begin{array}{ll}sc\\left(\\varphi \\right), & \\textrm {if }sc\\left(\\varphi \\right)\\textrm { is even or }\\infty ,\\\\sc\\left(\\varphi \\right)+1, & \\textrm {if }sc\\left(\\varphi \\right)\\textrm { is odd}.\\end{array}\\right.$ Also define $R & = & \\left\\lbrace \\varphi \\in C^{1}:\\,\\varphi \\left(0\\right)\\ne 0\\mbox{ or }\\dot{\\varphi }\\left(0\\right)\\varphi \\left(-1\\right)>0,\\right.\\\\& & \\,\\varphi \\left(-1\\right)\\ne 0\\mbox{ or }\\dot{\\varphi }\\left(-1\\right)\\varphi \\left(0\\right)<0,\\left.\\mbox{all zeros of }\\varphi \\mbox{ are simple}\\right\\rbrace .$ $V$ has the following lower semi-continuity and continuity property (for a proof, see [10], [16]).", "Lemma 2.2 For each $\\varphi \\in C\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace $ and $\\left(\\varphi _{n}\\right)_{0}^{\\infty }\\subset C\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace $ with $\\varphi _{n}\\rightarrow \\varphi $ as $n\\rightarrow \\infty $ , $V\\left(\\varphi \\right)\\le \\liminf _{n\\rightarrow \\infty }V\\left(\\varphi _{n}\\right)$ .", "For each $\\varphi \\in R$ and $\\left(\\varphi _{n}\\right)_{0}^{\\infty }\\subset C^{1}\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace $ with $\\left\\Vert \\varphi _{n}-\\varphi \\right\\Vert _{C^{1}}\\rightarrow 0$ as $n\\rightarrow \\infty $ , $V\\left(\\varphi \\right)=\\lim _{n\\rightarrow \\infty }V\\left(\\varphi _{n}\\right)<\\infty $ .", "The next result explains why $V$ is called a Lyapunov functional (for a proof, see [10], [16] again).", "For an interval $J\\subset \\mathbb {R}$ , we use the notation $J+\\left[-1,0\\right]=\\left\\lbrace t\\in \\mathbb {R}:\\, t=t_{1}+t_{2}\\mbox{ with }t_{1}\\in J,\\, t_{2}\\in \\left[-1,0\\right]\\right\\rbrace .$ Lemma 2.3 Assume that $\\mu \\ge 0$ , $J\\subset \\mathbb {R}$ is an interval, $a:J\\rightarrow \\mathbb {R}$ is positive and continuous, $z:J+\\left[-1,0\\right]\\rightarrow \\mathbb {R}$ is continuous, $z\\left(t\\right)\\ne 0$ for some $t\\in J+\\left[-1,0\\right]$ , and $z$ is differentiable on $J$ .", "Suppose that $\\dot{z}\\left(t\\right)=-\\mu z\\left(t\\right)+a\\left(t\\right)z\\left(t-1\\right)$ holds for all $t>\\inf J$ in $J$ .", "Then the following statements hold.", "(i) If $t_{1},t_{2}\\in J$ with $t_{1}<t_{2}$ , then $V\\left(z_{t_{1}}\\right)\\ge V\\left(z_{t_{2}}\\right)$ .", "(ii) If $t,t-2\\in J$ , $z\\left(t-1\\right)=z\\left(t\\right)=0$ , then either $V\\left(z_{t}\\right)=\\infty $ or $V\\left(z_{t-2}\\right)>V\\left(z_{t}\\right)$ .", "(iii) If $t\\in J$ , $t-3\\in J$ , and $V\\left(z_{t-3}\\right)=V\\left(z_{t}\\right)<\\infty $ , then $z_{t}\\in R$ .", "If $f$ is a $C^{1}$ -smooth function with $f^{\\prime }>0$ on $\\mathbb {R}$ , $x,\\hat{x}:J+\\left[-1,0\\right]\\rightarrow \\mathbb {R}$ are solutions of Eq.", "(REF ) and $c\\in \\mathbb {R}\\setminus \\left\\lbrace 0\\right\\rbrace $ , then Lemma REF can be applied for $z=\\left(x-\\hat{x}\\right)/c$ with the positive continuous function $a:J\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sx\\left(t-1\\right)+\\left(1-s\\right)\\hat{x}\\left(t-1\\right)\\right)\\mbox{d}s\\in \\left[0,\\infty \\right).$ A solution $x$ is oscillatory around an equilibrium $\\hat{\\xi }$ if $x^{-1}\\left(\\xi \\right)$ is not bounded from above, and it is slowly oscillatory around $\\hat{\\xi }$ if $t\\rightarrow x\\left(t\\right)-\\xi $ has one or two sign changes on each interval of length 1.", "$B\\left(\\varphi ,r\\right)$ , $\\varphi \\in C$ , $r>0$ , denotes the open ball in $C$ with center $\\varphi $ and radius $r$ .", "We use the notation $S_{\\mathbb {C}}^{1}$ for the set $\\left\\lbrace z\\in \\mathbb {C}:\\,\\left\|z\\right\|=1\\right\\rbrace $ .", "For a simple closed curve $c:\\left[a,b\\right]\\rightarrow \\mathbb {R}^{2}$ , $\\mbox{int}\\left(c\\left[a,b\\right]\\right)$ and $\\mbox{ext}\\left(c\\left[a,b\\right]\\right)$ denote the interior and exterior, i.e., the bounded and unbounded components of $\\mathbb {R}^{2}\\setminus c\\left(\\left[a,b\\right]\\right)$ , respectively.", "We use the same notations for closed curves $c:\\left[a,b\\right]\\rightarrow G_{2}$ , where $G_{2}$ is any two-dimensional real Banach space.", "We say $\\varphi \\le \\psi $ for $\\varphi ,\\psi \\in C$ if $\\varphi \\left(s\\right)\\le \\psi \\left(s\\right)$ for all $s\\in [-1,0]$ .", "Relation $\\varphi <\\psi $ holds if $\\varphi \\le \\psi $ and $\\varphi \\ne \\psi $ .", "In addition, $\\varphi \\ll \\psi $ if $\\varphi \\left(s\\right)<\\psi \\left(s\\right)$ for all $s\\in [-1,0]$ .", "Relations “$\\ge $ ”, “$>$ ” and “$\\gg $ ” are defined analogously.", "The semiflow $\\Phi $ is monotone in the following sense.", "Proposition 2.4 If $\\varphi ,\\psi \\in C$ with $\\varphi \\le \\psi $ $\\left(\\varphi \\ge \\psi \\right)$ , then $x_{t}^{\\varphi }\\le x_{t}^{\\psi }$ $\\left(x_{t}^{\\varphi }\\ge x_{t}^{\\psi }\\right)$ for all $t\\ge 0$ .", "If $\\varphi <\\psi $ $\\left(\\varphi >\\psi \\right)$ , then $x_{t}^{\\varphi }\\ll x_{t}^{\\psi }$ $\\left(x_{t}^{\\varphi }\\gg x_{t}^{\\psi }\\right)$ for all $t\\ge 2$ .", "If $\\varphi \\ll \\psi $ $\\left(\\varphi \\gg \\psi \\right)$ , then $x_{t}^{\\varphi }\\ll x_{t}^{\\psi }$ $\\left(x_{t}^{\\varphi }\\gg x_{t}^{\\psi }\\right)$ for all $t\\ge 0$ .", "The assertion follows easily from the variation-of-constant formula.", "For a proof we refer to [20].", "Note that Proposition REF guarantees the positive invariance of $C_{-2,0}$ , $C_{0,2}$ and $C_{-2,2}$ .", "The periodic solutions have nice monotone properties (see Theorem 7.1 in [17]) as follows.", "Proposition 2.5 Suppose $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is a periodic solution of Eq.", "(REF ) with minimal period $\\omega >0$ .", "Then $r$ is of monotone type in the following sense: if $t_{0}<t_{1}<t_{0}+\\omega $ are fixed so that $r\\left(t_{0}\\right)=\\min _{t\\mathbb {\\in R}}r(t)$ and $r\\left(t_{1}\\right)=\\max _{t\\mathbb {\\in R}}r(t)$ , then $\\dot{r}\\left(t\\right)>0$ for $t\\in \\left(t_{0},t_{1}\\right)$ and $\\dot{r}\\left(t\\right)<0$ for $t\\in \\left(t_{1},t_{0}+\\omega \\right)$ .", "We also need the next technical results.", "The first one is the direct consequence of Lemmas VI.4, VI.5 and VI.6 in [10].", "Lemma 2.6 Let $\\mu \\ge 0$ , $\\alpha _{0}>0$ and $\\alpha _{1}\\ge \\alpha _{0}$ .", "Let sequences of continuous real functions $a^{n}$ on $\\mathbb {R}$ and continuously differentiable real functions $z^{n}$ on $\\mathbb {R}$ , $n\\ge 0$ , be given such that for all $n\\ge 0$ , $\\alpha _{0}\\le a^{n}\\left(t\\right)\\le \\alpha _{1}$ for all $t\\in \\mathbb {R}$ , $z^{n}\\left(t\\right)\\ne 0$ for some $t\\in \\mathbb {R}$ , $V\\left(z_{t}^{n}\\right)\\le 2$ for all $t\\in \\mathbb {R}$ , and $z^{n}$ satisfies $\\dot{z}^{n}\\left(t\\right)=-\\mu z^{n}\\left(t\\right)+a^{n}\\left(t\\right)z^{n}\\left(t-1\\right)$ on $\\mathbb {R}$ .", "Let a further continuous real function $a$ on $\\mathbb {R}$ be given so that $a^{n}\\rightarrow a$ as $n\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ .", "Then a continuously differentiable function $z:\\mathbb {R}\\rightarrow \\mathbb {R}$ and a subsequence $\\left(z^{n_{k}}\\right)_{k=0}^{\\infty }$ of $\\left(z^{n}\\right)_{n=0}^{\\infty }$ can be given such that $z^{n_{k}}\\rightarrow z$ and $\\dot{z}^{n_{k}}\\rightarrow \\dot{z}$ as $k\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , moreover $\\dot{z}\\left(t\\right)=-\\mu z\\left(t\\right)+a\\left(t\\right)z\\left(t-1\\right)$ for all $t\\in \\mathbb {R}$ .", "The subsequent result shows that Lyapunov functionals can be used effectively to show that solutions of linear equations cannot decay too fast at $\\infty $ .", "For a proof, see Lemma VI.3 in [10].", "Lemma 2.7 Let $\\mu \\ge 0$ , $\\alpha _{0}>0$ and $\\alpha _{1}\\ge \\alpha _{0}$ .", "Assume that $t_{0}\\in \\mathbb {R}$ , $a:\\left[t_{0}-5,t_{0}\\right]\\rightarrow \\mathbb {R}$ is continuous with $\\alpha _{0}\\le a\\left(t\\right)\\le \\alpha _{1}$ for all $t\\in \\left[t_{0}-5,t_{0}\\right]$ , $z:\\left[t_{0}-6,t_{0}\\right]\\rightarrow \\mathbb {R}$ is continuous, differentiable for $t_{0}-5<t\\le t_{0}$ and satisfies (REF ) for $t_{0}-5<t\\le t_{0}$ .", "In addition, assume that $z_{t_{0}-5}\\ne 0$ and $V\\left(z_{t_{0}-5}\\right)\\le 2$ .", "Then there exists $K=K\\left(\\mu ,\\alpha _{0},\\alpha _{1}\\right)>0$ such that $\\left\\Vert z_{t_{0}-1}\\right\\Vert \\le K\\left\\Vert z_{t_{0}}\\right\\Vert .$ The last result of this section is Lemma I.8 in [10].", "It will be used to abbreviate proofs of smoothness of submanifolds.", "Proposition 2.8 Let $g$ be a $C^{1}$ -map from an $m$ -dimensional $C^{1}$ -manifold $M$ into a $C^{1}$ -manifold $N$ modeled over a Banach space.", "If for some $p\\in M$ , the derivative $Dg\\left(p\\right)$ of $g$ at $p$ is injective, then $p$ has an open neighborhood $U$ in $M$ so that $g\\left(U\\right)$ is an $m$ -dimensional $C^{1}$ -submanifold of $N$ ." ], [ "Floquet multipliers and a Poincaré return map", "In this section we give a brief introduction to the Floquet theory regarding periodic solutions which are slowly oscillatory around an equilibrium.", "Then we define a Poincaré map and collect the most important properties of its local invariant manifolds.", "At last we apply these results to $p$ , $q$ , $x^{1}$ and $x^{-1}$ .", "The section is closed by showing that the unstable space of the monodromy operator corresponding to the periodic orbit $\\mathcal {O}_{k}$ is one-dimensional for both $k\\in \\left\\lbrace -1,1\\right\\rbrace $ ." ], [ "Floquet multipliers", "Suppose $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is a periodic solution of Eq.", "(REF ) with minimal period $\\omega >0$ .", "If $r$ is slowly oscillatory around an equilibrium (as $p$ , $q$ , $x^{1}$ or $x^{-1}$ are), then Proposition REF implies that $\\omega \\in \\left(1,2\\right)$ .", "Assume that this is the case.", "Consider the period map $Q=\\Phi \\left(\\omega ,\\cdot \\right)$ with fixed point $r_{0}$ and its derivative $M=D_{2}\\Phi \\left(\\omega ,r_{0}\\right)$ at $r_{0}$ .", "Then $M\\varphi =u_{\\omega }^{\\varphi }$ for all $\\varphi \\in C$ , where $u^{\\varphi }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ is the solution of the linear variational equation $\\dot{u}(t) & =-u(t)+f^{\\prime }\\left(r\\left(t-1\\right)\\right)u\\left(t-1\\right)$ with $u_{0}^{\\varphi }=\\varphi $ .", "$M$ is called the monodromy operator.", "$M$ is a compact operator, 0 belongs to its spectrum $\\sigma =\\sigma \\left(M\\right)$ , and its eigenvalues of finite multiplicity – the so called Floquet multipliers – form $\\sigma \\left(M\\right)\\backslash \\left\\lbrace 0\\right\\rbrace $ .", "The importance of $M$ lies in the fact that we obtain information about the stability properties of the orbit $\\mathcal {O}_{r}=\\left\\lbrace r_{t}:\\, t\\in \\mathbb {R}\\right\\rbrace $ from $\\sigma \\left(M\\right)$ .", "As $\\dot{r}$ is a nonzero solution of the variational equation (REF ), 1 is a Floquet multiplier with eigenfunction $\\dot{r}_{0}$ .", "The periodic orbit $\\mathcal {O}_{r}$ is said to be hyperbolic if the generalized eigenspace of $M$ corresponding to the eigenvalue 1 is one-dimensional, furthermore there are no Floquet multipliers on the unit circle besides 1.", "The paper [16] of Mallet-Paret and Sell and Appendix VII of the monograph [10] of Krisztin, Walther and Wu confirm the subsequent properties.", "$\\mathcal {O}_{r}$ has a real Floquet multiplier $\\lambda _{1}>1$ with a strictly positive eigenvector $v_{1}$ .", "The realified generalized eigenspace $C_{<\\lambda _{1}}$ associated with the spectral set $\\left\\lbrace z\\in \\sigma :\\,\\left\|z\\right\|<\\lambda _{1}\\right\\rbrace $ satisfies $C_{<\\lambda _{1}}\\cap V^{-1}\\left(0\\right)=\\emptyset .$ Let $C_{\\le \\rho }$ , $\\rho >0$ , denote the realified generalized eigenspace of $M$ associated with the spectral set $\\left\\lbrace z\\in \\sigma :\\,\\left\|z\\right\|\\le \\rho \\right\\rbrace $ .", "The set $\\left\\lbrace \\rho \\in \\left(0,\\infty \\right):\\,\\sigma \\left(M\\right)\\cap \\rho S_{\\mathbb {C}}^{1}\\ne \\emptyset ,\\, C_{\\le \\rho }\\cap V^{-1}\\left(\\left\\lbrace 0,2\\right\\rbrace \\right)=\\emptyset \\right\\rbrace $ is nonempty and has a maximum $r_{M}$ .", "Then $C_{\\le r_{M}}\\cap V^{-1}\\left(\\left\\lbrace 0,2\\right\\rbrace \\right)=\\emptyset ,\\quad C_{r_{M}<}\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace \\subset V^{-1}\\left(\\left\\lbrace 0,2\\right\\rbrace \\right)\\mbox{ and }\\mbox{dim}C_{r_{M}<}\\le 3,$ where $C_{r_{M}<}$ is the realified generalized eigenspace of $M$ associated with the nonempty spectral set $\\left\\lbrace z\\in \\sigma :\\,\\left\|z\\right\|>r_{M}\\right\\rbrace $ .", "It will easily follow from the results of this paper that $\\mbox{dim}C_{r_{M}<}=3$ for the periodic solutions $p,q,x^{-1}$ and $x^{1}$ , see Remark REF .", "Recently Mallet-Paret and Nussbaum have shown that the equality $\\mbox{dim}C_{r_{M}<}=3$ holds in general [15].", "Let $C_{s}$ , $C_{c}$ and $C_{u}$ be the closed subspaces of $C$ chosen so that $C=C_{s}\\oplus C_{c}\\oplus C_{u}$ , $C_{s}$ , $C_{c}$ and $C_{u}$ are invariant under $M$ , and the spectra $\\sigma _{s}\\left(M\\right)$ , $\\sigma _{c}\\left(M\\right)$ and $\\sigma _{u}\\left(M\\right)$ of the induced maps $C_{s}\\ni x\\mapsto Mx\\in C_{s}$ , $C_{c}\\ni x\\mapsto Mx\\in C_{c}$ , and $C_{u}\\ni x\\mapsto Mx\\in C_{u}$ are contained in $\\left\\lbrace \\mu \\in \\mathbb {C}:\\,\\left\|\\mu \\right\|<1\\right\\rbrace $ , $\\left\\lbrace \\mu \\in \\mathbb {C}:\\,\\left\|\\mu \\right\|=1\\right\\rbrace $ and $\\left\\lbrace \\mu \\in \\mathbb {C}:\\,\\left\|\\mu \\right\|>1\\right\\rbrace $ , respectively.", "As $\\mathcal {O}_{r}$ has a real Floquet multiplier $\\lambda _{1}>1$ , $C_{u}$ is nontrivial.", "$C_{c}$ is also nontrivial because $\\dot{r}_{0}\\in C_{c}$ .", "It is easy to see that the monotone property of $r$ described in Proposition REF and $\\omega \\in \\left(1,2\\right)$ imply the existence of $t\\in \\mathbb {R}$ with $V\\left(\\dot{r}_{t}\\right)=2$ .", "As $\\mathbb {R}\\ni t\\rightarrow \\dot{r}_{t}\\in C$ is periodic, and $\\mathbb {R}\\ni t\\rightarrow V\\left(\\dot{r}_{t}\\right)$ is monotone decreasing by Lemma REF , it follows that $V\\left(\\dot{r}_{t}\\right)=2$ for all real $t$ .", "In particular, $V\\left(\\dot{r}_{0}\\right)=2$ .", "Hence (REF ) gives that $r_{M}<1$ , moreover (REF ) and (REF ) together give that $C_{c}\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace \\subset V^{-1}\\left(2\\right)$ .", "The nontriviality of $C_{u}$ and $\\mbox{dim}C_{r_{M}<}\\le 3$ in addition imply that $C_{c}$ is at most two-dimensional in our case: $C_{c}=\\left\\lbrace \\begin{array}{ll}\\mathbb {R}\\dot{r}_{0}, & \\mbox{if }\\mathcal {O}_{r}\\mbox{ is hyperbolic,}\\\\\\mathbb {R}\\dot{r}_{0}\\oplus \\mathbb {R}\\xi , & \\mbox{otherwise,}\\end{array}\\right.$ where $\\xi \\in C_{c}\\setminus \\mathbb {R}\\dot{r}_{0}$ provided that $\\mathcal {O}_{r}$ is nonhyperbolic." ], [ "A Poincaré return map", "As above, let $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ be any periodic solution of Eq.", "(REF ) which oscillates slowly around an equilibrium, and let $\\omega \\in \\left(1,2\\right)$ denote its minimal period.", "Fix a $\\xi \\in C_{c}\\setminus \\mathbb {R}\\dot{r}_{0}$ in case $O_{r}$ is nonhyperbolic and define $Y=\\left\\lbrace \\begin{array}{ll}C_{s}\\oplus C_{u}, & \\mbox{if }\\mathcal {O}_{r}\\mbox{ is hyperbolic,}\\\\C_{s}\\oplus \\mathbb {R}\\xi \\oplus C_{u}, & \\mbox{\\mbox{if }$\\mathcal {O}_{r}$\\mbox{ is nonhyperbolic}.", "}\\end{array}\\right.$ Then $Y\\subset C$ is a hyperplane with codimension 1.", "Choose $e^{}$ to be a continuous linear functional with null space $\\left(e^{}\\right)^{-1}\\left(0\\right)=Y$ .", "The Hahn–Banach theorem guarantees the existence of $e^{}$ .", "As $D_{1}\\Phi \\left(\\omega ,r_{0}\\right)1=\\dot{r}_{0}\\notin Y$ , and thus $e^{}\\left(D_{1}\\Phi \\left(\\omega ,r_{0}\\right)1\\right)\\ne 0$ , the implicit function theorem can be applied to the map $ $ $\\left(t,\\varphi \\right)\\mapsto e^{}\\left(\\Phi \\left(t,\\varphi \\right)-r_{0}\\right)$ in a neighborhood of $\\left(\\omega ,r_{0}\\right)$ .", "It yields a convex bounded open neighborhood $N$ of $r_{0}$ in $C$ , $\\varepsilon \\in \\left(0,\\omega \\right)$ and a $C^{1}$ -map $\\gamma :N\\rightarrow \\left(\\omega -\\varepsilon ,\\omega +\\varepsilon \\right)$ with $\\gamma \\left(r_{0}\\right)=\\omega $ so that for each $\\left(t,\\varphi \\right)\\in \\left(\\omega -\\varepsilon ,\\omega +\\varepsilon \\right)\\times N$ , the segment $x_{t}^{\\varphi }$ belongs to $r_{0}+Y$ if and only if $t=\\gamma (\\varphi )$ (see [2], Appendix I in [10], [13]).", "In addition, by continuity we may assume that $D_{1}\\Phi \\left(\\gamma (\\varphi ),\\varphi \\right)1\\notin Y$ for all $\\varphi \\in N$ .", "The Poincaré return map $P_{Y}$ is defined by $P_{Y}:N\\cap \\left(r_{0}+Y\\right)\\ni \\varphi \\mapsto \\Phi \\left(\\gamma (\\varphi ),\\varphi \\right)\\in r_{0}+Y.$ Then $P_{Y}$ is continuously differentiable with fixed point $r_{0}$ .", "It is convenient to have a formula not only for the derivative $DP_{Y}\\left(\\varphi \\right)$ of $P_{Y}$ at $\\varphi \\in N\\cap \\left(r_{0}+Y\\right)$ , but also for the derivatives of the iterates of $P_{Y}$ .", "For all $\\varphi $ in the domain of $P_{Y}^{j}$ , $j\\ge 1$ , set $\\gamma _{j}\\left(\\varphi \\right)=\\Sigma _{k=0}^{j-1}\\gamma \\left(P_{Y}^{k}\\left(\\varphi \\right)\\right).$ Then $DP_{Y}^{j}\\left(\\varphi \\right)\\eta =D_{1}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)\\gamma _{j}^{\\prime }\\left(\\varphi \\right)\\eta +D_{2}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)\\eta $ for all $\\eta \\in Y$ .", "Differentiation of the equation $e^{}\\left(\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)-r_{0}\\right)=0$ yields that $\\gamma _{j}^{\\prime }\\left(\\varphi \\right)\\eta =-\\frac{e^{}\\left(D_{2}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)\\eta \\right)}{e^{}\\left(D_{1}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)1\\right)},$ and therefore $DP_{Y}^{j}\\left(\\varphi \\right)\\eta =D_{2}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)\\eta -\\frac{e^{}\\left(D_{2}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)\\eta \\right)}{e^{}\\left(D_{1}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)1\\right)}D_{1}\\Phi \\left(\\gamma _{j}(\\varphi ),\\varphi \\right)1$ for all $\\eta \\in Y$ .", "Let $\\sigma \\left(P_{Y}\\right)$ and $\\sigma \\left(M\\right)$ denote the spectra of $DP_{Y}\\left(r_{0}\\right):Y\\rightarrow Y$ and the monodromy operator, respectively.", "We obtain the following result from Theorem XIV.4.5 in [2].", "Lemma 3.1 (i) $\\sigma \\left(P_{Y}\\right)\\setminus \\left\\lbrace 0,1\\right\\rbrace =\\sigma \\left(M\\right)\\setminus \\left\\lbrace 0,1\\right\\rbrace $ , and for every $\\lambda \\in \\sigma \\left(M\\right)\\setminus \\left\\lbrace 0,1\\right\\rbrace $ , the projection along $\\mathbb {R}\\dot{r}_{0}$ onto $Y$ defines an isomorphism from the realified generalized eigenspace of $\\lambda $ and $M$ onto the realified generalized eigenspace of $\\lambda $ and $DP_{Y}\\left(r_{0}\\right)$ .", "(ii) If the generalized eigenspace $G\\left(1,M\\right)$ associated with 1 and $M$ is one-dimensional, then $1\\notin \\sigma \\left(P_{Y}\\right)$ .", "(iii) If $\\dim G\\left(1,M\\right)>1$ , then $1\\in \\sigma \\left(P_{Y}\\right)$ , and the realified generalized eigenspaces $G_{\\mathbb {R}}\\left(1,M\\right)$ and $G_{\\mathbb {R}}\\left(1,P_{Y}\\right)$ associated with 1 and $M$ and with 1 and $DP_{Y}\\left(r_{0}\\right)$ , respectively, satisfy $G_{\\mathbb {R}}\\left(1,P_{Y}\\right)=Y\\cap G_{\\mathbb {R}}\\left(1,M\\right)\\quad \\mbox{and}\\quad G_{\\mathbb {R}}\\left(1,M\\right)=\\mathbb {R}\\dot{r}_{0}\\oplus G_{\\mathbb {R}}\\left(1,P_{Y}\\right).$ In our case, the special choice of $Y$ implies the following corollary.", "Corollary 3.2 (i) $C_{s}$ and $C_{u}$ are invariant under $DP_{Y}\\left(r_{0}\\right)$ , and the spectra $\\sigma _{s}\\left(P_{Y}\\right)$ and $\\sigma _{u}\\left(P_{Y}\\right)$ of the induced maps $C_{s}\\ni x\\mapsto DP_{Y}\\left(r_{0}\\right)x\\in C_{s}$ and $C_{u}\\ni x\\mapsto DP_{Y}\\left(r_{0}\\right)x\\in C_{u}$ are contained in $\\left\\lbrace \\mu \\in \\mathbb {C}:\\,\\left\|\\mu \\right\|<1\\right\\rbrace $ and $\\left\\lbrace \\mu \\in \\mathbb {C}:\\,\\left\|\\mu \\right\|>1\\right\\rbrace $ , respectively.", "(ii) If $M$ has an eigenfunction $v$ corresponding to a simple eigenvalue $\\lambda \\in \\sigma \\left(M\\right)\\setminus \\left\\lbrace 0,1\\right\\rbrace $ , then $v$ is an eigenfunction of $DP_{Y}\\left(r_{0}\\right)$ corresponding to the same eigenvalue.", "(iii) If $\\mathcal {O}_{r}$ is nonhyperbolic, then $\\xi $ is an eigenfunction of $DP_{Y}\\left(r_{0}\\right)$ , and it corresponds to an eigenvalue with absolute value 1.", "In particular, if $\\lambda _{1}$ is a simple Floquet multiplier, then the strictly positive eigenfunction $v_{1}$ of $M$ corresponding to $\\lambda _{1}$ is also an eigenfunction of $DP_{Y}\\left(r_{0}\\right)$ corresponding to $\\lambda _{1}$ .", "In case $\\mathcal {O}_{r}$ is hyperbolic, then according to Theorem I.3 in Appendix I of [10], there exist convex open neighborhoods $N_{s}$ , $N_{u}$ of $\\hat{0}$ in $C_{s}$ , $C_{u}$ , respectively, and a $C^{1}$ -map $w_{u}:N_{u}\\rightarrow C_{s}$ with range in $N_{s}$ so that $w_{u}\\left(\\hat{0}\\right)=\\hat{0}$ , $Dw_{u}\\left(\\hat{0}\\right)=0$ , and the submanifold $\\mathcal {W}_{loc}^{u}\\left(P_{Y},r_{0}\\right)=\\left\\lbrace r_{0}+\\chi +w_{u}\\left(\\chi \\right):\\,\\chi \\in N_{u}\\right\\rbrace $ of $r_{0}+Y$ is equal to the set $\\left\\lbrace \\varphi \\in r_{0}+N_{s}+N_{u}:\\,\\mbox{there is a trajectory }\\left(\\varphi _{n}\\right)_{-\\infty }^{0}\\mbox{ of }P_{Y}\\mbox{ with }\\varphi _{0}=\\varphi \\mbox{ such that }\\right.\\\\\\left.\\varphi _{n}\\in r_{0}+N_{s}+N_{u}\\mbox{ for all }n\\le 0\\mbox{ and }\\varphi _{n}\\rightarrow r_{0}\\mbox{ as }n\\rightarrow -\\infty \\right\\rbrace .$ $\\mathcal {W}_{loc}^{u}\\left(P_{Y},r_{0}\\right)$ is called a local unstable manifold of $P_{Y}$ at $r_{0}$ .", "The unstable set of the orbit $\\mathcal {O}_{r}$ is defined as the forward extension of $\\mathcal {W}_{loc}^{u}\\left(P_{Y},r_{0}\\right)$ in time: $\\mathcal {W}^{u}\\left(\\mathcal {O}_{r}\\right)=\\Phi \\left(\\left[0,\\infty \\right)\\times \\mathcal {W}_{loc}^{u}\\left(P_{Y},r_{0}\\right)\\right).$ If $\\mathcal {O}_{r}$ is hyperbolic, then $\\mathcal {W}^{u}\\left(\\mathcal {O}_{r}\\right)=\\left\\lbrace x_{0}:\\ x:\\mathbb {R}\\rightarrow \\mathbb {R}\\mbox{ is a solution of (\\ref {eq:eq_general}), }\\alpha \\left(x\\right)\\mbox{ exists and }\\alpha \\left(x\\right)=\\mathcal {O}_{r}\\right\\rbrace .$ If $\\mathcal {O}_{r}$ is hyperbolic, then by Theorem I.2 in [10], there are convex open neighborhoods $N_{s}$ , $N_{u}$ of $\\hat{0}$ in $C_{s}$ , $C_{u}$ , respectively, and a $C^{1}$ -map $w_{s}:N_{s}\\rightarrow C_{u}$ with range in $N_{u}$ such that $w_{s}\\left(\\hat{0}\\right)=\\hat{0}$ , $Dw_{s}\\left(\\hat{0}\\right)=0$ , and $\\mathcal {W}_{loc}^{s}\\left(P_{Y},r_{0}\\right)=\\left\\lbrace r_{0}+\\chi +w_{s}\\left(\\chi \\right):\\,\\chi \\in N_{s}\\right\\rbrace $ is equal to $\\left\\lbrace \\varphi \\in r_{0}+N_{s}+N_{u}:\\,\\mbox{there is a trajectory }\\left(\\varphi _{n}\\right)_{0}^{\\infty }\\mbox{ of }P_{Y}\\mbox{ in }\\right.\\\\\\left.r_{0}+N_{s}+N_{u}\\mbox{ with }\\varphi _{0}=\\varphi \\mbox{ and }\\varphi _{n}\\rightarrow r_{0}\\mbox{ as }n\\rightarrow \\infty \\right\\rbrace .$ $\\mathcal {W}_{loc}^{s}\\left(P_{Y},r_{0}\\right)$ is a local stable manifold of $P_{Y}$ at $r_{0}$ .", "It is a $C^{1}$ -submanifold of $r_{0}+Y$ with codimension $\\mbox{dim}C_{u}$ , and it is a $C^{1}$ -submanifold of $C$ with codimension $\\mbox{dim}C_{u}+1$ .", "In case $\\mathcal {O}_{r}$ is nonhyperbolic, we need a local center-stable manifold $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)$ of $P_{Y}$ at $r_{0}$ .", "According to Theorem II.1 in [10], there exist convex open neighborhoods $N_{sc}$ and $N_{u}$ of $\\hat{0}$ in $C_{s}\\oplus \\mathbb {R}\\xi $ and $C_{u}$ , respectively, and a $C^{1}$ -map $w_{sc}:N_{sc}\\rightarrow C_{u}$ such that $w_{sc}\\left(\\hat{0}\\right)=\\hat{0}$ , $Dw_{sc}\\left(\\hat{0}\\right)=0$ , $w_{sc}\\left(N_{sc}\\right)\\subset N_{u}$ and the local center-stable manifold $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)=\\left\\lbrace r_{0}+\\chi +w_{sc}\\left(\\chi \\right):\\,\\chi \\in N_{sc}\\right\\rbrace $ satisfies $\\bigcap _{n=0}^{\\infty }P_{Y}^{-1}\\left(r_{0}+N_{sc}+N_{u}\\right)\\subset \\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right).$ Note that $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)$ is also a $C^{1}$ -submanifold of $r_{0}+Y$ with codimension $\\mbox{dim}C_{u}$ , and it is a $C^{1}$ -submanifold of $C$ with codimension $\\mbox{dim}C_{u}+1$ .", "Proposition 3.3 One may choose the neighborhoods $N_{s}$ and $N_{sc}$ so small in the definitions of $\\mathcal {W}_{loc}^{s}\\left(P_{Y},r_{0}\\right)$ , $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)$ , respectively, such that for all $\\varphi $ in $\\mathcal {W}_{loc}^{s}\\left(P_{Y},r_{0}\\right)\\cap \\mathcal {A}$ and in $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)\\cap \\mathcal {A}$ , $\\dot{\\varphi }\\notin Y$ and $V\\left(\\dot{\\varphi }\\right)\\ge 2$ .", "Analogously, one may suppose that $\\dot{\\varphi }\\notin Y$ for all $\\varphi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},r_{0}\\right)\\cap \\mathcal {A}$ .", "Recall that the $C$ -norm and the $C^{1}$ -norm are equivalent on the global attractor $\\mathcal {A}$ .", "Hence for all $\\varphi \\in \\mathcal {A}$ with small $\\left\\Vert \\varphi -r_{0}\\right\\Vert $ , $\\dot{\\varphi }\\notin Y$ follows from $\\dot{r}_{0}\\notin Y$ , furthermore $V\\left(\\dot{\\varphi }\\right)\\ge 2$ follows from $V\\left(\\dot{r}_{0}\\right)=2$ and the lower semicontinuity of $V$ .", "The next result is an immediate consequence of Proposition I.7 in [10] combined with characterizations of the local stable and center-stable manifolds given by Theorems I.2 and II.1 in [10].", "Proposition 3.4 Let $\\mathcal {W}$ denote a local stable manifold $\\mathcal {W}_{loc}^{s}\\left(P_{Y},r_{0}\\right)$ if $\\mathcal {O}_{r}$ is hyperbolic, and let $\\mathcal {W}$ be a local center-stable manifold $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)$ otherwise.", "Let $\\varphi \\in C$ be given such that $\\Phi \\left(t,\\varphi \\right)\\rightarrow \\mathcal {O}_{r}$ as $t\\rightarrow \\infty $ .", "Then there exist $T\\ge 0$ and a trajectory $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ of $P_{Y}$ in $\\mathcal {W}$ such that $\\varphi ^{0}=\\Phi \\left(T,\\varphi \\right)$ and $\\varphi ^{n}\\rightarrow r_{0}$ as $n\\rightarrow \\infty $ ." ], [ "Examples", "Consider the case when $r$ is the LSOP solution $p$ given by Theorem A. Theorem A states that $\\mathcal {O}_{p}$ is hyperbolic, and has two real and simple Floquet multipliers outside the unit circle.", "Hence $C_{c}=\\mathbb {R}\\dot{p}_{0}$ and $C_{u}=\\left\\lbrace c_{1}v_{1}+c_{2}v_{2}:\\, c_{1},c_{2}\\in \\mathbb {R}\\right\\rbrace ,$ where $v_{1}$ is a positive eigenfunction corresponding to $M$ and the leading real eigenvalue $\\lambda _{1}>1$ , and $v_{2}$ is an eigenfunction corresponding to $M$ and the eigenvalue $\\lambda _{2}$ with $1<\\lambda _{2}<\\lambda _{1}$ .", "For the solution $u^{v_{2}}:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ of the linear variational equation (REF ) with initial segment $v_{2}$ , $V\\left(u_{t}^{v_{2}}\\right)=2$ for all $t\\ge 0$ .", "For both $i\\in \\left\\lbrace 1,2\\right\\rbrace $ , $\\lambda _{i}$ is an eigenvalue of $DP_{Y}\\left(p_{0}\\right)$ with the eigenvector $v_{i}$ .", "The local unstable manifold $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ of the Poincaré map $P_{Y}$ at $p_{0}$ is a two-dimensional $C^{1}$ -submanifold of $p_{0}+Y$ .", "We will use the subsequent technical result.", "Proposition 3.5 One may choose $N_{u}$ so small that the tangent space $T_{\\varphi }\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ has a strictly positive element for all $\\varphi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ .", "By decreasing $N_{u}$ if necessary, we can achieve that $v_{1}+Dw_{u}\\left(\\chi \\right)v_{1}\\gg \\hat{0}$ for all $\\chi \\in N_{u}$ , where $v_{1}$ is a fixed positive eigenfunction corresponding to the leading eigenvalue $\\lambda _{1}$ of $DP_{Y}\\left(p_{0}\\right)$ .", "Let $\\varphi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ be arbitrary and choose $\\chi ^{\\varphi }\\in N_{u}$ with $\\varphi =p_{0}+\\chi ^{\\varphi }+w_{u}\\left(\\chi ^{\\varphi }\\right)$ .", "Then for all $t$ in an open interval $I\\subset \\mathbb {R}$ containing 0, $\\gamma \\left(t\\right)=p_{0}+\\chi ^{\\varphi }+tv_{1}+w_{u}\\left(\\chi ^{\\varphi }+tv_{1}\\right)$ is defined.", "Moreover, $\\gamma :I\\rightarrow \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ is a $C^{1}$ -curve with $\\gamma \\left(0\\right)=\\varphi $ and $T_{\\varphi }\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\ni \\gamma ^{\\prime }\\left(0\\right)=v_{1}+Dw_{u}\\left(\\chi ^{\\varphi }\\right)v_{1}\\gg \\hat{0}.$ We plan to consider other periodic orbits oscillating slowly around an equilibrium, but keep the same notations for simplicity ($\\omega $ for the minimal period, $P_{Y}$ for the Poincaré map, $\\lambda _{i}$ , $i\\ge 1$ , for the Floquet multipliers, $v_{i}$ , $i\\ge 1$ , for eigenvectors, and so on).", "It will be clear from the context which periodic orbit we refer to.", "Theorem A gives a second LSOP solution $q:\\mathbb {R}\\rightarrow \\mathbb {R}$ .", "$\\mathcal {O}_{q}$ is hyperbolic, and it has exactly one simple Floquet multiplier outside the unit circle, which is real and greater than 1.", "This leading eigenvalue will be also denoted by $\\lambda _{1}$ , but it differs from the leading Floquet multiplier of $\\mathcal {O}_{p}$ .", "To $\\lambda _{1}$ there corresponds a positive eigenfunction $v_{1}$ (different from the previous $v_{1}$ ).", "Hence for $r=q$ , $C_{c}=\\mathbb {R}\\dot{q}_{0}$ and $C_{u}=\\mathbb {R}v_{1}$ .", "The local stable manifold $\\mathcal {W}_{loc}^{s}\\left(P_{Y},q_{0}\\right)$ of $P_{Y}$ at $q_{0}$ is a $C^{1}$ -submanifold of $q_{0}+Y$ with codimension 1, and a $C^{1}$ -submanifold of $C$ with codimension 2.", "We have the tangent space $T_{q_{0}}\\mathcal {W}_{loc}^{s}\\left(P_{Y},q_{0}\\right)=C_{s}$ at $q_{0}$ in $q_{0}+Y$ .", "Recall that there exist periodic solutions $x^{1}:\\mathbb {R}\\rightarrow \\mathbb {R}$ and $x^{-1}:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) oscillating slowly around $\\xi _{1}$ and $\\xi _{-1}$ with ranges in $\\left(0,\\xi _{2}\\right)$ and $\\left(\\xi _{-2},0\\right)$ , respectively, so that the ranges $x^{1}(\\mathbb {R})$ and $x^{-1}(\\mathbb {R})$ are maximal in the sense that $x^{1}(\\mathbb {R})\\supset x(\\mathbb {R})$ for all periodic solutions $x$ oscillating slowly around $\\xi _{1}$ with ranges in $(0,\\xi _{2})$ ; and analogously for $x^{-1}$ .", "We do not know whether the corresponding periodic orbits, $\\mathcal {O}_{1}$ and $\\mathcal {O}_{-1}$ , are hyperbolic or not.", "Proposition 3.6 For both periodic orbits $\\mathcal {O}_{1}$ and $\\mathcal {O}_{-1}$ , $\\dim C_{u}=1$ .", "We give a proof for $\\mathcal {O}_{1}$ .", "As $\\mathcal {O}_{1}$ has a Floquet multiplier $\\lambda _{1}>1$ , it is clear that $\\dim C_{u}\\ge 1.$ Let $\\mathcal {W}$ denote the local stable manifold $\\mathcal {W}_{loc}^{s}\\left(P_{Y},x_{0}^{1}\\right)$ if $\\mathcal {O}_{1}$ is hyperbolic, and let $\\mathcal {W}$ be the local center-stable manifold $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},x_{0}^{1}\\right)$ otherwise.", "Then $\\mathcal {W}$ is a $C^{1}$ -submanifold of $x_{0}^{1}+Y$ with $T_{x_{0}^{1}}\\mathcal {W}=C_{s}$ if $\\mathcal {O}_{1}$ is hyperbolic, and with $T_{x_{0}^{1}}\\mathcal {W}=C_{s}\\oplus \\mathbb {R}\\xi $ if $\\mathcal {O}_{1}$ is nonhyperbolic.", "By Theorem B, there exists $\\eta \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ so that $x_{t}^{\\eta }\\rightarrow \\mathcal {O}_{1}$ as $t\\rightarrow \\infty $ .", "Then Proposition REF guarantees the existence of a sequence $\\left(t_{n}\\right)_{n=0}^{\\infty }$ in $\\mathbb {R}$ with $t_{n}\\rightarrow \\infty $ as $n\\rightarrow \\infty $ such that $x_{t_{n}}^{\\eta }\\in \\mathcal {W}\\setminus \\left\\lbrace x_{0}^{1}\\right\\rbrace $ for all $n\\ge 0$ and $ $$x_{t_{n}}^{\\eta }\\rightarrow x_{0}^{1}$ as $n\\rightarrow \\infty $ .", "We introduce the notation $y^{n}:\\mathbb {R}\\rightarrow \\mathbb {R}$ , $n\\ge 0$ , for the function obtained from $x^{\\eta }$ by time shift so that $y_{0}^{n}=x_{t_{n}}^{\\eta }$ .", "Then $y^{n}\\left(t\\right)\\rightarrow x^{1}\\left(t\\right)$ as $n\\rightarrow \\infty $ for all $t\\in \\mathbb {R}$ by the continuity of the flow $\\Phi _{\\mathcal {A}}$ .", "Since $x^{\\eta }$ is a bounded solution of Eq.", "(REF ), the solutions $y^{n}$ are uniformly bounded on $\\mathbb {R}$ , and Eq.", "(REF ) gives a uniform bound for their derivatives.", "By applying the Arzelà–Ascoli theorem successively on the intervals $\\left[-j,j\\right]$ , $j\\ge 1$ , we obtain strictly increasing maps $\\chi _{j}:\\mathbb {N}\\rightarrow \\mathbb {N}$ , $1\\le j\\in \\mathbb {N}$ , so that for every integer $j\\ge 1$ , the subsequence $\\left(y^{\\chi _{1}\\circ \\ldots \\circ \\chi _{j}\\left(k\\right)}\\right)_{k=0}^{\\infty }$ converges uniformly on $\\left[-j,j\\right]$ .", "By diagonalization, set $\\chi \\left(k\\right)=\\chi _{1}\\circ \\ldots \\circ \\chi _{k}\\left(k\\right)$ and consider the subsequence $\\left(y^{n_{k}}\\right)_{k=0}^{\\infty }=\\left(y^{\\chi \\left(k\\right)}\\right)_{k=0}^{\\infty }$ .", "Then $y^{n_{k}}\\rightarrow x^{1}$ as $k\\rightarrow \\infty $ uniformly on all compact subsets of $\\mathbb {R}$ .", "Define $z^{k}\\left(t\\right)=\\frac{y^{n_{k}}\\left(t\\right)-x^{1}\\left(t\\right)}{\\left\\Vert x_{t_{n_{k}}}^{\\eta }-x_{0}^{1}\\right\\Vert }\\quad \\mbox{for all }k\\ge 0\\mbox{ and }t\\in \\mathbb {R}.$ Then $z^{k}$ , $k\\ge 0$ , satisfies the equation $\\dot{z}^{k}\\left(t\\right)=-z^{k}\\left(t\\right)+a_{k}\\left(t\\right)z^{k}\\left(t-1\\right)$ on $\\mathbb {R}$ , where the coefficient function $a_{k}$ is defined by $a_{k}:\\mathbb {R}\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sy^{n_{k}}\\left(t-1\\right)+\\left(1-s\\right)x^{1}\\left(t-1\\right)\\right)\\mbox{d}s\\in \\mathbb {R}^{+},\\quad k\\ge 0.$ Note that there are constants $\\alpha _{1}\\ge \\alpha _{0}>0$ independent of $k$ and $t$ such that $\\alpha _{0}\\le a_{k}\\left(t\\right)\\le \\alpha _{1}$ for all $ $$k\\ge 0$ and $t\\in \\mathbb {R}$ , moreover, $a_{k}\\rightarrow a$ as $k\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , where $a:\\mathbb {R}\\ni t\\mapsto f^{\\prime }\\left(x^{1}\\left(t-1\\right)\\right)\\in \\mathbb {R}^{+}.$ In addition, observe that for all $ $$k\\ge 0$ and $t\\in \\mathbb {R}$ , $z_{t}^{k}\\ne \\hat{0}$ because $y_{0}^{n_{k}}=x_{t_{n_{k}}}^{\\eta }\\ne x_{0}^{1}$ and the flow $\\Phi _{\\mathcal {A}}$ is injective.", "Hence $V\\left(z_{t}^{k}\\right)$ is defined and equals 2 for all $ $$k\\ge 0$ and $t\\in \\mathbb {R}$ by Proposition 8.3 in [8].", "Lemma REF then implies the existence of a continuously differentiable function $z:\\mathbb {R}\\rightarrow \\mathbb {R}$ and a subsequence $\\left(z^{k_{l}}\\right)_{l=0}^{\\infty }$ of $\\left(z^{k}\\right)_{k=0}^{\\infty }$ such that $z^{k_{l}}\\rightarrow z$ and $\\dot{z}^{k_{l}}\\rightarrow \\dot{z}$ as $k\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , moreover $\\dot{z}\\left(t\\right)=-z\\left(t\\right)+a\\left(t\\right)z\\left(t-1\\right)$ for all real $t$ .", "We claim that $z_{0}\\ne \\hat{0}$ and $z_{0}\\in T_{x_{0}^{1}}\\mathcal {W}=\\left\\lbrace \\begin{array}{ll}C_{s}, & \\mbox{if }\\mathcal {O}_{r}\\mbox{ is hyperbolic,}\\\\C_{s}\\oplus \\mathbb {R}\\xi , & \\mbox{otherwise.", "}\\end{array}\\right.$ Consider the map $w=w_{s}$ if $\\mathcal {O}_{1}$ is hyperbolic, and the map $w=w_{sc}$ otherwise.", "Choose $\\chi ^{l}\\in T_{x_{0}^{1}}\\mathcal {W}$ , $l\\ge 0$ , with $\\chi ^{l}\\rightarrow \\hat{0}$ as $l\\rightarrow \\infty $ so that $x_{t_{n_{k_{l}}}}^{\\eta }=x_{0}^{1}+\\chi ^{l}+w\\left(\\chi ^{l}\\right)$ for all $l\\ge 0$ .", "Then $z_{0}=\\lim _{l\\rightarrow \\infty }z_{0}^{k_{l}}=\\lim _{l\\rightarrow \\infty }\\frac{x_{t_{n_{k_{l}}}}^{\\eta }-x_{0}^{1}}{\\left\\Vert x_{t_{n_{k_{l}}}}^{\\eta }-x_{0}^{1}\\right\\Vert }=\\lim _{l\\rightarrow \\infty }\\frac{\\chi ^{l}+w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }.$ As $z_{0}$ is the limit of unit vectors, it is clearly nontrivial.", "$Dw\\left(\\hat{0}\\right)=0$ implies that $\\lim _{l\\rightarrow \\infty }w\\left(\\chi ^{l}\\right)/\\left\\Vert \\chi ^{l}\\right\\Vert =\\hat{0}$ and thus $\\lim _{l\\rightarrow \\infty }\\frac{w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }=\\lim _{l\\rightarrow \\infty }\\frac{\\frac{w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}\\right\\Vert }}{\\left\\Vert \\frac{\\chi ^{l}}{\\left\\Vert \\chi ^{l}\\right\\Vert }+\\frac{w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}\\right\\Vert }\\right\\Vert }=\\hat{0}$ and $ $ $\\lim _{l\\rightarrow \\infty }\\frac{\\left\\Vert \\chi ^{l}\\right\\Vert }{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }=\\lim _{l\\rightarrow \\infty }\\frac{1}{\\left\\Vert \\frac{\\chi ^{l}}{\\left\\Vert \\chi ^{l}\\right\\Vert }+\\frac{w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}\\right\\Vert }\\right\\Vert }=1.$ We obtain that $\\underset{{\\normalsize z_0}}{\\underbrace{\\frac{\\chi ^{l}+w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }}_{\\downarrow }} & =\\frac{\\chi ^{l}}{\\left\\Vert \\chi ^{l}\\right\\Vert } \\underset{1}{\\underbrace{\\frac{\\left\\Vert \\chi ^{l}\\right\\Vert }{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }}_{\\downarrow }} + \\underset{0}{\\underbrace{\\frac{w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }}_{\\downarrow }}$ as $l\\rightarrow \\infty $ .", "Then the limit $\\lim _{l\\rightarrow \\infty }\\chi ^{l}/\\left\\Vert \\chi ^{l}\\right\\Vert $ necessarily exists too, and $ $ $z_{0}=\\lim _{l\\rightarrow \\infty }\\frac{\\chi ^{l}+w\\left(\\chi ^{l}\\right)}{\\left\\Vert \\chi ^{l}+w\\left(\\chi ^{l}\\right)\\right\\Vert }=\\lim _{l\\rightarrow \\infty }\\frac{\\chi ^{l}}{\\left\\Vert \\chi ^{l}\\right\\Vert }\\in T_{x_{0}^{1}}\\mathcal {W}\\subset Y.$ Since $V\\left(z_{0}^{k_{l}}\\right)=2$ for all $l\\ge 0$ , the lower-semicontinuity of $V$ proved in Lemma REF implies that $V\\left(z_{0}\\right)\\le \\liminf _{l\\rightarrow \\infty }V\\left(z_{0}^{k_{l}}\\right)=2$ .", "Recall that $\\dot{x}_{0}^{1}\\in C_{c}$ also belongs to $V^{-1}\\left(\\left\\lbrace 0,2\\right\\rbrace \\right)$ , moreover, $\\dot{x}_{0}^{1}\\notin Y$ .", "Thus $\\dot{x}_{0}^{1}$ and $z_{0}$ are linearly independent elements of $\\left(C_{s}\\oplus C_{c}\\right)\\cap V^{-1}\\left(\\left\\lbrace 0,2\\right\\rbrace \\right)$ .", "In consequence, result (REF ) gives that $C_{u}$ is at most one-dimensional.", "The proof is analogous for $\\mathcal {O}_{-1}$ .", "The previous result implies that if $\\mathcal {O}_{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , is hyperbolic, then the local stable manifold $\\mathcal {W}_{loc}^{s}\\left(P_{Y},x_{0}^{k}\\right)$ of $P_{Y}$ at $x_{0}^{k}$ is a $C^{1}$ -submanifold of $x_{0}^{k}+Y$ with codimension 1 and with tangent space $T_{x_{0}^{k}}\\mathcal {W}_{loc}^{s}\\left(P_{Y},x_{0}^{k}\\right)=C_{s}$ at $x_{0}^{k}$ .", "It is a $C^{1}$ -submanifold of $C$ with codimension 2.", "Similarly, if $\\mathcal {O}_{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , is nonhyperbolic, then the local center-stable manifold $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},x_{0}^{k}\\right)$ of $P_{Y}$ at $x_{0}^{k}$ is a $C^{1}$ -submanifold of $x_{0}^{k}+Y$ with codimension 1 and with tangent space $T_{x_{0}^{k}}\\mathcal {W}_{loc}^{sc}\\left(P_{Y},x_{0}^{k}\\right)=C_{s}\\oplus \\mathbb {R}\\xi $ at $x_{0}^{k}$ .", "It is also a $C^{1}$ -submanifold of $C$ with codimension 2.", "Remark 3.7 We see from the proof of Proposition REF that for $r=x^{k}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , $C_{r_{M}<}$ admits at least three linearly independent elements: $v_{1}\\in C_{u}$ , $\\dot{x}_{0}^{k}\\in C_{c}$ and $z_{0}\\in C_{s}\\oplus C_{c}$ .", "As $C_{r_{M}}$ is at most three-dimensional by (REF ), we conclude that $\\dim C_{r_{M}<}=3$ .", "A similar reasoning confirms the same equality for $r=q$ .", "It is obviuos that the dimension of $C_{r_{M}<}$ is maximal also in the case $r=p$ , as $\\mathcal {O}_{p}$ has two Floquet-multipliers outside the unit circle.", "These observations are in accordance with the recent result [15] of Mallet-Paret and Nussbaum stating that $\\dim C_{r_{M}<}=3$ in more general situations." ], [ "The Proof of Theorem ", "Note that each $\\varphi $ in the unstable set $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ arises in the form $\\varphi =\\Phi \\left(t,\\psi \\right)$ , where $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ and $t>1$ .", "Indeed, $\\mathcal {W}^{u}\\left(\\mathcal {O}_{r}\\right)=\\Phi \\left(\\left[0,\\infty \\right)\\times \\mathcal {W}_{loc}^{u}\\left(P_{Y},r_{0}\\right)\\right),\\qquad \\mathrm {(3.5)}$ and from each $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ we can start a backward trajectory $\\left(\\psi ^{n}\\right)_{-\\infty }^{0}$ of $P_{Y}$ in $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ converging to $p_{0}$ as $n\\rightarrow -\\infty $ .", "As the first part of the proof of Theorem REF , we are going to show in Proposition REF that for all $t>1$ and $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ , $\\varphi =\\Phi \\left(t,\\psi \\right)$ belongs to a subset $W_{t,\\psi ,\\varepsilon }$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ that is a three-dimensional submanifold of $C$ .", "This implies that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is an immersed submanifold of $C$ .", "The proof of Proposition REF is based on (REF ), the differentiabilty of $\\Phi \|_{\\left(1,\\infty \\right)\\times C}$ and the injectivity of $D_{2}\\Phi \\left(t,\\varphi \\right)$ for $t\\ge 0$ .", "However, it does not follow immediately that $ $$\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is an embedded $C^{1}$ -submani$\\-$ fold of $C$ .", "We also need to show for any $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ the existence of a ball $B$ in $C$ centered at $\\varphi $ such that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap B=W_{t,\\psi ,\\varepsilon }\\cap B.$ To do this, we will give a sequence of further auxiliary results right after Proposition REF .", "We will introduce a projection $\\pi _{3}$ from $C$ into $\\mathbb {R}^{3}$ , and use the special properties of the Lyapunov fuctional $V$ to show that $\\pi _{3}$ is injective on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and on the tangent spaces of $W_{t,\\psi ,\\varepsilon }$ .", "These results will easily imply (REF ).", "Afterwards we offer a smooth global graph representation for $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ in order to indicate the simplicity of its structure.", "The smoothness of the sets $C_{-2}^{p},\\, C_{0}^{p}$ and $C_{2}^{p}$ then follows at once because they are open subsets of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "At last we show that the semiflow induced by the solution operator $\\Phi $ extends to a $C^{1}$ -flow on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "This property will be applied later in the proof of Theorem REF .", "Proposition 4.1 To each $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ and $t>1$ , there corresponds an $\\varepsilon =\\varepsilon \\left(\\psi ,t\\right)\\in \\left(0,t-1\\right)$ so that the subset $W_{t,\\psi ,\\varepsilon }=\\Phi \\left(\\left(t-\\varepsilon ,t+\\varepsilon \\right)\\times \\left(\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\cap B\\left(\\psi ,\\varepsilon \\right)\\right)\\right)$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is$ $ a three-dimensional $C^{1}$ -submanifold of $C$ .", "It is clear from (REF ) that $W_{t,\\psi ,\\varepsilon }$ defined as above is a subset of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ for all $\\varepsilon \\in \\left(0,t-1\\right).$ Consider the three-dimensional $C^{1}$ -submanifold $\\left(1,\\infty \\right)\\times \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ of $\\mathbb {R}\\times C$ and the continuously differentiable map $\\Sigma :\\left(1,\\infty \\right)\\times \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\ni \\left(s,\\varphi \\right)\\mapsto \\Phi \\left(s,\\varphi \\right)\\in C.$ It suffices to show by Proposition REF that for all $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ and $t>1$ , the derivative $D\\Sigma \\left(t,\\psi \\right)$ is injective on the tangent space $T_{\\left(t,\\psi \\right)}\\left(\\left(1,\\infty \\right)\\times \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\right)=\\mathbb {R}\\times T_{\\psi }\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ .", "This space is spanned by the tangent vectors of the following curves at 0: $\\left(-1,1\\right)\\ni s\\mapsto \\left(t+s,\\psi \\right)\\quad \\mbox{and}\\quad \\left(-1,1\\right)\\ni s\\mapsto \\left(t,\\gamma _{i}\\left(s\\right)\\right),\\, i\\in \\left\\lbrace 1,2\\right\\rbrace ,$ where $\\gamma _{i}:\\left(-1,1\\right)\\rightarrow \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\mbox{ is a }C^{1}\\mbox{-curve,}$ $\\gamma _{i}\\left(0\\right)=\\psi \\mbox{ and }\\mbox{ }D\\gamma _{i}\\left(0\\right)=\\eta _{i}\\mbox{ for both }i\\in \\left\\lbrace 1,2\\right\\rbrace ,$ with $\\eta _{1}$ and $\\eta _{2}$ forming a basis of the two-dimensional tangent space $T_{\\psi }\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ .", "As $\\eta _{1}\\in Y$ , $\\eta _{2}\\in Y$ and $\\dot{\\psi }\\notin Y$ by Proposition REF , the vectors $\\eta _{1}$ , $\\eta _{2}$ and $\\dot{\\psi }$ are linearly independent.", "Clearly, $\\frac{\\mbox{d}}{\\mbox{d}s}\\Sigma \\left(t+s,\\psi \\right)\|_{s=0}=\\frac{\\mbox{d}}{\\mbox{d}s}\\Phi \\left(t+s,\\psi \\right)\|_{s=0}=D_{1}\\Phi \\left(t,\\psi \\right)1=\\dot{x}_{t}^{\\psi }=D_{2}\\Phi \\left(t,\\psi \\right)\\dot{\\psi }$ and $\\frac{\\mbox{d}}{\\mbox{d}s}\\Sigma \\left(t,\\gamma _{i}\\left(s\\right)\\right)\|_{s=0}=\\frac{\\mbox{d}}{\\mbox{d}s}\\Phi \\left(t,\\gamma _{i}\\left(s\\right)\\right)\|_{s=0}=D_{2}\\Phi \\left(t,\\psi \\right)\\eta _{i},\\qquad i\\in \\left\\lbrace 1,2\\right\\rbrace .$ As $D_{2}\\left(t,\\psi \\right):C\\rightarrow C$ is injective (see Section 2) and $\\eta _{1}$ , $\\eta _{2}$ and $\\dot{\\psi }$ are linearly independent, we deduce that the range $D\\Sigma \\left(t,\\psi \\right)\\left(\\mathbb {R}\\times T_{\\psi }\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\right)$ is three-dimensional, and thus $D\\Sigma \\left(t,\\psi \\right)$ is injective.", "Next we characterize $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and its tangent vectors in terms of oscillation frequencies.", "Proposition 4.2 For all $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and $\\psi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ with $\\varphi \\ne \\psi $ , $V\\left(\\psi -\\varphi \\right)\\le 2$ .", "We distinguish three cases: (i) both $\\varphi \\in \\mathcal {O}_{p}$ and $\\psi \\in \\mathcal {O}_{p}$ ; (ii) $\\varphi \\in \\mathcal {O}_{p}$ and $\\psi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ (or vice verse); (iii) both $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ and $\\psi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ .", "Let $\\omega >1$ denote the minimal period of $p$ .", "It is easy to deduce from Proposition REF that $V\\left(p_{\\tau }-p_{\\sigma }\\right)=2\\mbox{ for all }\\tau \\in \\left[0,\\omega \\right)\\mbox{ and }\\sigma \\in \\left[0,\\omega \\right)\\mbox{ with }\\tau \\ne \\sigma .$ Hence the statement holds in case (i).", "Case (ii).", "By definition, there exist $\\sigma \\in \\left[0,\\omega \\right)$ and $\\left(t_{n}\\right)_{0}^{\\infty }\\subset \\mathbb {R}$ so that $t_{n}\\rightarrow -\\infty $ and $x_{t_{n}}^{\\psi }\\rightarrow p_{\\sigma }$ as $n\\rightarrow \\infty $ .", "As $x_{t_{n}}^{\\varphi }\\in \\mathcal {O}_{p}$ for all $n\\ge 0,$ we may also assume by compactness that $x_{t_{n}}^{\\varphi }\\rightarrow p_{\\tau }$ as $n\\rightarrow \\infty $ for some $\\tau \\in \\left[0,\\omega \\right)$ .", "As the $C$ -norm and $C^{1}$ -norm are equivalent on the global attractor, $x_{t_{n}}^{\\psi }\\rightarrow p_{\\sigma }$ and $x_{t_{n}}^{\\varphi }\\rightarrow p_{\\tau }$ as $n\\rightarrow \\infty $ also in $C^{1}$ -norm.", "By Lemma REF (iii) and property (REF ), $p_{\\sigma }-p_{\\tau }\\in R$ for all $\\tau \\in \\left[0,\\omega \\right)$ and $\\sigma \\in \\left[0,\\omega \\right)$ with $\\tau \\ne \\sigma $ .", "Hence if $\\sigma \\ne \\tau $ , then Lemma REF implies that $2=V\\left(p_{\\sigma }-p_{\\tau }\\right)=\\lim _{n\\rightarrow \\infty }V\\left(x_{t_{n}}^{\\psi }-x_{t_{n}}^{\\varphi }\\right).$ By the monotonicity of $V$ we conclude that $ $ $V\\left(x_{t}^{\\psi }-x_{t}^{\\varphi }\\right)\\le 2$ for all real $t$ .", "If $\\sigma =\\tau $ , then for all $\\varepsilon >0$ small, $\\sigma +\\varepsilon \\ne \\tau $ and $x_{t_{n}+\\varepsilon }^{\\psi }\\rightarrow p_{\\sigma +\\varepsilon }$ as $n\\rightarrow \\infty $ both in $C$ -norm and $C^{1}$ -norm.", "Therefore by Lemma REF and by our previous reasoning, $V\\left(x_{t}^{\\psi }-x_{t}^{\\varphi }\\right)\\le \\liminf _{\\varepsilon \\rightarrow 0+}V\\left(x_{t+\\varepsilon }^{\\psi }-x_{t}^{\\varphi }\\right)\\le 2$ for all $t\\in \\mathbb {R}$ .", "In particular, $V\\left(\\psi -\\varphi \\right)\\le 2$ .", "We omit the proof of case (iii), as it is analogous to the one given for (ii).", "As it is stated in the next proposition, the tangent vectors of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ have at most two sign changes.", "This result is a direct consequence of Proposition REF .", "Proposition 4.3 Assume $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , $\\gamma :\\left(-1,1\\right)\\rightarrow C$ is a $C^{1}$ -curve with $\\gamma \\left(0\\right)=\\varphi $ , and $\\left(s_{n}\\right)_{0}^{\\infty }$ is a sequence in $\\left(-1,1\\right)\\backslash \\left\\lbrace 0\\right\\rbrace $ so that $s_{n}\\rightarrow 0$ as $n\\rightarrow \\infty $ and $\\gamma \\left(s_{n}\\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ for all $n\\ge 0$ .", "Also assume that $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ .", "Then $V\\left(\\gamma ^{\\prime }\\left(0\\right)\\right)\\le 2$ .", "By Proposition REF , $V\\left(\\frac{\\gamma \\left(s_{n}\\right)-\\gamma \\left(0\\right)}{s_{n}}\\right)\\le 2\\quad \\mbox{for all }n\\ge 0.$ Since $\\left(\\gamma \\left(s_{n}\\right)-\\gamma \\left(0\\right)\\right)/s_{n}\\rightarrow \\gamma ^{\\prime }\\left(0\\right)$ in $C$ as $n\\rightarrow \\infty $ , the statement follows from the lower semi-continuity property of $V$ presented by Lemma REF .", "In order to get more information on the unstable set $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , we project it into the three-dimensional Euclidean space.", "Introduce the linear map $\\pi _{3}:C\\ni \\varphi \\mapsto \\left(\\varphi \\left(0\\right),\\varphi \\left(-1\\right),\\mathcal {I}\\left(\\varphi \\right)\\right)\\in \\mathbb {R}^{3},$ where $\\mathcal {I}\\left(\\varphi \\right)=_{-1}^{0}\\varphi \\left(s\\right)\\mbox{d}s$ .", "The next statement can be obtained also from Proposition REF .", "Proposition 4.4 $\\pi _{3}$ is injective on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Suppose that there exist $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and $\\psi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ so that $\\varphi \\ne \\psi $ and $\\pi _{3}\\varphi =\\pi _{3}\\psi $ .", "Consider the solutions $x^{\\varphi }:\\mathbb {R}\\rightarrow \\mathbb {R}$ and $x^{\\psi }:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ).", "The segments $x_{t}^{\\varphi }$ and $x_{t}^{\\psi }$ belong to $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , and the injectivity of the semiflow $\\Phi $ implies that $x_{t}^{\\varphi }\\ne x_{t}^{\\psi }$ for all $t\\in \\mathbb {R}$ .", "Hence $V\\left(x_{t}^{\\varphi }-x_{t}^{\\psi }\\right)\\le 2$ for all $t\\in \\mathbb {R}$ by Proposition REF .", "Since $\\varphi \\left(0\\right)-\\psi \\left(0\\right)=\\varphi \\left(-1\\right)-\\psi \\left(-1\\right)=0$ , Lemma REF (ii) gives that $V\\left(\\varphi -\\psi \\right)<V\\left(x_{-2}^{\\varphi }-x_{-2}^{\\psi }\\right)\\le 2,$ that is $V\\left(\\varphi -\\psi \\right)=0$ and $\\varphi \\le \\psi $ or $\\psi \\le \\varphi $ .", "Using $\\mathcal {I}\\left(\\varphi \\right)=\\mathcal {I}\\left(\\psi \\right)$ we conclude that $\\varphi =\\psi $ , which contradicts our initial assumption.", "We also need to know how $\\pi _{3}$ acts on the tangent vectors of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).$ Proposition 4.5 If $\\gamma :\\left(-1,1\\right)\\rightarrow C$ is a $C^{1}$ -curve with range in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ , then $\\pi _{3}\\gamma ^{\\prime }\\left(0\\right)\\ne \\left(0,0,0\\right)$ .", "Let $\\gamma :\\left(-1,1\\right)\\rightarrow C$ be a $C^{1}$ -curve with range in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and with $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ .", "Let $x:\\mathbb {R}\\rightarrow \\mathbb {R}$ be the unique solution of Eq.", "(REF ) with $x_{0}=\\gamma \\left(0\\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , and set $a:\\mathbb {R}\\ni t\\mapsto f^{\\prime }\\left(x\\left(t-1\\right)\\right)\\in \\mathbb {R}^{+}.$ 1.", "We claim that the problem ${\\left\\lbrace \\begin{array}{ll}\\dot{y}\\left(t\\right)=-y\\left(t\\right)+a\\left(t\\right)y\\left(t-1\\right), & t\\in \\mathbb {R},\\\\y_{0}=\\gamma ^{\\prime }\\left(0\\right)\\end{array}\\right.", "}$ has a unique solution $y:\\mathbb {R}\\rightarrow \\mathbb {R}$ .", "Fix a sequence $\\left(s_{n}\\right)_{n=0}^{\\infty }$ in $\\left(-1,1\\right)\\setminus \\left\\lbrace 0\\right\\rbrace $ with $s_{n}\\rightarrow 0$ as $n\\rightarrow \\infty $ .", "As $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ , we may assume that $\\gamma \\left(s_{n}\\right)\\ne \\gamma \\left(0\\right)$ for all $n\\ge 0$ .", "Consider the solutions $x^{n}=x^{\\gamma \\left(s_{n}\\right)}:\\mathbb {R}\\rightarrow \\mathbb {R}$ .", "Then $x_{t}^{n}\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ for all $n\\ge 0$ and $t\\in \\mathbb {R}$ , furthermore $x^{n}\\left(t\\right)\\rightarrow x\\left(t\\right)$ as $n\\rightarrow \\infty $ for all $t\\in \\mathbb {R}$ by the continuity of the flow $\\Phi _{\\mathcal {A}}$ .", "Since all their segments belong to the bounded global attractor, the solutions $x^{n}$ are uniformly bounded on $\\mathbb {R}$ , and Eq.", "(REF ) gives a uniform bound for their derivatives.", "Therefore by applying the Arzelà–Ascoli theorem successively on the intervals $\\left[-j,j\\right]$ , $j\\ge 1$ , and by using a diagonalization process, we obtain that $\\left(x^{n}\\right)_{n=0}^{\\infty }$ has a subsequence $\\left(x^{n_{k}}\\right)_{k=0}^{\\infty }$ such that the convergence $x^{n_{k}}\\rightarrow x$ is uniform on all compact subsets of $\\mathbb {R}$ .", "Set $y^{k}\\left(t\\right)=\\frac{x^{n_{k}}\\left(t\\right)-x\\left(t\\right)}{s_{n_{k}}}\\quad \\mbox{for all }k\\ge 0\\mbox{ and }t\\in \\mathbb {R}.$ Then for all $ $$k\\ge 0$ and $t\\in \\mathbb {R}$ , $y_{t}^{k}\\ne \\hat{0}$ by the injectivity of the flow $\\Phi _{\\mathcal {A}}$ , and $V\\left(y_{t}^{k}\\right)\\le 2$ by Proposition REF .", "In addition, $y^{k}$ , $k\\ge 0$ , satisfies the equation $\\dot{y}^{k}\\left(t\\right)=-y^{k}\\left(t\\right)+a_{k}\\left(t\\right)y^{k}\\left(t-1\\right)$ on $\\mathbb {R}$ , where $a_{k}:\\mathbb {R}\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sx^{n_{k}}\\left(t-1\\right)+\\left(1-s\\right)x\\left(t-1\\right)\\right)\\mbox{d}s\\in \\mathbb {R}^{+},\\quad k\\ge 0.$ It is clear that there are constants $\\alpha _{1}\\ge \\alpha _{0}>0$ independent of $k$ and $t$ such that $\\alpha _{0}\\le a_{k}\\left(t\\right)\\le \\alpha _{1}$ for all $ $$k\\ge 0$ and $t\\in \\mathbb {R}$ .", "Also note that $a_{k}\\rightarrow a$ as $k\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ .", "Therefore by Lemma REF , there exist a continuously differentiable function $y:\\mathbb {R}\\rightarrow \\mathbb {R}$ and a subsequence $\\left(y^{k_{l}}\\right)_{l=0}^{\\infty }$ of $\\left(y^{k}\\right)_{k=0}^{\\infty }$ such that $y^{k_{l}}\\rightarrow y$ and $\\dot{y}^{k_{l}}\\rightarrow \\dot{y}$ as $k\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , moreover $\\dot{y}\\left(t\\right)=-y\\left(t\\right)+a\\left(t\\right)y\\left(t-1\\right)$ for all real $t$ .", "It is clear from the construction that $y_{0}=\\lim _{l\\rightarrow \\infty }\\frac{x_{0}^{n_{k_{l}}}-x_{0}}{s_{n_{k_{l}}}}=\\lim _{l\\rightarrow \\infty }\\frac{\\gamma \\left(s_{n_{k_{l}}}\\right)-\\gamma \\left(0\\right)}{s_{n_{k_{l}}}}=\\gamma ^{\\prime }\\left(0\\right).$ The uniqueness of $y$ is guaranteed by Proposition REF .", "2.", "Next we claim that $\\left(-1,1\\right)\\ni s\\mapsto \\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(s\\right)\\right)$ is differentiable at $s=0$ , and $\\frac{\\mbox{d}}{\\mbox{d}s}\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(s\\right)\\right)\|_{s=0}=y_{-2}.$ If this is not true, then there exists a sequence $\\left(s_{n}\\right)_{n=0}^{\\infty }$ in $\\left(-1,1\\right)\\setminus \\left\\lbrace 0\\right\\rbrace $ with $s_{n}\\rightarrow 0$ as $n\\rightarrow \\infty $ such that for all $n\\ge 0$ , $\\frac{\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(s_{n}\\right)\\right)-\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(0\\right)\\right)}{s_{n}}$ remains outside a fixed neighborhood of $y_{-2}$ in $C$ .", "So to verify the claim, it suffices to show that any sequence $\\left(s_{n}\\right)_{n=0}^{\\infty }$ in $\\left(-1,1\\right)\\setminus \\left\\lbrace 0\\right\\rbrace $ with $s_{n}\\rightarrow 0$ as $n\\rightarrow \\infty $ admits a subsequence $\\left(s_{n_{l}}\\right)_{l=0}^{\\infty }$ for which $ $ $\\frac{\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(s_{n_{l}}\\right)\\right)-\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(0\\right)\\right)}{s_{n_{l}}}\\rightarrow y_{-2}\\quad \\mbox{as }l\\rightarrow \\infty .$ Indeed, by repeating the reasoning in the first part of the proof word by word, one can show that the sequence $\\left(x^{n}\\right)_{n=0}^{\\infty }$ formed by the solutions $x^{n}=x^{\\gamma \\left(s_{n}\\right)}:\\mathbb {R}\\rightarrow \\mathbb {R}$ , $n\\ge 0$ , has a subsequence $\\left(x^{n_{l}}\\right)_{l=0}^{\\infty }$ such that $\\left(x^{n_{l}}-x\\right)/s_{n_{l}}\\rightarrow y$ as $l\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ .", "In particular, $y_{-2}=\\lim _{l\\rightarrow \\infty }\\frac{x_{-2}^{n_{_{l}}}-x_{-2}}{s_{n_{l}}}=\\lim _{l\\rightarrow \\infty }\\frac{\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(s_{n_{l}}\\right)\\right)-\\Phi _{\\mathcal {A}}\\left(-2,\\gamma \\left(0\\right)\\right)}{s_{n_{_{l}}}}.$ 3.", "So $y_{-2}$ is a tangent vector of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ at $x_{-2}$ , and thus $V\\left(y_{-2}\\right)\\le 2$ by Proposition REF .", "4.", "To prove the assertion indirectly, suppose that $\\gamma ^{\\prime }\\left(0\\right)\\left(0\\right)=\\gamma ^{\\prime }\\left(0\\right)\\left(-1\\right)=\\mathcal {I}\\left(\\gamma ^{\\prime }\\left(0\\right)\\right)=0.$ Then as $ $$y\\left(0\\right)=\\gamma ^{\\prime }\\left(0\\right)\\left(0\\right)=0$ and $y\\left(-1\\right)=\\gamma ^{\\prime }\\left(0\\right)\\left(-1\\right)=0$ , $V\\left(\\gamma ^{\\prime }\\left(0\\right)\\right)<V\\left(y_{-2}\\right)\\le 2$ by Lemma REF (ii).", "So $V\\left(\\gamma ^{\\prime }\\left(0\\right)\\right)=0$ , that is $\\gamma ^{\\prime }\\left(0\\right)\\ge \\hat{0}$ or $\\gamma ^{\\prime }\\left(0\\right)\\le \\hat{0}$ .", "As we have also assumed that $\\mathcal {I}\\left(\\gamma ^{\\prime }\\left(0\\right)\\right)=0$ , necessarily $\\gamma ^{\\prime }\\left(0\\right)=\\hat{0}$ follows, a contradiction.", "The proof is complete.", "Now we can verify Theorem REF .", "1.The proof of the assertion that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a three-dimensional $C^{1}$ -submanifold of $C$.", "All $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ can be written in form $\\varphi =\\Phi \\left(t,\\psi \\right)$ , where $t>1$ and $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ .", "This property follows from relation (REF ) and the fact that to each $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ , there corresponds a trajectory $\\left(\\psi ^{n}\\right)_{-\\infty }^{0}$ of $P_{Y}$ in $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ with $\\psi ^{0}=\\psi $ and $\\psi ^{n}\\rightarrow p_{0}$ as $n\\rightarrow -\\infty $ .", "Hence Proposition REF guarantees the existence of $\\varepsilon >0$ so that the subset $W_{t,\\psi ,\\varepsilon }=\\Phi \\left(\\left(t-\\varepsilon ,t+\\varepsilon \\right)\\times \\left(\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)\\cap B\\left(\\psi ,\\varepsilon \\right)\\right)\\right)$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ containing $\\varphi $ is a three-dimensional $C^{1}$ -submanifold of $C$ .", "To show that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a three-dimensional $C^{1}$ -submanifold of $C$ , it suffices to exclude for all $t>1$ and $\\psi \\in \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ the existence of a sequence $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ in $ $$\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ so that $\\varphi ^{n}\\notin W_{t,\\psi ,\\varepsilon }$ for $n\\ge 0$ and $\\varphi ^{n}\\rightarrow \\varphi =\\Phi \\left(t,\\psi \\right)$ as $n\\rightarrow \\infty $ .", "According to Proposition REF , $D\\pi _{3}\\left(\\varphi \\right)=\\pi _{3}$ is injective on the three-dimensional tangent space $T_{\\varphi }W_{t,\\psi ,\\varepsilon }$ , i.e.", "it defines an isomorphism from $T_{\\varphi }W_{t,\\psi ,\\varepsilon }$ onto $\\mathbb {R}^{3}$ .", "Thus the inverse mapping theorem yields a constant $\\delta >0$ such that the restriction of $\\pi _{3}$ to $W_{t,\\psi ,\\varepsilon }\\cap B\\left(\\varphi ,\\delta \\right)$ is a diffeomorphism from $W_{t,\\psi ,\\varepsilon }\\cap B\\left(\\varphi ,\\delta \\right)$ onto an open set $U$ in $\\mathbb {R}^{3}$ .", "If a sequence $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ in $ $$\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ converges to $\\varphi $ as $n\\rightarrow \\infty $ , then $\\pi _{3}\\varphi ^{n}\\rightarrow \\pi _{3}\\varphi $ as $n\\rightarrow \\infty $ , and $\\pi _{3}\\varphi ^{n}\\in U$ for all sufficiently large $n$ .", "The injectivity of $\\pi _{3}$ on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ verified in Proposition REF then implies that $\\varphi ^{n}\\in W_{t,\\psi ,\\varepsilon }$ .", "2.", "Graph representation for $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$.", "Choose $\\varphi _{j}\\in C$ such that $\\pi _{3}\\varphi _{j}=e_{j}$ , $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , where $e_{1}=\\left(1,0,0\\right)$ , $e_{2}=\\left(0,1,0\\right)$ and $e_{3}=\\left(0,0,1\\right)$ .", "This is possible as $\\pi _{3}:C\\ni \\varphi \\mapsto \\left(\\varphi \\left(0\\right),\\varphi \\left(-1\\right),\\mathcal {I}\\left(\\varphi \\right)\\right)\\in \\mathbb {R}^{3}$ is injective on the 3-dimensional tangent spaces of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , and hence it is surjective.", "Clearly $\\varphi _{1}$ , $\\varphi _{2}$ and $\\varphi _{3}$ are linearly independent.", "Let $J_{3}:\\mathbb {R}^{3}\\rightarrow C$ be the injective linear map for which $J_{3}e_{j}=\\varphi _{j}$ , $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , and let $P_{3}=J_{3}\\circ \\pi _{3}$ .", "Then $P_{3}:C\\rightarrow C$ is continuous, linear and $P_{3}\\varphi _{j}=\\varphi _{j}$ for all $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ .", "In consequence, $P_{3}\\circ P_{3}=P_{3}$ , which means that $P_{3}$ is a projection.", "The space $G_{3}=P_{3}C=\\left\\lbrace c_{1}\\varphi _{1}+c_{2}\\varphi _{2}+c_{3}\\varphi _{3}:\\, c_{1},c_{2},c_{3}\\in \\mathbb {R}\\right\\rbrace $ is 3-dimensional, and with $E=P_{3}^{-1}\\left(0\\right)$ , we have $C=G_{3}\\oplus E$ .", "As the restriction of $P_{3}$ to $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is injective, the inverse $P_{3}^{-1}$ of the map $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\varphi \\mapsto P_{3}\\varphi \\in G_{3}$ exists.", "At last, introduce the map $w:P_{3}\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\chi \\mapsto \\left(\\mbox{id}-P_{3}\\right)\\circ P_{3}^{-1}\\left(\\chi \\right)\\in E.$ Then $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)=\\left\\lbrace \\chi +w\\left(\\chi \\right):\\,\\chi \\in P_{3}\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\right\\rbrace .$ It remains to show that $U_{3}=P_{3}\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is open in $G_{3}$ and $w$ is $C^{1}$ -smooth.", "Let $\\chi \\in P_{3}\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ be arbitrary.", "Then $\\chi =P_{3}\\varphi $ with some $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "As the restriction of $\\pi _{3}$ to $T_{\\varphi }\\mathcal {W}\\left(\\mathcal {O}_{p}\\right)$ is injective, $DP_{3}\\left(\\varphi \\right)=P_{3}$ defines an isomorphism from $T_{\\varphi }\\mathcal {W}\\left(\\mathcal {O}_{p}\\right)$ to $G_{3}$ .", "Consequently the inverse mapping theorem implies that an $\\varepsilon >0$ can be given such that $P_{3}$ maps $\\mathcal {W}\\left(\\mathcal {O}_{p}\\right)\\cap B\\left(\\varphi ,\\varepsilon \\right)$ one-to-one onto an open neighborhood $U\\subset U_{3}$ of $\\chi $ in $G_{3}$ , $P_{3}$ is invertible on $\\mathcal {W}\\left(\\mathcal {O}_{p}\\right)\\cap B\\left(\\varphi ,\\varepsilon \\right)$ , and the inverse $\\tilde{P}_{3}^{-1}$ of the map $\\mathcal {W}\\left(\\mathcal {O}_{p}\\right)\\cap B\\left(\\varphi ,\\varepsilon \\right)\\ni \\varphi \\mapsto P_{3}\\varphi \\in U$ is $C^{1}$ -smooth.", "As $w\\left(\\chi \\right)=\\left(\\mbox{id}-P_{3}\\right)\\circ P_{3}^{-1}\\left(\\chi \\right)=\\left(\\mbox{id}-P_{3}\\right)\\circ \\tilde{P}_{3}^{-1}\\left(\\chi \\right)$ for all $\\chi \\in U$ , the restriction of $w$ to $U$ is $C^{1}$ -smooth.", "3.", "The characterization of $C_{j}^{p}$ , $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ .", "Since the basin of attraction of a stable equilibrium is open in $C$ , the connecting set $C_{j}^{p}$ , $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ , is an open subset of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "It follows immediately that $C_{j}^{p}$ , $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ , is a three-dimensional $C^{1}$ -submanifold of $C$ and $C_{j}^{p}=\\left\\lbrace \\chi +w\\left(\\chi \\right):\\,\\chi \\in P_{3}C_{j}^{p}\\right\\rbrace $ for all $j\\in \\left\\lbrace -2,0,2\\right\\rbrace $ .", "As $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a $C^{1}$ -submanifold of $C$ , it makes sense to investigate the differentiability of the map $\\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}:\\mathbb {R}\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\left(t,\\varphi \\right)\\mapsto \\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).$ Suppose that $\\eta _{1}$ $\\eta _{2}$ and $\\eta _{3}$ form a basis of the three-dimensional tangent space $T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ at some $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Then for all $t\\in \\mathbb {R}$ , the tangent space $T_{\\left(t,\\varphi \\right)}\\left(\\mathbb {R}\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\right)$ of $\\mathbb {R}\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ at $(t,\\varphi )$ is spanned by the tangent vectors of the following curves at 0: $\\left(-1,1\\right)\\ni s\\mapsto \\left(t+s,\\varphi \\right)\\quad \\mbox{and}\\quad \\left(-1,1\\right)\\ni s\\mapsto \\left(t,\\gamma _{i}\\left(s\\right)\\right),\\, i\\in \\left\\lbrace 1,2,3\\right\\rbrace ,$ where $\\gamma _{i}:\\left(-1,1\\right)\\rightarrow \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a $C^{1}$ -curve with $\\gamma _{i}\\left(0\\right)=\\varphi $ and $D\\gamma _{i}\\left(0\\right)=\\eta _{i}$ for all $i\\in \\left\\lbrace 1,2,3\\right\\rbrace $ .", "We are going to apply the following assertion in the proof of Theorem REF .", "(ii).", "Proposition 4.6 The flow $\\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}$ is $C^{1}$ -smooth.", "For all $t\\in \\mathbb {R}$ and $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , $\\frac{\\mbox{d}}{\\mbox{d}s}\\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(t+s,\\varphi \\right)\|_{s=0}=\\dot{x}_{t}^{\\varphi }.$ For all $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and $\\eta \\in T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , the variational equation $\\dot{v}(t) & =-v(t)+f^{\\prime }\\left(x^{\\varphi }\\left(t-1\\right)\\right)v\\left(t-1\\right)$ has a unique solution $v^{\\eta }:\\mathbb {R}\\rightarrow \\mathbb {R}$ with $v_{0}^{\\eta }=\\eta $ .", "If $t\\in \\mathbb {R}$ and $\\gamma :\\left(-1,1\\right)\\rightarrow \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a $C^{1}$ -curve with $\\gamma \\left(0\\right)=\\varphi $ and $\\gamma ^{\\prime }\\left(0\\right)=\\eta ,$ then $\\frac{\\mbox{d}}{\\mbox{d}s}\\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(t,\\gamma \\left(s\\right)\\right)\|_{s=0}=v_{t}^{\\eta }.$ 1.", "To prove the smoothness of $\\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}$ , it is sufficient to show that for all $t\\in \\mathbb {R}$ , the map $\\left(t,\\infty \\right)\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\left(s,\\varphi \\right)\\mapsto \\Phi _{\\mathcal {A}}\\left(s,\\varphi \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is continuously differentiable.", "Let $t\\in \\mathbb {R}$ be given, and introduce the map $A_{t}:\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\varphi \\mapsto \\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).$ For $t\\ge 0$ , $A_{t}$ is clearly $C^{1}$ -smooth as $\\Phi \\left(t,\\cdot \\right)$ is $C^{1}$ -smooth and maps $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ into $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "For $t<0$ , the smoothness of $A_{t}$ follows from the smoothness of the map $\\Phi \\left(-t,\\cdot \\right)$ , the injectivity of its derivative, the inclusion $\\Phi \\left(-t,\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\right)\\subset \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and the inverse mapping theorem.", "For all $\\left(s,\\varphi \\right)\\in \\left(t,\\infty \\right)\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , $\\Phi _{\\mathcal {A}}\\left(s,\\varphi \\right)=\\Phi \\left(s+1-t,\\Phi _{\\mathcal {A}}\\left(t-1,\\varphi \\right)\\right)=\\Phi \\left(s+1-t,A_{t-1}\\left(\\varphi \\right)\\right).$ So the $C^{1}$ -smoothness of the maps $\\Phi \|_{\\left(1,\\infty \\right)\\times C}$ and $\\left(t,\\infty \\right)\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\left(s,\\varphi \\right)\\mapsto \\left(s+1-t,A_{t-1}\\left(\\varphi \\right)\\right)\\in \\left(1,\\infty \\right)\\times C$ guarantee that (REF ) is also continuously differentiable.", "2.", "Relation (REF ) is already known for $t>1$ .", "It can be easily obtained for $t\\le 1$ from the definition of the FrÃ©chet derivative.", "3.", "We already now that initial value problems corresponding to the variational equation $(2.2)$ exist and are unique in forward time, moreover relation (REF ) holds for $t\\ge 0$ .", "Fix $t<0$ .", "Note that if $\\gamma :\\left(-1,1\\right)\\rightarrow \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is a $C^{1}$ -curve with $\\gamma \\left(0\\right)=\\varphi $ and $\\gamma ^{\\prime }\\left(0\\right)=\\eta ,$ then $\\frac{\\mbox{d}}{\\mbox{d}s}\\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(t,\\gamma \\left(s\\right)\\right)\|_{s=0}=DA_{t}\\left(\\varphi \\right)\\eta .$ By part 1, the map $A_{t}$ is a $C^{1}$ -diffeomorphism with the inverse $A_{t}^{-1}=A_{-t}$ .", "Hence for all $\\eta \\in T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , $\\chi =DA_{t}\\left(\\varphi \\right)\\eta $ exists and belongs to $T_{\\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)}\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Then $\\eta =DA_{t}^{-1}\\left(\\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)\\right)\\chi =DA_{-t}\\left(\\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)\\right)\\chi =D_{2}\\Phi \\left(-t,\\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)\\right)\\chi =u_{-t}^{\\chi },$ where $u^{\\chi }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ is the solution of $\\dot{u}\\left(s\\right) & =-u\\left(s\\right)+f^{\\prime }\\left(x^{\\Phi _{\\mathcal {A}}\\left(t,\\varphi \\right)}\\left(s-1\\right)\\right)u\\left(s-1\\right)\\\\& =-u\\left(s\\right)+f^{\\prime }\\left(x^{\\varphi }\\left(t+s-1\\right)\\right)u\\left(s-1\\right)$ with $u_{0}^{\\chi }=\\chi $ .", "With transformation $v\\left(s\\right)=u\\left(s-t\\right)$ we obtain that the problem ${\\left\\lbrace \\begin{array}{ll}\\dot{v}\\left(s\\right)=-v\\left(s\\right)+f^{\\prime }\\left(x^{\\varphi }\\left(s-1\\right)\\right)v\\left(s-1\\right)\\\\v_{0}=\\eta \\end{array}\\right.", "}$ has a solution $v^{\\eta }$ on $\\left[t-1,\\infty \\right)$ satisfying $v_{t}^{\\eta }=\\chi =DA_{t}\\left(\\varphi \\right)\\eta $ .", "As this reasoning holds for any $t<0$ , (REF ) admits a solution $v^{\\eta }:\\mathbb {R}\\rightarrow \\mathbb {R}$ with $v_{t}^{\\eta }=DA_{t}\\left(\\varphi \\right)\\eta $ for any $t<0$ .", "By Proposition REF , $v^{\\eta }$ is unique.", "Relation (REF ) follows.", "The uniqueness of $v^{\\eta }$ and formula (REF ) guarantee the subsequent corollary.", "Corollary 4.7 For each fixed $t\\in \\mathbb {R}$ , the derivative of the map $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\varphi \\mapsto \\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(t,\\varphi \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ at any $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is injective on $T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ ." ], [ "The Proof of Theorem ", "Fix index $k\\in \\left\\lbrace -1,1\\right\\rbrace $ in the rest of the paper and consider the sets $C_{q}^{p}$ , $C_{k}^{p}$ and $S_{k}=C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}$ .", "5.1 Preliminary results on $\\overline{S_{k}}$ In this subsection we define a projection $\\pi _{2}$ from $C$ into $\\mathbb {R}^{2}$ and show that $\\pi _{2}$ is injective on the closure $\\overline{S_{k}}$ of $S_{k}$ in $C$ , see Proposition REF ).", "The proof of this assertion is based on the special properties of the discrete Lyapunov functional $V$ .", "The injectivity of $\\pi _{2}\|_{\\overline{S_{k}}}$ enables us to give a graph representation for $\\overline{S_{k}}$ (without smoothness properties): there is a linear isomorphism $J_{2}:\\mathbb {R}^{2}\\rightarrow C$ such that $P_{2}=J_{2}\\circ \\pi _{2}:C\\rightarrow C$ is a projection onto a two-dimensional subspace $G_{2}$ of $C$ , and a map $w_{k}:P_{2}\\overline{S_{k}}\\rightarrow P_{2}^{-1}\\left(0\\right)$ can be defined such that $\\overline{S_{k}}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}\\overline{S_{k}}\\right\\rbrace ,$ see Proposition REF .", "The differentiability of $w_{k}$ and the properties of its domain $P_{2}\\overline{S_{k}}\\subset G_{2}$ are studied only in Subsections 5.3 and 5.5.", "We also show at the end of this subsection that $\\pi _{2}\|_{\\overline{S_{k}}}$ is a homeomorphism onto its image (see Proposition REF ), moreover $\\pi _{2}$ is injective on the tangent spaces of $\\overline{S_{k}}$ (see Proposition REF ).", "Clearly, $S_{k}$ is invariant under $\\Phi _{\\mathcal {A}}$ .", "Then it easily follows that $\\overline{S_{k}}$ is invariant too.", "Indeed, let $\\varphi \\in \\overline{S_{k}}\\setminus S_{k}$ be arbitrary and choose a sequence $\\left(\\varphi _{n}\\right)_{n=0}^{\\infty }$ in $S_{k}$ converging to $\\varphi $ as $n\\rightarrow \\infty $ .", "As the global attractor $\\mathcal {A}$ is closed, $\\varphi \\in \\mathcal {A}$ .", "By the continuity of the flow $\\Phi _{\\mathcal {A}}$ on $\\mathbb {R}\\times \\mathcal {A}$ , $S_{k}\\ni x_{t}^{\\varphi _{n}}\\rightarrow x_{t}^{\\varphi }$ as $n\\rightarrow \\infty $ for all $t\\in \\mathbb {R}$ , which means that $\\overline{S_{k}}$ is invariant under $\\Phi _{\\mathcal {A}}$ .", "By Theorem B, $S_{k}=\\left\\lbrace \\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right):\\, x^{\\varphi }\\mbox{ oscillates around }\\xi _{k}\\right\\rbrace .\\qquad \\mathrm {(1.2)}$ Note that if $x^{\\varphi }$ is nonoscillatory around $\\xi _{k}$ for some $\\varphi \\in C$ (i.e.", "there exists $T\\ge 0$ so that $x_{T}^{\\varphi }\\gg \\hat{\\xi }_{k}$ or $x_{T}^{\\varphi }\\ll \\hat{\\xi }_{k}$ ), then $\\varphi $ has an open neighborhood $U_{\\varphi }$ in $C$ such that for all $\\psi \\in U_{\\varphi }$ , $x^{\\psi }$ is nonoscillatory around $\\xi _{k}$ .", "Hence it comes immediately from (REF ) that for all $\\varphi \\in \\overline{S_{k}}$ , $x^{\\varphi }$ oscillates around $\\xi _{k}$ .", "The next result states that the stable set of the unstable equilibrium $\\hat{\\xi }_{k}$ contains only nonordered elements with respect to the pointwise ordering.", "The proof follows the first part of the proof of Proposition 3.1 in [10].", "Proposition 5.1 There exist no $\\varphi \\in C$ and $\\psi \\in C$ with $\\varphi \\ll \\psi $ such that $x_{t}^{\\varphi }$ and $x_{t}^{\\psi }$ both converge to $\\hat{\\xi }_{k}$ as $t\\rightarrow \\infty $ .", "Suppose that $\\varphi \\in C$ , $\\psi \\in C$ , $\\varphi \\ll \\psi $ and both $x_{t}^{\\varphi }$ , $x_{t}^{\\psi }$ converge to $\\hat{\\xi }_{k}$ as $t\\rightarrow \\infty $ .", "Then $y:=x^{\\psi }-x^{\\varphi }$ is positive on $\\left[-1,\\infty \\right)$ by Proposition REF , it satisfies $\\dot{y}(t)=-y(t)+b\\left(t\\right)y(t-1)$ for all $t>0,$ where $b:\\left[0,\\infty \\right)\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sx^{\\psi }\\left(t-1\\right)+\\left(1-s\\right)x^{\\varphi }\\left(t-1\\right)\\right)\\mbox{d}s\\in \\left(0,\\infty \\right),$ furthermore $b\\left(t\\right)\\rightarrow f^{\\prime }\\left(\\xi _{k}\\right)$ as $t\\rightarrow \\infty $ .", "Since $f^{\\prime }\\left(\\xi _{k}\\right)>1$ by hypothesis (H1), the number $\\varepsilon =\\left(f^{\\prime }\\left(\\xi _{k}\\right)-1\\right)e^{-1}/2$ is positive.", "So there exists $T\\ge 0$ such that $b\\left(t\\right)\\ge f^{\\prime }\\left(\\xi _{k}\\right)-\\varepsilon $ for all $t\\ge T$ .", "Observe that the positivity of $y$ and $b$ implies that $\\frac{\\mbox{d}}{\\mbox{d}t}\\left(e^{t}y\\left(t\\right)\\right)=e^{t}b\\left(t\\right)y\\left(t-1\\right)>0\\quad \\mbox{for all }t>0.$ For this reason, $e^{t-1}y\\left(t-1\\right)<e^{t}y\\left(t\\right)$ for $t\\ge 1$ , and $\\dot{y}\\left(t\\right) & \\ge -y\\left(t\\right)+\\left(f^{\\prime }\\left(\\xi _{k}\\right)-\\varepsilon \\right)y\\left(t-1\\right)\\\\& \\ge -\\left(1+\\varepsilon e\\right)y\\left(t\\right)+f^{\\prime }\\left(\\xi _{k}\\right)y\\left(t-1\\right)$ for all $t\\ge T+1$ .", "The choice of $\\varepsilon $ ensures that $1+\\varepsilon e=\\frac{1}{2}+\\frac{1}{2}f^{\\prime }\\left(\\xi _{k}\\right)<f^{\\prime }\\left(\\xi _{k}\\right).$ Hence the equation $\\lambda +\\left(1+\\varepsilon e\\right)=f^{\\prime }\\left(\\xi _{k}\\right)e^{-\\lambda }$ has a positive real solution $\\lambda $ .", "Choose $\\delta >0$ so that $y(t)>\\delta e^{\\lambda t}$ on $[T,T+1]$ .", "Function $z(t)=\\delta e^{\\lambda t}$ is a solution of the equation $\\dot{z}\\left(t\\right)=-\\left(1+\\varepsilon e\\right)z\\left(t\\right)+f^{\\prime }\\left(\\xi _{k}\\right)z\\left(t-1\\right)$ on $\\mathbb {R}$ .", "Set $u=y-z.$ Then $u_{T+1}\\gg \\hat{0}$ and $\\dot{u}(t)\\ge -\\left(1+\\varepsilon e\\right)u(t)+f^{\\prime }\\left(\\xi _{k}\\right)u(t-1)\\textrm { for all }t\\ge T+1.$ If there existed $t^{}>T+1$ so that $u\\left(t^{}\\right)=0$ and $u$ is positive on $\\left[T,t^{}\\right)$ , then $\\dot{u}\\left(t^{}\\right)$ would be nonpositive.", "On the other hand, the inequality for $u$ combined with $u\\left(t^{}\\right)=0$ and $u\\left(t^{}-1\\right)>0$ would yield that $\\dot{u}\\left(t^{}\\right)>0$ .", "So $u(t)=y(t)-z(t)=y(t)-\\delta e^{\\lambda t}>0$ for all $t\\ge T$ , which contradicts the boundedness of $y$ .", "The next proposition is the analogue of Proposition 3.1 in [10].", "Proposition 5.2 (Nonordering of $\\overline{S_{k}}$ ) For all $\\varphi ,\\psi \\in C$ with $\\varphi <\\psi $$ $ , either $\\varphi \\in C\\backslash \\overline{S_{k}}$ or $ $$\\psi \\in C\\backslash \\overline{S_{k}}$ .", "If there are $\\tilde{\\varphi }\\in \\overline{S_{k}}$ and $\\tilde{\\psi }\\in \\overline{S_{k}}$ satisfying $\\tilde{\\varphi }<\\tilde{\\psi }$ , then by Proposition REF and the invariance of $\\overline{S_{k}}$ , $\\varphi =x_{2}^{\\tilde{\\varphi }}\\in \\overline{S_{k}}$ , $\\psi =x_{2}^{\\tilde{\\psi }}\\in \\overline{S_{k}}$ and $\\varphi \\ll \\psi $ .", "Theorem 4.1 in Chapter 5 of [20] proves that there is an open and dense set of initial functions in $C_{-2,2}$ so that the corresponding solutions converge to equilibria.", "Hence there exist $\\varphi ^{}\\in C$ and $\\psi ^{}\\in C$ with $\\varphi \\ll \\varphi ^{}\\ll \\psi ^{}\\ll \\psi $ such that both $x_{t}^{\\varphi ^{}}$ and $x_{t}^{\\psi ^{}}$ tend to equilibria as $t\\rightarrow \\infty $ .", "If $x_{t}^{\\psi ^{}}\\rightarrow \\hat{\\xi }$ as $t\\rightarrow \\infty $ , where $\\hat{\\xi }$ is any equilibrium with $\\xi >\\xi _{k}$ , then there exists $T>0$ such that $\\hat{\\xi }_{k}\\ll x_{T}^{\\psi ^{}}$ .", "Then $\\hat{\\xi }_{k}\\ll x_{T}^{\\psi ^{}}\\ll x_{T}^{\\psi }$ by Proposition REF , which contradicts the fact that the elements of $\\overline{S_{k}}$ oscillate around $\\xi _{k}$ .", "If $x_{t}^{\\psi ^{}}\\rightarrow \\hat{\\xi }\\ll \\hat{\\xi }_{k}$ as $t\\rightarrow \\infty $ , and there exists $T>0$ with $x_{T}^{\\psi ^{}}\\ll \\hat{\\xi }_{k}$ , then $x_{T}^{\\varphi }\\ll x_{T}^{\\psi ^{}}\\ll \\hat{\\xi }_{k}$ , which contradicts $\\varphi \\in \\overline{S}_{k}$ .", "Therefore, $\\omega \\left(\\psi ^{}\\right)=\\left\\lbrace \\hat{\\xi }_{k}\\right\\rbrace .$ Similarly, $\\omega \\left(\\varphi ^{}\\right)=\\left\\lbrace \\hat{\\xi }_{k}\\right\\rbrace $ .", "This is a contradiction to Proposition REF .", "Proposition 5.3 If $\\varphi \\in \\overline{S_{k}}$ , $\\psi \\in \\overline{S_{k}}$ and $\\varphi \\ne \\psi $ , then $V\\left(\\psi -\\varphi \\right)=2$ .", "If $\\varphi ,\\psi \\in S_{k}$ and $\\varphi \\ne \\psi $ , then $V\\left(\\psi -\\varphi \\right)\\le 2$ by Proposition REF .", "The lower-semicontinuity of $V$ (see Lemma REF ) hence implies that $V\\left(\\psi -\\varphi \\right)\\le 2$ for all $\\varphi ,\\psi \\in \\overline{S_{k}}$ satisfying $\\varphi \\ne \\psi $ .", "If $V\\left(\\psi -\\varphi \\right)=0$ , then $\\varphi <\\psi $ or $\\psi <\\varphi $ , which contradicts Proposition REF .", "The role of $\\pi _{3}$ in the proof of Theorem REF is now taken over by the linear map $\\pi _{2}:C\\ni \\varphi \\mapsto \\left(\\varphi \\left(0\\right),\\varphi \\left(-1\\right)\\right)\\in \\mathbb {R}^{2}.$ The next assertion is analogous to Proposition REF , and it will be used several times in the subsequent proofs.", "Proposition 5.4 $\\pi _{2}$ is injective on $\\overline{S_{k}}$ .", "Suppose that there exist $\\varphi \\in \\overline{S_{k}}$ and $\\psi \\in \\overline{S_{k}}$ so that $\\varphi \\ne \\psi $ and $\\pi _{2}\\varphi =\\pi _{2}\\psi $ .", "Consider the solutions $x^{\\varphi }:\\mathbb {R}\\rightarrow \\mathbb {R}$ and $x^{\\psi }:\\mathbb {R}\\rightarrow \\mathbb {R}$ .", "The invariance of $\\overline{S_{k}}$ implies that $x_{t}^{\\varphi }\\in \\overline{S_{k}}$ and $x_{t}^{\\psi }\\in \\overline{S_{k}}$ for all $t\\in \\mathbb {R}$ , and the the injectivity of the semiflow guarantees that $x_{t}^{\\varphi }\\ne x_{t}^{\\psi }$ for all $t\\in \\mathbb {R}$ .", "Hence $V\\left(x_{t}^{\\varphi }-x_{t}^{\\psi }\\right)=2$ for all real $t$ by Proposition REF .", "The initial assumption $\\varphi \\left(0\\right)-\\psi \\left(0\\right)=\\varphi \\left(-1\\right)-\\psi \\left(-1\\right)=0$ and Lemma REF (ii) however yield that $V\\left(\\varphi -\\psi \\right)<V\\left(x_{-2}^{\\varphi }-x_{-2}^{\\psi }\\right),$ which is a contradiction.", "The injectiviy of $\\pi _{2}\|_{\\overline{S_{k}}}$ is sufficient to give a graph representation for $\\overline{S_{k}}$ .", "Proposition 5.5 $\\overline{S_{k}}$ has a global graph representation: there exist a projection $P_{2}$ from $C$ onto a two-dimensional subspace $G_{2}$ of $C$ and a map $w_{k}:P_{2}\\overline{S_{k}}\\rightarrow P_{2}^{-1}\\left(0\\right)$ so that $\\overline{S_{k}}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}\\overline{S_{k}}\\right\\rbrace .$ Let $e_{1}=\\left(1,0,0\\right)$ and $e_{2}=\\left(0,1,0\\right)$ .", "Let $\\varphi _{1}$ and $\\varphi _{2}$ be the linearly independent elements of $C$ fixed in the proof of Theorem REF with the property that $\\pi _{3}\\varphi _{j}=e_{j}$ for $j\\in \\left\\lbrace 1,2\\right\\rbrace $ .", "Define $J_{2}:\\mathbb {R}^{2}\\rightarrow C$ to be the injective linear map for which $J_{2}\\left(1,0\\right)=\\varphi _{1}$ and $J_{2}\\left(0,1\\right)=\\varphi _{2}$ , and set $P_{2}=J_{2}\\circ \\pi _{2}:C\\rightarrow C$ .", "Then $P_{2}$ is continuous, linear and $P_{2}\\varphi _{j}=\\varphi _{j}$ for both $j\\in \\left\\lbrace 1,2\\right\\rbrace $ .", "Hence $P_{2}\\circ P_{2}=P_{2}$ , and $P_{2}$ is a projection.", "The 2-dimensional image space $G_{2}=P_{2}C=\\left\\lbrace c_{1}\\varphi _{1}+c_{2}\\varphi _{2}:\\, c_{1},c_{2}\\in \\mathbb {R}\\right\\rbrace $ is a subspace of $G_{3}$ and $C=G_{2}\\oplus P_{2}^{-1}\\left(0\\right)$ .", "(Note that $P_{2}$ and $G_{2}$ are both independent of $k$ .)", "As the restriction of $P_{2}$ to $\\overline{S_{k}}$ is injective by Proposition REF , the inverse $\\left(P_{2}\|_{\\overline{S_{k}}}\\right)^{-1}$ of the map $\\overline{S_{k}}\\ni \\varphi \\mapsto P_{2}\\varphi \\in G_{2}$ exists.", "With the map $w_{k}:P_{2}\\overline{S_{k}}\\ni \\chi \\mapsto \\left(\\mbox{id}-P_{2}\\right)\\circ \\left(P_{2}\|_{\\overline{S_{k}}}\\right)^{-1}\\left(\\chi \\right)\\in P_{2}^{-1}\\left(0\\right)$ we have (REF ).", "The smoothness of this representation will be verified later.", "Observe that $w_{-1}\|_{P_{2}\\left(\\overline{S_{-1}}\\cap \\overline{S_{1}}\\right)}=w_{1}\|_{P_{2}\\left(\\overline{S_{-1}}\\cap \\overline{S_{1}}\\right)}.$ Also note that now we have a global graph representation for any subset $W$ of $\\overline{S_{k}}$ : $W=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}W\\right\\rbrace .$ Let $\\pi _{2}^{-1}:\\pi _{2}\\left(\\overline{S_{k}}\\right)\\rightarrow C$ be the inverse of the injective map $\\overline{S_{k}}\\ni \\varphi \\mapsto \\pi _{2}\\varphi \\in \\mathbb {R}^{2}$ .", "Proposition 5.6 $\\pi _{2}^{-1}$ is Lipschitz-continuous.", "Suppose that $\\pi _{2}^{-1}$ is not Lipschitz-continuous, i.e., there are sequences of solutions $x^{n}:\\mathbb {R}\\rightarrow \\mathbb {R}$ and $y^{n}:\\mathbb {R}\\rightarrow \\mathbb {R}$ , $n\\in \\mathbb {N}$ , so that $x_{0}^{n}\\ne y_{0}^{n}$ for all $n\\ge 0$ , $x_{0}^{n},y_{0}^{n}\\in \\overline{S_{k}}$ for all $n\\ge 0$ , and $\\frac{\\left\|\\pi _{2}\\left(x_{0}^{n}-y_{0}^{n}\\right)\\right\|_{\\mathbb {R}^{2}}}{\\left\\Vert x_{0}^{n}-y_{0}^{n}\\right\\Vert }\\rightarrow 0\\mbox{\\quad }\\mbox{as }n\\rightarrow \\infty .$ By the compactness of $\\overline{S_{k}}$ , the solutions $x^{n}$ and $y^{n}$ are uniformly bounded, and Eq.", "(REF ) gives a uniform bound for their derivatives.", "Therefore we can use the Arzelà–Ascoli theorem successively on the intervals $\\left[-j,j\\right]$ , $j\\ge 1$ , and apply a diagonalization process to get subsequences $\\left(x^{n_{m}}\\right)_{m=0}^{\\infty }$ , $\\left(y^{n_{m}}\\right)_{m=0}^{\\infty }$ and continuous functions $x:\\mathbb {R}\\rightarrow \\mathbb {R}$ , $y:\\mathbb {R}\\rightarrow \\mathbb {R}$ so that $x^{n_{m}}\\rightarrow x$ and $y^{n_{m}}\\rightarrow y$ as $m\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ .", "Set functions $z^{m}:\\mathbb {R}\\ni t\\mapsto \\frac{x^{n_{m}}\\left(t\\right)-y^{n_{m}}\\left(t\\right)}{\\left\\Vert x_{0}^{n_{m}}-y_{0}^{n_{m}}\\right\\Vert }\\in \\mathbb {R},\\quad m\\in \\mathbb {N}.$ Then $V\\left(z_{t}^{m}\\right)=2$ for all $m\\ge 0$ and $t\\in \\mathbb {R}$ by Proposition REF , $\\left\\Vert z_{0}^{m}\\right\\Vert =1$ for all $m\\ge 0$ , and $\\left\|\\pi _{2}z_{0}^{m}\\right\|_{\\mathbb {R}^{2}}=\\frac{\\left\|\\pi _{2}\\left(x_{0}^{n_{m}}-y_{0}^{n_{m}}\\right)\\right\|_{\\mathbb {R}^{2}}}{\\left\\Vert x_{0}^{n_{m}}-y_{0}^{n_{m}}\\right\\Vert }\\rightarrow 0\\mbox{\\quad }\\mbox{as }m\\rightarrow \\infty .$ In addition, $\\dot{z}^{m}\\left(t\\right)=-z^{m}\\left(t\\right)+a_{m}\\left(t\\right)z^{m}\\left(t-1\\right)$ for all $m\\ge 0$ and $t\\in \\mathbb {R}$ , where the coefficient functions $a_{m}:\\mathbb {R}\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sx^{n_{m}}\\left(t-1\\right)+\\left(1-s\\right)y^{n_{m}}\\left(t-1\\right)\\right)\\mbox{d}s\\in \\mathbb {R}^{+},\\quad m\\ge 0,$ converge to $a:\\mathbb {R}\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sx\\left(t-1\\right)+\\left(1-s\\right)y\\left(t-1\\right)\\right)\\mbox{d}s\\in \\mathbb {R}^{+}$ uniformly on compact subsets of $\\mathbb {R}$ .", "It is also obvious that there are constants $\\alpha _{1}\\ge \\alpha _{0}>0$ such that $\\alpha _{0}\\le a_{m}\\left(t\\right)\\le \\alpha _{1}$ for all $m\\ge 0$ and $t\\in \\mathbb {R}$ .", "Therefore Lemma REF guarantees the existence of a subsequence $\\left(z^{m_{l}}\\right)_{l=0}^{\\infty }$ of $\\left(z^{m}\\right)_{m=0}^{\\infty }$ and a continuously differentiable function $z:\\mathbb {R}\\rightarrow \\mathbb {R}$ such that $z^{m_{l}}\\rightarrow z$ and $\\dot{z}^{m_{l}}\\rightarrow \\dot{z}$ as $l\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , and $z$ satisfies $\\dot{z}\\left(t\\right) & =-z\\left(t\\right)+a\\left(t\\right)z\\left(t-1\\right)\\quad \\mbox{for all }t\\in \\mathbb {R}.$ It is clear that $\\left\\Vert z_{0}\\right\\Vert =1$ , and thus $z_{0}\\ne \\hat{0}$ .", "In addition, $\\pi _{2}z_{0}=\\left(0,0\\right)$ .", "By Lemma REF , $V\\left(z_{t}\\right)\\le \\liminf _{l\\rightarrow \\infty }V\\left(z_{t}^{m_{l}}\\right)=2\\quad \\mbox{for all real }t.$ Hence Lemma REF (ii) and property $\\pi _{2}z_{0}=\\left(0,0\\right)$ together give that $V\\left(z_{0}\\right)=0$ .", "As $t\\mapsto V\\left(z_{t}\\right)$ is monotone nonincreasing, $V\\left(z_{3}\\right)=0$ .", "Lemma REF (iii) then implies that $z_{3}$ belongs to the function class $R$ , and $ $ the second statement of Lemma REF gives that $0=V\\left(z_{3}\\right)=\\lim _{l\\rightarrow \\infty }V\\left(z_{3}^{m_{l}}\\right),$ which contradicts $V\\left(z_{3}^{m_{l}}\\right)=2$ .", "We get the next result as a consequence, it is analogous to Proposition REF .", "Proposition 5.7 Suppose that $\\varphi \\in \\overline{S_{k}}$ , $\\gamma :\\left(-1,1\\right)\\rightarrow C$ is a $C^{1}$ -curve with $\\gamma \\left(0\\right)=\\varphi $ , and $\\left(s_{n}\\right)_{0}^{\\infty }$ is a sequence in $\\left(-1,1\\right)\\backslash \\left\\lbrace 0\\right\\rbrace $ so that $s_{n}\\rightarrow 0$ as $n\\rightarrow \\infty $ and $\\gamma \\left(s_{n}\\right)\\in \\overline{S_{k}}$ for all $n\\ge 0$ .", "If $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ , then $\\pi _{2}\\gamma ^{\\prime }\\left(0\\right)\\ne \\left(0,0\\right)$ .", "Let $K>0$ be a Lipschitz-constant for $\\pi _{2}^{-1}$ .", "Proposition REF guarantees that such $K$ exists.", "Then $\\left\\Vert \\frac{\\gamma \\left(s_{n}\\right)-\\gamma \\left(0\\right)}{s_{n}}\\right\\Vert \\le K\\left\|\\frac{\\pi _{2}\\gamma \\left(s_{n}\\right)-\\pi _{2}\\gamma \\left(0\\right)}{s_{n}}\\right\|_{\\mathbb {R}^{2}}$ for all $n\\ge 0$ .", "Letting $ $$n\\rightarrow \\infty $ we obtain that $\\left\\Vert \\gamma ^{\\prime }\\left(0\\right)\\right\\Vert \\le K\\left\|\\pi _{2}\\gamma ^{\\prime }\\left(0\\right)\\right\|_{\\mathbb {R}^{2}}$ .", "Therefore if $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ , then $\\pi _{2}\\gamma ^{\\prime }\\left(0\\right)\\ne \\left(0,0\\right)$ .", "5.2 The structure of $\\overline{S_{k}}$ It is obvious from the definition of $S_{k}$ that $\\left(\\mathcal {O}_{k}\\cup S_{k}\\cup \\mathcal {O}_{q}\\right)\\subset \\overline{S_{k}}$ .", "The converse inclusion is proved in this subsection based on the property that $\\pi _{2}$ maps $\\overline{S_{k}}$ injectively into $\\mathbb {R}^{2}$ .", "Then it will follow easily that $\\overline{C_{q}^{p}}=\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ and $\\overline{C_{k}^{p}}=\\mathcal {O}_{p}\\cup C_{k}^{p}\\cup \\mathcal {O}_{k}$ .", "Proposition REF implies that $\\pi _{2}$ maps periodic orbits with segments in $\\overline{S_{k}}$ into simple closed curves in $\\mathbb {R}^{2}$ , and the images of different periodic orbits are disjoint curves in $\\mathbb {R}^{2}$ .", "So $\\mathbb {R}\\ni t\\mapsto \\pi _{2}p_{t}\\in \\mathbb {R}^{2},\\ \\mathbb {R}\\ni t\\mapsto \\pi _{2}q_{t}\\in \\mathbb {R}^{2}$ and $\\mathbb {R}\\ni t\\mapsto \\pi _{2}x_{t}^{k}\\in \\mathbb {R}^{2}$ are pairwise disjoint simple closed curves.", "From $p\\left(\\mathbb {R}\\right)\\subsetneq q\\left(\\mathbb {R}\\right)\\subset \\left(\\xi _{-2},\\xi _{2}\\right)$ it follows that $\\pi _{2}\\mathcal {O}_{q}\\subset \\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)$ and $\\pi _{2}\\hat{\\xi }_{-2}$ , $\\pi _{2}\\hat{\\xi }_{2}$ belong to $\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{q}\\right)$ .", "It is also obvious that $\\pi _{2}\\hat{0}\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)$ and $\\pi _{2}\\mathcal {O}_{k}\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)$ .", "For the image of the unstable equilibrium $\\hat{\\xi }_{k}$ , we have $\\pi _{2}\\hat{\\xi }_{k}\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{k}\\right)$ .", "See Fig. 6.", "Let $A_{k}^{p}=\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{p}\\right),\\quad A_{q}^{p}=\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)\\cap \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{q}\\right)$ and $A_{k,q}=\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{q}\\right),$ see Fig. 6.", "Then by the Schönflies theorem [19], $A_{k}^{p}$ , $A_{q}^{p}$ and $A_{k,q}$ are homeomorphic to the open annulus $A^{\\left(1,2\\right)}=\\left\\lbrace u\\in \\mathbb {R}^{2}:\\,1<\\left\|u\\right\|<2\\right\\rbrace $ .", "For the closures $\\overline{A_{k}^{p}},$ $\\overline{A_{q}^{p}}$ and $\\overline{A_{k,q}}$ of $A_{k}^{p}$ , $A_{q}^{p}$ and $A_{k,q}$ in $\\mathbb {R}^{2}$ , respectively, we have $ $ $\\overline{A_{k}^{p}}=A_{k}^{p}\\cup \\pi _{2}\\mathcal {O}_{k}\\cup \\pi _{2}\\mathcal {O}_{p},\\quad \\overline{A_{q}^{p}}=A_{q}^{p}\\cup \\pi _{2}\\mathcal {O}_{p}\\cup \\pi _{2}\\mathcal {O}_{q}$ and $\\overline{A_{k,q}}=A_{k,q}\\cup \\pi _{2}\\mathcal {O}_{k}\\cup \\pi _{2}\\mathcal {O}_{q}.$ Observe that for all $\\varphi \\in C_{q}^{p}$ , $\\pi _{2}\\varphi \\in A_{q}^{p}$ because $t\\mapsto \\pi _{2}x_{t}^{\\varphi }$ is continuous, $\\pi _{2}x_{t}^{\\varphi }\\rightarrow \\pi _{2}\\mathcal {O}_{p}$ as $t\\rightarrow -\\infty $ , $\\pi _{2}x_{t}^{\\varphi }\\rightarrow \\pi _{2}\\mathcal {O}_{q}$ as $t\\rightarrow \\infty $ , $\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}\\subset \\overline{S_{k}}$ , and $\\pi _{2}$ is injective on $\\overline{S_{k}}$ .", "For the same reason, $\\pi _{2}C_{k}^{p}\\subseteq A_{k}^{p}$ .", "Then it is clear that $\\pi _{2}\\overline{C_{q}^{p}}=\\overline{\\pi _{2}C_{q}^{p}}\\subseteq \\overline{A_{q}^{p}}$ and $\\pi _{2}\\overline{C_{k}^{p}}=\\overline{\\pi _{2}C_{k}^{p}}\\subseteq \\overline{A_{k}^{p}}$ .", "As $\\mathcal {O}_{p}\\subseteq \\overline{C_{q}^{p}}\\cap \\overline{C_{k}^{p}}$ , we conclude that $\\pi _{2}\\mathcal {O}_{p}\\subseteq \\pi _{2}\\left(\\overline{C_{q}^{p}}\\cap \\overline{C_{k}^{p}}\\right)\\subseteq \\pi _{2}\\overline{C_{q}^{p}}\\cap \\pi _{2}\\overline{C_{k}^{p}}\\subseteq \\overline{A_{q}^{p}}\\cap \\overline{A_{k}^{p}}=\\pi _{2}\\mathcal {O}_{p},$ that is, $ $$\\pi _{2}\\mathcal {O}_{p}=\\pi _{2}\\left(\\overline{C_{q}^{p}}\\cap \\overline{C_{k}^{p}}\\right)$ .", "The injectivity of $\\pi _{2}$ on $\\overline{S_{k}}$ then implies that $\\mathcal {O}_{p}=\\overline{C_{q}^{p}}\\cap \\overline{C_{k}^{p}}.$ We also obtain from $\\pi _{2}C_{q}^{p}\\subseteq A_{q}^{p}$ and $\\pi _{2}C_{k}^{p}\\subseteq A_{k}^{p}$ that $\\pi _{2}S_{k}=\\pi _{2}C_{k}^{p}\\cup \\pi _{2}\\mathcal {O}_{p}\\cup \\pi _{2}C_{q}^{p}\\subseteq A_{k}^{p}\\cup \\pi _{2}\\mathcal {O}_{p}\\cup A_{q}^{p}=A_{k,q},$ and hence $\\pi _{2}\\overline{S_{k}}=\\overline{\\pi _{2}S_{k}}\\subseteq \\overline{A_{k,q}}$ .", "Note that this means that $\\hat{\\xi }_{k}\\notin \\overline{S_{k}}$ .", "Figure: The images of the equilibria and the periodic orbits under π 2 \\pi _{2},and the definitions of the open sets A 1 p A_{1}^{p}, A q p A_{q}^{p} andA 1,q A_{1,q}.It has been already verified that for all $\\varphi \\in \\overline{S_{k}}$ , $x^{\\varphi }$ oscillates around $\\xi _{k}$ .", "We claim that this oscillation is slow.", "Proposition 5.8 $V\\left(\\varphi -\\hat{\\xi }_{k}\\right)=2$ for all $\\varphi \\in \\overline{S_{k}}$ .", "1.", "First we prove the assertion for the elements of $S_{k}$ .", "Choose an arbitrary element $\\varphi \\in S_{k}$ and a sequence $\\left(t_{n}\\right)_{n=0}^{\\infty }$ with $t_{n}\\rightarrow -\\infty $ as $n\\rightarrow \\infty $ such that $x_{t_{n}}^{\\varphi }\\rightarrow p_{0}$ as $n\\rightarrow \\infty $ .", "As the $C$ -norm and $C^{1}$ -norm are equivalent on the global attractor, $x_{t_{n}}^{\\varphi }\\rightarrow p_{0}$ as $n\\rightarrow \\infty $ also in $C^{1}$ -norm.", "Note that $p$ is slowly oscillatory around $\\xi _{k}$ (see Proposition 8.2 in [8]), i.e., $V\\left(p_{t}-\\hat{\\xi }_{k}\\right)=2$ for all real $t.$ Hence Lemma REF .", "(iii) gives that $p_{0}-\\hat{\\xi }_{k}\\in R$ , and Lemma REF implies that $2=V\\left(p_{0}-\\hat{\\xi }_{k}\\right)=\\lim _{n\\rightarrow \\infty }V\\left(x_{t_{n}}^{\\varphi }-\\hat{\\xi }_{k}\\right).$ Then by the monotonicity of $V$ (see Lemma REF .", "(i)), $V\\left(x_{t}^{\\varphi }-\\hat{\\xi }_{k}\\right)\\le 2$ for all $t\\in \\mathbb {R}$ .", "If $V\\left(\\varphi -\\hat{\\xi }_{k}\\right)=0$ and $\\varphi <\\hat{\\xi }_{k}$ or $\\varphi >\\hat{\\xi }_{k}$ , then $x_{2}^{\\varphi }\\ll \\hat{\\xi }_{k}$ or $x_{2}^{\\varphi }\\gg \\hat{\\xi }_{k}$ by Proposition REF , which contradicts the fact that $x^{\\varphi }$ oscillates around $\\xi _{k}$ .", "2.", "Now choose any $\\varphi \\in \\overline{S_{k}}$ and fix a sequence $\\left(\\varphi _{n}\\right)_{n=0}^{\\infty }$ in $S_{k}$ with $\\varphi _{n}\\rightarrow \\varphi $ as $n\\rightarrow \\infty $ .", "Since $\\hat{\\xi }_{k}\\notin \\overline{S_{k}}$ , $V\\left(\\varphi -\\hat{\\xi }_{k}\\right)$ is defined.", "The lower semi-continuity of $V$ (see Lemma REF ) and part 1 yield that $V\\left(\\varphi -\\hat{\\xi }_{k}\\right)\\le \\liminf _{n\\rightarrow \\infty }V\\left(\\varphi _{n}-\\hat{\\xi }_{k}\\right)=2$ .", "Observe that assumption $V\\left(\\varphi -\\hat{\\xi }_{k}\\right)=0$ would lead to a contradiction just as in the previous step.", "So $V\\left(\\varphi -\\hat{\\xi }_{k}\\right)=2$ for all $\\varphi \\in \\overline{S_{k}}$ .", "Now we are ready to confirm the equalities regarding $\\overline{C_{k}^{p}}$ , $\\overline{C_{q}^{p}}$ and $\\overline{S_{k}}$ in Theorem REF .", "(ii).", "Proposition 5.9 $\\overline{S_{k}}=\\mathcal {O}_{k}\\cup S_{k}\\cup \\mathcal {O}_{q}=\\mathcal {O}_{k}\\cup C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ .", "Let us fix $k=1$ .", "It is clear from the definition of $S_{1}$ that $\\left(\\mathcal {O}_{1}\\cup \\mathcal {O}_{q}\\right)\\subset \\overline{S_{1}}$ , and thus we only need to verify the inclusion $\\overline{S_{1}}\\setminus S_{1}\\subseteq \\left(\\mathcal {O}_{1}\\cup \\mathcal {O}_{q}\\right)$ .", "Let $\\varphi \\in \\overline{S_{1}}\\setminus S_{1}$ be arbitrary.", "It is an immediate consequence of the oscillation of $x^{\\varphi }$ around $\\xi _{1}$ that $\\varphi \\notin \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , otherwise $\\varphi $ would also belong to $S_{1}$ by (REF ).", "It is also obvious that $\\varphi \\notin \\mathcal {A}_{-2,0}$ .", "There are two possibilities by Theorem B: either $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)$ or $\\varphi \\in \\mathcal {A}_{0,2}$ .", "If $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{q}\\right)$ , then necessarily $\\varphi \\in \\mathcal {O}_{q}$ , otherwise $x^{\\varphi }$ would converge to one of the equilibria $\\hat{\\xi }_{-2}$ , $\\hat{\\xi }_{2}$ by Theorem B.", "So it remains to show that the relation $\\varphi \\in \\mathcal {A}_{0,2}$ implies that $\\varphi \\in \\mathcal {O}_{1}$ .", "$\\mathcal {A}_{0,2}$ is a compact and invariant subset of $C$ , hence $\\varphi \\in \\mathcal {A}_{0,2}$ implies that $x_{t}^{\\varphi }\\in \\mathcal {A}_{0,2}$ for all real $t$ , moreover $\\alpha \\left(x^{\\varphi }\\right)$ and $\\omega \\left(\\varphi \\right)$ are also subsets of $ $$\\mathcal {A}_{0,2}$ .", "On the other hand, $\\overline{S_{1}}$ is also compact and invariant, so $\\alpha \\left(x^{\\varphi }\\right)\\cup \\omega \\left(\\varphi \\right)\\subset \\overline{S_{1}}$ , and $V\\left(\\psi -\\hat{\\xi }_{1}\\right)=2$ for all $\\psi \\in \\alpha \\left(x^{\\varphi }\\right)\\cup \\omega \\left(\\varphi \\right)$ by the previous proposition.", "The Poincaré–Bendixson Theorem (see Section ) then implies that $\\omega \\left(\\varphi \\right)$ is either a periodic orbit in $\\mathcal {A}_{0,2}$ oscillating slowly around $\\xi _{1}$ , or for each $\\psi \\in \\omega \\left(\\varphi \\right)$ , $\\alpha \\left(x^{\\psi }\\right)=\\omega \\left(\\psi \\right)=\\left\\lbrace \\hat{\\xi }_{1}\\right\\rbrace .$ As there are no homoclinic orbits to $\\hat{\\xi }_{1}$ (see Proposition 3.1 in [7]), $\\omega \\left(\\varphi \\right)=\\left\\lbrace \\hat{\\xi }_{1}\\right\\rbrace $ in the latter case.", "Similarly, $\\alpha \\left(x^{\\varphi }\\right)$ is either $\\left\\lbrace \\hat{\\xi }_{1}\\right\\rbrace $ or a periodic orbit in $\\mathcal {A}_{0,2}$ oscillating slowly around $\\xi _{1}$ .", "Recall that $x^{1}$ is defined so that the range $x^{1}(\\mathbb {R})$ is maximal in the sense that $x^{1}(\\mathbb {R})\\supset r(\\mathbb {R})$ for all periodic solutions $r$ oscillating slowly around $\\xi _{1}$ with range in $(0,\\xi _{2})$ .", "So if $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ is a periodic solution with segments in $\\alpha \\left(x^{\\varphi }\\right)\\cup \\omega \\left(\\varphi \\right)$ , then either $r$ is the time translation of $x^{1}$ , or $\\pi _{2}r_{t}\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{1}\\right)$ for all $t\\in \\mathbb {R}$ .", "Recall that $\\pi _{2}\\xi _{1}$ also belongs to $\\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{1}\\right)$ .", "On the other hand, $\\pi _{2}\\left(\\alpha \\left(x^{\\varphi }\\right)\\cup \\omega \\left(\\varphi \\right)\\right)\\subset \\pi _{2}\\overline{S_{k}}\\subseteq \\overline{A_{k,q}}\\subset \\mathbb {R}^{2}\\setminus \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{1}\\right).$ It follows that $\\pi _{2}\\left(\\alpha \\left(x^{\\varphi }\\right)\\cup \\omega \\left(\\varphi \\right)\\right)\\subseteq \\pi _{2}\\mathcal {O}_{1}$ and thus$ $ $\\alpha \\left(x^{\\varphi }\\right)=\\omega \\left(\\varphi \\right)=\\mathcal {O}_{1}$ .", "If $x^{\\varphi }$ is not the time translation of $x^{1}$ , then this is only possible if the curve $t\\rightarrow \\pi _{2}x_{t}^{\\varphi }$ is self-intersecting, which contradicts the injectivity of $\\pi _{2}$ on $\\overline{S_{1}}.$ Hence relation $\\varphi \\in \\mathcal {A}_{0,2}$ implies that $\\varphi \\in \\mathcal {O}_{1}$ .", "We have verified that each $\\varphi \\in \\overline{S_{1}}\\setminus S_{1}$ belongs to $\\mathcal {O}_{1}\\cup \\mathcal {O}_{q}$ , that is $\\overline{S_{1}}=\\mathcal {O}_{1}\\cup C_{1}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}.$ Handling the case $k=-1$ is completely analogous.", "Corollary 5.10 $\\overline{S_{-1}}\\cap \\overline{S_{1}}=\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ , $\\overline{C_{q}^{p}}=\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ and $\\overline{C_{k}^{p}}=\\mathcal {O}_{p}\\cup C_{k}^{p}\\cup \\mathcal {O}_{k}$ .", "The first equality follows immediately from Proposition REF .", "The second and third equalities come from $\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}\\subseteq \\overline{C_{q}^{p}}\\subseteq \\overline{S_{k}}=\\mathcal {O}_{k}\\cup C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q},$ $\\mathcal {O}_{k}\\cup C_{k}^{p}\\cup \\mathcal {O}_{p}\\subseteq \\overline{C_{k}^{p}}\\subseteq \\overline{S_{k}}=\\mathcal {O}_{k}\\cup C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q}$ and (REF ).", "5.3 The smoothness of $ $$C_{q}^{p}$ and $C_{k}^{p}$ Suppose $r$ is one of the periodic solutions $q$ or $x^{k}$ with minimal period $\\omega >1$ , and let $C_{r}^{p}$ be the heteroclinic connection from $\\mathcal {O}_{p}$ to $O_{r}=\\left\\lbrace r_{t}:\\, t\\in \\mathbb {R}\\right\\rbrace $ .", "Next we confirm that $C_{r}^{p}$ is a $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "First we verify that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ intersects transversally a local stable or a local center-stable manifold of a Poincaré map at a point of $\\mathcal {O}_{r}$ .", "It follows that the intersection is a one-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Then we apply the injectivity of the derivative of the flow induced by the solution operator on $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ (see Proposition REF and Corollary REF ) to confirm that each point $\\varphi $ in $C_{r}^{p}$ belongs to a “small” subset $W_{\\varphi }$ of $C_{r}^{p}$ that is a two-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "This means that $C_{r}^{p}$ is an immersed $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "In order to prove that $C_{r}^{p}$ is embedded in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , we have to show that for any $\\varphi $ in $C_{r}^{p}$ , there is no sequence in $C_{r}^{p}\\backslash W_{\\varphi }$ converging to $\\varphi $ .", "According to results of Subsection 5.1, $ $$\\pi _{2}$ is injective on $C_{r}^{p}$ and on the tangent spaces of $C_{r}^{p}$ , which implies that $\\pi _{2}W_{\\varphi }$ is open in $\\mathbb {R}^{2}$ .", "If a sequence $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ from the rest of the connecting set converges to $\\varphi $ as $n\\rightarrow \\infty $ , then $\\pi _{2}\\varphi ^{n}\\rightarrow \\pi _{2}\\varphi $ as $n\\rightarrow \\infty $ , and $\\pi _{2}\\varphi ^{n}\\in \\pi _{2}W_{\\varphi }$ for all $n$ large enough.", "The injectivity of $\\pi _{2}$ on $\\overline{S_{k}}$ then implies that $\\varphi ^{n}\\in W_{\\varphi }$ , which is a contradiction.", "So $C_{r}^{p}$ is embedded in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "With the projection $P_{2}$ and the map $w_{k}$ from Proposition REF , $C_{r}^{p}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}C_{r}^{p}\\right\\rbrace .$ Using the previously obtained result that $C_{r}^{p}$ is a $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , we prove at the end of this subsection that $w_{k}$ is continuously differentiable on the open set $P_{2}C_{r}^{p}$ , i.e., this representation for $C_{r}^{p}$ is smooth.", "Section has introduced a hyperplane $Y$ , a convex bounded open neighborhood $N$ of $r_{0}$ in $C$ , $\\varepsilon \\in \\left(0,\\omega \\right)$ and a $C^{1}$ -map $\\gamma :N\\rightarrow \\left(\\omega -\\varepsilon ,\\omega +\\varepsilon \\right)$ with $\\gamma \\left(r_{0}\\right)=\\omega $ so that for each $\\left(t,\\varphi \\right)\\in \\left(\\omega -\\varepsilon ,\\omega +\\varepsilon \\right)\\times N$ , the segment $x_{t}^{\\varphi }$ belongs to $r_{0}+Y$ if and only if $t=\\gamma (\\varphi )$ .", "A Poincaré return map $P_{Y}$ has been defined as $P_{Y}:N\\cap \\left(r_{0}+Y\\right)\\ni \\varphi \\mapsto \\Phi \\left(\\gamma (\\varphi ),\\varphi \\right)\\in r_{0}+Y.$ Let $\\mathcal {W}$ denote a local stable manifold $\\mathcal {W}_{loc}^{s}\\left(P_{Y},r_{0}\\right)$ of $P_{Y}$ at $r_{0}$ if $\\mathcal {O}_{r}$ is hyperbolic, and let $\\mathcal {W}$ be a local center-stable manifold $\\mathcal {W}_{loc}^{sc}\\left(P_{Y},r_{0}\\right)$ of $P_{Y}$ at $r_{0}$ otherwise.", "By Section , $\\mathcal {W}$ is a $C^{1}$ -submanifold of $r_{0}+Y$ with codimension 1, and it is a $C^{1}$ -submanifold of $C$ with codimension 2.", "The subsequent proposition is an important step toward the proof of the assertion that $C_{q}^{p}$ and $C_{k}^{p}$ are two-dimensional $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Proposition 5.11 $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}$ is a one-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "1.", "Theorem B and Proposition REF imply that $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}$ is nonempty.", "It suffices to verify that the inclusion map $i:\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\varphi \\mapsto \\varphi \\in C$ and $\\mathcal {W}$ are transversal.", "Then it follows that $i^{-1}\\left(\\mathcal {W}\\right)=\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}$ is a $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , furthermore it has the same codimension in $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ as $\\mathcal {W}$ in $C$ (see e.g.", "Corollary 17.2 in [1]).", "Accordingly we show that the inclusion map $i:\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\ni \\varphi \\mapsto \\varphi \\in C$ and $\\mathcal {W}$ are transversal.", "This means that for all $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ with $\\varphi =i\\left(\\varphi \\right)\\in \\mathcal {W}$ , (i) the inverse image $\\left(Di\\left(\\varphi \\right)\\right)^{-1}T_{i\\left(\\varphi \\right)}\\mathcal {W}=T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap T_{\\varphi }\\mathcal {W}$ splits in $T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ (it has a closed complementary subspace in $T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ ), and (ii) the space $Di\\left(\\varphi \\right)T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)=T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ contains a closed complement to $T_{i\\left(\\varphi \\right)}\\mathcal {W}=T_{\\varphi }\\mathcal {W}$ in $C$ .", "Property (i) holds because $\\mbox{dim}T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)=3<\\infty $ .", "In the following we confirm (ii).", "2.", "Let $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}$ .", "First note that the invariance of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ ensures that $\\dot{\\varphi }\\in T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "On the other hand, Proposition REF gives that $\\dot{\\varphi }\\notin Y$ can be assumed.", "Therefore $\\dot{\\varphi }\\in T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus T_{\\varphi }\\mathcal {W}.$ We claim that $T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ contains a sign-preserving element $\\chi $ .", "Let $Z$ be the hyperplane in $C$ with $C=\\mathbb {R}\\dot{p}_{0}\\oplus Z$ and define a Poincaré map $P_{Z}$ on a neighborhood of $p_{0}$ in $p_{0}+Z$ as in Section .", "(Here we use exceptionally the notation $Z$ and $P_{Z}$ to emphasize the difference from the above mentioned $Y$ and $P_{Y}$ .)", "Choose $\\psi $ from a local unstable manifold $\\mathcal {W}_{loc}^{u}\\left(P_{Z},p_{0}\\right)$ of $P_{Z}$ such that $\\varphi =\\Phi \\left(T,\\,\\psi \\right)$ for some $T\\ge 0$ .", "This is possible by (REF ).", "Choose $\\eta $ to be a strictly positive vector in $T_{\\eta }\\mathcal {W}_{loc}^{u}\\left(P_{Z},p_{0}\\right)$ .", "Proposition REF yields that the existence of such $\\eta $ may be supposed without loss of generality.", "Then $D_{2}\\Phi \\left(T,\\psi \\right)\\eta \\in T_{\\varphi }\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , and $D_{2}\\Phi \\left(T,\\psi \\right)\\eta =v_{T}^{\\eta }$ , where $v^{\\eta }:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ is the solution of the linear variational equation $\\dot{v}(t)=-v(t)+f^{\\prime }\\left(x^{\\psi }\\left(t-1\\right)\\right)v\\left(t-1\\right)$ with $v_{0}^{\\eta }=\\eta $ .", "The monotonicity of $V$ implies that $v_{T}^{\\eta }$ is also strictly positive.", "So set $\\chi =v_{T}^{\\eta }$ .", "Vectors $\\dot{\\varphi }$ and $\\chi $ are linearly independent because $V\\left(\\chi \\right)=0$ and we may assume by Proposition REF that $V\\left(\\dot{\\varphi }\\right)\\ge 2$ .", "3.", "As $T_{\\varphi }\\mathcal {W}$ is a subspace of $C$ with codimension 2, it suffices to confirm that $T_{\\varphi }\\mathcal {W}\\cap \\left(\\mathbb {R}\\dot{\\varphi }\\oplus \\mathbb {R}\\chi \\right)=\\left\\lbrace \\hat{0}\\right\\rbrace .$ Suppose that $a\\dot{\\varphi }+b\\chi \\in T_{\\varphi }\\mathcal {W}\\backslash \\left\\lbrace \\hat{0}\\right\\rbrace $ for some $a,b\\in \\mathbb {R}$ .", "Then $b\\ne 0$ as $\\dot{\\varphi }\\notin T_{\\varphi }\\mathcal {W}$ .", "Set $c=a/b$ and consider the vector $ $$c\\dot{\\varphi }+\\chi \\in T_{\\varphi }\\mathcal {W}\\backslash \\left\\lbrace \\hat{0}\\right\\rbrace $ .", "Let $v:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ be the solution of the linear variational equation $\\dot{v}(t) & =-v(t)+f^{\\prime }\\left(x^{\\varphi }\\left(t-1\\right)\\right)v\\left(t-1\\right)$ with $v_{0}=\\chi $ , and let $x=x^{\\varphi }$ .", "As $\\varphi \\in \\mathcal {W}$ , $\\gamma _{j}=\\Sigma _{i=0}^{j-1}\\gamma \\left(P_{Y}^{i}\\left(\\varphi \\right)\\right)$ is defined for all $j\\ge 1$ , and $\\gamma _{j}\\rightarrow \\infty $ as $j\\rightarrow \\infty $ .", "Then by formula (REF ), $T_{P_{Y}^{j}\\left(\\varphi \\right)}\\mathcal {W}\\ni DP_{Y}^{j}\\left(\\varphi \\right)\\left(c\\dot{\\varphi }+\\chi \\right) & =c\\dot{x}_{\\gamma _{j}}+v_{\\gamma _{j}}-\\frac{e^{}\\left(c\\dot{x}_{\\gamma _{j}}+v_{\\gamma _{j}}\\right)}{e^{}\\left(\\dot{x}_{\\gamma _{j}}\\right)}\\dot{x}_{\\gamma _{j}}\\\\& =v_{\\gamma _{j}}-\\frac{e^{}\\left(v_{\\gamma _{j}}\\right)}{e^{}\\left(\\dot{x}_{\\gamma _{j}}\\right)}\\dot{x}_{\\gamma _{j}}.$ An application of Lemma REF to the equation (REF ) and its strictly positive solution $v:\\left[-1,\\infty \\right)\\rightarrow \\mathbb {R}$ gives constants $K>0$ and $t\\ge 1$ such that $\\left\\Vert v_{s-1}\\right\\Vert \\le K\\left\\Vert v_{s}\\right\\Vert \\qquad \\mbox{for all }s\\ge t.$ Equation (REF ) with this estimate then gives a uniform bound for the derivatives $\\dot{v}_{\\gamma _{j}}/\\left\\Vert v_{\\gamma _{j}}\\right\\Vert $ , $j\\ge 1$ .", "So by the Arzelà–Ascoli Theorem, there exists a subsequence $\\left(\\frac{v_{\\gamma _{j_{n}}}}{\\left\\Vert v_{\\gamma _{j_{n}}}\\right\\Vert }\\right)_{n=0}^{\\infty }$ converging to a strictly positive unit vector $\\rho $ as $n\\rightarrow \\infty $ .", "As the $C$ -norm and the $C^{1}$ -norm are equivalent on $\\mathcal {A}$ , the convergence $x_{\\gamma _{j}}=P_{Y}^{j}\\left(\\varphi \\right)\\rightarrow r_{0}$ implies that $\\dot{x}_{\\gamma _{j}}\\rightarrow \\dot{r}_{0}$ as $j\\rightarrow \\infty .$ It follows that $\\frac{1}{\\left\\Vert v_{\\gamma _{j_{n}}}\\right\\Vert }DP_{Y}^{j_{n}}\\left(\\varphi \\right)\\left(c\\dot{\\varphi }+\\chi \\right)\\in T_{P_{Y}^{j_{n}}\\left(\\varphi \\right)}\\mathcal {W}$ converges to the vector $\\rho -\\frac{e^{}\\left(\\rho \\right)}{e^{}\\left(\\dot{r}_{0}\\right)}\\dot{r}_{0}\\in T_{r_{0}}\\mathcal {W}={\\left\\lbrace \\begin{array}{ll}C_{s}, & \\mbox{if }\\mathcal {O}_{r}\\mbox{ is hyperbolic,}\\\\C_{s}\\oplus \\mathbb {R}\\xi , & \\mbox{if }\\mathcal {O}_{r}\\mbox{ is nonhyperbolic.}\\end{array}\\right.", "}$ As $T_{r_{0}}\\mathcal {W}\\subseteq C_{\\le 1}$ and $\\dot{r}_{0}\\in C_{\\le 1}$ , this means that $C_{\\le 1}$ has a strictly positive element $\\rho $ .", "This is a contradiction since $\\mathcal {O}_{r}$ has a Floquet multiplier $\\lambda _{1}>1$ and $C_{<\\lambda _{1}}\\cap V^{-1}\\left(0\\right)=\\emptyset $ by (REF ).", "Now we can verify a part of Theorem REF .", "(i).", "Proposition 5.12 $C_{q}^{p}$ and $C_{k}^{p}$ are both two-dimensional $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Define $r$ , $\\mathcal {W}$ and $C_{r}^{p}$ as at the begining of this subsection.", "1.", "As a first step we confirm that to all $\\varphi \\in C_{r}^{p}$ , one can give a subset $W_{\\varphi }$ of $C_{r}^{p}$ so that $W_{\\varphi }$ is a two-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and contains $\\varphi $ .", "Let $\\varphi \\in C_{r}^{p}$ .", "Choose $T\\ge 0$ such that $\\psi =\\Phi \\left(T,\\varphi \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}$ and $\\dot{\\psi }\\notin Y$ .", "Propositions REF and REF guarantee that this is possible.", "Consider the two-dimensional $C^{1}$ -submanifold $\\mathbb {R}\\times \\left(\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\right)$ of $\\mathbb {R}\\times \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ and the map $\\Sigma :\\mathbb {R}\\times \\left(\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\right)\\ni \\left(t,\\eta \\right)\\mapsto \\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(t,\\eta \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).$ Proposition REF proves that $\\Sigma $ is $C^{1}$ -smooth and gives formulas for its derivatives.", "Note that the derivative of the map $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\ni \\eta \\mapsto \\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(-T,\\eta \\right)\\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ at $\\psi $ is injective on $T_{\\psi }\\left(\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\right)$ by Corollary REF .", "Also observe that $\\dot{\\psi }\\notin Y$ implies that $\\dot{\\psi }\\notin T_{\\psi }\\left(\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\right)$ .", "Using these two properties and a reasoning analogous to the one applied in Proposition REF , it is straightforward to show that $D\\Sigma \\left(-T,\\psi \\right)$ is injective on $\\mathbb {R}\\times T_{\\psi }\\left(\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\right)$ .", "Thus there exists an $\\varepsilon >0$ by Proposition REF such that the set $W_{\\varphi }=\\left\\lbrace \\Phi _{\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)}\\left(t,\\eta \\right):\\, t\\in \\left(-T-\\varepsilon ,-T+\\,\\varepsilon \\right),\\,\\eta \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\cap \\mathcal {W}\\cap B\\left(\\psi ,\\varepsilon \\right)\\right\\rbrace $ is a two-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "It is clear that $\\varphi \\in W_{\\varphi }$ .", "The invariance of $C_{r}^{p}$ implies that $W_{\\varphi }\\subseteq C_{r}^{p}$ .", "2.", "To complete the proof, it suffices to exclude for all $\\varphi \\in C_{r}^{p}$ the existence of a sequence $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ in $ $$C_{r}^{p}$ so that $\\varphi ^{n}\\notin W_{\\varphi }$ for $n\\ge 0$ and $\\varphi ^{n}\\rightarrow \\varphi $ as $n\\rightarrow \\infty $ .", "By Proposition REF , $D\\pi _{2}\\left(\\varphi \\right)=\\pi _{2}$ is injective on the two-dimensional tangent space $T_{\\varphi }W_{\\varphi }$ , hence it defines an isomorphism from $T_{\\varphi }W_{\\varphi }$ onto $\\mathbb {R}^{2}$ .", "Therefore there exists $\\tilde{\\varepsilon }>0$ such that the restriction of $\\pi _{2}$ to $W_{\\varphi }\\cap B\\left(\\varphi ,\\tilde{\\varepsilon }\\right)$ is a diffeomorphism from $W_{\\varphi }\\cap B\\left(\\varphi ,\\tilde{\\varepsilon }\\right)$ onto an open set $U$ in $\\mathbb {R}^{2}$ .", "If a sequence $\\left(\\varphi ^{n}\\right)_{n=0}^{\\infty }$ in $ $$C_{r}^{p}$ converges to $\\varphi $ as $n\\rightarrow \\infty $ , then $\\pi _{2}\\varphi ^{n}\\rightarrow \\pi _{2}\\varphi $ as $n\\rightarrow \\infty $ , and $\\pi _{2}\\varphi ^{n}\\in U$ for all $n$ large enough.", "The injectivity of $\\pi _{2}$ on $\\overline{S_{k}}$ verified in Proposition REF then implies that $\\varphi ^{n}\\in W_{\\varphi }$ .", "It is worth noting that the second part of the above proof confirms the following assertion.", "Proposition 5.13 $\\pi _{2}C_{q}^{p}$ and $\\pi _{2}C_{k}^{p}$ are open subsets of $\\mathbb {R}^{2}$ .", "We know from Proposition REF that there exist a projection $P_{2}$ from $C$ onto a two-dimensional subspace $G_{2}$ of $C$ and a map $w_{k}:P_{2}\\overline{S_{k}}\\rightarrow P_{2}^{-1}\\left(0\\right)$ so that $\\overline{S_{k}}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}\\overline{S_{k}}\\right\\rbrace .$ Then $C_{q}^{p}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}C_{q}^{p}\\right\\rbrace \\quad \\mbox{and}\\quad C_{k}^{p}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}C_{k}^{p}\\right\\rbrace .$ The next result implies that these representations of $C_{q}^{p}$ and $C_{k}^{p}$ are smooth.", "Proposition 5.14 $P_{2}C_{q}^{p}$ and $P_{2}C_{k}^{p}$ are open subsets of $G_{2}$ , and $w_{k}$ is continuously differentiable on $P_{2}C_{q}^{p}\\cup P_{2}C_{k}^{p}$ .", "The proof is based on the smoothness of $C_{q}^{p}$ and $C_{k}^{p}$ and applies an argument which is analogous to the one in the proof of Theorem REF .", "Let $C_{r}^{p}$ be any of the sets $C_{q}^{p}$ and $C_{k}^{p}$ .", "Let $\\chi \\in P_{2}C_{r}^{p}$ be arbitrary, and choose $\\varphi \\in C_{r}^{p}$ so that $\\chi =P_{2}\\varphi $ .", "As the restriction of $\\pi _{2}$ to $T_{\\varphi }C_{r}^{p}$ is injective, $J_{2}$ is a linear isomorhism and $P_{2}=J_{2}\\circ \\pi _{2}$ , $DP_{2}\\left(\\varphi \\right)=P_{2}$ defines an isomorphism from $T_{\\varphi }C_{r}^{p}$ to $G_{2}$ .", "The inverse mapping theorem implies that an $\\varepsilon >0$ can be given such that $P_{2}$ maps $C_{r}^{p}\\cap B\\left(\\varphi ,\\varepsilon \\right)$ one-to-one onto an open neighborhood $U\\subset P_{2}C_{r}^{p}$ of $\\chi $ in $G_{2}$ , $P_{2}$ is invertible on $C_{r}^{p}\\cap B\\left(\\varphi ,\\varepsilon \\right)$ , and the inverse $\\tilde{P}_{2}^{-1}$ of the map $C_{r}^{p}\\cap B\\left(\\varphi ,\\varepsilon \\right)\\ni \\varphi \\mapsto P_{2}\\varphi \\in U$ is $C^{1}$ -smooth.", "As $w_{k}\\left(\\chi \\right)=\\left(\\mbox{id}-P_{2}\\right)\\circ \\left(P_{2}\|_{\\overline{S_{k}}}\\right)^{-1}\\left(\\chi \\right)=\\left(\\mbox{id}-P_{2}\\right)\\circ \\tilde{P}_{2}^{-1}\\left(\\chi \\right)\\in P_{2}^{-1}\\left(0\\right)$ for all $\\chi \\in U$ , the restriction of $w_{k}$ to $U$ is $C^{1}$ -smooth.", "5.4 $C_{q}^{p}$ , $C_{k}^{p}$ and $S_{k}$ are homeomorphic to $A^{\\left(1,2\\right)}$ , and their closures are homeomorphic to $A^{\\left[1,2\\right]}$ Recall that $A_{q}^{p}=\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)\\cap \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{q}\\right),\\qquad A_{k}^{p}=\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)$ and $A_{k,q}=\\mbox{ext}\\left(\\pi _{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{q}\\right).$ We have already deduced that $\\pi _{2}C_{q}^{p}\\subseteq A_{q}^{p}$ and $\\pi _{2}C_{k}^{p}\\subseteq A_{k}^{p}$ .", "As a result, $\\pi _{2}S_{k}\\subseteq A_{k,q}$ .", "Proposition 5.15 The map $\\pi _{2}\|_{\\overline{S_{k}}}$ is a homeomorphism onto $\\overline{A_{k,q}}$ , furthermore $\\pi _{2}C_{k}^{p}=A_{k}^{p}$ , $\\pi _{2}C_{q}^{p}=A_{q}^{p}$ and $\\pi _{2}S_{k}=A_{k,q}$ .", "First we show that $\\pi _{2}C_{q}^{p}=A_{q}^{p}.$ By Proposition REF , $\\pi _{2}C_{q}^{p}$ is open in $A_{q}^{p}$ .", "We claim that $\\pi _{2}C_{q}^{p}$ is also closed in $A_{q}^{p}$ .", "So assume that $\\left(z_{n}\\right)_{n=0}^{\\infty }$ is a sequence in $\\pi _{2}C_{q}^{p}$ and $z_{n}\\rightarrow z\\in A_{q}^{p}$ as $n\\rightarrow \\infty $ .", "Let $\\varphi _{n}=\\pi _{2}^{-1}\\left(z_{n}\\right)\\in C_{q}^{p}$ , $n\\ge 0$ .", "By Proposition REF , $\\pi _{2}^{-1}$ is Lipschitz-continuous.", "Thus $\\left\\lbrace \\varphi _{n}\\right\\rbrace _{n=0}^{\\infty }$ is a Cauchy-sequence in $C_{q}^{p}$ and a $\\varphi \\in \\overline{C_{q}^{p}}$ can be given such that $\\varphi _{n}\\rightarrow \\varphi $ as $n\\rightarrow \\infty $ , moreover, $\\varphi =\\pi _{2}^{-1}\\left(z\\right)$ .", "It is clear that $\\varphi \\notin \\mathcal {O}_{p}$ and $\\varphi \\notin \\mathcal {O}_{q}$ because then $z=\\pi _{2}\\varphi \\notin A_{q}^{p}$ .", "Thus $\\varphi \\in \\overline{C_{q}^{p}}\\setminus \\left(\\mathcal {O}_{p}\\cup \\mathcal {O}_{p}\\right)=C_{q}^{p}$ (here we use Corollary REF ) and necessarily $z=\\pi _{2}\\varphi \\in \\pi _{2}C_{q}^{p}$ .", "In consequence, $\\pi _{2}C_{q}^{p}=A_{q}^{p}.$ It is analogous to verify that $\\pi _{2}C_{k}^{p}=A_{k}^{p}$ .", "It follows immediately that $\\pi _{2}S_{k}=\\pi _{2}\\left(C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\right)=A_{k}^{p}\\cup \\pi _{2}\\mathcal {O}_{p}\\cup A_{q}^{p}=A_{k,q}$ and $\\pi _{2}\\overline{S_{k}}=\\pi _{2}\\left(\\mathcal {O}_{k}\\cup S_{k}\\cup \\mathcal {O}_{q}\\right)=\\pi _{2}\\mathcal {O}_{k}\\cup A_{k,q}\\cup \\pi _{2}\\mathcal {O}_{q}=\\overline{A_{k,q}}.$ As both $\\pi _{2}\|_{\\overline{S_{k}}}:\\overline{S_{k}}\\rightarrow \\mathbb {R}^{2}$ and $\\pi _{2}^{-1}:\\pi _{2}\\overline{S_{k}}\\rightarrow C$ are continuous, we obtain that $\\pi _{2}\|_{\\overline{S_{k}}}$ defines a homeomorphism from $\\overline{S_{k}}$ onto $\\overline{A_{k,q}}$ .", "As a consequence we obtain that $C_{q}^{p}$ , $C_{k}^{p}$ , and $S_{k}$ are homeomorphic to the open annulus $A^{\\left(1,2\\right)}=\\left\\lbrace u\\in \\mathbb {R}^{2}:\\,1<\\left\|u\\right\|<2\\right\\rbrace .$ Since the above proposition implies that $\\pi _{2}\\overline{C_{k}^{p}}=\\overline{A_{k}^{p}}$ and $\\pi _{2}\\overline{C_{q}^{p}}=\\overline{A_{q}^{p}}$ , we also deduce that the closures $\\overline{C_{q}^{p}}$ , $\\overline{C_{k}^{p}}$ , and $\\overline{S_{k}}$ are homeomorphic to the closed annulus $A^{\\left[1,2\\right]}=\\left\\lbrace u\\in \\mathbb {R}^{2}:\\,1\\le \\left\|u\\right\|\\le 2\\right\\rbrace .$ Note that we have proven all the statements of Theorem REF .", "(i) regarding $C_{q}^{p}$ and $C_{k}^{p}$ (see propositions REF , REF and REF ).", "The smoothness of $S_{k}$ is considered in the next subsection.", "5.5 The smoothness of $S_{k}$ , $\\overline{C_{q}^{p}}$ , $\\overline{C_{k}^{p}}$ and $\\overline{S_{k}}$ Now we can round up the proofs of Theorem REF .", "(i) and (ii).", "Recall that $S_{k}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}S_{k}\\right\\rbrace ,\\qquad P_{2}S_{k}=P_{2}C_{k}^{p}\\cup P_{2}\\mathcal {O}_{p}\\cup P_{2}C_{q}^{p}$ and $w_{k}$ is continuously differentiable on the set $P_{2}C_{k}^{p}\\cup P_{2}C_{q}^{p}$ .", "Hence the smoothness of this representation for $S_{k}$ is proved by showing that $P_{2}S_{k}$ is open in $G_{2}$ and $w_{k}$ is smooth at the points of $P_{2}\\mathcal {O}_{p}$ .", "It follows at once that $S_{k}$ is a two-dimensional $C^{1}$ -submanifold of $C$ .", "Since $S_{k}$ is a subset of the three-dimensional $C^{1}$ -submanifold $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , it is obvious that $S_{k}$ is also a $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "We likewise verify that all points of $P_{2}\\mathcal {O}_{k}\\cup P_{2}\\mathcal {O}_{q}$ have open neighborhoods on which $w_{k}$ can be extended to $C^{1}$ -functions.", "As $P_{2}\\mathcal {O}_{k}\\cup P_{2}\\mathcal {O}_{q}$ is the boundary of $P_{2}\\overline{S_{k}}$ , this result shows that $\\overline{S_{k}}$ has a smooth representation with boundary, and thus $\\overline{S_{k}}$ is a two-dimensional $C^{1}$ -submanifold of the phase space $C$ with boundary.", "Similar reasonings yield the analogous results for $\\overline{C_{q}^{p}}$ and $\\overline{C_{k}^{p}}$ .", "Let $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ be any of the periodic solutions $x^{k}$ , $p$ or $q$ shifted in time so that $r\\left(0\\right)=\\xi _{k}$ and $\\dot{r}\\left(0\\right)>0$ .", "As $\\xi _{k}$ belongs to the ranges of $x^{k}$ , $p$ or $q$ , and $\\xi _{k}$ is not an extremum of them, the monotonicity property of periodic solutions in Proposition REF implies that this choice of $r$ is possible.", "Let $\\omega >0$ denote the minimal period of $r$ .", "By Eq.", "(REF ), $f\\left(r\\left(-1\\right)\\right)=\\dot{r}\\left(0\\right)+r\\left(0\\right)>\\xi _{k}=f\\left(\\xi _{k}\\right).$ As $f$ strictly increases, this means that $r\\left(-1\\right)>\\xi _{k}$ .", "Conversely, if there was $t_{}\\in \\left(0,\\omega \\right)$ such that $r\\left(t_{}\\right)=\\xi _{k}$ and $r\\left(t_{}-1\\right)>\\xi _{k}$ , then $\\dot{r}\\left(t_{}\\right)=-r\\left(t_{}\\right)+f\\left(r\\left(t_{}-1\\right)\\right)>-\\xi _{k}+f\\left(\\xi _{k}\\right)=0$ would follow, which would contradict Proposition REF .", "Therefore the half line $L_{k}=\\left\\lbrace \\left(\\xi _{k},x_{2}\\right)\\in \\mathbb {R}^{2}:\\, x_{2}>\\xi _{k}\\right\\rbrace $ and $\\pi _{2}\\mathcal {O}_{r}=\\left\\lbrace \\pi _{2}r_{t}:\\, t\\in \\left[0,\\omega \\right)\\right\\rbrace $ have exactly one point in common: $\\left(r\\left(0\\right),r\\left(-1\\right)\\right)=\\left(\\xi _{k},r\\left(-1\\right)\\right)$ .", "See Fig.", "7.", "Choose $s_{k},s_{p},s_{q}>\\xi _{k}$ so that $\\left\\lbrace \\left(\\xi _{k},s_{k}\\right)\\right\\rbrace =L_{k}\\cap \\pi _{2}\\mathcal {O}_{k},\\quad \\left\\lbrace \\left(\\xi _{k},s_{p}\\right)\\right\\rbrace =L_{k}\\cap \\pi _{2}\\mathcal {O}_{p}$ and $\\left\\lbrace \\left(\\xi _{k},s_{q}\\right)\\right\\rbrace =L_{k}\\cap \\pi _{2}\\mathcal {O}_{q}.$ As $s$ increases, $\\left(\\xi _{k},\\infty \\right)\\ni s\\mapsto \\left(\\xi _{k},s\\right)\\in \\mathbb {R}^{2}$ first intersects $\\pi _{2}\\mathcal {O}_{k}$ , then $\\pi _{2}\\mathcal {O}_{p}$ and finally $\\pi _{2}\\mathcal {O}_{q}$ because $\\left(\\xi _{k},s\\right)\\rightarrow \\pi _{2}\\hat{\\xi }_{k}=\\left(\\xi _{k},\\xi _{k}\\right)\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{k}\\right)\\mbox{ whenever }s\\rightarrow \\xi _{k}+,$ $\\pi _{2}\\mathcal {O}_{k}\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{p}\\right)$ and $\\pi _{2}\\mathcal {O}_{p}\\in \\mbox{int}\\left(\\pi _{2}\\mathcal {O}_{q}\\right)$ .", "So $\\xi _{k}<s_{k}<s_{p}<s_{q}$ , as it is shown by Fig.", "7.", "Figure: The definition of L 1 L_{1}, s 1 s_{1}, s p s_{p} and s q s_{q} in thecase k=1k=1.Consider the curve $h:\\left[s_{k},s_{q}\\right]\\ni s\\mapsto \\pi _{2}^{-1}\\left(\\xi _{k},s\\right)\\in C.$ Then $h$ is Lipschitz-continuous and injective.", "By Proposition REF , $h\\left(\\left[s_{k},s_{q}\\right]\\right)\\subset \\overline{S_{k}}$ .", "In detail, $h\\left(s_{k}\\right)\\in \\mathcal {O}_{k},\\quad h\\left(\\left(s_{k},s_{p}\\right)\\right)\\subset C_{k}^{p},\\quad h\\left(s_{p}\\right)\\in \\mathcal {O}_{p},\\quad h\\left(\\left(s_{p},s_{q}\\right)\\right)\\subset C_{q}^{p},\\quad \\mbox{and}\\quad h\\left(s_{q}\\right)\\in \\mathcal {O}_{q}.$ According to the next result, $h$ is $C^{1}$ -smooth on $\\left(s_{1},s_{q}\\right)\\setminus \\left\\lbrace s_{p}\\right\\rbrace $ .", "Proposition 5.16 $\\pi _{2}^{-1}\|_{\\pi _{2}\\left(C_{q}^{p}\\cup C_{k}^{p}\\right)}$ is $C^{1}$ -smooth.", "We know from Proposition REF that $\\pi _{2}\\left(C_{q}^{p}\\cup C_{k}^{p}\\right)$ is open in $\\mathbb {R}^{2}$ .", "For all $x\\in \\pi _{2}\\left(C_{q}^{p}\\cup C_{k}^{p}\\right)$ , the graph representation of $C_{q}^{p}\\cup C_{k}^{p}$ and the definition of $P_{2}$ together give that $C_{q}^{p}\\cup C_{k}^{p}\\ni \\pi _{2}^{-1}\\left(x\\right) & =P_{2}\\left(\\pi _{2}^{-1}\\left(x\\right)\\right)+w_{k}\\left(P_{2}\\left(\\pi _{2}^{-1}\\left(x\\right)\\right)\\right)\\\\& =J_{2}\\left(\\pi _{2}\\left(\\pi _{2}^{-1}\\left(x\\right)\\right)\\right)+w_{k}\\left(J_{2}\\left(\\pi _{2}\\left(\\pi _{2}^{-1}\\left(x\\right)\\right)\\right)\\right)\\\\& =J_{2}\\left(x\\right)+w_{k}\\left(J_{2}\\left(x\\right)\\right).$ As $J_{2}$ defines a linear isomorphism from $\\mathbb {R}^{2}$ to $G_{2}$ , it is continuously differentiable.", "In addition, $J_{2}\\left(\\pi _{2}\\left(C_{q}^{p}\\cup C_{k}^{p}\\right)\\right)=P_{2}\\left(C_{q}^{p}\\cup C_{k}^{p}\\right)$ , and $w_{k}$ is continuously differentiable on the open subset $P_{2}\\left(C_{q}^{p}\\cup C_{k}^{p}\\right)$ of $G_{2}$ by Proposition REF .", "Hence the statement follows.", "As a next step, we show the smoothness of $h$ at points $s_{k},s_{p}$ and $s_{q}$ .$ $ We will need the following technical result, which is part of Proposition 8.5 in [10].", "Proposition 5.17 (i) Let $v:\\mathbb {R}\\rightarrow \\mathbb {R}$ be a solution of Eq.", "(REF ) with $v_{0}\\ne \\hat{0}$ .", "If $V\\left(v_{t}\\right)=2$ for all $t\\in \\mathbb {R}$ , then $v_{0}\\in C_{r_{M}<}\\cap C_{\\le 1}$ .", "(ii) For every $\\varphi \\in C_{r_{M}<}\\cap C_{\\le 1}\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace $ , there is a solution $v:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) so that $v_{0}=\\varphi $ and $V\\left(v_{t}\\right)=2$ for all $t\\in \\mathbb {R}$ .", "Proposition 5.18 Let $\\in \\left\\lbrace k,p,q\\right\\rbrace $ and set $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ to be the periodic solution of Eq.", "(REF ) with $\\pi _{2}r_{0}=\\left(\\xi _{k},s_{}\\right)$ .", "(i) There exists a unique continuously differentiable function $z=z^{}:\\mathbb {R}\\rightarrow \\mathbb {R}$ satisfying ${\\left\\lbrace \\begin{array}{ll}\\dot{z}\\left(t\\right)=-z\\left(t\\right)+f^{\\prime }\\left(r\\left(t-1\\right)\\right)z\\left(t-1\\right), & t\\in \\mathbb {R},\\\\z\\left(-1\\right)=1,\\, z\\left(0\\right)=0,\\\\V\\left(z_{t}\\right)=2, & t\\in \\mathbb {R}.\\end{array}\\right.", "}$ (ii) For every $\\varepsilon >0$ , there exists $\\delta >0$ so that for all $\\chi \\in \\left[s_{k},s_{q}\\right]$ , $\\nu \\in \\left[s_{k},s_{q}\\right]$ with $\\left\|\\chi -s_{}\\right\|<\\delta $ , $\\left\|\\nu -s_{}\\right\|<\\delta $ and $\\chi \\ne \\nu $ , $\\left\\Vert \\frac{h\\left(\\chi \\right)-h\\left(\\nu \\right)}{\\chi -\\nu }-z_{0}\\right\\Vert <\\varepsilon .$ (iii) $z_{0}$ and $\\dot{r}_{0}$ are linearly independent.", "1.", "We prove that for all sequences $\\left(\\chi ^{n}\\right)_{n=0}^{\\infty }$ , $\\left(\\nu ^{n}\\right)_{n=0}^{\\infty }$ in $\\left[s_{k},s_{q}\\right]$ with $\\chi ^{n}\\ne \\nu ^{n}$ for all $n\\ge 0$ and $\\chi ^{n}\\rightarrow s_{}$ , $\\nu ^{n}\\rightarrow s_{}$ as $n\\rightarrow \\infty $ , there exist a strictly increasing sequence $\\left(n_{l}\\right)_{l=0}^{\\infty }$ and a continuously differentiable function $z=z^{}:\\mathbb {R}\\rightarrow \\mathbb {R}$ so that $z$ is a solution of the equation in (REF ), and $\\lim _{l\\rightarrow \\infty }\\frac{h\\left(\\chi ^{n_{l}}\\right)-h\\left(\\nu ^{n_{l}}\\right)}{\\chi ^{n_{l}}-\\nu ^{n_{l}}}=z_{0}.$ Consider the solutions $x^{n}:\\mathbb {R}\\rightarrow \\mathbb {R}$ and $y^{n}:\\mathbb {R}\\rightarrow \\mathbb {R}$ of Eq.", "(REF ) with $x_{0}^{n}=h\\left(\\chi ^{n}\\right)$ and $y_{0}^{n}=h\\left(\\nu ^{n}\\right)$ for all indices $n\\ge 0$ .", "Then $x^{n}\\left(-1\\right)=\\chi ^{n}$ , $y^{n}\\left(-1\\right)=\\nu ^{n}$ and $x^{n}\\left(0\\right)=y^{n}\\left(0\\right)=\\xi _{k}$ for all $n\\ge 0$ , moreover $x_{t}^{n}\\in \\overline{S_{k}}$ and $y_{t}^{n}\\in \\overline{S_{k}}$ for all $n\\ge 0$ and $t\\in \\mathbb {R}$ .", "Introduce the functions $z^{n}=\\frac{x^{n}-y^{n}}{\\chi ^{n}-\\nu ^{n}},\\qquad n\\ge 0.$ It is clear that $z^{n}\\left(0\\right)=0$ and $z^{n}\\left(-1\\right)=1$ for all $n\\ge 0$ .", "By Proposition REF , $V\\left(z_{t}^{n}\\right)=2$ for all $n\\ge 0$ and $t\\in \\mathbb {R}$ .", "In addition, $z^{n}$ , $n\\ge 0$ , satisfies the equation $\\dot{z}^{n}\\left(t\\right)=-z^{n}\\left(t\\right)+b^{n}\\left(t\\right)z^{n}\\left(t-1\\right)$ on $\\mathbb {R}$ , where the coefficient functions $b^{n}$ are defined as $b^{n}:\\mathbb {R}\\ni t\\mapsto \\int _{0}^{1}f^{\\prime }\\left(sx^{n}\\left(t-1\\right)+\\left(1-s\\right)y^{n}\\left(t-1\\right)\\right)\\mbox{d}s\\in \\left(0,\\infty \\right),\\quad n\\ge 0.$ Since $\\chi ^{n}\\rightarrow s_{}$ and $\\nu ^{n}\\rightarrow s_{}$ as $n\\rightarrow \\infty $ , $x_{0}^{n}\\rightarrow r_{0}$ and $y_{0}^{n}\\rightarrow r_{0}$ as $n\\rightarrow \\infty $ .", "It follows that $b^{n}\\rightarrow b$ as $n\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , where $b:\\mathbb {R}\\ni t\\mapsto f^{\\prime }\\left(r\\left(t-1\\right)\\right)\\in \\left(0,\\infty \\right).$ As the global attractor is bounded, there are constants $b_{1}>b_{0}>0$ so that $b_{0}<b^{n}\\left(t\\right)<b_{1}$ for all $n\\ge 0$ and $t\\in \\mathbb {R}$ .", "Thus Lemma REF ensures the existence of a continuously differentiable function $z:\\mathbb {R}\\rightarrow \\mathbb {R}$ and a subsequence $\\left(z^{n_{l}}\\right)_{l=0}^{\\infty }$ of $\\left(z^{n}\\right)_{n=0}^{\\infty }$ such that $z^{n_{l}}\\rightarrow z$ and $\\dot{z}^{n_{l}}\\rightarrow \\dot{z}$ as $l\\rightarrow \\infty $ uniformly on compact subsets of $\\mathbb {R}$ , and $z$ is a solution of the equation in (REF ).", "It is obvious that $z\\left(0\\right)=0$ and $z\\left(-1\\right)=1$ .", "By the first part of Lemma REF , $V\\left(z_{t}\\right)\\le 2$ for all real $t$ .", "Suppose $V\\left(z_{t^{}}\\right)=0$ for some $t^{}\\in \\mathbb {R}$ .", "Then $V\\left(z_{t}\\right)=0$ for all $t>t^{}$ and $V\\left(z_{t^{}+3}\\right)\\in R$ by Lemma REF .", "The $C^{1}$ -convergence of $z^{n_{l}}$ to $z$ and the second part of Lemma REF then imply that $V\\left(z_{t^{}+3}^{n_{l}}\\right)=0$ for all sufficiently large index $l$ , which is contradiction.", "So $V\\left(z_{t}\\right)=2$ for all real $t$ .", "2.", "Suppose that $\\hat{z}:\\mathbb {R}\\rightarrow \\mathbb {R}$ is also a continuously differentiable function satisfying (REF ), and $z\\ne \\hat{z}$ .", "Then Proposition REF yields that $z_{0}\\ne \\hat{z}_{0}$ .", "Function $d=z-\\hat{z}$ is a solution of ${\\left\\lbrace \\begin{array}{ll}\\dot{d}\\left(t\\right)=-d\\left(t\\right)+f^{\\prime }\\left(r\\left(t-1\\right)\\right)d\\left(t-1\\right), & t\\in \\mathbb {R},\\\\d\\left(-1\\right)=d\\left(0\\right)=0.\\end{array}\\right.", "}$ Since $z_{0},\\hat{z}_{0}\\in C_{r_{M}<}\\cap C_{\\le 1}$ by Proposition REF (i), $d_{0}\\in C_{r_{M}<}\\cap C_{\\le 1}\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace $ .", "Then it follows from Proposition REF (ii) that $V\\left(d_{t}\\right)=2$ for all $t\\in \\mathbb {R}$ .", "So $d_{0}\\in R$ by Lemma REF .", "(iii), which is impossible as $d\\left(-1\\right)=d\\left(0\\right)=0.$ These results imply both (i) and (ii).", "3.", "Solution $r$ has been defined to be a time translate of $x^{k}$ , $p$ or $q$ with $r\\left(0\\right)=\\xi _{k}$ .", "Hence $\\xi _{k}$ is not an extremum of $r$ , and thus $\\dot{r}\\left(0\\right)\\ne 0$ by Proposition REF .", "Consequently, $z_{0}\\notin \\mathbb {R}\\dot{r}_{0}\\setminus \\left\\lbrace \\hat{0}\\right\\rbrace $ , and $z_{0}$ and $\\dot{r}_{0}$ are linearly independent.", "Corollary 5.19 Function $h$ is $C^{1}$ -smooth on $\\left[s_{k},s_{q}\\right]$ .", "We extend the definition of $h$ to the half line $\\left(\\xi _{k},\\infty \\right)$ : we define $\\hat{h}:\\left(\\xi _{k},\\infty \\right)\\rightarrow C$ as $\\hat{h}\\left(s\\right)=h\\left(s\\right)$ for $s\\in \\left[s_{k},s_{q}\\right]$ , $\\hat{h}\\left(s\\right)=h\\left(s_{k}\\right)+\\left(s-s_{k}\\right)z_{0}^{k}\\quad \\mbox{for }s\\in \\left(\\xi _{k},s_{k}\\right)$ and $\\hat{h}\\left(s\\right)=h\\left(s_{q}\\right)+\\left(s-s_{q}\\right)z_{0}^{q}\\quad \\mbox{for }s>s_{q},$ where $z_{0}^{k}$ and $z_{0}^{q}$ are given by Proposition REF .", "Then $\\hat{h}$ is $C^{1}$ -smooth with $\\hat{h}^{\\prime }\\left(s_{k}\\right)=z_{0}^{k}$ , $\\hat{h}^{\\prime }\\left(s_{p}\\right)=z_{0}^{p}$ , and $\\hat{h}^{\\prime }\\left(s_{q}\\right)=z_{0}^{q}$ .", "According to the choice of $s_{k}<s_{p}<s_{q}$ and Proposition REF , $\\hat{h}\\left(s_{k}\\right)\\in \\mathcal {O}_{k},\\quad \\hat{h}\\left(\\left(s_{k},s_{p}\\right)\\right)\\subset C_{k}^{p},\\quad \\hat{h}\\left(s_{p}\\right)\\in \\mathcal {O}_{p},\\quad \\hat{h}\\left(\\left(s_{p},s_{q}\\right)\\right)\\subset C_{q}^{p}\\mbox{ and }\\hat{h}\\left(s_{q}\\right)\\in \\mathcal {O}_{q}.$ Observe that $\\pi _{2}\\hat{h}\\left(s\\right)=\\left(\\xi _{k},s\\right)$ for all $s>\\xi _{k}$ , hence the map $\\left(\\xi _{k},\\infty \\right)\\ni s\\mapsto \\pi _{2}\\hat{h}\\left(s\\right)\\in \\mathbb {R}^{2}$ is injective on $\\left(\\xi _{k},\\infty \\right)$ and has range in $L_{k}=\\left\\lbrace \\left(\\xi _{k},x_{2}\\right)\\in \\mathbb {R}^{2}:\\, x_{2}>\\xi _{k}\\right\\rbrace $ .", "So it follows from $\\pi _{2}\\overline{S_{k}}=\\overline{A_{k,q}}$ that $\\hat{h}\\left(\\left(\\xi _{k},s_{k}\\right)\\cup \\left(s_{q},\\infty \\right)\\right)\\cap \\overline{S_{k}}=\\emptyset .$ As $J_{2}:\\mathbb {R}^{2}\\rightarrow G_{2}$ is a linear isomorphism and $P_{2}=J_{2}\\circ \\pi _{2}$ , Proposition REF shows that $P_{2}C_{k}^{p}=\\mbox{ext}\\left(P_{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(P_{2}\\mathcal {O}_{p}\\right),\\quad P_{2}C_{q}^{p}=\\mbox{ext}\\left(P_{2}\\mathcal {O}_{p}\\right)\\cap \\mbox{int}\\left(P_{2}\\mathcal {O}_{q}\\right),$ $P_{2}S_{k}=\\mbox{ext}\\left(P_{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(P_{2}\\mathcal {O}_{q}\\right)$ and $P_{2}\\overline{S_{k}}=P_{2}\\mathcal {O}_{k}\\cup \\left(\\mbox{ext}\\left(P_{2}\\mathcal {O}_{k}\\right)\\cap \\mbox{int}\\left(P_{2}\\mathcal {O}_{q}\\right)\\right)\\cup P_{2}\\mathcal {O}_{q}.$ As $P_{2}\\mathcal {O}_{k}$ and $P_{2}\\mathcal {O}_{q}$ are the images of simple closed $C^{1}$ -curves, the boundary $P_{2}\\mathcal {O}_{k}\\cup P_{2}\\mathcal {O}_{q}$ of the domain $P_{2}\\overline{S_{k}}$ of $w_{k}$ is a one-dimensional $C^{1}$ -submanifold of $G_{2}$ .", "The next result shows that $w_{k}$ is continuously differentiable at the points of $P_{2}\\mathcal {O}_{p}$ , and it is smooth at the points of $P_{2}\\mathcal {O}_{k}\\cup P_{2}\\mathcal {O}_{q}$ in the sense that $w_{k}$ can be extended to continuously differentiable functions on open neighborhoods of the boundary points.", "Proposition 5.20 (i) To each $\\varphi \\in \\mathcal {O}_{k}\\cup \\mathcal {O}_{q}$ there corresponds an open neighborhood $U$ of $P_{2}\\varphi $ in $G_{2}$ and a continuously differentiable map $w_{k}^{e}:U\\rightarrow P_{2}^{-1}\\left(0\\right)$ such that $w_{k}^{e}\|_{U\\cap P_{2}\\overline{S_{k}}}=w_{k}\|_{U\\cap P_{2}\\overline{S_{k}}},$ and $U\\setminus \\left\\lbrace P_{2}x_{t}^{\\varphi }:\\, t\\in \\mathbb {R}\\right\\rbrace $ is the union of open connected disjoint subsets $U^{+}$ and $U^{-}$ with the following property: $U^{-}\\cap P_{2}\\overline{S_{k}}=\\emptyset $ and $U^{+}\\subset P_{2}C_{k}^{p}$ if $\\varphi \\in \\mathcal {O}_{k}$ , $U^{-}\\subset P_{2}C_{q}^{p}$ and $U^{+}\\cap P_{2}\\overline{S_{k}}=\\emptyset $ if $\\varphi \\in \\mathcal {O}_{q}$ .", "(ii) The map $w_{k}$ is continuously differentiable at the points of $P_{2}\\mathcal {O}_{p}$ .", "All $\\varphi \\in \\mathcal {O}_{p}$ has an open neighborhood $U$ of $P_{2}\\varphi $ in $G_{2}$ such that $U\\setminus P_{2}\\mathcal {O}_{p}$ is the union of open connected disjoint subsets $U^{+}$ and $U^{-}$ with $U^{-}\\subset P_{2}C_{k}^{p}$ and $U^{+}\\subset P_{2}C_{q}^{p}$ .", "The proof below verifies assertions (i) and (ii) simultaneously.", "1.", "Let $r:\\mathbb {R}\\rightarrow \\mathbb {R}$ be one of the periodic solutions $x^{k}$ , $p$ or $q$ shifted in time so that $r\\left(0\\right)=\\xi _{k}$ and $\\dot{r}\\left(0\\right)>0$ (that is $\\pi _{2}r_{0}\\in L_{k}$ ), and fix $\\in \\left\\lbrace k,p,q\\right\\rbrace $ accordingly.", "Set $s_{}=r\\left(-1\\right)$ .", "Let $\\varphi \\in \\mathcal {O}_{r}=\\left\\lbrace r_{t}:\\, t\\in \\mathbb {R}\\right\\rbrace $ and choose $T>1$ so that $\\varphi =\\Phi \\left(T,r_{0}\\right)$ .", "For all $0<\\varepsilon <\\min \\left\\lbrace T-1,s_{k}-\\xi _{k}\\right\\rbrace $ , the map $a:\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\ni \\left(t,s\\right)\\mapsto \\Phi \\left(T+t,\\hat{h}\\left(s_{}+s\\right)\\right)\\in C$ is $C^{1}$ -smooth with $Da\\left(0,0\\right)\\mathbb {R}^{2}=\\mathbb {R}\\dot{\\varphi }\\oplus \\mathbb {R}D_{2}\\Phi \\left(T,r_{0}\\right)z_{0}^{},$ where $z^{}:\\mathbb {R}\\rightarrow \\mathbb {R}$ is the solution of (REF ) given by Proposition REF .", "The vectors $\\dot{\\varphi }=D_{2}\\Phi \\left(T,r_{0}\\right)\\dot{r}_{0}$ and $D_{2}\\Phi \\left(T,r_{0}\\right)z_{0}^{}$ are linearly independent because $D_{2}\\Phi \\left(T,r_{0}\\right)$ is injective, and $\\dot{r}_{0}$ and $z_{0}^{}$ are linearly independent by Proposition REF (iii).", "Therefore Proposition REF implies that for all small $\\varepsilon >0$ , the set $a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\right)$ is a two-dimensional $C^{1}$ -submanifold of $C$ with $T_{\\varphi }a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\right)=Da\\left(0,0\\right)\\mathbb {R}^{2}.$ Then $a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,0\\right)\\right)$ and $a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(0,\\varepsilon \\right)\\right)$ are also two-dimensional $C^{1}$ -submanifolds of $C$ .", "2.", "Set $E_{1}=\\mathbb {R}\\dot{\\varphi }\\oplus \\mathbb {R}D_{2}\\Phi \\left(T,r_{0}\\right)z_{0}^{}$ and let $E_{2}$ be a closed complement of $E_{1}$ in $C$ .", "We claim that for small $\\varepsilon >0$ , there exist an open neighborhood $N_{\\varepsilon }$ of $\\hat{0}$ in $E_{1}$ and a continuously differentiable function $b:N_{\\varepsilon }\\rightarrow E_{2}$ so that $b\\left(\\hat{0}\\right)=0$ , $Db\\left(\\hat{0}\\right)=0$ and $a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\right)$ is the shifted graph of $b$ : $a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\right)=\\varphi +\\left\\lbrace \\chi +b\\left(\\chi \\right):\\,\\chi \\in N_{\\varepsilon }\\right\\rbrace .$ Let $\\mbox{Pr}_{E_{1}}$ denote the projection of $C$ onto $E_{1}$ along $E_{2}$ , and define $j:C\\rightarrow C$ by $j\\left(\\chi \\right)=\\chi -\\varphi $ for all $\\chi \\in C$ .", "Then $D\\left(\\mbox{Pr}_{E_{1}}\\circ j\\circ a\\right)\\left(0,0\\right)\\mathbb {R}^{2}=\\mbox{Pr}_{E_{1}}\\circ Da\\left(0,0\\right)\\mathbb {R}^{2}=E_{1}.$ Hence the inverse function theorem guarantees that $\\mbox{Pr}_{E_{1}}\\circ j\\circ a$ is a local $C^{1}$ -diffeomorphism, i.e.", "for for small $\\varepsilon >0$ , $\\mbox{Pr}_{E_{1}}\\circ j\\circ a$ maps $\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)$ injectively onto an open neighborhood $N_{\\varepsilon }$ of $\\hat{0}$ in $E_{1}$ , and the inverse $\\left(\\mbox{Pr}_{E_{1}}\\circ j\\circ a\\right)^{-1}$ of $\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\ni \\left(t,s\\right)\\mapsto \\mbox{Pr}_{E_{1}}\\circ j\\circ a\\left(t,s\\right)\\in N_{\\varepsilon }$ is $C^{1}$ -smooth.", "In consequence, $\\mbox{Pr}_{E_{1}}$ maps $j\\circ a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\right)$ onto $N_{\\varepsilon }$ injectively, and there exists a map $b:N_{\\varepsilon }\\rightarrow E_{2}$ so that $b\\left(\\hat{0}\\right)=0$ and $j\\circ a\\left(\\left(-\\varepsilon ,\\varepsilon \\right)\\times \\left(-\\varepsilon ,\\varepsilon \\right)\\right)=\\left\\lbrace \\chi +b\\left(\\chi \\right):\\,\\chi \\in N_{\\varepsilon }\\right\\rbrace .$ The smoothness of $b$ follows from $b=\\left(\\mbox{id}-\\mbox{Pr}_{E_{1}}\\right)\\circ j\\circ a\\circ \\left(\\mbox{Pr}_{E_{1}}\\circ j\\circ a\\right)^{-1}.$ $Db\\left(\\hat{0}\\right)=0$ because $Da\\left(0,0\\right)\\mathbb {R}^{2}=E_{1}$ .", "3.", "Next we show that the continuously differentiable map $c:E_{1}\\supset N_{\\varepsilon }\\ni \\chi \\mapsto P_{2}\\left(\\varphi +\\chi +b\\left(\\chi \\right)\\right)\\in G_{2}$ is a local $C^{1}$ -diffeomorphism.", "Note that $Dc\\left(\\hat{0}\\right)\\chi =P_{2}\\chi $ for all $\\chi \\in E_{1}$ .", "So it suffices to confirm that $P_{2}\|_{E_{1}}$ is injective.", "$E_{1}$ is spanned by the derivatives $D\\gamma \\left(0\\right)1$ of the curves $\\gamma :\\left(-1,1\\right)\\ni s\\mapsto a\\left(c_{1}s,c_{2}s\\right)\\in C,$ where $\\left(c_{1},c_{2}\\right)\\in \\mathbb {R}^{2}$ .", "From (REF ) and the invariance of $\\overline{S_{k}}$ it follows that if $s_{}+c_{2}s\\in \\left[s_{k},s_{q}\\right]$ , then $\\gamma \\left(s\\right)\\in \\overline{S_{k}}$ .", "Proposition REF gives that $\\pi _{2}\\gamma ^{\\prime }\\left(0\\right)\\ne \\left(0,0\\right)$ if $\\gamma ^{\\prime }\\left(0\\right)\\ne \\hat{0}$ .", "Thus $\\pi _{2}\|_{E_{1}}$ is injective.", "As $J_{2}$ is a linear isomorphism, $P_{2}=J_{2}\\circ \\pi _{2}$ is also injective on $E_{1}$ .", "In consequence, a positive constant $\\varepsilon _{0}$ can be given such that $c$ is a $C^{1}$ -diffeomorphism from $N_{\\varepsilon _{0}}$ onto an open neighborhood $U$ of $P_{2}\\varphi $ in $G_{2}$ .", "Define $c^{-1}$ to be the inverse of $N_{\\varepsilon _{0}}\\ni \\chi \\mapsto c\\left(\\chi \\right)\\in U$ .", "Constant $\\varepsilon _{0}$ can be chosen so that $\\varepsilon _{0}<\\min \\left\\lbrace T-1,s_{k}-\\xi _{k},s_{p}-s_{k},s_{q}-s_{p}\\right\\rbrace $ also holds.", "4.", "Notice that $U=P_{2}a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\right),$ and set $U^{-}= & P_{2}a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},0\\right)\\right),\\\\U^{0}= & P_{2}a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left\\lbrace 0\\right\\rbrace \\right),\\\\U^{+}= & P_{2}a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(0,\\varepsilon _{0}\\right)\\right).$ By steps 2 and 3 it is clear that $P_{2}$ restricted to $a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\right)$ defines a $C^{1}$ -diffeomorphism from $a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\right)$ onto $U$ .", "As $a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},0\\right)\\right)$ and $a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(0,\\varepsilon _{0}\\right)\\right)$ are two-dimensional $C^{1}$ -submanifolds of $C$ , the arcwise connected sets $U^{-}$ and $U^{+}$ are open in $G_{2}$ .", "5.", "As $\\hat{h}\\left(s_{}\\right)=r_{0}\\in \\mathcal {O}_{r}$ , we have $U^{0}\\subset P_{2}\\mathcal {O}_{r}$ .", "Assume that $\\varphi \\in \\mathcal {O}_{k}$ , that is $r$ is the time translate of $x^{k}$ , and $s_{}=s_{k}$ .", "Then the relations $\\varepsilon _{0}<s_{p}-s_{k}$ , $\\hat{h}\\left(\\left(s_{k},s_{p}\\right)\\right)\\subset C_{k}^{p}$ and the invariance of $C_{k}^{p}$ guarantee that $U^{+}\\subset P_{2}C_{k}^{p}\\subset \\mbox{ext}\\left(P_{2}\\mathcal {O}_{k}\\right)$ .", "As $U^{0}\\subset P_{2}\\mathcal {O}_{k}$ belongs to the boundary of both connected components of $G_{2}\\setminus P_{2}\\mathcal {O}_{k}$ , $U^{-}$ and $U^{+}$ belong to different connected components of $G_{2}\\setminus P_{2}\\mathcal {O}_{k}$ .", "So $U^{-}\\subset \\mbox{int}\\left(P_{2}\\mathcal {O}_{k}\\right)$ .", "Then (REF ) implies that $U^{-}\\cap P_{2}\\overline{S_{k}}=\\emptyset $ .", "In cases $\\varphi \\in \\mathcal {O}_{p}$ and $\\varphi \\in \\mathcal {O}_{q}$ , it is similar to show that $U^{-}\\subset P_{2}C_{k}^{p}$ , $U^{+}\\subset P_{2}C_{q}^{p}$ and $U^{-}\\subset P_{2}C_{q}^{p}$ , $U^{+}\\cap P_{2}\\overline{S_{k}}=\\emptyset $ , respectively.", "We omit the details.", "6.", "Introduce the $C^{1}$ -map $w_{k}^{e}:U\\ni \\eta \\mapsto \\varphi +c^{-1}\\left(\\eta \\right)+b\\left(c^{-1}\\left(\\eta \\right)\\right)-\\eta \\in C.$ For all $\\eta \\in U$ , $c^{-1}\\left(\\eta \\right)\\in N_{\\varepsilon _{0}}$ , and thus $P_{2}\\left(\\varphi +c^{-1}\\left(\\eta \\right)+b\\left(c^{-1}\\left(\\eta \\right)\\right)-\\eta \\right) & =P_{2}\\left(\\varphi +c^{-1}\\left(\\eta \\right)+b\\left(c^{-1}\\left(\\eta \\right)\\right)\\right)-P_{2}\\eta \\\\& =c\\left(c^{-1}\\left(\\eta \\right)\\right)-\\eta =0.$ So $w_{k}^{e}$ maps $U$ into $P_{2}^{-1}\\left(0\\right)$ .", "7.", "It remains to confirm (REF ).", "Let $\\eta \\in U\\cap P_{2}\\overline{S_{k}}$ be arbitrary.", "Then $\\eta =P_{2}a\\left(t,s\\right)=P_{2}\\Phi \\left(T+t,\\hat{h}\\left(s_{}+s\\right)\\right)$ for some $t\\in \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)$ and $s\\in \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)$ satisfying $s_{}+s\\in \\left[s_{k},s_{q}\\right]$ .", "As $\\hat{h}\\left(\\left[s_{k},s_{q}\\right]\\right)\\subset \\overline{S_{k}}$ and $\\overline{S_{k}}$ is invariant, $a\\left(t,s\\right)\\in \\overline{S_{k}}$ .", "Then due to the injectivity of $P_{2}$ on $\\overline{S_{k}}$ , $\\eta +w_{k}\\left(\\eta \\right)=a\\left(t,s\\right)$ follows.", "On the other hand, relation $\\eta +w_{k}^{e}\\left(\\eta \\right)=\\varphi +c^{-1}\\left(\\eta \\right)+b\\left(c^{-1}\\left(\\eta \\right)\\right)\\in a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\right)$ and the injectivity of $P_{2}$ on $a\\left(\\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\times \\left(-\\varepsilon _{0},\\varepsilon _{0}\\right)\\right)$ implies that $\\eta +w_{k}^{e}\\left(\\eta \\right)=a\\left(t,s\\right).$ Thus $w_{k}\\left(\\eta \\right)=w_{k}^{e}\\left(\\eta \\right)$ .", "Recall from Proposition REF ,that there exist a projection $P_{2}$ from $C$ onto a two-dimensional subspace $G_{2}$ of $C$ and a map $w_{k}:P_{2}\\overline{S_{k}}\\rightarrow P_{2}^{-1}\\left(0\\right)$ so that $\\overline{S_{k}}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}\\overline{S_{k}}\\right\\rbrace .$ This induces a global graph representation for any subset $W$ of $\\overline{S_{k}}$ : $W=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}W\\right\\rbrace .$ We already know from Propositions REF , REF and REF that the connecting sets $C_{q}^{p}$ and $C_{k}^{p}$ are two-dimensional $C^{1}$ -submanifolds of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ with smooth global graph representations, furthermore $C_{q}^{p}$ , $C_{k}^{p}$ and $S_{k}$ are homeomorphic to the open annulus $A^{\\left(1,2\\right)}$ .", "As $J_{2}:\\mathbb {R}^{2}\\rightarrow G_{2}$ is a linear isomorphism and $P_{2}=J_{2}\\circ \\pi _{2}$ , Proposition REF shows that $P_{2}S_{k}$ is open in $G_{2}$ .", "In addition, Propositions REF and REF .", "(ii) together give that $w_{k}$ is $C^{1}$ -smooth on $P_{2}S_{k}=P_{2}\\left(C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}\\right)$ .", "So the global graph representation $S_{k}=\\left\\lbrace \\chi +w_{k}\\left(\\chi \\right):\\,\\chi \\in P_{2}S_{k}\\right\\rbrace $ given for $S_{k}$ is smooth.", "This property with $S_{k}\\subset \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ guarantees that $S_{k}$ is a two-dimensional $C^{1}$ -submanifold of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ [12].", "Recall that Propositions REF and REF have confirmed the equalities $\\overline{C_{q}^{p}}=\\mathcal {O}_{p}\\cup C_{q}^{p}\\cup \\mathcal {O}_{q},\\qquad \\overline{C_{k}^{p}}=\\mathcal {O}_{p}\\cup C_{k}^{p}\\cup \\mathcal {O}_{k}$ and $\\overline{S_{k}}=\\mathcal {O}_{k}\\cup S_{k}\\cup \\mathcal {O}_{q}.$ As $J_{2}:\\mathbb {R}^{2}\\rightarrow G_{2}$ is a linear isomorphism and $P_{2}=J_{2}\\circ \\pi _{2}$ , Proposition REF yields that $P_{2}\\overline{S_{k}}$ is the closure of the open set $P_{2}S_{k}$ , and its boundary is $P_{2}\\left(\\mathcal {O}_{k}\\cup \\mathcal {O}_{q}\\right).$ The sets $P_{2}\\mathcal {O}_{k}$ and $P_{2}\\mathcal {O}_{q}$ are the images of simple closed $C^{1}$ -curves, hence the boundary is a one-dimensional $C^{1}$ -submanifold of $G_{2}$ .", "By the proof of Theorem REF .", "(i), $w_{k}$ is continously differentiable on $P_{2}S_{k}$ .", "Proposition REF .", "(i) in addition verifies that all points of $P_{2}\\left(\\mathcal {O}_{k}\\cup \\mathcal {O}_{q}\\right)$ have open neighborhoods in $G_{2}$ on which $w_{k}$ can be extended to $C^{1}$ -smooth functions.", "Summing up, the representation given for $\\overline{S_{k}}$ is a two-dimensional smooth global graph representation with boundary.", "It is analogous to show that the induced representations of $\\overline{C_{q}^{p}}$ and $\\overline{C_{k}^{p}}$ are two-dimensional global graph representations with boundary, therefore we omit the details.", "It follows immediately that $\\overline{C_{q}^{p}}$ , $\\overline{C_{k}^{p}}$ and $\\overline{S_{k}}$ are two-dimensional $C^{1}$ -submanifolds of $C$ with boundary [12].", "The assertion that $\\overline{C_{q}^{p}}$ , $\\overline{C_{k}^{p}}$ and $\\overline{S_{k}}$ are homeomorphic to the closed annulus $A^{\\left[1,2\\right]}$ follows from Proposition REF .", "5.6 $S_{1}$ and $S_{-1}$ are indeed separatrices To complete the proof of Theorem REF , it remains to show that $S_{-1}$ and $S_{1}$ are separatrices in the sense that $C_{2}^{p}$ is above $S_{1}$ , $C_{0}^{p}$ is between $S_{-1}$ and $S_{1}$ , furthermore $C_{-2}^{p}$ is below $S_{-1}$ .", "The underlying idea of the following proof is that the assertion restricted to a local unstable manifold $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ is true, and the monotonicity of the semiflow can be used to extend the statement for $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right).$ Recall that for the periodic orbit $\\mathcal {O}_{p}$ , the unstable space $C_{u}$ is two-dimensional: $C_{u}=\\left\\lbrace c_{1}v_{1}+c_{2}v_{2}:\\, c_{1},c_{2}\\in \\mathbb {R}\\right\\rbrace ,$ where $v_{1}$ is a positive eigenfunction corresponding the leading real Floquet multiplier $\\lambda _{1}>1$ , and $v_{2}$ is an eigenfunction corresponding the Floquet multiplier $\\lambda _{2}$ with $1<\\lambda _{2}<\\lambda _{1}$ .", "Also recall that a local unstable manifold $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ of $P_{Y}$ at $p_{0}$ is a graph of a $C^{1}$ -map: there exist convex open neighborhoods $N_{s}$ , $N_{u}$ of $\\hat{0}$ in $C_{s}$ , $C_{u}$ , respectively, and a $C^{1}$ -map $w_{u}:N_{u}\\rightarrow C_{s}$ with range in $N_{s}$ so that $w_{u}\\left(\\hat{0}\\right)=\\hat{0}$ , $Dw_{u}\\left(\\hat{0}\\right)=0$ and $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)=\\left\\lbrace p_{0}+\\chi +w_{u}\\left(\\chi \\right):\\,\\chi \\in N_{u}\\right\\rbrace .$ Choose $\\alpha \\in \\left(0,1\\right)$ so small that $\\left(-\\alpha ,\\alpha \\right)v_{1}+\\left(-\\alpha ,\\alpha \\right)v_{2}\\subset N_{u}$ and $\\sup _{\\chi \\in \\left(-\\alpha ,\\alpha \\right)v_{1}+\\left(-\\alpha ,\\alpha \\right)v_{2}}\\left\\Vert Dw_{u}\\left(\\chi \\right)\\right\\Vert <\\frac{1}{2}.$ Introduce the sets $A_{s}=\\left\\lbrace p_{0}+\\chi +w_{u}\\left(\\chi \\right):\\,\\chi \\in \\left(-\\alpha ,\\alpha \\right)v_{1}+sv_{2}\\right\\rbrace \\subset \\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right),\\quad s\\in \\left(-\\alpha ,\\alpha \\right).$ The elements of $A_{s}$ , $s\\in \\left(-\\alpha ,\\alpha \\right)$ , are ordered pointwisely.", "Indeed, if $s\\in \\left(-\\alpha ,\\alpha \\right)$ is fixed and $a,b\\in \\left(-\\alpha ,\\alpha \\right)$ are arbitrary with $a<b$ , then (REF ) implies that $\\left[b-a+\\int _{a}^{b}Dw_{u}\\left(uv_{1}+sv_{2}\\right)\\mbox{d}u\\right]v_{1}\\gg \\hat{0},$ and thus $p_{0}+\\left(av_{1}+sv_{2}\\right)+w_{u}\\left(av_{1}+sv_{2}\\right)\\ll p_{0}+\\left(bv_{1}+sv_{2}\\right)+w_{u}\\left(bv_{1}+sv_{2}\\right).$ Introduce the subsets $A_{s}^{k,+}=\\left\\lbrace \\varphi \\in A_{s}:\\, x_{t}^{\\varphi }\\gg \\hat{\\xi _{k}}\\mbox{ for some }t\\ge 0\\right\\rbrace $ and $A_{s}^{k,-}=\\left\\lbrace \\varphi \\in A_{s}:\\, x_{t}^{\\varphi }\\ll \\hat{\\xi _{k}}\\mbox{ for some }t\\ge 0\\right\\rbrace $ of $A_{s}$ for all $s\\in \\left(-\\alpha ,\\alpha \\right)$ .", "Then $A_{s}^{k,+}$ and $A_{s}^{k,-}$ are open and disjoint in $A_{s}$ .", "It is also clear from the monotonicity of the semiflow that for any $\\varphi ^{-}\\in A_{s}^{k,-}$ and $\\varphi ^{+}\\in A_{s}^{k,+}$ , $\\varphi ^{-}\\ll \\varphi ^{+}$ .", "We claim that there exists $\\beta \\in \\left(0,\\alpha \\right]$ so that $A_{s}^{k,+}$ and $A_{s}^{k,-}$ are nonempty for all $s\\in \\left(-\\beta ,\\beta \\right)$ .", "Choose $\\eta _{1}=p_{0}-\\frac{\\alpha }{2}v_{1}+w_{u}\\left(-\\frac{\\alpha }{2}v_{1}\\right)\\in A_{0}\\mbox{ and }\\eta _{2}=p_{0}+\\frac{\\alpha }{2}v_{1}+w_{u}\\left(\\frac{\\alpha }{2}v_{1}\\right)\\in A_{0}.$ Then $\\eta _{1}\\ll p_{0}\\ll \\eta _{2}$ .", "By Theorem 4.1 in Chapter 5 of [20], there is an open and dense set of initial functions in $C$ so that the corresponding solutions converge to equilibria.", "In consequence, there exist $\\eta _{1}^{+},\\eta _{1}^{-},\\eta _{2}^{+},\\eta _{2}^{-}\\in C$ such that $\\eta _{1}^{-}\\ll \\eta _{1}\\ll \\eta _{1}^{+}\\ll p_{0}\\ll \\eta _{2}^{-}\\ll \\eta _{2}\\ll \\eta _{2}^{+},$ and for both $i=1$ and $i=2$ , $x_{t}^{\\eta _{i}^{-}}$ and $x_{t}^{\\eta _{i}^{+}}$ converge to one of the equilibrium points as $t\\rightarrow \\infty $ .", "Since $\\max _{t\\in \\mathbb {R}}p\\left(t\\right)>\\xi _{1}$ , $\\min _{t\\in \\mathbb {R}}p\\left(t\\right)<\\xi _{-1}$ and $x_{t}^{\\eta _{1}^{-}}\\ll x_{t}^{\\eta _{1}^{+}}\\ll p_{t}\\ll x_{t}^{\\eta _{2}^{-}}\\ll x_{t}^{\\eta _{2}^{+}}\\quad \\mbox{for all }t\\ge 0$ by Proposition REF , we obtain that $x_{t}^{\\eta _{1}^{-}}\\rightarrow \\hat{\\xi }_{-2},\\ x_{t}^{\\eta _{1}^{+}}\\rightarrow \\hat{\\xi }_{-2},\\ x_{t}^{\\eta _{2}^{-}}\\rightarrow \\hat{\\xi }_{2}\\mbox{ and }x_{t}^{\\eta _{2}^{+}}\\rightarrow \\hat{\\xi }_{2}\\ \\mbox{as}\\ t\\rightarrow \\infty .$ Using again Proposition REF , we get that $x_{t}^{\\eta _{1}}\\rightarrow \\hat{\\xi }_{-2}$ and $x_{t}^{\\eta _{2}}\\rightarrow \\hat{\\xi }_{2}$ as $t\\rightarrow \\infty $ , therefore $x_{t_{1}}^{\\eta _{1}}\\ll \\hat{\\xi }_{k}$ and $x_{t_{2}}^{\\eta _{2}}\\gg \\hat{\\xi }_{k}$ for some $t_{1},t_{2}\\ge 0$ .", "The continuity of the semiflow $\\Phi $ implies that there exist open balls $B_{1},B_{2}$ centered at $\\eta _{1},\\eta _{2}$ , respectively, such that $x_{t_{1}}^{\\varphi }\\ll \\hat{\\xi }_{k}$ for all $\\varphi \\in B_{1}$ and $x_{t_{2}}^{\\varphi }\\gg \\hat{\\xi }_{k}$ for all $\\varphi \\in B_{2}$ .", "It follows that there exists $\\beta \\in \\left(0,\\alpha \\right]$ so that $A_{s}^{k,+}$ and $A_{s}^{k,-}$ are nonempty for all $s\\in \\left(-\\beta ,\\beta \\right)$ .", "Consequently, the set $A_{s}\\setminus \\left(A_{s}^{+}\\cup A_{s}^{-}\\right)$ is nonempty for all $s\\in \\left(-\\beta ,\\beta \\right)$ , i.e., $A_{s}$ has at least one element in $S_{k}$ .", "On the other hand, the nonordering property of $S_{k}$ stated in Proposition REF implies that $A_{s}\\cap S_{k}$ contains at most one element, i.e., $A_{s}\\cap S_{k}$ is a singleton for all $s\\in \\left(-\\beta ,\\beta \\right)$ .", "Note that for any $s\\in \\left(-\\beta ,\\beta \\right),$ $\\varphi ^{-}\\in A_{s}^{k,-}$ , $\\varphi ^{+}\\in A_{s}^{k,+}$ and $\\psi \\in A_{s}\\cap S_{k}$ , $\\varphi ^{-}\\ll \\psi \\ll \\varphi ^{+}$ .", "Also observe that if $\\left(\\varphi _{n}\\right)_{-\\infty }^{0}$ is a trajectory of $P_{Y}$ in $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ with $\\varphi _{n}\\rightarrow p_{0}$ as $n\\rightarrow -\\infty $ , then for all indeces with sufficiently large absolute value, $\\varphi _{n}\\in A_{s}$ for some $s\\in \\left(-\\beta ,\\beta \\right)$ .", "An element $\\varphi $ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is said to be above $S_{k}$ if $\\psi \\in S_{k}$ can be given with $\\psi \\ll \\varphi $ , and $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is said to be below $S_{k}$ if there exists $\\psi \\in S_{k}$ with $\\varphi \\ll \\psi $ .", "An element of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is between $S_{-1}$ and $S_{1}$ if it is below $S_{1}$ and above $S_{-1}$ .", "A subset $W$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is above (below) $S_{k}$ , if all elements of $W$ are above (below) $S_{k}$ .", "A subset $W$ of $\\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ is between $S_{-1}$ and $S_{1}$ if it is below $S_{1}$ and above $S_{-1}$ .", "Proposition 5.21 For each $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , exactly one of the following cases holds: (i) $\\varphi \\in S_{k}$ , (ii) $\\varphi $ is above $S_{k}$ , (ii) $\\varphi $ is below $S_{k}$ .", "It is clear that $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ cannot be below and above $S_{k}$ at the same time because then there would exist $\\psi _{1},\\psi _{2}\\in S_{k}$ with $\\psi _{1}\\ll \\varphi \\ll \\psi _{2}$ , which would contradict Proposition REF .", "For the same reason, $\\varphi \\in S_{k}$ cannot be above (or below) $S_{k}$ .", "So at most one of the above cases holds for all $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ .", "Let $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\backslash S_{k}$ be arbitrary.", "By (REF ) and the characterization of $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ , there exists a sequence $\\left(t_{n}\\right)_{-\\infty }^{0}$ with $t_{n}\\rightarrow -\\infty $ as $n\\rightarrow -\\infty $ so that $\\left\\lbrace x_{t_{n}}^{\\varphi }\\right\\rbrace _{-\\infty }^{0}$ is a trajectory of $P_{Y}$ in $\\mathcal {W}_{loc}^{u}\\left(P_{Y},p_{0}\\right)$ and $x_{t_{n}}^{\\varphi }\\rightarrow p_{0}$ as $n\\rightarrow -\\infty $ .", "So an index $n^{}\\le 0$ can be given with $t_{n^{}}<0$ such that $x_{t_{n^{}}}^{\\varphi }\\in A_{s}$ for some $s\\in \\left(-\\beta ,\\beta \\right)$ .", "Let $\\psi $ denote the single element of $A_{s}\\cap S_{k}$ .", "As the elements of $A_{s}$ are ordered pointwisely, we obtain that $x_{t_{n^{}}}^{\\varphi }\\ll \\psi $ or $x_{t_{n^{}}}^{\\varphi }\\gg \\psi $ or $x_{t_{n^{}}}^{\\varphi }=\\psi $ .", "Observe that $x_{t_{n^{}}}^{\\varphi }=\\psi $ is impossible: as $\\psi \\in S_{k}$ and $S_{k}$ is invariant, $x_{t_{n^{}}}^{\\varphi }=\\psi $ would imply that $\\varphi =x_{-t_{n^{}}}^{\\psi }\\in S_{k}$ , which contradicts the choice of $\\varphi $ .", "If $x_{t_{n^{}}}^{\\varphi }\\ll \\psi $ , then the invariance of $S_{k}$ and the monotonicity of the semiflow imply that $x_{-t_{n^{}}}^{\\psi }\\in S_{k}$ and $\\varphi \\ll x_{-t_{n^{}}}^{\\psi }$ , that is, $\\varphi $ is below $S_{k}$ .", "If $x_{t_{n^{}}}^{\\varphi }\\gg \\psi $ , then $\\varphi \\gg x_{-t_{n^{*}}}^{\\psi }$ and $\\varphi $ is above $S_{k}$ .", "Now we are able to complete the proof of Theorem REF .", "1.", "First we show that for any $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ , $\\varphi \\in C_{2}^{p}$ if and only if $\\varphi $ is above $S_{1}$ .", "Suppose that $\\varphi \\in C_{2}^{p}$ .", "Then $x_{t_{1}}^{\\varphi }\\gg \\hat{\\xi }_{1}$ for some $t_{1}>0$ .", "Choose $t_{2}>0$ in addition so that $x_{-t_{2}}^{\\varphi }\\in A_{s}$ for some $s\\in \\left(-\\beta ,\\beta \\right)$ .", "Necessarily $x_{-t_{2}}^{\\varphi }\\in A_{s}^{1,+}$ , and thereby $x_{-t_{2}}^{\\varphi }\\gg \\psi $ , where $\\psi $ is the single element of $A_{s}\\cap S_{1}$ .", "Then $x_{t_{2}}^{\\psi }\\in S_{1}$ and $\\varphi \\gg x_{t_{2}}^{\\psi }$ , that is, $\\varphi $ is above $S_{1}$ .", "Conversely, suppose that $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ is above $S_{1}$ , and choose $\\psi \\in S_{1}$ with $\\varphi \\gg \\psi $ .", "Recall that there is an open and dense set of initial functions in $C$ so that the corresponding solutions are convergent (Theorem 4.1 in Chapter 5 of [20]).", "Hence $\\eta _{1}\\in C$ , $\\eta _{2}\\in C$ and $\\eta _{3}\\in C$ can be given such that $\\psi \\ll \\eta _{1}\\ll \\eta _{2}\\ll \\varphi \\ll \\eta _{3},$ furthermore $x_{t}^{\\eta _{1}}$ , $x_{t}^{\\eta _{2}}$ and $x_{t}^{\\eta _{3}}$ converge to equilibria as $t\\rightarrow \\infty $ .", "By the monotonicity of the semiflow, $x_{t}^{\\psi }\\ll x_{t}^{\\eta _{1}}\\ll x_{t}^{\\eta _{2}}\\ll x_{t}^{\\varphi }\\ll x_{t}^{\\eta _{3}}\\quad \\mbox{for all }t\\ge 0,$ hence the oscillation of $x^{\\psi }$ around $\\hat{\\xi }_{1}$ implies that $\\omega \\left(\\eta _{i}\\right)$ is either $\\left\\lbrace \\hat{\\xi }_{1}\\right\\rbrace $ or $\\left\\lbrace \\hat{\\xi }_{2}\\right\\rbrace $ for all $i\\in \\left\\lbrace 1,2,3\\right\\rbrace $ .", "If $\\omega \\left(\\eta _{2}\\right)=\\left\\lbrace \\hat{\\xi }_{1}\\right\\rbrace $ , then necessarily $\\omega \\left(\\eta _{1}\\right)=\\omega \\left(\\eta _{2}\\right)=\\left\\lbrace \\hat{\\xi }_{1}\\right\\rbrace $ , which contradicts Proposition REF .", "So $\\omega \\left(\\eta _{2}\\right)=\\left\\lbrace \\hat{\\xi }_{2}\\right\\rbrace $ .", "Then (REF ) guarantees that $x_{t}^{\\eta _{3}}\\rightarrow \\hat{\\xi }_{2}$ and thus $x_{t}^{\\varphi }\\rightarrow \\hat{\\xi }_{2}$ as $t\\rightarrow \\infty $ .", "2.", "It is similar to show that for any $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ , $\\varphi \\in C_{-2}^{p}$ if and only if $\\varphi $ is below $S_{-1}$ .", "3.", "Relations $S_{k}=C_{k}^{p}\\cup \\mathcal {O}_{p}\\cup C_{q}^{p}$ , $k\\in \\left\\lbrace -1,1\\right\\rbrace $ , imply the equalities $C_{q}^{p}\\cup \\mathcal {O}_{p}=S_{-1}\\cap S_{1}$ and $C_{k}^{p}=S_{k}\\backslash S_{-k}$ for both $k\\in \\left\\lbrace -1,1\\right\\rbrace $ .", "4.", "It remains to verify that for $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ , $\\omega \\left(\\varphi \\right)=\\left\\lbrace \\hat{0}\\right\\rbrace $ if and only if $\\varphi $ is between $S_{-1}$ and $S_{1}$ .", "Recall that for both $k\\in \\left\\lbrace -1,1\\right\\rbrace $ and each $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)$ , $\\varphi $ is either below $S_{k}$ , or it is above $S_{k}$ , or it is an element of $S_{k}$ .", "For this reason, $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ is between $S_{-1}$ and $S_{1}$ if and only if all the following three properties hold: $\\varphi \\notin S_{-1}\\cup S_{1}$ , $\\varphi $ is not above $S_{1}$ and $\\varphi $ is not below $S_{-1}$ .", "So by the above results, $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\mathcal {O}_{p}$ is between $S_{-1}$ and $S_{1}$ if and only if $\\varphi \\in \\mathcal {W}^{u}\\left(\\mathcal {O}_{p}\\right)\\setminus \\left\\lbrace \\mathcal {O}_{p}\\cup C_{-2}^{p}\\cup C_{-1}^{p}\\cup C_{q}^{p}\\cup C_{1}^{p}\\cup C_{2}^{p}\\right\\rbrace =C_{0}^{p}.$" ] ]	1403.0444
[ [ "Universal Thermal Corrections to Single Interval Entanglement Entropy\n for Conformal Field Theories" ], [ "Abstract We consider single interval R\\'enyi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature.", "Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the R\\'enyi and entanglement entropy in a low temperature expansion.", "These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy.", "We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle." ], [ "=1 Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories John Cardy Oxford University, Rudolf Peierls Centre for Theoretical Physics 1 Keble Road, Oxford, OX1 3NP, United Kingdom All Souls College, Oxford Christopher P. Herzog C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Stony Brook University, Stony Brook, NY 11794 We consider single interval Rényi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature.", "Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the Rényi and entanglement entropy in a low temperature expansion.", "These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy.", "We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle.", "11.25.Hf,03.65.Ud,89.70.Cf The idea of entanglement, that a local measurement on a quantum system may instantaneously affect the outcome of local measurements far away, is a central concept in quantum mechanics.", "As such, it underlies the study of quantum information, communication and computation.", "However, entanglement also plays an increasing role in other areas of physics.", "To name three, measures of entanglement may detect exotic phase transitions in many-body systems lacking a local order parameter [1], [2]; such measures order quantum field theories under renormalization group flow [3], [4]; entanglement is a key concept in the black hole information paradox (see e.g.", "[5], [6]).", "Important measures of entanglement for a many-body system in its ground state are the Rényi and entanglement entropies.", "To define these quantities, we first partition the Hilbert space into pieces $A$ and complement $\\bar{A} = B$ .", "Typically (and hereafter in this letter) $A$ and $B$ correspond to spatial regions.", "Not all quantum systems may allow for such a partition.", "The reduced density matrix is defined as a partial trace of the full density matrix $\\rho $ over the degrees of freedom in $B$ : $\\rho _A \\equiv \\operatorname{tr}_B \\rho \\ .$ The entanglement entropy is then the von Neumann entropy of the reduced density matrix: $S_E \\equiv - \\operatorname{tr}\\rho _A \\log \\rho _A \\ .$ The Rényi entropies are $S_n \\equiv \\frac{1}{1-n} \\log \\operatorname{tr}(\\rho _A)^n \\ ,$ where $S_E = \\lim _{n \\rightarrow 1} S_n$ .", "The entanglement and Rényi entropies are well defined for excited and thermal states as well.", "One simply starts with the relevant excited state or thermal density matrix instead of the ground state density matrix $\\rho = \| 0 \\rangle \\langle 0 \|$ .", "For example, for thermal states, one would start with $\\rho = \\frac{e^{-\\beta H}}{\\operatorname{tr}( e^{-\\beta H})} \\ ,$ where $\\beta $ is the inverse temperature and $H$ is the Hamiltonian.", "However, it is well known that for mixed states, the entanglement entropy is no longer a good measure of quantum entanglement.", "The entanglement entropy is contaminated by the thermal entropy of region $A$ and in fact in the high temperature limit becomes dominated by it.", "It is of course never possible to take a strict zero temperature limit of real world systems.", "If we are to use entanglement entropy to measure quantum entanglement, then we need to know how to subtract off nonzero temperature contributions to $S_E$ and $S_n$ .", "Little has been said about these corrections in the literature thus far.", "Ref.", "[8] studied entanglement entropy for a massive relativistic scalar field in 1+1 dimensions.", "The authors provided numerical evidence that the corrections scale as $e^{-\\beta m}$ in the limit $\\beta m \\gg 1$ where $m$ is the mass of the scalar field.", "They further conjectured that for any many-body system with a mass gap $m_{\\rm gap}$ , such corrections should scale as $e^{-\\beta m_{\\rm gap}}$ when $\\beta m_{\\rm gap} \\gg 1$ .", "This conjecture more or less immediately follows from writing (REF ) as a Boltzmann sum over states.", "By assumption, the first excited state will come with contribution $e^{-\\beta m_{\\rm gap}}$ and give the dominant correction.", "Some possible but probably rare exceptions involve cases where for some reason the first excited state contribution to the entanglement entropy vanishes or where the degeneracy of states with energy $E = m_{\\rm gap}$ itself has an exponential dependence on temperature.", "While knowledge of the $e^{-\\beta m_{\\rm gap}}$ dependence is a good starting point, it would be even better to know the coefficient.", "In this letter we calculate the coefficient for 1+1 dimensional conformal field theories (CFTs) on a circle of circumference $L$ in the case where $A$ consists of a single interval of length $\\ell $ .", "The answer is $\\delta S_n &=& \\frac{g}{1-n} \\left[\\frac{1}{n^{2\\Delta -1}} \\frac{ \\sin ^{2\\Delta } \\left( \\frac{ \\pi \\ell }{L} \\right) }{\\sin ^{2\\Delta } \\left( \\frac{ \\pi \\ell }{n L} \\right)} -n\\right]e^{-2 \\pi \\Delta \\beta / L} +\\nonumber \\\\&& o(e^{-2 \\pi \\Delta \\beta / L})\\ , \\\\\\delta S_E &=& 2g \\Delta \\left[1 - \\frac{\\pi \\ell }{L} \\cot \\left( \\frac{ \\pi \\ell }{L} \\right) \\right] e^{-2 \\pi \\Delta \\beta / L} \\nonumber \\\\&& + o(e^{-2 \\pi \\Delta \\beta / L})\\ ,$ where $\\Delta $ is the smallest scaling dimension among the set of operators including the stress tensor and all primaries not equal to the identity and $g$ is their degeneracy.", "In order for this result to hold, the CFT needs to have a unique ground state separated from the first excited state by a nonzero mass gap (induced by the finite volume of the system).", "The correction () was obtained from the $n \\rightarrow 1$ limit of (REF ).", "Given the form of these corrections, another possible use for them is the determination of $\\Delta $ and $g$ .", "From the literature [9], [10], [11], [12], it is possible to deduce the correction (REF ) in two limiting cases – for free theories and for theories with AdS/CFT duals and large central charge.", "Using bosonization, ref.", "[9], [10] computed these corrections for massless free Dirac fermions.", "In this case, free fermionic operators with scaling dimension $\\Delta =1/2$ and $g=4$ yield the appropriate correction.", "Ref.", "[12] considered CFTs with large central charge $c$ and gravitational duals.", "In this case, the entanglement and Rényi entropies can be written as an asymptotic series in $c$ where the leading term is ${\\mathcal {O}}(c)$ .", "The authors computed the ${\\mathcal {O}}(1)$ corrections to the entanglement and Rényi entropies due to the stress tensor $T(z)$ , which has $\\Delta =2$ and $g=2$ .", "Although the authors do not write explicit formulae, it is trivial to extend their computation to operators of general conformal dimension.", "The computation involves a low temperature expansion of a Selberg-like zeta function.", "In all cases, their expression would match (REF ).", "For free CFTs more generally, the important observation is that this same Selberg-like zeta function appears in ref.", "[11] where the authors compute the determinant of the Laplacian for a $bc$ -system on an arbitrary genus Riemann surface.", "From ref.", "[11], one can then deduce that the same correction (REF ) holds for free CFTs The authors of ref.", "[11] demonstrate their formula only for a $bc$ -system with integer scaling dimension, but it is likely to hold more generally [13].. A check of these statements was performed explicitly for a free complex boson and free fermions in ref.", "[14].", "In this letter, we will show by explicit computation that the corrections (REF ) and () hold for a general 1+1 dimensional CFT.", "We then check that the corrections have expected behavior in the limits $\\ell \\ll L$ and $\\ell \\rightarrow L$ .", "Universal Thermal Corrections: Starting from the thermal density matrix (REF ), we compute the first few terms in a low temperature expansion of the Rényi entropies.", "Introducing a complete set of states, we can write the thermal density matrix as a Boltzmann sum over states: $\\rho = \\frac{1}{\\operatorname{tr}(e^{-\\beta H})} \\sum _{\|\\phi \\rangle } \|\\phi \\rangle \\langle \\phi \| \\, e^{-\\beta E_{\\phi } } \\ .$ For a CFT on a cylinder of circumference $L$ , coordinatized by $w= x - {\\rm i}t $ , the Hamiltonian is $H = \\left( \\frac{2\\pi }{L} \\right) \\left(L_0 + \\tilde{L}_0 - \\frac{c}{12}\\right) \\ ,$ where $L_0$ and $\\tilde{L}_0$ are the zeroth level left and right moving Virasoro generators and $c$ is the central charge.", "We will assume in what follows that placing the CFT on a cylinder produces a unique ground state and gaps the theory.", "Let $\\psi (w) \\ne 1$ be a Virasoro primary operator and $\|\\psi \\rangle = \\lim _{t \\rightarrow -\\infty } \\psi (w) \|0 \\rangle $ the corresponding state where $L_0 \|\\psi \\rangle = h \|\\psi \\rangle $ , $\\tilde{L}_0 \| \\psi \\rangle = \\tilde{h} \| \\psi \\rangle $ and $\\Delta = h + \\tilde{h}$ .", "As the $\|\\psi \\rangle $ are lowest weight states in the CFT, it follows – with one important exception – that the smallest nonzero $E_\\phi $ must correspond to a primary operator $\\psi (w)$ and moreover that $E_\\psi = \\frac{2 \\pi }{L} \\left( \\Delta - \\frac{c}{12} \\right)$ .", "The one important exception is when $\\Delta > 2$ and two descendants of the identity operator, the stress tensor $T(w)$ and its conjugate $\\tilde{T}(\\bar{w})$ , give the dominant correction.", "(The identity operator itself can be thought of as yielding the leading zero temperature contribution to the Rényi entropies.)", "We are assuming $\\Delta > 0$ , which may not be true for non-unitary CFTs, and no gravitational anomaly, i.e.", "$c_L = c_R$ .", "We divide up the spatial circle into regions $A$ and complement $B$ and compute the reduced density matrix $\\rho _A$ .", "Let us consider a low temperature expansion where $\\rho =\\frac{\\left( \|0\\rangle \\langle 0 \| + \| \\psi \\rangle \\langle \\psi \| e^{-2 \\pi \\Delta \\beta /L} + \\cdots \\right)}{1 + e^{-2 \\pi \\Delta \\beta / L} + \\cdots }\\ .$ It follows then that $\\operatorname{tr}(\\rho _A)^n &=& \\frac{\\operatorname{tr}\\left[ \\operatorname{tr}_B \\left( \|0\\rangle \\langle 0 \| + \| \\psi \\rangle \\langle \\psi \| e^{-2 \\pi \\Delta \\beta /L} + \\cdots \\right) \\right]^n}{\\left(1 + e^{-2 \\pi \\Delta \\beta / L} + \\cdots \\right)^n } \\\\&=&\\operatorname{tr}\\left( \\operatorname{tr}_B \|0 \\rangle \\langle 0 \| \\right)^n \\Biggl [ 1 + \\nonumber \\\\&& \\nonumber + \\left( \\frac{ \\operatorname{tr}\\left[ \\operatorname{tr}_B \| \\psi \\rangle \\langle \\psi \| \\left(\\operatorname{tr}_B \|0 \\rangle \\langle 0 \| \\right)^{n-1} \\right] }{\\operatorname{tr}\\left( \\operatorname{tr}_B \|0 \\rangle \\langle 0 \| \\right)^n}- 1 \\right) n e^{-2 \\pi \\Delta \\beta / L } \\\\&& + \\ldots \\Biggr ]$ The first term gives the zero temperature contribution to the Rényi entropy for a finite system [15] $\\log \\operatorname{tr}(\\operatorname{tr}_B \|0 \\rangle \\langle 0 \| )^n = c \\frac{1-n^2}{6n} \\log \\left[ \\frac{L}{\\pi } \\sin \\left( \\frac{ \\pi \\ell }{L} \\right) \\right] + \\ldots $ where $\\ell $ is the length of $A$ .", "Up to a normalization issue we will come to shortly, the expression $\\frac{ \\operatorname{tr}\\left[ \\operatorname{tr}_B \| \\psi \\rangle \\langle \\psi \| \\left(\\operatorname{tr}_B \|0 \\rangle \\langle 0 \| \\right)^{n-1} \\right] }{\\operatorname{tr}\\left( \\operatorname{tr}_B \|0 \\rangle \\langle 0 \| \\right)^n}$ is a two-point function of the operator $\\psi (w)$ .", "By the state operator correspondence, we can interpret the state $\|\\psi \\rangle $ as $\\lim _{t \\rightarrow -\\infty } \\psi (x,t) \|0 \\rangle $ and similarly $\\langle \\psi \| = \\lim _{t \\rightarrow \\infty } \\langle 0 \| \\psi (x,t)$ .", "In path integral language, the trace over an $n$ th power of the reduced density matrix becomes a partition function on an $n$ -sheeted copy of the cylinder, branched over the interval $A$ .", "The expression (REF ) is then the two-point function $\\langle \\psi (w_2, \\bar{w}_2) \\psi (w_1, \\bar{w}_1) \\rangle $ on this multisheeted cylinder for a particular choice of $w_1$ and $w_2$ .", "By the Riemann-Hurwitz theorem, this multisheeted cylinder has genus zero.", "There exists then a uniformizing map which takes the multisheeted cylinder to the plane, and on the plane, the two-point function (REF ) can be computed trivially.", "A uniformizing map from $n$ copies of a cylinder of circumference $L$ to the plane is $\\zeta ^{(n)} = \\left( \\frac{e^{2 \\pi {\\rm i}w/L} - e^{{\\rm i}\\theta _2}}{e^{2 \\pi i w/L} - e^{{\\rm i}\\theta _1}} \\right)^{1/n} \\ .$ We choose $\\theta _1$ and $\\theta _2$ such that the map is branched over the interval $A$ , which has endpoints on the cylinder such that $\\theta _2 - \\theta _1 = \\frac{2 \\pi \\ell }{ L}$ .", "Under this map, an operator inserted on the $j$ th cylinder at $t = -\\infty $ is located on the $\\zeta ^{(n)}$ plane at $\\zeta ^{(n)}_{-\\infty } = e^{{\\rm i}(\\theta _2 - \\theta _1)/n + 2 \\pi {\\rm i}j/ n}$ , while an operator inserted at $t = \\infty $ is located at $\\zeta ^{(n)}_\\infty = e^{2 \\pi {\\rm i}j/n}$ .", "The correlation function on the $\\zeta $ -plane between two quasi-primaries inserted at points $\\zeta ^{(n)}_2$ and $\\zeta ^{(n)}_1$ is $\\left\\langle \\psi \\left(\\zeta ^{(n)}_2,\\bar{\\zeta }^{(n)}_2 \\right) \\psi \\left(\\zeta ^{(n)}_1, \\bar{\\zeta }^{(n)}_1 \\right) \\right\\rangle =\\frac{1}{\\left( \\zeta ^{(n)}_{21} \\right)^{2h} \\left( \\bar{\\zeta }^{(n)}_{21} \\right)^{2\\tilde{h}}}$ using the standard CFT normalization for two point functions.", "Mapping back to the $n$ -sheeted cover of the $w$ -plane, we find that $\\left\\langle \\psi (w_2, \\bar{w}_2) \\psi (w_1, \\bar{w}_1) \\right\\rangle _n =\\frac{ \\left( \\frac{d \\zeta ^{(n)}_1}{dw_1} \\frac{d \\zeta ^{(n)}_2}{dw_2} \\right)^h}{ \\left(\\zeta ^{(n)}_{21} \\right)^{2h}}\\frac{ \\left( \\frac{d \\bar{\\zeta }^{(n)}_1}{d \\bar{w}_1}\\frac{d \\bar{\\zeta }^{(n)}_2}{d \\bar{w}_2} \\right)^{\\tilde{h}} }{ \\left(\\bar{\\zeta }^{(n)}_{21} \\right)^{2\\tilde{h}}} \\ $ where $\\psi (w_1, \\bar{w}_1)$ and $\\psi (w_2, \\bar{w}_2)$ are inserted on just one of the $n$ -sheets.", "We would like to work with a normalization such that $\\langle \\psi \| \\psi \\rangle = 1$ on the original $w$ -cylinder.", "To that end, we need to divide (REF ) by $\\langle \\psi (w_2, \\bar{w}_2) \\psi (w_1, \\bar{w}_1) \\rangle _1$ .", "The following identity is useful $\\frac{ d \\zeta ^{(n)} / d w}{d \\zeta ^{(1)} / dw} = \\frac{1}{n} \\frac{\\zeta ^{(n)}}{\\zeta ^{(1)}} \\ .$ We find then that ${\\frac{\\langle \\psi (w_2, \\bar{w}_2) \\psi (w_1, \\bar{w}_1) \\rangle _n}{\\langle \\psi (w_2, \\bar{w}_2) \\psi (w_1, \\bar{w}_1) \\rangle _1}=} \\nonumber \\\\&&f \\left(\\zeta ^{(n)}, \\zeta ^{(1)}, h \\right)f \\left(\\bar{\\zeta }^{(n)}, \\bar{\\zeta }^{(1)}, \\tilde{h} \\right)\\ ,$ where $f(x, y, h) \\equiv \\frac{1}{n^{2h}} \\left(\\frac{x_1 x_2}{y_1 y_2} \\right)^h \\left( \\frac{y_2-y_1}{x_2-x_1}\\right)^{2h} \\ .$ If we take $t_2 = \\infty $ and $t_1 = -\\infty $ , the expression simplifies, $\\frac{\\langle \\psi (\\infty ) \\psi (-\\infty ) \\rangle _n}{\\langle \\psi (\\infty ) \\psi (-\\infty ) \\rangle _1}&=&\\frac{ 1}{n^{2\\Delta }} \\left( \\frac{\\sin \\frac{\\theta _2 - \\theta _1}{2}}{\\sin \\frac{\\theta _2 - \\theta _1}{2n}}\\right)^{2\\Delta }\\ .$ The result (REF ) then follows immediately and () by taking the $n \\rightarrow 1$ limit of (REF ).", "The degeneracy factor $g$ comes from the fact that we can choose an orthonormal basis for operators $\\psi _i$ , $i=1, \\ldots , g$ with dimension $\\Delta $ where each $\\psi _i$ contributes equally to (REF ).", "A numerical confirmation of () is presented in figure REF for free fermions.", "We note in passing that a similar calculation was performed in ref.", "[17].", "In that paper, the authors consider Rényi entropies in excited states of the CFT, for which they need to insert two operators $\\psi (z, \\bar{z})$ on all of the $n$ sheets of the cylinder.", "This makes the actual computation impractical for $n>2$ .", "By contrast, the leading correction term in our calculation comes from inserting only two operators and may therefore be carried out for arbitrary $n$ .", "Figure: Thermal corrections to entanglement entropy δS E =S E (T)-S E (0)\\delta S_E = S_E(T)-S_E(0) plotted against ℓ/L\\ell /L for a free, massless 1+1 dimensional fermion.", "The points are numerically determined from a lattice model of size N=100N=100 grid points using the method described in , .", "LL is the size of the circle and ℓ\\ell of the interval.", "From top to bottom, the points correspond to LT=0.15,0.2,0.3,0.4,0.5LT = 0.15, 0.2, 0.3, 0.4, 0.5.", "The solid curve is the prediction ().", "Inset:δS E \\delta S_E Plotted against LTLT for ℓ=L\\ell = L. The curve is the thermal entropy correction 4(1+π/LT)4 ( 1 + \\pi / LT).Limiting Forms for the Corrections: In the limit $\\ell \\ll L$ , the correction (REF ) can be written as a power series in $\\ell /L$ , $\\delta S_n &=& \\left[ g \\Delta \\frac{1+n}{3n} \\frac{\\pi ^2 \\ell ^2}{L^2} + {\\mathcal {O}}\\left( \\frac{\\ell ^4}{L^4} \\right) \\right] e^{-2 \\pi \\Delta \\beta /L} \\nonumber \\\\&&+ o(e^{-2 \\pi \\Delta \\beta / L})\\ .$ We can understand this form of the correction from a twist operator formulation of the Rényi entropy.", "Instead of working on the uniformized Riemann surface $\\zeta ^{(n)}$ , we work on the individual cylinders coordinatized by $w$ but now with twist operators $P(\\ell /2)$ and $P(-\\ell /2)$ at the endpoints of the interval $A$ .", "We can write down a couple of terms in the OPE of the twist operators [18], [19], [20] We take there to be $2n$ twist operators, 2 on each cylindrical sheet of the Riemann surface.", "Note the coefficient of the stress tensor in the OPE and the exponent of the leading $\\ell $ are fixed by the conformal dimension of the twist operators, $h = \\tilde{h} = (c/24)(1-1/n^2)$ [15].", "Recall that the stress tensor $T(z)$ will appear with coefficient $2h/c$ in the OPE of two primary operators, provided that the primaries are normalized such that the two point function takes the standard form (REF ).", "Here there is instead an overall multiplicative factor $c_n$ .. ${P( \\ell /2) \\tilde{P}(- \\ell /2) = c_n \\ell ^{-(c/6) (1-1/n^2)} \\Biggl ( 1 + \\ldots }\\nonumber \\\\&&+ \\frac{1}{12}\\left(1 - \\frac{1}{n^2} \\right) \\ell ^2 (T(0) + \\tilde{T}(0) )+ \\ldots \\Biggr ) \\ .$ (A discussion of the normalization constants $c_n$ can be found in ref.", "[7].)", "The leading $(\\ell /L)^2$ correction (REF ) will then come from a three point function involving the stress tensor and the two quasi-primary fields $\\lim _{t \\rightarrow \\infty } \\psi (x,t)$ and $\\lim _{t \\rightarrow -\\infty } \\psi (x,t)$ producing, via the operator state correspondence, the first excited state in the Boltzmann sum (REF ).", "Recall the three point function of the stress tensor with two quasi-primary fields $\\psi (z, \\bar{z})$ of dimension $\\Delta = h + \\tilde{h}$ is $\\langle \\psi (z_2, \\bar{z}_2) T(z_3) \\psi (z_1, \\bar{z}_1) \\rangle = h\\frac{z_{12}^2}{z_{31}^2 z_{32}^2} \\frac{1}{\|z_{12}\|^{2 \\Delta } }\\ .$ If we now transform to the cylinder, $z = e^{2 \\pi {\\rm i}w / L}$ , choosing $z_1 = 0$ and $z_2 = \\infty $ , we find that $\\frac{\\langle \\psi (w_2, \\bar{w}_2) T(w_3) \\psi (w_1, \\bar{w}_1) \\rangle }{\\langle \\psi (w_2, \\bar{w}_2) \\psi (w_1, \\bar{w}_1) \\rangle } = -\\frac{4 \\pi ^2}{L^2} h + \\frac{\\pi ^2 c}{6 L^2}\\ .$ The small $\\ell $ correction should then be $g \\cdot \\frac{1}{1-n} \\cdot n \\cdot \\left(1 - \\frac{1}{n^2} \\right) \\frac{\\ell ^2}{12} \\cdot \\left( -\\frac{4 \\pi ^2 \\Delta }{L^2} \\right) \\cdot e^{-2 \\pi \\Delta \\beta / L}\\ ,$ which matches with (REF ).", "The $g$ is the degeneracy factor.", "The $1/(1-n)$ comes from the definition of the Rényi entropy.", "The $n$ is the number of different sheets where we can compute this three point function.", "The $(1-1/n^2)\\ell ^2/12$ comes from the OPE of the twist operators.", "Finally, we get contributions from three point functions involving $T(w)$ and $\\tilde{T}(\\bar{w})$ and a Boltzmann factor.", "Note the Schwarzian derivative contribution to the three point function (24) does not contribute to (25) but instead gets incorporated into the leading zero temperature part of the Rényi entropies.", "In the limt $\\ell \\rightarrow L$ , the correction (REF ) becomes $\\delta S_n &=& \\left[ -g\\frac{n}{1-n} + {\\mathcal {O}}\\left( 1 - \\frac{\\ell }{L} \\right)^{2 \\Delta } \\right] e^{-2 \\pi \\Delta \\beta / L} \\nonumber \\\\&&+ o(e^{-2 \\pi \\Delta \\beta /L} ) \\ .$ This expression traces its origin to the normalization of the thermal density matrix, i.e.", "the $e^{-2 \\pi \\Delta \\beta / L}$ in the denominator of (REF ), and leads to a peculiar singularity in the limit $n \\rightarrow 1$ .", "Interestingly, the analytic continuation $n \\rightarrow 1$ is sensitive to the order of limits $\\beta \\rightarrow \\infty $ and $\\ell \\rightarrow L$ .", "Above, we have taken $\\beta \\rightarrow \\infty $ first.", "If we instead first take $\\ell \\rightarrow L$ , and then take $n \\rightarrow 1$ , the coefficient of $e^{-2 \\pi \\Delta \\beta / L}$ now becomes singular in the limit $\\beta \\rightarrow \\infty $ .", "Let us see how this result comes about in greater detail.", "The $n$ th Rényi entropy can be computed from the partition function of an $n$ -sheeted copy of the torus, glued along the interval 0 to $\\ell $ .", "On each sheet of the Riemann surface, we insert the identity operator $\\operatorname{Id} = P P^{-1}$ along a spatial circle where $P$ shifts up a sheet and $P^{-1}$ shifts down.", "Then, we move $P^{-1}$ so that it overlaps and partially cancels with the interval 0 to $\\ell $ , leaving $P^{-1}$ along the interval $\\ell $ to $L-\\ell $ .", "In the limit $\\ell \\rightarrow L$ , we can replace the sewing prescription along the interval $\\ell $ to $L-\\ell $ with the OPE of twist operators that sit at $\\ell $ and $L-\\ell $ .", "The effect of the remaining $P^{-1}$ around the spatial circle is to glue the $n$ tori of length $\\beta $ into a single torus of length $n \\beta $ .", "Using the OPE (REF ), we find that $\\operatorname{tr}(\\rho _A)^n &=& c_n^n (L-\\ell )^{(-c/6)(n-1/n)}\\frac{ \\operatorname{tr}(\\rho ^n)}{(\\operatorname{tr}\\rho )^n} + \\ldots $ We will return to $L-\\ell $ dependent corrections in a moment.", "The leading $(L-\\ell )$ dependence yields the universal single interval Rényi entropy for a CFT on a line while the remaining ratio of thermal density matrices gives the thermal Rényi entropy for the whole system.", "The $n\\rightarrow 1$ limit of the thermal Rényi entropy will yield the ordinary thermal entropy.", "At low temperature for our system, the thermal Rényi entropy can be expanded, $\\frac{ \\operatorname{tr}(\\rho ^n)}{(\\operatorname{tr}\\rho )^n} =\\frac{1 + g e^{-2 \\pi \\Delta n \\beta /L} + \\ldots }{(1+ g e^{-2 \\pi \\Delta \\beta / L} + \\ldots )^n} \\ .$ From the $n\\rightarrow 1$ limit of (REF ), we may compute the entanglement entropy for the interval $A$ : $S_E &=& \\Biggl [\\frac{c}{3} \\log \\frac{L-\\ell }{\\epsilon } + g \\left(1 + \\frac{2 \\pi \\Delta \\beta }{L} \\right) e^{-2 \\pi \\Delta \\beta / L} + \\ldots \\Biggr ] \\nonumber \\\\&&+ {\\mathcal {O}}((L-\\ell )^2,(L-\\ell )^{2 \\Delta }) \\ .$ Note that the ${\\mathcal {O}}(e^{-2 \\pi \\Delta \\beta /L})$ correction matches well the result for free fermions, shown in the inset of figure REF .", "We expect an ${\\mathcal {O}}(L-\\ell )^2$ correction to (REF ) from expanding the universal leading $\\log \\sin (\\pi \\ell / L)$ piece of the entanglement and Rényi entropies on a circle.", "In terms of the OPE (REF ), this correction comes from the one point function of the stress tensor on a cylinder.", "Note that this correction is temperature independent and does not come with a $e^{-2 \\pi \\Delta \\beta / L}$ Boltzmann factor.", "The leading correction that does come with a $e^{-2 \\pi \\Delta \\beta / L}$ suppression should scale as ${\\mathcal {O}}(L-\\ell )^{2 \\Delta }$ .", "This correction will come from a $\\psi ^2$ term in the OPE of the two twist operators [20].", "By dimensional analysis, the coefficient of $\\psi ^2$ in the OPE contains the $(L-\\ell )^{2\\Delta }$ dependence while the Boltzmann suppression comes from two point functions of the $\\psi $ operators separated by a distance $\\beta $ on the torus of length $n \\beta $ and width $L$ .", "Discussion: For pure states, the Rényi entropy of a region and its complement are equal, $S_n(A) = S_n(B)$ , as follows from a Schmidt decomposition of the Hilbert space (see for example [16]).", "However, for thermal states, this symmetry is broken.", "Indeed for our corrections (REF ), $\\delta S_n(\\ell ) \\ne \\delta S_n(L-\\ell )$ .", "However, the Rényi entropies and our corrections do have the symmetry $S_n(\\ell ) = S_n(n L - \\ell )$ .", "To see this symmetry, imagine taking one of the twist operators around a spatial cycle on the torus $(n-1)$ times.", "Then, on the interval $(0,\\ell )$ we have a permutation $P_n^n=1$ , and on the complementary interval a permutation $P_n^{n-1}$ which is equivalent to $P_n$ .", "(For the correction (REF ), note that one should take the modulus of the ratio of sines.)", "The corrections (REF ) and () are very similar in spirit to universal corrections to two interval entanglement entropy on the plane [18], [19], [20].", "For two intervals of length $\\ell _1$ and $\\ell _2$ whose centers are separated by a distance $r$ , an OPE argument predicts a leading correction to the Rényi entropies of the form $\\delta S_n = C_n \\left( \\frac{\\ell _1 \\ell _2}{n^2 r^2} \\right)^{\\Delta } \\ ,$ where $\\Delta $ is again the lowest conformal scaling dimension among the primary operators and $C_n$ is known and calculable.", "It would be interesting to see if similar formulae can be found for CFTs in higher dimension.", "Acknowledgments: We would like to thank K. Balasubramanian, P. Calabrese, T. Faulkner, T. Hartman, S. Hartnoll, M. Headrick, M. Kulaxizi, A. Parnachev, T. Nishioka, L. Takhtajan, and E. Tonni for discussion.", "We thank the Kavli Institute for Theoretical Physics, Santa Barbara, where this research was carried out, for its hospitality.", "This work was supported by the National Science Foundation (NSF) under Grant No.", "PHY11-25915.", "J. C. was supported in part by the Simons Foundation, and C. H. by the NSF under Grants No.", "PHY08-44827 and PHY13-16617.", "C. H. thanks the Sloan Foundation for partial support." ] ]	1403.0578
[ [ "Network Traffic Decomposition for Anomaly Detection" ], [ "Abstract In this paper we focus on the detection of network anomalies like Denial of Service (DoS) attacks and port scans in a unified manner.", "While there has been an extensive amount of research in network anomaly detection, current state of the art methods are only able to detect one class of anomalies at the cost of others.", "The key tool we will use is based on the spectral decomposition of a trajectory/hankel matrix which is able to detect deviations from both between and within correlation present in the observed network traffic data.", "Detailed experiments on synthetic and real network traces shows a significant improvement in detection capability over competing approaches.", "In the process we also address the issue of robustness of anomaly detection systems in a principled fashion." ], [ "Introduction", "In its most abstract form, network traffic can be described by a time series $y(t)$ , where $y$ represents the observed state of the traffic.", "For example, $y(t)$ could simply be the total number of packets or could be a vector, where each component represents an active flow.", "A flow is an aggregation of packets by attributes like source and destination ip address.", "In order to detect anomalies in network traffic we must first model the generative process, which gives rise to the observable time series $<y(t)>$ .", "Assume that the latent variables $x(t)$ .", "The relationship between $y(t)$ and $x(t)$ can be abstractly represented by a model as $y(t)=f(x(t))$ .", "We can learn the model and obtain an estimation as $\\hat{y}=f(\\hat{x}(t))$ .", "Then an anomaly occurs of time $t$ if $y(t)-\\hat{y}(t)$ is greater than a pre-defined threshold.", "In order to design the generative model we have to capture different forms of correlation between variables of the system which we describe here.", "An important aspect that needs to be captured in any model of network traffic is the presence of between and within correlation in packet flows.", "For example, consider Figure REF (a), which shows the the time series of two flows, $f_{1}(t)$ and $f_{2}(t)$ .", "The point labeled $D$ is an example where the correlation within flow $f_{1}(t)$ flows has deviated from the expected norm.", "Similarly, the point labeled $P$ is where the correlation between the two flows $f_{1}$ and $f_{2}$ has deviated in a localized time window.", "The anomaly $D$ is an example of a Denial of Service (DoS) attack while an anomaly $P$ is an example of port scan.", "Discovering events like P and D is the focus of this paper.", "Figure: (a) An example of two flows f 1 f_1 and f 2 f_2 experiencing two different anomalies DoS attack (D) and port scan (P).", "(b) SVD finds D anomaly and misses P one as it is in its normal space.", "(c) and (d): mapping the f 1 f_1 and f 2 f_2 vector into a 2-dimensional space and applying SVD both P and D anomalies are detectable." ], [ "The Trajectory/Hankel Matrix", "A key tool that we will use to detect correlation deviation in network traffic, is the trajectory (or Hankel) matrix that will be constructed from the observed time series (see [28], [5]).", "For example, given two flows $\\lbrace f_{1}(i),f_{2}(i)\\rbrace _{i=1}^{T}$ , the Hankel matrix ($H$ ) of window length $L < T$ of the two flows is given by $\\left[\\begin{matrix}f_{1}(1) & \\hdots & f_{1}(L) \\\\f_{1}(2) & \\hdots & f_{1}(L+1) \\\\\\vdots & \\vdots & \\vdots \\\\f_{1}(T-L+1) & \\hdots &f_{1}(T) \\\\\\end{matrix}\\right\|\\left.\\begin{matrix}f_{2}(1) & \\hdots & f_{2}(L) \\\\f_{2}(2) & \\hdots & f_{2}(L+1) \\\\\\vdots & \\vdots & \\vdots \\\\f_{2}(T-L+1) & \\hdots & f_{2}(T) \\\\\\end{matrix}\\right]$ Now the key insight of the paper, is that the SVD of correlation (or covariance) matrix of the Hankel matrix ($H$ ), will capture both between and within correlation in network flows.", "Thus a low rank decomposition of $H$ will characterize the manifold structure $M$ between the flows as well as help identify the anomalies which deviate from the inferred manifold structure.", "For example, Figure REF (b), shows the relationship between the flows $f_{1}$ and $f_{2}$ and also the direction of the most dominant eigenvector of the standard correlation matrix (without the time lag).", "This decomposition is unable to capture the port scan (P) anomaly because, $P$ is not a simple violation of the between flow correlation but the existing correlation is violated only in a localized time window.", "In Figure REF (c), it is clear that a time window lag ($L=1$ ), captures the spatial correlation in a small time window and thus the $P$ anomaly is away from the main eigenvector.", "In Figure REF (d), there is no correlation violation within flow $f_{2}$ and thus the $P$ anomaly is in the direction of the main eigenvector.", "The remainder of this paper is structured as follows.", "Section explains the technique behind the singular spectrum analysis and its extension and compares both techniques with PCA.", "Section presents a validation of the different analysis algorithm based on SSA on a real traffic data and analyses their capability for anomaly detection.", "A brief background is presented in section and we discuss some conclusion remarks in section .", "We now justify the decomposition of the Hankel matrix based on a generative model of the data.", "In particular we will show that if data is generated by a Linear Dyanmical System (LDS), then the SVD decomposition of the Hankel matrix can be used to estimate the LDS.", "Assume data is generated from a Linear Dynamical System (LDS) given by: $\\begin{matrix}x(t + 1) & = Ax(t)+ w(t) \\\\y(t) & = Cx(t)+ v(t)\\end{matrix}$ where $x(t) \\in \\mathbb {R}^n$ is the system state vector, $A$ defines the system's dynamics, $w$ is the vector that captures the system error, e.g.", "a random vector from $\\mathcal {N}(0,Q)$ , $y(t)\\in \\mathbb {R}^m$ is the observation vector, $C)$ is the measurement function, $v$ is the vector that represents the measurement error, e.g.", "a random vector from $\\mathcal {N}(0,R)$ , Fig.REF presents a graphical model of LDS.", "Assume that data is generated from an LDS governed by the equation above.", "Given a sequence of observations $\\lbrace y_{i}\\rbrace _{i=1}^{n}$ , estimate $A,C,Q$ and $R$ .", "Figure: A linear dynamic latent model (LDS)To solve the above problem, we need to define the Hankel matrix of the observations as $H(t)= \\begin{pmatrix}y(t) & y(t+1) & y(t+2) & ... & y(n-\\ell +1)\\\\y(t+1)& y(t+2) & \\ddots & & \\vdots \\\\\\vdots &&&&\\\\y(t+\\ell ) & \\ldots & & & y(n)\\\\\\end{pmatrix}$ where y(t) is $m \\times n$ observation at time $t$ , and $H$ is a $\\ell \\times n^{\\prime }$ where $n^{\\prime }=n-\\ell +1$ .", "Equivalently, $H$ is a Hankel matrix if and only if there exists a sequence ,$s_1,s_2,...$ such that $H_{i,j}=s_{i+j-1}$ (see [12]).", "Therefore, every Hankel matrix uniquely determines a time series and every time series can be transferred into a Hankel matrix, i.e.", ": $H(t-i) \\Leftrightarrow y^i(t)$ where $y^i(t)=\\lbrace y(i), y(i+1),...,y(t),...\\rbrace $ .", "By replacing the entries of the Hankel matrix with their equivalent from the LDS: $\\small {H(1)= \\begin{pmatrix}CAx(0) & CAx(1) & CAx(2) & ... & CAx(n-\\ell )\\\\CAx(1) & CAx(2)& \\ddots & & \\vdots \\\\\\vdots &&&&\\\\CAx(\\ell -1)& \\ldots & & & CAx(n-1)\\\\\\end{pmatrix}=\\begin{pmatrix}CAx(0) & CA^2x(0) & CA^3x(0) & ... & CA^{n-\\ell +1}x(0)\\\\CA^2x(0) & CA^3x(0)& \\ddots & & \\vdots \\\\\\vdots &&&&\\\\CA^{\\ell }x(0)& \\ldots & & & CA^nx(0)\\\\\\end{pmatrix}}$ $=\\begin{array}{rll}\\begin{pmatrix}CA && CA^2 && CA^3 & ... && CA^{\\ell }\\end{pmatrix}^T & \\cdot &\\begin{pmatrix}x(0)& &Ax(0)&& A^2x(0)&&\\ldots && A^{n-\\ell -2}x(0)\\\\\\end{pmatrix}\\end{array}$ Define: $\\begin{array} {l}P=\\begin{pmatrix} CA & & CA^2 & &CA^3 & ... & &CA^{\\ell }\\end{pmatrix}^T \\\\Q=\\begin{pmatrix} x(0)& & Ax(0)& &A^2x(0)&\\ldots & &A^{n-\\ell -2}x(0) \\end{pmatrix}\\end{array}$ then: $\\begin{array}{rl}H(1)=& PQ\\end{array}$ The shifted Hankel matrices can be described by: $\\begin{array}{rl}H(i)=& PA^{i-1}Q\\end{array}$ To obtain the matrices $A$ and $B$ , perform singular value decomposition of $H(1)$ : $\\begin{array} {rl}H(1)& = U \\Sigma ^2 V^T \\\\\\end{array}$ where $\\Sigma ^2$ is a diagonal $\\ell \\times \\ell $ matrix containing the singular values and the $\\ell $ columns of $U$ are the singular vectors.", "Selecting the top-k $(1< k < \\ell )$ singular values from the matrix $\\Sigma ^2$ ,denoted by $\\Sigma _k$ , and $k$ associated singular vectors, denoted by $U_k$ , we define reduced rank matrices: $\\begin{array}{rl}P_k \\doteq & U_k \\Sigma _k\\end{array}$ $\\begin{array}{rl}Q_k \\doteq & \\Sigma _k V^T\\end{array}$ Using the 1-shifted Hankel matrix $H(2)$ and the reduced rank matrices $P_k$ and $Q_k$ : $\\begin{array} {rl}H(2)& = P_kA_kQ_k \\\\& = U_k \\Sigma A_k\\Sigma V^T \\\\\\end{array}$ Then the matrix $A_k$ can be approximated as: $\\begin{array} {rl}A_k & = (U_k \\Sigma _k)^{-1} H(2) (\\Sigma _k V^T)^{-1} \\\\\\end{array}$ Then, given $A_k$ we can estimate $C_k$ as: $\\begin{array} {rl}C_k & = P_1^{-1} A_k \\\\\\end{array}$ where $P_1$ is the first $m$ rows of the matrix $P$ .", "Given $A_k$ and $C_k$ , we can estimate $\\begin{array} {rl}\\delta _k=& y-\\hat{y}\\\\=& y-C_k\\hat{x}\\end{array}$ An outlier is reported whenever $\|\\delta _k\|$ exceeds a predefined threshold.", "In practice, we are able to use the decomposition of the Hankel matrix to identify outliers.", "Recall once again the SDV of the Hankel matrix: $\\begin{array} {rl}H(1)& = U \\Sigma ^2 V^T \\\\& = \\sum _{i=1}^{k} \\lambda _i^{1/2}U_iV_i^{\\prime } + \\sum _{i=k+1}^{\\ell } \\lambda _i^{1/2}U_iV_i^{\\prime } \\\\\\end{array}$ If we define $\\hat{H} \\doteq \\sum _{i=1}^{k} \\lambda _i^{1/2}U_iV_i^{\\prime }$ and $\\Delta _k \\doteq \\sum _{i=k+1}^{\\ell } \\lambda _i^{1/2}U_iV_i^{\\prime }$ then: $\\begin{array} {rl}\\Delta _k & = H(1) - \\hat{H} \\\\\\end{array}$ We know that every Hankel matrix is associated with a time series.", "Therefore if these matrices would be Hankel then we can obtain the error space.", "This can be performed by means of diagonal averaging procedure.", "The averaging over the diagonals $\\mathrm {i+j=const}$ of a matrix is called Hankelization.", "It transforms an arbitrary $\\ell \\times n^{\\prime }$ matrix to the form of a Hankel matrix, which can be subsequently converted to a time series.", "A Detailed procedure of Hankeliztion is given in Appendix ." ], [ "Multivariate Singular Spectrum Analysis", "The application of SVD to Hankel matrix is known as SSA or M-SSA.", "The key advantage of M-SSA is its ability to succinctly capture both between (spatial) and within (temporal) correlation in the underlying network traffic flows.", "Here we give a step-by-step introduction to SSA, as a method of discovering anomalies.", "Assume the network flow volume through a router at a pre-specified level of granularity (e.g.five minutes) is given by the time series.", "$y_{1},y_{2},\\ldots ,y_{m},w_{m+1},w_{m+2},\\ldots ,w_{n},y_{n+1},y_{n+2},\\ldots $ We have used both $y$ and $w$ to indicate that the nature of traffic has changed for $n-m+1$ time steps after $y_{m}$ .", "In practice we of course don't know where and when the traffic changes and is precisely what we want to infer.", "Choose an integer $\\ell < m$ , known as the embedding dimension and form the Hankel matrix for the $x$ part of the time series.", "${\\bf Y} =\\left(\\begin{array}{llll}y_{1} & y_{2} & \\ldots & y_{\\ell } \\\\y_{2} & y_{3} & \\ldots & y_{\\ell +1} \\\\\\ldots & \\ldots & \\ldots & \\ldots \\\\y_{m-\\ell +1} & y_{m-\\ell +2} & \\ldots & y_{m}\\end{array}\\right)$ Where each ${\\bf Y_{i}} = (y_{i},y_{i+1},\\ldots ,y_{i+\\ell })^{\\prime }$ , is of dimension $\\ell $ .", "In SSA, the assumption is that ${\\bf Y}$ captures the main dynamics of the network flow.", "We now apply the Singular Value Decomposition (SVD) of ${\\bf Y}$ as follows.", "For the $\\ell \\times \\ell $ covariance matrix of $Y$ give by $C = Y \\times Y^{\\prime }$ Compute the eigendecomposition of $C =[U,D]$ where $U$ is matrix where each column is a eigenvector and $D$ is the diagonal matrix of eigenvalues.", "The relationship between $C$ , $U$ and $D$ is given as $CU(:,i) = D(i,i)U(:,i) \\mbox{ for each $i$ }$ Form an $k$ -dimensional subspace $M$ of $R^{\\ell }$ where $k \\le \\ell $ , by using the top-k eigenvectors of $U$ , i.e., ${\\bf M} = U_{s}U_{s}^{\\prime }$ .", "The space ${\\bf M}$ is where the “normal” traffic lives and our objective is to look for changes in the flow which cannot be explained by ${\\bf M}$ .", "This is achieved by projecting a sliding window of $\\ell $ dimensional vectors on $M$ and raising an alarm whenever the deviation between a vector and its projection on $M$ becomes large.", "For example, consider a $\\ell $ -dim vector which contains parts of the changed traffic $y_{i}^{\\prime }s$ .", "${\\bf z} = (y_{m-1},y_{m},w_{1},\\ldots , w_{\\ell - m -1})^{\\prime }.$ Then, the deviation between ${\\bf z}$ and its projection on $M$ is given by ${\\bf e} = \\Vert {\\bf z - Mz}\\Vert $ .", "Assuming that the $w_{i}$ 's were generated by anomalous traffic, then the deviation ${\\bf e}$ will be large relative to deviations caused by normal traffic.", "To reconstruct the refined time series we proceed in a manner inverse to the step 2.", "On the other hand, if the objective is to reconstruct the original time series then we have to apply a hankelization (inverse) operator.", "The network anomaly detection process remains unaffected by the inverse operation.", "More details can be found in [29], [10], [11].", "Before we go into further details about SSA we illustrate the key steps using a simple example.", "Assume that a sample time series is given as ${\\bf y(t)} =\\left\\lbrace \\begin{array}{ll}sin(.2t) + \\varepsilon (t) & \\textit {if} \\quad 1 \\le t \\le 175 \\\\sin(.3t) + \\varepsilon (t) & \\textit {if} \\quad 176 \\le t \\le 375 \\\\sin(.2t) + \\varepsilon (t) & \\textit {if} \\quad 376 \\le t \\le 560 \\\\\\end{array}\\right.$ Here $\\varepsilon (t)$ is gaussian $\\mathcal {N}(0,1)$ noise.", "Notice that there is a change in the time series between $t=176$ and $t=376$ .", "Fig.", "REF (a and b) show the example time series without the noise and the time series with added noise.", "Fig.", "REF (c) shows the deviation of the signal for different values of $\\ell $ and $k$ .", "It is clear that the deviation becomes larger near time step 176 and then returns to its normal value after the change signal disappears around time step 376.", "Figure: An example of using SSA to detect changes in a time series forvarious combination of parameter values ℓ\\ell and kk.", "The timeseries changes in the middle which is reflected in the deviationin the bottom figure." ], [ "Choice of Parameters in SSA", "The key idea in SSA is the use of a trajectory matrix ${\\bf Y}$ which then factorized using SVD.", "The formal relationship between $Y$ and the underlying dynamics of the time series has been extensively researched in both the statistics and physics community.", "The key take away from the theoretical literature is that for an appropriate choice of $\\ell $ , the trajectory matrix will capture the appropriate dynamics of the underlying system (see [28], [5], [7], [6]).", "The choice of $\\ell $ along with $k$ (the dimensionality of the projected subspace) and the threshold $({\\bf e})$ are three important parameters that need to calibrated and set.", "These parameters are like “knobs” which a network administrator can use to adapt to specific network characteristics.", "Example 1 above already provides some indication of how the choices of $\\ell $ and $k$ have on time series monitoring.", "For example, for $\\ell =20$ , the deviation ${\\bf e}$ is less than for other values of $\\ell $ .", "This may surprising at first but notice the initial part of the time series has an intrinsic dimensionality of 1 (as it is composed of one $\\sin $ term).", "Thus a smaller value of $\\ell $ is better at capturing the dynamics of the time series than a larger value $\\ell =50,70$ .", "Now consider, the two cases where $L=50$ but $k=2$ or $k=4$ .", "Notice that the projected error (in the middle) is almost identical but at the tails the projection error is higher for $k=2$ than $k=4$ .", "This shows that while the choice of $k$ has a significant impact on the projection error of the normal traffic, when it comes to detecting the anomalous part the method is quite robust for different choices of $k$ .", "In fact this is one of the key strengths of SSA that we will exploit in the analysis of real network traffic data." ], [ "Network Anomaly Types", "A key contribution of our paper is that the approach based on M-SSA is able to detect almost all known types of network anomalies.", "In this section we describe the different types of common anomalies and explain why M-SSA provides subsumes other anomaly detectors.", "Table REF lists the common anomalies defined using the flow as a 5-tuple (source IP address, destination IP address, source port number, destination port number, transport protocol).", "More details can be found in [25], [26], [18], [16].", "A Denial of Service (DoS) attack occurs when the attacking hosts send a large number of small packets - typically TCP SYN segments - to the attacked host and service, i.e.", "a single IP address and port number, in order to deplete the system resources in the target host.", "The resulting traffic from DoS attack consists of a relatively small number of flows with large packet counts as DoS attack tools often forge the source port number.", "Note that the specific case of Distributed Denial of Service (DDoS) attacks is effectively the same attack, but with several source IP addresses.", "The number of attacking hosts however, is typically much smaller than the packet count.", "We thus consider DDoS to be a special case of a DoS attack, and label as such.", "Port scans are typically used by attackers to discover open ports on the target host.", "This is accomplished by sending small packets as connections requests to a large number of different ports on a single destination IP address.", "At the flow level, they are therefore characterized as an increase in the number of flows, each with a small packet count.", "Large file transfers are characterized by a few flows with packet counts which are significantly larger than what common applications use.", "Prefix outages occurs when part of the network becomes unreachable, they can be identified when traffic from one or more IP prefixes disappears, which translates in a drop in the number of flows.", "Link outage is in a way a more severe version of Prefix outage, where the number of flows on the link drop close to zero.", "Table: Network anomalies considered" ], [ "Experimental evaluation", "We have evaluated our proposed approach using both real and synthetic data sets.", "For comparison we have implemented well known network anomaly techniques based on wavelets, kalman filtering, fourier analysis and the more recent ASTUTE method.", "The use of synthetic data sets and simulation is a prerequisite for a rigorous evaluation strategy for network anomaly detection ([22], [27], [25]).", "Table: Alternative methods used in the experiments" ], [ "Detection Capability", "We evaluate the detection capability of M-SSA using two real network traces which we now describe.", "The first traffic trace if from the Abilene networkInternet2 - http://www.internet2.edu/ and has been used previously for network anomaly detection (see [25], [26], [18], [16]).", "The data set consists of a one month traffic trace from a backbone router in New York during August 2007.", "The Juniper router used to collect the data generated sampled J-flow statistics at the rate of 1/100.", "The flows were aggregated at five minute intervals.", "The key attributes of the flow are: number of packets, number of distinct source IP addresses, number of distinct destination IP addresses, number of distinct source port numbers and number of distinct destination ports numbers.", "The second, and more recent, traffic trace is from the MAWI (Measurement and Analysis on the WIDE Internet) archive project in Japanhttp://www.wide.ad.jp/project/wg/mawi.html.", "Here the data was sampled from a 150Mbps trans-pacific link between Japan and the United States for 63-hours in April 2012.", "Labelling traffic traces with anomalies is notoriously difficult.", "The commonly accepted method is to combine algorithmic detection with manual inspection of the data.", "We have followed the URCA (Unsupervised Root Cause Analysis) method proposed by [24] with a false positive rate of $2 \\times 10^{-9}$ , followed by a thorough manual inspection of the data set." ], [ "Results", "Table REF and Fig.", "REF show the results of the different methods including M-SSA.", "The following are the key take aways.", "M-SSA is capable of detecting a much wider range of anomalies regardless of their types.", "For the Abilene data, M-SSA was able to identify 100% of DoS attacks and over 95% port scans.", "Similarly on the MAWI data set the detection rate was 100% for DoS attacks and over 90% for port scans.", "All other techniques (which were compared) can be placed in two groups: Wavelets, Kalman and Fourier have high detection rates only for DoS attacks while ASTUTE performs exceedingly well only for port scan anomalies.", "In the Abilene data, around 7% of the anomalies are related to link outages.", "Here again, M-SSA has a 100% detection rate and except for Fourier, other techniques also have a high detection rate with Wavelets doing the best.", "Table: Number of anomalies per type found by each technique in two traffic traces from Abilene and WIDE networks.", "M-SSA is able to discover both DoS and port scan in both networks.Figure: Timeseries plots of measured and reconstructed data along with related residual vector squared magnitude; for one day of both traffic traces from Abilene and WIDE networks.", "Triggered alarms shown as red circles.To understand the results better we have carried out a deeper analysis by examining the characteristic features of the anomalies.", "In Fig.", "REF we plot the known Abilene anomalies using two features.", "The x-axis represents the change in packet counts between two consecutive time bins.", "The y-axis represents the number of distinct flows (5 tuples) in the time bin.", "The first observation is that the set of anomalies are clustered in distinct groups, with the set of anomalies detected by Wavelet and Kalman approximately common (Wavelet is slightly better in detecting some port scans).", "Secondly, the Kalman filter and Wavelet techniques are not able to find anomalies caused by large number of flows with small packet counts.", "These includes anomalies where the rate of change in packet count in individual flows over time is small, e.g.", "port scans, prefix outages and file transfers.", "Wavelet as a time-frequency technique is able to flag sudden changes in traffic, but will miss any small variations such as port scans and absorb them in the main trend.", "The Kalman filter technique is effective at detecting anomalies when the packet count variation over time is significant, such as DoS attacks.", "This is expected, as Kalman Filtering is essentially a forecasting technique in the time dimension.", "Another observation is that ASTUTE is not able to detect anomalies involving a few large flows (bottom right hand corner of Fig.", "REF ), such as DoS attacks.", "This is also expected, as ASTUTE is not able to detect large volume change in a few number of flows, because the $AAV$ process threshold is not violated (as the denominator of $AAV$ is the standard deviation which will be large) as mentioned by [26], [24].", "The results and analysis clearly suggest, as has been noted before by [25], that a hybrid approach consisting of ASTUTE and Kalman (or Wavelet) will capture most of the anomalies.", "Importantly, Fig.", "REF shows that the proposed M-SSA based approach is able to detect anomalies regardless of their location on the feature properties map.", "M-SSA is able to detect significant temporal changes in traffic as well as changes in the number of flows.", "M-SSA searches for correlation across flows properties (ASTUTE applies the same search concept between flows), while at the same time looking for temporal variation in a lag window dimension of $\\ell $ .", "ASTUTE is limited to two consecutive time bins.", "Figure: Anomalies feature map shows DoS attacks are associated with a small number of flows with large number of packets, while port scans are a larger number of flows correlated in same time.", "The coverage of M-SSA subsumes all the techniques.In order to evaluate the robustness and sensitivity of M-SSA we have designed a simulation set up where we inject artificial anomalies in real traces and measure the trade-off between the true positive and false positives using ROC curves.", "One of the biggest challenges in network anomaly detection systems, and which has limited their widespread adoption, is the high false positive rate exhibited by most existing techniques (see [22], [2])." ], [ "Simulation", "Our simulation is based on real trace data augmented with anomalous traffic injected in a similar fashion as in [25], [22], [2].", "However and in addition to previous work, we build a simulation model which captures several distinctive characteristics of anomalies.", "We consider the distribution of time between anomalies, duration, magnitude (packet count for DoS attacks, number of flows for port scans, etc), and the anomaly type distribution (DoS, port scan, etc).", "We first estimate the above parameters based on available observations in traffic traces.", "For example Fig.", "REF and Fig.", "REF show the histograms of these property values for DoS attacks and port scans respectively, as observed in the Abilene trace.", "We start the simulation assuming a non-anomalous time bin and choose the next attack time, by sampling from the empirical probability distribution of the time between anomalies.", "The anomaly type is then also chosen by sampling from the anomaly type distribution.", "At this point, a synthetic anomaly is generated by sampling from the anomaly duration and magnitude distribution, and injected into the synthetic trace.", "This process is repeated until the end of the simulation.", "The resulting trace therefore inherit the most significant statistical properties of the real data, e.g.", "the frequency of attacks and their magnitude.", "Figure: Illustration of the distribution histograms used to simulate DoS attacks.", "Distribution histograms characterize the duration of attacks and size of attack (e.g.", "number of flows involved in the attack plus the change in the packet volume)Figure: Illustration of the distribution histograms used to simulate port scans.", "Distribution histograms characterize the duration of attacks and size of attack (e.g.", "number of flows involved in the attack plus the change in the packet volume)" ], [ "Results", "The trade-off between false positive and true positive rate using the simulation data are captured using the ROC curve and are shown in Fig.", "REF .", "The simulation parameters for all algorithms are set as per Table REF .", "The ROC curves depicted in Fig.", "REF show that M-SSA has higher true positive rate for a given false positive rate, compared with all other techniques.", "For example, for a false positive rate of 0.01%, M-SSA detects 90% of anomalies, whereas Wavelet and ASTUTE only detect 77% and 81% respectively.", "A Hybrid detector including Wavelets, Kalman and ASTUTE shows slightly better trade-off for a false positive rate less than $10^{-5}$ but M-SSA is better for the rest of interval.", "The Area Under Curve (AUC) which measures the overall performance of the detector has been shown in Fig.", "REF (left).", "Figure: ROC curves: M-SSA has a better detection rate than alternative techniques.", "A Hybrid system shows slightly better trade-off for a false positive rate less than 10 -5 10^{-5}." ], [ "Configuration of Parameters", "We now evaluate the impact of the parameters: Lag Window Length ($\\ell $ ), the dimensionality $k$ of the projected space and the detection threshold $q_{\\beta }$ ." ], [ "Lag window length ($\\ell $ )", "The key take away from the theoretical literature is that for an appropriate choice of $\\ell $ , the Hankel matrix will capture the appropriate dynamics of the underlying system (see [28].", "According to [28], [5], [6] and [10], the choice of $\\ell $ must consider the trade-off between the maximum period (frequency) resolved and the statistical confidence of the result.", "A large value of $\\ell $ will potentially better capture the long range trends but the size of the covariance matrix will be larger which will have to be estimated from a time series of effective length $n-\\ell + 1$ .", "The choice of $\\ell $ has a significant impact on detection performance of different anomalies.", "DoS attacks and port scans are emblematic of two types of deviations in network traffic.", "DoS attacks are characterized by large changes in a (relatively) small number of flows as the attacking hosts send a large number of small packets to deplete system resources in the attacked host (see Fig.", "REF and Fig.", "REF ).", "Thus DoS like anomalies cause high temporal variation (within flows correlation) in the responsible flows and can be detected using techniques based on time series analysis.", "Port scans, in the other hand, are characterized as small increases in a large number of flows (see Fig.", "REF and Fig.", "REF ).", "This is required to detect for spatial correlation across flows (correlation between the flows) in order to find port scans.", "We run an experiment to discuss the impact of window length on capturing temporal/spatial correlation, i.e.", "whithin/between flow correlation, of the traffic data.", "ROC curves in Fig.", "REF and Fig.", "REF present DoS and port scan detection performance (separately) for varying window length.", "We describe the main findings learned from this experiment as follows.", "It is clear that the detection of DoS is almost independent of the window length, see Fig.", "REF .", "This is expected as DoS attacks cause high correlation within flows (temporal variations) and this can be always captured even if the window length is zero, i.e.", "the common PCA is able to report them.", "Across flows correlation is crucially dependent of window length as shown in Fig.", "REF .", "Thus the choice of window size has significant impact on detecting port scans.", "When the window length is zero the correlation across the flows can not be captured.", "when the window length is large across flows correlation is suppressed.", "What is required is a localized window where deviation from normal correlation can be detected.", "According to the experiment, detecting port scans is improved for window length of $\\ell =\\lbrace 4,8,12\\rbrace $ (hours) while it is worsen for smaller/larger window length.", "Figure: The impact of window length ℓ\\ell on detecting DoS attacks and port scans.", "Notice that ℓ\\ell has almost no impact on on DoS detection but significant impact on port scan detection." ], [ "Grouping indices (", "Another important parameter of M-SSA affecting results is the grouping indices, i.e.", "which components are grouped to provide the reconstructed data.", "The aim of our technique is to make a decomposition of the observed traffic into the sum of underlying traffic system (can be a number of interpretable components such as a slowly varying trend, oscillatory components) and a structureless noise, as ${ Y = X + E }$ .", "The decomposition of the series $Y$ into these two part is viable if the resulting additive components $X$ and $E$ are approximately separable from each other.", "Suppose the the full reconstructed components are denoted by $V_{i}=Mz$ for $i=\\lbrace 1,2,...,m \\times \\ell \\rbrace $ .", "To select which components to group, we compute the weighted correlation matrix (w-corr), where each element of the matrix $\\rho _{ij}$ is defined as: $\\rho _{ij}=\\frac{\\textsl {cov}_w(V_i,V_j)}{\\sigma _w(V_i)\\sigma _w(V_j)}$ using: $\\sigma _w^2(V_i)=W^{\\prime }V_i^{^{\\prime }}V_i\\quad , \\quad \\textsl {cov}_w(V_i,V_j)=W^{\\prime }V_i^{\\prime }V_j$ where $w_t=min\\lbrace t,\\ell ,n-\\ell \\rbrace $ for $t=\\lbrace 1:n\\rbrace $ is the weighting vector.", "If the absolute value of the w-correlations for two $V_i$ and $V_j$ is small (ideally zero), so the corresponding series are almost w-orthogonal and well separable.", "Fig.", "REF shows the absolute values of w-correlation for the first 50 reconstructed components.", "This is a grade matrix plot from red (corresponding to 1) to blue (corresponding to 0), which shows both the separability and dominance of components with highest eigenvalues values.", "This plot is useful to select how many components to select in the reconstruction phase, as we only need to select the first $k$ components with the largest w-corr values.", "From Fig.", "REF , we observe that the absolute value of the w-correlation for first 10 components are naturally grouped, a property that is observed for both the Abilene and MAWI datasets.", "We therefore suggest to use the first 10 components for the reconstruction when using M-SSA.", "So the $X=\\sum _{i=1}^{i=10}V_i$ and residual space ${E = Y - X }$ .", "In next section we will see that how the values of w-correlation can also be checked for adjusting the decision parameter ($\\emph {q}_\\beta $ ) so that a false positive rate can be met.", "Figure: Absolute values of w-correlation matrix plotted for the first 50 reconstructed components.", "The gaps in the scatter plot indicates how many components to select." ], [ "Decision Variable ($\\emph {q}_\\beta $ )", "For the decision threshold value (i.e., when to raise an alarm for any anomaly investigating $E$ space), we use the variables proposed in previous studies (see [17], [13], [14])in network anomaly detection but we address the problem associated with this criteria as discussed by [21].", "The threshold $\\emph {q}_{\\beta }$ is defined as $\\begin{array}{ll}\\emph {q}_\\beta & =Q(\\lambda _{k+1}:\\lambda _{\\ell \\times m},\\beta ) \\\\ [2ex]& =\\phi _1[\\frac{(1-\\beta ) \\sqrt{(}2\\phi _2h^2)}{\\phi _1}+1+\\frac{\\phi _2h(h-1)}{\\phi _1^2}]^{1/h}\\end{array}$ denotes the threshold for the 1$- \\beta $ confidence level, corresponds to a false alarm rate of $\\beta $ , and $h=1-\\frac{2\\phi _1\\phi _3}{3\\phi _2^2}, \\quad \\phi _i=\\sum _{j=k+1}^{\\ell m} \\lambda _i \\quad \\textit {for} \\quad i=1,2,3.$ Based on [14] the $Q$ in the above equation follows a gaussian distribution, and this convergence is robust even when the original data deviates from a gaussian distribution.", "[21] had questioned the robustness of the $Q$ metric - especially in the low false positive regime.", "[4] have shown that the main reason the metric is not robust is because the use of standard PCA results in a residual which exhibits temporal correlation.", "In principle the residual should correspond to noise and be completely uncorrelated.", "Thus by ensuring that temporal correlation (in the case of KL transform) and spatio-temporal correlation (in the case of M-SSA) is captured by the model, the $Q$ metric is robust.", "The w-correlation matrix computed above can help verify if the residual space, given by ${E = Y - X }$ where $X$ is the reconstructed space, contains correlated elements or not.", "For example, the w-correlation plot in Fig.", "REF clearly shows that that when $X$ is the space spanned by $V_i$ for $i>10$ , the reconstructed elements are strongly w-orthogonal in both Abilene and MAWI traffic, resulting in uncorrelated residuals." ], [ "Related Work", "Current network infrastructure is protected against malicious attacks by signature-based Intrusion Detection Systems (IDS) ([23], [20]).", "However, it is well known that attackers can circumvent these systems by generating small modifications of known signatures.", "In principle, anomaly-based detection systems (ADS) offer an attractive alterative to signature-based systems.", "ADS are based on the notion of ”statistical normality”, and malicious events are those that cause deviations from normal behavior.", "The major challenge is to characterize normal traffic subject to the constraint that network traffic exhibits non-stationary behavior.", "Existing techniques for ADS are based on decomposition methods of network time series.", "For example [17], [16], [18] has proposed the use of Principal Component Analysis (PCA) for detection of network wide anomalies.", "[32] has compared the use of Fourier, Wavelets and ARIMA methods for detection of link anomalies and then have used $\\ell _{1}$ optimization to recover the origin-destination pairs which may have caused the link anomalies to appear.", "Further refinements on PCA and state methods like Kalman Filtering have been extensively investigated for first extracting the normal behavior and then reporting deviations from normality as potential anomalies (see [3], [19], [32], [8], [15], [27]).", "The mathematical basis of Singular Spectrum Analysis (SSA) is the celebrated result in nonlinear dynamics due to [28].", "Taken's theorem asserts that the latent non-linear dynamics governing can be recovered using a delayed time embedding of the observable time series.", "The first practical use of Taken's theorem for time series analysis and the connection with spectral methods like singular value decomposition (SVD) was first proposed by [5], [7].", "Further application of the technique in climate and geophysical time series analysis has been extensively investigated in [30], [1], [11], [29], [9], [31], [10].", "" ], [ "Conclusions", "In this paper we have proposed a unified and robust method for network anomaly detection based on Multivariate Singular Spectrum Analysis (M-SSA).", "As M-SSA can detect deviations from both spatial and temporal correlation present in the data, it allows for the detection of both DoS and port scan attacks.", "A DoS attack is an example of temporal deviation while a port scan attack violates spatial correlation.", "Besides the use of M-SSA for network anomaly detection, we have carried out a comprehensive evaluation and compared M-SSA with other approaches based on wavelets, fourier analysis, kalman filtering and the recently introduced ASTUTE method.", "We have also carried out a rigorous analysis of the parameter configurations that accompany the use of M-SSA and address some of the important issues that have been raised in the networks community.", "Finally we have introduced a new labeled dataset from a large backbone link between Japan and the United States.", "This work is partially supported by NICTAhttp://nicta.com.au/.", "NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program." ], [ "Hankelization", "The averaging over the diagonals $\\mathrm {i+j=const}$ of the matrices $\\mathbf {X}_{I_i}$ is called Hankelization.", "The purpose of diagonal averaging is to transform a matrix to the form of a Hankel matrix, which can be subsequently converted to a time series.", "In other word, diagonal averaging maps matrices $\\mathbf {X}_{I_i}$ into a time series.", "to be continued ... Let Hankelization operator $\\mathcal {H}$ acting on any arbitrary matrix to turn it into a Hankle matrix in an optimal way.", "By applying the Hankelization procedure to all matrix components of $\\mathbf {X}_{I_i}$ the expansion will be: $\\mathbf {X}=\\mathbb {X}_{I_1}+...+\\mathbb {X}_{I_m}$ where $\\mathbb {X}_{I_1} = \\mathcal {H}X_{I_1}$ .", "Since all the matrices on the right-hand side of the expansion are Hankel matrices, each matrix uniquely specifies the time series and we thus obtain the decomposition of the original time series: $\\mathbb {X}_{I_1} \\equiv Y_1(t) \\quad ... \\quad \\mathbb {X}_{I_m} \\equiv Y_m(t)$ The complete original series is simply spume of the thus far obtained components.", "$Y(t)=Y_1(t)+Y_2(t)+...+Y_m(t)$" ] ]	1403.0157
[ [ "The Frobenius-Virasoro algebra and Euler equations" ], [ "Abstract We introduce an $\\mathfrak{F}$-valued generalization of the Virasoro algebra, called the Frobenius-Virasoro algebra $\\mathfrak{vir_F}$, where $\\mathfrak{F}$ is a Frobenius algebra over $\\mathbb{R}$.", "We also study Euler equations on the regular dual of $\\mathfrak{vir_F}$, including the $\\mathfrak{F}$-$\\mathrm{KdV}$ equation and the $\\mathfrak{F}$-$\\mathrm{CH}$ equation and the $\\mathfrak{F}$-$\\mathrm{HS}$ equation, and discuss their Hamiltonian properties." ], [ "Introduction", "Let $\\mathfrak {G}$ be a Lie algebra and $\\mathfrak {G}^$ (the regular part of) its dual, and let $\\langle ~,~\\rangle ^$ denote a natural pairing between $\\mathfrak {G}$ and $\\mathfrak {G}^$ .", "Definition 1.1 The Euler equation on $\\mathfrak {G}^$ is defined by the following system (e.g., [2], [7]): $\\frac{dm}{dt}=-ad^_{\\mathcal {A}^{-1}m}m,$ as an evolution of a point $m \\in \\mathfrak {G}^$ , where $\\mathcal {A}:\\mathfrak {G}\\rightarrow \\mathfrak {G}^$ is an invertible self-adjoint operator, called the inertia operator.", "It is well known that the KdV equation $u_t+3uu_x+c u_{xxx}=0\\nonumber $ and the Camassa-Holm (CH in brief) equation $m_t+2mu_x+m_xu+c u_{xxx}=0,\\quad m=u-u_{xx} \\nonumber $ and the Hunter-Saxton (HS in brief) equation $m_t+2mu_x+m_xu+c u_{xxx}=0,\\quad m=-u_{xx} \\nonumber $ could be regarded as Euler equations on the dual of Virasoro algebra $\\mathfrak {vir}$ with different inner products ([7], [9], [11], [12]).", "Let us remark that V.I.Arnold in [1] suggested a general framework for the Euler equation on an arbitrary Lie group $G$ , which is useful to characterize a variety of conservative dynamical systems, please see $e.g.$ , [2], [6], [7], [8], [9], [11], [12], [15], [17], [18] and references therein.", "If the corresponding Lie algebra is $\\mathfrak {G}$ , then the Euler equation (REF ) on $\\mathfrak {G}^$ could describe a geodesic flow w.r.t a suitable one-side invariant Riemannian metric on Lie group $G$ .", "In our recent works [16], [19], we studied the relation between Frobenius manifolds and Frobenius algebra-valued integrable systems.", "Definition 1.2 A Frobenius algebra $(\\mathfrak {F},g_\\mathfrak {F},{\\bf 1_\\mathfrak {F}},\\circ )$ over $\\mathbb {R}$ is a free $\\mathbb {R}$ -module $\\mathfrak {F}$ of finite rank $l$ , equipped with a commutative and associative multiplication $\\circ $ and a unit ${\\bf 1_\\mathfrak {F}}$ , and a $\\mathbb {R}$ -bilinear symmetric nondegenerate form $g_\\mathfrak {F}:\\mathfrak {F}\\times \\mathfrak {F}\\rightarrow \\mathbb {R}$ satisfying $g_\\mathfrak {F}(a\\circ b,c)=g_\\mathfrak {F}(a,b\\circ c)$ .", "Having this nondegenerate form $g_\\mathfrak {F}$ is equivalent to having a linear form $\\mathrm {tr}_\\mathfrak {F} \\,:\\mathfrak {F}\\rightarrow \\mathbb {R}$ whose kernel contains no trivial ideas.", "This linear form is often called a trace map.", "Indeed, given $g_\\mathfrak {F}$ , we put $\\mathrm {tr}_\\mathfrak {F} \\,(a):=g_\\mathfrak {F}(a,{\\bf 1_\\mathfrak {F}})$ .", "Conversely, given $\\mathrm {tr}_\\mathfrak {F} \\,$ , we could define $g_\\mathfrak {F}(a,b):=\\mathrm {tr}_\\mathfrak {F} \\,(a\\circ b)$ .", "Observe that an $\\mathfrak {F}$ -valued KdV ($\\mathfrak {F}$ -KdV) equation $u_t+3u\\circ u_x+\\zeta \\circ u_{xxx}=0, \\quad \\zeta \\in \\mathfrak {F} $ has been derived in [16], [19], where $u$ is a smooth $\\mathfrak {F}$ -valued function.", "A natural question is to ask: “Could the $\\mathfrak {F}$ -KdV equation (REF ) be regarded as a Euler equation on the regular dual of an infinite-dimensional Lie algebra $\\mathfrak {G}$ ?\"", "Our work is inspired by this question.", "This paper is to give an affirmative answer and organized as follows.", "Firstly, we introduce an infinite dimensional Lie algebra, called the Frobenius-Virasoro algebra $\\mathfrak {vir_F}$ , which is an $\\mathfrak {F}$ -valued generalization of the Virasoro algebra.", "Afterwards, we compute Euler equations on the regular dual $\\mathfrak {vir_F}^$ of $\\mathfrak {vir_F}$ for certain products, including the $\\mathfrak {F}$ -KdV equation, the $\\mathfrak {F}$ -CH equation and the $\\mathfrak {F}$ -HS equation.", "Moreover we show that all resulted Euler equations for the inner product $P_{\\alpha ,\\beta }$ are local bihamiltonian.", "Let us remark that in order to define the Euler equation on $\\mathfrak {vir_F}^$ , it is enough to require a commutative and associative algebra $(\\mathfrak {F},{\\bf 1_\\mathfrak {F}},\\circ )$ .", "In other words, we don't require the existence of trace map $\\mathrm {tr}_\\mathfrak {F} \\,$ .", "An interesting fact (also noted in [16]) is that if on $(\\mathfrak {F},{\\bf 1_\\mathfrak {F}},\\circ )$ , there are many different trace maps, then the corresponding $\\mathfrak {F}$ -valued Euler equation has many different (bi)hamiltonian structures.", "Finally we discuss some examples to illustrate our construction." ], [ "Euler equations on $\\mathfrak {vir_F}^$", "Throughout this paper, we assume that the Frobenius algebra $\\mathfrak {F}:=(\\mathfrak {F},\\mathrm {tr}_\\mathfrak {F} \\,,{\\bf 1_\\mathfrak {F}},\\circ )$ has the basis $e_1={\\bf 1_\\mathfrak {F}},\\, e_2,\\cdots , e_l$ ." ], [ "The Frobenius-Virasoro algebra $\\mathfrak {vir_F}$", "We begin with some definitions.", "Definition 2.1 We define an infinite-dimensional Lie algebra $(\\mathfrak {X},[~,~])$ over $\\mathbb {R}$ by $\\mathfrak {X}:=\\left\\lbrace u(x)\\frac{d}{dx}\| u\\in \\mathrm {C}^\\infty (\\mathbb {S}^1,\\mathfrak {F})\\right\\rbrace ,\\quad [u\\partial ,v\\partial ]:=(u{\\circ } v_x-u_x{\\circ } v)\\partial , \\quad \\partial =\\frac{d}{dx}.$ We remark that $\\mathfrak {X}$ is different from the loop algebra $L\\mathfrak {F}$ of $\\mathfrak {F}$ .", "As vector spaces, they are isomorphic under the map $\\Psi : L\\mathfrak {F}\\rightarrow \\mathfrak {X},\\quad \\Psi (u)=u\\partial .$ But as Lie algebras, $\\Psi $ is not a Lie algebra homomorphism.", "Lemma 2.2 The map $\\omega _\\mathfrak {F}: \\mathfrak {X}\\times \\mathfrak {X} \\rightarrow \\mathfrak {F}$ defined by $\\omega _\\mathfrak {F}(u\\partial , v\\partial )= \\int _{\\mathbb {S}^1} u{\\circ } v_{xxx}dx $ is a nontrivial 2-cocycle on $\\mathfrak {X}$ , called the $\\mathfrak {F}$ -valued Gelfand-Fuchs cocycle.", "Observe that the Frobenius algebra $\\mathfrak {F}$ is commutative and associative, then we have $\\mbox{(i).", "$\\omega _\\mathfrak {F}(u\\partial , v\\partial )=-\\omega _\\mathfrak {F}(v\\partial , u\\partial )$;\\quad (ii).", "$\\omega _\\mathfrak {F}(u\\partial , [v\\partial ,w\\partial ])+c.p.=0$},$ which follow the desired result.", "Definition 2.3 The central extension of $\\mathfrak {X}$ is called the Frobenius-Virasoro algebra, denoted by $\\mathfrak {vir_F}$ with the Lie bracket $[(u\\partial ,a),(v\\partial ,b)]:=\\left([u\\partial ,v\\partial ], \\,\\omega _\\mathfrak {F}(u\\partial ,v\\partial )\\right).", "$ When one chooses the Frobenius algebra $\\mathfrak {F}$ to be $\\mathbb {R}$ , $\\mathfrak {vir_F}$ is exactly the Virasoro algebra and $\\mathfrak {X}=\\mathrm {Vect}(\\mathbb {S}^1)$ .", "It is well known (e.g.", "[8], [13]) that the second continuous cohomology group $\\mathrm {H}^2(\\mathrm {Vect}(\\mathbb {S}^1),\\mathbb {R})\\cong \\mathbb {R}$ is generated by the Gefland-Fuchs cocycle.", "Generally when $\\rm {dim}\\mathfrak {F}>1$ , $\\mathrm {H}^2(\\mathfrak {X},\\mathfrak {F})$ is not generated by the $\\mathfrak {F}$ -valued Gelfand-Fuchs cocycle $\\omega _{\\mathfrak {F}}$ .", "An interesting problem is to compute $\\mathrm {H}^2(\\mathfrak {X},\\mathfrak {F})$ ." ], [ "Euler equations on $\\mathfrak {vir_F}^$", "We denote the regular dual of the Frobenius-Virasoro algebra $\\mathfrak {vir_F}$ by $\\mathfrak {vir_F}^=\\left\\lbrace (m(x,t)(dx)^2, \\zeta (t))\|\\mbox{$m(x,t)$ and $\\zeta (t)$ are smooth $\\mathfrak {F}$-valued functions } \\right\\rbrace \\nonumber $ with respect to the paring $\\langle (m dx^2, \\zeta ), (u\\partial , a) \\rangle ^=\\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathrm {S}^1}m{\\circ } u dx+\\mathrm {tr}_\\mathfrak {F} \\,(\\zeta {\\circ } a).$ Write $\\hat{m}=(m dx^2, \\zeta ) \\in \\mathfrak {vir_F}^$ and $\\hat{u}=(u\\partial , a),\\, \\hat{v}=(v\\partial , b)\\in \\mathfrak {vir_F}$ .", "By the definition, $\\langle ad^_{\\hat{u}}(\\hat{m}),\\hat{v} \\rangle ^=-\\langle \\hat{m},[\\hat{u},\\hat{v}]\\rangle ^= \\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathbb {S}^1}(2m{\\circ }u_x+m_x{\\circ }u+\\zeta {\\circ }u_{xxx}){\\circ }v dx \\nonumber $ which yields that the coadjoint action of $\\mathfrak {vir_F}$ on $\\mathfrak {vir_F}^$ is given by $ad^_{\\hat{u}} \\hat{m}=\\left((2m{\\circ }u_x+m_x{\\circ }u+\\zeta {\\circ }u_{xxx})(dx)^2,\\,0\\right).$ On $\\mathfrak {vir_F}$ , we introduce a two-parameter family of inner product $P_{\\alpha ,\\beta }$ , $\\alpha ,\\,\\beta \\in \\mathfrak {F}$ defined by $\\langle \\hat{u},\\hat{v}\\rangle = \\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathrm {S}^1}(\\alpha {\\circ }u{\\circ }v+\\beta {\\circ }u_x{\\circ }v_x) dx+\\mathrm {tr}_\\mathfrak {F} \\,(a{\\circ }~b).$ Observe that for the $P_{\\alpha ,\\beta }$ , the inertia operator $\\mathcal {A}:\\mathfrak {vir_F} \\longrightarrow \\mathfrak {vir_F}^$ is defined by $\\langle \\hat{u},\\hat{v} \\rangle =\\langle \\mathcal {A}(\\hat{u}),\\hat{v} \\rangle ^$ .", "In other words, $\\mathcal {A}(\\hat{u})=(\\Lambda (u),a)$ , where $\\Lambda =\\alpha -\\beta \\partial ^2$ is an $\\mathfrak {F}$ -valued differential operator.", "So we have Proposition 2.4 The Euler equation (REF ) on $\\mathfrak {vir_F}^$ for $P_{\\alpha ,\\beta }$ reads $m_t+2m{\\circ }u_x+m_x{\\circ }u+\\zeta {\\circ }u_{xxx}=0,\\quad \\zeta _t=0, $ where $m=\\Lambda (u)=\\alpha {\\circ }u-\\beta {\\circ }u_{xx}$ .", "When $\\zeta =0$ , the system (REF ) could be regarded as the Euler equation on $\\mathfrak {X}^$ .", "When the Frobenius algebra $\\mathfrak {F}$ is one-dimensional, i.e., $\\mathbb {R}$ , the system (REF ) is the Euler equation on $\\mathfrak {vir}^$ (e.g., [7]).", "Generally, when $\\alpha \\ne 0$ , $\\beta =0$ and $0\\ne \\zeta \\in \\mathfrak {F}$ , the system (REF ) reads the $\\mathfrak {F}$ -KdV equation $\\alpha {\\circ }u_t+3\\alpha {\\circ }u{\\circ }u_x+\\zeta {\\circ }u_{xxx}=0.", "$ When $\\alpha \\ne 0$ , $\\beta \\ne 0$ and $\\zeta \\in \\mathfrak {F}$ , the system (REF ) becomes the $\\mathfrak {F}$ -CH equation $\\quad m_t+2m{\\circ }u_x+m_x{\\circ }u+\\zeta {\\circ }u_{xxx}=0,\\quad m=\\alpha {\\circ }u-\\beta {\\circ }u_{xx}.$ When $\\alpha =0$ , $\\beta \\ne 0$ and $ \\zeta \\in \\mathfrak {F}$ , the system (REF ) reduces to the $\\mathfrak {F}$ -HS equation $\\beta {\\circ }(u_{xxt}+2u_{xx}{\\circ }u_x+u_{xxx}{\\circ }u)-\\zeta {\\circ }u_{xxx}=0,\\quad m=-\\beta {\\circ }u_{xx}.$ Example 2.5 Let $\\mathcal {Z}_2^{\\varepsilon }$ be a 2-dimensional commutative and associative algebra over $\\mathbb {R}$ with the basis $e_1={\\bf 1_\\mathfrak {F}}, e_2$ satisfying $e_1{\\circ } e_1=e_1,\\quad e_1{\\circ } e_2=e_2, \\quad e_2{\\circ } e_2=\\varepsilon e_1, \\quad \\varepsilon \\in \\mathbb {R}.", "\\nonumber $ Thus for any $A\\in \\mathcal {Z}^\\varepsilon _2$ , we could write $A=a_1e_1+a_2e_2$ , $a_k \\in \\mathbb {R}$ and define two “basic\" trace-type maps as follows $\\mathrm {tr}_{2,k}^\\varepsilon \\,(A)=a_k+a_2(1-\\delta _{k,2})\\delta _{\\varepsilon ,0} \\quad k=1,2.", "$ So $(\\mathcal {Z}_2^{\\varepsilon }, \\mathrm {tr}_{2,k}^\\varepsilon \\,, {\\bf 1_\\mathfrak {F}},{\\circ })$ for $k=1,2$ are the Frobenius algebras ([16]).", "The $\\mathcal {Z}_2^{\\varepsilon }$ -valued Euler equation with $\\zeta \\in \\mathcal {Z}_2^{\\varepsilon }$ is given by $m_t+2m\\circ u_x+m_x\\circ u+\\zeta \\circ u_{xxx}=0,\\quad m=\\alpha \\circ u-\\beta \\circ u_{xx}.", "$ (i).", "When $\\alpha =\\zeta ={\\bf 1_\\mathfrak {F}}$ and $\\beta =0$ , the system (REF ) reduces to the $\\mathcal {Z}_2^{\\varepsilon }$ -KdV equation ([16], [19]) $u_t+3u\\circ u_x+u_{xxx}=0,\\quad u=ve_1+we_2\\nonumber $ equivalently in componentwise forms, $v_t+3vv_x+v_{xxx}+3 \\varepsilon w w_x=0,\\quad w_t+3(vw)_x+w_{xxx}=0.$ When $\\varepsilon =0$ , the system (REF ) is the coupled KdV equation in [3], [4], [5].", "When $\\varepsilon =-1$ , the system (REF ) is a complexification of the KdV equation.", "(ii).", "When $\\alpha =\\beta ={\\bf 1_\\mathfrak {F}}$ and $\\zeta =0$ , the system (REF ) reduces to the $\\mathcal {Z}_2^{\\varepsilon }$ -CH equation $m_t+2m\\circ u_x+m_x\\circ u=0,\\quad m=u-u_{xx}, \\quad u=ve_1+we_2,\\nonumber $ equivalently in componentwise forms, $\\begin{array}{ll} p_t+2pv_x+p_xv+\\varepsilon (2qw_x+q_xw)=0, & p=v-v_{xx},\\\\q_t+2qv_x+q_xv+2pw_x+p_xw=0, & q=w-w_{xx}.\\end{array} $ When $\\varepsilon =-1$ , the system (REF ) is the complex-CH equation (e.g.,[14]).", "(iii).", "When $\\alpha =0$ and $\\beta =\\zeta ={\\bf 1_\\mathfrak {F}}$ , the system (REF ) reduces to the $\\mathcal {Z}_2^{\\varepsilon }$ -HS equation $m_t+2m\\circ u_x+m_x\\circ u=0,\\quad m=-u_{xx}, \\quad u=ve_1+we_2,\\nonumber $ equivalently in componentwise forms, $\\begin{array}{ll} p_t+2pv_x+p_xv+\\varepsilon (2qw_x+q_xw)=0, & p=-v_{xx},\\\\q_t+2qv_x+q_xv+2pw_x+p_xw=0, & q=-w_{xx}.\\end{array} $" ], [ "Hamiltonian structures of the Euler equation (", "Let us take two arbitrary smooth functionals $\\tilde{F}_i:\\mathfrak {vir_\\mathfrak {F}}^ \\rightarrow \\mathbb {R},\\quad \\tilde{F}_i(\\hat{m})=\\int _{\\mathbb {S}^1} \\mathrm {tr}_\\mathfrak {F} \\,F_i(m) dx=\\int _{\\mathbb {S}^1} f_i(m_1,\\cdots ,m_l) dx,\\quad i=1,2,$ where $m=\\displaystyle \\sum _{k=1}^lm_ke_k$ .", "The variational derivative $\\dfrac{\\delta \\tilde{F}_i }{\\delta \\hat{m}}$ is defined as $\\dfrac{\\delta \\tilde{F}_i }{\\delta \\hat{m}}=(\\dfrac{\\delta F_i }{\\delta m}\\partial ,\\, 0)\\in \\mathfrak {vir_\\mathfrak {F}},$ where $\\dfrac{\\delta F_i }{\\delta m}$ is implicitly determined by $\\tilde{F}_i(m+\\delta m)-\\tilde{F}_i(m)&=&\\int _{\\mathbb {S}^1} \\mathrm {tr}_\\mathfrak {F} \\,\\left(\\dfrac{\\delta F_i }{\\delta m}\\circ \\delta m+o(\\delta m)\\right)dx\\nonumber \\\\&=& \\int _{\\mathbb {S}^1} \\left(\\displaystyle \\sum _{k=1}^l\\dfrac{\\delta f_i }{\\delta m_k}\\delta m_k+o(\\delta m)\\right)dx $ and $\\dfrac{\\delta f_i }{\\delta m_k}$ is the usual variational derivative.", "This formula (REF ) is very crucial to construct the bihamiltonian representation of the Euler equation.", "On $\\mathfrak {vir_\\mathfrak {F}}^$ , there is a canonical Lie-Poisson bracket $\\mathcal {P}_2:=\\lbrace \\tilde{F}_1,\\tilde{F}_2\\rbrace _2(\\hat{m})= \\langle m, [\\dfrac{\\delta \\tilde{F}_1 }{\\delta \\hat{m}},\\dfrac{\\delta \\tilde{F}_2 }{\\delta \\hat{m}}] \\rangle ^=\\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathbb {S}^1} \\dfrac{\\delta {F}_1 }{\\delta {m}} \\circ \\mathcal {J}_2\\circ \\dfrac{\\delta {F}_2 }{\\delta {m}}\\,dx $ where $\\mathcal {J}_2=-(m\\partial +\\partial m+\\zeta \\partial ^3)$ and $\\hat{m}=(m dx^2, \\zeta ) \\in \\mathfrak {vir_F}^$ .", "Taking a fixed point $\\hat{m}_0=(\\dfrac{\\alpha }{2} dx^2, -\\beta )$ , we get another compatible Poisson bracket denoted by $\\mathcal {P}_1=\\lbrace \\tilde{F}_1,\\tilde{F}_2\\rbrace _1(\\hat{m}):=\\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathbb {S}^1} \\dfrac{\\delta {F}_1 }{\\delta {m}}\\circ \\mathcal {J}_1\\circ \\dfrac{\\delta {F}_2 }{\\delta {m}}\\,dx, \\quad \\mbox{i.e.,}\\quad \\mathcal {P}_1=\\mathcal {P}_2\|_{\\hat{m}=\\hat{m}_0}, $ where $\\mathcal {J}_1:=\\mathcal {J}_2\|_{\\hat{m}=\\hat{m}_0}=\\beta \\partial ^3-\\alpha \\partial =-\\partial \\Lambda $ .", "Theorem 2.6 The $\\mathfrak {F}$ -valued Euler equation (REF ) with $\\zeta \\in \\mathfrak {F}$ is local bihamiltonian with the freezing point $\\hat{m}_0=(\\dfrac{\\alpha }{2} dx^2, -\\beta )\\in \\mathfrak {vir_{\\mathfrak {F}}}^$ .", "Setting $H_1=\\frac{1}{2}\\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathbb {S}^1} m\\circ u dx,\\quad H_2=\\frac{1}{2} \\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathbb {S}^1}\\left({\\zeta }\\circ u\\circ u_{xx}+\\alpha \\circ u^3-\\frac{1}{2}{\\beta }\\circ u^2\\circ u_{xx}\\right) dx.$ With the formula (REF ), we get $\\frac{\\delta H_1}{\\delta u}=\\Lambda (u), \\quad \\frac{\\delta H_2}{\\delta u}=\\zeta \\circ u_{xx}+\\frac{3}{2}\\alpha \\circ u^2-\\frac{1}{2}{\\beta }\\circ u_x^2-\\beta \\circ u\\circ u_{xx}.\\nonumber $ By using $m=\\Lambda (u)$ , then $\\frac{\\delta H_1}{\\delta m}=\\Lambda ^{-1}\\circ \\frac{\\delta H_1}{\\delta u}=u,\\quad \\frac{\\delta H_2}{\\delta m}=\\Lambda ^{-1}\\circ \\frac{\\delta H_2}{\\delta u}.\\nonumber $ So the system (REF ) could be written as $m_t=\\mathcal {J}_1\\circ \\frac{\\delta H_2}{\\delta m}=\\mathcal {J}_2\\circ \\frac{\\delta H_1}{\\delta m}.$ Furthermore using the formula (REF ), in componentwise forms the Euler equation (REF ) has the following bihamlitonian representation $m_{k\\,t}=\\lbrace m_k, H_2\\rbrace _1= \\lbrace m_k, H_1\\rbrace _2,\\quad k=1,\\cdots l,$ where two compatible Poisson brackets $\\lbrace ~,~\\rbrace _i,\\, i=1,2 $ are defined in (REF ) and (REF ) respectively.", "Remark 2.7 Let us remark that when choose $\\mathfrak {F}$ as the Frobenius algebra $(\\mathcal {Z}_l, \\mathrm {tr}_l)$ ([3], [10], [19]), the Frobenius-Virasoro algebra $\\mathfrak {vir_F}$ coincides with the polynomial Virasoro algebra introduced by P.Casati and G.Ortenzi in [4].", "They also computed Euler equations on $\\mathfrak {vir_F}^$ and proved that they admitted a local bihamiltonian structure by using the trace-type map $\\mathrm {tr}_l$ .", "Actually in [16], it has been shown that there are at least $l$ “basic\" different ways to regard the algebra $\\mathcal {Z}_l$ as the Frobenius algebra $(\\mathcal {Z}_l, \\omega _{k})$ for $k=0,\\cdots ,l-1$ .", "We want to mention that the trace map $\\mathrm {tr}_l$ is a linear combination of “basic\" trace maps given by $\\mathrm {tr}_l=\\displaystyle \\sum _{k=0}^{l-1}\\omega _{k}-(l-1)\\,\\omega _{l-1}$ .", "Using Theorem REF , we thus obtain Corollary 2.8 The $\\mathcal {Z}_l$ -valued Euler equation (REF ) has at least $l$ “basic\" local bihamiltonian structures." ], [ "Examples", "According to Example REF , $(\\mathcal {Z}_2^{\\varepsilon }, \\mathrm {tr}_{2,k}^\\varepsilon \\,, {\\bf 1_\\mathfrak {F}},{\\circ }~)$ for $k=1,2$ are the Frobenius algebras.", "We thus have Corollary 2.9 The $\\mathcal {Z}_2^{\\varepsilon }$ -valued Euler equation (REF ) has at least two kinds of “basic\" local bihamiltonian structures.", "Naturally, we know that the $\\mathcal {Z}_2^{\\varepsilon }$ -CH equation (REF ) and the $\\mathcal {Z}_2^{\\varepsilon }$ -HS equation (REF ) have at least two kinds of “basic\" local bihamiltonian structures.", "For the $\\mathcal {Z}_2^{\\varepsilon }$ -KdV equation (REF ), two kinds of “basic\" local bihamiltonian structures have been obtained in [16], [19] by other methods.", "Based on our construction, more precisely we have Example 2.10 We consider the case: $[ \\varepsilon \\ne 0]$.", "(i).", "The $\\mathcal {Z}_2^{\\varepsilon }$ -KdV equation (REF ) $v_t+3vv_x+v_{xxx}+3 \\varepsilon w w_x=0,\\quad w_t+3(vw)_x+w_{xxx}=0\\nonumber $ could be rewritten as $\\left(\\begin{array}{c}v\\\\w\\end{array}\\right)_t=-\\left(\\begin{array}{cc}0& \\partial \\\\\\partial & 0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{2}}{\\delta v}\\\\\\frac{\\delta H_{2}}{\\delta w}\\end{array}\\right)=-\\left(\\begin{array}{cc}\\varepsilon J_1& J_0\\\\J_0 & J_1\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{1}}{\\delta v}\\\\\\frac{\\delta H_{1}}{\\delta w}\\end{array}\\right) \\nonumber $ with Hamiltonians $H_1=\\int _{\\mathbb {S}^1}vw dx,\\quad H_2=\\frac{1}{2} \\int _{\\mathbb {S}^1}(3v^2w+\\varepsilon w^3+2vw_{xx}) dx;$ and $\\left(\\begin{array}{c}v\\\\w\\end{array}\\right)_t=-\\left(\\begin{array}{cc}\\partial & 0\\\\0 & \\dfrac{1}{\\varepsilon }\\partial \\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{2}}{\\delta v}\\\\\\frac{\\delta \\widetilde{H}_{2}}{\\delta w}\\end{array}\\right)=-\\left(\\begin{array}{cc}J_0& J_1\\\\J_1 & \\frac{1}{\\varepsilon } J_0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{1}}{\\delta v}\\\\\\frac{\\delta \\widetilde{H}_{1}}{\\delta w}\\end{array}\\right) \\nonumber $ with Hamiltonians $\\widetilde{H}_1=\\frac{1}{2} \\int _{\\mathbb {S}^1}(v^2+\\varepsilon w^2) dx,\\quad \\widetilde{H}_2=\\frac{1}{2} \\int _{\\mathbb {S}^1}(v^3+vv_{xx}+3 \\varepsilon v w^2+\\varepsilon w w_{xx}) dx,$ where $J_0=\\partial ^3+v\\partial +\\partial v$ and $J_1=w\\partial +\\partial w$ .", "(ii).", "The $\\mathcal {Z}_2^{\\varepsilon }$ -CH equation (REF ) $\\begin{array}{ll} p_t+2pv_x+p_xv+\\varepsilon (2qw_x+q_xw)=0, & p=v-v_{xx},\\\\q_t+2qv_x+q_xv+2pw_x+p_xw=0, & q=w-w_{xx}, \\nonumber \\end{array}$ could be rewritten as $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}0& \\partial ^3- \\partial \\\\\\partial ^3-\\partial & 0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{2}}{\\delta p}\\\\\\frac{\\delta H_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}\\varepsilon K_1& K_0\\\\K_0 & K_1\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{1}}{\\delta p}\\\\\\frac{\\delta H_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $H_1=\\frac{1}{2}\\int _{\\mathbb {S}^1}(qv+pw) dx,\\quad H_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(2vw_{xx}+2wv_{xx}-2wvv_{xx}-v^2w_{xx}-\\varepsilon w^2w_{xx}\\right) dx,$ and $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\partial ^3-\\partial & 0\\\\0 & \\dfrac{1}{\\varepsilon }(\\partial ^3-\\partial )\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{2}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}K_0& K_1\\\\K_1 & \\dfrac{1}{\\varepsilon } K_0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{1}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $\\widetilde{H}_1=\\frac{1}{2} \\int _{\\mathbb {S}^1}(pv+\\varepsilon qw) dx,\\quad \\widetilde{H}_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(2vv_{xx}-v^2v_{xx}+\\varepsilon (ww_{xx}-w^2v_{xx}-2vww_{xx})\\right) dx,$ where $K_0=p\\partial +\\partial p$ and $K_1=q\\partial +\\partial q$ .", "(iii).", "The $\\mathcal {Z}_2^{\\varepsilon }$ -HS equation (REF ) $\\begin{array}{ll} p_t+2pv_x+p_xv+\\varepsilon (2qw_x+q_xw)=0, & p=-v_{xx},\\\\q_t+2qv_x+q_xv+2pw_x+p_xw=0, & q=-w_{xx}, \\end{array}\\nonumber $ could be rewritten as $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}0& \\partial ^3\\\\\\partial ^3& 0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{2}}{\\delta p}\\\\\\frac{\\delta H_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}\\varepsilon K_1& K_0\\\\K_0 & K_1\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{1}}{\\delta p}\\\\\\frac{\\delta H_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $H_1=\\frac{1}{2}\\int _{\\mathbb {S}^1}(qv+pw) dx,\\quad H_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(2wvp+v^2q+\\varepsilon w^2q \\right) dx,$ and $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\partial ^3& 0\\\\0 & \\dfrac{1}{\\varepsilon }\\partial ^3\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{2}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}K_0& K_1\\\\K_1 & \\dfrac{1}{\\varepsilon } K_0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{1}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $\\widetilde{H}_1=\\frac{1}{2} \\int _{\\mathbb {S}^1}(pv+\\varepsilon qw) dx,\\quad \\widetilde{H}_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(pv^2+\\varepsilon p w^2+2\\varepsilon vwq\\right) dx,$ where $K_0=p\\partial +\\partial p$ and $K_1=q\\partial +\\partial q$ .", "Example 2.11 We consider another case: $[ \\varepsilon =0]$.", "(i).", "The $\\mathcal {Z}_2^{0}$ -KdV equation (REF ) $v_t+3vv_x+v_{xxx}=0,\\quad w_t+3(vw)_x+w_{xxx}=0\\nonumber $ could be rewritten as $\\left(\\begin{array}{c}v\\\\w\\end{array}\\right)_t=-\\left(\\begin{array}{cc}0& \\partial \\\\\\partial & 0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{2}}{\\delta v}\\\\\\frac{\\delta H_{2}}{\\delta w}\\end{array}\\right)=-\\left(\\begin{array}{cc}0 & J_0\\\\J_0 & J_1\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{1}}{\\delta v}\\\\\\frac{\\delta H_{1}}{\\delta w}\\end{array}\\right) \\nonumber $ with Hamiltonians $H_1=\\int _{\\mathbb {S}^1}vw dx,\\quad H_2=\\frac{1}{2} \\int _{\\mathbb {S}^1}(3v^2w+2vw_{xx}) dx;$ and $\\left(\\begin{array}{c}v\\\\w\\end{array}\\right)_t=-\\left(\\begin{array}{cc}0 & \\partial \\\\\\partial & -\\partial \\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{2}}{\\delta v}\\\\\\frac{\\delta \\widetilde{H}_{2}}{\\delta w}\\end{array}\\right)=-\\left(\\begin{array}{cc}0& J_0\\\\J_0 & J_1-J_0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{1}}{\\delta v}\\\\\\frac{\\delta \\widetilde{H}_{1}}{\\delta w}\\end{array}\\right) \\nonumber $ with Hamiltonians $\\widetilde{H}_1=\\frac{1}{2} \\int _{\\mathbb {S}^1}(v^2+2vw) dx,\\quad \\widetilde{H}_2=\\frac{1}{2} \\int _{\\mathbb {S}^1}(v^3+vv_{xx}+3v^2 w+ 2v w_{xx}) dx,$ where $J_0=\\partial ^3+v\\partial +\\partial v$ and $J_1=w\\partial +\\partial w$ .", "(ii) The $\\mathcal {Z}_2^0$ -CH equation (REF ) $\\begin{array}{ll} p_t+2pv_x+p_xv=0, & p=v-v_{xx},\\\\q_t+2qv_x+q_xv+2pw_x+p_xw=0, & q=w-w_{xx} \\end{array}\\nonumber $ could be rewritten as $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}0& \\partial ^3-\\partial \\\\\\partial ^3-\\partial & 0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{2}}{\\delta p}\\\\\\frac{\\delta H_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}0 & K_0\\\\K_0 & K_1\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{1}}{\\delta p}\\\\\\frac{\\delta H_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $H_1=\\frac{1}{2}\\int _{\\mathbb {S}^1}(qv+pw) dx,\\quad H_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(2vw_{xx}+2wv_{xx}-2wvv_{xx}-v^2w_{xx}\\right) dx,$ and $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}0& \\partial ^3-\\partial \\\\\\partial ^3-\\partial & \\partial -\\partial ^3\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{2}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}0& K_0\\\\K_0 & K_1-K_0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{1}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $\\widetilde{H}_1=\\dfrac{1}{2}\\int _{\\mathbb {S}^1}(pv+qv+pw) dx$ and $\\widetilde{H}_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(2vw_{xx}+2wv_{xx}-2wvv_{xx}-v^2w_{xx}+2vv_{xx}-v^2v_{xx}\\right) dx,$ where $K_0=p\\partial +\\partial p$ and $K_1=q\\partial +\\partial q$ .", "(iii).", "The $\\mathcal {Z}_2^0$ -HS equation (REF ) $\\begin{array}{ll} p_t+2pv_x+p_xv=0, & p=-v_{xx},\\\\q_t+2qv_x+q_xv+2pw_x+p_xw=0, & q=-w_{xx}\\end{array}\\nonumber $ could be rewritten as $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}0& \\partial ^3\\\\\\partial ^3& 0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{2}}{\\delta p}\\\\\\frac{\\delta H_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}0 & K_0\\\\K_0 & K_1\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta H_{1}}{\\delta p}\\\\\\frac{\\delta H_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $H_1=\\frac{1}{2}\\int _{\\mathbb {S}^1}(qv+pw) dx,\\quad H_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(2wvp+v^2q \\right) dx,$ and $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)_t=\\left(\\begin{array}{cc}0& \\partial ^3-\\partial \\\\\\partial ^3-\\partial & \\partial -\\partial ^3\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{2}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{2}}{\\delta q}\\end{array}\\right)=-\\left(\\begin{array}{cc}0& K_0\\\\K_0 & K_1-K_0\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\delta \\widetilde{H}_{1}}{\\delta p}\\\\\\frac{\\delta \\widetilde{H}_{1}}{\\delta q}\\end{array}\\right) \\nonumber $ with Hamiltonians $\\widetilde{H}_1=\\frac{1}{2}\\int _{\\mathbb {S}^1}(pv+qv+pw) dx,\\quad \\widetilde{H}_2=\\frac{1}{4} \\int _{\\mathbb {S}^1}\\left(pv^2+2wvp+v^2q \\right) dx,$ where $K_0=p\\partial +\\partial p$ and $K_1=q\\partial +\\partial q$ ." ], [ "Euler equations on $\\mathfrak {vir_F}^$ for general product {{formula:e2a90e82-3b8e-450b-b7d9-d03160e8c33b}}", "To end up this section, on $\\mathfrak {vir_F}$ we introduce a general product $P_{\\alpha _0,\\cdots ,\\alpha _n}$ given by $\\langle \\hat{u},\\hat{v}\\rangle = \\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathrm {S}^1}\\Big (\\alpha _0{\\circ }u{\\circ }v+\\displaystyle \\sum _{k=1}^n\\alpha _k{\\circ }u^{(k)}{\\circ }v^{(k)}\\Big ) dx+\\mathrm {tr}_\\mathfrak {F} \\,(a{\\circ }~b),\\quad u^{(k)}=\\dfrac{d^k u}{dx^k}.$ By analogy with the above discussions, we have Proposition 2.12 The Euler equation (REF ) on $\\mathfrak {vir_F}^$ for $P_{\\alpha _0,\\cdots ,\\alpha _n}$ reads $m_t+2m{\\circ }u_x+m_x{\\circ }u+\\zeta {\\circ }u_{xxx}=0,\\quad \\zeta _t=0, $ where $m=\\alpha _0 {\\circ }u+\\displaystyle \\sum _{k=1}^n(-1)^k\\alpha _k {\\circ }u^{(2k)}$ .", "Moreover, the system (REF ) with $\\zeta \\in \\mathfrak {F}$ could be written as $m_{k,t}=\\lbrace m_k,H_1\\rbrace _2,\\quad H_1=\\frac{1}{2}\\mathrm {tr}_\\mathfrak {F} \\,\\int _{\\mathbb {S}^1} m\\circ u dx$ where $\\lbrace ~,~\\rbrace _2$ is defined in (REF ).", "Generally, when $n\\ge 2$ , the system (REF ) isn't a bihamiltonian system.", "But if there are many different ways to realize the algebra $(\\mathfrak {F}, {\\bf 1_\\mathfrak {F}},\\circ )$ as the Frobenius algebras, then it follows from Proposition REF that the system (REF ) has many different Hamiltonian structures.", "For instance, Corollary 2.13 The $\\mathcal {Z}_2^{\\varepsilon }$ -valued Euler equation (REF ) with $\\zeta \\in \\mathcal {Z}_2^{\\varepsilon }$ admits at least two “basic\" local Hamiltonian structures." ], [ "Conclusion", "In order to understand Eulerian nature of the $\\mathfrak {F}$ -valued KdV equation, we have introduced the Frobenius-Virasoro algebra $\\mathfrak {vir_\\mathfrak {F}}$ and also described Euler equations on $\\mathfrak {vir_\\mathfrak {F}}^$ under the product $P_{\\alpha _0,\\cdots ,\\alpha _n}$ and proved that all resulted Euler equations for $P_{\\alpha ,\\beta }$ are local bihamiltonian systems.", "Here we only studied the Euler equation associated with $\\mathfrak {vir_\\mathfrak {F}}$ .", "In subsequent publications we hope to address those problems related to algebraic properties of $\\mathfrak {vir_\\mathfrak {F}}$ , such as Q1.", "What is the second continuous cohomology group $\\mathrm {H}^2(\\mathfrak {X},\\mathfrak {F})$ ?", "Q2.", "How about the representation theory of $\\mathfrak {vir_\\mathfrak {F}}$ ?", "Q3.", "If exists, what is the corresponding Lie group $G_\\mathfrak {F}$ of $\\mathfrak {vir_\\mathfrak {F}}$ ?", "For instance, $G_\\mathbb {R}$ is the Bott-Virasoro group.", "Acknowledgements.", "The author is grateful to Professors Qing Chen, Yi Cheng and Youjin Zhang for constant supports and Professor Ian A.B.Strachan, Dr.Ying Shi for fruitful discussions.", "This work is partially supported by NCET-13-0550, NSFC (11271345, 11371138), SRF for ROCS, SEM and OATF,USTC." ] ]	1403.0027
[ [ "On differential systems with strongly indefinite variational structure" ], [ "Abstract We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals.", "There is no assumption to the effect that the nonlinear terms have to satisfy the Ambrosetti-Rabinowitz superquadratic condition.", "To establish the existence of solutions, a new version of the symmetric mountain pass theorem for strongly indefinite functionals is presented in this paper.", "This theorem is subsequently applied to deal with cases where all the Palais-Smale sequences of the energy functional may be unbounded." ], [ "This paper is concerned with the existence and the multiplicity of critical points of strongly indefinite even functionals which appear in the study of periodic solutions of Hamiltonian systems as well as solutions for some classes of elliptic systems.", "We recall that a functional $\\Phi :X\\rightarrow \\mathbb {R}$ defined on a Banach space is said to be strongly indefinite if it is neither bounded from above nor from below, even on subspaces of finite codimension.", "Its study then gives rise to an interesting and challenging variational problem, because the usual powerful critical point theorems in [1], [3], [8], [34] cannot be directly applied." ], [ "Critical Point Theory for strongly indefinite even functionals was studied in [4], [10], [12], [6], [13], [18], [22].", "In his seminal paper [13], Benci introduced various index and pseudo-index theories whose definitions, however, depend on the topology of the sublevel sets of the functional.", "In addition, a dimension property requirement significatively restricts the class of problems for which the results can be applied.", "In [4], [18], [22], the Galerkin type approximations were used to reduce the study of strongly indefinite functionals to a semidefinite situation where the basic idea of Lusternik-Schnirelmann theory applies.", "However, in order to control the critical points of the reduced functional, it is required in theses papers that the original functional satisfies a strong version of the usual compactness condition used in Critical Point Theory.", "In the recent papers [10], [12], the first two authors of this paper generalized the well known fountain theorem of Bartsch and Willem to strongly indefinite functionals.", "In contrast with [4], [18], [22], no reduction method was used and the proofs were directly carried out in the infinite-dimensional setting." ], [ "The first goal of this paper is to extend the results presented in [10], [12] to more general situations for which the Palais-Smale sequences of the functionals may be unbounded.", "This will have a crucial importance for the applications, since Palais-Smale type assumptions such as the Ambrosetti-Rabinowitz superlinear condition and its variations, extensively used in literature in the study of superlinear problems, can be avoided.", "We would like to stress that, in contrast with the common approach in the study of symmetric strongly indefinite functionals, ours is not based on any reduction method.", "The main ingredient is a weak-strong topology introduced by Kryszewski and Szulkin in [20].", "Our critical point theorems are stated in Section ." ], [ "In Section , we consider the applications to the problem of finding infinitely many large energy periodic solutions for the following first-order Hamiltonian system $\\left\\lbrace \\begin{array}{ll}\\partial _tu-\\Delta _x u+V(x)u=H_v(t,x,u,v)\\,\\, \\textnormal { in }\\mathbb {R}\\times \\Omega , & \\hbox{} \\\\-\\partial _tv-\\Delta _x v+V(x)v=H_u(t,x,u,v)\\,\\,\\textnormal { in }\\mathbb {R}\\times \\Omega , & \\hbox{}\\end{array}\\right.\\qquad \\mathrm {(HS)}$ where $\\Omega \\subset \\mathbb {R}^N$ ($N\\ge 1$ ) is a bounded smooth domain, and $H$ is a superquadratic $C^1$ -function which is $T$ -periodic ($T>0$ ) with respect to the $t$ -variable.", "By periodic solution, we mean a solution $z=(u,v):\\mathbb {R}\\times \\Omega \\rightarrow \\mathbb {R}^{2M}$ of (REF ) satisfying the conditions $z(t,x) &= z(t+T,x)\\quad \\forall (t,x)\\in \\mathbb {R}\\times \\Omega \\\\z(t,x) &= 0\\quad \\forall (t,x)\\in \\mathbb {R}\\times \\partial \\Omega .$ Setting $\\mathcal {J}=\\left(\\begin{array}{cc}0 & -I \\\\I & 0\\\\\\end{array}\\right),\\qquad \\mathcal {J}_0=\\left(\\begin{array}{cc}0 & I \\\\I & 0\\\\\\end{array}\\right),\\qquad z=(u,v),$ $A=\\mathcal {J}_0(-\\Delta _x+V), \\text{ and }\\mathcal {H}=-\\frac{1}{2}(Az,z)+H(t,x,z),$ system (REF ) simply reads: $\\mathcal {J}\\partial _t z=\\mathcal {H}_z(t,x,z),\\quad (t,x)\\in \\mathbb {R}\\times \\Omega ,$ which is the form of unbounded Hamiltonian systems or infinite-dimensional Hamiltonian systems in $L^2(\\Omega ,\\mathbb {R}^{2M})$ .", "This kind of systems were studied under various assumptions by Brezis and Nirenberg [14], and by Clément et al.", "[16]." ], [ "Usually, in the study of superquadratic Hamiltonian systems, the nonlinear terms satisfy the following condition named after Ambrosetti and Rabinowitz [1] $\\exists \\mu >2,\\,R>0\\quad ;\\quad 0<\\mu H(t,x,z)\\le H_z(t,x,z)z\\text{ for }\|z\|\\ge R,\\qquad \\mathrm {(AR)}$ see, for instance [2], [5], [7], [30] and references therein.", "It is well known that this condition is mainly used to obtain the boundedness of the Palais-Smale sequences of the energy functional and without it, the problem then becomes more complicated.", "Moreover without (REF ), it might happen that all the Palais-Smale sequences of the functional would be unbounded." ], [ "Let $\\widetilde{H}(t,x,u,v):=\\frac{1}{2}H_z(t,x,z)\\cdot z-H(t,x,z), \\quad S^1:=\\mathbb {R}/(T\\mathbb {Z}),\\,\\text{ and }\\,\\Theta =S^1\\times \\Omega .$ Our assumptions on the potential $V$ are the following: $V\\in C\\big (\\overline{\\Omega },\\mathbb {R}\\big )$ .", "$0\\notin \\sigma (S)$ , $S=-\\Delta _x+V$ .", "For the Hamiltonian $H$ we make the following assumptions: $H\\in C^1\\big (\\Theta \\times \\mathbb {R}^{2M},\\mathbb {R}\\big )$ is $T$ -periodic with respect to $t$ .", "$H(t,x,0)=0$ and $\|H_z(t,x,z)\|=\\circ (\|z\|)$ as $z\\rightarrow 0,$ uniformly in $(t,x)$ .", "$H(t,x,z)>0$ for $z\\ne 0$ and $\\frac{H(t,x,z)}{\|z\|^2}\\rightarrow \\infty $ as $\|z\|\\rightarrow \\infty ,$ uniformly in $(t,x)$ .", "$\\exists R>0$ such that $(1)$ $\\widetilde{H}(t,x,z)\\ge a_1\|z\|^2$ if $\|z\|\\ge R$ , $(2)$ $\\Big (\\frac{\|H_z(t,x,z)\|}{\|z\|}\\Big )^\\sigma \\le a_2\\widetilde{H}(t,x,z)$ if $\|z\|\\ge R$ , where $a_1,a_2>0$ , $\\sigma >1$ if $N=1$ and $\\sigma >\\frac{N}{2}+1$ if $N\\ge 2$ .", "Under these assumptions, Mao et al.", "[27] obtained a nontrivial periodic solution for (REF ) by using a local linking theorem.", "We will prove here that when the Hamiltonian is even with respect to $z$ , the number of periodic solutions is in fact infinite.", "This has already been observed by Bartsch and Ding in [5] where they studied (REF ) with $\\Omega $ bounded and with $\\Omega =\\mathbb {R}^N$ .", "However, they used condition (REF ) to verify the boundedness of the Palais-Smale sequences of the energy functional associated to (REF ), which was crucial for their argument.", "It is well known that (REF ) implies that $H(t,x,z)\\ge c\|z\|^\\mu $ for $\|z\|$ large, therefore it is stronger than $(H_3)$ .", "We would like to emphasize that in our situation, determining whether the Palais-Smale sequences of the energy functional are bounded or not can be a very difficult, if not impossible, task.", "Our result reads as follows: Theorem 1.1 Assume that $(V_1)$ , $(V_2)$ , and $(H_1)$ -$(H_4)$ are satisfied.", "If in addition $H$ is even in $z$ , then (REF ) has infinitely many pairs $\\pm z_k$ of $T$ -periodic solutions such that $\\Vert z_k\\Vert _\\infty \\rightarrow \\infty $ as $k\\rightarrow \\infty $ ." ], [ "Finally, as another application of our abstract result, we consider in Section the following elliptic system of Hamiltonian type $\\left\\lbrace \\begin{array}{ll}-\\Delta u=g(x,v)\\,\\, \\textnormal {in }\\Omega , & \\hbox{} \\\\-\\Delta v=f(x,u)\\,\\,\\textnormal {in }\\Omega , & \\hbox{} \\\\u=v=0\\textnormal { on }\\partial \\Omega , & \\hbox{}\\end{array}\\right.\\qquad \\mathrm {(ES)}$ where $\\Omega $ is a bounded smooth domain in $\\mathbb {R}^N$ , $N\\ge 3$ .", "The solutions of this problem describe steady states of some reaction-diffusion systems that derive from several applications, coming from mathematical biology or from the modelization of chemical reactions." ], [ "We study this problem under the following assumptions: $f,g\\in \\mathcal {C}(\\Omega \\times \\mathbb {R})$ and there is a constant $C>0$ such that $\|f(x,u)\|\\le C(1+\|u\|^{p-1}) \\textnormal { and } \|g(x,u)\|\\le C(1+\|u\|^{q-1})\\quad \\forall (x,u),$ $\\text{where } p,q>2\\text{ satisfy } \\frac{1}{p}+\\frac{1}{q}>1-\\frac{2}{N}.$ Furthermore, in case $N\\ge 5$ we impose $\\frac{1}{p}>\\frac{1}{2}-\\frac{2}{N}\\quad \\textnormal {and}\\quad \\frac{1}{q}>\\frac{1}{2}-\\frac{2}{N}.$ $f(x,u)=\\circ (u)$ and $g(x,u)=\\circ (u)$ as $u\\rightarrow 0$ , uniformly in $x$ .", "$F(x,u)/u^2\\rightarrow \\infty $ and $G(x,u)/u^2\\rightarrow \\infty $ , uniformly in $x$ , as $\|u\|\\rightarrow \\infty $ .", "$u\\mapsto f(x,u)/\|u\|$ and $u\\mapsto g(x,u)/\|u\|$ are increasing in $(-\\infty ,0)\\cup (0,+\\infty )$ .", "$f(x,-u)=-f(x,u)$ and $g(x,-u)=-g(x,u)$ for all $(x,u)$ .", "Before stating our result for this problem, we recall the following definition.", "Definition 1.2 We say that $(u,v)$ is a strong solution of (REF ) if $u\\in W^{2,p/(p-1)}(\\Omega )\\cap W_0^{1,p/(p-1)}(\\Omega )$ , $v\\in W^{2,q/(q-1)}(\\Omega )\\cap W_0^{1,q/(q-1)}(\\Omega )$ and $(u,v)$ satisfies $\\left\\lbrace \\begin{array}{ll}-\\Delta u=g(x,v)\\,\\, \\textnormal { a.e.", "in }\\Omega , & \\hbox{} \\\\-\\Delta v=f(x,u)\\,\\,\\textnormal { a.e.", "in }\\Omega .", "& \\hbox{}\\end{array}\\right.$ We will also prove the following Theorem 1.3 Under assumptions $(E_1)$ -$(E_5)$ , (REF ) has infinitely many pairs of strong solutions $\\pm (u_k,v_k)$ such that $\\Vert (u_k,v_k)\\Vert \\rightarrow \\infty $ as $k\\rightarrow \\infty $ , where $\\Vert \\cdot \\Vert $ represents the norm in the space $W^{2,p/(p-1)}(\\Omega )\\cap W_0^{1,p/(p-1)}(\\Omega )\\times W^{2,q/(q-1)}(\\Omega )\\cap W_0^{1,q/(q-1)}(\\Omega ).$ System (REF ) has been already studied from the variational point of view by many authors.", "Hulshof and van der Vorst [19], and de Figueiredo and Felmer [17] obtained positive solutions by requiring among others that the Hamiltonian $H(u,v)=F(u)+G(v)$ satisfies (REF ).", "In [18], Felmer and Wang considered the case where the full superquadratic range is not reached, and by using a variation of (REF ), they obtained infinitely many solutions when $H$ is even in $(u,v)$ by means of the Galerkin approximation.", "Recently, Szulkin and Weth [32] improved the above results by reducing the energy functional to the Nehari-Pankov manifold.", "To circumvent the difficulty that the Nehari-Pankov manifold is not necessary of class $C^1$ , they required the maps in assumption $(E_4)$ to be strictly increasing.", "In [9], the first author of this paper showed that it is sufficient to assume that these maps are only increasing.", "He applied the generalized fountain theorem obtained in [11] to a family of perturbed functionals.", "However, working with a family of modified functionals could make things unnecessary complicated.", "We considerably simplify this approach in the present paper." ], [ "Throughout the paper, we denote by $\|\\cdot \|_r$ the norm of the Lebesgue space $L^r$ ." ], [ "Critical point theorems for strongly indefinite functionals", "Let $X$ be a separable Hilbert space which admits an orthogonal decomposition $X=X^-\\oplus X^+$ , where $X^-$ is closed and $X^+=(X^-)^\\perp $ .", "We denote by $\\big <,\\big >$ the inner product of $X$ .", "Let $(a_j)_{j\\ge 0}$ be an orthonormal basis of $X^-$ .", "We define on $X$ a new norm by setting $u:=\\max \\Big (\\sum \\limits _{j=0}^{\\infty }\\frac{1}{2^{j+1}}\|\\big <P^-u,a_j\\big >\|,\\Vert P^+u\\Vert \\Big ),\\,\\, u\\in X,$ where $P^\\pm :X\\rightarrow X^\\pm $ are orthogonal projections, and we denote by $\\tau $ the topology generated by this norm.", "This topology was introduced by Kryszewski and Szulkin in [20] and is related to the topology on $X$ which is strong on $X^+$ and weak on bounded sets of $X^-$ .", "More precisely, if $(u_n)$ is a bounded sequence in $X$ then $u_n\\stackrel{\\tau }{\\rightarrow }u \\Longleftrightarrow P^-u_n \\rightharpoonup P^-u \\,\\ \\text{and} \\,\\ P^+u_n \\rightarrow P^+u.$ Now we recall some standard notations in Critical Point Theory: Let $\\Phi :X\\rightarrow \\mathbb {R}$ and $a,b\\in \\mathbb {R}$ .", "We denote by $\\Phi ^b:=\\big \\lbrace u\\in X\\,\\ ; \\,\\ \\Phi (u)\\le a\\big \\rbrace ,\\quad \\Phi _a:=\\big \\lbrace u\\in X\\,;\\, \\Phi (u)\\ge a\\big \\rbrace ,\\text{ and }\\Phi _a^b:=\\Phi _a\\cap \\Phi ^b.$ We say that a functional $\\Phi \\in C^1(X,\\mathbb {R})$ satisfies the Palais-Smale condition at the level $c\\in \\mathbb {R}$ if the following holds: $(PS)_c$ Any sequence $(u_n)\\subset X$ such that $\\Phi (u_n)\\rightarrow c$ and $\\Phi ^{\\prime }(u_n)\\rightarrow 0$ (Palais-Smale sequence at the level $c$ ) admits a convergent subsequence.", "This is the famous compactness condition in Critical Point Theory introduced by Palais and Smale [28].", "Since we would like to consider some situations where this condition does not hold, we will use its following weaker version due to Cerami [15]: $(Ce)_c$ Any sequence $(u_n)\\subset X$ such that $\\Phi (u_n)\\rightarrow c$ and $(1+\\Vert u_n\\Vert )\\Phi ^{\\prime }(u_n)\\rightarrow 0$ (Cerami sequence at the level $c$ ) admits a convergent subsequence.", "If $(Ce)_c$ holds we say that the functional $\\Phi $ satisfies the Cerami condition at the level $c$ ($(Ce)_c$ -condition for short)." ], [ "We now consider the class of $C^1$ functionals $\\Phi :X\\rightarrow \\mathbb {R}$ which satisfying: $(KS)$ $\\Phi (u)=\\frac{1}{2}\\Vert P^+u\\Vert ^2-\\frac{1}{2}\\Vert P^-u\\Vert ^2-\\Psi (u)$ , where $\\Psi \\in \\mathcal {C}^1(X,\\mathbb {R})$ is bounded below, weakly sequentially lower semicontinuous, and $\\Psi ^{\\prime }$ is weakly sequentially continuous." ], [ "The following deformation lemma will play a key role in the proof of our abstract results.", "Lemma 2.1 (Deformation lemma) Assume that $\\Phi $ satisfies $(KS)$ .", "Let $d\\ge b$ and $\\varepsilon >0$ such that $\\forall u\\in \\Phi ^{-1}([b-2\\varepsilon ,d+2\\varepsilon ]), \\, \\big (1+\\Vert u\\Vert \\big )\\Vert \\Phi ^{\\prime }(u)\\Vert \\ge 8\\varepsilon .$ Then there exists $\\eta \\in \\mathcal {C}([0,1]\\times \\Phi ^{d+2\\epsilon },X)$ such that: (i) $\\eta (0,u)=u$ for all $u\\in \\Phi ^{d+2\\varepsilon },$ (ii) $\\eta (1,\\Phi ^{d+\\varepsilon })\\subset \\Phi ^{b-\\varepsilon },$ (iii) $\\Phi (\\eta (\\cdot ,u))$ is non increasing, $\\forall u\\in \\Phi ^{d+2\\varepsilon }$ , (iv) each point $(t,u)\\in [0,1]\\times \\Phi ^{d+2\\varepsilon }$ has a $\\tau $ -neighborhood $N_{(t,u)}$ such that $\\big \\lbrace v-\\eta (s,v)\\, \\bigl \| \\, (s,v)\\in N_{(t,u)}\\cap ([0,1]\\times \\Phi ^{d+2\\varepsilon })\\bigr .\\big \\rbrace $ is contained in a finite-dimensional subspace of $X$ , (v) $\\eta $ is $\\tau $ -continuous, (vi) if $\\Phi $ is even then $\\eta (t,\\cdot )$ is odd $\\forall t\\in [0,1]$ .", "It goes back to an idea of Li and Szulkin [21]." ], [ "We define the vector field $w(u):=2\\nabla \\Phi (u)/\\Vert \\Phi ^{\\prime }(u)\\Vert ^2,\\quad u\\in \\Phi ^{-1}\\big ([b-2\\varepsilon ,d+2\\varepsilon ]\\big ).$ By assumption $(KS)$ we know that $\\Phi ^{\\prime }$ is weakly sequentially semicontinuous, and this implies that the function $v\\in \\Phi ^{d+2\\varepsilon }_{b-2\\varepsilon }\\mapsto \\big <\\Phi ^{\\prime }(v),w(u)\\big >\\in \\mathbb {R}$ is $\\tau $ -continuous.", "Hence every $u\\in \\Phi ^{d+2\\varepsilon }_{b-2\\varepsilon }$ has a $\\tau $ -neighborhood $N_u$ such that $\\big <\\Phi ^{\\prime }(v),w(u)\\big >>1\\quad \\forall v\\in N_u,$ $\\Vert u\\Vert <2\\Vert v\\Vert \\quad \\forall v\\in N_u,$ where (REF ) holds because the set $\\big \\lbrace z\\in X\\,;\\,\\Vert z\\Vert \\le \\alpha \\big \\rbrace $ is $\\tau $ -closed for any $\\alpha \\ge 0$ .", "Now since $(KS)$ implies that $\\Phi $ is $\\tau $ -upper semicontinuous, the set $\\widetilde{N}:=\\Phi ^{-1}\\big (]-\\infty ,b-2\\varepsilon [\\big )$ is $\\tau $ -open.", "It follows that $\\mathcal {N}:=\\widetilde{N}\\cup \\big \\lbrace N_v\\,;\\,b-2\\varepsilon \\le \\Phi (v)\\le d+2\\varepsilon \\big \\rbrace $ is a $\\tau $ -open covering of the metric space $\\big (\\Phi ^{d+2\\varepsilon },\\tau \\big )$ .", "We can then extract a $\\tau $ -locally finite $\\tau $ -open covering $\\mathcal {M}:=\\big \\lbrace M_i\\,;\\,i\\in I\\big \\rbrace $ of $\\Phi ^{c+2\\epsilon }$ finer than $\\mathcal {N}$ .", "Let $V:=\\bigcup \\limits _{i\\in I}M_i.$ For every $i\\in I$ we have either $M_i\\subset N_v$ for some $v$ or $M_i\\subset \\widetilde{N}$ .", "In the first case we define $v_i:=w(v)$ and in the second case $v_i:=0$ .", "Consider a $\\tau $ -Lipschitz continuous partition of unity $\\big \\lbrace \\lambda _i\\, ; \\, i\\in I \\big \\rbrace $ subordinated to $\\mathcal {M}$ and define on $V$ the vector field $f(u):=\\sum \\limits _{i\\in I}\\lambda _i(u)v_i.$ Clearly, the vector field $f$ is locally Lipschitz continuous and $\\tau $ -locally Lipschitz continuous.", "By using (REF ) and (REF ) we see that $\\Vert f(u)\\Vert \\le \\frac{1}{4\\varepsilon }\\big (1+2\\Vert u\\Vert \\big )\\quad \\forall u\\in V.$ It follows from Corollary $7.6$ in [31] that the problem $\\left\\lbrace \\begin{array}{ll}\\frac{d}{dt}\\sigma (t,u)=-f(\\sigma (t,u)) & \\hbox{} \\\\\\sigma (0,u)=u \\in \\Phi ^{d+2\\varepsilon } & \\hbox{}\\end{array}\\right.$ has a unique solution $\\sigma (\\cdot ,u)$ defined on ${\\mathbb {R}}^+$ .", "We define $\\eta :[0,1]\\times \\Phi ^{d+2\\varepsilon }$ by setting $\\eta (t,u):=\\sigma \\big ((d-b+2\\varepsilon )t,u\\big ).$ An argument similar to that in the proof of Lemma $6.8$ in [33] shows that $\\eta $ satisfies $(i)$ -$(v)$ .", "If $\\Phi $ is even, then we replace $f(u)$ with $\\frac{1}{2}\\big (f(u)-f(-u)\\big )$ .", "$(vi)$ is then a consequence of the existence and the uniqueness of the solution for the above Cauchy problem.", "Now we can state our first critical point theorem, that extends the generalized saddle point theorem in [25] to the case where the Palais-Smale sequences of $\\Phi $ may be unbounded.", "Theorem 2.2 (Saddle point theorem) Assume that $\\Phi $ satisfies $(KS)$ .", "If there exists $R>0$ such that $(A_0)\\quad \\quad \\quad b:=\\inf _{\\begin{array}{c}u\\in X^+\\end{array}}\\Phi (u)>\\sup _{\\begin{array}{c}u\\in X^-\\\\ \\Vert u\\Vert =R\\end{array}}\\Phi (u) \\quad \\text{and}\\quad d:=\\sup _{\\begin{array}{c}u\\in X^-\\\\ \\Vert u\\Vert \\le R\\end{array}}\\Phi (u)<\\infty ,\\quad $ then for some $c\\in [b,d]$ , there is a sequence $(u_n)\\subset X$ such that $\\Phi (u_n)\\rightarrow c\\quad \\text{and}\\quad \\big (1+\\Vert u_n\\Vert \\big )\\Phi ^{\\prime }(u_n)\\rightarrow 0\\,\\,\\text{as}\\,\\, n\\rightarrow \\infty .$ Moreover, if $\\Phi $ satisfies the $(Ce)_\\alpha $ condition for all $\\alpha \\in [b,d]$ , then $\\Phi $ has a critical value in $[b,d]$ .", "The proof follows the lines of [25].", "We assume by contradiction that for every $c\\in [b,d]$ there is no $(Ce)_c$ sequence for $\\Phi $ .", "Then there exists $\\varepsilon >0$ such that $u\\in \\Phi ^{-1}\\big ([b-2\\varepsilon ,d+2\\varepsilon ]\\big )\\Rightarrow \\big (1+\\Vert u\\Vert \\big )\\Vert \\Phi ^{\\prime }(u)\\Vert \\ge 8\\varepsilon .$ Let $M:=\\big \\lbrace u\\in X^-\\,;\\, \\Vert u\\Vert =R\\big \\rbrace .$ We apply Lemma REF , and we define $\\mu :[0,1]\\times M\\rightarrow X$ by setting $\\mu (t,u):=P^-\\eta (t,u)$ , where $\\eta $ is given by Lemma REF .", "Clearly, $(iv)$ and $(v)$ of Lemma REF imply that $\\mu $ is $\\tau $ -continuous and each $(t,u)\\in [0,1]\\times \\Phi ^{d+2\\varepsilon }$ has a $\\tau $ -neighborhood $N_{(t,u)}$ such that $\\big \\lbrace v-\\mu (s,v)\\, \\bigl \| \\, (s,v)\\in N_{(t,u)}\\cap ([0,1]\\times \\Phi ^{d+2\\varepsilon })\\big \\rbrace $ is contained in a finite-dimensional subspace of $X$ .", "We claim that $0\\notin \\mu \\big ([0,1]\\times \\partial M\\big )$ .", "Indeed, if there exists $(t_0,u_0)\\in [0,1]\\times \\partial M$ such that $\\mu (t_0,u_0)=0$ , then $\\eta (t_0,u_0)\\in X^+$ and by $(iii)$ of Lemma REF and assumption $(A_0)$ we have $\\Phi (u_0)=\\Phi \\big (\\eta (0,u_0)\\big )\\ge \\Phi \\big (\\eta (t_0,u_0)\\big )\\ge b>\\sup _{\\partial M}\\Phi ,$ which contradicts the fact that $u_0\\in \\partial M$ .", "$\\mu $ is then an admissible homotopy (in the sense of Kryszewski and Szulkin [20]) such that $0\\notin \\mu \\big ([0,1]\\times \\partial M\\big )$ .", "It follows from Theorem $2.4$ -$(iii)$ of [20] that the Kryszewski-Szulkin's degree (see [20]) $deg_{KS}\\big (\\mu (t,\\cdot ),int(M),0\\big )$ is well defined and does not depend on $t\\in [0,1]$ .", "Hence $deg_{KS}\\big (\\mu (1,\\cdot ),int(M),0\\big )=deg_{KS}\\big (\\mu (0,\\cdot ),int(M),0\\big )=deg_{KS}\\big (id,int(M),0\\big ).$ It follows from Theorem $2.4$ -$(i)$ of [20] that there is $u\\in int(M)$ such that $\\mu (1,u)=0$ , which implies that $\\eta (1,u)\\in X^+$ .", "By the definition of $b$ we have $b\\le \\Phi \\big (\\eta (1,u)\\big )$ .", "But $(ii)$ of Lemma REF implies, since $M\\subset \\Phi ^{d+\\varepsilon }$ , that $\\Phi \\big (\\eta (1,u)\\big )\\le b-\\varepsilon $ .", "This gives a contradiction.", "In order to obtain a multiplicity result we need to introduce some notations.", "We consider an orthonormal basis $(e_j)_{j\\ge 0}$ of $X^+$ and we set $X_k^-:=X^-\\oplus \\big (\\oplus _{j=0}^k\\mathbb {R}e_j\\big ) \\text{ and } X_k^+:=\\overline{\\oplus _{j=k}^\\infty \\mathbb {R}e_j}.$ Theorem 2.3 (Fountain theorem) Assume that $\\Phi $ satisfies $(KS)$ , that $\\Phi $ is even, and that there exist $\\rho _k>r_k>0$ such that: $(A_1) \\quad \\quad a_k:=\\sup _{\\begin{array}{c}u\\in X_k^-\\\\ \\Vert u\\Vert =\\rho _k\\end{array}}\\Phi (u)<\\min \\big (0,\\inf _{\\begin{array}{c}u\\in X\\\\ \\Vert u\\Vert \\le r_k\\end{array}}\\Phi (u)\\big ), \\quad d_k:=\\sup _{\\begin{array}{c}u\\in X_k^-\\\\ \\Vert u\\Vert \\le \\rho _k\\end{array}}\\Phi (u)<\\infty .\\quad \\quad \\quad $ $(A_2) \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad b_k:=\\inf _{\\begin{array}{c}u\\in X_k^+\\\\ \\Vert u\\Vert =r_k\\end{array}}\\Phi (u)\\rightarrow \\infty ,\\quad k\\rightarrow \\infty .\\quad \\quad \\quad \\quad \\quad \\quad \\quad $ Then, there exist a sequence $(u_k^n)_n\\subset X$ and a number $c_k\\ge b_k$ such that $\\Phi (u_k^n)\\rightarrow c_k\\quad \\text{and}\\quad \\big (1+\\Vert u_k^n\\Vert \\big )\\Phi ^{\\prime }(u_k^n)\\rightarrow 0\\,\\,\\text{as}\\,\\, n\\rightarrow \\infty .$ Moreover, if $\\Phi $ satisfies the $(Ce)_c$ condition for any $c>0$ , then $\\Phi $ has an unbounded sequence of critical values.", "Theorem REF generalizes Theorem 12 in [10] where the sequence $(u_k^n)$ was a Palais-Smale sequence.", "Let $B_k:=\\big \\lbrace u\\in X_k^-\\,;\\, \\Vert u\\Vert \\le \\rho _k\\big \\rbrace $ and let $\\Gamma _k$ be the set of maps $\\gamma :B_k\\rightarrow X$ such that: (a) $\\gamma $ is odd and $\\tau $ -continuous, (b) each $u\\in B_k$ has a $\\tau $ -neighborhood $\\mathcal {N}_u$ in $X^-_k$ such that $(id-\\gamma )\\bigl (\\mathcal {N}_u\\cap B_k\\bigr )$ is contained in a finite dimensional subspace of $X$ , (c) $\\Phi (\\gamma (u))\\le \\Phi (u)$ , $\\forall u\\in B_k$ .", "Let $\\gamma \\in \\Gamma _k$ and let $U=\\lbrace u\\in B_k \\, ; \\, \\Vert \\gamma (u)\\Vert <r_k\\bigr .\\rbrace $ .", "It obviously follows from $(a)$ and $(b)$ that $\\gamma $ is continuous (in the norm topology).", "If $u\\in B_k$ is such that $\\Vert u\\Vert =\\rho _k$ , then by $(c)$ we have $\\Phi (\\gamma (u))<\\Phi (u)$ , so it follows from $(A_1)$ that $u\\notin U$ .", "Hence $U$ is a symmetric $\\tau -$ open neighborhood of 0 in $X_k^-$ .", "It is clear that $B_k$ is $\\tau $ -closed, so we deduce from the $\\tau $ -continuity of $\\gamma $ that $\\overline{U}$ is also $\\tau $ -closed.", "Consider the map $P_{k-1}\\gamma :\\overline{U}\\rightarrow X^-_{k-1},$ where $P_{k-1}:X\\rightarrow X^-_{k-1}$ is the orthogonal projection.", "$P_{k-1}\\gamma $ is $\\tau $ -continuous.", "Let $u\\in U$ .", "From $(b)$ $u$ has a $\\tau $ -neighborhood $N_u$ such that $(id-\\gamma )(N_u\\cap U)\\subset W$ , where $W$ is a finite-dimensional subspace of $X$ .", "Let $v\\in N_u\\cap U\\subset X^-_{k}=X^-_{k-1}\\oplus {\\mathbb {R}} e_k$ , then $(id-P_{k-1}\\gamma )(v)=P_{k-1}(v-\\gamma (v))+\\lambda e_k\\in W+{\\mathbb {R}} e_k$ which is finite-dimensional.", "It follows from Theorem 5 in [10] that there exists $u_0\\in \\partial U$ such that $P_{k-1}\\gamma (u_0)=0$ .", "This implies that $\\gamma \\big (B_k\\big )\\cap \\big \\lbrace u\\in X^+_k\\,;\\, \\Vert u\\Vert =r_k\\big \\rbrace \\ne \\emptyset .$ Now $(A_1)$ and $(A_2)$ imply that there is $k_0$ big enough such that $b_k>a_k$ for every $k\\ge k_0$ .", "We fix $k\\ge k_0$ and we define $c_{k} := \\inf _{\\gamma \\in \\Gamma _{k}} \\sup _{u\\in B_{k}} \\Phi \\bigl ( \\gamma (u) \\bigr ).$ The intersection property (REF ) implies that $c_k\\ge b_k$ ." ], [ "We assume by contradiction that there is no Cerami sequence of $\\Phi $ at the level $c_k$ .", "Then there exists $\\varepsilon \\in (0,\\frac{c_k-a_k}{2})$ such that $u\\in \\Phi ^{-1}([c_k-2\\varepsilon ,c_k+2\\varepsilon ])\\quad \\Rightarrow \\quad \\big (1+\\Vert u\\Vert \\big )\\Vert \\Phi ^{\\prime }(u)\\Vert \\ge 8\\varepsilon ,$ where $\\gamma \\in \\Gamma _k$ is such that $\\sup _{B_{k}} \\Phi \\circ \\gamma \\le c_{k}+\\varepsilon .$ We apply Lemma REF with $b=d=c_k$ and we define on $B_k$ the map $\\beta (u):=\\eta (1,\\gamma (u)).$ It follows from $(iii),(iv),(v),(vi)$ of Lemma REF that $\\beta \\in \\Gamma _k$ .", "Now by using $(\\ref {e.6})$ and $(ii)$ of Lemma REF we obtain ${ \\displaystyle \\sup _{\\begin{array}{c}u \\in X^-_k \\\\ \\Vert u\\Vert = \\rho _k\\end{array}} \\Phi (\\beta (u)) = \\displaystyle \\sup _{\\begin{array}{c}u \\in X^-_k \\\\ \\Vert u\\Vert = \\rho _k\\end{array}} \\Phi (\\eta (1,\\gamma (u))) \\le c_k-\\varepsilon }$ giving a contradiction with the definition of $c_k$ .", "It then follows that $\\Phi $ has a Cerami sequence at level $c_k$ .", "Using the same argument as in the proof of Theorem 6 in [12], we can prove the following dual version of Theorem REF .", "Theorem 2.4 (Dual fountain theorem) Let $\\Phi (u)=\\frac{1}{2}\\Vert u^+\\Vert ^2-\\frac{1}{2}\\Vert u^-\\Vert ^2+\\Psi (u)$ , where $\\Psi \\in \\mathcal {C}^1(X,\\mathbb {R})$ is even, bounded below, and weakly sequentially lower semicontinuous, and $\\Psi ^{\\prime }$ is weakly sequentially continuous.", "If $\\forall k\\ge k_0$ , $\\exists \\rho _k>r_k>0$ such that $(B_1)$ $a^k \\, := \\, \\displaystyle \\inf _{\\begin{array}{c}u \\in X^+_{k}\\\\ \\Vert u\\Vert = \\rho _k\\end{array}} \\Phi (u) >\\max \\big (0, \\sup _{\\begin{array}{c}u\\in X\\\\ \\Vert u\\Vert \\le r_k\\end{array}}\\Phi (u)\\big )$ , $b^k \\, := \\, \\displaystyle \\sup _{\\begin{array}{c}u \\in X^{-}_k\\\\ \\Vert u \\Vert = r_k\\end{array}} \\Phi (u) <0$ , $(B_2)$ $d^k \\, := \\, \\displaystyle \\inf _{\\begin{array}{c}u \\in X^+_k\\\\ \\Vert u \\Vert \\le \\rho _k\\end{array}} \\Phi (u) \\rightarrow 0, \\, k \\rightarrow \\infty $ , Then there exist $c^k\\in [d^{k},0[$ and a sequence $(u_k^n)\\subset X$ such that $\\Phi (u_k^n)\\rightarrow c_k\\quad \\text{and}\\quad \\big (1+\\Vert u_n\\Vert \\big )\\Phi ^{\\prime }(u_k^n)\\rightarrow 0,\\quad \\text{as }n\\rightarrow \\infty .$ Moreover, if $\\Phi $ satisfies the $(Ce)_c$ condition for any $c\\in [d^{k_0},0[$ , then $\\Phi $ has a sequence $(u_k)$ of critical points such that $\\Phi (u_k)\\rightarrow 0^-$ as $k\\rightarrow \\infty $ ." ], [ "Periodic solutions of superquadratic Hamiltonian systems", "In this section, we apply our multiplicity result (Theorem REF ) to the Hamiltonian system $\\left\\lbrace \\begin{array}{ll}\\partial _tu-\\Delta _x u+V(x)u=H_v(t,x,u,v)\\,\\, \\textnormal { in }\\mathbb {R}\\times \\Omega , & \\hbox{} \\\\-\\partial _tv-\\Delta _x v+V(x)v=H_u(t,x,u,v)\\,\\,\\textnormal { in }\\mathbb {R}\\times \\Omega .", "& \\hbox{}\\end{array}\\right.\\qquad \\mathrm {(HS)}$ We assume here that assumptions $(V_1)$ , $(V_2)$ , and $(H_1)$ -$(H_4)$ are satisfied, and that $H$ is even in $z$ ." ], [ "By $(V_1)$ the operator $S=-\\Delta _x+V$ acting on $L^2(\\Omega ,\\mathbb {R}^{2M})$ is self-adjoint with domain $D(S)=H^2(\\Omega ,\\mathbb {R}^{2M})\\cap H^1_0(\\Omega ,\\mathbb {R}^{2M})$ .", "Let $L:=\\mathcal {J}\\partial _t+A$ .", "Observing that the operators $A$ and $\\mathcal {J}A$ acting on $L^2(\\Omega ,\\mathbb {R}^{2M})$ with domains $D(A)=D(\\mathcal {J}A)=H^2(\\Omega ,\\mathbb {R}^{2M})\\cap H^1_0(\\Omega ,\\mathbb {R}^{2M})$ are self-adjoint, then the operator $L$ acting on $L^2\\big (S^1,L^2(\\Omega ,\\mathbb {R}^{2M})\\big )$ is also self-adjoint.", "We know by [5] that $L^2\\big (S^1,L^2(\\Omega ,\\mathbb {R}^{2M})\\big )\\cong L^2\\big (S^1\\times \\Omega ,\\mathbb {R}^{2M}\\big ),\\quad 0\\notin \\sigma (A)\\cup \\sigma (\\mathcal {J}A),\\quad 0\\notin \\sigma (L),$ so there is an orthogonal decomposition $L^2(S^1\\times \\Omega ,\\mathbb {R}^{2M})=E^+\\oplus E^-,$ such that $L$ is negative on $E^-$ , and positive on $E^+$ .", "The space $X=D(\|L\|^{\\frac{1}{2}})$ is a Hilbert space with the inner product $\\big <w,z\\big >=\\big <\|L\|^{\\frac{1}{2}}w,\|L\|^{\\frac{1}{2}}z\\big >_{L^2}$ and norm $\\Vert z\\Vert ^2=\\big <z,z\\big >$ .", "Moreover, we have an orthogonal decomposition $X=X^-\\oplus X^+\\text{ with } X^\\pm =X\\cap E^\\pm .$ We have the following useful lemma: Lemma 3.1 ([5], Lemma $4.6$ ) The embedding of $X$ in $L^r\\big (S^1\\times \\Omega ,\\mathbb {R}^{2M}\\big )$ is compact for $r\\ge 2$ if $N=1$ , and for $r\\in [2,2(N+2)/2)$ if $N\\ge 2$ ." ], [ "Let's define the functional $\\Phi :X\\rightarrow \\mathbb {R}$ , $\\Phi (z)=\\frac{1}{2}\\Vert z^+\\Vert ^2-\\frac{1}{2}\\Vert z^-\\Vert ^2-\\Psi (z), \\,\\text{ where }\\,\\Psi (z)=\\int _\\Theta H(t,x,z),$ which is such that its critical points are also weak solutions of (REF ).", "Lemma 3.2 $\\Phi \\in \\mathcal {C}^1(X,\\mathbb {R})$ .", "$\\Psi $ is bounded below and is weakly sequentially lower semicontinuous.", "$\\Psi ^{\\prime }$ is weakly sequentially continuous.", "By $(H_4)$ -$(2)$ we have for $\|z\|\\ge R$ and for $(t,x)\\in \\Theta $ $\|H_z(t,x,z)\|^\\sigma &\\le a_2\\widetilde{H}(t,x,z)\|z\|^\\sigma \\\\& \\le a_2\\Big (\\frac{1}{2}H_z(t,x,z)z-H(t,x,z)\\Big )\|z\|^\\sigma \\\\&\\le \\frac{a_2}{2}\|H_z(t,x,z)\|\|z\|^{\\sigma +1}.$ Hence $\|H_z(t,x,z)\|\\le a_3\|z\|^{\\frac{\\sigma +1}{\\sigma -1}} \\quad \\forall \|z\|\\ge R,\\quad \\forall (t,x)\\in \\Theta ,$ where $a_3>0$ is constant.", "On the other hand, since $H_z$ is continuous, we have $\|H_z(t,x,z)\|\\le a_4$ , for $\|z\|\\le R$ and $(t,x)\\in \\Theta $ .", "We deduce that $\|H_z(t,x,z)\|\\le a_4+a_3\|z\|^{\\frac{\\sigma +1}{\\sigma -1}} \\quad \\forall z\\in \\mathbb {R}^{2M},\\quad \\forall (t,x)\\in \\Theta .$ Let $\\varepsilon >0$ .", "$(H_2)$ implies that there is $\\delta >0$ such that $\|H_z(t,x,z)\|\\le \\varepsilon \|z\|\\quad \\text{pour tout }\|z\|<\\delta .$ If $\|z\|\\ge \\delta $ , then we have in view of (REF ) $\|H_z(t,x,z)\|\\le a_4+a_3\|z\|^{\\frac{\\sigma +1}{\\sigma -1}}\\le a_4\\frac{\|z\|^p}{\\delta ^p}+a_3\|z\|^p=\\big (\\frac{a_4}{\\delta ^p}+a_3\\big )\|z\|^p,$ with $p=\\frac{\\sigma +1}{\\sigma -1}$ .", "Hence $\\forall \\varepsilon >0,\\,\\,\\exists C(\\varepsilon )>0\\,\\,;\\,\\, \|H_z(t,x,z)\|\\le \\varepsilon \|z\|+C(\\varepsilon )\|z\|^{p}\\quad \\forall z\\in \\mathbb {R}^{2M}.$ Now one can easily verify that $\\left\\lbrace \\begin{array}{ll}\\sigma >1\\text{ if }N=1 & \\hbox{} \\\\\\sigma >1+\\frac{N}{2}\\text{ if }N\\ge 2 & \\hbox{}\\end{array}\\right.\\Longrightarrow \\left\\lbrace \\begin{array}{ll}p+1>2\\text{ si }N=1 & \\hbox{} \\\\2<p+1<2(N+2)/N)\\text{ if }N\\ge 2.", "& \\hbox{}\\end{array}\\right.$ It then follows from Lemma REF that $\\Psi $ is well defined on $X$ .", "We can now use a standard argument to show that $\\Phi $ is of class $C^1$ on $X$ and $\\big <\\Phi ^{\\prime }(z),w\\big >=\\big <z^+-z^-,w\\big >-\\int _\\Theta wH_z(t,x,z),\\quad \\forall z,w\\in X.$ Since $H\\ge 0$ , the functional $\\Psi $ is bounded below.", "Let $z_n\\rightharpoonup z$ and $c\\in \\mathbb {R}$ such that $\\Psi (z_n)\\le c$ .", "By Lemma REF we have $z_n\\rightarrow z$ in $L^2$ and, up to a subsequence, $z_n(t,x)\\rightarrow z(x,t)$ a.e.", "in $\\Theta $ .", "Using Fatou's lemma we obtain $\\Psi (z)\\le c$ , which shows that $\\psi $ is weakly lower semicontinuous.", "Let $z_n\\rightharpoonup z$ .", "By Lemma REF we have $z_n\\rightarrow z$ in $L^{p+1}$ (where $p=\\frac{\\sigma +1}{\\sigma -1}$ ) which implies, using (REF ) and Theorem $A.2$ in [33], that $\|H(t,x,z_n)-H(t,x,z)\|_{\\frac{p+1}{p}}\\rightarrow 0$ .", "It is then easy to deduce, using Fatou's lemma, that $\\Psi ^{\\prime }$ is weakly sequentially continuous.", "We now prove that the functional $\\Phi $ satisfies the Cerami condition.", "Lemma 3.3 Let $(z_n)\\subset X$ be such that $d=\\sup \\Phi (z_n)<\\infty \\quad \\text{and}\\quad \\big (1+\\Vert z_n\\Vert \\big )\\Phi ^{\\prime }(z_n)\\rightarrow 0.$ Then $(z_n)$ admits a convergent subsequence.", "Let us first show that the sequence $(z_n)$ is bounded.", "Arguing by contradiction we suppose that $(z_n)$ is unbounded.", "Then, up to a subsequence, we have $\\Vert z_n\\Vert \\rightarrow \\infty $ .", "Observe that $\\big <\\Phi ^{\\prime }(z_n),z_n^+-z_n^-\\big >&=\\Vert z_n^+\\Vert ^2+\\Vert z_n^-\\Vert ^2-\\int _\\Theta (z_n^+-z_n^-)\\cdot H_z(t,x,z_n)\\\\&=\\Vert z_n\\Vert ^2\\Big (1-\\int _\\Theta \\frac{ (z_n^+-z_n^-)\\cdot H_z(t,x,z_n)}{\\Vert z_n\\Vert ^2}\\Big ).$ Since $\|\\big <\\Phi ^{\\prime }(z_n),z_n^+-z_n^-\\big >\|\\le \\Vert \\Phi ^{\\prime }(z_n)\\Vert \\Vert z_n^+-z_n^-\\Vert \\le 2\\Vert z_n\\Vert \\Vert \\Phi ^{\\prime }(z_n)\\Vert \\rightarrow 0,$ one must have $\\int _\\Theta \\frac{ (z_n^+-z_n^-)\\cdot H_z(t,x,z_n)}{\\Vert z_n\\Vert ^2}\\rightarrow 1.$ Set $w_n=z_n/\\Vert z_n\\Vert $ .", "Assumption $(H_4)$ -$(1)$ implies that $\\widetilde{H}(t,x,z)\\ge a_1\|z\|^2-c_1$ for all $(t,x,z)$ .", "We deduce that for $n$ big enough one has $1+d\\ge \\Phi (z_n)-\\frac{1}{2}\\big <\\Phi ^{\\prime }(z_n),z_n\\big >&=\\int _\\Theta \\widetilde{H}(t,x,z_n)\\\\&\\ge a_2\|z_n\|_2^2-c_1\|\\Theta \|,$ where $\|\\Theta \|$ denotes the Lebesgue measure of $\\Theta $ .", "() shows that $(\|z_n\|_2)$ is bounded.", "Hence $\|w_n\|_2=\\frac{\|z_n\|_2}{\\Vert z_n\\Vert }\\rightarrow 0.$ On the other hand we have $\\Big \|\\int _\\Theta \\frac{(z_n^+-z_n^-)\\cdot H_z(t,x,z_n)}{\\Vert z_n\\Vert ^2}\\Big \|&\\le \\int _\\Theta \\frac{\|z_n^+-z_n^-\|}{\\Vert z_n\\Vert }\\frac{\|H_z(t,x,z_n\|}{\\Vert z_n\\Vert }\\\\&=\\int _\\Theta \|w_n^+-w_n^-\|\|w_n\|\\frac{\|H_z(t,x,z_n\|}{\|z_n\|}\\\\\\le \\Big (\\int _\\Theta &\\big (\\frac{\|H_z(t,x,z_n\|}{\|z_n\|}\\big )^\\sigma \\Big )^{1/\\sigma }\\Big (\\int _\\Theta \\big (\|w_n^+-w_n^-\|\|w_n\|\\big )^{\\sigma ^{\\prime }}\\Big )^{1/\\sigma ^{\\prime }},$ with $\\frac{1}{\\sigma }+\\frac{1}{\\sigma ^{\\prime }}=1.$ Since $\|w_n^+-w_n^-\|\|w_n\|=\|w_n^+-w_n^-\|\|w_n^++w_n^-\|\\le (\|w_n^+\|+\|w_n^-\|)^2\\le 2(\|w_n^+\|^2+\|w_n^-\|^2),$ we have $\\int _\\Theta \\big (\|w_n^+-w_n^-\|\|w_n\|\\big )^{\\sigma ^{\\prime }}\\le c_2\\Big (\|w_n^+\|_{2\\sigma ^{\\prime }}^{2\\sigma ^{\\prime }}+\|w_n^-\|_{2\\sigma ^{\\prime }}^{2\\sigma ^{\\prime }}\\big ).$ By using the fact that $2\\sigma ^{\\prime }=p+1\\in [2,2(N+2)/N]$ , we have $\|w_n^\\pm \|_{2\\sigma ^{\\prime }}\\le \|w_n^\\pm \|_2^\\alpha \|w_n^\\pm \|_{2(N+2)/N}^{1-\\alpha }\\quad \\big (\\text{interpolation inequality}\\big ),$ where $\\alpha \\in [0,1]$ is such that $\\frac{\\alpha }{2}+\\frac{1-\\alpha }{2(N+2)/N}=1$ .", "Since $X$ continuously embeds in $L^{2(N+2)/N}$ , we have $\|w_n^\\pm \|_{2(N+2)/N}\\le c_3\\Vert w_n^\\pm \\Vert \\le c_3$ (because $\\Vert w_n^\\pm \\Vert \\le \\Vert w_n\\Vert =1$ ).", "Now, since the decomposition $X=X^-\\oplus X^+$ is also orthogonal with respect to the $L^2$ -norm, we have $\|w_n^\\pm \|_2\\le \|w_n\|_2$ .", "Hence $\|w_n^\\pm \|_{2\\sigma ^{\\prime }}\\le c_3\|w_n\|_2^\\alpha $ , which implies $\\Big \|\\int _\\Theta \\frac{(z_n^+-z_n^-)\\cdot H_z(t,x,z_n)}{\\Vert z_n\\Vert ^2}\\Big \|\\le c_4\\Big (\\int _\\Theta \\Big (\\frac{\|H_z(t,x,z_n\|}{\|z_n\|}\\Big )^\\sigma \\Big )^{1/\\sigma }\|w_n\|_2^{2\\alpha }.$ Let $\\varepsilon >0$ .", "By $(H_2)$ there exists $\\delta >0$ such that $\|H_z(t,x,z)\|\\le \\varepsilon \|z\|$ for $\|z\|<\\delta $ .", "Then $\\int _\\Theta \\Big (\\frac{\|H_z(t,x,z_n\|}{\|z_n\|}\\Big )^\\sigma &=\\int _{\\Theta \\cap \\lbrace \|z_n\|<\\delta \\rbrace } \\Big (\\frac{\|H_z\|}{\|z_n\|}\\Big )^\\sigma +\\int _{\\Theta \\cap \\lbrace \\delta \\le \|z_n\|\\le R\\rbrace } \\Big (\\frac{\|H_z\|}{\|z_n\|}\\Big )^\\sigma \\\\& \\qquad \\qquad \\qquad +\\int _{\\Theta \\cap \\lbrace \|z_n\|\\ge R\\rbrace } \\Big (\\frac{\|H_z\|}{\|z_n\|}\\Big )^\\sigma \\\\\\le \|\\Theta \|\\varepsilon ^\\sigma +&\\sup _{\\begin{array}{c}\\Theta \\cap \\lbrace \\delta \\le \|z_n\|\\le R\\rbrace \\end{array}}\\Big (\\frac{\|H_z\|}{\|z_n\|}\\Big )^\\sigma \|\\Theta \|+a_2\\int _{\\Theta \\cap \\lbrace \|z_n\|\\ge R\\rbrace }\\widetilde{H}(t,x,z_n),$ where we have used $(H_4)$ -$(2)$ for the last term in the RHS of the equality.", "We deduce from (REF ) that $\\Big (\\int _{\\Theta \\cap \\lbrace \|z_n\|\\ge R\\rbrace }\\widetilde{H}(t,x,z_n)\\Big )$ is bounded.", "Hence $\\Big (\\int _\\Theta \\big (\\frac{\|H_z(t,x,z_n\|}{\|z_n\|}\\big )^\\sigma \\Big )$ is bounded.", "It then follows from (REF ) and (REF ) that $\\int _\\Theta \\frac{(z_n^+-z_n^-)\\cdot H_z(t,x,z_n)}{\\Vert z_n\\Vert ^2}\\rightarrow 0,$ which contradicts (REF ).", "This contradiction leads to the conclusion that the sequence $(z_n)$ is bounded." ], [ "We may now suppose that $z_n\\rightharpoonup z$ in $X$ .", "Since $\\Vert z_n^\\pm -z^\\pm \\Vert ^2=\\pm \\big <\\Phi ^{\\prime }(z_n)-\\Phi ^{\\prime }(z),z_n^\\pm -z^\\pm \\big >\\pm \\int _\\Theta (z_n^\\pm -z^\\pm )\\cdot H_z(t,x,z_n),$ a standard argument using Lemma REF shows that $z_n\\rightarrow z$ .", "[Proof of Theorem REF ] Let $(e_i)$ be an orthonormal basis of $X^+$ .", "We recall that $X^-_k=X^-\\oplus \\big (\\oplus _{j=0}^k\\mathbb {R}e_j\\big )\\text{ and }X^+_k=\\overline{\\oplus _{j=k}^\\infty \\mathbb {R}e_j}.$ Let $z\\in X^-_k$ .", "Using $(H_3)$ we have $\\forall \\delta >0\\quad \\exists c_\\delta >0\\,\\,;\\,\\ H(t,x,z) \\ge \\delta \|z\|^2-c_\\delta .$ Then $\\Phi (z)=\\frac{1}{2}\\Vert z^+\\Vert ^2-\\frac{1}{2}\\Vert z^-\\Vert ^2-\\int _\\Theta H(t,x,z)\\le \\frac{1}{2}\\Vert z^+\\Vert ^2-\\frac{1}{2}\\Vert z^-\\Vert ^2-\\delta \|z\|_2^2+c_\\delta \|\\Theta \|.$ Since the decomposition $X=X^-\\oplus X^+$ is orthogonal with respect to the $L^2$ -norm, we have $\|z^+\|_2\\le \|z\|_2$ .", "Since all norms are equivalent on $\\oplus _{j=0}^k\\mathbb {R}e_j$ , there is $c_1>0$ such that $c_1\\Vert z^+\\Vert ^2\\le \|z\|_2^2$ .", "Hence $\\Phi (z)\\le \\big (\\frac{1}{2}-c_1\\delta \\big )\\Vert z^+\\Vert ^2-\\frac{1}{2}\\Vert z^-\\Vert ^2+c_1\|\\Theta \|.$ Let us choose $\\delta $ such that $c_1\\delta \\ge 1$ .", "Then we have $\\Phi (z)\\le -\\frac{1}{2}\\Vert z\\Vert ^2+c_\\delta \|\\Theta \|\\rightarrow -\\infty \\text{ as }\\Vert z\\Vert \\rightarrow \\infty .$" ], [ "Let $z\\in X^+_k$ .", "Then for every $\\varepsilon >0$ , we have $\\Phi (z)&=\\frac{1}{2}\\Vert z\\Vert ^2-\\int _\\Theta H(t,x,z)\\\\&\\ge \\frac{1}{2}\\Vert z\\Vert ^2-\\frac{\\varepsilon }{2}\|z\|_2^2-\\frac{c(\\varepsilon )}{p+1}\|z\|_{p+1}^{p+1}\\,\\,(\\text{by }(\\ref {deux}))\\\\&\\ge \\frac{1}{2}\\Vert z\\Vert ^2-c_2\\varepsilon \|z\|_2^2-\\frac{c(\\varepsilon )}{p+1}\|z\|_{p+1}^{p+1}\\,\\,(\\text{because } X\\hookrightarrow L^2)\\\\&\\ge \\big (\\frac{1}{2}-c_2\\varepsilon \\big )\\Vert z\\Vert ^2-\\frac{c(\\varepsilon )}{p+1}\\beta _k^{p+1}\\Vert z\\Vert ^{p+1},$ where $\\beta _k=\\sup _{\\begin{array}{c}w\\in X^+_k\\\\\\Vert w\\Vert =1\\end{array}}\|w\|_{p+1}$ .", "By choosing $\\varepsilon \\le \\frac{1}{2c_2}$ we obtain $\\Phi (z)\\ge \\frac{1}{2}\\Big (\\frac{1}{2}\\Vert u\\Vert ^2-c_3\\beta _k^{p+1}\\Vert z\\Vert ^{p+1}\\Big ).$ Hence, since $\\beta _k\\rightarrow 0$ as $k\\rightarrow \\infty $ (see [33], Lemma $3.8$ ), we have $\\Vert z\\Vert =r_k:=\\big (c_3(p+1)\\beta _k^{p+1}\\big )^{1/(1-p)}\\Rightarrow \\Phi (z)\\ge \\frac{1}{2}\\big (\\frac{1}{2}-\\frac{1}{p+1})r_k^2\\rightarrow \\infty ,\\,k\\rightarrow \\infty .$" ], [ "(REF ) and (REF ) show that the conditions $(A_1)$ and $(A_2)$ of Theorem REF are satisfied.", "In view of Lemmas REF and REF , we can apply Theorem REF and get the result." ], [ "Strong solutions of superquadratic elliptic systems", "In this section, we consider the problem $\\left\\lbrace \\begin{array}{ll}-\\Delta u=g(x,v)\\,\\, \\textnormal {in }\\Omega , & \\hbox{} \\\\-\\Delta v=f(x,u)\\,\\,\\textnormal {in }\\Omega , & \\hbox{} \\\\u=v=0\\textnormal { on }\\partial \\Omega , & \\hbox{}\\end{array}\\right.\\qquad \\mathrm {(ES)}$ and we assume in the sequel that the assumptions $(E_1)$ -$(E_5)$ are satisfied." ], [ "Consider the Laplacian as the operator $-\\Delta :H^2(\\Omega )\\cap H_0^1(\\Omega )\\subset L^2(\\Omega )\\rightarrow L^2(\\Omega ),$ and let $(\\varphi _j)_{j\\ge 1}$ a corresponding system of orthogonal and $L^2(\\Omega )$ -normalized eigenfunctions, with eigenvalues $(\\lambda _j)_{j\\ge 1}$ .", "Then, writing $u=\\sum \\limits _{j=1}^\\infty a_j\\varphi _j,\\quad \\textnormal {with } a_j=\\int _\\Omega u\\varphi _j ,$ we set, for $0\\le s\\le 2$ $E^s:=\\Big \\lbrace u\\in L^2(\\Omega )\\, \\big \|\\,\\sum \\limits _{j=1}^\\infty \\lambda _j^s\|a_j\|^2<\\infty \\Big \\rbrace $ and $A^s(u):=\\sum \\limits _{j=1}^\\infty \\lambda _j^{s/2}a_j\\varphi _j,\\quad \\forall u\\in D(A^s)=E^s.$ One can verify easily that $A^s$ is an isomorphism onto $L^2(\\Omega )$ .", "We denote $A^{-s}:=(A^s)^{-1}$ .", "It is well known (see [23]) that the space $E^s$ is a fractional Sobolev space with the inner product $\\big <u,v\\big >_s=\\int _\\Omega A^suA^sv.$ Lemma 4.1 ([29]) $E^s$ embeds continuously in $L^r(\\Omega )$ for $s>0$ and $r\\ge 1$ satisfying $\\frac{1}{r}\\ge \\frac{1}{2}-\\frac{s}{N}.$ Moreover, the embedding is compact when the inequality is strict.", "By assumption $(E_1)$ , there exist $s,t>0$ such that $s+t=2$ and $\\frac{1}{p}>\\frac{1}{2}-\\frac{s}{N}\\quad \\textnormal {and}\\quad \\frac{1}{q}>\\frac{1}{2}-\\frac{t}{N}.$ We consider the functional $\\Phi (u,v):=\\int _\\Omega A^suA^tv-\\Psi (u,v),\\quad (u,v)\\in E^s\\times E^t.$ where $\\Psi (u,v)=\\int _\\Omega \\Big (F(x,u)+G(x,v)\\Big ).$ It follows from Lemma REF that the inclusions $E^s\\hookrightarrow L^p(\\Omega )$ and $E^t\\hookrightarrow L^q(\\Omega )$ are continuous.", "This fact in conjunction with the estimate $\\Big \|\\int _\\Omega A^suA^tv\\Big \|\\le \|A^su\|_2\|A^tu\|_2=\\Vert u\\Vert _s\\Vert v\\Vert _t,$ imply that the preceding functional $\\Phi $ is well defined on $E:=E^s\\times E^t$ .", "Now a standard argument shows that if assumption $(E_1)$ holds, then the functional $\\Phi $ is of class $\\mathcal {C}^1$ on $E$ and its critical points are weak solutions of (REF ).", "We recall that $(u,v)\\in E^s\\times E^t$ is a weak solution of (REF ) if $\\int _\\Omega \\big (A^suA^tk+A^shA^tv\\big )-\\int _\\Omega \\Big (hf(x,u)+kg(x,v)\\Big )=0,\\quad \\forall (h,k)\\in E^s\\times E^t.$ The following regularity result is due to de Figueiredo and Felmer [17].", "Lemma 4.2 If $(u,v)\\in E^s\\times E^t$ is a weak solution of (REF ), then $(u,v)$ is a strong solution of (REF )." ], [ "We endow $E=E^s\\times E^t$ with the inner product $\\big <(u,v),(\\phi ,\\varphi )\\big >_{s\\times t}=\\big <u,\\phi \\big >_s+\\big <v,\\varphi \\big >_t,\\quad (u,v),(\\phi ,\\varphi )\\in E,$ and the associated norm $\\Vert (u,v)\\Vert _{s\\times t}^2=\\big <(u,v),(u,v)\\big >_{s\\times t}$ ." ], [ "In the following we may assume without loss of generality that $s\\ge t$ .", "One can easily verify that $E$ has the orthogonal decomposition (with respect to $\\big <\\cdot ,\\cdot \\big >_{s\\times t}$ ) $E=E^+\\oplus E^-$ , where $E^+:=\\Big \\lbrace (u,A^{s-t}u)\\,\|\\,u\\in E^s\\Big \\rbrace \\quad \\textnormal {and}\\quad E^-:=\\Big \\lbrace (u,-A^{s-t}u)\\,\|\\,u\\in E^s\\Big \\rbrace .$ If we denote by $P^\\pm :E\\rightarrow E^\\pm $ the orthogonal projections, then a direct calculation yields $P^\\pm (u,v)=\\frac{1}{2}\\big (u\\pm A^{t-s}v,v\\pm A^{s-t}u\\big ),\\quad \\forall (u,v)\\in E,$ and $\\Phi (u,v)=\\frac{1}{2}\\Vert P^+(u,v)\\Vert _{s\\times t}^2-\\frac{1}{2}\\Vert P^-(u,v)\\Vert _{s\\times t}^2-\\int _\\Omega \\Big (F(x,u)+G(x,v)\\Big ).$" ], [ "Let us recall the following technical result, which will play a crucial role in the verification of the Cerami condition.", "Proposition 4.3 ([24], Lemma $3.2$ ) Let $x\\in \\mathbb {R}^N$ , $u,v,s\\in \\mathbb {R}$ such that $s\\ge -1$ and $su+v\\ne 0$ .", "If $(E_2)$ -$(E_4)$ are satisfied then $f(x,u)\\Big [s(\\frac{s}{2}u+(1+s)v)\\Big ]&+F(x,u)-F(x,u+v)\\le 0,\\\\g(x,u)\\Big [s(\\frac{s}{2}u+(1+s)v)\\Big ]&+G(x,u)-G(x,u+v)\\le 0.$ Lemma 4.4 Let $(z_n=(u_n,v_n))\\subset E$ and $c>0$ such that $\\Phi (z_n)\\rightarrow c\\quad \\text{and}\\quad \\big (1+\\Vert z_n\\Vert _{s\\times t}\\big )\\Phi ^{\\prime }(z_n)\\rightarrow 0.$ Then $(z_n)$ has a convergent subsequence.", "Let us first show that $(z_n)$ is bounded.", "On the contrary if $(z_n)$ is unbounded, then we can assume that $\\Vert z_n\\Vert _{s\\times t}\\rightarrow \\infty $ .", "Set $w_n=z_n/\\Vert z_n\\Vert _{s\\times t}$ .", "Then, up to a subsequence, we have $w_n\\rightharpoonup w$ in $E$ .", "Since $F,G\\ge 0$ , we have for $n$ big enough $c-1\\le \\Phi (z_n)\\le \\frac{1}{2}\\Vert z_n^+\\Vert _{s\\times t}^2-\\frac{1}{2}\\Vert z_n^-\\Vert _{s\\times t}^2=\\frac{1}{2}\\Vert z_n\\Vert _{s\\times t}^2\\big (\\Vert w_n^+\\Vert _{s\\times t}^2-\\Vert w_n^-\\Vert _{s\\times t}^2\\big ).$ Recall that $\\Vert w_n^+\\Vert _{s\\times t}^2+\\Vert w_n^-\\Vert _{s\\times t}^2=1$ .", "Hence $\\Vert w_n^+\\Vert _{s\\times t}^2\\ge \\frac{1}{2}+\\frac{c-1}{\\Vert z_n\\Vert _{s\\times t}^2}$ .", "But since $\\Vert z_n\\Vert _{s\\times t}^2\\rightarrow \\infty $ , we have for $n$ big enough $\\frac{\|c-1\|}{\\Vert z_n\\Vert _{s\\times t}^2}\\le \\frac{1}{6}$ .", "It follows that $\\Vert w_n^+\\Vert _{s\\times t}^2\\ge \\frac{1}{3}$ .", "Now fix $r>0$ and set $s_n=\\frac{r}{\\Vert z_n\\Vert _{s\\times t}}-1$ and $y_n=-\\frac{r}{\\Vert z_n\\Vert _{s\\times t}}z_n^-=-rw_n^-$ .", "Then $s_n\\ge -1$ , $y_n\\in E^-$ and the sequences $(s_n)$ and $(y_n)$ are bounded.", "By setting $y_n=(a_n,b_n)$ , where $a_n\\in E^s$ and $b_n\\in E^t$ , we have $\\big <\\Phi ^{\\prime }(z_n),s_n(\\frac{s_n}{2}+1)z_n&+(1+s_n)y_n\\big >=\\big <z_n^+-z_n^-,s_n(\\frac{s_n}{2}+1)z_n+(1+s_n)y_n\\big >\\\\&-\\int _\\Omega \\big (s_n(\\frac{s_n}{2}+1)u_n+(1+s_n)a_n\\big )f(x,u_n)\\\\&-\\int _\\Omega \\big (s_n(\\frac{s_n}{2}+1)v_n+(1+s_n)b_n\\big )g(x,v_n).$ Since $(s_n)$ and $(y_n)$ are bounded, we obtain $\\big \|\\big <\\Phi ^{\\prime }(z_n),s_n(\\frac{s_n}{2}+1)z_n+(1+s_n)y_n\\big >\\big \|&\\le \\Vert \\Phi ^{\\prime }(z_n)\\Vert \\Vert s_n(\\frac{s_n}{2}+1)z_n+(1+s_n)y_n\\Vert \\\\&\\le C_1\\Vert \\Phi ^{\\prime }(z_n)\\Vert \\Vert z_n\\Vert _{s\\times t}+C_2\\Vert \\Phi ^{\\prime }(z_n)\\Vert \\rightarrow 0.", "$ We deduce that for $n$ big enough $\\nonumber \\big <z_n^+-z_n^-,s_n(\\frac{s_n}{2}+1)z_n&+(1+s_n)y_n\\big >\\le 1\\\\\\nonumber &+\\int _\\Omega \\big (s_n(\\frac{s_n}{2}+1)u_n+(1+s_n)a_n\\big )f(x,u_n)\\\\&+\\int _\\Omega \\big (s_n(\\frac{s_n}{2}+1)v_n+(1+s_n)b_n\\big )g(x,v_n).$ Noting that $rw_n^+=z_n+(s_nz_n+y_n)$ and $\\big <z_n^+-z_n^-,s_n(\\frac{s_n}{2}+1)z_n+(1+s_n)y_n\\big >=-\\frac{1}{2}\\Vert z_n^+\\Vert _{s\\times t}^2+\\frac{1}{2}\\Vert z_n^-\\Vert _{s\\times t}^2\\\\+\\frac{1}{2}r^2\\Vert w_n^+\\Vert _{s\\times t}^2+\\frac{1}{2}r^2\\Vert w_n^-\\Vert _{s\\times t}^2,$ we have $\\Phi (rw_n^+)-\\Phi (z_n)&=-\\frac{1}{2}\\Vert w_n^-\\Vert _{s\\times t}^2+\\big <z_n^+-z_n^-,s_n(\\frac{s_n}{2}+1)z_n+(1+s_n)y_n\\big >\\\\&+\\int _\\Omega \\Big (F(x,u_n)-F(x,(1+s_n)u_n+a_n)\\Big )\\\\&+\\int _\\Omega \\Big (G(x,u_n)-G(x,(1+s_n)u_n+a_n)\\Big ).$ We deduce by using (REF ) that for $n$ sufficiently large $\\Phi (rw_n^+)-\\Phi (z_n)&\\le 1+\\int _\\Omega \\big (s_n(\\frac{s_n}{2}+1)u_n+(1+s_n)a_n\\big )f(x,u_n)\\\\&\\qquad +\\int _\\Omega \\Big [F(x,u_n)-F\\big (x,(1+s_n)u_n+a_n\\big )\\Big ]\\\\&\\qquad + \\int _\\Omega \\big (s_n(\\frac{s_n}{2}+1)v_n+(1+s_n)b_n\\big )g(x,u_n)\\\\&\\qquad +\\int _\\Omega \\Big [G(x,u_n)-G\\big (x,(1+s_n)v_n+b_n\\big )\\Big ].$ We conclude by using Proposition REF that for $n$ big enough $\\Phi (rw_n^+)-1\\le \\Phi (z_n).$ If $w^+=0$ , then $w_n^+\\rightarrow 0$ in $L^p(\\Omega )\\times L^q(\\Omega )$ by Lemma REF .", "By $(E_1)$ and Theorem A.2 in [33] we have $\\int _\\Omega F\\big (x,(1+s_n)u_n+a_n\\big )\\rightarrow 0 \\text{ and } \\int _\\Omega G\\big (x,(1+s_n)v_n+b_n\\big )\\rightarrow 0.$ Since $\\Phi (z_n)\\rightarrow c$ , we then obtain for $n$ sufficiently large $c+1\\ge \\Phi (z_n)&\\ge \\Phi (rw_n^+)-1\\\\&= \\frac{1}{2}r^2\\Vert w_n\\Vert _{s\\times t}^2-\\int _\\Omega F\\big (x,(1+s_n)u_n+a_n\\big )\\\\&\\qquad -\\int _\\Omega G\\big (x,(1+s_n)v_n+b_n\\big )-1\\\\&\\ge \\frac{1}{2}r^2\\Vert w_n\\Vert _{s\\times t}^2-2\\\\&\\ge \\frac{r^2}{6}-2\\quad \\big (\\text{because }\\Vert w_n^+\\Vert _{s\\times t}^2\\ge \\frac{1}{3}\\big ).$ This leads to a contradiction if we take $r$ arbitrary large.", "Hence $w^+\\ne 0$ and then $w\\ne 0$ .", "Setting $w_n=(w_n^1,w_n^2)\\in E^s\\times E^t$ , we have $\\frac{\\Phi (z_n)}{\\Vert z_n\\Vert _{s\\times t}^2}&=\\frac{1}{2}\\Vert w_n^+\\Vert _{s\\times t}^2-\\frac{1}{2}\\Vert w_n^-\\Vert _{s\\times t}^2-\\int _\\Omega \\frac{F(x,u_n)}{\\Vert z_n\\Vert _{s\\times t}^2}-\\int _\\Omega \\frac{G(x,u_n)}{\\Vert z_n\\Vert _{s\\times t}^2}\\\\&=\\frac{1}{2}\\Vert w_n^+\\Vert _{s\\times t}^2-\\frac{1}{2}\\Vert w_n^-\\Vert _{s\\times t}^2-\\int _{w_n^1\\ne 0}\\frac{F(x,w^1_n\\Vert z_n\\Vert _{s\\times t})}{\\big \|w_n^1\\Vert z_n\\Vert _{s\\times t}\\big \|^2}\\\\&\\quad -\\int _{w_n^2\\ne 0}\\frac{G(x,w^2_n\\Vert z_n\\Vert _{s\\times t})}{\\big \|w_n^1\\Vert z_n\\Vert _{s\\times t}\\big \|^2}\\\\&\\le C-\\int _{w_n^1\\ne 0}\\frac{F(x,w^1_n\\Vert z_n\\Vert _{s\\times t})}{\\big \|w_n^1\\Vert z_n\\Vert _{s\\times t}\\big \|^2}-\\int _{w_n^2\\ne 0}\\frac{G(x,w^2_n\\Vert z_n\\Vert _{s\\times t})}{\\big \|w_n^1\\Vert z_n\\Vert _{s\\times t}\\big \|^2}.$ Since $(\\Phi (z_n))$ is bounded and $\\Vert z_n\\Vert _{s\\times t}\\rightarrow \\infty $ , the LHS converges to 0.", "By applying Fatou's lemma and using $(E_3)$ , we see that the RHS goes to $-\\infty $ , which is a contradiction.", "Consequently the sequence $(z_n)$ is bounded." ], [ "Now a standard argument based on Lemma REF shows that the bounded sequence $(z_n)$ has a convergent subsequence.", "[Proof of Theorem REF ] Let $(a_j)_{j\\ge 0}$ be an orthonormal basis of $E^s$ .", "We define an orthonormal basis $(e_j)_{j\\ge 0}$ of $E^+$ by setting $e_j:=\\frac{1}{\\sqrt{2}}\\big (a_j,A^{s-t}a_j\\big ).$ Let $E^-_k=E^-\\oplus \\big (\\oplus _{j=0}^k\\mathbb {R}e_j\\big )\\text{ and }E^+_k=\\overline{\\oplus _{j=k}^\\infty \\mathbb {R}e_j}.$ $(E_3)$ implies that for every $\\delta >0$ there is $C_\\delta >0$ such that $F(x,u)\\ge \\delta \|u\|^2-C_\\delta \\quad \\text{and}\\quad G(x,u)\\ge \\delta \|u\|^2-C_\\delta ,\\quad \\forall (x,u).$" ], [ "Let $z\\in E^-_k$ .", "Then $z=\\big (u,A^{s-t}u\\big )+\\big (v,-A^{s-t}v\\big )$ , where $v\\in E^s$ and $u\\in E^s_k:=\\oplus _{j=0}^k\\mathbb {R}a_j$ .", "Using (REF ), the fact that $E^{s-t}$ continuously embeds in $L^2(\\Omega )$ , and the parallelogram identity, we obtain $\\Phi (z)&=\\Vert u\\Vert ^2_s-\\Vert v\\Vert ^2_s-\\int _\\Omega \\Big (F(x,u+v)+G(x,A^{s-t}(u-v))\\Big )\\\\&\\le \\Vert u\\Vert ^2_s-\\Vert v\\Vert ^2_s-2C\\delta \\big (\|u\|^2_2+\|v\|^2_2\\big )+2C_\\delta \|\\Omega \| .$ Since all the norms are equivalent on the finite-dimensional subspace $E^s_k$ , there is a constant $c_1>0$ such that $c_1\\Vert u\\Vert _s\\le \|u\|_2$ .", "Hence $\\Phi (z)\\le (1-c_2\\delta )\\Vert u\\Vert ^2_s-\\Vert v\\Vert ^2_s-2C_\\delta \|\\Omega \|.$ Choose $\\delta >\\frac{1}{c_2}$ .", "Hence $\\Phi (z)\\rightarrow -\\infty $ as $\\Vert z\\Vert _{s\\times t}\\rightarrow \\infty $ , and consequently assumption $(A_1)$ of Theorem REF is satisfied." ], [ "Let $z\\in E^+_k$ .", "Then $z=(u,A^{s-t}u)$ with $u\\in \\overline{\\oplus _{j=k}^\\infty \\mathbb {R}a_j}$ , and $\\Phi (z)&=\\frac{1}{2}\\Vert z\\Vert _{s\\times t}^2-\\int _\\Omega \\Big (F(x,u)+G(x,A^{s-t}u)\\Big )\\\\&\\ge \\frac{1}{2}\\Vert z\\Vert _{s\\times t}^2-2\\int _\\Omega \\Big (F(x,u)+G(x,A^{s-t}u)\\Big ).$ By $(E_1)$ there is a constant $C_1>0$ such that $\|F(x,u)\|\\le C_1(1+\|u\|^p)\\quad \\text{and}\\quad \|G(x,A^{s-t}u)\|\\le C_1(1+\|A^{s-t}u\|^q).$ Hence $\\Phi (z)\\ge \\Vert u\\Vert ^2_s-2C_1\|u\|_p^p-2C_1\|A^{s-t}u\|^q_q-4C_1\|\\Omega \|.$ We define $\\beta _{1,k}:=\\sup _{\\begin{array}{c}u\\in \\overline{\\oplus _{j=k}^\\infty \\mathbb {R}a_j}\\\\\\Vert u\\Vert _s=1\\end{array}}\|u\|_p,\\quad \\beta _{2,k}:=\\sup _{\\begin{array}{c}v\\in \\overline{\\oplus _{j=k}^\\infty \\mathbb {R}(A^{s-t}a_j)}\\\\\\Vert v\\Vert _t=1\\end{array}}\|v\|_q,$ and $\\beta _k=\\max \\big \\lbrace \\beta _{1,k};\\beta _{2,k}\\big \\rbrace $ .", "Then $\\Phi (z)\\ge \\Vert u\\Vert ^2_s-2C_1\\beta _k^p\\Vert u\\Vert _s^p-2C_1\\beta _k^q\\Vert u\\Vert ^q_s-4C_1\|\\Omega \|.$ We may assume without loss of generality that $q\\le p$ and we set $r_k:=\\big (C_1p\\beta _k^p\\big )^{\\frac{1}{2-p}}.$ Then for $\\Vert z\\Vert _{s\\times t}=\\sqrt{2}\\Vert u\\Vert _s=r_k$ we have $\\Phi (z)\\ge \\widetilde{b_k}:=K\\beta _k^{\\frac{2p}{2-p}}\\Big [\\Big (\\frac{1}{4}-\\frac{1}{p(\\sqrt{2})^p}\\Big )-A\\beta _k^{\\frac{2(q-p)}{2-p}}\\Big ]-4C_1\|\\Omega \|,$ where $K,A>0$ are constant.", "We know by Lemma $3.8$ in [33] that $\\beta _{1,k}\\rightarrow 0$ and $\\beta _{2,k}\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "This implies that $\\widetilde{b_k}\\rightarrow \\infty $ as $k\\rightarrow \\infty $ , and then that assumption $(A_2)$ of Theorem REF is satisfied." ], [ "An argument similar to the one in the proof of Lemma REF shows that $\\Psi $ is weakly sequentially lower semicontinuous and that $\\Psi ^{\\prime }$ is weakly sequentially continuous.", "Since we have previously shown that the functional $\\Phi $ satisfies the Cerami condition at every nonnegative critical level, we can apply Theorem REF and get the desired result." ], [ "Acknowledgement", "This work was supported by a NSERC grant." ] ]	1403.0158
[ [ "Casimir interaction between spherical and planar plasma sheets" ], [ "Abstract We consider the interaction between a spherical plasma sheet and a planar plasma sheet due to the vacuum fluctuations of electromagnetic fields.", "We use the mode summation approach to derive the Casimir interaction energy and study its asymptotic behaviors.", "In the small separation regime, we confirm the proximity force approximation and calculate the first correction beyond the proximity force approximation.", "This study has potential application to model Casimir interaction between objects made of materials that can be modeled by plasma sheets such as graphene sheets." ], [ "Introduction", "Due to the potential impact to nanotechnology, the Casimir interactions between objects of nontrivial geometries have been under active research in recent years.", "Thanks to works done by several groups of researchers [1], [4], [5], [6], [7], [8], [9], [10], [2], [11], [12], [13], [14], [3], we have now a formalism to compute the exact functional representation (known as TGTG formula) for the Casimir interaction energy between two objects.", "Despite the seemingly different approaches taken, all the methods can be regarded as multiple scattering approach, which can also be understood from the point of view of mode summation approach [15], [16], [17].", "The basic ingredients in the TGTG formula are the scattering matrices of the two objects and the transition matrices that relate the coordinate system of one object to the other.", "In the case that the objects have certain symmetries that allow separable coordinate system to be employed, one can calculate these matrices explicitly.", "This has made possible the exact analytic and numerical analysis of the Casimir interaction between a sphere and a plate [18], [19], [20], [21], [22], [26], [23], [24], [25], [29], [28], [27], between two spheres [30], [31], [32], between a cylinder and a plate [2], [34], [33], between two cylinders [35], [36], [37], [38], between a sphere and a cylinder [39], [40], as well as other geometries [41], [42], [43].", "As is well known, the strength of the Casimir interaction does not only depend on the geometries of the objects, it is also very sensitive to the boundary conditions imposed on the objects.", "For the past few years, a lot of works have been done in the analysis of the quantum effect on objects with perfect boundary conditions such as Dirichlet, Neumann, perfectly conducting, infinitely permeable, etc.", "There are also a number of works which consider real materials such as metals modeled by plasma or Drude models [19], [20], [23], [24], [25], [27], [29], [31], [38], [40].", "In this work, we consider the Casimir interaction between a spherical plasma sheet and a planar plasma sheet.", "Plasma sheet model was considered in [33], [44], [45], [46], [47], [48], [49] to model graphene sheet, describing the $\\pi $ electrons in C$_{60}$ molecule.", "This model has its own appeal in describing a thin shell of materials that have the same attributes.", "In [33], the Casimir interaction between a cylindrical plasma sheet and a planar plasma sheet has been considered.", "Our work can be considered as a generalization of [33] where we consider a spherical plasma sheet instead of a cylindrical plasma sheet.", "One of the main objectives of the current work is to derive the TGTG formula for the Casimir interaction energy.", "As in [33], we are also going to study the asymptotic behaviors of the Casimir interaction in the small separation regimes.", "We would expect that the leading term of the Casimir interaction coincides with the proximity force approximation (PFA), which we are going to confirm.", "Another major contribution would be the exact analytic computation of the next-to-leading order term which determines the deviation from PFA." ], [ "The Casimir interaction energy", "In this section, we derive the TGTG formula for the Casimir interaction energy between a spherical plasma sheet and a planar plasma sheet.", "We follow our approach in [17].", "Assume that the spherical plasma sheet is a spherical surface described by $r=R$ in spherical coordinates $(r,\\theta ,\\phi )$ , and the planar plasma sheet is located at $z=L$ with dimension $H\\times H$ .", "It is assumed that $R<L\\ll H$ .", "The center of the spherical shell is the origin $O$ , and the center of the coordinate system about the plane $z=L$ is $O^{\\prime } =(0,0,L)$ .", "The electromagnetic field is governed by the Maxwell's equations: $\\begin{split}&\\nabla \\cdot \\mathbf {E}=\\frac{\\rho _f}{\\varepsilon _0},\\hspace{28.45274pt}\\nabla \\times \\mathbf {E}+\\frac{ \\partial \\mathbf {B} }{\\partial t}=\\mathbf {0},\\\\&\\nabla \\cdot \\mathbf {B}=0,\\hspace{28.45274pt} \\nabla \\times \\mathbf {B}-\\frac{1}{c^2}\\frac{\\partial \\mathbf {E}}{\\partial t}=\\mu _0\\mathbf {J}_f.\\end{split}$ The free charge density $\\rho _f$ and free current density $\\mathbf {J}_f$ are functions having support on the plasma sheets (boundaries).", "Let $\\mathbf {A}$ be a vector potential that satisfies the gauge condition $\\nabla \\cdot \\mathbf {A}=0$ and such that $\\mathbf {E}=-\\frac{\\partial \\mathbf {A}}{\\partial t},\\hspace{28.45274pt}\\mathbf {B}=\\nabla \\times \\mathbf {A}.$ $\\mathbf {A}(\\mathbf {x},t)$ can be written as a superposition of normal modes $\\mathbf {A}(\\mathbf {x},\\omega )e^{-i\\omega t}$ : $\\mathbf {A}(\\mathbf {x},t)=\\int _{-\\infty }^{\\infty }d\\omega \\,\\mathbf {A}(\\mathbf {x},\\omega )e^{-i\\omega t}.$ Maxwell's equations (REF ) imply that outside the boundaries, $\\nabla \\times \\nabla \\times \\mathbf {A}(\\mathbf {x},\\omega )=k^2\\mathbf {A}(\\mathbf {x},\\omega ),$ where $k=\\frac{\\omega }{c}.$ The boundary conditions are given by [44]: $\\begin{split}&\\mathbf {E}_{\\parallel }\\Bigr \|_{S_+}-\\mathbf {E}_{\\parallel }\\Bigr \|_{S_-}=\\mathbf {0},\\\\&\\mathbf {B}_{n}\\Bigr \|_{S_+}-\\mathbf {B}_{n}\\Bigr \|_{S_-}=0,\\\\&\\mathbf {E}_{n}\\Bigr \|_{S_+}-\\mathbf {E}_{n}\\Bigr \|_{S_-}=2\\Omega \\frac{c^2}{\\omega ^2}\\nabla _{\\parallel }\\cdot \\mathbf {E}_{\\parallel }\\Bigr \|_{ S},\\\\&\\mathbf {B}_{\\parallel }\\Bigr \|_{S_+}-\\mathbf {B}_{\\parallel }\\Bigr \|_{S_-}=-2i\\Omega \\frac{1}{\\omega }\\mathbf {n}\\times \\mathbf {E}_{\\parallel }\\Bigr \|_{S},\\end{split}$ where $S$ is the boundary, $S_+$ and $S_-$ are respectively the outside and inside of the boundary, $\\mathbf {n}$ is a unit vector normal to the boundary, and $\\Omega $ is a constant characterizing the plasma, having dimension inverse of length.", "The solutions of the equation (REF ) can be divided into transverse electric (TE) waves $\\mathbf {A}^{\\text{TE}}_{\\alpha }$ and transverse magnetic (TM) waves $\\mathbf {A}^{\\text{TM}}_{\\alpha }$ parametrized by some parameter $\\alpha $ and satisfy $\\frac{1}{k}\\nabla \\times \\mathbf {A}^{\\text{TE}}_{\\alpha }=\\mathbf {A}^{\\text{TM}}_{\\alpha },\\hspace{28.45274pt}\\frac{1}{k}\\nabla \\times \\mathbf {A}^{\\text{TM}}_{\\alpha }=\\mathbf {A}^{\\text{TE}}_{\\alpha }.$ Moreover, the waves can be divided into regular waves $\\mathbf {A}^{\\text{TE,reg}}_{\\alpha }$ , $\\mathbf {A}^{\\text{TM,reg}}_{\\alpha }$ that are regular at the origin of the coordinate system and outgoing waves $\\mathbf {A}^{\\text{TE,out}}_{\\alpha }$ , $\\mathbf {A}^{\\text{TM,out}}_{\\alpha }$ that decrease to zero rapidly when $\\mathbf {x}\\rightarrow \\infty $ and $k$ is replaced by $ik$ .", "In rectangular coordinates, the waves are parametrized by $\\alpha =\\mathbf {k}_{\\perp }=(k_x,k_y)\\in \\mathbb {R}^2$ , with $\\begin{split}\\mathbf {A}_{\\mathbf {k}_{\\perp }}^{\\text{TE}, \\begin{array}{c}\\text{reg}\\\\\\text{out}\\end{array}}(\\mathbf {x},\\omega )=& \\frac{1}{k_{\\perp }}e^{ik_xx+ik_yy\\mp i\\sqrt{k^2-k_{\\perp }^2}z}\\left(ik_y\\mathbf {e}_x-ik_x\\mathbf {e}_y\\right),\\\\\\mathbf {A}_{\\mathbf {k}_{\\perp }}^{\\text{TM}, \\begin{array}{c}\\text{reg}\\\\\\text{out}\\end{array}}(\\mathbf {x},\\omega )=& \\frac{1}{kk_{\\perp }}e^{ik_xx+ik_yy\\mp i\\sqrt{k^2-k_{\\perp }^2}z}\\left(\\pm k_x\\sqrt{k^2-k_{\\perp }^2}\\mathbf {e}_x \\pm k_y\\sqrt{k^2-k_{\\perp }^2}\\mathbf {e}_y+k_{\\perp }^2\\mathbf {e}_z\\right).\\end{split}$ Here $k_{\\perp }=\\sqrt{k_x^2+k_y^2}$ .", "In spherical coordinates, the waves are parametrized by $\\alpha =(l,m)$ , where $l=1,2,3,\\ldots $ and $-l\\le m\\le l$ , with $\\begin{split}\\mathbf {A}_{lm}^{\\text{TE},}(\\mathbf {x},\\omega )=&\\frac{\\mathcal {C}_{l}^}{\\sqrt{l(l+1)}}f_l^(kr)\\left(\\frac{im}{\\sin \\theta }Y_{lm}(\\theta ,\\phi )\\mathbf {e}_{\\theta }-\\frac{\\partial Y_{lm}(\\theta ,\\phi )}{\\partial \\theta }\\mathbf {e}_{\\phi }\\right),\\\\\\mathbf {A}_{lm}^{\\text{TM},}(\\mathbf {x},\\omega )=&\\mathcal {C}_{l}^\\left(\\frac{\\sqrt{l(l+1)}}{kr}f_l^(kr)Y_{lm}(\\theta ,\\phi )\\mathbf {e}_r+\\frac{1}{\\sqrt{l(l+1)}}\\frac{1}{kr}\\frac{d}{dr}\\left(rf_l^(kr)\\right)\\left[\\frac{\\partial Y_{lm}(\\theta ,\\phi )}{\\partial \\theta }\\mathbf {e}_{\\theta }+\\frac{im}{\\sin \\theta }Y_{lm}(\\theta ,\\phi )\\mathbf {e}_{\\phi }\\right]\\right).\\end{split}$ Here $=$ reg or out, with $f_l^{\\text{reg}}(z)=j_l(z)$ and $f_l^{\\text{out}}(z)=h_l^{(1)}(z)$ , $Y_{lm}(\\theta ,\\phi )$ are the spherical harmonics.", "The constants $\\mathcal {C}_{l}^{\\text{reg}}$ and $\\mathcal {C}_{l}^{\\text{out}}$ are chosen so that $\\mathcal {C}_{l}^{\\text{reg}} j_l(i\\zeta ) =\\sqrt{\\frac{\\pi }{2\\zeta }}I_{l+\\frac{1}{2}}(\\zeta ),\\quad \\mathcal {C}_{l}^{\\text{out}} h_l^{(1)}(i\\zeta ) =\\sqrt{\\frac{\\pi }{2\\zeta }}K_{l+\\frac{1}{2}}(\\zeta ).$ Now we can derive the dispersion relation for the energy eigenmodes $\\omega $ of the system.", "Inside the sphere ($r<R$ ), express $\\mathbf {A}(\\mathbf {x},t) $ in the spherical coordinate system centered at $O$ : $\\mathbf {A}(\\mathbf {x},t) =\\int _{-\\infty }^{\\infty } d\\omega \\sum _{l=1}^{\\infty }\\sum _{m=-l}^l\\left(A_1^{lm} \\mathbf {A}^{\\text{TE,reg}}_{lm}(\\mathbf {x},\\omega )+C_1^{lm}\\mathbf {A}^{\\text{TM,reg}}_{lm}(\\mathbf {x},\\omega )\\right)e^{-i\\omega t}.$ Outside the plane ($z>L$ ), express $\\mathbf {A} $ in the rectangular coordinate system centered at $O^{\\prime }$ : $\\mathbf {A}(\\mathbf {x}^{\\prime },t) =H^2\\int _{-\\infty }^{\\infty } d\\omega \\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi } \\left(B_2^{\\mathbf {k}_{\\perp }} \\mathbf {A}^{\\text{TE,out}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )+D_2^{\\mathbf {k}_{\\perp }}\\mathbf {A}^{\\text{TM,out}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )\\right)e^{-i\\omega t}.$ Here $\\mathbf {x}^{\\prime }=\\mathbf {x}-\\mathbf {L}$ , $\\mathbf {L}=L\\mathbf {e}_z$ .", "In the region between the sphere and the plane, $\\mathbf {A}$ can be represented in two ways: one is in terms of the spherical coordinate system centered at $O$ : $\\mathbf {A}(\\mathbf {x},t) =\\int _{-\\infty }^{\\infty } d\\omega \\sum _{l=1}^{\\infty }\\sum _{m=-l}^l\\left(a_1^{lm} \\mathbf {A}^{\\text{TE,reg}}_{lm}(\\mathbf {x},\\omega )+b_1^{lm} \\mathbf {A}^{\\text{TE,out}}_{lm}(\\mathbf {x},\\omega )+c_1^{lm} \\mathbf {A}^{\\text{TM,reg}}_{lm}(\\mathbf {x},\\omega )+d_1^{lm}\\mathbf {A}^{\\text{TM,out}}_{lm}(\\mathbf {x},\\omega )\\right)e^{-i\\omega t};$ and one is in terms of the rectangular coordinate system centered at $O^{\\prime }$ : $\\begin{split}\\mathbf {A}(\\mathbf {x}^{\\prime },t) =&H^2\\int _{-\\infty }^{\\infty } d\\omega \\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi }\\\\&\\hspace{56.9055pt}\\times \\left(a_2^{\\mathbf {k}_{\\perp } }\\mathbf {A}^{\\text{TE,reg}}_{\\mathbf {k}_{\\perp } }(\\mathbf {x}^{\\prime },\\omega )+b_2^{\\mathbf {k}_{\\perp } } \\mathbf {A}^{\\text{TE,out}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )+c_2^{\\mathbf {k}_{\\perp } } \\mathbf {A}^{\\text{TM,reg}}_{\\mathbf {k}_{\\perp } }(\\mathbf {x}^{\\prime },\\omega )+d_2^{\\mathbf {k}_{\\perp } }\\mathbf {A}^{\\text{TM,out}}_{\\mathbf {k}_{\\perp } }(\\mathbf {x}^{\\prime },\\omega )\\right)e^{-i\\omega t}.\\end{split}$ These two representations are related by translation matrices $\\mathbb {V}$ and $\\mathbb {W}$ : $\\begin{split}\\begin{pmatrix}\\mathbf {A}^{\\text{TE,reg}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )\\\\\\mathbf {A}^{\\text{TM,reg}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )\\end{pmatrix}=&\\sum _{l=1}^{\\infty }\\sum _{m=-l}^l \\begin{pmatrix}V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TE}} & V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TE}} \\\\V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TM}} & V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TM}} \\end{pmatrix}\\begin{pmatrix}\\mathbf {A}^{\\text{TE,reg}}_{lm}(\\mathbf {x},\\omega )\\\\\\mathbf {A}^{\\text{TM,reg}}_{lm}(\\mathbf {x},\\omega )\\end{pmatrix}\\\\\\begin{pmatrix}\\mathbf {A}^{\\text{TE,out}}_{lm}(\\mathbf {x},\\omega )\\\\\\mathbf {A}^{\\text{TM,out}}_{lm}(\\mathbf {x},\\omega )\\end{pmatrix}=&H^2\\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi } \\begin{pmatrix}W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TE,TE}} & W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TM,TE}} \\\\W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TE,TM}} & W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TM,TM}} \\end{pmatrix}\\begin{pmatrix}\\mathbf {A}^{\\text{TE,out}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )\\\\\\mathbf {A}^{\\text{TM,out}}_{\\mathbf {k}_{\\perp }}(\\mathbf {x}^{\\prime },\\omega )\\end{pmatrix}.\\end{split}$ Hence, $\\begin{split}\\begin{pmatrix}a_1^{lm}\\\\c_1^{lm}\\end{pmatrix}=&H^2\\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi }\\begin{pmatrix}V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TE}} & V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TM}} \\\\V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TE}} & V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TM}} \\end{pmatrix}\\begin{pmatrix}a_2^{\\mathbf {k}_{\\perp }}\\\\c_2^{\\mathbf {k}_{\\perp }}\\end{pmatrix},\\\\\\begin{pmatrix}b_2^{\\mathbf {k}_{\\perp }}\\\\d_2^{\\mathbf {k}_{\\perp }}\\end{pmatrix}=&\\sum _{l=1}^{\\infty }\\sum _{m=-l}^l\\begin{pmatrix}W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TE,TE}} & W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TE,TM}} \\\\W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TM,TE}} & W_{\\mathbf {k}_{\\perp }, lm}^{\\text{TM,TM}} \\end{pmatrix}\\begin{pmatrix}b_1^{lm}\\\\d_1^{lm}\\end{pmatrix}.\\end{split}$ These translation matrices have been derived in [10], [17].", "Their components are given by $\\begin{split}V_{lm,\\mathbf {k}_{\\perp }}^{\\text{TE,TE}} =V_{lm,\\mathbf {k}_{\\perp }}^{\\text{TM,TM}}=&-\\frac{4\\pi i }{\\sqrt{l(l+1)}} \\frac{\\partial Y_{l,-m}(\\theta _k,\\phi _k)}{\\partial \\theta _k}e^{i\\sqrt{k^2-k_{\\perp }^2}L},\\\\V_{lm,\\mathbf {k}_{\\perp }}^{\\text{TE,TM}} =V_{lm,\\mathbf {k}_{\\perp }}^{\\text{TM,TE}} =&\\frac{4\\pi i }{\\sqrt{l(l+1)}} \\frac{m}{\\sin \\theta _k} Y_{l,-m}(\\theta _k,\\phi _k) e^{i\\sqrt{k^2-k_{\\perp }^2}L},\\end{split}$ $\\begin{split}W_{\\mathbf {k}_{\\perp },lm}^{\\text{TE,TE}} =&\\frac{i }{H^2\\sqrt{l(l+1)}} \\frac{\\pi ^2}{k\\sqrt{k^2-k_{\\perp }^2}}\\frac{\\partial Y_{lm}(\\theta _k,\\phi _k)}{\\partial \\theta _k}e^{i\\sqrt{k^2-k_{\\perp }^2}L},\\\\W_{\\mathbf {k}_{\\perp },lm}^{\\text{TM,TE}} =&\\frac{i }{H^2\\sqrt{l(l+1)}} \\frac{\\pi ^2}{k\\sqrt{k^2-k_{\\perp }^2}}\\frac{m}{\\sin \\theta _k} Y_{lm}(\\theta _k,\\phi _k) e^{i\\sqrt{k^2-k_{\\perp }^2}L}.\\end{split}$ Here $\\theta _k$ and $\\phi _k$ are such that $k_{\\perp }=k\\sin \\theta _k$ , $k_x=k_{\\perp }\\cos \\phi _k$ and $k_y=k_{\\perp }\\sin \\phi _k$ .", "Let $\\Omega _s$ be the parameter characterizing the spherical plasma sheet.", "Matching the boundary conditions (REF ) on the sphere gives $\\begin{split}&a_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}j_l(kR)+b_1^{lm}\\mathcal {C}_{l }^{\\text{out}}h_l^{(1)}(kR)=A_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}j_l(kR),\\\\&a_1^{lm}\\mathcal {C}_{l }^{\\text{reg}} \\Bigl (j_l(kR)+kRj_l^{\\prime }(kR)\\Bigr )+b_1^{lm}\\mathcal {C}_{l }^{\\text{out}}\\left(h_l^{(1)}(kR)+kRh_l^{(1)\\prime }(kR)\\right)- A_1^{lm} \\mathcal {C}_{l }^{\\text{reg}}\\Bigl (j_l(kR)+kRj_l^{\\prime }(kR)\\Bigr )\\\\&\\hspace{113.81102pt}=2\\Omega _s RA_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}j_l(kR),\\\\&c_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}\\Bigl (j_l(kR)+kRj_l^{\\prime }(kR)\\Bigr )+d_1^{lm}\\mathcal {C}_{l }^{\\text{out}}\\left(h_l^{(1)}(kR)+kRh_l^{(1)\\prime }(kR)\\right)=C_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}\\Bigl (j_l(kR)+kRj_l^{\\prime }(kR)\\Bigr ),\\\\&c_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}j_l(kR)+d_1^{lm}\\mathcal {C}_{l }^{\\text{out}}h_l^{(1)}(kR)-C_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}j_l(kR)=-\\frac{2\\Omega _s c^2}{\\omega ^2R}C_1^{lm}\\mathcal {C}_{l }^{\\text{reg}}\\Bigl (j_l(kR)+kRj_l^{\\prime }(kR)\\Bigr ).\\end{split}$ Eliminating $A^{lm}$ and $C^{lm}$ , we obtain a relation of the form $\\begin{pmatrix} b_1^{lm}\\\\d_1^{lm}\\end{pmatrix}=-\\mathbb {T}_{lm}\\begin{pmatrix} a_1^{lm}\\\\c_1^{lm}\\end{pmatrix},$ where $\\mathbb {T}_{lm}$ is a diagonal matrix: $\\mathbb {T}_{lm}=\\begin{pmatrix} T_{lm}^{\\text{TE}}&0\\\\0& T_{lm}^{\\text{TM}}\\end{pmatrix}$ with $\\begin{split}T_{lm}^{\\text{TE}}(i\\xi )=& \\frac{2\\Omega _s R I_{l+\\frac{1}{2}}(\\kappa R)^2}{1+2\\Omega _s RI_{l+\\frac{1}{2}}(\\kappa R)K_{l+\\frac{1}{2}}(\\kappa R)},\\\\T_{lm}^{\\text{TM}}(i\\xi )=& -\\frac{ 2\\Omega _s \\left(\\frac{1}{2}I_{l+\\frac{1}{2}}(\\kappa R)+\\kappa RI_{l+\\frac{1}{2}}^{\\prime }(\\kappa R)\\right)^2}{\\kappa ^2R- 2\\Omega _s \\left(\\frac{1}{2}I_{l+\\frac{1}{2}}(\\kappa R)+\\kappa RI_{l+\\frac{1}{2}}^{\\prime }(\\kappa R)\\right)\\left(\\frac{1}{2}K_{l+\\frac{1}{2}}(\\kappa R)+\\kappa RK_{l+\\frac{1}{2}}^{\\prime }(\\kappa R)\\right)}.\\end{split}$ Here we have replaced $k$ by $i\\kappa $ and $\\omega $ by $i\\xi $ .", "Denote by $\\Omega _p$ be the parameter characterizing the planar plasma sheet.", "Matching the boundary conditions (REF ) on the plane gives $\\begin{split}&a_2^{\\mathbf {k}_{\\perp }}+ b_2^{\\mathbf {k}_{\\perp }}=B_2^{\\mathbf {k}_{\\perp }},\\\\&\\sqrt{k^2-k_{\\perp }^2}\\left(a_2^{\\mathbf {k}_{\\perp }}- b_2^{\\mathbf {k}_{\\perp }}+B_2^{\\mathbf {k}_{\\perp }}\\right)=-2i\\Omega _p B_2^{\\mathbf {k}_{\\perp }},\\\\&c_2^{\\mathbf {k}_{\\perp }}-d_2^{\\mathbf {k}_{\\perp }}=-D_2^{\\mathbf {k}_{\\perp }},\\\\&c_2^{\\mathbf {k}_{\\perp }}+d_2^{\\mathbf {k}_{\\perp }}-D_2^{\\mathbf {k}_{\\perp }}=\\frac{2i\\Omega _p c^2}{\\omega ^2}\\sqrt{k^2-k_{\\perp }^2}D_2^{\\mathbf {k}_{\\perp }}.\\end{split}$ From here, we find that $\\begin{pmatrix} a_2^{\\mathbf {k}_{\\perp }}\\\\c_2^{\\mathbf {k}_{\\perp }}\\end{pmatrix}=-\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }}\\begin{pmatrix} b_2^{\\mathbf {k}_{\\perp }}\\\\d_2^{\\mathbf {k}_{\\perp }}\\end{pmatrix},$ where $\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }}$ is a diagonal matrix with elements $\\begin{split}\\widetilde{T}_{\\mathbf {k}_{\\perp }}^{\\text{TE}}(i\\xi )=&\\frac{\\Omega _p}{\\Omega _p+\\sqrt{\\kappa ^2+k_{\\perp }^2}},\\\\\\widetilde{T}_{\\mathbf {k}_{\\perp }}^{\\text{TM}}(i\\xi )=&-\\frac{\\Omega _p \\sqrt{\\kappa ^2+k_{\\perp }^2}}{\\Omega _p \\sqrt{\\kappa ^2+k_{\\perp }^2}+\\kappa ^2}.\\end{split}$ The eigenmodes $\\omega $ are those modes where the boundary conditions give rise to nontrivial solutions of $(A_1^{lm}, C_1^{lm}, B_2^{\\mathbf {k}_{\\perp }}, D_2^{\\mathbf {k}_{\\perp }}, a_1^{lm}, b_1^{lm}, c_1^{lm}, d_1^{lm}, a_2^{\\mathbf {k}_{\\perp }}, b_2^{\\mathbf {k}_{\\perp }}, c_2^{\\mathbf {k}_{\\perp }}, d_2^{k_{\\perp }})$ .", "Now $\\begin{pmatrix} b_1^{lm}\\\\d_1^{lm}\\end{pmatrix}=&-\\mathbb {T}_{lm}\\begin{pmatrix} a_1^{lm}\\\\c_1^{lm}\\end{pmatrix}\\\\=&-\\mathbb {T}_{lm} H^2\\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi }\\begin{pmatrix}V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TE}} & V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TM}} \\\\V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TE}} & V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TM}} \\end{pmatrix}\\begin{pmatrix}a_2^{\\mathbf {k}_{\\perp }}\\\\c_2^{\\mathbf {k}_{\\perp }}\\end{pmatrix}\\\\=&\\mathbb {T}_{lm} H^2\\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi }\\begin{pmatrix}V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TE}} & V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TM}} \\\\V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TE}} & V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TM}} \\end{pmatrix}\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }}\\begin{pmatrix} b_2^{\\mathbf {k}_{\\perp }}\\\\d_2^{\\mathbf {k}_{\\perp }}\\end{pmatrix}\\\\=&\\mathbb {T}_{lm} H^2\\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi }\\begin{pmatrix}V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TE}} & V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TM}} \\\\V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TE}} & V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TM}} \\end{pmatrix}\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }}\\sum _{l^{\\prime }=1}^{\\infty }\\sum _{m^{\\prime }=-l^{\\prime }}^{l^{\\prime }}\\begin{pmatrix}W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TE,TE}} & W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TE,TM}} \\\\W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TM,TE}} & W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TM,TM}} \\end{pmatrix}\\begin{pmatrix}b_1^{l^{\\prime }m^{\\prime }}\\\\d_1^{l^{\\prime }m^{\\prime }}\\end{pmatrix}.$ This shows that the matrix $\\mathbb {B}$ with $(lm)$ component given by $\\begin{pmatrix} b_1^{lm}\\\\d_1^{lm}\\end{pmatrix}$ satisfies the relation $\\left(\\mathbb {I}-\\mathbb {M}\\right)\\mathbb {B}=\\mathbb {O},$ where the $(lm, l^{\\prime }m^{\\prime })$ -element of $\\mathbb {M}$ is given by $M_{lm, l^{\\prime }m^{\\prime }}=\\mathbb {T}_{lm} H^2\\int _{-\\infty }^{\\infty }\\frac{dk_x}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{dk_y}{2\\pi }\\begin{pmatrix}V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TE}} & V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TE,TM}} \\\\V_{ lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TE}} & V_{lm, \\mathbf {k}_{\\perp }}^{\\text{TM,TM}} \\end{pmatrix}\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }} \\begin{pmatrix}W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TE,TE}} & W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TE,TM}} \\\\W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TM,TE}} & W_{\\mathbf {k}_{\\perp }, l^{\\prime }m^{\\prime }}^{\\text{TM,TM}} \\end{pmatrix}.$ The condition for nontrivial solution of $\\mathbb {B}$ is thus given by $\\det \\left(\\mathbb {I}-\\mathbb {M}\\right)=0.$ Hence, the Casimir interaction energy between the spherical plasma sheet and the planar plasma sheet is $E_{\\text{Cas}}=\\frac{\\hbar }{2\\pi }\\int _0^{\\infty } d\\xi \\text{Tr}\\,\\ln \\left(\\mathbb {I}-\\mathbb {M}(i\\xi )\\right)=\\frac{\\hbar c}{2\\pi }\\int _0^{\\infty } d\\kappa \\text{Tr}\\,\\ln \\left(\\mathbb {I}-\\mathbb {M} \\right).$ Set $k_x=k_{\\perp }\\cos \\theta _k$ , $k_y=k_{\\perp }\\sin \\theta _k$ , integrate over $\\theta _k$ and make a change of variables $k_{\\perp }=\\kappa \\sinh \\theta $ , we find that $\\begin{split}\\mathbb {M}_{lm,l^{\\prime }m^{\\prime }}(i\\xi )=&\\delta _{m,m^{\\prime }} \\frac{(-1)^{m}\\pi }{2}\\sqrt{\\frac{(2l+1)(2l^{\\prime }+1)}{l(l+1)l^{\\prime }(l^{\\prime }+1)}\\frac{(l-m)!", "(l^{\\prime }-m)!}{(l+m)!", "(l^{\\prime }+m)!}}", "\\mathbb {T}_{lm}\\int _{0}^{\\infty }d\\theta \\sinh \\theta e^{-2\\kappa L\\cosh \\theta }\\\\&\\times \\left(\\begin{aligned} \\sinh \\theta P_l^{m\\prime }(\\cosh \\theta )\\hspace{14.22636pt} &-\\frac{m}{\\sinh \\theta }P_l^m(\\cosh \\theta )\\\\ -\\frac{m}{\\sinh \\theta }P_l^m(\\cosh \\theta )\\hspace{11.38092pt} & \\quad \\sinh \\theta P_l^{m\\prime }(\\cosh \\theta ) \\end{aligned}\\right)\\left(\\begin{aligned} \\frac{\\Omega _p}{\\Omega _p+\\kappa \\cosh \\theta } & ~\\hspace{28.45274pt} 0\\hspace{14.22636pt}\\\\\\hspace{14.22636pt} 0\\hspace{28.45274pt} & -\\frac{\\Omega _p \\cosh \\theta }{\\Omega _p \\cosh \\theta +\\kappa }\\end{aligned}\\right)\\\\&\\times \\left(\\begin{aligned} \\sinh \\theta P_{l^{\\prime }}^{m^{\\prime }\\prime }(\\cosh \\theta )\\hspace{14.22636pt} & \\frac{m^{\\prime }}{\\sinh \\theta }P_{l^{\\prime }}^{m^{\\prime }}(\\cosh \\theta )\\\\ \\frac{m^{\\prime }}{\\sinh \\theta }P_{l^{\\prime }}^{m^{\\prime }}(\\cosh \\theta )\\hspace{11.38092pt} & \\quad \\sinh \\theta P_{l^{\\prime }}^{m^{\\prime }\\prime }(\\cosh \\theta ) \\end{aligned}\\right).\\end{split}$ Here $P_l^m(z)$ is an associated Legendre function and $P_l^{m\\prime }(z)$ is its derivative, whereas $\\mathbb {T}_{lm}$ is given by (REF ).", "Notice that this approach has been formalized mathematically in [17].", "The self energy contributions from the sphere and the plane have automatically dropped out and (REF ) is the interaction energy between the sphere and the plane.", "In the limit $\\Omega _s\\rightarrow \\infty $ and $\\Omega _p\\rightarrow \\infty $ , we find from (REF ) and (REF ) that $\\begin{split}T_{lm}^{\\text{TE}}(i\\xi )=& \\frac{ I_{l+\\frac{1}{2}}(\\kappa R)}{K_{l+\\frac{1}{2}}(\\kappa R)},\\\\T_{lm}^{\\text{TM}}(i\\xi )=& \\frac{\\frac{1}{2}I_{l+\\frac{1}{2}}(\\kappa R)+\\kappa RI_{l+\\frac{1}{2}}^{\\prime }(\\kappa R)}{\\frac{1}{2}K_{l+\\frac{1}{2}}(\\kappa R)+\\kappa RK_{l+\\frac{1}{2}}^{\\prime }(\\kappa R)},\\end{split}$ $\\begin{split}\\mathbb {M}_{lm,l^{\\prime }m^{\\prime }}(i\\xi )=&\\delta _{m,m^{\\prime }} \\frac{(-1)^{m}\\pi }{2}\\sqrt{\\frac{(2l+1)(2l^{\\prime }+1)}{l(l+1)l^{\\prime }(l^{\\prime }+1)}\\frac{(l-m)!", "(l^{\\prime }-m)!}{(l+m)!", "(l^{\\prime }+m)!}}", "\\mathbb {T}_{lm}\\int _{0}^{\\infty }d\\theta \\sinh \\theta e^{-2\\kappa L\\cosh \\theta }\\\\&\\times \\begin{pmatrix} 1 & 0\\\\0& -1\\end{pmatrix}\\left(\\begin{aligned} \\sinh \\theta P_l^{m\\prime }(\\cosh \\theta )\\hspace{14.22636pt} &\\frac{m}{\\sinh \\theta }P_l^m(\\cosh \\theta )\\\\ \\frac{m}{\\sinh \\theta }P_l^m(\\cosh \\theta )\\hspace{11.38092pt} & \\quad \\sinh \\theta P_l^{m\\prime }(\\cosh \\theta ) \\end{aligned}\\right)\\left(\\begin{aligned} \\sinh \\theta P_{l^{\\prime }}^{m^{\\prime }\\prime }(\\cosh \\theta )\\hspace{14.22636pt} & \\frac{m^{\\prime }}{\\sinh \\theta }P_{l^{\\prime }}^{m^{\\prime }}(\\cosh \\theta )\\\\ \\frac{m^{\\prime }}{\\sinh \\theta }P_{l^{\\prime }}^{m^{\\prime }}(\\cosh \\theta )\\hspace{11.38092pt} & \\quad \\sinh \\theta P_{l^{\\prime }}^{m^{\\prime }\\prime }(\\cosh \\theta ) \\end{aligned}\\right),\\end{split}$ which recovers the Casimir interaction energy between a perfectly conducting spherical shell and a perfectly conducting plane [10], [17]." ], [ "Small separation asymptotic behavior", "In this section, we consider the asymptotic behavior of the Casimir interaction energy when $d\\ll R$ , where $d=L-R$ is the distance between the spherical plasma sheet and the planar plasma sheet.", "Let $\\varepsilon =\\frac{d}{R}$ be the dimensionless parameter, and we consider $\\varepsilon \\ll 1$ .", "There are also another two length parameters in the problem: $1/\\Omega _s$ and $1/\\Omega _p$ .", "Let $\\varpi _s=\\Omega _s d,\\hspace{28.45274pt}\\varpi _p=\\Omega _pd.$ They are dimensionless and we assume that they have order 1, i.e., $\\varpi _s\\sim 1,\\hspace{28.45274pt}\\varpi _p\\sim 1.$ First we consider the proximity force approximation to the Casimir interaction energy, which approximates the Casimir interaction energy by summing the local Casimir energy density between two planes over the surfaces.", "The Casimir interaction energy density between two planar plasma sheets with respective parameters $\\Omega _1$ and $\\Omega _2$ is given by the Lifshitz formula [50]: $\\mathcal {E}_{\\text{Cas}}^{\\parallel }(d)=\\frac{\\hbar c}{4\\pi ^2}\\int _0^{\\infty } d\\kappa \\int _{0}^{\\infty } dk_{\\perp } k_{\\perp } \\left[\\ln \\left(1-r_{\\text{TE}}^{(1)}r_{\\text{TE}}^{(2)}e^{-2d\\sqrt{k_{\\perp }^2+\\kappa ^2}}\\right)+\\ln \\left(1-r_{\\text{TM}}^{(1)}r_{\\text{TM}}^{(2)}e^{-2d\\sqrt{\\kappa ^2+k_{\\perp }^2}}\\right)\\right].$ Here $d$ is the distance between the two planar sheets, $r_{\\text{TE}}^{(i)}=&\\frac{\\Omega _i}{\\Omega _i+\\sqrt{\\kappa ^2+k_{\\perp }^2}},\\\\r_{\\text{TM}}^{(i)}=&-\\frac{\\Omega _i\\sqrt{\\kappa ^2+k_{\\perp }^2}}{\\Omega _i\\sqrt{\\kappa ^2+k_{\\perp }^2}+\\kappa ^2}$ are nothing but the components of the $\\mathbb {T}_2^{\\mathbf {k}_{\\perp }}$ given in (REF ).", "The proximity force approximation for the Casimir interaction energy between a sphere and a plate is then given by $E_{\\text{Cas}}^{\\text{PFA}}=&R^2\\int _0^{2\\pi } d\\phi \\int _0^{\\pi }d\\theta \\sin \\theta \\mathcal {E}^{\\parallel }_{\\text{Cas}}\\left(L+R\\cos \\theta \\right)\\\\\\sim &2\\pi R\\int _d^{\\infty } du \\mathcal {E}^{\\parallel }_{\\text{Cas}}(u)\\\\=&-\\frac{\\hbar c R}{2\\pi } \\int _0^{\\infty } d\\kappa \\int _0^{\\infty } dk_{\\perp } k_{\\perp }\\int _d^{\\infty }du\\sum _{n=1}^{\\infty }\\frac{1}{n}\\left(\\left[r_{\\text{TE}}^{(1)}r_{\\text{TE}}^{(2)}\\right]^n+\\left[r_{\\text{TM}}^{(1)}r_{\\text{TM}}^{(2)}\\right]^n\\right)e^{-2un\\sqrt{\\kappa ^2+k_{\\perp }^2}}\\\\=&-\\frac{\\hbar c R}{4\\pi } \\int _0^{\\infty } d\\kappa \\int _0^{\\infty } dk_{\\perp } \\frac{k_{\\perp }}{\\sqrt{\\kappa ^2+k_{\\perp }^2}} \\sum _{n=1}^{\\infty }\\frac{1}{n^2}\\left(\\left[r_{\\text{TE}}^{(1)}r_{\\text{TE}}^{(2)}\\right]^n+\\left[r_{\\text{TM}}^{(1)}r_{\\text{TM}}^{(2)}\\right]^n\\right)e^{-2dn\\sqrt{\\kappa ^2+k_{\\perp }^2}}\\\\=&-\\frac{\\hbar c R}{4\\pi } \\int _0^{\\infty } d\\kappa \\int _0^{\\infty } dk_{\\perp } \\frac{k_{\\perp }}{\\sqrt{\\kappa ^2+k_{\\perp }^2}} \\left(\\text{Li}_2\\left(r_{\\text{TE}}^{(1)}r_{\\text{TE}}^{(2)}e^{-2d\\sqrt{\\kappa ^2+k_{\\perp }^2}}\\right)+\\text{Li}_2\\left(r_{\\text{TM}}^{(1)}r_{\\text{TM}}^{(2)}e^{-2d\\sqrt{\\kappa ^2+k_{\\perp }^2}}\\right)\\right)\\\\=&-\\frac{\\hbar c R}{4\\pi }\\int _{0}^{\\infty } dq \\int _0^{q} d\\kappa \\left(\\text{Li}_2\\left(r_{\\text{TE}}^{(1)}r_{\\text{TE}}^{(2)}e^{-2dq}\\right)+\\text{Li}_2\\left(r_{\\text{TM}}^{(1)}r_{\\text{TM}}^{(2)}e^{-2dq}\\right)\\right).$ Here $\\displaystyle \\text{Li}_2(z)=\\sum _{n=1}^{\\infty } \\frac{z^n}{n^2}$ is a polylogarithm function of order 2.", "Making a change of variables $dq=t$ and $\\kappa =q\\sqrt{1-\\tau ^2}= t\\sqrt{1-\\tau ^2}/d$ , we finally obtain $E_{\\text{Cas}}^{\\text{PFA}}=-\\frac{\\hbar c R}{4\\pi d^2 }\\int _{0}^{\\infty } dt \\,t \\int _0^{1} \\frac{d\\tau \\,\\tau }{\\sqrt{1-\\tau ^2}} \\left(\\text{Li}_2\\left(r_{\\text{TE}}^{(1)}r_{\\text{TE}}^{(2)}e^{-2t}\\right)+\\text{Li}_2\\left(r_{\\text{TM}}^{(1)}r_{\\text{TM}}^{(2)}e^{-2t}\\right)\\right),$ where $r_{\\text{TE}}^{(i)}=&\\frac{\\Omega _i}{\\Omega _i+q}=\\frac{\\varpi _i}{\\varpi _i+t},\\\\r_{\\text{TM}}^{(i)}=&-\\frac{\\Omega _iq}{\\Omega _iq+ \\kappa ^2}=-\\frac{\\varpi _i}{\\varpi _i+t(1-\\tau ^2)}.$ Next, we consider the small separation asymptotic behavior of the Casimir interaction energy up to the next-to-leading order term in $\\varepsilon $ from the functional representation (REF ).", "In [29], we have considered the small separation asymptotic expansion of the Casimir interaction between a magnetodielectric sphere and a magnetodielectric plane.", "Our present scenario is similar to the one considered in [29].", "The major differences are the boundary conditions on the sphere and the plate that are encoded in the two matrices $\\mathbb {T}_{lm}$ and $\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }}$ .", "Hence, we do not repeat the calculations that have been presented in [29], but only present the final result and point out the differences.", "The leading term and next-to-leading term of the Casimir interaction energy $E_{\\text{Cas}}^0$ and $E_{\\text{Cas}}^1$ are given respectively by $\\begin{split}E_{\\text{Cas}}^0= &-\\frac{\\hbar c R}{4\\pi d^2}\\sum _{s=0}^{\\infty }\\frac{1}{(s+1)^2} \\int _0^{\\infty }dt\\,t \\int _0^1 \\frac{d\\tau \\,\\tau }{\\sqrt{1-\\tau ^2}} e^{-2t(s+1)}\\sum _{* =\\text{TE}, \\text{TM}}\\left[T^{* }_{0}\\widetilde{T}^{}_{0}\\right]^{s+1},\\end{split}$ $\\begin{split}E_{\\text{Cas}}^1 =&-\\frac{\\hbar c }{4\\pi d}\\sum _{s=0}^{\\infty }\\frac{1}{(s+1)^2} \\int _0^{\\infty }dt\\,t \\int _0^1 \\frac{d\\tau \\,\\tau }{ \\sqrt{1-\\tau ^2}}e^{-2t(s+1)} \\Biggl \\lbrace \\sum _{ =\\text{TE}, \\text{TM}}\\left[T^{* }_{0}\\widetilde{T}^{}_{0}\\right]^{s+1} \\left({A}+{C}^{}+{D}^{}\\right)+ {B}\\Biggr \\rbrace .\\end{split}$ Here $T^{\\text{TE} }_{0}=&\\frac{ \\varpi _s}{ \\varpi _s+ t },\\\\T^{\\text{TM} }_{0}=& \\frac{ \\varpi _s}{ \\varpi _s+ t\\left(1-\\tau ^2\\right)},\\\\\\widetilde{T}^{\\text{TE} }_{0}=&\\frac{ \\varpi _p}{ \\varpi _p+ t },\\\\\\widetilde{T}^{\\text{TM} }_{0}=& \\frac{ \\varpi _p}{ \\varpi _p+ t\\left(1-\\tau ^2\\right)},\\\\{A}=&\\frac{ t\\tau ^2}{3}\\left((s+1)^3 +2(s+1)\\right)+ \\frac{1}{3}\\left((\\tau ^2-2)(s+1)^2-3\\tau (s+1)+2\\tau ^2-1\\right)\\\\&+\\frac{\\tau ^4+\\tau ^2-12}{12t\\tau ^2}(s+1)+\\frac{(1+\\tau )(1-\\tau ^2)}{2t\\tau ^2}-\\frac{ (1-\\tau ^2)}{3t }\\frac{1}{s+1},\\\\{B}=&\\frac{ 1-\\tau ^2}{2t\\tau ^2 }\\left\\lbrace \\left(T^{\\text{TE} }_{0}\\widetilde{T}_0^{\\text{TM}}+T^{\\text{TM} }_{0}\\widetilde{T}_0^{\\text{TE}}\\right)\\frac{\\left[T^{\\text{TE} }_{0}\\widetilde{T}^{\\text{TE} }_{0}\\right]^{s+1 }-\\left[T^{\\text{TM} }_{0}\\widetilde{T}^{\\text{TM} }_{0}\\right]^{s+1} }{T_0^{\\text{TE}}\\widetilde{T}_0^{\\text{TE}}-T_0^{\\text{TM}}\\widetilde{T}_0^{\\text{TM}}}\\right.\\\\&\\left.\\hspace{56.9055pt}+2T^{\\text{TE} }_{0}\\widetilde{T}_0^{\\text{TE}}T^{\\text{TM} }_{0}\\widetilde{T}_0^{\\text{TM}} \\frac{\\left[T^{\\text{TE} }_{0}\\widetilde{T}^{\\text{TE} }_{0}\\right]^{s }-\\left[T^{\\text{TM} }_{0}\\widetilde{T}^{\\text{TM} }_{0}\\right]^{s} }{T_0^{\\text{TE}}\\widetilde{T}_0^{\\text{TE}}-T_0^{\\text{TM}}\\widetilde{T}_0^{\\text{TM}}}\\right\\rbrace ,\\\\{C}^{}=& C_{V } \\mathcal {K}^{}_1+C_J \\mathcal {W}^{}_1,\\\\{D}^{}=&D_{VV} \\mathcal {K}^{2}_1+D_{VJ} \\mathcal {K}^{* }_1\\mathcal {W}^{}_1+D_{JJ} \\mathcal {W}^{2}_1+ D_V \\mathcal {K}^{}_2 + D_J \\mathcal {W}^{}_2+ (s+1) \\mathcal {Y}^{}_2,$ with $C_{V}=&-\\frac{ \\tau }{3}\\left((s+1)^3+2(s+1)\\right)+\\frac{1-\\tau ^2}{6t\\tau }(s+1)^2+\\frac{1}{2t}(s+1)+\\frac{1-4\\tau ^2}{12t\\tau },\\\\C_J=&-\\frac{t\\tau }{3}\\left((s+1)^3-(s+1)\\right)+\\frac{1}{6\\tau }\\left((s+1)^2-1\\right),\\\\D_{VV}=&\\frac{1}{12t}\\left((s+1)^3-2(s+1)^2+2(s+1)-1\\right),\\\\D_{JJ}=&\\frac{t}{12}\\left((s+1)^3-2(s+1)^2-(s+1)+2\\right),\\\\D_{VJ}=&\\frac{1}{6}\\left((s+1)^3- (s+1) \\right),\\\\D_{V}=&\\frac{1}{6t}\\left(2(s+1)^2 +1\\right),\\\\D_{J}=&\\frac{t}{3}\\left( (s+1)^2- 1\\right),\\\\\\mathcal {K}^{\\text{TE}}_{1}=& - \\frac{t\\tau }{ \\varpi _p+ t },\\\\\\mathcal {K}^{\\text{TE}}_{2}=&-\\frac{t\\left( \\varpi _p+t\\left(1-2\\tau ^2\\right)\\right)}{2\\left( \\varpi _p+t \\right)^2},\\\\\\mathcal {K}^{\\text{TM}}_{1}=& \\frac{t\\left(1-\\tau ^2\\right) }{ \\varpi _p+ t\\left(1-\\tau ^2\\right)},\\\\\\mathcal {K}^{\\text{TM}}_{2}=& \\frac{t\\left(1-\\tau ^2\\right)\\left( \\varpi _p \\left(1-2\\tau ^2\\right)+ t\\left(1-\\tau ^2\\right) \\right)}{2\\left( \\varpi _p+ t\\left(1-\\tau ^2\\right)\\right)^2},\\\\\\mathcal {W}_{ 1}^{\\text{TE}}=&-\\frac{\\tau }{\\varpi _s+ t },\\\\\\mathcal {W}_{ 2}^{\\text{TE}}=&-\\frac{\\left(t(1-3\\tau ^2)+\\varpi _s\\left(1-\\tau ^2\\right)\\right)}{2t\\left(\\varpi _s+t\\right)^2},\\\\\\mathcal {Y}_{ 2}^{\\text{TE}}=&-\\frac{\\tau }{2\\left(\\varpi _s+t\\right)}+\\frac{1}{t}\\left(\\frac{1}{4}-\\frac{5\\tau ^2}{12}\\right),\\\\\\mathcal {W}_{ 1}^{\\text{TM}}=&\\frac{\\tau (1-\\tau ^2)}{ \\varpi _s+t\\left(1-\\tau ^2\\right)},\\\\\\mathcal {W}_{ 2}^{\\text{TM}}=&\\frac{\\left(1-\\tau ^2\\right)\\left(t(1-\\tau ^2)^2+\\varpi _s(1-3\\tau ^2)\\right)}{2t\\left(\\varpi _s+ t\\left(1-\\tau ^2\\right)\\right)^2},\\\\\\mathcal {Y}_{ 2}^{\\text{TM}}=&\\frac{\\tau \\left(1-\\tau ^2\\right)}{2\\left( \\varpi _s+t\\left(1-\\tau ^2\\right)\\right)}+\\frac{1}{t}\\left(\\frac{1}{4}+\\frac{7\\tau ^2}{12}\\right).$ We have replaced the $l$ in [29] with $t\\tau /\\varepsilon $ .", "The definition of ${B}$ , ${D}$ , $C_V, C_J, D_{VV}, D_{VJ}, D_{JJ}$ are slightly different than those in [29].", "For $$ =TE or TM, $\\mathcal {K}_1^, \\mathcal {K}_2^, \\mathcal {W}_1^, \\mathcal {W}_2^$ and $\\mathcal {Y}_2^$ are obtained from the asymptotic expansions of $\\mathbb {T}_{lm}$ and $\\widetilde{\\mathbb {T}}_{\\mathbf {k}_{\\perp }}$ .", "Hence, there are different than those obtained in [29].", "Using polylogarithm function, we can rewrite the leading term $E_{\\text{Cas}}^0$ (REF ) as $\\begin{split}E_{\\text{Cas}}^0= &-\\frac{\\hbar c R}{4\\pi d^2} \\int _0^{\\infty }dt\\,t \\int _0^1 \\frac{d\\tau \\,\\tau }{\\sqrt{1-\\tau ^2}}\\left(\\text{Li}_2\\left( T^{\\text{TE} }_{0}\\widetilde{T}^{\\text{TE}}_{0} e^{-2t}\\right)+\\text{Li}_2\\left( T^{\\text{TM} }_{0}\\widetilde{T}^{\\text{TM}}_{0} e^{-2t}\\right)\\right).\\end{split}$ It is easy to see that this coincides with the proximity force approximation (REF ) when $\\varpi _s=\\varpi _1$ and $\\varpi _p=\\varpi _2$ .", "Notice that the leading term $E_{\\text{Cas}}^0$ can be split into a sum of TE and TM contributions.", "However, because of the ${B}$ -term, the next-to-leading order term $E_{\\text{Cas}}^1$ (REF ) cannot be split into TE and TM contributions.", "In the limit $\\varpi _p, \\varpi _s\\rightarrow \\infty $ which corresponds to perfectly conducting boundary conditions on the sphere and the plate, we find that for $$ =TE or TM, $\\mathcal {K}^{}_{1}, \\mathcal {K}^{}_{2}, \\mathcal {W}^{}_{1}, \\mathcal {W}^{}_{2}$ vanishes, $T^{}_{0}=\\widetilde{T}^{}_0=1$ , $\\mathcal {B}=&\\frac{ \\left(1-\\tau ^2\\right)}{2t\\tau ^2 }(4s+2),\\\\\\mathcal {Y}_{ 2}^{\\text{TE}}=&\\frac{1}{t}\\left(\\frac{1}{4}-\\frac{5\\tau ^2}{12}\\right),\\\\\\mathcal {Y}_{ 2}^{\\text{TM}}=& \\frac{1}{t}\\left(\\frac{1}{4}+\\frac{7\\tau ^2}{12}\\right).$ Hence, $E_{\\text{Cas}}^0= &-\\frac{\\hbar c R}{2\\pi d^2}\\sum _{s=0}^{\\infty }\\frac{1}{(s+1)^2} \\int _0^{\\infty }dt\\,t \\int _0^1 \\frac{d\\tau \\,\\tau }{\\sqrt{1-\\tau ^2}} e^{-2t(s+1)}\\\\=&-\\frac{\\hbar c R}{8\\pi d^2}\\sum _{s=0}^{\\infty }\\frac{1}{(s+1)^4} \\\\=&-\\frac{\\hbar c\\pi ^3 R}{720 d^2},\\\\E_{\\text{Cas}}^1 =&-\\frac{\\hbar c }{4\\pi d}\\sum _{s=0}^{\\infty }\\frac{1}{(s+1)^2} \\int _0^{\\infty }dt\\,t \\int _0^1 \\frac{d\\tau \\,\\tau }{ \\sqrt{1-\\tau ^2}}e^{-2t(s+1)} \\left(2{A}+ {B}+(s+1)\\mathcal {Y}_2^{\\text{TE}}+(s+1)\\mathcal {Y}_2^{\\text{TM}}\\right)\\\\=&-\\frac{\\hbar c }{4\\pi d}\\sum _{s=0}^{\\infty }\\frac{1}{(s+1)^2} \\left(\\frac{1}{6(s+1)^2}-\\frac{2}{3}\\right)\\\\=&E_{\\text{Cas}}^0\\left(\\frac{1}{3}-\\frac{20}{\\pi ^2}\\right)\\frac{d}{R}.$ These recover the results for the case where both the sphere and the plane are perfectly conducting [26].", "Figure: The leading order term of the Casimir interaction energy normalized by E Cas PFA,PC E_{\\text{Cas}}^{\\text{PFA,PC}} (dashed line) and the sum of the leading and next-to-leading order terms normalized by E Cas PFA,PC E_{\\text{Cas}}^{\\text{PFA,PC}} (solid line) in the case both sphere and plane are graphene sheets.Figure: The ratio of the sum of the leading and next-to-leading order terms to the leading order term in the case both sphere and plane are graphene sheets.Figure: θ\\theta as a function of dd in the case both sphere and plane are graphene sheets.Next, we consider the special case where we have a spherical graphene sheet in front of a planar graphene sheet.", "The parameters $\\Omega _s$ and $\\Omega _p$ are both equal to $6.75\\times 10^5 \\text{m}^{-1}$ (see Ref.", "[50]).", "Assume that the radius of the spherical graphene sheet is $R=1$ mm.", "Let $E_{\\text{Cas}}^{\\text{PFA,PC}}=-\\frac{\\hbar c\\pi ^3 R}{720 d^2}$ be the leading term of the Casimir interaction between a perfectly conducting sphere and a perfectly conducting plane.", "In Fig.", "REF , we plot the ratio of the leading term of the Casimir interaction energy $E_{\\text{Cas}}^0$ to $E_{\\text{Cas}}^{\\text{PFA,PC}}$ , and the ratio of the sum of the leading term and next-to-leading order term $\\left(E_{\\text{Cas}}^0+E_{\\text{Cas}}^1\\right)$ to $E_{\\text{Cas}}^{\\text{PFA,PC}}$ .", "The ratio of $\\left(E_{\\text{Cas}}^0+E_{\\text{Cas}}^1\\right)$ to $E_{\\text{Cas}}^0$ is plotted in Fig.", "REF .", "From these graphs, we can see that the next-to-leading order term plays a significant correction when $d/R\\sim 0.1$ .", "Another important quantity that characterize the correction to proximity force approximation is $\\theta =\\frac{E^{1}_{\\text{Cas}}}{E_{\\text{Cas}}^0}\\frac{R}{d},$ so that $E_{\\text{Cas}}=E_{\\text{Cas}}^0\\left(1+\\frac{d}{R}\\theta +\\ldots \\right).$ In case of perfectly conducting sphere and plane, $\\theta $ is a pure number given by [26]: $\\theta =\\frac{1}{3}-\\frac{20}{\\pi ^2}=-1.69.$ In Fig.", "REF , we plot $\\theta $ as a function of $d$ for a spherical graphene sheet in front of a planar graphene sheet.", "We observe that its variation pattern is significantly different from the case of gold sphere and gold plane modeled by plasma model and Drude model which we studied in [29].", "Nevertheless, as $d$ is large enough, $\\theta $ approaches the limiting value (REF ).", "Figure: E Cas 0 /E Cas PFA,PC E_{\\text{Cas}}^0/E_{\\text{Cas}}^{\\text{PFA,PC}} as a function of dd.Figure: (E Cas 0 +E Cas 1 )/E Cas PFA,PC (E_{\\text{Cas}}^0+E_{\\text{Cas}}^1)/E_{\\text{Cas}}^{\\text{PFA,PC}} as a function of dd.Figure: θ\\theta as a function of dd.To study the dependence of the Casimir interaction energy on the parameters $\\Omega _s$ and $\\Omega _p$ , we plot in Fig.", "REF and Fig.", "REF respectively the ratio $E_{\\text{Cas}}^0/E_{\\text{Cas}}^{\\text{PFA,PC}}$ and the ratio $(E_{\\text{Cas}}^0+E_{\\text{Cas}}^1)/E_{\\text{Cas}}^{\\text{PFA,PC}}$ as a function of $d$ for various values of $\\Omega _s$ and $\\Omega _p$ .", "The variation of $\\theta $ is plotted in Fig.", "REF .", "It is observe that the larger $\\Omega $ is, the larger is the Casimir interaction energy.", "The behavior of $\\theta $ shown in Fig.", "REF is more interesting.", "It is observe that it has a minimum which appears at $d\\sim \\Omega ^{-1}$ when $\\Omega _s=\\Omega _p=\\Omega $ ." ], [ "Conclusion", "We study the Casimir interaction between a spherical object and a planar object that are made of materials that can be modeled as plasma sheets.", "The functional representation of the Casimir interaction energy is derived.", "It is then used to study the small separation asymptotic behavior of the Casimir interaction.", "The leading term of the Casimir interaction is confirmed to be agreed with the proximity force approximation.", "The analytic formula for the next-to-leading order term is computed based on a previously established perturbation analysis [29].", "The special case where the spherical object and planar object are graphene sheets are considered.", "The results are found to be quite different from the case of metallic sphere-plane configuration when the separation between the sphere and the plane is small.", "This may suggest a new experimental setup to test the Casimir effect.", "It also has potential application to nanotechnology.", "This work is supported by the Ministry of Higher Education of Malaysia under FRGS grant FRGS/1/2013/ST02/UNIM/02/2.", "I would like to thank M. Bordag for proposing this question." ] ]	1403.0314
[ [ "Search For The Higgs Boson Decaying Into Tau Pairs" ], [ "Abstract Talk given at PIC2013 summarizing the results of CMS-PAS-HIG-13-004." ], [ "Introduction", "We present a search for the Standard Model Higgs Boson decaying into tau-pairs in the invariant mass region 110-145 $\\textrm {GeV/c}^{2}$ performed using events recorded by the CMS experiment at the LHC in 2011 and 2012 corresponding to an integrated luminosity of 4.9 fb$^{-1}$ at a centre-of-mass energy of 7 TeV and 19.4 fb$^{-1}$ at 8 TeV TauPas.", "Since July 4th 2012 an experimental observation of a new Boson of a mass around 125 $\\textrm {GeV/c}^{2}$ and compatible with the Standard Model Higgs Boson hypothesis is established both by the Atlas and CMS collaborations Ref.", "CMSHiggs, AtlasHiggs, CMSHiggsSM.", "The observation of the Higgs-like Boson is predominantly based on its couplings to gauge bosons.", "To seek direct evidence for its fermionic couplings, the search for the Standard Model Higgs Boson decaying into tau-pairs is a distinguished channel to be analyzed with the CMS detector." ], [ "Analysis Strategy", "The search is simultaneously performed in several final states of the decaying di-tau system, henceforth denoted as: $\\tau _{h}\\tau _{h}$ , $e\\tau _{h}$ , $\\mu \\tau _{h}$ , $e\\mu $ and $\\mu \\mu $ , where $\\tau _{h}$ declares the hadronically decaying tau final states.", "For each final state the events are then divided into several disjoint event categories in order to enhance the overall sensetivity by exploiting the distinct event topologies of the Higgs production processes.", "Major Higgs production mechanisms considered are processes via gluon fusion, vector boson fusion and the vector boson associated production.", "The dedicated $H\\rightarrow \\tau \\tau $ analyses exploiting the associated Higgs productions (see Ref.", "CMSTauAssoPas) are not discussed in detail in this review, however are included in the results in figure REF and REF .", "The most sensitive category is the Vector Boson Fusion (VBF) category where the event topology of Standard Model Higgs production via Vector Boson Fusion is exploited which is characterized by two high $p_{T}$ jets in the forward regions of the detector.", "In addition to the event and lepton selection criteria events in the VBF category are required to have two jets with $p_{T}>30 \\textrm { GeV/c}$ and $\|\\eta \|>4.7$ , having an invariant mass of $m_{jj}>500 \\textrm { GeV/c}^{2}$ , with a seperation in the pseudo-rapdity $\\Delta \\eta _{jj} > 3.5$ and no addtional jet with $p_{T}>30 \\textrm { GeV/c}$ in the eta gap between the two corresponding jets.", "In addition a 1-Jet category is defined.", "Events with at least one jet with $p_{T}>30 \\textrm { GeV/c}$ and $\|\\eta \|>4.7$ are selected in this category, exluding events of the VBF category.", "The Standard Model Higgs production via gluon fusion dominates the signal contribution in the 1-Jet category.", "Further the 0-Jet category is defined for events with no jet with $p_{T}>30 \\textrm { GeV/c}$ and $\|\\eta \|>4.7$ .", "This category is largely dominated by background processes and is used to constrain the backgrounds in the VBF and 1-Jet categories.", "For all categories a b-tagged jet with $p_{T}>20 \\textrm { GeV/c}$ veto is applied to decouple the analysis from the MSSM Higgs search in the di-tau channel.", "Figure: Left: Distribution of the P ζ P_{\\zeta } variable in the eμe\\mu channel.", "Middle: Distribution of the m T m_{T} variable in the μτ h \\mu \\tau _{h}channel.", "Right: Distribution of the BDT discriminant in μμ\\mu \\mu channel as an example in the VBF category.The background compositions are channel and category dependent with the $Z\\gamma \\rightarrow \\tau \\tau $ processes representing typically the most dominant contributions over $t\\bar{t}$ , W+Jets, $Z\\rightarrow \\ell \\ell $ , QCD multijet-events and di-boson productions.", "To further reduce the various backgrounds additional topological event selection criteria depend on the final state are defined.", "For the $e\\mu $ final state the $t\\bar{t}$ background is significantly reduced with selection criteria on $D_\\zeta = p_\\zeta - 0.85 \\cdot p^{vis}_\\zeta > -20 \\textrm { GeV/c}$ , where $\\zeta $ is the bisector of the two leptons, $p_\\zeta = \\vec{p}_{T,1} \\cdot \\hat{\\zeta }+ \\vec{p}_{T,2} \\cdot \\hat{\\zeta }+ \\vec{E}^{miss}_{T} \\cdot \\hat{\\zeta }$ and $p^{vis}_\\zeta = \\vec{p}_{T,1} \\cdot \\hat{\\zeta }+ \\vec{p}_{T,2} \\cdot \\hat{\\zeta }$ .", "$D_{\\zeta }$ is a measure of how collinear the missing transverse energy vector is with the ransverse momentum of the di-lepton system (see figure REF (left)).", "For the $e\\tau _{h}$ and $\\mu \\tau _{h}$ final states the W+Jets background is reduced by selection criteria on the transverse mass variable $m_T = \\sqrt{2p_T E^{miss}_T (1-cos(\\Delta \\phi ))}$ (see figure REF (middle)).", "For the $\\mu \\mu $ final state the overwhelming $Z/\\gamma \\rightarrow \\mu \\mu $ background is reduced by selection criteria on dedicated multivariate Boosted Decision Trees in each category.", "(see figure REF (right)).", "Figure: Left: Combined m ττ m_{\\tau \\tau } distribution of τ h τ h \\tau _{h}\\tau _{h}, eτ h e\\tau _{h}, μτ h \\mu \\tau _{h} and eμe\\mu final states weightedby the ratio of the expected signal and background yields in each category.", "Right:Observed and expected 95% CL exclusion limit on the Standard Model Higgs Boson production rate" ], [ "Results", "The observed and expected 95% CL exclusion limit on the Standard Model Higgs Boson production rate is derived by a combined fit on the reconstructed di-tau mass $m_{\\tau \\tau }$ in each category for each channel and shown in figure REF (right).", "For the $\\mu \\mu $ final state the 2 dimensional distribution of $m_{\\tau \\tau }$ versus the invariant mass of the two muons is used.", "Figure REF (left) shows the combined $m_{\\tau \\tau }$ distribution obtained after summing up the $m_{\\tau \\tau }$ distribution of each category of the $\\tau _{h}\\tau _{h}$ , $e\\tau _{h}$ , $\\mu \\tau _{h}$ and $e\\mu $ final states weighted by the ratio of the expected signal and background yields.", "An excess of the observed limit when compared to the background-only hypothesis is found and the local p-value and signifiacance of this excess as a function of the Higgs mass is shown in figure REF (left).", "A maximum significance of 2.93 standard deviations is observed for a Higgs mass of $m_{H}=120 \\textrm { GeV/c}^{2}$ and a significance of 2.85$\\sigma $ at $m_{H}=125 \\textrm { GeV/c}^{2}$ compared to the Standard Model expectation of 2.63$\\sigma $ .", "The signal strength for $m_{H}$ is measured to be $1.1\\pm 0.4$ times the Standard Model Higgs production rate (see figure REF (right))and thus the excess is compatible with the hypothesis of the presence of the Standard Model Higgs Boson.", "Figure: Observed and expected p-values as a function of m H m_H.", "Right: Best-fit signal strength at m H =125 GeV /cm_H=125 \\textrm { GeV/c} relative to the standard model expectation." ] ]	1403.0365
[ [ "Finite generation and continuity of topological Hochschild and cyclic\n homology" ], [ "Abstract The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results.", "This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry.", "We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra $TR^r$, are finitely generated modules.", "We use this to establish the continuity of these homology theories for any given ideal.", "A consequence of such continuity results is the pro Hochschild-Kostant-Rosenberg theorem for topological Hochschild and cyclic homology.", "Finally, we show more generally that the aforementioned finite generation and continuity properties remain true for any proper scheme over such a ring." ], [ "Introduction", "The aim of this paper is to prove fundamental finite generation, continuity, and pro Hochschild–Kostant–Rosenberg theorems for the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings.", "As far as we are aware, these are the first general results on finite generation and continuity of topological Hochschild and cyclic homology, despite the obvious foundational importance of such problems in the subject.", "The fundamental hypothesis for the majority of our theorems is the classical notion of F-finiteness: Definition 0.1 A $\\mathbb {Z}_{(p)}$ -algebra (always commutative) is said to be F-finite if and only if the $\\mathbb {F}_p$ -algebra $A/pA$ is finitely generated over its subring of $p^{\\mbox{\\scriptsize th}}$ -powers.", "This is a mild condition: it is satisfied as soon as $A/pA$ is obtained from a perfect field by iterating the following constructions any finite number of times: passing to finitely generated algebras, localising, completing, or Henselising; see Lemma .", "To state our main finite generation result, we first remark that the Hochschild homology $H\\!H_n(A)$ of a ring $A$ is always understood in the derived sense (see Section REF ).", "Secondly, $T\\!H\\!H_n(A)$ denotes the topological Hochschild homology groups of a ring $A$ , while $T\\!R^r_n(A;p)$ denotes, as is now usual, the homotopy groups of the fixed point spectrum for the action of the cyclic group $C_{p^{r-1}}$ on the topological Hochschild homology spectrum $T\\!H\\!H(A)$ .", "It is known that $T\\!R^r_n(A;p)$ is naturally a module over the $p$ -typical Witt ring $W_r(A)$ .", "The obvious notation will be used for the $p$ -completed, or finite coefficient, versions of these theories, and for their extensions to quasi-separated, quasi-compact schemes following .", "Our main finite generation result is the following, where $A_p^{\\widehat{\\phantom{o}}}=\\varprojlim _sA/p^sA$ denotes the $p$ -completion of a ring $A$ : Theorem 0.2 (see Corol. )", "Let $A$ be a Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebra.", "Then: $H\\!H_n(A;\\mathbb {Z}_p)$ and $T\\!H\\!H_n(A;\\mathbb {Z}_p)$ are finitely generated $A_p^{\\widehat{\\phantom{o}}}$ -modules for all $n\\ge 0$ .", "$T\\!R_n^r(A;p,\\mathbb {Z}_p)$ is a finitely generated $W_r(A_p^{\\widehat{\\phantom{o}}})$ -module for all $n\\ge 0$ , $r\\ge 1$ .", "The key step towards proving Theorem REF is the following finite generation result for the André–Quillen homology of $\\mathbb {F}_p$ -algebras: Theorem 0.3 (see Thm. )", "Let $A$ be a Noetherian, F-finite $\\mathbb {F}_p$ -algebra.", "Then the André–Quillen homology groups $D_n^i(A/\\mathbb {F}_p)$ are finitely generated for all $n,i\\ge 0$ .", "Next we turn to “degree-wise continuity” for the homology theories $H\\!H$ , $T\\!H\\!H$ , and $T\\!R^r$ , by which we mean the following: given an ideal $I\\subseteq A$ , we examine when the natural map of pro $A$ -modules $\\lbrace H\\!H_n(A)\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace H\\!H_n(A/I^s)\\rbrace _s$ is an isomorphism, and similarly for $T\\!H\\!H$ and $T\\!R^r$ .", "This question was first raised by L. Hesselholt in 2001 , who later proved with T. Geisser the $T\\!H\\!H$ isomorphism in the special case that $A=R[X_1,\\dots ,X_d]$ and $R=\\langle X_1,\\dots ,X_s \\rangle $ for any ring $R$ .", "In Section we prove the following: Theorem 0.4 (see Thm. )", "Let $A$ satisfy the same conditions as Theorem REF , and let $I\\subseteq A$ be an ideal.", "Then the following canonical maps of pro $A$ -modules $\\lbrace H\\!H_n(A;\\mathbb {Z}/p^v)\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace H\\!H_n(A/I^s;\\mathbb {Z}/p^v)\\rbrace _s$ $\\lbrace T\\!H\\!H_n(A;\\mathbb {Z}/p^v)\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace T\\!H\\!H_n(A/I^s;\\mathbb {Z}/p^v)\\rbrace _s$ and the following canonical map of pro $W_r(A)$ -modules $\\lbrace T\\!R_n^r(A;p,\\mathbb {Z}/p^v)\\otimes _{W_r(A)}W_r(A/I^s)\\rbrace _s\\longrightarrow \\lbrace T\\!R_n^r(A/I^s;p,\\mathbb {Z}/p^v)\\rbrace _s$ are isomorphisms for all $n\\ge 0$ and $r,v\\ge 1$ .", "Applying Theorem REF simultaneously to $A$ and its completion $\\widehat{A}=\\varprojlim _sA/I^s$ with respect to the ideal $I$ , we obtain the following corollary: Corollary 0.5 (see Corol. )", "Let $A$ be a Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebra, and $I\\subseteq A$ an ideal.", "Then all of the following canonical maps (not just the compositions) are isomorphisms for all $n\\ge 0$ and $r,v\\ge 1$ : $&H\\!H_n(A;\\mathbb {Z}/p^v)\\otimes _A\\widehat{A}\\longrightarrow H\\!H_n(\\widehat{A};\\mathbb {Z}/p^v)\\longrightarrow \\varprojlim _sH\\!H_n(A/I^s;\\mathbb {Z}/p^v)\\\\&T\\!H\\!H_n(A)\\otimes _A\\widehat{A}\\longrightarrow T\\!H\\!H_n(\\widehat{A})\\longrightarrow \\varprojlim _sT\\!H\\!H_n(A/I^s;\\mathbb {Z}/p^v)\\\\&T\\!R^r_n(A;p,\\mathbb {Z}/p^v)\\otimes _{W_r(A)}W_r(\\widehat{A})\\longrightarrow T\\!R^r_n(\\widehat{A};p,\\mathbb {Z}/p^v)\\longrightarrow \\varprojlim _sT\\!R^r_n(A/I^s;p,\\mathbb {Z}/p^v)$ Of a more topological nature than such degree-wise continuity statements are spectral continuity, namely the question of whether the canonical map of spectra $T\\!H\\!H(A)\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!H\\!H(A/I^s)$ is a weak-equivalence, at least after $p$ -completion.", "The analogous continuity question for $K$ -theory was studied for discrete valuation rings by A. Suslin , I. Panin , and the first author , and for power series rings $A=R[[X_1,\\dots ,X_d]]$ over a regular, F-finite $\\mathbb {F}_p$ -algebra $R$ by Geisser and Hesselholt , with $I=\\langle X_1,\\dots ,X_d \\rangle $ .", "Geisser and Hesselholt proved the continuity of $K$ -theory in such cases by establishing it first for $T\\!H\\!H$ and $T\\!R^r$ .", "We use the previous degree-wise continuity results to prove the following: Theorem 0.6 (see Thm. )", "Let $A$ be a Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebra, and $I\\subseteq A$ an ideal; assume that $A$ is $I$ -adically complete.", "Then the following canonical maps are weak equivalences after $p$ -completion: $&T\\!H\\!H(A)\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!H\\!H(A/I^s)\\\\&T\\!R^r(A;p)\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!R^r(A/I^s;p)\\quad (r\\ge 1)$ and similarly for $T\\!R(-;p)$ , $T\\!C^r(-;p)$ , and $T\\!C(-;p)$ .", "There are two important special cases in which the results so far can be analysed further: when $p$ is nilpotent, and when $p$ generates $I$ .", "Firstly, if $p$ is nilpotent in $A$ , for example if $A$ is a Noetherian, F-finite, $\\mathbb {F}_p$ -algebra, then Theorem REF – Theorem REF are true integrally, without $p$ -completing or working with finite coefficients; see Corollaries and for precise statements.", "The second important special case is $I=pA$ , when Theorems REF and REF simplify to the following statements: Corollary 0.7 (see Corol. )", "Let $A$ be a Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebra.", "Then the following canonical maps are isomorphisms for all $n\\ge 0$ and $v,r\\ge 1$ : $H\\!H_n(A;\\mathbb {Z}/p^v)\\longrightarrow \\lbrace H\\!H_n(A/p^sA;\\mathbb {Z}/p^v)\\rbrace _s$ $T\\!H\\!H_n(A;\\mathbb {Z}/p^v)\\longrightarrow \\lbrace T\\!H\\!H_n(A/p^sA;\\mathbb {Z}/p^v)\\rbrace _s$ $T\\!R_n^r(A;p,\\mathbb {Z}/p^v)\\longrightarrow \\lbrace T\\!R_n^r(A/p^sA;p,\\mathbb {Z}/p^v)\\rbrace _s$ $T\\!C_n^r(A;p,\\mathbb {Z}/p^v)\\longrightarrow \\lbrace T\\!C_n^r(A/p^sA;p,\\mathbb {Z}/p^v)\\rbrace _s$ Moreover, all of the following maps (not just the compositions) are weak equivalences after $p$ -completion: $T\\!H\\!H(A)\\longrightarrow T\\!H\\!H(A_p^{\\widehat{\\phantom{o}}})\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!H\\!H(A/p^sA)$ $T\\!R^r(A;p)\\longrightarrow T\\!R^r(A_p^{\\widehat{\\phantom{o}}};p)\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!R^r(A/p^sA;p)\\quad (r\\ge 1)$ , and similarly for $T\\!R(-;p)$ , $T\\!C^r(-;p)$ , and $T\\!C(-;p)$ .", "Corollary REF was proved by Hesselholt and I. Madsen for finite algebras over the Witt vectors of perfect fields, and then by Geisser and Hesselholt for any (possibly non-commutative) ring $A$ in which $p$ is a non-zero divisor.", "Our result eliminates the assumption that $p$ be a non-zero divisor, but at the expense of requiring $A$ to be a Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebra.", "Next we present our pro Hochschild–Kostant–Rosenberg theorems, which can be found in Section , starting with algebraic Hochschild homology.", "Given a geometrically regular (e.g., smooth) morphism $k\\rightarrow A$ of Noetherian rings, the classical Hochschild–Kostant–Rosenberg theorem states that the antisymmetrisation map $\\Omega _{A/k}^n\\rightarrow HH_n^k(A)$ is an isomorphism of $A$ -modules for all $n\\ge 0$ .", "We establish its pro analogue in full generality: Theorem 0.8 (see Thm. )", "Let $k\\rightarrow A$ be a geometrically regular morphism of Noetherian rings, and let $I\\subseteq A$ be an ideal.", "Then the canonical map of pro $A$ -modules $\\lbrace \\Omega ^n_{(A/I^s)/k}\\rbrace _s\\longrightarrow \\lbrace HH_n^k(A/I^s)\\rbrace _s$ is an isomorphism for all $n\\ge 0$ .", "In the special case of finite type algebras over fields, the theorem was proved by G. Cortiñas, C. Haesemeyer, and C. Weibel .", "The stronger form presented here has recently been required in the study of infinitesimal deformations of algebraic cycles , .", "The analogue of the classical Hochschild–Kostant–Rosenberg theorem for topological Hochschild homology and the fixed point spectra $T\\!R^r$ were established by Hesselholt and state the following: If $A$ is a regular $\\mathbb {F}_p$ -algebra, then there is a natural isomorphism of $W_r(A)$ -modules $\\bigoplus _{i=0}^nW_r\\Omega _A^i\\otimes _{W_r(\\mathbb {F}_p)}TR_{n-i}^r(\\mathbb {F}_p;p)\\stackrel{\\simeq }{\\rightarrow }TR_n^r(A;p),$ where $W_r\\Omega _A^*$ denotes the de Rham–Witt complex of S. Bloch, P. Deligne, and L. Illusie.", "In the limit over $r$ , Hesselholt moreover showed that the contribution from the left side vanishes except in top degree $i=n$ , giving an isomorphism of pro abelian groups $\\lbrace W_r\\Omega _A^i\\rbrace _r\\cong \\lbrace T\\!R^r_n(A;p)\\rbrace _r$ which deserves to be called the Hochschild–Kostant–Rosenberg theorem for the pro spectrum $\\lbrace T\\!R^r\\rbrace _r$ .", "We prove the following pro versions of these HKR theorem with respect to powers of an ideal: Theorem 0.9 (see Thm.", "& Corol. )", "Let $A$ be a regular, F-finite $\\mathbb {F}_p$ -algebra, and $I\\subseteq A$ an ideal.", "Then: The canonical map $\\bigoplus _{i=0}^n\\lbrace W_r\\Omega _{A/I^s}^i\\otimes _{W_r(\\mathbb {F}_p)}T\\!R^r_{n-i}(\\mathbb {F}_p;p)\\rbrace _s\\longrightarrow \\lbrace T\\!R^r_n(A/I^s;p)\\rbrace _s$ of pro $W_r(A)$ -modules is an isomorphism for all $n\\ge 0$ , $r\\ge 1$ .", "The canonical map of pro pro abelian groups, i.e., in the category $\\operatorname{Pro}(\\operatorname{Pro}Ab)$ , $\\big \\lbrace \\lbrace W_r\\Omega _{A/I^s}^n\\rbrace _s\\big \\rbrace _r\\longrightarrow \\big \\lbrace \\lbrace T\\!R^r_n(A/I^s;p)\\rbrace _s\\big \\rbrace _r$ is an isomorphism for all $n\\ge 0$ .", "Finally, in Section , the earlier finite generation and continuity results are extended from Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebras to proper schemes over such rings.", "These are obtained by combining the results in the affine case with Zariski descent and Grothendieck's formal functions theorem for coherent cohomology.", "Our main finite generation result in this context is the following: Theorem 0.10 (see Corol. )", "Let $A$ be a Noetherian, F-finite, finite Krull-dimensional $\\mathbb {Z}_{(p)}$ -algebra, and $X$ a proper scheme over $A$ .", "Then: $H\\!H_n(X;\\mathbb {Z}_p)$ and $T\\!H\\!H_n(X;\\mathbb {Z}_p)$ are finitely generated $A_p^{\\widehat{\\phantom{o}}}$ -modules for all $n\\ge ~0$ .", "$T\\!R_n^r(X;p,\\mathbb {Z}_p)$ is a finitely generated $W_r(A_p^{\\widehat{\\phantom{o}}})$ -module for all $n\\ge 0$ , $r\\ge 1$ .", "Next we consider analogues of degree-wise continuity for proper schemes over $A$ : given an ideal $I\\subseteq A$ and a proper scheme $X$ over $A$ , we consider the natural map of pro $A$ -modules $\\lbrace H\\!H_n(X)\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace H\\!H_n(X_s)\\rbrace _s$ where $X_s:=X\\times _AA/I^s$ , and similarly for $T\\!H\\!H$ , $T\\!R$ .", "In this setup we establish the exact analogues of Theorem REF and Corollary REF ; see Theorem and Corollary for precise statements.", "As in the affine case, we use these degree-wise continuity statements to deduce continuity of topological cyclic homology for proper schemes over our usual base rings: Theorem 0.11 (see Thm. )", "Let $A$ be a Noetherian, F-finite $\\mathbb {Z}_{(p)}$ -algebra, $I\\subseteq A$ an ideal, and $X$ be a proper scheme over $A$ ; assume that $A$ is $I$ -adically complete.", "Then the following canonical maps of spectra are weak equivalences after $p$ -completion: $&T\\!H\\!H(X)\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!H\\!H(X_s)\\\\&T\\!R^r(X;p)\\longrightarrow \\operatornamewithlimits{\\operatornamewithlimits{holim}}_sT\\!R^r(X_s;p)\\quad (r\\ge 1),$ and similarly for $T\\!R(-;p)$ , $T\\!C^r(-;p)$ , and $T\\!C(-;p)$ .", "Also as in the affine case, it is particular interesting to consider the cases that $p$ is nilpotent or is the generator of $I$ ; see Corollaries and ." ], [ "Notation, etc.", "All rings from Section onwards are tacitly understood to be commutative; the strict exception is Proposition , which holds also in the non-commutative case.", "Modules are therefore understood to be symmetric bimodules whenever a bimodule structure is required for Hochschild homology; again, the strict exception is Proposition , where a non-symmetric module naturally appears.", "Given a positive integer $n$ , the $n$ -torsion of an abelian group $M$ is denoted by $M[n]$ .", "The second author would like to thank the Department of Mathematics at the University of Bergen for providing such a hospitable environment during two visits." ], [ "Pro abelian groups and Artin–Rees properties", "Here we summarise some results and notation concerning pro abelian group and pro modules which will be used throughout the paper.", "If $\\mathcal {A}$ is a category, then we denote by $\\operatorname{Pro}\\mathcal {A}$ the category of pro objects of $\\mathcal {A}$ indexed over the natural numbers.", "That is, an object of $\\operatorname{Pro}\\mathcal {A}$ is an inverse system $M_\\infty =\\lbrace M_s\\rbrace _s=``M_1\\leftarrow M_2\\leftarrow \\cdots \",$ where the objects $M_i$ and the transition maps belong to $\\mathcal {A}$ ; the morphisms are given by $\\operatorname{Hom}_{\\operatorname{Pro}\\mathcal {A}}(M_\\infty ,N_\\infty ):=\\varprojlim _r\\varinjlim _s\\operatorname{Hom}_\\mathcal {A}(M_s,N_r).$ If $\\mathcal {A}$ is abelian then so is $\\operatorname{Pro}\\mathcal {A}$ , and a pro object $M_\\infty \\in \\operatorname{Pro}\\mathcal {A}$ is isomorphic to zero if and only if it satisfies the trivial Mittag-Leffler condition: that is, that for all $r\\ge 1$ there exists $s\\ge r$ such that the transition map $M_s\\rightarrow M_r$ is zero.", "We will be particularly interested in the cases $\\mathcal {A}=Ab$ and $A\\operatorname{-mod}$ , where $A$ is a commutative ring, in which case we speak of pro abelian groups and pro $A$ -modules respectively.", "When it is unlikely to cause any confusion, we will occasionally use $\\infty $ notation in proofs for the sake of brevity; for example, if $I$ is an ideal of a ring $A$ and $M$ is an $A$ -module, then $M\\otimes _AA/I^\\infty =\\lbrace M\\otimes _AA/I^s\\rbrace _s,\\quad H\\!H_n(A/I^\\infty ,M/I^\\infty M)=\\lbrace H\\!H_n(A/I^s,M/I^sM)\\rbrace _s.$ For the sake of reference, we now formally state the fundamental Artin–Rees result which will be used repeatedly; this result appears to have been first noticed and exploited by M. André and D. Quillen : Theorem 1.1 (André, Quillen) Let $A$ be a commutative, Noetherian ring, and $I\\subseteq A$ an ideal.", "If $M$ is a finitely generated $A$ -module, then the pro $A$ -module $\\lbrace \\operatorname{Tor}_n^A(A/I^s,M)\\rbrace _s$ vanishes for all $n>0$ .", "The “completion” functor $-\\otimes _A A/I^\\infty \\,:A\\operatorname{-mod}&\\longrightarrow \\operatorname{Pro}A\\operatorname{-mod}\\\\M&\\mapsto \\lbrace M\\otimes _AA/I^s\\rbrace _s$ is exact on the subcategory of finitely generated $A$ -modules.", "[Sketch of proof] By picking a resolution $P_\\bullet $ of $M$ by finitely generated projective $A$ -modules and applying the classical Artin–Rees property to the pair $d(P_n)\\subseteq P_{n-1}$ , one sees that for each $r\\ge 1$ there exists $s\\ge r$ such that the map $\\operatorname{Tor}_n^A(A/I^s,M)\\rightarrow \\operatorname{Tor}_n^A(A/I^r,M)$ is zero.", "This proves (i).", "(ii) is just a restatement of (i).", "Corollary 1.2 Let $A$ be a commutative, Noetherian ring, $I\\subseteq A$ an ideal, and $M$ a finitely generated $A$ -module.", "Then the pro $A$ -module $\\lbrace \\operatorname{Tor}_n^A(A/I^s,M/I^s)\\rbrace _s$ vanishes for all $n>0$ .", "For each $r\\ge 1$ we may apply the previous theorem to the module $M/I^r$ to see that there exists $s\\ge r$ such that the second of the following arrows is zero: $\\operatorname{Tor}_n^A(A/I^s,M/I^s)\\rightarrow \\operatorname{Tor}_n^A(A/I^s,M/I^r)\\rightarrow \\operatorname{Tor}_n^A(A/I^r,M/I^r)$ Hence the composition is zero, completing the proof.", "Corollary 1.3 Let $A$ be a commutative, Noetherian ring, $I\\subseteq A$ an ideal, $M$ a finitely generated $A$ -module, and $G$ a finite group acting $A$ -linearly on $M$ .", "Then the canonical map of pro group homologies $\\lbrace H_n(G,M)\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace H_n(G,M/I^sM)\\rbrace _s$ is an isomorphism for all $n\\ge 0$ .", "Considering $\\mathbb {Z}$ as a left $\\mathbb {Z}[G]$ -module via the augmentation map, $A/I^s$ as a right $A$ -module, and $M$ as an $A-\\mathbb {Z}[G]$ -bimodule, there are first quadrant spectral sequences of $A$ -modules with the same abutement by : $E_{ij}^2(s)=\\operatorname{Tor}_i^A(A/I^s,\\operatorname{Tor}_j^{\\mathbb {Z}[G]}(M,\\mathbb {Z})),\\quad ^\\prime E_{ij}^2(s)=\\operatorname{Tor}_i^{\\mathbb {Z}[G]}(\\operatorname{Tor}_j^A(A/I^s,M),\\mathbb {Z}).$ These assemble to first quadrant spectral sequences of pro $A$ -modules with the same abutement: $E_{ij}^2(\\infty )=\\lbrace \\operatorname{Tor}_i^A(A/I^s,\\operatorname{Tor}_j^{\\mathbb {Z}[G]}(M,\\mathbb {Z}))\\rbrace _s,\\quad ^\\prime E_{ij}^2(\\infty )=\\lbrace \\operatorname{Tor}_i^{\\mathbb {Z}[G]}(\\operatorname{Tor}_j^A(A/I^s,M),\\mathbb {Z})\\rbrace _s.$ Since $\\operatorname{Tor}^{\\mathbb {Z}[G]}_j(M,\\mathbb {Z})$ is a finitely generated $A$ -module for all $j\\ge 0$ , Theorem REF (i) implies that $E_{ij}^2(\\infty )=0$ for $i>0$ ; so the $E(\\infty )$ -spectral sequence degenerates to edge map isomorphisms.", "Theorem REF (i) similarly implies that $\\operatorname{Tor}_j^A(A/I^s,M)=0$ for $j>0$ , and hence the $^\\prime E(\\infty )$ -spectral sequence also degenerates to edge map isomorphisms.", "Composing these edge map isomorphisms, we arrive at an isomorphism of pro $A$ -modules $\\lbrace \\operatorname{Tor}^{\\mathbb {Z}[G]}_n(M,\\mathbb {Z})\\otimes _AA/I^s\\rbrace _s\\stackrel{\\simeq }{\\rightarrow }\\lbrace \\operatorname{Tor}^{\\mathbb {Z}[G]}_n(M/I^sM,\\mathbb {Z})\\rbrace _s$ for all $n\\ge 0$ , which is exactly the desired isomorphism.", "Corollary 1.4 Let $A$ be a commutative, Noetherian ring, $I\\subseteq A$ an ideal, $M$ a finitely generated $A$ -module, and $m\\ge 1$ .", "Then the canonical maps $\\lbrace M[m]\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace M/I^sM\\,[m]\\rbrace _s,\\quad \\lbrace M/mM\\otimes _AA/I^s\\rbrace _s\\longrightarrow \\lbrace M/(mM+I^sM)\\rbrace _s$ are isomorphisms of pro $A$ -modules.", "These isomorphisms follow from the sequence $0\\rightarrow \\lbrace M[m]\\otimes _AA/I^s\\rbrace _s\\rightarrow \\lbrace M/I^sM\\rbrace \\xrightarrow{}\\lbrace M/I^sM\\rbrace \\rightarrow \\lbrace M/mM\\otimes _AA/I^s\\rbrace _s\\rightarrow 0,$ which is exact by Theorem REF (ii)." ], [ "André–Quillen and Hochschild homology", "Let $k$ be a commutative ring.", "Here we briefly review the André–Quillen and Hochschild homologies of $k$ -algebras, though we assume that the reader has some familiarity with these theories.", "We begin with André–Quillen homology , , .", "Let $A$ be a commutative $k$ -algebra, let $P_\\bullet \\rightarrow A$ be a simplicial resolution of $A$ by free commutative $k$ -algebras, and set $\\mathbb {L}_{A/k}:= \\Omega _{P_\\bullet /k}^1\\otimes _{P_\\bullet }A.$ Thus $\\mathbb {L}_{A/k}$ is a simplicial $A$ -module which is free in each degree; it is called the cotangent complex of the $k$ -algebra $A$ .", "Up to homotopy, the cotangent complex depends only on $A$ , since the free simplicial resolution $P_\\bullet \\rightarrow A$ is unique up to homotopy.", "Given simplicial $A$ -modules $M_\\bullet $ , $N_\\bullet $ , the tensor product and alternating powers are the simplicial $A$ -modules defined degreewise: $(M_\\bullet \\otimes _AN_\\bullet )_n=M_n\\otimes _AN_n$ and $(\\bigwedge _A^rM_\\bullet )_n=\\bigwedge _A^rM_n$ .", "In particular we set $\\mathbb {L}_{A/k}^i:=\\bigwedge _A^i\\mathbb {L}_{A/k}$ for each $i\\ge 1$ .", "The André–Quillen homology of the $k$ -algebra $A$ , with coefficients in an $A$ -module $M$ , is defined by $D_n^i(A/k,M):=\\pi _n(\\mathbb {L}_{A/k}^i\\otimes _A M).\\qquad \\mathrm {(n,i\\ge 0)}$ When $M=A$ the notation is simplified to $D_n^i(A/k):=D_n^i(A/k,A)=\\pi _n\\mathbb {L}_{A/k}^i$ ." ] ]	1403.0534
[ [ "The emergence of topologically protected surface states in epitaxial\n Bi(111) thin films" ], [ "Abstract Quantum transport measurements including the Altshuler-Aronov-Spivak (AAS) and Aharonov-Bohm (AB) effects, universal conductance fluctuations (UCF), and weak anti-localization (WAL) have been carried out on epitaxial Bi thin films ($10-70$ bilayers) on Si(111).", "The results show that while the film interior is insulating all six surfaces of the Bi thin films are robustly metallic.", "We propose that these properties are the manifestation of a novel phenomenon, namely, a topologically trivial bulk system can become topologically non-trivial when it is made into a thin film.", "We stress that what's observed here is entirely different from the predicted 2D topological insulating state in a single bilayer Bi where only the four side surfaces should possess topologically protected gapless states." ], [ "The emergence of topologically protected surface states in epitaxial Bi(111) thin films Kai Zhu Lin Wu Xinxin Gong Shunhao Xiao Shiyan Li Xiaofeng Jin [Corresponding author, E-mail:][email protected] Department of Physics, State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433,China Mengyu Yao Dong Qian Department of Physics and Astronomy, Key Laboratory of Artificial Structures and Quantum Control, Shanghai Jiao Tong University, Shanghai 200240, China Meng Wu Ji Feng Qian Niu International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing, China Fernando de Juan Dung-Hai Lee Materials Science Division, Lawrence Berkeley National Laboratories, Berkeley, CA 94720 Department of Physics, University of California at Berkeley, Berkeley, CA94720, USA Quantum transport measurements including the Altshuler-Aronov-Spivak (AAS) and Aharonov-Bohm (AB) effects, universal conductance fluctuations (UCF), and weak anti-localization (WAL) have been carried out on epitaxial Bi thin films ($10-70$ bilayers) on Si(111).", "The results show that while the film interior is insulating all six surfaces of the Bi thin films are robustly metallic.", "We propose that these properties are the manifestation of a novel phenomenon, namely, a topologically trivial bulk system can become topologically non-trivial when it is made into a thin film.", "We stress that what's observed here is entirely different from the predicted 2D topological insulating state in a single bilayer Bi where only the four side surfaces should possess topologically protected gapless states.", "73.20-r, 73.25.+i, 73.50.-h Historically Bi is a material where many interesting condensed matter phenomena were (first) observed.", "This includes the de Haas-van Alphen effect [1], quantum size confinement effect [2], quantum linear magnetoresistance [3], peculiar superconductivity [4], and possibly fractional quantum Hall effect [5].", "Despite of decades of studies, many fundamental properties of Bi remain poorly understood.", "For example, the Bi surfaces conduct much better than the bulk [6].", "Even more dramatically, the surface of Bi remains metallic after the bulk is turned into insulating by reducing the thickness of Bi films [7].", "The last phenomenon is reminiscent to the behavior of topological insulators.", "Given that many Bi-based compounds (e.g., Bi$_{x}$ Sb$_{1-x}$ , Bi$_{2}$ Se$_{3}$ , Bi$_{2}$ Te$_{3}$ ) are topological insulators [8], [9], “can reduction of thickness turn the Bi thin film into topological insulator from its topologically trivial bulk ?” This is the question we address in this letter.", "In the following we report the results of a number of quantum transport measurements and theoretical calculations.", "Combining these investigations we arrive at the the conclusion that Bi thin films are interesting examples where a topologically trivial system becomes non-trivial solely due to the reduction of thickness.", "Throughout this letter we focus on epitaxial Bi thin films grown on Si(111) substrate.", "Polished single crystal semi-insulating Si(111) substrates with high resistivity ($> 10^{6}$ $\\Omega $ cm at 250 K) were chemically cleaned before put into the ultrahigh vacuum (UHV) chamber.", "It is further flashed to 1300 K to obtain a well-ordered ($7\\times 7$ ) reconstructed surface, on which the epitaxial growth of Bi is carried out at 300 K [10].", "The details of our experimental set-up can be found in Ref [11], [12], [7], [13].", "A representative reflection high energy electron diffraction (RHEED) pattern of our MBE grown thin films is given in the inset of Fig.", "1(a).", "Both large angle and grazing angle x-ray diffraction measurements confirm that the Bi films investigated in this work are fully strain released with bulk lattice constants.", "Before the film is taken out of the UHV chamber we cap it with 5 nm MgO to protect it from oxidation.", "The transport measurements were carried out in an Oxford Cryofree magnet system with temperature down to 1.4 K and magnetic field up to 9 T. Fig.", "1(a) plots the longitudinal resistivity versus temperature.", "Two characteristic behaviors are observed: a low temperature metallic behavior ($d\\rho _{xx}/dT>0$ ) and a higher temperature insulating behavior ($d\\rho _{xx}/dT<0$ ).", "This result is consistent with the behavior of a thin film with an insulating interior but a metallic surface [7].", "More specifically we attribute the linearly rising resistivity at low temperatures to the transport behavior of the metallic surface (it is similar to the resistivity behavior of a monolayer indium film [14]).", "In contrast we attribute the high temperature behavior to the conduction due to thermally activated (across the interior energy gap) electrons and holes.", "Additional support of the low temperature conduction being due to the surface states comes from the fact that the low temperature total conductance of Bi films grown on two different substrates stays constant for thickness in the range of 15-27.5 nm.", "To our knowledge, due to the presence of bulk carriers, none of the as grown topological insulators Bi$_{x}$ Sb$_{1-x}$ , Bi$_{2}$ Se$_{3}$ , and Bi$_{2}$ Te$_{3}$ has exhibited the insulating to metallic crossover as shown in Fig.", "1(a).", "Figure: (a) 4 nm Bi film on Si(111): the resistivity as a function of temperature, and its representative RHEED pattern (inset).", "(b) The linear relationship between ln(σ interior )\\ln (\\sigma _{interior}) and 1/T1/T.To separate the surface and interior conduction we first fit the linearly increasing resistivity by $\\rho _{surface}(T)=\\rho _{surface}(0)+\\kappa $$T$ as marked by the red line in Fig.", "1(a).", "The corresponding surface conductivity is shown in the inset of Fig.", "1(b).", "By subtracting it from the total conductivity we obtain the temperature dependent conductivity of the film interior $\\sigma _{interior}(T)$ .", "In Fig.1(b) we plot $\\ln (\\sigma _{interior}(T))$ versus $1/T$ .", "The slope gives an estimate of the interior energy gap.", "The result of this estimate is $\\sim $ 62 meV.", "Our own angle resolved photoemission spectroscopic (ARPES) data on a 30 nm (75 BL) Bi(111) film confirms the existence of an indirect energy gap in the film interior[15].", "Due to the 62 meV gap the low temperature transport discussed below is completely dominated by the surface conduction.", "Figure: (a) SEM picture of the Bi film device, with a length of 18 μ\\mu m, width 220.7 nm, and thickness 16.0 nm.", "(b) Magneto-conductivity for 16.0 nm thick Bi, measured at 1.5 K with the magnetic field applied along the current direction.", "(c) Identified AAS and AB oscillations.", "(d) Temperature dependence of the AAS and AB effects.", "(e) Dominated AB oscillations for 8.0 nm Bi.", "(f) UCF with the magnetic field applied perpendicular to Bi film plane.Next we demonstrate all six (top, bottom and four side) surfaces of the film are metallic.", "Fig.", "2(a) shows a scanning electron microscope (SEM) picture of our device, in which both the current and magnetic field are applied along the x direction.", "The four probe method is used to measure the electric conductance.", "Fig.", "2(b) shows the raw conductance versus magnetic field data for a 16 nm Bi film at T=1.5 K .", "Besides the weak anti-localization peak around zero magnetic field reproducible fine structures are observed on top of a smooth background (marked by the dashed line).", "After a background subtraction, quasi-periodic oscillations with a magnetic field period 0.55 T are clearly identified (see Fig.", "2(c)).", "We attribute such oscillation to the Altshuler-Aronov-Spivak (AAS) $h/2e$ oscillation [16].", "Using $h/2e$ as the magnetic flux period, the above mentioned 0.55 T magnetic field period corresponds to a cross-section area of $3.8\\times 10^{-15}m^{2}$ , which is in excellent agreement with the measured cross-section area $S=w\\cdot d=3.5\\times 10^{-15}m^{2}$ ($w$ and $d$ are the width and the thickness of the film).", "As usual the amplitudes of the AAS oscillations decreases with increasing the temperature (see Fig.", "2(d)).", "In addition, as expected, for films with the same width and length but thinner thickness (8 nm film) the $h/e$ Aharonov-Bohm (AB) oscillation becomes more prominent (see Fig.", "2(e)).", "The observation of the AAS and/or AB effect attests for the fact that the films under study have an insulating interior but a metallic surface.", "In particular, in order for the threaded flux to affect transport not only the top and the bottom but also the side surfaces need to be metallic.", "These metallic surface states are apparently very robust as they survived the “brutal” e-beam lithography.", "Besides the AB and AAS oscillations, a careful examination of Fig.", "2(c) reveals other finer structures (e.g., the features between peaks -5 and -6, and 2 and 3).", "We attribute them to the universal conductance fluctuations (UCF) [17].", "Fig.", "2(f) shows a couple of magneto-conductance curves (with the magnetic field applied perpendicular to the Bi film) after background subtraction.", "For the red trace the magnetic field sweeps from -4 to 4 T, while for the blue trace the magnetic field is reversed (see the arrows).", "All the fine structures in Fig.", "2(f) (with the fluctuation amplitude $\\sim e^2/h$ ) are completely reproducible.", "They are the consequence of quantum interference of the electron matter waves in disordered media.", "Figure: (a) Representative magneto-conductivity curves for 4 nm Bi film with perpendicular magnetic field at varies temperature, the black line shows the fitting with the HNL equation.", "(b) α\\alpha and l ϕ l_{\\varphi } extracted form the HNL equation fitting plotted as a function of temperature.", "(c) α\\alpha and l ϕ l_{\\varphi } at 2 K plotted as a function of thickness.", "(d) Impurity effect on the weak anti-localization.Next we focus on the weak anti-localization (WAL) peak.", "In Fig.", "3(a) we show the magnetoresistance curves for several different temperatures.", "These curves are measured on a 4nm thick Bi film patterned into a Hall bar.", "The WAL peak is clearly observed.", "After detailed quantitative analysis we conclude that these magnetoresistance data are well described by the two dimensional Hikami-Larkin-Nagaoka (HLN) theory for transport in the presence of strong spin-orbit coupling $\\Delta \\sigma _{xx}(B)=\\alpha {e^{2}\\over 2\\pi ^{2}\\hbar }\\Big [\\Psi \\Big ({1\\over 2}+{\\hbar \\over 4eBl_{\\varphi }^{2}}\\Big )-\\ln \\Big ({\\hbar \\over 4eBl_{\\varphi }^{2}}\\Big )\\Big ],$ where $\\Delta \\sigma _{xx}(B)=\\sigma _{xx}(B)-\\sigma _{xx}(0)$ , $\\Psi $ is the digamma function [18].", "Treating $\\alpha $ (see later) and the phase coherence length $l_{\\varphi }$ as fitting parameters we fit the data to Eq.", "(REF ).", "An example of the quality of the fit is shown by the bottom data set (the black curve is the fit) of Fig.", "3(a).", "In Fig.", "3(b) we plot the fitted $\\alpha $ value (red dots) as a function of temperature, the result suggests $\\alpha $ is nearly temperature independent and is very close to -1/2.", "In contrast the fitted phase coherence length (blue dots) decreases monotonically with increasing temperature.", "In Fig.", "3(c) we plot the fitted $\\alpha $ as a function of film thickness for a fixed temperature.", "Again $\\alpha $ is approximately thickness independent and stays very close to -1/2.", "Theoretically $\\alpha $ = -1/2 is expected for a connected spin-orbit interacting 2D system [19].", "The fact that our fitted value is close to -1/2 again attests for the fact that the metallic surface states of our Bi film (top, bottom, and side surfaces together) forms a continuous 2D spin-orbit metal as expected for the surface states of topological insulator.", "In Fig.", "3(d) we show the effects of scalar and magnetic impurity on WAL.", "From the line shape of the curves it is clear that the WAL peak remains as a 0.5 ML of non-magnetic Cu are introduced to the top surface.", "In contrast when we add a 0.25 ML of magnetic Co the WAL peak is completely gone.", "This result supports the notion that the surface states of our Bi film are protected by the time reversal symmetry just like the surface states of topological insulators.", "Before turning to theory, we checked that the metallic surface states of Bi(111) thin films are robust against surface oxidation.", "In growing an 8 nm thick Bi thin film we deliberately expose half of the sample to air (at 300 K) before capping.", "Interestingly such “violent act” only caused a merely 10 percent change in the total conductance.", "In Fig.", "REF we show the effect of oxidization on the magnetoresistance curve in weak field.", "Clearly the WAL peak is robust against oxidization.", "These results are consistent with the picture that surface oxidation simply displaces the metallic surface toward the film interior.", "Figure: Magneto-conductivity for 8 nm Bi film with and without oxidation.Figure: (a) Schematic illustration of the physics which allows a trivial bulk system to becometopologically non-trivial when it is made into a thin film.", "(b) For the modeldiscussed in the text, wavefunctions of the edge modes at k=0k=0 and π\\pi for WW=40aa.", "(c) Spectrumfor WW=40aa.", "There are two pairs of counter-propagating Kramers pairs and hence the system istopologically trivial (note every mode is doubly degenerate).", "(d) Spectrum of the same insulatorwith WW=10aa.", "Only one pair of edge states remain gapless.All the experimental data presented up to this point supports the idea that when Bi (which is topologically trivial in the bulk) is made into thin films they become topological insulators.", "Here we demonstrate, as a matter of principle, that a topologically trivial system can become topologically non-trivial as it becomes sufficiently thin.", "For simplicity we focus on a two dimensional system (the generalization to 3D is straightforward).", "The basic physics is illustrated in Fig.", "REF (a) where each edge possesses two pairs of counter propagating helical modes (black and red) when $W$ is wide.", "With impurities backscattering between the red and black modes is allowed, hence the system is topologically trivial.", "However if the red edge modes decay into the bulk with a much larger length scale than the black ones there will be an intermediate range in $W$ where backscattering between the red edge modes from opposite edges is sufficiently strong but that between the black ones is still extremely weak.", "For that range of $W$ the red modes will be gapped out and the system becomes indistinguishable from a strong topological insulator.", "The above physics can be demonstrated by an explicit model $H = \\left(\\begin{smallmatrix}H_0({\\bf k}) & 0 \\\\0 & H_0^*(-{\\bf k})\\end{smallmatrix}\\right), {\\rm where~} H_0({\\bf k}) = \\epsilon _{{\\bf k}} + \\vec{\\sigma }\\cdot \\vec{d}({\\bf k}).$ Here $\\vec{\\sigma }$ are the Pauli matrices, $\\epsilon _{{\\bf k}} = A(\\cos k_x +\\cos k_y)$ , $d_x + i d_y = (\\sin k_x + i\\sin k_y)$ and $d_z = M - 4+2(1-\\delta )\\cos k_x +2(1+\\delta )\\cos k_y]$ .", "For $A=0.3,M=3.3,\\delta =0.2$ , this model describes a trivial insulator, featuring two gapped Dirac cones at $k=(0,\\pi )$ and $k=(\\pi ,0)$ with masses $m_{(0,\\pi )} \\gg m_{(\\pi ,0)}$ .", "For large $W$ (e.g., $W$ =40$a$ ), this model has two pairs of counter-propagating edge modes centered at edge momenta $k=0$ and $\\pi $ , see Fig.", "REF (c).", "Due to the difference in the masses the decay length of these two edge modes satisfy $L_0\\propto m_{(0,\\pi )}^{-1}<< L_{\\pi }\\propto m_{(\\pi ,0)}^{-1}$ as Fig.", "REF (b) shows.", "Reducing $W$ makes states from opposite edges hybridize.", "However if $L_0 < W << L_\\pi $ , only the $k=\\pi $ edge states hybridize significantly, causing them to open a gap (Fig.", "REF (d)).", "After that only one pair of edge states remain gapless and the system is indistinguishable from a strong topological insulator.", "It is well known that bulk Bi is topologically trivial but Sb is topologically non-trivial.", "The difference between the bulk band structure of these two materials lies in the fact that gap inversion occurs at three time reversal invariant momenta, the L points, for Bi but not Sb.", "It is also known that the spin orbit induced gap at L points in Bi is very small ($\\sim 15$ meV) [20] while the gap is large ($\\sim 0.72-0.81$ eV) at $\\Gamma $ point [21], [22].", "They play the roles of the small and large Dirac masses respectively in the simple model above.", "To further check the existence of topological thin film state discussed above we have performed first-principle calculations for a superlattice made up of Bi slabs of varying thickness stacked in the (111) direction (we also can tune the coupling between adjacent slabs by adjusting the vacuum thickness between slabs).", "The result show that for both 12 BL and 15 BL Bi(111) slabs the artificial 3D system is topologically non-trivial because the gap at the L points becomes de-inverted.", "The detailed results will be presented elsewhere [23].", "In summary we have carried out extensive quantum transport measurements and performed simple model as well as first-principle calculations for Bi thin films.", "All of our results suggest that when made thin enough Bi becomes topologically non-trivial.", "We explain this result in terms of a simple physical picture.", "We stress that what we discuss here is entirely different from the predicted two dimensional topological insulator when Bi film is only one bilayer thick.", "Our result should point out a new direction in the search of topologically interesting materials.", "The authors thank S. Blügel, G. Bihlmayer and I. Aguilera for their valuable discussions.", "XFJ is supported by MOST (No.", "2011CB921802) and NSFC (No.", "11374057).", "DQ is supported by NSFC ( No.", "11274228).", "JF is supported by NBRPC(2013CB921900).", "DHL and FdJ are supported by DOE Office of Basic Energy Sciences, Division of Materials Science, Material theory Program, grant DE-AC02-05CH11231." ] ]	1403.0066
[ [ "Distributed Cooperative Localization in Wireless Sensor Networks without\n NLOS Identification" ], [ "Abstract In this paper, a 2-stage robust distributed algorithm is proposed for cooperative sensor network localization using time of arrival (TOA) data without identification of non-line of sight (NLOS) links.", "In the first stage, to overcome the effect of outliers, a convex relaxation of the Huber loss function is applied so that by using iterative optimization techniques, good estimates of the true sensor locations can be obtained.", "In the second stage, the original (non-relaxed) Huber cost function is further optimized to obtain refined location estimates based on those obtained in the first stage.", "In both stages, a simple gradient descent technique is used to carry out the optimization.", "Through simulations and real data analysis, it is shown that the proposed convex relaxation generally achieves a lower root mean squared error (RMSE) compared to other convex relaxation techniques in the literature.", "Also by doing the second stage, the position estimates are improved and we can achieve an RMSE close to that of the other distributed algorithms which know \\textit{a priori} which links are in NLOS." ], [ "Introduction", "Wireless sensor network (WSN) localization has received great attention in recent years due to the large number of applications requiring accurate location information [1].", "Since the global positioning system (GPS) is not a reliable technology for localization of sensors in indoor place or dense urban areas, range measurements between pairs of neighbouring nodes, including sensors and anchors, may be used for the purpose of localization.", "Among the different technologies, ultra-wide band (UWB) signalling can yield accurate time of arrival (TOA) measurements in line of sight (LOS) scenarios, from which the range information can be extracted.", "The localization based on these measurements can be carried out for the entire network in a centralized or a distributed fashion.", "Among the popular centralized algorithms are semi-definite programming (SDP) [2] and second-order cone programming (SOCP) [3] convex relaxations.", "Distributed algorithms have also been proposed, including distributed SOCP [4], the iterative parallel projection method (IPPM) [5] and other localization approaches that alternate between convex and non convex optimization problems [6] [7].", "However, these approaches only consider the case where the pairwise range measurements are made under LOS condition.", "In practice, LOS measurements are limited and many links will face a non-line of sight (NLOS) condition.", "Due to the NLOS, the TOA measurements become positively biased [8], and consequently, the aforementioned techniques perform unsatisfactorily if the NLOS effects are not mitigated properly.", "In many of the localization techniques, the NLOS links have to be identified first.", "Various methods have been proposed for the identification of NLOS links in non-cooperative networks (see in [8] and the references therein), among which several are especially tailored for UWB applications [9], [10].", "After detecting the NLOS links through a suitable technique, the effect of NLOS error can be mitigated using different optimization techniques.", "A summary of the non-cooperative TOA-based NLOS mitigation techniques is given in [8].", "For cooperative localization, extension of the centralized SDP relaxation and the distributed IPPM to NLOS scenarios are considered in [11] and [12], respectively.", "NLOS identification remains however challenging for a large WSN with several pairwise measurements.", "Therefore, in many applications, it is impractical to assume that all the NLOS links can be identified accurately.", "In [13], an SDP relaxation is considered for non-cooperative localization in NLOS without prior detection of NLOS links.", "Although this technique is robust against NLOS errors, the computations need to be done centrally, thus it can not scale with the size of the network, and its extension to a distributed implementation remains an open topic for further study.", "In [14], a distributed cooperative projection onto convex sets (POCS) is employed to estimate the location of sensors, which is shown to be robust against NLOS errors.", "However, if only a portion of the measurements are affected by NLOS errors, the performance of POCS is far from being optimal.", "Another approach for robust estimation against outliers without prior outlier detection is to use Huber loss function, which offers a trade-off between $l_1$ and $l_2$ norm minimizations [15].", "In contrast to POCS, localization based on Huber cost function can achieve a good result only if it is well initialized and if a moderate or small portion of the measurements are contaminated by large errors, otherwise it may not necessarily give a good estimate.", "In this paper, to obtain accurate sensor location estimates under various NLOS scenarios, we propose a 2-stage algorithm based on Huber M-estimation for distributed cooperative localization in the presence of unidentified NLOS links.", "In the first stage, a convex relaxation similar to that in [6] is applied on the Huber cost function and sensor locations are then estimated iteratively.", "Since the performance may not be close to optimal when the ratios of NLOS to LOS links is low, in the second stage, the original Huber cost function is minimized iteratively with a suitable choice of tuning parameter.", "For the iterative optimization in both stages, we use a simple gradient descent technique since it can be easily implemented in a distributed manner.", "Through simulations, we first show that the proposed convex relaxation gives a robust estimate in different NLOS scenarios.", "Furthermore, we show that the position estimates are generally improved in the second stage as we minimize the original Huber cost function.", "The robustness of our algorithm to outliers is also evaluated by using a real set of sensor measurements obtained by the measurement campaign in [16].", "We consider a sensor network consisting of $N$ sensor nodes with unknown locations denoted by ${x}_i \\in \\mathbb {R}^2$ , for $i=1,\\ldots ,N$ , and $M$ anchors with known locations ${x}_i \\in \\mathbb {R}^2$ , for $i = N+1, \\ldots , N+M$ .", "We define ${\\cal S}$ as the set of all index pairs $(i,j)$ of all the neighbouring nodes that can communicate with each other, where we let $i<j$ to avoid repetition.", "We also define ${\\cal S}_i$ as the index set of all the neighbouring nodes of the $i$ -th sensor.", "We assume that a range measurement is obtained between each pair of neighbouring nodes with $(i,j) \\in {\\cal S}$ .", "For accurate TOA-based ranging, we either assume that the nodes are precisely synchronized over the network, or that the two-way ranging (TWR) protocol is employed to remove the clock error in the TOA measurements [17].", "The complete set of range measurements are modelled by the following equations: $r_{ij}={\\left\\lbrace \\begin{array}{ll}d_{ij} + n_{ij}, \\quad &(i,j) \\in {\\cal L} \\\\d_{ij} + b_{ij} + n_{ij}, \\quad &(i,j) \\in {\\cal N}\\end{array}\\right.", "}$ where we define $d_{ij}= \\Vert {x}_i-{x}_j \\Vert $ , along with the sets ${\\cal L} &= \\lbrace (i,j) \\in {\\cal S}: \\text{LOS link between $i$-th and $j$-th node} \\rbrace \\nonumber \\\\{\\cal N} &= \\lbrace (i,j) \\in {\\cal S}: \\text{NLOS link between $i$-th and $j$-th node} \\rbrace \\nonumber $ so that ${\\cal S}= {\\cal L} \\cup {\\cal N} $ .", "The measurement noise terms $n_{ij}$ are independent and identically distributed random variables with zero-mean and known variance $\\sigma _n^2$ .", "The terms $b_{ij}$ are the NLOS biases between the corresponding pair of nodes indexed by $(i,j) \\in {\\mathcal {N}}$ .", "In the literature, the NLOS biases have been modelled differently depending on the environment and wireless channel, for instance, exponential [9] or uniform [10] distributions are generally used.", "In this work, however, we do not assume any specific knowledge about the statistics of $b_{ij}$ , such as its mean and variance.", "Furthermore, no a priori knowledge about the status of a link, i.e., whether it is NLOS or LOS, is assumed to be available." ], [ "Problem Formulation", "The aim is to find estimates of the $N$ unknown sensor positions ${x}_i$ , denoted as $\\hat{{x}}_i$ , such that the corresponding errors in the estimated locations are small, ideally unbiased with small variances.", "Throughout this work we denote ${X}=[{x}_1,{x}_2,\\ldots ,{x}_N] \\in \\mathbb {R}^{2\\times N}$ as the unknown to be estimated.", "If there are no NLOS biases, due to the zero-mean Gaussian noise assumption, it can be easily shown that the maximum likelihood estimation (MLE) is equivalent to the $l_2$ norm minimization, so the cost function to be minimized is $f({X}) = \\sum _{(i,j)\\in {\\cal S} }^{} \\Big ( \\Vert {x}_j-{x}_i \\Vert -r_{ij} \\Big ) ^2$ which is a non-convex nonlinear least squares (NLS) problem with respect to ${X}$ [2].", "Since the NLOS biases exist in some measurements but cannot be identified, using $l_2$ norm minimization might not yield robust estimates.", "In the presence of outliers, Huber cost function provides a suitable replacement for $l_2$ norm minimization, by interpolating between $l_2$ and $l_1$ norm minimizations.", "Therefore, instead of (REF ) it is preferred to minimize $g( {{X}} ) = \\sum _{(i,j) \\in {\\cal S} } \\rho \\Big ( \\Vert {x}_i - {x}_j \\Vert - r_{ij} \\Big )$ where $\\rho (\\cdot )$ is the continuous and differentiable Huber function: $\\rho (u_{ij})={\\left\\lbrace \\begin{array}{ll}u_{ij}^2 , \\quad &\| u_{ij} \| < K \\\\2K\| u_{ij} \| - K^2 , \\quad &\|u_{ij}\| \\ge K \\\\\\end{array}\\right.", "}$ The argument $u_{ij}= \\Vert {x}_i - {x}_j \\Vert - r_{ij}$ , and $K$ is a fixed parameter which is chosen to be proportional to $\\sigma _n$ , e.g., $K=\\alpha \\sigma _n$ and $1.5 \\le \\alpha \\le 2$ [15].", "Although the Huber cost function is convex with respect to its argument, due to the non-convex nature of the range measurements with respect to the position coordinates, the final function is non-convex and hence initialization is crucial.", "Furthermore, the Huber M-estimation can perform well only if a small or moderate portion of the measurements are affected by outliers, otherwise it may not achieve a good estimation result.", "Therefore, in the following we propose a 2-stage algorithm that is robust in any NLOS scenario." ], [ "Robust Distributed Algorithm", "In this section, we first propose a convex relaxation of the Huber cost function.", "After converging to some stationary points, we then try to minimize the Huber cost function." ], [ "Stage I: Convex Relaxation", "A convex relaxation of the nonlinear least square problem in (REF ) has been proposed in [6] in the form of $\\tilde{f}({X}) = \\sum _{(i,j)\\in {\\cal S} }^{} \\Big ( (\\Vert {x}_j-{x}_i \\Vert -r_{ij})_{+} \\Big ) ^2 \\\\$ where $(\\Vert {x}_j-{x}_i \\Vert -r_{ij})_{+} = {\\left\\lbrace \\begin{array}{ll}0 , & \\Vert {x}_j-{x}_i \\Vert \\le r_{ij} \\\\\\Vert {x}_j-{x}_i \\Vert -r_{ij}, & \\Vert {x}_j-{x}_i \\Vert > r_{ij} \\\\\\end{array}\\right.", "}$ Further explanations about the convexity of this cost function are given in [7].", "The concept of this relaxation is similar to POCS proposed first in [18] and considered for cooperative localization in [14].", "Here, however, we propose to minimize $\\tilde{g}( {{X}} ) = \\sum _{(i,j) \\in {\\cal S} } \\tilde{\\rho } \\Big ( \\Vert {x}_i - {x}_j \\Vert - r_{ij} \\Big )$ where $\\tilde{\\rho }(\\cdot )$ is the convex relaxation of the Huber function with respect to ${X}$ , which is defined as $\\tilde{\\rho } (u_{ij})={\\left\\lbrace \\begin{array}{ll}0, \\quad & \\Vert {x}_i - {x}_j \\Vert \\le r_{ij} \\\\u_{ij}^2, \\quad & r_{ij} \\!<\\!", "\\Vert {x}_i - {x}_j \\Vert \\!<\\!", "r_{ij} + K_1 \\\\2K_1 u_{ij} - K_1^2, \\quad & \\Vert {x}_i - {x}_j \\Vert \\ge r_{ij} + K_1 \\\\\\end{array}\\right.", "}$ where $K_1=\\alpha _1 \\sigma _n$ is the parameter of the Huber loss function.", "The geometric interpretation of the original and relaxed Huber cost functions are illustrated in Fig.", "REF for the area between two nodes.", "Simulation result shows that, in many cases, this convex relaxation is more robust against large negative errors and gives a lower MSE for the network compared to the one in (REF ) or the cooperative POCS [14].", "The iterative gradient descent method for updating the position estimates can be stated at each node as ${x}_i^{(l+1)} = {x}_i^{(l)} - \\mu _1\\sum _{j \\in {\\cal S}_i} \\frac{\\partial \\tilde{\\rho }(u_{ij})}{\\partial {x}_i^{(l)}} , \\quad i = 1,\\ldots , N$ where $\\mu _1$ is a suitable step size and for every $j \\in {\\cal S}_i$ $\\frac{\\partial \\tilde{\\rho }(u_{ij})}{\\partial {x}_i^{(l)}} ={\\left\\lbrace \\begin{array}{ll}0 , &\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert \\le r_{ij} \\nonumber \\\\2u_{ij}^{(l)} \\frac{{x}_i^{(l)}-{x}_j^{(l)}}{\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert } , &r_{ij} \\!<\\!", "\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert \\!<\\!", "r_{ij} + K_1 \\nonumber \\\\2K_1 \\frac{{x}_i^{(l)}-{x}_j^{(l)}}{\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert } , &\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert \\ge r_{ij} + K_1 \\nonumber \\\\\\end{array}\\right.", "}$ After updating its location estimate using (REF ), each sensor sends the result to its neighbours.", "Therefore, every sensor uses the current estimate about its own position, the known positions of its neighbouring anchors, and the updated positions of its neighbouring sensors to find a new estimate of its position.", "After convergence, we find an estimate of ${X}$ which is the global minimum of the cost function in (REF ).", "The stopping criteria are either the maximum number of iterations or when the estimates of sensor positions at two consecutive iterations are smaller than a threshold, i.e., $\\Vert {x}_i^{(l+1)} - {x}_i^{(l)} \\Vert \\le \\nu _1$ for all $i=1,\\ldots ,N$ .", "The position estimates obtained at this stage are close to optimal if most of the measurements are NLOS.", "However, in other scenarios, these estimates may not be close to optimal, and minimizing (REF ) will give a better estimate as will be explained in the sequel." ], [ "Stage II: Position Refinement", "At this stage, we try to minimize the original Huber cost function in (REF ).", "The iterative gradient decent steps at each sensor node ${x}_i$ is ${x}_i^{(l+1)} = {x}_i^{(l)} - \\mu _2\\sum _{j\\in {\\cal S}_i} \\frac{\\partial \\rho (u_{ij}) }{\\partial {x}_i^{(l)}} , \\quad i = 1,\\ldots , N$ where $\\mu _2$ is a suitable step size and for every $j \\in {\\cal S}_i$ $\\frac{\\partial \\rho (u_{ij}) }{\\partial {x}_i^{(l)}} ={\\left\\lbrace \\begin{array}{ll}2u_{ij}^{(l)} \\frac{{x}_i^{(l)}-{x}_j^{(l)}}{\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert } ,\\quad \\Big \| \\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert - r_{ij} \\Big \| < K_2\\\\2K_2 \\frac{ {x}_i^{(l)} - {x}_j^{(l)} }{\\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert } , \\quad \\Big \| \\Vert {x}_i^{(l)} - {x}_j^{(l)} \\Vert - r_{ij} \\Big \| \\ge K_2\\end{array}\\right.}", "\\nonumber $ and $K_2=\\alpha _2 \\sigma _n$ is the parameter of the Huber cost function.", "The algorithm continues iteratively for a limited number of iterations similar to the first stage until convergence, i.e., $\\Vert {x}_i^{(l+1)}- {x}_i^{(l)} \\Vert \\le \\nu _2$ , for all $i=1,\\ldots , N$ .", "Figure: CDF of different methods and the proposed Huber relaxation: (a) P 𝒩 =0.95P_{\\cal N} =0.95; (b) P 𝒩 =0.5P_{\\cal N}=0.5; (c) P 𝒩 =0.05P_{\\cal N} =0.05.Figure: CDF of the proposed 2-stage algorithm and the IPPM in : (a) P 𝒩 =0.95P_{\\cal N} =0.95; (b) P 𝒩 =0.5P_{\\cal N}=0.5; (c) P 𝒩 =0.05P_{\\cal N} =0.05.Selecting a suitable $K_2$ is very important at this stage as it enables a trade off between robustness and accuracy.", "If the ratio of NLOS link is high, then selecting $K_2$ as done usually for Huber M-estimation, i.e., $1.5 \\sigma _n \\le K_2 \\le 2 \\sigma _n$ , might even result in deterioration of the position estimates.", "Thus, in this scenario, it is preferred to keep $\\alpha _2$ very small, so the second stage does not change the position estimates obtained in the first stage.", "On the other hand, if the ratio of the NLOS to LOS measurements is low, then the second stage can improve the positioning performance noticeably by selecting $1.5 \\le \\alpha _2 \\le 2$ .", "We note that the estimation performance is still improved when a smaller value of $\\alpha _2$ is chosen.", "Therefore, if we have an a priori estimate of the ratio of the NLOS to LOS measurements or the probability of a link being NLOS, then we can select $K_2$ according to the discussion above.", "However, if such information is not available, then we should select small $\\alpha _2$ , e.g., $\\alpha _2 = 0.1$ , to achieve robust estimation result in every scenario.", "Although the second stage might improve the localization accuracy, it requires a number of iterations to converge, which increases the computational cost and communication load over the network.", "Therefore, by tuning the stopping criteria in both stages of our algorithm we can have a trade off between computational cost and localization accuracy." ], [ "Simulation Results", "In this part, the performance of the proposed method is evaluated through simulations.", "We consider a network of $M=4$ anchors and $N=50$ sensors located on a 2D space.", "The sensors are randomly distributed on the plane while the anchors are at fixed locations ${x}_{N+1}=[0 , 0]^T$ , ${x}_{N+2}=[10, 0 ]^T$ , ${x}_{N+3}=[10 , 10]^T$ , and ${x}_{N+4}=[0 , 10]^T$ , where the units are in meters.", "The range measurements were generated according to the model in (REF ) with $\\sigma _n=0.5$ m and the NLOS bias is modelled as an exponential random variable with parameter $\\gamma =10$ m. The Monte Carlo (MC) simulations are done under 500 runs.", "As a performance metric, the network-average localization error for each noise realization, i.e., $ \\sqrt{ \\sum _{i=1}^{N} \\Vert \\hat{{x}}_i - {x}_i \\Vert ^2/N }$ is evaluated.", "We first consider the proposed convex relaxation and run this algorithm with $\\mu _1=0.04$ and $K_1=2\\sigma _n$ .", "We compare the proposed technique with the relaxation of the NLS in [6] with similar parameters and the same number of iterations, denoted by NLS relaxed.", "We also apply the mentioned iterative technique with the same parameters and iteration number on the original Huber cost function and denote it by Huber.", "Furthermore, we consider the cooperative POCS with parameter $\\lambda ^{l}=0$ , thus it becomes almost similar to the IPPM in [5], except that the projection is only implemented when $\\Vert {x}_i^{(l)} - {x}_j^{(l)}\\Vert \\ge r_{ij}$ .", "The initial sensor positions for all algorithms are selected to have a Gaussian distribution with mean equal to the true sensor positions and standard deviation of 10 meters.", "We define $P_{\\cal N}$ as the probability of a link being NLOS.", "We now consider three scenarios where the probability that a link is in NLOS is chosen to be $P_{\\cal N}=0.95$ , $P_{\\cal N}=0.5$ , and $P_{\\cal N}=0.05$ .", "In Fig.", "REF , the CDF of positioning error for different algorithms under various NLOS contamination level is shown after 50 iterations.", "As observed in Fig.", "REF , the relaxation of Huber cost function is slightly better than the relaxation of NLS, and it has almost the same performance as POCS.", "The original Huber cost function does not achieve a good result due to the lack of convexity and poor initialization.", "To do further position refinement, we also simulate the second phase of our algorithm with $\\mu _2=0.01$ and $K_2=0.1 \\sigma _n$ .", "For the initialization, we use the position estimates obtained at the first stage by our proposed algorithm using convex relaxation of Huber cost function.", "To have a lower bound on the performance of our algorithm, we implement the IPPM proposed in [12] with the knowledge of perfect NLOS identification and denote it by IPPM NLOS.", "Since the IPPM algorithm may not necessarily converge to a good solution because of lack of convexity, we use the position estimates obtained by cooperative POCS as initial points.", "The CDF of the error of our 2-stage algorithm is illustrated in Fig.", "REF along with the IPPM with prior NLOS identification, where in both, 50 iterations are considered.", "The CDF of the error of the proposed convex relaxation shown in Fig.", "REF is also plotted in Fig.", "REF .", "The results show that when the ratio of the NLOS to LOS measurements is high, the second stage of the algorithm might not improve the localization performance necessarily.", "However, when the ratio of the NLOS to LOS links decreases, the second stage can improve the estimates obtained in the first stage distinguishably.", "The performance of the proposed 2-stage algorithm is close to IPPM NLOS, which is based on perfect NLOS identification." ], [ "Experimental Results", "In this part, we consider localization of sensors using real data obtained by the measurement campaign reported in [16].", "The environment was an indoor office and there were 44 node locations where the transmitter and receiver were used at each location and pairwise range measurements were obtained.", "We consider four nodes in the corner as anchor nodes with perfect location information and the other 40 nodes as the sensors with unknown locations.", "Due to the scatterers and NLOS in the office, almost all of the measurements are affected by large positive errors as mentioned in [16].", "It is mentioned that the average amount of error is also calculated for these measurements, therefore, by subtracting that quantity from the measurements, a less unbiased set of measurement is obtained.", "To evaluate the performance of our algorithm in different conditions we consider these scenarios: The raw measurements are considered, hence many of the measurements have positive errors, i.e., $P_{\\cal N}$ is large.", "The positive bias is subtracted from half of the measurements randomly, hence $P_{\\cal N}$ is moderate.", "The average bias is subtracted from all the raw measurements, thus $P_{\\cal N} $ is small.", "Using the unbiased measurements, the standard deviation of measurement noise is estimated roughly to be $\\sigma _n=1$ m. By applying the iterative gradient descent technique on the proposed convex Huber cost function with $\\mu _1=0.04$ and $K_1=2\\sigma _n$ , an estimate of the positions of sensors are obtained iteratively for 50 iterations.", "The position estimates are also refined in the second stage with $K_2= 0.1 \\sigma _n $ and $\\mu _2=0.01$ for 50 iterations.", "The final estimates at the end of each stage of our algorithm and the estimates obtained by cooperative POCS are shown along with the true sensor positions in Fig.", "REF .", "The CDF of the positioning error in different NLOS scenarios are also illustrated in Fig.", "REF by running 500 MC trials.", "The results show that in general the relaxed Huber function achieves a better result compared to the other approaches.", "Moreover, the second stage of the algorithm noticeably improves the position estimates obtained in the first stage, especially when $P_{\\cal N}$ is small." ], [ "Conclusion", "A robust distributed cooperative localization technique has been proposed in this work.", "We first applied a convex relaxation on the Huber cost function and decent position estimates were obtained iteratively.", "In the second stage of our algorithm, by iteratively minimizing the Huber loss function, it was shown that further refinement of position estimates could be generally obtained.", "For iterative optimization in each stage, a gradient descent method was used.", "By testing real data set, the superiority of our algorithm was verified.", "We conclude that our 2-stage algorithm performs robustly against outliers; in particular it significantly outperforms other distributed techniques when the ratio of NLOS to LOS measurements is low." ], [ "Acknowledgment", "This work was supported by grants from the Natural Science and Engineering Research Council (NSERC) of Canada." ] ]	1403.0503
[ [ "A comprehensive scenario of the single crystal growth and doping\n dependence of resistivity and anisotropic upper critical fields in\n (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ ($0.22 \\leq x \\leq 1$)" ], [ "Abstract Large high-quality single crystals of hole-doped iron-based superconductor (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ were grown over a broad composition range $0.22 \\leq x \\leq 1$ by inverted temperature gradient method.", "We found that high soaking temperature, fast cooling rate, and adjusted temperature window of the growth are necessary to obtain single crystals of heavily K doped crystals (0.65$\\leq x \\leq$ 0.92) with narrow compositional distributions as revealed by sharp superconducting transitions in magnetization measurements and close to 100% superconducting volume fraction.", "The crystals were extensively characterized by x-ray and compositional analysis, revealing monotonic evolution of the $c$-axis crystal lattice parameter with K substitution.", "Quantitative measurements of the temperature-dependent in-plane resistivity, $\\rho(T)$ found doping-independent, constant within error bars, resistivity at room temperature, $\\rho(300K)$, in sharp contrast with significant doping dependence in electron and isovalent substituted BaFe$_2$As$_2$ based compositions.", "The shape of the temperature dependent resistivity, $\\rho(T)$, shows systematic doping-evolution, being close to $T^2$ in overdoped and revealing significant contribution of the $T$-linear component at optimum doping.", "The slope of the upper critical field, $d H_{c2}/dT$, scales linearly with $T_c$ for both $H\\parallel c$, $ H_{c2,c}$, and $H \\parallel ab$, $H_{c2,ab}$.", "The anisotropy of the upper critical field, $\\gamma \\equiv H_{c2,ab} / H_{c2,c}$ determined near zero-field $T_c$ increases from $\\sim$2 to 4-5 with increasing K doping level from optimal $x \\sim$0.4 to strongly overdoped $x$=1." ], [ "Introduction", "Superconductivity in (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ (BaK122 in the following) with transition temperature $T_c$ as high as 38 K was found by Rotter et al.", "[1] very soon after discovery of high-temperature superconductivity in LaFeAs(O,F) by Hosono group [2].", "It was found later that superconductivity in BaFe$_2$ As$_2$ can be also induced by electron doping on partial substitution of Fe atoms with aliovalent Co [3] and Ni [4], by isovalent substitution of Ru atoms at Fe sites [5] and P atoms at As sites [6], or by application of pressure [7].", "In both families of compounds the superconductivity has maximum $T_c$ close to a point where the antiferromagnetic order of the parent compounds BaFe$_2$ As$_2$ and LaFeAsO, respectively, is suppressed, prompting intense discussion about the relation of superconductivity and magnetism and potentially magnetic mechanism of superconducting pairing [8], [9], [10], [11], [12].", "A characteristic feature of the scenario, suggested for magnetically mediated superconductivity [13], [14], [15], is systematic doping evolution of all electronic properties, in particular of electrical resistivity.", "Superconducting $T_c$ has maximum at a point where line of the second order magnetic transition goes to $T=0$ (quantum critical point, QCP).", "Temperature dependent resistivity gradually transforms from $T^2$ expected in Fermi liquid theory of a metal away from QCP to $T$ -linear at the QCP.", "In the transformation range $\\rho (T)$ can be described with a second order polynomial, with the magnitude of $T$ -linear scaling with superconducting $T_c$ [12].", "In iron-based superconductors this scenario works very well in iso-electron doped BaP122 [16], [17], [18].", "Here maximum $T_c$ is indeed observed at $x$ =0.33, close to doping-tuned magnetic QCP, and signatures of QCP are found in both normal [16], [17], [18] and superconducting [19] states, with resistivity at optimal doping being $T$ -linear for both in-plane [16] and inter-plane [20] transport.", "Deviations from this scenario are not very pronounced in electron-doped BaCo122.", "Here maximum $T_c$ is observed close to a composition where $T_{N}(x)$ extrapolates to zero, though the actual line shows slope sign change on approaching $T$ =0 and reentrance of the tetragonal phase [21].", "The temperature-dependent in-plane resistivity is close to $T$ -linear at optimal doping and transforms to $T^2$ in the overdoped regime, while the inter-plane resistivity shows limited range of $T$ -linear dependence, terminated at high temperature by a broad crossover [22], [23], [24], [25] due to pseudogap.", "The resistivity anisotropy $\\gamma _{\\rho } \\equiv \\rho _c/\\rho _a$ scales with the anisotropy of the upper critical field $\\gamma _H \\equiv H_{c2,ab}/H_{c2,c}$ [22] with $\\gamma _{\\rho }=\\gamma _H^2$ .", "The $\\gamma _H(x)$ changes step-like between underdoped and overdoped regions of the dome [26], [27], due to Fermi surface topology change (Lifshits transition) [28].", "Contrary to the cases of iso-electron substitution and electron doping, no systematic studies of the temperature-dependent resistivity and anisotropic properties of hole-doped BaK122 system were reported so far.", "Studies were performed in the underdoped, $x<0.4$ [29], [30] compositions, for which high quality single crystals can be grown from FeAs flux [31], or in heavily overdoped range $x>0.76$ [32], where crystals were prepared from KAs flux [33].", "Crystals of BaK122 can be also grown from Sn flux [34], however, their properties are notably affected by Sn inclusions at sub-percent level and will not be discussed here.", "In BaK122 the superconductivity appears on sufficient suppression of antiferromagnetic order, for $x>\\approx 0.15$ , while magnetism is completely suppressed by $x \\approx $ 0.25 [35], [36], revealing a range of bulk coexistence.", "The doping edge of magnetism corresponds to $T_c\\sim $ 27 K [30], notably lower that the highest $T_c \\approx $ 38 K observed at optimal doping $x \\approx $ 0.4, away from concentration boundary of magnetism suppression.", "The $T_c(x)$ dependence for $x$ in the range 0.4 to 0.6 is nearly flat [37].", "The superconductivity is observed in the whole substitution range up to $x$ =1 with steady decrease of $T_c$ down to 3.7 K in the end member KFe$_2$ As$_2$ ($x$ =1).", "Broad crossover in the temperature dependent resistivity is observed in in-plane transport in single crystals of BaK122 at doping close to optimal [38], similar to pure stoichiometric KFe$_2$ As$_2$ (K122) [39], [40], [41], [42], [43].", "Explanation of the crossover was suggested as arising from multi-band effects [38], with contribution of two conductivity channels, as found in optical studies [44] with nearly temperature-independent and strongly temperature dependent resistivities, respectively.", "The maximum in $\\rho _a(T)$ of BaK122 was discussed by Gasparov et al.", "[45] as arising from phonon-assisted scattering between two Fermi-surface sheets.", "The information about the doping-evolution of the upper critical field in hole-doped BaK122 is scattered.", "Very high upper critical fields were reported for close to optimally doped compositions [46], [47], [45], in addition these compositions are characterized by rather small critical field anisotropy.", "In another doping regime, close to $x$ =1, very unusual behavior of the upper critical fields is found.", "In KFe$_2$ As$_2$ , the orbital $H_{c2}$ found in $H \\parallel c$ configuration, is close to $T$ -linear [40].", "The slope of the dependence does not depend on $T_c$ suppression with impurities [43].", "In configuration with magnetic field parallel to the plane, $H\\parallel ab$ , the upper critical field is Pauli limited, as suggested both by the difference in the shape of the phase diagram and quite sharp changes at $H_{c2}$ [48].", "Heat capacity study in $H \\parallel a$ configuration, however, had not found first order transition[49], but rather suggested multi-band Fulde-Ferrel-Larkin-Ovchinnikov (FFLO [50], [51]) state [52].", "In slightly less doped material with $x$ =0.93, ($T_c \\sim $ 8 K) hysteresis is observed in the field-tuned resistive transition curves in $H \\parallel ab$ configuration at temperatures below 1 K, which can be attributed to a first-order superconducting transition due to paramagnetic effect [53].", "More systematic studies of the anisotropic $H_{c2}$ in BaK122 system are desperately required.", "In this study, we report growth of high quality single crystals of (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ for all doping ranges ($0.22 \\le x \\le 1$ ) and report systematic study of their temperature-dependent resistivity and anisotropic upper critical fields.", "We found nearly doping independent resistivity value at high temperatures, which is in notable contrast to electron-doped BaCo122 [24] and iso-electron substituted BaP122 [16] materials.", "We find systematic evolution of the temperature dependent resistivity with doping and rapid decrease of residual resistivity towards $x$ =1.", "We also found that the slopes $ dH_{c2}/dT$ are proportional to $T_c$ for both $H \\parallel c$ and $H \\parallel ab$ configurations.", "The anisotropy $\\gamma \\equiv H_{c2,ab}/H_{c2,c}$ , increases from 2 to 4-5 with increasing K doping level.", "The doping dependence of anisotropy ratio might be linked with change of the topology of the Fermi surface and the evolution of the superconducting gap.", "We have previously described successful growth of the high quality single crystals of stoichiometric KFe$_2$ As$_2$ at $T$ =1157 K using KAs flux [43].", "One of the key elements of the growth technique was using a liquid-Sn sealing of alumina crucibles to suppress the evaporation of K and As.", "This technique allowed us to avoid use of quartz tubes in direct contact with K vapor and use of expensive sealed tantalum tubes.", "Analysis of the growth morphology in the case of KFe$_2$ As$_2$ [43] lead us to conclude that the crystals nucleate on the surface of the melt and grow by the reaction on the top surface of the crystal with K and As in the vapor phase.", "We were able to promote this reaction by developing an inverted-temperature-gradient method with the colder zone at the top of the crucible, as shown in Fig.", "REF (a).", "A temperature gap of 20 K was set between the top zone and the bottom zone.", "This method yielded higher quality crystals of KFe$_2$ As$_2$ with residual resistivity ratio of up to 3000 than obtained in traditional flux-method, as crystallization from the liquid top can expel impurity phases into the liquid during crystal growth.", "This method works very well for the growth of heavily K doped (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ single crystals.", "Small amount of Ba was added to the load with the ratio Ba:K:Fe:As=$y$ :5:2:6 ($y $ =0.1, 0.2, and 0.3) in the stating materials.", "The chemicals were weighed and loaded into an alumina crucible in a glove box under argon atmosphere.", "Because of use of higher soaking temperatures leading to higher vapor pressures Sn seal technique was not reliable enough, and we switched to tantalum tube sealing.", "The alumina crucibles were then sealed in a tantalum tube by arc welding.", "In Table REF we show the growth conditions of Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ single crystals.", "We started to grow heavily K doped crystals by following the same procedure that had worked well for the crystal growth of KFe$_2$ As$_2$ .", "For $y$ =0.1, we obtained single crystals with K doping level at around $x$ =0.90 using soaking temperature of 1193 K. The actual compositions of the crystals were determined by wavelength dispersive x-ray spectroscopy (WDS) electron-probe microanalysis.", "For $y$ =0.2 and 0.3, the single crystals obtained by cooling down from the soaking temperature of $T$ =1193 K display broad transitions, which suggests inhomogeneity of Ba and K distributions in the sample.", "We were able to improve sample quality by adjusting the composition of the starting load material and soaking temperatures, as shown in Table REF .", "We found that increase of the soaking temperature to $T$ =1273 K helps growth of the samples with $x$ =0.8 and 0.9 with sharp superconducting transition.", "The further increase of the soaking temperature up to 1323 K, leads to growth of the crystals showing multiple steps at the superconducting transition due to inhomogeneous K distribution.", "We found that higher soaking temperatures 1273K$\\le T \\le $ 1323 K and narrowed temperature window for crystal growth are similarly useful to grow the crystals within the doping range $0.6<x<0.9$ with sharp superconducting transition.", "Table: Growth conditions of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 single crystals.", "Soaking temperature corresponds to the set temperature of bottom zone, with the top zone temperature 20 K lower than the bottom zone.For the samples with K doping levels below $x$ =0.55, we turned to the FeAs flux method.", "The growth conditions can be found in Table REF .", "For the crystals within the optimal doping range ($0.3<x<0.5$ ), the growth using conditions as shown in Table REF yielded large and high quality crystals with sharp transition.", "Interestingly, to grow high quality underdoped crystals, a further increase of the soaking temperature to 1453 K and fast cooling rate of 2 K/h are needed.", "A series of large and high quality (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ single crystals ($0.22 \\le x \\le 1 $ ) with sizes up to $18 \\times 10\\times 1$ mm$^3$ , as shown in Fig.", "REF (b) for $x$ =0.39 and Fig.", "REF (c) for $x$ =0.92.", "In fact, the size of Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ single crystals was only limited by the size of alumina crucibles used." ], [ " Sample characterization", "XRD measurements were performed on a PANalytical MPD diffractometer using Co $K \\alpha $ radiation.", "The $K \\alpha 2$ radiation was removed with X'pert Highscore software.", "All BaK122 crystals are readily cleaved along the $ab$ plane, as shown in Figs.", "REF (b)-(c).", "The XRD patterns of BaK122 single crystals with $0.22 \\le x \\le 1$ are shown in Fig.", "REF .", "The traces of impurity phases close to the baseline are indicated by the asterisks, they are most likely caused by the flux inclusions.", "Figure REF (b) shows systematic shift of the (008) peak towards the lower angles with increasing K content.", "The $c$ -axis lattice parameter is estimated based on the (00$l$ ) diffractions and displayed as a function of K content in Fig.", "REF (c); it changes linearly with $x$ its values match well the results on polycrystalline samples [37].", "Figure: X-ray diffraction patterns of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 0.22≤x≤10.22 \\le x \\le 1 single crystals.", "The logarithmic plot reveals small amount of impurity phases indicated by the asterisks, which could result from the FeAs and KAs flux inclusions.", "(b) The (008) peak, seen in 61 ∘ <2Θ<67 ∘ ^{\\circ } <2 \\Theta <67 ^{\\circ } range, systematically shifts with increasing K doping level towards the low angles.", "(c) The cc lattice parameter changes linearly the K content xx.", "The dashed line is guide for eyes.Magnetic susceptibility $\\chi (T)$ was measured using PPMS Vibrating Sample Magnetometer (PPMS VSM, Quantum Design).", "Typical size of the single crystals used in magnetization measurements was 4$\\times $ 3$\\times $ 0.2 mm$^3$ , and their mass was $\\sim ~10$ mg. In-plane resistivity $\\rho _{a}$ was measured in four-probe configuration using Physical Property Measurement System (PPMS, Quantum Design).", "Samples were cleaved into bars with typical dimensions (1-2)$\\times $ 0(0.3-0.5)$\\times $ (0.02-0.05) mm$^3$ .", "Electrical contacts were made by soldering Ag wires using pure tin [54], [55] and had contact resistance typically in several $\\mu \\Omega $ range.", "Sample dimensions were measured using optical microscope with the accuracy of abou 10%.", "Quantitative characterization of resistivity was made on a big array of samples of each composition.", "Figure: (Color online) (a) Temperature dependence of the magnetic susceptibility χ(T)\\chi (T) of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 0.22≤x≤10.22 \\le x \\le 1 single crystals.", "Bulk superconducting transition temperature T c T_c was determined from the onset point of the rapid drop of χ(T)\\chi (T).", "(b) Doping phase diagram of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 as determined from magnetization measurements on single crystals 0.22≤x≤10.22 \\le x \\le 1.", "The superconducting transition temperature (red solid dots), T c (x)T_c(x), matches well that obtained on polycrystalline samples (blue dashes) , , .", "Solid line shows boundary of orthorhombic/antiferromagnetic phase from neutron scattering study on polycrystals , , .In Fig.", "REF we show the temperature dependence of magnetic susceptibility $\\chi (T)$ for BaK122 single crystals with $0.22 \\le x \\le 1$ .", "Sharp superconducting transition ($\\Delta T_c<$ 0.6 K) in magnetic susceptibility curves show high quality of crystals with $x$ =0.34, 0.39, 0.47, 0.53, 0.55, and 1.", "The transition width $\\Delta T_c$ was defined using 90% and 10% drop in $\\chi (T)$ of the full diamagnetic transition as the criterion.", "The samples with $x$ =0.82, 0.90, and 0.92 have $\\Delta T_c<$ 1 K. However, the samples with $x$ =0.65 and 0.80 have large $\\Delta T_c$ of 3 K and 5 K, respectively.", "As we mentioned in the Experimental section, we shifted the temperature windows and adjusted the starting load composition and materials to improve the sample quality and obtain sharper transitions.", "Using $T_c$ from magnetic susceptibility data of top panel of Fig.", "REF and $x$ values as obtained in WDS analysis, we constructed the doping phase diagram, as shown in bottom panel of Fig.", "REF .", "For reference we show the diagram as determined from measurements on high quality polycrystalline materials [36] and on high quality single crystals on the underdoped side [31].", "The three studies are in good agreement.", "We do not see any indications of the phase separation in our underdoped samples $x$ =0.22.", "Previous study of underdoped BaK122 samples grown from Sn flux with $x$ =0.28 found regions of antiferromagnetically (AF) ordered phase with size of 65 nm coexisting with nonmagnetic superconducting regions [56].", "Later study using three-dimensional (3D) atom probe tomography revealed that the separation is caused by inhomogeneous distributions of Ba and K elements [57], with a tendency for Ba and K atoms to form clusters.", "Thus we conclude that this problem is not characteristic of the growth technique we use.", "We do see, however, that strong inhomogeneity occurs during crystal growth of overdoped crystals ($0.65<x<0.8$ ).", "There is no intrinsic phase separation revealed for polycrystalline samples in this doping range.", "In our samples we do not see macroscopic inhomogeneity in WDS measurements with special resolution of about 1 $\\mu m$ .", "However, on finer scale two STM studies revealed ordered vortex lattice in single crystals of optimally doped BaK122 $x$ =0.40 [58] but a short-range hexagonal order (vortex glass phase) in single crystals of SrK122 $x=$ 0.25 [59].", "Song et al.", "suggested that mismatch between the size of the dopant K atom and of the host atoms Ba and Sr, Ba$^{2+}$ / K$^+$ and Sr$^{2+}$ / K$^+$ , respectively, causes dopant clustering, electronic inhomogeneity, and vortex glass phase in SrK122.", "K$^+$ ions should be less clustered in BaK122 than in SrK122 because ion size mismatch between K$^+$ and Ba$^{2+}$ is five times smaller than between K$^+$ and Ba$^{2+}$ [59].", "Thus one can expect that dopant (Ba$^{2+}$ / K$^+$ ) size mismatch diminishes for $x$ =0.50 doped sample, which is quite close to the optimally doped sample with $x$ =0.40.", "Detailed structure analysis [36] indicates that potassium substitution reduces the in-plane lattice parameters $a$ and $b$ and significantly increases the out-of-plane lattice parameter $c$ in BaK122 compounds, all measured at $T$ =1.7 K. The unit cell volume gradually decreases with increasing K content until $x\\approx $ 0.5, but slightly increases towards the end member KFe$_2$ As$_2$ .", "Although the evolution of the lattice parameters and of the unit cell volume does not provide us direct evidence about the lattice mismatch in the BaK122 compounds, the slight change of the unit cell volume for doping in the range 0.", "5$\\le x \\le $ 1 implies that Ba$^{2+}$ /K$^+$ ion clustering should be relatively easy to realize without too much disturbance in crystal structure in the overdoped samples.", "In other words, a broad distribution of K doping in the overdoped samples should not be challenged too much when considering weak lattice strains introduced by different K dopings.", "Therefore, it is important to search for suitable growth conditions to narrow K doping range when growing overdoped single crystals." ], [ " Doping evolution of the temperature-dependent resistivity ", "Temperature dependent in-plane resistivity $\\rho (T)$ of the samples with $x$ = 0.22 to 1.0 is shown in Fig.", "REF .", "The data are presented using normalized $\\rho (T)/\\rho (300K)$ plots and offset to avoid overlapping.", "The doping evolution of the actual resistivity values $\\rho (300K)$ shows significant scatter due to uncertainty of the geometric factors, which are strongly affected by hidden cracks in micacious crystals of iron pnictides [26], [22].", "Of note though that within statistical error, the resistivity $\\rho (300K)$ remains constant over the whole compositional range from heavily underdoped samples with $x$ =0.22 to heavily overdoped $x$ =1.0, which is distinctly different from electron doped BaCo122 [60], [24] and isoelectron substituted BaP122 [16], in which $\\rho (300K)$ decreases notably with doping.", "The first look at the temperature-dependent resistivity also does not show significant doping evolution.", "For all doping levels the $\\rho (T)$ curves show a broad crossover starting above 100 K and ending at around 200 K. The onset of this feature most clearly reveals itself as a maximum in the temperature-dependent resistivity derivative, see Fig.", "REF .", "The origin of the feature was discussed in terms of multi-band character of conductivity in which one of the bands has strongly temperature dependent contribution, while the other has nearly temperature independent conductivity [38], as contribution from phonon-assisted scattering between two Fermi-surface sheets [45] and as a feature associated with pseudogap, as suggested by its correlation with the maximum of the inter-plane transport $\\rho _c (T)$ in under-doped compositions [30], [24], [25].", "The position of the crossover does not change with doping, and since the Fermi surface topology reveals quite significant changes [61], the explanation of the maximum in term of special featu res of band structure [38], [45] is very unlikely.", "Figure: (Color online) (a) Top panel- fixed 40 to 60K range fit of the resistivity curves using second order polynomial ρ/ρ(300K)=α 0 +α 1 T+α 2 T 2 \\rho /\\rho (300K)=\\alpha _0+\\alpha _1T+\\alpha _2T^2, shown for selected dopings xx=0.22, 0.47, 0.65, 0.8, 1.", "The data are offset to avoid overlapping.", "Three panels at the bottom show doping evolution of the fit parameters α 0 \\alpha _0 (panel b), α 1 \\alpha _1 (panel c) and α 2 \\alpha _2 (panel d).Figure: (Color online) Top panel- fixed 40 to 60K range fit of the resistivity curves using power-law function ρ/ρ(300K)=ρ 0 +AT n \\rho /\\rho (300K)=\\rho _0+AT^n, shown for selected dopings xx=0.22, 0.47, 0.65, 0.8, 1.", "The data are offset to avoid overlapping.", "Bottom panel shows evolution of the power-law exponent nn with doping for fits over four different temperature ranges, 40 to 60 K as shown in top panel (red solid circles), 40 to 70 K (black up-triangles), 40 to 80 K ((blue down-triangles) and T c T_c to 60 K (magenta open circles).Figure: (Color online) Top panel- normalized resistivity curves ρ/ρ(300K)\\rho /\\rho (300K) plotted vs T 2 T^2 for all doping levels studied, x=x=0.22, 0.34, 0.39, 0.47, 0.53, 0.55, 0.65, 0.80, 0.82, 0.90, 0.92, 1.0.", "Bottom panel shows data over narrower temperature range in heavily overdoped compositions xx=0.80, 0.82, 0.90, 0.92, 1.0.", "The data does not show any significant dependence of the slope (proportional to T 2 T^2 coefficient α 2 \\alpha _2 even at low temperatures.At temperatures lower than 100 K, however, temperature-dependent resistivity shows some evolution.", "Because of high temperature of the superconducting transition, we cannot make correct analysis of the functional form of $\\rho (T)$ in the $T \\rightarrow 0$ limit over the whole dome.", "However, for the sake of comparison, we fitted the curves in a narrow range from 40 to 60 K, which was fixed for all compositions.", "These fits were done two ways.", "The first approach was using second order polynomial function, $\\rho (T)/\\rho (300K)=\\alpha _0 +\\alpha _1T+ \\alpha _2T^2$ , similar to the fit used by Doiron-Leyraud et al.", "[62] for electron-doped BaCo122.", "In the top panel of Fig.", "REF we show the fits over the range 40 to 60 K for $\\rho (T)$ curves for representative doping levels, three bottom panels show doping evolution of the fit parameters $\\alpha _0$ , $\\alpha _1$ and $\\alpha _2$ .", "This analysis reveals clearly that the dependence has highest linear contribution at $x$ =0.35 and 0.39, and that the $T^2$ contribution is minimum at $x$ =0.39, coinciding with maximum $T_c$ position but away from the doping border of the antiferromagnetic state at $x$ =0.26.", "The second approach was fitting the data using a power-law function, $\\rho /\\rho (300K)=\\rho _0+AT^n$ , as shown for selected compositions in Fig.", "REF .", "This approach is similar to the approach used by Shen et al.", "[29], however in their case the fitting range was extending to 80 K. For the sake of comparison, we did power-law analysis for the temperature ranges 40 to 70 K and 40 to 80 K, and from above $T_c$ to 60 K. The results of these fittings are shown in the bottom panel of Fig.", "REF .", "It can be seen that all ways of analysis find largest deviations from Fermi-liquid $T^2$ dependence at $x$ =0.39, which corresponds to a leading edge of maximum $T_c$ plateau of the $T_c(x)$ dome.", "Since $T_c(x)$ function is flat in 0.34 to 0.56 range, while both $T$ -linear contribution in the polynomial analysis, Fig.", "REF (c), and power-law exponent $n$ peak at $x$ =0.39, we conclude that $T_c$ and the amplitude of $T$ -linear contribution do not scale in BaK122, contrary to BaCo122 [12].", "Another interesting point is that exponent $n$ we observe in sample $x$ =0.39 is close to 1.5.", "This is notably higher than the lowest exponent $n$ =1.1 found in previous study [29].", "To further check the link between $T$ -linear contribution and maximum $T_c$ , further studies in high magnetic fields may be necessary.", "An interesting feature of these fits is that the residual resistivity takes negative values for most of the compositions.", "This fact is suggestive that at lower temperatures the $\\rho (T)$ curves should develop significant positive curvature, as is in fact observed for heavier doped compositions, in which broader temperature range can be studied.", "It also suggests that most of our samples have quite high residual resistivity ratio in $T \\rightarrow 0$ limit.", "On the other hand, the $T^2$ coefficient as determined from the polynomial fit for the range 40 to 60 K gradually increases towards $x$ =1.", "Since $T_c$ drops significantly in this range, we are able to make an analysis at lower temperatures.", "In Fig.", "REF we plot $\\rho (T)$ data for all samples using a $T^2$ plot, bottom panel shows expanded view for heavily overdoped samples.", "When plotted this way, the plots become linear right above $T_c$ , and the slopes of the curves do not show any noticeable doping evolution beyond error bars.", "This observation suggests that for all doping levels there is significant and non-critical $T^2$ coefficient, and indeed several contributions to conductivity are needed for correct account of its doping evolution." ], [ " Anisotropic upper critical fields", "The anisotropy of the upper critical field $\\gamma _H \\equiv \\frac{H_{c2ab}}{H_{c2c}}$ presents important information about the anisotropy of the electrical conductivity, $\\gamma _{\\rho } \\equiv \\frac{\\rho _c}{\\rho _a}$ .", "In a temperature range close to zero-field $T_c$ the two anisotropies are related as $\\gamma _H^2 = \\gamma _{\\rho }$ , a relation which was verified semi-quantitatively in KFe$_2$ As$_2$ [40].", "The angular dependent $H_{c2}(\\Theta )$ was also studied systematically in BaK122 with $x$ =0.92 [53], in which the authors found strong deviations from $\\cos (\\Theta )$ dependence expected in orbital limit [63].", "Scattered in $x$ measurements of $\\gamma _H$ were undertaken on samples close to optimal doping grown from Sn flux [34], [64], [45], [46] and from FeAs flux [65], [47].", "Here we study evolution of the $\\gamma _H (x)$ in BaK122 from resistive $H_{c2}$ measurements.", "Figure: (Color online) Temperature-dependent resistivity in single crystal of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 with xx=0.39 in the vicinity of the superconducting transition.", "The onset T c,onset T_{c,onset} of the transition is defined at the crossing point of the linear fits of the ρ(T)\\rho (T) in the normal state above T c T_c and at the sharp transition slope.", "The offset T c T_c corresponds to the extrapolation of the steep transition slope to zero resistance.In Fig.", "REF we show zoom of the $\\rho (T)$ curve in the vicinity of the superconducting transition in sample with $x$ =0.39.", "Here we show how we defined different criteria used to determine $T_c(H)$ dependence.", "We analyzed resistivity data by linear extrapolation of $\\rho (T)$ curves at the transition and above the transition.", "The onset $T_{c,onset}$ of the transition is defined at the crossing point of these linear fits.", "The offset $T_c$ corresponds to the crossing of the steep transition line with $\\rho =0$ line.", "Figure: (Color online)Temperature dependence of in-plane resistivity in single crystals of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 with xx=0.22 (a), 0.34 (b), 0.39 (c) and 0.47 (d) in magnetic fields H∥cH \\parallel c (top panels) and H∥abH \\parallel ab (middle panel) with magnetic fields (right to left) 0, 0.5, 1, 2, 3, ..., 9 T. Bottom panels show H c2 (T)H_{c2}(T) for two field orientations H∥cH \\parallel c (solid symbols) and H∥abH \\parallel ab (open symbols) as determined using onset (black squares) and offset (red circles) resistive transition criteria, see Fig.", ".Figure: (Color online) Temperature dependence of in-plane resistivity in single crystals of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 with xx=0.53 (a), 0.55 (b), 0.65 (c) and 0.80 (d) in magnetic fields H∥cH \\parallel c (top panels) and H∥abH \\parallel ab (middle panel) with magnetic fields (right to left) 0, 0.5, 1, 2, 3, ..., 9 T. Bottom panels show H c2 (T)H_{c2}(T) for two field orientations H∥cH \\parallel c (solid symbols) and H∥abH \\parallel ab (open symbols) as determined using onset (black squares) and offset (red circles) resistive transition criteria, see Fig.", ".Figure: (Color online) (Top panels) Temperature dependence of in-plane resistivity in single crystals of (Ba 1-x _{1-x}K x _x)Fe 2 _2As 2 _2 with xx=0.82 (a) (right to left 0, 0.5, 1, 2, 3, ..., 9 T), 0.90 (b) (right to left 0, 0.05, 0.1, 0.3, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5 T), 0.92 (c) (right to left 0, 0.5, 1, 2,..., 5 T) and 1 (d) (0, 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1 T) in magnetic fields H∥cH \\parallel c. Middle panels show the data for H∥abH \\parallel ab (a), xx=0.82, right to left 0, 0.5, 1, 2, 3, ..., 9 T, (b) xx=0.90, field values right to left 0, 0.05, 0.1, 0.3, 0.5, 1, 1.5, 2, 2.5, 3, 4, ..., 9 T, (c) xx=0.92 right to left 0, 0.5, 1, 2, 3, ..., 9 T and (d) xx=1 magnetic fields 0, 0.1, 0.2, 0.3, 0.5, 0.7, 1, 1.5, 2, 2.5, 3, 3.5, 4 T. Bottom panels show H c2 (T)H_{c2}(T) for two field orientations as determined using onset (squares) and offset (circles) of of resistive transition criteria, see Fig.", ".In Fig.", "REF we show resistivity data taken in magnetic fields parallel to $c$ -axis (top panels), parallel to the conducting $ab$ plane (middle panels) and temperature dependent $H_{c2}(T)$ for two field orientations determined using onset and offset criteria.", "The data are shown for BaK122 compositions with $x$ =0.22 (a), 0.34 (b), 0.39 (c), and 0.47 (d).", "Similar data for slightly to moderately overdoped compositions $x$ = 0.53 (a), 0.55 (b), 0.65 (c), 0.80 (d) are shown in Fig.", "REF , and for strongly overdoped compositions $x$ =0.82 (a), 0.90 (b), 0.92 (c) and 1.0 (d) in Fig.", "REF .", "Figure: (Color online) Summary of the H c2 (T)H_{c2}(T) curves, determined using onset criterion in temperature-dependent resistivity measurements, Figs.", ",, for BaK12 single crystals 0.22≤x≤1.00.22 \\le x \\le 1.0, in configurations H∥cH \\parallel c (top panel) and H∥abH \\parallel ab (bottom panel).", "Lines show linear fits of the data for fields close to zero-field T c (0)T_c(0) neglecting slight upturn in the lowest fields.", "The linear fits were used to determine slopes of the lines dH c2 (T)/dTdH_{c2}(T)/dT and evaluate zero-temperature H c2 (0)=-0.70T c (0)dH c2 /dTH_{c2}(0) =-0.70T_c(0) dH_{c2}/dT, as shown in Fig.", "below.For the samples with $x$ = 0.22, 0.34, 0.39, 0.47, 0.53, and 0.55, the $H_{c2}(T)$ curves show positive curvature close to $T_c(0)$ for lowest fields below $H$ =1 T. Going further below $T_c(0)$ , the $H_{c2}(T) $ gets practically $T$ -linear.", "This is exactly the range which we use for determination of the $dH_{c2}/dT$ slope (Fig.", "REF ) and evaluation of $H_{c2}(0)$ as $H_{c2}(0) =-0.70T_c(0) dH_{c2}/dT$ (as shown in Fig.", "REF ).", "For the heavily overdoped samples $x$ = 0.80, 0.82, 0.90, 0.92, and 1, the $H_{c2}(T)$ curves in configuration $H \\parallel ab$ show a clear decrease of slope on cooling with a tendency to saturation, whereas for $H \\parallel c$ the curves remain linear.", "The saturation in $H \\parallel ab$ reflects paramagnetic Pauli limiting [66].", "Similar saturation behavior is seen in underdoped samples [46].", "Figure: (Color online) Doping evolution of the slope of H c2 (T)H_{c2}(T) curves close to zero-field T c (0)T_c(0), dH c2 (T)/dTdH_{c2}(T)/dT (panel b), and of the extrapolated H c2 (0)=-0.70T c (0)dH c2 /dTH_{c2}(0) =-0.70T_c(0) dH_{c2}/dT (c), shown in comparison with doping evolution of the superconducting transition temperature T c (x)T_c (x) (a).In Fig.", "REF we summarize the doping evolution of the slope of the temperature dependent upper critical field for field orientations along $c$ -axis (open black circles) and along the plane (closed red circles) (middle panel).", "In the Werthamer-Helfand-Hohenberg (WHH) theory [67] of the upper critical field for orbital limiting mechanism, $H_{c2}(0)=-0.7T_c(0) dH_{c2}/dT $ .", "In the bottom panel of Fig.", "REF we plot $H_{c2}(0)$ estimated using WHH formula as $T_c(0) dH_{c2}/dT$ .", "Note huge values of $H_{c2,c} >$ 100 T for compositions close to optimal doping.", "Interesting, the $H_{c2,c}(x)$ and especially $H_{c2,ab}(x)$ dependence, middle panel of Fig.", "REF , peaks at 0.39 and is much sharper than $T_c(x)$ dependence." ], [ " Doping evolution of the anisotropy parameter $\\gamma $", "In Fig.", "REF we plot doping evolution of the anisotropy of the upper critical field $\\gamma (x)$ .", "It can be seen that $\\gamma $ increases approximately two times, from 2 to 4 to 5 (depending on criterion) with increasing K doping levels.", "The increase starts in heavily overdoped compositions $x>$ 0.82, not far from the point where Fermi surface topology change was found in angle-resolved photoelectron spectroscopy (ARPES) studies [61] and where magnetism of the compounds changes according to neutron scattering [68], [69] and NMR [70] studies.", "According to ARPES studies the electron sheet of the Fermi surface transforms to four tiny cylinders.", "Since electron sheets have largest contribution of $d_{z^2}$ orbitals, and are most warped, it is natural to expect anisotropy increase close to $x$ =1 end of the doping phase diagram, in line with the upper critical anisotropy increase with $x$ .", "Several previous studies of $H_{c2}$ anisotropy for selected $x$ close to optimal doping in BaK compounds were performed in high magnetic fields up to 60 T in samples with $T_c$ =28.2 K ($x$ =0.4) [46], $T_c$ =32 K ($x$ =0.45) [64], and $T_c$ =38.5 K ($x=$ 0.32) [45].", "They found anisotropy decreasing on cooling, which was presumably caused by contribution of paramagnetic effect for $H_{c2,ab}$ .", "Similar to high-field studies in single crystals of other iron-based superconductors BaCo122 $x$ =0.14 [71], [45], NdFeAsO$_{0.7}$ F$_{0.3}$ [72], LiFeAs [73], and FeTe$_{0.6}$ Se$_{0.4}$ [74], [75], we find rough linear increase of the $H_{c2,c}(T)$ , but concave dependence with a tendency for saturation for $H_{c2,ab}$ .", "For all compounds of iron based superconductors the anisotropy ratio $\\gamma $ at $T_c(0)$ is in the range 2 to 5, similar to our finding in BaK122, with Ca$_{10}$ (Pt$_3$ As$_8$ )(( Fe$_{1-x}$ Pt$_x$ )$_2$ As$_2$ )$_5$ with $x=$ 0.09 [76], SmFeAsO$_{0.85}$ F$_{0.15}$ [77] and LaFe$_{0.92}$ Co$_{0.08}$ AsO [78] being exceptions, with $\\gamma \\approx $ 7 to 8.", "Additional contribution to the doping evolution of the anisotropy of the upper critical field can come from evolution of the superconducting gap structure [79].", "Initial high-resolution ARPES study on optimally doped samples with $x$ =0.4 revealed a superconducting large gap ($\\Delta \\sim $ 12 meV) on the two small hole-like and electron-like Fermi surface sheets, and a small gap ($\\sim $ 6 meV) on the large hole-like Fermi surface [80].", "In heavily overdoped KFe$_2$ As$_2$ , the Fermi surface around the Brillouin-zone center is qualitatively similar to that of composition with $x$ =0.4, but the two electron pockets are absent due to an excess of the hole doping [81].", "ARPES study over a wide doping range of BaK122 discovered that the gap size of the outer hole Fermi surface sheet around the Brillouin zone center shows an abrupt drop with overdoping (for $x \\ge $ 0.6) while the gaps on the inner and middle sheets roughly scale with $T_c$ [82]." ], [ "Linear relation between $H_{c2}(T)$ slope and {{formula:5754442b-ee24-4942-a1d6-5d938ff92225}}", "The high values of the critical fields in iron pnictides are determined by their short coherence lengths in 1 to 3 nm range [83], due to their high $T_c$ and low Fermi velocities, $v$ , with $\\xi \\sim \\hbar v/2 \\pi k_B T_c$ .", "Discussing the reasons for remarkable proportionality of the slopes of $dH_{c2}/dT$ to $T_c$ for $H\\parallel c$ shown in Figs.", "REF , we recall that in clean isotropic s-wave materials, $H_{c2} = -\\frac{ \\phi _0 (1-T/T_c)}{ 2\\pi \\xi _0^2} \\,, \\quad \\xi _0\\sim \\frac{\\hbar v}{\\Delta _0}\\propto \\frac{v}{T_c} \\,,$ so that the slope $H_{c2}^\\prime \\propto T_c$ .", "For the dirty case $H_{c2}^\\prime $ is $T_c$ independent; indeed, $H_{c2} \\propto \\frac{ 1-T/T_c}{ \\xi _0\\ell } \\,,$ where $\\ell $ is the $T$ independent mean-free path.", "We should mention that a strong pair breaking could be another reason for $dH_{c2}/dT\\propto T_c$ .", "For a gapless uniaxial material, the slope of the upper critical field along the $c$ direction near $T_c$ is given by[84] $\\frac{dH_{c2,c}}{dT} = -\\frac{4\\pi \\phi _0 k_B^2 }{ 3\\hbar ^2\\langle \\Omega ^2 v_{ab}^2 \\rangle }\\,T_c \\,.$ Here ($\\Omega (\\mathbf {k}_f)$ describes the anisotropy of the order parameter and is assumed to have a zero Fermi surface average, $\\langle \\Omega \\rangle =0$ , which is the case for the d-wave or, approximately, for the $s^{\\pm }$ symmetry).", "In our view, the first reason, i.e.", "the long mean-free path, is a probable cause for $dH_{c2}/dT\\propto T_c$ .", "Studies of thermal conductivity [85] and London penetration depth [86] at optimal doping suggest full gap, which is inconsistent with the idea of gapless superconductivity.", "In Fig.", "REF we verify linear relation for BaK122 over a broad doping (and as a consequence $T_c$ ) range, using onset (top panel a) and offset (middle panel b) criteria.", "The relation indeed holds very well, especially for $H \\parallel c$ configuration where the $H_{c2,0}(x)$ curves extrapolate to zero on $T_C \\rightarrow 0$ .", "This suggests that there is no gross change in the Fermi velocity over the whole doping range.", "This quadratic $H_{c2}(T_c)$ relation is grossly violated in pure KFe$_2$ As$_2$ in which the relation is linear [43].", "For $H \\parallel ab$ the $H_{c2,0}(x)$ curve is also close to linear, but does not extrapolate to zero on $T_c \\rightarrow 0$ .", "This deviation may be suggestive that Fermi velocity for transport along $c$ -axis is strongly increasing in BaK compositions with lowest $T_c$ close to $x$ =1.", "Another way to check the linear relation between the slope of the upper critical field and $T_c$ is to plot their ratio, as shown in the bottom panel (c) of Fig.", "REF .", "Plotting data this way reveals one difficult to recognize feature.", "The data for $H \\parallel c$ indeed show constant and doping independent ratio $\\frac{dH_{c2}/dT}{T_c}$ .", "The ratio for $H \\parallel ab$ remains constant for most of the phase diagram and then increases rapidly for $x>$ 0.8, showing that the increase of the anisotropy in this range is caused by decrease of the Fermi velocity, as one would expect for more anisotropic materials.", "Using an inverted temperature gradient method we were able to grow large and high quality single crystals of (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ with doping range spanning from underdoped to heavily overdoped compositions ($0.22 \\le x \\le 1$ ).", "We show that high vapor pressure of K and As elements at the soaking temperature is an important factor in the growth of single crystals of BaK122.", "When setting the top zone as the cold zone, on cooling the nucleation starts from the surface layer of the liquid melt.", "It is also assisted by the vapor growth, because surface layer also saturates first due to the evaporation of K and As.", "The crystallization processes from the top of a liquid melt helps to expel impurity phases during, compared to the growth inside the flux.", "For the whole doping range $0.22 \\le x \\le 1$ , we harvested large crystals with in-plane size up to 18$\\times $ 10 mm$^2$ .", "The crystals show very sharp superconducting transitions (less than 1 K) in dc magnetic susceptibil ity measurements for the optimal doping 0.34$\\le x \\le $ 0.55 and extremely overdoping 0.82$\\le x \\le $ 1 regimes.", "Relatively broad transitions are observed in the samples $x$ =0.65 and 0.80, due to a broader distribution of Ba and K atoms and a tendency to K clustering in the lattice [57], [59].", "In-plane electrical resistivity shows systematic evolution with doping.", "It perfectly follows $T^2$ dependence in the overdoped compositions with doping-independent slope over the range 0.80 to 1.", "Close to optimal doping the dependence deviates from pure $T^2$ functional form and can be described either as a sum of $T$ -linear and $T^2$ contributions, similar to electron-doped materials [62], or using a power-law function with exponent $n \\approx $ 1.5.", "The anisotropy of the upper critical field shows rapid change in the heavily overdoped regime, concomitant with Fermi surface reconstruction.", "The slope of the $H_{c2}(T)$ curves scales with zero-field $T_c$ of the samples, suggesting nearly doping independent Fermi velocity." ], [ "Acknowledgements", "This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division.", "The research was performed at the Ames Laboratory, which is operated for the U.S. DOE by Iowa State University under contract DE-AC02-07CH11358.", "M. Rotter, M. Tegel, and D. Johrendt, Phys.", "Rev.", "Lett.", "101, 107006 (2008).", "Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J.", "Am.", "Chem.", "Soc.", "130, 3296 (2008).", "A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D. J. Singh, and D. Mandrus, Phys.", "Rev.", "Lett.", "101, 117004 (2008).", "L. J. Li, Y. K. Luo, Q.", "B. Wang, H. Chen, Z. Ren, Q. Tao, Y. K. Li, X. Lin, M. He, Z. W. Zhu, G. H. Cao, and Z.", "A. Xu, New J. Phys.", "11, 025008 (2009).", "W. Schnelle, A. Leithe-Jasper, R. Gumeniuk, U. Burkhardt, D. Kasinathan, and H. Rosner, Phys.", "Rev.", "B 79, 214516 (2009).", "S. Jiang, H. Xing, G. Xuan, C. Wang, Z. Ren, C. Feng, J. Dai, Z. Xu, and G. Cao, J.", "Phys.", ": Condens.", "Matter 21, 382203 (2009).", "E. Colombier, S. L. Bud'ko, N. Ni, and P. C. Canfield, Phys.", "Rev.", "B 79, 224518 (2009).", "J. Paglione and R. L. Greene, Nature Phys.", "6, 645 (2010).", "P. C. Canfield and S. L. Bud'ko, Ann.", "Rev.", "Cond.", "Mat.", "Phys.", "1, 27 (2010).", "D. C. Johnston, Adv.", "Physics.", "59, 803 (2010).", "G. R. Stewart, Rev.", "Mod.", "Phys.", "83, 1589 (2011).", "L. Taillefer, Ann.", "Rev.", "Cond.", "Matter Physics 1, 51 (2010).", "P. Monthoux, D. Pines, and G. G. Lonzarich, Nature 450, 1177 (2007).", "M. R. Norman, Science 332, 196 (2011).", "E. Abrahams and Q. Si, J.", "Phys.", ": Condens.", "Matter 23, 223201 (2011).", "S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Phys.", "Rev.", "B 81, 184519 (2010).", "S. Kasahara, H.J.", "Shi, K. Hashimoto, S. Tonegawa, Y. Mizukami, T. Shibauchi, K. Sugimoto, T. Fukuda, T. Terashima, Andriy H. Nevidomskyy, and Y. Matsuda, Nature 486, 382 (2012).", "Y. Nakai, T. Iye, S. Kitagawa, K. Ishida, H. Ikeda, S. Kasahara, H. Shishido, T. Shibauchi, Y. Matsuda, and T. Terashima, Phys.", "Rev.", "Lett.", "105, 107003 (2010).", "K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, Y. Matsuda, Science 336, 1554 (2012).", "M. A. Tanatar, K. Hashimoto, S. Kasahara, T. Shibauchi, Y. Matsuda, and R. Prozorov, Phys.", "Rev.", "B 87, 104506 (2013).", "S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, and A. I. Goldman, Phys.", "Rev.", "Lett.", "104, 057006 (2010).", "M. A. Tanatar, N. Ni, C. Martin, R. T. Gordon, H. Kim, V. G. Kogan, G. D. Samolyuk, S. L. Bud'ko, P. C. Canfield, and R. Prozorov, Phys.", "Rev.", "B 79, 094507 (2009).", "M. A. Tanatar, N. Ni, G. D. Samolyuk, S. L. Bud'ko, P. C. Canfield, and R. Prozorov, Phys.", "Rev.", "B 79, 134528 (2009).", "M. A. Tanatar, N. Ni, A. Thaler, S. L. Bud'ko, P. C. Canfield, and R. Prozorov, Phys.", "Rev.", "B 82, 134528 (2010).", "M. A. Tanatar, N. Ni, A. Thaler, S. L. Bud'ko, P. C. Canfield, and R. Prozorov, Phys.", "Rev.", "B 84, 014519 (2011).", "N. Ni, M. E. Tillman, J.-Q.", "Yan, A. Kracher, S. T. Hannahs, S. L. Bud'ko, and P. C. Canfield, Phys.", "Rev.", "B 78, 214515 (2008).", "J. Murphy, M. A. Tanatar, Hyunsoo Kim, W. Kwok, U. Welp, D. Graf, J. S. Brooks, S. L. Bud'ko, P. C. Canfield, and R. Prozorov, Phys.", "Rev.", "B 88, 054514 (2013).", "C. Liu, T. Kondo, R. M. Fernandes, A. D. Palczewski, E. D. Mun, N. Ni, A. N. Thaler, A. Bostwick, E. Rotenberg, J. Schmalian, S. L. Bud'ko, P. C. Canfield, and A. Kaminski, Nature Phys.", "6, 419 (2010).", "B. Shen, H. Yang, Z.-S. Wang, F. Han, B. Zeng, L. Shan, C. Ren, and H.-H. Wen, Phys.", "Rev.", "B 84, 184512 (2011).", "M. A. Tanatar, E. C. Blomberg, H. Kim, K. Cho,W.", "E. Straszheim, B. Shen, H.-H. Wen, and R. Prozorov, arXiv:1106.0533.", "H. Q. Luo, Z. S. Wang, H. Yang, P. Cheng, X. Zhu, and H.-H. Wen, Supercond.", "Sci.", "Technol.", "21, 125014 (2008).", "D. Watanabe, T. Yamashita, Y. Kawamoto, S. Kurata, Y. Mizukami, T. Ohta, S. Kasahara, M. Yamashita, T. Saito, H. Fukazawa, Y. Kohori, S. Ishida, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, A.", "B. Vorontsov, T. Shibauchi, and Y. Matsuda, arXiv:1307.3408.", "K. Kihou, T. Saito, S. Ishida, M. Nakajima, Y. Tomioka, H. Fukazawa, Y. Kohori, T. Ito, S. Uchida, A. Iyo, C.-H. Lee, and H. Eisaki, J. Phys.", "Soc.", "Jpn.", "79, 124713 (2010).", "N. Ni, S. L. Bud'ko, A. Kreyssig, S. Nandi, G. E. Rustan, A. I. Goldman, S. Gupta, J. D. Corbett, A. Kracher, and P. C. Canfield, Phys.", "Rev.", "B 78, 014507 (2008).", "S. Avci, O. Chmaissem, E. A. Goremychkin, S. Rosenkranz, J.-P. Castellan, D. Y. Chung, I. S. Todorov, J.", "A. Schlueter, H. Claus, M. G. Kanatzidis, A. Daoud-Aladine, D. Khalyavin, and R. Osborn, Phys.", "Rev.", "B 83, 172503 (2011).", "S. Avci, O. Chmaissem, D. Y. Chung, S. Rosenkranz, E. A. Goremychkin, J. P. Castellan, I. S. Todorov, J.", "A. Schlueter, H. Claus, A. Daoud-Aladine, D. D. Khalyavin, M. G. Kanatzidis, and R. Osborn, Phys.", "Rev.", "B 85, 184507 (2012).", "M. Rotter, M. Pangerl, M. Tegel, and D. Johrendt, Angew.", "Chem.", "Int.", "Ed.", "47, 7949 (2008).", "A.", "A. Golubov, O. V. Dolgov, A. V. Boris, A. Charnukha, D. L. Sun, C. T. Lin, A. F. Shevchun, A. V. Korobenko, M. R. Trunin, and V. N. Zverev, JETP Lett.", "94, 333 (2011).", "J. K. Dong, S. Y. Zhou, T. Y. Guan, H. Zhang, Y. F. Dai, X. Qiu, X. F. Wang, Y.", "He, X. H. Chen, and S. Y. Li, Phys.", "Rev.", "Lett.", "104, 087005 (2010).", "T. Terashima, M. Kimata, H. Satsukawa, A. Harada, K. Hazama, S. Uji, H. Harima, G.-F. Chen, J.-L. Luo, and N.-L. Wang, J. Phys.", "Soc.", "Jpn, 78, 063702 (2009).", "K. Hashimoto, A. Serafin, S. Tonegawa, R. Katsumata, R. Okazaki, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, H. Ikeda, Y. Matsuda, A. Carrington, and T. Shibauchi, Phys.", "Rev.", "B 82, 014526 (2010).", "J.-Ph.", "Reid, M. A. Tanatar, A. Juneau-Fecteau, R. T. Gordon, S. René de Cotret, N. Doiron-Leyraud, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, R. Prozorov, and L. Taillefer, Phys.", "Rev.", "Lett.", "109, 087001 (2012).", "Y. Liu, M. A. Tanatar, V. G. Kogan, Hyunsoo Kim, T. A. Lograsso, and R. Prozorov, Phys.", "Rev.", "B 87, 134513 (2013).", "Y. M. Dai, B. Xu, B. Shen, H. Xiao, H. H. Wen, X. G. Qiu, C. C. Homes, and R. P. S. M. Lobo, Phys.", "Rev.", "Lett.", "111, 117001 (2013).", "V. A. Gasparov, F. Wolff-Fabris, D. L. Sun, C. T. Lin, and J. Wosnitza, JETP Letters, 93, 26 (2011).", "H. Q. Yuan, J. Singleton, F. F. Balakirev, S. A. Baily, G. F. Chen, J. L. Luo, and N. L. Wang, Nature 457, 565 (2009).", "Z.-S. Wang, H.-Q.", "Luo, C. Ren, H.-H. Wen, Phys.", "Rev.", "B 78, 140501(R) (2008).", "D. A. Zocco, K. Grube, F. Eilers, T. Wolf, and H. v. Löhneysen, Phys.", "Rev.", "Lett.", "111, 057007 (2013).", "S. Kittaka, Y. Aoki, N. Kase, T. Sakakibara, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C.-H. Lee, A. Iyo, H. Eisaki, K. Deguchi, N. K. Sato, Y. Tsutsumi, K. Machida, J. Phys.", "Soc.", "Jpn 83, 013704 (2014).", "A. I. Larkin and Yu.", "N. Ovchinnikov, Sov.", "Phys.", "JETP 20, 762 (1965).", "P. Fulde and R. A. Ferrell, Phys.", "Rev.", "135, A550 (1964).", "M. Takahashi, T. Mizushima, and K. Machida, Phys.", "Rev.", "B 89, 064505 (2014).", "T. Terashima, K. Kihou, M. Tomita, S. Tsuchiya, N. Kikugawa, S. Ishida, C.-H. Lee, A. Iyo, H. Eisaki, and S. Uji, Phys.", "Rev.", "B 87, 184513 (2013).", "M. A. Tanatar, N. Ni, S. L. Bud'ko, P. C. Canfield, and R. Prozorov, Supercond.", "Sci.", "Technol.", "23, 054002 (2010).", "M. A.Tanatar, R. Prozorov, N. Ni, S. L. Bud'ko, and P. C. Canfield, U.S. Patent 8,450,246.", "J. T. Park, D. S. Inosov, Ch.", "Niedermayer, G. L. Sun, D. Haug, N. B. Christensen, R. Dinnebier, A. V. Boris, A. J.", "Drew, L. Schulz, T. Shapoval, U. Wolff, V. Neu, X. Yang, C. T. Lin, B. Keimer, and V. Hinkov, Phys.", "Rev.", "Lett.", "102, 117006 (2009).", "W. K. Yeoh, B. Gault, X. Y. Cui, C. Zhu, M. P. Moody, L. Li, R. K. Zheng, W. X Li, X. L. Wang, S. X. Dou, G. L. Sun, C. T. Lin, and S. P. Ringer, Phys.", "Rev.", "Lett.", "106, 247002 (2011).", "L. Shan, Y.-L. Wang, B. Shen, B. Zeng, Y. Huang, A. Li, D. Wang, H. Yang, C. Ren, Q.-H.Wang, S. H. Pan, and H.-H.Wen, Nature Phys.", "7, 325 (2011).", "C.-L. Song, Y. Yin, M. Zech, T. Williams, M. M. Yee, G.-F. Chen, J.-L. Luo, N.-L. Wang, E. W. Hudson, and J. E. Hoffman, Phys.", "Rev.", "B 87, 214519 (2013).", "F. Rullier-Albenque, D. Colson, A.", "Forget, and H. Alloul, Phys.", "Rev.", "Lett.", "103, 057001 (2009).", "N. Xu, P. Richard, X. Shi, A. van Roekeghem, T. Qian, E. Razzoli, E. Rienks, G.-F. Chen, E. Ieki, K. Nakayama, T. Sato, T. Takahashi, M. Shi, and H. Ding, Phys.", "Rev.", "B 88, 220508 (2013).", "N. Doiron-Leyraud, P. Auban-Senzier, S. René de Cotret, C. Bourbonnais, D. Jérome, K. Bechgaard, and L. Taillefer, Phys.", "Rev.", "B 80, 214531 (2009).", "J .", "Murphy, M. A. Tanatar, D. Graf, J. S. Brooks, S. L. Bud'ko, P. C. Canfield, V. G. Kogan, and R. Prozorov, Phys.", "Rev.", "B 87, 094505 (2013).", "M. M. Altarawneh, K. Collar, and C. H. Mielke, N. Ni, S. L. Bud'ko, and P. C. Canfield, Phys.", "Rev.", "B 78, 220505 (2008).", "U. Welp, R. Xie, A. E. Koshelev, W. K. Kwok, H. Q. Luo, Z. S. Wang, G. Mu, and H.-H. Wen, Phys.", "Rev.", "B 79, 094505 (2009).", "A. M. Clogston, Phys.", "Rev.", "Lett.", "9, 266 (1962); B.S.", "Chandrasekhar, Appl.", "Phys.", "Lett.", "1, 7 (1962).", "N. R. Werthamer, E. Helfand, and P.C.", "Hohenberg, Phys.", "Rev.", "147, 295 (1966).", "J.-P. Castellan, S. Rosenkranz, E. A. Goremychkin, D.Y.", "Chung, I. S. Todorov, M. G. Kanatzidis, I. Eremin, J. Knolle, A.V.", "Chubukov, S. Maiti, M. R. Norman, F. Weber, H. Claus, T. Guidi, R. I. Bewley, and R. Osborn, Phys.", "Rev.", "Lett.", "107, 177003 (2011).", "C. H. Lee, K. Kihou, H. Kawano-Furukawa, T. Saito, A. Iyo, H. Eisaki, H. Fukazawa, Y. Kohori, K. Suzuki, H. Usui, K. Kuroki, and K. Yamada, Phys.", "Rev.", "Lett.", "106, 067003 (2011).", "M. Hirano, Y. Yamada, T. Saito, R. Nagashima, T. Konishi, T. Toriyama, Y. Ohta, H. Fukazawa, Y. Kohori, Y. Furukawa, K. Kihou, C.-H. Lee, A. Iyo, H. Eisaki, J. Phys.", "Soc.", "Jpn., 81, 054704 (2012).", "M. Kano, Y. Kohama, D. Graf, F. Balakirev, A. S. Sefat, M. A. Mcguire, B. C. Sales, D. Mandrus, and S. W. Tozer, J. Phys.", "Soc.", "Jpn.", "78, 084719 (2009).", "J. Jaroszynski, F. Hunte, L. Balicas, Youn-jung Jo, I. Raičević, A. Gurevich, D. C. Larbalestier, F. F. Balakirev, L. Fang, P. Cheng, Y. Jia, and H. H. Wen, Phys.", "Rev.", "B 78, 174523 (2008).", "K. Cho, H. Kim, M. A. Tanatar, Y. J.", "Song, Y. S. Kwon, W. A. Coniglio, C. C. Agosta, A. Gurevich, and R. Prozorov, Phys.", "Rev.", "B 83, 060502(R) (2011).", "S. Khim, J. W. Kim, E. S. Choi, Y.", "Bang, M. Nohara, H. Takagi, and K. H. Kim, Physical Review B, 81, 184511 (2010).", "M. Fang, J. Yang, F. F. Balakirev, Y. Kohama, J. Singleton, B. Qian, Z. Q. Mao, H. Wang, and H. Q. Yuan, Phys.", "Rev.", "B 81, 020509 (2010).", "N. Ni, J. M. Allred, B. C. Chan, and R. J. Cava, Proc.", "Natl.", "Acad.", "Sci.", "(USA) 108, E1019 (2011).", "U. Welp, C. Chaparro, A. E. Koshelev, W. K. Kwok, A. Rydh, N. D. Zhigadlo, J. Karpinski, and S. Weyeneth, Phys.", "Rev.", "B 83, 100513(R) (2011).", "G. Li, G. Grissonnanche, J.-Q.", "Yan, R. W. McCallum, T. A. Lograsso, H. D. Zhou, and L. Balicas, Phys.", "Rev.", "B 86, 054517 (2012).", "V. G. Kogan and R. Prozorov, Rep. Prog.", "Phys.", "75, 114502 (2012).", "H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S. Souma, T. Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo, and N. L. Wang, Europhys.", "Lett.", "83, 47001 (2008).", "T. Sato, K. Nakayama, Y. Sekiba, P. Richard, Y.-M. Xu, S. Souma, T. Takahashi, G. F. Chen, J. L. Luo, N. L. Wang, and H. Ding, Phys.", "Rev.", "Lett.", "103, 047002 (2009).", "W. Malaeb, T. Shimojima, Y. Ishida, K. Okazaki, Y. Ota, K. Ohgushi, K. Kihou, T. Saito, C. H. Lee, S. Ishida, M. Nakajima, S. Uchida, H. Fukazawa, Y. Kohori, A. Iyo, H. Eisaki, C.-T. Chen, S. Watanabe, H. Ikeda, and S. Shin, Phys.", "Rev.", "B 86, 165117 (2012).", "A Gurevich, Rep. Prog.", "Phys.", "74, 124501 (2011).", "V. G. Kogan, Phys.", "Rev.", "B 80, 214532 (2009).", "J.-Ph.", "Reid, A. Juneau-Fecteau, R. T. Gordon, S. Rene de Cotret, N. Doiron-Leyraud, X. G. Luo, H. Shakeripour, J. Chang, M. A. Tanatar, H. Kim, R. Prozorov, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, B. Shen, H.-H. Wen, Louis Taillefer, Supercond.", "Sci.", "Technol.", "25, 084013 (2012).", "R. Prozorov, K. Cho, H. Kim and M. A.Tanatar, J. Phys.", "Cond.", "Matt.", "Conf.", "Series 449, 012020 (2013)." ] ]	1403.0227
[ [ "Lyman edges in supermassive black hole binaries" ], [ "Abstract We propose a new spectral signature for supermassive black hole binaries (SMBHBs) with circumbinary gas disks: a sharp drop in flux blueward of the Lyman limit.", "A prominent edge is produced if the gas dominating the emission in the Lyman continuum region of the spectrum is sufficiently cold (T < 20,000 K) to contain significant neutral hydrogen.", "Circumbinary disks may be in this regime if the binary torques open a central cavity in the disk and clear most of the hot gas from the inner region, and if any residual UV emission from the individual BHs is either dim or intermittent.", "We model the vertical structure and spectra of circumbinary disks using the radiative transfer code TLUSTY, and identify the range of BH masses and binary separations producing a Lyman edge.", "We find that compact supermassive binaries with orbital periods of ~0.1 - 10 yr, whose gravitational waves (GWs) are expected to be detectable by pulsar timing arrays (PTAs), could have prominent Lyman edges.", "Such strong spectral edge features are not typically present in AGN spectra and could serve as corroborating evidence for the presence of a SMBHB." ], [ "Introduction", "Supermassive black holes (SMBHs) are present in the centres of most, if not all, nearby galaxies (see reviews by e.g.", "[26]; [12]).", "If two galaxies containing SMBHs merge, this should then result in the formation of a SMBHB (e.g.", "[3]).", "Thus, given the hierarchical model for structure formation, in which galaxies are built up by mergers, one would naively expect SMBHBs to be quite common.", "Many candidates for binary BHs have been identified on kiloparsec scales, including two galaxies with spatially resolved active binary nuclei ([24], [10]; see, e.g.", "the review by [23] and [40] and references therein).", "At parsec scales, however, there is only one clear example: a radio observation of a BH pair with a projected separation of $\\sim $ 7 pc [34].", "There remain no confirmed binary black holes at subparsec separations.", "The lack of observational evidence for binaries at small separations suggests that the SMBHs either remain inactive during the merger, or that they merge within a small fraction of a Hubble time and are consequently rare ([15]).", "Another possibility is that the spectrum of a compact binary differs significantly from those of single-BH active galactic nuclei (AGN).", "A better understanding of the spectral energy distributions (SEDs) and lightcurves from circumbinary discs is necessary to determine whether binaries may therefore be missing from AGN surveys or catalogs [44].", "Gravitational waves (GWs) from a merging SMBHB may be detected in the next decade by pulsar timing arrays (PTAs; [4]).", "Identifying the gravitational wave source in EM bands would also have considerable payoffs for cosmology and astrophysics (e.g.", "[33]).", "Unfortunately, GWs yield limited precision on the sky position.", "For a PTA source, of order 10$^2$ (and perhaps as many as 10$^4$ ) plausible candidates may be present within the 3D measurement error box [43].", "Concurrent EM observations would then be necessary to identify the GW source.", "Many different EM signatures have been proposed for SMBHBs.", "These include periodic luminosity variations commensurate with the orbital frequency of the source ([16] and references therein) and broad emission lines that are double-peaked and/or offset in frequency ([40] and the references therein).", "Additionally, the evacuation of a central cavity by the binary could lead to a spectrum that is a distinctively soft [31], [42], and has unusually weak broad optical emission lines, compared to typical AGN [43].", "Other work [14], [35] describes spectral signatures of discs with partial cavities, which may show up in the SED as broad, shallow dips.", "We here propose that, in addition to the above signatures, the presence of a central cavity in a circumbinary disc could produce distinct absorption edges in the optical/UV (in particular, at the Lyman limit).", "This is analogous to the prominent Lyman break in galactic spectra a $\\mathrel {\\unknown.", "{\\hspace{1.0pt}\\sim }}$ >$$$ few$ 10$ Myr after a starburst, when thecomposite emission is dominated by less massive stars with cooleratmospheres \\cite {Leitherer+1999,Schaerer2003}.", "Physically, such anedge would be present if the disc is cold enough($ T$\\sim $ $<$ 20,000$ K) to have sufficient neutral (ground-state)hydrogen to absorb Lyman continuum photons.", "The disc should also behot enough (conservatively $ T$\\sim $ $>$ 10,000$ K); for yet coolerdiscs, the continuum emission redward of the Lyman limit may beobscured by metal absorption features.$ This Letter is organized as follows.", "In §, we describe the details of our disc and emission models.", "In §, we show examples of spectra for binary BH discs, and compare these to those of single BH discs.", "In §, we discuss caveats, including emission from mini-discs around each of the individual BHs that could mask the Lyman edge.", "We summarize our main conclusions and the implications of this work in §." ], [ "Disk Models", "In order to model disc spectra, we must begin with a model for the disc.", "In particular, we need the energetics (i.e.", "how much energy is dissipated from tidal torques and viscous heating throughout the disc).", "We here adopt the analytic models in [21].", "These are modified versions of a standard [39] accretion disc, to incorporate the angular momentum transfer and the corresponding heating of the disc by the binary torques.", "The models self-consistently track the co-evolution of the disc and the binary orbit, through a series of quasi-steady, axisymmetric configurations.", "There are several qualitatively different solutions for such a system, depending on the parameters of the binary (i.e.", "masses and orbital separation) and the disc (i.e.", "viscosity and accretion rate).", "For the purposes of this Letter, we concentrate on the case where a central cavity is opened and maintained, as the lack of the hot inner regions is responsible for the Lyman edges in the spectrum.", "Our conclusions on Lyman edges simply rely on emission from disc patches with effective temperature $T_{\\rm eff}\\approx 10,000-20,000$ K dominating the composite UV spectrum, and should not be sensitive to model details.", "Fig.", "REF shows illustrative examples of radial profiles of the effective temperature (defined by $\\sigma _{\\rm SB}T^4_{}\\equiv F$ , where $\\sigma _{\\rm SB}$ is the Stefan-Boltzmann constant, and $F$ is the total dissipation rate per unit disc surface area, including both viscous and tidal heating).", "The solid curve shows $T_{\\rm eff}(r)$ for a circumbinary disc around a $M_{\\rm tot}=M_1+M_2=10^8 $ binary, with mass ratio $q=M_2/M_1=0.05$ and accretion rate $\\dot{M}/\\dot{M}_{\\rm Edd}=0.25$ (assuming a radiative efficiency of 10%).", "The secondary is located at the radius $r_s =$ 230 $R_g $ ($R_g \\equiv G M_{\\rm tot}/c^2$ ), creating a cavity inside 470 $R_g$ .", "We assume in all of our models that the BHs are non-spinning, and adopt a viscosity parameter $\\alpha =0.1$ .", "For the other less important parameters, we use the same fiducial parameters as KHL12, except we set $f_{\\rm T}=3/8$ to be consistent with our assumption of vertically uniform dissipation The dashed curve shows, for comparison, $T_{\\rm eff}(r)$ for a disc around a single BH with the same mass.", "Most importantly, this figure shows that outside the cavity, the circumbinary disc is hotter (by a factor of $\\sim $ two; see also [29]) than the corresponding single-BH disc, but still not nearly as hot as the innermost regions of this single-BH disc.", "Figure: Temperature profiles for a circumbinary disc around a 10 8 ,q=0.05^8,\\; q=0.05 binary (solid) and for a thin disc around a single10 8 ^8 BH (dashed).", "Both discs assume an accretionrate M ˙/M ˙ Edd =0.25\\dot{M}/\\dot{M}_{\\rm Edd}=0.25, and the radial distance isshown in gravitational units.", "The solid curve is truncated at aradius of 470 R g R_g due to the presence of a cavity in the disc.", "The blackdot marks the radius of the secondary." ], [ "SEDs: previous models", "In the simplest model for the emerging spectrum, each disc annulus emits as a blackbody, with the effective temperature determined by the heating rate from viscous and tidal torques in that annulus.", "A more sophisticated version is a composite “greybody” spectrum.", "The blackbody emission in each disc annulus is modified by a correction factor, accounting for the effects of electron scattering opacity [31], [42], to obtain $F_\\nu \\sim \\pi \\frac{2 \\epsilon _{\\nu }^{1/2}}{1+\\epsilon _{\\nu }^{1/2}} B_{\\nu }.$ Here, $\\epsilon _{\\nu } \\equiv \\kappa _{\\rm abs, \\nu }/\\left(\\kappa _{\\rm abs, \\nu }+\\kappa _{\\rm es, \\nu }\\right)$ is the ratio of absorption to total opacity.", "For this model, we adopt the Kramers' bound-free opacity at solar metallicity for $\\kappa _{\\rm abs, \\nu }$ and $\\kappa _{\\rm es, \\nu }=0.4$ cm$^{2}$ g$^{-1}$ .", "We refer the reader to [42] for a detailed discussion of the greybody disc model." ], [ "SEDs: new RT models", "The greybody model ignores important radiative transfer (RT) effects.", "In particular, the opacity is assumed to be a smooth function of frequency.", "In reality, the bound-free opacities have sharp thresholds, corresponding to the onset of absorption from various species in the disc.", "For example, photons with energies above the Lyman limit $\\ge $ 13.6eV (or frequency 3.28 $\\times $ 10$^{15}$ Hz) can be absorbed by neutral (ground-state) hydrogen (H) within the disc.", "This sharp change in opacity can cause a corresponding prominent edge in the emerging disc spectrum.", "This may be understood as follows: in a plane-parallel infinite atmosphere, one sees the source function at optical depth $\\sim $ unity along the line of sight.", "Thus, the smaller opacity redwards of the Lyman limit means that we see the source function from deeper in the disc, where the temperature is (generally) higher.", "The higher temperature then translates into a greater flux redward of the Lyman limit (e.g.", "[18]).", "As the temperature increases, H is more ionized, decreasing the H bound-free opacity.", "Eventually, the combined electron scattering and free-free opacities overwhelm the discontinuity in the bound-free opacity, and the opacity on both sides of the Lyman edge becomes nearly equal, washing out any absorption edge in the spectrum.", "As the temperature continues to increase, non-local thermodybnamic equilibrium (NLTE) effects may cause an emission edge instead.", "To model the emission from the disc, we use the RT code TLUSTY ([19]).", "This code self-consistently solves the equations of vertical hydrostatic equilibrium, energy balance, RT, and the full non-LTE statistical equilibrium equations for all species that are present in the disc.", "Contributions from all bound-free and free-free transitions at all frequencies of interest are included, while bound-bound transitions are assumed to be in detailed balance.", "We model H as a nine-level atom and He as a four-level atom.", "Electron scattering, including Comptonization is also included.", "We assume that the disc is composed of H and He (at their Solar ratio).", "Metals are not included, but would make little difference to continuum spectra for $10^4$ K $\\lesssim \\lesssim 10^5$ K. For any given annulus, we calculate the vertical structure by specifying the vertical gravity $g_z$ , disc surface density $\\Sigma $ , and the total energy dissipation rate $$ .", "Spectra are insensitive to the surface density (provided the disc remains optically thick) and, for computational convenience, we fix $\\Sigma =2\\times 10^5~{\\rm g~cm^{-2}}$ throughout this Letter.", "Thus, specifying the radial profile $(r)$ and vertical gravity $g_z(r)$ fully determines the spectrum emerging from the disc annulus at radius $r$ .", "$g_z(r)$ is approximated to be linear in $z$ and proportional to the square of the local Keplerian angular frequency, $q_{g}\\equiv \\Omega _{\\rm K}^2$ .", "The composite spectrum may be computed by summing over all annuli.", "Although we have assumed that flux is radiatively transmitted through the disc, most of our model annuli have convectively unstable zones.", "Models with small density inversions (density increasing outward) also occur.", "These would be unstable.", "Finally, we assume that tidal and viscous torques dissipate energy locally, and uniformly in height (i.e.", "equal dissipation per unit column mass).", "The vertical energy distribution is poorly understood, and real discs may be advective, or the energy may be carried away by density waves [7], [9]." ], [ "Results: Edges in Disk Spectra", "In this section, we show examples of composite circumbinary disc spectra, and compare these to the simpler black– and greybody models, as well as to the corresponding single–BH disc spectra.", "In the left-hand panel of Fig.", "REF , the solid (dashed) curves show composite disc spectra corresponding to the circumbinary (single-BH) disc in Fig.", "REF .", "Realistically, the disc may extend to 2000 $R_g$ , where it would become gravitationally unstable, according to the Toomre criterion.", "However, due to practical issues with convergence, we were only able to get models within $\\sim $ 40 $R_g$ of the inner disc edge, and we only integrate the emission from this region.", "However, the excluded region is cooler than the inner edge of the disc and its emission should not mask the Lyman edge.", "The TLUSTY binary spectrum (in black), shows a prominent break at the Lyman limit ($3.28\\times 10^{15}$ Hz), which is not present in the black– or greybody disc models (shown in blue and red, respectively).", "Overall, the blackbody model overpredicts the flux at both low and high frequencies, but underpredicts it immediately below the Lyman limit.", "The greybody model does a better job at frequencies below the Lyman limit, but overpredicts the flux more significantly at higher frequencies.", "Figure: Left-hand panel: Spectra from discs with the temperatureprofiles shown in Figure .", "The black, blue andred solid curves are TLUSTY, blackbody, and greybodyspectra for circumbinary discs.", "The dashed curves with the samecolors refer to the same models for a single-BH disc.", "Rightpanel: Same as the left, except for discs around 2.5 timeslower-mass (M tot =4×10 7 M_{\\rm tot}=4 \\times 10^7 ) BHs.In contrast to the binary-BH spectrum, the single-BH spectrum has no sharp absorption edge at the Lyman limit (although there is a weak kink).", "This is because, as shown in Fig.", "REF , the innermost region of the single-BH disc is much hotter than the inner edge of the binary disc.", "Specifically, the binary disc has a maximum $T_{}$ of 11,000, whereas, the single-BH disc is considerably hotter, with a maximum $T_{}$ of 80,000 K (we have chosen a somewhat high $\\dot{M}/\\dot{M}_{\\rm Edd}$ so that inner edge of the circumbinary disc is above $10^4 K$ .", "However, we have verified that there is no Lyman edge in the single BH case for $\\dot{M}/\\dot{M}_{\\rm Edd}\\sim 0.1$ ).", "For a spinning BH, the inner edge is expected to be closer to the BH horizon (following the innermost stable circular orbit), which would further increase the maximum temperature in the single-BH case.", "However, winds may limit the maximum $T_{}$ in single-BH discs to 50,000 K [27].", "In the right-hand panel of Fig.", "REF , we show another set of illustrative disc spectra, with the same parameters as in the left panel, except for a lower mass ($M_{\\rm tot}=4\\times 10^7$ ).", "In this case, the binary disc is hotter ($=17,000$ K at the inner edge) and the Lyman edge feature is correspondingly weaker.", "The single-BH model looks similar to the one shown in the left." ], [ "Binary parameter space with edges", "We now discuss under what conditions binary discs are likely to have prominent Lyman edges.", "If the inner edge of the disc falls below a particular threshold temperature, the composite disc spectrum will have a prominent Lyman absorption edge.", "An example of a spectrum close to the threshold temperature is the one shown in the right-hand panel of Figure REF : if the disc were much hotter, the Lyman edge feature would be wiped out.", "The precise threshold will depend on the vertical gravity and other parameters.", "Physically, the vertical gravity sets the scale height, which determines the vertical density.", "The density, in turn, determines the ionization state, which then affects the opacity difference between the two sides of the Lyman limit.", "To establish a quantitative criterion for the threshold $$ , we find the maximum effective temperature such that there is at least an order of magnitude drop in the flux at the Lyman limit (for different gravity parameters).", "We then establish the following fitting formula between the threshold temperature as a function of the square of the local Keplerian angular frequency, $q_{g}\\equiv \\Omega _K^2$ : $\\begin{split}\\log \\left(\\frac{T_{\\rm thres}}{17,000\\text{K}}\\right)&=0.06~\\log \\left(\\frac{q_g}{10^{-12} s^{-2}}\\right).\\end{split}$ This is accurate to $\\sim 1\\%$Assuming $\\Sigma =2\\times 10^5$ g cm$^{-2}$ .", "In the optically thick limit changes in the $\\Sigma $ may affect the threshold by $\\sim 5\\%$ in the range $10^{-12}s^{-2}<q_g<10^{-9} s^{-2}$ .", "At higher $q_g$ , the threshold temperature increases less steeply with $q_g$ .", "However, $q_g>10^{-9} s^{-2}$ lies outside our parameter space of interest.", "At lower $q_g$ we were unable to construct models at the threshold, and simply extrapolated this fit.", "$q_g=10^{-12} s^{-2}$ at 160 $R_g$ for a 10$^8$ BH.", "Note that the spectrum in the right-hand panel of Fig.", "REF would not satisfy our conservative criterion.", "We also impose a low-temperature threshold of 10$^4$ K, below which metal absorption becomes important and possibly causes metal edges to appear in the spectrum.", "Comparing the maximum disc temperature to the fitting formula above, we identify regions in binary parameter space which are cool enough to have Lyman edges.Using the hottest annulus, as opposed to full composite spectra, is a good proxy to identify the presence/absence of the Lyman edge.", "In Fig.", "REF , we show the ranges of masses and separations for which a prominent Lyman edge is produced, for two different accretion rates and mass ratios.", "The corresonding ranges of orbital periods, and time-scale over which the edge would be visible, may be calculated (to within a factor of a few) by the following fitting formulas.", "$t_{\\rm orb} &\\simeq 0.04 - 0.09 \\; {\\rm yr} \\left(\\frac{\\dot{M}/\\dot{M}_{\\rm Edd}}{0.1}\\right)^{1/4} \\left(\\frac{M}{10^7 }\\right)^{1/3} \\left(\\frac{q}{1000}\\right)^{1/10}\\\\t_{\\rm ly} &\\simeq 10^4 \\; {\\rm yr} \\left(\\frac{\\dot{M}/\\dot{M}_{\\rm Edd}}{0.1}\\right)^{-1/2} \\left(\\frac{M}{10^7 }\\right)^{1/3} \\left(\\frac{q}{1000}\\right)^{3/4}$ Note that for a given $\\dot{M}/\\dot{M}_{\\rm Edd}$ and $q$ , there is a maximum total mass for the edge which is set by our low temperature threshold: $\\left(\\frac{M}{10^7 }\\right) \\lesssim 50 \\left(\\frac{\\dot{M}/\\dot{M}_{\\rm Edd}}{0.1}\\right)^{3/5} \\left(\\frac{q}{1000}\\right)^{-3/5}$ Figure: Regions of binary parameter space with a prominent Lymanedge.", "In both panels, the blue region corresponds to binaries witha prominent Lyman edge for q=0.05q=0.05 and M ˙/M ˙ Edd =0.25\\dot{M}/\\dot{M}_{\\rm Edd}=0.25.", "On the left,the red region corresponds to Lyman edges for q=1q=1 andM ˙/M ˙ Edd =0.25\\dot{M}/\\dot{M}_{\\rm Edd}=0.25.", "On the right, the red region corresponds toq=0.05q=0.05 and M ˙/M ˙ Edd =1\\dot{M}/\\dot{M}_{\\rm Edd}=1.", "The bends belowr∼r\\mathrel {\\unknown.", "{\\hspace{1.0pt}\\sim }}<few100 RgcorrespondtobinaryseparationswherethediscstructurehasdecoupledfromtheGW-drivenbinary.Aboveeachcoloredregiontheinnerdisctemperaturefallsbelow correspond to binary separationswhere the disc structure has decoupled from the GW-drivenbinary.", "Above each colored region the inner disc temperature fallsbelow 104K.Below K. Below 104KmetalabsorptioncouldmasktheLymanedge,asindicatedbythearrows.", "K metal absorption could mask the Lyman edge, as indicated by the arrows." ], [ "Mini-discs", "In general, we would not expect any central cavity to be completely empty.", "As first discussed in the smoothed-particle hydrodynamic simulations of [2] and confirmed by several recent works [17], [30], [6], [41], [36], [8], [11], [13] gas leaks into the cavity through non-axisymmetric streams.", "These streams can feed “mini-discs” around each individual BH, at rates set by the viscous time-scale of each mini-disc.", "The recent simulations suggest that accretion rates inside the cavity may be comparable to those onto a single-BH.", "The emission from hot mini-discs may mask the Lyman edge feature.", "To illustrate this, we calculate spectra for the discs shown in Fig.", "8b of [11] (reproduced here as the left panel of Fig.", "REF ).", "This figure shows a time averaged surface density profile for a $q=0.43$ binary.", "The simulation is scale-free so the masses, separation, and accretion rate are arbitrary.", "We adopt a $M_{\\rm tot}=5 \\times 10^7 $ and separation $r_s=460 R_g$ .", "For $q=0.43$ , the masses of the primary and secondary are then $3.5\\times 10^7 \\; $ and $1.5 \\times 10^7 \\; $ , and the system is on the verge of the GW inspiral stage.", "To model the spectrum of the circumbinary disc, we use $(r)$ from an axisymmetric KHL12 circumbinary disc model with inner radius at 460 $R_g$ (even though the simulated disc is in fact lopsided; note that the simulations are isothermal and do not predict $$ ).", "To obtain the $$ profile, we set $\\dot{M}/\\dot{M}_{\\rm Edd}=0.1$ .", "We likewise model the circumsecondary disc as a standard thin disc around a non-spinning BH (with truncation radius of 400 $R_g$ , where $R_g$ is defined in terms of just the secondary mass.", "This is roughly consistent with the size of the circumsecondary disc in Fig.", "REF ).", "Fig.", "REF shows spectra of the circumbinary and circumsecondary discs, assuming that 50% (purple), 10% (blue), and 5% (red) of the external $\\dot{M}$ fuels the secondary.", "For the 5% case, the Lyman edge is still prominent.", "However, it is greatly reduced for 10%, and for 50% it is completely obscured.", "Thus, if more than a few % of the $\\dot{M}$ in the cicumbinary disc leaks into the cavity and fuels a radiatively efficient, hot accretion flow, the Lyman edge can be obscured.", "As discussed above, [11] find circumsecondary accretion rates comparable to the rate onto a single BH, which would favor the $>50\\%$ case.", "There should also be some emission from the circumprimary disc, which we have not included.", "Figure: Left: Time average of the surface density distribution,reproduced with permission from.", "Right: spectra for thecircumbinary and circumsecondary discs.", "The solid (black) curve isfor the circumbinary disc, and the other curves are for themini-disc, corresponding to a different assumed fraction of thecircumbinary M ˙\\dot{M} fueling the secondary BH: 50% (purple), 10% (blue), and 5% (red).However, there are two reasons why Lyman edges could remain detectable, even with efficient fueling of the individual BHs.", "First, most of the gas entering the cavity may fuel the secondary (rather than the primary) BH.", "This could then lead to a radiatively inefficient, super-Eddington accretion flow, rather than a thin mini-disc; such discs have much fainter fluxes in the UV (see, e.g.", "[20]).", "Second, the fueling of the individual BHs may be intermittent.", "The simulations listed in the preceding paragaphs show that the rate at which the gas enters the cavity fluctuates strongly, tracking the binary's orbital period.", "Whether the BHs can accept this fuel depends on the viscous time-scale in their vicinity.", "There is some evidence from 3D magnetohdydrodynamic simulations that the effective $\\alpha $ may strongly increase inside the cavity [41], [32], [13].", "The streams from the circumbinary disc would then rapidly accrete onto the individual BHs, and mini-discs would either not form or would be intermittent [44].", "The relevant time-scale would be the viscous time of the mini-disc, which should generally extend to the tidal truncation radius as long as the specific angular momentum of the accreting streams exceeds that at the ISCO [35].", "Ultimately, the prominence of the Lyman edge features is tied to the nature of the mini-disks, and requires a better understanding of these flows." ], [ "Summary and Conclusions", "In this Letter, we have proposed that spectral edges, in particular at the Lyman limit, may be characteristic signatures of a circumbinary disc.", "Our conclusions can be summarized as follows.", "If binary torques clear a cavity in the circumbinary disc, the disc spectrum may exhibit a sharp drop at the Lyman limit.", "This is because the hottest region (i.e.", "the inner edge) of the disc is cool enough to have neutral H, absorbing nearly all flux blueward of the Lyman limit.", "This occurs below a critical $$ which generally lies in the range $\\sim $ 10,000 K-20,000 K, depending on vertical gravity and other parameters.", "At lower temperatures, absorption from metals (i.e.", "C) may cause spectral edges redward of the Lyman limit.", "Observationally, AGN spectra only show Lyman edges due to absorption by intervening neutral gas (see [1]).", "The inner regions of a single-BH AGN disc are hotter than for binaries (Fig.", "REF ), and can mask any edge produced in the outer disc, leaving only a small “kink”(Fig.", "REF ).", "Such kinks are not seen observationally, and understanding what would smear them (e.g.", "general relativistic effects or winds) is an open theoretical problem.", "For an overview of the Lyman edge problem in AGN spectral modeling see, e.g.", "[22] and [25].", "Neutral H in the binary's host galaxy, unrelated to the nuclear accretion disc itself, could cause a Lyman edge (as seen in a few AGNs).", "However, in the case of the disc, one could look for rotational broadening of the edge due to orbital motions in the disc, with velocities of order $10^4$ km/s.", "This may cause a $\\simeq $ 10% smearing on the edge.", "A portion of the binary parameter space could have a truncated circumbinary disc, with $$ in the critical range for a prominent Lyman edge (Fig.", "REF ).", "This parameter space partially overlaps with the expected typical parameters for individually resolvable PTA sources.", "These are very massive binaries, with $10^8 \\mathrel {\\unknown.", "{\\hspace{1.0pt}\\sim }}$ <$$ M $\\sim $ $<$ 109 $, andseparations ranging from 10^{\\prime }s to 1000^{\\prime }s of $ Rg$, and mass ratiospeaking at $ q1$ but with a long tail to lower values\\cite {Sesana+2012}.$ Efficient fueling of the BHs inside the central cavity could mask the Lyman edge feature in the circumbinary disc spectrum.", "For example, persistent emission from hot mini-discs (see [11]) would obscure the Lyman edge.", "However, if the accretion flows onto the individual BHs are radiatively inefficient and/or intermittent [44], the Lyman edge could remain visible, or appear periodically on the time-scale of the binary's orbit (which could be weeks to years; [15]).", "The proposed Lyman edge signature could be used in combination with other proposed EM signatures to refine the search for SMBHBs.", "We have conducted a preliminary search for the Lyman edge feature in x-ray weak quasars discussed in [5].", "In particular we looked at FUSE and IUE spectra for the ten objects in their Table 2, but found no sign of any Lyman edge feature.", "If detected in an AGN, a prominent Lyman edge would tighten the case for the presence of a compact binary BH.", "We thank Shane Davis for sharing a modified version of TLUSTY and for technical help, as well as for providing an initial table of spectral models.", "We also thank Shane Davis, Omer Blaes, Ivan Hubeny, Jules Halpern, and Frits Paerls for insightful conversations.", "ZH acknowledges support from NASA grant NNX11AE05G." ] ]	1403.0002
[ [ "Ergodic Sum-Rate Maximization for Fading Cognitive Multiple Access\n Channels without Successive Interference Cancellation" ], [ "Abstract In this paper, the ergodic sum-rate of a fading cognitive multiple access channel (C-MAC) is studied, where a secondary network (SN) with multiple secondary users (SUs) transmitting to a secondary base station (SBS) shares the spectrum band with a primary user (PU).", "An interference power constraint (IPC) is imposed on the SN to protect the PU.", "Under such a constraint and the individual transmit power constraint (TPC) imposed on each SU, we investigate the power allocation strategies to maximize the ergodic sum-rate of a fading C-MAC without successive interference cancellation (SIC).", "In particular, this paper considers two types of constraints: (1) average TPC and average IPC, (2) peak TPC and peak IPC.", "For the first case, it is proved that the optimal power allocation is dynamic time-division multiple-access (D-TDMA), which is exactly the same as the optimal power allocation to maximize the ergodic sum-rate of the fading C-MAC with SIC under the same constraints.", "For the second case, it is proved that the optimal solution must be at the extreme points of the feasible region.", "It is shown that D-TDMA is optimal with high probability when the number of SUs is large.", "Besides, we show that, when the SUs can be sorted in a certain order, an algorithm with linear complexity can be used to find the optimal power allocation." ], [ "Introduction", "The demand for frequency resources has dramatically increased due to the explosive growth of wireless applications and services in recent years.", "This poses a big challenge to the current fixed spectrum allocation policy.", "On the other hand, a report published by Federal Communications Commission (FCC) shows that the current scarcity of spectrum resource is mainly due to the inflexible spectrum regulation policy rather than the physical shortage of spectrum [1].", "Most of the allocated frequency bands are under-utilized, and the utilization of the spectrum varies in time and space.", "Similar observations have also been made in other countries.", "In particular, the spectrum utilization efficiency is shown to be as low as $5\\%$ in Singapore [2].", "The compelling need to improve the spectrum utilization and establish more flexible spectrum regulations motivates the advent of cognitive radio (CR).", "Compared to the traditional wireless devices, CR devices can greatly improve the spectrum utilization by dynamically adjusting their transmission parameters, such as transmit power, transmission rate and the operating frequency.", "Recently, FCC has agreed to open the licensed, unused television spectrum or the so-called white spaces to the new, unlicensed, and sophisticatedly designed CR devices.", "This milestone change of policy by the FCC indicates that CR is fast becoming one of the most promising technologies for the future radio spectrum utilization.", "This also motivates a wide range of research in the CR area, including the research work done in this paper.", "A popular model widely adopted in CR research is the spectrum sharing model.", "In a spectrum-sharing CRN, a common way to protect primary users (PU) is to impose an interference power constraint (IPC) at the secondary network, which requires the interference received at PU receiver to be below a prescribed threshold [3].", "Subject to such a IPC, the achievable rates of Additive White Gaussian Noise (AWGN) channels were investigated in [4].", "In [5], the authors studied the ergodic capacity of a single-user CRN under IPC in different fading environment.", "In [6], the authors studied the outage performance of such a single-user spectrum-sharing CRN under a IPC.", "In [7], the authors studied the capacity and power allocation for a spectrum-sharing fading CRN under both peak and average IPC.", "In [8], the optimal power allocation strategies to achieve the ergodic and outage capacity for a spectrum-sharing fading CRN under different combinations of the transmit power constraint (TPC) and the IPC were investigated.", "However, the aforementioned works only focused on the point-to-point secondary networks.", "In [9], from an information theoretic perspective, the authors investigated the achievable rate region of a Gaussian C-MAC.", "In [10] and [11], the authors investigated the optimal power allocation strategies for AWGN cognitive multiple access channels (C-MAC).", "In [12], the authors investigated the ergodic sum capacity for a fading C-MAC with multiple PUs.", "In [13], the authors studied the outage capacity region for a fading C-MAC.", "However, in these works, successive interference cancellation (SIC) decoders are assumed to be available, and thus no mutual interference among the secondary users (SU) is considered.", "Different from the aforementioned works, in this paper, we study the ergodic sum-rate and the corresponding optimal power allocation strategies of a fading C-MAC without SIC.", "Compared with the previous studies with SIC, the problem studied in this paper is much harder due to the existence of the mutual interference among SUs, which makes the problem a nonlinear, nonconvex constrained optimization problem.", "Another line of related research [14], [15], [16], [17] focused on the sum-rate maximization for MAC under non-CR setting (without IPC).", "In [14], the authors investigated the ergodic capacity region and its optimal power allocation for the fading MAC.", "In [15], the authors proposed the iterative water-filling algorithm to maximize the sum-rate of a multiple-input multiple-out (MIMO) MAC with SIC under individual power constraints.", "For sum-rate maximization of MAC without SIC, in [16], the authors were able to show the optimality of the binary power allocation for a two-user network.", "For arbitrary users, the authors only numerically illustrated the optimality of binary power allocation.", "While in [17], the authors analytically proved that binary power allocation is optimal for any number of users in terms of maximizing the sum-rate of the MAC without SIC.", "Compared with these works, the problem studied in this paper is more challenging due to the existence of the IPC, which changes the properties of the optimal power allocation.", "It is shown that binary power allocation is no longer optimal for our problem.", "The main contribution and the key results of this paper are listed as follows: We investigate the optimal power allocation strategies to maximize the ergodic sum-rate of a fading C-MAC without SIC under both TPC and IPC.", "In particular, we consider two types of constraints : (1) average TPC and average IPC, (2) peak TPC and peak IPC.", "For the average TPC and average IPC case, we prove that the optimal power allocation is dynamic time-division multiple-access (D-TDMA), which is exactly the same as the optimal power allocation given in [12] to maximize the ergodic sum-rate of the fading C-MAC with SIC under the same constraints.", "For the peak TPC and peak IPC case, we prove that the optimal solution must be at the extreme points of the feasible region.", "We show that D-TDMA is optimal when a certain condition is satisfied.", "It is also shown that D-TDMA is optimal with high probability when the number of SUs is large.", "Thus, we can solve the problem by searching the extreme points of the feasible region when the number of SUs is small, and by applying the D-TDMA scheme when the number of SUs is large.", "For the peak TPC and peak IPC case, we show that when the SUs can be sorted in a certain order, an algorithm with linear complexity can be developed to find the optimal power allocation of our problem.", "For the peak TPC and peak IPC case, we also show by simulations that the optimal power allocation to maximize the ergodic sum-rate of the fading C-MAC with SIC, which we refer to as SIC-OP, can be used as a good suboptimal power allocation for our problem.", "It is shown by simulations that SIC-OP is optimal or near-optimal for our problem when the D-TDMA is not optimal.", "The rest of the paper is organized as follows.", "The system model and power constraints are described in Section .", "The optimal power allocation strategies to maximize the ergodic sum-rate of the fading C-MAC without SIC are studied in Section .", "Then, the simulation results are presented and analyzed in Section .", "Section concludes the paper.", "In this paper, we consider a spectrum sharing CR network consists of one PU and a $K$ -user secondary multiple access network.", "The communication links between each SU and the PU receiver (PU-Rx) are referred as the interference links.", "The links between the SUs and the secondary base station (SBS) are referred as the secondary links.", "For the convenience of exposition, all the channels involved are assumed to be block-fading (BF) [18], i.e., the channels remain constant during each transmission block, but possibly change from one block to another.", "As shown in Fig.REF , the channel power gain of the interference link between SU-$i$ and the PU is denoted by $g_i$ .", "The channel power gain of the secondary link between SU-$i$ and the SBS is denoted as $h_{i}$ .", "All these channel power gains are assumed to be independent and identically distributed (i.i.d.)", "random variables (RVs) each having a continuous probability density function (PDF).", "All the channel state information (CSI) is assumed to be perfectly known at both SUs.", "CSI of the secondary links can be obtained at SUs by the classic channel training, estimation, and feedback mechanisms.", "CSI of the interference links between SUs and primary receivers can be obtained at SUs via the cooperation of the primary receivers.", "The noise at SBS is assumed to be circular symmetric complex Gaussian variable with zero mean and variance $\\sigma ^2$ denoted by ${\\mathcal {C}}{\\mathcal {N}}(0,\\sigma ^2)$ ." ], [ "Power Constraints", "In this paper, we denote the transmit power of SU-$i$ as $P_i$ , then the instantaneous interference received at PU-Rx from SU-$i$ is $g_iP_i$ .", "Then, the average and peak interference power constraint (IPC) can be described as $\\mbox{Average IPC:}~~&\\mathbb {E}\\left[\\sum _{i=1}^K g_iP_i\\right]\\le I^{av}, \\\\\\mbox{Peak IPC:}~~&\\sum _{i=1}^K g_iP_i\\le I^{pk},$ where $I^{av}$ denotes the limit of average received interference at the PU, and $I^{pk}$ denotes the maximum instantaneous interference that the PU can tolerate.", "$\\mathbb {E}[\\cdot ]$ denotes the statistical expectation over all the involved fading channel power gains.", "Usually, the average IPC is used to guarantee the long-term QoS of the PU when it provides delay-insensitive services.", "When the service provided by the PU has an instantaneous QoS requirement, the peak IPC is usually adopted.", "In this paper, we also consider the transmit power constraint (TPC) imposed at each SU.", "Same as the IPC, two types (both average and peak) of TPC are considered here.", "Let $P_i^{av}$ and $P_i^{pk}$ be the average and peak transmit power limit of SU-$i$ , respectively.", "Then, the average and peak TPC can be described as $\\mbox{Average TPC:}~~&\\mathbb {E}\\left[P_i\\right] \\le P_i^{av},~\\forall i, \\\\\\mbox{Peak TPC:}~~& P_i\\le P_i^{pk},~\\forall i,$ where $\\mathbb {E}[\\cdot ]$ denotes the statistical expectation over all the involved fading channel power gains.", "The peak power limitation is usually due to the nonlinearity of power amplifiers in practice.", "The average TPC is usually imposed to meet a long-term transmit power budget." ], [ "Ergodic Sum-rate Maximization for Fading\nC-MAC without SIC", "Without SIC decoders available at the SBS, the instantaneous transmission rate of each SU is given by $r_i=\\ln \\left(1+\\frac{h_{i}p_i}{\\sum _{j=1,j\\ne i}^Kh_{j}P_j+\\sigma ^2}\\right), ~\\forall i.", "$ For BF channels, ergodic rate is defined as the maximum achievable rate averaged over all the fading blocks.", "Then, the ergodic sum-rate of the fading C-MAC considered in this paper can be written as $\\mathbb {E}\\left[\\sum _{i=1}^K \\ln \\left(1+\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right)\\right].$ In the following, we study power allocation strategy to maximize the ergodic sum-rate of the fading C-MAC subject to the power constraints given in Section REF ." ], [ "Average TPC and Average IPC", "Under average TPC and average IPC, the optimal power allocation to maximize the ergodic sum-rate of the fading C-MAC can be obtained by solving the following optimization problem: Problem 1 $\\max _{P_i\\ge 0,\\forall i}~~ &\\mathbb {E}\\left[\\sum _{i=1}^K\\ln \\left(1+\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right)\\right],\\\\\\mbox{s.t.", "}~~~~ &(\\ref {Con-AIPC}),~(\\ref {Con-ATPC}).$ It is not difficult to observe that Problem REF is a non-convex optimization problem.", "Thus, we cannot solve it by the standard convex optimization techniques.", "To solve Problem REF , we first look at the following problem.", "Problem 2 $\\max _{P_i\\ge 0,\\forall i}~~ &\\mathbb {E}\\left[\\ln \\left(1+\\sum _{i=1}^K\\frac{h_{i}P_i}{\\sigma ^2}\\right)\\right],\\\\\\mbox{s.t.", "}~~~~ &(\\ref {Con-AIPC}),~(\\ref {Con-ATPC}).$ Problem REF gives the ergodic sum-rate for fading C-MAC with SIC, and it has been studied in [12].", "It is shown in [12] (Lemma 3.1) that the optimal solution of Problem REF is: at most one user is allowed to transmit in each fading block.", "Based on this fact, we obtain the following theorem.", "Theorem 1 The optimal solution of Problem REF is the same as that of Problem REF .", "It is observed that the constraints of Problem REF and Problem REF are exactly the same.", "Thus, the feasible sets of Problem REF and Problem REF are the same.", "Now, suppose $P=[P_1~ P_2~ \\cdots ~P_K]^T$ is a feasible solution of Problem REF .", "The rest of the proof consists of two steps.", "Step 1: Since $P$ is a feasible solution of Problem REF , it is also a feasible solution of Problem REF .", "Now, we show that the value of the objective function of Problem REF under $P$ is an upper-bound of that of Problem REF under the same $P$ , i.e., $\\mathbb {E}\\left[\\sum _{j=1}^K\\ln \\left(1+\\frac{ h_{j}P_j}{\\sigma ^2+\\sum _{i=1,i\\ne j}^Kh_{i}P_i}\\right)\\right] \\le \\mathbb {E}\\left[\\ln \\left(1+\\sum _{i=1}^K\\frac{h_{i}P_i}{\\sigma ^2}\\right)\\right]$ .", "Since the expectation operation is linear, it is equivalent to show that $\\sum _{j=1}^K \\ln \\left(1+\\frac{h_{j}P_j}{\\sigma ^2+\\sum _{i=1,i\\ne j}^K h_{i}P_i}\\right)\\le \\ln \\left(1+\\sum _{i=1}^K\\frac{h_{i}P_i}{\\sigma ^2}\\right)$ , which is given below.", "$\\ln \\left(1+\\sum _{i=1}^K\\frac{h_{i}P_i}{\\sigma ^2}\\right)=&\\ln \\left(\\frac{\\sigma ^2+\\sum _{i=1}^Kh_{i}P_i}{\\sigma ^2}\\right)\\nonumber \\\\=&\\ln \\left[\\left(\\frac{\\sigma ^2+\\sum _{i=1}^Kh_{i}P_i}{\\sigma ^2+\\sum _{i=2}^Kh_{i}P_i}\\right)\\left(\\frac{\\sigma ^2+\\sum _{i=2}^Kh_{i}P_i}{\\sigma ^2+\\sum _{i=3}^Kh_{i}P_i}\\right)\\cdots \\left(\\frac{\\sigma ^2+\\sum _{i=K}^Kh_{i}P_i}{\\sigma ^2}\\right)\\right]\\nonumber \\\\\\stackrel{a}{=}&\\sum _{j=1}^K\\ln \\left(\\frac{\\sigma ^2+\\sum _{i=j}^Kh_{i}P_i}{\\sigma ^2+\\sum _{i=j+1}^K h_{i}P_i}\\right)\\nonumber \\\\=&\\sum _{j=1}^K \\ln \\left(1+\\frac{h_{j}P_j}{\\sigma ^2+\\sum _{i=j+1}^K h_{i}P_i}\\right)\\nonumber \\\\\\stackrel{b}{\\ge }&\\sum _{j=1}^K \\ln \\left(1+\\frac{h_{j}P_j}{\\sigma ^2+\\sum _{i=1,i\\ne j}^K h_{i}P_i}\\right),$ where we introduce a dumb item $\\sum _{i=K+1}^K h_{i}P=0$ in the equality “a” for notational convenience.", "The inequality “b” follows from the fact that $\\sum _{i=1,i\\ne j}^K h_{i}P_i \\ge \\sum _{i=j+1}^K h_{i}P_i, \\forall j$ .", "Step 2: Now, we show that the optimal solution of Problem REF is the same as that of Problem REF .", "Since it is proved in [12] (Lemma 3.1) that the optimal solution of Problem REF is: at most one user is allowed to transmit in each fading block.", "It is easy to observe that the optimal solution of Problem REF is a feasible solution of Problem REF .", "Since we have shown in Step 1 that Problem REF provides an upper-bound of Problem REF for the same $P$ .", "Thus, it is easy to observe that the optimal solution of Problem REF must be the same as that of Problem REF , which is: at most one user is allowed to transmit in each fading block.", "Since in Theorem REF , we have shown that the optimal solution of Problem REF is the same as that of Problem REF .", "Thus, the optimal power allocation strategies for Problem REF can be obtained in the same way as [12].", "Interested readers can refer to Lemma 3.1 and 3.2 in [12] for details." ], [ "Peak TPC and Peak IPC", "Under peak TPC and peak IPC, the optimal power allocation to maximize the ergodic sum-rate of the fading C-MAC can be obtained by solving the following optimization problem: Problem 3 $\\max _{P_i\\ge 0, \\forall i}~~ &\\mathbb {E}\\left[\\sum _{i=1}^K\\ln \\left(1+\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right)\\right],\\\\\\mbox{s.t.", "}~~ & (\\ref {Con-PIPC}),~(\\ref {Con-PTPC}).$ Since all the constraints involved are instantaneous power constraints, Problem REF can be decomposed into a series of identical subproblems each for one fading state, which is Problem 4 $\\max _{P_i\\ge 0, \\forall i}~~ &\\sum _{i=1}^K \\ln \\left(1+\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right),\\\\\\mbox{s.t.", "}~~ & (\\ref {Con-PIPC}),~(\\ref {Con-PTPC}).$ It can be verified that Problem REF is non-convex.", "Thus, we cannot solve it directly by the standard convex optimization techniques.", "To solve Problem REF , we first investigate its properties.", "Lemma 1 The optimal solution $P^$ of Problem REF must be at the boundary of the feasible region of Problem REF .", "This can be proved by contradiction.", "Suppose the optimal solution $P^$ of Problem REF is in the interior of the feasible region, i.e., $0<P_i^<P_i^{pk}, \\forall i$ and $\\sum _{i=1}^K g_iP^_i< I$ .", "Now, we look at the power allocation $P_n$ of SU-$n$ .", "For convenience, we denote (REF ) as $f\\left(P\\right)$ .", "Then, $f\\left(P\\right)$ can be rewritten as $f\\left(P\\right)=\\ln \\left(1+\\frac{h_{n}P_n}{\\sigma ^2+\\sum _{j=1,j\\ne n}^Kh_{j}P_j}\\right)+\\sum _{i=1,i\\ne n}^K\\ln \\left(1+\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right).$ Taking the derivative of $f\\left(P\\right)$ with respect to $P_n$ , we have $\\frac{\\partial f\\left(P\\right)}{\\partial P_n}&=\\frac{1}{1+\\frac{ h_{n}P_n}{\\sigma ^2+\\sum _{j=1,j\\ne n}^Kh_{j}P_j}}\\frac{h_{n}}{\\sigma ^2+\\sum _{j=1,j\\ne n}^K h_{j}P_j} \\nonumber \\\\&+\\sum _{i=1,i\\ne n}^K\\frac{1}{1+\\frac{h_iP_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^Kh_{j}P_j}}\\left(-\\frac{h_iP_i}{\\left(\\sigma ^2+\\sum _{j=1,j\\ne i}^Kh_{j}P_j\\right)^2}\\right)h_n \\nonumber \\\\&=\\frac{h_n}{\\sigma ^2+\\sum _{j=1}^K h_{j}P_j}-\\sum _{i=1,i\\ne n}^K\\frac{ h_iP_i h_n}{\\left(\\sigma ^2+\\sum _{j=1}^Kh_{j}P_j\\right)\\left(\\sigma ^2+\\sum _{j=1,j\\ne i}^Kh_{j}P_j\\right)}\\nonumber \\\\&=\\frac{h_n}{\\sigma ^2+\\sum _{j=1}^K h_{j}P_j}\\left(1-\\sum _{i=1,i\\ne n}^K\\frac{ h_iP_i}{\\left(\\sigma ^2+\\sum _{j=1,j\\ne i}^Kh_{j}P_j\\right)}\\right).$ It is observed that $q(P_n)\\triangleq 1-\\sum _{i=1,i\\ne n}^K\\frac{h_iP_i h_n}{\\left(\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j\\right)}$ is a strictly increasing function with respect to $P_n$ .", "Then, the solution to $q(P_n)=0$ is unique.", "Consequently, the solution to $\\frac{\\partial f\\left(P\\right)}{\\partial P_n}=0$ is also unique since $\\frac{h_n}{\\sigma ^2+\\sum _{j=1}^K h_{j}P_j}$ is strictly positive.", "Denote the solution of $\\frac{\\partial f\\left(P\\right)}{\\partial P_n}=0$ as $\\tilde{P}_n$ , and we refer to $\\tilde{P}_n$ as the turning point.", "Then, on the left side of the turning point, $\\frac{\\partial f\\left(P\\right)}{\\partial P_n}$ is always negative, thus $f(0)>f(P_n), \\forall P_n \\in \\left[0,\\tilde{P}_n\\right]$ .", "On the right side of the turning point, $\\frac{\\partial f\\left(P\\right)}{\\partial P_n}$ is always positive, thus $f(P_n)<f(P_n^{mx}), \\forall P_n \\in \\left[\\tilde{P}_n,P_n^{mx}\\right]$ , where $P_n^{mx}=\\min \\left\\lbrace P_n^{pk},\\left(I-\\sum _{i=1,i\\ne n}^Kg_iP^_i\\right)\\big /{h_n}\\right\\rbrace $ .", "Thus, it is clear that the value of $f\\left(P\\right)$ can be increased by moving $P_n$ to the boundary.", "This contradicts with our assumption that $P^$ is the optimal solution.", "Thus, Lemma REF is proved.", "Based on the result of Lemma REF , we are able to obtain the following theorem.", "Theorem 2 The optimal solution $P^$ of Problem REF must be at the extreme point of the feasible region of Problem REF , i.e., at most one user's power allocation is fractional.", "Suppose the optimal solution is $P^$ .", "Thus, if $\\sum _{i=1}^K g_iP^_i< I$ , based the results of Lemma REF , it is clear that $P_i^, \\forall i$ is either equal to 0 or $P_i^{pk}$ .", "Thus, there is no fractional user.", "Now, we consider the case that $\\sum _{i=1}^K g_iP^_i=I$ .", "Suppose $P^_1$ and $P^_2$ are fractional, i.e., $0<P^_i<P^{pk}_i,\\forall i\\in \\left\\lbrace 1,2\\right\\rbrace $ .", "The interference constraint can be rewritten as $g_1P^_1+g_2P^_2+\\sum _{i=3}^Kg_iP^_i=I$ .", "For convenience, we define $Q\\triangleq I-\\sum _{i=3}^Kg_iP^_i$ .", "First, we consider the case that $\\frac{h_1}{g_1}>\\frac{h_2}{g_2}$ .", "Under this assumption, we write $P_2^$ as $P^_2=(Q-g_1P^_1)/g_2$ .", "For convenience, we denote (REF ) as $f\\left(P\\right)$ .", "Then, $f\\left(P\\right)$ can be rewritten as $f\\left(P^\\right)&=\\ln \\left(1+\\frac{h_{1}P^_1}{\\sigma ^2+\\sum _{j=3}^Kh_{j}P^_j+h_{2}(Q-g_1P^_1)/g_2}\\right)+\\ln \\left(1+\\frac{h_{2}(Q-g_1P^_1)/g_2}{\\sigma ^2+\\sum _{j=3}^Kh_{j}P^_j+h_{1}P^_1}\\right)\\nonumber \\\\&+\\sum _{i=3}^K\\ln \\left(1+\\frac{h_{i}P^_i}{\\sigma ^2+\\sum _{j=3, j\\ne i}^Kh_{j}P^_j+h_{1}P^_1+h_{2}(Q-g_1P^_1)/g_2}\\right).", "$ For notation convenience, define $C\\triangleq \\sigma ^2+\\sum _{j=3}^Kh_{j}P^_j$ and $D_i\\triangleq \\sigma ^2+\\sum _{j=3, j\\ne i}^Kh_{j}P^_j$ , then (REF ) can be rewritten as $f\\left(P^\\right)&=\\ln \\left(1+\\frac{h_{1}P^_1}{C+h_{2}(Q-g_1P^_1)/g_2}\\right)+\\ln \\left(1+\\frac{h_{2}(Q-g_1P^_1)/g_2}{C+h_{1}P^_1}\\right)\\nonumber \\\\&+\\sum _{i=3}^K\\ln \\left(1+\\frac{h_{i}P^_i}{D_i+h_{1}P^_1+h_{2}(Q-g_1P^_1)/g_2}\\right).$ Taking the derivative of $f\\left(P^\\right)$ with respect to $P^_1$ , we have $\\frac{\\partial f\\left(P^\\right)}{\\partial P^_1}&=\\frac{1}{C+\\frac{h_2}{g_2}Q+\\left(h_1-\\frac{h_2g_1}{g_2}\\right)P^_1}\\left(\\frac{h_1C+\\frac{h_1h_2}{g_2}Q}{C+\\frac{h_2}{g_2}Q-\\frac{h_2g_1}{g_2}P^_1}-\\frac{\\frac{h_2g_1}{g_2}C+\\frac{h_1h_2}{g_2}Q}{C+h_1P^_1}\\right.\\nonumber \\\\&\\left.-\\sum _{i=3}^K\\frac{\\left(h_1-\\frac{h_2g_1}{g_2}\\right)h_iP^_i}{D_i+\\frac{h_2}{g_2}Q+\\left(h_1-\\frac{h_2g_1}{g_2}\\right)P^_1}\\right),$ Since $\\frac{h_1}{g_1}>\\frac{h_2}{g_2}$ , $\\frac{\\partial f\\left(P^\\right)}{\\partial P^_1}$ is a strictly increasing function with respect to $P^_1$ .", "Thus, the solution to $\\frac{\\partial f\\left(P^\\right)}{\\partial P^_1}=0$ is unique.", "Denote the solution of $\\frac{\\partial f\\left(P^\\right)}{\\partial P^_1}=0$ as $\\tilde{P}^_1$ , and we refer to $\\tilde{P}^_1$ as the turning point.", "Then, on the left side of the turning point, $\\frac{\\partial f\\left(P^\\right)}{\\partial P^_1}$ is always negative, thus $f(0)>f(P_1), \\forall P_1 \\in \\left[0,\\tilde{P}^_1\\right]$ .", "On the right side of the turning point, $\\frac{\\partial f\\left(P^\\right)}{\\partial P^_1}$ is always positive, thus $f(P_1)<f(P_1^{pk}), \\forall P_1 \\in \\left[\\tilde{P}^_1,P_1^{pk}\\right]$ .", "Thus, it is clear that the value of $f\\left(P^\\right)$ can be increased by moving $P^_1$ to 0 or $P_1^{pk}$ .", "This contradicts with our assumption that $P^$ is the optimal solution.", "Now, we consider the case that $\\frac{h_1}{g_1}<\\frac{h_2}{g_2}$ .", "For this case, we can write $P_1^$ as $P_1^=(Q-g_2P^_2)/g_1$ .", "Then, using the same approach, we can show that the value of $f\\left(P^\\right)$ can be increased by moving $P^_2$ to 0 or $P_2^{pk}$ .", "Combining the above results, it is observed that at most one user's power allocation can be fractional.", "Theorem REF is thus proved.", "Based on Theorem REF , we can easily find the optimal solution $P^$ of Problem REF by searching the extreme points when the number of SUs is relatively small.", "However, when the number of SUs is large, this scheme may not be practical due to the high computing complexity.", "Fortunately, we are able to to show that with high probability, the optimal solution is D-TDMA when the number of SUs is large.", "This is given in Theorem REF .", "To prove Theorem REF , we need the following lemma.", "Lemma 2 The optimal solution $P^$ of Problem REF is $P_k^= \\min \\left\\lbrace P_k^{pk},\\frac{I^{pk}}{g_k}\\right\\rbrace $ where $k=\\mbox{argmax}_i~\\min \\left\\lbrace h_iP_i^{pk}, \\frac{h_i}{g_i}I^{pk}\\right\\rbrace $ , and $P_i^=0, \\forall i\\ne k$ , if the condition $\\ln \\left(1+h_{k}P_{k}^/\\sigma ^2\\right)\\ge 1$ holds.", "It is shown that in [17] (Theorem 4), the optimal solution for Problem REF without IPC is single-user transmission if at least one user satisfies $\\ln \\left(1+h_{i}P_{i}/\\sigma ^2\\right)\\ge 1$ , and the channel is assigned to the user with the largest $h_iP_i^{pk}$ at the current fading block.", "Our proof is mainly based on this result.", "Define $T_i\\triangleq \\min \\left\\lbrace P_i^{pk},\\frac{I^{pk}}{g_i}\\right\\rbrace $ .", "Suppose there exists at least one user satisfying the condition $\\ln \\left(1+h_{i}T_{i}/\\sigma ^2\\right)\\ge 1$ .", "Since the condition $\\ln \\left(1+h_{i}T_{i}/\\sigma ^2\\right)\\ge 1$ holds, it follows from [17] that the objective function of Problem REF is maximized when only one user transmits in each fading block.", "When there is only one user transmitting, the objective function of Problem REF reduces to $\\ln \\left(1+h_{i}P_{i}/\\sigma ^2\\right)$ , and the constraints reduces to $P_i \\le P_i^{pk}$ and $g_i P_i \\le I^{pk}$ .", "Clearly, the user with the largest $h_i T_i$ will maximize the objective function.", "Thus, the optimal allocation is $P_k^= T_k$ where $k=\\mbox{argmax}_i~ h_i T_i$ , and $P_i^=0, \\forall i\\ne k$ .", "Lemma REF is thus proved.", "Theorem 3 When the number of SUs is large, with high probability, the optimal solution of Problem REF is D-TDMA, i.e., one user transmitting in each fading block.", "From Lemma REF , it is known that if there exists at least one user satisfying the condition $\\ln \\left(1+h_{i}T_{i}/\\sigma ^2\\right)\\ge 1$ where $T_i\\triangleq \\min \\left\\lbrace P_i^{pk}, \\frac{I^{pk}}{g_i}\\right\\rbrace $ , the optimal solution of Problem REF is dynamic TDMA.", "Since all the channel power gains are i.i.d.", ", the probability of no user satisfying $\\ln \\left(1+h_{i}T_{i}/\\sigma ^2\\right)\\ge 1$ is $1-\\left(\\mbox{Prob}\\left\\lbrace \\ln \\left(1+h_{i}T_{i}/\\sigma ^2\\right)<1\\right\\rbrace \\right)^K$ .", "It is observed that this probability is a monotonic increasing function with respect to $K$ .", "Thus, when the number of SUs is large, with high probability, the condition will hold.", "Theorem REF is thus proved.", "Based on these results, we can solve Problem REF by searching the extreme points of the feasible region when the number of SUs is small, and by applying the D-TDMA scheme when the number of SUs is large.", "Readers may be interested in the number of SUs that is required to make the D-TDMA scheme optimal.", "We have investigated this issue in the simulation part given in Section .", "Please note that the condition given in Lemma REF is only a sufficient condition.", "In practice, the probability that D-TDMA is optimal is higher than $1-\\left(\\mbox{Prob}\\left\\lbrace \\ln \\left(1+h_{i}T_{i}/\\sigma ^2\\right)<1\\right\\rbrace \\right)^K$ .", "For the commonly used parameters, D-TDMA can achieve a near-optimal performance when the number of SUs is moderate (such as $K=5$ ).", "In the above, we have presented the approach to solve Problem REF in general.", "In the following, we show that if the SUs can be sorted in certain order according to their channel power gains, a simple algorithm with linear time complexity can be developed to solve Problem REF .", "Theorem 4 If the SUs can be sorted in the following order: $h_1>h_2>\\cdots >h_K$ and $\\frac{g_1}{h_1}<\\frac{g_2}{h_2}<\\cdots <\\frac{g_K}{h_K}$ .", "Then, there exists an optimal solution, for any two users indexed by $m$ and $n$ , if $m<n$ , their power allocation satisfies $P^_m \\ge P^_n$ .", "Assume that the users can be sorted in the following order: $h_1>h_2>\\cdots >h_K$ and $\\frac{g_1}{h_1}<\\frac{g_2}{h_2}<\\cdots <\\frac{g_K}{h_K}$ .", "Consider two users indexed by $m$ and $n$ with $m<n$ .", "Suppose at the optimal solution, $P^_m<P^_n$ .", "Now, we show this assumption does not hold by contradiction.", "For convenience, we define $P_i^\\prime \\triangleq h_iP_i$ .", "Then, Problem REF can be rewritten as Problem 5 $\\max _{P}~~ &\\sum _{i=1}^K \\ln \\left(1+\\frac{P_i^\\prime }{\\sigma ^2+\\sum _{j=1,j\\ne i}^K P_j^\\prime }\\right),\\\\\\mbox{s.t.", "}~~ &P_i^\\prime \\ge 0, ~\\forall i,\\\\&P_i^\\prime \\le h_iP^{pk},~\\forall i,\\\\&\\sum _{i=1}^K \\frac{g_i}{h_i}P_i^\\prime \\le I.$ In Theorem REF , we have proved that there is at most one fractional user.", "Thus, the value of $P^_m$ and $P^_n$ has the following two cases.", "Case 1: $0<P^_m<P^{pk}$ and $P^_{n}=P^{pk}$ .", "It follows that $P_m^\\prime =h_mP^_m$ and $P_{n}^\\prime =h_{n}P^{pk}$ .", "Then, based on the relationship between $P_m^\\prime $ and $P_n^\\prime $ , we have the following two subcases: Subcase 1: $P_m^\\prime <P_{n}^\\prime $ .", "Now, we swap the power allocation of these two users, i.e., $\\tilde{P}_m^\\prime =h_{n}P^{pk}$ and $\\tilde{P}_{n}^\\prime =h_mP^_m$ .", "Since $h_m>h_n$ , it is clear that $\\tilde{P}_m^\\prime =h_{n}P^{pk}<h_{m}P^{pk}$ .", "Since $P_m^\\prime <P_{n}^\\prime $ , it is clear that $\\tilde{P}_{n}^\\prime =h_mP^_m<h_nP^{pk}$ .", "At the same time, since $\\frac{g_m}{h_m}<\\frac{g_n}{h_n}$ , we have that $\\frac{g_m}{h_m}\\tilde{P}_m^\\prime +\\frac{g_n}{h_n}\\tilde{P}_n^\\prime <\\frac{g_m}{h_m}P_m^\\prime +\\frac{g_n}{h_n}P_n^\\prime $ .", "Thus, the power allocation $(\\tilde{P}_m^\\prime ,\\tilde{P}_n^\\prime )$ is a feasible solution of Problem REF .", "Besides, it is observed that the value of (REF ) under $(\\tilde{P}_m^\\prime ,\\tilde{P}_n^\\prime )$ is the same as that under $(P_m^\\prime ,P_n^\\prime )$ .", "Thus, $(\\tilde{P}_m^\\prime , \\tilde{P}_n^\\prime )$ is also an optimal solution of Problem REF .", "Subcase 2: $P_m^\\prime >P_{n}^\\prime $ .", "Now, we consider the power allocation $\\tilde{P}_m^\\prime =P_m^\\prime +\\Delta $ and $\\tilde{P}_{n}^\\prime =P_{n}^\\prime -\\Delta $ , where $\\Delta $ is a small constant such that $\\tilde{P}_m^\\prime \\le h_mP^{pk}$ and $\\tilde{P}_{n}^\\prime \\ge 0$ .", "Since $\\frac{g_m}{h_m}<\\frac{g_n}{h_n}$ , it is easy to verify that $\\frac{g_m}{h_m}\\tilde{P}_m^\\prime +\\frac{g_n}{h_n}\\tilde{P}_n^\\prime <\\frac{g_m}{h_m}P_m^\\prime +\\frac{g_n}{h_n}P_n^\\prime $ .", "Thus, the power allocation $(\\tilde{P}_m^\\prime ,\\tilde{P}_n^\\prime )$ is a feasible solution of Problem REF .", "Define (REF ) as $f\\left(P^\\prime \\right)$ .", "It follows that $f\\left(P^\\prime \\right)&=\\ln \\left(1+\\frac{P_m^\\prime }{\\sigma ^2+\\sum _{j\\ne m, n}^KP_j^\\prime +P_n^\\prime }\\right)+\\ln \\left(1+\\frac{P_n^\\prime }{\\sigma ^2+\\sum _{j\\ne m, n}^KP_j^\\prime +P_m^\\prime }\\right)\\nonumber \\\\&+\\sum _{i=1}^K\\ln \\left(1+\\frac{ P_i^\\prime }{\\sigma ^2+\\sum _{j\\ne i, m, n}^K P_j^\\prime +P_m^\\prime +P_n^\\prime }\\right).$ $f\\left(\\tilde{P}^\\prime \\right)&=\\ln \\left(1+\\frac{P_m^\\prime +\\Delta }{\\sigma ^2+\\sum _{j\\ne m, n}^KP_j^\\prime +P_n^\\prime -\\Delta }\\right)+\\ln \\left(1+\\frac{P_n^\\prime -\\Delta }{\\sigma ^2+\\sum _{j\\ne m, n}^KP_j^\\prime +P_m^\\prime +\\Delta }\\right)\\nonumber \\\\&+\\sum _{i=1}^K\\ln \\left(1+\\frac{ P_i^\\prime }{\\sigma ^2+\\sum _{j\\ne i, m, n}^KP_j^\\prime +P_m^\\prime +\\Delta +P_n^\\prime -\\Delta }\\right).$ For convenience, we define $Q^\\prime =\\sigma ^2+\\sum _{j\\ne m, n}^KP_j^\\prime $ .", "Then, it follows that $f\\left({P}^\\prime \\right)-f\\left(\\tilde{P}^\\prime \\right)&=\\ln \\left(\\frac{\\left(Q^\\prime +P_n^\\prime -\\Delta \\right)\\left(Q^\\prime +P_m^\\prime +\\Delta \\right)}{\\left(Q^\\prime +P_n^\\prime \\right)\\left(Q^\\prime +P_m^\\prime \\right)}\\right)\\nonumber \\\\&=\\ln \\left(\\frac{\\left(Q^\\prime +P_n^\\prime \\right)\\left(Q^\\prime +P_m^\\prime \\right)-\\left(P_m^\\prime -P_n^\\prime \\right)\\Delta -\\Delta ^2}{\\left(Q^\\prime +P_n^\\prime \\right)\\left(Q^\\prime +P_m^\\prime \\right)}\\right)\\nonumber \\\\&\\stackrel{a}{\\le } 0,$ where “a” results from the fact that $\\Delta >0$ and $P_m^\\prime >P_n^\\prime $ .", "This contradicts with our assumption.", "Case 2: $P^_m=0$ and $P^_{n}>0$ .", "It follows that $P_m^\\prime =0$ and $P_{n}^\\prime =h_{n}P^_{n}$ .", "We swap the power allocation of these two users, i.e., $\\tilde{P}_m^\\prime =h_{n}P^_{n}$ and $\\tilde{P}_{n}^\\prime =0$ .", "Since $h_m>h_n$ , it is clear that $\\tilde{P}_m^\\prime =h_{n}P^_{n}<h_{m}P^{pk}$ .", "At the same time, since $\\frac{g_m}{h_m}<\\frac{g_n}{h_n}$ , it can be verified that $\\frac{g_m}{h_m}\\tilde{P}_m^\\prime +\\frac{g_n}{h_n}\\tilde{P}_n^\\prime <\\frac{g_m}{h_m}P_m^\\prime +\\frac{g_n}{h_n}P_n^\\prime $ .", "Thus, the power allocation $(\\tilde{P}_m^\\prime ,\\tilde{P}_n^\\prime )$ is a feasible solution of Problem REF .", "Besides, it is observed that the value of (REF ) under $(\\tilde{P}_m^\\prime ,\\tilde{P}_n^\\prime )$ is the same as that under $(P_m^\\prime ,P_n^\\prime )$ .", "Thus, $(\\tilde{P}_m^\\prime , \\tilde{P}_n^\\prime )$ is also an optimal solution of Problem REF .", "Thus, combining the results Case 1 and Case 2, it is clear that there exists an optimal solution: for any two users indexed by $m$ and $n$ , if $m<n$ , their power allocation satisfies $P^_m \\ge P^_n$ .", "Theorem REF is thus proved.", "Based on this theorem, we can develop the Algorithm REF with linear complexity to solve Problem REF when the users can be sorted in the order stated in Theorem REF .", "[h!]", "Optimal power allocation for Problem REF with channel ordering [1] $g_1P^{pk}>{Q}$ $R^=\\log \\left(1+\\frac{h_1Q}{g_1\\sigma ^2}\\right)$ , $k^=1$ .", "Initialize $k=1$ .", "Find the largest $k$ that satisfies $\\sum _{i=1}^kg_iP^{pk}\\le I^{pk}$ and $k \\le K$ .", "Denote this $k$ as $k^L$ .", "Initialize $R(1)=\\log \\left(1+\\frac{h_1P^{pk}}{\\sigma ^2}\\right)$ , $k^=1$ , $R^=R(1)$ .", "$k=2$ to $k^L$ $R(k)=\\sum _{i=1}^k \\ln \\left(1+\\frac{h_{i}P^{pk}}{\\sigma ^2+\\sum _{j=1,j\\ne i}^k h_{j}P^{pk}}\\right)$ .", "$R(k)>R^$ $R^=R(k)$ , $k^=k$ .", "$R(k^L+1)=\\sum _{i=1}^{k^L+1} \\ln \\left(1+\\frac{h_{i}P_{i}}{\\sigma ^2+\\sum _{j=1,j\\ne i}^{k^L+1} h_{j}P_{j}}\\right)$ , where $P_{i}=P^{pk}, \\forall i \\le k^L$ , and $P_{k^L+1}=\\frac{I^{pk}-\\sum _{i=1}^{k^L}g_iP^{pk}}{g_{k^L+1}}$ .", "$R(k^L+1)>R^$ $R^=R(k^L+1)$ , $k^=k^L+1$ ." ], [ "Numerical results", "In this section, several numerical results are given to evaluate the performances of the proposed studies.", "All the channels involved are assumed to be Rayleigh fading, and thus the channel power gains are exponentially distributed.", "Unless specifically stated, we assume the mean of the channel power gains is one.", "The noise power $\\sigma ^2$ at SBS is also assumed to be 1.", "For convenience, the transmit power constraint at each SU is assumed to be the same.", "The numerical results presented here are obtained by taking average over 10000 rounds simulations.", "In this section, we only provide the simulation results for the ergodic sum-rate under peak TPC and peak IPC.", "No simulation results for the ergodic sum-rate under average TPC and average IPC are provided.", "This is due to the fact that we have shown that the optimal power allocation for the ergodic sum-rate with/without SIC under average TPC and average IPC is the same.", "As a result, the simulation results for this case are exactly the same as those shown in [12]." ], [ "Ergodic Sum-Rate with/without SIC", "First, we compare the ergodic sum-rate for the fading C-MAC with/without SIC under different combinations of TPC and IPC.", "In Fig.", "REF , Fig.", "REF and Fig.", "REF , we show the results for the fading C-MAC with $K=2,5,10$ , respectively.", "It is observed from all the figures that the ergodic sum-rate with SIC is always larger than that without SIC under the same TPC and IPC.", "This verifies our result that the ergodic sum-rate with SIC is a upper-bound of that without SIC.", "It is also observed that the gap between the ergodic sum-rate with SIC and that without SIC in general increases with the increasing of the number of SUs ($K$ ).", "The engineering insight behind this is that when the number of SUs is small in the C-MAC, it is not necessary to implement SIC at the SBS due to the cost and complexity.", "While when the number of SUs is large, it is worthwhile implementing SIC at the SBS to achieve a larger sum-rate.", "It is observed from all the curves that when the TPC of SU is large, the ergodic sum-rate gap with/without SIC is negligible.", "This is due to the following fact.", "When TPC is very large, TPC will not be the bottleneck, and the performance of the C-MAC will only depend on the IPC.", "It is proved in [19] that the ergodic sum-rate with/without SIC under only the IPC is the same.", "The optimal resource allocation for both cases are D-TDMA, and let the SU with the best $h_i/g_i$ to transmit in each fading block.", "Thus, from engineering design perspective, it is not necessary to implement SIC at the SBS when the TPC is relatively large as compared to the IPC." ], [ "Optimality of the D-TDMA", "In Fig.", "REF , we numerically compute the probability of D-TDMA being optimal for different number of SUs based on the condition given in Lemma REF .", "First, it is observed that the probability increases with the increasing of the number of SUs.", "It is also observed that the probability increases with the increasing of $P^{pk}$ for the same number of SUs.", "When $P^{pk}=5dB$ or $10dB$ , with only 10 SUs, the probability of D-TDMA being optimal is close to 1.", "When $P^{pk}=0dB$ , with 20 SUs, the probability of D-TDMA being optimal is more than $95\\%$ .", "These indicates that when the number of SUs is sufficiently large, the D-TDMA is optimal with a high probability.", "As pointed out previously, the condition given in Lemma REF is a sufficient condition.", "In practice, the probability that D-TDMA is optimal is higher than the probability shown in Fig.", "REF .", "In Fig.", "REF , we compare the ergodic sum-rate under the optimal power allocation and that under the D-TDMA when the number of SUs is 5.", "The results are obtained by averaging over 10000 rounds simulations.", "It is observed from the figure that when the TPC is larger than $0dB$ , D-TDMA can achieve the same ergodic sum-rate as the optimal power allocation.", "Even when the TPC is less than $0dB$ , the gap between the ergodic sum-rate under the optimal power allocation and that under the D-TDMA is not large.", "Thus, in general, we can use the D-TDMA scheme as a good suboptimal scheme when the number of SUs is larger than 5." ], [ "Ergodic Sum-Rate under the SIC-OP", "In this subsection, we compute the optimal power allocation for C-MAC with SIC first, and then apply the obtained power allocation to C-MAC without SIC (Problem REF ) as a suboptimal power allocation.", "For convenience, we denote the optimal power allocation for C-MAC with SIC as SIC-OP.", "We then compare the ergodic sum-rate (without SIC) under the SIC-OP with that under the D-TDMA.", "The ergodic sum-rate (without SIC) under the optimal power allocation are also included as a reference.", "In Fig.", "REF and Fig.", "REF , we assume that there are 5 SUs in the network, and the IPC is assumed to be $0 dB$ .", "In Fig.", "REF , we assume that the mean of the channel power gain is 1, i.e., $\\mathbb {E}\\lbrace h_i\\rbrace =\\mathbb {E}\\lbrace g_i\\rbrace =1, \\forall i$ .", "It is observed from Fig.", "REF that there exists one crossing-point, before which SIC-OP performs better than the D-TDMA.", "Actually, SIC-OP can achieve the same performance as the optimal power allocation when the TPC is sufficiently small.", "After the crossing point, D-TDMA performs better than the SIC-OP.", "When the TPC is sufficiently large, D-TDMA can achieve the same performance as the optimal power allocation.", "Similar results can be observed in Fig.", "REF , in which we assume that the mean of the channel power gain is 0.1, i.e., $\\mathbb {E}\\lbrace h_i\\rbrace =\\mathbb {E}\\lbrace g_i\\rbrace =0.1,\\forall i$ .", "The difference between Fig.", "REF and Fig.", "REF is that the crossing point of Fig.", "REF has a larger value of $P^{pk}$ as compared to the the crossing point of Fig.", "REF .", "This can be explained as follows.", "According to Lemma REF , the condition for D-TDMA being optimal is $\\ln \\left(1+h_{k}P_{k}^/\\sigma ^2\\right)\\ge 1$ .", "Thus, when the mean of $h_{k}$ is small, a larger $P_{k}^$ is needed to make D-TDMA optimal.", "In the following, we explain why SIC-OP can achieve the same performance as the optimal power allocation when the TPC is small.", "Now, we look at the sum-rate of MAC without SIC, which is $\\sum _{i=1}^K \\ln \\left(1+\\frac{ h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right)$ .", "When $\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}$ is small, it is equivalent to $\\sum _{i=1}^K \\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}$ , since $\\ln (1+x)\\approx x$ when $x$ is small.", "Further, since the TPC is small, $\\sum _{j=1,j\\ne i}^K h_{j}P_j\\approx \\sum _{j=1}^Kh_{j}P_j$ .", "Thus, $\\sum _{i=1}^K \\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\approx \\frac{ \\sum _{i=1}^K h_{i}P_i}{\\sigma ^2+\\sum _{j=1}^Kh_{j}P_j}=\\frac{1}{1+\\sigma ^2/\\sum _{i=1}^K h_{i}P_i}$ .", "Thus, maximizing $\\sum _{i=1}^K \\ln \\left(1+\\frac{h_{i}P_i}{\\sigma ^2+\\sum _{j=1,j\\ne i}^K h_{j}P_j}\\right)$ is equivalent to maximizing $\\sum _{i=1}^K h_{i}P_i$ .", "The sum-rate of MAC with SIC is obtained by maximizing $\\ln \\left(1+\\sum _{i=1}^Kh_{i}P_i\\right)$ , which is also equivalent to maximizing $\\sum _{i=1}^K h_{i}P_i$ , since the log function is a monotonic increasing function.", "Now, we explain why this observation is important.", "With this observation, we can solve Problem 3 by $\\max \\lbrace \\mbox{SIC-OP},\\mbox{D-TDMA}\\rbrace $ , which achieve the same performance as the optimal power allocation for most cases.", "Besides, the complexity is much lower than searching the extreme points, especially when the number of SUs is large." ], [ "Conclusions", "In this paper, we studied the ergodic sum-rate of a spectrum-sharing cognitive multiple access channel (C-MAC), where a secondary network (SN) with multiple secondary users (SUs) shares the spectrum band with a primary user (PU).", "We assumed an interference power constraint at the PU, individual transmit power constraints at the SUs, and to reduce decoding complexity, no successive interference cancellation (SIC) at the C-MAC.", "We investigated the optimal power allocation strategies for two types of power constraints: (1) average TPC and average IPC, and (2) peak TPC and peak IPC.", "For the average TPC and average IPC case, we proved that the optimal power allocation is dynamic time-division multiple-access (D-TDMA).", "For the peak TPC and peak IPC case, we proved that the optimal solution must be at the extreme points of the feasible region.", "We showed that D-TDMA is optimal with high probability when the number of SUs is large.", "We also showed through simulations that the optimal power allocation to maximize the ergodic sum-rate of the fading C-MAC with SIC is optimal or near-optimal for our setting when D-TDMA is not optimal.", "In addition, when some channel conditions are met, we gave a linear time complexity algorithm for finding the optimal power allocation." ] ]	1403.0355
[ [ "On deformations of Q-Fano threefolds II" ], [ "Abstract We investigate some coboundary map associated to a $3$-dimensional terminal singularity which is important in the study of deformations of singular $3$-folds.", "We prove that this map vanishes only for quotient singularities and a $A_{1,2}/4$-singularity, that is, a terminal singularity analytically isomorphic to a $\\mathbb{Z}_4$-quotient of the singularity $ (x^2+y^2 +z^3+u^2=0)$.", "As an application, we prove that a $\\mathbb{Q}$-Fano $3$-fold with terminal singularities can be deformed to one with only quotient singularities and $A_{1,2}/4$-singularities.", "We also treat the $\\mathbb{Q}$-smoothability problem on $\\mathbb{Q}$-Calabi--Yau $3$-folds." ], [ "Introduction", "We consider algebraic varieties over the complex number field $\\mathbb {C}$ .", "This paper is a continuation of [12].", "We study the $\\mathbb {Q}$ -smoothability of a $\\mathbb {Q}$ -Fano 3-fold $X$ via certain coboundary maps of local cohomology groups associated to the singularities on $X$ ." ], [ "$\\mathbb {Q}$ -smoothing of {{formula:858a2faf-461c-4ec2-ad1f-fc70249d692c}} -Fano 3-folds", "In this paper, a $\\mathbb {Q}$ -Fano 3-fold means a projective 3-fold with only terminal singularities whose anticanonical divisor is ample.", "A $\\mathbb {Q}$ -Fano 3-fold is an important object in the classification theory of algebraic 3-folds.", "It is one of the end products of the Minimal Model Program.", "Toward the classification of $\\mathbb {Q}$ -Fano 3-folds, it is fundamental to study their deformations.", "Locally, a 3-fold terminal singularity has a $\\mathbb {Q}$ -smoothing, that is, it can be deformed to a variety with only quotient singularities.", "In general, local deformations of singularities may not lift to a global deformation of a projective 3-fold as shown for Calabi–Yau 3-folds (cf.", "[8]).", "Nevertheless, Altınok–Brown–Reid ([1]) conjectured that a $\\mathbb {Q}$ -Fano 3-fold has a $\\mathbb {Q}$ -smoothing.", "(See Example REF for an example of a $\\mathbb {Q}$ -smoothing.)", "This conjecture aims to reduce the classification of $\\mathbb {Q}$ -Fano 3-folds to those with only quotient singularities.", "For example, there are several papers (cf.", "[2], [15]) on the classification of certain $\\mathbb {Q}$ -Fano 3-folds with only quotient singularities.", "Previously, deformations of $\\mathbb {Q}$ -Fano 3-folds are treated in several papers (cf.", "[9], [6], [16], [12]).", "In [12], the author proved that a $\\mathbb {Q}$ -Fano 3-fold with only “ordinary” terminal singularities has a $\\mathbb {Q}$ -smoothing.", "(See Definition REF for the ordinariness of the singularity.)", "In this article, we treat the remaining case, that is, a $\\mathbb {Q}$ -Fano 3-fold with non-ordinary terminal singularities.", "We can deform the non-ordinary terminal singularities except one special singularity as follows.", "Theorem 1.1 A $\\mathbb {Q}$ -Fano 3-fold can be deformed to one with only quotient singularities and $A_{1,2}/4$ -singularities.", "Here, an $A_{1,2}/4$ -singularity means a singularity analytically isomorphic to $0 \\in (x^2+y^2+z^3+u^2=0)/\\mathbb {Z}_4 \\subset \\mathbb {C}^4/\\mathbb {Z}_4 (1,3,2,1),$ where $x,y,z,u$ are coordinates on $\\mathbb {C}^4$ and $\\mathbb {C}^4/\\mathbb {Z}_4 (1,3,2,1)$ is the quotient of $\\mathbb {C}^4$ by an action of $\\mathbb {Z}_4=\\langle \\sigma \\rangle $ as follows: $\\sigma \\cdot (x,y,z,u) = (\\sqrt{-1} x, -\\sqrt{-1} y, -z, \\sqrt{-1} u).$ Although we do not know how to deal with $A_{1,2}/4$ -singularities, we believe that Theorem REF is useful for the classification.", "Remark 1.2 The author studied a deformation of a $\\mathbb {Q}$ -Fano 3-fold with its anticanonical element in [12] and [13].", "In [13], it is proved that, if a $\\mathbb {Q}$ -Fano 3-fold $X$ has a member $D \\in \|{-}K_X\|$ with only isolated singularities, then $X$ has a $\\mathbb {Q}$ -smoothing.", "In the proof, it is necessary to use [12] and Theorem REF in this paper.", "The existence of an elephant with mild singularities is discussed in [13] by showing several examples of $\\mathbb {Q}$ -Fano 3-folds." ], [ "Methods of the proof", "We use a method which is used in [12].", "Let $(U,p)$ be a germ of a 3-fold terminal singularity.", "The key tool of our method is the coboundary map $\\phi _{U}$ associated to some local cohomology group on a birational modification $\\tilde{U} \\rightarrow U$ .", "(See (REF ) for the definition of $\\phi _{U}$ .)", "If this map is nonzero, it is useful for finding a smoothing or a $\\mathbb {Q}$ -smoothing of a projective 3-fold.", "(cf.", "[10], [6], [12]) The following purely local statement is the main result of Section .", "Theorem 1.3 Let $(U,p)$ be a germ of a 3-fold terminal singularity which is not a quotient singularity.", "Then $\\phi _U =0$ if and only if $(U,p)$ is an $A_{1,2}/4$ -singularity.", "The map $\\phi _U$ is known to be nonzero when $(U,p)$ is Gorenstein ([10]) or $(U,p)$ is an ordinary singularity ([6], [12]).", "We calculate the coboundary map for a non-ordinary singularity.", "Let us mention about the proof of Theorem REF .", "Since a terminal singularity $(U,p)$ of index $r$ is a $\\mathbb {Z}_r$ -quotient of a hypersurface singularity $(V,q)$ , the set $T^1_{(U,p)}$ of first order deformations of $(U,p)$ is the $\\mathbb {Z}_r$ -invariant part of $T^1_{(V,q)}$ .", "The set $T^1_{(V,q)}$ can be written as $\\mathcal {O}_{V,q}/ J_{V,q}$ for the Jacobian ideal of $(V,q)$ .", "We calculate the map $\\phi _U$ by using this structure and the inequality (REF ) proved in [10].", "By Theorem REF (ii), the map $\\phi _U$ vanishes for a neighborhood $U$ of an $A_{1,2}/4$ -singularity.", "It seems that we need a new method to treat a $\\mathbb {Q}$ -Fano 3-fold with $A_{1,2}/4$ -singularities.", "(See Remark REF )" ], [ "$\\mathbb {Q}$ -smoothing of {{formula:fcc938f3-bd4f-4b06-90c4-636840929025}} -Calabi–Yau 3-folds", "As another corollary of Theorem REF , we obtain a similar result for $\\mathbb {Q}$ -Calabi–Yau 3-folds.", "Here, a $\\mathbb {Q}$ -Calabi–Yau 3-fold is a normal projective 3-fold with only terminal singularities whose canonical divisor is a torsion class.", "Let $r$ be the Gorenstein index of $X$ , that is, the minimal positive integer such that $\\mathcal {O}_X(rK_X) \\simeq \\mathcal {O}_X$ .", "The isomorphism $\\mathcal {O}_X(rK_X) \\simeq \\mathcal {O}_X$ determines the global index one cover $\\pi \\colon Y:= \\operatorname{Spec}\\oplus _{j=0}^{r-1} \\mathcal {O}_X(j K_X) \\rightarrow X$ .", "As a consequence of Theorem REF and the proof of [6], we obtain the following.", "Theorem 1.4 Let $X$ be a $\\mathbb {Q}$ -Calabi–Yau 3-fold.", "Assume that the global index one cover $Y \\rightarrow X$ is $\\mathbb {Q}$ -factorial.", "Then a $\\mathbb {Q}$ -Calabi–Yau 3-fold $X$ can be deformed to one with only quotient singularities and $A_{1,2}/4$ -singularities.", "Remark 1.5 Namikawa studied another invariant for terminal singularities and $\\mathbb {Q}$ -smoothability of $\\mathbb {Q}$ -Calabi–Yau 3-folds in his unpublished note.", "The invariant is $\\mu (X,x)$ defined in [10].", "It seems that this invariant also vanishes for a $A_{1,2}/4$ -singularity $(X,x)$ .", "So we do not know the $\\mathbb {Q}$ -smoothability of a $\\mathbb {Q}$ -Calabi-Yau 3-fold with $A_{1,2}/4$ -singularities." ], [ "Calculation of coboundary maps", "First, we introduce the coboundary map of local cohomology which is used in [12] to find a $\\mathbb {Q}$ -smoothing of a $\\mathbb {Q}$ -Fano 3-fold.", "(See also [10], [6].)", "Let $(U,p)$ be a germ of a 3-fold terminal singularity.", "Let $\\pi _U \\colon (V,q) \\rightarrow (U,p)$ be the index one cover.", "By the classification ([7], [11]), we see that $(V,q)$ is a hypersurface singularity and $\\pi _U$ is étale outside $p$ .", "Moreover, we have $(V,q) \\simeq ((f=0),0) \\subset (\\mathbb {C}^4, 0)$ for some $f \\in \\mathbb {C}[x,y,z,u]$ , where $x,y,z,u$ are coordinate functions on $\\mathbb {C}^4$ and $f$ satisfies $\\sigma \\cdot f = \\zeta _U f$ for the generator $\\sigma \\in G:= \\operatorname{Gal}(V/U) \\simeq \\mathbb {Z}_r$ and $\\zeta _U =\\pm 1$ .", "We define the ordinariness of a terminal singularity as follows.", "Definition 2.1 Let $(U,p)$ be a germ of a 3-fold terminal singularity.", "The germ $(U,p)$ is called ordinary (resp.", "non-ordinary) if $\\zeta _U=1$ (resp.", "$\\zeta _U= -1$ ).", "Remark 2.2 Let $(U,p)$ be a germ of a non-ordinary terminal singularity.", "By the classification ([7], [11]), we have $(U,p) \\simeq ((x^2 +y^2 +g(z,u)=0),0)/ \\mathbb {Z}_4 \\subset (\\mathbb {C}^4/ \\mathbb {Z}_4,0),$ where $g(z,u) \\in \\mathfrak {m}_{\\mathbb {C}^4,0}^2$ is some $\\mathbb {Z}_4$ -semi-invariant polynomial in $z, u$ and $\\sigma \\in \\mathbb {Z}_4$ acts on $\\mathbb {C}^4$ by $\\sigma \\cdot (x,y,z,u) \\mapsto (\\sqrt{-1}x, -\\sqrt{-1} y, -z, \\sqrt{-1}u)$ .", "Let $(U,p)$ be a germ of a 3-fold terminal singularity and $V$ its index one cover with the $\\mathbb {Z}_r$ -action as above.", "Let $\\nu \\colon \\tilde{V} \\rightarrow V$ be a $\\mathbb {Z}_r$ -equivariant resolution such that its exceptional divisor $F \\subset \\tilde{V}$ has SNC support and $\\tilde{V} \\setminus F \\simeq V \\setminus \\lbrace q \\rbrace $ .", "Let $V^{\\prime }:= V \\setminus \\lbrace q \\rbrace $ and $\\tau _V \\colon H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})) \\rightarrow H^2_{F}(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F-\\nu ^K_V))$ the coboundary map of the local cohomology.", "Note that the sheaf $\\mathcal {O}_{V}(-K_V)$ and $\\mathcal {O}_V$ are isomorphic as sheaves, but not isomorphic as $\\mathbb {Z}_r$ -equivariant sheaves.", "Let $\\tilde{\\pi } \\colon \\tilde{V} \\rightarrow \\tilde{U}:= \\tilde{V}/\\mathbb {Z}_r$ be the finite morphism induced by $\\pi $ and $E \\subset \\tilde{U}$ the exceptional locus of the birational morphism $\\mu \\colon \\tilde{U} \\rightarrow U$ induced by $\\nu $ .", "Let $U^{\\prime }:= U \\setminus \\lbrace p \\rbrace $ and $\\mathcal {F}_U^{(0)}$ the $\\mathbb {Z}_r$ -invariant part of $\\tilde{\\pi }_ \\Omega ^2_{\\tilde{V}}(\\log F)(-F -\\nu ^* K_V)$ .", "Then we have the coboundary map $\\phi _U \\colon H^1(U^{\\prime }, \\Omega ^2_{U^{\\prime }}(-K_{U^{\\prime }})) \\rightarrow H^2_{E}(\\tilde{U}, \\mathcal {F}_U^{(0)})$ which is the $\\mathbb {Z}_r$ -invariant part of $\\tau _V$ .", "We shall study these coboundary maps $\\tau _V$ and $\\phi _U$ in this section.", "For an ordinary terminal singularity, we can calculate the map $\\phi _U$ as follows.", "Theorem 2.3 (cf.", "[12]) Let $(U,p)$ be a germ of a 3-fold ordinary terminal singularity which is not a quotient singularity.", "Then we have $\\phi _U \\ne 0$ .", "In the following, we prepare ingredients for calculating $\\phi _U$ for a germ $(U,p)$ of a non-ordinary terminal singularities.", "We have $H^1_{F}( \\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F)) =0$ by the proof of [14].", "We also have $H^2(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F))=0$ by the Guillén–Navarro Aznar–Puerta–Steenbrink vanishing theorem.", "Thus we have an exact sequence $0 \\rightarrow H^1(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F) (-F -\\nu ^K_V)) \\rightarrow H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})) \\\\\\stackrel{\\tau _V}{\\rightarrow }H^2_{F}(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F - \\nu ^ K_V)) \\rightarrow 0$ The following inequality proved in [10] is useful for the calculation of the coboundary maps.", "Proposition 2.4 We have $\\dim \\operatorname{Ker}\\tau _V \\le \\dim \\operatorname{Im}\\tau _V.$ This is proved in Remark after [10].", "Let us recall the proof for the convenience of the reader.", "By the exact sequence (REF ), it is enough to show that $h^1(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F) (-F)) \\le h^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F)).$ We have a surjection $H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F)) \\rightarrow H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F))$ since we have $H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)\\otimes \\mathcal {O}_F) = \\operatorname{Gr}_F^2 H^5_{\\lbrace q\\rbrace }(V, \\mathbb {C}) =0$ .", "By the local duality, we have $H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F))^* \\simeq H^1(\\tilde{V}, \\Omega ^1_{\\tilde{V}}(\\log F)(-F)).$ Moreover we see that the differential homomorphism $d \\colon H^1(\\tilde{V}, \\Omega ^1_{\\tilde{V}}(\\log F)(-F)) \\rightarrow H^1(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F))$ is surjective by studying the spectral sequence $H^q(\\tilde{V}, \\Omega ^p_{\\tilde{V}}(\\log F)(-F)) \\Rightarrow \\mathbb {H}^{p+q}(\\tilde{V}, \\Omega ^{\\bullet }_{\\tilde{V}}(\\log F)(-F)) =0$ as in the proof of [10].", "Thus we obtain relations $h^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F)) \\ge h^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)) =h^1(\\tilde{V}, \\Omega ^1_{\\tilde{V}}(\\log F) (-F)) \\\\\\ge h^1(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F) (-F))$ and this implies (REF ).", "Let $T^1_{(V,q)}$ , $T^1_{(U,p)}$ be the sets of first order deformations of the germs $(V,q)$ and $(U,p)$ respectively.", "Recall that we have an isomorphism $T^1_{(V,q)} \\simeq \\mathcal {O}_{V,q}/ J_{V,q}$ of $\\mathcal {O}_{V,q}$ -modules for the Jacobian ideal $J_{V,q} \\subset \\mathcal {O}_{V,q}$ .", "Hence we have a surjective $\\mathcal {O}_{V,q}$ -module homomorphism $\\varepsilon \\colon \\mathcal {O}_{V,q} \\rightarrow T^1_{(V,q)}$ which sends $h \\in \\mathcal {O}_{V,q}$ to the corresponding deformation $\\varepsilon _h \\in T^1_{(V,q)}$ .", "Also we have a commutative diagram ${T^1_{(U,p)} [r]^{\\simeq \\ \\ \\ \\ \\ \\ \\ } @{^{(}->}[d] & H^1(U^{\\prime }, \\Omega ^2_{U^{\\prime }}(-K_{U^{\\prime }})) @{^{(}->}[d] \\\\T^1_{(V,q)} [r]^{\\simeq \\ \\ \\ \\ \\ \\ \\ } & H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})),}$ where the horizontal isomorphisms are restrictions by open immersions and the upper terms inject into the lower terms as the $\\mathbb {Z}_r$ -invariant parts.", "Note that we have the horizontal isomorphisms since $\\lbrace p \\rbrace \\hookrightarrow U$ and $\\lbrace q \\rbrace \\hookrightarrow V$ have codimensions 3, and the spaces $U$ and $V$ are Cohen-Macaulay.", "Thus we identify $T^1_{(V,q)}, T^1_{(U,p)}$ and $H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})), H^1(U^{\\prime }, \\Omega ^2_{U^{\\prime }}(-K_{U^{\\prime }}))$ respectively via these isomorphisms.", "We use the following notion of right equivalence ([4]).", "Definition 2.5 Let $\\mathbb {C}\\lbrace x_1,\\ldots , x_n \\rbrace $ be the convergent power series ring of $n$ variables.", "Let $f,g \\in \\mathbb {C} \\lbrace x_1,\\ldots , x_n \\rbrace $ .", "We say that $f$ is right equivalent to $g$ if there exists an automorphism $\\varphi $ of $\\mathbb {C} \\lbrace x_1, \\ldots , x_n \\rbrace $ such that $\\varphi (f) = g$ .", "We write this as $f \\overset{r}{\\sim } g$ .", "By using these ingredients, we calculate the coboundary map for a non-ordinary singularity.", "The following theorem and Theorem REF imply Theorem REF .", "Theorem 2.6 Let $(U,p)$ be a germ of a non-ordinary 3-fold terminal singularity which is not a quotient singularity.", "Assume that the index one cover $(V,q) \\lnot \\simeq ((x^2+y^2+z^3+u^2 =0),0)$ .", "Then we have $\\phi _U \\ne 0$ .", "Assume that $(V,q) \\simeq ((x^2+y^2+z^3+u^2 =0),0)$ .", "Then $\\phi _U =0$ .", "(i) Suppose that $\\phi _U =0$ .", "We show the claim by contradiction.", "We can write $g(z,u) = \\sum a_{i,j} z^i u^j \\in \\mathbb {C}[z,u]$ for some $a_{i,j} \\in \\mathbb {C}$ for $i,j \\ge 0$ .", "Since the generator $\\sigma \\in \\mathbb {Z}_4$ acts on $g$ by $\\sigma \\cdot g = -g$ and on $z^i u^j$ by $\\sigma \\cdot z^i u^j = \\sqrt{-1}^{2i +j} z^i u^j$ , we see that $a_{i,j} \\ne 0$ only if $2i+j \\equiv 2 \\mod {4}.$ Let $J_g:= (\\frac{\\partial g}{\\partial z}, \\frac{\\partial g}{\\partial u}) \\subset \\mathbb {C}[z,u]$ be the Jacobian ideal of the polynomial $g$ .", "Note that we have $T^1_{(V,q)} \\simeq \\mathbb {C}[z,u]/(g, J_g)$ since $\\varepsilon _x = \\varepsilon _y =0 \\in T^1_{(V,q)}$ .", "(Case 1) Assume that $a_{0,2} \\ne 0$ .", "We can write $g(z,u) = u^2(1+ h_1(z,u)) + h_2(z)$ for some polynomials $h_1(z,u) \\in (z,u) \\subset \\mathbb {C}[z,u]$ and $h_2(z) \\in (z) \\subset \\mathbb {C}[z]$ .", "Thus $g(z,u) \\in \\mathcal {O}_{\\mathbb {C}^2,0}$ is right equivalent to $u^2 + h_2(z)$ .", "We see that $h_2(z) \\in \\mathcal {O}_{\\mathbb {C},0}$ is right equivalent to $z^{2i_0+1}$ for some positive integer $i_0$ since $(g=0)$ has an isolated singularity and by the condition (REF ).", "Thus we have $(V,q) \\simeq ((x^2 +y^2 +z^{2i_0+1} + u^2=0),0).$ If $i_0 =1$ , it contradicts the assumption $(V,q) \\lnot \\simeq ((x^2+y^2+z^3+u^2 =0),0)$ .", "Hence we have $i_0 \\ge 2$ .", "By calculating the partial derivatives of $x^2+y^2 +z^{2 i_0 +1} +u^2$ , we see that $\\varepsilon _1, \\varepsilon _z, \\varepsilon _{z^2} \\in T^1_{(V,q)}$ are linearly independent and $\\dim T^1_{(V,q)} \\ge 3.$ On the other hand, we see that $\\tau _V(\\varepsilon _z) =0$ since we assumed $\\phi _U =0$ and $\\varepsilon _z \\in T^1_{(U,p)}$ .", "By this and the fact that $\\tau _V$ is an $\\mathcal {O}_{V,q}$ -module homomorphism, we obtain a surjection $\\mathbb {C}[z,u]/(z,u) \\rightarrow \\operatorname{Im}\\tau _V$ since $ \\varepsilon _u =0$ .", "By this surjection and $\\mathbb {C}[z,u]/(z,u) \\simeq \\mathbb {C}$ , we obtain $\\dim \\operatorname{Im}\\tau _V \\le 1$ .", "By this and the inequality (REF ), we obtain an inequality $\\dim T^1_{(V,q)} = \\dim \\operatorname{Im}\\tau _V + \\dim \\operatorname{Ker}\\tau _V \\le 1+1 =2$ and it is a contradiction.", "(Case 2) Assume that $a_{0,2} =0$ .", "Then we see that $a_{i,j} \\ne 0$ only if $2i +j \\ge 6$ by (REF ).", "Note that a monomial $z^i u^j$ with $2i+j \\ge 6$ is some multiple of either $z^3, z^2u^2, zu^4$ or $u^6$ .", "By computing partial derivatives of these monomials, we see that $(g,J_g) \\subset (z^2, z u^2, u^4)$ .", "Thus we see that $\\varepsilon _1, \\varepsilon _z, \\varepsilon _{zu}, \\varepsilon _u, \\varepsilon _{u^2}, \\varepsilon _{u^3} \\in T^1_{(V,q)}$ are linearly independent and we obtain $\\dim T^1_{(V,q)} \\ge 6.$ On the other hand, by the assumption $\\phi _U =0$ , we have $\\tau _V(\\varepsilon _z) =0, \\tau _V( \\varepsilon _{u^2}) =0$ since $\\varepsilon _z, \\varepsilon _{u^2} \\in T^1_{(U,p)}$ .", "Thus we have a relation $(z,u^2) \\subset \\operatorname{Ker}\\tau _V \\circ \\varepsilon \\subset \\mathcal {O}_{V,q}$ and obtain a surjection $\\mathbb {C}[z,u]/(z,u^2) \\rightarrow \\operatorname{Im}\\tau _V$ .", "This implies an inequality $\\dim \\operatorname{Im}\\tau _V \\le \\dim \\mathbb {C}[z,u]/(z,u^2) =2$ .", "By this inequality and the inequality (REF ), we have an inequality $\\dim T^1_{(V,q)} = \\dim \\operatorname{Ker}\\tau _V + \\dim \\operatorname{Im}\\tau _V \\le 2+2 =4.$ This contradicts (REF ).", "Hence we obtain $\\phi _U \\ne 0$ and finish the proof of (i).", "(ii) For non-negative integers $i,j$ , we set $b^{i,j}:= \\dim H^j(\\tilde{V}, \\Omega ^i_{\\tilde{V}}(\\log F)(-F)),$ $l^{i,j}:= \\dim H^j(F, \\Omega ^i_{\\tilde{V}}(\\log F) \\otimes \\mathcal {O}_F).$ Let $s_k(V,q)$ for $k=0,1,2,3$ be the Hodge number of the Milnor fiber of $(V,q)$ as in [14].", "By [14], we have $s_0=0, s_1= b^{1,1}, s_2 =b^{1,1} +l^{1,1}$ and $s_3 = l^{0,2}$ .", "We see that $l^{0,2} =0$ by [14].", "Since the sum $\\sum _{k=0}^3 s_k(V,q)$ is the Milnor number of $(V,q)$ , we obtain $2b^{1,1} +l^{1,1} = 2$ .", "Since $b^{1,1} \\ne 0$ by [10], we obtain $b^{1,1} =1, \\ \\ l^{1,1}=0.$ There exists an exact sequence $H^0(F, \\Omega ^1_{\\tilde{V}}(\\log F) \\otimes \\mathcal {O}_F) \\rightarrow H^1(\\tilde{V}, \\Omega ^1_{\\tilde{V}}(\\log F)(-F)) \\rightarrow H^1(\\tilde{V}, \\Omega ^1_{\\tilde{V}}(\\log F)) \\\\ \\rightarrow H^1(F, \\Omega ^1_{\\tilde{V}}(\\log F) \\otimes \\mathcal {O}_F).$ Since $l^{1,0} =0$ by [14], the both outer terms are zero and the homomorphism in the middle is an isomorphism.", "By this and (REF ), we have $\\mathbb {C} \\simeq H^1(\\tilde{V}, \\Omega ^1_{\\tilde{V}}(\\log F)) \\simeq H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F))^.$ Suppose that $\\tau _V(\\varepsilon _z) \\ne 0$ .", "Then $\\varepsilon _z \\notin \\operatorname{Ker}\\tau _V$ .", "This implies that $\\operatorname{Ker}\\tau _V =0$ since $T^1_{(V,q)} \\simeq \\mathbb {C}[z]/(z^2)$ as $\\mathbb {C}[z]$ -modules.", "Thus $\\mathbb {C}^2 \\simeq \\operatorname{Im}\\tau _V \\simeq H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F))$ .", "This contradicts (REF ).", "Thus we obtain $\\tau _V(\\varepsilon _z) =0$ .", "Since $T^1_{(U,p)} \\simeq \\mathbb {C}$ is generated by $\\varepsilon _z$ , we see that $\\phi _U=0$ .", "Thus we finish the proof of (ii).", "Now we prepare another coboundary map to study $\\mathbb {Q}$ -smoothability of a $\\mathbb {Q}$ -Calabi–Yau 3-fold.", "Let $(U,p)$ be a germ of a 3-fold terminal singularity and $V, \\tilde{V}, F, \\tilde{U}$ as before.", "We have the coboundary map $\\bar{\\tau }_V \\colon H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})) \\rightarrow H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(-\\nu ^ K_V))$ and this fits in the commutative diagram ${H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})) [r]^{\\bar{\\tau }_V} [d]^{\\tau _V} & H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(-\\nu ^* K_V)) \\\\H^2_F(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(\\log F)(-F - \\nu ^* K_V)), @{^{(}->}[ru]_{\\tau ^{\\prime }_V} &}$ where the injectivity of $\\tau ^{\\prime }_V$ is proved in the proof of [10].", "Let $\\bar{\\mathcal {F}}^{(0)}_U:= (\\tilde{\\pi }_* \\Omega ^2_{\\tilde{V}}(-\\nu ^* K_V))^{\\mathbb {Z}_r}$ be the $\\mathbb {Z}_r$ -invariant part.", "Let $\\bar{\\phi }_U \\colon H^1(U^{\\prime }, \\Omega ^2_{U^{\\prime }}(-K_{U^{\\prime }})) \\rightarrow H^2_E(\\tilde{U}, \\bar{\\mathcal {F}}_{U}^{(0)})$ be the coboundary map.", "It is the $\\mathbb {Z}_r$ -invariant part of $\\bar{\\tau }_V$ .", "As the $\\mathbb {Z}_r$ -invariant part of the diagram (REF ), we obtain the following diagram; ${H^1(U^{\\prime }, \\Omega ^2_{U^{\\prime }}(-K_{U^{\\prime }})) [r]^{\\bar{\\phi }_U} [d]^{\\phi _U} & H^2_E(\\tilde{U}, \\bar{\\mathcal {F}}_{U}^{(0)}) \\\\H^2_E(\\tilde{U},\\mathcal {F}_{U}^{(0)}).", "@{^{(}->}[ru]_{\\phi ^{\\prime }_U} &}$ By these arguments, we obtain the following result as a corollary of Theorem REF and Theorem REF .", "Corollary 2.7 Let $(U,p)$ be a germ of a 3-fold terminal singularity which is not a quotient singularity.", "Then $\\bar{\\phi }_U =0$ if and only if the germ $(U,p)$ is an $A_{1,2}/4$ -singularity.", "We use the blow-down morphism of deformations by a resolution $\\tilde{V} \\rightarrow V$ to find a $\\mathbb {Q}$ -smoothing.", "It is already used in several papers on deformations of singular 3-folds.", "(cf.", "[10], [9], [12]) Let $\\nu _* \\colon H^1(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(-K_{\\tilde{V}})) \\rightarrow H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }}))$ be the restriction homomorphism by the open immersion $V^{\\prime } \\hookrightarrow \\tilde{V}$ .", "We use this notation since there is a commutative diagram ${H^1(\\tilde{V}, \\Omega ^2_{\\tilde{V}}(-K_{\\tilde{V}})) [r]^{\\nu _} [d]^{\\simeq }& H^1(V^{\\prime }, \\Omega ^2_{V^{\\prime }}(-K_{V^{\\prime }})) [d]^{\\simeq } \\\\T^1_{\\tilde{V}} [r] & T^1_V,}$ where the lower horizontal homomorphism is the blow-down homomorphism of deformations ([17]).", "We can prove the relation $\\operatorname{Im}\\nu _ \\subset \\operatorname{Ker}\\tau _V = \\operatorname{Ker}\\bar{\\tau }_V$ by the same argument as in [12]." ], [ "Application to $\\mathbb {Q}$ -smoothing problems", "In [12], we proved the following.", "Theorem 3.1 Let $X$ be a $\\mathbb {Q}$ -Fano 3-fold.", "Then there exists a deformation $\\mathcal {X} \\rightarrow \\Delta ^1$ of $X$ over a unit disc $\\Delta ^1$ such that the general fiber $\\mathcal {X}_t$ for $t \\in \\Delta ^1 \\setminus \\lbrace 0 \\rbrace $ satisfies the following; For each singular point $p \\in \\mathcal {X}_t$ and its Stein neighborhood $U_p$ , the coboundary map $\\phi _{U_p}$ vanishes.", "As an application of this result and Theorem REF , we obtain a proof of Theorem REF as follows.", "By Theorem REF , we can deform a $\\mathbb {Q}$ -Fano 3-fold $X$ to one with only singularities $p_1, \\ldots , p_l$ such that $\\phi _{U_i} =0$ , where $U_i$ is a Stein neighborhood of $p_i$ for $i=1, \\ldots , l$ .", "By Theorem REF , such a terminal singularity is either a quotient singularity or an $A_{1,2}/4$ -singularity.", "Thus we finish the proof.", "Example 3.2 There exists an example of a $\\mathbb {Q}$ -Fano 3-fold with an $A_{1,2}/4$ -singularity.", "This example has a $\\mathbb {Q}$ -smoothing.", "Let $X:=X_{10}\\subset \\mathbb {P}(1,1,2,3,4)$ be a weighted hypersurface of degree 10 defined by the polynomial $f_{X_{10}}:= w^2(x_1^2 + x_2^2) +w(y^3+z^2) +x_1^{10} +x_2^{10} +y^5 + z^3 x_1,$ where $x_1, x_2, y, z, w$ are coodinates of weights $1,1,2,3,4$ , respectively.", "By perturbing the coefficients of the polynomial, we obtain that $\\operatorname{Sing}X = \\lbrace [0:0:0:1:0], [0:0:0:0:1] \\rbrace ,$ $p_z:= [0:0:0:1:0]$ is a $1/3(1,1,2)$ -singularity and $p_w:= [0:0:0:0:1]$ is an $A_{1,2}/4$ -singularity.", "Let $\\mathcal {X}:= (f_{X_{10}}+ t \\cdot yw^2=0) \\subset \\mathbb {P}(1,1,2,3,4) \\times \\mathbb {A}^1 \\rightarrow \\mathbb {A}^1$ be a deformation of $X$ , where $t$ is a coordinate of $\\mathbb {A}^1$ .", "Then we see that $\\mathcal {X}$ is a $\\mathbb {Q}$ -smoothing of $X$ .", "The general fiber $\\mathcal {X}_{t}$ has two $1/2(1,1,1)$ -singularities, a $1/3(1,1,2)$ -singularity and a $1/4(1,3,1)$ -singularity.", "Remark 3.3 We give a comment on a $\\mathbb {Q}$ -Fano 3-fold with $A_{1,2}/4$ -singularities.", "Let $X$ be a $\\mathbb {Q}$ -Fano 3-fold.", "The local-to-global spectral sequence of $\\operatorname{Ext}$ groups induces an exact sequence $\\operatorname{Ext}^1(\\Omega ^1_X, \\mathcal {O}_X) \\rightarrow H^0(X, \\underline{\\operatorname{Ext}}^1(\\Omega ^1_X, \\mathcal {O}_X) )\\rightarrow H^2(X, \\Theta _X),$ where $\\underline{\\operatorname{Ext}}^1$ is a sheaf of $\\operatorname{Ext}$ groups.", "Recall that $\\operatorname{Ext}^1(\\Omega ^1_X, \\mathcal {O}_X)$ and $ H^0(X, \\underline{\\operatorname{Ext}}^1(\\Omega ^1_X, \\mathcal {O}_X) )$ are the sets of first order deformations of $X$ and the singularities on $X$ , respectively.", "Thus, if we have $H^2(X, \\Theta _X) =0$ , we see that $X$ is $\\mathbb {Q}$ -smoothable.", "However, this approach does not work in general.", "Namikawa constructed an example of a Fano 3-fold $X$ with $A_{1,2}$ -singularities such that $H^2(X, \\Theta _X) \\ne 0$ ([9]).", "Here an $A_{1,2}$ -singularity is a hypersurface singularity locally isomorphic to $(x^2+y^2+z^3+u^2=0) \\subset \\mathbb {C}^4$ .", "This $X$ has a smoothing.", "The author expects that there also exists a $\\mathbb {Q}$ -Fano 3-fold $X$ with $A_{1,2}/4$ -singularities such that $H^2(X, \\Theta _X) \\ne 0$ .", "Thus we do not know $\\mathbb {Q}$ -smoothability of a $\\mathbb {Q}$ -Fano 3-fold with $A_{1,2}/4$ -singularities.", "As another application of Theorem REF , we obtain a proof of Theorem REF as follows.", "The proof is a modification of the proof of [6].", "We sketch the proof for the convenience of the reader.", "First we prepare notations to define the diagram (REF ).", "Let $p_1,\\ldots , p_l \\in X$ be the non-quotient singularities and $U_1, \\ldots , U_l$ their Stein neighborhoods.", "Let $\\nu \\colon \\tilde{Y} \\rightarrow Y$ be a $\\mathbb {Z}_r$ -equivariant resolution such that its exceptional divisor $F$ is a SNC divisor and $\\tilde{Y} \\setminus F \\simeq Y \\setminus \\nu ^{-1}(\\lbrace p_1, \\ldots , p_l \\rbrace )$ .", "Let $\\tilde{\\pi } \\colon \\tilde{Y} \\rightarrow \\tilde{X}:= \\tilde{Y}/ \\mathbb {Z}_r$ be the quotient morphism and $\\mu \\colon \\tilde{X} \\rightarrow X$ the induced birational morphism with the exceptional divisor $E$ .", "Let $V_i:= \\pi ^{-1}(U_i)$ , $\\tilde{V}_i:= \\nu ^{-1}(V_i)$ , $F_i:= F \\cap \\tilde{V_i}$ and $\\nu _i:= \\nu \|_{\\tilde{V}_i} \\colon \\tilde{V}_i \\rightarrow V_i$ be the restrictions.", "Let $\\tilde{U}_i:= \\mu ^{-1}(U_i)$ , $E_i := E \\cap \\tilde{U}_i$ and $\\tilde{\\pi }_i:= \\tilde{\\pi }\|_{\\tilde{V}_i} \\colon \\tilde{V}_i \\rightarrow \\tilde{U}_i$ the induced finite morphism.", "Let $\\bar{\\mathcal {F}}^{(0)}:= \\left( \\tilde{\\pi }_* \\Omega ^2_{\\tilde{Y}}(- \\nu ^* K_V) \\right)^{\\mathbb {Z}_r}$ be the $\\mathbb {Z}_r$ -invariant part and $\\bar{\\mathcal {F}}_i^{(0)}:= \\bar{\\mathcal {F}}^{(0)}\|_{\\tilde{U}_i}$ its restriction.", "Then we have the diagram ${H^1(X^{\\prime }, \\Omega ^2_{X^{\\prime }}(-K_{X^{\\prime }})) [r]^{\\oplus \\psi _i} [d]^{\\oplus p_{U_i}} &\\oplus _{i=1}^l H^2_{E_i}(\\tilde{X}, \\bar{\\mathcal {F}}^{(0)}) [r]^{\\oplus B_i} [d]_{\\oplus \\varphi _i}^{\\simeq } &H^2(\\tilde{X}, \\bar{\\mathcal {F}}^{(0)}) \\\\\\oplus _{i=1}^l H^1(U^{\\prime }_i, \\Omega ^2_{U^{\\prime }_i}(-K_{U^{\\prime }_i})) [r]^{\\bar{\\phi }_i} &\\oplus _{i=1}^l H^2_{E_i}(\\tilde{U}_i, \\bar{\\mathcal {F}}_i^{(0)}), &}$ where $X^{\\prime }:= X \\setminus \\lbrace p_1, \\ldots , p_l \\rbrace $ and $U_i^{\\prime } := U_i \\cap X^{\\prime }$ .", "Let $V^{\\prime }_i := \\pi ^{-1}(U^{\\prime }_i)$ .", "Note that $B_i \\circ \\varphi _i^{-1} \\circ \\bar{\\phi }_i$ is the $\\mathbb {Z}_r$ -invariant part of the composition $H^1(V^{\\prime }_i, \\Omega ^2_{V^{\\prime }_i}(-K_{V^{\\prime }_i})) \\rightarrow H^2_{F_i}(\\tilde{V}_i, \\Omega ^2_{\\tilde{V}_i}(-\\nu _i^* K_{V_i}))\\rightarrow H^2_{F_i}(\\tilde{Y}, \\Omega ^2_{\\tilde{Y}}(-\\nu ^* K_{Y})) \\\\\\rightarrow H^2(\\tilde{Y}, \\Omega ^2_{\\tilde{Y}}(-\\nu ^* K_{Y})).$ We see that this is zero by [10] since we assumed that $Y$ is $\\mathbb {Q}$ -factorial.", "Thus we also see that $B_i \\circ \\varphi _i^{-1} \\circ \\bar{\\phi }_i =0$ .", "There exists an element $\\eta _i \\in H^1(U^{\\prime }_i, \\Omega ^2_{U^{\\prime }_i}(-K_{U^{\\prime }_i}))$ such that $\\bar{\\phi }_i(\\eta _i) \\ne 0$ by Theorem REF .", "Since $B_i \\circ \\varphi _i^{-1} \\circ \\bar{\\phi }_i(\\eta _i) =0$ , there exists $\\eta \\in H^1(X^{\\prime }, \\Omega ^2_{X^{\\prime }}(-K_{X^{\\prime }}))$ such that $\\psi _i(\\eta )= \\varphi _i^{-1}(\\phi _i(\\eta _i))$ .", "By the relation (REF ) and $p_{U_i}(\\eta ) - \\eta _i \\in \\operatorname{Ker}\\bar{\\phi }_i$ , we see that $p_{U_i}(\\eta ) \\notin \\operatorname{Im}(\\nu _i)_*$ , where we use the inclusion $H^1(U^{\\prime }_i, \\Omega ^2_{U^{\\prime }_i}(-K_{U^{\\prime }_i})) \\subset H^1(V^{\\prime }_i, \\Omega ^2_{V^{\\prime }_i}(-K_{U^{\\prime }_i}))$ .", "By arguing as in the proof of [12], we can deform singularity $p_i \\in U_i$ as long as $\\bar{\\phi }_i \\ne 0$ .", "By Corollary REF , we obtain a required deformation since the deformations of a $\\mathbb {Q}$ -Calabi–Yau 3-fold are unobstructed ([8])." ], [ "Acknowledgments", "This paper is a part of the author's Ph.D thesis submitted to University of Warwick.", "The author would like to express deep gratitude to Prof.", "Miles Reid for his warm encouragement and valuable comments.", "He would like to thank Professor Yoshinori Namikawa for useful conversations.", "Part of this paper is written during the author's stay in Princeton university and the university of Tokyo.", "He would like to thank Professors János Kollár and Yujiro Kawamata for useful comments and nice hospitality.", "He thanks the referee for useful suggestions.", "He is partially supported by Warwick Postgraduate Research Scholarship." ] ]	1403.0212
[ [ "Sampling formulas for one-parameter groups of operators in Banach spaces" ], [ "Abstract We extend some results about sampling of entire functions of exponential type to Banach spaces.", "By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\\mathbf{B}_{\\sigma}(D),\\>\\>\\sigma>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line.", "This property allows to reduce sampling problems for $e^{tD}f$ with $f\\in \\mathbf{B}_{\\sigma}(D)$ to known sampling results for regular functions of exponential type $\\sigma$." ], [ "Introduction", "The goal of the paper is to extend some theorems about sampling of entire functions of exponential type to Banach spaces.", "Our framework starts with considering a generator $D$ of one-parameter strongly continuous group of operators $e^{tD}$ in a Banach space $E$ .", "The operator $D$ is used to define analogs of Bernstein subspaces $\\mathbf {B}_{\\sigma }(D)$ .", "The main property of vectors $f$ in $\\mathbf {B}_{\\sigma }(D)$ is that corresponding trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line.", "This fact allows to apply known sampling theorems to every function of the form $\\left<e^{tD}f, \\:g^{}\\right>\\in \\mathbb {B}_{\\sigma }^{\\infty }(\\mathbb {R}),\\:\\:\\:g^{}\\in E^{},$ or of the form $t^{-1}\\left<e^{tD}f-f, \\:g^{}\\right>\\in \\mathbb {B}_{\\sigma }^{2}(\\mathbb {R})$ , where $ \\mathbb {B}_{\\sigma }^{\\infty }(\\mathbb {R}),\\:\\: \\mathbb {B}_{\\sigma }^{2}(\\mathbb {R})$ are classical Bernstein spaces on $\\mathbb {R}$ .", "In remark REF we demonstrate that the assumption that $D$ generates a group of operators is somewhat essential if one wants to have non-trivial Bernstein spaces.", "In section 2 we give a few different descriptions of these spaces and one of them explores resent results in [7] which extend classical Boas formulas [4], [3] for entire functions of exponential type.", "Note, that in turn, Boas formulas are generalizations of Riesz formulas [19], [18] for trigonometric polynomials.", "Section contains two sampling-type formulas which explore regularly spaced samples and section is about two results in the spirit of irregular sampling.", "Section contains an application to inverse Cauchy problem for abstract Schrödinger equation.", "Note, that if $e^{tD} ,\\:\\:\\:t\\in \\mathbb {R}, $ is a group of operators in a Banach space $E$ then any trajectory $e^{tD}f,\\:\\:\\:f\\in E,$ is completely determined by any (single) sample $e^{\\tau D}f,$ because for any $t\\in \\mathbb {R}$ $e^{tD}f=e^{(t-\\tau )D} \\left(e^{\\tau D}f\\right).$ Our results in sections and have, however, a different nature.", "They represent a trajectory $e^{tD}f$ as a \"linear combination\" of a countable number of its samples.", "Such kind results can be useful when the entire group of operators $e^{tD}$ is unknown and only samples $e^{t_{k}D}f$ of a trajectory $e^{tD}f$ are given.", "It seems to be very interesting that not matter how complicated one-parameter group can be (think, for example, about a Schrödinger operator $D=-\\Delta +V(x)$ and the corresponding group $e^{itD}$ in $L_{2}(\\mathbb {R}^{d})$ ) the formulas (REF ), (REF ), (REF ), (REF ), (REF ) are universal in the sense that they contain the same coefficients and the same sets of sampling points.", "It was my discussions with Paul Butzer and Gerhard Schmeisser during Sampta 2013 in Jacobs University in Bremen of their beautiful work with Rudolf Stens [7], that stimulated my interest in the topic of the present paper.", "I am very grateful to them for this." ], [ "Bernstein vectors in Banach spaces", "We assume that $D$ is a generator of one-parameter group of isometries $e^{ tD}$ in a Banach space $E$ with the norm $\\Vert \\cdot \\Vert $ (for precise definitions see [5], [9]).", "The notations $\\mathcal {D}^{k}$ will be used for the domain of $D^{k}$ , and notation $\\mathcal {D}^{\\infty }$ for $\\bigcap _{k\\in \\mathbb {N}}\\mathcal {D}^{k}$ .", "Definition 2.1 The subspace of exponential vectors $\\mathbf {E}_{\\sigma }(D), \\:\\:\\sigma \\ge 0,$ is defined as a set of all vectors $f$ in $\\mathcal {D}^{\\infty }$ for which there exists a constant $C(f,\\sigma )>0$ such that $\\Vert D^{k}f\\Vert \\le C(f,\\sigma )\\sigma ^{k}, \\:\\:k\\in \\mathbb {N}.$ Note, that every $\\mathbf {E}_{\\sigma }(D)$ is clearly a linear subspace of $E$ .", "What is really important is the fact that union of all $\\mathbf {E}_{\\sigma }(D)$ is dense in $E$ (Corollary REF ).", "Remark 2.2 It is worth to stress that if $D$ generates a strongly continuous bounded semigroup then the set $\\bigcup _{\\sigma \\ge 0}\\mathbf {E}_{\\sigma }(D)$ may not be dense in $E$ .", "Indeed, consider a strongly continuous bounded semigroup $T(t)$ in $L_{2}(0,\\infty )$ defined for every $f\\in L_{2}(0,\\infty )$ as $T(t)f(x)=f(x-t),$ if $\\:x\\ge t$ and $T(t)f(x)=0,$ if $\\:\\:0\\le x<t$ .", "Inequality (REF ) implies that if $ f\\in \\mathbf {E}_{\\sigma }(D)$ then for any $g\\in L_{2}(0,\\infty )$ the function $\\left<T(t)f,\\:g\\right>$ is analytic in $t$ .", "Thus if $g$ has compact support then $\\left<T(t)f,\\:g\\right>$ is zero for all $t$ which implies that $f$ is zero.", "In other words in this case every space $\\mathbf {E}_{\\sigma }(D)$ is trivial.", "Definition 2.3 The Bernstein subspace $\\mathbf {B}_{\\sigma }(D), \\:\\:\\sigma \\ge 0,$ is defined as a set of all vectors $f$ in $E$ which belong to $\\mathcal {D}^{\\infty }$ and for which $\\Vert D^{k}f\\Vert \\le \\sigma ^{k}\\Vert f\\Vert , \\:\\:k\\in \\mathbb {N}.$ Lemma 2.4 ([11]) Let $D$ be a generator of an one parameter group of operators $e^{tD }$ in a Banach space $E$ and $\\Vert e^{tD}f\\Vert =\\Vert f\\Vert $ .", "If for some $f\\in E$ there exists an $\\sigma >0$ such that the quantity $\\sup _{k\\in N} \\Vert D^{k}f\\Vert \\sigma ^{-k}=R(f,\\sigma )$ is finite, then $R(f,\\sigma )\\le \\Vert f\\Vert .$ By assumption $\\Vert D^{r}f\\Vert \\le R(f,\\sigma )\\sigma ^{r}, r\\in \\mathbb {N}$ .", "Now for any complex number $z$ we have $ \\left\\Vert e^{zD}f\\right\\Vert =\\left\\Vert \\sum ^{\\infty }_{r=0}(z^{r}D^{r}g)/r!\\right\\Vert \\le R(f,\\sigma ) \\sum ^{\\infty }_{r=0}\|z\|^{r}\\sigma ^{r}/r!=R(f,\\sigma )e^{\|z\|\\sigma }.$ It implies that for any functional $h^{}\\in E^{}$ the scalar function $\\left<e^{zD}f,h^{}\\right>$ is an entire function of exponential type $\\sigma $ which is bounded on the real axis $\\mathbb {R}$ by the constant $\\Vert h^{}\\Vert \\Vert f\\Vert $ .", "An application of the Bernstein inequality gives $\\left\\Vert <e^{tD}D^{k}f,h^{}>\\right\\Vert _{C(\\mathbb {R})}=\\left\\Vert \\left(\\frac{d}{dt}\\right)^{k}\\left<e^{tD}f,h^{}\\right>\\right\\Vert _{C(\\mathbb {R})} \\le \\sigma ^{k}\\Vert h^{}\\Vert \\Vert f\\Vert .$ The last one gives for $t=0$ $ \\left\|\\left<D^{k}f,h^{}\\right>\\right\|\\le \\sigma ^{k} \\Vert h^{}\\Vert \\Vert f\\Vert .$ Choosing $h^{}$ such that $\\Vert h^{}\\Vert =1$ and $\\left<D^{k}f,h^{}\\right>=\\Vert D^{k}f\\Vert $ we obtain the inequality $\\Vert D^{k}f\\Vert \\le \\sigma ^{k} \\Vert f\\Vert , k\\in N$ , which gives $R(f,\\sigma )=\\sup _{k\\in \\mathbb {N} }(\\sigma ^{-k}\\Vert D^{k}f\\Vert )\\le \\Vert f\\Vert .$ Lemma is proved.", "Theorem 2.5 ([17]) Let $D$ be a generator of one-parameter group of operators $e^{tD }$ in a Banach space $E$ and $\\Vert e^{tD}f\\Vert =\\Vert f\\Vert $ .", "Then for every $\\sigma \\ge 0$ $\\mathbf {B}_{\\sigma }(D)= \\mathbf {E}_{\\sigma }(D) , \\:\\:\\:\\sigma \\ge 0,$ The inclusion $\\mathbf {B}_{\\sigma }(D)= \\mathbf {E}_{\\sigma }(D) , \\:\\:\\:\\sigma \\ge 0,$ is obvious.", "The opposite inclusion follows from the previous Lemma.", "Motivated by results in [7] we introduce the following bounded operators $\\mathcal {B}_{D}^{(2m-1)}(\\sigma )f=\\left(\\frac{\\sigma }{\\pi }\\right)^{2m-1}\\sum _{k\\in \\mathbb {Z}}(-1)^{k+1}A_{m,k}e^{\\frac{\\pi }{\\sigma }(k-1/2)D}f,\\:\\: f\\in E, \\:\\:\\sigma >0,\\:\\:\\:m\\in \\mathbb {N},$ $\\mathcal {B}_{D}^{(2m)}(\\sigma )f=\\left(\\frac{\\sigma }{\\pi }\\right)^{2m}\\sum _{k\\in \\mathbb {Z}}(-1)^{k+1}B_{m,k}e^{\\frac{\\pi k}{\\sigma }D}f, \\:\\:f\\in E,\\: \\sigma >0,\\:m\\in \\mathbb {N},$ where $A_{m,k}=(-1)^{k+1} \\rm sinc ^{(2m-1)}\\left(\\frac{1}{2}-k\\right)=\\frac{(2m-1)!", "}{\\pi (k-\\frac{1}{2})^{2m}}\\sum _{j=0}^{m-1}\\frac{(-1)^{j}}{(2j)!", "}\\left(\\pi (k-\\frac{1}{2})\\right)^{2j},\\:\\:\\:m\\in \\mathbb {N},$ for $k\\in \\mathbb {Z}$ and $B_{m,k}=(-1)^{k+1} \\rm sinc ^{(2m)}(-k)=\\frac{(2m)!", "}{\\pi k^{2m+1}}\\sum _{j=0}^{m-1}\\frac{(-1)^{j}(\\pi k)^{2j+1}}{(2j+1)!", "},\\:\\:\\:m\\in \\mathbb {N},\\:\\:\\:k\\in \\mathbb {Z}\\setminus {0},$ and $B_{m,0}=(-1)^{m+1} \\frac{\\pi ^{2m}}{2m+1},\\:\\:\\:m\\in \\mathbb {N}.$ Both series converge in $E$ due to the following formulas (see [7]) $\\left(\\frac{\\sigma }{\\pi }\\right)^{2m-1}\\sum _{k\\in \\mathbb {Z}}\\left\|A_{m,k}\\right\|=\\sigma ^{2m-1},\\:\\:\\:\\:\\:\\left(\\frac{\\sigma }{\\pi }\\right)^{2m}\\sum _{k\\in \\mathbb {Z}}\\left\|B_{m,k}\\right\|=\\sigma ^{2m}.$ Since $\\Vert e^{tD}f\\Vert =\\Vert f\\Vert $ it implies that $\\Vert \\mathcal {B}_{D}^{(2m-1)}(\\sigma )f\\Vert \\le \\sigma ^{2m-1}\\Vert f\\Vert ,\\:\\:\\:\\:\\:\\Vert \\mathcal {B}_{D}^{(2m)}(\\sigma )f\\Vert \\le \\sigma ^{2m}\\Vert f\\Vert ,\\:\\:\\:f\\in E.$ For the following theorem see [11], [12], [16], [17].", "Theorem 2.6 If $D$ generates a one-parameter strongly continuous bounded group of operators $e^{tD}$ in a Banach space $E$ then the following conditions are equivalent: $f$ belongs to $\\mathbf {B}_{\\sigma }(D)$ .", "The abstract-valued function $e^{tD}f$ is entire abstract-valued function of exponential type $\\sigma $ which is bounded on the real line.", "For every functional $g^{}\\in E^{}$ the function $\\left<e^{tD}f,\\:g^{}\\right>$ is entire function of exponential type $\\sigma $ which is bounded on the real line.", "The following Boas-type interpolation formulas hold true for $r\\in \\mathbb {N}$ $D^{r}f=\\mathcal {B}_{D}^{(r)}(\\sigma )f,\\:\\:\\:\\:\\:f\\in \\mathbf {B}_{\\sigma }(D).$ Corollary 2.1 Every $\\mathbf {B}_{\\sigma }(D)$ is a closed linear subspace of $E$ .", "Let's introduce the operator $\\:\\:\\Delta ^{m}_{s}f=(I-e^{sD})^{m}f, \\:\\:m\\in \\mathbb {N},$ and the modulus of continuity [5] $\\Omega _{m}(f,s)=\\sup _{\|\\tau \|\\le s}\\left\\Vert \\Delta ^{m}_{\\tau }f\\right\\Vert .$ The following theorem is proved in [14], [16], [17].", "Theorem 2.7 There exists a constant $C>0$ such that for all $\\sigma >0$ and all $f\\in \\mathcal {D}^{k}$ $inf_{ g\\in \\mathbf {B}_{\\sigma }(D) }\\Vert f-g\\Vert \\le C\\sigma ^{-k}\\Omega _{m-k}\\left(D^{k}f, \\sigma ^{-1}\\right),0\\le k\\le m.$ Corollary 2.2 The set $\\bigcup _{\\sigma \\ge 0}\\mathbf {B}_{\\sigma }(D)$ is dense in $E$ .", "Definition 2.8 For a given $f\\in E$ the notation $\\sigma _{f}$ will be used for the smallest finite real number (if any) for which $\\Vert D^{k}f\\Vert \\le \\sigma _{f}^{k}\\Vert f\\Vert ,\\:\\:\\:k\\in \\mathbb {N}.$ If there is no such finite number we assume that $\\sigma _{f}=\\infty $ .", "Now we are going to prove another characterization of Bernstein spaces.", "In the case of Hilbert spaces corresponding result was proved in [13], [15].", "Theorem 2.9 Let $f\\in E$ belongs to a space $\\mathbf {B}_{\\sigma }(D),$ for some $0<\\sigma <\\infty .$ Then the following limit exists $d_f=\\lim _{k\\rightarrow \\infty } \\Vert D^kf\\Vert ^{1/k} $ and $d_f=\\sigma _f.$ Conversely, if $f\\in \\mathcal {D}^{\\infty }$ and $d_f=\\lim _{k\\rightarrow \\infty }\\Vert D^k f\\Vert ^{1/k},$ exists and is finite, then $f\\in \\mathbf {B}_{d_f}(D)$ and $d_f=\\sigma _f .$ Let us introduce the Favard constants (see [1], Ch.", "V) which are defined as $K_{j}=\\frac{4}{\\pi }\\sum _{r=0}^{\\infty }\\frac{(-1)^{r(j+1)}}{(2r+1)^{j+1}},\\:\\:\\:j,\\:\\:r\\in \\mathbb {N}.$ It is known [1], Ch.", "V, that the sequence of all Favard constants with even indices is strictly increasing and belongs to the interval $[1,4/ \\pi )$ and the sequence of all Favard constants with odd indices is strictly decreasing and belongs to the interval $(\\pi /4, \\pi /2],$ i.e., $K_{2j}\\in [1,4/ \\pi ), \\; K_{2j+1}\\in (\\pi /4, \\pi /2].$ We will need the following generalization of the classical Kolmogorov inequality.", "It is worth noting that the inequality was first proved by Kolmogorov for $L^\\infty (\\mathbb {R})$ and later extended to $L^p(\\mathbb {R})$ for $1\\le p < \\infty $ by Stein [22] and that is why it is known as the Stein-Kolmogorov inequality.", "Lemma 2.10 Let $f\\in \\mathcal {D}^{\\infty } .$ Then, the following inequality holds $\\left\\Vert D^{k}f\\right\\Vert ^n \\le C_{k,n}\\Vert D^{n}f\\Vert ^{k}\\Vert f\\Vert ^{n-k},\\:\\:\\:0\\le k \\le n,$ where $C_{k,n}= (K_{n-k})^n/(K_{n})^{n-k}.$ Indeed, for any $h^{}\\in E^{}$ the Kolmogorov inequality [22] applied to the entire function $\\left<e^{tD}f,h^{}\\right>$ gives $\\left\\Vert \\left(\\frac{d}{dt}\\right)^{k}\\left<e^{tD}f,h^{}\\right>\\right\\Vert ^n_{C(\\mathbb {R}^{1})}\\le C_{k,n}\\left\\Vert \\left(\\frac{d}{dt}\\right)^{n}\\left<e^{tD}f,h^{}\\right>\\right\\Vert _{C(\\mathbb {R}^{1})}^{k}\\times $ $\\left\\Vert \\left<e^{tD}f,h^{}\\right>\\right\\Vert _{C(\\mathbb {R}^{1})}^{n-k},\\quad 0<k< n,$ or $\\left\\Vert \\left<e^{tD}D^{k}f,h^{}\\right>\\right\\Vert _{ C(\\mathbb {R}^{1})}^n\\le C_{k,n}\\left\\Vert \\left<e^{tD}D^{n}f,h^{}\\right>\\right\\Vert _{C(\\mathbb {R}^{1})}^{k}\\left\\Vert \\left<e^{tD}f,h^{}\\right>\\right\\Vert _{C(\\mathbb {R}^{1})}^{n-k}.$ Applying the Schwartz inequality to the right-hand side, we obtain $\\left\\Vert \\left<e^{tD}D^{k}f,h^{}\\right>\\right\\Vert _{ C(\\mathbb {R}^{1})}^n & \\le C_{k,n}\\Vert h^{}\\Vert ^{k}\\Vert D^{n}f\\Vert ^{k}\\Vert h^{}\\Vert ^{n-k}\\Vert f\\Vert ^{n-k}\\\\&\\le C_{k,n}\\Vert h^{}\\Vert ^n \\Vert D^{n}f\\Vert ^{k}\\Vert f\\Vert ^{n-k},$ which, when $t=0,$ yields $\\left\|\\left<D^{k}f,h^{}\\right>\\right\|^n\\le C_{k,n}\\Vert h^{}\\Vert ^n\\Vert D^{n}f\\Vert ^{k}\\Vert f\\Vert ^{n-k}.$ By choosing $h$ such that $\\left\|<D^{k}f,h^{}>\\right\|=\\Vert D^{k}f\\Vert $ and $\\Vert h^{}\\Vert =1$ we obtain (REF ).", "Proof of Theorem REF .", "From Lemma REF we have $ \\left\\Vert D^{k}f\\right\\Vert ^n \\le C_{k,n}\\Vert D^{n}f\\Vert ^{k}\\Vert f\\Vert ^{n-k}, \\quad 0\\le k \\le n.$ Without loss of generality, let us assume that $\\Vert f\\Vert =1.$ Thus, $ \\left\\Vert D^{k}f\\right\\Vert ^{1/k} \\le (\\pi /2)^{1/kn}\\Vert D^{n}f\\Vert ^{1/n}, \\quad 0\\le k \\le n.$ Let $k$ be arbitrary but fixed.", "It follows that $\\left\\Vert D^{k}f\\right\\Vert ^{1/k} \\le (\\pi /2)^{1/kn}\\Vert D^{n}f\\Vert ^{1/n}, \\mbox{ for all } n\\ge k,$ which implies that $\\left\\Vert D^{k}f\\right\\Vert ^{1/k}\\le \\underline{\\lim }_{n\\rightarrow \\infty }\\Vert D^{n}f\\Vert ^{1/n}.$ But since this inequality is true for all $k>0,$ we obtain that $\\overline{\\lim }_{k\\rightarrow \\infty }\\Vert D^{k}f\\Vert ^{1/k}\\le \\underline{\\lim }_{n\\rightarrow \\infty }\\Vert D^{n}f\\Vert ^{1/n},$ which proves that $d_f=\\lim _{k\\rightarrow }\\Vert D^{k}f\\Vert ^{1/k}$ exists.", "Since $f\\in \\mathbf {B}_{\\sigma }(D) $ the constant $\\sigma _{f}$ is finite and we have $\\Vert D^{k}f\\Vert ^{1/k}\\le \\sigma _f \\Vert f\\Vert ^{1/k},$ and by taking the limit as $k\\rightarrow \\infty $ we obtain $d_f\\le \\sigma _f.$ To show that $d_f= \\sigma _f,$ let us assume that $d_f< \\sigma _f.$ Therefore, there exist $M>0$ and $\\sigma $ such that $0<d_f<\\sigma < \\sigma _f$ and $\\Vert D^k f\\Vert \\le M \\sigma ^k, \\quad \\mbox{for all }k>0 .$ Thus, by Lemma REF we have $f\\in \\mathbf {B}_\\sigma (D) ,$ which is a contradiction to the definition of $\\sigma _f.$ Conversely, suppose that $d_f=\\lim _{k\\rightarrow \\infty } \\Vert D^kf\\Vert ^{1/k}$ exists and is finite.", "Therefore, there exist $M>0$ and $\\sigma >0$ such that $d_f<\\sigma $ and $\\Vert D^k f\\Vert \\le M \\sigma ^k, \\quad \\mbox{for all } k>0 ,$ which, in view of Lemma REF , implies that $f\\in \\mathbf {B}_{\\sigma }(D) .$ Now by repeating the argument in the first part of the proof we obtain $d_f=\\sigma _f ,$ where $\\sigma _f=\\inf \\left\\lbrace \\sigma : f\\in \\mathbf {B}_{\\sigma }(D)\\right\\rbrace .$ Theorem REF is proved.", "Now consider the following abstract Cauchy problem for the operator $D$ .", "$\\frac{d u(t)}{d t}=Du(t), \\:\\:u(0)=f,$ where $u: \\mathbb {R} \\rightarrow E$ is an abstract function with values in $E.$ Since solutions of this problem given by the formula $u(t)=e^{tD}f$ we obtain the following result.", "Theorem 2.11 A vector $f\\in E,$ belongs to $\\mathbf {B}_{\\sigma }(D)$ if and only if the solution $u(t)$ of the corresponding Cauchy problem (REF ) has the following properties: 1) as a function of $t,$ it has an analytic extension $u(z), z\\in \\mathbb {C}$ to the complex plane $\\mathbb {C}$ as an entire function; 2) it has exponential type $\\sigma $ in the variable $z$ , that is $\\Vert u(z)\\Vert _{E}\\le e^{\\sigma \|z\|}\\Vert f\\Vert _{E}.$ and it is bounded on the real line." ], [ "Sampling-type formulas for one-parameter groups", "We assume that $D$ generates one-parameter strongly cononuous bounded group of operators $e^{tD}, \\:\\:t\\in \\mathbb {R},$ in a Banach space $E$ .", "In this section we prove explicit formulas for a trajectory $e^{tD}f$ with $f\\in \\mathbf {B}_{\\sigma }(D)$ in terms of a countable number of equally spaced samples.", "Theorem 3.1 If $f\\in \\mathbf {B}_{\\sigma }(D)$ then the following sampling formulas hold for $t\\in \\mathbb {R}$ $e^{tD}f=f+tDf \\rm sinc\\left(\\frac{\\sigma t}{\\pi }\\right)+t\\sum _{k\\ne 0}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right),$ $f=e^{tD}f-t \\left(e^{t D}Df\\right)\\rm sinc\\left(\\frac{\\sigma t}{\\pi }\\right)-t\\sum _{k\\ne 0}\\frac{e^{\\left( \\frac{k\\pi }{\\sigma }+t \\right)D}f-e^{tD}f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }+k\\right).$ Remark 3.2 It is worth to note that if $\\:\\:\\:t\\ne 0,\\:\\:$ then right-hand side of (REF ) does not contain vector $f$ and we obtain a \"linear combination\" of $f$ in terms of vectors $e^{\\left( \\frac{k\\pi }{\\sigma }+t \\right)D}f,\\:\\:\\:k\\in \\mathbb {Z},$ and $e^{tD}Df$ .", "If $f\\in \\mathbf {B}_{\\sigma }(D)$ then for any $g^{}\\in E^{}$ the function $F(t)=\\left<e^{tD}f,\\:g^{}\\right>$ belongs to $B_{\\sigma }^{\\infty }(\\mathbb {R})$ .", "We consider $F_{1}\\in B_{\\sigma }^{2}( \\mathbb {R}),$ which is defined as follows.", "If $t\\ne 0$ then $F_{1}(t)=\\frac{F(t)-F(0)}{t}=\\left<\\frac{e^{tD}f-f}{t},\\:g^{}\\right>,$ and if $t=0$ then $F_{1}(t)=\\frac{d}{dt}F(t)\|_{t=0}=\\left<Df,\\:g^{}\\right>.$ We have $F_{1}(t)=\\sum _{k}F_{1}\\left(\\frac{k\\pi }{\\sigma }\\right)\\: \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right),$ which means that for any $g^{}\\in E^{}$ $\\left< \\frac{e^{tD}f-f}{t},\\:g^{} \\right>=\\sum _{k}\\left<\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }},\\:g^{}\\right>\\: \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right).$ Since $\\rm sinc ^{(n)} x=\\sum _{j=0}^{n}C_{n}^{j}(\\sin \\:\\pi x)^{(j)}\\left(\\frac{1}{\\pi x}\\right)^{(n-j)}=$ $\\frac{(-1)^{n}n!", "}{\\pi x^{n+1}}\\sum _{j=0}^{n}\\sin \\left(\\pi x+\\frac{jx}{2}\\right)\\frac{(-1)^{j}(\\pi x)^{j}}{j!", "}$ one has the estimate $\|\\rm sinc^{(n)}\\:x\|\\le \\frac{C}{\|x\|},\\:\\:\\:n=0,1,....$ which implies convergence in $E$ of the series $\\sum _{k}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right).$ It leads to the equality for any $g^{}\\in E^{}$ $\\left< \\frac{e^{tD}f-f}{t},\\:g^{} \\right>=\\left<\\sum _{k}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right),\\:g^{}\\right>,\\:\\:\\:t\\ne 0,$ and if $t= 0$ it gives the identity $\\left<Df,\\:g^{}\\right>=\\left<Df,\\:g^{}\\right>\\sum _{k} \\rm sinc \\:k.$ Thus, $\\frac{e^{tD}f-f}{t}=\\sum _{k}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right),\\:\\:\\:t\\ne 0,$ or for every $t\\in \\mathbb {R}$ $e^{tD}f=f+tDf \\rm sinc\\left(\\frac{\\sigma t}{\\pi }\\right)+t\\sum _{k\\ne 0}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right).$ Thus, (REF ) is proved.", "If in (REF ) we replace $f$ by $\\left(e^{\\tau D}f\\right)$ for a $\\tau \\in \\mathbb {R}$ we will have $e^{tD}\\left(e^{\\tau D}f\\right)=\\left(e^{\\tau D}f\\right)+t\\sum _{k\\ne 0}\\frac{e^{\\frac{k\\pi }{\\sigma }D}\\left(e^{\\tau D}f\\right)-\\left(e^{\\tau D}f\\right)}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma t}{\\pi }-k\\right)+tD\\left(e^{\\tau D}f\\right)\\rm sinc\\left(\\frac{\\sigma t}{\\pi }\\right).$ For $t=-\\tau $ we obtain the next formula which holds for any $\\tau \\in \\mathbb {R},\\:\\:f\\in \\mathbf {B}_{\\sigma }(D),$ $f=e^{\\tau D}f-\\tau \\sum _{k\\ne 0}\\frac{e^{\\left( \\frac{k\\pi }{\\sigma }+\\tau \\right)D}f-e^{\\tau D}f}{\\frac{k\\pi }{\\sigma }} \\rm sinc\\left(\\frac{\\sigma \\tau }{\\pi }+k\\right)-\\tau D\\left(e^{\\tau D}f\\right)\\rm sinc\\left(\\frac{\\sigma \\tau }{\\pi }\\right),$ which is the formula (REF ).", "Theorem is proved.", "The next Theorem is a generalization of what is known as Valiron-Tschakaloff sampling/interpolation formula [6].", "Theorem 3.3 For $f\\in \\mathbf {B}_{\\sigma }(D),\\:\\:\\:\\sigma >0,$ we have for all $z \\in \\mathbb {C}$ $e^{zD}f=z \\:\\rm sinc\\left(\\frac{\\sigma z}{\\pi }\\right)Df+\\rm sinc\\left(\\frac{\\sigma z}{\\pi }\\right)f+\\sum _{k\\ne 0}\\frac{\\sigma z}{k\\pi }\\rm sinc\\left(\\frac{\\sigma z}{\\pi }-k\\right)e^{\\frac{k\\pi }{\\sigma }D}f$ If $F\\in \\mathbf {B}_{\\sigma }(D),\\:\\:\\:\\sigma >0,$ then for all $z \\in \\mathbb {C}$ the following Valiron-Tschakaloff sampling/interpolation formula holds [6] $F(z)=z \\:\\rm sinc\\left(\\frac{\\sigma z}{\\pi }\\right)F^{^{\\prime }}(0)+\\rm sinc\\left(\\frac{\\sigma z}{\\pi }\\right)F(0)+\\sum _{k\\ne 0}\\frac{\\sigma z}{k\\pi }\\rm sinc\\left(\\frac{\\sigma z}{\\pi }-k\\right)F\\left(\\frac{k\\pi }{\\sigma }\\right)$ For $f\\in \\mathbf {B}_{\\sigma }(D),\\:\\:\\:\\sigma >0$ and $g^{}\\in E^{}$ we have $F(z)=\\left<e^{zD}f,\\:g^{}\\right>\\in \\mathbf {B}_{\\sigma }(D)$ and thus $\\left<e^{zD}f,\\:g^{}\\right>=z \\:\\rm sinc\\left(\\frac{\\sigma z}{\\pi }\\right)\\left<Df,\\:g^{}\\right>+\\rm sinc\\left(\\frac{\\sigma z}{\\pi }\\right)\\left<f,\\:g^{}\\right>+\\sum _{k\\ne 0}\\frac{\\sigma z}{k\\pi }\\rm sinc\\left(\\frac{\\sigma z}{\\pi }-k\\right)\\left<e^{\\frac{k\\pi }{\\sigma }D}f,\\:g^{}\\right>.$ Since the series $\\sum _{k\\ne 0}\\frac{\\sigma z}{k\\pi }\\rm sinc\\left(\\frac{\\sigma z}{\\pi }-k\\right)e^{\\frac{k\\pi }{\\sigma }D}f$ converges in $E$ for every fixed $z$ we obtain the formula (REF ).", "Theorem 3.4 If $f\\in \\mathbf {B}_{\\sigma }(D)$ then the following sampling formula holds for $t\\in \\mathbb {R}$ and $n\\in \\mathbb {N}$ $e^{tD}D^{n}f=\\sum _{k}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }}\\left\\lbrace n \\:\\rm sinc^{(n-1)}\\left(\\frac{\\sigma t}{\\pi }-k\\right) +\\frac{\\sigma t}{\\pi } \\rm sinc^{(n)}\\left(\\frac{\\sigma t}{\\pi }-k\\right)\\right\\rbrace .$ In particular, for $n\\in \\mathbb {N}$ one has $D^{n}f=\\mathcal {Q}_{D}^{n}(\\sigma )f,$ where the bounded operator $\\mathcal {Q}_{D}^{n}(\\sigma )$ is given by the formula $\\mathcal {Q}_{D}^{n}(\\sigma )f=n\\sum _{k}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }}\\left[ \\rm sinc^{(n-1)}\\left(-k\\right) +\\rm sinc^{(n)}\\left(-k\\right)\\right].$ Because $F_{1}\\in B_{\\sigma }^{2}(\\mathbb {R})$ we have $\\left(\\frac{d}{dt}\\right)^{n}F_{1}(t)=\\sum _{k}F_{1}\\left(\\frac{k\\pi }{\\sigma } \\right) \\rm sinc^{(n)}\\left(\\frac{\\sigma t}{\\pi }-k\\right)$ and since $\\left(\\frac{d}{dt}\\right)^{n}F(t)=n\\left(\\frac{d}{dt}\\right)^{n-1}F_{1}(t)+t\\left(\\frac{d}{dt}\\right)^{n}F_{1}(t)$ we obtain $\\left(\\frac{d}{dt}\\right)^{n}F(t)=n\\sum _{k}F_{1}\\left(\\frac{k\\pi }{\\sigma }\\right)\\: \\rm sinc^{(n-1)}\\left(\\frac{\\sigma t}{\\pi }-k\\right)+\\frac{\\sigma t}{\\pi }\\sum _{k}F_{1}\\left(\\frac{k\\pi }{\\sigma }\\right)\\: \\rm sinc^{(n)}\\left(\\frac{\\sigma t}{\\pi }-k\\right)$ Because $\\left(\\frac{d}{dt}\\right)^{n}F(t)=\\left<D^{n}e^{tD}f, g^{}\\right>,$ and $F_{1}\\left(\\frac{k\\pi }{\\sigma }\\right)=\\left<\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }}, g^{}\\right>$ we obtain that for $t\\in \\mathbb {R},\\:\\:n\\in \\mathbb {N},$ $D^{n}e^{tD}f=\\sum _{k}\\frac{e^{\\frac{k\\pi }{\\sigma }D}f-f}{\\frac{k\\pi }{\\sigma }}\\left[ n\\: \\rm sinc^{(n-1)}\\left(\\frac{\\sigma t}{\\pi }-k\\right) +\\frac{\\sigma t}{\\pi } \\rm sinc^{(n)}\\left(\\frac{\\sigma t}{\\pi }-k\\right)\\right].$ Theorem is proved." ], [ "Irregular sampling theorems", "In [8] the following fact was proved.", "Theorem 4.1 Let $\\lbrace t_{n}\\rbrace _{n\\in \\mathbb {Z}}$ be a sequence of real numbers such that $\\sup _{n\\in \\mathbb {Z}}\|t_{n}-n\|<1/4.$ Define the entire function $G(z)=(z-t_{0})\\prod _{n=1}^{\\infty }\\left(1-\\frac{z}{t_{n}}\\right)\\left(1-\\frac{z}{t_{-n}}\\right).$ Then for all $f\\in \\mathbf {B}_{\\pi }^{2}(\\mathbb {R})$ we have $f(t)=\\sum _{n\\in \\mathbb {Z}}f(t_{n})\\frac{G(t)}{G^{^{\\prime }}(t_{n})(t-t_{n})}$ uniformly on all compact subsets of $\\mathbb {R}$ .", "An analog of this result for Banach spaces is the following.", "Theorem 4.2 If $D$ generates a one-parameter strongly continuous group $e^{tD}$ of isometries in a Banach space $E$ .", "Suppose that assumptions of Theorem REF are satisfied and $t_{0}\\ne 0$ .", "Then for all $f\\in \\mathbf {B}_{\\pi }(D),\\:\\:g^{}\\in E^{}$ and every $t\\in \\mathbb {R}$ the following formulas hold $\\left<e^{tD}f,\\:g^{}\\right>=\\left<f,\\:g^{}\\right>+t\\sum _{n\\in \\mathbb {Z}}\\frac{\\left<e^{t_{n}D}f,\\:g^{}\\right>-\\left<f,\\:g^{}\\right>}{t_{n}}\\frac{G(t)}{G^{^{\\prime }}(t_{n})(t-t_{n})},$ and $\\left<f,g^{}\\right>=\\left<e^{t D}f,\\:g^{}\\right>+t\\sum _{n\\in \\mathbb {Z}}\\frac{\\left<e^{(t_{n}-t)D}f,\\:g^{}\\right>-\\left<e^{t D}f,\\:g^{}\\right>}{t_{n}}\\frac{G(-t)}{G^{^{\\prime }}(t_{n})(t+t_{n})}$ uniformly on all compact subsets of $\\mathbb {R}$ .", "Remark 4.3 The last formula represents a \"measurement\" $\\left<f,\\:g^{}\\right>$ through \"measurements\" $\\left<e^{(t_{n}-t)D}f,\\:g^{}\\right>$ and $\\left<e^{t D}f,\\:g^{}\\right>$ which are different from $\\left<f,\\:g^{}\\right>$ .", "If $f\\in \\mathbf {B}_{\\sigma }(D)$ then for any $g^{}\\in E^{}$ the function $F(t)=\\left<e^{tD}f,\\:g^{}\\right>$ belongs to $B_{\\sigma }^{\\infty }(\\mathbb {R})$ .", "We consider $F_{1}\\in B_{\\sigma }^{2}( \\mathbb {R}),$ which is defined as follows.", "If $t\\ne 0$ then $F_{1}(t)=\\frac{F(t)-F(0)}{t}=\\left<\\frac{e^{tD}f-f}{t},\\:g^{}\\right>,\\:\\:g^{}\\in E^{},$ and if $t=0$ then $F_{1}(t)=\\frac{d}{dt}F(t)\|_{t=0}=\\left<Df,\\:g^{}\\right>.$ We have $F_{1}(t)=\\sum _{n\\in \\mathbb {Z}}F_{1}(t_{n})\\frac{G(t)}{G^{^{\\prime }}(t_{n})(t-t_{n})}$ or $\\left<\\frac{e^{tD}f-f}{t},\\:g^{}\\right>=\\sum _{n\\in \\mathbb {Z}}\\left<\\frac{e^{t_{n}D}f-f}{t_{n}},\\:g^{}\\right>\\frac{G(t)}{G^{^{\\prime }}(t_{n})(t-t_{n})}$ uniformly on all compact subsets of $\\mathbb {R}$ .", "If we pick a non-zero $\\tau $ such that $\\tau \\ne t_{n}$ for all $t_{n}$ and set $f$ in (REF ) to $e^{\\tau D}f$ then for $t=-\\tau $ we will have the following formula which does not have vector $f$ on the right-hand side $\\left<f,g^{}\\right>=\\left<e^{\\tau D}f,\\:g^{}\\right>+\\tau \\sum _{n\\in \\mathbb {Z}}\\frac{\\left<e^{(t_{n}-\\tau )D}f,\\:g^{}\\right>-\\left<e^{\\tau D}f,\\:g^{}\\right>}{t_{n}}\\frac{G(-\\tau )}{G^{^{\\prime }}(t_{n})(\\tau +t_{n})}.$ Theorem is proved.", "In [21] the following result can be found.", "Theorem 4.4 Let $\\lbrace t_{n}\\rbrace _{n\\in \\mathbb {Z}}$ be a sequence of real numbers such that $\\sup _{n\\in \\mathbb {Z}}\|t_{n}-n\|<1/4.$ Define the entire function $G(z)=(z-t_{0})\\prod _{n=1}^{\\infty }\\left(1-\\frac{z}{t_{n}}\\right)\\left(1-\\frac{z}{t_{-n}}\\right).$ Let $\\delta $ be any positive number such that $0<\\delta <\\pi $ .", "Then for all $f\\in \\mathbf {B}_{\\pi -\\delta }(\\mathbb {R})$ we have $f(z)=\\sum _{n\\in \\mathbb {Z}}f(t_{n})\\frac{G(z)}{G^{^{\\prime }}(t_{n})(z-t_{n})}$ uniformly on all compact subsets of $\\mathbb {C}$ .", "From here we obtain the next fact.", "Theorem 4.5 Suppose that $D$ generates a one-parameter strongly continuous group $e^{tD}$ of isometries in a Banach space $E$ .", "Then in the same notations as in Theorem REF one has that for all $f\\in \\mathbf {B}_{\\pi -\\delta }(D)$ and all $g^{}\\in E^{}$ the following formula holds $\\left<e^{zD}f, g^{}\\right>=\\sum _{n\\in \\mathbb {Z}}\\left<e^{t_{n}D}f,\\:g^{}\\right>\\frac{G(z)}{G^{^{\\prime }}(t_{n})(z-t_{n})}$ uniformly on all compact subsets of $\\mathbb {C}$ .", "Proof follows from Theorem REF since for any $g^{}\\in E^{}$ the function $ \\left<e^{zD}f, g^{}\\right>$ belongs to $ \\mathbf {B}_{\\pi -\\delta }(\\mathbb {R})$ .", "Note that if in the last formula we will set $z$ to zero and assume that $t_{0}\\ne 0$ we will have a representation of $\\left<f,\\:g^{}\\right>$ in terms of samples $\\left<e^{t_{n}D}f,\\:g^{}\\right>\\ne \\left<f,\\:g^{}\\right>$ i.e.", "$\\left<f, g^{}\\right>= -\\sum _{n\\in \\mathbb {Z}}\\left<e^{t_{n}D}f,\\:g^{}\\right>\\frac{G(0)}{G^{^{\\prime }}(t_{n})t_{n}}.$" ], [ "An application to abstract Schrödinger equation", "We now assume that $E$ is a Hilbert space and $D$ is a selfadjoint operator.", "Then $e^{itD}$ is one-parameter group of isometries of $E$ .", "By the spectral theory [2], there exist a direct integral of Hilbert spaces $A=\\int A(\\lambda )dm (\\lambda )$ and a unitary operator $\\mathcal {F}_{D}$ from $E$ onto $A$ , which transforms the domain $\\mathcal {D}_{k}$ of the operator $ D^{k}$ onto $A_{k}=\\lbrace a\\in A\|\\lambda ^{k}a\\in A\\rbrace $ with norm $\\Vert a(\\lambda )\\Vert _{A_{k}}= \\left(\\int ^{\\infty }_{-\\infty } \\lambda ^{2k}\\Vert a(\\lambda )\\Vert ^{2}_{A(\\lambda )} dm(\\lambda ) \\right)^{1/2} $ and $\\mathcal {F}_{D}(Df)=\\lambda (\\mathcal {F}_{D}f), f\\in \\mathcal {D}_{1}.", "$ In this situation one can prove [11] the following.", "Theorem 5.1 A vector $f\\in E$ belongs to $ \\mathbf {B}_{\\sigma }(D)$ if and only if support of $\\mathcal {F}_{D}f$ is in $[-\\sigma , \\:\\sigma ]$ .", "We consider an abstract Cauchy problem for a self-adjoint operator $D$ which consists of funding an abstract-valued function $u: \\mathbf {R}\\rightarrow E$ which satisfies Schrödinger equation and has bandlimited initial condition $\\frac{du(t)}{dt}=iDu(t),\\:\\:\\:u(0)=f\\in \\mathbf {B}_{\\sigma }(D)$ (see [5], [9] for more details).", "In this case formula (REF ) can be treated as a solution to inverse problem associated with (REF ).", "Theorem 5.2 If conditions of Theorem REF are satisfied and $t_{0}\\ne 0$ then initial condition $f\\in \\mathbf {B}_{\\pi -\\delta }(D),\\:\\:0<\\delta <\\pi ,$ in (REF ) can be reconstructed (in weak sense) from the values of the solution $u(t_{n})$ by using the formula $\\left<f, g^{}\\right>=-\\sum _{n\\in \\mathbb {Z}}\\left<u(t_{n}), g^{}\\right>\\frac{G(0)}{G^{^{\\prime }}(t_{n})t_{n}},\\:\\:\\:g^{}\\in E^{*}.$ Similar results can be formulated by using formulas (REF ) and (REF )." ] ]	1403.0292
[ [ "Bose-Einstein condensation on quantum graphs" ], [ "Abstract We present results on Bose-Einstein condensation (BEC) on general compact quantum graphs, i.e., one-dimensional systems with a (potentially) complex topology.", "We first investigate non-interacting many-particle systems and provide a complete classification of systems that exhibit condensation.", "We then consider models with interactions that consist of a singular part as well as a hardcore part.", "In this way we obtain generalisations of the Tonks-Girardeau gas to graphs.", "For this we find an absence of phase transitions which then indicates an absence of BEC." ], [ "Introduction", "We present results on Bose-Einstein condensation (BEC) in many-particle systems on compact quantum graphs.", "[1], [2], [3] Quantum graphs are models of particles moving along the edges of a metric graph.", "They hence combine the simplicity of a one-dimensional model with the complexity of a graph.", "One of the major findings about quantum graphs was that the one-particle spectra display correlations that are well described by random-matrix theory.", "[4], [5] For that reason quantum graphs have become popular models in quantum chaos.", "Bose-Einstein condensation, on the other hand, is a quantum mechanical phenomenon of many-particle systems that can be well described within the framework of quantum statistical mechanics.", "Originally, this condensation was predicted by Einstein for a gas of non-interacting bosons in three dimensions.", "[6] The conclusion was that below some critical temperature $T_{crit}$ , or above some critical particle density $\\rho _{crit}$ , all additional particles brought into the system must condense into the one-particle ground state.", "In other words, the one-particle ground state becomes macroscopically occupied.", "As recognised later,[7] the macroscopic occupation of a single-particle state is indeed the underlying mechanism that gives rise to the condensation, also in interacting many-particle system We here introduce many-particle systems on graphs and study BEC in non-interacting systems as well as in systems with hardcore interactions, thus generalising the Tonks-Girardeau gas[8] to graphs." ], [ "Preliminaries", "In this section we briefly summarise relevant concepts of BEC as well as of many-particle quantum graphs.", "For more details on BEC see Refs.", "PO56,CCGOR11, on quantum graphs see Refs.", "KS06,Kuc04,GNUSMY06,BolEnd09, and on many-particle quantum graphs see Refs. BKSingular,BKContact.", "A definition of BEC that also applies to a system of interacting bosons was given in Ref.", "PO56 and employs the reduced one-particle density matrix obtained from the canonical density matrix (at inverse temperature $\\beta =\\frac{1}{T}$ ), $\\rho _{N}(x,y)=\\frac{1}{Z_{N}(\\beta )}\\sum _{n} e^{-\\beta E^{N}_{n}}\\bar{\\Psi }_{n}(x)\\Psi _{n}(y)\\ ,$ by tracing out all degrees of freedom except one.", "Here $\\Psi _{n}(x)$ is the $n$ -th eigenfunction of the $N$ -particle system with eigenvalues $E^{N}_{n}$ , and $Z_{N}(\\beta )=\\sum _{n}e^{-\\beta E^{N}_{n}}$ is the canonical partition function.", "Condensation is defined to occur when the largest eigenvalue of the reduced density matrix is asymptotically of order one as $N \\rightarrow \\infty $ .", "Note that this limit (the thermodynamic limit) is accompanied by the limit $V \\rightarrow \\infty $ , where $V$ is the volume of the one-particle configuration space, such that the particle density remains fixed.", "Unfortunately, it is in general very difficult to prove (or disprove) BEC in an interacting system in the sense of Penrose-Onsager.", "Instead we shall employ the connection between BEC and phase transitions in the case of interacting bosons on a graph.", "The classical configuration space for a particle on a graph is a compact metric graph, i.e., a finite, connected graph $\\Gamma = (\\mathcal {V},\\mathcal {E})$ with vertex set $\\mathcal {V} = \\lbrace v_1,\\dots ,v_{V}\\rbrace $ and edge set $\\mathcal {E}=\\lbrace e_1,\\dots ,e_{E}\\rbrace $ .", "The edges are identified with intervals $[0,l_e]$ , $e=1,\\dots ,E$ , thus assigning lengths to intervals.", "This then introduces a metric on the graph.", "Note that a graph is called compact when all lengths are finite.", "For the one-particle quantum system the Hilbert space is $\\mathcal {H}_1=L^{2}(\\Gamma )=\\bigoplus _{e=1}^{E}L^{2}(0,l_{e}).$ The quantum dynamics shall be generated by a self-adjoint realisation of the Laplacian $-\\Delta _{1}$ .", "As a differential expression this acts on each edge-component $f_e$ of a function $F=(f_1,\\dots ,f_E)\\in C^{\\infty }_{0}(\\Gamma )$ as the negative second derivative, $-\\Delta _1 F=(-f_1^{\\prime \\prime },\\dots ,-f_E^{\\prime \\prime })$ .", "As shown in Ref.", "Kuc04, one way to arrive at all possible self-adjoint realisations of the Laplacian uses quadratic forms, $Q_{1}[F] = \\sum _{e=1}^{E}\\int _0^{l_e}\|f^{\\prime }(x)\|^2 \\mathrm {d} x -\\langle F_{bv},L_1 F_{bv}\\rangle _{\\mathbb {C}^{2E}}\\ ,$ with domain $\\mathcal {D}_{Q_1} = \\lbrace F\\in H^1(\\Gamma ); P_1 F_{bv}=0 \\rbrace \\ .$ Here $F_{bv} = (f_1(0),\\dots ,f_E(0),f_1(l_1),\\dots ,f_E(l_E))^{T}\\in \\mathbb {C}^{2E}$ is the vector of boundary values, $P_1$ is an orthogonal projection on $\\mathbb {C}^{2E}$ and $L_1$ a self-adjoint endomorphism of $\\ker {P_1}$ .", "The $N$ -particle Hilbert space is the $N$ -fold tensor product $\\mathcal {H}_N=\\mathcal {H}_1 \\otimes \\cdots \\otimes \\mathcal {H}_1$ , $\\mathcal {H}_N =L^{2}(\\Gamma _{N})=\\bigoplus _{e_{1}e_{2}...e_{N}}L^{2}(D_{e_{1}e_{2}...e_{N}}),$ where $D_{e_{1}e_{2}...e_{N}}=(0,l_{e_{1}})\\times \\cdots \\times (0,l_{e_{N}})$ are $N$ -dimensional hypercubes.", "In order to introduce contact interactions that are formally given by a Hamiltonian $H_N =-\\Delta _N + \\alpha \\sum _{i\\ne j}\\delta (x_{e_i}-x_{e_j})$ we also need to dissect the $N$ -particle configuration space further by cutting the hypercubes along all hypersurplanes defined by $x_{e_{i}}=x_{e_{j}}$ when $e_{i}=e_{j}$ .", "We denote (REF ) as $L^2(\\Gamma ^{\\ast }_{N})$ when taking this further dissection into account.", "The bosonic subspace is denoted as $L^{2}_{B}(\\Gamma ^{\\ast }_{N})$ .", "As in the one-particle case, we require the $N$ -particle Hamiltonian to be a self-adjoint realisation of the Laplacian $-\\Delta _N$ .", "As a differential expression it acts as $(-\\Delta _{N}\\Psi )_{e_{1}\\dots e_{N}} =-\\left(\\frac{\\partial ^{2}}{\\partial x_{e_{1}}^{2}}+\\cdots +\\frac{\\partial ^{2}}{\\partial x_{e_{N}}^{2}}\\right)\\psi _{e_{1}\\dots e_{N}}\\ ,$ on functions $\\Psi \\in C^{\\infty }_{0}(\\Gamma _{N})$ .", "Self-adjoint (bosonic) realisations of $-\\Delta _{N}$ are obtained through suitable quadratic forms.", "For this, we first define vectors of boundary values $\\Psi _{bv}({y}) = \\begin{pmatrix} \\sqrt{l_{e_{2}}\\dots l_{e_{N}}}\\psi _{e_{1}\\dots e_{N}}(0,l_{e_{2}}y_{1},\\dots ,l_{e_{N}}y_{N-1}) \\\\\\sqrt{l_{e_{2}}\\dots l_{e_{N}}}\\psi _{e_{1}\\dots e_{N}}(l_{e_{1}},l_{e_{2}}y_{1},\\dots ,l_{e_{N}}y_{N-1}) \\end{pmatrix},$ with ${y} \\in [0,1]^{N-1}$ .", "The desired quadratic form now reads $\\begin{split}Q^{(N)}_{B}[\\Psi ]&= N \\sum _{e_{1}\\dots e_{N}}\\int _{0}^{l_{e_{1}}}\\dots \\int _{0}^{l_{e_{N}}}\|\\psi _{e_{1}\\dots e_{N},x_{e_1}}(x_{e_{1}},\\dots ,x_{e_{N}})\|^{2}\\ \\mathrm {d} x_{e_N}\\dots \\mathrm {d} x_{e_1} \\\\&\\quad -N\\int _{[0,1]^{N-1}}\\langle \\Psi _{bv},L_{N}({y})\\Psi _{bv}\\rangle _{\\mathbb {C}^{2E^{N}}} \\mathrm {d}{y},\\end{split}$ and is defined on $\\mathcal {D}_{Q^{(N)}_{B}} = \\lbrace \\Psi \\in H^{1}_{0,B}(\\Gamma ^{\\ast }_{N});\\ P_{N}({y})\\Psi _{bv}({y})=0\\ \\text{for a.e.", "}\\ {y}\\in [0,1]^{N-1}\\rbrace .$ Here $\\Psi \\in H^{1}_{0,B}(\\Gamma ^{\\ast }_{N})\\subset H^{1}_{B}(\\Gamma ^{\\ast }_{N})$ if each component $\\psi _{e_{1}e_{2}...e_{N}}$ is in $H^1$ and vanishes on the hyperplanes $x_{e_{i}}=x_{e_{j}}$ .", "Moreover, $P_N,L_N:[0,1]^{N-1} \\rightarrow M(2E^{N},\\mathbb {C})$ are bounded and measurable maps such that $P_{N}({y})$ is an orthogonal projection and $L_{N}({y})$ is a self-adjoint endomorphism of $\\ker {P_{N}({y})}$ .", "The self-adjoint realisation of $-\\Delta _{N}$ associated with $ Q^{(N)}_{B}[\\cdot ]$ is denoted as $(-\\Delta _{N},\\mathcal {D}^{\\infty }_{N}(P_{N},L_{N}))$ , see Ref. BKContact.", "Due to the Dirichlet conditions along the hyperplanes $x_{e_{i}}=x_{e_{j}}$ this operator is a rigorous version of the Hamiltonian (REF ) in the limit $\\alpha \\rightarrow \\infty $ .", "Therefore it represents hardcore interactions.", "It is important to note that the coordinate dependence of the maps $P_{N}$ and $L_{N}$ leads to (additional) singular many-particle interactions that are localised in the vertices of the graph.", "[1]" ], [ "BEC in non-interacting systems many-particle", "In this section we provide a complete classification of non-interacting many-particle systems on general compact quantum graph in terms of the presence or absence of BEC.", "Since non-interacting Hamiltonians are entirely determined by corresponding one-particle Hamiltonians $(-\\Delta _{1},\\mathcal {D}_{1}(P_1,L_1))$ , it is sufficient to refer to the latter only.", "Let $\\Gamma $ be a compact, metric graph with edge lengths $l_1,...,l_{E}$ .", "The thermodynamic limit (TL) consists of the scaling $l_{e}\\mapsto \\eta l_{e}$ and taking the limit $\\eta \\rightarrow \\infty $ .", "As a first result we obtain the following.", "[3] If the one-particle Laplacian $(-\\Delta _{1},\\mathcal {D}_{1}(P_1,L_1))$ is such that $L_1$ is negative semi-definite, no BEC occurs in the corresponding free Bose gas at finite temperature ($T > 0$ ) in the thermodynamic limit.", "The proof uses the formalism of the grand-canonical ensemble and, via a bracketing construction, compares the given systems with free Bose gases with Dirichlet- and Neumann boundary conditions in the vertices.", "The number of particles is not fixed and the particle density is adjusted via the chemical potential $\\mu $ .", "Our main result in this section is now the following.", "[3] Let a free Bose gas be given on a quantum graph with a one-particle Laplacian $(-\\Delta _{1},\\mathcal {D}_{1}(P_1,L_1))$ such that $L_1$ has at least one positive eigenvalue.", "Then, in the thermodynamical limit, there is a critical temperature $T_c>0$ such that BEC occurs below $T_c$ .", "In order to prove this statement one shows that the one-particle ground state energy (which is negative since $L_1$ has a positive eigenvalue, compare (REF )) remains negative in the thermodynamic limit.", "Indeed, one uses the lower bound for the one-particle ground state energy of Ref.", "KS06 as well as a suitable Rayleigh quotient to prove that the ground state energy converges to $-L^{2}_{max}$ , where $L_{max} > 0$ is the largest positive eigenvalue of $L_1$ .", "One also uses that the number of negative eigenstates is bounded from above [10] (even in the thermodynamic limit) and applies the trace formula for quantum graphs[12].", "With standard arguments the Theorem then follows." ], [ "Interacting many-particle systems", "In this final section we consider systems of bosons interacting via singular interactions localised at the vertices of the graph as well as hardcore contact interactions, i.e., we consider (bosonic) self-adjoint realisations $(-\\Delta _{N},\\mathcal {D}^{\\infty }_{N}(P_{N},L_{N}))$ of the $N$ -particle Laplacian.", "Our goal is to prove that no phase transitions (in the free energy density) are present, indicating an absence of condensation.", "For this recall that the free energy density of a sequence of $N$ -particle Hamiltonians $H_N$ with discrete spectra $\\lbrace E^{N}_{n}\\rbrace $ is defined as $f(\\beta ,\\mu )=-\\lim _{V \\rightarrow \\infty }\\frac{1}{\\beta V}\\log {\\sum _{N=1}^{\\infty }e^{N\\beta \\mu }\\operatorname{Tr}_{\\mathcal {H}_{N}}}e^{-\\beta H_N}\\ .$ The vanishing of functions in the domain $\\mathcal {D}^{\\infty }_{N}(P_{N},L_{N})$ along the hyperplanes $x_{e_{i}}=x_{e_{j}}$ allows to define a Fermi-Bose mapping, relating the bosonic free-energy density to a fermionic one (which is known explicitly).", "This leads to the following result.", "[3] Let $(-\\Delta _{N},\\mathcal {D}^{\\infty }_{N}(P_{N},L_{N}))_{N \\in \\mathbb {N}}$ be a family of bosonic Laplacians with repulsive hardcore interactions, indexed by the particle number $N$ .", "Assume that for this family the operator of multiplication with $L_N(\\cdot )$ on $L^2([0,1]^{N-1})$ is uniformly bounded with respect to $N$ .", "Then the bosonic, grand-canonical, free-energy density $f(\\beta ,\\mu )$ is given by $f(\\beta ,\\mu ) = -\\frac{1}{\\pi \\beta }\\int _{0}^{\\infty }\\log {\\left(1+e^{-\\beta (k^{2}-\\mu )}\\right)}\\ \\mathrm {d} k\\ .$ This function is smooth and, hence, there exists no phase transition.", "It is important to note that Theorem holds independently of the singular interactions in the vertices, i.e., independently of the maps $P_N$ and $L_N$ .", "Hence, even when BEC occurs without the hardcore interactions, switching on the latter destroys the condensation." ] ]	1403.0271
[ [ "Some mathematical remarks on the polynomial selection in NFS" ], [ "Abstract In this work, we consider the proportion of smooth (free of large prime factors) values of a binary form $F(X_1,X_2)\\in\\Z[X_1,X_2]$.", "In a particular case, we give an asymptotic equivalent for this proportion which depends on $F$.", "This is related to Murphy's $\\alpha$ function, which is known in the cryptographic community, but which has not been studied before from a mathematical point of view.", "Our result proves that, when $\\alpha(F)$ is small, $F$ has a high proportion of smooth values.", "This has consequences on the first step, called polynomial selection, of the Number Field Sieve, the fastest algorithm of integer factorization." ], [ "Introduction", "Smooth – or friable –numbers, defined as integers whose prime factors are smaller than a given bound, are a celebrated topic in analytic number theory and have a key importance in cryptography today.", "In this work we are motivated by the Number Field Sieve (NFS), the fastest algorithm of integer factorization [17].", "Briefly, if $N$ is an integer to be factored, NFS can be summarized as follows.", "In the first step, called polynomial selection, we select two irreducible polynomials with integer coefficients $f$ and $g$ , which have a common root $m$ modulo $N$ , i.e.", "$f(m)\\equiv 0\\equiv g(m)\\pmod {N}$ .", "In the next step, we fix a parameter $B$ and we search for $B$ pairs of coprime integers $(a,b)$ such that $F(a,b):=b^{\\deg f}f(a/b)$ and $G(a,b):=b^{\\deg g}g(a/b)$ are $B$ -smooth – an integer $n$ is $B$ -smooth if its greatest prime factor, denoted by $P(n)$ , satisfies $P(n)\\le B$ .", "The collected pairs allow us to obtain a $B\\times B$ linear system over ${\\mathbb {Z}}/2{\\mathbb {Z}}$ .", "Next, we compute a linear combination of the rows of the system.", "By a square root computation in a number field, we find a non-trivial solution of the equation $x^2\\equiv y^2\\bmod N$ , which gives a non-trivial factor of $N$ .", "Computing the complexity of the algorithm requires to find the distribution of coprime pairs $(a,b)$ which are smooth with respect to two binary forms $F$ and $G$ , i.e.", "$F(a,b)$ and $G(a,b)$ are smooth for two irreducible homogeneous polynomials $F$ and $G$ with integer coefficients.", "In the sequel, small caps letters $f$ and $g$ denote polynomials and capital letters denote the associated binary forms.", "The distribution of $B$ -smooth integers has made the object of abundant works (for an overview, we refer to [10] and [8]).", "For example, Hildebrand proved in [9] an asymptotic formula in the region $x\\ge 3,\\qquad \\exp \\left((\\log \\log x)^{5/3+\\varepsilon }\\right)\\le B\\le x.\\qquad \\mathrm {(H_{\\varepsilon })}$ Theorem A For any fixed $\\varepsilon >0$ and uniformly for $(x,B)$ in the region (REF ), we have $\\Psi (x,B):=\\#\\left\\lbrace n\\in [1,x]: P(n)\\le B \\right\\rbrace =x\\rho (u)\\left(1+O\\left(\\frac{\\log (u+1)}{\\log B}\\right)\\right),$ where $u:=\\frac{\\log x}{\\log B}$ and $\\rho $ denotes the Dickman function, namely the one defined by the delay differential equation {ll u'(u)+(u-1)=0 if u>1, (u)=1 if 0u 1. .", "A few years later, Saias refined this result by giving an asymptotic expansion of $\\Psi (x,B)$ .", "Theorem B (Main corollary,[23]) There exists $C>0$ such that, for any fixed $J\\ge 0$ , $\\varepsilon >0$ and uniformly for $(x,B)$ in the region (REF ) and such that 0<u<J+1(u-u) > C(J+1)BB, we have $\\Psi (x,B) =x\\left(\\sum _{j=0}^J\\gamma _j\\frac{\\rho ^{(j)}(u)}{(\\log B)^j}+O\\left(\\rho (u)\\left(\\frac{\\log (u+1)}{\\log B}\\right)^{J+1}\\right)\\right),$ where $\\gamma _j$ are the coefficients of the Taylor series in $s=0$ of $\\frac{s\\zeta (s+1)}{s+1}$ .", "In particular, we have $\\gamma _1=\\gamma -1.$ Let $x$ and $B$ be two given integers, $F(X_1,X_2)\\in {\\mathbb {Z}}[X_1,X_2]$ a binary form and $\\mathcal {K}$ a compact subset of ${\\mathbb {R}}^2$ whose boundary is a continuous closed curve with piecewise continuous derivatives.", "By $x\\mathcal {K}$ we denote the set $\\mathcal {K}$ rescaled by a factor $x$ .", "In order to study the distribution of the $B$ -smooth integers of the form $F(a,b)$ for coprime integers $a$ and $b$ , we consider the cardinal $\\Psi ^{(1)}_{F}(\\mathcal {K},x,B)$ defined by (1)F(K,x,B):=#{(a,b)xK:(a,b)=1 and P(F(a,b))B}.", "In [2], Balog, Blomer, Dartyge and Tenenbaum developed an argument which can be easily adapted to show the following result.", "Theorem C Let $\\mathcal {K}$ be a compact subset of ${\\mathbb {R}}^2$ whose boundary is a continuous closed curve with piecewise continuous derivatives, $k\\ge 1$ and $F_1(X_1,X_2),\\dots ,F_k(X_1,X_2)\\in \\mathbb {Z}[X_1,X_2]$ some integral and irreducible binary forms of degree $d_1\\ge \\dots \\ge d_k$ .", "There exists $u(d_1,\\dots ,d_k)$ in the interval $\\left(1/d_1,+\\infty \\right)$ with the following property.", "For any fixed $u<u(d_1,\\dots ,d_k)$ , there exists a constant $c_{F_1,\\dots ,F_k,\\mathcal {K}}(u)$ such that, for $B\\ge x^{1/u}\\ge 2$ , we have $ \\Psi ^{(1)}_{F_1\\dots F_k}(\\mathcal {K},x,B)\\ge c_{F_1,\\dots ,F_k,\\mathcal {K}}(u) x^2.$ More precisely, one can take $u(d_1,\\dots , d_k):=\\left\\lbrace \\begin{array}{ll}+\\infty &\\text{if }k\\ge 2\\text{ and } d_1+\\dots +d_k\\le 3,\\\\e^{\\frac{1}{2}}&\\text{if }k=1\\text{ and } d_1=3.\\end{array}\\right.$ It is common to make the assumption that integers represented by a given binary form have the same probability to be $B$ -smooth as arbitrary integers of the same size.", "Consequently, in the light of Theorem REF , we conjecture that, in a domain to be made precise, we have $\\Psi ^{(1)}_{F_1\\dots F_k}(\\mathcal {K},x,B)\\sim \\frac{6}{\\pi ^2}\\mathcal {A}(\\mathcal {K})x^2\\rho (d_1u)\\dots \\rho (d_ku),$ where $\\mathcal {A}(\\mathcal {K})$ denotes the area of $\\mathcal {K}$ .", "A similar formula was proven by the second author ([16] and [15]) when $d_1+\\dots +d_k\\le 3$ .", "Note that the right hand member of Equation REF does not depend on the binary forms $F_1$ , $\\ldots $ , $F_k$ .", "In the current state of research, it seems out of reach to obtain in the general case an equation in which both members depend on the binary forms.", "In Theorem REF we refine Theorem REF in the case $k=1$ and $d_1=2$ by making explicit the first approximation term.", "Since this term depends on the polynomial $f$ , it can be used in the polynomial selection stage of NFS, which is done as follows.", "Using one of the two methods of Kleinjung ([13],[1] and [14],[1]), one generates a large number of pairs of polynomials $f$ and $g$ , such that $f$ is irreducible and $g$ linear.", "For each pair of polynomials, one computes Murphy's $\\mathbb {E}(F,G)$ or Murphy's $\\alpha (f)$ for the associated binary forms, as defined in [19].", "Hence one can make a model of the polynomial selection as a random trial of polynomials from a set $E(d,\\textbf {I})=\\left\\lbrace f=\\sum _{i=0}^df_iX^i\\in {\\mathbb {Z}}[X]\\mid f\\text{ is irreducible},\\forall i, f_i\\in I_i\\right\\rbrace ,$ where $\\textbf {I}=\\prod _{i=0}^dI_i$ is a $(d+1)$ -tuple of intervals.", "Murphy's $\\alpha $ is the main object in this article.", "It is hard to determine when it was proposed in the cryptographic community, but it was known to Montgomery in 1996 [3].", "In his thesis, Murphy [19] introduced $\\alpha (f)$ as the sum of a series and gave evidence that, when $\\alpha (f)$ is small, $F$ has a high proportion of smooth values.", "It is computed using the number of roots of $f$ modulo each prime power $p^k$ .", "Based on $\\alpha (f)$ , one can compute Murphy's $\\mathbb {E}(F,G)$ , which takes into account the real roots of $f$ and $g$ , but it is more costly to compute and not much more accurate than $\\alpha (f)$ .", "Also note that, $\\alpha $ does not depend on the linear polynomial $g$ since, based on experiments, one can make the conjecture that $g$ has a small influence on the formula of Equation (REF ).", "A thorough development on the polynomial selection from a cryptographic perspective is due to Bai [1]." ], [ "Outline", "In Section , we give a rigorous definition of $\\alpha (f)$ .", "The mean value of $\\alpha (f)$ over $E(d,\\textbf {I})$ will be the main goal of Section .", "In the last section, we introduce a modification of NFS.", "It allows us to obtain a rigorous result on the proportion of smooth elements in number fields of arbitrary degree and then to show that $\\alpha (f)$ effectively occurs in the proportion of smooth values of a binary form of degree 2." ], [ "Notation", "In what follows, $K$ stands for a number field and $d_K$ , $\\mathcal {O}_K$ , $U_K$ , $G_K$ , $\\zeta _K$ and $\\lambda _K$ denote respectively its degree, ring of integer, unit group, class group, Dedekind zeta function and residue of $\\zeta _K$ .", "The letters $p$ , $\\mathfrak {p}$ and $\\mathfrak {I}$ denote respectively a rational prime, a prime ideal and an arbitrary ideal of $\\mathcal {O}_K$ ." ], [ "Definition and convergence of Murphy's $\\alpha (f)$", "From a cryptographic point of view, Theorem REF , proved in Section REF , states that $\\alpha (f)$ is a good indicator of a polynomial's efficiency for NFS when $f$ is quadratic.", "In this section we show that it has two properties which are equally important: it has an easy-to-compute formula and it is defined by a series with a high speed of convergence." ], [ "Definition of $\\alpha (f)$", "Murphy introduced $\\alpha $ explicitly for arbitrary polynomials, but he gives credit to Montgomery for using the formula in the case of quadratic polynomials [3].", "One can find the formula of $\\alpha $ by the following heuristic argument.", "For any integer $n$ and bound $C$ , the $C$ -sifted part of $n$ is the largest divisor of $n$ without prime factors less than $C$ .", "For a bound $B$ , the $B$ -smooth part of $n$ is the largest $B$ -smooth divisor of $n$ .", "Experiments show that one can obtain a good guess of $\\Psi ^{(1)}_{F}(\\mathcal {K},x,B)$ by the following empirical method: Choose a large constant $C$ and compute the average value $\\text{cont}(F,C)$ of the logarithm of the $C$ -smooth part of the values of $F$ .", "Define $\\alpha (F,C)$ as the average value of the logarithm of the $C$ -smooth part of a random integer minus $\\text{cont}(F,C)$ .", "Approximate $\\Psi ^{(1)}_{F}(\\mathcal {K},x,B)$ by the cardinality of $x\\mathcal {K}$ times the probability of a random $C$ -sifted integer of size $\\left(\\max _{(a,b)\\in x\\mathcal {K}}\|F(a,b)\|+\\alpha (f,C)\\right)$ to be $B$ -smooth.", "This suggests to define $\\alpha $ as in the definition below.", "In the sequel, $f$ is a polynomial in ${\\mathbb {Z}}[X]$ such that $\\operatorname{Disc}(f)\\ne 0$ and $p$ is a prime.", "The associated binary form $F$ is defined by $F(X_1,X_2)=X_2^{\\deg (f)}f(X_1/X_2)$ .", "Definition 2.1 For any prime $p$ we define, if it exists, $\\alpha _p(f)=(\\log p)\\left(\\frac{1}{p-1}-\\text{cont}_p(f)\\right), $ with $\\text{cont}_p(f)=\\lim _{x\\rightarrow \\infty }\\frac{\\sum _{(a,b)\\in [1,x]^2,\\gcd (a,b,p)=1}\\operatorname{val}_p F(a,b)}{\\#\\left\\lbrace (a,b)\\in [1,x]^2: \\gcd (a,b,p)=1\\right\\rbrace }.$ Under the reserve of proving the convergence of the series below, we define $\\alpha (f)=\\sum _{p\\text{ prime}}\\alpha _p(f).$ To get an other expression for $\\text{cont}_p(f)$ , we can split the region {(a,b)[1,x]2: (a,b,p)=1} in congruence classes modulo $p^k$ and try to approximate #{(a,b)[1,x]2: (a,b,p)=1, pk\|F(a,b)} by x2p2k#{(a,b)[1,pk]2: (a,b,p)=1, pk\|F(a,b)} This procedure is essentially the object of Lemma REF .", "Before doing this, we can remark that $\\#\\left\\lbrace (a,b)\\in [1,p^k]^2: \\gcd (a,b,p)=1, p^k\|F(a,b)\\right\\rbrace =\\varphi (p^k)n_{p^k}(f),$ where $n_{p^k}(f)&=&\\#\\left\\lbrace a\\in [0,p^k-1]:f(a)\\equiv 0\\mod {p}^k\\right\\rbrace \\\\&+&\\#\\left\\lbrace b\\in [0,p^k-1]:b\\equiv 0\\mod {p},~F(1,b)\\equiv 0\\mod {p}^k\\right\\rbrace .$ Nagell [20] proved what survives of Hensel's lemma when the hypothesis on the derivative fails.", "We adapt his result to obtain an upper bound of $n_{p^k}$ in a similar way one would in the case when Hensel's lemma applies.", "Lemma 2.2 If $p$ does not divide $\\operatorname{Disc}(f)$ , then $n_{p^k}(f)=n_p(f)$ .", "In the general case, for any prime $p$ and $k\\ge 1$ , we have npk(f)2(f)p(2valp(Disc(f)),k).", "The first assertion is a direct consequence of [[20],Theorem 1] which asserts that #{ a[0,pk-1]: f(a)0 ppk}= #{ a[0,p-1]: f(a)0 pp}.", "In the proof of [[20],Theorem 2], it is shown that #{ a[0,pk-1]: f(a)0 ppk}(f)p(2valp(Disc(f)),k).", "When applied to $f(x)=F(x,1)$ and $\\overline{f}=F(1,x)$ , this implies the second assertion.", "Proposition 2.3 We have, for every prime $p$ , p(f)=p(1p-1-pp+1k1npk(f)pk).", "We first focus on the numerator of $\\text{cont}_p(f)$ .", "Let $x$ be a sufficiently large integer.", "One can choose $k_0$ such that $x^{2/3} \\le p^{k_0}\\le px^{2/3}$ .", "We write 1a,bx (a,b)=1valp(F(a,b))= 1(p,x)+2(p,x) with 1(p,x)=kk01a,bx#{ 1a,bx:(a,b,p)=1 and pk\|F(a,b)} and 2(p,x)= kk0+11a,bx#{ 1a,bx:(a,b,p)=1 and pk\|F(a,b)}.", "In view of the formula (REF ), we can use Lemma 3.2 of [5] to deduce that 1(p,x)= kk0 #{(a,b)[1,pk]2: (a,b,p)=1, pk\|F(a,b)}p2kx2 +O( xpk0/2(k0p)f+pk0(k0 p)2f)$\\displaystyle +O\\left( xp^{k_0/2}(k_0\\log p)^{\\nu _f}+p^{k_0}(k_0 \\log p)^{2\\deg f}\\right)$ =(1-1p) kk0 npk(f)pkx2+O( x4/3(x)f), where $\\nu _f=\\deg f\\left( 1+2\\deg f\\right)^{\\deg f+1}$ .", "On the other hand, since $\\operatorname{val}_p\\left(F(a,b)\\right)\\ll \\log x$ , we can use Lemma REF and again Lemma 3.2 of [5] to deduce that 2(p,x)x1a,bx#{ 1a,bx:(a,b,p)=1 and pk0\|F(a,b)} (x)x2#{(a,b)[1,pk0]2: (a,b,p)=1, pk0\|F(a,b)}p2k0+O( x4/3(x)f) x4/3(x)f .", "Finally, we note that #{1a,bx:(a,b,p)=1}= #{1a,bx:pa}+ #{1a,bx:p\|a and pb} = (1-1p2)x2+O(x).", "The result follows when $x$ tends to infinity since then $k_0$ tends to infinity." ], [ "Convergence of $\\alpha (f)$", "The formula of $\\alpha _p(f)$ gets a simple form when $p$ does not divide $\\operatorname{Disc}(f)$ nor the leading coefficient of $f$ .", "Indeed, Lemma REF and Proposition REF imply that, for such primes $p$ , we have $\\alpha _p(f)=\\log p\\left(\\frac{1}{p-1}-\\frac{n_p(f)}{p-1}\\left(\\frac{p}{p+1}\\right)\\right).$ Let $\\omega $ be a root of $f$ , $K$ the rupture field of $f$ and $\\tilde{\\omega }:=F(1,0)\\omega $ an integer of $K$ .", "It follows from a result of Dedekind [6] that, for any prime $p$ which not divide $F(1,0)$ nor the index $[\\mathcal {O}_K:{\\mathbb {Z}}[\\tilde{\\omega }]]$ , $n_p(f)$ is the number of ideals $\\mathfrak {p}$ such that $N(\\mathfrak {p})=p$ .", "This suggests to put $p_0=\\max \\left\\lbrace p\\text{ prime}: p\|F(1,0)F(0,1)\\text{ or }p\|\\operatorname{Disc}(F)\\text{ or }p\|[\\mathcal {O}_K,{\\mathbb {Z}}[\\tilde{\\omega }]]\\right\\rbrace .$ After the previous discussion, the problem of convergence of $\\alpha (f)$ is reduced to showing the convergence of the series pp(1p-1-np(K)p-1(pp+1)) where $n_p(K)$ denotes the number of ideals $\\mathfrak {p}$ such that $N(\\mathfrak {p})=p$ .", "We first remark that, for any $X\\ge 2$ , we can write pXp(1p-1-np(K)p-1(pp+1)) =pX pp(1-np(K)) +pXpp(p-1)(1-np(K)p+1).", "On the one hand, from the trivial estimation $\|n_p(K)\|\\le n_K$ and the Chebyshev estimation $\\sum _{p\\le X}\\log p\\le eX$ with $e=1.01624 $ (see Theorem 9 of [22]), we can use a summation by parts to get, for any $X_2\\ge X_1\\ge n_K$ , X1<pX2pp(p-1)\|1-np(K)p+1\| X1<pX2pp(p-1) 3eX1-1.", "On the other hand, we can write, again with a summation by parts, X1<pX2 pp(1-np(K)) = R(X2)X2- R(X1)X1+ X1X2 R(t)t2dt, where $R$ is the rest term defined by R(t):=pt(1-np(K))p. Therefore, it suffices to use a sufficiently sharp estimation of $R(t)$ , which is the object of the next theorem.", "On the one hand, we can obtain a very sharp estimation using the Riemann hypothesis for $\\zeta _K$ and $\\zeta _{\\mathbb {Q}}$ .", "But on the other hand, we have a good estimation relying on no assumptions.", "Theorem 2.4 (Theorem 9.2 of [18]) There exists an absolute effectively computable constant $c_1>0$ such that, if $X\\ge \\exp \\left(4d_K(\\log \\operatorname{Disc}(K))^2\\right)$ , then \|N(pk)XN(p)-X+ X(K)(K)\| X(-c1dK-1/2(X)1/2), where $\\beta (K)$ denotes the largest real zero of $\\zeta _K$ in the interval $(0,1)$ if it exists and $1/2$ otherwise.", "Moreover, if the Riemann Hypothesis holds for $\\zeta _K$ , there exist explicit constants $a_K$ , $b_K$ and $c_K$ such that, for $X\\ge 2$ , we have \|N(pk)XN(p)-X\| X1/2(aK+bKX+cK(X)2).", "Remark 2.5 Some effective bounds for $\\beta _K$ are contained in Theorem 1.4 of [18].", "Numerical values for $a_K$ , $b_K$ and $c_K$ are given without proof in [21].", "The values $a_K=\\frac{4781}{96}\\log (d_K)+\\frac{58681}{113}n_K$ , $b_K=\\frac{23}{3}\\log (d_K)+\\frac{68}{3}n_K$ and $c_K=\\frac{863}{31}n_K$ can be rigorously obtained from Theorem $8.1$ of [26].", "In order to use Theorem REF , we have to study the contribution of powers of prime ideals.", "Using the Chebyshev estimation (REF ), we get, for any $X\\ge 2$ , N(pk)X k2 or N(p) not prime N(p) e dKk2pX1kp edK(X12+X2X1/3).", "Consequently, we have, for $t\\ge \\exp \\left(4d_K(\\log \\operatorname{Disc}(K))^2\\right)$ , $R(t)\\ll d_Kt^{\\frac{1}{2}}+\\frac{t^{\\beta (K)}}{\\beta (K)}+t\\exp \\left(-c_1(d_K)^{-1/2}\\left(\\log t\\right)^{1/2}\\right)$ By a straightforward calculation of primitive, we deduce from these estimations that we have, for $X_2\\ge X_1\\ge \\exp \\left(4d_K(\\log \\operatorname{Disc}(K))^2\\right)$ , \|X1<pX2pp(1-np(K))\| \|R(X1)\|X1+\|R(X2)\|X2 +X1X2\|R(t)\|t2dt dKX1-12+X1(K)-1(K)+(-c1(dK)-1/2 (X1)1/2), which implies the convergence of $\\alpha (f)$ .", "In order to get a good estimation of the convergence speed, we now assume that the Riemann Hypothesis holds for $\\zeta _{\\mathbb {Q}}$ and $\\zeta _K$ .", "It follows from Theorem REF that we have, for $t\\ge 2$ , \|R(t)\|edKt1/2+edKt1/3t/2+aKt1/2+bKt1/2(t)+cKt1/2(t)2.", "As a consequence of the previous discussion, we can get that, for $X\\ge 2$ , \|X<ppp(1-np(K))\| \|R(X)\|X+X+\|R(t)\|t2dt X-1/2((3aK+3edK+4bK+16cK+(3bK+8cK)X+3cK(X)2) +X-1/6edK4(92+5X)).", "It follows that the speed of convergence is given, for $X\\ge \\max (p_0,n_K)$ , by \|(f)-pXp(f)\|X-1/2(3 eX1/2X-1+X-1/6 edK4(92+5X) +(3aK+3edK+4bK+16cK+(3bK+8cK)X+3cK(X)2)).", "Example 2.6 Using the best numerical values in Remark REF we can certify effective bounds on $\\alpha (F)$ for given binary forms $F$ .", "Consider for example $F(X_1,X_2)=X_1^2+qX_2^2$ with $q=10^{30}+57$ .", "By computing the partial sum of $\\alpha (F)$ for primes less than $X=40096176099$ we obtain: $\\left\|\\alpha (F)-2.39\\right\|<1.$ This emphasizes the importance of obtaining small effective constants in Theorem REF ." ], [ "Towards the average of $\\alpha $ on a set of polynomials", "The polynomial selection stage of NFS consists in enumerating polynomials $f=\\sum _{i=0}^df_i x^i$ of a given degree and with a bound on each coefficient $f_i$ and in selecting those with the best value of $\\alpha $ .", "Some variants restrict the enumeration to a subset and a short list of polynomials with a good $\\alpha $ can be further tested with longer tests or by direct sieving.", "In any case, by computing the average of $\\alpha $ we guarantee a value of $\\alpha $ for the best polynomials.", "During the polynomial selection in NFS, it is common to restrict the search to a set of polynomials $f$ given by $\\deg f$ and the size of each coefficient.", "For each pair $(m,d)$ of integers and each $d$ -tuple $\\mathbf {I}=I_0\\times \\cdots \\times I_{d-1}$ of intervals such that, for all $i$ , $ I_i\\subset [-m,m]$ , we put $E^{(1)}(m,d,\\textbf {I})=\\left\\lbrace f=x^d+\\sum _{i=0}^{d-1} f_i x^i:(f_0,f_1,\\ldots ,f_{d-1})\\in \\textbf {I},\\operatorname{Disc}(f)\\ne 0 \\right\\rbrace .$ Due to technical reasons, we now now study the average of $\\alpha (f)$ on $E^{(1)}(m,d,\\textbf {I})$ rather than $E(d,\\textbf {I})$ defined by (REF ).", "Theorem 3.1 For any given prime $p$ , uniformly with respect to I, one has $\\lim _{{m\\rightarrow \\infty \\\\ \\min _j\|I_j\|/d(\\log d+\\log m)\\rightarrow \\infty }}\\frac{1}{\\#E^{(1)}(m,d,\\textbf {I})} \\sum _{f\\in E^{(1)}(m,d,\\textbf {I})}\\alpha _p(f)=\\alpha _p(X).$ In view of Proposition REF and Theorem REF , we can suppose that $d\\ge 2$ and write, for any prime $p$ , p(f)-p(x)=ppp+1k11-npk(f)pk.", "For any pair $k$ , we put $S_p(k,m,d,\\textbf {I})=\\sum _{{f\\in E(d,m,\\textbf {I})}}(1-n_{p^k}(f)).$ Then we have fE(1)(m,d,I)(p(f)-p(x))=p(1)(m,d,I)+p(2)(m,d,I), where $\\Sigma _p^{(1)}(m,d,\\textbf {I})&=&\\frac{p\\log p}{p+1}\\sum _{k\\le k_0}\\frac{S_p(k,m,d,\\textbf {I})}{p^k},\\\\\\Sigma _p^{(2)}(m,d,\\textbf {I})&=&\\frac{p\\log p}{p+1}\\sum _{k\\ge k_0} \\frac{S_p(k,m,d,\\textbf {I})}{p^k}.$ Using the definition of the discriminant, for any $f$ in $E^{(1)}(m,d,\\textbf {I})$ , we have the upper bound \|Disc(f)\|(2d-1)!", "m2d-1.", "Consider $k_0(p)= \\left\\lceil \\log _p \\left((2d-1)!m^{2d-1}\\right)\\right\\rceil +\\lceil \\log _p(md)\\rceil $ .", "Case $k\\le k_0(p)$ .", "Since the elements of $E^{(1)}(m,d,\\textbf {I})$ are monic, we have #{fE(1)(m,d,I), pjdf(p-j)0( pk)pk }=0.", "Consequently, we can write p(1)(m,d,I)=ppp+1kk01pk(#E(1)(m,d,I)- r=0pk-1 #{fE(1)(m,d,I), f(r)0( pk)pk }).", "We consider first the cardinality of $E^{(1)}(m,d,\\textbf {I})$ .", "Given $(f_1,\\ldots ,f_{d-1})\\in I_1\\times \\cdots I_{d-1}$ , the polynomial $dx^{d-1}+\\sum _{i=1}^{d-1} if_ix^{i-1}$ has at most $d-1$ complex roots.", "For each such root $z$ , there is exactly one value of $f_0\\in I_0$ such that $\\sum _{i=0}^d f_i z^i=0$ .", "Hence there are at most $d\|\\mathbf {I}\|/\|I_0\|$ polynomials $f$ of zero discriminant and coefficients in $\\textbf {I}$ .", "It follows that E(1)(m,d,I)=#{(f0,...,fd-1)I} -#{(f0,...,fd-1)I: Disc(xd+i=0d-1fixi)=0} =\|I\|(1+O(dj\|Ij\|)) .", "Let $k\\le k_0(p)$ be an integer and $r\\in [0,p^k-1]$ .", "For each $(d-1)$ -tuple $(f_1,\\ldots ,f_{d-1})\\in I_1\\times \\cdots \\times I_{d-1}$ , the number of values $f_0$ such that $f(r)\\equiv 0\\pmod {p^k}$ is $\\left\\lfloor \\frac{\|I_0\|}{p^k}\\right\\rfloor +\\epsilon $ with $\\epsilon =0$ or 1.", "Hence, it follows that #{fE(1)(m,d,I), f(r)0( pk)pk } = #{(f0,...,fd-1)I: f(r)0( pk)pk} +O(#{(f0,...,fd-1)I: Disc(xd+i=0d-1fixi)=0}) = (\|I0\|pk+O(1))\|I\|\|I0\|(1+O(dj\|Ij\|))+O(d\|I\|j\|Ij\|) =\|I\|pk+O(\|I\|dj\|Ij\|).", "It results that p(1)(m,d,I)pkk0(p)\|I\|dj\|Ij\| k0(p)p \|I\|dj\|Ij\|.", "Case $k\\ge k_0(p)$ .", "Due to the choice of $k_0(p)$ , we have $k_0\\ge 2\\operatorname{val}_p \\operatorname{Disc}(f)$ for all polynomials $f$ in $E(m,d,\\textbf {I})$ .", "By Lemma REF , for all $k\\ge k_0(p)$ , we have npk(f)dDisc(f)2, which is further upper bounded by $ (2d-1)!dm^{2d-1}\\le p^{k_0(p)}/m$ .", "We deduce that p(2)(m,d,I)d((2d-1)!m2d-1)2pk>k0(p)p-k\|E(1)(m,d,I)\| d((2d-1)!m2d-1)2\|I\| ppk0(p) \|I\|m.", "When combining the bounds on $\\sum ^{(1)}_p(m,d,\\textbf {I})$ and $\\sum ^{(2)}_p(m,d,\\textbf {I})$ , we obtain that, uniformly for $p\\ge 1$ , we have fE(1)(m,d,I)(p(f)-p(x))\|I\|(1m+d(d+m)j\|Ij\|).", "In view of the previous theorem, it seems to be interesting to compute the value of $\\alpha (X)$ .", "This is the aim of the following proposition.", "Proposition 3.2 Let $g=aX+b\\in {\\mathbb {Z}}[X]$ be a polynomial with $\\gcd (a,b)=1$ .", "Then we have (g)=12A--(2)0.56 .", "where $A$ denotes the Glaisher-Kinkelin constant and $\\gamma $ denotes the Euler-Mascheroni constant.", "Since $f$ has degree 1 and $\\gcd (a,b)=1$ , we have, for every prime $p$ and $k\\ge 1$ , npk(f)=1.", "Consequently, it follows from Proposition REF that (f)=ppp-1(1-pp+1)= ppp2-1.", "From the formula Q'(s)Q(s)=-ppps-1, which holds for any complex $s$ such that $\\Re (s)>1$ , we deduce that (f)=ppp2-1=-Q'(2)Q(2).", "The result is then a direct consequence of the formulas Q(2)=26 and Q'(2)=26(+(2)-12A).", "We can remark that this proposition asserts that $\\alpha (g)=\\alpha (X)$ for any linear polynomial $g$ .", "This observation is a new argument towards the direction that the polynomial selection is essentially not influenced by the linear polynomial." ], [ "The algorithm", "The main goal of this section is to prove smoothness results for binary forms of degree 2.", "This case can be treated with multiplicative methods since the values of a quadratic binary form are norms of arbitrary integer elements of a quadratic field.", "The same theorems apply to binary forms of higher degrees if we modify the algorithm as below.", "By doing so, we transfer the difficulty from the field of analytic number theory to that of algorithmic number theory.", "In short, in our modification of NFS, instead of considering elements $a-b\\theta $ of ${\\mathbb {Q}}(\\theta )$ , we consider arbitrary elements $a_0+a_1\\omega +\\cdots a_{d-1} \\omega ^{d-1}$ of norm bounded by a constant, where $d$ is the degree of the defining polynomial $f$ .", "In more detail, the new version of the algorithm is as follows.", "We select two polynomials $f$ and $g$ , with $g$ linear such that there exists an integer $m$ such that $f(m)\\equiv g(m)\\equiv 0\\mod {N}$ .", "We use the same factor base as in the classical version of NFS, i.e.", "if $B$ is the smoothness bound, the factor base includes degree-1 ideals in the number field of $f$ and primes up to $B$ .", "Let $\\omega $ be a root of $f$ in its number field.", "We set $X_f$ and $X_g$ to the maximal value of $\\operatorname{N}(a_0+a_1\\omega )$ and $\|a_0+a_1m\|$ respectively when $a_0$ and $b_0$ are bounded by the constant used in NFS.", "Next we collect primitive polynomials $P(x)=a_0+a_1x+\\cdots +a_{d-1}x^{d-1}$ such that $(a_0,\\dots ,a_{d-1})=1$ $\|\\operatorname{N}(P(\\omega ))\|\\le X_f\\text{ and }\|P(m)\|\\le X_g.$ $\\operatorname{N}(P(\\omega ))$ and $\|P(m)\|$ are $B$ -smooth.", "Each polynomial $P$ allows us to obtain a relation as explained by Joux, Lercier, Smart and Vercauteren in [12].", "Finally, we use the linear system to obtain a non-trivial solution of equation $X^2\\equiv Y^2\\pmod {N}$ by following step by step the classical variant of NFS.", "The practicality of this modification will be investigated by the first author in a future work.", "The main difficulty is to enumerate the ideals whose norm is bounded by a given constant." ], [ "The smoothness probability : general case", "Let $\\omega $ be an algebraic integer, non rational, and $K={\\mathbb {Q}}(\\omega )$ .", "In view of the previous discussion, we now focus on the study of the cardinality of {(a0,...,ad-1)Zd: (a0,...,ad-1)=1, N(a0+...+ad-1d-1)x and P(N(a0+...+ad-1d-1))B}$\\displaystyle \\text{ and }P\\left(N(a_0+\\dots +a_{d-1}\\omega ^{d-1})\\right)\\le B\\Bigg \\rbrace $ .", "If the unit group $U_K$ is infinite (this is the case when $d_K\\ge 3$ or $K$ is a real quadratic field), such a set is infinite.", "However, we can remark that the ideals $\\mathfrak {I}$ generated by its elements are primitive, namely that, for any prime $p$ , $p\\mathcal {O}_K\\nmid \\mathfrak {I}$ .", "Consequently, it makes sense to concentrate ourself to the cardinality K(1)(x,B):=#{I primitive: N(I)x and P(N(I))B}.", "A standard way – the one followed here – to get an asymptotic formula for $\\Psi _K^{(1)}(x,B)$ consists to apply to the Dirichlet series $\\mathcal {F}_K(s)$ defined by FK(s):=I primitive1N(I)s some results of complex analysis, such as Perron's formula.", "It is consistent to take a look at the shape of $\\mathcal {F}_K(s).$ Using the inclusion–exclusion principle, we first remark that we have, for $\\Re (s)>1$ , FK=m1(m)mOK\|I1N(I)s=K(s)Q(dKs)-1.", "Moreover, using the properties of the Riemann zeta function, it is immediate that $\\zeta _{\\mathbb {Q}}(d_Ks)^{-1}$ is absolutely convergent for $\\Re (s)>\\frac{1}{d_K}$ .", "In view of the previous discussion, we are now in capacity to use asymptotic results of Hanrot, Tenenbaum and Wu [11].", "We obtain the following theorem.", "Theorem 4.1 Let $K$ be a number field of degree $d_K\\ge 2$ .", "Then, there exists $C>0$ such that, for any $J\\ge 0$ and $\\varepsilon >0$ , we have, uniformly for $\\exp \\left((\\log \\log x)^{5/3+\\varepsilon }\\right)\\le B\\le x$ and 0<u<J+1{u}>C(J+1)2 BB $\\Psi _K^{(1)}(x,B) =x\\left(\\sum _{j=0}^J\\gamma _j(K)\\frac{\\rho ^{(j)}(u)}{(\\log B)^j}+O\\left(\\rho (u)\\left(\\frac{\\log (u+1)}{\\log B}\\right)^{J+1}\\right)\\right),$ where j(K)j1+j2=j1j1!j2!", ".", "j1(1-s-1)K(s)sj1\|s=1 .j2Q(dKs)-1sj2\|s=1.", "In particular, we have 0(K)=KQ(dK) and 1(K)=0(K)( -1+ pp( 1p-1-contp(K))), with contp(K)=(k1k#\|{I primitive,N(I)=pk}pk ) (k0#{I primitive,N(I)=pk}pk )-1.", "In view of Equation (REF ), it is immediate that $\\mathcal {F}_K(s)$ satisfies the Condition $(1.7)$ of [11].", "Moreover, as it is noted in Section $2.3$ of [11], Theorem II$.1.13$ of [25] implies that, for any $\\frac{1}{d(K)}<\\delta <1$ and uniformly for $\\Re (s)\\ge \\delta $ , we have P(n)B(n)ndKs=n(n)ndKs+O(1B1-).", "Consequently, we can apply successively Theorem 1.2 and Theorem 1.1 of [11] to deduce (REF ).", "The statement on the values $\\gamma _0(K)$ and $\\gamma _1(K)$ follows from the fact that .", "( FK(s)sFK(s) - Q(s)sQ(s))\|s=1= pp( 1p-1-contp(K) )." ], [ "The smoothness probability : imaginary quadratic case", "Let $f$ be an irreducible quadratic polynomial.", "Its discriminant $\\operatorname{Disc}(f)$ is a fundamental discriminant if it satisfies one of the following conditions : $\\operatorname{Disc}(f) \\equiv 1 \\pmod {4}$ and is square-free, $\\operatorname{Disc}(f) = 4m$ where $m \\equiv 2\\text{ or }3 \\pmod {4}$ and $m$ is square-free.", "We now apply the previous result to get an asymptotic estimation related to the proportion of smooth values of quadratic binary forms with fundamental negative discriminant.", "Theorem 4.2 Let $F(X_1,X_2)\\in {\\mathbb {Z}}[X_1,X_2]$ be a primitive and irreducible quadratic form such that $\\operatorname{Disc}(F)$ is negative and fundamental.", "Let $\\mathcal {K}_F$ the compact defined by $\\mathcal {K}_F:=\\left\\lbrace (x_1,x_2)\\in {\\mathbb {R}}^2:\|F(x_1,x_2)\|\\le 1\\right\\rbrace $ Then, there exists $\\kappa >0$ such that, for any $\\varepsilon >0$ , we have, uniformly for $\\exp \\left((\\log \\log x)^{5/3+\\varepsilon }\\right)\\le B\\le x(\\log x)^{-\\kappa }$ , (1)F(KF,x,B)(1)F(KF,x,x)= (xe(f),B)xe(f)(1+O(((u+1))2(B)2)).", "Let $\\omega $ be a root of $f(X)=F(X,1)$ and $K:={\\mathbb {Q}}(\\omega )$ .", "Since $\\operatorname{Disc}(f)$ is a fundamental discriminant, we have $\\operatorname{Disc}(K)=\\operatorname{Disc}(f)$ .", "Moreover, there exists a basis $(\\omega _1,\\omega _2)$ of $\\mathcal {O}_K$ such that, for any integers $a$ and $b$ , one has $F(a,b)=\\operatorname{N}(a\\omega _1+b\\omega _2).$ Since $U_K$ is finite, we have (1)F(KF,x,B)=#{=(a1+b2)OK : (a,b)=1, N(a1+b2)\|x, P(N((a1+b2))B} = \|U(K)\| #{I principal ideal : I is primitive, N(I)x, P(N(I))B} In order to pick up ideals from the class $\\text{Cl}(\\mathcal {O}_K)$ , i.e.", "principal ideals, we can consider the group $\\widehat{G_K}$ of the multiplicative characters of the class group $G_K$ .", "By the orthogonality property of characters, we have #{ICl(OK):N(I)x,I primitive,P(N(I)))B} =1\|GK\|GK(1)(x,B;), where (1)(x,B;)= I primitive N(I)x P(N(I))B(I).", "Contribution of nontrivial characters: Since $\\text{Cl}(p\\mathcal {O}_K)$ is the identity element of the class group $G_K$ , the inclusion-exclusion principle implies that I primitive(J)N(J)s = I(I)N(I)s (p(1-1p2s))-1.", "Consequently, we can adapt, step by step, the proof of Theorem REF to deduce that, for any $\\varepsilon $ and uniformly for x3 and ((x)5/3+)Bx, we have I primitive(I)N(I)sx(u)(-(B)3/5-).", "This procedure is essentially made in [24] and [7].", "Contribution from the trivial character : For the principal character, denoted by $\\chi _0$ , we use Theorem REF .", "There exists $c>0$ such that, for any $\\varepsilon >0$ , we have, uniformly for $x\\ge 3\\text{ and }\\exp \\left((\\log \\log x)^{5/3+\\varepsilon }\\right)\\le B\\le x(\\log x)^{-c},$ (1)(x,B;0)= x(0(K)(u)+1(K)'(u)B+ O((u)((u+1)B)2 )), where $\\gamma _0(K)=\\frac{6\\lambda _{K}}{\\pi ^2}$ and 1(K)=0(K)( -1+ pp( 1p-1-contp(K))).", "Using the decomposition of rational primes into ideals of $\\mathcal {O}_K$ (see for example the discussion in Section $6.4$ of [4]), we can note that #{I primitive,N(I)=pk}= {ll 0 if pDisc(K) and k2, np(K) if k=1 or pDisc(K), .", "and therefore $\\operatorname{cont}_p(K)=\\left\\lbrace \\begin{array}{ll}\\frac{1}{p+1}&\\text{ if }p\|\\operatorname{Disc}(K),\\\\\\frac{p}{p+1}\\frac{n_p(K)}{p-1}&\\text{ otherwise}.\\end{array}\\right.$ A careful study of $\\operatorname{cont}_p(f)$ implies that we have actually contp(K)=contp(f) To see this, assume first that $p\|\\operatorname{Disc}(K)$ .", "In view of the hypothesis on $\\operatorname{Disc}(K)$ , a straightforward computation implies that $n_{p}(f)=1$ and $n_{p^k}(f)=0$ for $k\\ge 2$ , and therefore Equation (REF ) holds.", "We consider now primes $p$ which do not divide $\\operatorname{Disc}(K)$ , for which we must show that $n_p(f)=n_K(f)$ (Hensel's Lemma allows to obtain $n_{p^k}(f)=n_{p^k}(K)$ for $k\\ge 2$ ).", "If $p$ does not divide $2F(1,0)F(0,1)$ , since the index is 1 or 2, Dedekind's result states that $n_p(f)=n_p(K)$ .", "If $p$ is an odd prime which divide $F(1,0)F(0,1)$ , it is not difficult, using the decomposition of $p$ in $\\mathcal {O}_K$ , to see that $n_p(f)=n_p(K)=2$ .", "If $p=2$ and (at least) one of $F(1,0)$ and $F(0,1)$ is even, then $\\operatorname{Disc}(K)\\equiv 1\\pmod {8}$ , which implies that $n_2(K)=2$ .", "But then $F(0,1)$ and $F(1,1)$ are even and one obtains $n_2(f)=2=n_2(K)$ .", "Finally, if $p=2$ does not divide $F(0,1)$ nor $F(1,0)$ , all the coefficients of $F$ are odd and then $n_2(f)=0$ .", "Since, in this case, $\\operatorname{Disc}(K)\\equiv 5\\pmod {8}$ , we have also $n_2(K)=0=n_2(f)$ .", "For the remaining primes, we have by Lemma REF that $n_{p^k}(f)=n_p(K)$ for any $k\\ge 1$ which implies (REF ).", "From this discussion, it finally follows that (1)F(KF,x,B)=6K2\|GK\|x((u)+(-1+(f) )'(u)B+O((u)((u+1))2(B)2)).", "Using the standard Selberg-Delange's method instead of Theorem REF (see [25]), we can also prove that, for any $\\varepsilon >0$ , we have (1)F(KF,x,x):=#{(a,b)Z2:(a,b)=1, \|F(a,b\|x } =62\|GK\|x+O(x(-x)3/5-)).", "From Theorem REF , we see also that for any $\\varepsilon >0$ and uniformly for $ x\\ge 3\\text{ and }\\exp \\left((\\log \\log x)^{5/3+\\varepsilon }\\right)\\le B\\le x(\\log x)^{-c},$ we have (x,B)=x((u)+(-1)'(u)B+O(((u+1))2(B)2)).", "This enables us to estimate the right-hand term of Equation REF and to deduce the result.", "Remark 4.3 The theorem above encompasses a large set of binary forms.", "For example, since the quadratic binary form $F=X_1^2+qX_2^2$ defined in Example REF has fundamental discriminant and $\\alpha (F)$ is positive, we know that asymptotically it has less smooth values than the random integers of same size.", "Nevertheless, many examples of binary forms $F^{\\prime }$ with good values of $\\alpha (F^{\\prime })$ have non fundamental disciminants." ], [ "Conclusion and open questions", "The results in this article establish a rigorous connection between Murphy's $\\alpha $ and a polynomial's efficiency in NFS.", "On can improve the speed of the algorithm by studying $\\alpha $ and, in particular, the following questions: What is the maximum value of $\\alpha $ on a given set $E(d,\\textbf {I})$ ?", "Indeed, if a polynomial with a good value of $\\alpha $ is found, one can end the polynomial selection phase, reducing therefore the time spent in this phase of the algorithm.", "Can one define a variance of $\\alpha $ ?", "Indeed, experiments indicate that, uniformly on the ideals products $\\textbf {I}$ , the distribution of the values of $\\alpha $ on a set $E(d,m,\\textbf {I})$ converges to a Gaussian distribution when $m$ tends to infinity.", "If one can define and compute the variance of $\\alpha $ , one will be able to find a good trade-off between the time spent to select a good polynomial and the time used to collect relations using that polynomial." ] ]	1403.0184
[ [ "Bayesian Density Estimation via Multiple Sequential Inversions of 2-D\n Images with Application in Electron Microscopy" ], [ "Abstract We present a new Bayesian methodology to learn the unknown material density of a given sample by inverting its two-dimensional images that are taken with a Scanning Electron Microscope.", "An image results from a sequence of projections of the convolution of the density function with the unknown microscopy correction function that we also learn from the data.", "We invoke a novel design of experiment, involving imaging at multiple values of the parameter that controls the sub-surface depth from which information about the density structure is carried, to result in the image.", "Real-life material density functions are characterised by high density contrasts and typically are highly discontinuous, implying that they exhibit correlation structures that do not vary smoothly.", "In the absence of training data, modelling such correlation structures of real material density functions is not possible.", "So we discretise the material sample and treat values of the density function at chosen locations inside it as independent and distribution-free parameters.", "Resolution of the available image dictates the discretisation length of the model; three models pertaining to distinct resolution classes are developed.", "We develop priors on the material density, such that these priors adapt to the sparsity inherent in the density function.", "The likelihood is defined in terms of the distance between the convolution of the unknown functions and the image data.", "The posterior probability density of the unknowns given the data is expressed using the developed priors on the density and priors on the microscopy correction function as elicitated from the Microscopy literature.", "We achieve posterior samples using an adaptive Metropolis-within-Gibbs inference scheme.", "The method is applied to learn the material density of a 3-D sample of a real nano-structure and of simulated alloy samples." ], [ "Introduction", "Nondestructive learning of the full three-dimensional material density function in the bulk of an object, using available two dimensional images of the object, is an example of a standard inverse problem , , , , .", "The image results from the projection of the three dimensional material density onto the image plane.", "However, inverting the image data does not lead to a unique density function in general; in fact to render the inversion unique, further measurements need to be invoked.", "For instance, the angle at which the object is imaged or viewed is varied to provide a series of images, thereby expanding available information.", "This allows to achieve ditinguishability (or identifiability) amongst the solutions for the material density.", "A real-life example of such a situation is presented by the pursuit of material density function using noisy 2-dimensional (2-D) images taken with electron microscopy techniques .", "Such non-invasive and non-destructive 3-D density modelling of material samples is often pursued to learn the structure of the material in its depth ndt, with the ulterior aim of controlling the experimental conditions under which material samples of desired qualities are grown.", "Formally, the projection of a density function onto a lower dimensional image space is referred to as the Radon Transform; see radon, radonxrays.", "The inverse of this projection is also defined but requires the viewing angle as an input and secondly, involves taking the spatial derivative of the density function, rendering the computation of the inverse projection numerically unstable if the image data is not continuous or if the data comprises limited-angle images , or if noise contaminates the data .", "Furthermore, in absence of measurements of the viewing angle, the implementation of this inverse is not directly possible, as panaretos suggested.", "Even when the viewing angle is in principle measurable, logistical difficulties in imaging at multiple angles result in limited-angle images.", "For example, in laboratory settings, such as when imaging with Scanning Electron Microscopes (SEMs), the viewing angle is varied by re-mounting the sample on stubs that are differently inclined each time.", "This mounting and remounting of the sample is labour-intensive and such action leads to the breaking of the vacuum within which imaging needs to be undertaken, causing long waiting periods between two consecutive images.", "When vacuum is restored and the next image is ready to be taken, it is very difficult to identify the exact fractional area of the sample surface that was imaged the last time, in order to scan the beam over that very area.", "In fact, the microscopist would prefer to opt for an imaging technique that allows for imaging without needing to break the vacuum in the chamber at all.", "Such logistical details are restrictive in that this allows imaging at only a small number of viewing angles, which can cause the 3-D material density reconstruction to be numerically unstable.", "This problem is all the more acute when the data is discontinuous and heterogeneous in nature.", "Indeed, with the help of ingenious imaging techniques such as compressive sensing , the requirement of a large number of image data points is potentially mitigated.", "If implemented, compressive imaging would imply averaging the image data over a chosen few pixels, with respect to a known non-adaptive kernel.", "Then the inversion of such compressed data would require the computation of one more averaging or projection (and therefore one more integration) than are otherwise relevant.", "While this is easily done, the implementation of compressive sensing of electron microscopy data can be made possible only after detailed instrumentational additions to the imaging paraphernalia is made.", "Such instrumentational additions are outside the scope of our work.", "We thus invoke a novel design of imaging experiment that involves imaging at multiple values of some relevant parameter that is easily varied in a continuous way over a chosen range, unlike the viewing angle.", "In this paper, we present such an imaging strategy that allows for multiple images to be recorded, at each value of this parameter, when a cuboidal slab of a material sampe is being imaged with an SEM.", "In imaging with an SEM, the projection of the density function is averaged over a 3-D region inside the material, the volume of which we consider known; such a volume is indicated in the schematic diagram of this imaging technique shown in Figure REF .", "Within this region, a pre-fixed fraction of the atomistic interactions between the material and the incident electron beam stay confined, , .", "The 2-D SEM images are taken in one of the different types of radiation that are generated as a result of the atomistic interactions between the molecules in the material and a beam of electrons that is made incident on the sample, as part of the experimental setup that characterises imaging with SEM, , , .", "Images taken with bulk electron microscopy techniques potentially carry information about the structure of the material sample under its surface, as distinguished from images obtained by radiation that is reflected off the surface of the sample, or is coming from near the surface.", "Since the radiation generated in an elemental three-dimensional volume inside the bulk of the material sample, is due to the interaction of the electron beam and the material density in that volume, it is assumed that the material density is proportional to the radiation density generated in an elemental 3-D volume.", "The radiation generated in an elemental volume inside the bulk of the material, is projected along the direction of the viewing angle chosen by the practitioner to produce the observed 2-D image.", "In practice, the situation is rendered more complicated by the fact that for the 2-D images to be formed, it is not just the material density function, but the convolution of the material density function with a kernel that is projected onto the image space.", "The nature of the modulation introduced by this kernel is then also of interest, in order to disentangle its effect from that of the unknown density function that is responsible for generating the measured image.", "As for the modulation, both enhancement and depletion of the generated radiation is considered to occur in general.", "It is to be noted that this kernel is not the measure of blurring of a point object in the imaging technique, i.e.", "it is not the point spread function (PSF) of the observed image, but is characteristic of the material at hand.", "The kernel really represents the unknown microscopy correction function that convolves with the material density function–this convolution being projected to form the image data, which gets blurred owing to specifics of the imaging apparataus.", "Learning the de-blurred image from the PSF-convolved image data is an independent pursuit, referred to as Blind Deconvolution , , .", "Here our aim is to learn the material density and the microscopy correction function using such noisy image data.", "The convolution of the unknown kernel with the unknown density, is projected first onto the 2-D surface of the system, and the resulting projection is then successively marginalised over the 2 spatial coordinates that span the plane of the system surface.", "Thus, three successive projections result in the image data.", "Here we deal with the case of an image resulting from a composition of a sequence of three independent projections of the convolution of the material density function and the unknown kernel or microscopy function.", "So only a sequence of inversions of the image allow us to learn this density.", "Such multiple inversions are seldom addressed in the literature, and here, we develop a Bayesian method that allows for such successive inversions of the observed noisy, PSF-convolved 2-D images, in order to learn the kernel as well as the density function.", "The simultaneous pursuit of both the unknown density and the kernel, is less often addressed than are reported attempts at density reconstruction, under an assumed model for the kernel, goldstein,correpm,heinrich,pichou85.", "We focus here on the learning of a density function that is not necessarily convex, is either sparse or dense, is often multimodal - with the modes characterised by individualised, discontinuous substructure and abrupt bounds, resulting in the material density function being marked by isolated islands of overdensity and sharp density contrasts; we shall see this below in examples of reconstructed density functions presented in Section and Section .", "Neither the isolated modes of the material density function in real-life material samples nor the adjacent bands of material over-density, can be satisfactorily modelled with a mixture model.", "Again, modelling of such a real-life trivariate density function with a high-dimensional Gaussian Process (GP) is not possible in lieu of training data, given that the covariance function of the invoked GP will then have to be non-stationary and exhibit non-smooth spatial variation.", "Here by “training data” is implied data comprising a set of known values of the material density, at chosen points inside the sample; such known values of the density function are unknown.", "Even if the covariance function were varying smoothly over the 3-D spatial domain its parametrisation is $ad hoc$ in lieu of training data but with an abruptly evolving covariance function–as in the problem of inverting SEM image data–such parametrisation becomes impossible , especially when there is no training data available.", "At the same time, the blind implementation of the inverse Radon Transform would render the solution unstable given that the distribution of the image data in real-life systems is typically, highly discontinuous.", "In the following section, (Section ) we describe the experimental setup delineating the problem of inversion of the 2-D images taken with Electron Microscopy imaging techniques, with the aim of learning the sub-surface material density and the kernel.", "In addition, we present the novel design of imaging experiment that achieves multiple images.", "In Section we discuss the outstanding difficulties of multiple and successive inversions and qualitatively explain the relevance of the advanced solution to this problem.", "This includes a discussion of the integral representation of the multiple projections (Section REF ), the outline of the Bayesian inferential scheme adopted here (Section REF ) and a section on the treatment of the low-rank component of the unknown material density (section REF ).", "The details of the microscopy image data that affect the modelling at hand are presented in the subsequent section.", "This includes a description of the 3 models developed to deal with the three classes of image resolution typically encountered in electron microscopy (Section REF ) and measurement uncertainties of the collected image data (Section REF ).", "Two models on the kernel, as motivated by small and high levels of information available on the kernel given the material at hand are discussed in Section ; priors on the parameters of the respective models is discussed here as well.", "Section is devoted to the development of priors on the unknown density function such that the priors are adaptive to the sparsity in the density.", "In Section , the discretised form of the successive projections of the convolution of the unknown functions is given for the three different models.", "Inference is discussed in Section .", "Section discusses with salient aspects of the uniqueness of the solutions.", "Application of the method to the analysis of real Scanning Electron Microscope image data is included in Section while results from the inversion of simulated image data are presented in Section .", "Relevant aspects of the methodology and possible other real-life applications in the physical sciences are discussed in Section ." ], [ "Application to Electron Microscopy Image Data", "For our application, the system i.e.", "the slab of material, is modelled as a rectangular cuboid such that the surface at which the electron beam is incident is the $Z$ =0 plane.", "Thus, the surface of the system is spanned by the orthogonal $X$ and $Y$ -axes, each of which is also orthogonal to the $Z$ -axis that spans the depth of the slab.", "In our problem, the unknown functions are the material density $\\rho (x,y,z)$ and the kernel $\\eta (x,y,z)$ .", "Here we learn the unknown functions by inverting the image data obtained with a Scanning Electron Microscope.", "Though the learning of the convolution $\\rho \\ast \\eta $ of $\\rho (x,y,z)$ and $\\eta (x,y,z)$ , is in principle ill-posed, we suggest a novel design of imaging experiment that allows for an expansion of the available information leading to $\\rho \\ast \\eta $ being learnt uniquely when noise in the data is small (Section ).", "This is achieved by imaging each part of the material sample at $N_{{\\textrm {eng}}}$ different values of a parameter $E$ such that images taken at different values $\\epsilon $ of this parameter $E$ , carry information about the density function from different depths under the sample surface.", "Such is possible if $E$ represents the energy of the electrons in the incident electron beam that is used to take the SEM image of the system; since the sub-surface penetration depth of the electron beam increases with electron energy, images taken with beams of different energies inherently bear information about the structure of the material at different depths.", "The imaging instrument does not image the whole sample all at the same time.", "Rather, the imaging technique that is relevant to this problem is characterised by imaging different parts of the sample, successively in time.", "At each of these discrete time points, the $i$ -th part, ($i=1,\\ldots ,N_{{\\textrm {data}}}$ ) of the sample is viewed by the imaging instrument, along the viewing vector ${v}_i$ to record the image datum in the $i$ -th pixel on the imaging screen.", "Thus, every pixel of image data represents information about a part of the sample.", "The $i$ -th viewing vector corresponds to the $i$ -th point of incidence of the electron beam on the surface of the material sample.", "The image datum recorded in the $i$ -th pixel then results from this viewing, and harbours information about the sample structure inside the $i$ -th interaction-volume, which is the region that bounds atomistic interactions between the incident electron beam and molecules of the material (see Figure REF ).", "The point of incidence of the $i$ -th viewing vector, i.e.", "the centre of the $i$ -th interaction volume is marked in the diagram.", "The volume of any such 3-D interaction-volume is known from microscopy theory and is a function of the energy $E$ of the beam electrons.", "In fact, motivated by the microscopy literature, , we model the shape of the $i$ -th interaction-volume as hemispherical, centred at the incidence point of $v_i$ , with the radius of this hemisphere modelled as $\\propto E^{1.67}$ .", "We image each part of the sample at $N_{{\\textrm {eng}}}$ different values of $E$ , such that the $k$ -th value of $E$ is $\\epsilon _k$ , $k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "To summarise, the data comprise a $N_{{\\textrm {eng}}}$ 2-D images, each image being a square spatial array of ${N_{{\\textrm {data}}}}$ number of pixels such that at the $i$ -th pixel, for the $k$ -th value of $E$ , the image data is $\\tilde{I}_i^{(k)}$ where $i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "Consideration of this full set of images will then suggest the sub-surface density function of the sample, in a fully discretised model.", "Convolution $\\rho \\ast \\eta $ of $\\rho (x,y,z)$ and $\\eta (x,y,z)$ is projected onto the system surface, i.e.", "the $Z$ =0 plane and this projection is then further projected onto one of the $X$ or $Y$ axes, with the resulting projection being projected once again, to the central point of the interaction-volume created by the $i$ -th incidence of the beam of energy $\\epsilon _k$ , to give rise to the image datum in the $i$ -th pixel in the $k$ -th image.", "Expanding information by imaging at multiple values of $E$ is preferred to imaging at multiple viewing angles for logistical reason as discussed above in Section .", "Further, the shape of the interaction-volume is rendered asymmetric about the line of incidence of the beam when the beam is tilted to the vertical , where the asymmetry is dependent on the tilt angle and the material.", "Then it is no longer possible to have confidence in any parametric model of the geometry of the interaction-volume as given in microscopy theory.", "Knowledge of the symmetry of the interaction-volume is of crucial importance to any attempt in inverting SEM image data with the aim of learning the material density function.", "So in this application, varying the tilt angle will have to be accompanied by approximations in the modelling of the projection of $\\rho \\ast \\eta $ .", "We wish to avoid this and therefore resort to imaging at varying values of $E$ .", "Physically speaking, the unknown density $\\rho (x,y,z)$ would be the material density representing amount or mass of material per unit 3-D volume.", "The measured image datum could be the radiation in Back Scattered Electrons or X-rays, in each 2-D pixel, as measured by a Scanning Electron Microscope.", "Figure: Left: schematic diagram of the imagingexperiment relevant to the application of inverting image datataken with a Scanning Electron Microscope, to learn sub-surfacematerial density and the kernel.", "The interaction-volume createdby the incidence of the electron beam (shown in thick arrow) ismodelled as a hemisphere (shown in blue) centred at the point ofincidence of this beam with radius given as a function of theenergy EE of the beam.", "At each of the N data N_{{\\textrm {data}}} incidences ofthe beam, N eng N_{{\\textrm {eng}}} images are taken with beams that aredistinguished by the energy EE of the electrons in them.", "TheXX, YY and ZZ-axes characterising the 3-dimensional grid usedin the modelling are marked.", "The ii-th incidence of the beam ison the point (x i ,y i ,0)(x_i,y_i,0) on the sample surface.", "Right:The jj-th and ii-th beam incidences are depicted.", "At theii-th incidence, the interaction-volumes resulting frominteractions of the material with beams of energy E=ϵ k E=\\epsilon _k(larger interaction-volume in outline only) andE=ϵ k-1 E=\\epsilon _{k-1} (smaller interaction-volume in blue) areschematically depicted." ], [ "Modelling", "The general inverse problem is $I = {\\cal P}(\\rho ) + \\varepsilon $ where the data $I:{\\mathbb {R}}^m\\longrightarrow {\\cal D}\\subseteq {\\mathbb {R}}$ , while the unknown function $\\rho :{\\mathbb {R}}^n\\longrightarrow {\\cal H}\\subseteq {\\mathbb {R}}$ , with $m\\le n$ .", "In particular, 3-D modelling involves the case of $n=3$ , $m=2$ .", "The case of $n > m$ is fundamentally an ill-posed problem , .", "Here $\\varepsilon $ is the measurement noise, the distribution of which, we assume known.", "In our application, the data is represented as the projection of the convolution of the unknown functions as: ${\\tilde{I}}={\\cal C}[\\rho \\ast \\eta ] + \\varepsilon , \\nonumber $ where the projection operator ${\\cal C}$ is a contractive projection from the space that $\\rho \\ast \\eta $ lives in - which is $\\subseteq {\\mathbb {R}}^3$ - onto the image space ${\\cal D}$ .", "${\\cal C}$ itself is a composition of 3 independent projections in general, ${\\cal C}=P_1\\circ P_2\\circ P_3, \\nonumber $ where $P_1$ is the projection onto the $Z$ =0 plane, followed by $P_2$ - the projection onto the $Y$ =0 axis, followed by $P_3$ - projection onto the centre of a known material-dependent three dimensional region inside the system, namely the interaction volume.", "These 3 projections are commutable, resulting in an effective contractive projection of $\\rho \\ast \\eta $ onto the centre of this interaction volume.", "Thus, the learning of $\\rho \\ast \\eta $ requires multiple (three) inversions of the image data.", "In this sense, this is a harder than usual inversion problem.", "The interaction volume and its centre - at which point the electron beam is incident - are shown in Figure REF .", "It is possible to reduce the learning of $\\rho \\ast \\eta $ to a least-squares problem in the low-noise limit, rendering the inverse learning of $\\rho \\ast \\eta $ unique by invoking the Moore Penrose inverse of the matrix ${\\bf C}$ that is the matrix representation of the ${\\cal C}$ operator (Section ).", "The learning of $\\rho ({x,y,z})$ and $\\eta ({x,y,z})$ individually, from the uniquely learnt $\\rho \\ast \\eta $ is still an ill-posed problem.", "In a non-Bayesian framework, a comparison of the number of unknowns with the number of measured parameters is relevant; imaging at $N_{{\\textrm {eng}}}$ number of values of $E$ suggests that $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}$ parameters are known while at most $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}} +N_{{\\textrm {eng}}}$ are unknown.", "Thus, for the typical values of $N_{{\\textrm {eng}}}$ and $N_{{\\textrm {data}}}$ used in applications, ratio of known to unknown parameters is $\\ge 0.99$ , using the aforementioned design of experiment (see Section REF ).", "The unknown $N_{{\\textrm {eng}}}$ parameters still renders the individual learning of $\\rho ({x,y,z})$ and $\\eta ({x,y,z})$ non-unique.", "Such considerations are however relevant only in the absence of Bayesian spatial regularisation , , and as we will soon see, the lack of smooth variation in the spatial correlation structure underlying typically discontinuous real-life material density functions, render the undertaking of such regularisation risky.", "However in the Bayesian framework we seek the posterior probability of the unknowns given the image data.", "Thus, in this approach there is no need to directly invert the sequential projection operator ${\\cal C}$ ; the variance of the posterior probability density of the unknowns given the data is crucially controlled by the strength of the priors on the unknowns , .", "We develop the priors on the unknowns using as much of information as is possible.", "Priors on the kernel can be weak or strong depending on information available in the literature relevant to the imaged system.", "Thus, in lieu of strong priors that can be elicited from the literature, a distribution-free model for $\\eta ({x,y,z})$ is motivated, characterised by weak priors on the shape of this unknown function.", "On the contrary, if the shape is better known in the literature for the material at hand, the case is illustrated by considering a parametric model for $\\eta ({x,y,z})$ , in which priors are imposed on the parameters governing this chosen shape.", "Also, the material density function can be sparse or dense.", "To help with this, we need to develop a prior structure on $\\rho ({x,y,z})$ that adapts to the sparsity of the density in its native space.", "It is in this context, that such a problem of multiple inversions is well addressed in the Bayesian framework." ], [ "Defining a general voxel", "Lastly, we realise that the data at hand is intrinsically discrete, owing to the imaging mechanism.", "Then, for a given beam energy, collected image data is characterised by a resolution that depends on the size of the electron beam that is used to take the image, as well as the details of the particular instrument that is employed for the measurements.", "The SEM cannot “see” variation in structure at lengths smaller than the beam size.", "In practice, resolution $\\omega $ is typically less than the beam size; $\\omega $ is given by the microscopist.", "Then only one image datum is available over an interval of width $\\omega $ along each of the $X$ and $Y$ -axes, i.e.", "only one datum is available from a square of edge $\\omega $ , located at the beam pointing.", "This implies that using such data, no variation in the material density can be estimated within any square (lying on the $X-Y$ plane) of side $\\omega $ , located at a beam pointing, i.e.", "$\\rho (x,y,z)=\\rho (x+\\delta _1,y+\\delta _2,z)$ , where $\\delta _1,\\delta _2\\in [0,\\omega )$ , $\\forall \\:x,y,z$ .", "The reason for not being able to predict the density over lengths smaller than $\\omega $ , in typical real-life material samples, is discussed below.", "image data are recorded at discrete (chosen) values of the beam energy $E$ such that for a given beam pointing, a beam at a given value $\\epsilon _k$ of $E$ carries information about the material density from within the sub-surface depth of $h^{(k)}$ .", "Then 2 beams at successive values $\\epsilon _{k-1}$ and $\\epsilon _k$ of $E$ allow for estimation of the material density within the depth interval $[h^{(k-1)}, h^{(k)})$ that is bounded by the beam penetration depths.", "Here $k=1,2,\\ldots ,N_{{\\textrm {eng}}}$ , $h^{(0)}$ =0.", "This implies that for a given beam pointing, over any such interval, no variation in the material density can be learnt from the data, i.e.", "$\\rho (x,y,z)=\\rho (x,y,z+\\delta _3)$ where $\\delta _3\\in [h^{(k-1)}, h^{(k)})$ , $\\forall \\:x,y,z$ .", "Again, the reason for inability to predict the density at any $z$ is explained below.", "As is evident in these 2 bulletted points above, we are attempting to learn the density at discrete points identified by the resolution in the data.", "The inability to predict the density at any point inside the sample could be possible in alternative models, for example when a Gaussian Process (GP) prior is used to model the trivariate density function that is typically marked by disjoint support in ${\\mathbb {R}}^3$ and/or by sharp density contrasts (seen in a real density as in Figure REF and those learnt from simulated image data, as in Figure REF ).", "While the covariance structure of such a GP would be non-stationary (as motivated in the introductory section), the highly discontinuous nature of real-life density functions would compel this non-stationary covariance function to not vary smoothly.", "Albeit difficult, the parametrisation of such covariance kernels can in principle be modelled using a training data–except for the unavailability of such training data (comprising known values of density at chosen points inside the given 3-D sample).", "As training data is not at hand, parametrisation of the non-stationary covariance structure is not possible, leading us to resort to a fully discretised model in which we learn the density function at points inside the sample, as determined by the data resolution.", "In fact, the learnt density function could then be used as training data in a GP-based modelling scheme to help learn the covariance structure.", "Such an attempt is retained for future consideration.", "Thus, prediction would involve (Bayesian) spatial regularisation which demands modelling of the abruptly varying spatial correlation structure that underlies the highly discontinuous, trivariate density function of real life material samples (see Figure REF ).", "The spatial variation in the correlation structure of the image data cannot be used as a proxy for that of the material density function, since the former is not a pointer to the latter, given that compression of the density information within a 3-D interaction-volume generates an image datum.", "Only with training data comprising known material density at chosen points, can we model the correlation structure of the density function.", "In lieu of training data, as in the current case, modelling of such correlation is likely to fail.", "Hence we learn the density function at the grid points of a 3-D Cartesian grid set up in the space of $X$ , $Y$ and $Z$ .", "No variation in the material density inside the $ik$ -th grid-cell can be learnt, where this grid-cell is defined as the “$ik$ -th voxel” which is the cuboid given by square cross-sectional area (on a $Z=$ constant plane) of size $\\omega ^2$ , with edges parallel to the $X$ and $Y$ axes, located at the $i$ -th beam incidence, depth along the $Z$ -axis ranging from $h^{(k-1)}\\le z< h^{(k)}$ , with $i=1,2,\\ldots ,N_{{\\textrm {data}}}$ , $k=1,2,\\ldots ,N_{{\\textrm {eng}}}$ , $h^{(0)}=0$ .", "Then one pair of opposite vertices of the $ik$ -th voxel are respectively at points $(x_i,y_i,h^{(k-1)})$ and $(x_i,y_i,h^{(k)})$ , where the $i$ -th beam incidence is at the point $(x_i,y_i,0)$ and ${x_i:= i\\:{\\textrm {modulo}}\\:(\\sqrt{N_{{\\textrm {data}}}})\\quad y_i:={\\textrm {int}}(i/\\sqrt{N_{{\\textrm {data}}}})+1.", "}.$ See Figure REF for a schematic depiction of a general voxel.", "Here $\\sqrt{N_{{\\textrm {data}}}}$ is the number of intervals of width $\\omega $ along the $X$ -axis, as well as along the $Y$ -axis.", "Thus, on the part of the $X-Y$ plane that is imaged, there are $N_{{\\textrm {data}}}$ squares of size $\\omega $ .", "Formally, the density function that we can learn using such discrete image data is represented as $\\xi ^{(k)}_{i} := \\rho (x,y,z)\\quad \\mbox{for}\\quad x\\in [x_i,x_i+\\omega ),y\\in [y_i,y_i+\\omega ), z\\in [h^{(k-1)},h^{(k)}),$ This representation suggests that a discretised version of the sought density function is what we can learn using the available image data.", "We do not invoke any distribution for the unknown parameters $\\xi ^{(1)}_1,\\xi ^{(2)}_1,\\ldots ,\\xi ^{(N_{{\\textrm {eng}}})}_1,\\xi ^{(1)}_2,\\ldots ,\\xi ^{(N_{{\\textrm {eng}}})}_2,\\ldots ,\\xi ^{(1)}_{N_{{\\textrm {data}}}},\\ldots ,\\xi ^{(N_{{\\textrm {eng}}})}_{N_{{\\textrm {data}}}}$ .", "It is to be noted that the number of parameters that we seek is deterministically driven by the image data that we work with–the discretisation of the data drives the discretisation of the space of $X$ , $Y$ and $Z$ and the density in each of the resulting voxels is sought.", "Thus, when the image has a resolution $\\omega $ and there are $N_{{\\textrm {eng}}}$ such image data sets recorded at a value of $E$ , the number of sought density parameters is $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}$ where $N_{{\\textrm {data}}}$ is the number of squares of size $\\omega $ that comprise part of the surface of the material slab that is imaged.", "Here the number of parameters $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}$ is typically large; in the application to real data that we present later, this number is 2250.", "We set up a generic model that allows for the learning of such large, data-driven number of distribution-free density parameters in a Bayesian framework.", "Figure: Figure to bring out the general (ikik-th) voxel, asdistinguished from the general (ikik-th) interaction-volume.", "Forthe value ϵ k \\epsilon _k of the imaging parameter EE, the ii-thviewing vector v i v_i impinges the ZZ=0 surface at the materialslab at point (x i ,y i ,0)(x_i,y_i,0), creating the ikik-thinteraction-volume (in blue) that is modelled as a hemisphere ofradius R0 (k) R0^{(k)}.", "The maximal depth that this interaction-volumeextends to, is h (k) h^{(k)}.", "Thus, under the assumption ofhemi-spherical shape, R0 (k) =h (k) R0^{(k)}=h^{(k)}.", "Here,i=1,...,N data i=1,\\ldots ,N_{{\\textrm {data}}} and k=1,...,N eng k=1,\\ldots ,N_{{\\textrm {eng}}}.", "In the figure,the beam is incident on the 14-th square on the surface, where the origin is at the marked vertex O.", "Also, the depicted interaction-volume goes down to the 4-th ZZ-bin, i.e.", "has a radius=R0 (4) =h (4) R0^{(4)}=h^{(4)}.", "Then the depicted interaction-volume is the ikik-th one, with i=14i=14 andkk=4.", "The example voxel the outermost face of which on the X=0X=0plane is shown in red, lies between depths h (3) h^{(3)} andh (4) h^{(4)}, and has i=2i=2; this is then the ikik-th voxel, withii=2, kk=4.", "The grid-cells on the ZZ=constant planes aresquares of side ω\\omega , the resolution length of the imaginginstrumentation.", "The gridding along the ZZ-axis isnon-uniform.", "Material density inside the ikik-th voxel is theconstant ξ i (k) \\xi _i^{(k)} and for all i=1,...,N data i=1,\\ldots ,N_{{\\textrm {data}}},correction function in this voxel is η (k) \\eta ^{(k)},k=1,...,N eng k=1,\\ldots ,N_{{\\textrm {eng}}}." ], [ "Defining a general interaction-volume", "Under the consideration of a hemi-spherical shape of the interaction-volume, the $ik$ -th interaction-volume is completely specified by pinning its centre to the $i$ -th beam pointing at $(x_i,y_i,0)$ on the $Z$ =0 plane and fixing its radius to $R0^{(k)}\\propto {\\epsilon _k}^{1.67}$ , where the constant of proportionality is known from microscopy theoryand $x_i$ and $y_i$ are defined in Equation REF .", "In this hemispherical geometry, the maximal depth that the $ik$ -th interaction volume goes down to is $h^{(k)}=R0^{(k)}$ .", "In the context of the application to microscopy, atomic theory studies undertaken by kanaya suggest that the maximal depth to which an electron beam can penetrate inside a given material sample is $h^{(k)}$ (measured in $\\mu $ m), for a beam of electrons of energy $E=\\epsilon _k$ (measured in kV), inside material of mass density of $d$ (measured in gm cm$^{-3}$ ), atomic number ${\\cal Z}$ and atomic weight $A$ (per gm per mole), is $h^{(k)} = \\displaystyle {\\frac{0.0276 A \\epsilon _k^{1.67}}{d {\\cal Z}^{0.89}}}\\quad k\\in {\\mathbb {Z}}_{+}.$ Here ${\\cal Z}$ is an integer valued atomic number of the material, ${\\cal Z} > 0$ while $A$ and $d$ are positive-definite real valued constants.", "As $E$ increases, the depth and radial extent of the interaction-volume increases.", "The measured image datum in the $i$ -th pixel, created at $E=\\epsilon _k$ , is ${\\tilde{I}}_i^{(k)}$ .", "This results when the convolution $(\\rho \\ast \\eta )_i^{(k)}$ , of the unknown density and kernel in the voxels that lie inside the $ik$ -th interaction-volume, is sequentially projected (as mentioned in Section ), onto the centre of the $ik$ -th interaction-volume, i.e.", "onto the point $(x_i,y_i,0)$ .", "This projection is referred to as ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ .", "The integral representation of such a projection suggests 3 integrals, each over the three spatial coordinates that define the $ik$ -th interaction-volume, according to ${\\cal C}(\\rho \\ast \\eta )_i^{(k)} = \\displaystyle {\\frac{\\int _{R=0}^{R0^{(k)}}\\int _{\\theta =0}^{\\theta _{max}} RdR d\\theta \\int _{z=0}^{z=z_{max}(R,\\theta )}\\rho ({\\bf s})\\ast \\eta ({\\bf s})dz}{ \\int _{R=0}^{R_{max}}\\int _{\\theta =0}^{\\theta _{max}} RdR d\\theta }}$ for $i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "Here the vector ${\\bf s}$ represents value of displacement from the point of incidence $(x_i,y_i,0)$ , to a general point $(x,y,z)$ , inside the interaction-volume, i.e.", "${\\bf s}:= (x-x_i, y-y_i, z)^T &=& (R\\cos \\theta , R\\sin \\theta , z)^T\\nonumber \\\\R = \\sqrt{(x-x_i)^2+(y-y_i)^2} &&\\quad \\tan \\theta = \\displaystyle {\\frac{y-y_i}{x-x_i}}.", "\\nonumber \\\\$" ], [ "Cases classed by image resolution", "We realise that the thickness of the electron beam imposes an upper limit on the image resolution, i.e.", "on the smallest length $\\omega $ over which structure can be learnt in the available image data.", "For example, the resolution is finer when the SEM image is taken in Back Scattered Electrons ($\\omega \\lesssim $ 0.01$\\mu $ m) than in X-rays ($\\omega \\sim $ 1$\\mu $ m).", "The comparison between the cross-sectional areas of a voxel and of an interaction-volume, on the $Z$ =constant plane is determined by $\\omega $ since the area of the voxel at any $z$ is $\\omega ^2$ .", "the atomic-number parameter ${\\cal Z}$ of the material at hand (see Equation REF ) since the radius $R0^{(k)}$ of the hemi-spherical interaction-volume is a monotonically decreasing function of ${\\cal Z}$ , at $E=\\epsilon _k$ .", "Then we can think of 3 different resolution-driven models such that voxel cross-sectional area on the $Z$ =0 plane exceeds that of interaction-volumes attained at all $E$ , i.e.", "$\\pi (R0^{(k)})^2\\le \\omega ^2,\\:\\forall \\:k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "voxel cross-sectional area on the $Z$ =0 plane exceeds that of interaction-volumes at some values of $E$ but is lower than that of interaction-volumes attained at higher $E$ values, i.e.", "$\\pi (R0^{(k)})^2\\le \\omega ^2,\\:\\forall \\:k=1,\\ldots ,k_{in}$ and $\\pi (R0^{(k)})^2 > \\omega ^2,\\:\\forall \\:k=k_{in}+1,\\ldots ,N_{{\\textrm {eng}}}$ .", "voxel cross-sectional area on the $Z$ =0 plane is lower than that of interaction-volumes attained at all $E$ , i.e.", "$\\pi (R0^{(k)})^2 > \\omega ^2,\\:\\forall \\:k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "The first two models are realised for coarse resolution of the imaging technique, i.e.", "for large $\\omega $ .", "This is contrasted with the 3rd model, which pertains to fine resolution, i.e.", "low $\\omega $ .", "Since $R0^{(k)}$ is a monotonically decreasing function of ${\\cal Z}$ (Equation REF ), the 1st model is feasible for “high-${\\cal Z}$ materials”, while the 2nd model is of relevance when the material at hand is a “low-${\\cal Z}$ material”.", "To make this more formal, we introduce two classes of materials as follows.", "For fixed values of all parameters but ${\\cal Z}$ , ${\\textrm {if}}\\quad \\omega ^2 \\ge \\pi [R0^{(N_{{\\textrm {eng}}})}]^2\\vert {\\cal Z}, & & \\quad {\\textrm {``high-{\\cal Z}^{\\prime \\prime } material}}\\\\{\\textrm {if}}\\quad \\omega ^2 < \\pi [R0^{(N_{{\\textrm {eng}}})}]^2\\vert {\\cal Z}, & & \\quad {\\textrm {``low-{\\cal Z}^{\\prime \\prime } material}}.\\nonumber $ The computation of the sequential projections involve multiple integrals (see Equation REF ).", "The required number of such integrals can be reduced by invoking symmetries in the material density in any interaction-volume.", "Such symmetries become avaliable, as the resolution in the imaging technique becomes coarser.", "For example, for the 1st model, when the resolution is the coarsest, the minimum length $\\omega $ learnt in the data exceeds the diameter of the lateral cross-section of the largest interaction-volume achieved at the highest value of $E$ , i.e for $E=\\epsilon _{N_{{\\textrm {eng}}}}$ .", "(Here “lateral cross section” implies the cross-section on the $Z$ =0 plane).", "Then for the coarsest resolution, $2R0^{(N_{{\\textrm {eng}}})} \\le \\omega $ (see Figure REF ) and so, $\\displaystyle {\\pi [R0^{(N_{{\\textrm {eng}}})}]^2} \\le \\omega ^2$ which means the lateral cross-section of the $ik$ -th interaction-volume is contained inside that of the $ik$ -th voxel.", "Now in our discretised representation, the material density inside any voxel is a constant.", "Therefore it follows that when resolution is the coarsest, material density at any depth inside an interaction-volume, is a constant independent of the angular coordinate $\\theta $ (see Equation REF ).", "Thus, the projection of $\\rho \\ast \\eta $ onto the centre of the interaction-volume, does not involve integration over this angular coordinate.", "However, such will not be possible for the 3rd model at finer imaging resolution.", "When the resolution is the finest possible with an SEM, $\\omega $ is the smallest.", "In this case, the lateral cross-section of many voxels fill up the cross-sectional area of a given interaction volume.", "Thus, the density is varying within the interaction-volume.", "Thus, the projection of $\\rho \\ast \\eta $ cannot forgo integrating over $\\theta $ .", "The last case that we consider falls in between these extremes of resolution; this is the 2nd model.", "Isotropy is valid $\\forall \\:k=1,\\ldots ,k_{in}$ but not for $k=k_{in}+1,\\ldots ,N_{{\\textrm {eng}}}$ ; (see middle panel in Figure REF ).", "Figure: A schematic diagram to depict the three models that wework with.", "When the instrumental resolution is the coarsest out ofthe three cases, i.e.", "ω\\omega is the largest, (for high-𝒵{\\cal Z} materials), the size of the ikik-th interaction-volumes(depicted in red) is such that their cross-sectionalareas on a ZZ=constant plane fall short of the area of a voxel(outlined in black), on this plane.", "Here k=1,...,N eng k=1,\\ldots ,N_{{\\textrm {eng}}}though in the representation on the left, a value of 2 has beenused for N eng N_{{\\textrm {eng}}}.", "Thus, the projection 𝒞(ρη) i (k) {\\cal C}(\\rho \\ast \\eta )_i^{(k)} onto the centre of aninteraction-volume is done by considering an isotropic densityfunction, since density inside a voxel is a constant.", "Middlepanel: (for low-𝒵{\\cal Z} materials) the cross-sectional areaπ(R0 (k) ) 2 \\pi (R0^{(k)})^2 of the ikik-th interaction-volume and ω 2 \\omega ^2of any voxel, on the ZZ=0 plane are such thatπ(R0 (k) ) 2 ≤ω 2 \\pi (R0^{(k)})^2\\le \\omega ^2 for k=1,...,k in k=1,\\ldots ,k_{in}.", "On theother hand, for k=k in +1,...,N eng k= k_{in}+1,\\ldots ,N_{{\\textrm {eng}}},π(R0 (k) ) 2 >ω 2 \\pi (R0^{(k)})^2>\\omega ^2 and therefore for such kk, the ikik-thinteraction-volume spills into the neighbouring voxels on theZZ=0 plane.", "Thus, the contribution to projection 𝒞(ρη) i (k) {\\cal C}(\\rho \\ast \\eta )_i^{(k)} from the ikik-thinteraction-volume includes contribution from these neighbouringvoxels as well.", "Such contribution is modelled using anearest-neighbour weighted averaging.", "Left: for the finestinstrumental resolution, (obtained for example with Back ScatteredElectron images taken with SEM), for all k=1,...,N eng k=1,\\ldots ,N_{{\\textrm {eng}}} ingeneral, the interaction volume exceeds a voxel in cross-sectionalarea, on the ZZ=0 surface." ], [ "Noise in discrete image data", "As discussed above in Section , noise in the data can crucially influence how well-conditioned the inverse problem is.", "Image data is invariably subjected to noise, though for images taken with Scanning Electron Microscopes (SEM), the noise is small.", "Also, imaging with SEM being a controlled experiment, this noise can be reduced by the microscopist by increasing the length of time taken to scan the sample with the electron beam that is made incident on the sample in this imaging procedure.", "As illustrations of the noise in real SEM data, scanlength suggest that in a typical 20 second scan of the SEM, the signal to noise is about 20:1, which can be improved to 120:1 for a longer acquisition time of 10 minutes.", "Importantly, this noise could arise due to to variations in the beam configurations, detector irregularities, pressure fluctuations, etc.", "In summary, such noise is considered to be due to random variations in parameters that are intrinsically continuous, motivating a Gaussian noise distribution.", "Thus, the noise in the 2-D image data, created by the $i$ -th beam pointing for $E=\\epsilon _k$ , (i.e.", "the noise in ${\\tilde{I}}_i^{(k)}$ ) is modelled as a random normal variable, drawn from the normal with standard deviation $\\sigma _i^{(k)}\\lesssim $ 0.05$\\tilde{I}_i^{(k)}$ , $k=1,\\ldots ,N_{{\\textrm {eng}}},\\:i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "In the illustration with real data, a scan speed of 50 s was used, which implies a noise of less than 5$\\%$ .", "In modern SEMs, noise reduction is alternatively handled using pixel averaging (\"Stereoscan 430 Operator Manual\", Leica Cambridge Ltd. Cambridge, England, 1994)." ], [ "Identifying low-rank and spatially-varying components of density", "It is known in the literature that the general, under-determined deconvolution problem is solvable only if the unknown density is intrinsically, “sufficiently” sparse donohotanner,ma.", "Here we advance a methodology, within the frame of a designed experiment, to learn the density - sparse or dense.", "In the following subsection, we will see that priors on the sparsity that we develop here, bring in relatively more information into the models when the density structure is sparse.", "With this in mind, the density is recognised to be made of a constant $\\rho _0$ (the low-rank component in the limiting sense), and the spatially varying component $\\rho _1(x,y,z)$ that may be sparse or dense in ${\\mathbb {R}}^3$ .", "We view the constant part of the density as $\\rho _0=\\rho _0\\delta (x-x_i, y-y_i, z)$ where $\\delta (\\cdot ,\\cdot ,\\cdot )$ is the Dirac delta function on ${\\mathbb {R}}^3$ , centred at the centre of the $ik$ -th interaction volume, $\\forall \\:k=1,2,\\ldots ,N_{{\\textrm {eng}}}$ .", "Then in our problem, the contribution of the constant part of the density to the projection onto the centre of the $ik$ -th interaction-volume is ${\\cal C}(\\rho _0\\ast {\\eta }({x,y,z})_i^{(k)})&\\equiv &\\rho _0{\\cal C}(\\delta (x-x_i,y-y_i,z)\\ast {\\eta }(x,y,z))_i^{(k)}) \\nonumber \\\\&=& I0^{(k)},$ a constant independent of the beam pointing location $i$ , if $\\eta (x,y,z)$ is restricted to be a function of the depth coordinate $Z$ only.", "As is discussed in Section , this is indeed what we adopt in the model for the kernel.", "Then, $I0^{(k)}$ depends only on the known morphological details of the interaction-volume for a given value of $E$ , $\\forall \\:i=1,2,\\ldots ,N_{{\\textrm {data}}}$ .", "Thus, $\\lbrace {\\tilde{I}}_i^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}= \\lbrace {\\tilde{I}1}_i^{(k)} +I0^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}$ , where ${\\tilde{I}1}^{(k)}_i$ is the spatially-varying component of the image data.", "The identification of the constant component of the density is easily performed as due to the constant component of the measurable.", "In our inversion exercise, it is the $\\lbrace {\\tilde{I}1}_i^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}$ field that is actually implemented as data, after $I0^{(k)} := \\inf \\lbrace {\\tilde{I}}_i^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}$ is subtracted from $\\lbrace {\\tilde{I}}_i^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}$ , for each $k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "Hereafter, when we refer to the data, the spatially-varying part of the data will be implied; it is this part of the data that will hereafter be referred to as $\\lbrace {\\tilde{I}}_i^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}$ , at each value $\\epsilon _k$ of $E$ , $k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "Its inversion will yield a spatially varying sparse/dense density, that we will from now, be referred to as $\\rho (x,y,z)$ that in general lies in a non-convex subset of ${\\mathbb {R}}_{\\ge 0}$ .", "Thus, we see that in this model, it is possible for $\\rho (x,y,z)$ to be 0.", "The construction of the full density, inclusive of the low-rank and spatially-varying parts, is straightforward once the latter is learnt." ], [ "Basics of algorithm leading to learning of unknowns", "The basic scheme of the learning of the unknown functions is as follows.", "First, we perform multiple projections of the convolution of the unknown functions in the forward problem, onto the incidence point of the $ik$ -th interaction-volume, $\\forall \\:i,k$ .", "The likelihood is defined as a function of the Euclidean distance between the spatially-averaged projection ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ , and the image datum ${\\tilde{I}}_i^{(k)}$ .", "We choose to work with a Gaussian likelihood (Equation REF ).", "Since the imaging at any value of $E$ along any of the viewing vectors is done independent of all other values of $E$ and all other viewing vectors, we assume $\\lbrace {\\tilde{I}}_i^{(k)}\\rbrace $ to be $iid$ conditionally on the values of the model parameters.", "We develop adaptive priors on sparsity on the density and present strong to weak priors on the kernel, to cover the range of high to low information about the kernel that may be available.", "In concert with these, the likelihood leads to the posterior probability of the unknowns, given all the image data.", "The posterior is sampled from using adaptive Metropolis-Hastings within Gibbs.", "The unknown functions are updated using respective proposal densities.", "Upon convergence, we present the learnt parameters with 95$\\%$ -highest probability density credible regions.", "At the same time, discussions about the identifiability of the solutions for the 2 unknown functions and of uniqueness of the solutions are considered in detail." ], [ "Kernel function", "Identifiability between the solutions for the unknown material density and kernel is aided by the availability of measurement of the kernel at the surface of the material sample and the model assumption that the kernel function depends only on the depth coordinate $Z$ and is independent of $X$ and $Y$ .", "Thus, the correction function is $\\eta (z)$ .", "In the discretised space of $X$ , $Y$ and $Z$ , the kernel is then defined as $\\eta ^{(k)}:=\\eta (z) \\quad \\mbox{for}\\quad z\\in [h^{(k-1)},h^{(k)}), \\quad {k=1,\\ldots ,N_{{\\textrm {eng}}}}.$ Then $\\eta {(0)}$ is the value of the kernel function on the material surface, i.e.", "at $Z=0$ and this is available from microscopy theory .", "On occasions when the shape of the kernel function is known for the material at hand, only the parameters of this shape need to be learnt from the data.", "Given the measured value $\\eta (0)$ , i.e.", "the measured $\\eta ^{(1)}$ , the total number of kernel parameters is then 2.", "In this case, the total number of parameters that we attempt learning from the data is $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}} +2$ , i.e.", "the 2 kernel shape parameters and $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}$ number of material density parameters.", "We refer to such a model for the kernel as parametric.", "For other materials, the shape of the kernel function may be unknown.", "In that case, a distribution-free model of the $N_{{\\textrm {eng}}}-1$ parameters of the vectorised kernel function are learnt.", "In this case, the total number of parameters that we learn is $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}+N_{{\\textrm {eng}}}-1$ .", "We fall back on elicitation from the literature on electron microscopy to obtain the priors on the kernel parameters.", "In microscopy practice, the measured image data collected along an angle $\\chi $ to the vertical is given by ${\\displaystyle {Q\\int _{0}^{\\infty } \\Psi (\\rho z)\\exp [-f(\\chi )\\rho z]d(\\rho z)}}$ , where $Q\\in {\\mathbb {R}}_{> 0}$ and $f(\\chi )\\propto 1/\\sin (\\chi )$ with a proportionality constant that is not known apriori but needs to be estimated using system-specific Monte Carlo simulations or approximations based on curve-fitting techniques; see goldstein.", "$\\Psi (\\rho z)$ is the distribution of the variable $\\rho z$ and is again not known apriori but the suggestion in the microscopy literature has been that this can be estimated using Monte Carlo simulations of the system or from atomistic models.", "However, these simulations or model-based calculations are material-specific and their viability in inhomogeneous material samples is questionable.", "Given the lack of modularity in the modelling of the relevant system parameters within the conventional approach, and the existence in the microscopy literature of multiple models that are distinguished by the underlying approximations , , it is meaningful to seek to learn the correction function.", "Our construct differs from this formulation in that we construct an infinitesimally small volume element of depth $\\delta z$ inside the material, at the point $(x,y,z)$ .", "In the limit $\\delta z\\longrightarrow $ 0, the density of the material inside this infinitesimally small volume is a constant, namely $\\rho (x,y,z)$ .", "Thus, the image datum generated from within this infinitesimally small volume - via convolution of the material density and the kernel - is $\\rho (x,y,z)\\eta (z)\\delta z$ .", "Thus, over this infinitesimal volume, our formulation will tie in with the representation in microscopic theory, if we set $\\eta (z)\\propto \\Psi (\\rho z)\\exp [-f(\\chi )\\rho z]$ .", "It then follows that $\\eta (z)$ is motivated to have the same shape as $\\Psi (\\rho z)\\exp (-cz)$ , where $c$ is a known, material dependent constant.", "When information about this shape is invoked, the model for $\\eta ^{(k)}$ is referred to as parametric; however, keeping general application contexts in mind, a less informative prior for the unknown kernel is also developed.", "$c$ depends upon whether the considered image is in Back Scattered Electrons (BSE) or X-rays.", "In he former case $c$ represents the BSE coefficient, often estimated with Monte-Carlo simulations and in the latter, it is the linear attenuation coefficient)." ], [ "Parametric model for correction function", "Information is sometimes available on the shape of the function that represents the kernel.", "For example, we could follow the aforementioned suggestion of approximating the form of $\\eta (z)$ as $\\Psi (\\rho z)\\exp (-cz)$ , multiplied by the scale-factor $Q$ , as long as $c$ is known for the material and imaging technique at hand.", "In fact, for the values of beam energy $E$ that we work at, for most materials $c\\lesssim 10^{-2}$ , so that $\\exp (-cz)\\approx $ 1 ( http://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.html, Goldstein et.", "al 2003).", "Thus, we approximate the shape of $\\eta (z)$ to resemble that of $\\Psi (\\rho z)$ , as given in microscopy literature.", "This shape is akin to that of the folded normal distribution , so that we invoke the folded normal density function to write $\\eta (z) \\equiv \\displaystyle {Q\\left[\\exp \\left(-\\frac{(z-\\eta _0)^2}{2s^2}\\right) +\\exp \\left(-\\frac{(z+\\eta _0)^2}{2s^2}\\right)\\right] }.$ Thus, in this model, $\\eta (z)$ is deterministically known if the parameters $Q$ , $\\eta _0$ and $s$ are, where, $\\eta _0$ and $s$ are the mean and dispersion that define the above folded normal distribution.", "Out of these three parameters, only two are independent since $\\eta (z)$ is known on the surface, i.e.", "$\\eta ^{(1)}$ is known (see Section ).", "Thus, by setting $\\eta ^{(1)}=2Q\\exp [-\\eta _0^2/2s^2]$ , we relate $\\eta _0$ deterministically to $s$ and $Q$ .", "In this model of the correction function, we put folded normal priors on $Q$ and $s$ .", "Thus, this model of the correction function is parametric in the sense that $\\eta (z)$ is parametrised, and priors are put on its unknown parameters." ], [ "Distribution-free model for correction function", "In this model for the correction function, we choose folded normal priors for $\\eta ^{(k)}$ , i.e.", "$\\pi (\\eta ^{(k)})={\\cal N}_F(m_\\eta ^{(k)},s)$ with $m_\\eta ^{(k)}=\\displaystyle {Q\\left[\\exp \\left(-\\frac{(h^{(k)}-\\eta _0)^2}{2s^2}\\right) + \\exp \\left(-\\frac{(h^{(k)}+\\eta _0)^2}{2s^2}\\right)\\right]}$ .", "The folded normal priors on $\\eta ^{(\\cdot )}$ underlie the fact that $\\eta ^{(k)}\\ge 0,\\:\\forall \\:k=1,\\ldots ,N_{{\\textrm {eng}}}$ and that there is a non-zero probability for $\\eta ^{(k)}$ to be zero.", "Thus, folded normal and truncated normal priors for $\\eta ^{(k)}$ would be relevant but gamma priors would not be.", "Also, microscopy theory allows for the kernel at the surface to be known deterministically, i.e.", "$\\eta ^{(1)}$ is known (Section ).", "Then setting $\\pi (\\eta ^{(1)})=$ 1, we relate $Q, \\eta _0$ and $s$ .", "This relation is used to compute $s$ , while uniform priors are assigned to the hyper-parameters $\\eta _0$ , $Q$ .", "We refer to this as the “distribution-free model of $\\eta (z)$ ”.", "We illustrate the effects of both modelling strategies in simulation studies that are presented in Section .", "The correction function is normalised by ${\\widehat{\\eta }}^{(1)}/\\eta ^{(1)}$ , where ${\\widehat{\\eta }}^{(1)}$ is the estimated unscaled correction function in the first $Z$ -bin, i.e.", "for $z\\in [0,h^{(1)})$ and $\\eta ^{(1)}$ is the measured value of the correction function at the surface of the material, given in microscopy literature.", "It is to be noted that the available knowledge of $\\eta ^{(1)}$ allows for the identifiability of the amplitudes of $\\rho (x,y,z)$ and $\\eta (z)$ ." ], [ "Priors on sparsity of the unknown density", "In this section we present the adaptive priors on the sparsity in the density, developed by examining the geometrical aspects of the problem.", "As we are trying to develop priors on the sparsity, we begin by identifying the voxels in which density is zero, i.e.", "$\\xi _i^{(k)}=0$ .", "This identification will be made possible by invoking the nature of the projection operator ${\\cal C(\\cdot )}$ .", "For example, we realise that it is possible for the measured image datum ${\\tilde{I}}_i^{(k)}$ collected from the $ik$ -th interaction-volume to be non-zero, even when density in the $ik$ -th voxel is zero, owing to contributions to ${\\tilde{I}}_i^{(k)}$ from neighbouring voxels that are included within the $ik$ -th interaction-volume.", "Such understanding is used to identify the voxels in which density is zero.", "For a voxel in which density is non-zero, we subsequently learn the value of the density.", "The constraints that lead to the identification of voxels with null density can then be introduced into the model via the prior structure in the following ways.", "We could check if ${\\cal C}({\\rho }\\ast {\\eta })_i^{(k)}$ = ${\\cal C}({\\rho }\\ast {\\eta })_{-im}^{(k)}$ , (where ${\\cal C}(\\cdot )_{-im}^{(k)} :=$ projection onto the centre of the $ik$ -th interaction-volume without including density from the $mk$ -th voxel, i.e.", "without including ${\\xi }_m^{(k)}$ ).", "If so, then ${\\xi }_m^{(k)}=0$ .", "However, this check would demand the computation of ${\\cal C}({\\rho }\\ast {\\eta })_i^{(k)}$ over a given interaction-volume, as many times as there are voxels that lie fully or partially inside this interaction-volume.", "Such computation is avoided owing to its being computationally intensive.", "Instead, we opt for a probabilistic suggestion for when density in a voxel is zero.", "Above, $i=1,2,\\ldots ,N_{{\\textrm {data}}}$ , $k=1,2,\\ldots ,N_{{\\textrm {eng}}}$ .", "We expect that the projection of $\\rho \\ast \\eta $ onto the centre $(x_i,y_i,0)$ of the interaction-volume achieved at $E=\\epsilon _k$ will in general exceed the projection onto the same central point of a smaller interaction volume (at $E=\\epsilon _{k-1}$ ).", "However, if the density of the $ik$ -th voxel is zero or very low, the contributions from the interaction-volume generated at $E=\\epsilon _k$ may not exceed that from the same generated at $E=\\epsilon _{k-1}$ .", "Thus, it might be true that ${\\cal C}({\\rho }\\ast {\\eta })_i^{(k)} \\le {\\cal C}({\\rho }\\ast {\\eta })_i^{(k-1)} \\Longrightarrow \\xi _i^{(k)}=0$ $\\forall \\:i,k$ .", "This statement is true with some probability.", "An alternate representation of the statement REF is achieved as follows.", "We define the random variable $\\tau _i^{(k)}$ , with $0< \\tau _i^{(k)}\\le 1$ , such that $\\tau _i^{(k)} &:=& \\displaystyle {\\frac{{\\cal C}(\\rho \\ast \\eta )_i^{(k)}}{{\\cal C}(\\rho \\ast \\eta )_i^{(k-1)}}},\\:\\:{\\textrm {if}}\\quad {\\cal C}(\\rho \\ast \\eta )_i^{(k)} \\le {\\cal C}(\\rho \\ast \\eta )_i^{(k-1)}\\quad \\mbox{and}\\quad {\\cal C}(\\rho \\ast \\eta )_i^{(k-1)}\\ne 0 \\\\\\tau _i^{(k)} &:=& 1 \\quad {\\textrm {otherwise}} \\nonumber $ Then the statement REF is the same as the statement: “it might be true that $\\tau _i^{(k)} \\le 1\\Longrightarrow \\xi _i^{(k)}=0$ or $\\xi _i^{(k)}$ is very low”, with some probability $\\nu (\\tau _i^{(k)})$ .", "In fact, as the projection from the bigger interaction-volume is in general in excess of that from a smaller interaction-volume, we understand that closer ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ is to ${\\cal C}(\\rho \\ast \\eta )_i^{(k-1)}$ , higher is the probability that $\\xi _i^{(k)}$ is close to 0.", "In other words, the smaller is $\\tau _i^{(k)}$ , higher is the probability $\\nu (\\tau _i^{(k)})$ that $\\xi _i^{(k)}$ is close to 0, where $\\nu (\\tau _i^{(k)})=p^{\\tau _i^{(k)}}(1-p)^{1-\\tau _i^{(k)}},$ with the hyper-parameter $p$ controlling the non-linearity of response of the function $\\nu (\\tau _i^{(k)})$ to increase in $\\tau _i^{(k)}$ .", "The advantage of the chosen form of $\\nu (\\cdot )$ is that it is monotonic and its response to increasing value of its argument is controlled by a single parameter, namely $p$ .", "We assign $p$ a a hyper prior that is uniform over the experimentally-determined range of [0.6, 0.99], to ensure that $\\nu (\\tau _i^{(k)})$ is flatter for lower $p$ and steeper for higher $p$ , (as $\\tau _i^{(k)}$ moves across the range $(0,1]$ ).", "The prior on the density parameter $\\xi _i^{(k)}$ is then defined as $\\pi _0(\\xi _i^{(k)})= \\displaystyle {{\\exp \\left[-\\left(\\xi _i^{(k)}\\nu (\\tau _i^{(k)})\\right)^2\\right]}}.$ Thus, $\\pi _0(\\xi _i^{(k)})\\in $ [0,1] $\\forall \\:i,k$ .", "Any normalisation constant on the prior can be subsumed into the normalisation of the posterior of the unknowns given the data; had we used a normalisation of $\\sqrt{4\\pi /p^2}$ , for all $i$ , at those $k$ for which ${\\cal C}(\\rho \\ast \\eta )_i^{(k)} > {\\cal C}(\\rho \\ast \\eta )_i^{(k-1)}$ or ${\\cal C}(\\rho \\ast \\eta )_i^{(k-1)}=0$ , the prior on $\\xi _i^{(k)}$ would have reduced to being a normal ${\\cal N}(0,2/p^2)$ .", "This is because, for such $\\xi _i^{(k)}$ , $\\tau _i^{(k)}=1$ so that $\\nu (\\tau _i^{(k)})=p$ .", "However, for $\\tau _i^{(k)} < 1$ the prior on $\\xi _i^{(k)}$ is non-normal as the dispersion is itself dependent on $\\xi _i^{(k)}$ .", "This prior then adapts to the sparsity of the material density distribution.", "We do not aim to estimate the degree of sparsity in the material density in our work but aim to develop a prior $\\pi _0(\\xi _i^{(k)})$ $\\forall \\:i,k$ so that $\\lbrace \\xi _i^{(k)}\\rbrace _{k=1}^{N_{{\\textrm {eng}}}}$ sampled from this prior represents the material density distribution at the given $i$ , however sparse the vector $(\\xi _i^{(1)},\\xi _i^{(2)},\\ldots ,\\xi _i^{(N_{{\\textrm {eng}}})})^T$ is.", "Evidently, we do not use a mixture prior but opt for no mixing as in greenshteinpark.", "Indeed, in our prior structure, the term $\\left(\\xi _i^{(k)}\\nu (\\tau _i^{(k)}\\right)^2$ could have been replaced by $\\left\|\\xi _i^{(k)}\\nu (\\tau _i^{(k)}\\right\|$ , (as in parametric Laplace priors suggested by parkcasella, hans, johnstonesilverman), i.e.", "by $\\xi _i^{(k)}\\nu (\\tau _i^{(k)})$ since $\\xi _i^{(k)}$ is non-negative, but as far as sparsity tracking is concerned–which is our focus in developing this prior here–the prior in Equaion REF suffices.", "That such a prior probability density sensitively adapts to the sparsity in the material density distribution, is brought about in the results of 2 simulation studies shown in Figure REF .", "In these studies, the density parameter values in the $ik$ -th voxel are simulated from 2 simplistic toy models that differ from each other in the degree of sparsity of the true material density distribution: $\\xi _i^{(k)}=u_1^{10}/u_2$ , and $\\xi _i^{(k)}=u_3^{10}$ respectively, (where $u_1, u_2, u_3$ are uniformly distributed random numbers in $[0,1]$ ), at a chosen $i$ and energy indices $k=1,2,\\ldots ,10$ .", "In the simulations we specify the beam penetration depth $h^{(k)}\\propto \\epsilon _k^{1.67}$ as suggested by kanaya; as any interaction-volume is hemispherical, its radius $R0^{(k)}=h^{(k)}$ .", "The kernel parameters $\\eta ^{(k)}$ are generated from a quadratic function of $h^{(k)}$ with noise added.", "In the simulations, the material is imaged at resolution $\\omega $ such that $\\pi [R0^{(10)}]^2 \\le \\omega ^2$ , i.e.", "the “1st model” is relevant (see Section REF ).", "This allows for simplification of the computation of ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ according to Equation REF .", "Then at this $i$ , for $k=1,2,\\ldots ,10$ , $\\xi _i^{(k)}$ are plotted in Figure REF against $k$ , as is the logarithm of the prior $\\pi _0(\\xi _i^{(k)})$ computed according to Equation REF , with $p$ held as a random number, uniform in [0.6,0.99].", "Logarithm of the priors are also plotted as a function of the material density parameter.", "We see from the figure that the prior developed here tracks the sparsity of the vector $(\\xi _i^{(1)},\\xi _i^{(2)},\\ldots ,\\xi _i^{(N_{{\\textrm {eng}}})})^T$ well.", "Figure: Top: in the left panel black filled circlesdepict values of simulated material density parametersξ i (k) =u 1 10 /u 2 \\xi _i^{(k)}=u_1^{10}/u_2, u 1 ,u 2 ∼𝒰[0,1]u_1,u_2\\sim {\\cal U}[0,1], at anarbitrary beam position index ii, as a function of the energyindex kk, for k=1,2,...,10k=1,2,\\ldots ,10.", "Log of the priorπ 0 (ξ i (k) )\\pi _0(\\xi _i^{(k)}), as given in Equation ,is shown in the middle panel as a function of kk for p∼𝒰[0.6,0.99]p\\sim {\\cal U}[0.6,0.99].", "The log prior is plotted against the true valuesof ξ i (k) \\xi _i^{(k)} in black filled circles in the rightpanel.Bottom: Asin the top panels, except that this simulation is of a sparsermaterial density distribution with density parameters generated asξ i (k) =u 1 10 \\xi _i^{(k)}=u_1^{10}." ], [ "Three models for image inversion", "In this section we develop the projection of the convolution of the unknown density and kernel onto the centre of the $ik$ -th interaction volume, for each of the 3 models discussed in Section REF , motivated by the 3 different classes of image data classified by their resolution.", "The difference between the size of a voxel and that of an interaction-volume determines the difficulty in the inversion of to the image data.", "As explained in Section REF , we attempt to identify means of dimensionality reduction, i.e.", "reducing the number of integrals involved in the sequential projection of $\\rho \\ast \\eta $ , (see Equation REF ).", "We do this by identifying isotropy in the distribution of the material density within the interaction-volume when possible, leading to elimination of the requirement of averaging over the angular coordinate." ], [ "1st and 2nd-models - Low resolution", "This class of image resolutions ($\\omega \\sim 1\\mu m$ ), pertain to the case when the system is imaged by an SEM in X-rays." ], [ "High-${\\cal Z}$ Systems", "In the 1st model, for the high-${\\cal Z}$ materials, the cross-sectional area of an interaction-volume attained at any $E$ will fit wholly inside that of a voxel, where the cross-section is on a $Z$ -constant plane.", "Then at a given depth, the density inside this interaction-volume is that inside the voxel, i.e.", "is a constant.", "Thus, dimensionality reduction is most easily achieved in this case with the density function isotropic inside an interaction-volume, bearing no dependence on the angular coordinate $\\theta $ of Equation REF .", "Then when we revisit Equation REF in its discretised form, the discrete convolution $\\rho \\ast \\eta $ within the $k$ -th $Z$ -bin and the $i$ -th beam pointing gives $(\\rho \\ast \\eta )_i^{(k)} = \\displaystyle {\\sum _{m=1}^k \\xi _i^{(m)}\\eta ^{(k-m)}}$ , so that the projection onto the centre of the $ik$ -th interaction-volume is discretised to give in Equation REF , gives the following.", "$\\displaystyle {{\\cal C}(\\rho \\ast \\eta )_i^{(k)}} =\\displaystyle {\\frac{1}{(R0^{(k)})^2}}\\displaystyle {{\\LARGE {\\sum _{q=0}^{k}}}\\left[\\frac{(R0^{(q)})^2 -(R0^{(q-1)})^2}{2}\\left\\lbrace \\sum _{t=0}^{q}\\left(\\left(h^{(t)} - h^{(t-1)}\\right)\\displaystyle {\\sum _{m=1}^t \\xi ^{(m)}_{i} {\\eta }^{(t-m)}}\\right)\\right\\rbrace \\right]}.$" ], [ "Low-${\\cal Z}$ Systems", "In the 2nd model, for low-${\\cal Z}$ materials, for any $i\\in \\lbrace 1,\\ldots ,N_{{\\textrm {data}}}\\rbrace $ , and $k > k_{in}$ , the cross-sectional area of the $ik$ -th interaction-volume on the $Z$ =0 plane will spill out of the $i1$ -th voxel into the neighbouring voxels.", "Then only for at $k=1,2,\\ldots ,k_{in}$ , isotropy within the $ik$ -th interaction volume holds good.", "In general, the projection ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ onto the centre of the $ik$ -th interaction-volume includes contributions from all those voxels that lie wholly as well as partly inside this interaction volume.", "This projection is then computed by first learning the weighted average of the contributions from all the relevant voxels and then distributing this learnt weighted average over the identified voxels in the proportion of the weights which in turn are in proportion of the voxel volume that lies within the $ik$ -th interaction-volume.", "Thus, for the $i^{\\prime } k^{\\prime }$ -th voxel that is a neighbour of the $ik^{\\prime }$ -th voxel that lies wholly inside the $ik$ -th interaction-volume, let this proportion be $w^{(k^{\\prime })}_{i^{\\prime }\\vert i}$ , where $k^{\\prime }\\le k$ .", "In general, at a given $Z$ , any bulk voxel has 8 neighbouring voxels and when the voxel lies at the corner or edge of the sample, number of nearest neighbours is less than 8.", "Then, at a given $Z$ , there will be contribution from at most 9 voxels towards ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ .", "At $Z=z\\in [h^{(k^{\\prime }-1)},h^{(k^{\\prime })}]$ , for any $i$ , let the maximum number of nearest neighbours be $i_{max}\\vert i,k$ so that $i_{max}\\vert i,k \\le 9$ .", "The notation for this number bears its dependence on both $i$ and $k$ .", "We define ${\\bar{\\xi }}_i^{(k^{\\prime })}$ as the weighted average of the densities in the $ik^{\\prime }$ -th voxel and its nearest neighbours that are fully or partially included within the $ik$ -th interaction-volume.", "Here $k^{\\prime } \\le k$ , $k=1,\\ldots ,N_{{\\textrm {eng}}}$ , $i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "Thus, ${\\bar{\\xi }}_i^{(k^{\\prime })}:=\\displaystyle {\\sum _{i^{\\prime }=1}^{i_{max}\\vert i,k^{\\prime }}\\xi _{i^{\\prime }\\vert i}^{(k^{\\prime })}w^{(k^{\\prime })}_{i^{\\prime }\\vert i}},$ where the $i^{\\prime }$ -th neighbour of the $ik^{\\prime }$ -th voxel at the same depth, harbours the density $\\xi _{i^{\\prime }\\vert i}^{(k^{\\prime })}$ and there is a maximum of $i_{max}\\vert i,k^{\\prime }$ such neighbours.", "The effect of this averaging over the nearest neighbours at this depth, is equivalent to averaging over the angular coordinate $\\theta $ and results in the angular averaged density ${\\bar{\\xi }}_i^{(k^{\\prime })}$ at this $Z$ , which by definition, is isotropic, i.e.", "independent of the angular coordinate.", "Then the projection onto the centre of the $ik$ -th interaction-volume, for $k > k_{in}$ , is computed as in Equation REF with the material density term $\\xi _i^{(\\cdot )}$ on the RHS of this equation replaced by the isotropic angular averaged density ${\\bar{\\xi }}_i^{(\\cdot )}$ .", "However, for $k \\le k_{in}$ , the projection is computed as in Equation REF ." ], [ "3", "In certain imaging techniques, such as imaging in Back Scattered Electrons by an SEM or FESEM, the resolution $\\omega \\ll R0_i^{(N_{{\\textrm {eng}}})}$ , $\\forall \\: i = 1,\\ldots , N_{{\\textrm {data}}}$ .", "In this case, at a given $Z$ , material density from multiple voxels contribute to ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ and the three integrals involved in the computation of this projection, as mentioned in Equation REF , cannot be avoided.", "Knowing the shape of the interaction-volume, it is possible to identify voxels that live partly or wholly inside the $ik$ -th interaction-volume as well as compute the fractional or full volume of each such voxel inside the $ik$ -th interaction volume.", "For this model, the projection equation is written in terms of the coordinates $(x,y,z)$ of a point instead of the polar coordinate representation of this point, where the point in question lies inside the $ik$ -th interaction-volume that is centred at $(x_i,y_i,0)$ .", "Then inside the $ik$ -th interaction-volume, at a given $x$ and $y$ , $z\\in \\left[0,\\sqrt{\\left(R0^{(k)}\\right)^2 - (x-x_i)^2 -(y-y_i)^2}\\right]$ .", "For $x-x_i\\in [(u-1)\\omega , u\\omega ] \\quad u=\\displaystyle {-(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)+1,-(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)+2,\\ldots ,(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)},\\nonumber $ the index $p_u(k)$ of the $Y$ -bin of voxels lying fully inside the $ik$ -th interaction volume, with respect to the centre of this interaction-volume, are $p_u(k)=-q_u(k),-q_u(k)+1,\\ldots ,0,1,2,\\ldots ,q_u(k)-1,q_u(k), \\nonumber $ where $q_u(k):=\\displaystyle {(int)\\left(\\frac{\\sqrt{(R0^{(k)})^2 - u^2\\omega ^2}}{\\omega }\\right)}.\\nonumber $ Then using the definition of the beam-pointing index in terms of the $X$ -bin and $Y$ -bin indices of voxels (see Equation REF ), we get the beam-pointing index $\\varrho _u(i,k)$ of voxels lying wholly inside the $ik$ -th interaction-volume, for a given $u$ is $\\varrho _u(i,k) = i-q_u(k)\\sqrt{N_{{\\textrm {data}}}}+u,\\:i-\\left(q_u(k)-1\\right)\\sqrt{N_{{\\textrm {data}}}}+u,\\:\\ldots , \\:i-\\left(q_u(k)-2q_u(k)\\right)\\sqrt{N_{{\\textrm {data}}}}+u, \\nonumber $ i.e.", "for a given $u$ , $\\varrho _u(i,k)=i+p_u(k)\\sqrt{N_{{\\textrm {data}}}}+u$ .", "The depth coordinate of voxels with beam-pointing index $\\varrho _u(i,k)$ lying inside the $ik$ -th interaction-volume are $z\\in \\displaystyle {\\left[0, {\\sqrt{(R0^{(k)})^2 - (p_u(k))^2\\omega ^2 - u^2\\omega ^2}}\\right]}$ so that the energy index of voxels lying fully inside at $Y$ -bin $p_u(k)$ and $x-x_i\\in [(u-1)\\omega ,u\\omega )$ are $\\in [1,t_{max}(u)]$ where $t_{max}(u)\\in {\\mathbb {Z}}_{> 0}$ such that $t_{max}(u)=\\max \\lbrace 1,2,\\ldots ,N_{{\\textrm {eng}}}\\rbrace $ that satisfies $\\displaystyle {h^{(t_{max}(u))}}\\le {\\sqrt{(R0^{(k)})^2 - (p_u(k))^2\\omega ^2 - u^2\\omega ^2}}.\\nonumber $ At this $Y$ -bin index $p_u(k)$ , there will also exist a voxel lying partly inside the $ik$ -th interaction-volume, at the $(t_{max}(u)+1)$ -th $Z$ -bin, between depths $h^{t_{max}(u)}$ and ${\\sqrt{(R0^{(k)})^2 - (p_u(k))^2 \\omega ^2 - u^2\\omega ^2}}$ .", "In addition, the projection ${\\cal C}(\\rho \\star \\eta )_i^{(k)}$ will include contributions from voxels at the edge of this interaction-volume, lying partly inside it; the beam-pointing indices of such voxels will be $i-\\left(q_u(k)+1\\right)\\sqrt{N_{{\\textrm {data}}}}+u$ and $i+\\left(q_u(k)+1\\right)\\sqrt{N_{{\\textrm {data}}}}+u$ for $x-x_i\\in [(u-1)\\omega ,u\\omega ]$ with $u$ and $q(u)$ defined as above.", "Lastly, parts of voxels at beam-pointing indices $i-\\displaystyle {(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}-1$ and $i+\\displaystyle {(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}+1$ will also be contained inside the $ik$ -th interaction-volume.", "These voxels at the edges extend into the 1st $Z$ -bin.", "We can compute the fraction $r_{a}^{(b)}(i,k)$ of the volume of the $ab$ -th voxel contained partly within the $ik$ -th interaction-volume by tracking the geometry of the system.", "Then using the discretised version of Equation REF , we write, $\\displaystyle {\\omega ^{-2}(R0^{(k)})^2 {\\cal C}(\\rho \\ast \\eta )_i^{(k)}} =$ $&& \\displaystyle {\\sum _{u=-(int)(R0^{(k)}/\\omega )}^{(int)(R0^{(k)}/\\omega )}\\:\\:\\sum _{p_u(k)=-q_u(k)}^{q_u(k)}\\sum _{t=1}^{t_{max}(u)}\\left[\\left(h^{(t)}-h^{(t-1)}\\right)\\sum _{m=1}^t {\\xi }_{\\varrho _u(i,k)}^{(m)} {\\eta }^{(t-m)}\\right]} + \\nonumber \\\\&&\\displaystyle {\\sum _{u=-(int)(R0^{(k)}/\\omega )}^{(int)(R0^{(k)}/\\omega )}\\:\\:\\sum _{p_u(k)=-q_u(k)}^{q_u(k)}\\left[\\left(\\sqrt{(R0^{(k)})^2-((q_u(k))^2+u^2)\\omega ^2} - h^{(t_{max}(u))}\\right)\\sum _{m=1}^{t_{max}(u)+1} \\chi _{\\varrho _u}^{(m)}(i,k){\\eta }^{(t_{max}(u)+1-m)}\\right]} + \\nonumber \\\\&& \\displaystyle {\\sum _{\\ell (i,k)}\\left[\\left(h^{(1)}\\right){r}_{\\ell }^{(1)}(i,k) {\\xi }_{\\ell (i,k)}^{(1)} {\\eta }^{(0)}\\right]}$ where $\\ell (i,k)=i-\\displaystyle {(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}-1, i+\\displaystyle {(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}+1, i-\\left(q_u(k)+1\\right)\\sqrt{N_{{\\textrm {data}}}}+u, i+\\left(q_u(k)+1\\right)\\sqrt{N_{{\\textrm {data}}}}+u$ , for $u=\\displaystyle {-(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}+1,\\displaystyle {-(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}+2,\\ldots ,\\displaystyle {(int)\\left(\\frac{R0^{(k)}}{\\omega }\\right)}$ , $\\chi _{\\varrho _u}^{(m)}(i,k):=r_{\\varrho _u}^{(m)}(i,k){\\xi }_{\\varrho _u(i,k)}^{(m)}$ , and $\\eta ^{(1)}$ is the measured value of the kernel on the system surface (see Section )." ], [ "Inference", "In this work, we learn the unknown material density and kernel parameters using the mismatch between the data $\\lbrace {\\tilde{I}}_i^{(k)}\\rbrace ^{k=N_{{\\textrm {eng}}};\\:i=N_{{\\textrm {data}}}}_{k=1;\\:i=1}$ and $\\lbrace {\\cal C}(\\rho \\ast \\eta )_i^{(k)}\\rbrace ^{k=N_{{\\textrm {eng}}};\\:i=N_{{\\textrm {data}}}}_{k=1;\\:i=1}$ , in terms of which, the likelihood is defined.", "The material density and kernel are convolved, and this convolution is sequentially projected onto the centre of the the $ik$ -th interaction volume, in the model (out of the 3 models, depending on the resolution of the image data at hand).", "Thus, if for the given material, the available data is such that $\\omega ^2 \\ge \\pi [R0^{(N_{{\\textrm {eng}}})}]^2$ , then we use Equation REF to implement ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ .", "If $\\omega ^2 \\ge \\pi [R0^{(k_{in})}]^2$ but $\\omega ^2 < \\pi [R0^{(k_{in}+1)}]^2$ , Equation REF is relevant while for $\\omega ^2 \\le \\pi [R0^{(k)}]^2$ the nearest neighbour averaging is invoked; see Section REF .", "We choose to work with a Gaussian likelihood: $\\\\&&{\\cal L}\\left(\\xi _1^{(1)},\\ldots ,\\xi _1^{(N_{{\\textrm {eng}}})},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})},\\eta ^{(1)},\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\vert {\\tilde{I}}_1^{(1)}, {\\tilde{I}}_1^{(2)},\\ldots ,{\\tilde{I}}_1^{(N_{{\\textrm {eng}}})},{\\tilde{I}}_2^{(1)},\\ldots ,{\\tilde{I}}_2^{(N_{{\\textrm {eng}}})},\\ldots ,{\\tilde{I}}_{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\right) = \\nonumber \\\\&&\\displaystyle {\\prod _{k=1}^{N_{{\\textrm {eng}}}}\\prod _{i=1}^{N_{{\\textrm {data}}}}\\frac{1}{\\sqrt{2\\pi }\\sigma _i^{(k)}}\\exp \\left[-\\frac{\\left({\\cal C}(\\rho \\ast \\eta )_i^{(k)} - {\\tilde{I}}_i^{(k)}\\right)^2}{2\\left(\\sigma _i^{(k)}\\right)^2}\\right]},$ where the noise in the image datum ${\\tilde{I}}_i^{(k)}$ is $\\sigma _i^{(k)}$ ; it is discussed in Section REF .", "Towards the learning of the unknown functions, the joint posterior probability density of the unknown parameters, given the image data, is defined using Bayes rule as $\\\\&&\\displaystyle {\\pi \\left(\\xi _1^{(1)},\\ldots ,\\xi _1^{(N_{{\\textrm {eng}}})},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})},\\eta ^{(1)},\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\vert {\\tilde{I}}_1^{(1)},\\ldots ,{\\tilde{I}}_{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\right)} \\propto \\nonumber \\\\&&\\displaystyle {{\\cal L}\\left(\\xi _1^{(1)},\\ldots ,\\xi _1^{(N_{{\\textrm {eng}}})},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})},\\eta ^{(1)},\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\vert {\\tilde{I}}_1^{(1)},\\ldots ,{\\tilde{I}}_1^{(N_{{\\textrm {eng}}})},\\ldots ,{\\tilde{I}}_2^{(N_{{\\textrm {eng}}})},\\ldots ,{\\tilde{I}}_{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\right)\\times } \\nonumber \\\\&& \\displaystyle {\\pi _0\\left(\\xi _1^{(1)},\\xi _1^{(2)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\right)\\nu _0\\left(\\eta ^{(1)},\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\right)} \\nonumber $ where $\\pi _0(\\xi _1^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})})$ is the adaptive prior probability on the sparsity of the density function, as discussed in Section .", "Also, $\\nu _0(\\eta ^{(1)},\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})})$ is the prior on the kernel, discussed in Section .", "Once the posterior probability density of the material density function and kernel, given the image data is defined, we use the adaptive Metropolis within Gibbs to generate posterior samples.", "At the $n$ -th iteration, $n=1,\\ldots ,N_{max}$ , $\\xi _i^{(k)}$ is proposed from a folded normal density.", "This choice of the proposal density is motivated by a non-zero probability for $\\xi _i^{(k)}$ to be zero.", "The latter constraint rules out a gamma or beta density that $\\xi _i^{(k)}$ is proposed from but truncated and folded normal densities are acceptable $k=1\\ldots ,N_{{\\textrm {eng}}},\\:i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "Of these we choose the easily computable folded normal proposal density .", "The proposed density in the $n$ -th iteration, in the $ik$ -th voxel is ${\\tilde{\\xi }}_i^{(k)}\\vert _n \\sim {\\cal N}_F(\\mu _i^{(k)}\\vert _n, \\varsigma _i^{(k)}\\vert _n)$ while the current density in this voxel at the $n$ -th iteration is defined as ${\\xi }_i^{(k)}\\vert _{n}$ .", "The distribution ${\\cal N}_F(a, b)$ is the folded normal distribution with mean $a\\in {\\mathbb {R}},\\: a > 0$ and standard deviation $b\\in {\\mathbb {R}},\\:b>0$ .", "We choose the mean and variance of this proposal density to be $\\\\\\mu _i^{(k)}\\vert _n &=& \\xi _i^{(k)}\\vert _{n-1},\\quad \\forall \\:n=1,\\ldots ,N_{max} \\nonumber \\\\\\left(\\varsigma _i^{(k)}\\vert _n\\right)^2 &=&{\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{\\sum _{p=n_0}^{n-1}\\left(\\xi _i^{(k)}\\vert _p\\right)^2}{n-n_0} -\\left[\\frac{\\sum _{p=n_0}^{n-1}\\left(\\xi _i^{(k)}\\vert _p\\right)}{n-n_0}\\right]^2}\\quad \\textrm {if}\\quad n\\ge n_0 \\nonumber \\\\T\\xi _i^{(k)}\\vert _0 \\quad {\\textrm {if}}\\quad n<n_0 \\nonumber \\end{array}\\right.", "}$ The random variable $T$ is considered to be uniformly distributed, i.e.", "$T\\sim U(0,1]$ .", "Thus, for $n \\ge n_0$ , the proposal density is adaptive, haario.", "We choose $n_0=10^4$ and $N_{max}$ is of the order of 8$\\times 10^{5}$ .", "We choose $\\xi _i^{(k)}\\vert _0$ by assigning constant density to the voxels that constitute the $ik$ -th interaction-volume, $k=1\\ldots ,N_{{\\textrm {eng}}},\\:i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "When a distribution-free model for the kernel is used, in the $n$ -th iteration, $\\eta ^{(k)}$ is proposed from exponential proposal density with a constant rate parameter $s_1$ .", "When the parametric model for the kernel is used, $\\eta (z)$ is calculated as given in Equation REF , conditional on the values of 2 the parameters $Q$ and $\\eta _0$ .", "The proposed parameters at the $n$ -th iteration are ${\\tilde{Q}}_n$ and $({\\tilde{\\eta }_0})_n$ .", "${\\tilde{Q}}_n$ and $({\\tilde{\\eta }_0})_n$ are each proposed from independent exponential proposal densities with constant rate parameters.", "Inference is performed by sampling from the high dimensional posterior (Equation REF ) using Metropolis-within-Gibbs block update, gilks96,chibgreenberg95.", "Let the state at the $n$ iteration be $\\varepsilon _n=(\\xi _1^{(1)}\\vert _n,\\ldots ,\\xi _1^{(N_{{\\textrm {eng}}})}\\vert _n,\\ldots ,\\xi _2^{(N_{{\\textrm {eng}}})}\\vert _n, \\ldots , \\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\vert _n,\\eta ^{(1)}\\vert _n,\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\vert _n)^T.$ For the implementation of the block Metropolis-Hastings, we partition the state vector $\\varepsilon _n$ as: $\\varepsilon _n^T = ((\\varepsilon _n^{(\\xi )})^T, (\\varepsilon _n^{(\\eta )})^T),\\nonumber $ where $\\varepsilon _n^{(\\xi )} &=& (\\xi _1^{(1)}\\vert _n,\\ldots ,\\xi _1^{(N_{{\\textrm {eng}}})}\\vert _n,\\ldots ,\\xi _2^{(N_{{\\textrm {eng}}})}\\vert _n, \\ldots , \\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\vert _n)^T,\\nonumber \\\\\\varepsilon _n^{(\\eta )} &=& (\\eta ^{(1)}\\vert _n,\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\vert _n)^T.$ Here $n=1,\\ldots ,N_{burn_{in}},\\ldots ,N_{max}$ .", "We typically use $N_{max} >$ 8$\\times 10^5$ and $N_{burn_{in}}$ =1$\\times 10^5$ .", "Then, the state $\\varepsilon _{n+1}$ is given by the successive updating of the two blocks: ${\\varepsilon }_{n+1}^{(\\xi )}$ and ${\\varepsilon }_{n+1}^{(\\eta )}$ ." ], [ "Posterior probability measure in small noise limit", "Here we check for uniqueness in the learnt $\\rho \\ast \\eta $ .", "With the aim of investigating the posterior probability density in the small noise limit, we recall the chosen priors for the material density (Section ), kernel parameters (Section ), and the likelihood function (Equation REF ).", "Theorem 8.1 In the limit of small noise, $\\sigma _i^{(k)}\\longrightarrow 0$ , the joint posterior probability of the density and kernel, given the image data, for all beam-pointing vectors ($i=1,\\ldots ,N_{{\\textrm {data}}}$ ) and all $\\epsilon _k,\\:k=1,\\ldots ,N_{{\\textrm {eng}}}$ , reduces to a product of $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}$ Dirac measures, with the $ik$ -th measure centred at the solution to the equation ${\\tilde{I}}_i^{(k)} = {\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ , Logarithm of the posterior probability of the discretised distribution-free model is $&&\\log \\pi \\left(\\xi _1^{(1)},\\ldots ,\\xi _1^{(N_{{\\textrm {eng}}})},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})},\\eta ^{(1)},\\ldots ,\\eta ^{(N_{{\\textrm {eng}}})}\\vert {\\tilde{I}}_1^{(1)},\\ldots ,{\\tilde{I}}_{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})}\\right)= \\nonumber \\\\&&\\displaystyle {\\sum _{i=1}^{N_{{\\textrm {data}}}}\\sum _{k=1}^{N_{{\\textrm {eng}}}}\\left[-\\log {\\sigma _i^{(k)}}- \\left(\\frac{({\\tilde{I}}_i^{(k)} -{\\cal C}(\\rho \\ast \\eta )_i^{(k)})^2}{2(\\sigma _i^{(k)})^2}\\right)\\right]}- \\displaystyle {\\sum _{k=1}^{N_{{\\textrm {eng}}}}\\left[\\frac{(\\eta ^{(k)}+\\eta _0^{(k)})^2}{2N(s^{(k)})^2}\\right]}- \\sum _{i=1}^{N_{{\\textrm {data}}}}\\sum _{k=1}^{N_{{\\textrm {eng}}}}\\left[{\\left(\\xi _i^{(k)}\\nu (\\tau _i^{(k)})\\right)^2}\\right] + A, \\nonumber \\\\&&$ where $A\\in {\\mathbb {R}}$ is a finite constant.", "Thus, $\\displaystyle {\\lim _{\\sigma _i^{(k)}\\longrightarrow 0} \\pi (\\xi _1^{(1)},\\ldots ,\\xi _{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})},\\eta ^{(1)},\\ldots ,\\eta ^{(m)}\\vert {\\tilde{I}}_1^{(1)},\\ldots ,{\\tilde{I}}_{N_{{\\textrm {data}}}}^{(N_{{\\textrm {eng}}})})} \\propto && \\nonumber \\\\\\displaystyle {\\lim _{\\sigma _i^{(k)}\\rightarrow 0}\\left[\\prod \\limits _{i=1}^{N_{{\\textrm {data}}}}\\prod \\limits _{k=1}^{N_{{\\textrm {eng}}}}\\frac{1}{\\sigma _i^{(k)}}\\exp \\left(-\\frac{({\\tilde{I}}_i^{(k)} -{\\cal C}(\\rho \\ast \\eta )_i^{(k)})^2}{2(\\sigma _i^{(k)})^2}\\right)\\right]}.", "&&$ The right hand side of this equation is the product of Dirac delta functions centred at ${\\tilde{I}}_i^{(k)} = {\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ , for $i=1,\\ldots ,N_{{\\textrm {data}}},\\:k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "Thus, the posterior probability density reduces to a product of Dirac measures for each $i,k$ , with each measure centred on the solution of the equation ${\\tilde{I}}_i^{(k)} = {\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ .", "On the basis of Theorem REF , we arrive at the following important results.", "$(\\rho \\ast \\eta )_i^{(k)}$ is the least-squares solution to ${\\tilde{I}}_i^{(k)} = {\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ .", "The three contractive projections that act successively to project $\\rho \\ast \\eta $ onto the space of functions defined over the point of incidence of any interaction-volume, are commutable.", "Thus, for example, the result is invariant to whether the projection onto the $Z$ =0 plane happens first and then the projection onto the $Y$ -axis happens or whether first the projection onto the $Y$ =0 plane is performed, followed by the projection onto the $Z$ =0 axis.", "Each of these projections is also orthogonal and is therefore represented by a projection matrix ${\\bf P}_i$ , $i=1,2,3$ , that is idempotent (${\\bf P}_i={\\bf P}_i^2$ ) and symmetric.", "Then, the composition of these projections, i.e.", "the ${\\cal C}$ operator, which in the matrix representation is ${\\bf C}$ , is a product of three idempotent and symmetric matrices that commute with each other.", "This implies that ${\\bf C}$ is idempotent and symmetric, implying that ${\\cal C}$ is an orthogonal projection.", "In the small noise limit, $(\\rho \\ast \\eta )_i^{(k)}$ is the unique solution to the least-squares problem ${\\tilde{I}}_i^{(k)}={\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ .", "The justification behind this results is as follows.", "In the matrix representation, ${\\bf C}$ is a square matrix of dimensionality ${N_{{\\textrm {eng}}}\\times N_{{\\textrm {eng}}}}$ , where the image data is a $N_{{\\textrm {eng}}}\\times N_{{\\textrm {data}}}$ dimensional matrix and $\\rho \\ast \\eta $ is also represented by a $N_{{\\textrm {eng}}}\\times N_{{\\textrm {data}}}$ dimensional matrix.", "${\\cal C}$ being a composition of three commutable orthogonal projections, the matrix ${\\bf C}^{(N_{{\\textrm {eng}}}\\times N_{{\\textrm {eng}}})}$ is idempotent and symmetric.", "Then the Moore-Penrose pseudo-inverse of ${\\bf C}$ exists as ${\\bf C}^{+}$ , and is unique for given ${\\bf C}$ .", "Then $(\\rho \\ast \\eta )_i^{(k)}$ is the unique solution to the least squares problem ${\\tilde{I}}_i^{(k)} = {\\bf C}(\\rho \\ast \\eta )_i^{(k)}$ achieved in the low noise limit, such that $(\\rho \\ast \\eta )_i^{(k)} ={\\bf C}^{+} {\\tilde{I}}_i^{(k)}$ .", "The establishment of the uniqueness of the learnt $\\rho \\ast \\eta $ in the small noise limit is due to our designing of the imaging experiment to ensure that the dimensionality of the image space coincides with that of the space of the unknown parameters.", "In the presence of noise in the data, the learnt $\\rho \\ast \\eta $ is no longer unique.", "The variation in this learnt function is then given by the condition number $\\kappa ({\\bf C})$ of ${\\bf C}$ .", "Now, ${\\kappa }({\\bf C})=\\parallel {\\bf C}\\parallel \\parallel {\\bf C}^{+}\\parallel $ , where $\\parallel \\cdot \\parallel $ here refers to the 2-norm.", "For orthogonal projection matrices, norm is 1, i.e.", "$\\kappa ({\\bf C})$ =1.", "Therefore, fractional deviation of uniqueness in learnt value of $\\rho \\ast \\eta $ is the same as the noise in the image data, which is at most 5$\\%$ (see Section REF )." ], [ "Quantification of deviation from uniqueness of learnt functions", "While $\\rho \\ast \\eta $ is learnt uniquely in the small noise limit, the learning of $\\rho (x,y,z)$ and $\\eta (z)$ from this unique convolution is classically, an ill-posed problem since then we know $N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}$ parameters but $(N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}) + N_{{\\textrm {eng}}}$ parameters are unknown in the implementation of the distribution-free model of $\\eta (z)$ ; again $(N_{{\\textrm {data}}}\\times N_{{\\textrm {eng}}}) + 2$ are unknown in case the parametric model of $\\eta (z)$ is used.", "Then the ratios of the known to unknown parameters in these two cases are as high as $r_{nonpram}\\approx 0.990$ and $r_{param} \\approx 0.998$ respectively, for typical values of $N_{{\\textrm {data}}}$ =100, $N_{{\\textrm {eng}}}$ =10.", "In a Bayesian framework, the problem is addressed via the priors on the unknown parameters presented above.", "The prior on the material density parameters is adaptive to the sparsity of the density in its native space.", "Thus, this prior should expectedly perform equally well when the density function is intrinsically dense as well as for density functions marked by high degrees of sparsity.", "This is evident in the quality of comparison of the true densities and the densities learnt from simulated image data sets obtained by sampling from two density functions that are equivalent in all respect except for their inherent sparsity, as shown in Figure REF .", "At the same time, the parametric model for the kernel is more informed than the distribution-free model.", "We find corroboration of improved inference on the unknowns when the parametric, rather than distribution-free model for the kernel parameters is implemented, in learning with simulated data, (see Figures REF and REF )." ], [ "Inversion of simulated microscopy image data", "We describe the application of the inverse methodology described above to simulated images, as a test of the method, i.e.", "compare the true material density and true kernel parameters (or correction function) with the estimated density and kernel respectively.", "Here, the “true” density and kernel are the chosen density and kernel functions using which the simulated image data are constructed.", "The discussed examples include inversion of simulated image data of an example of a high-${\\cal Z}$ material, namely Copper-Tungsten alloys, the true density function of which is dense (Sample I-CuW), sparse (Sample II-CuW), such that the density structure is characterised by isolated modes with sharp edges.", "of a low-${\\cal Z}$ material, namely a Ni-Al alloy which is dense (Sample I-NiAl), sparse (Sample II-NiAl).", "The (simulated) images are produced, in some cases at 18 beam energy values and in other cases at 10 different values of $E$ , corresponding to $\\epsilon _k=n+k$ , in real physical units of energy, namely kiloVolts (or kV), $k=1,\\ldots ,N_{{\\textrm {eng}}}$ , $n\\in {\\mathbb {R}}_{\\ge 0}$ .", "We work with $N_{{\\textrm {data}}}$ =225, i.e.", "there are 15 pixels along the $X$ and 15 along the $Y$ -axis corresponding to 15 beam pointings along each of these axes.", "Then at any $k$ , the image data in 225 pixels, are arranged in a 15$\\times $ 15 square array.", "The image data in each pixel is computed from a chosen density and chosen correction function by allowing the sequential projection operator ${\\cal C}$ to operate upon $\\rho \\ast \\eta $ where the material density parameters are chosen as $\\xi _i^{(k)} = \\displaystyle {\\Upsilon \\frac{A_i}{\\displaystyle {\\left[\\epsilon ^2 +\\frac{x_i^2}{B_i^2} +\\frac{y_i^2}{B_i^2} +\\frac{(h^{(k)})^2}{B_i^2 (1 - Q_i^2)}\\right]}}}&&\\quad {\\textrm {where}}$ $\\Upsilon = ({\\textrm {int}})(N_{{\\textrm {eng}}} U_1),\\quad U_1\\sim {\\cal U}[0,1]\\quad {\\textrm {for the sparse Sample~II-CuW and Sample~II-NiAl}}$ , $\\Upsilon = 1 \\quad {\\textrm {for the dense Sample~I-CuW and Sample I-NiAl}}$ , $A_i = U_2,\\quad U_2\\sim {\\cal U}[0,1]$ , $B_i = U_3, \\quad U_3\\sim {\\cal U}[0,n\\omega ]\\quad n\\in {\\mathbb {Z}}^{+}$ , $Q_i = U_4,\\quad U_4\\sim {\\cal U}[0,1], \\forall \\:i=1,\\ldots ,N_{{\\textrm {data}}}$ .", "Here ${\\cal U}[0,\\cdot ]$ is the uniform distribution over the range $[0,\\cdot ]$ .", "The true correction function is chosen to emulate a folded normal distribution with a mean of $\\gamma $ and dispersion $d_s$ .", "$\\eta ^{(k)} = \\displaystyle {\\exp \\left[\\frac{(h^{(k)} - \\gamma )^2}{2d_s^2}\\right] +\\exp \\left[\\frac{(h^{(k)} + \\gamma )^2}{2d_s^2}\\right] }$ We recall from Section , that for materials at a high value of the atomic number ${\\cal Z}$ , when imaged at low resolutions, the unknown functions are learnt as per the prescription of the 1st model.", "See Equation REF for definition of “high” and “low” ${\\cal Z}$ materials.", "In this connection we mention that for the low resolution imaging techniques (such as imaging in X-rays), the smallest resolved length in the observed images is $\\omega $ =1.33$\\mu $ m while for the illustration of the high resolution technique (such as imaging with Back Scattered Electrons), a much finer resolution of $\\omega $ =9 nm, is considered.", "Materials for which ${\\cal Z}$ is low, when imaged at low resolutions, are modelled using the 2nd model.", "Any material that is imaged at high resolution, i.e.", "for small values of $\\omega $ , will be modelled using the 3rd-model.", "The true density and correction function are then used in the model appropriate equation (Equation REF for the 3rd model and Equation REF for the other 2 models) to generate the sequential projections of the convolution $\\rho \\ast \\eta $ to give us the simulated image data $\\lbrace \\tilde{I_i^{(k)}}\\rbrace _{i=1;\\:k=1}^{N_{{\\textrm {data}}},\\:N_{{\\textrm {eng}}}}$ .", "The simulated images are produced with noise of 5$\\%$ to 3$\\%$ , i.e.", "$\\sigma _i^{(k)}$ =0.05 to 0.03 times ${\\tilde{I}}_i^{(k)}$ .", "We start MCMC chains with a seed density $\\xi _{i,seed}^{(k)}$ in the $ik^{th}$ voxel defined as $\\xi _{i,seed}^{(k)}={\\tilde{I}}_i^{(k)}/\\Phi _{seed}^{(k)}$ , where the starting correction function $\\lbrace \\eta _{seed}^{(k)}\\rbrace _{k=1}^{N_{{\\textrm {eng}}}}$ helps define $\\Phi _{seed}^{(k)} =\\sum _{j=1}^{k}\\eta _{seed}^{(j)}(h^{(j)}-h^{(j-1)})$ .", "The initial seed for the correction function is chosen to be motivated by the forms suggested in the literature; in fact, we choose the initial correction function to be described by a half-normal distribution, with a standard deviation of 5$\\mu $ m and mean of 5$\\mu $ m. We use adaptive Metropolis within Gibbs for our inference, as discussed above in Section ." ], [ "Low resolution, high-${\\cal Z}$ system", "These illustrations are performed with intrinsic densities that are sparse (Sample II-CuW) and dense (Sample I-CuW).", "The idea behind the modelling in this case is that the cross-sectional area of the surface of the interaction-volume at any $E$ , is less than the cross-sectional area of the voxel on this $Z$ =0 surface, rendering the inversion the simplest out of the three illustrations we discuss.", "An example resolution of $\\omega $ =1.33 $\\mu $ m of such low-resolution imaging techniques, allows for 15 beam pointings over an interval of 20 $\\mu $ m, from $x$ =-10$\\mu $ m to $x$ =10$\\mu $ m at given $y$ , at spatial intervals where $\\omega $ .", "Again, at a given $x$ , there are 15 beam pointings, at steps of $\\omega $ , from $y$ =-10$\\mu $ m to $y$ =10$\\mu $ m. $N_{{\\textrm {eng}}}$ was chosen as 18.", "We invert the image data thus generated, both when we consider a distribution-free model for $\\eta (z)$ as well as the parametric model.", "Figure: Estimated material density parameters over-plotted in theX-ZX-Z plane, ∀y\\forall y, in solid contour lines, over the truedensity (in filled coloured contours), given simulated images ofCopper-Tungsten alloy - Sample I-CuW (left) and Sample II-CuW(right).", "The simulated image data corresponds to a low-resolutionimaging technique (such as when the system is imaged by an SEM inX-rays), at different beam energy values.", "The material densityestimate at the medial level of the posterior probability isplotted.", "The parametric model for the correction function wasused for these runs.Figure REF represents the estimated density functions for the two simulated samples, represented as contour plots in the $X-Z$ plane, $\\forall y$ , superimposed over the respective true densities which are shown as filled contour plots, when the parametric model for $\\eta (z)$ is used.", "We set $\\sigma _i^{(k)}=0.05{\\tilde{I}}_i^{(k)}$ .", "The panel on the left describes a true density function that is dense in its native space while on the right, the true density function is sparse.", "Figure REF represents results of implementation of the distribution-free models for the kernel, using $\\sigma _i^{(k)}=0.03{\\tilde{I}}_i^{(k)}$ .", "This figure includes the plot of ${\\cal C}(\\rho \\ast \\eta )_i^{(k)}$ as a function of the pixel location along the $X$ axis, over all $k=1,\\ldots ,N_{{\\textrm {eng}}}$ .", "This data is over-plotted on the image data ${\\tilde{I}}_i^{(k)}$ , plotted against $\\lambda _i$ for all $k$ .", "Similarly, the estimated ${\\hat{\\eta }}(z)$ and true correction function are also compared.", "Trace plots for the chains are also included.", "It is to be noted that the definition of the likelihood in terms of the mismatch between data and the projection of the learnt density and kernel parameters, i.e.", "${\\cal C}({\\hat{\\rho }}\\ast {\\hat{\\eta }})_i^{(k)}$ , compels ${\\cal C}({\\hat{\\rho }}\\ast {\\hat{\\eta }})_{\\cdot }^{(\\cdot )}$ and ${\\tilde{I}}_{\\cdot }^{(\\cdot )}$ to coincide, even if ${\\hat{\\rho }}({\\bf x})$ is poorly estimated, though such a comparison serves as a zeroth order check on the involved coding.", "Figure: Left: - projection 𝒞(ρ ^η ^) i (k) {\\cal C}({\\hat{\\rho }}\\ast {\\hat{\\eta }})_i^{(k)} plotted in black, in modelunits, as a function of beam location on the XX-axis, (where thebeam pointing index i=1,...,225i=1,\\ldots , 225), over the image data (inred), for all beam energy values ϵ k =2+k\\epsilon _k=2+k (in kiloVolts),k=1,...,18k=1,\\ldots ,18, for Sample I-CuW (upper panel) and Sample II-CuWwith the relatively sparser material density (lower).", "Middle:estimated correction function, in black, superimposed on the trueη (·) \\eta ^{(\\cdot )} for the two samples (in red).", "The error bars in theabove plots, as well as in all plots that follow, correspond to95%\\% highest probability density credible regions, while themedial level of the posterior is marked by a symbol (filledcircle).", "The trace of the joint posterior probability density of theunknown parameters, given the image data is shown in the rightpanels.", "The distribution-free model for the correction function wasimplemented." ], [ "Low resolution, low-${\\cal Z}$ system", "Distinguished from the last section, in this section we deal with the case of a low atomic number material imaged at coarse or low imaging resolution.", "In this case, the surface cross-sectional area of an interaction-volume exceeds that of a voxel only for $E=\\epsilon _{k} \\le \\epsilon _{k_{in}}$ , for $k=1,\\ldots ,k_{in},k_{in+1},\\ldots ,N_{{\\textrm {eng}}}$ .", "The illustration of this case is discussed here for Sample II-NiAl, that is an alloy of Nickel, Aluminium, Tallium, Rhenium; see dsouza.", "The average atomic weight of our simulated sample is such that $k_{in}$ =13 but the extent of interaction-volumes attained at $E=\\epsilon _{k}$ (=$k$ +2 kiloVolts) for $k=14,\\ldots ,18$ , is in excess of the image resolution 1.33 $\\mu $ m. All other details are as in the previous illustration.", "Figure REF depicts the learnt density structure; visualisation to this effect is provided for the Ni-Al alloy sample in terms of the plots of $\\xi _i^{(k)}$ against $h^{(k)}$ , at chosen values of $X$ and $Y$ inside the sample.", "Since there are 225 voxels used in these exercises, the depiction of the density at each of these voxels cannot be accommodated in the paper, but Figure REF shows the density structure at values of beam pointing index $i$ corresponding to 15 values of $Y$ , at one value of the $X$ (=6.67$\\mu $ m).", "Figure: For the simulated (sparse and low-𝒵{\\cal Z})Sample II-NiAl, the estimated density parameters ξ i (k) \\xi _i^{(k)}plotted in model units as a function of h (k) h^{(k)} (in μ\\mu m), atX=6.67X=6.67μ\\mu m anddistinct values of the beam pointing locations along the YY-axis, inthe range of -10μ\\mu m to 10μ\\mu m, at intervals of 1.33μ\\mu m wherethe panel in the lower left-hand corner is at the smallest value of YY and that in the upper right-hand most panel is at the highest value of YY.", "The true variation of the learnt material density withdepth is shown in red in each panel.", "This is compared to the material densityestimated at the lower and upper bounds of the 95%\\% HPD credibleregion, as plotted in cyan and black respectively (in modelunits).", "The parametric model for η (·) \\eta ^{(\\cdot )} was used." ], [ "High resolution", "The challenge posed by the high resolution ($\\omega \\lesssim $ 10 nm=0.01 $\\mu $ m) of imaging, to our modelling is logistical.", "This logistical problem lies in fact that for $\\omega \\lesssim $ 10 nm the run-time involved in the reconstruction of the density over the length scales of $\\sim $ 1 $\\mu $ m, is prohibitive; for an image resolution of $\\omega $ =10 nm, there are 100 voxels included over a length interval of 1 $\\mu $ m, while at example lower resoutions relevant to the aforementioned illustrations there would be ${\\mbox{int}}(100/15)$ .", "In light of this problem, for a high resolution of $\\omega $ =2 nm that we work with, we choose 20 voxels (each of cross-sectional size 2 nm), across the $X$ range of -20 nm to 20 nm.", "A similar range in $Y$ is considered.", "The simulated system is imaged at 10 energy values $\\epsilon _k=k+1.5$ kV, $k=1,\\ldots ,10$ .", "The atomic number parameter ${\\cal Z}$ of the used material is such that the radius $R0^{(1)}=h^{(1)}$ of the interaction-volume attained at $k=1$ is about 21 nm, so that all the studied voxels live inside this and all other (larger) interaction-volumes.", "It is possible to run multiple parallel chains with data from distinct parts of the observed image, in order to cover this available image.", "That way, the material density function of the whole sample will be learnt.", "The plots of the learnt density for the intrinsically dense and sparse true density functions are shown as a function of depth $Z$ , for all $X$ and $Y$ , in Fig REF , which display results of runs done with the distribution-free as well as parametric models for the correction function (using $\\sigma _i^{(k)}=0.05{\\tilde{I}}_i^{(k)}$ , $i=1,\\ldots ,N_{{\\textrm {data}}}$ , $k=1,\\ldots ,N_{{\\textrm {eng}}}$ ).", "The multiscale structure of the heterogeneity in these simulated density functions is brought out by presenting the logarithms of the estimated densities parameters alongside the logarithms of the chosen or true density parameters.", "Figure: Top row: results of inversion of simulated image dataof the intrinsically sparse low-𝒵{\\cal Z} Sample II-NiAl when adistribution-free model for the correction function is used.", "Thetrue material density structure is presented in the panels on theleft, as a function of zz, for all X,YX, Y, in model units.", "Thelearnt material density, at the medial level of the posterior, issimilarly presented in the middle panels while the learntη (k) \\eta ^{(k)} over 95%\\% HPD credible region is shown in black as afunction of h (k) h^{(k)} in the right panels, in model units,superimposed on the true kernel parameters in red.", "Middle row:the intrinsically dense, true and learnt (at the posterior median)material density of simulated sample Sample I-NiAl in the left andmiddle panels respectively with the learnt η (·) \\eta ^{(\\cdot )}.", "Aparametric model for the correction function is employed.", "Bottom row: results from implementing simulated images of thesparse Sample II-NiAl when a parametric model for the correctionfunction is employed." ], [ "Application to the analysis of real SEM image data of Nickel and Silver nano-composite", "In this section we discuss the application of the methodology advanced above towards the learning of the 3-D material density and the microscopy correction function (or kernel) by inverting 11 images taken with a real SEM, in a kind of radiation called Back Scattered Electron (BSE).", "The imaged sample is a brick of Nickel (Ni) and Silver (Ag) nanoparticles, taken with the SEM (the Leica, Stereoscan 430 brand), at 11 distinct values of the beam energy parameter $E$ such that the values that $E$ takes are $10,11,\\ldots ,20$ kV.", "The brick of nanoparticles was prepared by the drop-cast method in the laboratory.", "The resolution of BSE imaging with the used SEM is $\\approx $ 50 nm=0.05 $\\mu $ m, i.e.", "the smallest length over which sub-structure is depicted in the image is about 50 nm.", "While this resolution can be coarser than that for images taken in the radiation of Secondary Electrons , an image taken in Secondary Electrons will however not bear information coming from the bulk of the material.", "Hence the motivation behind using BSE image data.", "We sample two distinct areas from each of the 2-D images taken at values $10,11,\\ldots ,20$ kV of $E$ , resulting in two different image data sets.", "The first data $D_1$ comprises a square area of size 101pixels$\\times $ 101pixels, with this square seated in a different location on an image than the smaller square that contain the pixels that define the image data set $D_2$ .", "This second data $D_2$ comprises a much smaller square of area 41pixels$\\times $ 41pixels.", "Given the imaging resolution length of 0.05$\\mu $ m, i.e.", "given that the width of each square pixel is 0.05$\\mu $ m, in image data set $D_1$ the 101 pixels along either the $X$ or $Y$ -axes are set to accommodate the length interval [-2.5 $\\mu $ m, 2.5 $\\mu $ m] while in data $D_2$ , the pixels occupy the interval [-1 $\\mu $ m, 1 $\\mu $ m].", "Then in data set $D_1$ , at a given value of $E$ , image datum from a total of 100$^2$ =10,000 pixels are recorded, i.e.", "$N_{{\\textrm {data}}}$ for this data set is 10,000.", "For $D_2$ , $N_{{\\textrm {data}}}$ =41$^2$ =1,764.", "Both $D_1$ and $D_2$ comprise 11 sets of BSE image data $\\lbrace {\\tilde{I}}_i^{(k)}\\rbrace _{i=1}^{N_{{\\textrm {data}}}}$ , with each such set imaged with a beam of energy $\\epsilon _k$ , $k=1,\\ldots ,11$ such that $\\epsilon _k=k+9$ kiloVolts.", "For this material, the mean of the atomic number, atomic mass and physical density in gm cm$^{-3}$ yields average atomic number parameter ${\\cal Z}$ of 37.5, and the numerical values of the parameters $A$ and $d$ in Equation REF as 83.28 and about 9.7 respectively; then the smallest interacion-volume attained at $E=\\epsilon _1$ is $R0^{(1)}=h^{(1)}\\approx $ 0.44 $\\mu $ m and the largest at $E=\\epsilon _{10}$ is about 1.4 $\\mu $ m. The learning of the material density of a nanostructure is very useful for device engineers engaged in employing such structures in the realisation of electronic devices ; lack of consistency among measured electronic characteristics of the formed devices is tantamount to deviation from a standardised device behaviour and such can be predicted if heterogeneity in the depth distribution of nanoparticles is identified.", "Only upon the receipt of quantified information about the latter, is the device engineer able to motivate adequate steps to remedy the device realisation.", "In addition, such information holds potential to shed light on the physics of interactions between nanoparticles.", "If the calibration of the intensity $I_i^{(k)}$ of the image datum is available in physical units (of surface density of the BSE), then we could express the measured image data - as manifest in the recorded image - in relevant physical units.", "In that case, the learnt density could be immediately expressed in physical units.", "However, such a calibration is not available to us.", "In fact, in this method, the learnt density is scaled in the following way.", "The measured value of $\\eta ^{(1)}$ from microscopy theory for this material suggests using the normalisation factor for $\\eta ^{(\\cdot )}$ is $\\approx {\\hat{\\eta }}^{(1)}/0.325$ , so that the $\\eta ^{(\\cdot )}$ parameters, thus normalised, are in the physical units of $\\mu $ m. (The arithmetic mean of the atomistic parameters of Ni and Ag yield the value of 0.325).", "This in turn would imply that the material density parameters are each in the physical unit of $\\mu $ m$^{-3}$ since the product of the density and kernel parameters appears in the projection which is measured in physical units of “per unit area”, i.e.", "in “per length unit$^2$ ” or ($\\mu $ m)$^{-2}$ .", "Here ${\\hat{\\eta }}^{(1)}$ is the learnt value of the kernel in the first $Z$ -bin.", "Physically speaking, here we learn the number density in each voxel.", "Figure: The top panels display two of the 11 images of theprepared blend of Nickel and Silver nanoparticles, taken with anSEM, at beam energy valuesof 20 kV (right) and 10 kV (left).", "A 5 μ\\mu m×\\times 5 μ\\mu marea was identified in each of the 11 images taken in the kind ofradiation called Back Scattered Electrons (BSE), to form the dataD 1 D_1.", "The colourised squares in red in each image represent theareas that contribute to D 1 D_1 at energies 20 kV and 10 kV.", "Thedistribution of the measured image data over these square areas inred (rotated clockwise by 90 ∘ ^\\circ ), for beam energies of 10 kVand 20 kV are shown in the left and right panels repectively, on the lowerrow.The configuration that the constituent nanoparticles are expected to relax into, is due to several factors, including contribution from the surface effects - the surfaces of nanoparticles are active - this encourages interactions, resulting in a clustered configuration.", "Additionally, gravitational forces that the nanoparticle aggregates sediment under, are also active, with the nanoparticle diameter responsible for determining the relative importance of the different physical influences .", "Thus, in general, along the $Z$ -axis, we expect a clustered configuration, embedded within layers.", "This is what we see in the representation of the learnt density in the $Y=0$ plane, when the image data $D_1$ and $D_2$ are inverted; see Figure REF .", "In Figure REF we present the learnt kernel parameters; the kernels learnt by using data from two distinct parts of the image are expected to overlap given that in our model, the kernel function is a function of $Z$ alone with no dependence on $X$ and $Y$ .", "Such is recovered using data $D_1$ and $D_2$ , as shown in the left panel of this figure.", "The sequential projections of the convolution of the learnt density and learnt correction function, onto the image space, using the data set $D_1$ is overplotted on the image data in Figure REF .", "The parametric model for the correction function is implemented here.", "Other relevant effects include viscosity and Brownian motion.", "Figure: The left and right panels show the slice of the learntthree dimensional material density in the Y=0Y=0 plane frominversion of image data D 1 D_1 and D 2 D_2 respectively.", "In fact, toconstruct these figures we use the density parameter value that isthe median of the posterior of the parameter given the image data.", "Thedata are obtained by imaging a brick of Nickel and Silvernanoparticles in a radiation called Back Scattered Electrons, atresolution of 0.05μ\\mu m, at 11 different values of EE, from 10 kV,...\\ldots , 20 kV.", "D 1 D_1 and D 2 D_2 comprise 101×\\times 101 pixelsand 41×\\times 41 pixels respectively.", "The labels on the colour-barare number density values, in physical units (of μ\\mu m -3 ^{-3})." ], [ "MCMC convergence diagnostics", "In this section we include various diagnostics of an MCMC chain that was run until convergence, using the image data $D_1$ .", "These include trace of the likelihood (Figure REF ), and histograms of multiple learnt parameters - $\\xi _{50}^{(1)}$ , $\\eta ^{(1)}$ - from 1000 steps, in two distinct parts of the chain, namely, for step number $N\\in $ [1599001,1600000] and $N\\in $ [799001,800000], respectively (Figure REF ).", "The histograms of the likelihood over these two separate parts of the chain are also presented in this figure.", "Figure: The top panel shows the spatially-averaged projectionof convolution of learnt material density and correction function,onto the image space, learnt using image data D 1 D_1 in filledcontours.", "The real image data D 1 D_1 is overlaid in black solidcontours.", "The lower panel shows the correction functions learntfrom data D 1 D_1 and D 2 D_2 in black and red respectively.", "The errorbars in this plot represent the learnt 95%\\% highest probabilitydensity region.Figure: The trace of the joint posterior probability densityof all uknown model parameters given data D 1 D_1, from a chain thatis run with these data.Figure: The histograms of the learnt material density in the voxelmarked by the viewing vector v i v_i, i=50i=50 and the lowest valueof EE, (E=ϵ 1 E=\\epsilon _1), from two disjoint parts (middle and end) of theMCMC chain that is 1.6×\\times 10 6 ^6 steps long, are depicted in redand black respectively, in the right panel.", "The histograms of thelearnt correction function for k=1k=1 , obtained for these two partsof the chain are displayed in the left panel.", "The histograms of thevalue of the likelihood in these two parts of the chain are plottedin red and black in the middle panel.", "The chain was run with realimage data D 1 D_1 ." ], [ "Discussion", "In this work we have advanced a Bayesian methodology that performs ditribution-free reconstruction of the material density of a material sample and either a parametric or distribution-free reconstruction of the microscopy correction function, given 2-D images of the system taken in any kind of radiation that is generated inside the bulk of the system as a consequence of the material being impinged upon by an electron beam.", "This methodology is advanced as capable of learning the unknowns even when the the inverse Radon transform is not stable; in addition to noise in the data, such instability can also arise when the underlying distribution of the image data is not continuous.", "This in turn arises from a material density function that is heterogeneous and is marked by a dense or sparse modal structure, characterised by abruptly declining modal strengths.", "Given such sharp density contrasts that typify real-life material samples (see Figure REF and Figure REF ), mixture models cannot work satisfactorially in this situation.", "Also, the density field may not be necessarily convex in real-life material samples.", "Crucially, this inverse problem connects the measured 2-D radiation in any pixel, with multiple sequential projections of the convolution of the unknown density and the unknown kernel.", "The novelty of our solution to this harder-than-usual inverse problem includes the imaging of the system at distinct beam energies, adoption of a fully discretised model, identifying priors on the sparsity of the density and borrowing existing knowledge from the domain of application.", "In this application in which images are taken with electron microscopes, the kernel is the microscopy correction function, the shapes of which are motivated by the microscopy literature.", "In particular, there is deterministic information available in the literature about the correction at the surface, allowing for identifiability of the global scales of the unknown material density and correction function.", "From the point of view of 3-D structure modelling using images taken with bulk microscopic imaging techniques, (such as Scanning Electron Microscopy, Electron Probe Microscopy), our aim here supercedes mere identification of the geometrical distribution of the material i.e.", "the microstructure.", "In fact, we aim to estimate the very material density at chosen points inside the bulk of the material sample.", "Conventionally, Monte Carlo simulation studies of microstructure are undertaken; convolution of such simulated microstructure, with a chosen luminosity density function is then advanced as a model for the density.", "We advance a methodology that is a major improvement upon this.", "One key advantage of our approach is that estimates of the 3-D material density are derived from non-invasive and nondestructive bulk imaging techniques.", "This feature sets our approach beyond standard methodology that typically relies on experimental designs involving the etching away of layers of the sample material at specific depths.", "Though the microstructure, at this depth, can in principle be identified this way, a measure of $\\rho (x,y,z)$ is not achievable.", "That too, only constraints on the microstructure at such specific depths are possible this way, and interpolation between the layers - based on assumptions about the linearity of the microstructure distribution - are questionable in complex real-life material samples.", "Of course, such a procedure also damages the sample in the process.", "Thus, the scope of the non-destructive methodology that we advance is superior.", "An added flexibility of our model is that it allows for the learning of the correction function from the image data.", "The expansion of the information is made possible in this work by suggesting multiple images of the same sample taken with radiation characterised by different values of the parameter $E$ that controls the sub-surface depth from which information is carried, to result in the image.", "Given the differential penetration depths at different values of $E$ , the images taken in this way are realisations of the 3-D structure of the sample to different depths.", "While a number of attempts at density modelling that use multiple viewing angles have been reported in the literature, the imaging of the system at different $E$ is less common.", "Logistical advantages of imaging with SEM in this way exist over the conventionally undertaken multi-angle imaging.", "One example of an imaging technique that may appear to share similarity with our strategy of collecting information from contiguous slices at successive values of the depth variable is volume imaging with MRI, i.e.", "Magnetic Resonance Imaging , .", "However, unlike in our imaging method, in this technique, the image data at any such slice is obtained via multiple angle imaging; the imaging parameter whose value is varied to procure the image data is still the viewing angle, except this variation is implemented at each individual slice.", "In our method, the parameter that is varied is itself the sub-surface depth, achieved by varying the electron beam energy $E$ , since depth has a one-to-one correspondence with $E$ .", "The method that we have discussed above is indeed developed to solve an unconventionally difficult deprojection problem, but the method is equally capable of estimating the unknown density in an integral equation of the 1st kind - Fredholm or Volterra - and thereby be applied towards density reconstruction in a wide variety of contexts, when image synthesis is possible.", "The scope of such applications is of course ample, including the identification and quantification of the density of the metallic molecules that have infused into a piece of polymer that is employed for charge storage purposes, degrading the quality of the device as a result, or the learning of the density of a heterogeneous nanostructure leading to increased understanding of lack of robustness of device behaviour, or identification of the distribution of multiple phases of an alloy in the depth of a metallic sample meant to be used in industry and even estimation of density of luminous matter in astronomical objects, viewed with telescopes, at different wavelengths.", "In fact, in the case of self-emitting systems that are studied in the emitted particles, the problem is simpler, (as in several other applications), since the interaction-volume is not relevant and the integral reduces to the (easier in general) Volterra integral equation of the first kind.", "In either case, domain-specific details need to be invoked to attain dimensionality reduction." ], [ "Acknowledgments", "We are thankful to Dr. John Aston, Dr. Sourabh Bhattacharya and Prof. Jim Smith for their helpful suggestions.", "DC acknowledges the Warwick Centre for Analytical Sciences fellowship.", "FR acknowledges no conflict of interest with any ongoing research work at NVD." ] ]	1403.0510
[ [ "Possible Triplet $p+ip$ Superconductivity in Graphene at Low Filling" ], [ "Abstract We study the Hubbard model on the honeycomb lattice with nearest-neighbor hopping ($t>0$) and next-nearest-neighbor one ($t'<0$).", "When $t'<-t/6$, the single-particle spectrum is featured by the continuously distributed Van-Hove saddle points at the band bottom, where the density of states diverges in power-law.", "We investigate possible unconventional superconductivity in such system with Fermi level close to the band bottom by employing both random phase approximation and determinant quantum Monte-Carlo approaches.", "Our study reveals a possible triplet $p+ip$ superconductivity in this system with appropriate interactions.", "Our results might provide a possible route to look for triplet superconductivity with relatively-high transition temperature in a low-filled graphene and other similar systems." ], [ "introduction", "Graphene, a single layer of carbon atoms forming a honeycomb lattice, has been among the most exciting research fields since sythesized[1].", "Enormous attentions on this remarkable material have been focused on exploring physics related to its Dirac-cone band structure[2].", "For graphene close to half-filling, the density of states (DOS) at the Fermi level is almost vanishing; as a consequence, relatively weak/intermediate short-range repulsive interactions in general do not induce phase transitions at low temperature[2].", "Nonetheless, exotic phases might be induced by repulsive interactions when the Fermi level is finitely away from the Dirac point.", "For instance, it was shown by renormalization group (RG) calculations that unconventional/topological superconductivity (SC) is induced by weak repulsive interactions in honeycomb Hubbard models finitely away from half-filling[3], [4].", "More recently, exotic phases such as $d+id$ [3], [5], [6], [7], [8], [9], [10], [11] topological superconductivity[12], [13] and Chern band insulators with spin density waves[14], [15] near the type-I Van-Hove singularity (VHS) at 1/4 electron or hole doping, where the DOS at Fermi level diverges logarithmically.", "Such logarithmically diverging DOS close the VHS may significantly raise superconducting transition temperature.", "More recently, it was shown by RG analysis that topological triplet $p+ip$ superconductivity can generically occur in systems at type-II VHS where the saddle points are not at time-reversal-invariant momenta[16], [17].", "In 2D, for a Fermi surface with discrete Van-Hove saddle points, the DOS at Fermi level diverges only logarithmically.", "It would be interesting to study phases in systems with a power-law diverging DOS.", "Indeed, it was shown that for the hopping parameters satisfying $t^{\\prime }<-t/6$ , an inverse-square-root diverging DOS occurs close to band bottom of the lower band, where the band bottom is a closed line instead of discrete points as shown in Fig.", "REF (b).", "In the graphene, such hopping parameters are possible[18], [19], [20], and high levels of doping are experimentally accessible recently[21].", "Note that the band bottom occurring at a closed line only when no third-neighbor or longer-range hopping is considered.", "This kind of line band bottom may be considered as a set of continuously distributed VH saddle points.", "Recent determinant quantum Monte-Carlo (DQMC) study has revealed ferromagnetic-like spin-correlations in such system[22], which implies possibility of a dominant triplet pairing state in this system with repulsive interactions.", "In this paper, we report both random phase approximation (RPA) analysis and DQMC studies of pairing symmetries of possible SC induced by weak or intermediate repulsive interactions in graphene at low fillings whose DOS at Fermi level is significantly enhanced by the power-law singularity at the band bottom.", "Both numerical approaches obtain the $p+ip$ triplet pairing as the leading instability of the system in different parameter regimes.", "For $t^{\\prime }=-0.2t$ , $U/t=3.0$ , and filling $n=0.2$ , the transition temperature $T_{c,\\textrm {triplet}}$ into the triplet pairing state is estimated to be in the order of $10^{-2}t$ .", "For graphene $t\\sim 2.0$ eV, this implies that the $T_{c,\\textrm {triplet}}$ in graphene might be as high as 200K when the Fermi level is tuned appropriately close to the band bottom.", "These results might provide a possible route to look for triplet superconductivity with relatively-high transition temperature in graphene at low filling." ], [ "Model and approach", "We start from the following Hubbard model on the honeycomb lattice $H=-t\\sum _{\\left\\langle i,j\\right\\rangle }c^{\\dagger }_{i\\sigma }c_{j\\sigma }-t^{\\prime }\\sum _{\\left\\langle \\langle i,j\\right\\rangle \\rangle }c^{\\dagger }_{i\\sigma }c_{j\\sigma }+U\\sum _{i}n_{i\\uparrow }n_{i\\downarrow }, $ where $c^\\dag _{i\\sigma }$ is the electron creation operator at site $i$ and with spin polarization $\\sigma =\\uparrow ,\\downarrow $ and $U$ labels the on-site repulsive interaction.", "Here the $t$ and $t^{\\prime }$ terms describe the nearest neighbor (NN) and next nearest neighbor (NNN) hoppings, respectively.", "We consider the case of $t>0$ and $t^{\\prime }<0$ , which is supported by recent first principle calculations[18] and experiments[19].", "As the ratio $\|\\frac{t^{\\prime }}{t}\|$ varies from around 0.1[19] to around 0.3[20] in different experiments, we focus on the possible cases with $t^{\\prime }<-t/6$ and take $t^{\\prime }=-0.2t$ in our calculations unless stated otherwise.", "The band structure is shown in Fig.", "REF (a), together with the Fermi levels for filling $n=0.2$ per site.", "We notice one remarkable feature of this band structure: the band bottom of this system does not locate at the $\\Gamma $ -point; instead it consists two closed lines around $\\Gamma $ .", "As a consequence, the DOS is divergent in an inverse-square-root fashion near the band bottom, as shown in Fig.", "REF (b).", "The Fermi surface (FS) of the system at $n=0.2$ is shown in Fig.", "REF (c), which contains an inner hole-pocket and an outer electron-pocket.", "Such a Hubbard-model with only on-site interaction has been widely engaged[5], [6], [7], [9], [10] to describe the graphene doped to near the VH points because at such dopings, the divergent DOS on the FS leads to strong screening of the Coulomb interaction.", "In the following, we adopt perturbative RPA analysis for weak $U$ interactions and the DQMC calculations for relatively strong $U$ to investigate the pairing symmetries of the possible SC at low filling.", "Figure: (Color online)(a) The energy band along high symmetry line in the first Brillouin Zone; (b) The DOS as function of energy with t ' =-0.2tt^{\\prime }=-0.2t; and (c) The Fermi surface at filling n=0.2n=0.2." ], [ "RPA treatment", "We adopted the standard multi-orbital RPA approach[23], [24], [25], [26], [27], [28] in our study for the small $U$ ($=0.1t$ ) case.", "Various susceptibilities of non-interacting electrons of this system are defined as $\\chi ^{(0)l_{1},l_{2}}_{l_{3},l_{4}}\\left(\\mathbf {q},\\tau \\right)\\equiv \\frac{1}{N}\\sum _{\\mathbf {k_{1},k_{2}}}\\left<T_{\\tau }c^{\\dagger }_{l_{1}}(\\mathbf {k_{1}},\\tau )c_{l_{2}}(\\mathbf {k_{1}+q},\\tau )\\right.\\nonumber \\\\\\left.c^{+}_{l_{3}}(\\mathbf {k_{2}+q},0)c_{l_{4}}(\\mathbf {k_{2}},0)\\right>_0,$ where $l_{i}$ $(i=1,2)$ denotes orbital (sublattice) index.", "Largest eigenvalues of the susceptibility matrix $\\chi ^{(0)}_{l,m}\\left(\\mathbf {q}\\right)\\equiv \\chi ^{(0)l,l}_{m,m}\\left(\\mathbf {q},i\\nu =0\\right)$ is shown in Fig.", "REF for filling $n=0.1$ , which shows dominant distributions on a small circle around the $\\Gamma $ -point.", "This suggests strong ferromagnetic-like intra-sublattice spin fluctuations in the system.", "Generally, it is found that at low fillings, the radius of the circle scales with filling.", "At low fillings, the eigenvector of the susceptibility matrix reveals that the inter-sublattice spin fluctuations in the system are also ferromagnetic-like, although somewhat weaker than the intra-sublattice ones.", "Such ferromagnetic-like spin fluctuations are consistent with the ferromagnetic spin correlations revealed by the DQMC calculations[22].", "With weak Hubbard-$U$ , the spin ($\\chi ^{s}$ ) or charge ($\\chi ^{c}$ ) susceptibilities in the RPA level are given by $\\chi ^{s\\left(c\\right)}\\left(\\mathbf {q},i\\nu \\right)=\\left[I\\mp \\chi ^{(0)}\\left(\\mathbf {q},i\\nu \\right)\\bar{U}\\right]^{-1}\\chi ^{(0)}\\left(\\mathbf {q},i\\nu \\right),$ where $\\bar{U}^{\\mu \\nu }_{\\mu ^{\\prime }\\nu ^{\\prime }}$ ($\\mu \\nu =1,2$ ) is a 4$\\times $ 4 matrix, whose only two nonzero elements are $\\bar{U}^{11}_{11}=\\bar{U}^{22}_{22}=U$ .", "Clear, the repulsive Hubbard-$U$ suppresses $\\chi ^{c}$ but enhances $\\chi ^{s}$ .", "Thus, the spin fluctuations take the main role of mediating the cooper pairing in the interacting system[23].", "In the RPA level, the cooper pairs near the FS acquire an effective interaction $V_\\textrm {eff}$[23], [24], [28] via exchanging the spin fluctuations represented by the spin susceptibilities.", "From this effective interaction, one obtains the linearized gap equation near the superconducting critical temperature $T_c$ , solving which one obtains the leading pairing symmetry (symmetries) of the system.", "Figure: (Color online) Largest eigenvalues of the susceptibility matrix in non-interacting limit in the first Brillouin-Zone.Our results for $n$ =0.1 and $n$ =0.2 reveal that the leading pairing symmetries of the system at these low fillings are degenerate $p_x$ and $p_y$ doublets, as shown in Fig.", "REF (a) and (b), which should be further mixed as $p_x\\pm i p_y$ to minimize the ground state energy, as suggested by our further mean-field calculations on the effective Hamiltonian.", "Such a triplet pairing is mediated by the ferromagnetic-like spin fluctuations in the system, as shown in Fig.", "REF .", "The subleading pairing symmetries of the system at these low fillings are triplet $f$ -wave shown in Fig.", "REF (a) for $n=0.1$ and singlet $d_{xy}$ and $d_{x^{2}-y^{2}}$ doublets (which should further be mixed as $d_{xy} \\pm id_{x^{2}-y^{2}}$ to lower the energy) shown in Fig.", "REF (a) and (b) for $n=0.2$ .", "Note that we have chosen such a small $U$ as $U=0.1t$ in our RPA calculations.", "For larger $U$ beyond its critical value $U_c$ , the divergence of the spin susceptibility invalidate our RPA calculations for superconductivity.", "Physically, such a divergent spin susceptibility for $U>U_c$ may not necessarily lead to a magnetically-ordered state since the distribution of the susceptibility shown in Fig.", "REF does not possess a sharply peaked structure at particular momentum.", "Instead, the competition among different wave vectors may lead to paramagnetic behavior or short-ranged spin correlations which provide basis for the cooper pairing.", "We leave the study for the case of $U>U_c$ to the following DQMC approach, which is suitable for strong coupling problems.", "Figure: (Color online) (a) and (b) show the p x p_x and p y p_y pairing symmetries in the kk-space and (c) shows the phase of the p+ipp+ip pairing symmetry on the honeycomb lattice in the real space.Figure: (a) shows the ff pairing in the kk-space and (b) shows the phase of the ff pairing symmetries in the real space.Figure: (Color online) (a) and (b) show the d xy d_{xy} and d x 2 -y 2 d_{x^{2}-y^{2}} pairing symmetries in the kk-space and (c) shows the phase of the d+idd+id pairing symmetries in the real space." ], [ "DQMC simulations", "The DQMC simulation is a powerful unbiased numerical tool to study the physical properties of such strongly-correlated electronic systems as the Hubbard model.", "The basic strategy of DQMC is to express the partition function as a high-dimensional integral over a set of random auxiliary fields.", "The integral is then accomplished by Monte Carlo techniques.", "For more technique details, we refer to Refs.", "[29], [22], [30].", "Figure: The lattice geometries for the 2×1082\\times 108 (a), 2×752\\times 75 (b) and 2×482\\times 48 (c) honeycomb lattices.To investigate the SC property, we compute the pairing susceptibility, $P_{\\alpha }\\equiv \\frac{1}{N_s}\\sum _{i,j}\\int _{0}^{\\beta }d\\tau \\langle \\Delta _{\\alpha }^{\\dagger }(i,\\tau )\\Delta _{\\alpha }^{\\phantom{\\dagger }}(j,0)\\rangle .$ Here $\\alpha $ stands for the pairing symmetry, and the corresponding pairing order parameter $\\Delta _{\\alpha }^{\\dagger }(i)$ is defined as $\\Delta _{\\alpha }^{\\dagger }(i)\\ \\equiv \\sum _{l}f_{\\alpha }^{*}(\\delta _{l})(c_{{i}\\uparrow }c_{{i+\\delta _{l}}\\downarrow }\\pm c_{{i}\\downarrow }c_{{i+\\delta _{l}}\\uparrow })^{\\dagger },$ where $f_{\\alpha }(\\delta _{l})$ is the form factor of the pairing function, the vectors $\\delta _{l}$ denote the bond connections, and “$\\pm $ \" labels triplet/singlet symmetries respectively.", "Figure: (Color online) Pairing susceptibility P α P_{\\alpha } as a functionof temperature for different pairing symmetries with U=3tU=3t at nn=0.2 (a) and nn=0.1 (b) on a 2×752 \\times 75 lattice (solid line).", "The P p+ip P_{p+ip} at nn=0.2 on a 2×482 \\times 48 lattice (dash red line) and a 2×1082 \\times 108 lattice are also shown (dotted red line) in (a).", "Here the units of temperature is tt.Guided by the RPA results, three different pairing symmetries were investigated in the following DQMC studies, i.e.", "$p+ip$ , $f$ , and $d+id$ symmetries, whose form factors are illustrated in Fig.", "REF (c), Fig.", "REF (b), and Fig.", "REF (b) respectively.", "These different pairing symmetries can be distinguished by their different phase shifts upon each 60$^\\circ $ rotation, which are $\\pi /3$ , $2\\pi /3$ and $\\pi $ respectively.", "The NNN-bond $p+ip$ and $f$ wave triplet pairings shown possess the following form factors, $\\ f_{p+ip}(\\delta _{l})=e^{i(l-1)\\frac{\\pi }{3}},\\ f_{f}(\\delta _{l})=(-1)^{l},~l=1,\\cdots ,6,$ and the NN-bond singlet $d+id$ pairing shown possesses the form factor $\\ f_{d+id}(\\delta _{l})=e^{i(l-1)\\frac{2\\pi }{3}},~l=1,2,3.$ Note that the NN-bond pairing is prohibited in the $f$ -symmetry.", "As for the $p+ip$ and $d+id$ ones, although pairings on both the NN-bond and the NNN-bond are allowed, our DQMC calculations show they are weaker (stronger) on the former than on the latter for the $p+ip$ ($d+id$ ) symmetry, reflecting the fact that the spin-fluctuations on the former are less ferromagnetic-like than those on the latter, consistent with our RPA calculations.", "We have also studied longer-range pairings by adding third and forth bond pairings in former factors, which turn out be much weaker than that of the NN-bond and NNN-bond presented above.", "Our DQMC simulations of the system were performed at finite temperatures on a $2\\times 48$ , a $2\\times 75$ and a $2\\times 108$ lattices with periodic boundary conditions.", "Here, each lattice we employed in simulations consists of two interpenetrating triangular sublattices with hexagonal shape such that it preserves most geometric symmetries of graphene, as shown in Fig.", "REF .", "In each case, the total number of unit cells is $3L^2$ and the total number of lattice sites is $2\\times 3L^2$ with $L=$ 6, 5, or 4 in Fig.", "REF (a), (b) and (c) respectively.", "Fig.", "REF shows the temperature dependence of the pairing susceptibilities for different pairing symmetries with electron filling $n$ =0.2 (a) and $n$ =0.1 (b) with $U=3t$ .", "Within the parameter range investigated, the pairing susceptibilities for various symmetries increase as the temperature is lowered, and most remarkably, the $p+ip$ pairing symmetry dominates other ones at relatively low temperatures, consistent with the RPA results.", "In Fig.REF (a), the pairing susceptibility $P_{p+ip}$ on a $2\\times 48$ and a $2\\times 108$ lattices are also shown, in comparison with that on the $2 \\times 75$ lattice, from which one verifies negligible finite size effects.", "Figure: (Color online) Pairing susceptibility P p+ip P_{p+ip} as a functionof temperature with U=3tU=3t and nn=0.2 for a 2×752 \\times 75 lattice (a) and a 2×122 \\times 12 lattice (b) (solid line).", "The fitting data are also shown as dashed lines.The superconducting transition occurs as the pairing susceptibility diverges.", "However, DQMC simulations encounter the notorious minus problem in this doped system as well; consequently the lower the temperature used in DQMC, the larger the error bar is.", "In Fig.", "REF , we have simulated the system to the lowest temperature at our best while keep a reasonable error bar.", "The lowest temperature for the $2 \\times 75$ lattice is $t/12$ and the lowest temperature for the $2 \\times 12$ lattice is $t/15$ .", "Within our numerical results, We fit the DQMC data with a formula of $P=a/(T-T_c)+b$ , as shown (dashed lines) in Fig.", "REF and then we extrapolate to obtain the $T_c$ .", "The fitting agrees with the DQMC data reasonably well.", "From this fitting, one may estimate a $T_c$ of about $\\sim 0.01t$ , which is roughly $\\sim 200$ K. Figure: (Color online) The intrinsic pairing interactionP p+ip -P ˜ p+ip P_{p+ip}-\\widetilde{P}_{p+ip} as a function of temperature for different UU (a) and different nn (b) on a 2×752 \\times 75 lattice.In order to extract the intrinsic pairing interaction in our finite system, one should subtract from $P_{\\alpha }$ its uncorrelated single-particle contribution $\\widetilde{P}_{\\alpha }$ , which is achieved by replacing $\\langle c_{{i}\\downarrow }^{\\dag }c_{{j}\\downarrow }c_{i+\\delta _{l}\\uparrow }^{\\dag }c_{j+\\delta _{l^{\\prime }}\\uparrow }\\rangle $ in Eq.", "(REF ) with $\\langle c_{{i}\\downarrow }^{\\dag }c_{{j}\\downarrow }\\rangle \\langle c_{i+\\delta _{l}\\uparrow }^{\\dag }c_{j+\\delta _{l^{\\prime }}\\uparrow }\\rangle $ .", "Clearly in Fig.", "REF , the intrinsic pairing interaction $P_{p+ip}-\\widetilde{P}_{p+ip}$ shows qualitatively the same temperature dependence as that of $P_{p+ip}$ , which is positive and increases with the lowering of temperature.", "Such a temperature dependence of $P_{\\alpha }-\\widetilde{P}_{\\alpha }$ suggests effective attractions generated between electrons and the instability toward SC in the system at low temperatures.", "Moreover, Fig.", "REF (a) shows that the intrinsic pairing interaction for $p+ip$ symmetry enhances with larger $U$ , indicating the enhanced pairing strength with the enhancement of the electron correlations.", "As for the other two pairing symmetries shown, our DQMC results yield negative intrinsic pairing interactions, reflecting the fact that the realization of the $p+ip$ symmetry at low temperatures will suppress other competing pairing channels." ], [ "Conclusions and discussions", "We have performed combined RPA analysis and DQMC calculations for the low-filled honeycomb Hubbard model with weak and strong repulsive U respectively.", "Both studies show that the triplet $p+ip$ SC occurs as the ground state of our model system of low-filled graphene.", "Besides graphene, the results obtained here also apply to other isostructure materials, such as silicene[31] and germanene[32].", "Furthermore, by trapping some fermionic cold atoms into an optical lattice, one may also be able to simulate the Hubbard-model on a honeycomb lattice studied here[33], [34], [35], with tunable parameters and dopings, which is expected to realize the triplet $p+ip$ superfluidity.", "Acknowledgement: We would like to thank Zhong-Bin Huang, Yuigui Yao, Yu-Zhong Zhang, and Su-Peng Kou for stimulating discussions.", "This work is supported in part by NSFC (Grant Nos.", "11104014, 11274041, 11374034, and 11334012), by Research Fund for the Doctoral Program of Higher Education of China 20110003120007 and SRF for ROCS (SEM) (T.M.", "), by the NCET program under the grant No.", "NCET-12-0038 (F.Y.", "), and by the Thousand-Yound-Talent Program of China (H.Y.", ")." ] ]	1403.0295
[ [ "Chandra X-ray spectroscopy of a clear dip in GX 13+1" ], [ "Abstract The source GX 13+1 is a persistent, bright Galactic X-ray binary hosting an accreting neutron star.", "It shows highly ionized absorption features, with a blueshift of $\\sim$ 400 km s$^{-1}$ and an outflow-mass rate similar to the accretion rate.", "Many other X-ray sources exhibit warm absorption features, and they all show periodic dipping behavior at the same time.", "Recently, a dipping periodicity has also been determined for GX 13+1 using long-term X-ray folded light-curves, leading to a clear identification of one of such periodic dips in an archival Chandra observation.", "We give the first spectral characterization of the periodic dip of GX 13+1 found in this archival Chandra observation performed in 2010.", "We used Chandra/HETGS data (1.0-10 keV band) and contemporaneous RXTE/PCA data (3.5-25 keV) to analyze the broadband X-ray spectrum.", "We adopted different spectral models to describe the continuum emission and used the XSTAR-derived warm absorber component to constrain the highly ionized absorption features.", "The 1.0-25 keV continuum emission is consistent with a model of soft accretion-disk emission and an optically thick, harder Comptonized component.", "The dip event, lasting $\\sim$ 450 s, is spectrally resolved with an increase in the column density of the neutral absorber, while we do not find significant variations in the column density and ionization parameter of the warm absorber with respect to the out-of-dip spectrum.", "We argue that the very low dipping duty-cycle with respect to other sources of the same class can be ascribed to its long orbital period and the mostly neutral bulge, that is relatively small compared with the dimensions of the outer disk radius." ], [ "Introduction", "Low-mass X-ray binary (LMXB) dipping sources are characterized by periodic (or quasi-periodic) dips in their light-curves that are evidence for a fixed structure in the reference frame of the binary system.", "These dips may also be related to super-orbital periodicities, which are more difficult to constrain when their appearance is transient [23].", "To date, 13 LMXBs hosting a neutron star (NS) and 6 LMXBs hosting a black hole have shown clear dips in their light curves.", "In Table REF , we show an updated list of these sources with some basic data and references to the literature.", "Table: LMXB dipping sourcesTwo main physical models are widely discussed in the literature to explain the occurrence of dips: [67] proposed a variable, azimuthal-dependent height of the accretion disk's outer rim and a large system-inclination angle.", "According to the orbital phase, our line-of-sight partially or totally intercepts the rim that causes local absorption of X-rays produced in the innermost parts of the system.", "The rim geometry was empirically adjusted by matching synthesized geometries and the regular patterns observed in the light curves [40].", "Some dippers also show periods without dips, which points to a strong variability of the occulting regions [58].", "Alternatively, another explanation was proposed by [21]: if matter from the companion star is able to skim across the thickness of the outer accretion disk, part of the incoming stream may impact the disk at a much closer radius [38]; when part of this stream collides with the disk, it is quickly dynamically and thermally virialized; but a fraction of it (which is a tunable parameter of the model) receives energy from the impact shock and splits into a two-zone medium, forming blobs of cold, condensed gas, surrounded by a lower density hotter plasma at large scale-heights above the disk [34].", "This scenario is able to partially account for many empirical facts such as the dip's periodic occurrence, the dependence on orbital phase, and the duration and time scales of the single dips.", "Both scenarios involve the common ingredient of a high inclination angle.", "For low-mass companion stars with short orbital periods ($<$ 1 day), the inferred inclination angle, $i$ , is constrained between 65$^{\\circ }$ and 85$^{\\circ }$ , while for higher inclinations eclipses are also expected.", "In these eclipsing binaries, direct emission from the NS is blocked by the disk thickness and only scattered emission from an accretion-disk corona (ADC) may be observed [30].", "Together with these physical scenarios, many studies have been focused on deriving geometrical and physical constraints by spectrally resolving the dip events.", "Spectra from dipping sources have been fitted using a two-component spectral decomposition consisting of a thermal black-body emission from the surface of the NS and a Comptonized emission (usually fitted with a cut-off power-law).", "Seed photons of the Comptonized spectrum come from the accretion disk and the Comptonization is thought to occur at large disk radii in an extended corona, whose radius is $>>$ 10$^{9}$ cm.", "Using the ingress and egress times of the deep dips (where emission is totally blocked at the dip bottom), it has been shown that the corona emission is gradually covered (progressive covering approach) and therefore extended, with a disk-like geometry, while the black-body emission is point-like and attributed to the NS emission [9].", "The main assumption in deriving the estimates for the ADC radius is that the dip is caused by the bulge located at the outer accretion disk [67], whose main effect is a progressive photoelectric absorption of the primary incident source flux.", "Detection in the past decade of resonant absorption features of highly ionized elements in the X-ray spectrum (see Table REF ) has provided new clues for separating the spectral formation.", "Local absorption features often appear to be blue-shifted, which points to a disk-wind or generally out-flowing, photoionized plasma.", "The ionization state of the optically thick absorbing plasma is variable and the time scales can be as short as a few ks [63], with a wind velocity of thousands of km/s.", "In all cases, the most clearly resolved lines are from H-like and He-like transitions of iron, which implies that the ionization parameter, $\\xi $ , of the warm absorber is $>$ 100 [33].", "[7] first advanced the hypothesis that during dipping there might be a tight relation between cold and warm absorption because the overall X-ray variability during dipping would be driven by fast changes in the column density and ionization state of the warm absorbing medium along our line of sight.", "In this scenario, there is no more need for a partial covering of an extended continuum corona because the soft excess observed during dipping is naturally accounted for by a combination of strong increase in the column density of the warm medium and a decrease of its ionization parameter.", "Outside dips, a warm absorber has been always observed, which implies that the medium has a cylindrical distribution and is not confined to the locus of the bulge.", "In light of these findings, the continuum decomposition has also been questioned, because an extended corona was felt to be less necessary [19]." ], [ "The source GX13+1", "The source GX 13+1 is a persistent X-ray binary system belonging to the so-called class of GX bright bulge sources, with an estimated distance of 7 $\\pm $ 1 kpc.", "The compact object is an accreting NS that has sporadically shown type-I X-ray bursts [41], orbiting an evolved mass-donor giant star of spectral class KIII V [5].", "The system has peculiar characteristics, being in between the classification of low-mass and high-mass systems; this is also testified by the long orbital period of $\\sim $ 24 days [11], [29], which makes it the LMXB with the second-longest orbital period after GRS 1915+105 (30.8 d period; it has a black hole as accreting compact object).", "The 3–20 keV Rossi-XTE (RXTE) spectrum of GX 13+1 has been investigated by [26], in connection with its radio emission.", "The spectrum was deconvolved according to the Eastern interpretation, that is the sum of a softer multicolored accretion-disk emission and a thermal Comptonized, harder optically thick emission in the boundary layer.", "The low-resolution RXTE/PCA spectrum also needed some local features (broad iron Gaussian line and a 9 keV absorption edge) to obtain a satisfactory fit.", "More recently, emission of higher than 20 keV has been observed with INTEGRAL/ISGRI data [46], and was subsequently analyzed according to a thermal plus bulk Comptonization model [39].", "Using narrower bands such as the 1–10 keV CCD typical range, the general continuum adopted was found to be well approximated with the sum of soft disk emission and black-body harder emission [60], [55], [63], [19].", "The black-body emission approximates an optically thick Comptonized emission, which we deduce from the difficulty in constraining both the optical depth and the electron temperature with a limited energy range and the high optical depths characteristics of the very soft spectra of bright accreting NS LMXBs.", "Analysis of K-$\\alpha $ edge depths has also shown that in the direction of the source the ISM composition (or absorbing, circumbinary cold matter) is significantly overabundant in elements heavier than oxygen [62], with X-ray fine-structure absorption features (XAFS) around the Si and S K-$\\alpha $ edges.", "High-resolution spectroscopy with the Chandra HETGS revealed a radiatively/thermally driven disk wind with an outflow velocity of $\\sim $ 400 km s$^{-1}$ and multiple absorption features from highly ionized elements [63].", "The wind probably carries a significant fraction of the total mass-accretion rate, up to 10$^{18}$ g s$^{-1}$ .", "Observations with XMM-Newton also revealed a broad (equivalent width $\\gg $ 100 eV) iron emission line, whose origin is associated to a disk-reflection component, and the broad width is ascribed to Compton broadening in the warm corona.", "A global spectral account of the total variability has also been proposed, where the main drivers for the spectral variability are neutral cold absorption and variability associated with a reflection component [19].", "Periodic dipping in the LMXB GX 13+1 was suspected for a long time, on the basis of an energy-dependent modulation observed in long-term light curves [11], [19].", "[29] systematically searched in archived X-ray observations for clear signatures of periodic dips.", "Applying timing techniques to long-term folded X-ray light curves provided a successful method that led to a refined orbital-period estimate (24.5274(2) d) and to the first ephemeris for the dip passage times.", "The only periodic dip that could be assigned on the basis of this ephemeris for a pointed X-ray observation was in an archival Chandra observation performed in 2010.", "This corroborates that inclination is a key factor to spot warm absorbing winds in LMXBs and that they are optically thick to radiation only close to the plane of the accretion disk.", "This relation has recently also been pointed out by [49] for Galactic LXMBs hosting black-holes.", "We present in this article the results of the spectroscopic analysis of the $Chandra$ dip event, showing that the main driver of the dipping in this source is an increase in cold photoelectric absorption." ], [ "Observation and data reduction", "We used CIAO 4.5 for the Chandra data extraction and analysis, CALDB 4.4.7 for the calibration data files, the software package HeaSOFT version 6.13 for the RXTE/PCA data extraction, reduction, and scientific analysis, the Interactive Spectral Interpretation System (ISIS) 1.6.2 [27] for the spectral fitting, and Xspec v.12.8.1 for spectral models.", "Observation times are given in Coordinated Universal Time (UTC)." ], [ "The source GX 13+1 was observed multiple times with the Chandra observatory from 2004 to 2012.", "For the present work we used the observation with sequence number 11814 from the Chandra archive.", "The observation started at 2010-08-01 00:32:37 UTC and lasted 28.12 ks.", "At the same time the Rossi-XTE satellite observed the source, which provided overlapping, although not complete, monitoring.", "The observation performed in (faint) timed exposure mode used the High Transmission Grating Spectrometer (HETGS) to diffract the spectrum, and a (350 rows) subarray of the ACIS-S detector to mitigate the effects of photon pile-up, with a CCD frame time of 1.24104 s. The brightness of the source ($\\sim $ 6$\\times $ 10$^{-9}$ erg cm$^{-2}$ s$^{-1}$ ) prevented studying the zeroth-order events since these are strongly affected by pile-up.", "The location of the center of the zeroth-order image was therefore determined using the tg_findzo scripthttp://space.mit.edu/cxc/analysis/findzo/, as recommended by the Chandra team in case of zeroth-order pile-up.", "We derived a source position at R.A. = 18$^{h}$ 14$^m$ 31$^{s}$ .08, Dec (J2000) = -17$$ 09$$ 26$$ .1 (J2000, 0.$$ 6 uncertainty), compatible with the coordinates reported in [29].", "Data were extracted from regions around the medium- and high-energy grating arms (MEG and HEG) with the tg_create_mask tool, manually setting the zeroth-order position as derived by the tg_findzo script and choosing a width for HEG and MEG arms of 25 sky pixels.", "We extracted only first-order positive (HEG+1, MEG+1) and negative (HEG-1, MEG-1) spectra, as they provide the best signal-to-noise ratio and higher orders are mostly affected by pile-up.", "We used standard CIAO tools to derive all the other related spectral products.", "Spectra were finally re-binned to have at least 25 counts per energy channel to allow the use of $\\chi ^2$ statistics.", "We used the HEG spectra in the 1.0–10.0 keV range and the MEG spectra in the 1.0–5.0 keV range; both spectra are background-dominated below 1 keV." ], [ "For RXTE/PCA spectra we used source and background spectra, matrices and ancillary responses generated according to standard pipelines and selection criteriahttp://heasarc.nasa.gov/docs/xte/recipes/pca_spectra.htmlanalysis.", "We used only events from PCU2 data, as this PCU more completely and uniformly overlapped with the Chandra observation, and limited our spectral analysis to the top-layer events as these provide the best signal-to-noise spectra in the 3.5–25.0 keV energy range, where the response matrix is best calibrated; a systematic error of 1% was added in quadrature to the statistical error.", "For light curves and hardness ratios we exploited the broader range 2.0–30.0 keV.", "The details of the $RXTE$ observations used for the analysis are summarized in Table REF .", "Table: Log of the RXTE/PCU observations.Net count rates and exposures for PCU2 top-layer spectraare given.", "Start time (TSTART) is in UTC." ], [ "Light curves and time-selected spectral analysis", "In Fig.", "REF we show the Chandra light curve in the 1.0–10.0 keV range, extracted from the ACIS-S HEG first-order diffraction arm and the hardness ratio defined from the (4.0–10.0 keV)/(1.0–4.0 keV) count ratio.", "The dip is clearly chromatic, the hardening is smooth, and the center of the dip falls at 2010-08-01 05:23:22.848 UTC; the dip, which is neatly resolved, is quite symmetrical with ingress and egress times that are nearly coincidental, although the smooth ingress/egress times and the correlated variation of the persistent emission does not allow us to strictly define its temporal duration.", "Fitting the dip shape with a Gaussian in a local (750 s wide) neighborhood of the dip center, we derived a full width at half maximum of $\\sim $ 450 s. Before and after the dip, the spectrum shows an overall smooth and moderate softening.", "In the pre-dip part of the observation, the average count rate and hardness ratio are 24.4$\\pm $ 0.1 cts s$^{-1}$ and 0.798$\\pm $ 0.003, in the post-dip part these are 20.2$\\pm $ 0.1 cts s$^{-1}$ and 0.773$\\pm $ 0.003.", "Figure: Chandra/HEG light curve and hardness ratio.", "Timeis in hours since the start of the day 2010-08-01 00:32:37UTC.", "The soft rate refers to the 1.0–4.0 keV range, while thehard rate refers to the 4.0–10.0 keV range.", "Bin time is 50s.", "Re-adapted from .In Fig.", "REF , we show the PCU2 count rates in the selected energy band 2–6 and 6–10 keV and the hardness ratio.", "While there is modest variability in the first three pointings, a significant increase in the hardness is evident in the last two.", "But the dip event, observed in the continuous Chandra pointing, was unfortunately missed by $RXTE$ .", "Figure: Light curves and hardness ratio for the RXTERXTE/PCU2observations.", "Soft and hard rates computed in the 2–6 keV and inthe 6–10 keV range, respectively.", "Time is in hours since2010-08-01 00:00 UTC.On the basis of these hardness ratios, keeping the number of spectra reasonably within the need to have sufficient statistics to constrain the main spectral parameters, we created the good-time intervals (Table REF ) and then extracted the corresponding energy spectra.", "RXTE/PCA observations were summed with the mathpha tool, after we verified their spectral consistency.", "In Fig.", "REF we show the PCU2 pointed observations overimposed to the Chandra light curve (the HEG rate is multiplied by a factor of 24 to visually match the PCU2 data), and the time selections through alternating white and gray overlays.", "Two main longer intervals identify the pre-dip and post-dip spectra; one interval is centred around the dip bottom and two shorter intervals trace the ingress and egress passages (we did not use a specific cut on the count rate to define these intervals) for a total of five time-selected spectra.", "Figure: 2–30 keV PCU2 top layer count rate (green points) and1–10 keV Chandra/HEG rate (red points).", "Chandrarate is multiplied by a factor of 24 to approximately match the PCU2rate.", "Bin time is 64s.", "Alternated shaded areas are based on thetime-selections of Table .", "Time is in hourssince 2010-08-01 00:00 UTCTable: Time-selected intervals used for the spectral analysis ofChandra and RXTE observations." ], [ "Pile-up treatment", "Because of the intense source flux the Chandra grating spectra are moderately affected by pile-up.", "To assess its impact, we used in the spectral fits the convolution component simple_gpile2 developed by [24] from the original code in [44].", "The main effect of pile-up for first-order spectra is to reduce the effective count rate in each detector pixel.", "The simple_gpile2 uses the free fitting parameter $\\gamma $ , which corrects the detector count rate according to the equation $C^{\\prime } (\\lambda ) = C (\\lambda ) \\cdot exp(- \\gamma \\cdot C_{tot}(\\lambda )),$ where $C (\\lambda )$ is the observed detector count rate, $C_{tot}(\\lambda )$ is the count rate computed according to the total source fluxes (summed over all orders), and $\\gamma $ is expressed in units of s Å/cts.", "The $\\gamma $ parameter is free to vary for each HEG/MEG dataset, with an expected value that is the function of the time frame and detector wavelength accuracy, as expressed by the simple relation $\\gamma _0 = 3 \\Delta \\lambda \\times t_{frame}$ where $\\Delta \\lambda $ corresponds to 5.5 mÅ for the HEG grating arm, and 11 mÅ for the MEG arm, and $t_{frame}$ = 1.24 s for this observation.", "The factor 3 takes into account that the detection cell is constituted by a three-pixel array.", "In Fig.REF , we show the pile-up fraction as a function of energy according to the best-fitting model discussed in Section REF .", "We found that the $\\gamma $ parameters did not strongly depend on the continuum choice and showed little variation for the time-selected spectra.", "The pile-up fraction, as expected, reached higher values for the MEG spectra (peak values $\\gtrsim $ 15% between 1.54 keV and 1.74 keV) than for the HEG spectra (peak values $\\gtrsim $ 10% between 1.4 keV and 1.8 keV).", "The pile-up fraction is negligible in the K$\\alpha $ iron range.", "Figure: Pile-up fraction as a function of energy for the ChandraChandra/HETGS data.Black line HEG+1 data, red line MEG+1 data.", "Negative orders show a very similar dependence." ], [ "Absorption line detections in the non-dip $Chandra$ data", "The $Chandra$ X-ray spectrum is rich in local absorption features, similarly to the Chandra observation analyzed in [63].", "In this section, we focus on the detection and identification of these local features, while the details on the continuum model are addressed in the next section.", "To increase the signal-to-noise ratio and assess the detection level of these features we considered the time-averaged spectrum, filtering out only the interval with the dip.", "We considered HEG$\\pm $ 1 and MEG$\\pm $ 1 data as independent datasets, adopting a continuum model consisting of soft disk emission and thermal Comptonized component (the choice of the exact continuum has marginal influence on the parameter values) and locally searched for narrow absorption features around the expected values for the resonant transitions H-like and He-like of the most abundant elements, fitting the line profiles with Gaussians.", "In Table REF we report the detected lines (normalization values not compatible with 0 at 2.7 $\\sigma $ ) associated to the element transitions, the measured shift with respect to the laboratory rest-frame and the widths.", "Two lines close to the Si K$\\alpha $ edge are possibly not related to the warm medium but to X-ray absorption fine structures (XAFS) close to the Si-edge structure [62].", "Best-fitting values for the pile-up $\\gamma $ correction factors applied for each grating arm are reported in Table REF .", "Table: Absorption line detections for ChandraChandra non-dipspectrum.", "An absorbed (𝚝𝚋𝚟𝚊𝚛𝚊𝚋𝚜\\texttt {tbvarabs}) continuum model ofthermal disk emission and thermal Comptonization is used.Inspecting the results from Gaussian fitting of the absorption lines, we noted that the lines are mostly produced by resonant transitions of H-like ions with a common blue-shift (weighted average 490 km s$^{-1}$ ).", "Under the assumption that all these features could then be described by only one photoionized medium, we adopted a tabular spectral model derived from the XSTAR code, warmabshttp://heasarc.gsfc.nasa.gov/xstar/docs/html/node99.html to self-consistently fit all these features.", "We set as free fitting parameters the log of the ionization factor $\\xi $ , the relative hydrogen column density (in units of log(N$_{\\textrm {H}}/10^{22}$ )), the blue-shift of all the lines (using the $z$ red-shift parameters and allowing for negative values) and the turbulent broadening (in km s$^{-1}$ ).", "We adopted as electron density a value of 10$^{12}$ cm$^{-3}$ , although we note that this choice has only a marginal impact on the best-fitting values of all the other parameters.", "We used the default value of the table model for the irradiating flux, a power-law with spectral index 2.", "This is a good approximation of the 1–10 keV spectrum of the source, since fitting with a simple absorbed power-law would result in a photon-index of 2.07.", "The best-fitting values for the warmabs component and the continuum model adopted are discussed and reported in the following section (last column of Table REF ), while the best-fitting values of the pile-up $\\gamma $ parameters for the different grating arms are reported in Table REF .", "Table: Best-fitting values of the averaged ChandraChandranon-dip spectrum for the γ\\gamma pile-up parameters.In Fig.", "REF , we compare model and residuals for the two approaches (local Gaussians and warmabs model) for the most interesting energy ranges.", "The strongest detected features are satisfactorily fitted with the warmabs component, with the exception of the local features around the K$\\alpha $ Si-edge (see upper right panel of Fig.REF ).", "For the ionization parameter derived from the best-fitting model, the possible contribution of the Ly$\\alpha $ transition of He-like Si at 1.865 keV is negligible, and therefore we conclude that the most likely origin of these features is the presence of XAFS.", "The overall reduced $\\chi ^2$ ($\\chi ^2_{red}$ ) of the model adopting local Gaussians is 1.074 (7793 degrees of freedom, [d.o.f.", "]), while the model with the warmabs gave 1.102 (7818 d.o.f.).", "Adding two Gaussians with energies and widths corresponding to the XAFS values of Table REF gave a satisfactory account of the residuals and a $\\chi ^2_{red}$ of 1.080, for 7812 d.o.f.", "Although the statistical difference in the two fits is significant, the matching between the position, widths, and broadening of the highly ionized absorption lines in the two approaches is remarkable, and we conclude that the key physical characteristics of the absorption features are satisfactorily accounted for by a uniformly photoionized wind.", "Figure: Data, best-fit model, and residuals for detected absorptionlocal features of the Chandra non-dip average spectrum(HEG+MEG data combined for clarity), using a continuum model of anabsorbed accretion disk and thermal Comptonized component.", "Leftpanels: lines fitted with Gaussian profiles; right panels: linesfitted with the XSTAR warm-absorber model." ], [ "Broad-band continuum model", "To fit the broad-band continuum, we modeled the interstellar absorption with the tbvarabs spectral component, with the abundance table of [1] and the cross section of [66].", "We forced the Mg, S and Fe abundance to assume the same value as that of the Si abundance and left this parameter free to vary with respect to all the other elements (which effectively implies a different abundance ratio of the high-Z elements with respect to the low-Z ones), according to the possible overabundance in heavy-Z metals reported in the study of [62].", "In addition to the resonant absorption lines, the warm absorber is also responsible for the continuum optically thin Compton scattering.", "To account for this effect, we multiplied the model by an optically thin Compton scattering component (cabs in Xspec), tying the cabs electron density to 1.3 times the value of the warmbs column density, because we assumed a metallicity ($Y+Z$ =0.3) for the warm absorbing medium.", "We used a multiplication constant between the $Chandra$ and RXTE/PCA models to take into account possible calibration differences in the observed flux.", "To determine the continuum emission we used the $Chandra$ /HEG and RXTE/PCA spectra; we tried two different spectral broad-band models, but we limited our comparison to the pre-dip spectrum.", "The first model follows the so-called extended coronal model, which has been extensively adopted for dipping sources and more recently also for bright Z-sources [4], [10].", "This modelconstantsimple_gpile2tbvarabscabs warmabs(cutoffpl + blackbody + gaussian) in Xspec language is composed of a cut-off power-law to take into account emission from the harder coronal environment, a thermal black-body component to take into account direct emission from the NS, and a broad iron emission line, with an upper limit to the width at 0.5 keV, to prevent unphysical continuum distortions, and a line energy constrained to take values between 6.4 keV and 7 keV.", "This model gave a rather flat value for the photon-index of the cut-off power-law, however, with a best-fitting value of 1.1$\\pm $ 0.3 and a cut-off energy of 4.4$\\pm $ 0.4 keV, while the radius from the thermal black-body component is $\\sim $ 4 km, and might be compatible with a narrow equatorial strip of the NS surface at a temperature of 1.00$\\pm $ 0.05 keV.", "The reduced $\\chi ^2$ for this fit was 1.06 (2416 d.o.f) and no other significant residual pattern was evident.", "If the index of the power-law is associated to a Comptonized spectrum, the low index value becomes difficult to interpret, because the cut-off energy is $\\sim $ 4.4 keV.", "For a spectral index $\\alpha $ $\\sim $ 0, there is no physical solution that can be converted in terms of physical parameters [51].", "We then adopted a model according to the so-called Eastern decomposition, composed of a multitemperature accretion-disk model with a zero-torque boundary condition [69] to model the softer energies and a Comptonization model to fit the high-energy band [59].", "In the Comptonization component we assume spherical geometry.", "The spectral parameters that determine the shape of the Comptonized spectrum are the soft seed-photon temperature ($kT_s$ ), the temperature of the electron cloud ($kT_e$ ), and the optical depth ($\\tau $ ).", "For this model constantsimple_gpile2* tbvarabscabswarmabs*(ezdiskbb+comptt+gaussian) in Xspec, we found a strong correlation between the disk temperature, and the soft seed-photon temperature.", "This is expected since for X-ray sources at high inclination angle the observed disk emission, which scales as the cosine of the angle, becomes weaker with respect to the more isotropic boundary layer emission.", "However, for similar accretion rates, in low-inclination and less absorbed LMXB, where disk emission is clearly separated, the two temperatures are usually found to have very similar values, with the disk temperature typically 20%-30% lower than kT$_s$ [16], because the Comptonization is thought to act mainly on the hotter photons produced at the surface of the NS.", "Because of the overlapping range of values for the two parameters, we decided to force them to assume the same value to make the fit more stable.", "For this model we derived an inner radius for the accretion disk compatible with a truncation radius at the boundary layer ($\\sim $ 15 km), which for high-accretion rates can extend to a distance similar to the radius of the NS [12].", "The highest disk temperature (set equal to the photon-seed temperature) value is $\\sim $ 1 keV, while the high optical depth ($\\tau $ $\\sim $ 8.5) and low electron temperature ($\\sim $ 3 keV) are consistent with the general softness of the spectrum.", "A moderately broad iron line in emission is required.", "The line equivalent width is $\\sim $ 50 eV, significantly lower than the values reported in XMM-Newton observations [19].", "We note, however, that the line detection is mostly driven by residuals of the RXTE/PCA, while using $Chandra$ /HEG data alone the detection would be less constraining.", "We show the broad-band $RXTE$ /PCA unfolded spectra together with the contributions from the additive components and residuals in Fig.", "REF .", "Figure: RXTERXTE/PCA pre-dip (3.5–25 keV range) unfolded spectrum, with contributions from spectralcomponents and residuals, adopting the Eastern decomposition: green, red, and blue lines show contribution fromaccretion disk, Comptonized component, and Gaussian line, respectively.Because the best-fitting parameters of the model and the general spectral decomposition agree with the results of other LMXBs that accrete at similar rates, we hold this continuum model to be adequately solid and adopted it for the other time-selected intervals and to obtain the best-fitting averaged continuum emission of the whole out-of-dip $Chandra$ observation.", "In Table REF , we report the best-fitting values and associated errors for the two time-selected pre-dip and post-dip spectra and for the average $Chandra$ observation.", "The most significant changes from pre-dip to post-dip parameters are on the fluxes of the two components (a change of $\\sim $ 20%, while the statistical error on a single measure is $\\lesssim $ 3%), and a change in the optical depth of the Comptonized component, while the other spectral parameters are found consistent within the statistical uncertainties.", "The average out-of-dip $Chandra$ observation presents a higher value for the N$_{\\textrm {H}}$ parameter than that of the RXTE+HEG fit, but the error on the parameter is underestimated because we kept the Mg/Si/S/Fe abundance frozen to the reference value of 1.3, as the two parameters were found to be strongly correlated.", "Table: Fitting results for the time-selected out-of-dip spectra and for theChandraChandra average spectrum." ], [ "Spectral changes during dipping", "To study the spectral changes during the dip, we used only Chandra HEG and MEG data, as there was no strictly contemporaneous RXTE observation of source during the dip.", "Because of the short duration of the dip and the low effective area of the $Chandra$ /HETGS, no clear absorption feature was detected, so that to better constrain spectral variations we coarsely re-binned each grating arm data up to 100 counts per channel.", "We assumed that during the dip the continuum spectral shape is consistent with the out-of-dip $Chandra$ average spectrum, because the dip falls approximately at the center of the observation.", "We therefore kept all the spectral parameters frozen to the best-fitting average values of the spectrum analyzed in Sect.", "REF with the warmabs component (i.e.", "$Chandra$ average column of Table REF ).", "We first tried to model the time-selected dip spectra allowing only for a variation of the neutral absorption column, keeping the parameters of the warmabs component frozen to the average values.", "A multiplicative constant before the model was used to evaluate possible flux variations between this time-selected spectrum and the averaged one.", "The residual pattern can be satisfactorily flattened, and we noted no other significant residual.", "Intrinsic flux variations with respect to the averaged spectrum are only of a few percent, as the multiplicative constant is very close to unity in all examined spectra (see Table REF ).", "We also tested the possible presence of a neutral partial covering effect.", "The covering fraction was always compatible with total coverage, and only lower limits could be assessed, without significant improvements in the $\\chi ^2$ of the fits.", "The ingress and egress spectra also show that the variation on the cold column density is possibly smooth, with values slightly higher than the corresponding values reported in Table REF for the pre-dip and post-dip persistent spectra.", "This is consistent with the hypothesis that the smooth dip profile is caused at first by an increase and then a decrease of this parameter alone.", "In a second round of fits, we tested whether the spectral change is compatible with a variation of the column density and the ionization state of the warmabs component alone (keeping the neutral N$_{\\textrm {H}}$ fixed to the average value).", "If the multiplicative constant is kept fixed to unity, no acceptable fit is obtained ($\\chi _{red}$ $>>$ 2), whereas when we allowed a variable constant, we noted a statistically acceptable fit ($\\chi ^2_{red}$ 1.08) for the dip spectrum with a strong decrease in both the column densities (N$_{H, wa}$ $\\sim $ 8 $\\times $ 10$^{22}$ cm$^{-2}$ ) and in the ionization state (log($\\xi $ ) $\\sim $ -0.3), in contrast with a general increase in warm column densities during dips in all other dipping sources [17].", "However, the model constant correspondingly decreases by $\\sim $ 20% in correlation with the decrease of the cabs value, which is tied to the warmabs column density, so that we conclude that this fit artificially reproduces the former model, without being physically consistent.", "Finally, we evaluated whether a change in the ionization parameter and in the absorption column of the warm absorber could be assessed at the same time, but we found marginal or no fit improvements compared with the best-fitting model shown in Table REF .", "For the dip spectrum, we found that the column density of the warm absorber is the same as in the averaged spectrum, while for the ionization parameter we found a lower limit compatible with the out-of-dip value (log($\\xi $ ) = 3.6), as shown by the contour plots of the two warmabs components in the right panel of Fig.", "REF .", "Similar results were also obtained for the ingress and egress spectra.", "Table: Best-fitting spectral parameters for ingress, egress, and dip time-selectedspectra.In summary, our results indicate that the main driver of the dip event is a rise in the column density of a neutral (or very mildly ionized) component, without requiring any change of the warmabs properties with respect to the out-of-dip interval.", "Keeping the neutral column density fixed at the out-of-dip averaged value, no physical or statistically acceptable solution is found in terms of variation of the warm absorber properties.", "Figure: Left panel: data, unfolded best-fitting spectral model, and residuals for the time-selecteddip spectrum (HEG+MEG data combined).", "Right panel: contour plots for the logarithm of warmabs columndensity (in units of 10 22 ^{22} cm -2 ^{-2}) and log of ionization parameter.", "Red, green, and bluecontours indicate confidence levels at 68%, 90%, and 99%, respectively." ], [ "Continuum formation and warm absorption", "We investigated the 1.0–25.0 keV spectrum of GX 13+1 by exploiting the high-resolution Chandra/HETGS spectrum, which has allowed tight control of the physical characteristics of the warm absorber, and RXTE/PCA spectrum, that has allowed to constrain the broad-band continuum.", "Comparing different spectral models, we found that the parameters of the warm absorber, as expected, are not strongly influenced by the choice of the underlying continuum.", "The continuum spectrum is extremely soft, closely resembling typical Z-sources spectra.", "The most physically plausible model is consistent with a hot accretion disk, whose apparent inner radius is compatible with the extension of the boundary layer for such high accretion rates [50], and an optically thick thermal Comptonization component.", "The total luminosity of the source is close to 50% of the Eddington limit for a canonical 1.4 M$_{\\odot }$ NS, and it is similar to past observations, pointing to a certain long-term accretion stability.", "The short-term smooth spectral variability that we observed in the $\\sim $ 30 ks Chandra observation mostly reflects a change in the flux of the components, while the general spectral shape remains unvaried.", "Moreover, energetically the overall emission is probably strongly dominated by coronal power, and it is unclear how the accretion power can be efficiently transferred at very large radii, while the possible power extractable from the accretion disk may be only a small fraction of the energy carried by the disk [64].", "The continuum is absorbed by a warm optically thick plasma.", "Its characteristics are similar to what was shown in past Chandra observations, and we found no strong variability in the present observation, because the best-fitting parameters between the pre-dip and post-dip spectra are all consistent within the associated uncertainties.", "Inspecting the landscape of the absorption features more closely, we found no evidence of the H-like transition of Mn, Cr, and Ar reported in [63], but we significantly detected the Ly$\\alpha $ transition of Alxiii and Ly$\\beta $ transitions from Mgxii and Fexxvi, together with two absorption lines close to the Si K-$\\alpha $ edge related to interstellar absorption [62].", "The self-consistent model warmabs yielded a satisfactory representation of all the local features.", "The average escape velocity of the photoionized medium is $\\sim $ 420$\\pm $ 60 km s$^{-1}$ ; a similar value was found in the observation of [63], while the lines appear broadened by the same order.", "This is expected if the wind is thermally driven and pushed by pressure gradients.", "In this case, the thermal motion of the ions would be similar to the sound speed $c_s$ , that would be similar to the escape velocity.", "Roughly speaking, the radius at which the wind is launched should be larger than the radius for which the escape velocity is the value we measured, so that R$_{wind}$ $>$ 2$\\times $ 10$^{11}$ cm.", "A similar value is obtained by a first-order calculation, taking the ionization parameter derived by the fit ($\\xi $ $\\sim $ 4000), a density of 10$^{12}$ cm$^{-3}$ , and observed luminosity (L$_{x}$ $\\simeq $ 10$^{38}$ erg s$^{-1}$ for a distance of 7 kpc), holding a distance $r=\\sqrt{L_x/(n \\cdot \\xi )}$ $\\sim $ 1.4 $\\times $ 10$^{11}$ cm." ], [ "Nature of the dip in the light curve", "A comprehensive study on the dipping activity of the source during a time span of more than 14 years was presented in [29], where evidence was given for a long-term modulation of dip passages at an inferred period of 24.5274(2) d. The phase of the $Chandra$ dip corresponds to the zero-phase of the periodic modulation, and therefore this is the first spectroscopic study of the periodic dip in GX 13+1.", "The lack of past dip evidence in GX 13+1 may be attributed to the very long orbital period of the source, which exceeds the rest of the dipping class by almost two orders of magnitude, and the possible low duty-cycle of the dips.", "In most dipping sources the dip duty-cycle may be as large as half of the entire orbital cycle [2], while, for longer-period dippers (P$_{orb}$ $>$ 1 day) the duty cycle drops to a few percentage as in the case of GRO J1655-40, and 4U 1639-47, or Cir X-1, which implyies that structure that causes the dip does not simply scale with the system dimensions.", "If the structure is identified with a bulge, formed at the impact point between the accretion stream and the outer disk radius, then its physical dimensions are probably determined by local conditions.", "In this case, the duty cycle would inversely scale with the accretion-disk radius.", "Indeed, searching all the RXTE PCA pointed observations, [29] found no clear dip episode, the dip spotted in the Chandra observation has a duration (FWHM) of only $\\sim $ 450 s, with very regular and smooth ingress and egress times.", "At the dip bottom the observed 1–10 keV flux is $\\sim $ 2/3 of the pre-dip flux, and the covering fraction is compatible with being total.", "No substructures are present within the dip, which points to a homogeneous absorber.", "If we consider the orbital period of GX 13+1 ($\\sim $ 24.5 d) and the possible mass range for the companion star (between 1 and 5 M$_{}$ ), we can estimate the Roche lobe radius of the primary according to the Eggleton formula [20]: $R_{RL} = a \\frac{0.49 q^{2/3}}{0.6 q^{2/3} + ln(1 + q^{1/3})},$ where $q=M_{NS}/M_{comp}$ is the mass ratio, $a$ is orbital semi-major axis, and the mass of the NS is assumed to be 1.4 M$_{sun}$ .", "We obtained Roche-lobe radii in the (1.2–3.3)$\\times $ 10$^{12}$ cm range.", "If the truncation radius ($R_{\\textrm {tr}}$ ) for the accretion disk is taken to be $\\sim $ 80% of the Roche-lobe radius, we estimate $R_{\\textrm {tr}}$ = (0.96–2.64)$\\times $ 10$^{12}$ cm.", "From the same calculations, we derived that the angle subtended by the companion star's radius is $\\sim $ 20, assuming it completely fills its Roche lobe.", "Because eclipses are absent, the inclination angle of the system must be $\\lesssim $ 70.", "If matter causing the dip is assumed for simplicity to have a spherical shape of diameter $D_{blob}$ and fixed in position, we can estimate its diameter from the duration of the dip (we conservatively assumed as dip duration the entire dip episode, shoulders and deep dip, of 1400 s) and the orbital period $D_{blob} = \\frac{\\Delta T_{dip}}{P_{orb}} \\times R_{\\textrm {tr}},$ which gives $D_{blob}$ = (6.3–17.4) $\\times $ 10$^{8}$ cm.", "If the accretion disk height-to-radius ratio were close to 0.1, as in the standard $\\alpha $ -disk case, we immediately note that the disk height is about three orders of magnitude greater than the blob radius (as calculated from a variety of different possible prescriptions), and this low ratio makes it unlikely that the blob is associated to a thickening of the disk or some unstable local structure because in this case we would expect strong variability and multiple dip substructures which are not observed.", "Alternatively, we might assume that only a tiny fraction of the top of the bulge causes the shallow dip, and thus the peculiarity of this source might be that it is an almost limiting case among the high-inclination systems, which is also supported by the general low upper value on the inclination from the eclipse absence.", "We have shown that during the dip the observed hardening is mostly due to a rising of a neutral or very mildly ionized absorption column.", "A change of only the warm absorber characteristics, with an increase in the column density and a decrease of the ionization state, does not lead to acceptable fit results.", "If we consider that the column density due to the ISM, as derived from the out-of-dip spectrum, is $\\sim $ 4.2$\\times $ 10$^{22}$ cm$^{-2}$ , at the dip bottom we find an intervening local column density $\\Delta N_{\\textrm {H}}$ of 4.5$\\times $ 10$^{22}$ atoms cm$^{-2}$ .", "If the occulting region is placed at the minimum outer disk radius (10$^{12}$ cm) and the density $n_{cold}$ is on the order of $\\Delta N_{\\textrm {H}}/D_{blob}$ $\\sim $ 3.7 $\\times $ 10$^{13}$ cm$^{-3}$ , calculated using the blob diameter, we derive a maximum ionization parameter $\\xi $ = $L_x/(n_{cold}r_{min}^2)$ $\\sim $ 3.", "Thus we conclude that the general picture of an occulting, almost neutral blob, set at a distance equal to the outer disk radius is self-consistent with a plasma in a low ionization state." ], [ "GX 13+1 in context", "The nature of the dipping phenomenon is still not firmly established, neither is the physical connection with the geometry of the system.", "GX 13+1 is the most luminous accreting dipping system, and its broad-band spectrum closely resembles the typical softness of the so-called Z-sources and bright atoll group [14], [15], with a hot ($\\sim $ 1 keV) disk spectrum and an optically thick Comptonized component with a low ($\\sim $ 3 keV) electron temperature.", "In addition to the high luminosity, GX 13+1 also has the longest orbital period and hence the largest outer-disk radius.", "We have shown that if the dipping is caused by a stable structure at the disk's outer rim, even this intense flux is not able to strongly ionize it.", "X 1624-490 is the second dipping source for brightness and longest orbital period ($L_x$ $\\sim $ 0.25 L$_{Edd}$ , P$_{orb}$ 21 hr), which also shows a contribution from neutral absorption during dipping [17].", "In this source an equal and large (a factor of 6) increase both in the column density of the ionized and neutral matter has been observed.", "The short duration of the dip and the small $Chandra$ effective area compared with $XMM-Newton$ does not allow us to constrain the variability of the warm absorber properties well, which seems to be modest compared with the significant change of the column density of the cold absorber, however.", "For GX 13+1 the ubiquitous presence of the warm absorber and the possible formation region at a distance of $\\sim $ 10$^{11}$ cm suggest that cold absorption (possibly located at a distance $>$ 10$^{12}$ cm) and warm absorption (disk-wind) are not physically connected.", "New observations with higher statistics will eventually provide more constraints on this matter." ], [ "Conclusions", "We have reported the first spectroscopic time-resolved investigation of the periodic dip of GX 13+1.", "The broad-band spectrum derived by a combined fit of Chandra/HEG and RXTE/PCA allowed us to consistently determine the continuum and discrete emission features of the source.", "Chandra data confirm an out-flowing optically thick warm absorber.", "Because of the short duration of the dip, we were unable to firmly constrain possible changes in the properties of the absorbing wind.", "The observed spectral hardening during the dip is mostly due to an increase in the column density of a neutral absorber, while the warm-absorber component is not modified with respect to the out-of-dip spectrum.", "Simple estimates on the dimensions of the structure that cause the dip indicate a very small occulting region when compared with the expected scale-heights at the outer radius, while simple geometric considerations on the system point to a possible inclination of $\\lesssim $ 70.", "The authors thank the anonymous referee for the helpful comments and suggestions.", "A. D. thanks M. Hanke for useful discussions about the $simple\\_gpile(2)$ model and the use of ISIS.", "Authors acknowledge support from INAF/PRIN 2012-06.", "A. D. , T. D. , R. I. N. R. acknowledge support from Fondo Finalizzato alla Ricerca 2012/13 from the University of Palermo.", "A. R. gratefully acknowledges the Sardinia Regional Government for the financial support (P. O. R. Sardegna F.S.E.", "Operational Programme of the Autonomous Region of Sardinia, European Social Fund 2007-2013 - Axis IV Human Resources, Objective l.3, Line of Activity l.3.1).", "Work in Cagliari was partially funded by the Regione Autonoma della Sardegna through POR-FSE Sardegna 2007-2013, L.R.", "7/2007, Progetti di ricerca di base e orientata, Project N CRP-60529.", "This research has made use of a collection of ISIS scripts provided by the Dr. Karl Remeis observatory, Bamberg, Germany at http://www.sternwarte.uni-erlangen.de/isis/." ] ]	1403.0071
[ [ "Vertex Operators, $\\mathbb{C}^3$ Curve and Topological Vertex" ], [ "Abstract In this article, we prove the conjecture that Kodaira-Spencer theory for the topological vertex is a free fermion theory.", "By dividing the $\\mathbb{C}^3$ curve into core and asymptotic regions and using Boson-Fermion correspondence, we construct a generic three-leg correlation function which reformulates the topological vertex in a vertex operator approach.", "We propose a conjecture of the correlation function identity which in a degenerate case becomes Zhou\\rq{}s identity for a Hopf link." ], [ "Introduction", "It has been proposed that the Chern-Simons theory of a gauge group $U(N)$ in the large $N$ limit is dual to A-model topological string theory [1].", "[2] provided a brane configuration of the knot of Chern-Simons theory.", "[3] discovered a quantum structure of Chern-Simons theory and from the well-known Wess-Zumino-Witten (WZW) model it even discovered the deeper relation between knot invariants in Chern-Simons theory with $SU(2)$ gauge group and characters of WZW model.", "Later the gauge group has been generalized to $U(N)$ [4].", "In the large $N$ limit some knot invariants such as the loop and the Hopf link are shown to be directly related to symmetric functions such as Schur and skew Schur functions.", "In [5] the authors discovered some interesting vertex structure for some geometrical and physical invariants such as the Donaldson-Thomas invariants of ${\\mathbb {C}}^3$ or in physical language, BPS invariants of D0-D6 branes on ${\\mathbb {C}}^3$ .", "From statistical point of view it is the partition function of a crystal melting model.", "[6] extended this structure to a general case where there are asymptotic boundaries of those crystals and related the partition function to a topological vertex.", "They managed to build up this connection because of Zhou's identity [26].", "In [7], the authors achieved a B-model approach to the topological vertex based on the observation of the mirror curve of $\\mathbb {C}^3$ and the related symmetries.", "However, an explicit correspondence between A-model and B-model is still an open question.", "We try to find a more obvious relation between A- and B-model in this article.", "In our approach, the curve of $\\mathbb {C}^3$ is essential.", "In A-model description, $\\mathbb {C}^3$ could be seen as a cotangent bundle $T^S^3$ with a single $S^3$ as the base.", "However, as the CS/WZW correspondence [3] saying, topological invariants in A-model becomes correlation functions in B-model, where there is an modular $S$ transformation inserted between bra and ket states.", "This correspondence strongly implies the mirror curve of $\\mathbb {C}^3$ can not be expressed simply in one coordinate chart.", "We need at least two coordinate charts , one being related to another by $S$ transformation.", "Thus a complete B-model mirror curve of $\\mathbb {C}^3$ has an asymptotic region (near infinity where the bra state is inserted in) and a core region (near origin where the ket state is inserted in) with point 1 the fix point.", "This means the A-model theory is a union of CFTs in two regions with a defect inserted at point 1.", "Surprisingly, if we introduce an excitation at point 1, the Hamiltonian blows up the excitation and forms a distribution corresponding to the representation of the excitation.", "This is very much like the so-called projective representation of affine algebra as in [23].", "Also in [12], this structure had been introduced without proof.", "The CFT considered at hand is a Kodaira-Spencer theory as explained in [7], which by definition is a bosonic theory, equipped with a broken $\\mathcal {W}_{1+\\infty }$ symmetry.", "A conjecture also was proposed in [7] that the corresponding fermionic theory is a free fermion theory.", "We prove in this article that for the topological vertex case, where the unbroken $W$ symmetry is $W^3_0$ , this conjecture is true.", "For other cases, for example, $W^4$ , if one would like to keep the integrable structure, the corresponding fermionic theory is still a free one.", "It is compatible with Dijkgraaf's work [17] two decades ago.", "Free fermion has been used in many research areas of physics.", "In [13] two-dimensional Yang-Mills theory of $U(N)$ gauge group the Vandermonde of group measure implies there is a fermionic structure.", "In two dimensions, due to Boson-Fermion correspondence, vertex operator is a very useful tool.", "In B-model [7] provided a beautiful explanation of a B-brane insertion as a fermion field and symmetries of the Riemann surface as the sources of transition function of sections of fiber bundles of fermionic fields.", "Because the duality between A-model and B-model, we would expect a similar structure in the A-model side.", "In this paper we aim to discover this structure and approach this topic via a vertex operator formalism.", "The structure of this paper is following.", "In sec.", "we clarify some notation to be used in this paper and make some preparation.", "In sec.", "we provide a generating function for the vertex operator.", "In sec.", "we obtain the fermionic expression for $W^n_0$ and prove the free fermion conjecture for $W^3_0$ .", "We also examine the curve of $\\mathbb {C}^3$ in different regions and related symplectic transformations.", "In sec.", "we solve Hamiltonian equations for two different coordinate charts and obtain wave functions.", "We construct the correlation function with three generic representations inserted at three points: 0,1,$\\infty $ in a single patch.", "The cyclic symmetry of the vertex becomes a conjecture of an identity for the correlation function.", "In the limitation situation, this correlation function identity becomes Zhou's identity of Hopf link.", "In sec.", "we point out the future working direction." ], [ "Notations and Preliminaries", "A partition $\\lambda $ is any sequence $\\lambda = (\\lambda _1\\,, \\lambda _2\\,, \\lambda _3\\,, \\cdots )$ of non-negative integers in weakly decreasing order: $\\lambda _1\\ge \\lambda _2\\ge \\lambda _3\\ge \\cdots .$ The diagram of a partition $\\lambda $ may be formally defined as the set of points $(i, j)\\in {\\mathbb {Z}}^2$ such that $1\\le j\\le \\lambda _i$ .", "More often it is convenient to replace the points by squares.", "It is also called a Young diagram.", "The conjugate of a partition $\\lambda $ is denoted by $\\lambda ^t$ whose diagram is obtained by reflection in the main diagonal of $\\lambda $ .", "A Schur function $s_{\\lambda }(z_i)$ is a symmetric function of $z_i\\,,i=1,2,\\cdots ,\\infty $ 's and labeled by partition $\\lambda $ .", "Especially, when $z_i = q^{\\rho _i} =q^{-i+\\frac{1}{2}}$ there is a very useful product formula for $s_{\\lambda }$ $s_{\\lambda }(q^{-\\rho }) = \\frac{q^{\|\|\\lambda ^t\|\|/2}}{\\prod _{(i,j)\\in \\lambda } 1-q^{h(i,j)}}\\,,$ where $\|\|\\lambda ^t\|\| = \\sum _i{(\\lambda ^t_i)^2}$ and $h(i,j)$ is the hook length of square $(i,j)$ .", "Sometimes it is more useful to represent the hook length as $h(i,j) = a(i,j)+l(i,j)+1$ where $a(i,j)$ and $l(i,j)$ are arm-length and leg-length respectively and $a(i,j) = \\lambda _i - j \\,, \\quad l(i,j) = \\lambda ^t_j -i .$" ], [ "Zhou's Hopf link identity", "The Hopf link is defined by $W_{\\lambda \\mu }=W_{\\lambda }\\,s_{\\mu }(q^{\\lambda +\\rho }),$ where $W_{\\lambda }=(-1)^{\\lambda }q^{\\kappa _\\lambda /2}\\,s_{\\lambda }(q^{-\\rho })=s_\\lambda (q^\\rho ),$ $\\rho =-\\frac{1}{2},-\\frac{3}{2},-\\frac{5}{2},\\cdots $ , and $q=e^{g_s}$ .", "According to the duality between Chern-Simons and A-model topological string theory, $g_s$ is the string coupling constant.", "$\\kappa _{\\mu }$ is the \"energy\" of the representation $\\mu $ and $\\kappa _\\mu = \\sum _i{\\mu _i^2}-\\sum _j{\\mu ^t_j}^2=\\sum _{i\\le \\ell (\\mu )}\\left((\\mu +\\rho )_i^2-\\rho _i^2\\right) $ where $\\ell (\\mu )$ is the length of the partition $\\mu $ , that is $\\mu _1^t$ .", "Zhou's Hopf link identity can be written as $q^{\\kappa _{\\mu ^t}/2}s_{\\lambda }(q^{-\\rho })s_{\\mu ^t}(q^{-\\lambda ^t-\\rho })=\\sum _{\\eta }s_{\\lambda /\\eta }(q^{-\\rho })s_{\\mu /\\eta }(q^{-\\rho })\\,,$ where $s_{\\lambda /\\eta }(q^{-\\rho })$ is the skew Schur function, see [21] for a detailed definition.", "For a special case $\\lambda =\\phi $ it gives rise to an interesting identity $\\phi $ denotes the empty Young diagram: $s_{\\mu }(q^{-\\rho }) = q^{\\kappa _{\\mu ^t}/2}s_{\\mu ^t}(q^{-\\rho })\\,.$ With the help of the useful identity [5] and [27] obtained a different formula $s_{\\mu /\\eta }(q^{-\\nu -\\rho }) = (-)^{\|\\mu \|+\|\\eta \|}s_{\\mu ^t/\\eta ^t}(q^{\\nu +\\rho })\\,,$ which is not correct.", "In [27] there was only a minor typo in the last line in the derivation of eq.", "(28).", "In [5] there was no derivation.", "We provide an independent derivation in App.", ".", "We thank Professor Guo-ce Xin for pointing out the problem for us.", "[5], [27]: $s_{\\mu /\\eta }(q^{-\\nu -\\rho }) = (-)^{\|\\mu \|+\|\\eta \|}s_{\\mu ^t/\\eta ^t}(q^{\\nu ^t+\\rho })$ we obtain $W_{\\lambda \\mu } = (-)^{\|\\lambda \|+\|\\mu \|}q^{\\kappa _\\lambda /2+\\kappa _\\mu /2}\\sum _{\\eta }s_{\\lambda /\\eta }(q^{-\\rho })s_{\\mu /\\eta }(q^{-\\rho }).$ [26] provided a mathematical proof of this Hopf link identity.", "Since this identity has a lot of applications in both mathematics and physics, see, e.g.", "[6], [28], we expect to uncover its origin from a physical point of view.", "However, a concrete physical proof is still an open problem for us." ], [ "Vertex Operators and Generating Functions", "Let us first introduce some basic ingredients of conformal field theory (CFT) of holomorphic boson and also chiral fermion fields." ], [ "Bosonization and Fermionization", "For a fermion-antifermion system defined on a complex plane, we have the following chiral fermion fields (Neuve-Schwarz fermions): $\\psi (z) = \\sum _{r\\in \\mathbb {Z}+\\frac{1}{2}}\\psi _r z^{-r-1/2}\\,, \\\\\\nonumber \\psi ^(z) = \\sum _{s\\in \\mathbb {Z}+\\frac{1}{2}}\\psi ^_s z^{-s-1/2}.$ They have the following operator product expansions (OPEs): $\\psi (z)\\psi ^(z^{\\prime }) &=&\\frac{1}{z-z^{\\prime }}:\\psi (z)\\psi ^(z^{\\prime }):+\\cdots \\, \\\\\\nonumber \\psi ^(z)\\psi (z^{\\prime }) &=& \\frac{1}{z-z^{\\prime }}:\\psi ^(z)\\psi (z^{\\prime }): +\\cdots \\,\\,,$ and also the anti-commutation relations: $\\lbrace \\psi _n , \\psi ^_m\\rbrace = \\delta _{n+m,0}\\,, \\quad \\text{others} = 0\\,,\\quad \\quad (\\psi _n)^=\\psi ^_{-n}\\,.$ In another side, the holomorphic bosonic field $\\varphi (z)$ is given as following $\\varphi (z) &=& q_0+ p_0\\ln z+ \\sum _{n\\ne 0}\\frac{a_n}{-n}z^{-n}, \\\\\\nonumber [a_n, a_m] &=& n\\delta _{n+m,0}\\,\\,,\\, [p_0,q_0]=1\\\\\\varphi (z) \\varphi (w) &=& \\ln (z-w):\\varphi (z) \\varphi (w): \\,,$ If $\\bar{\\varphi }$ denotes the corresponding anti-holomorphic bosonic field then a free bosonic field $\\varphi (z,\\bar{z})$ can be constructed as: $\\varphi (z,\\bar{z}) = \\varphi (z)-\\bar{\\varphi }(\\bar{z})\\,.$ In this article, we only make the holomorphic part of the bosonic field dynamic and leave the anti-holomorphic part non-dynamic.", "While this boson is called a chiral boson and the corresponding vertex operators are called chiral vertex operators.", "A chiral vertex operator may be written as $V_\\alpha (z) = e^{\\alpha \\varphi (z)},$ its conjugation is $(V_{\\alpha }(z))^* = e^{-\\alpha \\varphi (z^)}= V_{-\\alpha }(z^).$ However, we just consider the case that $\\alpha =1$ and denote $V(z) = e^{\\varphi (z)}\\,\\,,\\,\\,V^(z) = e^{-\\varphi (z)}.$ From modes expansion and Heisenberg algebra (REF ), we can calculate the OPEs of $V(z)$ and $V^{}(z^{\\prime })$ : $:V(z)::V(z^{\\prime }): &=& :VV(z^{\\prime }{}):(z-z^{\\prime })+reg.\\\\\\nonumber :V^{}(z)::V^{}(z^{\\prime }):&=&:V^{}V^{}(z^{\\prime }{}):(z-z^{\\prime })+reg.\\\\:V^{}(z)::V(z^{\\prime }):&=&:V^{}V(z^{\\prime }):\\frac{1}{(z-z^{\\prime })}-\\partial {\\varphi (z^{\\prime })}+reg.\\\\\\nonumber :V(z)::V^{}(z^{\\prime }):&=&:V V^{}(z^{\\prime }):\\frac{1}{(z-z^{\\prime })}+\\partial {\\varphi (z^{\\prime })}+reg.,$ here $reg.$ means regular terms.", "The singular parts of these OPEs are the same as those of chiral fermions in eq.", "(REF ).", "However chiral fermions have no self-contractions they are not completely the same as $V$ and $V^$ .", "Nevertheless according to Pauli's exclusive principle, namely the fermionic statistics, the correlation function of fermions is related to Slater determinant $\\langle vac\|\\prod _{i=1}^N \\psi (z_i)\\prod _{j=1}^N\\psi ^(w_j) \|vac\\rangle & = &\\langle 0\|\\prod _{i=1}^N V(z_i) \\prod _{j=1}^NV^(w_j)\|0\\rangle \\nonumber \\\\={\\rm Det}(\\frac{1}{z_i - w_j})&=&\\frac{\\prod _{i<j}(z_i - z_j)(w_i-w_j)}{\\prod _{i,j =1}^N(z_i-w_j)} \\,.$ It reminds us the miraculous boson/fermion correspondence.", "Therefore during the calculation of the correlation function, we can replace all fermionic fields by bosonic vertex operators such that $\\psi (z)\\sim V(z) = e^{\\varphi (z)}\\,,\\quad \\psi ^(z)\\sim V^(z) = e^{-\\varphi (z)}\\,.$ However, we need to bear it in mind that chiral fermions are not exactly these vertex operators because of the different self OPEs.", "Secondly there could be different numbers of $V$ and $V^$ in the correlation function by carefully choosing the charges of bra and ket vacua.", "But if there are different number of $\\psi $ and $\\psi ^$ , the correlation function is automatically vanishing.", "In another way, since both fermionic and bosonic theories have the same $U(1)$ symmetry, the charge is measured by the number of $\\partial \\varphi (z)$ in the bosonic theory and $\\psi \\psi ^(z)$ in the fermionic theory.", "Hence we have the fermionization as follows: $\\partial \\varphi (z)=(\\psi \\psi ^)(z)\\,,$ in terms of the modes expansion that is $a_n =\\sum _{r\\in {\\mathbb {Z}}+1/2} :\\psi _{n-r}\\psi ^_r:\\,.$" ], [ "Generating Functions", "In the following we denote $V_+$ and $V_-$ as the positive and negative modes part of $e^\\varphi $ and $V_+^$ and $V_-^$ as the corresponding part of $e^{-\\varphi }$ , that is, $V_+(z)=\\exp \\left\\lbrace \\sum _{n>0}\\frac{a_n}{-n}z^{-n}\\right\\rbrace ,\\quad V_-(z)=\\exp \\left\\lbrace \\sum _{n>0}\\frac{a_{-n}}{n}z^{n}\\right\\rbrace ,\\\\\\nonumber V_+^{}(z)=\\exp \\left\\lbrace \\sum _{n>0}\\frac{a_n}{n}z^{-n}\\right\\rbrace ,\\quad V_-^(z)=\\exp \\left\\lbrace \\sum _{n>0}\\frac{a_{-n}}{-n}z^{n}\\right\\rbrace \\,.$ They form four types of generating functions of Schur functions, namely $\\prod _i V_-(z_i)&=&\\sum _\\lambda s_\\lambda (z_i)s_\\lambda (a_-),\\\\\\prod _i V_+^(z_i)&=&\\sum _\\lambda s_\\lambda (z_i^{-1})s_\\lambda (a_+),\\\\\\prod _i V^_-(z_i)&=&\\sum _\\lambda (-1)^{\|\\lambda \|}s_\\lambda (z_i)s_{\\lambda ^t}(a_-),\\\\\\prod _i V_+(z_i)&=&\\sum _\\lambda (-1)^{\|\\lambda \|}s_\\lambda (z_i^{-1})s_{\\lambda ^t}(a_+).$ We will use these generating functions frequently in our calculation.", "They also can be deduced from four basic generating functions such that: $V_-(z) &=& \\sum _{r\\in \\mathbb {Z}^+} s_{(r)}(z)s_{(r)}(a_-)\\\\\\nonumber V_-^(z)&=&\\sum _{r\\in \\mathbb {Z}^+}(-)^r s_{(r)}(z)s_{(1^r)}(a_-)\\\\\\nonumber V_+(z) &=& \\sum _{r\\in \\mathbb {Z}^+}(-)^r s_{(r)}(1/z)s_{(1^r)}(a_+)\\\\\\nonumber V_-^(z)&=&\\sum _{r\\in \\mathbb {Z}^+} s_{(r)}(1/z)s_{(r)}(a_+)\\,,$ where $(r)$ ($(1^r)$ ) denotes a length-$r$ horizontal (vertical) Young diagram.", "Actually, the Schur polynomials of the horizontal and vertical Young diagrams are the same as the complete (homogeneous) and elementary symmetric polynomials respectively, that is, $s_{(r)}(z) = h_r(z),\\quad s_{(1^r)} =e_r(z)\\,.$" ], [ "Fermionic Vacua and Maya/Young Correspondence", "The vacuum of free fermion theory corresponds to a filled Dirac sea.", "Firstly, for the ket vacuum, we denote $\|\\Omega \\rangle $ as the 'fake' vacuum of the theory, which is annihilated by all modes of $\\psi $ but not $\\psi ^$ .", "Since $\\psi ^$ is the anti-particle field of $\\psi $ , the positive modes of $\\psi ^$ should be understood as creation operators of anti-particles.", "The 'real' (physical) ket vacuum should have a natural Dirac sea structure and is denoted by $\|vac\\rangle $ and defined as $\|vac\\rangle = \\psi _{1/2}^\\psi _{3/2}^\\psi _{5/2}^\\cdots \|\\Omega \\rangle \\,.$ The bra vacuum could be defined by conjugation of ket vacuum $\\langle vac\| = \\langle \\Omega \|\\cdots \\psi _{-5/2}\\psi _{-3/2}\\psi _{-1/2}\\,.$ A unitary excitation on the ket vacuum $\|vac\\rangle $ always contains pairs of particle-anti-particle.", "To track the sign of the excited state, we would better define an excited state as follows $(-)^{\\sum _i^n s_i-1/2}\\prod _{i=1}^n \\psi _{-r_i}\\psi ^_{-s_i}\|vac\\rangle \\,,$ where the subscripts $r_i, s_i$ are positive half integers ($\\mathbb {Z}_{>0} -1/2$ ).", "Since the choice of particle or anti-particle is arbitrary we could choose the 'fake' vacuum $\|\\Omega ^{\\prime }\\rangle $ as the one annihilated by all modes of $\\psi ^$ .", "Therefore the definition of bra and ket vacua should be changed as $&&\|vac^{\\prime }\\rangle = \\psi _{1/2}\\psi _{3/2}\\psi _{5/2}\\cdots \|\\Omega ^{\\prime }\\rangle \\\\&&\\langle vac^{\\prime }\| = \\langle \\Omega ^{\\prime } \|\\cdots \\psi ^_{-5/2}\\psi ^_{-3/2}\\psi ^_{-1/2}\\,.$ Similarly we can define corresponding excited states.", "There is a transformation switching these two choices of vacua which is called an involution and denoted by $\\omega $ .", "It acts on the states and operators as follows $\\omega : \|\\Omega \\rangle \\rightarrow \|\\Omega ^{\\prime }\\rangle \\,, \\quad \\psi ^_n\\rightarrow (-)^n\\psi _n \\,, \\quad \\psi _n\\rightarrow (-)^n\\psi ^_n\\,.$ For states defined by eq.", "(REF ), we can use Maya diagram to demonstrate the excitations.", "For example, the Maya diagram corresponding to the vacuum is shown in fig.", "REF a.", "An excited state can be obtained by exchanging certain white and black dots of the vacuum Maya diagram.", "For example Fig.", "REF b. shows an excited state denoted by $(r_1=3/2, r_2=1/2, s_1=5/2, s_2=1/2)$ .", "There is an amazing correspondence between Maya diagrams and Young diagrams.", "For each white/black dot, we assign a unit leftward/downward line segment connected end to end.", "Hence a Maya diagram corresponds to a unique Young diagram in this setup.", "Thus the subscripts $r_i, s_i$ are Frobenius coordinates defining the Young diagram.", "Fig.", "REF c. shows the corresponding Young diagram of Maya diagram in Fig.", "REF b. which is $\\lbrace 2,2,1\\rbrace $ .", "As shown in [14], the excitation states defined in this way are Schur states in fermionic operator representation.", "Figure: a.", "The Maya diagram for the vacuum, b. the Maya diagram for the excited state (r 1 =3/2,r 2 =1/2,s 1 =5/2,s 2 =1/2)(r_1=3/2, r_2=1/2, s_1=5/2, s_2=1/2), c. the corresponding Young diagram of b." ], [ "Curve driving Patch-Shifting", "Previously, we have considered the definition of bosonic and fermionic theory on the complex plane.", "However, we are more interested in theory on special Riemann surface with punctures and also defects [15].", "By special we mean the Riemann surface is obtained by gluing various regions, with defects inserted at fix points.", "To obtain Riemann Surface with more complicated topology, we need to define theories on tubes and pants.", "A conformal field theory defined on a tube is a boundary CFT (BCFT) [11] with two boundary conditions.", "While on a pants, the corresponding CFT is a BCFT with three boundary conditions.", "However, the problem at hand differs from a BCFT problem because the theory is not defined on a simple Riemann surface but with defect inserted.", "Precisely, we have a core region in the toric structure of $\\mathbb {C}^3$ and three asymptotic regions which are local patches and can be transited among themselves.", "A defect is inserted at point $z=1$The defect could be anywhere on the complex plane, however, by global conformal transformation, it can be fixed at point 1 without loss of generality..", "Therefore a BCFT analysis may fail in this case.", "As shown in [7], the Riemann surface corresponding to toric Calabi-Yau could be defined patch by patch and there are some symplectic transformations of coordinates of local patches.", "A simpler case for two-punctured theory can be obtained by joining two patches together.", "That is to say, if we define a theory on a local patch associated with one of these two punctures, then another theory on another local patch, these two theories can be related to each other by patch-shifting transformation, which is a symplectic transformation.", "The cut and join operation should preserve the symplectic transformation from one patch to another.", "Symplectic transformation is an area-preserving operation which is compatible with the measure on Riemann surface." ], [ "The $\\mathcal {W}_{1+\\infty }$ Algebra", "Now for an infinite cylinder, the symplectic form is $\\Omega = d x\\wedge d p\\,, $ hence the symplectic transformation of $x$ is $x\\rightarrow x+\\epsilon (x) = x+f(p) = x +\\sum _{n=0}^{\\infty } f_n p^n\\,.$ The transformation of a quantum chiral scalar field associated with the change of the local coordinate $\\delta x = \\epsilon (x)$ is implemented by the operator $\\oint T(x)\\epsilon (x)\\frac{d x}{2\\pi i},$ where the stress tensor for chiral boson theory is $T(x) = \\frac{1}{2} [\\partial \\varphi \\partial \\varphi ](x)\\,.$ The observables of the chiral bosonic theory correspond to variations of the complex structure at infinity.", "On each patch this is described by the modes of a chiral boson $\\varphi (x)$ , defined by $\\partial \\varphi (x) =p(x)\\,.$ Now the symplectic transformation has an operator expression, namely $\\oint T(x)\\epsilon (x)\\frac{dx}{2\\pi i} &=& \\lim _{x^{\\prime }{} \\rightarrow x}\\oint \\frac{1}{4\\pi i} [\\partial \\varphi \\partial \\varphi ](x)\\sum _{n\\ge 0}f_n[(\\partial \\varphi )^n](x^{\\prime }{}) d x \\\\\\nonumber &=&\\frac{1}{2}\\sum _{n\\ge 0}(n+2)!f_n W_0^{n+2}(x)+ terms \\,\\,involve\\,\\, (\\partial ^3\\varphi )\\,.$ $W_0^{n}$ is the zero mode of free (non-interacting) $\\mathcal {W}^n$ transformation which is defined by $W^n(z) &=& \\frac{1}{n!", "}(z\\partial _z \\varphi (z))^n\\\\W^n_m &=& \\oint \\frac{1}{ 2\\pi i z}\\frac{1}{n!}", "z^{-m+n}(\\partial _{z}\\varphi (z))^n\\\\\\nonumber &=& \\frac{1}{n!", "}\\sum _{k_i=-\\infty }^{\\infty }\\delta \\left((\\sum _{i=1}^n k_i)-m\\right)\\left(:a_{-k_1}a_{-k_2}\\cdots a_{-k_n}:\\right) \\,,$ up to some constant ground energy due to normal ordering.", "In the derivation, it is useful to apply the coordinate transformation from the cylindrical coordinates to the complex plane ones $z = e^{x}$ .", "Thus $z ^{-1}d z = d x, \\partial _x = z\\partial _z\\,, \\partial \\varphi (x) = z\\partial _z\\varphi (z) =\\sum _n a_n z^n, \\,.$ The appearance of a term containing $\\partial ^3\\varphi $ in eq.", "(REF ) reflects the non-associativity (and also non-commutation) of operator expansion product meaning $[A[BC]](z)\\ne [[AB]C](z)\\,.$ This is crucial in the derivation of operator formalism of a given integral formula.", "Next we will use $W^3_0$ and $W^4_0$ as two examples to explain it.", "Firstly, we have $&&[\\partial _x\\varphi \\partial _x\\varphi \\partial _x\\varphi ](x) = z^3[(\\partial _z\\varphi )^3](z) \\\\\\nonumber &=&\\frac{z^3}{2\\pi i} \\oint _z \\frac{1}{z^{\\prime }{}-z}\\partial _{z^{\\prime }{}}\\varphi (z^{\\prime }{})[\\partial _{z}\\varphi \\partial _z\\varphi ](z)d z^{\\prime }{}$ while $&&[(\\partial _x\\varphi )^3](x)+[\\partial _x^3\\varphi ](x)\\\\\\nonumber &=& \\frac{z^3}{2\\pi i}\\oint _z\\frac{1}{z^{\\prime }{}-z}[\\partial _{z^{\\prime }{}}\\varphi \\partial _{z^{\\prime }{}}\\varphi ](z^{\\prime }{})\\partial _z \\varphi (z)dz^{\\prime }{}\\,.$ Secondly, by using the modes expansion as we defined before, we have the bosonic operator formalism of (REF ) such that $W^3_0 &=&\\frac{1}{6} \\oint _x dx \\frac{1}{2\\pi i}[\\partial _x\\varphi \\partial _x\\varphi \\partial _x\\varphi ](x) \\\\\\nonumber &=& \\frac{1}{2}\\sum _{n,m>0}(a_{-n-m} a_m a_n + a_{-n}a_{-m}a_{n+m}) +\\frac{1}{2}\\sum _{n>0}a_0a_{-n}a_{n}+\\frac{1}{6}a_0^3\\,.$ In another way, the bosonic operator formalism of (REF ) gives rise to $\\widetilde{W}^3_0 &=&\\frac{1}{6}\\oint _x dx \\frac{1}{2\\pi i} \\left([(\\partial _x\\varphi )^3](x)+[\\partial _x^3\\varphi ](x)\\right) = W^3_0 + \\frac{1}{3}a_0\\,.$ Here $a_0=p_0$ is the momentum of the center of mass.", "Hence the last term is not an important correction for the spectrum of $W^3_0$ .", "But it is fascinating for us that $\\frac{1}{3}a_0$ also appears in the fermionic side which seems to provide further evidence for Boson-Fermion correspondence and we will proceed to that point later.", "The non-associative property leads to a severe problem, that is $[W^n,W^m]\\ne 0, \\,\\,for\\,\\,n\\ne m\\,.$ This is a displeasing result since we would expect a $\\mathcal {W}_{1+\\infty }$ symmetry to generate the integrability, which means there are infinite many conserved currents commuting with each other.", "Another problem is that it is difficult to construct the exact form of $W^n$ in terms of bosonic fields.", "However, the difference of $W^3_0$ and $\\widetilde{W}^3_0$ reveals a simple fact: the difference is just a total derivative!", "Actually, $\\widetilde{W}_0^3$ rather than $W^3_0$ is the one included in the $\\mathcal {W}_{1+\\infty }$ algebra.", "If we proceed to the fourth $W$ generator, there are three different choices.", "An arduous way to find the correct one is to apply the commutation on those choices with the lower order $W$ generators to check which one gives rise to zero simultaneously.", "However, it turns out to be simpler to approach this problem from the fermionic picture and we will elaborate it next.", "Up to a total derivative, we see that $W^3$ is the same as $\\widetilde{W}^3$ .", "Moreover we can use a $\\partial $ -cohomology definition of $\\mathcal {W}_{1+\\infty }$ algebra which means the algebra is closed upon modulo all total derivatives.", "Then we get a good definition of $W^n$ algebra in terms of the bosonic formalism $W^n(z) = \\frac{1}{n!", "}(\\partial \\varphi )^n \\,(mod\\,\\,\\partial )\\,.$ This observation was known long ago since Dijkgraaf's paper [17].", "However, the derivation is not exactly the same.", "Actually it is quite astonishing for us.", "Since we are only considering how free chiral boson CFT goes from one patch to another patch keeping some symplectic symmetries, then we obtain the chiral boson theory which turns out to be a Kodaira-Spencer-like one as the previous work [17] by Dijkgraaf." ], [ "Fermionic Representation ", "We want to check the non-associative property in a fermionic picture.", "Firstly we consider the $W^3_0$ case.", "Since there is no significant difference between $W^3_0$ and $\\widetilde{W}^3_0$ it is sufficient to examine $W^3_0$ .", "From fermionization $\\partial _z\\varphi (z) = \\psi \\psi ^(z)\\,, $ we can write down the fermionic formalism of $W^3_0$ as $6 W^3(w) &=& \\oint _z \\frac{dz}{2\\pi i (z-w)}[\\psi \\psi ^](z)[-\\psi \\partial _w\\psi ^-\\psi ^\\partial _w\\psi ](w)\\\\\\nonumber &=& -\\oint _z \\left(\\frac{\\psi (z)\\partial _w\\psi ^(w)}{(z-w)^2}+\\frac{\\psi ^(z)\\psi (w)}{(z-w)^3}\\right)\\frac{dz}{2\\pi i}\\\\\\nonumber &+&\\oint _z \\left(\\frac{\\psi ^(z)\\partial _w\\psi (w)}{(z-w)^2}+\\frac{\\psi (z)\\psi ^(w)}{(z-w)^3}\\right)\\frac{dz}{2\\pi i}\\\\\\nonumber &=& 2[\\partial \\psi ^\\partial \\psi ](w)+\\frac{1}{2}[\\partial ^2\\psi \\psi ^](w)-\\frac{1}{2}[\\partial ^2\\psi ^ \\psi ](w)\\,.$ To get the first equality, we used the well-known result that $[\\partial \\varphi \\partial \\varphi ](w) = -[\\psi \\partial \\psi ^+\\psi ^\\partial \\psi ](w)\\,.$ Actually, it can be treated as the first generalization of Boson-Fermion correspondence.", "The fermionic modes expansion leads to an operator formalism of $W_0^3$ , namely $W_0^3 = \\frac{1}{2}\\sum _{r\\in \\mathbb {Z}_{>0}-\\frac{1}{2}}\\left(r^2+\\frac{1}{12}\\right)(\\psi _{-r}\\psi ^_r-\\psi ^_{-r}\\psi _r)\\,.$ Similarly, $\\widetilde{W}_0^3$ has a fermionic expression $6\\widetilde{W}^3(w) = \\frac{3}{2}[\\partial ^2\\psi \\psi ^-\\partial ^2\\psi ^ \\psi ](w)\\,,$ and the operator formalism $\\widetilde{W}_0^3 = \\frac{1}{2}\\sum _{r\\in \\mathbb {Z}^+-1/2}\\left(r^2+\\frac{3}{4}\\right)(\\psi _{-r}\\psi ^_r-\\psi ^_{-r}\\psi _r) = W^3_0 +\\frac{1}{3}a_0\\,,$ where we substituted $a_0 =\\sum _r :\\psi _{-r}\\psi ^_r:$ .", "In the last equality of (REF ), the first term could be rewritten as $2\\partial \\psi ^\\partial \\psi =\\partial (\\psi ^\\partial \\psi -\\psi \\partial \\psi ^)-\\psi ^\\partial ^2\\psi +\\psi \\partial ^2 \\psi ^\\,.$ The operator formalism perfectly matches with previous bosonic result.", "A byproduct of this result is the generalization of Boson-Fermion correspondence to higher derivatives.", "For example we have the second generalization formula: $[\\partial ^2\\psi \\psi ^* - \\partial ^2\\psi ^\\psi ](z)&=&\\frac{2}{3}[(\\partial \\varphi )^3+\\partial ^3\\varphi ](z)\\,.$ Secondly the OPE method could be easily generalized to $W^4_0$ case while there are three ways of multiplication $[\\partial \\varphi ][(\\partial \\varphi )^3],\\,\\,[(\\partial \\varphi )^2][\\partial \\varphi )^2]\\,,\\,\\,[(\\partial \\varphi )^3][\\partial \\varphi ]\\,.$ Hence there are three kinds of fourth-level Boson-Fermion correspondence as follows: $[(\\partial \\varphi )^4]&=&\\left([\\frac{3}{2}\\partial \\psi \\partial ^2\\psi ^+\\frac{1}{6}\\partial ^3\\psi \\psi ^]+\\lbrace \\psi \\leftrightarrow \\psi ^\\rbrace \\right)\\\\ & & \\quad + [2\\partial \\psi ^\\partial \\psi \\psi \\psi ^]\\\\\\,[(\\partial \\varphi )^2][\\partial \\varphi )^2]&=&[2\\partial ^3\\varphi \\partial \\varphi +(\\partial \\varphi )^4]\\\\&=&\\left([\\frac{4}{3}\\partial ^3\\psi \\psi ^-\\partial ^2\\psi \\partial \\psi ^]+\\lbrace \\psi \\leftrightarrow \\psi ^\\rbrace \\right)\\\\ & & \\quad -[2\\partial \\psi ^\\partial \\psi \\psi \\psi ^]\\\\\\,[(\\partial \\varphi )^3][\\partial \\varphi ]&=& [3\\partial ^3\\varphi \\partial \\varphi + 3(\\partial ^2\\varphi )^2+ (\\partial \\varphi )^4]\\\\&=&\\frac{5}{3}\\left[\\partial ^3\\psi \\psi ^+ \\partial ^3\\psi ^\\psi \\right]\\\\ & &\\quad + [2\\partial \\psi ^\\partial \\psi \\psi \\psi ^]\\,.$ These results have not modulo total derivatives yet.", "It is quite difficult to write down the operator formalism.", "All of them contain four-fermion terms which in general will spoil the integrable structure.", "If we use the $\\partial $ -cohomology definition, these three are the same up to total derivatives and constant factors.", "In fermionic picture, the $W^4$We do not distinguish $W$ and $\\tilde{W}$ explicitly from now on.", "could be rewritten as $W^4(w) &=& \\frac{1}{4}\\oint \\frac{dz}{2\\pi i (z-w)}W^3(z)\\psi \\psi ^(w)\\\\\\nonumber &=&\\frac{1}{24}\\oint \\frac{dz}{2\\pi i (z-w)}\\frac{3}{2}[\\partial ^2\\psi \\psi ^-\\partial ^2\\psi ^\\psi ](z)[\\psi \\psi ^](w)\\\\\\nonumber &=& \\frac{1}{12}(\\partial ^3\\psi \\psi ^+\\partial ^3\\psi ^\\psi )(w)\\,.$ A recursive derivation shows that the generic $W^n$ can be expressed as $W^n(z)= \\frac{1/2}{(n-1)!", "}\\left(\\partial ^{n-1}\\psi \\psi ^+(-)^n\\partial ^{n-1}\\psi ^\\psi \\right)\\,.$ Therefore $W^n_0$ have distinct expressions for odd and even $n$ 's.", "For odd $n$ , $W^n_0 \\propto \\sum _{r\\in \\mathbb {Z}^+-1/2}(\\text{even power polynomial of} \\,\\,r)(\\psi _{-r}\\psi ^_r-\\psi ^_{-r}\\psi _r)\\,,$ and for even $n$ $W^n_0 \\propto \\sum _{r\\in \\mathbb {Z}^+-1/2}(\\text{odd power polynomial of} \\,\\,r)(\\psi _{-r}\\psi ^_r+\\psi ^_{-r}\\psi _r)\\,\\,.$ Especially, the exact form of $W^4_0$ gives rise to $W^4_0 = \\left(\\frac{r^3}{6}+\\frac{23 r}{24}\\right)(\\psi _{-r}\\psi ^_r+\\psi ^_{-r}\\psi _r)\\,.$ A bonus of this fermionic operator formalism is that it reveals the integrable structure explicitly inherited from free fermions.", "The reason is that an eigenvector of these $W$ operators is formed by those pair excitations of fermions (REF ) above Dirac sea as discussed before.", "This argument actually provides a proof of a conjecture proposed in [7] where they pointed out that the Kodaira-Spencer chiral bosonic CFT exactly act like free chiral fermions.", "We have a stronger result that for a chiral boson action involving $(\\partial \\varphi )^n$ ($n$ arbitrary) interactions, the fundamental theory is a free fermion theory.", "Although the bosonic theory is in general quite difficult to deal with, the corresponding fermionic one is rather simple.", "Moreover the integrable structure is explicit." ], [ "Quantum Curve and Patch-Shifting", "Now we consider the quantum curve for $\\mathbb {C}^3$ , which dominants the behavior of patch-shifting.", "Actually, the curve under consideration has distinct representation as a core and an asymptotic part, which are related by an $S$ transformation.", "In [7], [6], [9], [19] and [25], $\\mathbb {C}^3$ has a mirror manifold defined by the algebraic equation $zw-e^p-e^x-1=0\\,,$ the core curve of $\\mathbb {C}^3$ is understood as $e^{p} + e^{x} + 1 =0\\,.$ A quantization of this curve is to require a basic commutation relation $[p,x]=g_s\\,,\\quad p = g_s \\partial _x\\,.$ A toric Calabi-Yau three-fold can be treated as gluing of various local $\\mathbb {C}^3$ .", "Therefore it is not sufficient to know the core region geometry of $\\mathbb {C}^3$ without knowing the asymptotic one of it.", "In [2], [22], the Ooguri-Vafa operator actually does the work of gluing core and asymptotic region.", "The asymptotic region can be obtained by the $S$ transformation of core region geometry.", "This $S$ transformation is a generator of the modular group $PSL(2,\\mathbb {Z})$ .", "Another generator of the modular group is $T$ transformation, which plays a role of framing changing, see [7] for a detailed analysis.", "Acting on canonical doublet $(x,p)$ , $S$ and $T$ have the matrix representation: $S=\\left(\\begin{array}{cc} 0&1\\\\-1&0\\end{array}\\right)\\,\\,\\,,T=\\left(\\begin{array}{cc} 1&1\\\\0&1\\end{array}\\right)\\,.$ It is easy to check the relation that $ST =\\left(\\begin{array}{cc} 0&1\\\\-1&-1\\end{array}\\right)\\,,\\quad (ST)^3 =1\\,.$ It is well known that $ST$ transformation generates a $Z_3$ subgroup of $PSL(2,\\mathbb {Z})$ .", "From $S$ transformation we obtain the curve in asymptotic region $ e^{-x}+e^{p}+1 =0\\,.$ However, there are actually three different asymptotic regions of $\\mathbb {C}^3$ , reflecting the fact that the toric diagram of $\\mathbb {C}^3$ has three legs.", "Therefore the curve (REF ) should be triply degenerate.", "It is easy to check the invariance of (REF ) under $ST$ .", "If we denote these three patches as $u, v, w$ -patch respectively and define a cyclic relation $u=p_w=g_s\\partial _w\\,,\\quad v=p_u=g_s\\partial _u\\,,\\quad w=p_v=g_s\\partial _v \\,,$ then the $\\mathbb {Z}_3$ cyclic symmetry is explicit.", "From the asymptotic curve in $u$ -patch, when $u$ goes to infinity, $v$ should become $i\\pi $ .", "It bothers a lot in further analysis.", "A more convenient way is to throw away the $i\\pi $ dependence in all three patches, that is, to reparameterize $x\\rightarrow x+i\\pi , \\,\\, p\\rightarrow p+i\\pi \\,.$ It changes the core geometry to $e^{x}+e^p-1=0\\,,$ and the asymptotic geometry to $e^{-x}+e^p-1=0\\,.$ Hence the $\\mathbb {Z}_3$ symmetry is not generated by $ST$ but by the following $U$ -transformation [7], $U\\left(\\begin{array}{c}u\\\\v\\end{array}\\right) = ST\\left(\\begin{array}{c}u\\\\v\\end{array}\\right) +\\left(\\begin{array}{c}0\\\\i\\pi \\end{array}\\right)\\,.$ It is straightforward to check that $U$ transformation satisfies $U^3 =1\\,.$ We claim core curve (REF ) and asymptotic curve (REF ) play important roles in patch-shifting." ], [ " $W^3_0$ as the generator of {{formula:d02e0057-3dba-46e3-ac10-a967c88f73dc}}", "Previously we studied $W$ algebra.", "However in this article the related symmetry is $W^3_0$ , since the local patches are joint by $T$ transformation and $W^3_0$ is the generator of $T$ .", "$T$ acts as follows $T:\\left(\\begin{array}{c}u\\\\v\\end{array}\\right)=\\left(\\begin{array}{c}u+v\\\\v\\end{array}\\right)\\,.$ We first notice that $v=g_s\\partial _u$ is expressed as $v = g_s\\partial _u\\varphi (u)$ in a chiral boson theory.", "Then all arguments follow as we have discussed in previous section.", "The current related to the transformation is simply $W^3$ .", "Excitated modes of $W^3$ do not contribute because we only consider the asymptotic region on the $u$ -patch (or $v$ -patch) as $u\\rightarrow \\infty $ (or $v\\rightarrow \\infty $ ).", "Therefore only vacuum state contributes otherwise the theory will not be unitary.", "Thus only $W^3_0$ survives.", "In summary we conclude that $T$ transformation is generated by $W^3_0$ multiplied by $g_s$ .", "A $T^n$ transformation of an eigenfunction $f(u)$ has the standard expression $e^{ n g_s W^3_0} f(u) e^{- n g_s W^3_0} = f(u+n v)\\,.$" ], [ "$S$ transformation on the base", "Now let us consider $S$ transformation on the patches.", "As we argued before, the $S$ transformation interchanges the canonical pair $(x,p)$ .", "$x$ and $p$ form a canonical bundle, with the symplectic form as defined previously.", "The $S$ transformation also preserves the symplectic structure.", "If we treat $x$ as the base and $p$ the fiber, $S$ transformation actually bends $x$ to its normal direction.", "From a physical viewpoint, this can be understood as an insertion of a loop defect which bends the base and fiber simultaneously.", "Therefore the Hamiltonians on both sides of the defect should also be related to each other by $S$ transformation.", "This is very important in our further analysis." ], [ "Zhou's Identity and the Topological Vertex", "In this section, we consider the problem how to obtain partition functions from curves and the underlining symplectic transformations." ], [ "Vacuum Partition Function and the Curve of ${\\mathbb {C}}^3$", "Now we look for an eigenfunction of Hamiltonian in the core region $H_c(p,x) = e^{x}+e^p-1\\,,$ where the subindex $c$ denotes the core.", "In the local $u$ -patch with $u$ being a coordinate on a cylinder, we have asymptotic curve $e^{-u}+e^v-1=0\\,,$ the Hamiltonian in this region in the complex plane coordinates is $H_a(L_0,u) = \\frac{1}{z_1}+e^{g_s L_0} -1\\,,$ where $L_0 = z_1\\partial _{z_1} ,\\, z_1 = e^u,\\, z_2 = e^v, z_3 =e^w$ .", "It drives the evolution in the asymptotic region that is $z_1>1$ region, while in $z_1<1$ region, the core curve $H_c$ drives the evolution.", "Now we introduce an anti-B-brane at the infinity of $z_1$ -plane.", "The next step is to move it into the core region.", "This can be treated as an evolution of Hamiltonian.", "Unfortunately, there are infinitely many evolution paths in the spirit of path integral.", "Moreover it is quite difficult to approach this problem from a standard Hamiltonian analysis since the Hamiltonian is highly nonlinear.", "Therefore we need to find a new description of Hamiltonian evolution.", "Notice the CFT implied by the curve is a Kodaira-Spencer theory, which is equivalent to a free fermion theory.", "Since a brane (anti-brane) could be understood as a fermion (anti-fermion) insertion in local patch [7], the bosonized fermion (anti-fermion) field will have a representation (ignore the zero modes) $e^{\\pm \\varphi (z)} = \\exp \\left(\\pm \\sum _{n\\ne 0} \\frac{a_{-n}}{n}z^n\\right)\\,.$ It means the evolution of Hamiltonian can be replaced by infinitely many branes insertions in between $z_1=1$ and $z_1=\\infty $ .", "This is due to the fact radial ordering on a complex plane is a time ordering on cylinder while the later is controlled by the Hamiltonian.", "The OPEs of branes and anti-branes now can be understood as propagators.", "However, the positions where B-branes are inserted are arbitrary according to path integral.", "We may expect a classical equation of motion to determine the orbit completely.", "However, it is difficult to deal with a quantum system where there are different Hamiltonians in different regions.", "To simplify the problem a bit in this article we divide the space into the asymptotic region and the core region associate an Hamiltonian with each coordinate charts.", "We may choose the two coordinate charts to be $U_a = \\lbrace z_1 = e^u\\in (0,\\infty ]\\rbrace \\,,\\,\\,U_c = \\lbrace z_1^{\\prime }{} = e^{u^{\\prime }{}}\\in [0,\\infty )\\rbrace $ where $U_a$ is dominated by asymptotic Hamiltonian and $U_c$ is dominated by core Hamiltonian.", "In $U_a$ chart, we propose the following ansatz equation $\\langle 0\|H_a \\exp \\left(\\sum _{n>0}\\frac{a_{n}}{n}z^{-n}\\right)\\prod _{i\\ge 1}^\\infty V_-(w_i)\|0\\rangle \\equiv 0\\,.$ Applying (REF ) and moving $q^{L_0}$ out of the correlation function, we obtain $&& \\langle 0\|(q^{L_0} + \\frac{1}{z} -1) \\exp \\left(\\sum _{n>0,i\\ge 1}\\frac{(w_i/z)^n}{n}\\right)\|0\\rangle \\\\&=& (q^{L_0}+\\frac{1}{z}-1)\\prod _{i=1}^\\infty (1-w_i/z)^{-1}.$ The vanishing condition gives rise to $\\prod _{i=1}^\\infty (1-\\frac{w_i}{q z})^{-1} = \\left(1-\\frac{1}{z}\\right)\\prod _{i=1}^\\infty (1-w_i/z)^{-1}\\,.$ Suppose $w_1 = 1$ and a recursion relation $w_{i+1} = q^{-1} w_{i}\\,.$ We can prove that it is the solution of the eigen-equation (REF ).", "The resulting wave function turns out be a quantum dilogarithm [16], namely $\\Psi ^(u) = \\exp \\left(\\sum _{n>0}\\frac{ -q^n e^{-nu}}{n(n)_q}\\right)\\,,$ where $(n)_q = 1-q^n\\,.$ The orbit of the branes insertions are then a set of discrete points at $\\lbrace w_i = q^{-i+1}\\rbrace $ , $(i\\ge 1)$ .", "This analysis can be generalized to including the evolution of many anti-branes as well.", "The insertions of these anti-branes can be understood as the generating function for bra Schur states, namely $\\langle 0\|\\prod _i V_+^(z_i) = \\sum _\\lambda \\langle \\lambda \|s_{\\lambda }(z_i)\\,.$ We have chosen anti-brane inserted at infinity.", "Certainly we can consider brane inserted at infinity where a similar analysis leads to the following ansatz equation $\\langle 0\| H_a V_+(z)\\prod _i V_-^(w_i)\|0\\rangle =0\\,.$ Solve the equation we can locate the positions of branes $V_-^$ 's on the orbit $\\lbrace w_i = q^{-i+1},\\,i\\ge 1\\rbrace \\,.$ It is then clear how to determine positions of anti-branes (branes) in $[1,\\infty )$ .", "A similar analysis can be done for $U_c$ coordinate chart where the branes insertions are near origin ($e^u=0$ ).", "Hence it will locate the orbit points of anti-branes in the region $(0,1]$ by the ansatz equation $&&\\langle 0\| \\prod _{i=1}^\\infty V_+^(w_i)\\exp \\left(\\sum _{n>0}\\frac{a_{-n}}{n}z^n\\right) H_c\|0\\rangle =0\\,.$ The solution of this equation gives rise to a set of $w_i$ 's $\\lbrace w_i = q^{i-1},\\,\\,i\\ge 1\\rbrace \\,.$ The next step is to join these two coordinate charts into a single $u$ -patch as we noted.", "The anti-brane from infinity and the brane from origin meet at point 1 and annihilate each other identically.", "It actually gives rise to a vacuum partition function in $u$ -patch, we get $&&\\langle 0\|\\prod _{i=1}^\\infty V_-(q^{-i+1})q^{L_0}\\prod _{j=1}^{\\infty }V_+^(q^{j-1})\|0\\rangle \\\\\\nonumber &=&\\langle 0\|\\prod _{i=1}^\\infty V_-(q^{\\rho _i})\\prod _{j=1}^{\\infty }V_+^(q^{-\\rho _j})\|0\\rangle =1\\,.$ It is what we expect because when we glue two cylinders into a torus, the torus vacuum partition function can be chosen as 1 due to normalization.", "However, this result is quite different from the one obtained in [5], where the vacuum partition function is chosen to be MacMahon function.", "There is a subtle feature need to be clarified.", "For the vertex operators $V$ and $V^$ , if there are no zero modes, it is not a faithful correspondence between fermion and boson.", "However in this article, zero modes will not play significant roles in many calculations.", "Only if two charts are joined into a single patch with a defect at point 1, must the contribution of zero modes be retrieved.", "In that case the contribution will highly depend on the representation of the defect.", "It is worth comparing the vertex operator formalism with the definition fermionic vacuum.", "An observation is that suppose we define a correspondence $\\psi _{\\rho _i}\\rightarrow V_-(q^{\\rho _i})\\,,\\,\\,\\psi ^_{-\\rho _j}\\rightarrow V_+^(q^{-\\rho _i})\\,,$ the Dirac sea structure corresponding to fermionic vacuum now becomes $\\cdots \\psi _{-5/2}\\psi _{-3/2}\\psi _{-1/2}\\psi ^_{1/2}\\psi ^_{3/2}\\psi ^_{5/2}\\cdots \\,.$ Thus it corresponds to inner product of the fermionic vacuum $\\langle vac\|vac\\rangle $ as we obtained in sec.", "where the fermionic vacuum corresponds to the insertions of branes at infinity and anti-branes at origin.", "The last paragraph is only a rough idea about the projective relation from vertex operators to fermions.", "We shall have a more concrete derivation of it in next subsection." ], [ "Excited States and the Profile of a Young Diagram", "Now we consider excited states in $u$ -patchfor excited states in $v$ - or $w$ -patch, the same argument follows.", "Firstly, suppose there is an excited state labeled by a Young diagram $\\lambda $ inserted at infinity of $u$ -patch and there are no excitations on the other two patches.", "The partition function is $\\langle \\lambda \|\\prod _{i>0} V_-(q^{\\rho _i})\\prod _{j>0} V_+^(q^{-\\rho _j})\|0\\rangle = s_{\\lambda }(q^{\\rho })\\,.$ A slightly more complicated case is that besides $\\lambda $ there is also an excited state labeled by $\\mu $ at the origin of $u$ -patch.", "The the partition function is $\\langle \\lambda \|\\prod _{i\\ge 1}V_-(q^{\\rho _i})\\prod _{j\\ge 1} V_+^(q^{-\\rho _j})\|\\mu \\rangle = \\sum _\\eta s_{\\lambda /\\eta }(q^{\\rho })s_{\\mu /\\eta }(q^{\\rho })\\,.$ To obtain the equality we have used $\\sum _\\eta \\langle \\lambda \|\\prod _j V_-(z_j)\|\\eta \\rangle &=&\\sum _{\\eta ,\\xi }\\langle 0\|s_{\\lambda }(a_+)s_{\\eta }(a_-)s_{\\xi }(a_-)\|0\\rangle s_\\xi (z_j)\\\\\\nonumber &=&\\sum _{\\eta ,\\theta ,\\xi }c_{\\eta \\theta }^\\lambda \\langle \\theta \|\\xi \\rangle s_\\xi (z_j) = \\sum _{\\eta ,\\xi } c_{\\eta \\xi }^\\lambda s_{\\xi }(z_j) = \\sum _{\\eta }s_{\\lambda /\\eta }(z_j)\\,,$ where $s_{\\lambda /\\eta }$ is a skew Schur polynomial and $c_{\\eta \\xi }^\\lambda $ is the Littlewood-Richardson coefficient defined by $s_{\\lambda /\\eta } = \\sum _{\\theta }c_{\\eta \\theta }^\\lambda s_{\\theta }\\,.$ Since we argued in previous section, branes and anti-branes can be inserted not only at the infinity of a given patch, but also at the point 1 (more precisely, on the unit circle).", "Although we have calculated the simple case for excitations near the origin and infinity on a complex plane it would be quite interesting to ask the question about excitations in the bulk near point 1.", "It corresponds to the case of joining two charts into a single patch with some defect inserted at point 1.", "In a local patch, the unit circle does not belong to either the asymptotic region or the core region.", "Previously we considered the insertion of vacuum at point 1 the fermionic vacuum becomes products of $V_-$ and $V_+^$ 's, with all $V_-$ 's ($V_+^$ 's) located to the left (right) side of point 1, namely $V_-$ 's are in the asymptotic region corresponding to the outgoing modes and $V_+^$ 's are in the core region corresponding to the incoming modes.", "Now we consider a fermionic excited state labeled by Young diagram $\\nu $ .", "In the fermionic picture, it is an excited state from Dirac sea.", "The $\\nu $ state can be written down according to the profile of the Young diagram $\\nu $ .", "As in fig.", "REF , where $\\nu = \\lbrace 5,4,2,1\\rbrace $ the corresponding fermionic excitations are $\\cdots \\psi _{-11/2}\\psi ^_{-9/2}\\psi _{-7/2}\\psi ^_{-5/2}\\psi _{-3/2}\\psi _{-1/2}\\psi ^_{1/2}\\psi _{3/2}\\psi ^_{5/2}\\psi _{7/2}\\psi ^_{9/2}\\cdots \\,\\,.$ Figure: An example of a fermionic excited state and the corresponding Young diagramFor a general $\\nu $ , the modes of $\\psi $ (white dots) belong to the set $\\left\\lbrace \\cdots ,\\,\\,\\nu ^t_3 -3+\\frac{1}{2}\\,,\\,\\nu ^t_2-2+\\frac{1}{2},\\,\\,\\nu ^t_1-1+\\frac{1}{2}\\right\\rbrace \\equiv \\lbrace \\nu ^t+\\rho \\rbrace \\,.$ Similarly the modes of $\\psi ^$ (black dots) belong to the set $\\left\\lbrace -\\nu _1+1-\\frac{1}{2},\\,\\,-\\nu _2+2-\\frac{1}{2},\\,\\,-\\nu _3 +3-\\frac{1}{2},\\,\\cdots \\right\\rbrace \\equiv \\lbrace -\\nu -\\rho \\rbrace \\,.$ In this fermionic picture, it is clear that presumably, there is an infinity height fermionic tower at point $e^u=1$ .", "This tower will expand to elsewhere in $u$ -patch due to quantum shift.", "For the case $\\nu =\\phi $ , the empty set, we have already seen this quantum shift changes the vacuum Dirac sea to an infinite products of $V$ and $V^$ 's.", "Actually, it is very simple to deduce from the curve.", "At point $z_1=1$ , the Hamiltonian just becomes $H(L_0,1) = q^{L_0}\\,.$ The fermionic modes expansion becomes $\\psi (1) = \\sum _{r\\in \\mathbb {Z}-\\frac{1}{2}}\\psi _r\\,,\\,\\,\\psi ^(1) = \\sum _{r\\in \\mathbb {Z}-\\frac{1}{2}}\\psi ^_r\\,.$ In a quantum manner, all excitations are including in multi-products of these fields.", "For the physical vacuum $vac$ , it is a multi-product in sequence as $\\cdots \\psi _{-5/2}\\psi _{-3/2}\\psi _{-1/2}\\psi ^_{1/2}\\psi ^_{3/2}\\psi ^_{5/2}\\cdots \\,\\,.$ Hamiltonian at point 1 is a transport operator moving all $\\psi $ - fields to the left of 1 and $\\psi ^$ -fields to the right of 1.", "Further according to the bosonization formula, we reproduce the vacuum partition function as $\\langle 0\|\\prod (q^{L_0}V_-(1))q^{L_0}\\prod (V_+^(1)q^{L_0})\|0\\rangle \\,.$ It is just another expression of (REF ).", "Here the left (right) transporting behavior is transferred to left (right) action on the vertex operators.", "It proves the projective relation as we mentioned in eq.", "(REF ).", "If we want to generalize the analysis to a generic $\\nu $ state, then we just need to reshuffle (REF ) according to the profile here profile means the sequence of $V_-$ and $V^_+$ 's is determined according to the profile by the projective relation of the Young diagram of $\\nu $ .", "Hence it gives rise to $\\langle 0\|\\prod _{\\text{profile}\\,\\,\\nu }V_-(q^{\\nu ^t+\\rho })V_+^(q^{-\\nu -\\rho })\|0\\rangle \\,.$ Moving all $V_+^$ 's to the right side of all $V_-$ 's we get $&\\langle 0&\\hspace{-5.69054pt}\|\\prod _{\\text{profile}\\,\\,\\nu }V_-(q^{\\nu ^t+\\rho })V_+^(q^{-\\nu -\\rho })\|0\\rangle \\\\\\nonumber &=& \\prod _{(i,j)\\in \\nu }\\frac{1}{1-q^{h(i,j)}}\\equiv Z_{\\nu }(q)\\,,$ where $Z_\\nu (q) &:= &\\prod _{i,j\\in \\nu }\\frac{1}{1-q^{h(i,j)}} = Z_{\\nu ^t}(q)\\nonumber \\\\&=&(-)^{\|\\nu \|}\\prod _{(i,j)\\in \\nu }\\frac{q^{-h(i,j)}}{1-q^{-h(i,j)}}\\\\\\nonumber &=&(-)^{\|\\nu \|}q^{-\|\|\\nu \|\|/2-\|\|\\nu ^t\|\|/2}\\prod _{(i,j\\in \\nu )}\\frac{1}{1-q^{-h(i,j)}}\\,,$ with $h(i,j)$ being the hook length of square $(i,j)$ in $\\nu $ .", "Notice that $Z_\\nu $ is neither $s_{\\nu }(q^{-\\rho })$ nor $s_{\\nu ^t}(q^{-\\rho })$ .", "Schur polynomial in variables $\\lbrace q^{1/2},q^{3/2},q^{5/2},\\cdots \\rbrace $ is $s_{\\nu }(q^{-\\rho }) = q^{\\frac{\|\|\\nu ^t\|\|}{2}}\\prod _{i,j\\in \\nu }\\frac{1}{1-q^{h(i,j)}}= (-)^{\|\\nu \|}s_{\\nu ^t}(q^{\\rho })\\,.$ Now we consider $V_-$ and $V^_+$ 's insertions respectively.", "For the $V_-$ 's insertions, we have $\\langle 0\| \\prod _{i\\ge 1} V_-(q^{\\nu ^t+\\rho _i})\\,.$ A Young diagram $\\nu $ in terms of Frobenius notation is $\\lbrace r_1,r_2,\\cdots ,r_d\|s_1,s_2,\\cdots ,s_d\\rbrace $ where $r_i = \\nu _i-i+\\frac{1}{2},\\,\\,s_i = \\nu ^t_i-i+\\frac{1}{2}\\,.$ According to the projective relation the fermionic bra state can be represented as $&\\langle &\\hspace{-9.95845pt} \\Omega \|\\cdots \\psi _{\\nu ^t_i-i+\\frac{1}{2}}\\psi _{\\nu ^t_{i-1}-i+\\frac{3}{2}}\\cdots \\psi _{\\nu ^t_1-\\frac{1}{2}}\\\\\\nonumber &=&\\langle vac\|(-)^{\\sum _{i=1}^{d}(s_i-\\frac{1}{2})}\\prod _i^d\\psi _{s_i}\\psi ^_{r_i}=\\langle \\nu \|\\,.$ Similarly, for $V_-$ 's insertions, the corresponding fermionic ket state is $(-)^{\\sum _i^d(r_i-\\frac{1}{2})}\\prod _i^d\\psi _{-s_i}\\psi ^_{-r_i}\|vac\\rangle = \|\\nu ^t\\rangle \\,.$ The states are compatible with the geometrical observation from infinity to the origin on one local patch.", "The $S$ transformation which exchanges canonical variables (position and momentum) “bends” the project line to its normal at point 1.", "Then near infinity, we see the profile of $\\nu $ , while near the origin, we find that it reflects to $\\nu ^t$ .", "This observation defines the following rules: 1.", "From infinity to 1, the representation has not been changed.", "2.", "From 1 to 0, the representation becomes its transpose.", "In summary we can consider the patch-shifting and its impacts on the vertex operator formalism.", "We propose a configuration $\\langle \\lambda ,\\nu ,\\mu \\rangle & \\equiv &(-)^{\|\\nu \|}q^{\\frac{\|\|\\nu \|\|}{2}}\\langle \\lambda \|\\prod _{\\text{profile}\\,\\, \\nu }V_-(q^{\\nu ^t+\\rho })V_+^(q^{-\\nu -\\rho })\|\\mu \\rangle \\\\\\nonumber &=& s_{\\nu }(q^{\\rho })\\sum _{\\eta }s_{\\lambda /\\eta }(q^{\\nu ^t+\\rho })s_{\\mu /\\eta }(q^{\\nu +\\rho }).$ The factor $(-)^{\|\\nu \|} q^{\|\|\\nu \|\|/2}$ comes from zero modes of $V$ and $V^$ .", "Actually, if we keep the Boson-Fermion correspondence being exact, we should include the contribution of zero modes.", "The result of normal ordering now becomes: $\\prod _{(i,j)\\in \\nu }\\frac{1}{q^{-\\nu _i-\\rho _i}-q^{\\nu ^t_j+\\rho _j}}&=&(-)^{\|\\nu \|}q^{\|\|\\nu \|\|/2-\|\|\\nu ^t\|\|}\\prod _{(i,j)\\in \\nu }\\frac{1}{1-q^{-h(i,j)}}\\\\\\nonumber &=(-)^{\|\\nu \|}&q^{\\kappa _{\\nu }/2}s_{\\nu }(q^{\\rho })\\,.$ Then up to a framing factor $(-)^{\|\\nu \|}q^{\\kappa _\\nu /2}$ , the Schur function $s_{\\nu }(q^\\rho )$ occurs as desired.", "The states under the shifting from a $u$ -patch to a $v$ -patch are compatible with the corresponding curves of different charts on patches.", "For example, the insertion of the bra state $\\lambda $ at infinity on $u$ -patch is an insertion at point 1 in $v$ -patch.", "Thus patch-shifting leads to bringing a $\\lambda $ state from infinity of $u$ to 1 of $v$ .", "Then a $\\nu $ insertion at point 1 in the $u$ -patch becomes a ket state $\\nu ^t$ inserted at the core region in $v$ -patch.", "Similarly a ket state $\\mu $ inserted in the core region determined by $e^{u}+e^v-1 = 0$ in the $u$ -patch should be transformed to the asymptotic region in $v$ -patch by $S$ -transformation, and the $T$ transformation is required to cancel the divergence.", "For example $e^{-u-v}+e^{-v}-1=0$ as $u$ goes to $-\\infty $ , $v$ becomes $\\infty $ , this operation moves $\\mu $ ket state to a $\\mu ^t$ bra state along with a factor $q^{\\kappa _{\\mu }/2}$ due to the $T$ transformation.", "To join the asymptotic region and the core region together into a T-transformed $v$ -patch, we need $T$ -transform the core region (with $\\nu ^t$ inserted on) and also the defect (representation $\\lambda $ ).", "It results in a further $q^{\\kappa _{\\nu }/2}$ factor in the expression in $v$ -patch.", "Notice that there is no further factor corresponding to a $\\lambda $ insertion at point 1 since $q^{\\kappa _{\\lambda }/2}q^{\\kappa _{\\lambda ^t}/2}=1\\,.$ Now we have the following conjecture $\\langle \\lambda , \\nu ,\\mu \\rangle &=& q^{\\frac{\\kappa _{\\mu }+\\kappa _{\\nu }}{2}}\\langle \\mu ^t ,\\lambda ,\\nu ^t\\rangle \\\\\\nonumber &=&q^{\\frac{\\kappa _{\\lambda }+\\kappa _{\\mu }}{2}}\\langle \\nu ,\\mu ^t,\\lambda ^t\\rangle \\,.$ It is our major observation from the curve of $\\mathbb {C}^3$ .", "It is difficult to verify this conjecture directly.", "However, if we let one of the representations $\\lambda $ , $\\mu $ and $\\nu $ be an empty representation $\\phi \\equiv 0$ , then the resulting identities are just Zhou's identities [26].", "For example, let $\\nu =0$ .", "We have $\\langle \\lambda ,0,\\mu \\rangle &=& \\sum _{\\eta }s_{\\lambda /\\eta }(q^{\\rho })s_{\\mu /\\eta }(q^{\\rho })\\\\\\nonumber &=&q^{\\kappa _{\\mu }/2}\\langle \\mu ^t, \\lambda , 0\\rangle = q^{\\kappa _{\\mu }/2}s_{\\lambda }(q^{\\rho })s_{\\mu ^t}(q^{\\lambda ^t+\\rho })\\\\\\nonumber &=& q^{(\\kappa _{\\lambda }+\\kappa _{\\mu })/2}\\langle 0,\\mu ^t,\\lambda ^t\\rangle \\\\\\nonumber &=& q^{(\\kappa _{\\lambda }+\\kappa _{\\mu })/2}s_{\\mu ^t}(q^{\\rho })s_{\\lambda ^t}(q^{\\mu ^t+\\rho })\\,.$ It is nothing but Zhou's identity.", "We can verify other degenerate cases of (REF ) in detail.", "Consequently we get Zhou's identities in all cases." ], [ "The relation with the Topological Vetex", "It would be interesting to compare eq.", "(REF ) with the famous topological vertex proposed in [6] and further the topological vertex in terms of symmetric polynomials in [5] and [26].", "The topological vertex in [5], [26] is defined as $C(\\lambda ,\\,\\mu ,\\,\\nu ) = q^{\\kappa _\\lambda /2}s_{\\nu }(q^\\rho )\\sum _{\\eta }s_{\\mu /\\eta }(q^{\\nu ^t+\\rho })s_{\\lambda ^t/\\eta }(q^{\\nu +\\rho })$ In our configuration $C(\\lambda ,\\,\\mu ,\\,\\nu )=q^{\\kappa _{\\lambda }/2}\\langle \\mu ,\\,\\nu ,\\,\\lambda ^t\\rangle \\,.$ It means what we have obtained is a reformulation of the topological vertex.", "However, the approach here is quite different from that in [6] and [5].", "An direct observation is that our definition as in eq.", "(REF ) has a very clear patch meaning rather than a unified topological vertex.", "The cyclic symmetry of the topological vertex now becomes the shifting of patches." ], [ "Conclusions", "We find an explicit correspondence between A- and B-model for the case of topological vertex.", "In our opinion, the mirror curve of $\\mathbb {C}^3$ is not a global ly defined chart but a union of two coordinate charts within defects inserting at point 1.", "It is crucial for deriving B-model correlation function, which becomes A-model topological invariant.", "A new vertex operator approach to the topological vertex is proposed.", "On the way of doing this, we prove the conjecture proposed in [7].", "The vertex operator approach can be treated as an application of projective representation introduced in [23].", "Finally, we propose a conjecture on the topological vertex (or in B-model, a three-leg correlation function) identity (REF ), which becomes Zhou's identities of Hopf links in degenerate cases.", "There are many further works in this direction.", "We just list three of them for instance.", "Firstly, the identity (REF ) is new and a mathematical proof is not known to the authors.", "Secondly, the vertex operator approach could be generalized to other curves associated to many toric Calabi-Yau manifolds.", "Due to the identity (REF ), it is quite free to glue topological vertices to formulate complicated toric Calabi-Yau's.", "This calculation is working in progress and a future article will contain some applications.", "Thirdly, it is natural to ask for a refined version of this approach.", "However, this is quite difficult since there the refined curve Eynard and Kozcaz provided a mirror curve for refine topological vertex in [18], the curve has no simple expression as the topological vertex.", "is very complicated and related symplectic transformations are not well-known.", "Maybe a simpler case could be considered first.", "For example, when a background charge is introduced into the Kodaira-Spencer theory the resulting theory is hence the Feign-Fuchs bosonic theory.", "The underlining integrability is controlled by the Calogero-Sutherland model [10], [24].", "In this case, two refined parameters($t$ and $q$ ) are related by $t=q^{\\alpha }$ (twisted case) and the eigenfunctions are Jack symmetric functions in the limit $q\\rightarrow 1$ .", "A very similar analysis could be done for this case.", "We expect a Jack symmetric function expression for the twisted topological vertex." ], [ "Acknowledgments", "We would like to thank Professor Guoce Xin, Professor Ming Yu and Professor Jian Zhou for valuable comments.", "The authors are grateful to Morningside Center of Chinese Academy of Sciences and Kavli Institute for Theoretical Physics China at the Chinese Academy of Sciences for providing excellent research environment.", "This work is also partially supported by Beijing Municipal Education Commission Foundation (KZ201210028032, KM201210028006), Beijing Outstanding Person Training Funding (2013A005016000003)." ], [ "Some Notations on Symmetric Polynomials", "In this appendix we just provide a brief review of some symmetric functions.", "For detailed description please look up the book by Macdonald [21].", "Definition 1 An elementary symmetric polynomial is defined by $e_r(x_1, x_2, \\cdots )=\\sum _{i_1<i_2<\\cdots <i_r}x_{i_1}x_{i_2}\\cdots x_{i_r},$ for $r\\ge 1$ and $e_0=1$ .", "The generating function for the $e_r$ is $E(t)=\\sum _{r\\ge 0}e_rt^r=\\prod _{i\\ge 1}(1+x_it).$ Definition 2 A complete (homogenous) symmetric polynomial is defined by $h_r(x_1, x_2, \\cdots )=\\sum _{i_1\\le i_2\\le \\cdots \\le i_r}x_{i_1}x_{i_2}\\cdots x_{i_r},$ for $r\\ge 1$ and $h_0=1$ .", "The generating function for the $h_r$ is $H(t)=\\sum _{r\\ge 0}h_rt^r=\\prod _{i\\ge 1}\\frac{1}{1-x_it}.$ Definition 3 A Schur polynomial $s_\\lambda $ as a symmetric polynomial in variables $x_1, x_2, \\cdots $ corresponding to a partition $\\lambda $ is defined by $s_{\\lambda }(x_1,\\cdots ,x_N):= \\sum _{T}{x}^T$ where $T$ is a semi-standard tableau of shape $\\lambda $ and ${x}^T=\\prod _ix_i^{n_i}$ with $n_i$ the number of $i$ filling in $T$ .", "Definition 4 (Jacobi-Trudi) The Schur polynomial can be calculated from the elementary or complete polynomials by $s_\\nu (x_1, x_2, \\cdots , x_n)=\\det (h_{\\nu _i-i+j})=\\det (e_{\\nu ^t_i-i+j}).$ Now suppose the variables ($x_1, x_2, \\cdots $ ) appear in a formal power series $E(t)=\\prod _i(1+x_it)$ .", "We simply denote the Schur function by $s_\\nu (E(t)).$ For example $E(t)=\\prod _{i=0}^\\infty (1+q^it)=\\sum _{r=0}^\\infty e_rt^r$ where $e_r = \\prod _{i=1}^r\\frac{q^{i-1}}{1-q^i}.$ Hence the corresponding Schur function is written as $s_\\lambda (1, q, q^2, \\cdots )$ .", "In the q-number notation $[x]=q^{x/2}-q^{-x/2}$ $s_\\nu (q^{-\\rho }) =(-1)^{\|\\nu \|}q^{-\\kappa (\\nu )/4}\\prod _{x\\in \\nu }\\frac{1}{[h(x)]}$ where $h(x)$ is the hook length of the square $x$ and $\\kappa (\\nu )=2(n(\\nu ^t)-n(\\nu ))$ with $n(\\nu )=\\sum _i \\nu _i(i-1)$ .", "Now let us generalize the formal power series to a more complicated case $E_\\mu (t)=\\prod _{i=1}^\\infty (1+q^{\\mu _i-i+1/2}t)=\\prod _{i=1}^\\ell \\frac{1+q^{\\mu _i-i+1/2}t}{1+q^{-i+1/2}t}\\prod _{i=1}^\\infty (1+q^{-i+1/2}t).$ Recall a very useful identification between multisets of number $\\lbrace \\mu _i-i, (d<i\\le \\ell )\\rbrace =\\lbrace -1,\\cdots , -\\ell \\rbrace -\\lbrace -\\mu ^t_i+i-1, (1\\le i\\le d)\\rbrace $ where $d$ is the diagonal of $\\nu $ .", "According to Frobenius notation $\\nu =(\\alpha _1, \\cdots , \\alpha _d\|\\beta _1,\\cdots , \\beta _d)$ , it can be written as $\\lbrace \\mu _i-i, (d<i\\le \\ell )\\rbrace =\\lbrace -1,\\cdots , -\\ell \\rbrace -\\lbrace -\\beta _i-1, (1\\le i\\le d)\\rbrace .$ (REF ) becomes $E_\\mu (t)=\\prod _{i=1}^{d(\\mu )}\\frac{1+q^{\\alpha _i+1/2}t}{1+q^{-\\beta _i-1/2}t}\\prod _{i=1}^\\infty (1+q^{-i+1/2}t).$ Therefore $s_\\nu (E_\\mu (t))=s_\\nu (q^{\\mu _1-1+1/2}, q^{\\mu _2-2+1/2}, \\cdots )\\,,$ or it can be put in a simple notation $s_\\nu (q^{\\mu +\\rho })$ where $\\rho =-\\frac{1}{2}, -\\frac{3}{2}, \\cdots $ .", "In the Frobenius notation $s_\\nu (E_\\mu (t))=s_\\nu (q^{\\alpha _1+\\frac{1}{2}}, \\cdots , q^{\\alpha _{d(\\mu )}+\\frac{1}{2}}, q^{-\\frac{1}{2}}, \\cdots , \\widehat{q^{-\\beta _1-\\frac{1}{2}}}, \\cdots , \\widehat{q^{-\\beta _{d(\\mu )}-\\frac{1}{2}}}, q^{-\\beta _{d(\\mu )}-\\frac{3}{2}},\\cdots ).$ Definition 5 A skew Schur polynomial $s_{\\lambda /\\mu }$ as a symmetric function in variables $x_1, x_2, \\cdots $ is defined by $s_{\\lambda /\\mu }(x_1, x_2, \\cdots )=\\sum _{T} {x}^T$ where $T$ is a semi-standard tableau of shape $\\lambda -\\mu $ .", "The skew Schur function has a property $s_{\\lambda /\\mu }(x, y)=\\sum _\\nu s_{\\lambda /\\nu }(x)s_{\\nu /\\mu }(y).$ Therefore it can be generalized to $n$ sets of variables $x^{(1)}, \\cdots , x^{(n)}$ $s_{\\lambda /\\mu }(x^{(1)}, \\cdots , x^{(n)})=\\sum _{(\\nu )}\\prod _{i=1} s_{\\nu ^{(i)}/\\nu ^{(i-1)}}(x^{(i)})$ summed over all sequences $(\\nu )=(\\nu ^{(0)}, \\cdots , \\nu ^{(n)})$ of partitions such that $\\nu ^{(0)}=\\mu $ , $\\nu ^{(n)}=\\lambda $ , and $\\nu ^{(0)}\\subset \\cdots \\subset \\nu ^{(n)}$ .", "Definition 6 (Jacobi-Trudi) The skew Schur polynomial also can be calculated from the elementary or complete polynomials by $s_{\\lambda /\\mu }(x_1, x_2, \\cdots , x_n)=\\det (h_{\\lambda _i-\\mu _j-i+j})=\\det (e_{\\lambda _i^t-\\mu ^t_j-i+j}).$" ], [ " The identity", "In this appendix we provide a combinatoric proof of the identity $s_{\\lambda /\\mu }( q^{\\nu +\\rho }) = (-1)^{\|\\lambda \|-\|\\mu \|}s_{\\lambda ^t /\\mu ^t}( q^{-\\nu ^t -\\rho }).$ According to the definition of $s_{\\lambda /\\mu }$ (REF ) we only need to prove $h_r(q^{\\nu +\\rho })=(-1)^re_r(q^{-\\nu ^t-\\rho }).$ Now we use the Frobenius notation of a partition $\\nu =(\\alpha _1, \\cdots , \\alpha _d\|\\beta _1,\\cdots , \\beta _d)$ .", "Suppose $\\nu _1=N$ , $\\nu _1^t=k$ $E(t, q^{-(\\nu ^t+\\rho )})&=&(1+q^{-(\\nu _1^t-1+1/2)}t)\\cdots (1+q^{-(\\nu ^t_j-j+1/2)}t)\\cdots (1+q^{-(\\nu ^t_N-N+1/2)}t)\\times \\nonumber \\\\&&\\times (1+q^{-(-(N+1)+1/2)}t)\\cdots \\nonumber \\\\&=&\\prod _{i=1}^d\\frac{1+q^{-(\\beta _i+1/2)}t}{1+q^{\\alpha _i+1/2}t}E_0(t)$ where $E_0(t)=\\prod (1+q^{-\\rho }t)$ .", "We have used an identity among multisets of number $\\lbrace 1, 2, \\cdots , N\\rbrace =\\lbrace j-\\nu ^t_j, (N\\ge j>d)\\rbrace \\cup \\lbrace \\alpha _i+1 (i=1, \\cdots , d) \\rbrace .$ Similarly $H(t, q^{\\nu +\\rho })&=&\\frac{1}{1-q^{\\nu _1-1+1/2}t}\\cdots \\frac{1}{1-q^{\\nu _i-i+1/2}t}\\cdots \\frac{1}{1-q^{\\nu _k-k+1/2}t}\\frac{1}{1-q^{-(k+1)+1/2}t}\\nonumber \\\\&=&\\prod _{i=1}^d\\frac{1-q^{-(\\beta _i+1/2)}t}{1-q^{\\alpha _i+1/2}t}H_0(t)$ where $H_0(t)=\\prod (1-q^{-\\rho }t)^{-1}$ and $\\lbrace 1, 2, \\cdots , k\\rbrace =\\lbrace i-\\nu _i, (k\\ge i>d)\\rbrace \\cup \\lbrace \\beta _i+1 (i=1, \\cdots , d) \\rbrace .$ The first factor in (REF ) and (REF ) are almost the same except the $+$ and $-$ sign in front of $q$ .", "In addition $\\prod (1-q^{-\\rho } t)$ and $\\prod (1-q^{\\rho }t)^{-1}$ have the same power expansion of $t$ .", "The difference can be resolved by $E(-t, q^{-\\nu ^t-\\rho })=H(t, q^{\\nu +\\rho }).$ Therefore we obtain the result we want $e_r(q^{-\\nu ^t-\\rho })=(-1)^rh_r(q^{\\nu +\\rho }).$" ] ]	1403.0181
[ [ "Factorization Property of Generalized s-self-decomposable measures and\n class $L^f$ distributions" ], [ "Abstract The method of \\emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades.", "In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \\emph{factorization property}.", "These are the classes $\\mathcal{U}^f_{\\be}$ and $L^f$, respectively." ], [ "0 0 Abstract.", "The method of random integral representation, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades.", "In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the factorization property.", "These are the classes $\\mathcal {U}^f_{\\beta }$ and $L^f$ , respectively Mathematics Subject Classifications(2000): Primary 60F05 , 60E07, 60B11; Secondary 60H05, 60B10.", "Key words and phrases: Generalized s-selfdecomposable distributions; selfdecomposable distributions; factorization property; class $L^f$ ; infinite divisibility; Lévy-Khintchine formula; Euclidean space; Lévy process; Brownian motion; random integral; Banach space .", "Abbrivated title: Factorization property In probability theory, from its very beginning, characteristic functions (Fourier transforms) were used to describe measures and to prove limiting distributions theorems.", "In the past few decades many classes of probability measures (e.g.", "selfdecomposable measures , n-times selfdecomposable, s-selfdecomposable, type G distribution, etc.)", "were characterized in terms of distributions of some random integrals; cf.", "Jurek (1985, 1988) , Jurek and Vervaat (1983), Jurek and Mason (1993), Jurek and Yor (2004), Iksanov, Jurek and Schreiber (2004) and recently Aoyama and Maejima (2007).", "More precisely, for each of those classes one integrates a fixed deterministic function with respect to a class of Lévy processes, with possibly a time scale change.", "Moreover, what we must emphasize here is that from the random integral representations easily follow those in terms of characteristic functions, and also one can infer from them new convolution factorizations or decompositions.", "Thus the random integral representations provide a new method in the area called the arithmetic of probability measures; cf.", "Cuppens (1975) or Linnik and Ostrovskii (1977).", "In this note we consider more specific situations.", "Namely, for a convolution semigroup $\\mathcal {C}$ of distributions of some random integrals and a measure $\\mu \\in \\mathcal {C}$ we are interested in decompositions of the form $\\mu =\\mu _1\\ast \\rho , \\ \\ \\mu _1\\in \\mathcal {C},$ for some probability measure $\\rho $ that is intimately related to the measure $\\mu _1$ .", "This paper was inspired by questions related to the class $L^f$ of selfdecomposable measures having the so called factorization property that was introduced and investigated in Iksanov, Jurek and Schreiber (2004).", "Finally, let us note that the random integral representations for classes $\\mathcal {U}^f_{\\beta }$ (Corollary 1(a)) and $L^f$ (Corollary 3) provide more examples for the conjectured \"meta-theorem\" in The Conjecture on www.math.uni.wroc.p/$\\sim $ zjjurek or see Jurek (1985) and (1988).", "1.", "Notation and the results.", "Our results are presented for probability measures on Euclidean space $\\mathbb {R}^d$ .", "However, our proofs are such that they hold true for measures on infinite dimensional real separable Banach space $E$ with the scalar product replaced by the bilinear form between $E^{\\prime }\\times E$ and $\\mathbb {R}$ ; $E^{\\prime }$ denotes the topological dual of $E$ and, of course, $(\\mathbb {R}^d)^{\\prime }=\\mathbb {R}^d$ ; cf.", "Araujo-Giné (1980), Chapter III.", "In particular, one needs to keep in mind Remark 1, below.", "Let $ID$ and $ID_{\\log }$ denote all infinitely divisible probability measures (on $\\mathbb {R}^d$ or $E$ ) and those that integrate the logarithmic function $\\log (1+\|\|x\|\|)$ , respectively.", "Let $Y_{\\nu }(t),t\\ge 0$ denote an $\\mathbb {R}^d$ (or $E$ ) - valued Lévy process, i.e., a process with stationary independent increments, starting from zero, and with paths that continuous from the right and with finite left limits, such that $\\nu $ is its probability distribution at time 1: $\\mathcal {L}(Y_{\\nu }(1))=\\nu $ , where $\\nu $ can be any $ID$ probability measure.", "Throughout the paper $\\mathcal {L}(X)$ will denote the probability distribution of an $\\mathbb {R}^d$ -valued random vector (or a Banach space E-valued random elements if the Reader is interested in that generality).", "Definition 1 For $\\beta >0$ and a Lévy process $Y_{\\nu }$ , let us define $\\mathcal {J}^{\\beta }(\\nu ):\\,=\\mathcal {L}\\bigl (\\int _0^1t^{1/\\beta }\\;dY_{\\nu }(t)\\bigr )=\\mathcal {L}\\bigl (\\int _0^1t\\;dY_{\\nu }(t^{\\beta })\\bigr ), \\ \\ \\mathcal {U}_{\\beta }:\\,=\\mathcal {J}^{\\beta }(ID).$ To the distributions from $\\mathcal {U}_{\\beta }$ we refer to as generalized s-selfdecomposable distributions.", "The classes $\\mathcal {U}_{\\beta }$ were already introduced in Jurek (1988) as the limiting distributions in some schemes of summing independent variables.", "The terminology has its origin in the fact that distributions from the class $\\mathcal {U}_{1} \\equiv \\mathcal {U}$ were called s-selfdecomposable distribution (the \"s-\" , stands here for the shrinking operations that were used originally in the definition of $\\mathcal {U}$ ); cf.", "Jurek (1985), (1988) and references therein.", "2Proposition A factorization of generalized s-selfdecomposable distribution.", "In order that a generalized s-selfdecomposable distribution $\\mu =J^{\\beta }(\\rho )$ , from the class $\\mathcal {U}_\\beta $ , convoluted with its background measure $\\rho $ is again in the class $\\mathcal {U}_\\beta $ it is sufficient and necessary that $\\rho \\in \\mathcal {U}_{2\\beta }.$ More explicitly, $[\\,J^\\beta (\\rho )\\rho =J^\\beta (\\nu )\\,] \\Longleftrightarrow [\\,\\rho =J^{2\\beta }(\\nu ^{\\tfrac{1}{2}})\\,]$ Furthermore, for each $\\tilde{\\mu }\\in \\mathcal {U}_\\beta $ there exists a unique $\\tilde{\\rho }\\in \\mathcal {U}_{2\\beta }$ such that $\\tilde{\\mu }=J^\\beta (\\tilde{\\rho })* \\tilde{\\rho }$ and $J^{2\\beta }\\bigl (\\tilde{\\mu }\\bigr )=J^{\\beta }\\bigl ((\\tilde{\\rho })^{{\\displaystyle }2}\\bigr )$ Let us denote by $\\mathcal {U}_{\\beta }^f$ the class of generalized s-selfdecomposable admitting the factorization property, i.e, $\\mu :=J^\\beta (\\rho )\\in \\mathcal {U}_{\\beta }$ has the factorization property if $J^\\beta (\\rho )\\ast \\rho \\in \\mathcal {U}_{\\beta }$ .", "Corollary 1 For $\\beta >0$ we have equalities $(a) \\ \\ \\mathcal {U}^f_{\\beta }=\\mathcal {J}^{2\\beta }(\\mathcal {U}_{\\beta })=\\mathcal {J}^{2\\beta }(\\mathcal {J}^\\beta (ID))=\\\\ = \\lbrace \\mathcal {L}(\\int ^{1}_{0}(1-\\sqrt{t})^{1/\\beta }\\,dY_{\\nu }(t)): \\nu \\in ID\\rbrace .", "\\qquad \\qquad \\qquad $ (b) $\\mathcal {U}_{\\beta }=\\lbrace \\mathcal {J}^{\\beta }(\\rho )\\ast \\rho : \\rho \\in \\mathcal {U}_{2\\beta }\\rbrace $ .", "Taking in Proposition 1 $\\beta =1$ we get the following Corollary 2 Factorization of s-selfdecomposable distributions.", "An s-selfdecomposable distribution $\\mu =J(\\rho )$ convoluted with $\\rho $ is again s-selfdecomposbale if and only if $\\rho \\in \\mathcal {U}_2$ .", "Thus we have $\\mathcal {U}^f=\\mathcal {J}^{2}(\\mathcal {U})$ .", "More explicitly $[\\,J(\\rho )\\rho =J(\\nu )\\,] \\Longleftrightarrow [\\,\\rho =J^{2}\\bigl (\\nu ^{{\\displaystyle }\\tfrac{1}{2}}\\bigr )\\,].$ Moreover, for each $\\tilde{\\mu }\\in \\mathcal {U}$ there exist a unique $\\rho \\in \\mathcal {U}_2$ such that $\\tilde{\\mu }=J(\\tilde{\\rho })\\tilde{\\rho }$ and $J^{2}\\bigl (\\tilde{\\mu }\\bigr )=J\\bigl ((\\tilde{\\rho })^{2})\\bigr )$ .", "Consequently, $\\mathcal {U}=\\lbrace \\mathcal {J}^{2}(\\rho )\\ast \\rho :\\rho \\in \\mathcal {U}\\rbrace $ .", "Following Jurek-Vervaat (1983) or Jurek (1985) we recall the following Definition 2 For a measure $\\nu \\in ID_{\\log }$ and a Lévy process $Y_{\\nu }$ let us define $\\mathcal {I}(\\nu ):= \\mathcal {L}\\bigl (\\int _0^{\\infty }e^{-s}\\,d\\,Y_{\\nu }(s)\\bigr ), \\ \\ \\ L:=\\mathcal {I}(ID_{\\log })$ and distributions from $L$ are called selfdecomposable or Lévy class L distributions.", "In classical probability theory the selfdecomposability ( or in other words, the Lévy class $L$ distributions) is usually defined via some decomposability property or by scheme of limiting distributions.", "However, since Jurek-Vervaat (1983) we know that the class $L$ coincides with the class of distributions of random integrals given in (5) and thus it is used in this note as its definition.", "Before going further, let us recall the following example that led to, and justified interest in, that kind of investigations/factorizations.", "Example.", "For two dimensional Brownian motion $\\textbf {B}_t:= (B^1_t, B^2_t)$ , the process $\\mathcal {A}_t:=\\int _0^tB^1_s\\,dB^2_s-B^2_s\\,dB^1_s,\\ \\ \\ t>0,$ called Lévy's stochastic area integral, admits the following factorization $\\chi (t):=E[ e^{it\\mathcal {A}_u}\|\\textbf {B}_u=(\\sqrt{u},\\sqrt{u})]=\\frac{tu}{\\sinh tu}\\cdot \\exp [-(tu\\,\\cosh tu -1)],$ cf.", "P. Lévy (1951) or Yor (1992), p. 19.", "Iksanov-Jurek-Schreiber (2004), p. 1367, proved that the factorization (6) may be interpreted as follows: if $\\nu $ is the probability measure with the characteristic function $t\\rightarrow \\exp [-(tu\\,\\cosh tu-1)]$ then $\\mathcal {I}(\\nu )$ has the characteristic function $t\\rightarrow \\frac{tu}{\\sinh tu}$ , and also $\\mathcal {I}(\\nu )\\ast \\nu = \\mathcal {I}(\\rho ), \\ \\ \\mbox{ for some} \\ \\ \\rho \\in ID_{\\log };$ i.e., $\\mathcal {I}(\\nu )$ is selfdecomposable and when convoluted with its background driving probability measure $\\nu $ we again get a distribution from the class $L$ .", "Let us note that the convolution factorizations (7), (3) and (4) are of the form described in (1), with different semigroups $\\mathcal {C}$ .", "7Proposition Random integral representation of $I\\bigl (J^\\beta \\bigl (\\textsl {ID}_{\\log }\\bigr )\\bigr )$.", "For $\\nu \\in \\textsl {ID}_{\\log }$ and $\\beta >0$ $I\\bigl (J^\\beta \\bigl (\\nu \\bigr )\\bigr )=\\mathcal {L}\\bigl (\\int _{0}^{\\infty }{\\displaystyle e^{\\displaystyle -s}}\\,dY_{\\nu }\\bigl (\\sigma _{\\beta }(s)\\bigr )\\bigr ) ,$ where $Y_{\\nu }(t), t\\ge 0$ is a Lévy process such that $\\mathcal {L}(Y_{\\nu }(1))=\\nu $ and the deterministic inner clock $\\sigma _{\\beta }$ is given by $\\sigma _{\\beta }(s):= s+\\tfrac{1}{\\beta }e^{-\\beta s}-\\tfrac{1}{\\beta }, \\ s\\ge 0$ .", "From Proposition 1 (ii) in Iksanov-Jurek-Schreiber (2004) and taking $\\beta =1$ in Proposition 2 we get Corollary 3 For the class, $L^f$ , of selfdecomposable distributions with factorization property, we have the following random integral representation $L^f = \\big \\lbrace \\mathcal {L}\\bigl (\\int _{0}^{\\infty } e^{-s}\\,dY_{\\nu }(s +e^{-s}-1))\\,: \\ \\nu \\in ID_{\\log }\\big \\rbrace .$ 2.", "Proofs.", "For a probability Borel measures $\\mu $ on $\\mathbb {R}^d$ , its characteristic function $\\hat{\\mu }$ is defined as $\\hat{\\mu }(y):=\\int _{\\mathbb {R}^d} e^{i<y,x>}\\mu (dx), \\ y\\in \\mathbb {R}^d,$ where $<\\cdot ,\\cdot >$ denotes the scalar product; (in case one wants to have results on Banach spaces $<\\cdot ,\\cdot >$ is the bilinear form on $E^{\\prime }\\times E$ and $y\\in E^{\\prime }$ ).", "Recall that for infinitely divisible measures $\\mu $ their characteristic functions admit the following Lévy-Khintchine formula $\\hat{\\mu }(y)= e^{\\Phi (y)}, \\ y \\in \\mathbb {R}^d, \\ \\ \\mbox{and theexponents $\\Phi $ are of the form} \\\\ \\Phi (y)=i<y,a>-\\frac{1}{2}<y,Sy> + \\qquad \\qquad \\qquad \\\\ \\int _{\\mathbb {R}^d\\backslash \\lbrace 0 \\rbrace }[e^{i<y,x>}-1-i<y,x>1_B(x)]M(dx),$ where $a$ is a shift vector, $S$ is a covariance operator corresponding to the Gaussian part of $\\mu $ and $M$ is a Lévy spectral measure.", "Since there is a one-to-one correspondence between a measure $\\mu \\in ID$ and the triples $a$ , $S$ and $M$ in its Lévy-Khintchine formula (10) we will write $\\mu =[a,S,M]$ .", "Finally, let recall that $M \\ \\mbox{is Lévy spetral measure on $\\mathbb {R}^d$ iff} \\ \\ \\int _{\\mathbb {R}^d}\\min (1, \|\|x\|\|^2)M(dx)<\\infty $ (For infinite divisibility of probability measures on Banach spaces we refer to the monograph by Araujo-Giné (1980), Chapter 3, Section 6, p. 136.", "Let us stress that the characterization (11), of Lévy spectral measures, is in general NOT true in infinite dimensional Banach spaces !", "However, it holds true in Hilbert spaces; cf.", "Parthasarathy (1967), Chapter VI, Theorem 4.10.)", "Before proving Proposition 1, let us note the following auxiliary facts.", "11Lemma (a) For the mapping $J^\\beta $ and $\\nu \\in ID$ we have $\\widehat{\\mathcal {J}^{\\beta }(\\nu )}(y)=\\exp \\int _0^1\\log \\widehat{\\nu }(t^{1/\\beta }y)\\,dt =\\exp \\mathbf {E}[\\log \\widehat{\\nu }(U^{1/\\beta }y)], \\ \\ y\\in \\mathbb {R}^d \\ (\\mbox{or}\\ E^{\\prime }).$ and $U$ is a random variable uniformly distributed over the unit interval $(0,1)$ .", "(b) The mapping $\\mathcal {J}^{\\beta }$ is one-to-one.", "More explicitly we have that $\\frac{d}{ds}[s\\log \\widehat{\\mathcal {J}^{\\beta }(\\nu )}(s^{1/{\\beta }}y)]\|_{s=1}=\\log \\hat{\\nu }(y), \\ \\ \\mbox{for all} \\ y\\in \\mathbb {R}^d \\ (\\mbox{or}\\ E^{\\prime }).$ (c) The mappings $\\mathcal {J}^\\beta , \\beta >0$ commute, i.e., for $\\beta _1, \\beta _2 >0$ and $\\nu \\in ID$ , $\\mathcal {J}^{\\beta _1}(\\mathcal {J}^{\\beta _2}(\\nu ))=\\mathcal {J}^{\\beta _2}(\\mathcal {J}^{\\beta _1}(\\nu ))$ .", "(d) For probability measures $\\nu _1, \\nu _2$ and $c>0$ we have that $\\mathcal {J}^{\\beta }(\\nu _1\\ast \\nu _2)= \\mathcal {J}^{\\beta }(\\nu _1)\\ast \\mathcal {J}^{\\beta }(\\nu _2); \\ \\ (\\mathcal {J}^{\\beta }(\\nu ))^{\\ast c}=\\mathcal {J}^{\\beta }(\\nu ^{\\ast c})$ (e) For $\\beta >0$ and $\\rho \\in ID$ we have the identity $\\mathcal {J}^{2\\beta }(\\mathcal {J}^{\\beta }(\\rho ) \\ast \\rho ) =\\mathcal {J}^{\\beta }(\\rho ^{\\ast 2})$ Proof of Lemma 1.", "Part (a) follows from the definition of the random integrals and is a particular form (take matrix $Q=I$ ) of Theorem 1.3 (a) in Jurek (1988).", "For the claim (b) note that for each fixed $y$ we have $\\log \\widehat{\\mathcal {J}^{\\beta }(\\nu )}(s^{1/{\\beta }}y)=s^{-1}\\int _0^s\\log \\hat{\\nu }(r^{1/{\\beta }}y)dr,\\ \\ s \\in \\mathbb {R}^+.$ This gives the formula in (b), similarly as in Jurek (1988), p. 484.", "Equalities in (c) and (d) are also consequences of (a); cf.", "Jurek(1988), Theorem 1.3 (a) and (c).", "Finally, for the identity in (e) note, using (14) that $\\log \\Big (\\mathcal {J}^{2\\beta }\\big (\\mathcal {J}^{\\beta }(\\rho )\\ast \\rho \\big )\\Big )^{\\widehat{}}(y)=\\int _0^1\\log \\big (\\mathcal {J}^{\\beta }(\\rho )\\ast \\rho \\big )\\Big )^{\\widehat{}}(s^{1/2\\beta }y)=\\\\ \\int _0^1 \\int _0^1\\log \\hat{\\rho }(t^{1/\\beta }s^{1/2\\beta }y)dt\\,ds +\\int _0^1\\log \\hat{\\rho }(s^{1/2\\beta }y)ds \\ \\ \\ \\ (\\mbox{put} \\ t^2s=:u) \\ \\ \\ \\ \\ \\\\= \\int _0^1 1/2 \\int _0^s\\log \\hat{\\rho }(u^{1/2\\beta }y)(us)^{-1/2}du\\,ds + \\int _0^1\\log \\hat{\\rho }(s^{1/2\\beta }y)ds \\\\=\\int _0^1\\log \\hat{\\rho }(u^{1/2\\beta }y)\\,u^{-1/2} \\big ( 1/2\\int _u^1s^{-1/2}ds \\big )\\,du + \\int _0^1\\log \\hat{\\rho }(s^{1/2\\beta }y)ds= \\\\\\int _0^1 u^{-1/2}\\log \\hat{\\rho }(u^{1/2\\beta }y)du= 2 \\int _0^1\\log \\hat{\\rho }(u^{1/2\\beta }y)d(u^{1/2})= \\\\ \\int _0^1\\log \\hat{\\rho ^{\\ast 2}}(s^{1/\\beta }y)ds = \\log \\,(\\mathcal {J}^\\beta (\\rho ^{\\ast 2})){\\hat{}}\\,(y), \\ \\ \\ \\ \\ \\ \\ $ which completes the proof of Lemma 1.", "Proof of Proposition 1.", "Suppose we have that $J^\\beta (\\rho )\\rho =J^\\beta (\\nu )$ .", "Then by the properties described in Lemma 1, $J^{\\beta }\\bigl (J^{2\\beta }(\\nu )\\bigr )=J^{2\\beta }\\bigl (J^\\beta (\\nu )\\bigr )=J^{2\\beta }\\bigl (J^\\beta (\\rho )\\rho \\bigr )=J^\\beta (\\rho ^{\\ast 2}),$ and hence $\\rho ^{\\ast 2}= J^{2\\beta }(\\nu )$ , i.e., $\\rho =(J^{2\\beta }(\\nu ))^{\\ast 1/2}= J^{2\\beta }(\\nu ^{\\ast 1/2})$ , which proves the necessity.", "The converse claim also follows from the above reasoning.", "For the last part, let us note that if $\\tilde{\\mu }=J^{\\beta }(\\nu )\\in \\mathcal {U}_{\\beta }$ then taking $\\rho :=J^{2\\beta }(\\nu ^{\\ast 1/2})\\in \\mathcal {U}_{2\\beta }$ one gets the required equality.", "Proof of Corollary 1.", "Note that $\\nu =\\mathcal {J}^\\beta \\in \\mathcal {U}^f_\\beta \\ \\mbox{iff} \\ J^\\beta (\\rho )\\ast \\rho \\in \\mathcal {U}_\\beta \\ \\ \\mbox{iff} \\ \\rho \\in \\mathcal {U}_{2\\beta }$ , by (3) in Proposition 1.", "Last equality is from the Example (a) from Czyďż̋ewska-Jankowska and Jurek (2008).", "Similarly one gets part (b) using Proposition 1 and Lemma 1 (e).", "Proposition 1 can be expressed in terms of characteristic functions as follows: Corollary 4 In order that $\\exp {\\int ^{1}_{0}\\log {\\hat{\\rho }\\left(t^{1/\\beta }y\\right)}dt}\\;\\cdot \\;\\hat{\\rho }\\left(y\\right)=\\exp {\\int ^{1}_{0}\\log {\\hat{\\nu }\\left( t^{1/\\beta }y\\right)}dt},\\qquad y\\in \\mathbb {R}^d \\ (\\mbox{or}\\ E^{\\prime })$ for some $\\mu $ and $\\rho $ in ID it is necessary and sufficient that $\\hat{\\rho }\\left(y\\right)=\\exp {\\int ^{1}_{0}\\tfrac{1}{2}\\log {\\hat{\\nu }\\left( t^{1/(2\\beta )}y\\right)}dt};$ or in terms of the Lévy spectral measures as: Corollary 5 In order to have the equality $\\int ^{1}_{0}M(t^{-1/\\beta }A)\\;dt+M(A)=\\int ^{1}_{0}G(t^{-1/\\beta }A)\\,dt,\\qquad \\textrm {for each Borel }A\\in \\mathcal {B}_0 ,$ for some Lévy spectral measures $M$ and $G$ , it is necessary and sufficient that $M(A)=\\int ^{1}_{0}\\tfrac{1}{2}G(t^{-1/(2\\beta )}A)\\,dt, \\qquad \\textrm {for each} \\ \\ A\\in \\mathcal {B}_0 ,$ because if $\\rho =[a, S, M]$ then the left hand side in the Corollary is the Lévy spectral measure of $J^\\beta (\\rho )\\rho $ .", "For references let state the following 5Lemma (i) If $\\nu = [a,R,M]$ and $J^\\beta (\\nu )=[a^{(\\beta )},R^{(\\beta )},M^{(\\beta )}]$ then $a^{(\\beta )}:=\\tfrac{\\beta }{(1+\\beta )}\\;a+\\int ^{1}_{0}t^{1/\\beta }\\!\\!\\int _{\\left\\lbrace 1<\|\\!\|x\|\\!\|\\le t^{-1/\\beta }\\right\\rbrace }x\\;M(dx)\\;dt \\\\=\\frac{\\beta }{\\beta +1}\\,(a+\\int _{(\|\|x\|\|>1)}x\\,\|\|x\|\|^{-1-\\beta }M(dx)\\,); \\ \\ \\ \\ \\ \\ R^{(\\beta )}:=\\tfrac{\\beta }{2+\\beta }\\,R; \\qquad \\qquad \\\\M^{(\\beta )}(A):=\\int ^{1}_{0}T_{t^{1/\\beta }}\\;M(A)\\;dt,\\ \\ \\textrm {foreach}\\,\\,A\\in \\mathcal {B}_0 .", "\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad $ (ii) For $\\beta >0$ , we have that $J^\\beta (\\nu )\\in ID_{\\log }$ if and only if $\\nu \\in ID_{\\log }$ .", "Proof of Lemma 2.", "(i) Uniqueness of the triplets: a shift vector a , Gaussian covariance R and Lévy spectral measure M in the Lévy-Khintchine formula and equation (12) in Lemma 1 give the expressions for $a^{(\\beta )}, R^{(\\beta )}$ and for $M^{(\\beta )}$ ; for details cf.", "formulas (1.10), (1.11) and (1.12) in Jurek (1988), with the matrix $Q=I$ .", "For part (ii), note that since we have $\\int \\limits _{\\left\\lbrace \\left\\Vert x\\right\\Vert >1\\right\\rbrace }\\!\\!\\!\\!\\!\\log {\|\\!\|x\|\\!\|}\\,M^{(\\beta )}(dx)=\\int \\limits ^{1}_{0}\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>1\\right\\rbrace }\\!\\!\\!\\!\\!\\log {\|\\!\|x\|\\!\|}\\,T_{t^{1/\\beta }}M(dx)\\;dt=\\nonumber \\\\=\\int \\limits ^{1}\\limits _{0}\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|t^{1/\\beta }x\|\\!\|>1\\right\\rbrace }\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\log {\|\\!\|t^{1/\\beta }x\|\\!\|}\\,M(dx)\\,dt=\\int \\limits ^{1}\\limits _{0}\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>\\tfrac{1}{t^{1/\\beta }}\\right\\rbrace }\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\log {(t^{1/\\beta }\|\\!\|x\|\\!\|)}\\,M(dx)\\,dt=$ $=\\!\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>1\\right\\rbrace }\\int \\limits ^{1}\\limits _{\|\\!\|x\|\\!\|^{-1/\\beta }}\\!\\!\\!\\!\\!\\log {(t^{1/\\beta }\|\\!\|x\|\\!\|)}\\,dt\\,M(dx)=\\!\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>1\\right\\rbrace }\\!\\!\\!\\!\\!\\tfrac{1}{\|\\!\|x\|\\!\|^\\beta }\\!\\!\\int \\limits ^{\|\\!\|x\|\\!\|}\\limits _{\|\\!\|x\|\\!\|^{1-1/\\beta ^2}}\\!\\!\\!\\!\\!\\!\\!\\beta w^{\\beta -1}\\,\\log {w}\\,dw\\,M(dx)=\\nonumber \\\\=\\!\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>1\\right\\rbrace } \\!\\!\\!\\!\\!\\tfrac{1}{\|\\!\|x\|\\!\|^\\beta }\\biggl [w^\\beta \\log {w}-\\tfrac{1}{\\beta }w^\\beta \\biggr \|^{w=\|\\!\|x\|\\!\|}_{w=\|\\!\|x\|\\!\|^{1-1/\\beta ^2}}\\biggr ]\\,M(dx)=\\nonumber \\\\=\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>1\\right\\rbrace }\\!\\!\\!\\!\\!\\log {\|\\!\|x\|\\!\|}\\,M(dx)-\\!\\!\\!\\!\\!\\int \\limits _{\\left\\lbrace \|\\!\|x\|\\!\|>1\\right\\rbrace }[ \\tfrac{1}{\\beta }+\\tfrac{1}{\|\\!\|x\|\\!\|^{1/\\beta }}\\bigl ((1-\\tfrac{1}{\\beta ^2})\\log {\|\\!\|x\|\\!\|}-\\tfrac{1}{\\beta }\\bigr )]\\,M(dx)\\;\\nonumber $ and the last integral is finite (the integrand function is bounded on $(\|\|x\|\|>1)$ and Lévy spectral measures M are finite on the complements of all neighborhoods of zero; comp.", "(11)), therefore from the above we conclude that $[\\,\\int \\limits _{\\left\\lbrace \\left\\Vert x\\right\\Vert >1\\right\\rbrace }\\!\\!\\!\\!\\!\\log {\|\\!\|x\|\\!\|}\\,M^{(\\beta )}(dx)< \\infty ] \\ \\ \\mbox{iff} \\ \\ [\\int \\limits _{\\left\\lbrace \\left\\Vert x\\right\\Vert >1\\right\\rbrace }\\!\\!\\!\\!\\!\\log {\|\\!\|x\|\\!\|}\\,M(dx)< \\infty ].$ But since the function $u\\rightarrow \\log (1+u)$ , for $u>0$ , is sub-additive therefore we may apply Proposition 1.8.13 in Jurek-Mason (1993) and infer the claim (ii).", "This completes the proof of Lemma 2.", "Proof of Proposition 2.", "If $\\nu \\in ID_{\\log }$ then, by Lemma 2, $\\mathcal {J}^{\\beta }(\\mu )\\in ID_{\\log }$ and thus the improper random integral $\\int _0^{\\infty }e^{-s}dY_{\\mathcal {J}^\\beta (\\nu )}(s)$ converges (is well-defined) almost surely (in probability and in distribution); cf.", "Jurek-Vervaat (1983), Lemma 1.1 or Jurek (1985).", "Hence and Lemma 1(a) we get that $\\log {\\bigl (I\\left(J^\\beta \\left(\\nu \\right)\\right)}\\bigr )^{\\widehat{}}\\left(y\\right) =\\int _0^{\\infty }\\log \\widehat{\\mathcal {J}^{\\beta }(\\nu )}(e^{-s}y)ds =\\int _0^{\\infty }\\int _0^1 \\log \\hat{\\nu }(v^{1/\\beta }e^{-s}y) dv ds \\\\= \\int _0^1\\int _0^{v^{1/\\beta }}\\log \\hat{\\nu }(uy)u^{-1}du= \\int _0^1(\\int _{u^\\beta }^1dv)\\log \\hat{\\nu }(uy)u^{-1}du dv =\\\\\\int ^{1}_{0}\\log {\\hat{\\nu }\\left(uy\\right)}\\bigl (u^{-1}-u^{\\beta -1}\\bigr )\\,du=\\int ^{\\infty }_{0}\\log {\\hat{\\nu }\\bigl (e^{\\displaystyle -s}y\\bigr )}\\bigl (1-e^{-\\beta s}\\bigr )\\,ds=\\nonumber \\\\= \\int ^{\\infty }_{0}\\log {\\hat{\\nu }\\bigl (e^{{\\displaystyle -s}}y\\bigr )\\,d\\sigma _{\\beta }(s)}.$ On the other hand, the random integral $\\int _{0}^{\\infty } e^{-s}\\,dY_{\\nu }(\\sigma _{\\beta }(s)):\\,=\\,\\lim _{b\\rightarrow \\infty } \\int _0^b e^{-s}\\,dY_{\\nu }(\\sigma _{\\beta }(s))\\ \\mbox{exists indistribution},$ (or in probability or almost surely) because the function $y\\rightarrow \\lim _{b\\rightarrow \\infty } {\\Bigl ({\\mathcal {L}\\bigl (\\int _{0}^b{e^{-s}}\\,dY_{\\nu }(\\sigma _{\\beta }(s))}\\Bigr )^{\\widehat{}}(y)}\\\\= \\lim _{b\\rightarrow \\infty }\\exp \\int ^{b}_{0}\\log {\\hat{\\nu }(e^{-s}}y)\\,d\\sigma _{\\beta }(s)=\\exp \\int ^{\\infty }_{0}\\log {\\hat{\\nu }(e^{-s}}y)\\,d\\sigma _{\\beta }(s),$ is a characteristic function.", "Moreover, we have that $I\\bigl (J^\\beta \\bigl (\\nu \\bigr )\\bigr )=\\mathcal {L}\\bigl (\\int _{0}^{\\infty }{e^{-s}}\\,dY_{\\nu }\\bigl (\\sigma _{\\beta }(s)\\bigr )\\bigr ),$ which completes a proof of Proposition 2.", "17Remark Our argument above is valid for infinite dimensional Banach spaces, although one should be aware that in that generality convergence of characteristic functions to a characteristic function does not guarantee weak convergence of corresponding distributions ( probability measures); cf.", "Araujo-Gine (1980), Theorem 4.19 on p. 29.", "Proof of Corollary 3.", "Recall that by definition $L^f=\\lbrace \\mathcal {I}(\\mu ): \\mathcal {I}(\\mu )\\ast \\mu \\in L\\rbrace $ .", "However, in view of Proposition 1 (ii) in Iksanov-Jurek-Schreiber (2004) we have $L^f=\\mathcal {I}(\\mathcal {J}(ID_{\\log })$ .", "Consequently, taking $\\beta =1$ in Proposition 2 we get the corollary.", "References [1] T. Aoyama and M. Maejima (2007).", "Characterizations of subclasses otf type G distributions on $\\mathbb {R}^d$ by stochastic random integral representation,Bernoulli, vol.", "13, pp.", "148-160.", "[2] A. Araujo and E. Gine (1980).", "The central limit theorem for real and Banach valued random variables.", "John Wiley & Sons, New York.", "[3] R. Cuppens (1975).", "Decomposition of multivariate probabilities.", "Academic Press, New York.", "[4] A. Czyzewska-Jankowska and Z. J Jurek (2008).", "A note on a composition of two random integral mappings $J^\\beta $ and some examples, preprint.", "[5] A. M. Iksanov, Z. J. Jurek and B. M. Schreiber (2004).", "A new factorization property of the selfdecomposable probability measures, Ann.", "Probab.", "vol.", "32, Nr 2, str.", "1356-1369.", "[6] Z. J. Jurek (1985).", "Relations between the s-selfdecomposable and selfdecomposable measures.", "Ann.", "Probab.", "vol.13, Nr 2, str.", "592-608.", "[7] Z. J. Jurek (1988).", "Random Integral representation for Classes of Limit Distributions Similar to Lavy Class $L_{0}$ , Probab.", "Th.", "Fields.", "78, str.", "473-490.", "[8] Z. J. Jurek and J. D. Mason (1993).", "Operator-limit distributions in probability theory.", "John Wiley &Sons, New York.", "[9] Z. J. Jurek and W. Vervaat (1983).", "An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrscheinlichkeitstheorie verw.", "Gebiete, 62, pp.", "247-262.", "[10] Z. J. Jurek and M. Yor (2004).", "Selfdecomposable laws associated with hyperbolic functions, Probab.", "Math.", "Stat.", "24, no.1, pp.", "180-190.", "[11] P. Lévy (1951).", "Wiener's random functions, and other Laplacian random functions, Proc.", "Second Berkeley Symposium Math.", "Statist.", "Probab.", "str. 171-178.", "Univ.", "California Press, Berkeley.", "[12] Ju.", "V. Linnik and I. V. Ostrovskii (1977).", "Decomposition of Random Variables and Vectors.", "American Mathematical Society, Providence, Rhode Island.", "[13] K. R. Parthasarathy (1967).", "Probability measures on metric spaces.", "Academic Press, New York and London.", "Institute of Mathematics University of Wrocław Pl.Grunwaldzki 2/4 50-384 Wrocław, Poland e-mail: [email protected] or [email protected] www.math.uni.wroc.pl/$^{\\sim }$ zjjurek" ] ]	1403.0089
[ [ "Infinitely many solutions to a fractional nonlinear Schr\\\"{o}dinger\n equation" ], [ "Abstract This paper considers the fractional Schr\\\"{o}dinger equation \\begin{equation}\\label{abstract} (-\\Delta)^s u + V(\|x\|)u-u^p=0, \\quad u>0, \\quad u\\in H^{2s}(\\R^N) \\end{equation} where $0<s<1$, $1<p<\\frac{N+2s}{N-2s}$, $V(\|x\|)$ is a positive potential and $N\\geq 2$.", "We show that if $V(\|x\|)$ has the following expansion: \\[ V(\|x\|)=V_0 + \\frac{a}{\|x\|^m} + o\\left(\\frac{1}{\|x\|^m}\\right) \\qquad \\mbox{as} \\ \|x\| \\rightarrow +\\infty, \\] in which the constants are properly assumed, then (\\ref{abstract}) admits infinitely many non-radial solutions, whose energy can be made arbitrarily large.", "This is the first result for fractional Schr\\\"{o}dinger equation.", "The $s=1$ case corresponds to the known result in Wei-Yan \\cite{WY}." ], [ "Introduction and main results", "The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics.", "The nonlinear fractional nonlinear Schrödinger equation is as follows: $i\\psi _t=(-\\Delta )^s\\psi + \\widetilde{V}(x) \\psi -\|\\psi \|^{p-1}\\psi $ where $(-\\Delta )^s$ ($0<s<1$ ) denotes the classical fractional Laplacian, $\\tilde{V}$ is a bounded potential and $p>1$ .", "We are interested in finding standing wave solutions, which are solutions of the form $\\psi (x,t)=u(x)e^{i\\lambda t}$ with the function $u$ real-valued.", "Let $V(x)=\\widetilde{V}(x) + \\lambda $ , then $\\psi $ is a solution of (REF ) if and only if $u$ solves the following equation $(-\\Delta )^su + V(x)u -\|u\|^{p-1}u=0 \\qquad \\mbox{in }\\mathbb {R}^N.$ A similar problem to (REF ) is the following fractional scalar field equation $(-\\Delta )^s u + u=Q(x)u^p, \\qquad u>0 \\qquad \\mbox{in} \\ \\mathbb {R}^N.$ It is also absorbing to study the singularly perturbed problem $\\varepsilon ^{2s}(-\\Delta )^su + V(x)u -\|u\|^{p-1}u=0 \\qquad \\mbox{in} \\ \\mathbb {R}^N$ or $\\varepsilon ^{2s}(-\\Delta )^s u + u=Q(x)u^p, \\qquad u>0 \\qquad \\mbox{in} \\ \\mathbb {R}^N$ where $\\varepsilon >0$ is a small parameter.", "The natural place to look for solutions that decay at infinity is the space $H^{2s}(\\mathbb {R}^N)$ of all functions $u\\in L^2(\\mathbb {R}^N)$ such that $\\int _{\\mathbb {R}^N} (1+\|\\xi \|^{4s})\|\\hat{u}(\\xi )\|^2 \\mathrm {d}\\xi < \\infty ,$ where $\\widehat{}$ denotes the Fourier transform.", "The fractional Laplacian $(-\\Delta )^s u$ for $u\\in H^{2s}(\\mathbb {R}^N)$ is defined by $\\widehat{(-\\Delta )^s u} (\\xi ) = \|\\xi \|^{2s} \\hat{u}(\\xi ) .$ For (REF )–(REF ), an interesting problem is to find solutions with a spike pattern concentrating around some points.", "As for the standard case $s=1$ of (REF ) or (REF ), this has been the topic of many works relating the concentration points with critical points of the potential, starting in 1986 from the pioneering work Floer-Weinstein [17].", "Later many works show that the number of the critical points of $V(x)$ (or $Q(x)$ ) (see for example [1], [6], [7], [11], [12], [13], [14], [24], [27]), the type of the critical points of $V(x)$ (or $Q(x)$ ) (see for example [9], [21], [23], [30]), and the topology of the level set $V(x)$ (or $Q(x)$ ) (see for example [2], [3], [8], [16]), can effect the number of solutions of (REF ) (or (REF )).", "It is now known that when the parameter $\\varepsilon $ goes to zero, the number of the solutions may tend to infinity.", "For the $s=1$ case of (REF ) or (REF ), in 2010 Wei-Yan [28] get a multiplicity result under some symmetry assumption of $V(x)$ near the infinity.", "Recently we are told that del Pino-Wei-Yao [15] get the similar result with a weaker symmetry assumption on $V(x)$ .", "As to the fractional case $0< s<1$ , very few is known.", "Recently Dávila-del Pino-Wei [10] obtained the first result of spike pattern for the fractional Schrödinger equation (REF ) with $1 <p < \\frac{N+2s}{N-2s}$ .", "A natural question is can we get multiplicity result for (REF ) (or(REF )) with $0< s < 1$ ?", "What is the situation in the fractional case?", "In this paper we will give an affirmative answer!", "This paper is concerned about the following fractional Laplacian problem $(-\\Delta )^su + V(\|x\|)u -\|u\|^{p-1}u=0, \\quad u>0, \\qquad u\\in H^{2s}(\\mathbb {R}^N)$ where $0<s<1$ , $1<p<\\frac{N+2s}{N-2s}$ and $N\\ge 2$ .", "We suppose that $V(x)$ satisfies the following assumption.", "Assumption $\\mathbf {\\mathcal {V}}$ .", "$V$ is positive and radially symmetric, i.e.", "$V(x)=V(\|x\|)>0$ and there are constants $a>0$ and $V_0>0$ such that $V(\|x\|)=V_0 + \\frac{a}{\|x\|^m} + o\\left( \\frac{1}{\|x\|^m}\\right), \\qquad \\mbox{as} \\ \|x\| \\rightarrow +\\infty ,$ where $\\max \\left\\lbrace 0,\\ (N+2s)\\left[1-(p-1)N-2ps + \\max \\left\\lbrace s, \\ p-\\frac{N}{2}\\right\\rbrace \\right]\\right\\rbrace < m < N+2s.$ Without loss of generality, we may assume $V_0=1$ for the sake of simplicity.", "It's easy to see that $\\left[1-(p-1)N-2ps + \\max \\left\\lbrace s, \\ p-\\frac{N}{2}\\right\\rbrace \\right]<1 \\qquad \\mbox{for any } p>1, \\ s \\in (0, 1).$ By direct computations we find that in three dimension case, if $1 + \\frac{1-s}{3+2s}<p<\\frac{3+2s}{3-2s}, \\qquad \\frac{1}{6} < s \\le \\frac{1}{2},$ then we just need that $m\\in (0, N+2s)$ .", "The aim of this paper is to obtain infinitely many non-radial positive solutions to (REF ), whose energy may be arbitrarily large.", "Our main result in this paper is stated in the following theorem.", "Theorem 1.1 If $V(\|x\|)$ satisfies the assumption $\\mathcal {V}$ , then the problem (REF ) admits infinitely many non-radial positive solutions.", "Moreover, the energy of these solutions may be arbitrarily large.", "Remark 1.1 The condition on potential $V(x)$ is more general than that of $V$ in [28] for $s=1$ .", "The main reason is that in our case we can deduce the exact relationship between the radius and the number of spikes, while in [28], the authors can't solve it exactly using the leading terms of energy.", "We believe that the symmetry on $V$ is technical and then make the following conjecture.", "Conjecture 1.1 Problem (REF ) has infinitely many solutions if there are constants $a>0, m \\in (0, N+2s)$ and $V_0>0$ , such that $V(x)=V_0 + \\frac{a}{\|x\|^m} + o\\left( \\frac{1}{\|x\|^m}\\right), \\qquad \\mbox{as} \\ \|x\| \\rightarrow +\\infty .$ Remark 1.2 Using the same argument, we can prove that if $Q(\|x\|)=Q_0 - \\frac{a}{\|x\|^m}+ o\\left( \\frac{1}{\|x\|^m}\\right) \\qquad \\mbox{as} \\ \|x\| \\rightarrow +\\infty ,$ where the constants are similarly assumed, then problem (REF ) has infinitely many positive non-radial solutions.", "Before close this introduction, let us outline the main idea in the proof of Theorem REF .", "Our aim is to construct solutions with a large number of bumps near the infinity.", "Since $\\lim _{\|x\| \\rightarrow +\\infty } V(\|x\|)=1,$ we will use the solution of $(-\\Delta )^su +u -\|u\|^{p-1}u=0, \\qquad u>0, \\qquad u\\in H^{2s}(\\mathbb {R}^N)$ to build up the approximate solution for problem (REF ).", "It is known (see for instance [20]) the existence of a positive, radial least energy solution $w(x)$ , which gives the lowest possible value for the energy $J_1(v)=\\frac{1}{2}\\int _{\\mathbb {R}^N} v(-\\Delta )^s v + \\frac{1}{2}\\int _{\\mathbb {R}^N} v^2 - \\frac{1}{p+1}\\int _{\\mathbb {R}^N} \|v\|^{p+1}$ among all nontrivial solutions of (REF ).", "An important property, which has been proven recently by Frank-Lenzman-Silvestre [20] (see also [4], [19]), is that there exists a radial least energy solution which is non-degenerate, in the sense that the space of solutions of the equation $(-\\Delta )^s \\phi + \\phi -pw^{p-1}\\phi =0, \\qquad \\phi \\in H^{2s}(\\mathbb {R}^N)$ consists exactly of the linear combinations of the translation-generators $\\frac{\\partial w}{\\partial x_j}, j=1, \\ldots , N.$ Also we have the following behavior for $w(x)$ ([20]): $w^{\\prime }(\|x\|)<0; \\qquad w(\|x\|)=\\frac{A}{\|x\|^{N+2s}}\\left(1+o(1)\\right), \\qquad A>0, \\qquad \\mbox{ as} \\quad \|x\| \\rightarrow +\\infty .$ Let $q_j = \\left(r\\cos \\frac{2(j-1)\\pi }{k}, r\\sin \\frac{2(j-1)\\pi }{k}, \\mathbf {0}\\right), \\qquad j=1, \\ldots , k,$ where $\\mathbf {0}$ is the zero vector in $\\mathbb {R}^{N-2}, r \\in \\left[\\frac{1}{C_0}k^{\\frac{N+2s}{N+2s-m}}, C_0k^{\\frac{N+2s}{N+2s-m}} \\right]$ for large positive constant $C_0$ independent of $k$ .", "Set $x=(x^{\\prime }, x^{\\prime \\prime })$ , $x \\in \\mathbb {R}^2$ , $x^{\\prime \\prime } \\in \\mathbb {R}^{N-2}$ .", "Define $H_s=\\Big \\lbrace u ~\\Big \|~ u \\in H^{2s}(\\mathbb {R}^N),\\ u \\ \\mbox{is even in}\\ x_h\\ (h=2, \\ldots , N)\\ \\text{and}\\\\u\\left(r\\cos \\theta , r\\sin \\theta , x^{\\prime \\prime }\\right)=u\\left(r\\cos \\left(\\theta + \\frac{2\\pi j}{k}\\right), r\\sin \\left(\\theta + \\frac{2\\pi j}{k}\\right), x^{\\prime \\prime }\\right),\\ j=1, \\ldots , k-1 \\Big \\rbrace .$ Define $W(x)=\\sum \\limits _{j=1}^k w(x-q_j),$ then Theorem REF is a direct consequence of the following result.", "Theorem 1.2 Suppose $V(\|x\|)$ satisfies the assumption $\\mathcal {V}$ .", "Then there is an integer $k_0>0$ , such that for any integer $k \\ge k_0$ , Problem (REF ) has a solution $u_k$ of the form $u_k(x) =W(x) + \\varphi (x),$ where $\\varphi (x) \\in H_s$ and the energy at $u_k$ goes to infinity as $k $ goes to infinity.", "Remark 1.3 Note that there is no parameter in the problem (REF ).", "Using the number of spikes as parameter, we get the first multiplicity result for fractional nonlinear Schrödinger equation, which seems a new phenomenon for fractional nonlinear Schrödinger equation.", "Remark 1.4 Since the approximate solution has polynomial decay, we should deal with every term carefully in the calculous which makes our proof a little bit complicated.", "By the way, in [28], the approximation has exponential decay.", "The paper is organized as follows.", "In Section , we introduce some preliminaries.", "In Section , the ansatz is established.", "In Section , we deal with the corresponding linearized problem.", "In Section , the nonlinear problem is considered and the proof of Theorem REF is given.", "Finally some important estimates and the expansion of the energy are stated in Section .", "Notations.", "In what follows, the symbol $C$ always denotes a various constant independent of $k$ ." ], [ "Preliminaries", "In this section, we get a useful a-priori estimate for a related linear equation.", "Let $0 < s <1$ .", "Various definitions of the fractional Laplacian $(-\\Delta )^s \\varphi $ of a function $\\varphi $ defined in $\\mathbb {R}^N$ are available, depending on its regularity and growth properties, see for example [10].", "A useful (local) representation given by Caffarelli and Silvestre [5], is via the following boundary value problem in the half space $\\mathbb {R}_+^{N+1}=\\lbrace (x, y)~\|~ x\\in \\mathbb {R}^N, y>0 \\rbrace $ : $\\nabla \\cdot \\left(y^{1-2s} \\nabla \\tilde{\\varphi }\\right)=0 \\quad \\mbox{in} \\ \\mathbb {R}_+^{N+1}, \\qquad \\tilde{\\varphi }(x, 0)=\\varphi (x) \\quad \\mbox{on} \\ \\mathbb {R}^N.$ Here $\\tilde{\\varphi }$ is the $s$ -harmonic extension of $\\varphi $ , explicitly given as a convolution integral with the $s$ -Poisson kernel $p_s(x, y)$ , $\\tilde{\\varphi }(x, y)=\\int _{\\mathbb {R}^N} p_s(x-z, y)\\varphi (z) dz,$ where $p_s(x, y)=c_{N,s}\\frac{y^{4s-1}}{(\|x\|^2 + \|y\|^2)^{\\frac{N-1+4s}{2}}}$ and $c_{N,s}$ achieves $\\int _{\\mathbb {R}^N} p_s(x, y)dx=1$ .", "Then under suitable regularity, $(-\\Delta )^s \\varphi $ is the Dirichlet-to-Neumann map for this problem, that is $(-\\Delta )^s \\varphi (x) =\\lim _{y \\rightarrow 0^+} y^{1-2s}\\partial _y \\tilde{\\varphi }(x,y).$ For $m>0$ and $g \\in L^2(\\mathbb {R}^N)$ , let us consider now the equation $(-\\Delta )^s \\varphi +m \\varphi =g \\qquad \\mbox{in} \\ \\mathbb {R}^N.$ Then in terms of Fourier transform, for $\\varphi \\in L^2(\\mathbb {R}^N)$ , this problem reads $\\left(\|\\xi \|^{2s} + m \\right)\\hat{\\varphi }=\\hat{g}$ and has a unique solution $\\varphi \\in H^{2s}(\\mathbb {R}^N)$ given by the convolution $\\varphi (x)=T_m(g):=\\int _{\\mathbb {R}^N} k(x-z)g(z)dz$ where the Fourier transform of $k$ is $\\hat{k}(\\xi ) =\\frac{1}{\|\\xi \|^{2s}+m}.$ Then we have the following main properties of the fundamental solution $k(x)$ (see for example [20], [18]): $k(x)$ is radially symmetric and positive, $k \\in C^{\\infty }(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ satisfying $\\begin{split}&(i) \\qquad \|k(x)\| + \|x\|\|\\nabla k(x)\| \\le \\frac{C}{\|x\|^{N-2s}} \\qquad \\quad \\mbox{for all} \\quad \|x\| \\le 1;\\\\&(ii) \\qquad \\lim _{\|x\|\\rightarrow \\infty } k(x)\|x\|^{N+2s}=\\alpha >0;\\\\&(iii) \\qquad \|x\|\|\\nabla k(x)\| \\le \\frac{C}{\|x\|^{N+2s}} \\qquad \\qquad \\qquad \\mbox{for all} \\quad \|x\| \\ge 1.\\end{split}$ Using (REF ) written in weak form, $\\varphi $ can be characterized by $\\varphi (x) =\\tilde{\\varphi }(x, 0)$ in trace sense, where $\\tilde{\\varphi } \\in H$ is the unique solution of $\\iint _{\\mathbb {R}_+^{N+1}} \\nabla \\tilde{\\varphi }(x, y) \\cdot \\nabla \\phi (x, y) y^{1-2s}\\mathrm {d}x\\mathrm {d}y + m\\int _{\\mathbb {R}^N} \\varphi (x) \\phi (x,0) dx=\\int _{\\mathbb {R}^N} g(x)\\phi (x,0)dx,$ for all $ \\phi \\in H$ , where $H$ is the Hilbert space of functions $\\phi \\in H^1_\\text{loc}(\\mathbb {R}_+^{N+1})$ such that $\\Vert \\phi \\Vert _H^2:=\\iint _{\\mathbb {R}_+^{N+1}} \| \\nabla \\phi (x, y)\|^2 y^{1-2s}dxdy + m\\int _{\\mathbb {R}^N} \|\\phi (x,0)\|^2 dx < +\\infty ,$ or equivalent the closure of the set of all functions in $C_c^\\infty (\\overline{\\mathbb {R}_+^{N+1}})$ under this norm.", "For our purpose, we need the following four lemmas, see [10]: Lemma 2.1 Let $g \\in L^2(\\mathbb {R}^N)$ .", "Then the unique solution $\\tilde{\\varphi } \\in H$ of the problem (REF ) is given by the s-harmonic extension of the function $\\varphi =T_m(g)$ .", "Lemma 2.2 Let $0 \\le \\mu < N+2s$ .", "Then there exists a positive constant $C$ such that $\\Vert (1 + \|x\|)^\\mu T_m(g)\\Vert _{L^\\infty (\\mathbb {R}^N)} \\le C \\Vert (1 + \|x\|)^\\mu g\\Vert _{L^\\infty (\\mathbb {R}^N)}.$ Lemma 2.3 Assume that $g \\in L^2(\\mathbb {R}^N)\\cap L^\\infty (\\mathbb {R}^N)$ .", "Then the following holds: if $\\varphi =T_m(g)$ then there is a $C>0$ such that $\\sup \\limits _{x \\ne y} \\frac{\\left\|\\varphi (x) - \\varphi (y)\\right\|}{\|x-y\|^\\beta } \\le C \\Vert g\\Vert _{L^\\infty (\\mathbb {R}^N)}$ where $\\beta =\\min \\lbrace 1, 2s \\rbrace $ .", "Lemma 2.4 Let $\\varphi \\in H^{2s}$ be the solution of $(-\\Delta )^s \\varphi + W(x) \\varphi =g \\qquad \\mbox{in}\\ \\mathbb {R}^N$ with bounded potential $W$ .", "If $\\inf _{x \\in \\mathbb {R}^N} W(x)=: m>0$ , $g \\ge 0$ .", "Then $\\varphi \\ge 0$ in $\\mathbb {R}^N$ .", "Using these lemmas, we obtain an a-priori estimate for any solution $\\varphi \\in L^2(\\mathbb {R}^N)\\cap L^\\infty (\\mathbb {R}^N)$ of (REF ).", "Lemma 2.5 Let $W$ be a continuous function, and assume that for $k$ points $q_1, \\ldots , q_k$ , there is an $R>0$ and $B=\\cup _{j=1}^k B_R(q_j)$ such that $\\inf \\limits _{x \\in \\mathbb {R}^N \\backslash B} W(x)=:m >0.$ Then given any number $\\frac{N}{2} < \\mu < N+2s$ , there exists a uniform positive constant $C=C(\\mu , R)$ independent of $k$ such that for any $\\varphi \\in H^{2s}(\\mathbb {R}^N)\\cap L^\\infty (\\mathbb {R}^N)$ and $g$ satisfying (REF ) with $\\Vert \\rho ^{-1} g\\Vert _{L^\\infty (\\mathbb {R}^N)} < +\\infty , \\qquad \\text{where }\\rho (x)=\\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^\\mu },$ we have the validity of the estimate $\\Vert \\rho ^{-1} \\varphi \\Vert _{L^\\infty (\\mathbb {R}^N)} \\le C \\left(\\Vert \\varphi \\Vert _{L^\\infty (B)} + \\Vert \\rho ^{-1} g\\Vert _{L^\\infty (\\mathbb {R}^N)}\\right).$ We rewrite (REF ) as $(-\\Delta )^s \\varphi + \\widetilde{W} \\varphi =\\tilde{g},$ where $\\tilde{g}=(m-W)\\chi _B \\varphi + g$ , $\\widetilde{W}=m\\chi _B + W(1-\\chi _B)$ and $\\chi _B$ is the characteristic function on $B$ .", "By careful calculation, it is deduced that $\|\\tilde{g}(x)\| \\le C\\Vert \\varphi \\Vert _{L^\\infty (B)} + \\Vert \\rho ^{-1}g\\Vert _{L^\\infty (\\mathbb {R}^N)} \\rho \\le M\\rho $ where $\\begin{split}M=&\\ C\\Vert \\varphi \\Vert _{L^\\infty (B)}\\sup \\limits _{x \\in B}\\left( \\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^\\mu } \\right)^{-1} + \\Vert \\rho ^{-1}g\\Vert _{L^\\infty (\\mathbb {R}^N)}\\\\\\le &\\ C \\Vert \\varphi \\Vert _{L^\\infty (B)}\\max _{1\\le j\\le k} \\sup \\limits _{x \\in B_R(q_j)}\\left( \\frac{1}{(1 + \|x-q_j\|)^\\mu } \\right)^{-1}+ \\Vert \\rho ^{-1}g\\Vert _{L^\\infty (\\mathbb {R}^N)}\\\\\\le &\\ C \\Vert \\varphi \\Vert _{L^\\infty (B)}(1 + R^\\mu )+ \\Vert \\rho ^{-1}g\\Vert _{L^\\infty (\\mathbb {R}^N)}\\\\\\le &\\ C(\\mu , R) \\left(\\Vert \\varphi \\Vert _{L^\\infty (B)}+ \\Vert \\rho ^{-1}g\\Vert _{L^\\infty (\\mathbb {R}^N)} \\right).\\end{split}$ From Lemma REF with $0 < \\mu < N+2s$ , the positive solution $\\varphi _0$ to the problem $(-\\Delta )^s \\varphi _0 + m \\varphi _0 = \\frac{1}{(1 + \|x\|)^\\mu }$ satisfies $\\varphi _0 = O(\|x\|^{-\\mu })$ as $\|x\| \\rightarrow +\\infty $ .", "Since $\\inf \\limits _{x \\in \\mathbb {R}^N} \\widetilde{W}(x) \\ge m$ obviously, we have $\\left((-\\Delta )^s + \\widetilde{W} \\right) \\bar{\\varphi } \\ge M \\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^\\mu }$ where $\\bar{\\varphi }(x) =M \\sum \\limits _{j=1}^k \\varphi _0(x -q_j)$ .", "Setting $\\psi =\\varphi - \\bar{\\varphi }$ , one find that $(-\\Delta )^s \\psi + \\widetilde{W} \\psi =\\bar{g} \\le 0.$ Using Lemma REF we get that $\\varphi \\le \\bar{\\varphi }$ .", "Arguing similarly for $-\\varphi $ , we get that $\|\\varphi \| \\le \\bar{\\varphi }$ .", "Then it holds that $\\begin{split}\\Vert \\rho ^{-1}\\varphi \\Vert _{L^\\infty (\\mathbb {R}^N)} & \\le \\Vert \\rho ^{-1}\\bar{\\varphi }\\Vert _{L^\\infty (\\mathbb {R}^N)}=M\\left\\Vert \\left( \\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^\\mu } \\right)^{-1} \\sum \\limits _{i=1}^k \\varphi _0(x-q_i)\\right\\Vert _{L^\\infty (\\mathbb {R}^N)}\\\\& \\le C M \\left\\Vert \\left(\\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^\\mu } \\right)^{-1} \\sum \\limits _{i=1}^k \\frac{1}{(1 + \|x-q_i\|)^\\mu }\\right\\Vert _{L^\\infty (\\mathbb {R}^N)}\\\\& \\le C M.\\end{split}$ The desired estimate follows right now.", "Examining the above proof, we can deduce the following immediately.", "Corollary 2.1 Let $\\rho (x)$ be defined as in the previous lemma.", "Assume that $\\varphi \\in H^{2s}(\\mathbb {R}^N)$ satisfies the problem (REF ) and that $\\inf \\limits _{x \\in \\mathbb {R}^N} W(x)=:m >0.$ Then we have that $\\varphi \\in L^\\infty (\\mathbb {R}^N)$ and it satisfies $\\Vert \\rho ^{-1} \\varphi \\Vert _{L^\\infty (\\mathbb {R}^N)} \\le C \\Vert \\rho ^{-1} g\\Vert _{L^\\infty (\\mathbb {R}^N)}$ where $C=C(\\mu )$ independent of $k$ .", "Remark 2.1 We build these results for any $\\frac{N}{2} < \\mu < N+2s$ , but for our purpose, from now on we choose $\\mathbf {\\mu = \\frac{N}{2}-\\frac{m}{N+2s}+1+\\sigma } \\in (\\frac{N}{2}, N+2s).$ Here $\\sigma >0$ is small enough.", "Due to the symmetry, we define $\\Omega _j$ as follows $\\Omega _j=\\left\\lbrace y =(y^{\\prime },y^{\\prime \\prime }) \\in \\mathbb {R}^2 \\times \\mathbb {R}^{N-2}: \\quad \\left\\langle \\frac{y^{\\prime }}{\|y^{\\prime }\|}, \\frac{q_j}{\|q_j\|}\\right\\rangle \\ge \\cos \\frac{\\pi }{k}\\right\\rbrace ,$ and introduce the following estimate for later use.", "For any $\\beta \\ge \\frac{N+2s-m}{N+2s}$ and fixed $\\ell $ , as $k\\rightarrow \\infty $ , it holds that $\\sum _{i\\ne \\ell }\\frac{1}{\|q_i-q_\\ell \|^\\beta }=\\frac{1}{2^\\beta }\\sum _{i\\ne \\ell } \\frac{1}{r^\\beta \\sin ^\\beta \\frac{\|i-\\ell \|\\pi }{k}}\\le \\frac{C k^\\beta }{r^\\beta }\\sum _{i=1}^k \\frac{1}{i^\\beta }\\le {\\left\\lbrace \\begin{array}{ll}\\frac{Ck^\\beta }{r^\\beta }=O(r^{-\\frac{m\\beta }{N+2s}}) \\qquad & \\beta >1, \\smallskip \\\\\\frac{Ck^\\beta \\ln k}{r^\\beta }=O(r^{-\\frac{m\\beta }{N+2s}}\\ln r) \\quad & \\beta =1, \\smallskip \\\\\\frac{C k}{r^\\beta }=O(r^{-(\\beta -\\frac{N+2s-m}{N+2s})}) \\quad & \\beta <1.\\end{array}\\right.", "}$ Remark 2.2 It holds that $\\rho (x) \\le C + C\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{\\frac{N}{2}-\\frac{m}{N+2s}+1+\\sigma }} \\le C + C\\left(\\frac{k}{r}\\right)^{\\frac{N}{2}-\\frac{m}{N+2s}+1+\\sigma } \\le C.$ according to Lemma REF .", "Also we easily have $\\int _{\\Omega _1} \\rho ^2 \\le \\int _{\\Omega _1} \\left( \\frac{1}{(1 + \|x-q_1\|)^{\\frac{N}{2}-\\frac{m}{N+2s}+1+\\sigma }} + \\frac{1}{(1 + \|x-q_1\|)^{\\frac{N}{2}+\\sigma }}\\sum _{j=2}^k\\frac{1}{\|q_1-q_j\|^{1-\\frac{m}{N+2s}}} \\right)^2 \\mathrm {d}x \\le C.$ In what follows, we use $\\Vert f\\Vert _$ to mean $\\Vert \\rho ^{-1} f\\Vert _{L^\\infty (\\mathbb {R}^N)}$ for convenience, i.e.", "$\\Vert f\\Vert _ = \\Vert \\rho ^{-1} f\\Vert _{L^\\infty (\\mathbb {R}^N)} = \\sup _{x\\in \\mathbb {R}^N} \\left(\\sum _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{\\frac{N}{2}-\\frac{m}{N+2s}+1+\\sigma }}\\right)^{-1}f(x) .$ A useful fact is that if $f$ , $g \\in L^2(\\mathbb {R}^N)$ and $F=T_m(f), G=T_m(g)$ , then the following holds $\\int _{\\mathbb {R}^N} G (-\\Delta )^s F - \\int _{\\mathbb {R}^N} F (-\\Delta )^s G = -\\int _{\\mathbb {R}^N} T_m(f) g + \\int _{\\mathbb {R}^N} f T_m(g) =0$ since the kernel $k$ is radially symmetric." ], [ "Ansatz", "In this section, we set up the approximation solution and estimate the corresponding error term.", "By a solution of the problem $(-\\Delta )^s u + V u - u^p=0 \\qquad \\mbox{in} \\ \\mathbb {R}^N,$ we mean a $u \\in H^{2s}(\\mathbb {R}^N) \\cap L^\\infty (\\mathbb {R}^N)$ such that the above equation is satisfied.", "Let us observe that it suffices to solve $(-\\Delta )^s u + V u - u_+^p=0 \\qquad \\mbox{in} \\ \\ \\mathbb {R}^N$ where $u_+ =\\max \\lbrace u , 0\\rbrace $ with the help of Lemma REF .", "We look for a solution $u$ of the form $u=W +\\varphi , \\quad W =\\sum \\limits _{j=1}^k W_j, \\quad W_j =w(x-q_j)$ where $\\varphi \\in H_s$ is a small function, disappearing as $k \\rightarrow +\\infty $ .", "In terms of $\\varphi $ , the equation (REF ) becomes $(-\\Delta )^s \\varphi (x) + V(\|x\|)\\varphi (x)-pW^{p-1}\\varphi (x) = E + N(\\varphi ) \\qquad \\mbox{in} \\ \\ \\mathbb {R}^N,$ where $N(\\varphi ) = (W + \\varphi )^p_+ - W^p- pW^{p-1}\\varphi ,\\\\E = \\sum \\limits _{j=1}^k \\left( 1 - V(\|x\|) \\right)W_j + \\left(\\sum \\limits _{j=1}^k W_j\\right)^p - \\sum \\limits _{j=1}^k W_j^p.$ Rather than solving the problem (REF ) directly, we shall first solve a projected version of it, precisely, ${\\left\\lbrace \\begin{array}{ll}(-\\Delta )^s \\varphi (x) + V(\|x\|)\\varphi (x)-pW^{p-1}\\varphi (x) = E + N(\\varphi ) + c\\sum \\limits _{j=1}^k Z_j \\qquad \\mbox{in} \\ \\ \\mathbb {R}^N,\\\\\\ \\varphi \\in H_s, \\smallskip \\\\\\displaystyle \\int _{\\mathbb {R}^N} Z_j \\varphi =0, \\qquad j=1, \\ldots , k,\\end{array}\\right.", "}$ for some pair $(\\varphi , c)$ where $ \\varphi \\in H^{2s}(\\mathbb {R}^N)\\cap L^\\infty (\\mathbb {R}^N), c$ is a constant, $Z_j = \\frac{\\partial W_j}{\\partial r} \\qquad \\text{for } j=1, \\ldots , k, \\qquad \\text{and} \\qquad \|Z_j\| \\le \\frac{C}{(1 + \|x-q_j\|)^{N+2s}} \\text{ obviously}.$ After the problem (REF ) solved, a variational process will carry out to find a suitable $r$ and then make the constant $c$ in (REF ) be zero, i.e.", "we solve the problem (REF ).", "At the end of this section, we give the estimate of $E$ .", "Lemma 3.1 It holds that $\\Vert E\\Vert _* \\le C\\left( \\frac{k}{r} \\right)^{\\min \\left\\lbrace N+2s,(N+2s)p-\\mu \\right\\rbrace }+ \\frac{C}{r^{N+2s-\\mu }} + \\frac{C}{r^m} =o\\left(\\frac{1}{r^{m/2}}\\right).$ By symmetry, we just assume that $x \\in \\Omega _1$ in the following proof.", "Obviously we know $\|x-q_j\|\\ge \|x-q_1\| \\qquad \\text{for } j=2, \\ldots , k.$ If $\|x\| \\ge \|q_1\|/2 = r/2$ , then $V(\|x\|) - 1= O\\left( \\frac{1}{\|x\|^m} \\right)=O\\left( \\frac{1}{r^m} \\right)$ and in this region $\\begin{split}&\\ \\left\|\\rho ^{-1}\\sum \\limits _{j=1}^k \\left( 1 - V(\|x\|) \\right)W_j\\right\| \\le \\frac{C}{r^m}\\left(\\sum \\limits _{i=1}^k \\frac{1}{(1 + \|x-q_i\|)^{\\mu }}\\right)^{-1}\\sum \\limits _{j=1}^k W_j\\\\\\le &\\ \\frac{C}{r^m}\\left(\\sum \\limits _{i=1}^k \\frac{1}{(1 + \|x-q_i\|)^{\\mu }}\\right)^{-1}\\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s}}\\\\\\le &\\ \\frac{C}{r^m} \\left(\\sum _{i=1}^k \\frac{1}{(1 + \|x-q_i\|)^{\\mu }}\\right)^{-1}\\sum _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{\\mu }} \\\\\\le &\\ \\frac{C}{r^m}.\\end{split}$ While for $\|x\| \\le r/2$ , then $\|x -q_1\| \\ge \|q_1\| - \|x\| \\ge \\frac{r}{2} \\quad \\text{and}\\quad \|x-q_j\|\\ge \\frac{r}{2} \\qquad \\text{for}\\ j=2, \\ldots , k.$ Hence $\\begin{split}&\\ \\left\|\\rho ^{-1}\\sum \\limits _{j=1}^k \\left( 1 - V(\|x\|) \\right)W_j\\right\| \\le C\\rho ^{-1}\\sum \\limits _{j=1}^k W_j\\le C\\rho ^{-1}\\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s}}\\\\\\le &\\ C\\rho ^{-1}\\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{\\mu }} \\cdot \\frac{1}{(1 + \|x-q_j\|)^{N+2s-\\mu }}\\\\\\le &\\ C\\left(\\sum \\limits _{i=1}^k \\frac{1}{(1 + \|x-q_i\|)^{\\mu }}\\right)^{-1}\\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{\\mu }} \\cdot \\frac{1}{r^{N+2s-\\mu }}\\\\\\le &\\ \\frac{C}{r^{N+2s-\\mu }}.\\end{split}$ For the other part in $E$ , we observe that $\\left\| \\left(\\sum \\limits _{j=1}^k W_j\\right)^p - \\sum \\limits _{j=1}^k W_j^p \\right\| \\le CW_1^{p-1}\\sum \\limits _{j=2}^k W_j +C\\sum \\limits _{j=2}^k W_j^p + C\\left(\\sum \\limits _{j=2}^k W_j\\right)^p.$ In the case of $\\mu \\le (N+2s)(p-1)$ , $\\begin{split}&\\ \\rho ^{-1}W_1^{p-1}\\sum \\limits _{j=2}^k W_j \\le C\\rho ^{-1}\\frac{1}{(1 + \|x-q_1\|)^{(N+2s)(p-1)}}\\sum \\limits _{j=2}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s}} \\\\\\le &\\ C (1 + \|x-q_1\|)^{\\mu }\\frac{1}{(1 + \|x-q_1\|)^{(N+2s)(p-1)}}\\sum \\limits _{j=2}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s}} \\\\\\le &\\ C \\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^{N+2s}} \\le C \\left(\\frac{k}{r}\\right)^{N+2s};\\end{split}$ Otherwise if $\\mu > (N+2s)(p-1)$ , then $\\begin{split}&\\ \\rho ^{-1}W_1^{p-1}\\sum \\limits _{j=2}^k W_j \\le C\\rho ^{-1}\\frac{1}{(1 + \|x-q_1\|)^{(N+2s)(p-1)}}\\sum \\limits _{j=2}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s}} \\\\\\le &\\ C \\rho ^{-1}\\frac{1}{(1 + \|x-q_1\|)^{\\mu }}\\sum \\limits _{j=2}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s -\\mu +(N+2s)(p-1)}} \\\\\\le &\\ C \\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^{(N+2s)p-\\mu }} \\le C \\left(\\frac{k}{r}\\right)^{(N+2s)p-\\mu },\\end{split}$ where we used Lemma REF .", "It is easy to deduce that $\\begin{split}\\rho ^{-1}\\sum \\limits _{j=2}^k W_j^p &\\le C \\rho ^{-1}\\sum \\limits _{j=2}^k \\frac{1}{(1 + \|x-q_j\|)^{(N+2s)p-\\mu }}\\frac{1}{(1 + \|x-q_1\|)^{\\mu }}\\\\&\\le C \\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^{(N+2s)p-\\mu }} \\le C \\left(\\frac{k}{r}\\right)^{(N+2s)p-\\mu }\\end{split}$ and $\\begin{split}\\rho ^{-1}\\left(\\sum \\limits _{j=2}^k W_j\\right)^p &\\le C\\rho ^{-1}\\left( \\sum \\limits _{j=2}^k \\frac{1}{(1 + \|x-q_j\|)^{N+2s-\\frac{\\mu }{p}}} \\frac{1}{(1 + \|x-q_1\|)^{\\frac{\\mu }{p}}}\\right)^p \\\\&\\le C \\left(\\sum \\limits _{j=2}^k\\frac{1}{\|q_j-q_1\|^{N+2s-\\frac{\\mu }{p}}}\\right)^p \\le C(\\frac{k}{r})^{(N+2s)p-\\mu }.\\end{split}$ The condition (REF ) of $m$ leads obviously to $N+2s-\\frac{\\mu }{p}>1,\\ N+2s-\\mu >\\frac{m}{2}$ and $(N+2s)p-\\mu >\\frac{N+2s}{2}$ .", "Thus we get the desired result by combining these above estimates." ], [ "Linearized theory", "This section is devoted to solve a projected linear problem.", "We consider the linear problem of finding $\\varphi \\in H^{2s}(\\mathbb {R}^N)$ such that for certain constant $c$ , we have ${\\left\\lbrace \\begin{array}{ll}(-\\Delta )^s \\varphi + V(\|x\|)\\varphi -pW^{p-1}\\varphi = g + c\\sum \\limits _{j=1}^k Z_j \\qquad \\qquad \\mbox{in} \\ \\mathbb {R}^N,\\\\\\ \\varphi \\in H_s, \\smallskip \\\\\\displaystyle \\int _{\\mathbb {R}^N} Z_j \\varphi =0, \\qquad j=1, \\ldots , k.\\end{array}\\right.", "}$ The constant $c$ is uniquely determined in terms of $\\varphi $ and $g$ when $k$ is sufficient large from the equation $\\begin{split}c\\int _{\\mathbb {R}^N} \\sum \\limits _{j=1}^k Z_j Z_1 &= \\int _{\\mathbb {R}^N}\\left[ (-\\Delta )^s \\varphi + V(\|x\|)\\varphi -pW^{p-1}\\varphi \\right]Z_1 - \\int _{\\mathbb {R}^N} gZ_1\\\\&=\\int _{\\mathbb {R}^N}\\left[ (-\\Delta )^s Z_1 + V(\|x\|)Z_1-pW^{p-1}Z_1\\right]\\varphi + O(\\Vert g\\Vert _)\\int _{\\mathbb {R}^N} \\rho \|Z_1\|\\\\&=\\int _{\\mathbb {R}^N} \\left[(V-1) + p(W_1^{p-1}- W^{p-1}) \\right]Z_1\\varphi +O(\\Vert g\\Vert _),\\end{split}$ where we use Lemma REF to obtain $\\int _{\\mathbb {R}^N} \\rho \|Z_1\| \\le C\\left(1 + \\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{\\mu }}\\right)\\int _{\\mathbb {R}^N} \\frac{1}{(1+\|x\|)^{N+2s}} dx \\le C.$ By direct calculation, it is easy to see that $\\int _{\\mathbb {R}^N} Z_1^2 = \\int _{\\mathbb {R}^N} \\left(\\frac{\\partial w(x-q_1)}{\\partial r}\\right)^2 dx=\\int _{\\mathbb {R}^N} \\left(\\frac{\\partial w(x-q_1)}{\\partial x_1}\\right)^2 dx =\\frac{1}{N}\\int _{\\mathbb {R}^N} (w^{\\prime }(\|x\|))^2dx,$ and $&\\ \\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N} Z_jZ_1 = \\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N} w^{\\prime }(\|x-q_1\|) \\frac{x-q_1}{\|x-q_1\|} \\cdot (-\\frac{q_1}{r})w^{\\prime }(\|x-q_j\|) \\frac{x-q_j}{\|x-q_j\|} \\cdot (-\\frac{q_j}{r})\\mathrm {d}x \\nonumber \\\\=&\\ \\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N} w^{\\prime }(\|x\|) \\frac{x^1}{\|x\|} w^{\\prime }(\|x+q_1-q_j\|) \\frac{x+q_1-q_j}{\|x+q_1-q_j\|} \\cdot \\frac{q_j}{r} \\mathrm {d}x \\nonumber \\\\=&\\ \\sum \\limits _{j=2}^k \\left(\\int _{\\left\\lbrace \|x\| \\le \\frac{1}{2}\|q_1 -q_j\|\\right\\rbrace } + \\int _{\\left\\lbrace \|x\| \\ge \\frac{1}{2}\|q_1 -q_j\|\\right\\rbrace } \\right)w^{\\prime }(\|x\|)\\frac{x^1}{\|x\|} w^{\\prime }(\|x+q_1-q_j\|) \\frac{x+q_1-q_j}{\|x+q_1-q_j\|} \\cdot \\frac{q_j}{r} \\mathrm {d}x \\nonumber \\\\\\le &\\ C\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s}} \\int _{\\mathbb {R}^N} \|w^{\\prime }(\|x\|)\|dx \\le C\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s}} = O\\left((\\frac{k}{r})^{N+2s}\\right), $ where $x=(x^1, \\ldots , x^N)$ .", "It implies that $\\lbrace Z_j\\rbrace _{j=1}^k$ is approximately orthogonal provided $k$ large enough because of the symmetry.", "As to the first term in the right hand side of (REF ), we do the following analysis.", "$&\\ \\left\| \\int _{\\Omega _1} (V(\|x\|)-1)Z_1 \\varphi \\right\| \\le C\\Vert \\varphi \\Vert _* \\int _{\\Omega _1} \|V(\|x\|)-1\| \\rho \\frac{1}{(1 + \|x-q_1\|)^{N+2s}} \\mathrm {d}x \\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _* \\left(\\int _{\\lbrace x\\in \\Omega _1\| \|x\| \\ge \|q_1\|/2\\rbrace } + \\int _{\\lbrace x\\in \\Omega _1\| \|x\| \\le \|q_1\|/2\\rbrace } \\right)\|V(\|x\|)-1\| \\rho \\frac{1}{(1 + \|x-q_1\|)^{N+2s}}\\mathrm {d}x \\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _* \\int _{\\lbrace x\\in \\Omega _1\| \|x\| \\ge \|q_1\|/2\\rbrace } \\frac{1}{\|x\|^m}\\frac{1}{(1 + \|x-q_1\|)^{N+2s}} \\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{\\mu }} \\mathrm {d}x \\nonumber \\\\& + C\\Vert \\varphi \\Vert _* \\int _{\\lbrace x\\in \\Omega _1\| \|x\| \\le \|q_1\|/2\\rbrace }\\frac{1}{(1 + \|x-q_1\|)^{N+2s}} \\sum \\limits _{j=1}^k \\frac{1}{(1 + \|x-q_j\|)^{\\mu }}\\mathrm {d}x \\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _* \\frac{1}{r^m} \\int _{\\lbrace x\\in \\Omega _1 \| \|x\| \\ge \|q_1\|/2\\rbrace } \\frac{1}{(1 + \|x-q_1\|)^{N+2s}} \\left(\\frac{1}{(1 + \|x-q_1\|)^{\\mu }} +\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{\\mu }} \\right) \\mathrm {d}x \\nonumber \\\\&+ C\\Vert \\varphi \\Vert _\\int _{\\lbrace x \\in \\Omega _1\| \|x\| \\le \|q_1\|/2\\rbrace } \\frac{1}{r^{N+2s}}\\left(\\frac{1}{(1 + \|x-q_1\|)^{\\mu }} +\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{\\mu }} \\right)\\mathrm {d}x \\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _\\left( \\frac{1}{r^m}+ \\frac{1}{r^{m}} (\\frac{k}{r})^{\\mu } + \\frac{1}{r^{\\mu +2s }} + \\frac{1}{r^{2s}} (\\frac{k}{r})^{\\mu }\\right)=o(\\Vert \\varphi \\Vert _), \\qquad \\mbox{as} \\ k \\rightarrow +\\infty .", "$ In addition, note that, for any $j\\ne 1$ , $\\ell \\ne 1$ and $j\\ne \\ell $ , $&\\ \\int _{\\Omega _\\ell } \\frac{\\mathrm {d}x}{(1+\|x-q_1\|)^{N+2s}(1+\|x-q_j\|)^\\mu } \\\\\\le &\\ \\frac{C}{\|q_j-q_1\|^{1-\\frac{m}{N+2s}}}\\int _{\\Omega _\\ell } \\left[ \\frac{1}{(1+\|x-q_1\|)^{\\frac{3}{2}N+2s+\\sigma }} + \\frac{1}{(1+\|x-q_j\|)^{\\frac{3}{2}N+2s+\\sigma }}\\right] \\mathrm {d}x \\\\\\le &\\ \\frac{C}{\|q_j-q_1\|^{1-\\frac{m}{N+2s}}}\\left[ \\frac{1}{\|q_\\ell -q_1\|^{\\frac{N}{2}+2s+\\sigma }} + \\frac{1}{\|q_\\ell -q_j\|^{\\frac{N}{2}+2s+\\sigma }} \\right],$ where Lemma REF is used in the first inequality.", "It is checked that $&\\ \\left\| \\int _{\\mathbb {R}^N\\setminus \\Omega _1} (V-1)Z_1 \\varphi \\right\| \\le C\\Vert \\varphi \\Vert _ \\sum _{\\ell =2}^k \\int _{\\Omega _\\ell } \|V(\|x\|)-1\| \\frac{\\rho (x)}{(1 + \|x-q_1\|)^{N+2s}} \\mathrm {d}x \\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _* \\sum _{\\ell =2}^k \\int _{\\Omega _\\ell } \\frac{1}{(1 + \|x-q_1\|)^{N+2s}}\\left(\\frac{1}{(1 + \|x-q_1\|)^{\\mu }}+\\frac{1}{(1 + \|x-q_\\ell \|)^{\\mu }} + \\sum _{\\begin{array}{c}j=2 \\\\ j\\ne \\ell \\end{array}}^k \\frac{1}{(1 + \|x-q_j\|)^{\\mu }}\\right) \\mathrm {d}x\\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _\\left( \\sum _{\\ell =2}^k \\frac{1}{\|q_\\ell -q_1\|^{\\mu +2s}} + \\sum _{\\ell =2}^k\\frac{1}{\|q_\\ell -q_1\|^{\\mu +2s-\\sigma }} + \\sum _{\\begin{array}{c}\\ell ,j=2 \\\\ j\\ne \\ell \\end{array}}^k \\frac{C}{\|q_j-q_1\|^{1-\\frac{m}{N+2s}}} \\frac{1}{\|q_\\ell -q_1\|^{\\frac{N}{2}+2s+\\sigma }}\\right) \\nonumber \\\\& + C\\Vert \\varphi \\Vert _\\sum _{\\begin{array}{c}\\ell ,j=2 \\\\ j\\ne \\ell \\end{array}}^k \\frac{C}{\|q_j-q_1\|^{1-\\frac{m}{N+2s}}} \\frac{1}{\|q_\\ell -q_j\|^{\\frac{N}{2}+2s+\\sigma }} \\nonumber \\\\\\le &\\ C\\Vert \\varphi \\Vert _* \\left(\\frac{k}{r}\\right)^{\\frac{N}{2}+2s}.", "$ Thus from (REF ) and (REF ) we get that $\\int _{\\mathbb {R}^N} (V-1)Z_1 \\varphi = o(\\Vert \\varphi \\Vert _) .$ When $1<p\\le 2$ , it holds that $&\\ \\left\| \\int _{\\Omega _1} (W_1^{p-1}- W^{p-1}) Z_1\\varphi \\mathrm {d}x \\right\| \\le C \\Vert \\varphi \\Vert _ \\int _{\\Omega _1} \\left( \\sum \\limits _{j=2}^k W_j\\right)^{p-1} \\rho \|Z_1\| \\mathrm {d}x \\\\\\le &\\ C \\Vert \\varphi \\Vert _* \\left( \\sum _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s}}\\right)^{p-1} \\int _{\\Omega _1} \\left[ \\frac{1}{(1+\|x-q_1\|)^\\mu } + \\sum _{j=2}^k \\frac{1}{\|q_1-q_j\|^\\mu } \\right] \\frac{1}{(1+\|x-q_1\|)^{N+2s}} \\mathrm {d}x \\\\\\le &\\ C \\Vert \\varphi \\Vert _* \\left(\\frac{k}{r}\\right)^{(N+2s)(p-1)},$ and, similar to (REF ), $&\\ \\left\| \\int _{\\mathbb {R}^N\\setminus \\Omega _1} (W_1^{p-1}- W^{p-1}) Z_1\\varphi \\right\| dx \\le C \\Vert \\varphi \\Vert _* \\sum _{\\ell =2}^k \\int _{\\Omega _\\ell } \\left( \\sum \\limits _{j=2}^k W_j\\right)^{p-1} \\rho \|Z_1\|dx \\\\\\le &\\ C \\Vert \\varphi \\Vert _* \\sum _{\\ell =2}^k \\int _{\\Omega _\\ell } \\left[\\frac{1}{(1+\|x-q_\\ell \|)^{(N+2s)(p-1)}}+\\left(\\frac{k}{r}\\right)^{(N+2s)(p-1)}\\right] \\\\&\\ \\qquad \\qquad \\qquad \\cdot \\left[\\frac{1}{(1+\|x-q_1\|)^\\mu }+\\frac{1}{(1+\|x-q_\\ell \|)^\\mu }+\\sum _{\\begin{array}{c}j=2 \\\\j \\ne \\ell \\end{array}}^k\\frac{1}{(1+\|x-q_j\|)^\\mu }\\right]\\frac{\\mathrm {d}x}{(1+\|x-q_1\|)^{N+2s}} \\\\\\le &\\ C \\Vert \\varphi \\Vert _* \\left(\\frac{k}{r}\\right)^{\\frac{N}{2}+2s}.$ Thus we obtain, from the above two estimates, that $\\left\| \\int _{\\mathbb {R}^N} (W_1^{p-1}- W^{p-1}) Z_1\\varphi \\mathrm {d}x \\right\| \\le C \\Vert \\varphi \\Vert _* \\left(\\frac{k}{r}\\right)^{\\min \\left\\lbrace (N+2s)(p-1),\\frac{N}{2}+2s\\right\\rbrace }.$ For the case $p >2$ , with Lemma REF , $& \\left\| \\int _{\\Omega _1} (W_1^{p-1}- W^{p-1}) Z_1\\varphi \\mathrm {d}x \\right\| \\le C \\Vert \\varphi \\Vert _* \\int _{\\Omega _1} \\left( W_1^{p-2} \\sum \\limits _{j=2}^k W_j + ( \\sum \\limits _{j=2}^k W_j)^{p-1} \\right)\\rho \|Z_1\| dx \\nonumber \\\\\\le &\\ C \\Vert \\varphi \\Vert _* \\left[ \\sum _{j=2}^k \\frac{1}{\|q_j-q_1\|^{N+2s}} + \\left(\\sum _{j=2}^k \\frac{1}{\|q_j-q_1\|^{N+2s}}\\right)^{p-1}\\right] \\le C \\Vert \\varphi \\Vert _* \\left(\\frac{k}{r}\\right)^{N+2s}, $ and, also similar to (REF ), $\\begin{split}\\left\| \\int _{\\mathbb {R}^N\\setminus \\Omega _1} (W_1^{p-1}- W^{p-1}) Z_1\\varphi \\right\| &\\le C \\Vert \\varphi \\Vert _* \\sum _{\\ell =2}^k \\int _{\\Omega _\\ell } \\left( W_1^{p-2} \\sum \\limits _{j=2}^k W_j + ( \\sum \\limits _{j=2}^k W_j)^{p-1} \\right)\\rho \|Z_1\| \\\\&\\le C \\Vert \\varphi \\Vert _* \\left(\\frac{k}{r}\\right)^{\\frac{N}{2}+2s}, \\end{split}$ on account that, in $\\Omega _\\ell $ , $W_1^{p-2} \\sum _{j=2}^k W_j &\\le \\frac{C}{\|q_\\ell -q_1\|^{(N+2s)(p-2)}}\\left[ \\frac{1}{(1+\|x-q_\\ell \|)^{N+2s}} + \\sum _{j=2, j\\ne \\ell }^k \\frac{1}{\|q_j-q_\\ell \|^{N+2s}} \\right] ,\\\\\\left(\\sum _{j=2}^k W_j\\right)^{p-1} &\\le \\frac{C}{(1+\|x-q_\\ell \|)^{(N+2s)(p-1)}} + C\\left(\\sum _{j=2, j\\ne \\ell }^k \\frac{1}{\|q_j-q_\\ell \|^{N+2s}}\\right)^{p-1}.$ So it is concluded from (REF ) and (REF ) that $\\left\| \\int _{\\mathbb {R}^N} (W_1^{p-1}- W^{p-1}) Z_1\\varphi \\mathrm {d}x \\right\| \\le C \\Vert \\varphi \\Vert _* \\left( \\frac{k}{r} \\right)^{\\frac{N}{2}+2s}.$ Combining the above inequalities leads to the following lemma right now.", "Lemma 4.1 If $(\\varphi , c)$ solves the problem (REF ), then $c=o(\\Vert \\varphi \\Vert _) + O(\\Vert g\\Vert _).$ In the rest of this section we shall build a solution to the problem (REF ).", "Proposition 4.1 Given $k$ large enough, the exists a solution $\\varphi =T(g)$ to (REF ) which defines a linear operator of $g$ , provided that $\\Vert g\\Vert _* < +\\infty $ .", "Moreover, $\\Vert \\varphi \\Vert _* \\le C \\Vert g\\Vert _* \\qquad \\text{and} \\qquad c \\le C \\Vert g\\Vert _,$ where the positive constant $C$ is independent of $k$ .", "The key difference of the proof between this proposition and Proposition 4.1 in [10] is that now we should build an a priori estimate which is independent of $k$ , see the coming Lemma REF .", "Once we get such estimate, the remaining is just the same as that in [10].", "Lemma 4.2 Under the assumptions of Proposition REF , there exists a positive constant $C$ independent of $k$ such that for any solution $\\varphi $ with $\\Vert \\varphi \\Vert _ < +\\infty $ , we have the following an a priori estimate $\\Vert \\varphi \\Vert _* \\le C \\Vert g\\Vert _.$ We argue by contradiction.", "Suppose that there are $g_k$ , $r_k \\in \\left[\\frac{1}{C_0}k^{\\frac{N+2s}{N+2s-m}}, C_0k^{\\frac{N+2s}{N+2s-m}}\\right]$ and $\\varphi _k$ solving (REF ) for $g=g_k, r=r_k$ with $\\Vert g_k\\Vert _ \\rightarrow 0$ and $\\Vert \\varphi _k\\Vert _* \\ge C^{\\prime } >0$ .", "We may assume that $\\Vert \\varphi _k\\Vert _* =1$ .", "For simplicity, we drop the subscript $k$ .", "From the conditions of potential $V$ , obviously $\\inf _{\\mathbb {R}^N} V >0$ .", "On the other hand, in the equation of $\\varphi $ , $(-\\Delta )^s \\varphi + (V-pW^{p-1})\\varphi =g + c\\sum \\limits _{j=1}^k Z_j,$ we find that $\\begin{split}V(x) - pW^{p-1}(x) &\\ge V(x) -C\\left( \\frac{1}{(1 + \|x-q_1\|)^{N+2s}} + \\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^{N+2s}} \\right)^{p-1}\\\\&\\ge V(x) - C\\left( \\frac{1}{(1 + \|x-q_1\|)^{N+2s}} + (\\frac{k}{r})^{N+2s} \\right)^{p-1}\\ge \\frac{1}{2} V(x)\\end{split}$ for any $x \\in \\Omega _1\\setminus B_R(q_1)$ , which leads to $\\inf _{\\mathbb {R}^N \\setminus \\underset{j=1}{\\overset{k}{\\cup }} B_R(q_j)} (V(x) - pW^{p-1}(x)) \\ge \\frac{1}{2} \\inf _{\\mathbb {R}^N} V(x) >0.$ Accordingly, by Lemma REF and Lemma REF , it holds that $\\Vert \\varphi \\Vert _* \\le C\\left(\\Vert \\varphi \\Vert _{L^\\infty (\\underset{j=1}{\\overset{k}{\\cup }} B_R(q_j))} + \\Vert g\\Vert _* + \|c\| \\bigg \\Vert \\sum \\limits _{j=1}^k Z_j\\bigg \\Vert _* \\right)\\\\\\le C\\Vert \\varphi \\Vert _{L^\\infty (\\underset{j=1}{\\overset{k}{\\cup }} B_R(q_j))} + o(1),$ from which we may assume that, up to a subsequence, $\\Vert \\varphi \\Vert _{L^\\infty (B_R(q_1))} \\ge \\gamma >0.$ Let us set $\\tilde{\\varphi }(x)=\\varphi (x+q_1)$ , then $\\tilde{\\varphi }$ satisfies $(-\\Delta )^s \\tilde{\\varphi } + V(\|x+q_1\|)\\tilde{\\varphi } - pw^{p-1}(x)\\tilde{\\varphi }=\\tilde{g}$ where $\\begin{split}\\tilde{g}(x)=&g(x+q_1) + c\\left(Z_1(x+q_1) + \\sum \\limits _{j=2}^k Z_j(x+q_1) \\right) \\\\&+ p\\left[\\left(w(x) + \\sum \\limits _{j=2}^k w(x+q_1-q_j)\\right)^{p-1} - w^{p-1}(x) \\right]\\tilde{\\varphi }.\\end{split}$ For any point $x$ in an arbitrarily compact set of $\\mathbb {R}^N$ , we have, from Remark REF , that $\|g(x+q_1)\| \\le \\Vert g\\Vert _\\rho (x+q_1) \\le C \\Vert g\\Vert _ =o(1).$ It is easy to see that $V(x +q_1) \\rightarrow 1$ , $c=o(\\Vert \\varphi \\Vert _) + O(\\Vert g\\Vert _) \\rightarrow 0$ and $\\begin{split}\\left\|Z_1(x+q_1) + \\sum \\limits _{j=2}^k Z_j(x+q_1)\\right\| &\\le \\frac{C}{(1 + \|x+q_1\|)^{N+2s}} + C\\sum \\limits _{j=2}^k \\frac{1}{\|x + q_1-q_j\|^{N+2s}}\\\\&\\le C + C\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s}} \\le C.\\end{split}$ For the last term in (REF ), as $1 < p\\le 2$ , $\\begin{split}& \\left\| \\left[\\left(w(x) + \\sum \\limits _{j=2}^k w(x+q_1-q_j)\\right)^{p-1} - w^{p-1}(x) \\right]\\tilde{\\varphi } \\right\|\\\\\\le & C \\left(\\sum \\limits _{j=2}^k w(x+q_1-q_j)\\right)^{p-1} \\le C \\left(\\sum \\limits _{j=2}^k \\frac{1}{\|x+q_1 -q_j\|^{N+2s}})\\right)^{p-1}\\\\\\le & C \\left(\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s}}\\right)^{p-1} \\le C \\left(\\frac{k}{r}\\right)^{(N+2s)(p-1)},\\end{split}$ while for $p >2$ , $& \\left\| \\left[\\left(w(x) + \\sum \\limits _{j=2}^k w(x+q_1-q_j)\\right)^{p-1} - w^{p-1}(x) \\right]\\tilde{\\varphi } \\right\|\\\\\\le & C w^{p-2}\\sum \\limits _{j=2}^k w(x+q_1-q_j) + C \\left(\\sum \\limits _{j=2}^k w(x+q_1-q_j)\\right)^{p-1}\\\\\\le &\\ \\sum \\limits _{j=2}^k \\frac{C}{\|x+q_1 -q_j\|^{N+2s}} + C \\left(\\sum \\limits _{j=2}^k \\frac{1}{\|x+q_1 -q_j\|^{N+2s}}\\right)^{p-1}\\\\\\le & \\sum \\limits _{j=2}^k \\frac{C}{\|q_1-q_j\|^{N+2s}} + C \\left(\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s}}\\right)^{p-1}\\\\\\le & C(\\frac{k}{r})^{N+2s}+ C (\\frac{k}{r})^{(N+2s)(p-1)} \\le C\\left(\\frac{k}{r}\\right)^{N+2s}.$ Hence $\\tilde{g} \\rightarrow 0$ uniformly on any compact set of $\\mathbb {R}^N$ as $k \\rightarrow \\infty $ .", "Meanwhile, from $(-\\Delta )^s \\tilde{\\varphi } + \\tilde{\\varphi } = \\left(1-V(x+q_1)\\right)\\tilde{\\varphi } + pw^{p-1}\\tilde{\\varphi } + \\tilde{g},$ and Lemma REF , we obtain that $\\sup _{x\\ne y}\\frac{\\left\|\\tilde{\\varphi }(x) - \\tilde{\\varphi }(y)\\right\|}{\|x-y\|^\\beta } \\le C\\left(\\Vert (1-V)\\tilde{\\varphi }\\Vert _{L^\\infty } + \\Vert w^{p-1}\\tilde{\\varphi }\\Vert _{L^\\infty } +\\Vert \\tilde{g}\\Vert _{L^\\infty } \\right)\\\\\\le C(\\Vert \\varphi \\Vert _* + \\Vert \\tilde{g}\\Vert _{L^\\infty }) \\le C$ where $\\beta =\\min \\lbrace 1, 2s\\rbrace $ .", "Hence up to a subsequence, we may assume that $\\tilde{\\varphi } \\rightarrow \\varphi _0$ uniformly on any compact set.", "It is easy to observe that $\\varphi _0$ satisfies ${\\left\\lbrace \\begin{array}{ll}(-\\Delta )^s \\varphi _0 + \\varphi _0-pw^{p-1}\\varphi _0 =0 \\qquad &\\mbox{in}\\ \\mathbb {R}^N, \\smallskip \\\\\\ \\varphi _0 \\in H_s, \\medskip \\\\\\displaystyle \\int _{\\mathbb {R}^N} \\frac{\\partial w}{\\partial x^1} \\varphi _0 =0,\\end{array}\\right.", "}$ where $x=(x^1, \\ldots , x^N)$ .", "Besides, we know, from Remark REF , that $\\int _{B_R(0)} \\varphi _0^2 \\le \\int _{B_R(0)}\\tilde{\\varphi }_k^2 = \\int _{B_R(q_1)}\\varphi _k^2 \\le \\Vert \\varphi _k\\Vert _^2 \\int _{B_R(q_1)}\\rho ^2 \\le C,$ which means that $\\varphi _0 \\in L^2(\\mathbb {R}^N)$ .", "Then the non-degeneracy result in [20] implies that $\\varphi _0$ must be a linear combination of the partial derivatives $\\frac{\\partial w}{\\partial x^i}, i=1, \\ldots , N$ .", "But the symmetry and orthogonality condition yield that $\\varphi _0 \\equiv 0$ , which is a contradiction to (REF ).", "The lemma is then proved." ], [ "The variational reduction and the proof of Theorem ", "In this section we first solve the intermediate nonlinear problem (REF ), i.e.", "${\\left\\lbrace \\begin{array}{ll}(-\\Delta )^s \\varphi (x) + V(\|x\|)\\varphi (x)-pW^{p-1}\\varphi (x) = E + N(\\varphi ) + c\\sum \\limits _{j=1}^k Z_j \\qquad \\mbox{in} \\ \\ \\mathbb {R}^N,\\\\\\ \\varphi \\in H_s, \\smallskip \\\\\\displaystyle \\int _{\\mathbb {R}^N} Z_j \\varphi =0 \\qquad \\text{for any } j=1, \\ldots , k.\\end{array}\\right.", "}$ Then we solve the final nonlinear problem (REF ) variationally.", "Proposition 5.1 Assume that $k$ large enough, for any $r \\in \\left[\\frac{1}{C_0}k^{\\frac{N+2s}{N+2s-m}}, C_0k^{\\frac{N+2s}{N+2s-m}} \\right]$ , the problem (REF ) has a unique small solution $\\varphi =\\Phi (r)$ with $\\Vert \\varphi \\Vert _ \\le C\\left( \\frac{k}{r} \\right)^{\\min \\lbrace N+2s, (N+2s)p-\\mu \\rbrace }+ \\frac{C}{r^{N+2s-\\mu }} + \\frac{C}{r^m}= o\\left(\\frac{1}{r^{m/2}}\\right).$ Furthermore, the map $r \\rightarrow \\Phi (r)$ is of class $C^1$ , and $\\Vert \\Phi ^{\\prime }(r)\\Vert _* \\le C\\left( \\frac{k}{r} \\right)^{\\min \\lbrace N+2s, (N+2s)p-\\mu \\rbrace }+ \\frac{C}{r^{N+2s-\\mu }} + \\frac{C}{r^m}.$ Problem (REF ) can be written as the fixed point problem $\\varphi =T(E + N(\\varphi ))=:\\mathcal {A}(\\varphi ) \\qquad \\text{for } \\varphi \\in H_s.$ Let $\\mathfrak {F}=\\lbrace \\varphi \\in H_s \\mid \\Vert \\varphi \\Vert _* \\le s_0\\rbrace ,$ where $s_0>0$ is a small number determined later.", "If $\\varphi \\in \\mathfrak {F}$ , either $1 < p \\le 2$ , $\\Vert N(\\varphi )\\Vert _* \\le C\\Vert \\varphi ^p\\Vert _* \\le C \\Vert \\varphi \\Vert _^p \\Vert \\rho \\Vert ^{p-1}_{L^{\\infty }(\\mathbb {R}^N)} \\le C\\Vert \\varphi \\Vert _^p;$ or $p >2$ , $\\Vert N(\\varphi )\\Vert _* \\le C \\Vert \\varphi ^2 W^{p-2}\\Vert _* + C \\Vert \\varphi ^p\\Vert _\\le C \\Vert \\varphi \\Vert _^2 \|\\rho W\|_{L^{\\infty }(\\mathbb {R}^N)} + C\\Vert \\varphi \\Vert _^p \\Vert \\rho ^{p-1}\\Vert _{L^{\\infty }(\\mathbb {R}^N)} \\le C \\Vert \\varphi \\Vert _^2 .$ By Proposition REF and Lemma REF , $\\Vert \\mathcal {A}(\\varphi )\\Vert _* \\le C\\left(\\Vert E\\Vert _* + \\Vert N(\\varphi )\\Vert _* \\right) \\le C \\Vert E\\Vert _* + C(\\Vert \\varphi \\Vert _* + \\Vert \\varphi \\Vert _^{p-1})\\Vert \\varphi \\Vert _ \\le s_0$ if we choose $C(s_0+s_0^{p-1}) \\le \\frac{1}{2}$ and $k$ large enough such that $C\\left( \\frac{k}{r} \\right)^{\\min \\lbrace N+2s, (N+2s)p-\\mu \\rbrace }+ \\frac{C}{r^{N+2s-\\mu }} + \\frac{C}{r^m} \\le \\frac{1}{2}s_0.$ On the other hand, for any $\\varphi _i \\in H_s$ , $i=1, 2$ , $\|N(\\varphi _1) - N(\\varphi _2)\|=\|N^{\\prime }(t)(\\varphi _1 - \\varphi _2)\|$ where $t $ lies between $\\varphi _1$ and $\\varphi _2$ .", "For $1 < p\\le 2$ , $\|N^{\\prime }(t)\| \\le C\|t\|^{p-1} \\le C (\|\\varphi _1\|^{p-1} + \|\\varphi _2\|^{p-1})$ which tells us that $\\begin{split}\\Vert N(\\varphi _1)-N(\\varphi _2) \\Vert _* & \\le C\\Vert \\varphi _1 - \\varphi _2\\Vert _* (\\Vert \\varphi _1\\Vert ^{p-1}_{L^{\\infty }(\\mathbb {R}^N)} + \|\\varphi _2\|^{p-1}_{L^{\\infty }(\\mathbb {R}^N)})\\\\& \\le C\\Vert \\varphi _1 - \\varphi _2\\Vert _* (\\Vert \\varphi _1\\Vert ^{p-1}_* + \\Vert \\varphi _2\\Vert ^{p-1}_)\\Vert \\rho \\Vert ^{p-1}_{L^{\\infty }(\\mathbb {R}^N)})\\\\& \\le C\\Vert \\varphi _1 - \\varphi _2\\Vert _ (\\Vert \\varphi _1\\Vert ^{p-1}_* + \\Vert \\varphi _2\\Vert ^{p-1}_)\\\\& \\le Cs_0^{p-1}\\Vert \\varphi _1 - \\varphi _2\\Vert _\\le \\frac{1}{2} \\Vert \\varphi _1 - \\varphi _2\\Vert _\\end{split}$ provided $s_0$ small enough.", "And for $p>2$ , $\|N^{\\prime }(t)\| \\le C(W^{p-2}\|t\| + \|t\|^{p-1})$ , from which we can deduce that $\\begin{split}&\\ \\Vert N(\\varphi _1)-N(\\varphi _2) \\Vert _ \\\\\\le &\\ C\\Vert \\varphi _1 - \\varphi _2\\Vert _\\left[ \\Vert \\rho W\\Vert _{L^{\\infty }(\\mathbb {R}^N)})(\\Vert \\varphi _1\\Vert _ + \\Vert \\varphi _2\\Vert _) + (\\Vert \\varphi _1\\Vert ^{p-1}_ +\\Vert \\varphi _2\\Vert ^{p-1}_)\\Vert \\rho \\Vert ^{p-1}_{L^{\\infty }(\\mathbb {R}^N)})\\right]\\\\\\le &\\ C \\Vert \\varphi _1 - \\varphi _2\\Vert _\\left( \\Vert \\varphi _1\\Vert _* + \\Vert \\varphi _2\\Vert _+ \\Vert \\varphi _1\\Vert ^{p-1}_ +\\Vert \\varphi _2\\Vert ^{p-1}_* \\right)\\\\\\le &\\ C(s_0 + s_0^{p-1})\\Vert \\varphi _1 - \\varphi _2\\Vert _* \\le \\frac{1}{2} \\Vert \\varphi _1 - \\varphi _2\\Vert _\\end{split}$ with $s_0$ small enough.", "Thus we obtain that $\\mathcal {A}$ is a contraction mapping and the problem (REF ) has a unique solution $\\varphi $ .", "Obviously according to Lemma $\\ref {ee}$ , $\\Vert \\varphi \\Vert _ \\le C\\left( \\frac{k}{r} \\right)^{\\min \\lbrace N+2s, (N+2s)p-\\mu \\rbrace }+ \\frac{C}{r^{N+2s-\\mu }} + \\frac{C}{r^m}.$ For the proof of $\\Phi (r) \\in C^1$ , please refer to [10].", "Here we don't repeat it.", "Next, we will use the above introduced ingredients to find existence results for the nonlinear problem (REF ), i.e.", "the equation $(-\\Delta )^s u + V(x)u - u_+^p=0.$ Set the following energy functional $J(u)=\\frac{1}{2}\\int _{\\mathbb {R}^N} u(-\\Delta )^s u + V(x)u^2 - \\frac{1}{p+1}\\int _{\\mathbb {R}^N} u_+^{p+1}$ whose nontrivial critical points are solutions to (REF ).", "We want to find a solution of (REF ) with the form $U =W+\\varphi $ where $\\varphi =\\Phi (r)$ is found in Proposition REF .", "Then it is easy to observe that $(-\\Delta )^s U + VU - U^p_+ = c\\sum \\limits _{j=1}^k Z_j.$ Hence we need to find suitable $r$ such that the coefficient $c=0$ .", "The problem can be formulated variationally as follows.", "Lemma 5.1 Let $F(r) = J(U)=J(W + \\Phi (r))$ , then $c=0$ if and only if $F^{\\prime }(r)=0$ .", "Assume that $\\tilde{U}$ is the unique $s$ -harmonic extension of $U=W + \\Phi (r)$ , then the well-known computation by Caffarelli and Silvestre [5] shows that $F(r)=\\frac{1}{2}\\int _{\\mathbb {R}^{N+1}_+} \|\\nabla \\tilde{ U}\|^2 y^{1-2s}\\mathrm {d}x \\mathrm {d}y + \\frac{1}{2}\\int _{\\mathbb {R}^N} V(\|x\|) U^2 -\\frac{1}{p+1}\\int _{\\mathbb {R}^N} U_+^{p+1}.$ So with $\\partial _r U = \\partial _r W + \\Phi ^{\\prime }(r)=\\sum \\limits _{i=1}^k Z_i + \\Phi ^{\\prime }(r)$ , (REF ), (REF ) and Proposition REF , $\\begin{split}F^{\\prime }(r)&=\\int _{\\mathbb {R}^{N+1}_+} \\nabla \\tilde{U} \\cdot \\nabla (\\partial _r \\tilde{ U}) y^{1-2s} + \\int _{\\mathbb {R}^N} V(\|x\|) U \\partial _r U - \\int _{\\mathbb {R}^N} U_+^p \\partial _r U\\\\\\ &=\\int _{\\mathbb {R}^N} \\left((-\\Delta )^sU + V(\|x\|)U -U_+^p \\right)\\partial _r U =c\\sum \\limits _{j=1}^k \\int _{\\mathbb {R}^N} Z_j \\partial _r U\\\\&=c\\sum \\limits _{i, j=1}^k \\int _{\\mathbb {R}^N} Z_jZ_i + c\\sum \\limits _{ j=1}^k \\int _{\\mathbb {R}^N} Z_j \\Phi ^{\\prime }(r)\\\\&=c\\left(\\frac{k}{N}\\int _{\\mathbb {R}^N} (w^{\\prime }(\|x\|))^2dx +k O((\\frac{k}{r})^{N+2s}) + O(\\Vert \\Phi ^{\\prime }(r)\\Vert _\\sum \\limits _{j=1}^k \\int _{\\mathbb {R}^N} \|Z_j\|\\rho ) \\right)\\\\&=ck\\left(\\frac{1}{N}\\int _{\\mathbb {R}^N} (w^{\\prime }(\|x\|))^2dx + o(1)\\right),\\end{split}$ with $k$ large enough.", "The proof is finished.", "Now our task is to find a critical of the functional $F(r)$ .", "We have the following expansion of $F(r)$ .", "Proposition 5.2 There exists $k_0$ such that for any $k \\ge k_0, r \\in I_0$ , the following expansion holds $F(r)=k\\left[A_1 + \\frac{B_1}{r^m}- \\frac{B_2k^{N+2s}}{r^{N+2s}} + o\\left( r^{-m} \\right)\\right]$ where $A_1$ , $B_1$ , $B_2$ are universal positive constants defined in Proposition REF and the interval $I_0$ is given by $I_0=\\left[\\frac{1}{C_0}k^{\\frac{N+2s}{N+2s-m}}, C_0k^{\\frac{N+2s}{N+2s-m}} \\right].$ Since $U=W+\\varphi $ , let us expand $J(U)$ at $W$ and get that $\\begin{split}J(U)=&\\ \\frac{1}{2}\\int _{\\mathbb {R}^N} U(-\\Delta )^sU + V U^2 - \\frac{1}{p+1}\\int _{\\mathbb {R}^N} U_+^{p+1}\\\\=& J(W) + \\int _{\\mathbb {R}^N} \\left[(-\\Delta )^s U + VU - U_+^p \\right]\\varphi - \\frac{1}{2}\\int _{\\mathbb {R}^N} \\left(\\varphi (-\\Delta )^s\\varphi + V \\varphi ^2 - p W^{p-1}\\varphi ^2\\right)\\\\& -\\frac{1}{p+1}\\int _{\\mathbb {R}^N} \\left((W+\\varphi )_+^{p+1} - W^{p+1} - (p+1)W^p\\varphi -\\frac{p(p+1)}{2} W^{p-1}\\varphi ^2 \\right)\\\\& + \\int _{\\mathbb {R}^N}\\left(U_+^p-W^p-pW^{p-1}\\varphi \\right)\\varphi .\\end{split}$ Since $\\int _{\\mathbb {R}^N} \\varphi Z_j=0$ for all $j=1, \\ldots , k$ , the second term disappears.", "From Remark REF , we have $\\begin{split}&\\left\|\\int _{\\mathbb {R}^N} \\left(\\varphi (-\\Delta )^s\\varphi + V \\varphi ^2 - p W^{p-1}\\varphi ^2\\right)\\right\|\\\\= &\\int _{\\mathbb {R}^N} \\left\|E+N(\\varphi )\\right\| \|\\varphi \|\\le C k\\left(\\Vert E\\Vert _ + \\Vert N(\\varphi )\\Vert _\\right) \\Vert \\varphi \\Vert _ \\int _{\\Omega _1} \\rho ^2 \\\\\\le & C k\\left(\\Vert E\\Vert _* + \\Vert N(\\varphi )\\Vert _\\right) \\Vert \\varphi \\Vert _ = k o(r^{-m}).\\end{split}$ Similarly, it is easy to see that $\\begin{split}&\\left\| \\int _{\\mathbb {R}^N} \\left((W+\\varphi )_+^{p+1} - W^{p+1}- (p+1)W^p\\varphi -\\frac{p(p+1)}{2} W^{p-1}\\varphi ^2 \\right) \\right\| \\\\\\le & C \\int _{\\mathbb {R}^N} \|\\varphi \|^{\\min \\lbrace p+1,3\\rbrace }\\le C k \\Vert \\varphi \\Vert ^{\\min \\lbrace p+1,3\\rbrace }_\\int _{\\Omega _1} \\rho ^2 =k o\\left(r^{-m} \\right),\\end{split}$ and $\\left\|\\int _{\\mathbb {R}^N}\\left(U_+^p-W^p-pW^{p-1}\\varphi \\right)\\varphi \\right\|=O\\left(\\int _{\\mathbb {R}^N}W^{p-1}\\varphi ^2 \\right)=kO\\left( \\Vert \\varphi \\Vert _^2\\right)=ko(r^{-m}).$ The proof is completed.", "[Proof of Theorem REF ] Now consider the following problem $\\max _{r \\in I_0} F(r)$ .", "We want to verify that the maximum points lie in the interior of the interval $ I_0$ .", "For this, let $r_0=\\left( \\frac{(N+2s)B_2}{mB_1} \\right)^{\\frac{1}{N+2s-m}}k^{\\frac{N+2s}{N+2s-m}} \\in I_0$ for large positive constant $C_0$ .", "It is easy to show that for large $k$ , $F(r_0)= kA_1 + k^{1-\\frac{(N+2s)m}{N+2s-m}}B_1\\left( \\frac{mB_1}{(N+2s)B_2}\\right)^{\\frac{m}{N+2s-m}}\\frac{N+2s-m}{N+2s} + o\\left( k^{1-\\frac{(N+2s)m}{N+2s-m}} \\right).$ On the other hand, it always holds that $\\begin{split}F\\left(\\frac{1}{C_0}k^{\\frac{N+2s}{N+2s-m}}\\right)= &\\ kA_1 + k^{1-\\frac{(N+2s)m}{N+2s-m}}\\left(B_1C_0^m-B_2C_0^{N+2s} \\right) +o\\left( k^{1-\\frac{(N+2s)m}{N+2s-m}} \\right)\\\\<&\\ kA_1 + k^{1-\\frac{(N+2s)m}{N+2s-m}}B_1\\left( \\frac{mB_1}{(N+2s)B_2}\\right)^{\\frac{m}{N+2s-m}}\\frac{N+2s-m}{2(N+2s)}\\end{split}$ and $\\begin{split}F(C_0k^{\\frac{N+2s}{N+2s-m}})=&\\ kA_1 + k^{1-\\frac{(N+2s)m}{N+2s-m}}\\left(\\frac{B_1}{C_0^m}-\\frac{B_2}{C_0^{N+2s}} \\right) +o\\left( k^{1-\\frac{(N+2s)m}{N+2s-m}} \\right)\\\\<&\\ kA_1 + k^{1-\\frac{(N+2s)m}{N+2s-m}}\\frac{B_1}{C_0^m}+o\\left( k^{1-\\frac{(N+2s)m}{N+2s-m}} \\right)\\\\<&\\ kA_1 + k^{1-\\frac{(N+2s)m}{N+2s-m}}B_1\\left( \\frac{mB_1}{(N+2s)B_2}\\right)^{\\frac{m}{N+2s-m}}\\frac{N+2s-m}{2(N+2s)},\\end{split}$ if we choose $C_0$ large enough such that $B_1C_0^m-B_2C_0^{N+2s}<0, \\qquad \\frac{B_1}{C_0^m} < B_1\\left( \\frac{mB_1}{(N+2s)B_2}\\right)^{\\frac{m}{N+2s-m}}\\frac{N+2s-m}{4(N+2s)}$ which can be done because of $0 <m< N+2s.", "$ If we let $F(r_1)=\\max _{r \\in I_0} F(r)$ , then $r_1$ is an interior point of $I_0$ and thus $F^{\\prime }(r_1)=0$ , which gives a critical point of $F(r)$ .", "Therefore Lemma REF implies Theorem REF ." ], [ "Appendix: Energy expansion", "In this section, the important expansion of the energy at $W$ is given.", "First we list the following lemmas, whose proofs can be found in [25].", "Lemma 6.1 For any $\\alpha >0$ , $\\sum _{j=1}^k \\frac{1}{(1+\|x-x_j\|)^\\alpha }\\le C + C \\sum _{j=2}^k \\frac{1}{\|x_1-x_j\|^\\alpha }, \\qquad \\forall \\ \\ x\\in \\mathbb {R}^N$ where $C>0$ is a constant independent of $k$ .", "Lemma 6.2 For any constant $0<\\sigma <N-2$ , there is a constant $C>0$ , such that $\\int _{\\mathbb {R}^N} \\frac{1}{\|y-z\|^{N-2}}\\frac{1}{(1+\|z\|)^{2+\\sigma }}\\mathrm {d}z\\le \\frac{C}{(1+\|y\|)^{\\sigma }}.$ The proof of this lemma, a more general one actually, can also be found in [22], [29].", "Lemma 6.3 For any constant $0\\le \\sigma \\le \\min \\lbrace \\alpha ,\\ \\beta \\rbrace $ , there is a constant $C > 0$ such that, for any $i\\ne j$ , $\\frac{1}{(1+\|x-q_i\|)^\\alpha }\\frac{1}{(1+\|x-q_j\|)^\\beta } \\le \\frac{C}{\|q_i-q_j\|^\\sigma } \\left[\\frac{1}{(1+\|x-q_i\|)^{\\alpha +\\beta -\\sigma }}+\\frac{1}{(1+\|x-q_j\|)^{\\alpha +\\beta -\\sigma }}\\right].$ The proof of the above lemma may be found in [26].", "Next we focus on the expansion of energy at $W$ .", "Recall the positive least energy solution $w$ to (REF ).", "Proposition 6.1 It holds that $J(W)=k \\left[A_1 + \\frac{B_1}{r^m}-\\frac{B_2k^{N+2s}}{r^{N+2s}} + o\\left( r^{-m} + (\\frac{k}{r})^{N+2s} \\right)\\right],$ where $A_1 =\\left(\\frac{1}{2}-\\frac{1}{p+1}\\right)\\int _{\\mathbb {R}^N} w^{p+1}(x)dx, \\quad B_1=\\frac{a}{2}\\int _{\\mathbb {R}^N} w^2(x) dx$ and $B_2$ are all positive numbers.", "Recall that $q_j = \\left(r\\cos \\frac{2(j-1)\\pi }{k}, r\\sin \\frac{2(j-1)}{k}, \\mathbf {0}\\right), \\qquad j=1, \\ldots , k,$ where $\\mathbf {0}$ is the zero vector in $\\mathbb {R}^{N-2}$ , $r \\in \\left[\\frac{1}{C_0}k^{\\frac{N+2s}{N+2s-m}}, C_0k^{\\frac{N+2s}{N+2s-m}} \\right]$ for a large positive constant $C_0$ .", "By direct calculous, we get that $\|q_1-q_j\|=2r\\sin \\frac{(j-1)\\pi }{k}, \\qquad 0 < c^{\\prime } \\le \\frac{\\sin \\frac{(j-1)\\pi }{k}}{\\frac{(j-1)\\pi }{k}} \\le c^{\\prime \\prime },$ from which we can find that for any $\\ell >1$ , $\\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^\\ell }=\\frac{1}{(2r)^\\ell } \\sum \\limits _{j=2}^k \\frac{1}{(\\sin \\frac{(j-1)\\pi }{k})^\\ell } = C_{\\ell } \\left(\\frac{k}{r}\\right)^\\ell + o\\left(\\left(\\frac{k}{r}\\right)^\\ell \\right),$ where $C_l >0.$ Denote $W_j(x)=w(x-q_j), \\quad j=1, \\ldots , k, \\qquad W(x)=\\sum \\limits _{j=1}^k W_j(x).$ $\\begin{split}J(W_1)&=\\frac{1}{2}\\int _{\\mathbb {R}^N} \\left[w(x-q_1)(-\\Delta )^sw(x-q_1)+V(\|x\|)w^2(x-q_1)\\right]\\mathrm {d}x-\\frac{1}{p+1}\\int _{\\mathbb {R}^N}w^{p+1}(x-q_1)\\mathrm {d}x\\\\&=J_1(w)+\\frac{1}{2}\\int _{\\mathbb {R}^N}\\left(V(\|x\|)-1\\right)w^2(x-q_1)dx\\\\&=\\left(\\frac{1}{2}-\\frac{1}{p+1} \\right)\\int _{\\mathbb {R}^N}w^{p+1}dx +\\frac{1}{2}\\int _{\\mathbb {R}^N}\\left(V(\|x-q_1\|)-1\\right)w^2(x)dx.\\end{split}$ For any $\\alpha >0$ , $x \\in B_{r/2}(0)$ , since $\\frac{1}{\|x-q_1\|^\\alpha } =\\frac{1}{\|q_1\|^\\alpha }\\left[1+O\\left(\\frac{\|x\|}{\|q_1\|}\\right) \\right],$ we deduce that $&\\ \\frac{1}{2}\\int _{\\mathbb {R}^N}\\left(V(\|x-q_1\|)-1\\right)w^2(x) \\mathrm {d}x=\\left(\\int _{\\lbrace x\| \|x\|< \\frac{r}{2}\\rbrace } + \\int _{\\lbrace x\| \|x\|\\ge \\frac{r}{2}\\rbrace } \\right)\\left(V(\|x-q_1\|)-1\\right)w^2(x) \\mathrm {d}x \\nonumber \\\\=&\\ \\frac{1}{2}\\int _{\\lbrace x\| \|x\|< \\frac{r}{2}\\rbrace } \\left[\\frac{a}{\|x-q_1\|^m} + o\\left(\\frac{1}{\|x-q_1\|^{m}} \\right) \\right] w^2(x)dx + O\\left( \\int _{\\lbrace x\| \|x\|\\ge \\frac{r}{2}\\rbrace } w^2(x)dx \\right) \\nonumber \\\\=&\\ \\frac{a}{2\|q_1\|^m}\\int _{\\lbrace x\| \|x\|< \\frac{r}{2}\\rbrace } w^2(x)dx + O\\left(r^{-(m+1)}\\int _{\\lbrace x\| \|x\|< \\frac{r}{2}\\rbrace } \|x\|w^2(x)dx + r^{-(N+4s)} \\right) +o(r^{-m}) \\nonumber \\\\=&\\ \\frac{B_1}{r^m} + o(r^{-m})+ O\\left(r^{-(N+4s)}\\right)=\\frac{B_1}{r^m} + o(r^{-m}),$ where the positive constant $B_1 = \\frac{a}{2}\\int _{\\mathbb {R}^N} w^2(x)\\mathrm {d}x$ .", "Recall that $\\Omega _j=\\left\\lbrace y =(y^{\\prime },y^{\\prime \\prime }) \\in \\mathbb {R}^2 \\times \\mathbb {R}^{N-2}: \\quad \\left\\langle \\frac{y^{\\prime }}{\|y^{\\prime }\|}, \\frac{q_j}{\|q_j\|}\\right\\rangle \\ge \\cos \\frac{\\pi }{k}\\right\\rbrace .$ By symmetry, we can deduce that $\\begin{split}J(W)=&\\frac{1}{2}\\int _{\\mathbb {R}^N} W(-\\Delta )^s W + V(\|x\|)W^2 - \\frac{1}{p+1}\\int _{\\mathbb {R}^N} W^{p+1}\\\\=& \\frac{1}{2}\\int _{\\mathbb {R}^N} W\\left((-\\Delta )^sW + W \\right) + \\frac{1}{2}\\int _{\\mathbb {R}^N} \\left(V(\|x\|)-1\\right)W^2- \\frac{1}{p+1}\\int _{\\mathbb {R}^N} W^{p+1}\\\\=&\\frac{1}{2}\\int _{\\mathbb {R}^N} W \\sum \\limits _{j=1}^k W^p_j + \\frac{k}{2} \\int _{\\Omega _1}\\left(V(\|x\|)-1\\right)W^2- \\frac{k}{p+1}\\int _{\\Omega _1} W^{p+1}\\\\=&\\frac{k}{2}\\int _{\\mathbb {R}^N} w^{p+1} + \\frac{k}{2}\\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N}W_1^pW_j + \\frac{k}{2} \\int _{\\Omega _1}\\left(V(\|x\|)-1\\right)W^2- \\frac{k}{p+1}\\int _{\\Omega _1} W^{p+1}.\\end{split}$ Now let us do the computations term by term.", "With (REF ) and (REF ) at hand, we find that $&\\ \\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N}W_1^pW_j =\\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N} w^p(\|x-q_1\|) w(\|x-q_j\|)dx \\nonumber \\\\=&\\ \\sum \\limits _{j=2}^k \\int _{\\mathbb {R}^N} w^p(\|x\|) w(\|x + q_1-q_j\|)dx \\nonumber \\\\=&\\ \\sum \\limits _{j=2}^k \\int _{\\lbrace x \| \|x\|\\le \|q_1-q_j\|/2 \\rbrace } w^p(\|x\|) \\left[ \\frac{A}{\|x+q_1-q_j\|^{N+2s}} + o\\left(\\frac{1}{\|x+q_1-q_j\|^{N+2s}} \\right)\\right]dx \\nonumber \\\\& + \\sum \\limits _{j=2}^k \\int _{\\lbrace x \| \|x\|\\ge \|q_1-q_j\|/2 \\rbrace } O\\left(\\frac{1}{\|q_1-q_j\|^{(N+2s)p}} \\right)w(\|x + q_1-q_j\|) dx \\nonumber \\\\=&\\ \\sum \\limits _{j=2}^k \\frac{A}{\|q_1-q_j\|^{N+2s}} \\int _{\\mathbb {R}^N} w^p(x) dx +o\\left(\\sum \\limits _{j=2}^k\\frac{1}{\|q_1-q_j\|^{N+2s}} \\right)+ O\\left(\\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{(N+2s)p}}\\right) \\nonumber \\\\=&\\ \\sum \\limits _{j=2}^k \\frac{\\widetilde{B}_2}{\|q_1-q_j\|^{N+2s}}+ o\\left((\\frac{k}{r} )^{N+2s} \\right)$ where the positive constant $\\widetilde{B}_2 = A\\int _{\\mathbb {R}^N} w^p.$ For any $x \\in \\Omega _1$ , it is obvious that $\|x-q_j\| \\ge \|x-q_1\|$ and $\|x-q_j\| \\ge \|q_j-q_1\|/2$ for $j=2, \\ldots , k$ .", "Then for any $0\\le \\alpha \\le N+2s$ , $W_j(x) \\le \\frac{C}{\\left(1 + \|x-q_j\|\\right)^{N+2s}} \\le \\frac{C}{\\left(1 + \|x-q_1\|\\right)^{\\alpha }\|q_1-q_j\|^{N+2s-\\alpha }}.$ Hence for any $0\\le \\alpha < N+2s-1$ , $\\sum \\limits _{j=2}^k W_j(x)=O\\left(\\frac{1}{\\left(1 + \|x-q_1\|\\right)^{\\alpha }} \\sum \\limits _{j=2}^k \\frac{1}{\|q_1-q_j\|^{N+2s-\\alpha }} \\right)=O\\left(\\frac{1}{\\left(1 +\|x-q_1\|\\right)^{\\alpha }}\\left(\\frac{k}{r}\\right)^{N+2s-\\alpha }\\right).$ Now we can deduce that $&\\ \\frac{1}{2} \\int _{\\Omega _1}\\left(V(\|x\|)-1\\right)W^2 =\\frac{1}{2} \\int _{\\Omega _1}\\left(V(\|x\|)-1\\right) \\left(W_1 + \\sum \\limits _{j=2}^k W_j \\right)^2 \\nonumber \\\\=&\\ \\frac{1}{2} \\int _{\\Omega _1}\\left(V(\|x\|)-1\\right)W_1^2 + O\\left( \\int _{\\Omega _1}\\left\|V(\|x\|)-1 \\right\|W_1\\sum \\limits _{j=2}^k W_j + (\\frac{k}{r})^{2N+4s-2\\alpha }\\int _{\\Omega _1} \\frac{1}{(1 +\|x-q_1\|)^{2\\alpha }} \\right) \\nonumber \\\\=&\\ \\frac{B_1}{r^m} + o(r^{-m})+ O\\left(\\left(\\frac{k}{r}\\right)^{N+2s}\\int _{\\Omega _1} \\left\|V(\|x\|)-1 \\right\|W_1 +(\\frac{k}{r})^{N+3s}\\right) \\nonumber \\\\=&\\ \\frac{B_1}{r^m} + o\\left(r^{-m}+ (\\frac{k}{r})^{N+2s}\\right)+(\\frac{k}{r})^{N+2s}O\\left(\\int _{\\Omega _1} \\left\|V(\|x\|)-1 \\right\|w(x-q_1) \\right) \\nonumber \\\\=&\\ \\frac{B_1}{r^m} + o\\left(r^{-m}+ (\\frac{k}{r})^{N+2s}\\right)+ (\\frac{k}{r})^{N+2s}O \\left(\\left(\\int _{\\lbrace x \| \|x\| \\le \\frac{r}{2}\\rbrace } + \\int _{\\lbrace x \| \|x\| \\ge \\frac{r}{2}\\rbrace }\\right) \\left\|V(x)-1 \\right\|w(x-q_1)\\right) \\nonumber \\\\=&\\ \\frac{B_1}{r^m} + o\\left(r^{-m}+ (\\frac{k}{r})^{N+2s}\\right)+ (\\frac{k}{r})^{N+2s}O\\left(\\frac{1}{r^{s}} + \\frac{1}{r^m} \\right) \\nonumber \\\\=&\\ \\frac{B_1}{r^m} + o\\left(r^{-m}+ (\\frac{k}{r})^{N+2s}\\right), $ where we choose $\\alpha =\\frac{N+s}{2}$ .", "For the last term in the energy $J(W)$ , it is not difficult to check that $\\begin{split}&\\frac{1}{p+1}\\int _{\\Omega _1} W^{p+1}=\\frac{1}{p+1}\\int _{\\Omega _1} \\left(W_1 + \\sum \\limits _{j=2}^k W_j \\right)^{p+1}\\\\=&\\frac{1}{p+1}\\int _{\\Omega _1}W_1^{p+1} + \\int _{\\Omega _1}W_1^p\\sum \\limits _{j=2}^k W_j +O\\left(\\int _{\\Omega _1} W_1^{p-1}(\\sum \\limits _{j=2}^k W_j)^2 \\right) +O\\left(\\int _{\\Omega _1}(\\sum \\limits _{j=2}^k W_j)^{p+1} \\right)\\\\=&\\frac{1}{p+1}\\int _{\\Omega _1}W_1^{p+1} + \\int _{\\Omega _1}W_1^p\\sum \\limits _{j=2}^k W_j +O\\left(\\left(\\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^{\\frac{N}{2}+2s}}\\right)^2\\right) \\\\&+ O\\left(\\left(\\sum \\limits _{j=2}^k \\frac{1}{\|q_j-q_1\|^{N+2s-\\frac{N+(p-1)s}{p+1}}}\\right)^{p+1}\\right)\\\\=&\\frac{1}{p+1}\\int _{\\mathbb {R}^N} w^{p+1} + \\sum _{j=2}^k \\frac{\\widetilde{B}_2}{\|q_1-q_j\|^{N+2s}} + O\\left( (\\frac{k}{r})^{N+4s} \\right) .\\end{split}$ Combining (REF ), (REF ) and (REF ), we get the desired expansion of energy $J(W)=k\\left[A_1 + \\frac{B_1}{r^m}- \\frac{1}{2}\\sum _{j=2}^k \\frac{\\widetilde{B}_2}{\|q_1-q_j\|^{N+2s}} + o\\left( r^{-m} + (\\frac{k}{r})^{N+2s} \\right)\\right],$ where $A_1 =\\left(\\frac{1}{2}-\\frac{1}{p+1}\\right)\\int _{\\mathbb {R}^N} w^{p+1}(x)\\mathrm {d}x, \\quad B_1=\\frac{a}{2}\\int _{\\mathbb {R}^N} w^2(x) \\mathrm {d}x, \\quad \\widetilde{B}_2 = A\\int _{\\mathbb {R}^N} w^p.$ With (REF ) at hand, we finished the proof.", "Acknowledgement.", "Wang is supported by NSFC (Project11371254).", "Zhao is supported by NSFC (Project 11101155) and by the Fundamental Research Funds for the Central Universities." ] ]	1403.0042
[ [ "Automating Fault Tolerance in High-Performance Computational Biological\n Jobs Using Multi-Agent Approaches" ], [ "Abstract Background: Large-scale biological jobs on high-performance computing systems require manual intervention if one or more computing cores on which they execute fail.", "This places not only a cost on the maintenance of the job, but also a cost on the time taken for reinstating the job and the risk of losing data and execution accomplished by the job before it failed.", "Approaches which can proactively detect computing core failures and take action to relocate the computing core's job onto reliable cores can make a significant step towards automating fault tolerance.", "Method: This paper describes an experimental investigation into the use of multi-agent approaches for fault tolerance.", "Two approaches are studied, the first at the job level and the second at the core level.", "The approaches are investigated for single core failure scenarios that can occur in the execution of parallel reduction algorithms on computer clusters.", "A third approach is proposed that incorporates multi-agent technology both at the job and core level.", "Experiments are pursued in the context of genome searching, a popular computational biology application.", "Result: The key conclusion is that the approaches proposed are feasible for automating fault tolerance in high-performance computing systems with minimal human intervention.", "In a typical experiment in which the fault tolerance is studied, centralised and decentralised checkpointing approaches on an average add 90% to the actual time for executing the job.", "On the other hand, in the same experiment the multi-agent approaches add only 10% to the overall execution time." ], [ "Introduction", "The scale of resources and computations required for executing large-scale biological jobs are significantly increasing [1], [2].", "With this increase the resultant number of failures while running these jobs will also increase and the time between failures will decrease [3], [4], [5].", "It is not desirable to have to restart a job from the beginning if it has been executing for hours or days or months [6].", "A key challenge in maintaining the seamless (or near seamless) execution of such jobs in the event of failures is addressed under research in fault tolerance [7], [8], [9], [10].", "Many jobs rely on fault tolerant approaches that are implemented in the middleware supporting the job (for example [6], [11], [12], [13]).", "The conventional fault tolerant mechanism supported by the middleware is checkpointing [14], [15], [16], [17], which involves the periodic recording of intermediate states of execution of a job to which execution can be returned if a fault occurs.", "Such traditional fault tolerant mechanisms, however, are challenged by drawbacks such as single point failures [18], lack of scalability [19] and communication overheads [20], which pose constraints in achieving efficient fault tolerance when applied to high-performance computing systems.", "Moreover, many of the traditional fault tolerant mechanisms are manual methods and require human administrator intervention for isolating recurring faults.", "This will place a cost on the time required for maintenance.", "Self-managing or automated fault tolerant approaches are therefore desirable, and the objective of the research reported in this paper is the development of such approaches.", "If a failure is likely to occur on a computing core on which a job is being executed, then it is necessary to be able to move (migrate) the job onto a reliable core [21].", "Such mechanisms are not readily available.", "At the heart of this concept is mobility, and a technique that can be employed to achieve this is using multi-agent technologies [22].", "Two approaches are proposed and implemented as the means of achieving both the computation in the job and self-managing fault tolerance; firstly, an approach incorporating agent intelligence, and secondly, an approach incorporating core intelligence.", "In the first approach, automated fault tolerance is achieved by a collection of agents which can freely traverse on a network of computing cores.", "Each agent carries a portion of the job (or sub-job) to be executed on a computing core in the form of a payload.", "Fault tolerance in this context can be achieved since an agent can move on the network of cores, effectively moving a sub-job from one computing core which may fail onto another reliable core.", "In the second approach, automated fault tolerance is achieved by considering the computing cores to be an intelligent network of cores.", "Sub-jobs are scheduled onto the cores, and the cores can move processes executed on them across the network of cores.", "Fault tolerance in this context can be achieved since a core can migrate a process executing on it onto another core.", "A third approach is proposed which combines both agent and core intelligence under a single umbrella.", "In this approach, a collection of agents freely traverse on a network of virtual cores which are an abstraction of the actual hardware cores.", "The agents carry the sub-jobs as a payload and situate themselves on the virtual cores.", "Fault tolerance is achieved either by an agent moving off one core onto another core or the core moving an agent onto another core when a fault is predicted.", "Rules are considered to decide whether an agent or a core should initiate the move.", "Automated fault tolerance can be beneficial in areas such as molecular dynamics [23], [24], [25], [26].", "Typical molecular dynamics simulations explore the properties of molecules in gaseous, liquid and solid states.", "For example, the motion of molecules over a time period can be computed by employing Newton's equations if the molecules are treated as point masses.", "These simulations require large numbers of computing cores that run sub-jobs of the simulation which communicate with each other for hours, days and even months.", "It is not desirable to restart an entire simulation or to loose any data from previous numerical computations when a failure occurs.", "Conventional methods like periodic checkpointing keep track of the state of the sub-jobs executed on the cores, and helps in restarting a job from the last checkpoint.", "However, overzealous periodic checkpointing over a prolonged period of time has large overheads and contributes to the slowdown of the entire simulation [27].", "Additionally, mechanisms will be required to store and handle large data produced by the checkpointing strategy.", "Further, how wide the failure can impact the simulation is not considered in checkpointing.", "For example, the entire simulation is taken back to a previous state irrespective of whether the sub-jobs running on a core depend or do not depend on other sub-jobs.", "One potential solution to mitigate the drawbacks of checkpointing is to proactively probe the core for failures.", "If a core is likely to fail, then the sub-job executing on the core is migrated automatically onto another core that is less likely to fail.", "This paper proposes and experimentally evaluates multi-agent approaches to realising this automation.", "Genome searching is considered as an example for implementing the multi-agent approaches.", "The results indicate the feasibility of the multi-agent approaches; they require only one-fifth the time compared to that required by manual approaches.", "The remainder of this paper is organised as follows.", "The Methods section presents the three approaches proposed for automated fault tolerance.", "The Results section highlights the experimental study and the results obtained from it.", "The Discussion section presents a discussion on the three approaches for automating fault tolerance.", "The Conclusions section summarises the key results from this study." ], [ "Methods", "Three approaches to automate fault tolerance are presented in this section.", "The first approach incorporates agent intelligence, the second approach incorporates core intelligence, and in the third a hybrid of both agent and core intelligence is incorporated." ], [ "Approach 1: Fault Tolerance incorporating Agent Intelligence", "A job, $J$ , which needs to be executed on a large-scale system is decomposed into a set of sub-jobs $J_{1}, J_{2} \\cdots J_{n}$ .", "Each sub-job $J_{1}, J_{2} \\cdots J_{n}$ is mapped onto agents $A_{1}, A_{2} \\cdots A_{n}$ that carry the sub-jobs as payloads onto the cores, $C_{1}, C_{2} \\cdots C_{n}$ as shown in Figure 1.", "The agents and the sub-job are independent of each other; in other words, an agent acts as a wrapper around a sub-job to situate the sub-job on a core.", "Figure: The job, sub-jobs, agents, virtual cores and computing cores in the two approaches proposed for automated fault toleranceThere are three computational requirements of the agent to achieve successful execution of the job: (a) the agent needs to know the overall job, $J$ , that needs to be achieved, (b) the agent needs to access data required by the sub-job it is carrying and (c) the agent needs to know the operation that the sub-job needs to perform on the data.", "The agents then displace across the cores to compute the sub-jobs.", "Intelligence of an agent can be useful in at least four important ways for achieving fault tolerance while a sub-job is executed.", "Firstly, an agent knows the landscape in which it is located.", "Knowledge of the landscape is threefold which includes (a) the knowledge of the computing core on which the agent is located, (b) knowledge of other computing cores in the vicinity of the agent and (c) knowledge of agents located in the vicinity.", "Secondly, an agent identifies a location to situate within the landscape.", "This is possible by gathering information from the vicinity using probing processes and is required when the computing core on which the agent is located is anticipated to fail.", "Thirdly, an agent predicts failures that are likely to impair its functioning.", "The prediction of failures (for example, due to the failure of the computing core) is along similar lines to proactive fault tolerance.", "Fourthly, an agent is mobile within the landscape.", "If the agent predicts a failure then the agent can relocate onto another computing core thereby moving off the job from the core anticipated to fail (refer Figure 2).", "Figure: Agent-Core interaction in Approach 1.", "Agents A 1 ,A 2 A_{1}, A_{2} and A 3 A_{3} are situated on cores C 1 ,C 2 C_{1}, C_{2} and C 3 C_{3} respectively.", "A failure is predicted on core C 1 C_{1}.", "The agent A 1 A_{1} moves onto core C a C_{a}.The intelligence of agents is incorporated within the following sequence of steps that describes an approach for fault tolerance: Agent Intelligence Based Fault Tolerance Decompose a job, $J$ , to be executed on the landscape into sub-jobs, $J_{1}, J_{2} \\cdots J_{n}$ Each sub-job provided as a payload to agents, $A_{1}, A_{2} \\cdots A_{n}$ Agents carry jobs onto computing cores, $C_{1}, C_{2} \\cdots C_{n}$ For each agent, $A_{i}$ located on computing core $C_{i}$ , where $i = 1$ to $n$ Periodically probe the computing core $C_{i}$ if $C_{i}$ predicted to fail, then Agent, $A_{i}$ moves onto an adjacent computing core, $C_{a}$ Notify dependent agents Agent $A_{i}$ establishes dependencies Collate execution results from sub-jobs A failure scenario is considered for the agent intelligence based fault tolerance concept.", "In this scenario, while a job is executed on a computing core that is anticipated to fail any adjacent core onto which the job needs to be reallocated can also fail.", "The communication sequence shown in Figure 3 is as follows.", "The hardware probing process on the core anticipating failure, $C_{PF}$ notifies the failure prediction to the agent process, $P_{PF}$ , situated on it.", "Since the failure of a core adjacent to the core predicted to fail is possible it is necessary that the predictions of the hardware probing processes on the adjacent cores be requested.", "Once the predictions are gathered, the agent process, $P_{PF}$ , creates a new process on an adjacent core and transfers data it was using onto the newly created process.", "Then the input dependent (${P_{ID}}_{1} \\cdots {P_{ID}}_{n}$ ) and output dependent (${P_{OD}}_{1} \\cdots {P_{OD}}_{n}$ ) processes are notified.", "The agent process on $C_{PF}$ is terminated thereafter.", "The new agent process on the adjacent core establishes dependencies with the input and output dependent processes.", "Figure: Communication sequence in the failure scenario of agent intelligence based fault tolerance" ], [ "Approach 2: Fault Tolerance incorporating Core Intelligence", "A job, $J$ , which needs to be executed on a large-scale system is decomposed into a set of sub-jobs $J_{1}, J_{2} \\cdots J_{n}$ .", "Each sub-job $J_{1}, J_{2} \\cdots J_{n}$ is mapped onto the virtual cores, $VC_{1}, VC_{2} \\cdots VC_{n}$ , an abstraction over $C_{1}, C_{2} \\cdots C_{n}$ respectively as shown in Figure 4.", "The cores referred to in this approach are virtual cores which are an abstraction over the hardware computing cores.", "The virtual cores are a logical representation and may incorporate rules to achieve intelligent behaviour.", "Figure: Job-Virtual Core interaction in Approach 2.", "Jobs J 1 ,J 2 J_{1}, J_{2} and J 3 J_{3} are situated on virtual cores VC 1 ,VC 2 VC_{1}, VC_{2} and VC 3 VC_{3} respectively.", "A failure is predicted on core C 1 C_{1} and VC 1 VC_{1} moves the job J 1 J_{1} onto virtual core VC a VC_{a}.Intelligence of a core is useful in a number of ways for achieving fault tolerance.", "Firstly, a core updates knowledge of its surrounding by monitoring adjacent neighbours.", "Independent of what the cores are executing, the cores can monitor each other.", "Each core can ask the question `are you alive?'", "to its neighbours and gain information.", "Secondly, a core periodically updates information of its surrounding.", "This is useful for the core to know which neighbouring cores can execute a job if it fails.", "Thirdly, a core periodically monitors itself using a hardware probing process and predicts if a failure is likely to occur on it.", "Fourthly, a core can move a job executing on it onto an adjacent core if a failure is expected and adjust to failure as shown in Figure 4.", "Once a job has relocated all data dependencies will need to be re-established.", "The following sequence of steps describe an approach for fault tolerance incorporating core intelligence: Core Intelligence Based Fault Tolerance Decompose a job, $J$ , to be executed on the landscape into sub-jobs, $J_{1}, J_{2} \\cdots J_{n}$ Each sub-job allocated to cores, $VC_{1}, VC_{2} \\cdots VC_{n}$ For each core, $VC_{i}$ , where $i = 1$ to $n$ until sub-job $J_{i}$ completes execution Periodically probe the computing core $C_{i}$ if $C_{i}$ predicted to fail, then Migrate sub-job $J_{i}$ on $VC_{i}$ onto an adjacent computing core, $VC_{a}$ Collate execution results from sub-jobs" ], [ "Core Intelligence Failure Scenario", "Figure 5 shows the communication sequence of the core failure scenario considered for the core intelligence based fault tolerance concept.", "The hardware probing process on the core predicted to fail, $C_{PF}$ notifies a predicted failure to the core.", "The job executed on $VC_{PF}$ is then migrated onto an adjacent core $VC_{1} \\cdots VC_{n}$ once a decision based on failure predictions are received from the hardware probing processes of adjacent cores." ], [ "Approach 3: Hybrid Fault Tolerance", "The hybrid approach acts as an umbrella bringing together the concepts of agent intelligence and core intelligence.", "The key concept of the hybrid approach lies in the mobility of the agents on the cores and the cores collectively executing a job.", "Decision-making is required in this approach for choosing between the agent intelligence and core intelligence approaches when a failure is expected.", "A job, $J$ , which needs to be executed on a large-scale system is decomposed into a set of sub-jobs $J_{1}, J_{2} \\cdots J_{n}$ .", "Each sub-job $J_{1}, J_{2} \\cdots J_{n}$ is mapped onto agents $A_{1}, A_{2} \\cdots A_{n}$ that carry the sub-jobs as payloads onto the virtual cores, $VC_{1}, VC_{2} \\cdots VC_{n}$ which are an abstraction over $C_{1}, C_{2} \\cdots C_{n}$ respectively as shown in Figure 1.", "The following sequence of steps describe the hybrid approach for fault tolerance incorporating both agent and core intelligence: Hybrid Intelligence Based Fault Tolerance Decompose a job, $J$ , to be executed on the landscape into sub-jobs, $J_{1}, J_{2} \\cdots J_{n}$ Each sub-job provided as a payload to agents, $A_{1}, A_{2} \\cdots A_{n}$ Agents carry jobs onto virtual cores, $VC_{1}, VC_{2} \\cdots VC_{n}$ For each agent, $A_{i}$ located on virtual core $VC_{i}$ , where $i = 1$ to $n$ Periodically probe the computing core $C_{i}$ if $C_{i}$ predicted to fail, then if `Agent Intelligence' is a suitable mechanism, then Agent, $A_{i}$ , moves onto an adjacent computing core, $VC_{a}$ Notify dependent agents Agent $A_{i}$ establishes dependencies else if `Core Intelligence' is a suitable mechanism, then Core $VC_{i}$ migrates agent, $A_{i}$ onto an adjacent computing core, $VC_{a}$ Collate execution results from sub-jobs Figure: Communication sequence in core intelligence based fault toleranceWhen a core failure is anticipated both an agent and a core can make decisions which can lead to a conflict.", "For example, an agent can attempt to move onto an adjacent core while the core on which it is executing would like to migrate it to an alternative adjacent core.", "Therefore, an agent and the core on which it is located need to negotiate before either of them initiate a response to move (see Figure 6).", "The rules for the negotiation between the agent and the core in this case are proposed from the experimental results presented in this paper (presented in the Decision Making Rules sub-section).", "Figure: Conflict negotiation and resolution in Approach 3.", "Agents A 1 ,A 2 A_{1}, A_{2} and A 3 A_{3} are situated on virtual cores VC 1 ,VC 2 VC_{1}, VC_{2} and VC 3 VC_{3} which are mapped onto computing cores C 1 ,C 2 C_{1}, C_{2} and C 3 C_{3} respectively.", "A failure is predicted on core C 1 C_{1}.", "The agent A 1 A_{1} and VC 1 VC_{1} negotiate to decide who moves the sub-job onto core VC a VC_{a}.In this section, the experimental platform is considered followed by the experimental studies and the results obtained from experiments." ], [ "Platform", "Four computer clusters were used for the experiments reported in this paper.", "The first was a cluster available at the Centre for Advanced Computing and Emerging Technologies (ACET), University of Reading, UK.", "Thirty three compute nodes connected through Gigabit Ethernet were available, each with Pentium IV processors and 512 MB-2 GB RAM.", "The remaining three clusters are compute resources, namely Brasdor, Glooscap and Placentia, all provided by The Atlantic Computational Excellence Network (ACEnet) [28], Canada.", "Brasdor comprises 306 compute nodes connected through Gigabit Ethernet, with 932 cores and 1-2 GB RAM.", "Glosscap comprises 97 nodes connected through Infiniband, with 852 cores and 1-8 GB RAM.", "Placentia comprises 338 compute nodes connected through Infiniband, with 3740 cores and 2-16 GB RAM.", "The cluster implementations in this paper are based on the Message Passing Interface (MPI).", "The first approach, incorporating agent intelligence, is implemented using Open MPI [29], an open source implementation of MPI 2.0.", "The dynamic process model which supports dynamic process creation and management facilitates control over an executing process.", "This feature is useful for implementing the first approach.", "The MPI functions useful in the implementation are (i) MPI_COMM_SPAWN which creates a new MPI process and establishes communication with an existing MPI application and (ii) MPI_COMM_ACCEPT and MPI_COMM_CONNECT which establishes communication between two independent processes.", "The second approach, incorporating core intelligence, is implemented using Adaptive MPI (AMPI) [30], developed over Charm++ [31], a C++ based parallel programming language.", "The aim of AMPI is to achieve dynamic load balancing by migrating objects over virtual cores and thereby facilitating control over cores.", "Core intelligence harnesses this potential of AMPI to migrate a job from a core onto another core.", "A strategy to migrate a job using the concepts of processor virtualisation and dynamic job migration in AMPI and Charm++ is reported in [32]." ], [ "Experimental Studies", "Parallel reduction algorithms [38], [39] which implement the bottom-up approach (i.e., data flows from the leaves to the root) are employed for the experiments.", "These algorithms are of interest for three reasons.", "Firstly, the algorithm is used in a large number of scientific applications including computational biological applications in which optimizations are performed (for example, bootstrapping).", "Incorporating self-managing fault tolerant approaches can make these algorithms more robust and reliable [40].", "Secondly, the algorithm lends itself to be easily decomposed into a set of sub-jobs.", "Each sub-job can then be mapped onto a computing core either by providing the sub-job as a payload to an agent in the first approach or by providing the job onto a virtual core incorporating intelligent rules.", "Thirdly, the execution of a parallel reduction algorithm stalls and produces incorrect solutions if a core fails.", "Therefore, parallel reduction algorithms can benefit from local fault-tolerant techniques.", "Figure 7 is an exemplar of a parallel reduction algorithm.", "In the experiments reported in this paper, a generic parallel summation algorithm with three sets of input is employed.", "Firstly, $I_{(1,1)}$ , $I_{(1,2)}$ $\\cdots $ $I_{(1,x)}$ , secondly, $I_{(2,1)}$ , $I_{(2,2)}$ $\\cdots $ $I_{(2,y)}$ , and thirdly, $I_{(3,1)}$ $\\cdots $ $I_{(3,z)}$ .", "The first level nodes which receive the three sets of input comprise three set of nodes.", "Firstly, ${N_{1}}_{(1,1)}$ , ${N_{1}}_{(1,2)}$ $\\cdots $ ${N_{1}}_{(1,x)}$ , secondly, ${N_{1}}_{(2,1)}$ , ${N_{1}}_{(2,2)}$ $\\cdots $ ${N_{1}}_{(2,y)}$ , and thirdly, ${N_{1}}_{(3,1)}$ , ${N_{1}}_{(3,2)}$ $\\cdots $ ${N_{1}}_{(3,z)}$ .", "The next level of nodes, ${N_{2}}_{(1,1)}$ , ${N_{2}}_{(2,1)}$ and ${N_{3}}_{(3,1)}$ receive inputs from the first level nodes.", "The resultant from the second level nodes is fed in to the third level node ${N_{3}}_{(1,1)}$ .", "The nodes reduce the input through the output using the parallel summation operator ($\\oplus $ ).", "Figure: Generic parallel summation algorithm.", "The inputs are denoted by II and the three levels of nodes are denoted by NN.", "The inputs are passed to the nodes N 1 N_{1} which are then reduced and passed to nodes N 2 N_{2} and finally onto N 3 N_{3} for the output.The parallel summation algorithm can benefit from the inclusion of fault tolerant strategies.", "The job, $J$ , in this case is summation, and the sub-jobs, $J_{1}, J_{2} \\cdots J_{n}$ are also summations.", "In the first fault tolerant approach, incorporating mobile agent intelligence, the data to be summed along with the summation operator is provided to the agent.", "The agents locate on the computing cores and continuously probe the core for anticipating failures.", "If an agent is notified of a failure, then it moves off onto another computing core in the vicinity, thereby not stalling the execution towards achieving the summation job.", "In the second fault tolerant approach, incorporating core intelligence, the sub-job comprising the data to be summed along with the summation operator is located on the virtual core.", "When the core anticipates a failure, it migrates the sub-job onto another core.", "A failure scenario is considered for experimentally evaluating the fault tolerance strategies.", "In the scenario, when a core failure is anticipated the sub-job executing on it is relocated onto an adjacent core.", "Of course this adjacent core may also fail.", "Therefore, information is also gathered from adjacent cores as to whether they are likely to fail or not.", "This information is gathered by the agent in the agent-based approach and the virtual core in the core-based approach and used to determine which adjacent core the sub-job needs to be moved to.", "This failure scenario is adapted to the two strategies giving respectively the agent intelligence failure scenario and the core intelligence failure scenario (described in the Methods section)." ], [ "Experimental Results", "Figures 8 through 13 are a collection of graphs plotted using the parallel reduction algorithm as a case study for both the first (agent intelligence - Figure 8, Figure 10 and Figure 12) and second (core intelligence - Figure 9, Figure 11 and Figure 13) fault tolerant approaches.", "Each graph comprises four plots, the first representing the ACET cluster and the other three representing the three ACEnet clusters.", "The graphs are also distinguished based on the following three factors that can affect the performance of the two approaches: Figure: No.", "of dependencies vs time taken for reinstating execution after failure in the agent intelligent approachFigure: No.", "of dependencies vs time taken for reinstating execution after failure in the core intelligent approachFigure: Size of data vs time taken for reinstating execution after failure in the agent intelligent approachFigure: Size of data vs time taken for reinstating execution after failure in the core intelligent approachFigure: Process size vs time taken for reinstating execution after failure in the agent intelligent approachFigure: Process size vs time taken for reinstating execution after failure in the core intelligent approach(i) The number of dependencies of the sub-job being executed denoted as $Z$ .", "If the total number of input dependencies is $d_{i}$ and the total number of output dependencies is $d_{o}$ , then $Z = d_{i} + d_{o}$ .", "For example, in a parallel summation algorithm incorporating binary trees, each node has two input dependencies and one output dependency, and therefore $Z = 3$ .", "In the experiments, the number of dependencies is varied between 3 and 63, by changing the number of input dependencies of an agent or a core.", "The results are presented in Figure 8 and Figure 9.", "(ii) The size of the data communicated across the cores denoted as $S_{d}$ .", "In the experiments, the input data is a matrix for parallel summation and its size is varied between $2^{19}$ to $2^{31}$ KB.", "The results are presented in Figure 10 and Figure 11.", "(iii) The process size of the distributed components of the job denoted as $S_{p}$ .", "In the experiments, the process size is varied between $2^{19}$ to $2^{31}$ KB which is proportional to the input data.", "The results are presented in Figure 12 and Figure 13.", "Figure 8 is a graph of the time taken in seconds for reinstating execution versus the number of dependencies in the agent intelligence failure scenario.", "The mean time taken to reinstate execution for 30 trials, ${{\\Delta T}_{A}}_{2}$ , is computed for varying numbers of dependencies, $Z$ ranging from 3 to 63.", "The size of the data on the agent is $S_{d}=2^{24}$ kilo bytes.", "The approach is slowest on the ACET cluster and fastest on the Placentia cluster.", "In all cases the communication overheads result in a steep rise in the time taken for execution until $Z=10$ .", "The time taken on the ACET cluster rises once again after $Z=25$ .", "Figure 9 is a graph of the time taken in seconds for reinstating execution versus the number of dependencies in the core intelligence failure scenario.", "The mean time taken to reinstate execution for 30 trials, ${{\\Delta T}_{C}}_{2}$ , is computed for varying number of dependencies, $Z$ ranging from 3 to 63.", "The size of the data on the core is $S_{d}=2^{24}$ kilo bytes.", "The approach requires almost the same time on the four clusters for reinstating execution until $Z=10$ , after which there is divergence in the plots.", "The approach lends itself well on Placentia and Glooscap.", "Figure 10 is a graph showing the time taken in seconds for reinstating execution versus the size of data in kilobytes (KB), $S_{d} = 2^n$ , where $n = 19, 19.5 \\cdots 31$ , carried by an agent in the agent intelligence failure scenario.", "The mean time taken to reinstate execution for 30 trials, ${{\\Delta T}_{A}}_{2}$ , is computed for varying sizes of data ranging from $2^{19}$ to $2^{31}$ KB.", "The number of dependencies $Z$ is 10 for the graph plotted.", "Placentia and Glooscap outperforms ACET and Brasdor in the agent approach for varying size of data.", "Figure 11 is a graph showing the time taken in seconds for reinstating execution versus the size of data in kilobytes (KB), $S_{d} = 2^n$ , where $n = 19, 19.5 \\cdots 31$ , on a core in the core intelligence failure scenario.", "The mean time taken to reinstate execution for 30 trials, ${{\\Delta T}_{C}}_{2}$ , is computed for varying sizes of data ranging from $2^{19}$ to $2^{31}$ KB.", "The number of dependencies $Z$ is 10 for the graph plotted.", "In this graph, nearly similar time is taken by the approach on the four clusters with the ACET cluster requiring more time than the other clusters for $n > 24$ .", "Figure 12 is a graph showing the time taken in seconds for reinstating execution versus process size in kilobytes (KB), $S_{p} = 2^n$ , where $n = 19, 19.5 \\cdots 31$ , in the agent intelligence failure scenario.", "The mean time taken to reinstate execution for 30 trials, ${{\\Delta T}_{A}}_{2}$ , is computed for varying process sizes ranging from $2^{19}$ to $2^{31}$ KB.", "The number of dependencies $Z$ is 10 for the graph plotted.", "The second scenario performs similar to the first scenario.", "The approach takes almost similar times to reinstate execution after a failure on the four clusters, but there is a diverging behaviour after $n > 26$ .", "Figure 13 is a graph showing the time taken in seconds for reinstating execution versus process size in kilobytes (KB), $S_{p} = 2^n$ , where $n = 19, 19.5 \\cdots 31$ , in the core intelligence failure scenario.", "The mean time taken to reinstate execution for 30 trials, ${{\\Delta T}_{C}}_{2}$ , is computed for varying process sizes ranging from $2^{19}$ to $2^{31}$ KB.", "The number of dependencies $Z$ is 10 for the graph plotted.", "The approach has similar performance on the four clusters, though Placentia performs better than the other three clusters for a process size of more than $2^{26}$ KB." ], [ "Decision Making Rules", "Parallel simulations in molecular dynamics model atoms or molecules in gaseous, liquid or solid states as point masses which are in motion.", "Such simulations are useful for studying the physical and chemical properties of the atoms or molecules.", "Typically the simulations are compute intensive and can be performed in at least three different ways [26].", "Firstly, by assigning a group of atoms to each processor, referred to as atom decomposition.", "The processor computes the forces related to the group of atoms to update their position and velocities.", "The communication between atoms is high and effects the performance on large number of processors.", "Secondly, by assigning a block of forces from the force matrix to be computed to each processor, referred to as force decomposition.", "This technique scales better than atom decomposition but is not a best solution for large simulations.", "Thirdly, by assigning a three dimensional space of the simulation to each processor, referred to as spatial decomposition.", "The processor needs to know the positions of atoms in the adjacent space to compute the forces of atoms in the space assigned to it.", "The interactions between the atoms are therefore local to the adjacent spaces.", "In the first and second decomposition techniques interactions are global and thereby dependencies are higher.", "Agent and core based approaches to fault tolerance can be incorporated within parallel simulations in the area of molecular dynamics.", "However, which of the two approaches, agent or core intelligence, is most appropriate?", "The decomposition techniques considered above establish dependencies between blocks of atoms and between atoms.", "Therefore the degree of dependency affects the relocation of a sub-job in the event of a core failure and reinstating it.", "The dependencies of an atom in the simulation can be based on the input received from neighbouring atoms and the output propagated to neighbouring atoms.", "Based on the number of atoms allocated to a core and the time step of the simulation the intensity of numerical computations and the data managed by a core vary.", "Large simulations that extend over long periods of time generate and need to manage large amounts of data; consequently the process size on a core will also be large.", "Therefore, (i) the dependency of the job, (ii) the data size and (iii) the process size are factors that need to be taken into consideration for deciding whether an agent-based approach or a core-based approach needs to come into play.", "Along with the observations from parallel simulations in molecular dynamics, the experimental results provide an insight into the rules for decision-making for large-scale applications.", "From the experimental results graphed in Figure 8 and Figure 9, where dependencies are varied, core intelligence is superior to agent intelligence if the total dependencies $Z$ is less than or equal to 10.", "Therefore, If the algorithm needs to incorporate fault tolerance based on the number of dependencies, then if $Z \\le 10$ use core intelligence, else use agent or core intelligence.", "From the experimental results graphed in Figure 10 and Figure 11, where the size of data is varied, agent intelligence is more beneficial than core intelligence if the size of data $S_{d}$ is less than or equal to $2^{24}$ KB.", "Therefore, If the algorithm needs to incorporate fault tolerance based on the size of data, then if $S_{d} \\le 2^{24}$ KB, then use agent intelligence, else use agent or core intelligence.", "From the experimental results graphed in Figure 12 and Figure 13, where the size of the process is varied, agent intelligence is more beneficial than core intelligence if the size of the process $S_{p}$ is less than or equal to $2^{24}$ KB.", "Therefore, If the algorithm needs to incorporate fault tolerance based on process size, then if $S_{p} \\le 2^{24}$ KB, then use agent intelligence, else use agent or core intelligence.", "The number of dependencies, size of data, and process size are the three factors taken into account in the experimental results.", "The results indicate that the approach incorporating core intelligence takes lesser time than the approach incorporating agent intelligence.", "There are two reasons for this.", "Firstly, in the agent approach, the agent needs to establish the dependency with each agent individually, where as in the core approach as a job is migrated from a core onto another its dependencies are automatically established.", "Secondly, agent intelligence is a software abstraction of the sub-job, thereby adding a virtualised layer in the communication stack.", "This increases the time for communication.", "The virtual core is also an abstraction of the computing core but is closer to the computing core in the communication stack.", "The above rules can be incorporated to exploit both agent-based and core-based intelligence in a third, hybrid approach.", "The key concept of the hybrid approach combines the mobility of the agents on the cores and the cores collectively executing a job.", "The approach can select whether the agent-based approach or the core-based approach needs to come to play based on the rules for decision-making.", "The key observation from the experimental results is that the cost of incorporating intelligence at the job and core levels for automating fault tolerance is less than a second, which is smaller than the time taken by manual methods which would be in the order of minutes.", "For example, in the first approach, the time for reinstating execution with over 50 dependencies is less than 0.55 seconds and in the second approach, is less than 0.5 seconds.", "Similar results are obtained when the size of data and the process are large." ], [ "Genome Searching using Multi-Agent approaches", "The proposed multi-agent approaches and the decision making rules considered in the above sections are validated using a computational biology job.", "A job that fits the criteria of reduction algorithms is considered.", "In reduction algorithms, a job is decomposed to sub-jobs and executed on multiple nodes and the results are further passed onto other node for completing the job.", "One popular computational biology job that fits this criteria is searching for a genome pattern.", "This has been widely studied and fast and efficient algorithms have been developed for searching genome patterns (for example, [33], [34] and [35]).", "In the genome searching experiment performed in this research multiple nodes of a cluster execute the search operation and the output produced by the search nodes are then combined by an additional node.", "The focus of this experimental study is not parallel efficiency or scalability of the job but to validate the multi-agent approaches and the decision making rules in the context of computational biology.", "Hence, a number of assumptions are made for the genome searching job.", "First, redundant copies of the genome data are made on the same node to obtain a sizeable input.", "Secondly, the search operation is run multiple times to span long periods of time.", "Thirdly, the jobs are executed such that they can be stopped intentionally by the user at any time and gather the results of the preceding computations until the execution was stopped.", "The Placentia cluster is chosen for this validation study since it was the best performing cluster in the empirical study presented in the previous sections.", "The job is implemented using R programming which uses MPI for exploiting computation on multiple nodes of the Placentia cluster.", "Bioconductor packageshttp://bioconductor.org/ are required for supporting the job.", "The job makes use of BSgenome.Celegans.UCSC.ce2, BSgenome.Celegans.UCSC.ce6 and BSgenome.Celegans.UCSC.ce10 as input data which are the ce2, ce6 and ce10 genome for chromosome I of Caenorhabditis elegans [36], [37].", "A list of 5000 genome patterns each of which is a short nucleotide sequence of 15 to 25 bases is provided to be searched against the input data.", "The forward and reverse strands of seven Caenorhabditis elegans chromosomes named as chrI, chrII, chrIII, chrIV, chrV, chrX, chrM are the targets of the search operation.", "When there is a target hit the search nodes provide to the node that gathers the results the name of the chromosome where the hit occurs, two integers giving the starting and ending positions of the hit, an indication of the hit either in the forward or reverse strand, and unique identification for every pattern in the dictionary.", "The results are tabulated in an output file in the combining node.", "A sample of the output is shown in Figure 14.", "Figure: Sample output from searching genome pattern.", "The output shows the name of the chromosome where the target hit occurs, followed by two integers giving the starting and ending positions of the hit, an indication of the hit either in the forward or reverse strand, and unique identification for every pattern in the dictionary.Redundant copies of the input data are made to obtain 512 MB (which is $2^{19}$ KB) and the job is executed for one hour.", "In a typical experiment the number of dependencies, $Z$ was set to 4; three nodes of the cluster performed the search operation while the fourth node combined the results passed on to it from the three search nodes.", "In the agent intelligence based approach the time for predicting the fault is 38 seconds, the time for reinstating execution is 0.47 seconds, the overhead time is over 5 minutes and the total time when one failure occurs per hour is 1 hour, 6 minutes and 17 seconds.", "In the core intelligence based approach the time for predicting the single node failure is similar to the agent intelligence approach; the time for reinstating execution is 0.38 seconds, the overhead time is over 4 minutes and the total time when one failure occurs per hour is 1 hour, 5 minutes and 8 seconds.", "In another experiment for 512 MB size of input data the number of dependencies was set to 12; eleven nodes for searching and one node for combining the results provided by the eleven search nodes.", "In the agent intelligence based approach the time for reinstating execution is 0.54 seconds, the overhead time is over 6 minutes and the total time when one failure occurs per hour is 1 hour, 7 minutes and 34 seconds.", "In the core intelligence based approach the time for reinstating execution is close to 0.54 seconds, the overhead time is over 6 minutes and the total time when one failure occurs per hour is 1 hour, 7 minutes and 48 seconds.", "The core intelligence approach requires less time than the agent intelligence approach when $Z=3$ , but the times are comparable when $Z=12$ .", "So, the above two experiments validate Rule 1 for decision making considered in the previous section.", "Experiments were performed for different input data sizes; in one case $S_{d} = 2^{19}$ KB and in the other $S_{d} = 2^{25}$ KB.", "The agent intelligence approach required less time in the former case than the core intelligence approach.", "The time was comparable for the latter case.", "Hence, the genome searching job in the context of the experiments validated Rule 2 for decision making.", "Similarly, when process size was varied Rule 3 was found to be validated.", "The genome searching job is used as an example to validate the use of the multi-agent approaches for computational biology jobs.", "The decision making rules empirically obtained were satisfied in the case of this job.", "The results obtained from the experiments for the genome searching job along with comparisons against traditional fault tolerance approaches, namely centralised and decentralised checkpointing are considered in the next section." ], [ "Discussion", "All fault tolerance approaches initiate a response to address a failure.", "Based on when a response is initiated with respect to the occurrence of the failure, approaches can be classified as proactive and reactive.", "Proactive approaches predict failures of computing resources before they occur and then relocate a job executing on resources anticipated to fail onto resource that are not predicted to fail (for example [32], [43], [44]).", "Reactive approaches on the other hand minimise the impact of failures after they have occurred (for example checkpointing [16], rollback recovery [45] and message logging [46]).", "A hybrid of proactive and reactive, referred to as adaptive approaches, is implemented so that failures that cannot be predicted by proactive approaches are handled by the reactive approaches [47], [48], [49].", "The control of a fault tolerant approach can be either centralised or distributed.", "In approaches where the control is centralised, one or more servers are used for backup and a single process responsible for monitoring jobs that are executed on a network of nodes.", "The traditional message logging and checkpointing approach involves the periodic recording of intermediate states of execution of a job to which execution can be returned if faults occur.", "Such approaches are susceptible to single point failure, lack scalability over a large network of nodes, have large overheads, and require large disk storage.", "These drawbacks can be minimised or avoided when the control of the approaches is distributed (for example, distributed diagnosis [50], distributed checkpointing [41] and diskless checkpointing [51]).", "In this paper two distributed proactive approaches towards achieving fault tolerance are proposed and implemented.", "In both approaches a job to be computed is decomposed into sub-jobs which are then mapped onto the computing cores.", "The two approaches operate at the middle levels (between the sub-jobs and the computing cores) incorporating agent intelligence.", "In the first approach, the sub-jobs are mapped onto agents which are released onto the cores.", "If an agent is notified of a potential core failure during execution of the sub-job mapped onto it, then the agent moves onto another core thereby automating fault tolerance.", "In the second approach the sub-jobs are scheduled on virtual cores, which are an abstraction of the computing cores.", "If a virtual core anticipates a core failure then it moves the sub-job on it to another virtual core, in effect onto another computing core.", "The two approaches achieve automation in fault tolerance using intelligence in agents and using intelligence in cores respectively.", "A third approach is proposed which brings together the concepts of both agent intelligence and core intelligence from the first two approaches." ], [ "Overcoming the problems of Checkpointing", "The conventional approaches to fault tolerance such as checkpointing have large communication overheads based on the periodicity of checkpointing.", "High frequencies of checkpointing can lead to heavy network traffic since the available communication bandwidth will be saturated with data transferred from all computing nodes to the a stable storage system that maintains the checkpoint.", "This traffic is on top of the actual data flow of the job being executed on the network of cores.", "While global approaches are useful for jobs which are less memory and data intensive and can be executed over short periods of time, they may constrain the efficiency for jobs using big data in limited bandwidth platforms.", "Hence, local approaches can prove useful.", "In the case of the agent based approaches there is high periodicity for probing the cores in the background but very little data is transferred while probing unlike in checkpointing.", "Hence, communication overhead times will be significantly lesser.", "Lack of scalability is another issue that affects efficient fault tolerance.", "Many checkpointing strategies are centralised (with few exceptions, such as [41], [42]) thereby limiting the scale of adopting the strategy.", "This can be mitigated by using multiple centralised checkpointing servers but the distance between the nodes and the server discounts the scalability issue.", "In the agent based approaches, all communications are short distance since the cores only need to communicate with the adjacent cores.", "Local communication therefore increases the scale to which the agent based approaches can be applied.", "Checkpointing is susceptible to single point failures due to the failure of the checkpoint servers.", "The job executed will have to be restarted.", "The agent-based approaches are also susceptible to single point failures.", "While they incorporate intelligence to anticipate hardware failure the processor core may fail before the sub-job it supports can be relocated to an adjacent processor core, before the transfer is complete, or indeed the core onto which it is being transferred may also fail.", "However, the incorporation of intelligence on the processor core, specifically the ability to anticipate hardware failure locally, means that the numbers of these hardware failures that lead to job failure can be reduced when compared to traditional checkpointing.", "But since there is the possibility of agent failure the retention of some level of human intervention is still required.", "Therefore, we propose combining checkpointing with the agent-based approaches, the latter acting as a first line of anticipatory response to hardware failure backed up by traditional checkpointing as a second line of reactive response." ], [ "Predicting potential failures", "Figure 15 shows the execution of a job between two checkpoints, $C_{n}$ and $C_{n+1}$ , where $PF$ is the predicted failure and $F$ is the actual failure of the node on which a sub-job is executing.", "Figure 15(a) shows when there are no predicted failures or actual failures that occur on the node.", "Figure 15(b) shows when a failure occurs but could not be predicted.", "In this case, the system fails if the multi-agent approaches are solely employed.", "One way to mitigate this problem is by employing the multi-agent approaches in conjunction with checkpointing as shown in the next section.", "Figure 15(c) shows when the approaches predict a failure which does not happen.", "If a large number of such predictions occur then the sub-job needs to be shifted often from one node to the other which adds to the overhead time for executing the job.", "This is not an ideal case and makes the job unstable.", "Figure 15(d) shows the ideal case in which a fault is predicted before it occurs.", "Figure: Fault prediction between two checkpoints, C n C_{n} and C n+1 C_{n+1}.", "(a) Ideal state of the job when no faults occur.", "(b) Failure state of the job when a fault occurs but is not predicted.", "(c) Unstable state of the job when a fault is predicted but does not occur.", "(d) Ideal prediction state of the job when a fault is predicted and occurs thereafter.Failure prediction is based on a machine learning approach that is incorporated within multi-agents.", "This prediction is based on a log that is maintained on the health of the node and its adjacent nodes.", "Each agent sends out 'are you alive' signals to adjacent nodes to gather the state of the adjacent node.", "The machine learning approach is constantly evaluating the state of the system against the log it maintains, which is different across the nodes.", "The log can contain the state of the node from past failures, work load of the nodes when it failed previously and even data related to patterns of periodic failures.", "However, this prediction method cannot predict a range of faults due to deadlocks, hardware and power failures and instantaneously occurring faults.", "Hence, the multi-agent approaches are most useful when used along with checkpointing.", "It was observed that nearly 29% of all faults occurring in the cluster could be predicted.", "Although this number is seemingly small it is helpful to not have to rollback to a previous checkpoint when a large job under time constraints is executed.", "Accuracy of the predictions were found to be 64%; the system was found to be stable in 64 out of the 100 times a prediction was made.", "To increase the impact of the multi-agent approaches more faults will need to be captured.", "For this extensive logging and faster methods for prediction will need to be considered.", "These approaches will have to be used in conjunction with checkpointing for maximum effectiveness.", "The instability due to the approaches shifting jobs between nodes when there is a false prediction will need to be reduced to improve the overall efficiency of the approaches.", "For this, the state of the node can be compared with other nodes so that a more informed choice is made by the approaches." ], [ "Comparing traditional and multi-agent approaches", "Table 1 shows a comparison between a number of fault tolerant strategies, namely centralised and decentralised checkpointing and the multi-agent approaches.", "An experiment was run for a genome searching job that was executed multiple times on the Placentia cluster.", "Data in the table was obtained to study the execution of the genome searching job between two checkpoints ($C_{n}$ and $C_{n+1}$ ) which are one hour apart.", "The execution is interrupted by failure $F$ as shown in Figure 16.", "Two types of single node failure are simulated in the execution.", "The first is a periodic node failure which occurs at 15 minutes after $C_{n}$ and 45 minutes before $C_{n+1}$ (refer Figure 16(a)), and the second is a random node failure which occurs $x$ minutes after $C_{n}$ and $60-x$ minutes before $C_{n+1}$ (refer Figure 16(b)).", "The average time when a random failure occurs is found to be 31 minutes and 14 seconds for 5000 trials.", "The size of data, $S_{d} = 2^{19}$ KB and the number of dependencies, $Z=4$ .", "Figure: Fault occurrences between two checkpoints, C n C_{n} and C n+1 C_{n+1}.", "(a) Periodic failure that occurs 14 minutes after C n C_{n} and 46 minutes before C n+1 C_{n+1}.", "(b) Random failure that occurs xx minutes after C n C_{n} and 60-x60-x minutes before C n+1 C_{n+1}.In Table 1, the average time taken for reinstating execution, for the overheads and for executing the job between the checkpoints is considered.", "The time taken for reinstating execution is for bringing execution back to normal after a failure has occurred.", "The reinstating time is obtained for one periodic single node failure and one random single node failure.", "The overhead time is for creating the checkpoints and transferring data for the checkpoint to the server.", "The overhead time is obtained for one periodic single node failure and one random single node failure.", "The execution time without failures, when one periodic failure occurs per hour and when five random failures occur per hour is obtained.", "Centralised checkpointing using single and multiple servers is considered when the frequency of checkpointing is once every hour.", "In the case of both single and multiple server checkpointing the time taken for reinstating execution regardless of whether it was a periodic or random failure is 14 minutes and 8 seconds.", "On a single server the overhead is 8 minutes and 5 seconds where as the overhead to create the checkpoint is 9 minutes and 14 seconds which is higher than overheads on a single server and is expected.", "The average time taken for executing the job when one failure occurs includes the elapsed execution time (15 minutes for periodic failure and 31 minutes and 14 seconds for random failure) until the failure occurred and the combination of the time for reinstating execution after the failures and the overhead time.", "For one periodic failure that occurs in one hour the penalty of execution when single server checkpointing is 62% more than executing without a failure; in the case of a random failure that occurs in one hour the penalty is 89% more than executing without a failure.", "If five random failure occur then the penalty is 445%, requiring more than five times the time for executing the job without failures.", "Centralised checkpointing with multiple servers requires more time than with a single server.", "This is due to the increase in the overhead time for creating checkpoints on multiple servers.", "Hence, checkpointing with multiple servers requires 64% and 91% more time than the time for executing the job without any failures for one periodic and one random failure per hour respectively.", "On the other hand executing jobs when decentralised checkpointing on multiple servers is employed requires similar time to that taken by centralised checkpointing on a single server.", "The time for reinstating execution is higher than centralised checkpointing methods due to the time required for determining the server closest to the node that failed.", "However, the overhead times are lower than other checkpoint approaches since the server closest to the node that failed is chosen for creating the checkpoint which reduces data transfer times.", "The multi-agent approaches are proactive and therefore the average time taken for predicting single node failures are taken into account which is nearly 38 seconds.", "The time taken for reinstating execution after one periodic single node failure for the agent intelligence approach is 0.47 seconds and for the core intelligence approach is 0.38 seconds.", "Since $Z \\le 10$ the core intelligence approach is selected.", "In this case, the core intelligence approach is faster than the agent intelligence approach in the total time it takes for executing the job when there is one periodic or random fault and when there are five faults that occur in the job.", "The multi-agent approaches only require one-fifth the time taken by the checkpointing methods for completing execution.", "This is because the time for reinstating and the overhead times are significantly lower than the checkpointing approaches.", "Table 2 shows a comparison between centralised and decentralised checkpointing and the multi-agent approaches for a genome searching job that is executed on the Placentia cluster for five hours.", "The checkpoint periodicity is once every one, two and four hours as shown in Figure 17.", "Similar to Table 1 periodic and random failures are simulated.", "Figure 17(a) shows the start and completion of the job without failures or checkpoints.", "When the checkpoint periodicity is one hour there are four checkpoints, $C_{1}$ , $C_{2}$ , $C_{3}$ and $C_{4}$ (refer Figure 17(b)); a periodic node failure is simulated after 14 minutes from a checkpoint and the average time at which a random node failure occurs is found to be 31 minutes and 14 seconds from a checkpoint for 5000 trials.", "When checkpoint periodicity is two hours there are two checkpoints, $C_{1}$ and $C_{2}$ (refer Figure 17(c)); a periodic node failure is simulated after 28 minutes from a checkpoint and the average time a random node failure occurs is found to be after 1 hour, 3 minutes and 22 seconds from a checkpoint for 5000 trials.", "When checkpoint periodicity is four hours there is only one checkpoint $C_{1}$ (refer Figure 17(d)); a periodic node failure is simulated after 56 minutes from a checkpoint and the average time at which a random failure occurs is found to be after 2 hours, 8 minutes and 47 seconds from each checkpoint for 5000 trials.", "Figure: Execution of the five hour job with and without checkpoints.", "(a) When no checkpoints are placed from the start to completion of the job.", "(b) When four checkpoints each one hour apart are placed from start to completion of the job.", "(c) When two checkpoints each two hours apart are placed from start to completion of job.", "(d) When one checkpoint is placed after four hours of starting the job.Similar to Table 1, in Table 2, the average time taken for reinstating execution, for the overheads and for executing the job from the start to finish with and without checkpoints is considered.", "The time to bring execution back to normal after a failure has occurred is referred to as reinstating time.", "The time to create checkpoints and transfer checkpoint data to the server is referred to as the overhead time.", "The execution of the job when one periodic and one random failure occurs per hour and when five random failures occur per hour is considered.", "Without checkpointing the genome searching job is run such that a human administrator monitors the job from its start until completion.", "In this case, if a node fails then the only option is to restart the execution of the job.", "Each time the job fails and given that the administrator detected it using cluster monitoring tools as soon as the node failed approximately, then at least ten minutes are required for reinstating the execution.", "If a periodic failure occurred once every hour from the 14th minute from execution then there are five periodic faults.", "Once a failure occurs the execution will always have to come back to its previous state by restarting the job.", "Hence, the five hour job, with just one periodic failure occurring every hour will take over 21 hours.", "Similarly, if a random failure occurred once every hour (average time of occurrence is 31 minutes and 14 seconds after execution starts), then there are five failure points, and over 23 hours are required for completing the job.", "When five random failures occur each hour of the execution then more than 80 hours are required; this is nearly 16 times the time for executing the job without a failure.", "Centralised checkpointing on a single server and on multiple servers and decentralised checkpointing on multiple servers for one, two and four hour periodicity in the network are then considered in Table 2.", "For checkpointing methods when one hour frequency is chosen more than five times the time taken for executing the job without failures is required.", "When the frequency of checkpointing is every two hours then just under four times the time taken for executing the job without failures is required.", "In the case when the checkpoint is created every four hours just over 3 times the time taken for executing the job without failures is required.", "The multi-agent approaches on the other hand take only one-fourth the time taken by traditional approaches for the job with five single node faults that occur each hour.", "This is significant time saving for running jobs that require many hours for completing execution." ], [ "Similarities and differences between the approaches", "The agent and core intelligence approaches are similar in at least four ways.", "Firstly, the objective of both the approaches is to automate fault tolerance.", "Secondly, the job to be executed is broken down into sub-jobs which are executed.", "Thirdly, fault tolerance is achieved in both approaches by predicting faults likely to occur in the computing core.", "Fourthly, technology enabling mobility is required by both the approaches to carry the sub-job or to push the sub-job from one core onto another.", "These important similarities enable both the agent and core approaches to be brought together to offer the advantages as a hybrid approach.", "While there are similarities between the agent and core intelligence approaches there are differences that reflect in their implementation.", "These differences are based on: (i) Where the job is situated - in the agent intelligence approach, the sub-job becomes the payload of an agent situated on a computing core.", "In the core intelligence approach, the sub-job is situated on a virtual core, which is an abstraction of the computing core.", "(ii) Who predicts the failures - the agent constantly probes the compute core it is situated on and predicts failure in the agent approach, whereas in the core approach the virtual core anticipates the failure.", "(iii) Who reacts to the prediction - the agent moves onto another core and re-establishes its dependencies in the agent approach, whereas the virtual core is responsible for moving a sub-job onto another core in the core approach.", "(iv) How dependencies are updated - an agent requires to carry information of its dependencies when it moves off onto another core and establishes its dependencies manually in the agent approach, whereas the dependencies of the sub-job on the core do not require to be manually updated in the core approach.", "(v) What view is obtained - in the agent approach, agents have a global view as they can traverse across the network of virtual cores, which is in contrast to the local view of the virtual cores in the core approach." ], [ "Conclusions", "The agent based approaches described in this paper offer a candidate solution for automated fault tolerance or in combination with checkpointing as proposed above offer a means of reducing current levels of human intervention.", "The foundational concepts of the agent and core based approaches were validated on four computer clusters using parallel reduction algorithms as a test case in this paper.", "Failure scenarios were considered in the experimental studies for the two approaches.", "The effect of the number of dependencies of a sub-job being executed, the volume of data communicated across cores, and the process size are three factors considered in the experimental studies for determining the performance of the approaches.", "The approaches were studied in the context of parallel genome searching, a popular computational biology job, that fits the criteria of a parallel reduction algorithm.", "The experiments were performed for both periodic and random failures.", "The approaches were compared against centralised and decentralised checkpointing approaches.", "In a typical experiment in which the fault tolerant approaches are studied in between two checkpoints one hour apart when one random failure occurs, centralised and decentralised checkpointing on an average add 90% to the actual time for executing the job without any failures.", "On the other hand, in the same experiment the multi-agent approaches add only 10% to the overall execution time.", "The multi-agent approaches cannot predict all failures that occur in the computing nodes.", "Hence, the most efficient way of incorporating these approaches is to use them on top of checkpointing.", "The experiments demonstrate the feasibility of such approaches for computational biology jobs.", "The key result is that a job continues execution after a core has failed and the time required for reinstating execution is lesser than checkpointing methods.", "Future work will explore methods to improve the accuracy of prediction as well as increase the number of faults that can be predicted using the multi-agent approaches.", "The challenge to achieve this will be to mine log files for predicting a wide range of faults and predict them as quickly as possible before the fault occurs.", "Although the approaches can reduce human administrator intervention they can be used independently only if a wider range of faults can be predicted with greater accuracy.", "Until then the multi-agent approaches can be used in conjunction with checkpointing for improving fault tolerance.", "The authors would like to thank the administrators of the compute resources at the Centre for Advanced Computing and Emerging Technologies (ACET), University of Reading, UK and the Atlantic Computational Excellence Network (ACEnet).", "Comparing fault tolerant approaches between checkpoints with one hour periodicity.", "The average time taken (a) for predicting one single node failure, (b) for reinstating execution after one periodic single node failure, (c) for reinstating execution after one random single node failure, (d) for the overheads related to one periodic single node failure, (e) for the overheads related to one random single node failure, and (f) for executing the job without failures and checkpoints, with one periodic failure per hour and with five periodic failures per hour are tabulated for centralised checkpointing with single and multiple servers, decentralised checkpointing with multiple servers and the multi-agent approaches.", "Table: NO_CAPTIONComparing fault tolerant approaches for a five hour job with checkpoints having one, two and four hour periodicity.", "The average time taken (a) for predicting one single node failure, (b) for reinstating execution after one periodic single node failure, (c) for reinstating execution after one random single node failure, (d) for all the overheads related to one periodic single node failure, (e) for all the overheads related to one random single node failure, and (f) for executing the job without failures and checkpoints, with one periodic failure per hour and with five periodic failures per hour are tabulated for cold restart with no fault tolerance, centralised checkpointing with single and multiple servers, decentralised checkpointing with multiple servers and the multi-agent approaches.", "Table: NO_CAPTION" ] ]	1403.0500
[ [ "A regularization approach to non-smooth symplectic geometry" ], [ "Abstract We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions.", "This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry.", "For mildly singular symplectic forms, including the continuous non-differentiable case, we prove the existence of generalized Darboux coordinates in the sense of a local non-smooth pull-back to the canonical symplectic form on the cotangent bundle." ], [ "Introduction", "Regularization approaches to non-smooth differential geometry and its applications to mathematical physics have been successfully developed in the context of Colombeau-type generalized functions and tensor fields (cf., e.g., [4], [5], [22], [23], [9]).", "In the present paper we take up the study of generalized symplectic structures based on previous investigations of linear symplectic structures on modules over Colombeau generalized numbers in [12].", "Our main motivations for the systematic development of a non-smooth symplectic differential geometry are driven by deeper applications in microlocal analysis, classical mechanics or general relativity in terms of analysis on semi-Riemannian manifolds: First, modern research on propagation of singularities for (pseudo-) differential operators with non-smooth principal symbol on a manifold is based on an understanding of the corresponding non-smooth Hamiltonian vector field and its generalized bicharacteristic flow (cf.", "[6], [24]) as well as on an analysis of microlocal mapping properties of generalized Fourier integral solution operators in terms of the wave front sets of their kernels, which are to be generalized Lagrangian submanifolds (cf.", "[7], [8]).", "Second, non-smooth symplectic structures arise in the study of the geodesic flow in classical mechanics or in general relativity in the context of non-smooth metrics or space-times, unified in models of generalized semi-Riemannian manifolds $(M,g)$ .", "The geodesic flow can then be described in terms of symplectomorphisms on a generalized symplectic manifold $(TM, \\sigma )$ .", "The basic construction is as follows: The geodesic flow is defined by the non-smooth geodesic spray $G$ , given as a vector field on $TM$ in coordinates $(x,v)$ by $G(x,v) = \\sum _{1 \\le j \\le n} v_j\\, \\partial _{x_j} -\\sum _{1 \\le j,k,l \\le n} \\Gamma ^j_{k l}\\, v_k\\, v_l\\, \\partial _{v_j}$ with the Christoffel symbols $\\Gamma ^j_{k l}$ (cf.", "[1]).", "Non-degeneracy of the metric provides a `non-smooth diffeomorphism' $g^\\flat \\colon TM \\rightarrow T^* M$ .", "The latter allows one to pull-back the canonical symplectic form to define a non-smooth symplectic form $\\sigma $ on $TM$ , which locally reads $\\sigma = \\sum _{1 \\le i,j \\le n} g_{ij}\\, dx_i \\wedge dv_j +\\sum _{i,j,k} \\frac{\\partial g_{ij}}{\\partial {x_k}} \\, v_i\\, dx_j \\wedge dx_k.$ Note that in a sense, here the `generalized symplectomorphism' $(g^\\flat )^{-1}$ provides Darboux coordinates for $(TM, \\sigma )$ ." ], [ "Generalized differential geometry on smooth manifolds", "In this section we briefly recall some notions from Colombeau's theory of nonlinear generalized functions and non-smooth differential geometry in this setting.", "For details we refer to [9].", "Let $M$ be a smooth (Hausdorff and second countable) manifold of dimension $n$ .", "Colombeau generalized functions on $M$ are introduced as equivalence classes $u=[(u_\\varepsilon )_\\varepsilon ]$ of moderate modulo negligible nets in ${\\mathcal {C}}^\\infty (M)$ , where moderateness, resp.", "negligibility, are characterized by ${\\mathcal {E}}_M(M)&:=&\\lbrace (u_\\varepsilon )_\\varepsilon \\in {\\mathcal {C}}^\\infty (M)^{]0,1]}:\\ \\forall K\\subset \\subset M,\\ \\forall L\\in {\\mathcal {P}}(M)\\ \\exists N\\in \\mathbb {N}:\\sup _{x\\in K}\|Lu_\\varepsilon (x)\|=O(\\varepsilon ^{-N})\\rbrace , \\\\{\\mathcal {N}}(M)&:=& \\lbrace (u_\\varepsilon )_\\varepsilon \\in {\\mathcal {E}}_M(M):\\ \\forall K\\subset \\subset M,\\ \\forall m \\in \\mathbb {N}_0:\\ \\sup _{x\\in K}\|u_\\varepsilon (x)\|=O(\\varepsilon ^{m})\\rbrace ,$ and ${\\mathcal {P}}(M)$ is the space of linear differential operators on $M$ .", "Then the Colombeau algebra ${\\mathcal {G}}(M)$ of generalized functions on $M$ is defined as is the quotient ${\\mathcal {E}}_M(M)/{\\mathcal {N}}(M)$ ; it is a fine sheaf of differential algebras with respect to the Lie derivative along smooth vector fields.", "Colombeau generalized functions on $M$ are uniquely determined by their values on compactly supported generalized points on $M$ , which are denoted by $\\widetilde{M}_c$ and defined as follows.", "In the space $M_c$ of nets $(x_\\varepsilon )_\\varepsilon \\in M^{]0,1]}$ with the property that $x_\\varepsilon $ stays in a fixed compact set for $\\varepsilon $ small, one introduces an equivalence relation $\\sim $ : $(x_\\varepsilon )_\\varepsilon \\sim (y_\\varepsilon )_\\varepsilon :\\Leftrightarrow d_h(x_\\varepsilon ,y_\\varepsilon )=O(\\varepsilon ^m)$ , for all $m>0$ , $(x_\\varepsilon )_\\varepsilon ,\\, (y_\\varepsilon )_\\varepsilon \\in M_c$ , and distance function $d_h$ induced on $M$ by any Riemannian metric $h$ .", "Then the space of compactly supported generalized points on $M$ is the quotient space $\\widetilde{M}_c:=M_c/\\sim $ , with elements $\\tilde{x}=[(x_\\varepsilon )_\\varepsilon ]$ .", "Generalized numbers are equivalence classes $r=[(r_\\varepsilon )_\\varepsilon ]$ of moderate nets of real numbers $\\lbrace (r_\\varepsilon )_\\varepsilon \\in \\mathbb {R}^{]0,1]} : \\exists N\\in \\mathbb {N}: \|r_\\varepsilon \| = O(\\varepsilon ^{-N})\\rbrace $ modulo the set of negligible nets $\\lbrace (r_\\varepsilon )_\\varepsilon \\in \\mathbb {R}^{]0,1]} : \\forall m\\in \\mathbb {N}_0: \|r_\\varepsilon \| = O(\\varepsilon ^{m})\\rbrace $ .", "$\\widetilde{\\mathbb {R}}$ is the ring of constants in the Colombeau algebra of generalized functions on $\\mathbb {R}$ .", "A generalized number $r\\in \\widetilde{\\mathbb {R}}$ is called strictly nonzero if $\|r_\\varepsilon \|\\ge \\varepsilon ^m$ for some $m\\in \\mathbb {N}$ and $\\varepsilon $ small; it is called strictly positive if $r_\\varepsilon \\ge \\varepsilon ^m$ for some $m\\in \\mathbb {N}$ and $\\varepsilon $ small.", "Invertible elements of the ring $\\widetilde{\\mathbb {R}}$ are precisely those which are strictly nonzero.", "Similarly, $u\\in {\\mathcal {G}}(M)$ has a multiplicative inverse if and only if it is strictly nonzero in the sense that for any $K\\Subset M$ there exists some $m$ such that $\\inf _{x\\in K}\|u_\\varepsilon (x)\|\\ge \\varepsilon ^m$ for $\\varepsilon $ small.", "In turn, this is equivalent to $u(\\tilde{x})$ being invertible in $\\widetilde{\\mathbb {R}}$ for any compactly supported generalized point $\\tilde{x}$ .", "Let $E$ be a vector bundle over $M$ , and denote by $\\Gamma (E)$ the space of smooth sections of $E$ .", "The space of Colombeau generalized sections of $E$ , $\\Gamma _{\\mathcal {G}}(E)$ , is defined as the quotient $\\Gamma _{{\\mathcal {E}}_M}(E)/\\Gamma _{{\\mathcal {N}}}(E)$ , where $\\Gamma _{{\\mathcal {E}}_M}(E)&:=& \\lbrace (s_\\varepsilon )_{\\varepsilon }\\in \\Gamma (E)^{]0,1]} :\\ \\forall L\\in {\\mathcal {P}}(E)\\, \\forall K\\subset \\subset M \\, \\exists N\\in \\mathbb {N}:\\sup _{x\\in K}\\Vert Lu_\\varepsilon (x)\\Vert _h = O(\\varepsilon ^{-N})\\rbrace ,\\\\\\Gamma _{{\\mathcal {N}}}(E)&:=& \\lbrace (s_\\varepsilon )_{\\varepsilon }\\in \\Gamma _{{\\mathcal {E}}_M}(E) :\\ \\forall K\\subset \\subset M \\, \\forall m\\in \\mathbb {N}:\\sup _{x\\in K}\\Vert u_\\varepsilon (x)\\Vert _h = O(\\varepsilon ^{m})\\rbrace ,$ Here ${\\mathcal {P}}(E)$ is the space of linear differential operators $\\Gamma (E)\\rightarrow \\Gamma (E)$ , and $\\Vert \\,\\Vert _h$ is the norm on the fibers of $E$ induced by any Riemannian metric $h$ on $M$ .", "The ${\\mathcal {C}}^\\infty (M)$ -module of Colombeau generalized sections of $E$ is projective and finitely generated, and can be characterized by the ${\\mathcal {C}}^\\infty (M)$ -module isomorphisms: $\\Gamma _{{\\mathcal {G}}}(E)={\\mathcal {G}}(M)\\otimes _{{\\mathcal {C}}^\\infty (M)}\\Gamma (E)= L_{{\\mathcal {C}}^\\infty (M)}(\\Gamma (E^\\ast ),{\\mathcal {G}}(M))$ .", "In case $E$ is a tensor bundle $T_s^r(M)$ we use the notation ${\\mathcal {G}}_s^r(M)$ for $\\Gamma _{{\\mathcal {G}}}(T_s^r(M))$ .", "Moreover, in the case of the tangent bundle $TM$ the generalized sections are the generalized vector fields, and will be denoted by $\\mathfrak {X}_{\\mathcal {G}}(M)$ , while in the case of the cotangent bundle $T^\\ast M$ we write $\\Omega _{\\mathcal {G}}^1(M)$ for the corresponding generalized sections (generalized one-forms).", "Also, when $E$ is the vector bundle of exterior $k$ -forms on $TM$ , i.e., $E=\\Lambda ^kT^\\ast M$ , then the generalized sections are the generalized $k$ -forms on $M$ , and are denoted by $\\Omega _{\\mathcal {G}}^k(M)$ .", "Finally, we will also make use of a particular feature of Colombeau's approach, namely manifold-valued generalized functions ([14], [16]).", "Given manifolds $M$ , $N$ , the space of c-bounded generalized functions is the quotient space ${\\mathcal {G}}[M,N]:={\\mathcal {E}}_M[M,N]/\\sim $ .", "Here, ${\\mathcal {E}}_M[M,N]$ is the set of all nets $(u_\\varepsilon )_{\\varepsilon \\in ]0,1]}$ such that $(f\\circ u_\\varepsilon )_\\varepsilon $ is moderate for every $f\\in {\\mathcal {C}}^\\infty (N)$ .", "The equivalence relation $\\sim $ is defined as $(u_\\varepsilon )_\\varepsilon \\sim (v_\\varepsilon )_\\varepsilon $ if for any Riemannian metric $h$ on $N$ , any $m\\in \\mathbb {N}$ and every $K\\Subset M$ , $\\sup _{p\\in K}d_h(u_\\varepsilon (p),v_\\varepsilon (p))=O(\\varepsilon ^m)$ for $\\varepsilon \\rightarrow 0$ .", "These generalized functions are called c-bounded since any representative is bounded, uniformly in $\\varepsilon $ , on any compact subset of $M$ for $\\varepsilon $ small.", "They can be composed unrestrictedly, and invertible c-bounded generalized functions serve as non-smooth analogues of diffeomorphisms in the smooth category (cf.", "[3], [26]).", "The paper is organized as follows.", "For the sake of completeness we first review in Section 2 some results from [12] on symplectic forms on $\\widetilde{\\mathbb {R}}$ -modules, symplectic bases, maps, and submodules.", "Then we turn to the manifold setting, study various conditions for skew-symmetry and nondegeneracy of generalized 2-forms, and provide an equivalent characterization of a generalized symplectic form on a manifold.", "Section 3 is devoted to the Darboux theorem and its generalization for a generalized symplectic form.", "We state conditions which imply that a generalized symplectic form on a manifold looks locally like the canonical symplectic form on $T^\\ast (\\widetilde{\\mathbb {R}}^n)$ .", "In the last Section 4 we introduce notions of generalized Hamiltonian vector fields and Poisson structures." ], [ "Review of symplectic modules over the ring of generalized numbers", "Here we briefly recall basic notions and results about symplectic $\\widetilde{\\mathbb {R}}$ -modules that are essential for the study of generalized symplectic structures on manifolds, and refer to [12] for an in-depth analysis.", "On an $\\widetilde{\\mathbb {R}}$ -module $V$ we define a symplectic form $\\sigma :V \\times V \\rightarrow \\widetilde{\\mathbb {R}}$ to be an $\\widetilde{\\mathbb {R}}$ -bilinear form that is skew-symmetric ($\\sigma (v,w) = - \\sigma (w,v), \\forall v, w \\in V$ ), and non-degenerate ($\\sigma (v,w) = 0, \\forall w \\Rightarrow v = 0$ ).", "We call the pair $(V,\\sigma )$ a symplectic $\\widetilde{\\mathbb {R}}$ -module.", "As the standard model space for a symplectic $\\widetilde{\\mathbb {R}}$ -module we take $(T^({\\widetilde{\\mathbb {R}}}^n), \\tilde{\\omega })$ , where $T^({\\widetilde{\\mathbb {R}}}^n)= \\widetilde{\\mathbb {R}}^n \\times \\widetilde{\\mathbb {R}}^n$ , and the symplectic form $\\tilde{\\omega }$ is defined as $\\widetilde{\\omega }((x,\\xi ),(y,\\eta )) = \\sum _{j=1}^n y_j \\xi _j -\\sum _{j=1}^n x_j \\eta _j= \\langle y, \\xi \\rangle - \\langle x, \\eta \\rangle \\qquad \\forall (x,\\xi ), (y,\\eta ) \\in T^(\\widetilde{\\mathbb {R}}^n).$ For this symplectic form, the vectors $\\lbrace e_1,\\ldots ,e_n,f_1,\\ldots ,f_n\\rbrace $ , $e_j := (\\delta _j,0)$ , $f_j = (0,\\delta _j)$ ($\\delta _j$ is the $j^\\text{th}$ standard unit vector, $1 \\le j \\le n$ ), form a basis which turns $T^(\\widetilde{\\mathbb {R}}^n)$ into a free module of rank $2n$ .", "Moreover, one has $\\widetilde{\\omega }(e_j,e_l) = 0 = \\widetilde{\\omega }(f_j,f_l),\\quad \\widetilde{\\omega }(f_j,e_l) = \\delta _{jl}\\quad (1 \\le j,l \\le n),$ with $\\delta _{jl}$ the Kronecker delta.", "A basis satisfying this property is called a symplectic basis.", "In [12] we proved that any symplectic free module of finite rank possesses a symplectic basis, which further implies that its rank has to be even.", "Also, we showed that any given “partial symplectic basis” of the free symplectic $\\widetilde{\\mathbb {R}}$ -module $(V,\\sigma )$ of finite rank can be extended to a full one, i.e., any free set $B:= \\lbrace e_i\\in V \\mid i \\in I \\rbrace \\cup \\lbrace f_j\\in V \\mid j \\in J\\rbrace $ ($I, J \\subseteq \\lbrace 1,\\ldots n\\rbrace $ ) satisfying $\\sigma (e_i,e_k) = 0 = \\sigma (f_j,f_l), \\quad \\sigma (f_j,e_i) = \\delta _{ji}\\quad (i,k \\in I; j,l \\in J),$ can be extended by vectors $e_i \\in V$ ($i \\in \\lbrace 1, \\ldots n\\rbrace \\setminus I$ ) and $f_j \\in V$ ($j \\in \\lbrace 1, \\ldots n\\rbrace \\setminus J$ ) to a symplectic basis $\\lbrace e_1, \\ldots , e_n, f_1, \\ldots , f_n\\rbrace $ of $(V,\\sigma )$ .", "One of the main results of [12] is that any free symplectic $\\widetilde{\\mathbb {R}}$ -module of finite rank is symplectomorphic to $(T^(\\widetilde{\\mathbb {R}}^n),\\widetilde{\\omega })$ .", "A symplectomorphism between symplectic $\\widetilde{\\mathbb {R}}$ -modules $(V_1,\\sigma _1)$ and $(V_2,\\sigma _2)$ is an $\\widetilde{\\mathbb {R}}$ -linear isomorphism $f:V_1\\rightarrow V_2$ that preserves symplectic structures, i.e., $\\sigma _2(f(v_1),f(v_2)) = \\sigma _1(v_1,v_2), \\quad \\forall v_1, v_2 \\in V.$ In case $f$ is an $\\widetilde{\\mathbb {R}}$ -linear map but not an isomorphism, it is called a symplectic map.", "Every symplectic map is injective.", "A symplectic map $f:V_1\\rightarrow V_2$ is a symplectomorphism if, in addition, $(V_1,\\sigma _1)$ and $(V_2,\\sigma _2)$ are free and of equal finite rank." ], [ "Manifolds with generalized symplectic forms", "The following definition is the natural extension of the notion of a smooth symplectic form on a manifold to the setting of Colombeau generalized functions.", "Definition 2.1 A generalized symplectic form on the smooth $d$ -dimensional manifold $M$ is a closed generalized 2-form $\\sigma \\in \\Gamma _{\\mathcal {G}}(\\Lambda ^2T^M)$ that is non-degenerate, i.e., for every chart $(W,\\psi )$ and $\\tilde{z} \\in {\\psi (W)}^{\\sim }_c$ the $\\widetilde{\\mathbb {R}}$ -bilinear form $ \\psi _* \\sigma (\\tilde{z}) \\colon \\widetilde{\\mathbb {R}}^d \\times \\widetilde{\\mathbb {R}}^d \\rightarrow \\widetilde{\\mathbb {R}}$ is non-degenerate .", "As was already mentioned in the introduction, the fact that $ \\psi _* \\sigma $ is a symplectic form on $\\widetilde{\\mathbb {R}}^d$ implies that $d$ is even, say $d=2n$ .", "Given any $(0,2)$ -tensor field $\\alpha $ on $M$ and a coordinate system $(\\psi =(x^1,\\dots ,x^d),U)$ we may write $\\alpha \|_{U} = \\alpha _{ij} dx^i\\otimes dx^j$ and set $\\mathrm {vol}(\\alpha )\|_U:=\\sqrt{\|\\det (\\alpha _{ij})\|}$ .", "As this quantity transforms by multiplication with the Jacobian determinant of the chart transition functions, we obtain a well-defined 1-density $\\mathrm {vol}(\\alpha )$ on $M$ .", "In the case where $\\alpha $ is a Riemannian metric on $M$ , $\\mathrm {vol}(\\alpha )$ is the Riemannian volume density of $\\alpha $ .", "A component-wise application of the above procedure to a generalized $(0,2)$ -tensor $\\alpha $ yields a corresponding generalized one-density $\\mathrm {vol}(\\alpha )\\in \\Gamma _{\\mathcal {G}}(\\mbox{Vol\\,}(M))$ (with $\\mbox{Vol\\,}(M)$ the 1-density bundle over $M$ ).", "Proposition 2.2 Let $\\sigma \\in {\\mathcal {G}}_2^0(M)$ .", "Then the following are equivalent: (i) $\\sigma :\\mathfrak {X}(M)\\times \\mathfrak {X}(M)\\rightarrow {\\mathcal {G}}(M)$ is skew-symmetric $($ equivalently, $\\sigma \\in \\Gamma _{\\mathcal {G}}(\\Lambda ^2T^M))$ and $\\mathrm {vol}(\\sigma )$ is strictly positive.", "(ii) For each chart $(W,\\psi )$ and for each $\\tilde{z}\\in \\psi (W)^\\sim _c$ , the map $\\psi _\\sigma (\\tilde{z}):\\widetilde{\\mathbb {R}}^{d}\\times \\widetilde{\\mathbb {R}}^{d} \\rightarrow \\widetilde{\\mathbb {R}}$ is skew-symmetric and non-degenerate.", "(iii) $\\mathrm {vol}(\\sigma )$ is strictly positive and for each relatively compact open set $V\\subseteq M$ there exist a representative $(\\sigma _\\varepsilon )_\\varepsilon $ of $\\sigma $ and $\\varepsilon _0>0$ such that $\\sigma _\\varepsilon \|_V$ is skew-symmetric for all $\\varepsilon <\\varepsilon _0$ .", "Proof.", "(i) $\\Rightarrow $ (ii) By the point-value characterization of invertibility of generalized functions ([9]), strict positivity of $\\mathrm {vol}(\\sigma )$ implies that $\\det (\\psi _\\sigma (\\tilde{z}))$ is invertible in $\\widetilde{\\mathbb {R}}$ for each $\\tilde{z}\\in M^\\sim _c$ .", "Hence by [9], $\\psi _\\sigma (\\tilde{z}):\\widetilde{\\mathbb {R}}^{d}\\times \\widetilde{\\mathbb {R}}^{d} \\rightarrow \\widetilde{\\mathbb {R}}$ is non-degenerate.", "Skew-symmetry of $\\psi _\\sigma (\\tilde{z})$ follows by inserting basis vector fields in a chart and evaluating.", "(ii) $\\Rightarrow $ (i) Given $X_1,X_2\\in {X}(M)$ it follows from (ii) and the point value characterization of Colombeau functions that $\\sigma (X_1,X_2)\|_W=-\\sigma (X_2,X_1)\|_W$ on any chart domain $W$ .", "This gives skew-symmetry since ${\\mathcal {G}}$ is a sheaf.", "By [9], for any $\\tilde{z}\\in \\psi (W)^\\sim _c$ , $\\det (\\psi _\\sigma (\\tilde{z}))$ is invertible, hence ($\\psi _\\sigma (\\tilde{z})$ being skew-symmetric) strictly positive, in $\\widetilde{\\mathbb {R}}$ .", "Therefore $\\mathrm {vol}(\\sigma )\|_W$ is strictly positive.", "Since $W$ was any chart domain, $\\mathrm {vol}(\\sigma )$ is strictly positive on $M$ .", "(i) $\\Rightarrow $ (iii) Using a partition of unity the problem can be reduced to the case of $M=\\mathbb {R}^d$ .", "Now pick any representative $(\\sigma _\\varepsilon )_\\varepsilon $ of $\\sigma $ .", "Then $(\\tilde{\\sigma }_\\varepsilon )_{ij}:= \\frac{1}{2}((\\sigma _\\varepsilon )_{ij}-(\\sigma _\\varepsilon )_{ji})$ gives a skew-symmetric representative of $\\sigma $ (cf.", "[12]).", "(iii) $\\Rightarrow $ (i) Let $X_1$ , $X_2\\in {X}(M)$ and let $V$ be any relatively compact open subset of $M$ .", "Picking a representative as in (iii) it is clear that $\\sigma (X_1,X_2)=-\\sigma (X_2,X_1)$ on $V$ .", "Then the sheaf property of ${\\mathcal {G}}$ gives skew-symmetry.", "$\\Box $ To obtain a characterization of symplectic generalized forms from this result, we will use the following generalized Poincaré lemma (see [9]): Theorem 2.3 Let $\\alpha \\in \\Gamma _{\\mathcal {G}}(\\Lambda ^kT^M)$ be closed.", "If $p\\in M$ and $U$ is a neighborhood of $p$ that is diffeomorphic to an open ball in $\\mathbb {R}^d$ then there exists $\\beta \\in \\Gamma _{\\mathcal {G}}(\\Lambda ^{k-1}T^M)$ such that $\\alpha \|_U=d\\beta \|_U$ .", "Corollary 2.4 Let $\\sigma \\in {\\mathcal {G}}_2^0(M)$ .", "Then the following are equivalent: (i) $\\sigma $ is a generalized symplectic form on $M$ .", "(ii) $\\mathrm {vol}(\\sigma )$ is strictly positive and for every $p\\in M$ there exists an open neighborhood $U$ of $p$ and a representative $(\\sigma _\\varepsilon )_\\varepsilon $ of $\\sigma \|_U$ such that each $\\sigma _\\varepsilon $ is a symplectic form on $U$ .", "Proof.", "(ii) $\\Rightarrow $ (i) is immediate from Prop.", "REF (iii).", "(i) $\\Rightarrow $ (ii) Given $p\\in M$ , let $U$ be a relatively compact neighborhood of $p$ diffeomorphic (via some chart $\\psi $ ) to an open ball in $\\mathbb {R}^d$ .", "Then Prop.", "REF (iii) provides a representative $(\\tilde{\\sigma }_\\varepsilon )_\\varepsilon $ of $\\sigma $ as well as some $\\varepsilon _0>0$ and $m\\in \\mathbb {N}$ such that $\\det (\\psi _\\tilde{\\sigma }_\\varepsilon (x))>\\varepsilon ^m$ for $\\varepsilon <\\varepsilon _0$ and all $x\\in \\psi (U)$ , so $\\tilde{\\sigma }_\\varepsilon \|_U$ is a non-degenerate 2-form.", "Since $\\sigma $ is closed, there exists a negligible net $(n_\\varepsilon )_\\varepsilon $ of 3-forms such that for all $\\varepsilon $ we have $d\\tilde{\\sigma }_\\varepsilon = n_\\varepsilon $ .", "Thus $dn_\\varepsilon =0$ for all $\\varepsilon $ .", "By the classical Poincaré lemma we may write $n_\\varepsilon =dm_\\varepsilon $ on $U$ , and (the proof of) Th.", "REF shows that $(m_\\varepsilon )_\\varepsilon $ is a negligible net of 2-forms on $U$ .", "Thus $\\sigma _\\varepsilon :=\\tilde{\\sigma }_\\varepsilon - m_\\varepsilon $ is a representative of $\\sigma \|_U$ with $d\\sigma _\\varepsilon =0$ for all $\\varepsilon $ .", "Finally, non-degeneracy of $\\sigma _\\varepsilon $ for $\\varepsilon $ small follows since $\\mathrm {vol}(\\sigma )$ is strictly positive by Prop.", "REF .", "$\\Box $ Note that $\\sigma _\\varepsilon $ is non-degenerate if and only if the $n$ -fold exterior product $\\sigma _\\varepsilon ^n := \\sigma _\\varepsilon \\wedge \\sigma _\\varepsilon \\wedge \\ldots \\wedge \\sigma _\\varepsilon $ provides a volume form on $M$ (with $\\dim (M)=d=2n$ )." ], [ "A generalized Darboux theorem for non-smooth symplectic forms", "In his fundamental work on the distributional approach to non-smooth mechanics [18], J. E. Marsden states: “It is meaningful to talk about generalized symplectic forms although this does not lead to a satisfactory theory.", "Clearly Darboux's theorem cannot hold in that case.” The problem we address in this section is: let $\\sigma $ be a generalized symplectic form on the manifold $M$ (in the sense described in the previous section).", "Can we find generalized Darboux coordinates?", "That is, for any $p\\in M$ we seek an open neighborhood $U$ of $p$ and a generalized diffeomorphism $\\Phi \\in {\\mathcal {G}}[U,\\mathbb {R}^{2n}]$ such that $\\sigma = \\Phi ^* \\widetilde{\\omega },$ where $\\widetilde{\\omega }$ is the canonical 2-form on $T^(\\widetilde{\\mathbb {R}}^n)$ .", "Weinstein's proof (based on an isotopy method by Moser from [21]) outlines the following basic steps to construct Darboux coordinates (cf.", "[27]; see also [1], [19], or [17]) — we add remarks on the key issues in extending this to the non-smooth situation below: We are dealing with a local question, hence we may assume without loss of generality that $M= \\mathbb {R}^{2n}$ and $p=0$ .", "Define the constant symplectic form $\\beta $ on $\\mathbb {R}^{2n}$ by $\\beta {(x)} := \\sigma {(0)}$ for every $x \\in \\mathbb {R}^{2n}$ and put $\\mu _t := \\sigma + t \\, (\\beta - \\sigma ) \\quad (0 \\le t \\le 1).$ Note that $d \\mu _t = 0$ , since $\\sigma $ is closed and $\\beta $ is constant; moreover, $\\mu _0= \\sigma $ and $\\mu _1 = \\beta $ .", "On some open ball around 0, say $B_R(0)$ , we have that $\\mu _t$ is non-degenerate for every $t \\in [0,1]$ .", "In the case of generalized 2-forms we will need a condition, called condition $(\\star )$ below, ensuring that the same holds uniformly for small values of the parameter $\\varepsilon $ , if $(\\mu ^\\varepsilon _t)_{\\varepsilon \\in \\,]0,1]}$ is a family representing $\\mu _t$ .", "Applying Poincaré's lemma on $B_R(0)$ there is a 1-form $\\alpha $ such that $d \\alpha = \\beta - \\sigma $ and $\\alpha (0) = 0$ .", "Define a vector field $X_t$ on $B_R(0)$ by requiring $\\mu _t(X_t,.)", "= - \\alpha $ .", "Since this corresponds to a pointwise inversion of the matrices representing $\\mu _t$ , the non-smooth analogue will depend on the precise information from the “non-degeneracy” condition $(\\star )$ in a crucial way.", "Since ${X_t}(0) = 0$ we have an evolution $\\theta _{t,s}$ for the time-dependent vector field $X_t$ defined up to $t=1$ on some possibly smaller neighborhood $U$ of 0 (note that $\\theta _{t,0}(0) = 0$ for all $t$ and $\\theta _{0,0}$ is the identity).", "In the non-smooth case, again exploiting details from condition ($\\star $ ) will be essential to establish a generalized evolution correspondingly.", "The result of the following calculation (using [17] to obtain the first equality) $\\frac{d}{dt} (\\theta _{t,0}^\\, \\mu _t) =\\theta _{t,0}^(L_{X_t}\\, \\mu _t) + \\theta _{t,0}^\\, \\frac{d}{dt} \\mu _t\\\\= \\theta _{t,0}^(d\\, i_{X_t}\\, \\mu _t + i_{X_t}\\, d \\mu _t) + \\theta _{t,0}^(\\beta - \\sigma ) = \\theta _{t,0}^(- d \\alpha + \\beta - \\sigma ) = 0$ implies $\\theta _{1,0}^\\, \\beta = \\theta _{1,0}^\\, \\mu _1 =\\theta _{0,0}^\\, \\mu _0 = \\sigma $ , hence in coordinates corresponding to the diffeomorphism $\\theta _{1,0}$ the symplectic form $\\sigma $ is the constant form $\\beta $ .", "Finally, we may map $\\beta $ into the canonical 2-form (or $\\widetilde{\\omega }$ in the generalized setting) by the Darboux-analogue of symplectic linear algebra ([12]), which in combination with the previous step yields the desired transformation of $\\sigma $ .", "Turning now to the generalized setting, let $\\sigma $ be a generalized symplectic form on the smooth $2n$ -dimensional manifold $M$ .", "The basic condition on $\\sigma $ , which guarantees the existence of generalized Darboux coordinates, is that $\\sigma $ should possess a representative $(\\sigma _\\varepsilon )_\\varepsilon $ satisfying: ($\\star $ ) The family $(\\sigma _\\varepsilon )_{\\varepsilon \\in \\, ]0,1]}$ of maps $M \\rightarrow \\Lambda ^2 T^M$ is equicontinuous and satisfies the following on any chart $(W,\\psi )$ with domain $W \\subseteq M$ and matrix representation $\\Omega ^\\varepsilon \\colon W \\rightarrow M(2n, \\mathbb {R})$ of $\\sigma _\\varepsilon $ with respect to this chart: $\\forall K \\Subset W\\; \\exists C_1, C_2 > 0\\; \\exists \\eta > 0\\; \\forall \\varepsilon \\in \\, ]0,\\eta ]$ $\\;\\forall q \\in K$ $0 < C_1 \\le \\min {\\mathcal {A}}(\\Omega ^\\varepsilon (q)) \\le \\max {\\mathcal {A}}(\\Omega ^\\varepsilon (q)) \\le C_2.$ Here, for a matrix $B \\in M(2n,\\mathbb {R})$ we let ${\\mathcal {A}}(B) := \\lbrace \|\\lambda \| \\mid \\lambda \\text{ is an eigenvalue of } B \\rbrace $ .", "Remark 3.1 (i) Condition ($\\star $ ) holds for the typical convolution-type regularization of a uniformly continuous symplectic form.", "(ii) ($\\star $ ) requires that the $\\varepsilon $ -parametrized family of volume forms $\\sigma _\\varepsilon ^n := \\sigma _\\varepsilon \\wedge \\sigma _\\varepsilon \\wedge \\ldots \\wedge \\sigma _\\varepsilon $ has uniform bounds on how a $2n$ -dimensional volume is squeezed or stretched.", "(iii) If $\\sigma $ satisfies ($\\star $ ) and, in addition, is associated to some $\\sigma _0\\in \\Omega ^2_{\\mathcal {D}^{\\prime }}(M)$ , then by Arzela-Ascoli there exists some sub-sequence $(\\sigma _{\\varepsilon _k})_{k\\in \\mathbb {N}}$ that converges locally uniformly to a continuous two-form $\\sigma _1$ .", "Since $\\sigma _\\varepsilon \\rightarrow \\sigma _0$ in distributions it follows that $\\sigma _0=\\sigma _1$ is continuous.", "(iv) The generalized 2-forms $\\varepsilon dx\\wedge d\\xi $ and $\\frac{1}{\\varepsilon } dx\\wedge d\\xi $ are simple examples of generalized symplectic forms that do not satisfy ($\\star $ ), yet can be transformed to the canonical symplectic form.", "However, no such transformation can be c-bounded.", "(v) Writing $H$ and $y_+$ for embeddings of the Heaviside function resp.", "its primitive into the Colombeau algebra, consider $\\sigma := (1+H(x))dx\\wedge d\\xi $ .", "Then (by (iii)), $\\sigma $ does not satisfy ($\\star $ ).", "Nevertheless, setting $\\Psi (y,\\eta ):=(y-\\frac{1}{2}y_+,\\eta )$ we obtain $\\Psi _\\sigma = dx\\wedge d\\xi $ .", "Moreover, $\\Psi $ is a generalized diffeomorphism, and both $\\Psi $ and $\\Psi ^{-1}$ are associated to Lipschitz continuous transformations.", "Theorem 3.2 Let $\\sigma $ be a generalized symplectic form on the $2n$ -dimensional smooth manifold $M$ with representative $(\\sigma _\\varepsilon )_{\\varepsilon \\in \\,]0,1]}$ such that $(\\star )$ holds.", "Then every $p \\in M$ possesses a neighborhood $U$ and a generalized diffeomorphism $\\Phi \\in {\\mathcal {G}}[U,V]$ with $V$ open in $T^(\\mathbb {R}^n)$ such that $\\Phi _ (\\sigma \\!\\mid _U) = \\widetilde{\\omega }$ .", "Proof.", "Step 1: This can be carried out as in the above scheme: we may assume that $M=\\mathbb {R}^{2n}$ and that $p=0$ .", "Then each $\\sigma _\\varepsilon $ is a skew-symmetric and non-degenerate matrix.", "In particular, $\\Omega ^\\varepsilon =\\sigma _\\varepsilon $ in ($\\star $ ).", "Step 2: Setting $\\beta :=\\sigma (0)$ , $\\beta $ is a symplectic form on the $\\widetilde{\\mathbb {R}}$ -module $T^({\\widetilde{\\mathbb {R}}}^n) = \\widetilde{\\mathbb {R}}^n \\times \\widetilde{\\mathbb {R}}^n$ and clearly $d\\beta =0$ .", "For $t\\in [0,1]$ , set $\\mu _t^\\varepsilon := \\sigma _\\varepsilon + t \\, (\\beta _\\varepsilon - \\sigma _\\varepsilon )$ .", "Then $\\mu _0 = \\sigma $ and $\\mu _1 = \\beta $ in $\\Gamma _{\\mathcal {G}}(\\Lambda ^2T^\\mathbb {R}^{n}) = \\Omega _{\\mathcal {G}}(\\mathbb {R}^{2n})$ , and $d \\mu _t = 0$ .", "Step 3: By assumption ($\\star $ ), the family of matrix-valued maps $\\mu ^\\varepsilon :=(t,q) \\mapsto \\mu _t^\\varepsilon (q), \\quad [0,1] \\times \\mathbb {R}^{2n} \\rightarrow M(2n, \\mathbb {R}) \\qquad (\\varepsilon \\in ]0,1])$ is equicontinuous.", "We show that for small $\\varepsilon > 0$ , $\\mu ^\\varepsilon $ maps $[0,1]$ times some fixed neighborhood of 0 into the set of invertible matrices (corresponding to the non-degeneracy of the associated 2-forms) with uniform bounds on the operator norms of the matrices and their inverses.", "The precise claim is as follows: ($\\star \\star $ ) $\\exists R > 0$ $\\exists \\varepsilon _0 > 0$ $\\exists D > 0$ such that $\\forall \\varepsilon \\in \\, ]0,\\varepsilon _0]$ $\\forall t \\in [0,1]$ $\\forall q \\in B_R(0)$ the matrix $\\mu _t^\\varepsilon (q)$ is invertible and $\\Vert \\mu _t^\\varepsilon (q)\\Vert _{\\text{op}} \\le D, \\quad \\Vert \\mu _t^\\varepsilon (q)^{-1}\\Vert _{\\text{op}} \\le D.$ To see this, note first that uniform boundedness of $\\mu _t^\\varepsilon (q)$ for $q \\in B_1(0)$ (or any relatively compact subset of $\\mathbb {R}^{2n}$ ) and small $\\varepsilon $ follows from $\\Vert \\mu _t^\\varepsilon (q)\\Vert _{\\text{op}} \\le (1-t) \\Vert \\sigma _\\varepsilon (q)\\Vert _{\\text{op}} + t \\,\\Vert \\sigma _\\varepsilon (0)\\Vert _{\\text{op}}\\le \\Vert \\sigma _\\varepsilon (q)\\Vert _{\\text{op}} + \\Vert \\sigma _\\varepsilon (0)\\Vert _{\\text{op}},$ recalling that the operator norm equals the spectral radius for (skew-symmetric hence) normal operators, and finally applying ($\\star $ ) with $K = \\overline{B_1(0)}$ .", "Hence there is some $\\eta \\in \\,]0,1]$ and $C_2 > 0$ such that $\\Vert \\mu _t^\\varepsilon (q)\\Vert _{\\text{op}} \\le 2\\, C_2 \\qquad \\forall \\varepsilon \\in \\,]0,\\eta ]\\; \\forall t \\in [0,1]\\;\\forall q \\in B_1(0).$ We will also establish invertibility and uniform boundedness of the family of inverses on some ball $B_R(0)$ with $0< R \\le 1$ and for $0 < \\varepsilon \\le \\eta $ .", "To this end, we note that $- \\sigma _\\varepsilon (0)^2$ is a (self-adjoint) positive-definite operator, hence $\\inf \\limits _{\\Vert v\\Vert =1} \\Vert \\sigma _\\varepsilon (0) v\\Vert ^2 =\\inf \\limits _{\\Vert v\\Vert =1} v^T \\cdot (- \\sigma _\\varepsilon (0)^2) \\cdot v =\\min {\\mathcal {A}}(-\\sigma _\\varepsilon (0)^2) = \\big ( \\min {\\mathcal {A}}(\\sigma _\\varepsilon (0)) \\big )^2.$ Let $C_1$ denote the (positive) lower bound in ($\\star $ ) applied as above with $K = \\overline{B_1(0)}$ , then the previous observation gives $\\Vert \\sigma _\\varepsilon (0) v\\Vert \\ge \\min {\\mathcal {A}}(\\sigma _\\varepsilon (0)) \\Vert v\\Vert \\ge C_1 \\Vert v\\Vert $ and therefore $\\Vert \\sigma _\\varepsilon (0)^{-1}\\Vert _{\\text{op}} \\le \\frac{1}{C_1} \\quad \\forall \\varepsilon \\in \\,]0,\\eta ].$ In completing the proof of $(\\star \\star $ ) we will make use of the following well-known fact about invertibility in normed algebras with unit (e.g., [13]): If $A$ is invertible and $\\Vert B - A\\Vert < 1/\\Vert A^{-1}\\Vert $ , then $B$ is invertible and $\\Vert B^{-1} - A^{-1}\\Vert \\le \\Vert B - A\\Vert \\Vert A^{-1}\\Vert ^2/(1 - \\Vert A^{-1}\\Vert \\Vert B - A\\Vert )$ .", "We will apply this to the situation with $A = \\sigma _\\varepsilon (0)$ and $B = \\mu _t^\\varepsilon (q)$ .", "Since we have shown $C_1 \\le 1/\\Vert \\sigma _\\varepsilon (0)^{-1}\\Vert _{\\text{op}}$ above, proving a uniform estimate of the form $\\Vert \\mu _t^\\varepsilon (q) - \\sigma _\\varepsilon (0)\\Vert _{\\text{op}} \\le C_1/2 \\qquad \\forall \\varepsilon \\in \\, ]0,\\eta ] \\;\\forall t \\in [0,1] \\; \\forall q \\in B_R(0)\\qquad \\mathrm {(\\Delta )}$ will suffice to establish invertibility of $\\mu _t^\\varepsilon (q)$ and a uniform bound $\\Vert \\mu _t^\\varepsilon (q)^{-1}\\Vert _{\\text{op}} \\le \\Vert \\mu _t^\\varepsilon (q)^{-1} - \\sigma _\\varepsilon (0)^{-1}\\Vert _{\\text{op}} +\\Vert \\sigma _\\varepsilon (0)^{-1}\\Vert _{\\text{op}} \\le \\frac{\\frac{C_1}{2} \\frac{1}{C_1^2}}{1 - \\frac{1}{C_1} \\frac{C_1}{2}} +\\frac{1}{C_1} = \\frac{2}{C_1}.$ To argue that ($\\Delta $ ) holds for some $0 < R \\le 1$ we simply call on the equicontinuity of $(\\sigma _\\varepsilon )_{\\varepsilon \\in \\,]0,1]}$ to establish the last inequality in the following chain: $\\Vert \\mu _t^\\varepsilon (q) - \\sigma _\\varepsilon (0)\\Vert _{\\text{op}} =\\Vert (1-t) \\big (\\sigma _\\varepsilon (q) - \\sigma _\\varepsilon (0)\\big )\\Vert _{\\text{op}} \\le \\Vert \\sigma _\\varepsilon (q) - \\sigma _\\varepsilon (0)\\Vert _{\\text{op}} \\le \\frac{C_1}{2}.$ Therefore ($\\star \\star $ ) holds with $D := \\max (2 C_2, 2/C_1)$ .", "Step 4: By Th.", "REF we may construct a generalized 1-form $\\alpha $ on $B_R(0)$ such that $d\\alpha = \\beta - \\sigma $ .", "Moreover, the proof of [9] shows that $\\alpha $ has a representative given by $\\alpha _\\varepsilon (q)(v) = \\int _0^1 t(\\beta _\\varepsilon - \\sigma _\\varepsilon )(tq)(q,v)\\,dt \\qquad (v\\in \\mathbb {R}^{2n})$ In particular, $\\alpha _\\varepsilon (0)=0$ for all $\\varepsilon $ .", "Step 5: By non-degeneracy of $\\mu _t^\\varepsilon $ , for each $\\varepsilon $ there exists a unique vector field $X^\\varepsilon _t$ on $B_R(0)$ such that $\\mu _t^\\varepsilon (X_t^\\varepsilon ,\\,.\\,)=-\\alpha _\\varepsilon $ on $B_R(0)$ for all $t\\in [0,1]$ and all $\\varepsilon $ .", "Moderateness of $\\alpha _\\varepsilon $ and $\\mu _t^\\varepsilon $ , together with boundedness of $(\\mu _t^\\varepsilon )^{-1}$ imply that the net $(X^\\varepsilon _t)_\\varepsilon $ defines a generalized time-dependent vector field on $B_R(0)$ , satisfying global bounds with respect to $t\\in [0,1]$ .", "Next, let $R^{\\prime }< R$ and pick a smooth, compactly supported plateau function $\\varphi : \\mathbb {R}^{2n}\\rightarrow \\mathbb {R}$ such that $\\varphi \\equiv 1$ on a neighborhood of $B_{R^{\\prime }}(0)$ .", "Then $Y^\\varepsilon _t(q):=\\varphi (q)X^\\varepsilon _t(q)$ defines a global time-dependent generalized vector field $Y_t$ that coincides with $X_t$ on $B_{R^{\\prime }}(0)$ .", "Step 6: Denote by $\\theta ^\\varepsilon $ the flow of $Y^\\varepsilon _t$ , i.e., $\\frac{d}{dt}\\theta ^\\varepsilon (t,s,q) &= Y^\\varepsilon _t(\\theta ^\\varepsilon (t,s,q))\\\\\\theta ^\\varepsilon (s,s,q) &= q$ Since by $(\\star )$ and the above construction $Y^\\varepsilon _t$ is globally bounded, uniformly in $t$ and $\\varepsilon $ , each $\\theta ^\\varepsilon $ is defined on all of $[0,1]\\times [0,1]\\times \\mathbb {R}^{2n}$ .", "Thus we obtain a c-bounded generalized function $\\theta =[(\\theta ^\\varepsilon )_\\varepsilon ]\\in {\\mathcal {G}}[[0,1]\\times [0,1]\\times \\mathbb {R}^{2n},\\mathbb {R}^{2n}]$ , and for each fixed $(t,s)\\in [0,1]^2$ an invertible generalized map $\\theta _{t,s} := \\theta (t,s,\\,.\\,)$ .", "By condition ($\\star $ ) it follows that the family $X^\\varepsilon _t$ ($\\varepsilon \\in ]0,1]$ ) is equicontinuous, uniformly in $t$ .", "The same therefore is true of $Y^\\varepsilon _t$ ($\\varepsilon \\in ]0,1]$ ).", "Together with the global boundedness of $Y^\\varepsilon _t$ and the continuous dependence of $\\theta ^\\varepsilon $ on the right hand side of its defining equation this implies that also $\\theta ^\\varepsilon $ ($\\varepsilon \\in ]0,1]$ ) is equicontinuous.", "Since $Y^\\varepsilon _t(0)=0$ , and thereby $\\theta ^\\varepsilon (t,0,0) = 0$ for all $t$ , it follows that there exists some $R^{\\prime \\prime }<R^{\\prime }$ such that $\\theta ^\\varepsilon (t,0,q)\\in B_{R^{\\prime }}(0)$ for all $\\varepsilon $ , all $q\\in B_{R^{\\prime \\prime }}(0)$ and all $t\\in [0,1]$ .", "Step 7: By the above preparations, on $B_{R^{\\prime \\prime }}(0)$ we can calculate as follows: $\\frac{d}{dt} ((\\theta ^\\varepsilon _{t,0})^\\, \\mu ^\\varepsilon _t) =(\\theta ^\\varepsilon _{t,0})^(L_{Y^\\varepsilon _t}\\, \\mu ^\\varepsilon _t) + (\\theta ^\\varepsilon _{t,0})^\\, \\frac{d}{dt} \\mu ^\\varepsilon _t =(\\theta ^\\varepsilon _{t,0})^(L_{X^\\varepsilon _t}\\, \\mu ^\\varepsilon _t) + (\\theta ^\\varepsilon _{t,0})^\\, \\frac{d}{dt} \\mu ^\\varepsilon _t\\\\= (\\theta ^\\varepsilon _{t,0})^(d\\, i_{X^\\varepsilon _t}\\, \\mu ^\\varepsilon _t + i_{X^\\varepsilon _t}\\, d \\mu ^\\varepsilon _t) + (\\theta ^\\varepsilon _{t,0})^(\\beta _\\varepsilon - \\sigma _\\varepsilon ) =(\\theta ^\\varepsilon _{t,0})^(- d \\alpha _\\varepsilon + \\beta _\\varepsilon - \\sigma _\\varepsilon ).$ Integrating, we obtain $(\\theta ^\\varepsilon _{1,0})^\\beta _\\varepsilon = (\\theta ^\\varepsilon _{1,0})^\\, \\mu ^\\varepsilon _1 = (\\theta ^\\varepsilon _{0,0})^\\mu ^\\varepsilon _0 + \\nu _\\varepsilon = \\sigma _\\varepsilon + \\nu _\\varepsilon $ with $(\\nu _\\varepsilon )_\\varepsilon $ a negligible 2-form.", "Thus $\\sigma = (\\theta ^\\varepsilon _{1,0})^\\beta $ as a generalized 2-form.", "Consequently, the generalized diffeomorphism $\\theta _{1,0}$ transforms $\\sigma $ into the constant symplectic form $\\beta $ .", "Step 8: Finally, by choosing (according to [12]) a symplectic basis on ${\\widetilde{\\mathbb {R}}}^{2n}$ corresponding to $\\beta $ we transform $\\beta $ to the canonical symplectic form $\\widetilde{\\omega }$ .", "$\\Box $" ], [ "Generalized Hamiltonian vector fields and Poisson structures", "The natural next step in the development of generalized symplectic geometry is the introduction of Hamiltonian vector fields and, building on this, Poisson structures.", "This final section is devoted to providing these notions.", "To begin with, we analyze the purely algebraic setup, based on [12].", "Given an $\\widetilde{\\mathbb {R}}$ -module $V$ , its dual $\\widetilde{\\mathbb {R}}$ -module $\\mathrm {L}(V,\\widetilde{\\mathbb {R}})$ is denoted by $V^{\\prime }$ .", "Lemma 4.1 Let $(V,\\sigma )$ be a symplectic $\\widetilde{\\mathbb {R}}$ -module that is free and of finite rank.", "Then for any $\\varphi \\in V^{\\prime }$ there exists a unique $h_\\varphi \\in V$ , the Hamiltonian vector corresponding to $\\varphi $ , such that $\\forall v\\in V: \\quad \\varphi (v) = \\sigma (h_\\varphi ,v).$ Furthermore, the map $\\varphi \\mapsto h_\\varphi $ , $V^{\\prime }\\rightarrow V$ is a linear isomorphism Uniqueness is immediate by the non-degeneracy of $\\sigma $ .", "To prove existence, picking any basis of $V$ and the corresponding dual basis on $V^{\\prime }$ we can rewrite the defining equation for $h_\\varphi $ in matrix form as $\\varphi \\cdot v = h_\\varphi ^T \\cdot \\sigma \\cdot v$ .", "Hence $h_\\varphi ^T = \\varphi \\cdot \\sigma ^{-1}.$ More explicitly, in terms of a symplectic basis $(e_1,\\dots ,e_n,f_1,\\dots ,f_n)$ of $V$ , we have $h_\\varphi = \\sum _{i=1}^n (\\varphi (f_i)e_i - \\varphi (e_i)f_i).$ The final claim is immediate from the construction.", "Based on this result, we can introduce the Poisson bracket $\\lbrace \\,\\,,\\,\\rbrace :V^{\\prime }\\times V^{\\prime } &\\rightarrow \\widetilde{\\mathbb {R}}\\\\\\lbrace \\varphi ,\\psi \\rbrace &= \\sigma (h_\\varphi ,h_\\psi )$ As in the vector space setting it is easily seen that $\\lbrace \\,\\,,\\,\\rbrace $ is skew-symmetric, non-degenerate and satisfies the Jacobi-identity.", "Thus we obtain: Proposition 4.2 Let $(V,\\sigma )$ be a symplectic $\\widetilde{\\mathbb {R}}$ -module that is free and of finite rank.", "Then $(V^{\\prime },\\lbrace \\,\\,,\\,\\rbrace )$ is a symplectic $\\widetilde{\\mathbb {R}}$ -module that is free and of the same rank as $V$ .", "Turning now to the manifold setting, we have: Theorem 4.3 Let $(M,\\sigma )$ be a generalized symplectic manifold.", "Then the mapping $\\sigma _\\flat : \\mathfrak {X}_{\\mathcal {G}}(M) &\\rightarrow \\Omega ^1_{\\mathcal {G}}(M) \\\\\\sigma _\\flat (X)(Y) &:= \\sigma (X,Y) \\quad (X,Y \\in \\mathfrak {X}_{\\mathcal {G}}(M))$ is a ${\\mathcal {G}}(M)$ -linear isomorphism.", "Its inverse will be denoted by $\\sigma ^\\sharp $ .", "Since $Y\\mapsto \\sigma (X,Y)$ is ${\\mathcal {G}}(M)$ -linear, $\\sigma _\\flat (X)\\in \\Omega ^1_{\\mathcal {G}}(M)$ for any $X\\in \\mathfrak {X}_{\\mathcal {G}}(M)$ (see [9]), and clearly $\\sigma _\\flat $ is ${\\mathcal {G}}(M)$ -linear.", "To prove injectivity, suppose that $\\sigma (X,Y)=0$ for all $Y\\in \\mathfrak {X}_{\\mathcal {G}}(M)$ .", "Then for any chart $(W,\\psi )$ it follows that $\\psi _(\\sigma )(\\tilde{z})(\\psi _(X)(\\tilde{z}),w)=0$ for any $\\tilde{z}\\in \\psi (W)^\\sim _c$ and any $w\\in \\widetilde{\\mathbb {R}}^{2n}$ .", "Thus non-degeneracy gives $\\psi _*(X)(\\tilde{z})=0$ , implying that $X=0$ .", "To show surjectivity, by the sheaf property it suffices to consider the case $M=\\mathbb {R}^{2n}$ .", "Let $\\alpha \\in \\Omega ^1_{\\mathcal {G}}(\\mathbb {R}^{2n})$ .", "Then, using (REF ), in matrix notation we may set $X_\\alpha ^T := \\alpha \\cdot \\sigma ^{-1}$ .", "By the positivity of $\\det (\\sigma )$ in ${\\mathcal {G}}(\\mathbb {R}^{2n})$ (Cor.", "REF ), this defines a generalized vector field $X_\\alpha \\in \\mathfrak {X}_{\\mathcal {G}}(M)$ , and by construction $\\sigma _\\flat (X_\\alpha )=\\alpha $ .", "Remark 4.4 Note that the previous result did not make use of generalized Darboux coordinates (which in general may not be available), but is valid for arbitrary generalized symplectic manifolds.", "Based on this result, we may now define: Definition 4.5 Let $(M,\\sigma )$ be a generalized symplectic manifold.", "For any $f\\in {\\mathcal {G}}(M)$ , the generalized vector field $H_f:=\\sigma ^\\sharp (df)\\in \\mathfrak {X}_{\\mathcal {G}}(M)$ is called the Hamiltonian vector field of $f$ .", "Moreover, for $f$ , $g\\in {\\mathcal {G}}(M)$ , the Poisson bracket of $f$ and $g$ is given by $\\lbrace f,g\\rbrace :=\\sigma (H_f,H_g)$ .", "Analogous to the smooth setting, the Poisson bracket induces an $\\widetilde{\\mathbb {R}}$ -Lie-algebra structure on ${\\mathcal {G}}(M)$ .", "Summing up, the above constructions provide the foundations for a regularization-based approach to non-smooth symplectic geometry.", "Contrary to the distributional setting, the additional flexibility of Colombeau's theory allows one to retain the basic structure of the smooth setting.", "Building on these constructions one may now systematically explore applications, in particular in non-smooth mechanics.", "In particular, previous work in this direction (e.g., [10], [15]) can now be viewed from a unifying perspective.", "Finally, we hope that this approach will be useful in studying the propagation of singularities for pseudo-differential operators with non-smooth principal symbol on differentiable manifolds." ], [ "Acknowledgment", "This work was supported by project P25236 of the Austrian Science Fund and projects 174024 of the Serbian Ministry of Science, and 114-451-3605 of the Provincial Secretariat for Science." ] ]	1403.0234
[ [ "The tunneling model of laser-induced ionization and its failure at low\n frequencies" ], [ "Abstract The tunneling model of ionization applies only to longitudinal fields: quasistatic electric fields that do not propagate.", "Laser fields are transverse: plane wave fields that possess the ability to propagate.", "Although there is an approximate connection between the effects of longitudinal and transverse fields in a useful range of frequencies, that equivalence fails completely at very low frequencies.", "Insight into this breakdown is given by an examination of radiation pressure, which is a unique transverse-field effect whose relative importance increases rapidly as the frequency declines.", "Radiation pressure can be ascribed to photon momentum, which does not exist for longitudinal fields.", "Two major consequences are that the near-universal acceptance of a static electric field as the zero frequency limit of a laser field is not correct; and that the numerical solution of the dipole-approximate Schr\\\"{o}dinger equation for laser effects is inapplicable as the frequency declines.", "These problems occur because the magnetic component of the laser field is very important at low frequencies, and hence the dipole approximation is not valid.", "Some experiments already exist that demonstrate the failure of tunneling concepts at low frequencies." ], [ "Introduction", "The quantum phenomenon of tunneling through a potential barrier has been known and fruitfully employed since its introduction in 1928 to describe nuclear alpha decay[1] and, in the same year, to calculate the ionization of the hydrogen atom by a constant electric field[2].", "Both of those early applications involved static electric fields.", "The same concept has been applied in more recent years[3], [4], [5] to laser-induced ionization when the laser field is approximated as a quasistatic electric field.", "The concept of tunneling ionization is that an impenetrable potential barrier is rendered penetrable by the superposition of an oscillatory electric field: $\\mathbf {E}\\left( t\\right) $ .", "It is usual to describe the applied field by a scalar potential, $\\phi \\left( t\\right) =-\\mathbf {r\\cdot E}\\left( t\\right) ,\\;\\mathbf {A}=0,$ since this leads to the familiar graphical illustration where a potential well representing the force binding an electron to an atom is periodically depressed to allow the electron to escape by tunneling through the depressed barrier.", "The potentials in Eq.", "(REF ) describe a longitudinal field.", "Longitudinal fields can oscillate in time, but they do not propagate.", "The scalar-potential-only nature of Eq.", "(REF ) is usual but not essential.", "It is always possible to find a gauge transformation that replaces the scalar potential $\\phi \\left( t\\right) $ in Eq.", "(REF ) by a vector potential $\\mathbf {A}\\left( t\\right) $ .", "Gauge equivalence means that such a vector-potential description also represents a longitudinal field.", "Actual laser fields are true propagating plane-wave (PW) fields, also known as transverse fields.", "The name comes from the fact that a PW field has electric and magnetic components of equal magnitude (in Gaussian units) that are perpendicular to each other, and the plane they define is perpendicular to the direction of propagation.", "That is, propagation is in a direction transverse to the electric and magnetic fields.", "Laser fields are vector fields that cannot be completely described by a scalar potential, nor by any potential that is gauge-equivalent to a scalar potential.", "The simplest way to describe transverse fields (see, for example, the textbook of Jackson[6]) is by a vector potential function alone $\\phi =0,\\;\\mathbf {A}=\\mathbf {A}\\left( \\varphi \\right) $ that depends on spacetime coordinates $x^{\\mu }$ only in the combination $\\varphi & =k^{\\mu }x_{\\mu }=\\omega t-\\mathbf {k\\cdot r,}\\\\k^{\\mu } & :\\left( \\frac{\\omega }{c},\\mathbf {k}\\right) ,\\quad x^{\\mu }:\\left( ct,\\mathbf {r}\\right) .", "$ The quantity $\\varphi $ is the phase of a propagating field, and $k^{\\mu }$ and $\\mathbf {k}$ are the 4-vector propagation vector and its 3-vector component.", "This prescription for the potentials is known as the radiation gauge (or Coulomb gauge).", "Longitudinal and transverse fields are fundamentally different electromagnetic phenomena.", "These differences are explored in depth in this article.", "For some purposes it is possible to neglect the dependence of $\\mathbf {A}$ on the spatial coordinate $\\mathbf {r}$ , in which case (known as the dipole approximation) there is a gauge transformation due to Göppert-Mayer[7] (GM) that provides a gauge equivalence between the dipole-approximation vector potential $\\mathbf {A}\\left( t\\right) $ and the scalar potential of Eq.", "(REF ).", "It is of fundamental importance to maintain awareness that fields described within the GM gauge (also known as the length gauge) cannot be anything more than longitudinal fields.", "A longitudinal field can never be gauge-equivalent to a transverse field, but only to an approximation to a transverse field.", "This means that the tunneling concept, dependent on the scalar potential (REF ) or any potential gauge-equivalent to (REF ), can never be more than a limited approximation for laser phenomena.", "It is the elucidation of these limitations that is a major focus of this article.", "Tunneling is a concept applicable only to longitudinal fields, and is only a very limited approximation for transverse (e.g.", "laser) fields.", "The focus of attention is now shifted to an examination of parameters wherein tunneling can be a meaningful approximation for laser fields.", "There is an upper limit on the field frequency for which the dipole approximation is applicable that was pointed out by Göppert-Mayer[7].", "When the field wavelength is less than the size of the atom, it can act as a probe of the structure of the atom.", "That is not possible for $\\lambda \\gtrsim 1\\;a.u.$ , which limits the field frequency to $\\omega \\lesssim 2\\pi c $ in atomic units.", "It is not generally recognized that there is a lower limit to the frequency at which the dipole approximation can be applied.", "This lower limit is of far more practical importance than the upper limit of Eq.().", "The reason for this oversight may be that no such lower limit exists for QSE fields but, importantly, it does for PW fields Section II gives a qualitative insight into the atomic domain as it appears for QSE fields; that is for fields describable by Eq.", "(REF ), and Section III repeats the analysis for PW fields.", "The contrast between the two types of fields is striking.", "This comes about because the strength of a longitudinal field is judged only by the magnitude of the electric field, whereas it is the strongly frequency-dependent ponderomotive potential that is required for appraisal of transverse fields[8], [9].", "For instance, a parameter domain that applies to very weak QSE fields is shown to consist of very strong PW fields, and vice versa.", "Important matters elaborated in Section III include the criteria for the onset of nondipole effects at low frequencies.", "Section IV concentrates explicitly on low-frequency behavior.", "The analytical methods available for the description of ionization by strong laser fields are described in Section V. These include the tunneling model, the strong-field approximation (SFA), and the numerical solution of the time-dependent Schrödinger equation (TDSE).", "The domains of applicability have some overlap, but this overlap is far more limited than is evident from the current literature.", "In particular, the literature exhibits widespread dependence on the “exactness” of the TDSE without regard for the failure of the dipole approximation for low-frequency laser fields.", "Finally, the results are summarized and evaluated in Section VI.", "Qualitative assessments are made, including the conclusion that the tunneling approximation is useful for laser-induced ionization problems in only a very small region in the frequency and intensity domain of laser fields." ], [ "Quasistatic electric fields", "The simplest electromagnetic field is a static electric field.", "This can be described by the potentials $\\phi =-\\mathbf {r\\cdot E}_{0},\\;\\mathbf {A}=0, $ where the subscript on the electric field vector $\\mathbf {E}_{0}$ is a reminder that the field so described is constant.", "A simple but important generalization is a field configuration in which there is no magnetic field and the electric field is time-dependent, as in Eq.", "(REF ).", "This a quasistatic electric (QSE) field.", "Another indicator of this identification is the Lorentz invariant $\\mathbf {E}^{2}-\\mathbf {B}^{2}=-\\frac{1}{2}F^{\\mu \\nu }F_{\\mu \\nu }, $ where the inner product of the two electromagnetic-field tensors $F^{\\mu \\nu }$ on the right-hand side of the equation shows the reason why this quantity is a Lorentz scalar; that is, its value is invariant under any Lorentz transformation.", "For QSE fields, it is always true that $\\mathbf {E}^{2}-\\mathbf {B}^{2}>0.", "$ By contrast, a laser field is a PW field for which it is always true that $\\mathbf {E}^{2}-\\mathbf {B}^{2}=0.", "$ The important conclusion is that the GM gauge, that employs the potentials (REF ) (or any potentials gauge-equivalent to (REF )) for the description of a laser field, approximates the laser field by a QSE field.", "There is no limit in which the GM gauge is exact.", "Figure: A QSE field contains field information solely in the form of thedirection and amplitude of the electric field 𝐄\\mathbf {E}.", "For atomicproblems, 𝐄≫1a.u.\\left\|\\mathbf {E}\\right\|\\gg 1a.u.", "indicates a strong fieldand 𝐄≪1a.u.\\left\|\\mathbf {E}\\right\|\\ll 1a.u.", "indicates a weak field.", "Noother field information appears in the figure.", "The location of the atomicbinding energy E B E_{B} (selected to be 0.5a.u.0.5a.u.)", "is shown, since a tunnelingmodel applies only if the field frequency is such that ℏω≪E B \\hbar \\omega \\ll E_{B}.The location of the frequency corresponding to wavelength λ=1a.u.\\lambda =1a.u.", "isalso shown, since it is known that the dipole approximation will not be validbeyond that frequency.A QSE field is a longitudinal field, which means that there is only one spatial direction that is important: the electric field direction.", "A QSE field does not possess a propagation capability; it can oscillate with time, but it cannot propagate.", "The electromagnetic field enters into the potentials through the direction and the magnitude of the electric field.", "In atomic applications, the electric field is strong when $\\left\|\\mathbf {E}\\right\|\\gg 1a.u.$ , and it is weak when $\\left\|\\mathbf {E}\\right\|\\ll 1a.u.$ This is indicated in Fig.", "REF , which is a plot showing field frequency on the $x-axis$ and field intensity on the $y-axis$ .", "Intrinsic frequency considerations do not occur, but two frequency limits of atomic origin are shown.", "One is the well-known upper limit on the dipole approximation, where Fig.REF shows the frequency that is given in Eq.", "(REF ).", "Another upper limitation on the frequency comes from the tunneling method itself, since tunneling is only meaningful when an uncountably large number of photons participate.", "In practical terms, “uncountably large” in an ionization event may be satisfied by a number of the order of 10.", "Figure REF shows this tunneling limit in terms of the binding energy of an electron in ground-state hydrogen at $0.5a.u.$ Among the qualitative properties just listed about QSE fields, there is nothing to serve as an indicator that the dipole approximation is inapplicable to laser fields at low frequencies.", "This is apparently the underlying reason why a low-frequency limit has escaped notice in the Atomic, Molecular, and Optical (AMO) community.", "Even a modern book entitled Atoms in Intense Laser Fields[10] asserts (see pp.267-289) that the GM gauge is the preferred gauge for laser problems since it is well-behaved as the frequency approaches zero.", "There is no awareness that there is a low-frequency limit for the applicability of the GM gauge to laser problems.", "The line of reasoning followed by the authors is based on the concept of adiabaticity, and the entire discussion depends on the validity of the dipole approximation.", "The fact that a constant electric field emerges as $\\omega \\rightarrow 0$ is prima facie evidence that the discussion in Ref.", "[10] relates only to longitudinal fields and not to laser fields.", "Another clear indication of the problem that exists in the AMO community can be found in a recent paper in a prestigious rapid-publication journal, where two consecutive sentences that are contradictory are viewed as if they were mutually supportive[11].", "The first sentence is: “In adiabatic tunneling the laser field is treated as if it were a static field, time serving only as a parameter.” That is, the authors state that they are treating the laser field as if it were a QSE field, where Eq.", "(REF ) is valid.", "The next sentence is: “It is rigorously valid for long wavelengths ...” The authors thereby state that they view the laser field as if it has a zero-frequency limit, and they view that limit as a static electric field, despite the previous statement that they are concerned with laser fields, where Eq.", "(REF ) applies.", "Equations (REF ) and (REF ) are incompatible, not equivalent.", "References [10] and [11] are not singled out for special criticism, but they are cited because they are especially visible representations of the prevailing view.", "The following Section further elaborates the fundamental differences between QSE fields and laser fields." ], [ "Plane wave fields", "Plane wave (PW) fields are transverse fields.", "(See Chapter 7 in the text by Jackson[6].)", "An essential property of PW fields is that any occurrence of the spacetime 4-vector $x^{\\mu }$ must occur as the scalar product with the propagation 4-vector $k^{\\mu }$[12], [13], [14], as shown in Eq.", "(REF ).", "This product, which is the phase of a propagating field, will be called $\\varphi $ .", "The descriptions “PW”, “transverse”, and “propagating” are here viewed as equivalent designations of the type of fields that lasers produce.", "A basic property of such fields is that, once generated, they propagate indefinitely in vacuum without the need for sources.", "This last property is very important.", "The GM gauge does not have that feature of a freedom from sources.", "That is, no field can exist in the GM gauge without sources to sustain it[15].", "They are “virtual sources” in the sense that they do not actually exist in the laboratory.", "Because of the gauge equivalence between the GM gauge and a dipole approximation to a PW, the properties of those sources do not normally intrude in a calculation.", "There are special cases, however, when the virtual sources of the GM gauge can produce unintended consequences[15] even when the dipole approximation is valid.", "The best indicator of the strength of coupling of a PW field to a charged particle is the ponderomotive potential $U_{p},$ given by $U_{p}=I/\\left( 2\\omega \\right) ^{2} $ in atomic units, where $I$ is the field intensity.", "In the radiation gauge, this quantity is a true potential energy[9] that depends on the local values of $I$ and $\\omega $ .", "Its dimensionless form also serves as the coupling constant between a strong PW field and an electron, replacing the fine structure constant $\\alpha $ of perturbation theory[16], [8], [9].", "The analog of $U_{p}$ in the GM gauge is a “quiver energy” that corresponds to the oscillation of the charged particle as it is driven by the QSE field.", "The magnitude of $U_{p}$ is the same in both gauges, but the interpretation is different.", "In the GM gauge, $U_{p}$ is a kinetic energy of the charged particle that results from the virtual sources described above.", "In the radiation gauge, there are no sources to drive the particle, and $U_{p}$ is a potential energy, not a kinetic energy.", "Figure: A PW field is a propagating field that has many more physicalfeatures than a QSE field.", "The most important properties to note are the majorqualitative differences between Fig.", "and Fig.. Relativisticconditions are clearly present here, including regions such as in the lowerleft of the figure where relativistically-strong-field effects occur inregions that are labeled “Weak Field” inFig..", "The “Weak Field” designationin the upper right of this figure contrasts with the “StrongField” label in Fig..", "The onset of low-frequencynondipole behavior is marked either by the displacement in the fieldpropagation direction caused by the magnetic field (β 0 =1a.u.\\beta _{0}=1a.u.)", "or byradiation-pressure-caused contributions to energy and momentum (KE radp =0.5a.u.KE_{radp}=0.5a.u.).", "Neither of these low-frequency indicators exists in Fig..The overall conclusion is that QSE fields and PW fields are fundamentallydifferent electromagnetic phenomena with major qualitative differences.In the nonperturbative theory of the interaction of charged free particles with strong laser fields, as in Compton scattering[17], photon-multiphoton pair production[16], or pair annihilation[18], a single intensity parameter occurs that is the same for all free-particle interactions.", "Many different notations occur in the literature, but it will be designated here as $z_{f}$ , where $z_{f}=2U_{p}/mc^{2}.", "$ The quantity $z_{f}$ can be viewed as a dimensionless statement of the ponderomotive potential.", "When $z_{f}=1$ , then $2U_{p}$ equals the rest energy of the particle, and the process is unequivocally relativistic.", "This means that the dipole approximation has no validity.", "Figure REF is the PW analog of Fig.REF for QSE fields, and the line corresponding to $z_{f}=1$ shows immediately that there is a drastic difference between the behavior of QSE and PW fields.", "For example, the lower left region of Fig.REF corresponds to a very weak QSE field, whereas the same region in Fig.REF represents a very strong PW field.", "The line corresponding to $z_{f}=1$ is given by $I_{rel}=2c^{2}\\omega ^{2} $ in atomic units.", "The condition $z_{f}=1$ refers to a strongly relativistic environment, but the onset of nondipole behavior can occur at significantly lower intensities.", "One way to estimate the lower-frequency limit of the dipole approximation is to examine the well-known “figure-8” motion of a free charged particle in a PW field[19], [13].", "This is shown in Fig.REF in the frame of reference where the particle is at rest when averaged over a full cycle.", "At low field intensity, the figure-8 reduces to a straight-line oscillation of amplitude $\\alpha _{0}$ .", "Departure from straight-line behavior occurs at increasing intensity since the coupled action of the electric and magnetic fields of the PW causes a motion in the direction of propagation of the field.", "When the amplitude in the propagation direction is of the order of one atomic unit, $\\beta _{0}\\approx \\frac{U_{p}}{2mc\\omega }=1a.u., $ this signals an important contribution from the magnetic field that will certainly influence the nature of the interaction with the ion, and the dipole approximation is not valid.", "The line determined by the condition (REF ) is $I_{fig8}=8c\\omega ^{3} $ in atomic units, and is shown in Fig.REF .", "Figure: A free electron in a PW field executes a “figure-8” orbit in a frame of reference where the electronis at rest on average.", "The amplitude α 0 \\alpha _{0} is in the direction of theelectric field.", "The amplitude of motion in the direction of propagation of thePW, , β 0 \\beta _{0}, caused by the combined action of the electricand magnetic fields, is defined as shown here.", "When β 0 \\beta _{0} is of theorder of one a.u.a.u., this is an indication of the low frequency failure of thedipole approximation.An alternative way to assess the importance of nondipole effects is to consider the momentum and energy of a motion induced by radiation pressure.", "The momentum in the direction of field propagation resulting from radiation pressure is[20], [21], [22] $p_{}=U_{p}/c.", "$ The kinetic energy corresponding to this momentum is $KE_{radp}=\\frac{p_{}^{2}}{2m}=\\frac{U_{p}^{2}}{2mc^{2}},$ where the nonrelativistic form on the left-hand side is sufficient since this limitation occurs well before the fully relativistic limit is reached at $2U_{p}=mc^{2}.$ The condition (REF ) gives the line in Fig.REF determined by $I_{radp}=4c\\omega ^{2} $ in atomic units.", "This is parallel to the $I_{rel}$ of Eq.", "(REF ), but smaller by the factor $2/c$ a.u.", "That is, the onset of a relativistic effect like radiation pressure makes its presence felt well before the fully relativistic condition of Eq.", "(REF ).", "By using very sensitive measurement techniques, radiation pressure effects have already been observed in the laboratory[23] at 800 nm wavelength.", "An intensity of $8\\times 10^{14}W/cm^{2}$ was specifically analyzed[21], [22].", "In atomic units, the intensity was $0.023$ , whereas Eq.", "(REF ) predicts $I_{radp}=1.78a.u.$ for that wavelength.", "Thus the experiment detected radiation pressure at little more than one percent of the condition stated in Eq.", "(REF ).", "This confirms that Eq.", "(REF ) gives a realistic assessment of conditions where the dipole approximation will fail." ], [ "Low Frequencies", "Figures REF and REF make it very clear that longitudinal fields and transverse fields are different electromagnetic phenomena.", "That major distinction arises from the unique properties of a propagating field.", "The failure of correspondence is most striking at low frequencies.", "From the point of view of longitudinal fields, low frequencies are viewed as being in the so-called tunneling limit, where QSE fields approach static electric fields.", "For transverse fields, low frequencies correspond to the completely different domain of Extreme Low Frequency (ELF) radio waves.", "The qualitative features of these two fundamentally different domains of electromagnetic phenomena are outlined here." ], [ "“Tunneling limit” and the\nKeldysh parameter", "The tunneling view of ionization leads to a single controlling intensity parameter known as the Keldysh parameter[3] that can be written as $\\gamma _{K}=\\sqrt{E_{B}/2U_{p}}, $ where $E_{B}$ is the binding energy of the electron in the atom or molecule.", "This quantity is also called the ionization potential, and designated by $IP$ or $I_{p}$ .", "The putative tunneling domain is defined by $\\gamma _{K}<1, $ and the tunneling limit is $\\gamma _{K}\\rightarrow 0.", "$ The opposite case to the tunneling domain is the so-called multiphoton domain: $\\gamma _{K}>1$ .", "It has become standard practice in the description of laser-induced processes to specify whether an experiment or theory relates to one or the other of these two longitudinal-field domains, without regard to the qualitatively different behavior of actual laser (transverse) fields.", "With Eq.", "(REF ) used to introduce field intensity and frequency into Eq.", "(REF ), the dividing line between these two domains will be called $I_{tun}$ , and is given by $I_{tun}=2E_{B}\\omega ^{2}, $ which leads to the tunneling limit expressed as $I_{tun}\\gg 2E_{B}\\omega ^{2}$ .", "Equation (REF ) has the same $\\omega ^{2}$ dependence as $I_{rel}$ in Eq.", "(REF ) and $I_{radp}$ in Eq.", "(REF ), although the prefactors are such that $I_{tun}=1$ lies well below the other two lines in a diagram such as Fig.REF .", "As the frequency declines, however, the physical system heads inexorably towards the relativistic domain where the dipole approximation certainly fails and the tunneling approximation has no applicability to laser effects[24].", "On the other hand, from the point of view of longitudinal fields, declining frequency is simply a matter of an ac field approaching a dc limit.", "This appears to present no conceptual problems[25], and the AMO community routinely evaluates analytical methods by the limit as $\\omega \\rightarrow 0$ .", "If that limit corresponds to static-electric-field results, this is regarded as evidence of an acceptable theory.", "The fact that $\\mathbf {E}^{2}-\\mathbf {B}^{2}>0$ for such a limit seems never to be noticed." ], [ "ELF radio waves", "Transverse fields always retain their propagation property as $\\omega \\rightarrow 0,$ meaning that $\\mathbf {E}^{2}-\\mathbf {B}^{2}=0$ is sustained and the magnetic field is always present.", "Maintenance of the propagation property means that decreasing frequency implies a progression from ultraviolet to visible to infrared to microwave to radio phenomena.", "The zero frequency limit can be approached, but never reached.", "The divergence of $U_{p}$ as $\\omega \\rightarrow 0$ , shown in Eq.", "(REF ), is symptomatic of the energy demands of producing ELF radio transmissions.", "The lowest ELF frequency of which this writer is aware is the 76 $Hz$ system that the U.S. Navy proposed as a means of communicating with submerged submarines.", "The project was named “Project Sanguine”[26], and the original design (never built) required a massive 600 $MW$ of power to produce a signal with such small bandwidth that only simple coded messages could be sent.", "An ELF radio wave is conceptually as distant from a constant electric field as would be a pure magnetic field when judged by the relevant $\\mathbf {E}^{2}-\\mathbf {B}^{2}$ value.", "Nevertheless, the hallmark of success valued from a QSE point of view in the AMO community is that theoretical predictions should match the properties of constant electric fields.", "This is completely inappropriate for laser fields." ], [ "Analytical methods", "A brief survey is given here of some of the consequences of the foregoing considerations for theoretical techniques employed to describe atomic systems subjected to very intense laser fields." ], [ "Tunneling", "Time-independent potential barriers that are penetrable by quantum tunneling processes are treated in essentially all textbooks on quantum mechanics, and that problem is thoroughly understood.", "Tunneling methods were later applied to QSE fields[3], [4], [5].", "Many investigators who depend on tunneling methods to solve problems in laser-induced ionization processes regard the existence of a static-field limit of their method as a reassurance of accuracy.", "This should be worrisome rather than reassuring.", "Transverse fields do not have a physically attainable zero frequency limit, which is evident in several ways.", "The radiation pressure result would be divergent were there a zero frequency limit, as is clear from Eqs.", "(REF ) and (REF ).", "Figure REF shows that there is no access to zero frequency of a PW field without entering into the relativistic domain where the dipole approximation necessarily fails.", "Under relativistic conditions the magnetic component of a transverse field becomes as important as the electric component.", "Theories of relativistic tunneling have been published[27], but they relate only to extremely strong QSE fields.", "Relativistically strong laser fields cannot be treated by QSE methods.", "Figure: This figure shows the domain of applicability of the tunneling modelfor laser-induced ionization – dubbed “TunnelOasis” – that follows from applying the constraints shownin Fig..", "The Tunnel Oasis is enclosed in the triangular domainbounded by solid lines.", "The Tunnel Oasis is a subset of the substantiallylarger “Dipole Oasis”, enclosed within thedashed lines in the figure.", "The Dipole Oasis is the domain in which the dipoleapproximation is applicable.", "The bounds at low frequencies all arise from theconstraints evaluated by examining radiation pressure effects in terms ofdisplacements due to the magnetic component of the laser field (β 0 =1a.u.\\beta _{0}=1a.u.)", "in Fig.", "or from the energy directly due to radiationpressure effects (KE radp =0.5a.u.KE_{radp}=0.5a.u.)", "shown in Fig.. TheDipole Oasis also represents that domain of field parameters in which thereis a gauge correspondence between the length and velocity gauges.", "Both Oasesexpand considerably at lower intensities – the domain of traditional AMOphysics.", "The point labeled “AA” locates theexperiments done with a CO 2 CO_{2} laser, and the point“BB” marks some of theexperiments done by the Ohio State University group.", "Both pointslie outside the Tunnel Oasis, and both sets of experiments show features thatare not explicable in terms of a tunneling process.Figure REF shows the domain in an intensity-frequency diagram where the tunneling method is applicable for laser-induced processes.", "This domain is dubbed the “Tunnel Oasis”, since that is where a tunneling model can be applied successfully without concern for the broader limitations of the tunneling approximation.", "The lower limits on frequency come entirely from transverse-field effects as shown in Fig.REF .", "Such effects are not present in a longitudinal-field analysis, as Fig.REF clearly shows.", "The upper limit on frequency is specific to tunneling, since the basic premise of a tunneling theory is that it is a field-emission effect, not explicable in terms of a limited number of photons.", "Experiments done with a $CO_{2}$ laser in the 1980s[28] showed a striking departure from the properties of tunneling behavior[29].", "The $CO_{2}$ -laser parameters are shown as the point labeled “$A$ ” in Fig.REF .", "This is plainly outside the Tunnel Oasis.", "The pioneering “low-energy-structure” (LES) experiments[30], [31] of the group at Ohio State University also possess spectrum features that are not explicable within the tunneling approximation, which is to be expected from the location marked “$B$ ” in Fig.REF .", "Both of these sets of experiments with linearly polarized light show clear departures from tunneling behavior, where the spectrum peaks sharply at zero energy[32], [33].", "A remarkable feature of the Tunnel Oasis is how small it is when compared to the overall parameter space which is, or will become, the range of laser parameters.", "One may call it an “accident of Nature” that the most commonly used source of strong laser fields operates at about $800nm$ , within the Oasis except at extremely large intensities where saturation will occur, and the failure of the tunneling model is obscured.", "Another aspect of Fig.REF is that the lowest intensity shown is $10^{-4}a.u.$ ($3.5\\times 10^{12}W/cm^{2}$ ).", "Traditional AMO physics is conducted at much lower intensities than that, where the triangular Tunnel Oasis expands considerably.", "This is the underlying reason why the limitations of the GM gauge and its low-frequency failure have not become visible until recently." ], [ "Strong-Field Approximation", "The Strong-Field Approximation (SFA) is based on the idea that after a photoelectron has been ionized from an atom by a very intense field, its behavior is dominated by the field that caused the ionization, rather than by the residual effects of the binding potential[34].", "Discussion of the SFA is difficult because its definition has become muddled.", "The analytical approximation method of Ref.", "[34] is not a tunneling method, so the name “Strong-Field Approximation” was proposed[35] for this method to distinguish it from GM-gauge methods that are all basically tunneling approximations.", "Unfortunately, the purpose of Ref.", "[35] was not understood, and so the designation “SFA” began to be applied to all approximations where the field in the final state dominates residual Coulomb effects.", "The remarks that follow pertain only to the non-tunneling SFA of 1980[34].", "The 1990 paper[35] showed that the method of Ref.", "[34] is actually based on a completely relativistic formalism, when that can be subjected to the dipole approximation.", "The distinction that is vital is that the magnetic field is always present but, apart from the essential propagation property it imparts, its direct effect can otherwise be ignored when nonrelativistic conditions obtain.", "This confines the 1980 SFA to the portion of Fig.REF labeled “Dipole Oasis”, but not to the much smaller Tunnel Oasis.", "(Actually, the SFA appears to retain some of its relativistic character even into the low-frequency domain below the Tunnel Oasis[29].)", "This equivalence to a full transverse-field approximation was demonstrated explicitly in another paper, also in 1990[20], by a completely relativistic calculation that reduced in the nonrelativistic case to the 1980 SFA directly with no resort to any tunneling-type approximations.", "The applicability of the 1980 SFA to domains outside the Tunnel Oasis, but within the Dipole Oasis has been verified by successful high frequency comparisons with the TDSE[36] and with the High-Frequency Approximation (HFA) of Gavrila[37]." ], [ "Time-Dependent Schrödinger Equation", "Direct numerical solution of the time-dependent Schrödinger equation for laser-induced processes has come to be called “TDSE”.", "Numerical methods have advanced to the point that accurate TDSE calculations can be performed over a wide range of laboratory parameters.", "TDSE is regarded as “exact”, and it is frequently applied to verify the accuracy of analytical approximations.", "A disadvantage of TDSE is that it does not give clear physical insights as to why particular types of behavior occur in physical systems, whereas analytical approximations can give rise to instructive physical interpretations.", "The perception of exactness of the TDSE approach has been carried too far.", "Many investigators (see, for example, Ref.", "[10]) fail to notice the low-frequency limit of the dipole approximation, and make the assumption that TDSE can comfortably be extended all the way to zero frequency.", "The range of applicability of the TDSE numerical method is indicated by the Dipole Oasis domain in Fig.REF .", "Numerical methods applied to frequencies lower than the Dipole Oasis would require eschewing the dipole approximation for frequencies somewhat less than the low frequency limit of the Tunnel Oasis, and full solution of the Dirac equation in three spatial coordinates for still lower frequencies.", "Those capabilities do not exist at present." ], [ "Length gauge and velocity gauge", "The meaning of “length gauge” is unambiguous: it is identical to the GM gauge.", "That is, it refers to the representation of an electromagnetic field of laser origin by the $\\mathbf {r\\cdot E}\\left( t\\right) $ scalar potential of Eq.", "(REF ).", "In other words, it approximates a PW field by the conceptually simpler QSE field.", "The term “velocity gauge” can be misinterpreted, as detailed in the above discussion about the SFA.", "The simplest solution to this situation is to confine the meaning of velocity gauge to that use of the dipole approximation that is exactly gauge-equivalent to the GM (or length) gauge.", "The basic interaction Hamiltonian for the velocity gauge is, in atomic units and for a single particle: $H_{I}^{VG}=\\mathbf {A}\\left( t\\right) \\mathbf {\\cdot p}+\\frac{1}{2}\\mathbf {A}^{2}\\left( t\\right) .", "$ As the dipole approximation is employed in the SFA of Refs.", "[34], [35], [38], [39], a suitable nomenclature is to call it “radiation gauge” or “radiation gauge in the dipole approximation” when that modifier is appropriate.", "The terminology “velocity gauge” can convey the wrong impression." ], [ "Summary", "For a charged particle in a laser field (or in any propagating field), it is at low frequencies that the magnetic component of the field becomes most important.", "This stands in sharp contrast to charged particle behavior in any field describable by a scalar potential of the form $\\mathbf {r\\cdot E}\\left(t\\right) $ (a longitudinal field) for which no magnetic field exists.", "The tunneling model of ionization is confined entirely to longitudinal fields, so it has no relevance for low-frequency laser fields.", "Quality indices based on the $\\mathbf {r\\cdot E}\\left( t\\right) $ potential are actually counter-indicative.", "A frequently-employed example is the concept that an analytical method should reproduce static-electric-field properties as the frequency declines.", "This violates the very concept of a propagating field that, to maintain its propagation property, must always retain the appropriate time dependence given by the phase of the propagating field.", "The low-frequency failure of dipole-approximation methods applies to all such techniques employed for laser fields, including numerical methods for solution of the Schrödinger equation (TDSE) and all length-gauge and velocity-gauge treatments.", "The tunneling model is a length-gauge approximation that has the additional limitation imposed by the high-frequency constraint that the energy of a single photon of the field must be much less than the binding potential of a prospective detached electron.", "The result is that the “Tunnel Oasis” of Fig.REF , defining the domain of laser parameters for which the tunneling model can be applied, is far smaller than the overall “Dipole Oasis” in Fig.REF within which the dipole approximation for laser-field effects has validity.", "The Oases shown in Fig.REF continue to expand as the intensity falls below the $3.5\\times 10^{12}W/cm^{2}$ lowest intensity of the figure.", "Traditional AMO physics is conducted at much lower intensities where the low-frequency failures of the GM gauge and of the tunneling model are not in evidence.", "This is the most likely explanation for why the low-frequency failure of the dipole approximation has not previously attracted notice.", "That situation is changing.", "Existing laser experiments at low frequencies already show the failure of the tunneling approximation." ] ]	1403.0568
[ [ "A 1/n Nash equilibrium for non-linear Markov games of mean-field-type on\n finite state space" ], [ "Abstract We investigate mean field games for players, who are weakly coupled via their empirical measure.", "To this end we investigate time-dependent pure jump type propagators over a finite space in the framework of non-linear Markov processes.", "We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics which leads to the well-known Mckean-Vlasov dynamics in the limit as the number N of players goes to infinity.", "The case where one player has an individual preference different to the ones of the remaining players is also covered.", "The limiting system represents a 1/N-Nash Equilibrium for the approximating system of N players." ], [ " We investigate mean field games from the point of view of a large number of indistinguishable players which eventually converges to infinity.", "The players are weakly coupled via their empirical measure.", "The dynamics of the individual players is governed by pure jump type propagators over a finite space.", "Investigations are conducted in the framework of non-linear Markov processes.", "We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics.", "In the limit as the number $N$ of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics.", "The case where one player has an individual preference different from the ones of the remaining players is also covered.", "The two results combined reveal a ${\\tfrac{1}{N}}$-Nash Equilibrium for the approximating system of $N$ players.", "section1.7plus.5Introduction Mean field game theory is a type of dynamic Game theory where the agents are coupled with each other by their individual dynamics and their empirical mean.", "The objective of each agent, given in terms of the so called cost function, does not only depend on her own preference and decision but also on the decisions of the other players.", "All in all it is a mathematical tool to describe a control problem with a large number $N$ of agents where the impact of the individual decisions of the other agents is becoming extremely weak compared to the overall impact as $N$ increases to infinity.", "The limiting model emerges from the fact that each agent constructs her strategy from her own state and from the state of the empirical mean of an infinite number of co-agents of hers and results in a decoupled dynamics and objective which depend on the law of her dynamics.", "The mean field approach has been independently developed by J.-M. Lasry and P.-L. Lions in a series of papers see [12] and the references therein using nonlinear PDE's and by M. Huang, P. Caines, Malhamé, see [14] [15] in the setting of stochastic processes, see also [6].", "The investigations in this work are carried out in the framework of non-linear Markovian propagators, respectively time inhomogeneous nonlinear Feller processes, which was developed by Vassili Kolokoltsov [18] [19].", "We focus on propagators related to processes of pure jump type with finite intensity measure on a finite set $\\mathbb {X}=\\lbrace 1,\\ldots ,k\\rbrace $ , $k\\in I\\!\\!N$ .", "The elements of this set can be identified with the possible decisions of the players, respectively with the (financial) positions in the financial instruments of a finite market.", "Our starting point of the so called closed-loop construction including an optimal control is the following forward Kolmogorov equation written in the weak form: $\\frac{\\partial f_s}{\\partial s} - \\left( \\mathfrak {A}[s,\\rho _s,u_s]f_s,\\mu \\right)&=& 0,\\quad 0\\le t < s\\le T \\\\f(t,\\mathbf {x}) &=& \\Phi (\\mathbf {x}),\\quad \\mathbf {x}\\in \\hat{\\mathbb {X}}\\nonumber $ where $\\mu $ is a finite measure in $\\hat{\\mathbb {X}}$ and $f$ is an element of the dual space and a differentiable function in time, the set of bounded continuous functions $\\mathbf {C}([0,T]\\times \\hat{\\mathbb {X}})$ .", "The set is defined by $\\hat{\\mathbb {X}}:= \\cup _{N=1}^\\infty \\mathbb {X}^N$ , where $\\mathbb {X}^N$ is the $N$ -fold direct product of the set $\\mathbb {X}$ , $\\mathbf {\\rho }$ is a function on $[0,T]$ taking values in the set of finite measures $\\mathbb {M}(\\hat{\\mathbb {X}})$ .", "Finally the generator $\\mathfrak {A}$ is of the form ${\\mathfrak {A}[s,\\mathbf {x},\\rho _s,u_s]f(s,\\mathbf {x})= \\sum _{i=1}^{\\left\|\\mathbf {x}\\right\|} \\mathbf {A}^{i}[s,\\mathbf {x},\\rho _s,u_s]f(s,\\mathbf {x})}\\nonumber \\\\&=& \\sum _{i=1}^{\\left\|\\mathbf {x}\\right\|}\\int _{\\mathbb {X}} \\left(f_{i^{\\prime }}(s,y)-f_{i^{\\prime }}(s,x_i)\\right)u_s\\nu (s,x_i,\\rho _s,dy).$ Here we introduce the notation $f_{i^{\\prime }}$ in order to describe that $\\mathbf {A}^i$ acts on the component $x_i\\in \\mathbb {X}$ only.", "In fact $f_{i^{\\prime }}(x_i) = f_{\\mathbf {x}_i^{\\prime }}(x_i)$ where $\\mathbf {x}_i^{\\prime }\\in \\mathbb {X}^{\\left\|\\mathbf {x}\\right\|-1}$ is derived by removing the variable corresponding to the $i^{th}$ agent from $\\mathbf {x}$ .", "The length of the vectors describing the number of players is denoted by $\\left\|\\mathbf {x}\\right\|$ .", "Hypothesis A We assume $\\nu _i(s, j, \\rho , u)$ to be linear in the parameter $u=u_s$ and postulate $\\nu _i(s, j, \\rho )$ to be a bounded kernel in all parameters uniformly in $s$ , $0\\le t < s\\le T$ , and vanishing for $i=j$ .", "The choice of the space $\\mathbb {X}$ means that the integral is a sum.", "The parameters of the generator $\\mathfrak {A}$ are subject to the assumptions that the control law $\\mathbf {u}\\in \\mathcal {U}$ satisfies $u_s\\in U$ with bounded convex set $U$ having a smooth boundary, and that $\\mathbf {\\rho }$ is a Lipschitz continuous measure valued function on $[0,T]$ such that for all $s\\in [t,T]$ we have $\\rho _s\\in \\mathbf {P}_\\delta (\\hat{\\mathbb {X}})$ , the linear hull of Dirac probability measures.", "The natural domain of the operator $\\mathfrak {D}(\\mathfrak {A}[s,\\rho _s,u_s])\\subset \\mathbf {C_0}(\\hat{\\mathbb {X}})$ and $\\mathbf {C_0}(\\hat{\\mathbb {X}})$ is the set of continuous functions vanishing at infinity on the discrete space $\\hat{\\mathbb {X}}$ which will be restricted according to technical constraints.", "For the sake of completeness we add that $\\mathbf {P}_\\delta (\\mathbb {X})\\subset \\mathbb {M}(\\mathbb {X})$ , with $\\mathbb {M}(\\mathbb {X})$ being the set of finite measures on $\\mathbb {X}$ .", "An analogous statement holds when replacing $\\mathbb {X}$ by $\\hat{\\mathbb {X}}$ .", "We also introduce the set of continuous measure valued functions $\\mathbf {C}([0,T],\\,\\mathbb {M}(\\mathbb {X}))$ , respectively, $\\mathbf {C}_\\mu =\\lbrace \\mathbf {\\rho }\\in \\mathbf {C}([0,T],\\,\\mathbb {M}(\\mathbb {X}))\\mid \\rho _0=\\mu \\rbrace $ for later purposes.", "We mention that there exists an injection from $\\hat{\\mathbb {X}}$ into $I\\!\\!N$ , which rises the question why we are using the notion $C(\\hat{\\mathbb {X}})$ of continuous functions.", "In fact, it seems advantageous at this stage to keep the analogy to jump processes on continuous spaces.", "Later we shall identify $C(\\hat{\\mathbb {X}})$ and $I\\!\\!R^k$ .", "We also mention that $\\mathbb {M}(\\mathbb {X})$ is isomorphic to $I\\!\\!R^k$ .", "The sensitivity analysis is carried out on an open neighborhood $M\\subset I\\!\\!R^k$ of the origin.", "As mentioned above the construction involves a mean-field type limit consistent with a given optimal control problem.", "This is a particular example of measure valued limits from the theory of interacting particle systems.", "A key role within the toolbox of this theory plays the injection from the equivalence class $S\\hat{\\mathbb {X}}$ of vectors $\\mathbf {x}\\in \\hat{\\mathbb {X}}$ , which are identical up to a permutation of players, into the set of point measures on $\\mathbb {X}$ , defined by $\\mathbf {x}= (\\jmath _1,\\ldots ,\\jmath _N)\\quad \\longrightarrow \\quad {\\tfrac{1}{N}}(\\delta _{\\jmath _1}+ \\ldots + \\delta _{\\jmath _N})=: {\\tfrac{1}{N}}\\delta _{\\mathbf {x}}\\ .$ More precisely, for arbitrary $N\\in I\\!\\!N$ the mapping constitutes a bijection between $S\\mathbb {X}^N$ and the subset $\\mathbf {P}^N_\\delta (\\mathbb {X})=\\lbrace \\mu \\in \\mathbb {M}(\\mathbb {X}),$ $\\mu =\\frac{1}{N} \\sum _{k=1}^{N} \\delta _k\\rbrace $ of $N$ -point measures in $\\mathbb {X}$ .", "Implicitly we identify $\\mathbf {P}^N_\\delta (\\mathbb {X})$ in $\\mathbb {X}$ with the set of Dirac measure in $S\\mathbb {X}^N$ .", "For each $N\\in I\\!\\!N$ the space $\\mathbf {C_0}^{sym}(\\mathbb {X}^N)$ of $I\\!\\!R$ -valued continuous functions which are invariant under component-wise permutations of their arguments is equivalent with the space of $I\\!\\!R$ -valued continuous functions $\\mathbf {C}(S\\mathbb {X}^N)$ .", "Moreover, $\\mathbf {C_0}^{sym}(\\hat{\\mathbb {X}})$ is a core of the operator $\\mathfrak {A}[s,\\rho _s,u_s]$ .", "The restriction $\\mathfrak {A}_N[s,\\rho _s,u_s]$ of the operator $\\mathfrak {A}[s,\\rho _s,u_s]$ to $\\mathbf {C}(S\\mathbb {X}^N)$ generates a time inhomogeneous Markov process $X^N(s) = (X^N_1(s),\\ldots ,X^N_N(s))$ , $s\\in [t,T]$ , in $\\mathbb {X}^N$ , see e.g.", "[7], [16].", "Since all agents are assumed to be subject to the same equation, the generator $\\mathfrak {A}$ being of special form (REF ), one investigates the dynamics for one representative of $N$ agents given by the time inhomogeneous Markov process $\\mathbf {X}^N:= \\mathbf {X}^N_i$ , $1\\le i\\le N$ , in $\\mathbb {X}$ .", "As the number $N$ of agents tends to infinity the dynamics of the representative player depends on her own state and distribution only.", "Similar results from mathematical physics exist and physicists phrase this phenomenon: ”the individual dynamics in the mean field model separate as $N\\rightarrow \\infty $ ”.", "By assumption the objective for each of the $N$ players is to find the value function $V^N(t,x)= \\sup _{\\mathbf {u}}{I\\!\\!E}_{x}\\left[\\int _t^TJ(s,X^N(s),\\rho _s^N,u_s)\\, ds + V^T(X^N(T),\\rho _T)\\right]$ on $[0,T]\\times \\mathbb {X}$ , i.e.", "to maximize her expected payoff over a suitable class of admissible control processes $\\mathbf {u}=\\lbrace u(s,X^N(s))\\mid 0\\le s\\le T\\rbrace \\in \\mathcal {U}$ .", "Here the cost function $J: [0,T]\\times \\mathbb {X}\\times \\mathbf {P}_\\delta ^N(\\mathbb {X})\\times U \\rightarrow I\\!\\!R$ and the terminal cost function $V^T:\\mathbb {X}\\times \\mathbf {P}_\\delta ^N(\\mathbb {X})\\rightarrow I\\!\\!R$ , as well as the final time $T$ are given.", "With a particular choise we insure that the cost function is concave.", "An explicit expression for the value function can be derived by dynamic programming as solution of the HJB equation (REF ).", "For admissible control processes the HJB equation is well posed and the resulting optimal feedback control function $\\hat{\\mathbf {u}}^N$ is unique for given start value $x\\in \\mathbb {X}$ and given $\\mathbf {\\rho }$ .", "The so-called kinetic equation which leads to the nonlinear Markov process in the sense of V. Kolokoltsov with control law $\\mathbf {u}$ is derived by making an Ansatz motivated by the weak form of the one player evolution with an intrinsic choice of the parameter $\\mathbf {\\rho }$ in the generator: $\\frac{d}{ds}(g,\\mu _s)= (\\mathbf {A}[s,\\rho _s,u] g,\\mu _s)\\vert _{\\mathbf {\\mu }=\\mathbf {\\rho }}\\footnote {The precise meaning of this equation in the setting of this paper isgiven in Theorem \\ref {b}}$ for arbitrary $g\\in \\mathbf {C}(\\mathbb {X})$ and arbitrary finite measures $\\mu _s\\in \\mathbb {M}(\\mathbb {X})$ which are differentiable in $s\\in [0,T]$ .", "To this end the corresponding differential equation for the adjoint operator and the Koopman propagator to the nonlinear flow given by the solution are investigated.", "The construction exhibits the order of convergence to be ${\\tfrac{1}{N}}$ .", "The associated control problem reveals an optimal feedback control $\\mathbf {u}$ .", "Finally MFG consistency is said to hold if the fixed measure valued function $\\mathbf {\\rho }$ in the objective function can be replaced by the empirical measures $\\mu _s^N = {\\tfrac{1}{N}}(\\delta _{X_1^N(s)}+ \\ldots + \\delta _{X_N^N(s)}), \\qquad t\\le s \\le T,$ of the underlying process while well-posedness of the optimal control problem and uniqueness of the optimal control parameter are conserved - as a result of what could be called a closed loop construction.", "This is realized by a fix point argument which establishes the ${\\tfrac{1}{N}}$ -Nash equilibrium.", "We conclude the introduction with an overview of how the paper is organized.", "In Section 2 the dynamics of the game is introduced, in particular the Markovian propagator or time inhomogeneous semi group and the continuous in time Markov chain for one representative player.", "In Section 3 the limiting dynamics is set up and the generator of the corresponding Koopman propagator is explicitly derived.", "The sensitivity analysis for the two associated control problems is discussed in Section 4.", "In the subsequent Section 5 the limit when the number of players tends to infinity is investigated.", "Bounds for the approximation error are derived for the dynamics as well as for the value functions.", "In the concluding section the ${\\tfrac{1}{N}}$ -Nash equilibrium is established.", "section1.7plus.5Pure Jump Markov Processes In the entire section let us assume that the generator $\\mathfrak {A}$ decomposes as given in (REF ).", "Hence we consider a single player $i$ .", "In order to simplify notations we even drop the index $i$ , i.e.", "$\\mathbf {A}:= \\mathbf {A}^i$ whence $1\\le i\\le N$ .", "At the same time the values of all agents different from $i$ are kept fix, i.e.", "$f(s,i):= f_{i^{\\prime }}(s,i)$ , with the notation introduced above.", "Real valued functions on $\\mathbb {X}=\\lbrace 1,\\ldots , k\\rbrace $ can be represented as $k$ -vectors and consequenly the generator $\\mathbf {A}$ as a $k\\times k$ -matrix.", "We assume that $\\mathbf {A}$ is a time inhomogeneous $Q$ -matrix on $\\mathbb {X}$ , i.e.", "$-\\infty < \\nu _{i}(s,i,\\rho _s)\\le 0$ for all $i\\in \\mathbb {X}$ ; $\\nu _{j}(s,i,\\rho _s)\\ge 0$ for $i\\ne j$ , $i,j\\in \\mathbb {X}$ ; $\\sum _{j} \\nu _{j}(s,i,\\rho _s) = 0$ for all $i$ .", "$-\\infty < \\nu _{i}(s,i,\\rho _s)\\le 0$ for all $i\\in \\mathbb {X}$ ; $\\nu _{j}(s,i,\\rho _s)\\ge 0$ for $i\\ne j$ , $i,j\\in \\mathbb {X}$ ; $\\sum _{j} \\nu _{j}(s,i,\\rho _s) = 0$ for all $i$ .", "Thus we have $\\nu _{i}(s,\\rho _s):=\\nu _{i}(s,i,\\rho _s)=-\\sum _{i\\ne j} \\nu _{j}(s,i,\\rho _s)$ since the row sum vanishes.", "We find using matrix form ${\\mathbf {A}[s,\\rho _s,u_s]\\mathbf {f}(s)}\\\\&=&\\!\\!\\begin{pmatrix}\\nu _1(s,\\rho _s)u_1 & \\nu _2(s,1,\\rho _s)u_2 & \\cdots & \\nu _k(s,1,\\rho _s)u_k \\\\\\nu _1(s,2,\\rho _s)u_1 & \\nu _2(s,\\rho _s)u_2 & \\cdots & \\nu _k(s,2,\\rho _s)u_k \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\nu _1(s,k,\\rho _s)u_1 & \\nu _2(s,k,\\rho _s)u_2 & \\cdots & \\nu _k(s,\\rho _s)u_k\\end{pmatrix}\\mathbf {f}(s)\\nonumber $ where $\\mathbf {f}(s)=(f(s,1),\\ldots ,f(s,k))^t\\in C([0,T])$ is $I\\!\\!R^k$ -valued.", "Since $\\mathbf {A}$ is a finite dimensional matrix valued function of time we have the following: $\\left\|\\mathbf {A}f\\right\|\\le C \\left\\Vert f\\right\\Vert ,$ $f\\in I\\!\\!R^k$ which means that the matrix valued function $\\mathbf {A}$ constitutes a bounded linear operator.", "Proposition 0.1 Let $M$ be a subset in the unit ball $B_1(0)\\subset I\\!\\!R^k$ and $U\\subset I\\!\\!R^k$ a convex bounded open control set.", "Assume that the matrix valued function $\\mathbf {A}(s,\\rho )$ in (REF ) is continuous in $t > t_0$ for some $t_0\\in I\\!\\!R$ , that it is of type $C^q$ in the parameters $\\rho \\in I\\!\\!R^k$ .", "Then so is the unique linear flow induced by $\\nu $ .", "The proof is a direct consequence of the results on linear ordinary differential equations in [24] CH.", "XVIII section 4 and [1] CH 2.", "Since $\\mathbf {A}x$ satisfies a linear growth condition the unique global flow in the above theorem exists on the whole space.", "The solution to the Kolmogorov equation given by the matrix (REF ) possesses the cocycle property, which replaces the semi group property of autonomous systems see [28], [3].", "Intimately related to the cocycle property is the notion of a propagator to be found e.g.", "in physics publications or in works of Reed and Simon respectively V. Kolokoltsov.", "A family of mappings $U^{t,s}$ , $t\\le s\\le T$ , in a set $S$ is called a (forward) propagator (resp.", "backward propagator) in $S$ if $U^{t,t} = id_S$ and the following iteration equation holds: $U^{t,r} = U^{t,s}U^{s,r}$ for $t\\le s\\le r$ .", "Here $U^{t,s}U^{s,r}$ is to be interpreted as the iteration of mappings.", "For linear propagators or evolutions it means the application of linear operators, see [2] and [7].", "Remark 0.2 The matrix valued functions $\\Lambda (s,r,\\cdot )$ constitute bounded linear operators.", "The family $\\lbrace \\Lambda (t,s,\\cdot ) \\mid 0\\le t\\le s\\le T\\rbrace $ generated by the operator $\\mathbf {A}$ defines a positive, strongly continuous linear propagator or evolution on the set of Euclidean $k$ -vectors which trivially coincides with the set of (continuous) real valued functions on the discrete set $\\mathbb {X}$ .", "We now recall the connection between linear propagators or evolutions and non-autonomous Markov processes which we intend to use for solving the control problem.", "We adopt the notation in [2] to the time dependent case.", "Let $(\\mathcal {E},\\mathfrak {E})$ be a measurable space and $U^{t,r}$ an arbitrary linear propagator.", "Assume that $x\\in \\mathcal {E}, \\, \\mathrm {E}\\in \\mathfrak {E}$ .", "We say that $\\lbrace p(t,x,r,\\mathrm {E}) :=(U^{t,r}\\chi _\\mathrm {E})$ , where $\\,0\\le t\\le r<\\infty \\rbrace $ and $\\chi _E$ is the indicator function of the set, is a normal transition family if the maps $x\\rightarrow p(t,x,r,\\mathrm {E})$ are measurable for each $\\mathrm {E}\\in \\mathfrak {E}$ ; the Chapman Kolmogorov equation holds; $p(t,x,r,\\cdot )$ is a probability measure $\\mathfrak {E}$ .", "the maps $x\\rightarrow p(t,x,r,\\mathrm {E})$ are measurable for each $\\mathrm {E}\\in \\mathfrak {E}$ ; the Chapman Kolmogorov equation holds; $p(t,x,r,\\cdot )$ is a probability measure $\\mathfrak {E}$ .", "For the finite measurable space $(\\mathbb {X}, \\mathfrak {P}(\\mathbb {X}))$ measurability in $x$ is trivially satisfied and the cocycle property together with the existence of a kernel reveal the Chapman Kolmogorov equation.", "The Markov property follows from the following proposition which is a straight forward adaption from [26].", "Proposition 0.3 A time inhomogeneous matrix $Q(t),\\, 0\\le t\\le s\\le T$ , on a finite set $I$ is a $Q$ -matrix if and only if $p(t,s)= T\\!\\exp \\int _t^s Q(\\tau )\\, d\\tau $ is a stochastic matrix for all $ 0\\le t\\le s\\le T$ .", "We thus have: Lemma 0.4 The family $\\Lambda $ generated by the matrix valued function $\\mathbf {A}u$ is a normal transition family.", "As done in [16] the notion of a projective family for time inhomogeneous transition probabilities see [2] Theorem 3.1.7 holds for general measurable spaces $(E,\\mathfrak {E})$ and trivially also to finite sets $\\mathbb {X}$ , and arbitrary probability measures $\\mu $ in $\\mathbb {X}$ .", "The existence of a process is then guaranteed by the Kolmogorov existence theorem.", "We only state the existence of a process in the following Proposition 0.5 Given a normal transition family $\\lbrace p_{t,r}(x,K),\\, 0\\le t\\le r<\\infty \\rbrace $ and a fixed probability measure $\\mu $ on the finite measurable space $(\\mathbb {X},\\mathfrak {P}(\\mathbb {X}))$ , then there exists a probability space $(\\Omega ,\\, \\mathcal {F},\\, {I\\!\\!P}_{\\mu })$ , a filtration $(\\mathcal {F}_t, t\\ge 0)$ and a Markov process $(X_t, t\\ge 0)$ on that space such that: $\\begin{array}{lc}I\\!\\!P\\left[X(r)\\in A \\mid X(t) = x\\right] = p_{t,r}(x,A) \\mbox{ for each } 0\\le t\\le r,\\ x\\in \\mathbb {X},\\ A\\in \\mathfrak {P}(\\mathbb {X})\\ .\\\\X(0) \\mbox{ has law } \\mu \\ .\\end{array}$ Since $\\mathbb {X}$ is compact the proof follows the line of arguments in Ethier Kurtz Theorem and [2] Theorem 3.1.7.", "The result holds in general for Polish spaces.", "In Section 5 we shall see that the solution of the kinetic equation (REF ) is the limit of the linear $N$ -mean field evolutions as $N$ tends to infinity which arises when restricting the generator $\\mathfrak {A}$ in (REF ) to $C^{sym}(\\mathbb {X}^N)$ and replacing the parameter $\\rho $ by the empirical distribution.", "The corresponding adjoint is denoted by $\\mathfrak {A}^{}_N$ .", "In order to prove the mean field limit we need to unify spaces.", "This is possible since the factor spaces $S\\hat{\\mathbb {X}}$ and the spaces of $N$ -point measures $\\mathbf {P}^N_\\delta (\\mathbb {X})$ on the one hand as well as the corresponding larger spaces $\\mathbb {M}(\\mathbb {X})$ and $I\\!\\!R^k$ on the other hand can be identified.", "We consider existence and uniqueness and the sensitivity analysis of the solutions of the Kolmogorov equation and the optimal control problems for the $N$ -player on the larger space $I\\!\\!R^k$ .", "Consequently we replace or identify in a first step $\\mathfrak {A}^{}_N$ with the linear operator $\\mathfrak {\\hat{A}}^N[t,\\mathbf {x},\\delta _{\\mathbf {x}},u] F(\\delta _{\\mathbf {x}}):=\\mathfrak {A}^{}_N[t,\\mathbf {x},\\delta _{\\mathbf {x}},u] f(\\mathbf {x})$ on $C(\\mathbb {M})$ for elements $\\delta _{\\mathbf {x}}\\in \\mathbf {P}^N_\\delta (\\mathbb {X})\\subset \\mathbb {M}$ .", "The operator reads in more detail: ${{\\hat{\\mathfrak {A}}}^N [t,\\delta _{\\mathbf {x}},u] F(\\delta _{\\mathbf {x}}) = \\sum _{i=1}^N {\\mathbf {A}^i}^{}[t,\\delta _{\\mathbf {x}},u] F(\\delta _{\\mathbf {x}})}\\\\&=& \\sum _{\\ell =1}^k \\sum _{\\ell ^{\\prime } =1}^k n_{\\ell ^{\\prime }}\\nu _{\\delta _{\\ell ^{\\prime }}}(t, \\delta _{\\ell }) u_{\\ell ^{\\prime }}\\left[ F \\left( \\sum _{\\ell =1}^k n_\\ell \\delta _\\ell + \\delta _{\\ell ^{\\prime }} - \\delta _\\ell \\right)-F\\left( \\sum _{\\ell =1}^k n_\\ell \\delta _\\ell \\right)\\right]$ where $\\nu _{\\delta _{\\ell ^{\\prime }}}(t, \\delta _{\\ell },u)$ is the transpose of the matrix $\\nu _{\\delta _\\ell }(t, \\delta _{\\ell ^{\\prime }},u)$ and $n_\\ell $ describes how often the value $l$ appears and $n_{\\ell ^{\\prime }}$ is specified by the kernel $\\nu $ .", "In a second step, specific to the case of a finite set $\\mathbb {X}$ , we identify $\\lbrace \\delta _1,\\ldots , \\delta _k \\rbrace $ with the standard basis $\\lbrace e_1,\\ldots , e_k \\rbrace $ in $I\\!\\!R^k$ to find that $\\mathbf {x}\\in S\\hat{\\mathbb {X}}$ with $\\left\|\\mathbf {x}\\right\|= N$ corresponds to $x^N = \\sum _{\\ell =1}^k n_\\ell e_\\ell \\in I\\!\\!R^k$ with $\\sum _{\\ell = 1}^k n_\\ell = N$ and: ${{\\hat{\\mathfrak {A}}}^N [t,{\\tfrac{1}{N}}x^N,u] F({\\tfrac{1}{N}}x^N):= \\sum _{i=1}^N {\\mathbf {A}^i}^{}[t,{\\tfrac{1}{N}}x^N,u] F({\\tfrac{1}{N}}x^N)}\\\\&=& \\sum _{\\ell =1}^k \\sum _{\\ell ^{\\prime } =1}^k n_{\\ell ^{\\prime }} \\nu _{\\ell }(t, e_{\\ell ^{\\prime }}) u_\\ell \\left[F \\left({\\tfrac{1}{N}}\\sum _{\\ell =1}^k n_\\ell e_\\ell + {\\tfrac{1}{N}}(e_{\\ell ^{\\prime }} - e_\\ell )\\right)-F\\left({\\tfrac{1}{N}}\\sum _{\\ell =1}^k n_\\ell e_\\ell \\right)\\right].\\nonumber $ Hypothesis B For the rest of the paper we complement the assumptions made on the domains of the variables and parameters of operator $\\mathbf {A}$ by regularity conditions, namely: We assume that $\\nu $ is uniformly Lipschitz continuous in the parameter $x\\in I\\!\\!R^k$ , and continuous in $t$ .", "Moreover, the partial derivatives $\\nabla _x \\nu (t,i,x)$ are assumed to exist in $C_\\infty (\\mathbb {R}^k)$ as functions of $x$ and to be uniformly Lipschitz continuous, uniformly in the other variables.", "Finally let the cost function $J(s,i,x,u)$ , $0\\le t< s\\le T$ , to the control problems in Section 4 be quadratic concave in u and satisfy the same properties as $\\nu $ regarding $s,i,x$ with $s$ replacing $t$ .", "In the sequel we give an alternative representation of the linear operator $\\hat{\\mathfrak {A}}_t^N$ .", "For practical reasons we introduce a scaling parameter $h \\in \\mathbb {R}^+$ .", "For differentiable functions $F$ on $\\mathbb {M}(\\mathbb {X})$ a variational derivative $\\frac{\\delta F(Y)}{\\delta Y(x)}$ of $F$ is the Gateaux derivative $D_{\\delta _x}$ of $F$ in the direction of $\\delta _x$ , $x\\in \\mathbb {X}$ .", "Since $\\mathbb {M}(\\mathbb {X})$ , and $I\\!\\!R^k$ are isomorphic we are able to work with directional derivatives $\\partial _{x}$ on $I\\!\\!R^k$ .", "Proposition 0.6 Assume that $\\nu $ satisfies the Hypothesis A and that $M$ is a bounded open subset of $I\\!\\!R^k$ .", "Let $F\\in C^2(I\\!\\!R^k)$ then the operator $\\mathfrak {\\hat{A}}^N[t,hx^N,u]$ , where $t\\in [0,T]$ , $x^N = \\sum _{i=1}^k n_i e_i\\in I\\!\\!R^k$ such that $\\sum _{\\ell =1}^k n_\\ell = N$ , $u\\in U$ , and $h \\in \\mathbb {R}^+$ , has the representation: ${ \\mathfrak {\\hat{A}}^N[t,h x^N,u] F(h x^N)= Nh\\left( \\mathbf {A}^[t, h x^N,u]\\partial _{h e_\\ell }F(h x^N), e_\\ell \\right)+ h^2 \\int _0^1\\!", "ds(1-s)}\\\\&\\times &\\!\\!", "\\sum _{\\ell ,\\ell ^{\\prime },\\ell ^{\\prime \\prime }}\\!\\!\\!", "\\eta _\\ell \\nu _{\\ell }(t,e_\\ell )u_\\ell \\left(\\partial _{h e_\\ell }F( h x^N + h s(e_{\\ell ^{\\prime }} + e_{\\ell ^{\\prime \\prime }} - e_{\\ell })),((e_{\\ell ^{\\prime }}\\otimes e_{\\ell ^{\\prime \\prime }}))\\right)\\ .$ Due to the fact that the agents are indistinguishable the sum with respect to $\\ell $ becomes a factor $N.$ For $Y$ and $Y+\\zeta $ such that the whole line $\\lbrace Y+ \\theta \\zeta \\mid 0\\le \\theta \\le 1\\rbrace $ is in $M\\subset I\\!\\!R^k$ and $F\\in C^2(M)$ the Taylor theorem gives the following representation, see [25], [[18] Cor 13 of Lemma 12.6.1]: $F(Y+\\zeta ) - F(Y) = \\left(\\frac{\\partial F(Y)}{\\partial \\zeta } ,\\zeta \\right)+\\int _0^1 ds(1-s)\\left(\\frac{\\partial ^2 F(Y)}{\\partial \\zeta ^2},\\zeta \\otimes \\zeta \\right)\\ .$ Inserting the Taylor expansion of order 2 into (REF ) under the integral for the choice $Y=hx^N$ and $\\zeta = h((e_\\ell +e_{\\ell ^{\\prime }}) - e_\\ell )$ finishes the proof.", "We gladly anticipate that the first term coincides with the generator of the Koopman propagator constructed in Proposition (3.4) for the choice $h={\\tfrac{1}{N}}$ .", "For $Y$ and $Y+\\zeta $ such that the whole line $\\lbrace Y+ \\theta \\zeta \\mid 0\\le \\theta \\le 1\\rbrace $ is in $M\\subset I\\!\\!R^k$ and $F\\in C^2(M)$ the Taylor theorem gives the following representation, see [25], [[18] Cor 13 of Lemma 12.6.1]: $F(Y+\\zeta ) - F(Y) = \\left(\\frac{\\partial F(Y)}{\\partial \\zeta } ,\\zeta \\right)+\\int _0^1 ds(1-s)\\left(\\frac{\\partial ^2 F(Y)}{\\partial \\zeta ^2},\\zeta \\otimes \\zeta \\right)\\ .$ Inserting the Taylor expansion of order 2 into (REF ) under the integral for the choice $Y=hx^N$ and $\\zeta = h((e_\\ell +e_{\\ell ^{\\prime }}) - e_\\ell )$ finishes the proof.", "We gladly anticipate that the first term coincides with the generator of the Koopman propagator constructed in Proposition (3.4) for the choice $h={\\tfrac{1}{N}}$ .", "Since the jump type operator $\\mathfrak {\\hat{A}}^N[t, x^N,u]$ is a linear combination of copies of the operator $\\mathbf {A}[t,h x^N]$ all properties previously investigated are conserved, in particular existence and sensitivity results, the fact that $\\mathfrak {\\hat{A}}^N[t, x^N,u]$ generates a strongly continuous contraction propagator $\\psi ^{t,s}_N$ and that there exists a corresponding Markov process.", "We also note that all statements in this section hold for $\\mathbf {A}^$ as well.", "section1.7plus.5Properties of the Non linear Evolution In the sequel we investigate the nonlinear Kinetic equation which was motivated by the weak equation (REF ), namely $\\dot{\\mu _s}=A^[s,\\mu _s,u_s] \\mu _s, \\ \\ \\mu _t=\\mu , \\ \\ s \\in [t,T]\\ ,$ $0 < t < T$ .", "For the finite set $\\mathbb {X}$ the set of real valued functions on $\\mathbb {X}$ , the set of bounded measurable and and the set of bounded continuous functions $C_\\infty (\\mathbb {X})$ coincide and are isomorphic to $I\\!\\!R^k$ .", "Consequently the dual space, the space of bounded measures $\\mathbb {M}(\\mathbb {X})$ , is isomorphic to $I\\!\\!R^k$ .", "When identifying $\\mathbb {M}(\\mathbb {X})$ and $I\\!\\!R^k$ the Kinetic equation, is the following nonlinear differential equation in $I\\!\\!R^k$ : $\\dot{x_s}=A^[s,x_s,u] x_s, \\qquad x_t = x, \\ \\ s \\in [t,T]\\ ,$ $0 < t < T$ .", "Under the conditions of Hypothesis A the subsequent theorem gives the existence of a corresponding flow which is continuously differentiable in all variables, parameters and initial conditions.", "Proposition 0.7 Let $M$ be a subset in $B_1(0)\\subset I\\!\\!R^k$ and $U\\subset I\\!\\!R^k$ a convex bounded open control set.", "Assume that the matrix valued function $\\mathbf {A}^(s,x)$ in (REF ) is continuous in $t > t_0$ for some $t_0\\in I\\!\\!R$ and of type $C^q$ , for $q\\ge 1$ , in the variable $x\\in M$ .", "Then so is the unique nonlinear global flow $\\alpha (t_0,t,x_0,u),\\ t_0\\le t$ , arising from the solution of the Kinetic equation above.", "The unique global flow is defined on the whole space $I\\!\\!R^k$ .", "Since $\\nu $ is bounded and the set of admissible controls $U\\subset I\\!\\!R^k$ is bounded, the vector valued function $\\nu (s,x)ux$ satisfies a uniform Lipschitz condition on $B_1(0)\\subset I\\!\\!R^k$ .", "Hence $\\mathbf {A}^{}(s,x)$ is bounded in $M$ .", "Moreover, the conditions of Theorems 2) and 7) as well Remarks 3) in [24] CH.", "XVIII and Theorem 2.9 in [1] CH 2 apply which finishes the proof.", "Since the differential equation (REF ) satisfies a linear growth condition The unique global flow extends to $I\\!\\!R^k$ .", "This automatically implies that the solution $x_t$ of (REF ) is Lipschitz continuous in the initial conditions: Corollary 0.8 For all $x,\\, y \\in M \\subset I\\!\\!R^k $ the unique solution to equation (REF ) given by Proposition REF is Lipschitz continuous in the initial data i.e: $\\left\|\\alpha (0,t,x,u) - \\alpha (0,t,y,u)\\right\| \\le C(T)\\left\|x- y\\right\|$ where $\\left\|\\cdot \\right\|$ is the Euclidean norm in $I\\!\\!R^k$ .", "We summarize our findings by concluding that the initial value problem (REF ) is well-posed.", "Definition 0.9 Let $\\beta (t,s), \\quad 0\\le t \\le s \\le T,$ be a nonsingular flow in a set $K$ of a given Banach space, then $(\\Phi ^{t,s} F)(b):=F(\\beta (t,s,b)) \\qquad \\beta _t = b\\in K$ defines a linear operator on $C(K)$ which we call the Koopman propagator with respect to $\\beta (t,s)$ .", "The notion coincides with the Koopman operator in [23].", "From the general theory, see [23], it follows that the Koopman propagator has the following properties: $\\Phi ^{t,s}$ is a linear propagator.", "$\\Phi ^{t,s}$ is a contraction on $C(K)$ , i.e.", "$\\left\\Vert \\Phi ^{t,s} F\\right\\Vert < \\left\\Vert F\\right\\Vert _{C(K)} \\qquad \\forall F \\in C(K)$ , $\\Phi ^{t,s}$ is a linear propagator.", "$\\Phi ^{t,s}$ is a contraction on $C(K)$ , i.e.", "$\\left\\Vert \\Phi ^{t,s} F\\right\\Vert < \\left\\Vert F\\right\\Vert _{C(K)} \\qquad \\forall F \\in C(K)$ , where $\\left\\Vert \\cdot \\right\\Vert $ denotes the norm in the Banach space.", "Being a contraction the Koopman propagator is bounded.", "For $M$ as in Proposition REF we introduce the set $C^1 (M)$ of functionals $F=F(x)$ such that the gradient $\\nabla _{x} F$ is continuous.", "This space becomes a Banach space when equipped with the norm $\\left\\Vert F\\right\\Vert _{C^1(M)}:= \\sup _{x \\in K} \\left\|(\\nabla _{x} F)(x)\\right\|.$ Proposition 0.10 Under the conditions given in Hypotheses A and B we have: i) The time inhomogeneous global flow $\\alpha $ corresponding to the solution $x_s$ , $0\\le t < s\\le T$ of the kinetic equation defines the time inhomogeneous Koopman propagator: $(\\phi ^{t,s} F)(x):=F(\\alpha (t,s,x,u)) \\qquad x\\in I\\!\\!R^k,\\ 0\\le t \\le s \\le T.$ ii) The generator of the Koopman propagator is defined by $\\mathcal {A}[t,x,u ]F(x)=\\sum _{i=1}^{k}\\frac{\\partial F}{\\partial x_i} a_i^[t,x,u]x ,$ where $a_i^[t,x]$ corresponds to row $i$ of the matrix valued function $\\mathbf {A}^ [t, x]$ .", "iii) The Koopman propagator constitutes a strongly continuous family of bounded linear operators on $C^1(M)$ .", "The time inhomogeneous global flow $\\alpha $ corresponding to the solution $x_s$ , $0\\le t < s\\le T$ of the kinetic equation defines the time inhomogeneous Koopman propagator: $(\\phi ^{t,s} F)(x):=F(\\alpha (t,s,x,u)) \\qquad x\\in I\\!\\!R^k,\\ 0\\le t \\le s \\le T.$ The generator of the Koopman propagator is defined by $\\mathcal {A}[t,x,u ]F(x)=\\sum _{i=1}^{k}\\frac{\\partial F}{\\partial x_i} a_i^[t,x,u]x ,$ where $a_i^[t,x]$ corresponds to row $i$ of the matrix valued function $\\mathbf {A}^* [t, x]$ .", "The Koopman propagator constitutes a strongly continuous family of bounded linear operators on $C^1(M)$ .", "For the sake of a more comprehensive notation we drop the control parameter $u$ .", "First we shall prove that the global flow induced by the solution to the nonlinear kinetic equation is nonsingular.", "i) Let $x_t = x$ .", "Since $\\mathbf {A}^[s,x,u]x$ is uniformly Lipschitz continuous hence satisfies a linear growth condition the unique global flow $\\alpha $ given by Proposition 3.1 is defined on the whole space.", "Consequenly the flow constitutes a nonsingular transformation.", "ii) Under the assumptions at the beginning of Section 2 $(\\nabla _{x} A^) (x)$ exists and for every $x_0 \\in \\mathbb {R}^k$ the solution $x_s = \\alpha (0,s,x_0)$ exists for all $s\\in [0,T]$ .", "By inserting into the definition we find $\\frac{(\\phi ^{t,s}F)(x)-F(x)}{s-t} = \\frac{F(\\alpha (t,s,x,u))-F(x)}{s-t}= \\frac{F(x_s)-F(x)}{s-t}\\ .$ where $x=x_t$ and $0\\le t\\le s\\le T$ .", "For $F \\in C^1(M)$ with compact support the mean value theorem reveals $\\frac{(\\phi ^{t,s}F)(x)-F(x)}{s-t}&=&\\sum _{i=1}^k F_{x_i}(x_{\\theta }) \\dot{x_{\\theta }}= \\sum _{i=1}^k F_{x_i}(x_{\\theta })a_i^[\\theta ,x,u] x_{\\theta }\\\\&=& \\sum _{i=1}^k F_{x_i}(\\alpha (t,\\theta ,x,u))a_i^[\\theta ,x,u] \\alpha (t,\\theta ,x, u)$ where $t \\le \\theta \\le s$ .", "Since the derivatives $F_{x_i}$ have compact support by introducing the flow $\\alpha (t,s,x_t)$ given by the solution $x_s$ into the kinetic equation (REF ) with $x_t = x$ we obtain for $t\\le \\theta \\le s$ : $\\lim _{(s-t) \\rightarrow 0} F_{x_i}(\\alpha (t,\\theta ,x))\\cdot a_i^[t,x,u]\\alpha (t,\\theta ,x)= F_{x_i}(x)a_i^[t,x,u] x$ uniformly $ \\forall x\\in M$ and $\\forall F \\in C^1(M)$ thus (REF ) has a strong limit in $C^1(M)$ and the infinitesimal generator $\\bf \\mathcal {A}$ is giving by ${\\bf \\mathcal {A}}[t,x,u]F(x)=\\sum _{i=1}^{k}\\frac{\\partial F}{\\partial x_i} a_i^[t, x,u]x\\ .$ iii) To show the strong continuity we insert the definition of the Koopman propagator and exploit the properties of the flow given by Proposition REF , i.e.we have: $\\lim _{(t_0,s_0) \\rightarrow (t,s)} \\left\\Vert \\phi ^{t,s} F - \\phi ^{t_0,s_0} F\\right\\Vert =0$ for every $F \\in C^1(M)$ .", "The fact that the set of continuously differentiable functions with compact support form a dense subset of $C^1(M)$ concludes the proof.", "For the sake of a more comprehensive notation we drop the control parameter $u$ .", "First we shall prove that the global flow induced by the solution to the nonlinear kinetic equation is nonsingular.", "i) Let $x_t = x$ .", "Since $\\mathbf {A}^[s,x,u]x$ is uniformly Lipschitz continuous hence satisfies a linear growth condition the unique global flow $\\alpha $ given by Proposition 3.1 is defined on the whole space.", "Consequenly the flow constitutes a nonsingular transformation.", "ii) Under the assumptions at the beginning of Section 2 $(\\nabla _{x} A^) (x)$ exists and for every $x_0 \\in \\mathbb {R}^k$ the solution $x_s = \\alpha (0,s,x_0)$ exists for all $s\\in [0,T]$ .", "By inserting into the definition we find $\\frac{(\\phi ^{t,s}F)(x)-F(x)}{s-t} = \\frac{F(\\alpha (t,s,x,u))-F(x)}{s-t}= \\frac{F(x_s)-F(x)}{s-t}\\ .$ where $x=x_t$ and $0\\le t\\le s\\le T$ .", "For $F \\in C^1(M)$ with compact support the mean value theorem reveals $\\frac{(\\phi ^{t,s}F)(x)-F(x)}{s-t}&=&\\sum _{i=1}^k F_{x_i}(x_{\\theta }) \\dot{x_{\\theta }}= \\sum _{i=1}^k F_{x_i}(x_{\\theta })a_i^[\\theta ,x,u] x_{\\theta }\\\\&=& \\sum _{i=1}^k F_{x_i}(\\alpha (t,\\theta ,x,u))a_i^[\\theta ,x,u] \\alpha (t,\\theta ,x, u)$ where $t \\le \\theta \\le s$ .", "Since the derivatives $F_{x_i}$ have compact support by introducing the flow $\\alpha (t,s,x_t)$ given by the solution $x_s$ into the kinetic equation (REF ) with $x_t = x$ we obtain for $t\\le \\theta \\le s$ : $\\lim _{(s-t) \\rightarrow 0} F_{x_i}(\\alpha (t,\\theta ,x))\\cdot a_i^[t,x,u]\\alpha (t,\\theta ,x)= F_{x_i}(x)a_i^[t,x,u] x$ uniformly $ \\forall x\\in M$ and $\\forall F \\in C^1(M)$ thus (REF ) has a strong limit in $C^1(M)$ and the infinitesimal generator $\\bf \\mathcal {A}$ is giving by ${\\bf \\mathcal {A}}[t,x,u]F(x)=\\sum _{i=1}^{k}\\frac{\\partial F}{\\partial x_i} a_i^[t, x,u]x\\ .$ iii) To show the strong continuity we insert the definition of the Koopman propagator and exploit the properties of the flow given by Proposition REF , i.e.we have: $\\lim _{(t_0,s_0) \\rightarrow (t,s)} \\left\\Vert \\phi ^{t,s} F - \\phi ^{t_0,s_0} F\\right\\Vert =0$ for every $F \\in C^1(M)$ .", "The fact that the set of continuously differentiable functions with compact support form a dense subset of $C^1(M)$ concludes the proof.", "section1.7plus.5Controlled Jump Markov Process In this subsection we shall describe the principle of dynamic programming and the corresponding HJB equation for the finite state Markov Jump Processes corresponding to the $N$ -mean field dynamics and the Koopman propagator.", "Two types of control problems with game theoretic applications will be covered: One simplified preliminary cost function $J$ which does not depend on an individual player but on the dynamics associated with the $N$ -mean-field respectively Koopman propagator only.", "The other case where an individual player is introduced whose dynamics is given by the operator $\\mathbf {A}$ and who is subject to the $N$ -mean-field respectively mean field and both appear as measure valued parameters $\\mathbf {y}$ respectively $\\mathbf {y}^N$ in the cost function.", "In the first case the cost function $J:[0,T]\\times I\\!\\!R^k\\times U \\rightarrow I\\!\\!R$ defines $\\int _t^T J(s,X^N_s,u_s)\\, ds + {V^{\\prime }}^T(X^N_T) ,$ where ${V^{\\prime }}^T(X^N_T)$ describes a terminal cost and $X^N$ is the Markov process generated by ${\\hat{\\mathfrak {A}}}^N$ in (2.4).", "In the second case the optimal payoff for one player is represented by the value function $V:[0,T]\\times \\mathbb {X}\\times I\\!\\!R^k \\rightarrow I\\!\\!R$ : $V(t,j,y):= \\sup _{\\mathbf {u}\\in \\mathcal {U}}\\mathbb {E}_j \\left[ \\int _t^T J(s,X_s^{1},y_s,u_s)\\, ds + V^T(X_T,y_T)\\right]$ starting at time $t$ and position $j$ .", "The process $X^{1}$ with generator $\\mathbf {A}[s,j,y_s]$ is associated with the dynamics of the player.", "The measure valued parameter $\\mathbf {y}$ is replaced by the mean field respectively the $N$ -mean field in the end.", "In the latter case we use the notation $V^N$ for the value function.", "In order to guarantee a unique optimal control law in the set of Lipschitz continuous functions of the solutions, we confine to quadratic cost functions $J(s,j,y,u) = \\sum _{\\ell = 1}^m J_{j,\\ell } (s,y) u_\\ell - \\left\|u\\right\|^2,$ for $J_{j,\\ell }\\in I\\!\\!R_{+}$ and $s\\in [0,T], j\\in \\mathbb {X}, y\\in M, u\\in U$ see [29].", "We emphasize that the assumptions of Hypotheses A and B hold even for this section.", "The methodology in a standard setting reveals the HJB equation i.e.", "the following system of ordinary differential equations $\\frac{\\partial V}{\\partial t}+\\max _{u}\\left[\\sum _{\\ell = 1}^m J_{j,\\ell }(t,y) u_\\ell - \\left\|u\\right\|^2 + \\mathbf {A}[t,y,u]V\\right]=0\\ .$ Remark 0.11 Due to Hypothesis B, $\\nu _j (t,i,y,u)\\in C_0^1(I\\!\\!R^k)$ in the variable $y$ .", "The derivatives $(\\frac{\\partial }{\\partial y} A[t,y])(x)$ are bounded and continuous.", "For any $t \\in [0,T], \\mathbf {A}[t,y]$ is $C^1_0-$ differentiable with respect to the vector $y$ , also there exists a constant $c_1$ such that: $\\sup _{(t,u)}\\left\|\\frac{\\partial \\mathbf {A}[t,y]}{\\partial y}\\right\|_{\\mathbb {R}^k}\\le c_1\\left\|y\\right\|$ $J(t,y)$ is $C^1-$ differentiable with respect to the vector $y$ , also there exists a constant $c_2$ such that: $\\sup _{(t,u)}\\left\|\\frac{\\partial J[t,y]}{\\partial y}\\right\|_{\\mathbb {R}^k} \\le c_2\\left\|y\\right\|$ For any $t \\in [0,T], \\mathbf {A}[t,y]$ is $C^1_0-$ differentiable with respect to the vector $y$ , also there exists a constant $c_1$ such that: $\\sup _{(t,u)}\\left\|\\frac{\\partial \\mathbf {A}[t,y]}{\\partial y}\\right\|_{\\mathbb {R}^k}\\le c_1\\left\|y\\right\|$ $J(t,y)$ is $C^1-$ differentiable with respect to the vector $y$ , also there exists a constant $c_2$ such that: $\\sup _{(t,u)}\\left\|\\frac{\\partial J[t,y]}{\\partial y}\\right\|_{\\mathbb {R}^k} \\le c_2\\left\|y\\right\|$ Corollary 0.12 Assume that Hypotheses A and B hold.", "Suppose that $\\mathbf {A}[t,y]$ is as in (REF ).", "Then the solution of the ordinary differential equation (REF ) is well posed for all terminal data $V^T \\in \\mathbb {R}^k$ and the solution $V_t$ is of class $C^1$ in the parameters $y$ and $u$ for all $t \\in [0,T]$ .", "Suppose that $J_{j,\\ell } (s,\\alpha ,y)$ is $C^1$ -differentiable with respect to the additional parameter $\\alpha \\in I\\!\\!R$ , then the solution $V_t$ is $C^1$ -differentiable with respect to the parameters $y$ , $u$ , and $\\alpha $ .", "The result is a direct consequence of Proposition REF .", "In order to cover the control problem (4.1) we need to generalize Corollary 4.2 in such a way that the parameter $y$ is replaced by a curve $\\mathbf {y}$ .", "We study smooth dependence of the solution of the HJB equation (REF ) above when replacing the parameter $y\\in I\\!\\!R^k$ by a curve $\\mathbf {y}= y(t)$ , $t\\in [0,T]$ , in $I\\!\\!R^k$ , i.e.", "$\\frac{\\partial V(t,\\mathbf {y})}{\\partial t}+ \\max _{u}\\left[ \\sum _{\\ell = 1}^m J_{j,\\ell } (t,\\mathbf {y}) u_\\ell - \\left\|u\\right\|^2 + \\mathbf {A}[t,j,\\mathbf {y},u] V(t,j)\\right] = 0,$ respectively $H(t,j,V,\\mathbf {y})=\\max _{u}\\left[\\sum _{\\ell = 1}^m J_{j,\\ell } (t,\\mathbf {y}) u_\\ell - \\left\|u\\right\|^2+ \\mathbf {A}[t,j,\\mathbf {y},u] V(t,j)\\right]\\, .$ Let us introduce a curve $\\mathbf {y}$ in the form of a piece of a straight line into the value function.", "For any $(t,j) \\in [0,T]\\times \\mathbb {X}$ and $\\mathbf {y}^1, \\mathbf {y}^2 \\in C([0,T],M)$ we define: $V(t,\\alpha ,j):= V(t,j,\\mathbf {y}^1+\\alpha (\\mathbf {y}^2-\\mathbf {y}^1)) , \\quad \\alpha \\in [0,1].$ Hence the smooth dependence on the solutions of the HJB equation on the functional vector valued parameter $\\mathbf {y}$ reduces to dependence on the real parameter $\\alpha $ .", "If the directional derivative $\\partial _{\\mathbf {y}^2-\\mathbf {y}^1} V$ of $V(t,j,\\mathbf {y})$ exists and is continuous we have: $V(t,j,\\mathbf {y}^2)-V(t,j,\\mathbf {y}^1)=\\int _0^1{\\partial _{\\mathbf {y}^2-\\mathbf {y}^1} V(t,j,\\mathbf {y}^1+\\alpha ((\\mathbf {y}^2-\\mathbf {y}^1)))d\\alpha .}", "\\nonumber $ We adopt the assumptions we made when we studied smooth dependence on the real parameter $y$ .", "Theorem 0.13 Under the previous conditions and Hypotheses A and B we have that for any $\\mathbf {y}\\in C([0,T],I\\!\\!R^k)$ the solution of equation (REF ) is Lipschitz continuous in $\\mathbf {y}$ uniformly, i.e.", "for $\\mathbf {y}^1, \\mathbf {y}^2 \\in C([0,T],I\\!\\!R^k)$ , there exists a constant $K \\ge 0$ such that $\\sup \\limits _{(t,j) \\in [0,T]\\times \\mathbb {X}}\\left\|{V(t,j,\\mathbf {y}^1)-V(t,j,\\mathbf {y}^2)}\\right\| \\le K \\sup _{t \\in [0,T]} \\left\|\\mathbf {y}^1-\\mathbf {y}^2\\right\|.$ For measure valued functions $\\mathbf {y}\\in \\mathbb {M}([0,T])$ the Euklidean $k$ -norm on the right hand side can be replaced by a weak-norm, namely: $\\sup _{(t,j) \\in [0,T]\\times \\mathbb {X}} \\left\|{V(t,j,\\mathbf {y}^1)-V(t,j,\\mathbf {y}^2)}\\right\|\\le K \\left\\Vert \\mathbf {y}^1-\\mathbf {y}^2\\right\\Vert ^{}_2.$ where $\\left\\Vert \\mathbf {y}^1-\\mathbf {y}^2\\right\\Vert ^{}_2:= \\sup _{f \\in C^2([0,T],I\\!\\!R^k)\\vert \\left\\Vert f\\right\\Vert \\le 1}(f,\\mathbf {y}^1-\\mathbf {y}^2)$ .", "The first result follows from Proposition 3.1 where the parameter $\\mathbf {y}$ is chosen from the Banach space of continuous functions with the supremum norm.", "The second result follows since the norm is weaker.", "We point out that $C^2$ is dense in $C$ .", "An analogous result holds for the value function $V$ associated with the Koopman dynamics.", "Moreover, there exists a unique optimal Feedback control law $\\hat{u}$ to the value function $V$ in (4.1).", "Proposition 0.14 Under Hypotheses A and B, given a final payoff $V^{\\prime T}$ the optimal control $\\hat{u}$ defined by the cost function (4.1), is of feedback form $\\hat{u}=\\Gamma (t,\\cdot )$ and is Lipschitz continuous i.e, for any $\\mathbf {\\eta },\\mathbf {x}\\in C_{y}\\left([0,T],I\\!\\!R^k\\right)$ $\\Gamma (t,\\eta _t)-\\Gamma (t,x_t)\\le k_1 \\sup _{s \\in [0,T]}\\left\|\\eta _s - x_s\\right\|_{\\mathbb {R}^k}, \\ \\forall t \\in [0,T],$ A proof by Xu may be found in [30].", "A proof by Xu may be found in [30].", "The line of arguments and results presented above for the control problem with value function $V$ carries over to the modifications considered in this paper.", "We shall not repeat it.", "section1.7plus.5Convergence of N-particle Approximations In Physics and Biology scaling limits and analyzing scaling limits are well established techniques which allow to focus on particular aspects of the system under consideration.", "Scaling empirical measures by a small parameter $h$ in such a way that the measure $h(\\delta _{x_1}+\\ldots + \\delta _{x_N})$ remains finite when the number $N$ of particles or species tends to infinity and the individual contribution becomes negligible allows to treat the ensemble as continuously distributed.", "Scaling $k^{th}$ -order interactions by $h^{k-1}$ reflects the idea that they are more rare than $k-\\ell $ order ones for $1\\le \\ell < k$ and makes them neither negligible nor overwhelming.", "This scaling transforms an arbitrary generator $\\Lambda _k$ of a $k^{th}$ -order interaction into $\\Lambda _k^hF(h \\delta _{\\mathbf {x}})= h^{k-1} \\sum _{I \\subset \\left\\lbrace 1, \\cdots ,n\\right\\rbrace ,\\left\|I\\right\|=k}\\int _{\\mathbb {X}^k}{}\\left[F(h \\delta _{\\mathbf {x}} - h\\delta _{\\mathbf {x}_I}+h\\delta _y)-F(h\\delta _{\\mathbf {x}})\\right] \\times P(x_{\\mathbf {x}_I};dy)$ with positive kernel $P(x_{\\mathbf {x}_I};dy)$ .", "The $N$ -mean field limit is a law of large numbers for the first order interactions given by the $N$ -mean field evolutions.", "For the special case of pure jump type $N$ -mean field evolutions, cf.", "(REF ), we prove weak convergence to the solution of the kinetic equation (REF ) by exploiting properties of the corresponding propagators.", "The procedure consists of introducing the scale $h= \\frac{1}{N}$ and as explained in Section 3 by unifying space, i.e.", "it is pursued by substituting $f(\\mathbf {x})$ by $F(\\frac{1}{\\left\|\\mathbf {x}\\right\|} \\delta _\\mathbf {x})$ where $\\left\|\\mathbf {x}\\right\|$ denotes the length of the vector.", "In this section we adopt the representation on $I\\!\\!R^k$ which was introduced in (2.4).The property exploited in the construction proving the $N$ -mean field limit is: Proposition 0.15 Let $L^i:\\mathbb {R}^k\\rightarrow \\mathbb {R}^k , i=1,2, t\\ge 0,$ be two families of arbitrary bounded matrix valued functions which are continuous in time.", "Assume moreover, that $U^{t,r}_i$ are two linear propagators in $\\mathbb {R}^k$ with $\\left\\Vert U^{t,r}_i\\right\\Vert \\le C_1, i=1,2$ , such that for any $f \\in \\mathbb {R}^k$ the equation $\\frac{d}{ds}U^{t,s}f= U^{t,s}L_s f, \\quad \\frac{d}{ds}U^{s,r} f=-L U^{s,r} f, \\ \\ t\\le s\\le r,$ holds in $\\mathbb {R}^k$ for both pairs $\\left(L^i,U_i\\right)$ .", "Then we have $\\begin{array}{lc}i)&U^{t,r}_2-U^{t,r}_1=\\int ^r_t{U^{t,s}_2(L_s^{2}-L_s^{1})}U^{s,r}_1ds\\\\ii)&\\left\\Vert U^{t,r}_2-U^{t,r}_1\\right\\Vert _{B_1(0)\\rightarrow \\mathbb {R}^k} \\le C_1^2 (r-t) \\sup \\limits _{t\\le s\\le r}\\left\\Vert L_s^{2}-L_s^{1}\\right\\Vert _{B_1(0)\\rightarrow \\mathbb {R}^k}\\ .\\end{array}$ The result is adopted from a well known result on bounded linear operators see e.g.", "[10], and [18].", "The representation is used to derive the subsequent properties.", "The propagator $\\Lambda (t,s,\\cdot )$ generated by the operator $\\mathbf {A}^$ in (REF ) is bounded.", "As in Section 3 let $\\psi _N^{t,s}, t \\le s,$ be the $N$ -mean field propagator generated by (REF ) and assume that $\\phi ^{t,s}$ is the Koopman propagator defined in (REF ).", "Since the linear combination $\\mathfrak {A}^N$ of copies of the bounded operators $\\mathbf {A}^$ in (REF ) is also bounded, Remark REF implies that $\\psi _N^{t,s}$ is bounded.", "In the first step let us compare the Koopman dynamics with the one of the $N$ -mean field.", "Exploiting Proposition REF i) we derive an estimate for the deviation of the propagator $\\psi ^{t,s}_N$ from the Koopman propagator $\\phi ^{t,s}$ .", "We first study the unrealistic case of a common initial condition.", "Let us fix the control parameter and set $x^N = {\\tfrac{1}{N}}\\sum _{\\ell =1}^k n_\\ell e_\\ell $ with $\\sum _{\\ell =1}^k n_\\ell = N$ .", "By construction $\\mathfrak {A}^N$ and $\\psi ^{t,s}_N$ satisfy equation (REF ) then Proposition REF reveals: $\\left[(\\psi _N^{t,s}-\\phi ^{t,s})F\\right](x^N)\\hspace{-2.0pt}=\\hspace{-2.0pt} \\int _t^s \\!\\!", "{\\left[\\psi _N^{t,r}(\\hat{\\mathfrak {A}}^N[r,\\alpha (t,r,x^N)]-{\\bf \\mathcal {A}}[r,\\alpha (t,r,x^N)])\\phi ^{r,s}F\\right](x^N) dr}$ for $F \\in C^2 (M)$ and the flow $\\alpha $ as in Proposition 3.1, independent of the control parameter $u\\in U$ .", "We continue by estimating ${\\sup _{y\\in M}\\left\|\\left[(\\psi _N^{t,s}-\\phi ^{t,s})F\\right]\\!", "(y\\!", ")\\right\|\\!\\le \\!\\!", "\\int _t^s \\!", "\\left\\Vert \\psi _N^{t,r}\\right\\Vert \\sup _{\\genfrac{}{}{0.0pt}{}{y,y^{\\prime } \\in M}{r\\in [0,T]}}\\!\\left\|(\\hat{\\mathfrak {A}}^N[r,y^{\\prime }] -{\\bf \\mathcal {A}}[r,y^{\\prime }]) \\phi ^{r,s} F(y\\!", ")\\right\|\\!", "ds} \\nonumber \\\\&\\le &\\frac{(s-t)}{N}\\left\\Vert \\psi _N^{t,r}\\right\\Vert \\left\\Vert \\hat{\\mathfrak {A}}^N -{\\bf \\mathcal {A}}\\right\\Vert \\left\\Vert \\phi ^{r,s}\\right\\Vert \\left\\Vert F\\right\\Vert _{C^2(M)}\\le \\frac{C(T)}{N} \\left\\Vert F\\right\\Vert _{C^2(M)} \\qquad \\qquad $ for $0 \\le t \\le r \\le s \\le T,$ and Proposition 2.6 was applied in the last step.", "The Sobolev type norm $\\left\\Vert F\\right\\Vert _{C^2(M)}$ combines the supremum norm of $F$ and its second derivative.", "The constant $C(T)$ summarizing the three operator norms and integration with respect to time.", "This estimate will in a further step be applied to estimate the order of convergence in the mean field limit.", "The initial values are chosen to suit the operators and hence differ while $N$ changes.", "In fact, we shall assume that the initial conditions $x_0^N=\\frac{1}{N}\\sum _{\\ell =1}^k \\eta _{0,\\ell } e_{\\ell }\\ \\in I\\!\\!R^k,$ with $\\sum _{\\ell =1}^k \\eta _{0,\\ell } = N$ , of the Kolmogorov equation for the generators $\\hat{\\mathfrak {A}}^N$ converge in $I\\!\\!R^k$ , as $N\\rightarrow \\infty $ , to a vector $x_0 \\in I\\!\\!R^k$ in such a way that $\\left\|x_0^N-x_0\\right\| \\le \\frac{k_1}{N}\\mbox{ with a constant } k_1 \\ge 0\\ .$ Theorem 0.16 Let the assumptions of Hypotheses A, B, and Proposition 5.1 be satisfied and let the initial conditions $ x_0^N \\in I\\!\\!R^k$ be subject to (REF ).", "Assume a fixed control parameter $\\gamma \\in U$ .", "Then the following bounds hold: i) For $t\\in [0,T]$ with arbitrary $T\\ge 0,$ we have $\\left\|(\\psi ^{0,t}_{N,\\gamma } F)(x_0^N)-(\\phi ^{0,t}_\\gamma F)(x_0)\\right\|\\le \\frac{C(T)}{N} \\left(T\\left\\Vert F\\right\\Vert _{C^2 (M)}+k_1\\right)$ with a constant $C(T)$ independent of $\\gamma $ ; ii) For $J(t,x,\\gamma )$ on $[0,T]\\times I\\!\\!R^k\\times U$ there holds $\\left\|{\\int _t^T {\\!\\!J(s, x_{\\gamma ,s}^N, \\gamma )ds}-\\int _t^T {\\!\\!J(s,x_{\\gamma ,s},\\gamma ) ds}}\\right\| \\\\\\le \\frac{C(T)}{N} \\left(T\\left\\Vert J\\right\\Vert ^{}_2 + k_1\\right)$ For $t\\in [0,T]$ with arbitrary $T\\ge 0,$ we have $\\left\|(\\psi ^{0,t}_{N,\\gamma } F)(x_0^N)-(\\phi ^{0,t}_\\gamma F)(x_0)\\right\|\\le \\frac{C(T)}{N} \\left(T\\left\\Vert F\\right\\Vert _{C^2 (M)}+k_1\\right)$ with a constant $C(T)$ independent of $\\gamma $ ; For $J(t,x,\\gamma )$ on $[0,T]\\times I\\!\\!R^k\\times U$ there holds $\\left\|{\\int _t^T {\\!\\!J(s, x_{\\gamma ,s}^N, \\gamma )ds}-\\int _t^T {\\!\\!J(s,x_{\\gamma ,s},\\gamma ) ds}}\\right\| \\\\\\le \\frac{C(T)}{N} \\left(T\\left\\Vert J\\right\\Vert ^{}_2 + k_1\\right)$ where $x_{\\gamma ,t}^N $ is the law of the Markov process specified by the propagator $\\psi _{N,\\gamma }^{0,t}$ and $x_{\\gamma ,s}$ is the law of the Markov process given by the Koopman propagator $\\phi ^{0,t}_\\gamma $ or equivalently $x_{\\gamma ,s}=\\alpha (t,s,x_0,\\gamma )$ .", "The norm $\\left\\Vert J\\right\\Vert ^_2$ was introduced in (REF ).", "i) The chain of inequalities (5.3) holds uniformly for all $\\gamma \\in U$ .", "For $F \\in {C^2 (M)}$ we have $\\left\|(\\psi _{N,\\gamma }^{0,t}F)(x_0^N)\\!-\\!", "(\\phi _\\gamma ^{0,t}F)(x_0)\\right\|\\!\\!\\le \\!\\!", "\\left\|(\\psi _{N,\\gamma }^{0,t}F\\!-\\!\\phi _\\gamma ^{0,t}F)(x_0^N)\\right\|+ \\left\|(\\phi _\\gamma ^{0,t}F)(x_0^N)\\!-\\!", "(\\phi _\\gamma ^{0,t}F)(x_0)\\right\|.$ Estimating the first term by (REF ) and the second one by exploiting the Lipschitz continuity of the nonlinear flow guaranteed by Proposition REF reveals the estimate.", "ii) Let us represent the integral $\\int _t^T {J(s, {\\tfrac{1}{N}}X_{\\gamma ,\\gamma ,s}^N,\\gamma )ds}$ as the limit of Riemannian sums.", "Then the wanted inequality is obtained by applying Theorem 4.3 and part i) term by term and passing to the limit.", "This finishes the proof.", "i) The chain of inequalities (5.3) holds uniformly for all $\\gamma \\in U$ .", "For $F \\in {C^2 (M)}$ we have $\\left\|(\\psi _{N,\\gamma }^{0,t}F)(x_0^N)\\!-\\!", "(\\phi _\\gamma ^{0,t}F)(x_0)\\right\|\\!\\!\\le \\!\\!", "\\left\|(\\psi _{N,\\gamma }^{0,t}F\\!-\\!\\phi _\\gamma ^{0,t}F)(x_0^N)\\right\|+ \\left\|(\\phi _\\gamma ^{0,t}F)(x_0^N)\\!-\\!", "(\\phi _\\gamma ^{0,t}F)(x_0)\\right\|.$ Estimating the first term by (REF ) and the second one by exploiting the Lipschitz continuity of the nonlinear flow guaranteed by Proposition REF reveals the estimate.", "ii) Let us represent the integral $\\int _t^T {J(s, {\\tfrac{1}{N}}X_{\\gamma ,\\gamma ,s}^N,\\gamma )ds}$ as the limit of Riemannian sums.", "Then the wanted inequality is obtained by applying Theorem 4.3 and part i) term by term and passing to the limit.", "This finishes the proof.", "In the next section we shall assume that all agents are following a common strategy $\\gamma (t,j)$ but one player, for instance the first one, who applies a different control $u_{1,t}=\\tilde{\\gamma }(t,j)$ .", "section1.7plus.5Mean field limits as an $\\epsilon $ -Nash equilibrium A strategy portfolio $\\Gamma $ in a game of $N$ agents with payoffs $V_i(\\Gamma ), i=1,...,N,$ is called an $\\epsilon $ -Nash equilibrium if, for each player $i$ and an acceptable individual strategy $u_i$ $V_i(\\Gamma ) \\ge V_i(\\Gamma _{-i},u_i)- \\epsilon ,$ where $(\\Gamma _{-i},u_i)$ denotes the profile obtained from $\\Gamma $ by substituting the strategy of player $i$ with $u_i.$ Consequently we present two couples, consisting of a family of N-mean field games, where one player has a different preference than all others, and a game with the corresponding Koopman dynamics.", "For the sake of a shorter notation we use again $x^n:= {\\tfrac{1}{N}}\\sum _{\\ell }^k \\eta _{\\ell } e_{\\ell }$ .", "In the first model the $N$ -mean field acts as a single player and the differing preference $\\tilde{\\gamma }$ is part of it and in the second model this player is kept separate.", "In the first setting the dynamics of $N$ interacting agents will be generated by the following operator $\\hat{\\mathfrak {A}}^N [t, x^N,\\gamma ,\\tilde{\\gamma }] F( x^N)= \\left[ \\hat{\\mathfrak {A}}^N [t, x^N,\\gamma ] + A^1[t, x^N,\\tilde{\\gamma }]- A^1[t,x^N,\\gamma ]\\right] F( x^N)$ where $\\hat{\\mathfrak {A}}^N $ and $A^1=\\mathbf {A}^$ were defined in (REF ) respectively (REF ), and $F \\in C_\\infty ^2 (I\\!\\!R^k)$ .", "Remark 0.17 Let $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ the $N$ -mean field propagator on $C_\\infty ^1(I\\!\\!R^k)$ generated by $\\hat{\\mathfrak {A}}^N [t, x,\\gamma ,\\tilde{\\gamma }]$ .", "Since $\\mathfrak {A}^N[t, x,\\gamma ,\\tilde{\\gamma }]$ is a linear combination of the linear operator $\\mathbf {A}^$ , $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ possesses the same properties as the propagator $\\Lambda $ which is generated by $\\mathbf {A}^$ , i.e.", "it is linear, for $T$ sufficiently small it is a contraction operator hence bounded, and it is Lipschitz continuous in the initial condition, i.e.", "it possesses the Feller property.", "Theorem 0.18 Suppose Hypotheses A and B hold for the family of Markov jump type operators $\\mathbf {A}^[t,y,\\gamma (t,.", ")]$ with a class of functions $\\gamma : \\mathbb {R}^+ \\times \\mathbb {\\mathbb {X}}\\rightarrow U,$ which are continuous in the first variable and Lipschitz continuous in the second one, let $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ be as above, and let $\\phi ^{0,t}_\\gamma $ the Koopman propagator (REF ).", "Then the following bounds hold: i) Let $t \\in [0,T]$ with any $T \\ge 0$ .", "For $F\\in C^2(M)$ : $\\left\|(\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}F)(x_0^N)-(\\phi ^{0,t}_\\gamma F)(x_0)\\right\|\\le \\frac{C(T)}{N} \\left(T\\left\\Vert F\\right\\Vert _{C^2 (M)} + k_1\\right);$ with a constant $C(T)$ independent on $\\tilde{\\gamma }$ .", "ii) Let $J(t,x,\\gamma )$ be defined on $ [0,T]\\times I\\!\\!R^k\\times U$ .", "Then the following bounds exist: $\\left\|{\\int _t^T {\\!\\!J(s, x_{\\gamma ,\\tilde{\\gamma },s}^N,\\tilde{\\gamma })ds}-\\int _t^T {\\!\\!J(s,x_{\\gamma ,s},\\gamma )ds}}\\right\| \\\\\\le \\frac{C(T)}{N} \\left(T\\left\\Vert J\\right\\Vert ^{}_2 + k_1\\right)$ Let $t \\in [0,T]$ with any $T \\ge 0$ .", "For $F\\in C^2(M)$ : $\\left\|(\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}F)(x_0^N)-(\\phi ^{0,t}_\\gamma F)(x_0)\\right\|\\le \\frac{C(T)}{N} \\left(T\\left\\Vert F\\right\\Vert _{C^2 (M)} + k_1\\right);$ with a constant $C(T)$ independent on $\\tilde{\\gamma }$ .", "Let $J(t,x,\\gamma )$ be defined on $ [0,T]\\times I\\!\\!R^k\\times U$ .", "Then the following bounds exist: $\\left\|{\\int _t^T {\\!\\!J(s, x_{\\gamma ,\\tilde{\\gamma },s}^N,\\tilde{\\gamma })ds}-\\int _t^T {\\!\\!J(s,x_{\\gamma ,s},\\gamma )ds}}\\right\| \\\\\\le \\frac{C(T)}{N} \\left(T\\left\\Vert J\\right\\Vert ^{}_2 + k_1\\right)$ where $x_{\\gamma ,\\tilde{\\gamma },t}^N$ is the law of the process specified by the propagator $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ and $x_{\\gamma , t}$ is the solution of the kinetic equation (REF ) with initial value $x_0$ .", "i) Having applied the representation (REF ) the proof follows along the same lines as the one of Theorem (REF ).", "ii) Let us represent the integral $\\int _t^T {J(s, x_{\\gamma ,\\tilde{\\gamma },s}^N,\\tilde{\\gamma })ds}$ as the limit of Riemannian sums.", "Then the wanted inequality is obtained by applying part i) term by term and passing to the limit.", "i) Having applied the representation (REF ) the proof follows along the same lines as the one of Theorem (REF ).", "ii) Let us represent the integral $\\int _t^T {J(s, x_{\\gamma ,\\tilde{\\gamma },s}^N,\\tilde{\\gamma })ds}$ as the limit of Riemannian sums.", "Then the wanted inequality is obtained by applying part i) term by term and passing to the limit.", "Let us tend to the second setting.", "Let the first player have a differing preference and assume that in this case $J$ depends on the (unscaled) position of a tagged player, her differing strategy, and the empirical mean.", "So we have to look at the process of pairs $(X^{N,1}_{t},x_t^N)$ , which refers to a chosen tagged agent and an overall mass.", "The generator of the pair $(X^{N,1}_{t},X_t^N)$ of processes are defined on the space $C_\\infty ^1 (\\mathbb {X}\\times I\\!\\!R^k)$ and take the form $\\hat{\\mathfrak {A}}^N_{tag}[t,j_1, x^N,\\gamma ,\\tilde{\\gamma }]F(j_1, x^N):= \\left(A^1[t,j_1, x^N,\\bar{\\gamma }] \\right.", "\\\\\\left.", "+\\hat{\\mathfrak {A}}^N[t, x^N,\\gamma ,\\tilde{\\gamma }]\\right) F(j_1, x^N),$ with $\\hat{\\mathfrak {A}}^N[t,j_1, x,\\gamma ,\\tilde{\\gamma }]$ as in (REF ).", "The corresponding propagator will be denoted by, $\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ .", "Remark 0.19 Since $\\hat{\\mathfrak {A}}_{tag}^N$ is a finite sum of copies of the operator $\\mathbf {A}^$ the corresponding propagator $\\xi ^{0,t}_{N,\\gamma ,\\tilde{\\gamma }}$ possesses the same properties as the propagator $\\Lambda $ , generated by $\\mathbf {A}^$ , and $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ i.e.", "it is linear, for $T$ sufficiently small it is a contraction operator hence bounded, and it is Lipschitz continuous in the initial condition, i.e.", "it possesses the Feller property.", "Remark 0.20 Let $\\phi _{\\gamma }^{0,t}$ the propagator generated by the family $\\mathbf {A}^[t,j_1,y,\\tilde{\\gamma }] + Id\\,{\\bf \\mathcal {A}}[t,y,\\gamma ]$ on $C_\\infty ^1 (\\mathbb {X}\\times I\\!\\!R^k)$ .", "Since the operator $\\mathbf {A}^$ is bounded and more regular in the parameters than $\\mathcal {A}$ , the propagator $\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }}$ inherits the properties of the Koopman propagator $\\phi ^{0,t}$ .", "Here we mention in particular that $\\phi _{\\gamma ,\\tilde{\\gamma }}^{0,t}$ is a strongly continuous contraction.", "By inserting (REF ) into the definition and by applying Proposition REF we find: $\\hat{\\mathfrak {A}}^N_{tag}[t,j_1, x^N,\\gamma ,\\tilde{\\gamma }]F(j_1, x^N )= \\left(A^1[t,j_1, x^N,\\tilde{\\gamma }]+ Id{\\bf \\mathcal {A}}[t,y,\\gamma ]\\right)F(j_1, x^N)+O({\\tfrac{1}{N}}x).", "$ For the Kolmogorov equation corresponding to this generator we make the assumptions on the initial conditions that $x_{1,0}^N \\in M$ converges, as $N \\rightarrow \\infty $ to a point $x_{1,0} \\in M$ such that $\\left\|x_{1,0}^N-x_{1,0}\\right\| \\le \\frac{k_2}{N}\\hbox{ with a constant }k_2\\ .$ Theorem 0.21 Under the assumptions of the Theorem REF , let $\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }}$ , $\\xi ^{0,t}_{N,\\gamma ,\\tilde{\\gamma }}$ , and the cost function $J$ be as above.", "Then the following bounds exist for $t \\in [0,T]$ , $T \\ge 0$ : i) For $F\\in C^2 (\\mathbb {X}\\times M)$ we have $ \\left\|(\\xi ^{0,t}_{N,\\gamma ,\\tilde{\\gamma }}F)(x_{1,0}^N,x_0^N)-(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }}F)(x_{1,0},x_0)\\right\|\\le \\frac{C(T)}{N} \\left(t\\left\\Vert F\\right\\Vert _{C^2(\\mathbb {X}\\times M)}+k_1\\right)$ with a constant $C(T)$ not depending on $\\gamma ,\\tilde{\\gamma }$ ; ii) for $J(t,j,y,u)$ on $[0,T]\\times \\mathbb {X}\\times I\\!\\!R^k\\times U$ : $\\begin{split}&\\left\|I\\!\\!E\\left[\\int _t^T J(s,X_{\\tilde{\\gamma },s}^{N,1}, x_{\\gamma ,\\tilde{\\gamma },t}^{N,1},\\tilde{\\gamma }(s))ds-\\int _t^T J(s,X_{\\gamma ,s}^1,x_{\\gamma ,s},\\gamma (s))ds\\right] \\right\| \\\\&\\le \\frac{C(T)}{N} \\left((T+k_2)\\left\\Vert J\\right\\Vert +k_1\\right)\\end{split}$ For $F\\in C^2 (\\mathbb {X}\\times M)$ we have $ \\left\|(\\xi ^{0,t}_{N,\\gamma ,\\tilde{\\gamma }}F)(x_{1,0}^N,x_0^N)-(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }}F)(x_{1,0},x_0)\\right\|\\le \\frac{C(T)}{N} \\left(t\\left\\Vert F\\right\\Vert _{C^2(\\mathbb {X}\\times M)}+k_1\\right)$ with a constant $C(T)$ not depending on $\\gamma ,\\tilde{\\gamma }$ ; for $J(t,j,y,u)$ on $[0,T]\\times \\mathbb {X}\\times I\\!\\!R^k\\times U$ : $\\begin{split}&\\left\|I\\!\\!E\\left[\\int _t^T J(s,X_{\\tilde{\\gamma },s}^{N,1}, x_{\\gamma ,\\tilde{\\gamma },t}^{N,1},\\tilde{\\gamma }(s))ds-\\int _t^T J(s,X_{\\gamma ,s}^1,x_{\\gamma ,s},\\gamma (s))ds\\right] \\right\| \\\\&\\le \\frac{C(T)}{N} \\left((T+k_2)\\left\\Vert J\\right\\Vert +k_1\\right)\\end{split}$ where $\\left\\Vert J\\right\\Vert = \\left\\Vert J\\right\\Vert _{C(\\mathcal {U})} + \\left\\Vert J\\right\\Vert ^{}_2$ .", "The pair $\\left(X_{\\tilde{\\gamma },s}^{N,1}, {\\tfrac{1}{N}}X_{\\gamma ,\\tilde{\\gamma },t}^N\\right)$ is the Markov process specified by the propagator $\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}$ , and the process $X_{\\gamma ,s}^{1}$ is generated by $\\mathbf {A}[t,x_{\\gamma ,s},\\gamma ]$ .", "Here $x_{\\gamma ,s}$ corresponds to the solution to the kinetic equation (REF ) with initial condition $x_0$ .", "The basic idea is to insert definitions and to exploit the properties of the propagators $\\Lambda ^{t,s}$ , $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{t,s}$ , and $\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{t,s}$ summarized in Remarks 6.1 and 6.2 and the processes corresponding to the two first ones.", "i) For $F \\in C_{\\infty }^2 (\\mathbb {X}\\times M)$ , we have $\\begin{split}& \\left\|\\left(\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t} F \\right)(x_{1,0}^N,x_0 ^N)- \\left(\\phi ^{0,t}_{\\gamma \\tilde{\\gamma }} F\\right)(x_{1,0},x_0)\\right\|\\\\& \\qquad \\le \\left\|\\left(\\left(\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}- \\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} \\right) F\\right)(x^N_{1,0}, x_0^N) \\right\|+ \\left\| \\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x^N_{1,0},x_0 ^N)\\right.\\\\& \\qquad \\quad \\left.- \\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x_{1,0},x_0^N)\\right\|+ \\left\|\\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x_{1,0},x_0 ^N)- \\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x_{1,0},x_0) \\right\|.\\end{split}$ We estimate the first term by (REF ) using the operator norm and the second and the third ones by using the assumptions (REF ) and (REF ) respectively.", "This finishes the proof of part i).", "ii) Let us represent both integrals $\\int _t^T {J(s,X_{\\tilde{\\gamma },s}^{N,1}, x_{\\gamma ,\\tilde{\\gamma },t}^N,\\tilde{\\gamma }(s ))ds}$ and $\\int _t^T J(s,X_{\\gamma ,s}^1, x_{\\gamma ,s},\\gamma (s))ds$ as the limits of Riemannian sums.", "Then the result ii) is obtained by proceeding as in the proof of Theorem 5.2 and by applying Theorem 4.3 together with part i) term-by-term and passing to the limit.", "The basic idea is to insert definitions and to exploit the properties of the propagators $\\Lambda ^{t,s}$ , $\\psi _{N,\\gamma ,\\tilde{\\gamma }}^{t,s}$ , and $\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{t,s}$ summarized in Remarks 6.1 and 6.2 and the processes corresponding to the two first ones.", "i) For $F \\in C_{\\infty }^2 (\\mathbb {X}\\times M)$ , we have $\\begin{split}& \\left\|\\left(\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t} F \\right)(x_{1,0}^N,x_0 ^N)- \\left(\\phi ^{0,t}_{\\gamma \\tilde{\\gamma }} F\\right)(x_{1,0},x_0)\\right\|\\\\& \\qquad \\le \\left\|\\left(\\left(\\xi _{N,\\gamma ,\\tilde{\\gamma }}^{0,t}- \\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} \\right) F\\right)(x^N_{1,0}, x_0^N) \\right\|+ \\left\| \\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x^N_{1,0},x_0 ^N)\\right.\\\\& \\qquad \\quad \\left.- \\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x_{1,0},x_0^N)\\right\|+ \\left\|\\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x_{1,0},x_0 ^N)- \\left(\\phi ^{0,t}_{\\gamma ,\\tilde{\\gamma }} F\\right)(x_{1,0},x_0) \\right\|.\\end{split}$ We estimate the first term by (REF ) using the operator norm and the second and the third ones by using the assumptions (REF ) and (REF ) respectively.", "This finishes the proof of part i).", "ii) Let us represent both integrals $\\int _t^T {J(s,X_{\\tilde{\\gamma },s}^{N,1}, x_{\\gamma ,\\tilde{\\gamma },t}^N,\\tilde{\\gamma }(s ))ds}$ and $\\int _t^T J(s,X_{\\gamma ,s}^1, x_{\\gamma ,s},\\gamma (s))ds$ as the limits of Riemannian sums.", "Then the result ii) is obtained by proceeding as in the proof of Theorem 5.2 and by applying Theorem 4.3 together with part i) term-by-term and passing to the limit.", "The results of this and the previous two sections, and Theorem REF in particular are based on a fixed control parameter, depending on time however, and thus hold independently on the MFG methodology.", "Theorem 0.22 Let $ \\lbrace A[t,j, y, u] \\mid t \\ge 0,j\\in \\mathbb {X}, y \\in M, u \\in \\mathcal {U}\\rbrace $ be the family of jump type operators given in (REF ) and $\\mathbf {x}$ be the solution to equation (REF ).", "Assume the following i) The kernel $\\nu (t,j,y,u_t)$ satisfies the Hypotheses A and B; ii) The time-dependent Hamiltonian $H_t$ is of the form (REF ); iii) The terminal function $V^T $ is in $C^1_\\infty (\\mathbb {X}\\times I\\!\\!R^k)$ .", "iv) The initial conditions $x_0^N$ of an $N$ players game converge to $x_0$ in $I\\!\\!R^k$ in a way that (REF ) is satisfied and (REF ) holds.", "Then the strategy profile $u = \\Gamma (t, x^N_{1,0},\\alpha (0,t,x^N_0))$ , defined via HJB (REF ) and (REF ) is an $\\epsilon $ -Nash equilibrium in a $N$ players game, with $\\epsilon = \\frac{C(T)}{N}(\\left\\Vert J\\right\\Vert _{C(\\mathcal {U})} + \\left\\Vert J\\right\\Vert ^{}_2+ \\left\\Vert V^T\\right\\Vert _{C^2_\\infty (\\mathbb {X}\\times I\\!\\!R^k) }+1).$ Due to Assumption ii) the unique solution to the HJB equation admits a unique optimal control parameter given by (REF ).", "The optimal feedback control law $\\Gamma =\\Gamma (t,\\alpha (0,t,x_{1,0},x_0))$ of the game with one player and the mean field with the Koopman dynamics and the cost function (REF ) is applied to all players of the N-mean field possibly without one who applies $\\tilde{\\gamma }$ .", "Then the operator difference in (5.2) at the point $(x^N_{1,0},x^N_0)$ reads $(\\hat{\\mathfrak {A}}^N[t,\\alpha (t,r),\\tilde{\\gamma }, \\Gamma [r,\\alpha (t,r,x_{1,0},x_0))]-{\\bf \\mathcal {A}}[t,\\alpha (t,r),\\Gamma [r,\\alpha (t,r,x_{1,0},x_0)])\\phi ^{r,s}$ for the flow $\\alpha (t,r)=\\alpha (t,r,x^N_{1,0},x^N_0)$ as in Proposition 3.1 with an additional player.", "Since the optimal feedback control $\\Gamma $ depends on the flow $\\alpha $ in a Lipschitz continuous way only, while the operator difference in (5.3) is estimated by the second order term of a Taylor expansion, we uniformly approximate by twice continuously differentiable functions using the Stone Weierstass Theorem.", "The assumptions are satisfied since the evolution is a contraction in a subset of a Euclidean unit ball.", "Alternatively assume that the first player chooses a different strategy $\\tilde{\\gamma }$ .", "The state dynamics of the first player, who is subject to an $N$ -mean field, is described in terms of the process $X^{N,1}$ .", "Let $(x^N_{1,0},x^N_0)$ be the initial condition.", "Then, we have ${\\left\| V^N (0,x^N_{1,0},x_{0}^N,\\Gamma )-V^N (0,x^N_{1,0},x_{0}^N,\\tilde{\\gamma })\\right\|}\\\\&\\le &\\left\| I\\!\\!E_j \\int _0^T \\!\\!", "J (s,X_{\\Gamma ,s}^{N,1},x^N_{\\Gamma ,s},\\Gamma )ds- I\\!\\!E_j \\int _0^T \\!\\!J (s,X^{N,1}_{\\tilde{\\gamma },s},x^N_{\\Gamma _{-1},s},\\tilde{\\gamma })ds\\right\|\\\\&& \\qquad + \\left\|I\\!\\!E\\left[V^T (X_{\\Gamma ,T}^{N,1})\\right] -I\\!\\!E\\left[V^T (X_{\\tilde{\\gamma },T}^{N,1})\\right]\\right\|.$ For the proof the difference is rewritten such that the estimates in Theorems 6.2 and 6.5 can be applied taking into account the regularization mentioned above.", "We find $\\left\|V^1 (0,X_{\\Gamma ,0}^N) -\\!V^1 (0,X_{\\Gamma _{-1}, \\tilde{\\gamma },0} ^N ) \\right\|\\!\\!", "\\le \\!", "\\frac{C(T)}{N}(\\left\\Vert J\\right\\Vert + \\left\\Vert V^T\\right\\Vert + \\!1)$ where $\\left\\Vert J\\right\\Vert $ is the supremum norm in the components.", "It is clear that these estimates hold if we start the game at any time $t \\in [0, T]$ .", "This completes the proof and the construction of the mean-field game in this paper.", "Due to Assumption ii) the unique solution to the HJB equation admits a unique optimal control parameter given by (REF ).", "The optimal feedback control law $\\Gamma =\\Gamma (t,\\alpha (0,t,x_{1,0},x_0))$ of the game with one player and the mean field with the Koopman dynamics and the cost function (REF ) is applied to all players of the N-mean field possibly without one who applies $\\tilde{\\gamma }$ .", "Then the operator difference in (5.2) at the point $(x^N_{1,0},x^N_0)$ reads $(\\hat{\\mathfrak {A}}^N[t,\\alpha (t,r),\\tilde{\\gamma }, \\Gamma [r,\\alpha (t,r,x_{1,0},x_0))]-{\\bf \\mathcal {A}}[t,\\alpha (t,r),\\Gamma [r,\\alpha (t,r,x_{1,0},x_0)])\\phi ^{r,s}$ for the flow $\\alpha (t,r)=\\alpha (t,r,x^N_{1,0},x^N_0)$ as in Proposition 3.1 with an additional player.", "Since the optimal feedback control $\\Gamma $ depends on the flow $\\alpha $ in a Lipschitz continuous way only, while the operator difference in (5.3) is estimated by the second order term of a Taylor expansion, we uniformly approximate by twice continuously differentiable functions using the Stone Weierstass Theorem.", "The assumptions are satisfied since the evolution is a contraction in a subset of a Euclidean unit ball.", "Alternatively assume that the first player chooses a different strategy $\\tilde{\\gamma }$ .", "The state dynamics of the first player, who is subject to an $N$ -mean field, is described in terms of the process $X^{N,1}$ .", "Let $(x^N_{1,0},x^N_0)$ be the initial condition.", "Then, we have ${\\left\| V^N (0,x^N_{1,0},x_{0}^N,\\Gamma )-V^N (0,x^N_{1,0},x_{0}^N,\\tilde{\\gamma })\\right\|}\\\\&\\le &\\left\| I\\!\\!E_j \\int _0^T \\!\\!", "J (s,X_{\\Gamma ,s}^{N,1},x^N_{\\Gamma ,s},\\Gamma )ds- I\\!\\!E_j \\int _0^T \\!\\!J (s,X^{N,1}_{\\tilde{\\gamma },s},x^N_{\\Gamma _{-1},s},\\tilde{\\gamma })ds\\right\|\\\\&& \\qquad + \\left\|I\\!\\!E\\left[V^T (X_{\\Gamma ,T}^{N,1})\\right] -I\\!\\!E\\left[V^T (X_{\\tilde{\\gamma },T}^{N,1})\\right]\\right\|.$ For the proof the difference is rewritten such that the estimates in Theorems 6.2 and 6.5 can be applied taking into account the regularization mentioned above.", "We find $\\left\|V^1 (0,X_{\\Gamma ,0}^N) -\\!V^1 (0,X_{\\Gamma _{-1}, \\tilde{\\gamma },0} ^N ) \\right\|\\!\\!", "\\le \\!", "\\frac{C(T)}{N}(\\left\\Vert J\\right\\Vert + \\left\\Vert V^T\\right\\Vert + \\!1)$ where $\\left\\Vert J\\right\\Vert $ is the supremum norm in the components.", "It is clear that these estimates hold if we start the game at any time $t \\in [0, T]$ .", "This completes the proof and the construction of the mean-field game in this paper.", "Acknowledgements The authors are expressing their deep gratitude to Sergio Albeverio for many years of fruitful collaboration, and for an outstanding course on the physical aspects of mean field theory.", "The PhD students from Linnaeus University are deeply indebted to Andreas Ioannidis for his beautiful, deep course on semi group theory.", "Moreover, we would like to thank Diogo Gomes, Minyi Huang, Andrei Khrennikov and Torsten Linstöm for stimulating discussions.", "The authors from Linneaus university gratefully achknowledge financial support by FTK, Linnaeus University." ] ]	1403.0426
[ [ "Monte Carlo Studies of Identified Two-particle Correlations in p-p and\n Pb-Pb Collisions" ], [ "Abstract Azimuthal particle correlations have been extensively studied in the past at various collider energies in p-p, p-A, and A-A collisions.", "Hadron-correlation measurements in heavy-ion collisions have mainly focused on studies of collective (flow) effects at low-$p_T$ and parton energy loss via jet quenching in the high-$p_T$ regime.", "This was usually done without event-by-event particle identification.", "In this paper, we present two-particle correlations with identified trigger hadrons and identified associated hadrons at mid-rapidity in Monte Carlo generated events.", "The primary purpose of this study was to investigate the effect of quantum number conservation and the flavour balance during parton fragmentation and hadronization.", "The simulated p-p events were generated with PYTHIA 6.4 with the Perugia-0 tune at $\\sqrt{s}=7$ TeV.", "HIJING was used to generate $0-10\\%$ central Pb-Pb events at $\\sqrt{s_{\\rm NN}}=2.76$ TeV.", "We found that the extracted identified associated hadron spectra for charged pion, kaon, and proton show identified trigger-hadron dependent splitting.", "Moreover, the identified trigger-hadron dependent correlation functions vary in different $p_T$ bins, which may show the presence of collective/nuclear effects." ], [ "Introduction", "Charged hadron-hadron correlations have been extensively studied since the ISR and S$p\\bar{p}$ S experiments in the mid '70's [1].", "These measurements have provided a good basis for investigating fragmentation processes.", "Well before jet reconstruction methods were born, these correlation measurements have been applied more recently in relativistic heavy-ion collisions where full jet reconstruction is challenging, especially at low jet energies.", "The focus of these studies was to investigate medium effects such as back-to-back jet suppression and modifications of the fragmentation process.", "Additionally, the discovery of long-range correlation structures in p-p, p-Pb and Pb-Pb collisions at the LHC was based on two-particle correlation measurements [2], [2], [3].", "Various hadronization and hydrodynamical models of the space-time evolution of the hot and dense color medium, which have been employed to describe these phenomena, rely heavily on assumed parton and hadron distributions.", "Nevertheless, the measured hadron spectra and extracted fragmentation functions are integrated distributions which mix up all possible contributions of parton-hadron channels via the convolution of initial-state, QCD and final state effects of high energy collisions [4].", "The only way to decouple the specific hadronization contributions is to test the conservation of quantum numbers by tracing the baryon number and flavor-specific emission patterns of hadrons from the partonic medium.", "This study focuses on the identified-hadron correlations, which is related to the hadronization mechanisms.", "Here, the baryon-to-meson ratio (for light and strange particles) is enhanced by more than a factor of three in Pb-Pb collisions compared to p-p collisions [5].", "Since this unique kinematic range separates the low momentum thermal bulk production from the parton fragmentation dominated regime at high momenta, the expected production mechanism is of particular interest.", "On the other hand, the question of quantum number conservation during the phase transition from the partonic to the hadronic world can be addressed only through correlations beyond the simple baryon-to-meson ratios.", "In this short review we investigate the strength of the expected effects revealed by PID-triggered hadron-hadron correlations in high-energy p-p and Pb-Pb collisions at LHC energies in the mid-rapidity region ($\|\\eta \|<1$ ).", "This region represents the sensitive geometrical areas of the RHIC/LHC detectors, that are common to STAR, PHENIX, ALICE, ATLAS and CMS." ], [ "Monte Carlo samples used for the analysis", "In this study we focused on the generated identified particles: $\\pi ^{\\pm }$ , $K^{\\pm }$ , $p$ , $\\bar{p}$ , and the charged hadrons $(h^\\pm )$ as a reference.", "Event generators PYTHIA 6.4 [6] and HIJING 1.36 [7] were used for the production of p-p and Pb-Pb event samples.", "The Perugia-0 tune [8] was chosen for PYTHIA.", "HIJING event generation included the simulated effects of quenching and shadowing.", "The generated data include 200 million p-p events at 7 TeV and 4 million central ($0-10\\%$ ) Pb-Pb events at $2.76$ $A$ TeV.", "All charged final-state particles were kept for analysis from the Monte Carlo events, nevertheless the trigger particles and associated particles used for the two-particle correlations measurements were limited in pseudorapidity.", "Trigger particles with $\|\\eta _{trig}\|>0.5$ and the associated particles with $\|\\eta _{assoc}\|>1$ were rejected.", "In order to exclude the low-momentum region from the bulk effects, the transverse momenta of the trigger particles were considered above $p_{T,trig}>2$ GeV/c in all the cases.", "The associated particle momenta were chosen such that $p_{T,assoc} < p_{T,trig}$ to avoid double counting.", "The uncorrelated background was subtracted by the ZYAM method [9]." ], [ "The definition of the associated per trigger yield", "Measuring quantum number conservation requires constraints both on the trigger and the associated particle directions to tighten the sensitivity of the measurement.", "The associated per trigger yield is defined with the following quantity as a function of pseudorapidity difference $\\Delta \\eta = \\eta _{trig}-\\eta _{assoc}$ and azimuthal angle difference $\\Delta \\phi = \\phi _{trig}-\\phi _{assoc}$ : $\\frac{\\mathrm {d^{2}}N}{\\mathrm {d}(\\Delta \\eta )\\mathrm {d}(\\Delta \\phi )}(\\Delta \\phi ,\\Delta \\eta ,p_{T,trig},p_{T,assoc}) = \\frac{1}{N_{\\mathrm {trig}}}\\cdot \\frac{\\mathrm {d}N_{\\mathrm {assoc}}}{\\mathrm {d}(\\Delta \\eta )\\mathrm {d}(\\Delta \\phi )}~,$ where $N_{\\mathrm {trig}}$ and $N_{\\mathrm {assoc}}$ is the number of trigger and associated particles, respectively.", "This quantity was measured in several $p_{T,trig}$ and $p_{T,assoc}$ intervals.", "The azimuthal angle correlations in this study are always projected within the pseudorapidity difference $\|\\Delta \\eta \|<1$ .", "The associated per trigger yield on the near-side was extracted from the $\\Delta \\phi $ interval $\|\\Delta \\phi \|<\\pi /2$ and the away-side from the $\\Delta \\phi $ interval $\\pi /2 < \\Delta \\phi < 3\\pi /2$ ." ], [ "The PID-associated spectra", "To test the quantum number conservation with $p_{T,assoc}$ the identified associated spectra of the associated particles (hereafter called PID-associated spectra) have been plotted.", "We have investigated the PID-associated raw spectra using $\\pi ^{\\pm }$ , $K^{\\pm }$ , $p$ , $\\bar{p}$ , and charged hadron $h^{\\pm }$ triggers.", "Figure REF shows the raw (unnormalized) identified particle yields $\\mathrm {d} N^{PID}_{assoc}/ \\mathrm {d} p_{T,assoc}$ for the identified associated particles up to $p_{T,assoc}<25$ GeV/c.", "The PID-associated spectra have been plotted for positively charged pions, as trigger particles, selected from the transverse momentum range 2 GeV/c $ < p_{T,trig} < 25$ GeV/c.", "The upper row (Figure REF $a$ and $b$ ) shows the 7 TeV p-p PYTHIA results and the lower row (Figure REF $c$ and $d$ ) shows the $0-10\\%$ central Pb-Pb collisions at 2.76 $A$ TeV from HIJING.", "The left column shows the spectra extracted from the near-side and the right column shows the spectra extracted from the away-side.", "The PID-associated spectra of the different associated particle species show the expected evolution with the transverse momentum.", "The yields of the PID-associated spectra significantly decrease with the selection of charged pion, kaon and proton triggers.", "Figure: (Color online) Identified associated particle spectra in correlation with positively charged pions (π + \\pi ^{+}).", "Open symbols indicate the positive particles and the filled symbols indicate the negative particles.", "The upper row (aa and bb) shows the PYTHIA simulation in p-p and the lower row (cc and dd) shows the 0-10%0-10\\% central Pb-Pb HIJING events.", "The left column corresponds to the near side (π/2<\|Δφ\|\\pi /2 < \|\\Delta \\phi \|) correlations and the right column corresponds to the away-side (π/2<Δφ<3π/2\\pi /2 < \\Delta \\phi < 3\\pi /2) correlations." ], [ "Identified particle ratios", "At the Monte Carlo event generator level, the basic conservation laws of the quantum numbers – such as charge ($C$ ), baryon number ($B$ ), and strangeness ($S$ ) – are fulfilled and reflected in the PID-associated spectra after the hadronization process.", "To extract and enhance the expected quantum number conservation effects, we use the following ratio: $\\frac{\\mathrm {d}N^{PID}_{assoc}}{\\mathrm {d}p_{T,assoc}}~\\Big /~\\frac{\\mathrm {d}N^{hadron}_{assoc}}{\\mathrm {d}p_{T,assoc}}~$ which is the PID-triggered to charged hadron-triggered ratio of the PID-associated particle spectra, that is the case when we plot the associated spectra having an identified trigger compared to the one when we have a unidentified (charged hadron) trigger.", "Furthermore, it can be understood as a specific normalization to the hadron-triggered associated spectra.", "Figure: (Color online) PID-triggered associated particle yields relative to the charged hadron-triggered associated yields.", "Open symbols indicate the positive particles and the filled symbols indicate the negative particles.", "The three different cases for the trigger/associated particle species are the unlike-sign pairs of π + /π - \\pi ^{+}/\\pi ^{-}, K + /K - K^{+}/K^{-} and p/p ¯p/\\bar{p}.", "The upper row (aa and bb) shows the PYTHIA simulation in p-p and the lower row (cc and dd) shows the 0-10%0-10\\% central Pb-Pb HIJING events.", "The left column corresponds to the near side (π/2<\|Δφ\|\\pi /2 < \|\\Delta \\phi \|) correlations and the right column corresponds to the away-side (π/2<Δφ<3π/2\\pi /2 < \\Delta \\phi < 3\\pi /2) correlations.In Figure REF the ratios for six identified trigger / identified associated particle species are shown: $\\pi ^{+}/\\pi ^{+}$ , $\\pi ^{+}/\\pi ^{-}$ , $K^{+}/K^{+}$ , $K^{+}/K^{-}$ , $p/p$ , and $p/\\bar{p}$ .", "The Fig.", "REF $a$ shows the near-side ratios in p-p events.", "A visible splitting effect is observed for all the trigger species starting from the first $p_{T,assoc}$ bin up to $10-12$ GeV/c.", "The largest splitting is observed in the unlike-sign correlation, $p/\\bar{p}$ , where there is approximately 10-times more anti-protons than protons above a specific $p_{T,assoc}$ bin, that is the splitting strength is about an order of magnitude larger than the like-sign correlation $p/p$ .", "The splitting, in case of unlike-sign pairs, for $K^{+}/K^{-}$ is weaker than in the $p/\\bar{p}$ but still comparable with it but the $\\pi ^{+}/\\pi ^{-}$ shows very weak splitting.", "In contrast, the away-side p-p ratios, in Fig.", "REF $b$ , show an equal ratio for the pions and protons (compared to the near-side in Fig.", "REF $a$ ).", "Conversely, kaons show a weak splitting in a limited transverse momentum range with respect to the near-side.", "The baryon number and charge is conserved and leads to highly correlated distributions in the same cone, since the fragmentation process was not modified by the media.", "These qualitative observations show that the strength of the quantum number conservation effects is increasing in the order of charge ($C$ ), strangeness ($S$ ) and baryon number ($B$ ) respectively.", "In case of hadron correlations with multiple quantum number conservation ($C$ , $B$ , $S$ , ... etc.)", "this effect will be stronger.", "Fig.", "REF $c$ and $d$ exhibit a peculiar pattern for the case of HIJING on the near- and away-side in the $p_{T,assoc}=2-8$ GeV/c range for charged kaons and protons, where the baryon/meson anomaly was observed at RHIC and at the LHC [10], followed by the reversal of the splitting trend at higher transverse momenta.", "The charged pions show the splitting pattern only on the near-side.", "The lack of the splitting effect below 2 GeV/c may be attributed to the fact that in HIJING the embedded mini-jet production starts to appear around this scale.", "On the away-side (Fig.", "REF $d$ ), it can be seen that the effect is opposite to the PYTHIA (Fig.", "REF $b$ ), there are more like-sign particles coming from that region.", "Although it is much smaller than in the near-side, but not negligible.", "Moreover, we believe that the observed splitting of the protons and anti-protons presumably related to the embedded parametrization of baryon number and charge in HIJING.", "Further studies are ongoing to characterize better the observed splitting effects that provide an interesting contrast point between various MC event generators and experimental results from the LHC." ], [ "Identified particle ratios in different trigger $p_T$ bins", "In the previous subsection the associated particle spectra were studied in the case that the trigger particles have $p_{T}$ in a wide range (2 GeV/c $ < p_{T,trig} < 25$ GeV/c).", "In this section we investigate the $p_{T,trig}$ dependence of the spectra.", "We plot the differences between the like-sign and unlike-sign particle pairs for the normalized yields $(\\mathrm {d}N^{PID}_{assoc}/\\mathrm {d}p_{T,assoc})/(\\mathrm {d}N^{hadron}_{assoc}/\\mathrm {d}p_{T,assoc})$ as plotted in Fig.", "REF .", "Since positively charged triggers were choosen we came to the definition below in order to plot these ratios higher than one: Figure: (Color online) Yield differences between the oppositely charged trigger/associated particle pairs.", "The yield enhancement is relative to the positively charged trigger particles.", "The upper row (aa and bb) shows the PYTHIA simulation in p-p and the lower row (cc and dd) shows the 0-10%0-10\\% central Pb-Pb HIJING events.", "The left and right panels correspond to the near- and away-side cases, respectively.$\\scalebox {1.75}{\\frac{\\left(\\frac{\\mathrm {d}N^{PID}_{assoc}}{\\mathrm {d}p_{T,assoc}}~\\Big /~\\frac{\\mathrm {d}N^{hadron}_{assoc}}{\\mathrm {d}p_{T,assoc}}\\right)^{(+/-)}}{\\left(\\frac{\\mathrm {d}N^{PID}_{assoc}}{\\mathrm {d}p_{T,assoc}}~\\Big /~\\frac{\\mathrm {d}N^{hadron}_{assoc}}{\\mathrm {d}p_{T,assoc}}\\right)^{(+/+)}} = \\frac{\\left(\\frac{\\mathrm {d}N^{PID}_{assoc}}{\\mathrm {d}p_{T,assoc}}\\right)^{(+/-)}}{\\left(\\frac{\\mathrm {d}N^{PID}_{assoc}}{\\mathrm {d}p_{T,assoc}}\\right)^{(+/+)}}}$ where $(+/-)$ and $(+/+)$ denote the unlike-sign and like-sign trigger/associated particle pairs, respectively.", "This ratio gives a more detailed insight into the evolution of the particle yield differences between the positive and negative associated particles as a function of $p_{T,assoc}$ .", "The ratios defined by Eq.", "(REF ) are plotted for four different $p_{T,trig}$ trigger bins on the horizontal axis of Fig.", "REF .", "In Fig.", "REF $a$ it can be observed how the splits – in agreement with Fig.", "REF – evolve as a function of $p_{T,assoc}$ in the near-side cone in PYTHIA events.", "In general, the splitting effect is the largest and growing for the case when we look at the differences between the $p$ and $\\bar{p}$ .", "Concerning strange particles it seems to saturate at higher $p_{T,assoc}$ .", "For pions the relative difference is about factor of $2-3$ at most.", "There is almost no change in the ordering of the species in the other $p_{T,trig}$ bins, $3-7$ GeV/c, $7-10$ GeV/c, and $10-20$ GeV/c, and the shapes remain at the same level.", "Concerning the away-side, in Fig.", "REF $b$ , pions and protons with opposite charges are generated with the same yield but there can be a marginal difference for the kaons as it was shown in Fig.", "REF $b$ .", "For HIJING-generated Pb-Pb events in the $0-10\\%$ central region, Fig.", "REF $c$ and $d$ , the kaons and the protons show an reverse trend with respect to the PYTHIA events.", "In the near-side case, the spectrum is different for the first $p_{T,assoc}$ bins in all cases for the trigger $p_{T,trig}$ bins and after that rises above unity.", "The number of $K$ 's produced (compared to $\\pi $ 's) is proportional to $p_{T,assoc}$ .", "This trend is unexpected and completely opposite to the PYTHIA on the away-side for all the $p_{T,trig}$ bins.", "Moreover, the effect manifests with similar magnitude within any $p_{T,trig}$ ranges.", "In summary, these effects are observed in the central Pb-Pb collisions within the $3-10$ GeV/c transverse momentum range, where the Cronin effect [11] and the baryon/meson anomaly act.", "Conversly these effects are not seen in the bulk, as indicated by the plot." ], [ "Testing conservation laws by $\\Delta \\phi $ correlations", "In Fig.", "REF the $\\Delta \\phi $ -distributions given by Eq.", "(REF ) of identified two-particle correlations are shown for unlike-sign trigger/associated particle pairs in different $p_{T,trig}$ , $p_{T,assoc}$ bins extracted from PYTHIA events.", "Recall that the spectra, mentioned previously, are baseline-subtracted by applying the ZYAM-method.", "The columns refer to the different species of the trigger/associated particle pairs, going from left to right we plotted $\\pi ^{+}/\\pi ^{-}$ , $K^{+}/K^{-}$ and $p/\\bar{p}$ .", "The rows correspond to the different settings of the $p_{T,trig}$ bins, going from top to bottom they are the following: 3 GeV/c $ < p_{T,trig} < 7$ GeV/c, 7 GeV/c $ < p_{T,trig} < 10$ GeV/c, and 10 GeV/c $ < p_{T,trig} < 20$ GeV/c.", "Figure: (Color online) PYTHIA simulation of the p-p collisions at s=7\\sqrt{s}=7 TeV.", "The associated per trigger yields are plotted for unlike-sign trigger/associated particle pairs: π + /π - \\pi ^{+}/\\pi ^{-}, K + /K - K^{+}/K^{-} and p/p ¯p/\\bar{p} in different p T,trig p_{T,trig} – p T,assoc p_{T,assoc} bins.", "The extracted distributions are integrated in the pseudorapidity difference range \|Δη\|<1\|\\Delta \\eta \|<1.", "The different columns and rows correspond to different species of trigger/associated particle pairs and trigger momentum bins p T,trig p_{T,trig}, respectively.", "The insets correspond to magnified portion of the y-axis of the plotted distribution for better visibility.The near- and away-side peaks show the expected evolution of $p_{T,assoc}$ .", "Higher trigger momentum $p_{T,trig}$ yields a narrower near-side peak, whereas higher associated momentum particles $p_{T,assoc}$ result in a lower associated particle yield.", "Nevertheless, the amplitude of the correlation peak stays approximately constant on the away-side as a function of $p_{T,trig}$ .", "If we compare the columns, that is the different trigger/associated particle pairs, we can observe that on the near-side, the ratio of the number of the associated particles to the trigger particles is less for strange particles with respect to the baryons compared to the pions.", "There is a hint that the same effect is visible on the away-side, but a more thorough study is needed.", "In Fig.", "REF the distributions for the Pb-Pb events, generated by HIJING, show the same trend that is seen on the near-side for the PYTHIA events.", "The difference between PYTHIA and HIJING in terms of the chosen trigger/assoc-iated species is evident, practically there is only a small flavour and charge correlation effect in HIJING at all studied $p_{T,trig}$ momentum bin.", "Moreover, the number of associated particles for kaons and protons has the same order of magnitude on the near-side as well as on the away-side.", "Figure: (Color online) HIJING simulation of Pb-Pb collisions at s NN =2.76\\sqrt{s_{\\rm NN}}=2.76 TeV.", "The associated per trigger yields are plotted for unlike-sign trigger/associated particle pairs: π + /π - \\pi ^{+}/\\pi ^{-}, K + /K - K^{+}/K^{-} and p/p ¯p/\\bar{p} in different p T,trig p_{T,trig}–p T,assoc p_{T,assoc} bins.", "The extracted distributions are integrated in the pseudorapidity difference range \|Δη\|<1\|\\Delta \\eta \|<1.", "The different columns and rows correspond to different species of trigger/associated particle pairs and p T,trig p_{T,trig} trigger momentum bins, respectively.", "The insets correspond to magnified portion of the y-axis of the plotted distribution for better visibility." ], [ "Summary and Outlook", "Monte Carlo studies of two-particle azimuthal correlations have been performed with identified trigger particles and identified associated particles in p-p collisions at $\\sqrt{s} = 7$ TeV and in Pb-Pb collisions at $\\sqrt{s_{\\rm NN}} = 2.76~A$ TeV at mid-rapidity.", "As for the first verification of the generated samples the identified hadrons were extracted from the raw associated particle spectra: $\\pi ^{+}$ , $\\pi ^{-}$ , $K^{+}$ , $K^{-}$ , $p$ and $\\bar{p}$ with fixed positively charged pion ($\\pi ^{+}$ ) trigger in two separated $\\Delta \\phi $ regions, near- and away-side.", "Further investigation was needed to look deeper into the validity of the quantum number conservations of correlated particle pairs.", "For this reason we plotted the identified associated spectra for like-sign and unlike-sign trigger/associated particle pairs compared to the charged hadron identified associated spectra.", "We obtained conclusive results for PYTHIA: the PID-triggered-to-hadron-triggered associated spectra on the near-side show a significant splitting between the particles and anti-particles which have the largest effect for the baryon triggers, but is absent on the away-side.", "Qualitatively the difference between the baryon triggers and the pions is about one order of magnitude.", "For the HIJING simulation the results show an opposite trend to the PYTHIA up to $p_{T,assoc} = 7-10$ GeV/c and these effects are not vanishing on the away-side except for the pions.", "The splitting in Pb-Pb collisions starts at slightly larger $p_{T,assoc}$ threshold than in p-p because of the embedded physics in HIJING (for example the minijet production), but further investigation is needed with real data to confirm this observation.", "To conclude: collective/nuclear effects are visible in the studied $3-10$ GeV/c momentum range.", "There is no significant change between the yields of the unlike-sign particle pairs as we change the trigger $p_{T,trig}$ bins.", "Finally, we showed the PID-triggered azimuthal $\\Delta \\phi $ correlation spectra for the different trigger species.", "We have seen more associated protons than kaons on the near-side of the p-p collisions which does not depend on the applied trigger momentum $p_{T,trig}$ cuts.", "In Pb-Pb collisions the probability to generate protons or kaons as associated particles on the near-side is about equal." ], [ "Acknowledgement", "This work was supported by Hungarian OTKA grants NK778816, NK106119, K104260, and TET 12 CN-1-2012-0016.", "Author Gergely Gábor Barnaföldi also thanks the János Bolyai Research Scholarship of the Hungarian Academy of Sciences." ] ]	1403.0117
[ [ "Improper Ferroelectricity and Piezoelectric Responses in Rhombohedral\n ($A$,$A^{\\prime}$)$B_2$O$_6$ Perovskite Oxides" ], [ "Abstract High-temperature electronic materials are in constant demand as the required operational range for various industries increases.", "Here we design $(A,A^\\prime)B_2$O$_6$ perovskite oxides with [111] ``rock salt\" $A$-site cation order and predict them to be potential high-temperature piezoelectric materials.", "By selecting bulk perovskites which have a tendency towards only out-of-phase $B$O$_6$ rotations, we avoid possible staggered ferroelectric to paraelectric phase transitions while also retaining non-centrosymmetric crystal structures necessary for ferro- and piezoelectricity.", "Using density functional theory calculations, we show that (La,Pr)Al$_2$O$_6$ and (Ce,Pr)Al$_2$O$_6$ display spontaneous polarizations in their polar ground state structures; we also compute the dielectric and piezoelectric constants for each phase.", "Additionally, we predict the critical phase transition temperatures for each material from first-principles to demonstrate that the piezoelectric responses, which are comparable to traditional lead-free piezoelectrics, should persist to high temperature.", "These features make the rock salt $A$-site ordered aluminates candidates for high-temperature sensors, actuators, or other electronic devices." ], [ "Introduction", "Ferroelectric materials are of great scientific and technological importance, especially for use in electronic devices such as non-volatile memory [1], [2], tunable capacitors [3], [4], and tunnel junctions.", "[5] The fact that there are so few mechanisms capable of generating spontaneous polarizations with high Curie temperatures in crystalline systems, however, suggests that there are limited opportunities to improve upon existing devices.", "Understanding how ferroelectricity arises, as well as how to control and purposefully induce it in novel ways, is therefore critical to engineering new technologies based on ferroic transitions.", "The family of $AB$ O$_3$ perovskite oxides is ideal for investigations of this nature, as the crystal family can accommodate a wide range of chemistries and exhibits strong electron-lattice coupling, which allow for highly tunable ground states through atomic substitution and strain.", "[6], [7], [8] Typically, ferroelectricity in transition metal perovskite oxides is realized by having a $d^0$ $B$ -site cation or lone-pair active $A$ -site cation, [9], [10] which undergo collective polar displacements through the so-called second order Jahn-Teller effect [11], [12].", "An alternative strategy receiving considerable interest lately relies on an improper “geometric route,” whereby a ferroelectric polarization is produced as a by-product of some complex lattice distortion(s), as opposed to being directly driven by inversion lifting displacements originating in the chemistries of the cations.", "[13], [14] Electric polarizations can arise in this manner through, e.g., rotations of corner-connected $B$ O$_6$ octahedra that are coupled to and induce polar displacements of the $A$ and $B$ cations in perovskites.", "If the compound is ordered as ($A$ ,$A^\\prime $ )$B_2$ O$_6$ , then two chemically distinct $A$ -sites will displace by different amounts due to a size differential, which leads to an observed electric dipole.", "[15] Previous efforts have defined the octahedral rotational patterns which, when combined with alternating $A$ and $A^{\\prime }$ atoms along the [001] crystallographic axis, lift the symmetries preventing spontaneous polarizations.", "In the case of these `layered' superlattices, only a combination of in-phase (+) and out-of-phase (–) $B$ O$_6$ octahedral rotations ($a^+b^-b^-$ in Glazer notation) satisfies the required criteria put forth.", "[16] Energetically, this occurs because the modes describing the in-phase and out-of-phase rotations (identified by the irreducible representations $M_3^+$ and $R_4^+$ of $Pm\\bar{3}m$ , respectively) couple to a polar mode,[17] resulting in an anharmonic term in the free energy expansion that ultimately stabilizes the polar structure.", "Materials with spontaneous polarizations resulting from this coupling of two zone-boundary modes are known as “hybrid improper\" ferroelectrics.", "[18] The fact that [001]-ordering is conducive to layer-by-layer growth of double perovskites through pulsed laser deposition or molecular beam epitaxy,[19] and that the required $a^+b^-b^-$ tilt pattern is one of the most commonly adopted by perovskite oxides, make these compounds attractive for investigation.", "However, the fact that the polarization arises from a trilinear coupling leads to an often overlooked problem: mode condensation.", "There is the real possibility that the two rotational modes condense at different temperatures, leading to a staggered paraelectric to ferroelectric phase transition.", "Although an avalanche transition (in which the modes condense at the same temperature) is possible in ferroelectric oxides [20], non-avalanche transitions are also likely to occur.", "[21] This makes it difficult to predict whether or not there will be two phase transitions from the paraelectric to ferroelectric state, i.e., a cell-doubling antidistortive transition, owing to the rotations, followed by an inversion lifting one, in the material without detailed and careful experimental study.", "However, it is possible to avoid this problem if the spontaneous polarization is induced by only a single rotational mode, making such materials improper rather than hybrid-improper ferroelectrics.", "Materials of this nature could find use in high-temperature electronic applications such as sensors.", "[22] Our previous work has shown that although both in-phase and out-of-phase rotations are required to lift inversion symmetry in [001]-ordered double perovskites, only out-of-phase rotations, which are described by a single mode, are needed if the $A$ -sites are ordered along the [111] direction (the difference in cation ordering is shown in fig:orderings).", "[23] In this work, we examine the feasibility of this symmetry result using density functional theory calculations, and subsequently design a series of [111]-ordered $(A,A^{\\prime })B_2$ O$_6$ double perovskites exhibiting only out-of-phase rotations with modest electronic polarizations.", "We also investigate their piezoelectric response and estimate the phase transition temperatures based on first-principles total energy differences.", "We find that while two compounds, (La,Pr)Al$_2$ O$_6$ and (Ce,Pr)Al$_2$ O$_6$ , exhibit spontaneous polarizations in their polar ground state, they undergo a transition to a chiral phase at low temperatures accompanied by a disappearance of the polarization; albeit, they then exhibit an order of magnitude increase in the piezoelectric response.", "These materials then transform to a centrosymmetric phase at 1300 K. While the third compound, (La,Nd)Al$_2$ O$_6$ , does not have a polar ground state, we predict it remains non-centrosymmetric to even higher temperatures.", "Figure: An (AA,A ' A^{\\prime })B 2 B_2O 6 _6 superlattice ordered along the (a) [001]-direction (`layered') and (b) [111]-direction (`rock salt').", "AA and A ' A^\\prime atoms are in green and orange, respectively, while BBO 6 _6 octahedra are given in blue." ], [ "Computational Methods", "All investigations were performed using density functional theory[24] as implemented in the Vienna ab-initio Simulation Package (VASP).", "[25], [26] We used projector augmented-wave (PAW) potentials[27] with the PBEsol functional[28], with the $5s^25p^65d^16s^2$ valence electron configuration for La , $5s^25p^64f^16s^2$ for Nd, $5s^25p^66s^25d^1$ for Pr, $5s^25p^65d^16s^2$ for Ce, $3s^23p^1$ for Al, and $2s^22p^6$ for O.", "A plane-wave cutoff of 550 eV and a 6$\\times $ 6$\\times $ 6 Monkhorst-Pack mesh[29] were used during the structural relaxations.", "The phonon band structure of each compound was computed using density functional perturbation theory[30] with an increased plane-wave cutoff of 800 eV.", "The ground states were determined by performing a full structural relaxation on 15 candidate structures, which were generated using linear combinations of unstable phonons of each compound.", "The electric polarization was calculated using the Berry phase method[31], [32] as implemented in VASP.", "Finally, the piezoelectric and elastic tensors were computed within density functional perturbation theory.", "[33], [34] We note that DFT calculations on Ce-based compound often treat the $4f$ electron explicitly, while also including a Hubbard-$U$ correction of 6-10 eV on those states.", "Here we use a Ce PAW which does not include the $4f$ states, instead placing these electrons in the core to eliminate any adjustable $U$ parameters in our calculations.", "We evaluated this approximation by performing a structural relaxation of bulk CeAlO$_3$ , treating the $f$ -electron both in the core and as valence electrons with $U=0$ -10 eV in increments of 1 eV.", "The relaxed lattice parameters and rotation angles of CeAlO$_3$ with the $f$ -electrons in the core and treated explicitly as valence electrons with $U=8$ both agree with the experimental data to within 1%, showing that there is essentially no error if the $f$ -electrons are not included.", "Owing to the structural agreement, all presented results use the former PAW." ], [ "Bulk Rare Earth Aluminate Ground States", "We first determined the ground state structures of four non-polar aluminate dielectrics which contain only out-of-phase AlO$_6$ octahedral rotations: LaAlO$_3$ , NdAlO$_3$ , PrAlO$_3$ , and CeAlO$_3$ .", "The computationally and experimentally determined space group, lattice constants, and octahedral rotation angles and tilt patterns are summarized in tab:bulk.", "We find that LaAlO$_3$ and NdAlO$_3$ both exhibit the $a^-a^-a^-$ tilt pattern (space group $R\\bar{3}c$ ), in good agreement with experimental results.", "[35], [36] Next, we find that PrAlO$_3$ exhibits the $a^0b^-b^-$ tilt pattern (space group $Imma$ ) in its ground state.", "Although this compound is found to exhibit space group $C2/m$ with an $a^0b^-c^-$ tilt pattern at low temperatures experimentally, we find this phase is 10 meV per formula unit (f.u.)", "higher in energy than the $Imma$ phase; however, PrAlO$_3$ is known to undergo a continuous phase transition to $Imma$ at 150 K. [37] Finally, we find that CeAlO$_3$ displays the $a^0a^0c^-$ tilt pattern (space group $I4/mcm$ ), which is consistent with the experimentally known low-temperature structure.", "[38] Table: Structural details of the bulk aluminate superlattice constituents.", "Each has a centrosymmetric space group (S.G.) and only out-of-phase rotations about one-, two-, or three-axes; the magnitude of which is given by Θ\\Theta and determined by measuring the Al-O-Al bond angle θ\\theta , Θ=(180 ∘ -θ)/2\\Theta = (180^{\\circ }-\\theta )/2.", "The tolerance factor (τ\\tau )is computed using bond lengths obtained from the bond valence model.", "The calculated lattice parameters (in Å) and rotational angle (in degrees) are compared to experimental (Exp.)", "data (cf.", "text for references)." ], [ "$A$ -Cation Ordered Ground States Structures", "LaAlO$_3$ is then ordered with both NdAlO$_3$ and PrAlO$_3$ along the cubic [111] direction, Note that throughout we choose to write the stoichiometric formula in a manner that emphasizes each compounds similarity with double perovskites, but each phase may be equivalently written as an ultra-short period 1/1, for example, (La,Nd)Al$_2$ O$_6$ is equivalent to an (LaAlO$_3$ )$_1$ /(NdAlO$_3$ )$_1$ .", "resulting in two rock salt double perovskites: (La,Nd)Al$_2$ O$_6$ and (La,Pr)Al$_2$ O$_6$ .", "CeAlO$_3$ and PrAlO$_3$ were ordered in the same way resulting in a third superlattice: (Ce,Pr)Al$_2$ O$_6$ .", "By employing a symmetry-restricted soft-phonon search, we determined the ground state structures of each compound, which are summarized in tab:gs.", "All three ordered aluminates contain only out-of-phase rotations [fig:LPAOenergy(a)], which in combination with the [111]-cation ordering, results in non-centrosymmetric (chiral or polar) space groups.", "However, only (La,Pr)Al$_2$ O$_6$ and (Ce,Pr)Al$_2$ O$_6$ display a spontaneous polarization, as they are polar ($Imm2$ ), where as (La,Nd)Al$_2$ O$_6$ is chiral and non-polar.", "We now carry out a detailed examination of each compounds' atomic structure to understand the origin for the non-centrosymmetric symmetries, the different space groups adopted, as well as the microscopic mechanism for the emergence of an electric polarization from the combination of two bulk non-polar dielectrics.", "Table: Structural details and ferroelectric properties of the [111]-ordered perovskite aluminates.", "All compounds exhibit out-of-phase rotations in their ground states, with the rotation magnitude and electric polarization specified by Θ\\Theta (degrees) and 𝒫\\mathcal {P} (μ\\mu C/cm 2 ^2), respectively.Figure: (a) The atomic displacements constituting the out-of-phase octahedral rotations described by the R 4 + R_4^+ irreducible representation in (La,Pr)Al 2 _2O 6 _6 and (Ce,Pr)Al 2 _2O 6 _6.", "(b) Energy as a function of atomic displacements (inset) described by the R 5 + R_5^+ (upper) and R 4 + R_4^+ (lower) modes in (La,Pr)Al 2 _2O 6 _6.", "(c) The coupling of R 4 + R_4^+ with R 5 + R_5^+ results in a cooperative lowering of the total energy, stabilizing the non-centrosymmetric structure (grey lines obtained from a fit of the Landau potential, see text).", "Note that there exists an equivalent energy minimum at R 5 + ≃-0.1R_5^+\\simeq -0.1 for R 4 + R_4^+ imposed with opposite sense (negative mode amplitude)." ], [ "Chiral (La,Nd)Al$_2$ O{{formula:eb2043cf-63c0-4be2-bf2e-1613fa8f4bb7}}", "In its ground state, (La,Nd)Al$_2$ O$_6$ displays an $a^-a^-a^-$ tilt pattern, which is similar to its two bulk $AB$ O$_3$ constituents, and the $R32$ space group.", "The distortion relating the undistorted structure to the ground state transforms as a single irreducible representation (irrep), $R_{4}^{+}$ , of the undistorted 5-atom $Pm\\bar{3}m$ , which describes collective oxygen displacements producing out-of-phase rotations along each Cartesian axis.", "This may be equivalently described as a single out-of-phase rotation about the trigonal (3-fold) axis of the structure.", "The Landau theory of displacive phase transitions allows for the free energy of a system to be expanded in terms of an order parameter; in these superlattices, the order parameter best capturing the symmetry reduction from the cubic to antiferrodistortive phase is the AlO$_6$ octahedral rotation angle (given by $\\Theta $ in tab:gs), which can be mapped onto the amplitude of the zone-boundary mode $R_{4}^{+}$ .", "This permits us to write a free energy expression for (La,Nd)Al$_2$ O$_6$ as $\\mathcal {F} = \\alpha _1Q_{R_4^+}^{2} + \\beta _1Q_{R_4^+}^4\\,,$ with coefficients obtained from fits to our DFT computed mode amplitude–energy plots (tab:freeenergy).", "This compound has a null polarization, despite the fact that it exhibits a non-centrosymmetric space group, owing to the chiral structure.", "The out-of-phase rotations along all three crystallographic axes (in addition to rock salt ordering) removes all possibility for mirror symmetry but is compatible with three-fold and 2-fold rotation axes.", "As previously reported for hybrid improper ferroelectrics, it is essentially the anti-polar displacements of the different $A$ -site cations which results in the electric polarization (a ferr$i$ electric type mechanism).", "[17], [42] However, although out-of-phase rotations alone can lift inversion symmetry in [111]-ordered superlattices, the $A$ -site atoms cannot displace perpendicular to any direction which contain out-of-phase rotations.", "These features make the compound chiral and not polar.Indeed, in the $R32$ space group, the $A$ and $A^{\\prime }$ cations occupy the $3a$ and $3b$ Wyckoff positions, both of which exhibit site symmetry 32.", "This symmetry constraint suggests that out-of-phase rotations along one or two directions could be compatible with mirror planes in [111] $A$ -site ordered perovskites since those directions without any rotations would serve as unique anisotropic axes for which a loci of points defining the reflection plane may exist." ], [ "Polar (La,Pr)Al$_2$ O{{formula:ad33daec-d24e-4fb3-840c-ae69a972bd4b}} and (Ce,Pr)Al{{formula:f3cb8ed0-7b5b-4225-8180-e4f6235789d7}} O{{formula:696797fa-a853-48df-aefc-65b13cd2af6b}}", "We next investigate ordered aluminates which contain out-of-phase rotations along only two crystallographic axes.", "Because the bulk phases of the constituents in (La,Pr)Al$_2$ O$_6$ and (Ce,Pr)Al$_2$ O$_6$ superlattices display different octahedral rotation patterns (tab:bulk), there is a competition between having out-of-phase rotations along one, two, or three axes.", "We find that the ground state structures of both $A$ -site ordered aluminates exhibit the $a^0b^-b^-$ rotational pattern with the polar $Imm2$ space group.", "This rotation pattern allows for displacements of the $A$ -sites perpendicular to the $a$ axis, resulting in small electric polarizations of 1.80 $\\mu $ C/cm$^2$ in (La,Pr)Al$_2$ O$_6$ and 1.78 $\\mu $ C/cm$^2$ in (Ce,Pr)Al$_2$ O$_6$ .", "Unlike (La,Nd)Al$_2$ O$_6$ , the symmetry of these polar ground state structures requires two irreps for a complete description: $Q_1$ , which describes the out-of-phase rotations and transforms like the irrep $R_{4}^{+}$ [identical to that in (La,Nd)Al$_2$ O$_6$ , fig:LPAOenergy(a)], and $Q_2$ , which describes $A$ -site displacements and transforms like $R_{5}^{+}$ [cf.", "inset in fig:LPAOenergy(b)].", "Both irreps are given relative to the $Pm\\bar{3}m$ 5-atom perovskite.", "fig:LPAOenergy(b) shows the evolution of the total energy of (La,Nd)Al$_2$ O$_6$ with respect to increasing amplitude of these two modes.", "We find that while $R_{4}^{+}$ is a soft mode (negative quadratic-like curvature about the origin) and results in a large gain, $R_5^+$ alone leads to an energy penalty (positive curvature).", "On the other hand, when the two modes coexist [fig:LPAOenergy(c)], we find that with increasing amplitude of the nominally hard $R_{5}^{+}$ mode any non-zero amplitude of the non-polar out-of-phase rotations ($R_{4}^{+}$ ) leads to increased stability of the $Imm2$ (La,Nd)Al$_2$ O$_6$ structure without a change in curvature of the energy surface.", "This behavior is similar to that seen in the improper ferroelectric YMnO$_3$ ,[44] whereby a non-polar mode also stabilizes a hard polar mode with non-zero amplitude.", "Note, the free energy evolution of (Ce,Pr)Al$_2$ O$_6$ exhibits identical behavior (data not shown).", "Table: Free energy expansion coefficients to eq:landau for each [111]-ordered AA-site perovskite aluminate.To understand the anharmonic lattice interactions which provide the stability of the polar phase, we expand the free energy (with respect to $Pm\\bar{3}m$ ) as $\\mathcal {F} &=& \\alpha _1Q_{R_4^+}^{2} + \\alpha _2Q_{R_5^+}^{2} + \\beta _1Q_{R_4^+}^4 +\\beta _2Q_{R_5^+}^4\\\\\\nonumber &+&\\gamma _1Q_{R_4^+}^{3}Q_{R_5^+}+\\gamma _2Q_{R_4^+}Q_{R_5^+}^3+\\delta Q_{R_5^+}^2Q_{R_4^+}^2\\,.$ By fitting the free energy expression truncated to quartic order to the calculated total energies in Figure fig:LPAOenergy(c), the coefficients of eq:landau are obtained (tab:freeenergy).", "We find that the most important anharmonic term coupling $Q_{R_4^+}$ to $Q_{R_5^+}$ is given by $\\gamma _1Q_{R_4^+}^3Q_{R_5^+}$ with a negative coefficient that leads to the cooperative stability and coexistence of the out-of-phase rotations and polar displacements.", "The $\\gamma _2Q_{R_4^+}Q_{R_5^+}^3$ term contributes to the asymmetry of the energy surface, but is not the key interaction stabilizing the non-zero polar displacements in the $Imm2$ structure.", "Similarly, the mixed bi-quadratic term renormalizes the homogeneous quadratic terms, and leads to the decrease in radius of curvature of the free energy curve near the minimum with increasing ${R_4^+}$ , yet it is not responsible for the minimum.", "Thus, the requirement of two coupled modes as $\\sim \\!Q_{R_4^+}^3Q_{R_5^+}$ , with the primary $Q_{R_4^+}$ octahedral rotation mode, reveals that the aluminates behave as conventional improper ferroelectrics and not hybrid-improper ferroelectrics; the latter are susceptible to staggered phase transitions owing to the necessity of two primary modes.", "Figure: The experimentally and computationally determined phase transitions of (La,Nd)Al 2 _2O 6 _6, (La,Pr)Al 2 _2O 6 _6, and (Ce,Pr)Al 2 _2O 6 _6.", "The phases are defined according to crystal system, with the space group symmetries specified in tab:symmetry.", "Note that the unordered solid solutions for each aluminate are centrosymmetric, whereas the [111] AA-cation ordering leads to a non-centrosymmetric structure which persist to high temperature (≥1300\\ge 1300 K), allowing for the presence of ferro- and piezoelectricity." ], [ "Structural Phase Transitions", "We next investigate the critical temperatures associated with the paraelectric to ferroelectric transitions to explore the potential of these aluminates for high-temperature applications.", "Recent work has shown that $T_c$ in compounds exhibiting soft-mode driven phase transitions is strongly correlated to the energy difference between the two symmetry related structures.", "[45] By using the fact that $\\Delta E=k_BT_c$ , where $k_B$ is Boltzmann's constant, we are able to estimate phase transition temperatures using DFT total energies.", "The phase diagrams of the rare earth aluminate solid solutions (effectively disordered $A$ -site phases) of all compounds investigated in this work have also been experimentally determined,[46] and thus will serve as a useful comparison to the predicted critical temperatures of the [111] ordered phases.", "Because the symmetries of an ordered and disordered compound with the same tilt pattern are different, we refer to the corresponding phases by the crystal system generated by the octahedral rotation pattern to more easily draw comparisons.", "The abbreviations used for each phase, along with the corresponding tilt pattern and space group are listed in tab:symmetry.", "Table: The possible crystal systems, octahedral rotation patterns, and space groups (S.G.) that may be found in (La,Nd)Al 2 _2O 6 _6, (La,Pr)Al 2 _2O 6 _6, or (Ce,Pr)Al 2 _2O 6 _6, along with the abbreviations used for each.", "Note that not every aluminate displays every phase tabulated below (see fig:transitions).Experimentally, solid solution (La,Nd)Al$_2$ O$_6$ is found to undergo a rhombohedral (R, $a^-a^-a^-$ ) to cubic (C, $a^0a^0a^0$ ) phase transition at 1471 K. Our calculations of the ordered [111]-system estimate this transition to occur at 1338 K, in good agreement with the experimental results (fig:transitions, top panel).Note that other ordered phases (such as [001]) could display different transition temperatures.", "Solid solution (La,Pr)Al$_2$ O$_6$ displays a more complicated series of phase transitions: $\\mathrm {M} \\xrightarrow{} \\mathrm {O} \\xrightarrow{} \\mathrm {R} \\xrightarrow{} \\mathrm {C}.$ The monoclinic (M) phase exhibits the $a^0b^-c^-$ tilt pattern, while the orthorhombic (O) phase exhibits the $a^0b^-b^-$ pattern.", "In our calculations of the [111]-ordered phase, however, the monoclinic $Cm$ structure was dynamically unstable to the orthorhombic $Imm2$ phase; no local minimum could be found.", "Thus the experimentally known sequence of phase transitions in the solid solution are not reproduced in the ordered phase (fig:transitions, center panel).", "This result may either be a consequence of the cation order or a limitation of the exchange-correlation functional used in the DFT calculations.", "However, we estimate the $\\mathrm {O} \\rightarrow \\mathrm {R}$ and $\\mathrm {R} \\rightarrow \\mathrm {C}$ transitions to occur at 130 K and 1328 K, respectively, in good agreement with the experimental results of 93 K and 1306 K, which suggests that the monoclinic phase is suppressed by the cation order.", "Additionally, we found an orthorhombic $Pmc2_1$ ($a^+b^-b^-$ ) that is $\\sim $ 20 meV higher in energy than the ground state $Imm2$ structure; although this phase is not reported experimentally in the solid solution phase diagram, it could manifest during growth of [111]-ordered (La,Pr)Al$_2$ O$_6$ .", "Finally, solid solution (Ce,Pr)Al$_2$ O$_6$ displays the most complicated phase diagram, having five experimental transitions:[48] $\\mathrm {T} \\xrightarrow{} \\mathrm {M} \\xrightarrow{} \\mathrm {O}\\xrightarrow{}\\mathrm {R} \\xrightarrow{} \\mathrm {C}.$ In contrast, we find the orthorhombic $Imm2$ structure to be the ground state phase and the tetragonal structure to be unstable (fig:transitions).", "Based on energetics, we estimate an $\\mathrm {O} \\rightarrow \\mathrm {R}$ transition at 34 K and an $\\mathrm {R} \\rightarrow \\mathrm {C}$ transition at 1281 K. We conjecture that many of the discrepancies between the ordered aluminates and the experimental solid solutions, e.g., the loss of the tetragonal phase in (Ce,Pr)Al$_2$ O$_6$ , may be attributed to the $A$ -site rock salt ordering pattern and its compatibility with the single antiferrodistortive AlO$_6$ rotation mode.", "Because ordering along [111] results in a three dimensional pattern (alternating $A$ -sites along each axis), it is more compatible with a three dimensional octahedral rotation pattern ($a^-a^-a^-$ , i.e., out-of-phase rotations along each axis), rather than a one- or two-dimensional tilt pattern which would create a unique axis in the structure.", "This geometric argument is corroborated by examining rock salt $B$ -site ordered perovskites: while rock salt $A$ -site ordered perovskite oxides are rare, [111]-$B$ -site ordering is by far the most common type.", "[49], [50], [51] The three-dimensional $a^+b^-b^-$ tilt system is the most common octahedral rotation pattern adopted by $B$ -site ordered perovskites (in addition to being the most common overall).", "The $a^-a^-a^-$ and $a^0b^-b^-$ patterns are found to be the next most frequently observed, with $a^0a^0c^-$ being the least common.", "[49], [52] By considering this distribution of structures, it seems reasonable that the ground states of the $A$ -site ordered aluminates investigated here should also exhibit either the $a^-a^-a^-$ or $a^0b^-b^-$ rotational pattern.", "Additionally, it appears that PrAlO$_3$ ($a^0b^-b^-$ ) influences the tilt pattern of the ordered superlattice more so than the other bulk rare earth aluminates; it turns off a rotation about one Cartesian axis in LaAlO$_3$ ($a^-a^-a^-$ ), stabilizing the $a^0b^-b^-$ rotation pattern.", "Whereas it turns on a rotation when ordered with CeAlO$_3$ ($a^0a^0c^-$ ) to also give the $a^0b^-b^-$ rotation pattern.", "This is due to the fact that the phonon mode which transforms as $R_4^+$ is much more unstable in PrAlO$_3$ ($\\omega =177.5i$ cm$^{-1}$ ) than in LaAlO$_3$ ($\\omega =127.1i$ cm$^{-1}$ ) or CeAlO$_3$ ($\\omega =148.4i$ cm$^{-1}$ ), leading to control of the superlattice structure by PrAlO$_3$ .", "This suggests that the rotation pattern adopted by the ground state structure could be directly designed by selection of the rotational mode instability strengths (or mismatch) of the bulk $AB$ O$_3$ perovskite oxides interleaved to form the superlattice.", "Table: The computed frozen-ion (ϵ elec \\epsilon ^{elec}), relaxed-ion (ϵ ion \\epsilon ^{ion}), and total (ϵ\\epsilon ) dielectric, relaxed-ion piezoelectric stress (ee) and strain (dd), and Born effective charge (Z * Z^) tensors for (La,Nd)Al 2 _2O 6 _6, (La,Pr)Al 2 _2O 6 _6, and (Ce,Pr)Al 2 _2O 6 _6.", "Because (La,Nd)Al 2 _2O 6 _6 does not exhibit any other phases besides R32R32 and Fm3 ¯mFm\\bar{3}m, only the tensor properties of the R32R32 phase are presented.", "Point group 32 exhibits 3 dielectric and 5 piezoelectric coefficients, but only 2 of which are independent in each case: ϵ 11 =ϵ 22 ≠ϵ 33 \\epsilon _{11}=\\epsilon _{22}\\ne \\epsilon _{33}, and d 11 =-d 12 =-1 2d 26 d_{11}=-d_{12}=-\\frac{1}{2}d_{26} and d 14 =-d 25 d_{14}=-d_{25}.", "The mm2mm2 point group exhibits 3 independent dielectric and 5 independent piezoelectric coefficients.", "Although the Al and O Born effective charge tensors contain small off-diagonal elements, only the main diagonal coefficients are given." ], [ "Dielectric and Acentric Properties", "As discussed previously, we find the ground state structures of (La,Pr)Al$_2$ O$_6$ and (Ce,Pr)Al$_2$ O$_6$ are improper ferroelectrics with small electric polarizations of 1.80 and 1.78 $\\mu $ C/cm$^2$ , respectively, resulting from $A$ -site displacements.", "Our estimate of the critical transition temperatures for each shows that these aluminates would only retain this spontaneous polarization to 93 K and 34 K, at which point they transition from a polar to chiral structure.", "However, all three compounds retain non-centrosymmetric structures up to high temperatures (>1300 K, see fig:transitions).", "To investigate their viability for use in high-temperature applications, we compute the dielectric and piezoelectric tensors, as well as the Born effective charges, for the non-centrosymmetric phases of each ordered compound (tab:piezo).", "Although these properties are all computed at 0 K, they can still be used to draw useful comparisons across different phases.", "The total dielectric constant $\\epsilon $ consists of electronic (frozen-ion) and ionic (relaxed-ion) contributions.", "We first find that the electronic dielectric tensor is approximately isotropic, as well as nearly equivalent across all chemical compositions (a well-known phenomenon in perovskite oxides[54]) and structural phases.", "The ionic contributions are much larger; we also find that on average they increase when transitioning from the polar $Imm2$ to chiral $R32$ structure.", "The total dielectric constant of each aluminate also falls well within the known range for perovskite oxides; (La,Nd)Al$_2$ O$_6$ in particular displays a dielectric constant near the top of this range.", "[54] The relaxed-ion piezoelectric stress ($e_{ij}$ ) and strain ($d_{ij}$ ) coefficients are related through the compliance tensor $C_{ij}$ (the inverse of the elastic tensor) by $d_{ij}=S_{ik}e_{kj}$ .", "Note that due to symmetry, the $R32$ piezoelectric tensors have 5 components, but only 2 of which are independent; the $Imm2$ phase has 5 independent coefficients.", "First, the chiral $R32$ phases of (La,Nd)Al$_2$ O$_6$ and (La,Pr)Al$_2$ O$_6$ exhibit relatively large piezoelectric coefficients, comparable to those of common lead-free piezoelectric materials such as BaTiO$_3$ and LiNbO$_3$ .", "[55], [56] Interestingly, the polar $Imm2$ phase of (La,Pr)Al$_2$ O$_6$ has piezoelectric coefficients that are an order of magnitude smaller.", "Because the chiral phases have no net dipole due to the $A$ -sites sitting on high symmetry positions, an applied stress will generate a much larger induced polarization than in the polar phases (which already have a spontaneous polarization).", "Additionally, the piezoelectric response of (Ce,Pr)Al$_2$ O$_6$ is much smaller than either of the La-based aluminates; it does, however, also show the same increase in response across the $Imm2$ -to-$R32$ structural phase boundary (tab:piezo).", "Thus, while the polarization goes to zero across the $Imm2$ -to-$R32$ transition in (La,Pr)Al$_2$ O$_6$ and (Ce,Pr)Al$_2$ O$_6$ , there is a large increase in the piezoelectric response in the chiral structures.", "Finally, the Born effective charges were computed for the five phases.", "Typically, proper $AB$ O$_3$ ferroelectrics exhibit anomalously large Born effective charges; for example, the nominal charge for Ba, Ti, and O in BaTiO$_3$ are $+2$ , $+4$ , and $-2$ , respectively, while $Z_\\mathrm {Ba}^ = +2.56$ , $Z_\\mathrm {Ti}^* = +7.26$ , and $Z^*_\\mathrm {O} = -5.73$ .", "[57] In these ordered aluminates, however, the Born effective charges are close to their nominal values, with only the $A$ -site cations displaying any significant deviation.", "This clearly highlights the difference in mechanism between proper ferroelectrics such as BaTiO$_3$ , where polarization arises from $B$ -site displacements due to enhanced covalency with the surrounding oxygen anions, and improper ferroelectrics such as these ordered aluminates, where unequal anti-aligned $A$ -site displacements result in layer dipoles.", "We showed that the ordering of $A$ -site cations along the [111]-crystallographic axis in perovskite oxide superlattices, in conjunction with any rotational pattern containing only out-of-phase rotations results in a non-centrosymmetric space group.", "By selecting compounds which contain ground state rotational patterns described by only one lattice mode (irreducible representation $R_4^+$ ), we were able to design three rock salt ordered superlattices, two of which display a spontaneous electric polarization originating from an improper ferroelectric mechanism.", "This type of design strategy prevents staggered paraelectric to ferroelectric phase transitions, possible in hybrid improper ferroelectrics where the most frequently pursued design route requires the $a^+b^-b^-$ tilt pattern, without sacrificing the desired electronic properties.", "Additionally, we predicted the phase transition temperatures of each superlattice from first-principles, which showed that these compounds remain non-centrosymmetric up to 1300 K. Across the ferroelectric phase transition, there is an order of magnitude increase in the piezoelectric coefficients comparable to conventional piezoelectric materials such as BaTiO$_3$ .", "This indicates that these materials may find potential use in high-temperature applications, as they have reasonable piezoelectric responses that persist over a broad temperature range.", "We hope the framework presented here encourages the synthesis of ultra-short perovskite superlattices on [111]-oriented substrates, and that it can be extended beyond these aluminates to other perovskite oxides with out-of-phase rotations.", "J.Y.", "and J.M.R.", "acknowledge funding support from the Army Research Office under grant number W911NF-12-1-0133.", "J.Y.", "thanks members of the Rondinelli group for useful discussions.", "DFT calculations were performed using the CARBON cluster at the Center for Nanoscale Materials [Argonne National Laboratory, supported by the U.S. DOE, Office of Basic Energy Sciences (BES), DE-AC02-06CH11357], and the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575." ] ]	1403.0151
[ [ "On the Intersection Property of Conditional Independence and its\n Application to Causal Discovery" ], [ "Abstract This work investigates the intersection property of conditional independence.", "It states that for random variables $A,B,C$ and $X$ we have that $X$ independent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies $X$ independent of $(A,B)$ given $C$.", "Under the assumption that the joint distribution has a continuous density, we provide necessary and sufficient conditions under which the intersection property holds.", "The result has direct applications to causal inference: it leads to strictly weaker conditions under which the graphical structure becomes identifiable from the joint distribution of an additive noise model." ], [ "Application to Causal Inference", "Inferring causal relationships is a major challenge in science.", "In the last decades considerable effort has been made in order to learn causal statements from observational data.", "Causal discovery methods make assumptions that relate the joint distribution with properties of the causal graph.", "Constraint-based or independence-based methods [7], [10] and some score-based methods [1], [5] assume the Markov condition and faithfulness.", "A distribution is said to be Markov with respect to a directed acyclic graph (DAG) $G$ if each d-separation in the graph implies the corresponding (conditional) independence; the distribution is faithful with respect to $G$ if the reverse statement holds.", "These two assumptions render the Markov equivalence class of the correct graph identifiable from the joint distribution, i.e.", "the skeleton and the v-structures of the graph can be inferred from the joint distribution [11].", "Methods like LiNGAM [9] or additive noise models [6], [8] assume the Markov condition, too, but do not require faithfulness; instead, these methods assume that the structural equations come from a restricted model class (e.g.", "linear with non-Gaussian noise or non-linear with additive Gaussian noise).", "In order to prove that the directed acyclic graph (DAG) is identifiable from the joint distribution [8] require a strictly positive density.", "Their proof makes use of the intersection property of conditional independence (Definition REF ) which is known to hold for positive densities [7]." ], [ "Main Contributions", "In Section we provide a sufficient and necessary condition on the density for the intersection property to hold (Corollary REF ).", "This result is of interest in itself since the developed condition is weaker than strict positivity.", "As mentioned above, some causal discovery methods based on structural equation models require the intersection property for identification; they therefore rely on the strict positivity of the density.", "This can be achieved by fully supported noise variables, for example.", "Using the new characterization of the intersection property we can now replace the condition of strict positivity.", "In fact, we show in Section that noise variables with a path-connected support are sufficient for identifiability of the graph (Proposition REF ).", "This is already known for linear structural equation models [9] but not for non-linear models.", "As an alternative, we provide a condition that excludes constant functions and leads to identifiability, too (Proposition REF ).", "In Section , we provide an example of a structural equation model that violates the intersection property (but satisfies causal minimality).", "Its corresponding graph is not identifiable from the joint distribution.", "In correspondence to the theoretical results of this work, some noise densities in the example are do not have a path-connected support and the functions are partially constant.", "We are not aware of any causal discovery method that is able to infer the correct DAG or the correct Markov equivalence class; the example therefore shows current limits of causal inference techniques.", "It is non-generic in the case that it violates all sufficient assumptions mentioned in Section ." ], [ "Conditional Independence and the Intersection Property", "We now formally introduce the concept of conditional independence in the presence of densities and the intersection property.", "Let therefore $A, B, C$ and $X$ be (possibly multi-dimensional) random variables that take values in metric spaces $\\mathcal {A, B, C}$ and $\\mathcal {X}$ respectively.", "We first introduce assumptions regarding the existence of a density and some of its properties that appear in different parts of this paper.", "(A0) The distribution is absolutely continuous with respect to a product measure of a metric space.", "We denote the density by $p(\\cdot )$ .", "This can be a probability mass function or a probability density function, for example.", "(A1) The density $(a,b,c) \\mapsto p(a,b,c)$ is continuous.", "(A2) For each $c$ with $p(c)>0$ the set $\\mathrm {supp}_c(A,B) := \\lbrace (a,b)\\,:\\, p(a,b,c)>0\\rbrace $ contains only one path-connected component (see Definition REF ).", "(A2') The density $p(\\cdot )$ is strictly positive.", "Condition (A2') implies (A2).", "We assume (A0) throughout the whole work.", "In this paper we work with the following definition of conditional independence.", "Definition 1 (Conditional Independence) We call $X$ independent of $A$ conditional on $B$ and write $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A \\,\|\\,B$ if and only if $ p(x,a \\,\|\\,b) = p(x\\,\|\\,b)p(a \\,\|\\,b)$ for all $x,a,b$ such that $p(b)>0$ .", "The intersection property of conditional independence is defined as follows [7].", "Definition 2 (Intersection Property) We say that the joint distribution of $X,A,B,C$ satisfies the intersection property if $ X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A \\,\|\\,B, C \\; \\text{ and } \\; X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}B \\,\|\\,A, C \\quad \\Rightarrow \\quad X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}(A,B) \\,\|\\,C\\ .$ The intersection property (REF ) has been proven to hold for strictly positive densities [7].", "It is also known that the intersection property does not necessarily hold if the joint distribution does not have a density [3].", "[4] provides measure-theoretic necessary and sufficient conditions for the intersection property.", "In this work we assume the existence of a density (A0) and provide more detailed conditions under which the intersection property holds." ], [ "Counter Example", "We now give an example of a distribution that does not satisfy the intersection property (REF ).", "Since the joint distribution is absolutely continuous with respect to the Lebesgue measure, the example shows that the intersection property requires further restrictions on the density apart from its existence.", "We will later use the same idea to prove Proposition REF that shows the necessity of our new condition.", "Example 1 Consider a structural equation model for random variables $X,A,B$ : $A &= N_A\\,,\\\\B &= A + N_B\\,,\\\\X &= f(B) + N_X\\,,$ with $N_A \\sim \\mathcal {U}([-2;-1] \\cup [1;2])$ , $N_B, N_X \\sim \\mathcal {U}([-0.3;0.3])$ being jointly independent.", "Let the function $f$ be of the form $f(b) = \\left\\lbrace \\begin{array}{cl}+10& \\text{if } b>0.5\\,,\\\\0& \\text{if } b<-0.5\\,,\\\\g(b)& \\text{ else,}\\end{array} \\right.$ where the function $g$ can be chosen to make $f$ arbitrarily smooth.", "Some parts of this structural equation model are summarized in Figure REF .", "We clearly have $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A \\,\|\\,B$ and $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}B \\,\|\\,A$ but $X \\mbox{${}\\lnot \\!\\perp \\hspace{-6.111pt}\\perp {}$}A$ and $X \\mbox{${}\\lnot \\!\\perp \\hspace{-6.111pt}\\perp {}$}B$ .", "A formal proof of this statement is provided in the more general setting of Proposition REF .", "It will turn out to be important that the two connected components of the support of $A$ and $B$ cannot be connected by an axis-parallel line.", "In the notation introduced in Definition REF below, this means $Z_1$ and $Z_2$ are not equivalent.", "Within each component, however, that is if we consider the areas $A,B>0$ and $A,B<0$ separately, we do have the independence statement $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A,B$ that is predicted by the intersection property.", "This observation will be formalized as the weak intersection property in Proposition REF .", "Figure: Example .", "The plot on the left hand side shows the support of variables AA and BB in black.", "In the areas filled with dark gray and light gray the function ff takes values ten and minus ten, respectively.", "The structural equation model corresponds to the top graph but the distribution can also be generated by a structural equation model with the bottom graph.Example REF has the following important implication for causal inference.", "The distribution satisfies causal minimality with respect to two different graphs, namely $A \\rightarrow B \\rightarrow X$ and $X \\leftarrow A \\rightarrow B$ (see Figure REF ).", "Since it violates faithfulness and the intersection property, we are not aware of any causal inference method that is able to recover the correct graph structure based on observational data only.", "Recall that [8] assume strictly positive densities in order to assure the intersection property.", "More precisely, the example shows that Lemma 37 in [8] does not hold anymore when the positivity is violated." ], [ "Necessary and sufficient condition for the intersection property", "This section characterizes the intersection property in terms of the joint density over the corresponding random variables.", "In particular, we state a weak intersection property (Proposition REF ) that leads to a necessary and sufficient condition for the classical intersection property, see Corollary REF .", "For these results, the notion of path-connectedness becomes important.", "A continuous mapping $\\lambda : [0,1] \\rightarrow \\mathcal {X}$ into a metric space $\\mathcal {X}$ is called a path between $\\lambda (0)$ and $\\lambda (1)$ in $\\mathcal {X}$ .", "A subset $\\mathcal {S} \\subseteq \\mathcal {X}$ is called path-connected if every pair of points from $\\mathcal {S}$ can be connected by a path in $\\mathcal {S}$ .", "We require the following definition.", "Definition 3 (i) For each $c$ with $p(c)>0$ we consider the (not necessarily closed) support of $A$ and $B$ : $\\mathrm {supp}_c(A,B) := \\lbrace (a,b)\\ : \\ p(a,b,c) > 0 \\rbrace \\,.$ We further write for all sets $M \\subset \\mathcal {A} \\times \\mathcal {B}$ $\\mathrm {proj}_A(M) &:= \\lbrace a \\in \\mathcal {A}\\ : \\ \\exists b \\text{ with } (a,b) \\in M \\rbrace \\; \\text{ and}\\\\\\mathrm {proj}_B(M) &:= \\lbrace b \\in \\mathcal {B}\\ : \\ \\exists a \\text{ with } (a,b) \\in M \\rbrace \\,.$ (ii) We denote the path-connected components of $\\mathrm {supp}_c(A,B)$ by $(Z^c_i)_i$ .", "Two path-connected components $Z^c_{i_1}$ and $Z^c_{i_2}$ are said to be coordinate-wise connected if $\\mathrm {proj}_A (Z^c_{i_1}) \\cap \\mathrm {proj}_A (Z^c_{i_2}) &\\ne \\emptyset \\qquad \\text{ or}\\\\\\mathrm {proj}_B (Z^c_{i_1}) \\cap \\mathrm {proj}_B (Z^c_{i_2}) &\\ne \\emptyset $ We then say that $Z^c_i$ and $Z^c_j$ are equivalent if and only if there is a sequence $Z^c_{i} = Z^c_{i_1}, \\ldots , Z^c_{i_m} = Z^c_{j}$ with two neighbours $Z^c_{i_k}$ and $Z^c_{i_{k+1}}$ being coordinate-wise connected.", "We represent these equivalence classes by the union of all its members.", "These unions we denote by $(U^c_{i})_i$ .", "We further introduce a deterministic function $U^c$ of the variables $A$ and $B$ .", "We set $U^c := \\left\\lbrace \\begin{array}{cl}i & \\text{ if } \\;(A,B) \\in U^c_i\\\\0 & \\text{ if } \\;p(A,B) = 0\\end{array}\\right..$ We have that $U^c = i$ if and only if $A \\in \\mathrm {proj}_A(U^c_{i})$ if and only if $B \\in \\mathrm {proj}_B(U^c_{i})$ .", "Note that the projections $\\mathrm {proj}_A(U^c_{i})$ are disjoint (for different $i$ ); similarly for $\\mathrm {proj}_B(U^c_{i})$ .", "(iii) The case where there is no variable $C$ can be treated as if $C$ was deterministic: $p(c) = 1$ for some $c$ .", "In Example REF there is no variable $C$ .", "Figure REF shows the support $\\mathrm {supp}_c(A,B)$ in black.", "It contains two path-connected components.", "Since they cannot be connected by axis-parallel lines, they are not equivalent; thus, one of them corresponds to $U_1^C$ and the other to $U_2^c$ .", "Figure REF shows another example that contains three equivalence classes of path-connected components; again, there is no variable $C$ ; we formally introduce a deterministic variable $C$ that always takes the value $c$ .", "Figure: Each block represents one path-connected component Z i c Z_i^c of the support of p(a,b)p(a,b).", "All blocks with the same filling are equivalent since they can be connected by axis-parallel lines.", "There are three different fillings corresponding to the equivalence classes U 1 c U_1^c, U 2 c U_2^c and U 3 c U_3^c.Using Definition REF we are now able to state the two main results, Propositions REF and REF .", "As a direct consequence we obtain Corollary REF which generalizes the condition of strictly positive densities.", "Proposition 1 (Weak Intersection Property) Assume (A0), (A1) and that $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A \\,\|\\,B, C$ and $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}B \\,\|\\,A, C$ .", "Consider now $c$ with $p(c)>0$ and the variable $U^c$ as defined in Definition REF (ii).", "We then have the weak intersection property: $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}(A,B) \\,\|\\,C = c, U^c\\ .$ This means that $p(x\\,\|\\,a,b,c,u^c) = p(x \\,\|\\,c, u^c)$ for all $x,a,b$ with $p(a,b,c) > 0$ .", "Proposition 2 (Failure of Intersection Property) Assume (A0), (A1) and that there are two different sets $U^{c^}_{1} \\ne U^{c^}_2$ for some $c^$ with $p(c^)>0$ .", "Then there is a random variable $X$ such that the intersection property (REF ) does not hold for the joint distribution of $X,A,B,C$ .", "As a direct corollary from these two propositions we obtain a characterization of the intersection property in the case of continuous densities.", "Corollary 1 (Intersection Property) Assume (A0) and (A1).", "Then $&\\text{The intersection property~(\\ref {eq:inters}) holds for all variables } X.", "\\\\\\Longleftrightarrow \\quad &\\text{All components } Z^c_i \\text{ are equivalent, i.e.", "there is only one set } U^c_1.$ In particular, this is the case if (A2) holds (there is only one path-connected component) or (A2') holds (the density is strictly positive)." ], [ "Application to Causal Discovery", "We now define what we mean by identifiability of the graph in continuous additive noise models.", "Assume that a joint distribution over $X_1, \\ldots , X_p$ is generated by a structural equation model (SEM) $ X_i = f_i(X_{{\\mathbf {PA}}^{}_{i}}) + N_i\\,,$ with continuous, non-constant functions $f_i$ , additive and jointly independent noise variables $N_i$ with mean zero and sets ${\\mathbf {PA}}^{}_{i}$ that are the parents of $i$ in a directed acyclic graph $\\mathcal {G}$ .", "To simplify notation, we identify variables $X_i$ with its index (or node) $i$ .", "We consider the following statement $() \\quad \\begin{array}{l}\\mathcal {G} \\text{ is identifiable from the joint distribution, i.e.", "it cannot}\\\\\\text{ be generated by an SEM with different graph } \\mathcal {H} \\ne \\mathcal {G}\\,.\\end{array}$ [8] prove this identifiability by extending the identifiability from graphs with two nodes to graphs with an arbitrary number of variables.", "Because they require the intersection property, it is shown only for strictly positive densities.", "But since Corollary REF provides weaker assumption for the intersection property, we can use it to obtain new identifiability results.", "Proposition 3 Assume that a joint distribution over $X_1, \\ldots , X_p$ is generated by a structural equation model (REF ).", "If all densities of $N_1, \\ldots , N_p$ are path-connected, then the density of $X_1, \\ldots , X_p$ is path-connected, too.", "Thus, the intersection property (REF ) holds for any disjoint sets of variables $X,A,B,C \\in \\lbrace X_1, \\ldots , X_p\\rbrace $ (see Corollary REF ).", "Therefore, statement $()$ holds if the noise variables have continuous densities and path-connected support.", "Example REF violates the assumption of Proposition REF since the support of $A$ is not path-connected.", "It satisfies another important property, too: the function $f$ is constant on some intervals.", "The following proposition shows that this is necessary to violate identifiability.", "Proposition 4 Assume that a joint distribution over $X_1, \\ldots , X_p$ is generated by a structural equation model (REF ) with graph $\\mathcal {G}$ .", "Let us denote the non-descendants of $X_i$ by ${\\mathbf {ND}}^{\\mathcal {G}}_{i}$ .", "Assume that the structural equations are non-constant in the following way: for all $X_i$ , for all its parents $X_j \\in {\\mathbf {PA}}^{}_{i}$ and for all $X_{\\mathbf {C}} \\subseteq {\\mathbf {ND}}^{\\mathcal {G}}_{i} \\setminus \\lbrace X_j\\rbrace $ , there are $(x_j,x_j^{\\prime },x_k,x_c)$ such that $f_i(x_j,x_k) \\ne f_i(x_j^{\\prime },x_k)$ and $p(x_j,x_k,x_c)>0$ and $p(x_j^{\\prime },x_k,x_c)>0$ .", "Here, $x_k$ represents the value of all parents of $X_i$ except $X_j$ .", "Then for any ${\\mathbf {PA}}^{}_{i} \\setminus \\lbrace j\\rbrace \\subseteq \\mathbf {S} \\subseteq {\\mathbf {ND}}^{\\mathcal {G}}_{i} \\setminus \\lbrace j\\rbrace $ , it holds that $X_i \\mbox{${}\\lnot \\!\\perp \\hspace{-6.111pt}\\perp {}$}X_j \\, \\,\|\\,\\, \\mathbf {S}$ .", "Therefore, statement $()$ follows.", "Proposition REF provides an alternative way to prove identifiability.", "The results are summarized in Table REF .", "Table: This table shows conditions for continuous additive noise models (ANMs) that lead to identifiability of the directed acyclic graph from the joint distributions.", "Using the characterization of the intersection property we could weaken the condition of a strictly positive density." ], [ "Conclusion", "It is possible to prove the intersection property of conditional independence for variables whose distributions do not have a strictly positive density.", "A necessary and sufficient condition for the intersection property is that all path-connected components of the support of the density are equivalent, that is they can be connected by axis-parallel lines.", "In particular, this condition is satisfied for densities whose support is path-connected.", "In the general case, the intersection property still holds conditioning on any equivalence class of path-connected components, we call this the weak intersection property.", "This insight has a direct application in causal inference.", "For continuous additive noise models we can prove identifiability of the graph from the joint distribution using strictly weaker assumptions than before." ], [ "Proof of Proposition ", "We require the following well-known lemma [2].", "Lemma 1 We have $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A \\,\|\\,B$ if and only if $p(x \\,\|\\,a,b) = p(x \\,\|\\,b)$ for all $x,a,b$ such that $p(a,b) > 0$ and $p(b)>0$ .", "(of Proposition REF ) We have by Lemma REF $ p(x \\,\|\\,b,c) = p(x\\,\|\\,a,b,c) = p(x \\,\|\\,a,c)$ for all $x,a,b,c$ with $p(a,b,c)>0$ .", "As the main argument we show that $ p(x \\,\|\\,b,c) = p(x \\,\|\\,\\tilde{b},c)$ for all $x,b,\\tilde{b}, c$ with $b, \\tilde{b} \\in \\mathrm {proj}_B(U^c_i)$ for the same $i$ .", "Step 1, we prove equation (REF ) for $b, \\tilde{b} \\in Z^c_i$ , that is there is a path $(a(t),b(t))$ , such that $p(a(t),b(t),c)>0$ for all $0 \\le t \\le 1$ , and $b(0) = b$ and $b(1) = \\tilde{b}$ .", "Since the interval $[0,1]$ is compact and $p$ is continuous, the path $\\lbrace (a(t),b(t))\\ : \\ 0 \\le t \\le 1\\rbrace $ is compact, too.", "Define for each point $(a(t),b(t))$ on the path an open ball with radius small enough such that all $(a,b)$ in the ball satisfy $p(a,b,c)>0$ .", "Since this is an open cover of the space, choose a finite subset, of size $n$ say, of all those balls that still provide an open cover of the path.", "Without loss of generality generality let $(a(0),b(0))$ be the center of ball 1 and $(a(1),b(1))$ be the center of ball $n$ .", "It suffices to show that equation (REF ) holds for the centres of two neighbouring balls, say $(a_1,b_1)$ and $(a_2, b_2)$ .", "Choose one point $(a^,b^)$ from the non-empty intersection of those two balls.", "Since $d((a_1,b_1), (a^,b_1)) < d((a_1,b_1), (a^,b^))$ and $d((a_2,b_2), (a_2,b^)) < d((a_2,b_2), (a^,b^))$ for the Euclidean metric $d$ , we have that $p(a_1,b_1,c)$ , $p(a^,b_1,c)$ , $p(a^,b^,c)$ , $p(a_2,b^,c)$ and $p(a_2,b_2,c)$ are all greater zero.", "Therefore, using equation (REF ) several times, $p(x \\,\|\\,b_1,c) &= p(x \\,\|\\,a_1,c) = p(x \\,\|\\,a^,c)\\\\&= p(x \\,\|\\,b^,c) = p(x \\,\|\\,a_2,c) =p(x \\,\|\\,b_2,c)$ This shows equation (REF ) for $b, \\tilde{b} \\in Z^c_i$ .", "Step 2, we prove equation (REF ) for $b \\in Z^c_i$ and $\\tilde{b} \\in Z^c_{i+1}$ , where $Z^c_i$ and $Z^c_{i+1}$ are coordinate-wise connected (and thus equivalent).", "If $b^ \\in \\mathrm {proj}_B(Z^c_i) \\cap \\mathrm {proj}_B(Z^c_{i+1})$ , we know that $p(x \\,\|\\,b,c) = p(x \\,\|\\,b^,c) = p(x \\,\|\\,\\tilde{b},c)$ from the argument given in step 1 above.", "If $a^ \\in \\mathrm {proj}_A(Z^c_i) \\cap \\mathrm {proj}_A(Z^c_{i+1})$ , then there is a $b_i, b_{i+1}$ such that $(a^,b_i) \\in Z^c_i$ and $(a^,b_{i+1}) \\in Z^c_{i+1}$ .", "By equation (REF ) and the argument from step 1 we have $p(x \\,\|\\,b,c) = p(x \\,\|\\,b_i,c) = p(x \\,\|\\,b_{i+1},c) = p(x \\,\|\\,\\tilde{b},c)$ We can now combine these two steps in order to prove the original claim from equation (REF ).", "If $b, \\tilde{b} \\in \\mathrm {proj}_B(U^c_i)$ then $b \\in \\mathrm {proj}_B(Z^c_1)$ and $\\tilde{b} \\in \\mathrm {proj}_B(Z^c_n)$ , say.", "Further, there is a sequence $Z^c_1, \\ldots , Z^c_n$ coordinate-connecting these components.", "Combining steps 1 and 2 proves equation (REF ).", "Consider now $x,b,c$ such that $p(b,c)>0$ (which implies $p(c)>0$ ) and consider $u^c = i$ , say.", "Observe further that $p(a,c)>0$ for $a \\in \\mathrm {proj}_A(U^c_i)$ .", "We thus have $p(x,u^c \\,\|\\,c) &= \\int _{a} p(x,a,u^c \\,\|\\,c) \\ da= \\int _{a \\in \\mathrm {proj}_A(U^c_i)} p(x,a \\,\|\\,c) \\ da\\\\&= \\int _{a \\in \\mathrm {proj}_A(U^c_i)} \\frac{p(x,a,c) p(a ,c)}{p(c)p(a,c)} \\ da \\\\&= \\int _{a \\in \\mathrm {proj}_A(U^c_i)} p(x \\,\|\\,a,c) p(a \\,\|\\,c) \\ da \\\\&= \\int _{a \\in \\mathrm {proj}_A(U^c_i), p(a,b,c)>0} p(x \\,\|\\,a,c) p(a \\,\|\\,c) \\ da \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad + \\int _{a \\in \\mathrm {proj}_A(U^c_i),p(a,b,c)=0} p(x \\,\|\\,a,c) p(a \\,\|\\,c) \\ da \\\\&= p(x \\,\|\\,b,c) \\int _{a \\in \\mathrm {proj}_A(U^c_i), p(a,b,c)>0} p(a \\,\|\\,c) \\ da + \\int _{\\mathcal {A}_b} p(x \\,\|\\,a,c) p(a \\,\|\\,c) \\ da\\\\&=: (\\#)$ with $\\mathcal {A}_b = \\lbrace a \\in \\mathrm {proj}_A(U^c_i)\\ : \\ p(a,b,c)=0\\rbrace $ .", "It is the case, however, that for all $a \\in \\mathcal {A}_b$ there is a $\\tilde{b}(a) \\in \\mathrm {proj}_B(U^c_i)$ with $p(a,\\tilde{b}(a),c)>0$ .", "But since also $b \\in \\mathrm {proj}_B(U^c_i)$ we have $p(x\\,\|\\,\\tilde{b},c) = p(x\\,\|\\,b,c)$ by equation (REF ).", "Ergo, $(\\#) &= p(x \\,\|\\,b,c) \\int _{a \\in \\mathrm {proj}_A(U^c_i), p(a,b,c)>0} p(a \\,\|\\,c) \\ da + \\int _{\\mathcal {A}_b} p(x \\,\|\\,a,\\tilde{b}(a), c) p(a \\,\|\\,c) \\ da\\\\&= p(x \\,\|\\,b,c) \\int _{a \\in \\mathrm {proj}_A(U^c_i),p(a,b,c)>0} p(a \\,\|\\,c) \\ da + p(x \\,\|\\,b, c) \\int _{\\mathcal {A}_b} p(a \\,\|\\,c) \\ da\\\\&= p(x \\,\|\\,b, c) \\int _{a \\in \\mathrm {proj}_A(U^c_i)} p(a \\,\|\\,c) \\ da\\\\&= p(x\\,\|\\,b,c)\\ p(u^c\\,\|\\,c)$ This implies $p(x\\,\|\\,c,u^c) = p(x\\,\|\\,b,c)\\ .$ Together with equation (REF ) this leads to $p(x \\,\|\\,a,b,c,u^c) = p(x \\,\|\\,a,b,c) = p(x \\,\|\\,c, u^c)\\,.$" ], [ "Proof of Proposition ", "Define $X$ according to $X = g(C,U^C) + N_X$ where $N_X \\sim \\mathcal {U}([-0.1,0.1])$ is uniformly distributed with $(N_X, A, B, C)$ being jointly independent.", "Define $g$ according to $g(c,u^c) = \\left\\lbrace \\begin{array}{cl}10 & \\text{if } C = c^* \\text{ and } u^{c^} = 1\\\\0 & \\text{ otherwise}\\end{array} \\right.$ Fix a value $c$ with $p(c)>0$ .", "We then have for all $a, b$ with $p(a, b, c)>0$ that $p(x \\,\|\\,a, b, c) = p(x \\,\|\\,c, u^c) = p(x \\,\|\\,a, c) = p(x \\,\|\\,b, c)$ because $U^c$ can be written as a function of $A$ or of $B$ .", "We therefore have that $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}A \\,\|\\,B, C$ and $X \\mbox{${}\\perp \\hspace{-6.111pt}\\perp {}$}B \\,\|\\,A, C$ .", "Depending on whether $b$ is in $\\mathrm {proj}_B(U^{c^}_1)$ or not we have $p(x=0 \\,\|\\,b,c^)=0$ or $p(x=10 \\,\|\\,b,c^)=0$ , respectively.", "Thus, $&p(x=10 \\,\|\\,b,c^) \\cdot p(x=0 \\,\|\\,b,c^)& = 0\\,\\text{, whereas}\\\\&p(x=10 \\,\|\\,c^) \\cdot p(x=0 \\,\|\\,c^)& \\ne 0\\,.$ This shows that $X \\mbox{${}\\lnot \\!\\perp \\hspace{-6.111pt}\\perp {}$}B \\,\|\\,C = c^$ ." ], [ "Proof of Proposition ", "Since the true structure corresponds to a directed acyclic graph, we can find a causal ordering, i.e.", "a permutation $\\pi :\\lbrace 1, \\ldots , p\\rbrace \\rightarrow \\lbrace 1, \\ldots , p\\rbrace $ such that ${\\mathbf {PA}}^{}_{\\pi (i)} \\subseteq \\lbrace \\pi (1), \\ldots , \\pi (i-1)\\rbrace \\,.$ In this ordering, $\\pi (1)$ is a source node and $\\pi (p)$ is a sink node.", "We can then rewrite the structural equation model in (REF ) as $X_{\\pi (i)} = \\tilde{f}_{\\pi (i)}(X_{\\pi (1), \\ldots , \\pi (i-1)}) + N_{\\pi (i)}\\,,$ where the functions $\\tilde{f}_i$ are the same as $f_i$ except they are constant in the additional input arguments.", "The statement of the proposition then follows by the following argument: consider a one-dimensional random variable $N$ with mean zero and a (possibly multivariate) random vector $X$ both with path-connected support and a continuous function $f$ .", "Then, the support of the random vector $(X,f(X)+N)$ is path-connected, too.", "Indeed, consider two points $(x_0,y_0)$ and $(x_1,y_1)$ from the support of $(X,f(X)+N)$ .", "The path can then be constructed by concatenating three sub-paths: (1) the path between $(x_0,y_0)$ and $(x_0,f(x_0))$ ($N$ 's support is path-connected), (2) the path between $(x_0,f(x_0))$ and $(x_1,f(x_1))$ on the graph of $f$ (which is path-connected due to the continuity of $f$ ) and (3) the path between $(x_1,f(x_1))$ and $(x_1,y_1)$ , analogously to (1).", "Therefore the statements of Lemma 37 and thus Proposition 28 from [8] remain correct, which proves $()$ for noise variables with continuous densities and path-connected support." ], [ "Proof of Proposition ", "The proof is immediate.", "Since $p(x_i \\,\|\\,x_j,x_k,x_c) \\ne p(x_i \\,\|\\,x_j^{\\prime },x_k,x_c)$ (the means are not the same) the statement follows from Lemma REF .", "In this case, Lemma 37 might not hold but more importantly Proposition 28 does [8].", "This proves $(*)$ ." ] ]	1403.0408
[ [ "Approximating Persistent Homology in Euclidean Space Through Collapses" ], [ "Abstract The \\v{C}ech complex is one of the most widely used tools in applied algebraic topology.", "Unfortunately, due to the inclusive nature of the \\v{C}ech filtration, the number of simplices grows exponentially in the number of input points.", "A practical consequence is that computations may have to terminate at smaller scales than what the application calls for.", "In this paper we propose two methods to approximate the \\v{C}ech persistence module.", "Both are constructed on the level of spaces, i.e.", "as sequences of simplicial complexes induced by nerves.", "We also show how the bottleneck distance between such persistence modules can be understood by how tightly they are sandwiched on the level of spaces.", "In turn, this implies the correctness of our approximation methods.", "Finally, we implement our methods and apply them to some example point clouds in Euclidean space." ], [ "Introduction", "Topological data analysis in general, and persistent homology in particular, have shown great promise as tools for analyzing real-world data arising in the sciences.", "Examples of successful applications range from image analysis , , to cancer research , virology and sensor networks .", "Central to persistent homology are standard constructions for recovering the homology of an underlying topological space from a finite sample set, chiefly the Čech and Vietoris–Rips complexes.", "Unfortunately, due to the inclusive nature of their filtrations, the number of simplices grows exponentially in the number of sample points.", "This may be unfortunate as simplices added at small scales may contribute little to homology at larger, possibly more interesting, scales.", "An extreme example may be a constant region in a measurement signal (perhaps from faulty equipment or downtime) under time-delay embedding .", "In such a case, a large proportion of the point cloud may lie in, say, a dense lump of $N$ points that contributes nothing to the cloud's overall homology, yet introduces $\\binom{N}{k+1}$ $k$ -simplices in the complex from an early scale.", "Preprocessing of the point cloud may sometimes rectify the situation, but such schemes are often decidedly “off-line” in the sense that they require a one-off decision about which sparsifications to effectuate ahead of persistence computations.", "We propose more “on-line” methods wherein a decision to attempt a simplification of the simplicial complex may be made at any time during computations when it is deemed necessary.", "The simplification operation itself requires only that the point cloud comes supplied with its complete linkage hierarchical clustering, which may be computed ahead of time once and for all, or the computation of nets." ], [ "Contributions", "The well-known Nerve lemma allows one to capture the topology of a continuous space using discrete structures.", "However, the lemma works under the assumption of a good cover, i.e.", "a cover wherein every finite intersection of covering sets is contractible.", "This means that whenever we have a parametrized sequence of good covers, connected by maps of covers, the persistence diagram captured by the nerves equals the persistence diagram computed by singular homology on the level of spaces.", "A central result in this paper is a way to bound the bottleneck distance between these two persistence diagrams when the covers are not necessarily good.", "Using this result we provide an approximation to the Čech persistence module built on a finite sample from Euclidean space.", "The method enjoys several favorable properties: it approximates the Čech persistence module with provable error bounds and allows for size reduction on a heuristic basis, i.e.", "only when the complex becomes too large to store.", "Unfortunately, computing the weights of the simplices turns out to be expensive, making it inapplicable in most settings.", "To mend this we propose an easy to compute approximation which performs surprisingly well on real data sets.", "Using our aforementioned result we also show that the net-tree construction as introduced by Sheehy and Dey et al.", "works well for the Čech complex in Euclidean space.", "This approach enjoys very powerful theoretical bounds, e.g.", "a linear growth in the number of simplices as a function of sampled points.", "In practice, however, it is difficult to prevent the complex from growing too large.", "Having implemented an algorithm to compute persistence diagrams of simplicial complexes connected by simplicial maps we conclude the paper by applying our approximations to a variety of point samples in Euclidean space.", "To the best of our knowledge, this is the first paper where persistence computations are performed on simplicial complexes connected by more general simplicial maps than inclusions." ], [ "Outline", "In Section we review background material and Dey et al.", "'s algorithm for computing persistent homology of simplicial complexes connected by simplicial maps.", "In particular, we introduce the concept of sequences of covers, and in Section we give a homotopy colimit argument which relates the persistence module associated to a sequence of covers to that formed by the covering sets on the level of spaces.", "This relation is used in Section REF to prove a sandwich type theorem for sequences of covers.", "We give two approaches to approximating the Čech persistence module in Section .", "The paper concludes with Section where we compute the persistence diagrams of example point clouds in Euclidean space using the aforementioned approximations." ], [ "Related work", "In low-dimensional Euclidean space the alpha complex offers a memory efficient way to compute the persistence diagrams of a point cloud.", "Unfortunately, the number of simplices grows exponentially in the ambient dimension, making it inefficient in high-dimensional space.", "The witness complex is a simplicial complex built on a subset of the sample, called landmarks.", "Unfortunately, the persistence diagrams of the associated filtration may depend heavily on the choice of landmarks.", "Sheehy and later Dey et al.", "approximate the Vietoris–Rips complex using net-trees, and Kerber and Sharathkumar arrive at similar results for the Čech complex in Euclidean space using quadtrees.", "Our constructions in Section REF is an adaption on the work of Dey et al.", "to the Čech complex in Euclidean space.", "The construction in Section REF can be viewed as a particular type of a graph induced complex .", "Chazal and Oudot prove the results in Section for the case where all the simplicial maps are inclusions.", "Recent research , , provides methods to reduce the size of simplicial complexes after being stored, e.g.", "to provide faster persistence computations.", "Such reductions are not discussed in this paper as we seek to compute persistence diagrams of point clouds whose filtered complexes are too large to be stored to begin with." ], [ "Background material", "In this section we survey prerequisite background material and fix notation.", "We assume familiarity with basic concepts from algebraic topology, and basic knowledge of persistent homology.", "For introductions see and , respectively.", "Throughout the paper, all simplicial complexes are assumed to be finite and unoriented.", "A simplex is considered a set of vertices, and we write a $k$ -simplex $\\lbrace i_0,\\cdots ,i_k\\rbrace $ as $[i_0,\\cdots ,i_k]$ .", "For a simplicial complex $K$ , we will denote its geometric realization by $\|K\|$ .", "Moreover, if $f: K\\rightarrow L$ is a simplicial map between simplicial complexes, then $\|f\|: \|K\| \\rightarrow \|L\|$ denotes the continuous map between their geometric realizations defined by $f$ on the vertices and extended linearly using barycentric coordinates.", "The $p$ -th singular homology vector space of a topological space $X$ with coefficients in the field $\\mathbb {Z}_2$ will be denoted by $H_p(X)$ , and for a continuous map $f: X\\rightarrow Y$ we denote its induced map on homology by $f_: H_p(X) \\rightarrow H_p(Y)$ .", "When $X = \|K\|$ is the geometric realization of a simplicial complex, we will make no distinction between the $p$ -th simplicial homology vector space of $K$ and the $p$ -th singular homology vector space of $\|K\|$ .", "Cohomology vector spaces over $\\mathbb {Z}_2$ are similarly denoted by $H^p(X)$ .", "A collection of open sets $\\mathcal {U} = \\lbrace U_i \\, \\mid \\,i\\in I\\rbrace $ indexed by a finite set $I$ is said to be a (finite) cover of $\\cup _{i\\in I} U_i$ .", "The nerve $N\\mathcal {U}$ of the cover $\\mathcal {U}$ is the simplicial complex with vertex set $I$ and a $k$ -simplex $[i_0, \\ldots , i_k]\\in N\\mathcal {U}$ if $U_{i_0} \\cap \\cdots \\cap U_{i_k} \\ne \\emptyset $ .", "Let $\\mathcal {U} = \\lbrace U_i \\, \\mid \\,i\\in I\\rbrace $ and $\\mathcal {V} = \\lbrace V_j\\, \\mid \\,j \\in J\\rbrace $ be covers of topological spaces $U\\subseteq V$ .", "A map of sets $F: I\\rightarrow J$ is said to be a map of covers if $U_{i}\\subseteq V_{F(i)}$ for all $i\\in I$ .", "It is easy to check that $F$ extends to a simplicial map $F: N\\mathcal {U} \\rightarrow N\\mathcal {V}$ between the nerves of the covers.", "By a sequence of covers we will mean a collection of covers $\\lbrace \\mathcal {U}(\\alpha )\\, \\mid \\,\\alpha \\in A\\subset [0,\\infty )\\rbrace $ , each indexed respectively by $I(\\alpha )$ , together with maps of covers $F^{\\alpha , \\alpha ^\\prime } : I(\\alpha ) \\rightarrow I(\\alpha ^\\prime )$ such that $F^{\\alpha ,\\alpha } = \\mathrm {id}$ and $F^{{\\alpha },{\\alpha ^{\\prime \\prime }}} =F^{{\\alpha ^\\prime },{\\alpha ^{\\prime \\prime }}}\\circ F^{{\\alpha },{\\alpha ^\\prime }}$ for all $\\alpha ^{\\prime \\prime } \\ge \\alpha ^\\prime \\ge \\alpha $ .", "Such a sequence will be denoted by a pair $(\\mathcal {U}, F)$ .", "Similarly, for any sequence of covers we have an induced sequence of nerves which will be denoted by $(N\\mathcal {U}, F)$ ." ], [ "Persistence modules", "A persistence module $\\mathbb {V}$ over $A\\subseteq \\mathbb {R}$ is a collection of $\\mathbf {k}$ -vector spaces $\\lbrace V(\\alpha ) \\, \\mid \\,\\alpha \\in A \\rbrace $ and linear maps $v^{\\alpha ,{\\alpha ^\\prime }}: V(\\alpha ) \\rightarrow V(\\alpha ^\\prime )$ for all $\\alpha \\le \\alpha ^\\prime $ such that $v^{\\alpha ,\\alpha } = \\mathrm {id}$ and $v^{{\\alpha },{\\alpha ^{\\prime \\prime }}} =v^{{\\alpha ^\\prime },{\\alpha ^{\\prime \\prime }}}\\circ v^{{\\alpha },{\\alpha ^\\prime }}$ .", "The direct sum of two persistence modules $\\mathbb {U}$ and $\\mathbb {W}$ , both indexed over the same set, is the persistence module $\\mathbb {V} = \\mathbb {U}\\oplus \\mathbb {W}$ where $V(\\alpha ) = U(\\alpha )\\oplus W(\\alpha )$ and $v^{\\alpha ,{\\alpha ^\\prime }} = u^{\\alpha ,{\\alpha ^\\prime }}\\oplus w^{\\alpha ,{\\alpha ^\\prime }}$ .", "We say that $\\mathbb {V}$ is indecomposable if the only decompositions of $\\mathbb {V}$ are the trivial decompositions $0\\oplus \\mathbb {V}$ and $\\mathbb {V}\\oplus 0$ .", "Definition 1 Let $J\\subseteq A$ be an interval, i.e.", "if $s,t\\in J$ and $s<r<t$ then $r\\in J$ .", "The interval module over $J$ is the persistence module $\\mathbb {I}^J$ defined by $I^J(\\alpha ) ={\\left\\lbrace \\begin{array}{ll}\\mathbf {k} & \\text{if } \\alpha \\in J\\\\0 & \\text{otherwise}\\end{array}\\right.", "}$ and $i^{\\alpha ,{\\alpha ^\\prime }} = \\mathrm {id}:I^J(\\alpha )\\rightarrow I^J(\\alpha ^{\\prime })$ whenever $\\alpha \\le \\alpha ^\\prime \\in J$ and 0 otherwise.", "It is not difficult to show that $\\mathbb {I}^J$ is indecomposable, and the Krull–Remak–Schmidt–Azumaya theorem tells us that if $&\\mathbb {V}\\cong \\bigoplus _{l\\in L} \\mathbb {I}^{J_l} &\\mathbb {V}\\cong \\bigoplus _{m\\in M}\\mathbb {I}^{K_m},$ then there is a bijection $\\sigma : L\\rightarrow M$ such that $J_l =K_{\\sigma (l)}$ for all $l\\in L$ .", "So whenever $\\mathbb {V}$ admits such a decomposition we can characterize it by the multiset $\\lbrace J_l\\, \\mid \\,l \\in L\\rbrace $ of intervals called the persistence diagram $\\mathbf {D}(\\mathbb {V})$ of $\\mathbb {V}$ .", "An interval $(b,d)\\in \\mathbf {D}(\\mathbb {V})$ represents a feature of $\\mathbb {V}$ with birth and death time $b$ and $d$ , respectively.", "A persistence diagram is usually depicted as a collection of points in $(\\mathbb {R}\\cup \\lbrace \\pm \\infty \\rbrace )^2$ .", "A recent theorem by Crawley-Boevey asserts that $\\mathbb {V}$ admits a decomposition into interval modules if $V_\\alpha $ is finite-dimensional for all $\\alpha \\in \\mathbb {R}$ .", "For an example of a persistence module which does not admit an interval decomposition, see .", "To every sequence of covers $(\\mathcal {U}, F)$ we have an associated persistence module $(H_p(N\\mathcal {U}), F_)$ with vector spaces $\\lbrace H_p(N\\mathcal {U}(\\alpha )) \\, \\mid \\,\\alpha \\in A\\subseteq [0,\\infty ) \\rbrace $ and maps $(F^{\\alpha ,{\\alpha ^\\prime }})_$ .", "As the covers are finite, all the homology vector spaces will have finite dimension, and thus the persistence diagrams are well-defined.", "In particular, if $P\\subseteq M$ is a finite set of points in a metric space $M$ , and $B(p; \\alpha )$ is the open ball of radius $\\alpha $ centered at $p$ , we get a sequence of covers by defining $B(p;0) = \\lbrace p\\rbrace $ , $\\mathcal {U}(P;\\alpha ) = \\lbrace B(p; \\alpha ) \\, \\mid \\,p \\in P\\rbrace $ and $F=\\mathrm {id}$ .", "The induced sequence of nerves is known as the Čech filtration and the associated persistence module is the Čech persistence module.", "In the remainder of this paper $\\mathcal {C}(P; \\alpha )$ denotes the nerve of the Čech filtration of $P$ at scale $\\alpha $ .", "Another popular construction is the Vietoris–Rips complex $\\mathcal {R}(P;\\alpha )$ which is defined as the largest simplicial complex with the same 1-skeleton as $\\mathcal {C}(P; \\alpha )$ .", "By definition, it follows that $\\mathcal {C}(P; \\alpha ) \\subseteq \\mathcal {R}(P; \\alpha )$ , and for $P\\subseteq \\mathbb {R}^n$ , it is also true that $\\mathcal {R}(P; \\alpha ) \\subseteq \\mathcal {C}(P;\\sqrt{2}\\alpha )$ ." ], [ "Metrics and approximations", "Let $\\Delta $ denote the multiset of all pairs $(x,x)\\in (\\mathbb {R}\\cup \\lbrace \\pm \\infty \\rbrace )^2$ , each with countably infinite multiplicity.", "A partial matching between two persistence diagrams $D$ and $D^{\\prime }$ is a bijection $\\gamma : B\\cup \\Delta \\rightarrow B^{\\prime }\\cup \\Delta $ , and we denote all such by $\\Gamma (D,D^{\\prime })$ .", "The following defines a metric on persistence diagrams: Definition 2 The bottleneck distance between two persistence diagrams $B$ and $B^{\\prime }$ is $\\operatorname{d_\\mathrm {B}}(B,B^{\\prime }) = \\inf _{\\gamma \\in \\Gamma (D,D^{\\prime })}\\sup _{(b,d)\\in B} \|\|(b,d)-\\gamma ((b,d))\|\|_\\infty $ where $\|\|(b_1,d_1)-(b_2,d_2)\|\|_\\infty = \\max (\|b_1-b_2\|, \|d_1-d_2\|).$ The theory of interleavings offers a generalization of the bottleneck distance to persistence modules that do not admit a decomposition into indecomposables.", "Importantly, if there exists an $\\epsilon $ -interleaving between two persistence modules, then their bottleneck distance is at most $\\epsilon $ .", "In this paper we adopt the conventions of , and use a slight reformulation of the ordinary theory of interleavings.", "Definition 3 Two persistence modules $\\mathbb {U}$ and $\\mathbb {V}$ indexed over $[0,\\infty )$ are said to be $c$ -approximate if there exist a constant $c\\ge 1$ and two families of homomorphisms $\\lbrace \\phi _\\alpha : U(\\alpha )\\rightarrow V(c\\alpha )\\rbrace _{\\alpha \\ge 0}$ and $\\lbrace \\psi _\\alpha :V(\\alpha ) \\rightarrow U(c\\alpha )\\rbrace _{\\alpha \\ge 0}$ such that the following four diagrams commute for all $\\alpha \\le \\alpha ^\\prime $ : $\\begin{tikzcd}U(\\frac{\\alpha }{c}) {dr}{rrr} &~&~&U(c\\alpha ^\\prime )& U(c\\alpha ) {r}&U(c\\alpha ^\\prime ) \\\\~& V(\\alpha ){r}& V(\\alpha ^\\prime ){ur}& V(\\alpha ) {r}{ur}& V(\\alpha ^\\prime ){ur} & ~\\\\~&U(\\alpha ){r} & U(\\alpha ^\\prime ) {dr} & U(\\alpha ){r}{dr} & U(\\alpha ^\\prime ){dr}& ~\\\\V(\\frac{\\alpha }{c}){ru}{rrr}&~&~&V(c\\alpha ^\\prime ) & V(c\\alpha ){r} & V(c\\alpha ^\\prime )\\end{tikzcd}$ The following theorem is immediate from the theory of interleavings .", "Theorem 4 If $\\mathbb {U}$ and $\\mathbb {V}$ are $c$ -approximate, then their bottleneck distance is bounded by $\\log c$ on the $\\log $ -scale.", "The above result can be seen as a general version of the relationship between the Čech and Vietoris–Rips filtrations.", "Indeed, while the bottleneck distance between their persistence diagrams may be arbitrarily large, the inclusions $\\mathcal {C}(P;\\alpha ) \\subseteq \\mathcal {R}(P;\\alpha ) \\subseteq \\mathcal {C}(P;\\sqrt{2}\\alpha )$ ensure that a feature $(b,d)$ in the Vietoris–Rips persistence module is also a feature in the Čech persistence module if $d-b\\ge \\sqrt{2}b$ , and vice versa." ], [ "Computing persistent homology using\nannotations", "Many widely implemented and used algorithms for computing persistent homology assume that the maps in the persistence module are induced by inclusions of simplicial complexes, i.e.", "that the underlying sequence is a filtration.", "As shall become clear, we will need to compute in the setting of general simplicial maps.", "Definition 5 A surjective simplicial map $f:K\\rightarrow K^{\\prime }$ with the property that there exist distinct $[a],[b]\\in K$ such that $f(\\sigma ) = {\\left\\lbrace \\begin{array}{ll}\\sigma \\setminus \\lbrace b\\rbrace &\\quad \\text{ if $a,b\\in \\sigma $ } \\\\\\lbrace a\\rbrace \\cup \\sigma \\setminus \\lbrace b\\rbrace &\\quad \\text{ if $a\\notin \\sigma $, $b\\in \\sigma $ } \\\\\\sigma &\\quad \\text{ otherwise}\\end{array}\\right.", "}$ is called an edge contraction of $[a,b]$ to $[a]$.", "Simplices $\\sigma ,\\sigma ^{\\prime }\\in K$ are called mirror simplices (for $f$ ) if $f(\\sigma )=f(\\sigma ^{\\prime })$ .", "We will often refer to an edge contraction like that above by $[a,b]\\mapsto [a]$ .", "Since up to isomorphism any simplicial map $K\\rightarrow K^{\\prime }$ decomposes into a finite sequence of inclusions and edge contractions, we only need to deal with those two types and adjust the persistence module indices accordingly to reflect the addition of extra maps.", "Likewise, as is normal, we decompose inclusions into ones of the form $K\\rightarrow K\\cup \\lbrace \\sigma \\rbrace $ and refer to these as “adding a simplex $\\sigma $ ”.", "We will use Dey et al.", "'s method of persistence annotations to compute (the persistence diagrams of) persistence modules with simplicial maps, and now quickly review their algorithm and our implementation details.", "The method of annotation tracks homology with $\\mathbb {Z}_2$ coefficients across a persistence module by storing the value of all cohomology generators at each simplex and updating these “annotations” to reflect the inclusion of a simplex or the contraction of an edge.", "Care should be taken to notice a slight difference in terminology: our definition of annotations reflects Dey's valid annotations.", "Definition 6 An annotation for a simplicial complex $K$ is a linear map $\\Phi _p : C_p(K)\\rightarrow \\mathbb {Z}_2^n$ with the property that $\\varphi _1 = [c\\mapsto \\Phi _p(c)_1],\\cdots , \\varphi _n = [c\\mapsto \\Phi _p(c)_n]$ is a basis for $H^p(K)$ .", "Here $\\Phi _p(c)_i$ denotes the $i$ 'th component of $\\Phi _p(c)\\in \\mathbb {Z}_2^n$ .", "A key observation is the following: the persistent homology of a sequence of simplicial complexes can be obtained by dualizing on the level of chains and taking cohomology.", "This is true since when working over $\\mathbb {Z}_2$ (or any field), the map $\\alpha :H^p(K) \\rightarrow {\\rm Hom}(H_p(K), \\mathbb {Z}_2)$ defined by $\\alpha ([f])([c]) = f(c)$ is an isomorphism.", "Thus, intervals in persistent cohomology are dual to intervals in persistent homology.", "Therefore, we shall interchangeably speak of a homology class born at persistence index $i$ as a cohomology class in the opposite direction dying at persistence index $i$ .", "By storing the value of $\\Phi _p$ at each $p$ -simplex, that simplex' contribution to the (co)homology vector space is known and so allows us to only make changes to homology near the site of a contraction.", "This “locality” of the changes introduced by an edge contraction is summarized in the following definition , proposition and lemmas.", "Definition 7 The link of a simplex $\\sigma $ in a simplicial complex $K$ is the set $\\operatorname{lk}_K\\sigma = \\lbrace \\tau \\setminus \\sigma \\, \\mid \\,\\sigma \\subseteq \\tau \\in K\\rbrace .$ An edge $[a,b]\\in K$ satisfies the link condition if $\\operatorname{lk}_K[a]\\cap \\operatorname{lk}_K[b] = \\operatorname{lk}_K[a,b]$ .", "When the simplicial complex in question is clear, we shall simply write $\\operatorname{lk}$ for $\\operatorname{lk}_K$ .", "Proposition 8 The contraction $f:K\\rightarrow K^{\\prime }$ of an edge that satisfies the link condition induces a homotopy equivalence $\|f\|:\|K\|\\rightarrow \|K^{\\prime }\|$ , and hence an isomorphism $f_\\ast :H_\\ast (K)\\rightarrow H_\\ast (K^{\\prime })$ .", "Lemma 9 If $[a,b]\\in K$ , then $\\operatorname{lk}_K[a,b]\\subseteq \\operatorname{lk}_K[a]\\cap \\operatorname{lk}_K[b]$ .", "Suppose $\\eta \\in \\operatorname{lk}[a,b]$ .", "Then there exists a $\\tau \\in K$ with $[a,b]\\subseteq \\tau $ and $\\eta =\\tau \\setminus [a,b]$ .", "Since $K$ is a simplicial complex, it also contains $\\tau ^{\\prime }=\\tau \\setminus [a]$ and $\\tau ^{\\prime \\prime }=\\tau \\setminus [b]$ .", "We have $[b]\\subseteq \\tau ^{\\prime }$ and $\\eta =\\tau ^{\\prime }\\setminus [b]$ , so $\\eta \\in \\operatorname{lk}[b]$ .", "The same argument using $\\tau ^{\\prime \\prime }$ gives that $\\eta \\in \\operatorname{lk}[a]$ .", "For the following lemma we shall write $L_K(a,b)=(\\operatorname{lk}_{K}[a]\\cap \\operatorname{lk}_{K}[b])\\setminus \\operatorname{lk}_{K}[a,b]$ .", "Lemma 10 If $\\eta \\in L_K(a,b)$ , then $K^{\\prime }=K\\cup \\lbrace \\eta \\cup [a,b]\\rbrace $ is also a simplicial complex, and moreover $L_{K^{\\prime }}(a,b) = L_K(a,b)\\setminus \\lbrace \\eta \\rbrace $ .", "Observe that $[a,b]\\lnot \\subseteq \\eta $ .", "$K^{\\prime }$ is still a simplicial complex, as all faces of $\\eta \\cup [a,b]$ are present in $K$ by the assumption that $\\eta \\in L_K(a,b)$ .", "Note that by definition $&\\operatorname{lk}_{K^{\\prime }}[a] = \\operatorname{lk}_K[a]\\cup \\lbrace \\eta \\cup [b]\\rbrace &&\\operatorname{lk}_{K^{\\prime }}[b] = \\operatorname{lk}_K[b]\\cup \\lbrace \\eta \\cup [a]\\rbrace ,&$ so $\\operatorname{lk}_{K^{\\prime }}[a]\\cap \\operatorname{lk}_{K^{\\prime }}[b]=\\operatorname{lk}_{K}[a]\\cap \\operatorname{lk}_{K}[b]$ .", "It also follows from the definition that $\\operatorname{lk}_{K^{\\prime }}[a,b] = \\operatorname{lk}_{K}[a,b]\\cup \\lbrace \\eta \\rbrace ,$ so $L_{K^{\\prime }}(a,b) = L_K(a,b)\\setminus \\lbrace \\eta \\rbrace $ .", "In summary, we see that to contract an edge we only need to change the simplicial complex in the vicinity of that edge.", "Suppose $K=(K_0\\xrightarrow{}{}K_1 \\xrightarrow{} \\cdots \\xrightarrow{} K_m)$ is a sequence of simplicial complexes (with the $f_i$ 's simplicial maps) whose persistence module $H_\\ast (K)=(H_\\ast (K_0)\\xrightarrow{}{}H_\\ast (K_1) \\xrightarrow{} \\cdots \\xrightarrow{} H_\\ast (K_m))$ has been computed, and write $\\Phi _p^i$ for the annotation of $H^p(K_i)$ and $n$ for its dimension.", "To compute the persistence module of $K^{\\prime }=(K_0\\xrightarrow{}{} \\cdots \\xrightarrow{} K_m \\xrightarrow{} K_{m+1}),$ there are four cases to handle: $f_{m}$ adds a single $p$ -simplex $\\sigma $ , and... $\\Phi _{p-1}^m(\\partial \\sigma )=0$ .", "This corresponds to a generator of $H_p(K^{\\prime })$ being born at persistence index $m+1$ , or equivalently to a generator of $H^p(K^{\\prime })$ dying at $m$ going left (see Proposition 5.2 in ).", "Define $\\Phi _p^{m+1}:C_p(K_{m+1})\\rightarrow \\mathbb {Z}_2^{n+1}$ by $\\Phi _p^{m+1}(\\tau ) = {\\left\\lbrace \\begin{array}{ll}(\\Phi _p^m(\\tau )_1,\\cdots ,\\Phi _p^m(\\tau )_n, 0) &\\quad \\text{ if } \\tau \\ne \\sigma \\\\(0,\\cdots ,0, 1) &\\quad \\text{ if } \\tau = \\sigma \\end{array}\\right.", "}$ and extending linearly.", "In other dimensions $q\\ne p$ , we set $\\Phi _q^{m+1}=\\Phi _q^m$ .", "$\\Phi _{p-1}^m(\\partial \\sigma )_{i_1}=\\cdots =\\Phi _{p-1}^m(\\partial \\sigma )_{i_l}=1$ for some $l\\ge 1$ .", "In this case $\\sigma $ kills a class in $H_{p-1}(K^{\\prime })$ at $m+1$ , or equivalently gives birth to one of the generators $\\varphi _{i_1},\\cdots ,\\varphi _{i_l}$ of $H^{p-1}(K^{\\prime })$ in the reverse direction (see Proposition 5.2 in ).", "We kill the youngest homology class, say the one numbered $u$ (so $\\varphi _u$ is born in the reverse direction).", "Note that $\\gamma :K_{m+1}\\rightarrow \\mathbb {Z}_2^n$ defined by $\\gamma (\\tau ) = {\\left\\lbrace \\begin{array}{ll}\\Phi _{p-1}^m(\\tau ) + \\Phi _{p-1}^m(\\partial \\sigma ) &\\quad \\text{ if } \\Phi _{p-1}^m(\\tau )_u = 1 \\\\\\Phi _{p-1}^m(\\tau ) &\\quad \\text{ otherwise}\\end{array}\\right.", "}$ has 0 in component $u$ of all its values.", "Define $\\Phi _{p-1}^{m+1}:C_{p-1}(K_{m+1})\\rightarrow \\mathbb {Z}_2^{n-1}$ as $\\gamma $ with the $u$ -th component removed, and extend linearly.", "In other dimensions $q\\ne p-1$ , we set $\\Phi _q^{m+1}(\\tau ) = {\\left\\lbrace \\begin{array}{ll}(0,\\cdots ,0) &\\quad \\text{ if } \\tau = \\sigma , q=p\\\\\\Phi _q^m(\\tau ) &\\quad \\text{ otherwise}.\\end{array}\\right.", "}$ $f_{m}$ contracts $[a,b]$ to $[a]$ , and... $[a,b]$ satisfies the link condition.", "Let $M_{p-1}=\\lbrace \\sigma \\in K_m \\, \\mid \\,\\dim \\sigma = p-1 , a\\in \\sigma \\text{ and }\\sigma \\text{ has a mirror under } f_m\\rbrace ,$ and note that to any $\\tau \\in M_{p-1}$ , there is a unique $g_\\tau \\in K_m$ with $\\tau \\subseteq g_\\tau $ , $\\dim g_\\tau = p$ and $[a,b]\\subseteq g_\\tau $ .", "Define $\\Phi _p^{m+1}$ on the $p$ -simplices of $K_{m+1}$ by $\\Phi _p^{m+1}(\\sigma ) = \\Phi _p^m(\\sigma ) + \\underset{\\begin{array}{c}\\sigma \\supseteq \\tau \\in M_{p-1}\\end{array}}{\\sum }\\Phi _p^m(g_\\tau ),$ noting that the sum may be empty.", "This corresponds to Dey's “annotation transfers” — see Proposition 4.4 and 4.5 of for a more detailed explanation.", "$[a,b]$ does not satisfy the link condition.", "Lemma REF tells us which simplices to add, repeatedly hitting the cases REF and REF , until the link condition becomes fulfilledThis must happen after a finite number of steps since Lemma REF shows that the size of $(\\operatorname{lk}[a]\\cap \\operatorname{lk}[b])\\setminus \\operatorname{lk}[a,b]$ is reduced by one every time one of the new simplices is added.", "Moreover, one can in practice expect the number of simplices added to be small compared to the size of $K_m$ since only cofaces of $[a,b]$ are added..", "Afterwards contracting $[a,b]$ is handled by case REF .", "Some bookkeeping is of course required if one wants to consider the potentially many homology changes from the inclusions as occurring at persistence index $m+1$ .", "Dey et al.", "show in (Proposition 5.1) that $\\Phi _\\ast ^{m+1}$ as constructed above is an annotation for $H^\\ast (K_{m+1})$ .", "With $K_0=\\emptyset $ and the associated empty annotation $\\Phi _\\ast ^0$ , then, the above is a correct algorithm for computing persistent homology." ], [ "Some implementation details", "As suggested in , , the simplex tree is a data structure that is well-suited for storing the simplicial complex in the above algorithm.", "A simplex tree is a trie (also called a prefix tree), which is a tree $T$ that stores a simplicial complex $K$ whose vertices $V$ have a total ordering $\\le $ by the following rules: $T$ contains a distinguished root.", "Every non-root node $n\\in T$ carries the data of a label $L(n)\\in V$ .", "The root is labelled by a distinguished symbol, say $\\ast $ , and we extend the ordering to $\\ast < v$ for all $v\\in V$ to ease notation.", "Nodes have zero or more children.", "If $n$ is a child of $p$ , then $L(n)>L(p)$ .", "If $n$ and $m$ both are children of $p$ , then $L(n)\\ne L(m)$ .", "The simplicial complex $K$ to be encoded corresponds to all paths to the root of $T$ , and we write $S(n)\\in K$ for the simplex corresponding to the path from $n\\in T$ .", "We will also refer to the root having depth 0, and in general a node as having depth $k+1$ if its parent has depth $k$ .", "Thus $\\operatorname{depth}(n) = \\dim S(n) + 1$ .", "In terms of implementation, every node holds a pointer to its parent and a dictionaryA dictionary is here any data structure with logarithmic lookup time complexity for keys.", "of pointers to its children, keyed on their labels.", "Furthermore, we augment the tree by adding to each node a “cousin pointer”: We call $m$ a cousin of $n$ if $\\operatorname{depth}(m)=\\operatorname{depth}(n)$ and $L(m) = L(n)$ .", "Every node holds a pointer to one of its cousins in such a way that they form a cyclic linked list that visits every cousin at the same depth precisely once (per cycle).", "In addition, an arbitrary representative of each such cyclic linked list is maintained in a dictionary keyed on labels and depths.", "Figure REF shows an example of the basic part of a simplex tree, along with an example of annotations (intermediate data structures are dropped from the figure, and annotations are attached directly to the simplices for ease of visualization).", "Figure: A somewhat simplified simplex tree representation of asimplicial complex.", "Annotation values on the 1-simplices areincluded for a persistence module in which the simplices are addedin the order⋯,[1,5],[4,5],[2,5],[3,5],[1,4],[2,3],[1,4,5],[2,3,5],[3,4],[1,2]\\cdots ,[1,5],[4,5],[2,5],[3,5],[1,4],[2,3],[1,4,5],[2,3,5],[3,4],[1,2],leading up to the top row situation.", "To contract the edge [1,2][1,2],the link condition must be fulfilled, requiring the inclusion of[1,2,5][1,2,5] (middle row).", "The situation after contraction is shownin the bottom row.Boissonnat and Maria show that this data structure allows us to efficiently insert and remove simplices, and compute their faces and cofaces.", "For details, see .", "To tie the simplex tree to the annotations discussed earlier, we want to associate to each node (i.e.", "each simplex) its annotation value.", "Since multiple simplices are likely to share the same annotation value, we go by way of a union find structure.", "Each node thus contains a pointer to a node in a forest, wherein each tree represents an annotation value shared by multiple cohomologous simplices.", "The root of each tree in the forest points to the actual annotation value of the simplices pointing to nodes in that tree.", "The annotation values themselves are also kept referenced in a dictionary (keyed on the annotation values) for easy access and updating as used in the algorithm outlined earlier." ], [ "Persistent homology of sequences of\ncovers", "In the following we assume that all covering sets are subsets of some metric space and that every cover is finite.", "In particular, this means that all our spaces are paracompact.", "Moreover, the constructions in this section can be seen as special cases of the much more general construction of a homotopy colimit of a diagram of topological spaces.", "To any open covering $\\mathcal {U} = \\lbrace U_i \\, \\mid \\,i\\in I\\rbrace $ of an open set $U$ we assign a topological space $\\Delta U_\\mathcal {U}\\subset \|N\\mathcal {U}\|\\times U$ defined as the disjoint union $\\bigsqcup _{S \\in N\\mathcal {U}}\|S\|\\times \\bigcap _{i\\in S}U_i$ under the equivalence relation $(s, x) \\sim (t,x)$ if $s\\in \|S\|, t\\in \|T\|, S\\subseteq T$ and $s=t$ .", "This construction comes equipped with continuous projection maps $\\pi _1: \\Delta U_\\mathcal {U} \\rightarrow \|N\\mathcal {U}\|$ and $\\pi _2:\\Delta U_\\mathcal {U}\\rightarrow U$ given by projecting onto the first and second factor, respectively.", "Lemma 11 The fiber projecting map $\\pi _2:\\Delta U_\\mathcal {U}\\rightarrow U$ is a homotopy equivalence.", "[sketch.]", "As $U$ is assumed to be paracompact we can choose a partition of unity $\\lbrace \\phi _i\\rbrace _{i\\in I}$ subordinate to $\\mathcal {U}$ and define $g: U \\rightarrow \\Delta U_\\mathcal {U}$ by $g(x) = \\sum _{i\\in I}\\left(\\phi _i(x)v_i, x\\right),$ where $v_i$ is the vertex corresponding to $U_i$ .", "Then $\\pi _2\\circ g = \\text{id}_U$ and it is not difficult to show that $g\\circ \\pi _2 \\simeq \\text{id}_{\\Delta (\\mathcal {U})}$ .", "For a complete proof see .", "Now let $\\mathcal {V} = \\lbrace V_j \\, \\mid \\,j\\in J\\rbrace $ be a finite cover of $V\\supseteq U$ and $F: I\\rightarrow J$ a map of covers.", "Recall that $\|F\|: \|N\\mathcal {U}\|\\rightarrow \|N\\mathcal {V}\|$ denotes the continuous map defined on the vertices by the induced simplicial map between the nerves.", "If we let ${\\rm inc}_U^V : U\\hookrightarrow V$ denote the inclusion of $U$ into $V$ we get the commutative diagram U rincUV V UUr\|F\|incUVu2[swap]d[swap]1 VVu[swap]2d1 \|NU\|r[swap]\|F\| \|NV\| .", "By passing to (singular) homology and using that $\\pi _2$ is a homotopy equivalence we can reverse arrows to find the following commutative diagram: $\\begin{tikzcd}H_(U) {r}{\\left(\\text{inc}_U^V\\right)_}{d}{}[swap]{(\\pi _1)_\\circ (\\pi _2)_^{-1}}&H_(V){d}{(\\pi _1)_\\circ (\\pi _2)_^{-1}}\\\\H_(\|N\\mathcal {U}\|){r}[swap]{\|F\|_}& H_(\|N\\mathcal {V}\|)\\end{tikzcd}$ Example 12 Note that Diagram (REF ) does not commute on the level of spaces: let $\\mathcal {U} = \\lbrace U\\rbrace $ and $\\mathcal {V} = \\lbrace U, V\\rbrace $ where $U\\cap V\\ne \\emptyset $ .", "If $x\\in U\\cap V$ then $(\|F\|\\circ \\pi _1\\circ g)(x)$ is a point in $\|N\\mathcal {U}\|$ whereas $(\\pi _1\\circ g \\circ {\\rm inc})(x)$ can be any point along the edge $\|[U,V]\|$ , depending on the choice of partition of unity.", "See Figure REF .", "Figure: This diagram is an example of the diagram from Example not commuting on the level of spaces.Definition 13 A cover is said to be good if every finite intersection of its sets is contractible.", "The following theorem is one of the great pillars of computational algebraic topology.", "It allows us to use discrete information to capture the topology of a continuous space.", "For a proof see Section 4.G.", "of .", "Theorem 14 If $\\mathcal {U}$ is a good cover, then the base projection map $\\pi _1: \\Delta U_\\mathcal {U} \\rightarrow \|N\\mathcal {U}\|$ is a homotopy equivalence.", "Corollary 15 If $\\mathcal {U}$ is a good cover, then the composition $(\\pi _1)_\\circ (\\pi _2)_^{-1}$ is an isomorphism." ], [ "A sandwich theorem for sequences of covers", "We will use the results from the previous section to prove a sandwich type theorem for sequences of covers.", "The idea is that if a sequence of covers can be sandwiched between two sequences of good covers, then the persistence module associated to the middle sequence approximates the persistence modules associated to the good covers.", "Let $\\left(\\mathcal {U}, F_\\mathcal {U}\\right), \\left(\\mathcal {V},F_{\\mathcal {V}}\\right)$ and $\\left(\\mathcal {W}, F_{\\mathcal {W}}\\right)$ be sequences of covers satisfying $U(\\alpha )\\subseteq V(\\alpha )\\subseteq W(\\alpha ) \\subseteq U(c\\alpha )$ together with maps of covers $F_{\\mathcal {V},\\mathcal {W}}^{\\alpha , \\alpha ^\\prime }&: \\mathcal {V}(\\alpha ) \\rightarrow \\mathcal {W}(\\alpha ^\\prime ) &F_{\\mathcal {W},{\\mathcal {V}}}^{\\alpha , c\\alpha ^\\prime }: \\mathcal {W}(\\alpha ) \\rightarrow \\mathcal {V}(c\\alpha ^\\prime )$ for all $\\alpha ^\\prime \\ge \\alpha $ and a fixed constant $c\\ge 1$ .", "Moreover, we assume that the maps of covers satisfy the following coherence relations: $F_{\\mathcal {W}}^{\\alpha ^\\prime , \\alpha ^{\\prime \\prime }}\\circ F_{{\\mathcal {V}},{\\mathcal {W}}}^{\\alpha , \\alpha ^\\prime } =F_{\\mathcal {V},\\mathcal {W}}^{\\alpha ^\\prime , \\alpha ^{\\prime \\prime }}\\circ F_{\\mathcal {V}}^{\\alpha , \\alpha ^\\prime }& ~ & F_{\\mathcal {W},\\mathcal {V}}^{\\alpha /c, c\\alpha ^\\prime }\\circ F_{\\mathcal {V},\\mathcal {W}}^{\\alpha /c, \\alpha /c} = F_{\\mathcal {V}}^{\\alpha /c, c\\alpha ^\\prime }$ for all $\\alpha ^{\\prime \\prime }\\ge \\alpha ^\\prime \\ge \\alpha $ .", "For notational simplicity we let $\\eta _{\\mathcal {U},\\mathcal {V}}^{\\alpha ,\\alpha ^\\prime } = \|F_{\\mathcal {U},\\mathcal {V}}^{\\alpha ,\\alpha ^\\prime }\|_$ and accordingly for the other maps of covers above.", "From Corollary REF we know that if $\\left(\\mathcal {U}, F_{\\mathcal {U}}\\right)$ and $\\left(\\mathcal {W}, F_{\\mathcal {W}}\\right)$ are sequences of good covers, then there exist unique linear maps $\\eta _{\\mathcal {U},{\\mathcal {V}}}^{\\alpha , \\alpha ^\\prime }$ , $\\eta _{\\mathcal {U},\\mathcal {W}}^{\\alpha ,\\alpha ^\\prime }$ and $\\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha , c\\alpha ^\\prime }$ making the following diagrams commute: $\\begin{tikzcd}H_p(U(\\alpha )){d}{(\\pi _1)_\\circ (\\pi _2)_^{-1}}[swap]{\\cong }{r}{\\left({\\rm inc}_{U(\\alpha )}^{V(\\alpha ^\\prime )}\\right)_}& H_p(V(\\alpha ^\\prime )){d}[swap]{(\\pi _1)_\\circ (\\pi _2)_^{-1}} & H_p(U(\\alpha )){d}{(\\pi _1)_\\circ (\\pi _2)_^{-1}}[swap]{\\cong }{r}{\\left({\\rm inc}_{U(\\alpha )}^{W(\\alpha ^\\prime )}\\right)_} & H_p(W(\\alpha ^\\prime )){d}{\\cong }[swap]{(\\pi _1)_\\circ (\\pi _2)_^{-1}} \\\\H_p(\|N\\mathcal {U}(\\alpha )\|){r}[swap]{\\eta _{\\mathcal {U},{\\mathcal {V}}}^{\\alpha , \\alpha ^\\prime }} & H_p(\|N\\mathcal {V}(\\alpha ^\\prime )\|)& H_p(\|N\\mathcal {U}(\\alpha )\|){r}[swap]{\\eta _{\\mathcal {U},\\mathcal {W}}^{\\alpha ,\\alpha ^\\prime }} & H_p(\|N\\mathcal {W}(\\alpha ^\\prime )\|)\\end{tikzcd}$ $\\begin{tikzcd}~&H_p(W(\\alpha )){d}{(\\pi _1)_\\circ (\\pi _2)_^{-1}}[swap]{\\cong }{r}{\\left({\\rm inc}_{W(\\alpha )}^{U(c\\alpha ^\\prime )}\\right)_} & H_p(U(c\\alpha ^\\prime )){d}{\\cong }[swap]{(\\pi _1)_\\circ (\\pi _2)_^{-1}} \\\\& H_p(\|N\\mathcal {W}(\\alpha )\|){r}[swap]{\\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha , c\\alpha ^\\prime }} & H_p(\|N\\mathcal {U}(c\\alpha ^\\prime )\|)\\end{tikzcd}$ Hence, there are well-defined linear maps $\\phi _\\alpha = \\eta _{\\mathcal {U},\\mathcal {V}}^{\\alpha , c\\alpha }&: H_p(\|N\\mathcal {U}(\\alpha )\|)\\rightarrow H_p(\|N\\mathcal {V}(c\\alpha )\|) \\\\\\psi _\\alpha = \\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha ,c\\alpha }\\circ \\eta _{\\mathcal {V},\\mathcal {W}}^{\\alpha ,\\alpha }&:H_p(\|N\\mathcal {V}(\\alpha )\|) \\rightarrow H_p(\|N\\mathcal {U}(c\\alpha )\|)$ Also, note that the map $\\eta _{\\mathcal {W},\\mathcal {V}}^{\\alpha ,c\\alpha ^\\prime }$ is the unique map that makes Diagram (REF ) commute.", "Theorem 16 If $\\left(\\mathcal {U}, F_{\\mathcal {U}}\\right)$ and $\\left(\\mathcal {W}, F_{\\mathcal {W}}\\right)$ are sequences of good covers, then the families of homomorphisms $\\lbrace \\phi _\\alpha \\rbrace _{\\alpha \\in [0, \\infty )}$ and $\\lbrace \\psi _\\alpha \\rbrace _{\\alpha \\in [0, \\infty )}$ defined in Equation (REF ) satisfy the diagrams of Definition 3.", "In particular, the persistence modules $\\left(H_p(\|N\\mathcal {U}\|), \\eta _{\\mathcal {U}}\\right) \\text{ and } \\left(H_p(\|N\\mathcal {V}\|), \\eta _{\\mathcal {V}}\\right)$ are $c$ -approximate.", "We need to show that the following four relations in Definition REF are satisfied for all $\\alpha \\le \\alpha ^\\prime $ : $&\\psi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}}^{\\alpha , \\alpha ^\\prime }\\circ \\phi _{\\alpha /c} = \\eta _{\\mathcal {U}}^{\\alpha /c, c\\alpha ^\\prime }&&\\psi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}}^{\\alpha , \\alpha ^\\prime } = \\eta _{\\mathcal {U}}^{c\\alpha , c\\alpha ^\\prime }\\circ \\psi _\\alpha \\nonumber \\\\&\\phi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {U}}^{\\alpha , \\alpha ^\\prime }\\circ \\psi _{\\alpha /c} = \\eta _{\\mathcal {V}}^{\\alpha /c, c\\alpha ^\\prime }&&\\phi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {U}}^{\\alpha , \\alpha ^\\prime } = \\eta _{\\mathcal {V}}^{c\\alpha , c\\alpha ^\\prime }\\circ \\phi _\\alpha $ It follows from the uniqueness of the above linear maps, and the associativity of the maps in a sequence of covers, that any map composed out of the maps $\\eta _\\mathcal {U}^{-,-}, \\eta _{\\mathcal {V}}^{-,-}, \\eta _\\mathcal {W}^{-,-}, \\eta _{\\mathcal {U}, \\mathcal {W}}^{-,-}, \\eta _{\\mathcal {W}, \\mathcal {U}}^{-, -}, \\eta _{\\mathcal {W}, \\mathcal {V}}^{-, -} \\hspace{5.69046pt}{\\rm and}\\hspace{5.69046pt} \\eta _{\\mathcal {U}, \\mathcal {V}}^{-,-}$ is uniquely defined by its domain and co-domain.", "That, together with the coherence relations of Equation (REF ), will prove the theorem.", "We will do the top left case of Equation (REF ) in full detail whereas we will refer to uniqueness arguments in the other three cases.", "Top left: $& \\psi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}}^{\\alpha , \\alpha ^\\prime }\\circ \\phi _{\\alpha /c} \\\\&= \\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha ^\\prime ,c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V},\\mathcal {W}}^{\\alpha ^\\prime ,\\alpha ^\\prime }\\circ \\eta _\\mathcal {V}^{\\alpha , \\alpha ^\\prime }\\circ \\eta _{\\mathcal {U}, \\mathcal {V}}^{\\alpha /c, \\alpha } \\\\&= \\eta _{\\mathcal {W}, \\mathcal {U}}^{\\alpha ^\\prime , c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}, \\mathcal {W}}^{\\alpha , \\alpha ^\\prime }\\circ \\eta _{\\mathcal {U}, \\mathcal {V}}^{\\alpha /c, \\alpha } \\hspace{28.45274pt} \\\\&= \\eta _{\\mathcal {W}, \\mathcal {U}}^{\\alpha ^\\prime , c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}, \\mathcal {W}}^{\\alpha , \\alpha ^\\prime }\\circ (\\pi _1\\circ \\pi _2^{-1})_\\circ \\left({\\rm inc}_{U(\\alpha /c)}^{V(\\alpha )}\\right)_\\circ (\\pi _1\\circ \\pi _2^{-1})_^{-1}\\\\&= \\eta _{\\mathcal {W}, \\mathcal {U}}^{\\alpha ^\\prime , c\\alpha ^\\prime }\\circ (\\pi _1\\circ \\pi _2^{-1})_\\circ \\left({\\rm inc}_{V(\\alpha )}^{W(\\alpha ^\\prime )}\\right)_ \\left({\\rm inc}_{U(\\alpha /c)}^{V(\\alpha )}\\right)_\\circ (\\pi _1\\circ \\pi _2^{-1})_^{-1}\\\\&= (\\pi _1\\circ \\pi _2^{-1})_\\circ \\left({\\rm inc}_{U(\\alpha /c)}^{U(c\\alpha ^\\prime )}\\right)_\\circ (\\pi _1\\circ \\pi _2^{-1})_^{-1} \\\\& =\\eta _{\\mathcal {U}}^{\\alpha /c, c\\alpha ^\\prime }$ The second equality follows from the coherence relations.", "Bottom left: By definition, $\\phi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {U}}^{\\alpha , \\alpha ^\\prime }\\circ \\psi _{\\alpha /c}= \\eta _{\\mathcal {U}, \\mathcal {V}}^{\\alpha ^\\prime , c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {U}}^{\\alpha , \\alpha ^\\prime }\\circ \\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha /c, \\alpha }\\circ \\eta _{\\mathcal {V},\\mathcal {W}}^{\\alpha /c, \\alpha /c}$ .", "Using that the composition of the three leftmost maps has same domain and co-domain as $\\eta _{\\mathcal {W},\\mathcal {V}}^{\\alpha /c, c\\alpha ^\\prime }$ , we are left with $\\eta _{\\mathcal {W}, \\mathcal {V}}^{\\alpha /c,c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V},\\mathcal {W}}^{\\alpha /c,\\alpha /c}=\\eta _{\\mathcal {V}}^{\\alpha /c, c\\alpha ^\\prime }$ .", "Here the last equality follows from Equation (REF ).", "Top right: From the coherence relations in Equation (REF ) we find $\\psi _{\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}}^{\\alpha , \\alpha ^\\prime }= \\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha ^\\prime , c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V},\\mathcal {W}}^{\\alpha ^\\prime ,\\alpha ^\\prime }\\circ \\eta _{\\mathcal {V}}^{\\alpha , \\alpha ^\\prime }= \\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha ^\\prime , c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {W}}^{\\alpha , \\alpha ^\\prime }\\circ \\eta _{\\mathcal {V},\\mathcal {W}}^{\\alpha , \\alpha }$ .", "Lastly, we note that $\\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha ^\\prime ,c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {W}}^{\\alpha , \\alpha ^\\prime }$ and $\\eta _{\\mathcal {U}}^{c\\alpha , c\\alpha ^\\prime }\\circ \\eta _{\\mathcal {W},\\mathcal {U}}^{\\alpha , c\\alpha }$ are equal by uniqueness.", "Bottom right: Both sides of the equation are composed out of maps from (REF ).", "The following is a corollary of the proof.", "Corollary 17 Any two of the persistence modules $\\left(H_p(\|N\\mathcal {U}\|),\\eta _{\\mathcal {U}}\\right), \\left(H_p(\|N\\mathcal {V}\|), \\eta _{\\mathcal {V}}\\right) \\text{ and }\\left(H_p(\|N\\mathcal {W}\|), \\eta _{\\mathcal {W}}\\right)$ are $c$ -approximate.", "Note that we do not require the covers in the sequence $(\\mathcal {V},F_{\\mathcal {V}})$ to be good.", "One application of this, which will be pursued in the next section, is the following.", "Let $(\\mathcal {U}, F_{\\mathcal {U}})$ be a sequence of good covers and $(\\mathcal {V}, F_\\mathcal {V})$ another sequence of covers where each open set in $\\mathcal {V}(\\alpha )$ is a union of open sets in $\\mathcal {U}(\\alpha )$ .", "Thus, we have a map of covers $\\mathcal {U}(\\alpha ) \\rightarrow \\mathcal {V}(\\alpha ^\\prime )$ , and more interestingly, a linear map $H_p(\|N\\mathcal {U}(\\alpha )\|)\\rightarrow H_p(\|N\\mathcal {V}(\\alpha ^\\prime )\|)$ .", "However, we do not have a map of covers the other way around, so it is a priori not clear how to define the interleaving map in the opposite direction.", "This is illustrated in Figure REF .", "The previous theorem tells us that such a map can be constructed and gives an upper bound on the bottleneck distance between the associated persistence modules.", "Figure: The map of covers ff is defined as sending a ball to theunion it belongs to, and gg as the obvious map of covers arisingfrom U⊆U ' U\\subseteq U^{\\prime } and V⊆V ' V\\subseteq V^{\\prime }.", "There is no map of covers hh makingthe diagram commute on the level of covers." ], [ "Approximating the Čech complex in Euclidean space", "In this section we construct two different approximation schemes for the Čech persistence module built on a finite set of points in Euclidean space.", "It is clear that for any $c$ -approximation of the Čech persistence module, a $\\sqrt{2}c$ -approximation can be had via the Vietoris–Rips complex built on its 1-skeleton.", "For a treatment of approximate Vietoris–Rips complexes in general metric spaces see , ." ], [ "Linear-size approximation of the Čech persistence module", "This section is an adaption of the work in to the case of Čech complexes in Euclidean space.", "Throughout this section, $P\\subseteq \\mathbb {R}^n$ .", "Definition 18 For a set of points $P$ , we say that $P^\\prime \\subseteq P$ is a $\\delta $ -net of $P$ if for every $p\\in P$ there exists a $p^\\prime \\in P^{\\prime }$ such that $\|\|p-p^\\prime \|\| \\le \\delta $ for any $p,q\\in P^{\\prime }$ , $\|\|p-q\|\|>\\delta $ .", "Choose parameters $\\alpha _0,\\epsilon \\ge 0$ and define a sequence of point sets $P_k$ for $k=0,1, \\cdots , m$ such that $P_0 =P$ and $P_{k+1}$ is an $\\alpha _0\\epsilon ^2(1+\\epsilon )^{k-1}$ -net of $P_k$ .", "We refer to such a collection $P_0,\\cdots ,P_m$ as a net-tree.", "Furthermore, let $\\mathcal {C}(P_k;\\alpha )$ be the Čech complex at scale $\\alpha $ built upon the vertex set $P_k$ , and $U(P_k; \\alpha )$ the union of open balls of radius $\\alpha $ centered at each point in $P_k$ .", "We clearly have maps $\\pi _k: P_k\\rightarrow P_{k+1}$ which send a vertex $p\\in P_k$ to its most nearby vertex in $P_{k+1}$ .", "Lemma 19 For every $k=0, \\ldots , m-1$ we have inclusions $U(P; \\alpha _0(1+\\epsilon )^k) \\subseteq U(P_{k+1}; \\alpha _0(1+\\epsilon )^{k+1}).$ Let $p\\in P=P_0$ and $x\\in \\mathbb {R}^n$ be any point such that $\|\|p -x\|\|<\\alpha _0(1+\\epsilon )^k$ .", "Since $P_{1}$ is an $\\alpha _0\\epsilon ^2(1+\\epsilon )^{-1}$ -net of $P$ we can find $\\pi _0(p)\\in P_1$ such that $\|\|\\pi _0(p) - p\|\| \\le \\alpha _0\\epsilon ^2(1+\\epsilon )^{-1}$ .", "Similarly, we can find $p^\\prime = (\\pi _k\\circ \\cdots \\circ \\pi _0) (p)\\in P_{k+1}$ such that $\|\|p^\\prime - x\|\| &= \|\|\\pi _k\\circ \\cdots \\circ \\pi _0(p) - x\|\|\\\\&\\le \|\|p - x \|\| + \\sum _{i=0}^k \\alpha _0\\epsilon ^2(1+\\epsilon )^{i-1}\\\\&\\le \|\|p-x\|\| + \\frac{\\alpha _0\\epsilon ^2}{1+\\epsilon }\\cdot \\frac{(1+\\epsilon )^{k+1}-1}{\\epsilon }\\\\&< \\alpha _0(1+\\epsilon )^k + \\alpha _0\\epsilon (1+\\epsilon )^{k} = \\alpha _0(1+\\epsilon )^{k+1}.$ In particular, for $p\\in P_k$ we have that $B(p; \\alpha _0(1+\\epsilon )^k)\\subseteq B(\\pi _k(p); \\alpha _0(1+\\epsilon )^{k+1})$ , and thus $\\pi _k: P_k \\rightarrow P_{k+1}$ is a map of covers $\\pi _k: \\mathcal {U}(P_k; \\alpha _0(1+\\epsilon )^k) \\rightarrow \\mathcal {U}(P_{k+1};\\alpha _0(1+\\epsilon )^{k+1}).$ Using this we define a sequence of covers associated to the net tree by defining $\\mathcal {U}^{\\rm net}(P;\\alpha )= \\mathcal {U}(P_k; \\alpha _0(1+\\epsilon )^k)$ where $k$ is the greatest integer such that $\\alpha _0(1+\\epsilon )^k \\le \\alpha $ .", "The maps between the covers are given by compositions of $\\pi _k$ 's.", "We will denote the induced sequence of nerves by $\\mathcal {C}^{\\rm net}(P)$ and the associated persistence module by $(H_p(\\mathcal {C}^{\\rm net}(P)), \\pi _)$ .", "Recall that with this notation we have that $U^{\\rm net}(P; \\alpha ) = U(P_k; \\alpha _0(1+\\epsilon )^k) = \\bigcup _{p\\in P_k} B(p; \\alpha _0(1+\\epsilon )^k)$ Proposition 20 The persistence modules $(H_p(\\mathcal {C}^{\\rm net}(P)), \\pi _)$ and $(H_p(\\mathcal {C}(P)), {\\rm id}_)$ are $(1+\\epsilon )^2$ -approximate.", "Using that $U^{\\rm net}(P; \\alpha ) = U(P_k; \\alpha _0(1+\\epsilon )^k)$ together with Lemma REF we have the chain of inclusions $U^{\\rm net}(P;\\alpha ) \\subseteq U(P; \\alpha ) \\subseteq U(P; \\alpha _0(1+\\epsilon )^{k+1}) &\\subseteq U(P_{k+2}; \\alpha _0(1+\\epsilon )^{k+2}) \\\\&= U^{\\rm net}(P;\\alpha (1+\\epsilon )^2).$ The rest of the proof follows by applying Theorem REF with $\\mathcal {U} = \\mathcal {U}^{\\rm net}(P)$ and $\\mathcal {V} = \\mathcal {W}= \\mathcal {U}(P)$ .", "Proposition 21 Let $P\\subseteq \\mathbb {R}^n$ be a set of $m$ points.", "Then the number of $p$ -simplices in $\\mathcal {C}^{\\rm net}(P;\\alpha _0(1+\\epsilon )^k)$ is $\\mathcal {O}\\left((\\frac{1}{\\epsilon })^{\\mathcal {O}(np)}m\\right)$ .", "This is Theorem 6.3 in together with the fact that the doubling dimension of $\\mathbb {R}^n$ is $\\mathcal {O}(n)$ The net-tree construction exhibits great theoretical properties both with regards to approximating the Čech persistence module and in terms of size complexity.", "In practice however, as we shall see in Section , the complex often grows too large to be stored.", "Not doing a single collapse between scale $\\alpha _0(1+\\epsilon )^k$ and scale $\\alpha _0(1+\\epsilon )^{k+1}$ will in many situations introduce too many new simplices.", "To mend this we introduce a complex which allows for more numerous collapses, at the expense of computation time and poorer error bounds." ], [ "Approximations through non-good covers", "We propose a general framework to approximate persistence modules associated to sequences of good covers.", "Using this framework we give an explicit approximation of the Čech persistence module in Euclidean space.", "Let $(\\mathcal {U},F)$ be a sequence of covers with index sets $\\lbrace I(\\alpha )\\rbrace _{\\alpha \\ge 0}$ and $J(I(\\alpha ))$ a partition of $I(\\alpha )$ .", "We make the following assumption on the partitions: if $J\\in J(I(\\alpha ))$ then for all $\\alpha ^\\prime \\ge \\alpha $ there exists $J^\\prime \\in J(I(\\alpha ^\\prime ))$ such that $J\\subseteq J^\\prime $ .", "In other words, if two elements are partitioned together at some scale $\\alpha $ , they will be partitioned together at all scales $\\alpha ^\\prime \\ge \\alpha $ .", "Moreover, if $J\\in J(I(\\alpha ))$ then $F^{\\alpha ,\\alpha ^\\prime }(J)$ denotes the set $J^\\prime \\in J(I(\\alpha ^\\prime ))$ such that $J\\subseteq J^\\prime $ .", "Lemma 22 For each $\\alpha \\ge 0$ , let $J(I(\\alpha ))$ be a partition of $I(\\alpha )$ as described above.", "Then the pair $(\\widetilde{\\mathcal {U}}, F)$ with $\\widetilde{\\mathcal {U}}(\\alpha ) = \\left\\lbrace \\widetilde{U}_J(\\alpha ) = \\bigcup _{j \\in J} U_j(\\alpha ) \\, \\mid \\,J\\in J(I(\\alpha ))\\right\\rbrace $ is a sequence of covers.", "This follows from that $J\\subseteq F^{\\alpha ,\\alpha ^\\prime }(J)$ for all $J\\in J(I(\\alpha ))$ .", "For such a choice of partitions we say that $(\\widetilde{\\mathcal {U}}, F)$ is coarsening of $(\\mathcal {U},F)$ .", "Let $(\\widetilde{\\mathcal {U}}(P), {\\rm id})$ be any coarsening of the Čech sequence of covers $\\mathcal {U}(P)$ on a finite point set $P\\subseteq \\mathbb {R}^n$ .", "Furthermore, define an associated sequence of good covers $(\\operatorname{CH}\\widetilde{\\mathcal {U}}, {\\rm id})$ where $\\operatorname{CH}{\\widetilde{\\mathcal {U}}}(\\alpha ) = \\left\\lbrace \\operatorname{CH}\\left(\\widetilde{U}_k(\\alpha )\\right) \\, \\mid \\,\\widetilde{U}_k(\\alpha )\\in \\widetilde{\\mathcal {U}}(\\alpha ) \\right\\rbrace ,$ and $\\operatorname{CH}(-)$ denotes the convex hull.", "In the following proposition $(\\widetilde{\\mathcal {C}}(P),\\mathrm {id})$ denotes the induced sequence of nerves of $(\\widetilde{\\mathcal {U}}(P),\\mathrm {id})$ .", "Proposition 23 If there exists a constant $c\\ge 1$ such that $\\operatorname{CH}\\left(\\widetilde{U}_J(\\alpha )\\right) \\subseteq \\bigcup _{j\\in J} U_j(c\\alpha )$ for all $\\alpha \\ge 0$ and all $J\\in J(\\alpha )$ , then the persistence modules $\\left(H_p(\\mathcal {C}(P)),{\\rm id}_\\right)$ and $\\left(H_p(\\widetilde{\\mathcal {C}}(P)), {\\rm id}_\\right)$ are $c$ -approximate.", "We will use Theorem REF .", "We see that the inclusion condition is satisfied by assumption: $U(\\alpha ) \\subseteq U(\\alpha ) \\subseteq \\bigcup _{J\\in J(\\alpha )}\\operatorname{CH}\\left(\\widetilde{U}_J(\\alpha )\\right) \\subseteq U(c\\alpha ).$ Moreover, $\\widetilde{\\mathcal {U}}(P; \\alpha )$ and $\\operatorname{CH}\\widetilde{\\mathcal {U}}(P;\\alpha )$ have the same indexing set, so the coherence relations of Equation (REF ) are trivially satisfied.", "We see that every time we make our cover coarser, the number of 0-simplices in the nerve is reduced, and hence so is the size of the simplicial complex." ], [ "An explicit approximation", "In the previous section we provided a general framework for constructing $c$ -approximations to the Čech persistence module.", "We now give an explicit construction using Proposition REF .", "Lemma 24 Let $P=\\lbrace p_0,p_1, \\ldots , p_k\\rbrace \\subset \\mathbb {R}^n$ where $p_0 = 0$ and $\|\|p_i\|\| \\le \\alpha $ for all $i$ .", "Then for any point $x\\in \\operatorname{CH}(P)$ there exists $p_i\\in P$ such that $\|\|x-p_i\|\|\\le \\alpha /\\sqrt{2}$ .", "By definition of $p_0$ we may assume without loss of generality that $x = (x_1, 0, \\ldots , 0)$ where $x_1 > \\alpha /\\sqrt{2}$ .", "Let $p_i = (p_{i,1}, p_{i,2}, \\ldots p_{i,n})$ be a point in $P$ such that $p_{i,1} \\ge p_{j,1}$ for every other $j$ , and assume that $\|\|x-p_i\|\| > \\alpha /\\sqrt{2}$ .", "Using the law of cosines: $\\alpha ^2 \\ge \|\|p_i\|\|^2 &= \|\|(p_i-x) + x\|\|^2 = \|\|p_i-x\|\|^2 + \|\|x\|\|^2 - 2\|\|p_i-x\|\|\\cdot \|\|x\|\|\\cos (\\angle p_0xp_i)\\\\&> \\frac{\\alpha ^2}{2} + \\frac{\\alpha ^2}{2} - 2\|\|p_i-x\|\|\\cdot \|\|x\|\|\\cos (\\angle p_0xp_i)\\\\&= \\alpha ^2 - 2\|\|p_i-x\|\|\\cdot \|\|x\|\|\\cos (\\angle p_0xp_i)$ implying that $\\cos (\\angle p_0xp_i)>0$ .", "By application of the Euclidean inner product we find $(p_i -x)\\cdot (-x) = -p_{i,1}\\cdot x_1 + x_1^2 = \|\|p_i-x\|\|\\cdot \|\|x\|\|\\cdot \\cos (\\angle p_0xp_i) > 0$ and therefore $p_{i,1} < x_1$ , contradicting that $x$ was enclosed in the convex hull of $P$ .", "Figure REF shows an extreme case of the previous Lemma.", "Figure: The vertices p 0 ,p 1 ,p 2 p_0, p_1, p_2 of an isosceles triangle T=CH{p 0 ,p 1 ,p 2 }T=\\operatorname{CH}\\lbrace p_0,p_1,p_2\\rbrace with legs of length α\\alpha and base of length 2α\\sqrt{2}\\alpha form an extreme case of Lemma as x∈Tx\\in T lies a distance α/2\\alpha /\\sqrt{2} from every vertex.Proposition 25 Let $\\alpha \\ge 0$ and $\\epsilon \\ge 0$ .", "If $P=\\lbrace p_0, p_1, \\cdots , p_k\\rbrace \\subset \\mathbb {R}^n$ is a set of points such that $\|\|p_i-p_j\|\| \\le \\epsilon \\alpha $ , then the following relation holds: $\\operatorname{CH}\\left(\\bigcup _{0\\le i \\le k}B(p_i; \\alpha ) \\right) \\subseteq \\bigcup _{0\\le i \\le k}B\\left(p_i; \\alpha \\sqrt{1+\\frac{\\epsilon ^2}{2}}\\right).$ First, observe that we have the equality $\\operatorname{CH}\\left(\\bigcup _{0\\le i \\le k}B(p_i; \\alpha ) \\right) = \\left\\lbrace x\\in \\mathbb {R}^n \\, \\mid \\,\\exists y\\in \\operatorname{CH}(P), \|\|x- y\|\| < \\alpha \\right\\rbrace .$ Any point $x\\in \\operatorname{CH}(P)$ is contained in the union $\\cup _{0\\le i\\le k} B(p_i; \\epsilon \\alpha /\\sqrt{2})$ by Lemma REF .", "Thus, what remains to be shown is that the proposition holds true for any $x\\in \\mathbb {R}^n$ for which there is a $p\\in \\operatorname{CH}\\lbrace p_{i_0}, \\ldots , p_{i_k}\\rbrace $ , $k\\le n-1$ , such that $\|\|x-p\|\|<\\alpha $ .", "The last inequality follows since $x$ is in the exterior of the convex hull and the most nearby point cannot be strictly inside an $n$ -simplex.", "Denote by $x^\\prime $ the orthogonal projection of $x$ down on the affine space spanned by $\\lbrace p_{i_0}, \\ldots , p_{i_k}\\rbrace $ .", "If $x^\\prime \\in \\operatorname{CH}\\lbrace p_{i_0}, \\ldots , p_{i_k}\\rbrace $ it follows from Lemma REF that there exists a $p_{i_j}$ such that $\|\|p_{i_j} - x\|\|^2 = \|\|p_{i_j} - x^\\prime \|\|^2 + \|\|x-x^\\prime \|\|^2 \\le \\frac{\\epsilon ^2\\alpha ^2}{2} + \\alpha ^2 = \\alpha ^2\\left(1+\\frac{\\epsilon ^2}{2}\\right).$ If $x^\\prime \\notin \\operatorname{CH}\\lbrace p_{i_0}, \\cdots , p_{i_k}\\rbrace $ it implies the existence of a point $p^\\prime $ on the boundary of $\\operatorname{CH}\\lbrace p_{i_0}, \\cdots , p_{i_k}\\rbrace $ such that $\|\|x-p^\\prime \|\| \\le \|\|x-p\|\| < \\alpha $ and we can repeat the process for that point.", "This completes the proof as the proposition is trivially true if $k=0$ .", "Figure: Left: the convex hull of a union of three balls.", "Right: By increasingthe radii of the balls their union eventually covers the convex hull.The previous proposition is illustrated in Figure REF .", "By combining Propositions REF and REF we have shown the following.", "Proposition 26 Let $\\epsilon \\ge 0$ .", "Suppose ${\\widetilde{\\mathcal {U}}}(P)$ is a coarsening of $\\mathcal {U}(P)$ with the property that for every $\\alpha \\ge 0$ and every pair of indices $i, j\\in J\\in J(I(\\alpha ))$ , the inequality $\|\|p_i-p_j\|\|\\le \\alpha \\cdot \\epsilon $ holds.", "Then $H_p(\\widetilde{\\mathcal {C}}(P), {\\rm id}_)$ is a $\\sqrt{1+\\epsilon ^2/2}$ -approximation of the Čech persistence module built on $P$ .", "The previous proposition allows us to build good approximations to the Čech persistence module with far fewer simplices.", "A problem with this approach is that such a memory efficient construction comes at the expense of computing weights of simplices.", "As an example, if $J(I(\\alpha ))$ consists of $k$ partitions, each with $m$ elements, then computing the smallest $\\alpha $ at which they have a $k$ -intersection has time complexity $\\mathcal {O}(m^k)$ .", "To mend this we seek methods to approximate this persistence module by ones that are less computationally expensive.", "The next section details one method for doing so." ], [ "Choosing a representative", "Let $(\\widetilde{\\mathcal {U}}(P), {\\rm id})$ be a coarsening of the Čech sequence of covers and for every $\\alpha \\ge 0$ and every $J\\in J(\\alpha )$ choose a representative $p_j\\in P$ , where $j\\in J$ .", "Denote the set of representatives at scale $\\alpha $ by $P_\\alpha $ .", "For every $\\alpha \\ge 0$ we define the subcomplex $\\mathcal {C}^{\\rm rep}(P; \\alpha )\\subseteq \\widetilde{\\mathcal {C}}(P; \\alpha )$ to be the smallest simplicial complex such that: $\\mathcal {C}(P_\\alpha ; \\alpha )\\subseteq \\mathcal {C}^{\\rm rep}(P; \\alpha )$ ${\\rm id}^{\\alpha , \\alpha ^\\prime }: \\widetilde{\\mathcal {C}}(P;\\alpha )\\rightarrow \\widetilde{\\mathcal {C}}(P;\\alpha ^\\prime )$ restricts to a simplicial map $\\mathcal {C}^{\\rm rep}(P; \\alpha ) \\rightarrow \\mathcal {C}^{\\rm rep}(P; \\alpha ^\\prime )$ The idea is to choose a set of representatives, one for each element $J(I(\\alpha ))$ , and use those representatives to approximate the persistent homology computation.", "However, to get a well-defined sequence of simplicial complexes and simplicial maps, we need to make sure that the image of a simplex spanned by one set of representatives is a simplex at a later filtration time, where the set of representatives may be different.", "Thus, our approximate complex contains the simplicial complex built on the set of representatives and, in addition, the images of simplices spanned by representatives at earlier filtration times.", "Proposition 27 The persistence modules $(H_p(\\widetilde{\\mathcal {C}}(P)), \\mathrm {id}_)$ and $(H_p(\\mathcal {C}^{\\rm rep}(P)), \\mathrm {id}_*)$ are $\\frac{1}{1-\\epsilon }$ -approximate.", "The simplicial complexes $\\mathcal {C}^{\\rm rep}(P_\\alpha ; \\alpha )$ and $\\widetilde{\\mathcal {C}}(P; \\alpha )$ are defined over the same indexing set $J(I(\\alpha ))$ for every $\\alpha \\ge 0$ .", "This follows from having chosen one representative for each covering set of $\\widetilde{\\mathcal {U}}(\\alpha )$ .", "Now choose $x\\in U_J\\in \\widetilde{\\mathcal {U}}(\\alpha )$ , where $\|\|x-p_j\|\|<\\alpha $ for some $j\\in J$ , and let $p$ be the representative of $\\mathrm {id}^{\\alpha ,\\alpha /(1-\\epsilon )}(J)\\in J(I(\\alpha /(1-\\epsilon )))$ .", "Then $\|\|p-x\|\| \\le \|\|p-p_j\|\| + \|\|x-p_j\|\| < \\frac{\\alpha \\epsilon }{1-\\epsilon } + \\alpha = \\frac{\\alpha }{1-\\epsilon }.$ Hence, we have a map of covers $\\widetilde{\\mathcal {U}}(P;\\alpha )\\rightarrow \\mathcal {U}(P_{\\alpha /(1-\\epsilon )}; \\frac{\\alpha }{1-\\epsilon })$ which induces the first map of the composition $\\widetilde{\\mathcal {C}}(P; \\alpha ) \\rightarrow \\mathcal {C}\\left(P_{\\alpha /(1-\\epsilon )}; \\frac{\\alpha }{1-\\epsilon }\\right)\\subseteq \\mathcal {C}^{\\rm rep}\\left(P_{\\alpha /(1-\\epsilon )}; \\frac{\\alpha }{1-\\epsilon }\\right)\\subseteq \\widetilde{\\mathcal {C}}\\left(P; \\frac{\\alpha }{1-\\epsilon }\\right)$ The proof follows from application of Definition REF and Theorem REF ." ], [ "Relationship to graph induced complexes", "We conclude this section by briefly discussing a related construction introduced in by Dey et al.", "Definition 28 Let $G(V)$ be a graph with vertex set $V$ and let $\\nu : V\\rightarrow V^\\prime $ be a vertex map where $\\nu (V) = V^\\prime \\subseteq V$ .", "The graph induced complex $\\mathcal {G}(V, V^\\prime , \\nu )$ is defined as the simplicial complex where a $k$ -simplex $[v_1^\\prime , \\ldots , v_{k+1}^\\prime ]$ is in $\\mathcal {G}(V, V^\\prime , \\nu )$ if and only if there exists a $(k+1)$ -clique $\\lbrace v_1, \\ldots , v_{k+1}\\rbrace \\subseteq V$ such that $\\nu (v_i) = v_i^\\prime $ for each $i\\in \\lbrace 1, \\ldots , k+1\\rbrace $ .", "First we note that a coarsening of a cover as defined at the beginning of Section REF induces a graph induced complex.", "Indeed, just choose a representative for each partition and let $\\nu $ be the map which takes a vertex to its representative.", "This, together with a net-tree construction as in Section REF , is utilized in to construct a linear-size approximation to the Vietoris–Rips persistence module.", "Constructing the analogue Čech approximation is straightforward and it can be shown that it enjoys error bounds similar to what we proved in Section REF .", "In fact, the analogue Čech construction is nothing more than forming a coarsening of the Čech sequence of covers where the process of partitioning covering sets is determined by a net-tree.", "Unfortunately, as discussed at the end of Section REF , computing the $k$ -intersections needed for this construction is very time consuming." ], [ "Computational experiments", "This section details our implementation of the approximation schemes described above, as well as some computational examples examining their efficacy and practical applicability." ], [ "Implementation", "We realize an implementation of the approximation schemes detailed in Section REF as a C++ program in the following way.", "The program takes as parameters $\\epsilon \\ge 0$ (as in Section REF ), a maximal scale $\\alpha _{\\mathrm {max}}>0$ (as usual when computing persistence), a maximal simplex dimension $D>0$ (as usual) and $L\\in \\mathbb {N}$ (to be explained later).", "Given an input point cloud $P=\\lbrace p_1,\\cdots ,p_N\\rbrace \\subseteq \\mathbb {R}^d$ , we first use Müllner's fastcluster to compute its hierarchical clustering $\\operatorname{HC}(P)$ with the complete linkage criterion.", "This is considered a preprocessing step.", "A cluster is a pair $(p, X)$ with $p\\in X\\subseteq P$ , wherein $p$ will be called the cluster's representative and $X$ its members.", "At initialization time, we begin with $N$ clusters $c^0_1=(p_1,\\lbrace p_1\\rbrace ), c^0_2=(p_2,\\lbrace p_2\\rbrace ),\\cdots ,c^0_N=(p_N, \\lbrace p_N\\rbrace ).$ and denote their enumeration by $C^0=\\lbrace 1,\\cdots ,N\\rbrace $ .", "We shall regard $\\operatorname{HC}(P)$ as the data of a series of linkage events of the form $(s, i,j)\\in \\mathbb {R}\\times \\mathbb {N}\\times \\mathbb {N}$ ordered by the first component, and (arbitrarily) with the convention that $i<j$ .", "An event like this signifies the linking of clusters $c^l_i=(p^l_i, X^l_i)$ and $c^l_j=(p^l_j, X^l_j)$ at scale $s$ , from which we form a new cluster $c^{l+1}_i=(p^{l+1}_i, X^l_i\\cup X^l_j)$ where $p^{l+1}_i\\in X^l_i\\cup X^l_j$ ; in principle the new representative $p^{l+1}_i$ can be chosen arbitrarily from $X^l_i\\cup X^l_j$ , but for heuristic reasons we pick the point in the member set $X^l_i\\cup X^l_j$ closest to that set's centroid.", "We maintain a priority queue $Q$ of simplices prioritized by their persistence time.", "At initialization, the queue contains the 0-simplices $[1],\\cdots ,[N]$ all at persistence time 0.", "A simplex tree, along with associated annotations and other data structures as described in Section REF , are also initialized empty.", "These data structures that track homology will jointly be referred to as $\\operatorname{PH}$ below, and we shall abuse language and speak of a simplex as “belonging to $\\operatorname{PH}$ ” when the simplex is present in the simplicial complex.", "We also initialize $\\alpha ^{\\prime }=0$ and $l=0$ to begin with.", "The implementation code then proceeds in the following steps: If $Q$ is empty, we are done and go to step REF .", "If not, pop a simplex $\\sigma $ and its persistence scale $\\alpha $ from the front, and continue.", "If $\\alpha > \\alpha _{\\mathrm {max}}$ , we are done and go to step REF .", "Otherwise continue.", "If $\\sigma $ is not already in $\\operatorname{PH}$ , add it according to Section REF .", "In both cases, continue.", "If $\\dim \\sigma >D$ , go to step REF .", "Otherwise, for each simplex $\\tau \\in \\lbrace \\sigma \\cup \\lbrace i\\rbrace \\, \\mid \\,i\\in C^l\\rbrace $ : computeOur implementation uses Gärtner's Miniball for this computation.", "the radius $r_\\tau $ of the smallest enclosing ball of the set $\\lbrace p^l_i\\, \\mid \\,i\\in \\tau \\rbrace \\subseteq P$ , and add $\\tau $ to $Q$ at persistence scale $r_\\tau $ .", "Go to step REF .", "If at least $L$ simplices have been added to $\\operatorname{PH}$ since the last time this step was reached, we (possibly) perform a simplification by going to step REF .", "Otherwise go to step REF .", "For each linkage event $(s,i,j)\\in \\operatorname{HC}(P)$ for which $s\\in [\\alpha ^{\\prime },\\epsilon \\alpha )$ , perform the edge contraction $[i,j]\\mapsto [i]$ according to Section REF , taking care to adjust persistence times to reflect a (possible) series of inclusions to satisfy the link condition.", "If there were no linkage events in the given interval, go to step REF .", "Otherwise, denote the clusters present after handling the linkage event, as explained earlier in this section, by $\\lbrace C^{l+1}_{i_1},\\cdots ,C^{l+1}_{i_{N_{l+1}}}\\rbrace \\subseteq \\lbrace C^{l}_{j_1},\\cdots ,C^{l}_{j_{N_{l}}} \\rbrace $ and go to step REF .", "Clear $Q$ and reset it to contain the 0-simplices $[i_1],\\cdots ,[i_{N_l}]$ , all at persistence scale 0.", "Update $l$ to $l+1$ and $\\alpha ^{\\prime }$ to $\\epsilon \\alpha $ , and go to step REF .", "We are done.", "Any persistent homology generators not yet killed off are recorded as on the form $(b,\\infty )$ .", "The algorithm above may be summarized as follows: Compute Čech persistence until the underlying simplicial complex has at least $L$ simplices.", "When that is the case, walk up the complete linkage dendrogram of the point cloud until scale $\\epsilon \\alpha $ is reached, where $\\alpha $ is the persistence scale.", "Any linkage event encountered corresponds to an edge contraction, which is performed.", "After that, computation of Čech persistence resumes as before, albeit on a reduced and changed point cloud, and collapses may happen again when $L$ more simplices have been added.", "We terminate upon reaching $\\alpha _{\\mathrm {max}}$ , and ignore simplices of dimension above $D$ (thus computing homology in dimensions $0,\\cdots ,D-1$ ).", "Note that $L$ is merely a parameter to reduce computational overhead involved in the collapses, as a higher value postpones contractions until the simplicial complex is denser.", "In principle, $L$ can be thought of as zero.", "Also observe that $\\epsilon =0$ corresponds to computing ordinary Čech persistence." ], [ "Experiments", "This section describes three experiments designed to test the feasibility of our implementation.", "A calculation ranging from scale 0 to scale $\\alpha _{\\mathrm {max}}$ will have its resulting persistence diagram drawn as the region above the diagonal in $[0,\\alpha _{\\mathrm {max}}]^2$ .", "Generators still alive at $\\alpha _{\\mathrm {max}}$ will be referred to as on the form $(b,\\infty )$ and plotted as triangles, while generators of the form $(b,d)$ with $d\\le \\alpha _{\\mathrm {max}}$ will be plotted as dots.", "See Figure REF for an example of drawing conventions." ], [ "Wedge of six circles enclosing each other", "We produced a point cloud by randomly (uniformly) sampling 100 points from a circle of radius 1 centered at $(0,1)$ , 200 points from a circle of radius 2 centered at $(0,2)$ and so forth up to 600 points from a circle of radius 6 centered at $(0,6)$ .", "Each point in the circle of radius $r$ was perturbed by radial noise sampled from the uniform distribution on $[(1-0.02)r, (1+0.02)r)$ .", "The very dense region near the origin where all the circles meet (see Figure REF ) contributes nothing to homology, but significantly adds to the number of simplices if no collapse is done.", "Running to $\\alpha _{\\mathrm {max}}=2$ , our implementation clearly limited the number of simplices — see Figure REF and note especially the rapid increases between collapses, the regimes where the ordinary Čech filtration is formed — while producing a highly correct persistence diagram, as is shown in Figure REF .", "Figure: The point cloud from the example in Section .Figure: Persistence diagrams of the (noisy) wedge of six circles inSection with ϵ=3/4\\epsilon =3/4 andα max =2\\alpha _{\\mathrm {max}}=2.Figure: The simplex count while computing persistence for theexample in Section .", "The net tree computations were run with α 0 =10 -3 \\alpha _0=10^{-3} and ϵ=0.7\\epsilon =0.7 in the notation of Section ." ], [ "The real projective plane", "We sampled $\\mathbb {R}P^2$ by randomly selecting 5000 points on $\\mathbb {S}^2$ and embedding them in $\\mathbb {R}^4$ under $(x,y,z)\\mapsto (xy, xz, y^2 -z^2, 2yz)$ as a test of how well our scheme handles higher dimensions.", "Figure REF shows that the expected persistence diagram resulted when computing to $\\alpha _{\\mathrm {max}}=0.54$ at $\\epsilon =1.0$ .", "Figure REF compares our scheme (at $\\epsilon =1$ ) with the very beginning an ordinary Čech filtration.", "Our implementation keeps the number of simplices manageable, peaking at just above $3\\cdot 10^5$ simplices near the end (scale $0.54$ ), while still recovering the correct persistence diagram.", "The figure also shows the simplex count for the net tree construction; notice that we were unable to correctly choose $\\alpha _0$ and $\\epsilon $ so as to make computations with it feasible, unlike for the example in Section REF .", "Figure: Persistence diagrams for the 5000 point random sample ofℝP 2 \\mathbb {R}P^2 embedded in ℝ 4 \\mathbb {R}^4 as described inSection , with ϵ=1.0\\epsilon =1.0 andα max =0.54\\alpha _{\\mathrm {max}}=0.54.Figure: The simplex count for the ℝP 2 \\mathbb {R}P^2 example fromSection compared to that of an ordinaryČech filtration and the net tree approach (with α 0 =10 -3 \\alpha _0=10^{-3} and ϵ=0.7\\epsilon =0.7 in the notation of Section )." ], [ "Time-delay embedding", "We solved the Lorenz system (with parameters $\\sigma =10$ , $r=28$ , $b=8/3$ in the notation of ) and created a time series $y\\in \\mathbb {R}^{15000}$ by adding together all three of the solution's coordinates at each of 15000 points in time.", "Let $A(i)$ denote the (discrete) correlation of $y$ and $y$ shifted $i$ places to the right.", "The first local minimum of $A$ occurs at 15, so that was used as delay to embed $y$ in $\\mathbb {R}^3$ by delay-embedding.", "The resulting point cloud, with $15000-(3-1)\\cdot 15 = 14970$ points, reconstructs the Lorenz attractor as seen in Figure REF .", "Observe that there are regions that have a very high density of points.", "Our implementation computes the expected persistence diagram (Figure REF ) while keeping the number of simplices low (Figure REF ).", "Figure: Lorenz system scalar measurements (parts shown on the left)and delay-embedding reconstructed attractor (right), as detailedin Section .Figure: Persistence diagrams for the Lorenz attractor described in Section .Figure: Simplex count for the Lorenz attractor computations described in Section ." ], [ "Conclusions and future work", "We have presented two approximation schemes for the Čech filtration in Euclidean space.", "One construction uses a net-tree to build the Čech complex at fewer and fewer simplices as we increase the scale parameter.", "The other approach forms a coarsening of the Čech filtration by using covering sets formed by unions of open balls.", "Computing $k$ -intersections of such covering sets is computationally expensive, so we approximated the persistence module by choosing a representative at each scale.", "In practice we experienced far better results with this method than the net-tree approach.", "This contrasts with the superior theoretical guarantees enjoyed by the net-tree construction.", "By approximating the Čech filtration through representatives we lose much of the theoretical guarantees, but the frequent collapses allow for much greater maximum scales.", "We believe that an interesting direction for future work is to find other approximations than choosing a representative for each covering set.", "This could be done either by choosing multiple representative points, or by using the embedding to approximate the covering sets by sets for which computing $k$ -intersections is tractable.", "The proofs in this paper also rely heavily on the notion of good covers.", "In general metric spaces a cover by a union of balls may fail to be good, and the Nerve lemma is lost.", "It would be interesting to see if there are similar results without this precondition.", "We believe it should be so, as the net-tree construction for the Vietoris–Rips filtration extends to general metric spaces." ] ]	1403.0533
[ [ "Steady-state Mechanical Squeezing in an Optomechanical System via\n Duffing Nonlinearity" ], [ "Abstract Quantum squeezing in mechanical systems is not only a key signature of macroscopic quantum effects, but can also be utilized to advance the metrology of weak forces.", "Here we show that strong mechanical squeezing in the steady state can be generated in an optomechanical system with mechanical nonlinearity and red-detuned monochromatic driving on the cavity mode.", "The squeezing is achieved as the joint effect of nonlinearity-induced parametric amplification and cavity cooling, and is robust against thermal fluctuations of the mechanical mode." ], [ "Introduction ", "Enormous progress has been achieved in the field of cavity optomechanics in the past few years [1].", "Examples include the preparation of mechanical modes to their quantum ground state, the demonstration of strong optomechanical coupling in the microwave and optical regimes, and the coherent state conversion between cavity and mechanical modes [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].", "Given these technological advances, the effective quantum manipulation of mechanical modes becomes a promising goal.", "Quantum squeezing of mechanical modes is one of the key macroscopic quantum effects that can be utilized to study the quantum-to-classical transition and to improve the precision of quantum measurements [12], [13], [14], [15], [16], [17], [18].", "Thermal squeezing of mechanical modes using parametric processes and measurement-based ideas has been demonstrated in recent experiments [19], [20], [21], [22], [23].", "In simple schemes using parametric amplification, squeezing is limited by the so-called 3 dB limit – quantum noise cannot be reduced below half of the standard quantum limit – due to the instability of the mechanical systems [24].", "In recent years, a number of schemes have been proposed to generate mechanical squeezing that can go beyond the 3 dB limit, including methods based on parametric processes, measurement- and feedback-based schemes, as well as approaches utilizing the concept of quantum reservoir engineering [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43].", "However, quantum squeezing of mechanical modes has not been observed experimentally.", "Note that in recent experiments, squeezing in optical fields has been achieved in optomechanical systems [44], [45], [46].", "These experiments have the potential to reach a squeezing level well below the quantum limit.", "Figure: The schematic of an optomechanical system with mechanical mode bb (nonlinearity η\\eta ), main cavity aa, and an ancilla cavity a s a_{s}.", "The pump field on cavity aa (a s a_{s}) is indicated by amplitude Ω d \\Omega _d (Ω p \\Omega _p) and frequency ω d \\omega _d (ω p \\omega _p).", "The detection circuit is enclosed by gray-dashed lines.Here we present a method to generate strong steady-state mechanical squeezing in an optomechanical system via mechanical nonlinearity and cavity cooling.", "The mechanical nonlinearity required in this scheme is achieved by coupling the mechanical mode to an ancilla system, such as an external electrode or a qubit, and its magnitude far exceeds that of the intrinsic mechanical nonlinearity [47], [48].", "The driving on the cavity is a red-detuned monochromatic source which generates strong optomechanical coupling between the cavity and the mechanical modes and greatly reduces the thermal fluctuations of the mechanical mode.", "This driving, when combined with the nonlinearity of the mechanical mode, also induces a parametric-amplification process which plays a key role in generating squeezing.", "We find that near an optimal detuning point, strong squeezing well below the standard quantum limit can be reached even at high temperatures.", "Meanwhile, the red-detuned driving serves to protect the system from instability.", "The mechanical squeezing can be detected by homodyning the output field of an ancilla cavity mode driven by a second pump pulse.", "Compared with previous works, our proposal only requires one driving source on the main cavity and is robust against thermal fluctuations.", "The parametric-amplification process induces a huge increase in the effective mechanical frequency which strongly suppresses the quantum backaction noise.", "Our proposal could help the generation of strong quantum squeezing in mechanical systems.", "This paper is organized as follows.", "In Sec.", ", we introduce an optomechanical system with mechanical nonlinearity and derive its effective Hamiltonian under strong driving.", "In Sec.", ", we study the steady-state squeezing of the mechanical mode and identify the optimal parameter regime for the squeezing.", "Analytical solutions of two limiting cases are presented in Sec.", ", and the detection of the mechanical squeezing is discussed in Sec. .", "In Sec.", ", we discuss the validity of the linearization procedure and the effect of the detection on the proposed squeezing scheme.", "Conclusions are given in Sec.", "." ], [ "System ", "Consider the optomechanical system depicted in Fig.", "REF with the Hamiltonian ($\\hbar =1$ ) $H_{t}&=&H_{c}+H_{m}-g_{0}a^{\\dagger }a(b^{\\dagger }+b), \\\\H_{c}&=&\\delta _{a}a^{\\dagger }a +\\Omega _{d}(a^{\\dagger }+a), \\\\H_{m}&=&\\omega _{m}b^{\\dagger }b+(\\eta /2)(b^{\\dagger }+b)^{4},$ where $a$ ($a^{\\dagger }$ ) and $b$ ($b^{\\dagger }$ ) are the annihilation (creation) operators of the cavity mode and the mechanical mode, respectively.", "The cavity mode (with frequency $\\omega _{a}$ ) is described by the Hamiltonian $H_{c}$ written in the rotating frame of a monochromatic driving field with detuning $\\delta _{a}$ and amplitude $\\Omega _{d}$ .", "The Hamiltonian of the mechanical mode $H_{m}$ (with frequency $\\omega _{m}$ ) contains a Duffing nonlinear term with amplitude $\\eta $ .", "The last term in Eq.", "(REF ) describes the radiation-pressure interaction between the cavity and the mechanical modes with coupling strength $g_{0}$ [49].", "For mechanical modes in the sub-gigahertz range, the intrinsic nonlinearity is usually very weak with nonlinear amplitude smaller than $10^{-15}\\,\\omega _m$ [47].", "A strong nonlinearity can be produced by coupling the mechanical mode to an ancilla system [50], [51], [52], [53].", "For example, by coupling the mechanical mode to a qubit, a nonlinear amplitude of $\\eta =10^{-4}\\,\\omega _m$ can be obtained (see Appendix for details).", "Other approaches can also be applied to enhance the nonlinearity, such as by softening the mechanical mode [35], [54].", "Note that nonlinearity in other forms, such as cubic potential $\\eta (b+b^{\\dag })^{3}$ , can also be utilized to implement our scheme (see Appendix for details).", "When including the dissipation caused by the system-bath coupling, the full dynamics of this optomechanical system is described by the master equation $\\dot{\\rho }=-i[H_{t},\\rho ]+\\kappa \\mathcal {D}[a]\\rho +\\gamma (\\bar{n}_{\\textrm {th}}+1)\\mathcal {D}[b]\\rho +\\gamma \\bar{n}_{\\textrm {th}}\\mathcal {D}[b^{\\dagger }]\\rho .", "$ Here $\\mathcal {D}[o]\\rho =o\\rho o^{\\dagger }-(o^{\\dagger }o\\rho +\\rho o^{\\dagger }o)/2$ is the standard Lindblad superoperator for the damping of the cavity and the mechanical modes, $\\kappa $ and $\\gamma $ are the cavity and the mechanical damping rates, respectively, and $\\bar{n}_{\\rm th}$ is the thermal phonon occupation number.", "Figure: (a) The steady-state amplitudes \|α\|\|\\alpha \| and β\\beta versus the driving power PP.", "(b) The squeezed mechanical frequency ω m ' \\omega _{m}^{\\prime } and the coupling constant G ' G^{\\prime } versus PP.", "The asterisks in (a) are obtained with the detection circuit included (see Eq. ()).", "The frequencies of the cavity modes aa and a s a_s are ω a /2π=500 THz \\omega _a/2\\pi =500\\,\\textrm {THz} and ω s /2π=1000 THz \\omega _s/2\\pi =1000\\,\\textrm {THz}, respectively.", "The driving amplitudes are Ω d =2Pκ/ω a \\Omega _d=\\sqrt{2P\\kappa /\\omega _{a}} and Ω p =2P s κ s /ω s \\Omega _p=\\sqrt{2P_s\\kappa _s/\\omega _{s}}.", "Other parameters are ω m /2π=2 MHz \\omega _m/2\\pi =2\\,\\textrm {MHz}, g 0 =g s =10 -4 ω m g_0=g_s=10^{-4}\\,\\omega _m, η=10 -4 ω m \\eta =10^{-4}\\,\\omega _m, κ=κ s =0.1ω m \\kappa =\\kappa _s=0.1\\,\\omega _m, γ=10 -6 ω m \\gamma =10^{-6}\\,\\omega _m, and P s =0.1P_s=0.1 μ\\mu W.Strong red-detuned driving on the cavity generates large steady-state amplitudes in both the cavity and the mechanical modes.", "Let $\\alpha $ ($\\beta $ ) be the steady-state amplitude of the cavity (mechanical) mode under the red-detuned driving.", "Using the standard linearization procedure, the steady-state amplitudes can be derived by solving the following equations: $\\left[-i(\\delta _a-2g_0\\beta )-\\kappa /2\\right]\\alpha -i\\Omega _{d}=0, \\\\16\\eta \\beta ^3+(12\\eta +\\omega _{m})\\beta -g_0\|\\alpha \|^2=0, $ where we have dropped $\\gamma $ -dependent terms because $\\gamma \\ll \\kappa , \\eta $ .", "With (moderately) strong driving on the cavity, these amplitudes satisfy $\|\\alpha \|,\\,\\beta \\gg 1$ , as shown in Fig.", "REF (a).", "At a driving power of $P=0.1\\,\\textrm {mW}$ , $\|\\alpha \|\\approx 10^{3}$ and $\\beta \\approx 40$ , consistent with our assumptions for linearization.", "In the vicinity of the steady-state amplitudes, the master equation of our optomechanical system has the same form as that in Eq.", "(REF ) but with $H_{t}$ replaced by a shifted Hamiltonian $H_{\\textrm {sh}} = H_{\\textrm {eff}}+H_{\\textrm {nl}}$ .", "Here $H_{\\textrm {eff}}=\\Delta _{a}a^{\\dagger }a+\\tilde{\\omega }_{m}b^{\\dagger }b+\\Lambda (b^{2}+b^{\\dagger 2})-G(a+a^{\\dagger })(b+b^{\\dagger }),$ only containing linear and bilinear terms with the following coefficients $& \\Delta _a =\\delta _a-2g_0\\beta ,\\quad \\tilde{\\omega }_m =\\omega _m+2\\Lambda ,\\nonumber \\\\& \\Lambda =3\\eta (4\\beta ^2+1),\\quad G=g_0\|\\alpha \|; $ at the same time $H_{\\textrm {nl}}=&- g_0a^{\\dagger }a\\left(b+b^{\\dag }\\right) + \\frac{1}{2}\\eta (b^{\\dagger 4}+4b^{\\dagger 3}b+3b^{\\dagger 2}b^2 \\nonumber \\\\+& 8\\beta b^{\\dagger 3}+24\\beta b^{\\dagger 2}b+h.c.", "),$ composed of all the nonlinear terms generated by the radiation-pressure interaction and the Duffing nonlinearity.", "The operator $a$ ($b$ ) here and hereafter is the shifted operator defined relative to the steady-state amplitude $\\alpha $ ($\\beta $ ).", "With $g_0,\\, \\eta \\beta \\ll \\Lambda , G$ , these nonlinear terms in $H_{\\rm nl}$ are much weaker than the linear and bilinear terms in $H_{\\rm eff}$ .", "After neglecting the nonlinear terms, the master equation becomes $\\dot{\\rho }=- i[H_{\\textrm {eff}},\\rho ]+\\kappa \\mathcal {D}[a]\\rho +\\gamma (\\bar{n}_{\\textrm {th}}+1)\\mathcal {D}[b]\\rho +\\gamma \\bar{n}_{\\textrm {th}}\\mathcal {D}[b^{\\dagger }]\\rho ,$ governed by the effective Hamiltonian $H_{\\rm eff}$ and the damping terms.", "The third term in $H_{\\rm eff}$ describes a parametric-amplification process induced by the Duffing nonlinearity and plays a key role in squeezing generation [55].", "This term can also be viewed as an increase of the spring constant of the mechanical mode.", "The last term in $H_{\\rm eff}$ describes an effective optomechanical coupling that causes cooling and heating of the mechanical mode [56], [57], [58], [59], [60].", "Parametric-amplification processes induce instability.", "Applying the Routh-Hurwitz criterion [61], we derive the stability condition for this system: $16G^2<(\\omega _m+4\\Lambda )(4\\Delta _a+\\kappa ^2/\\Delta _a),$ for red-detuned driving with $\\Delta _a>0$ .", "This condition is satisfied in all relevant parameter regimes in our scheme (see Appendix for details).", "Interestingly, at the optimal detuning point for squeezing (see below), this condition can be simplified to be $g_{0}<\\sqrt{27\\omega _m\\eta }$ , independent of the driving power $P$ .", "Meanwhile, our parameter regimes are well separated from the bistability threshold for a Duffing oscillator." ], [ "Mechanical squeezing ", "Apply the squeezing transformation $S(r) = \\exp [r(b^{2}-b^{\\dagger 2} )/2]$ with squeezing parameter $r=(1/4)\\ln (1+4\\Lambda /\\omega _{m})$ to the effective Hamiltonian $H_{\\textrm {eff}}$ [36].", "Under this transformation, $S^{\\dagger }(r)bS(r) = b\\cosh (r) - b^{\\dag }\\sinh (r)$ and $S^{\\dagger }(r)aS(r) = a$ .", "The Hamiltonian is hence transformed to be $H_{\\textrm {eff}}^{\\prime }=S^{\\dagger }(r)H_{\\textrm {eff}}S(r)$ with $H_{\\textrm {eff}}^{\\prime }=\\Delta _{a}a^{\\dagger }a+\\omega ^{\\prime }_{m}b^{\\dagger }b-G^{\\prime }(a+a^{\\dagger })(b^{\\dagger }+b),$ where $\\omega _{m}^{\\prime }=\\omega _{m}\\sqrt{1+4\\Lambda /\\omega _{m}}$ is the transformed mechanical frequency and $G^{\\prime }=G(1+4\\Lambda /\\omega _{m})^{-1/4}$ is the transformed optomechanical coupling.", "In Fig.", "REF (b), we plot $\\omega _{m}^{\\prime }$ and $G^{\\prime }$ as functions of the driving power $P$ , both of which increase monotonically with $P$ .", "At a driving power of $P=0.1\\,\\textrm {mW}$ , we have $\\omega _{m}^{\\prime }\\approx 3\\,\\omega _{m}$ and $G^{\\prime }\\approx \\,0.6 G$ .", "We then apply the squeezing transformation $S(r)$ to the master equation in Eq.", "(REF ) and define the transformed density matrix $\\rho ^{\\prime }=S^{\\dagger }(r)\\rho S(r)$ .", "It can be shown that $S^{\\dagger }(r) \\mathcal {D}[a]\\rho S(r) = \\mathcal {D}[a]\\rho ^{\\prime }$ and $&S^{\\dagger }(r) \\mathcal {D}[b]\\rho S(r) = \\cosh ^{2}(r) \\mathcal {D}[b] \\rho ^{\\prime } + \\sinh ^{2}(r) \\mathcal {D}[b^{\\dag }] \\rho ^{\\prime } & \\nonumber \\\\& - \\cosh (r)\\sinh (r)\\left( \\mathcal {G}[b] +\\mathcal {G}[b^{\\dag }]\\right) \\rho ^{\\prime }& $ with $\\mathcal {G}[o]\\rho =o\\rho o-(oo\\rho +\\rho oo)/2$ .", "Similar result can be obtained for the term $S^{\\dagger }(r) \\mathcal {D}[b^{\\dag }]\\rho S(r)$ .", "With the condition $\\Delta _{a}, \\omega _{m}^{\\prime } \\gg G^{\\prime }, \\gamma (\\bar{n}_{th}+1)$ , the $\\mathcal {G}[b]\\rho ^{\\prime }$ and $\\mathcal {G}[b^{\\dag }]\\rho ^{\\prime }$ terms in the above equation are fast oscillating with factors $\\sim e^{\\pm 2 i \\omega _{m}^{\\prime } t }$ and can be neglected under the rotating-wave approximation (RWA).", "The validity of this approximation is manifested in Fig.", "REF , where numerical result calculated from the transformed master equation agrees accurately with the result from the original master equation.", "Hence under the RWA, the transformed master equation for the density matrix $\\rho ^{\\prime }$ has the same form as Eq.", "(REF ) with $H_{\\textrm {eff}}$ replaced by $H_{\\textrm {eff}}^{\\prime }$ and $\\bar{n}_{\\textrm {th}}$ replaced by $\\bar{n}_{\\textrm {th}}^{\\prime }=\\bar{n}_{\\textrm {th}}\\cosh (2r)+\\sinh ^{2}(r).$ Note that the mechanical damping rate is not affected by the squeezing transformation.", "As the Hamiltonian $H_{\\textrm {eff}}^{\\prime }$ only contains linear and bilinear couplings between the cavity and the mechanical modes, the transformed master equation for $\\rho ^{\\prime }$ describes a standard cavity cooling process with thermal phonon number $\\bar{n}_{\\textrm {th}}^{\\prime }$ [57], [58], [59].", "Figure: (Color online) The squeezing of XX (in units of dB) versus Δ a \\Delta _a and Λ\\Lambda at n ¯ th =0\\bar{n}_{\\textrm {th}}=0.", "Parameters are the same as in Fig. .", "The dashed and solid lines correspond to squeezing at the optimal detuning (Δ a =ω m ' \\Delta _{a}=\\omega _{m}^{\\prime }) and at 3 dB, respectively.The squeezing of the mechanical mode can be calculated by solving the above master equation.", "The steady-state density matrix $\\rho _{\\rm ss}^{\\prime }$ in the transformed frame can be derived by solving Eq.", "(REF ) numerically.", "The steady-state average of an arbitrary operator $A$ in the original frame (before the transformation) is $\\langle A\\rangle =\\textrm {Tr}[S^{\\dagger }(r)AS(r)\\rho _{\\rm ss}^{\\prime }]$ .", "For the displacement quadrature $X=(b+b^{\\dag })/\\sqrt{2}$ of the mechanical mode, its steady-state variance can then be derived as $\\langle \\delta X^{2}\\rangle _{\\rm ss}=\\left(\\bar{n}_{\\textrm {eff}}^{\\prime }+\\frac{1}{2}\\right)e^{-2r},$ where $\\bar{n}_{\\textrm {eff}}^{\\prime }$ is the steady-state phonon number of the transformed system and is determined by the cooling process.", "Best cooling in the transformed system occurs at the optimal detuning $\\Delta _{a}=\\omega _{m}^{\\prime }$ .", "Hence Eq.", "(REF ) shows that at a given driving power (given $r$ and $\\Lambda $ ), squeezing is strongest at the optimal detuning.", "This is clearly illustrated by the dashed contour in Fig.", "REF .", "For comparison, we also plot the contour of the 3 dB limit where $\\langle \\delta X^{2}\\rangle _{\\rm ss}=1/4$ .", "Figure: (Color online) (a) The variance 〈δX 2 〉 ss \\langle \\delta X^{2}\\rangle _{\\rm ss} versus n ¯ th \\bar{n}_{\\textrm {th}} at selected driving powers.", "(b) The variance 〈δX 2 〉 ss \\langle \\delta X^{2}\\rangle _{\\rm ss} versus PP at selected n ¯ th \\bar{n}_{\\textrm {th}}.", "All plots are at the optimal detuning.", "Other parameters are the same as in Fig. .", "The shadowed blue bottom region corresponds to squeezing beyond the 3 dB limit.", "The solid curves (circles) correspond to exact numerical solution (analytical solution in the strong-coupling limit).In Fig.", "REF (a), we plot $\\langle \\delta X^{2}\\rangle _{\\rm ss}$ as a function of the average thermal phonon number $\\bar{n}_{\\textrm {th}}$ .", "The variance is proportional to $\\bar{n}_{\\textrm {th}}$ with a slope that decreases with the driving power.", "This can be explained by Eq.", "(REF ) where the variance increases with the effective phonon number $\\bar{n}_{\\textrm {eff}}^{\\prime }$ which is proportional to $\\bar{n}_{\\textrm {th}}$ .", "Our result also shows that as the driving power reaches a threshold value, squeezing exceeding 3 dB can be reached.", "Even at a high temperature with $\\bar{n}_{\\textrm {th}}\\sim 10^{4}$ , strong steady-state squeezing can still be achieved by increasing the driving power.", "The dependence of the variance on the driving power is shown in Fig.", "REF (b), where the mechanical squeezing becomes stronger as the driving power increases." ], [ "Cooling limit ", "To better understand the proposed squeezing scheme, we study limiting cases that have analytical solutions.", "First, consider the limit of $G^{\\prime }\\ll \\kappa \\ll \\omega _{m}^{\\prime }$ , where a cooling equation for the mechanical mode can be derived from the master equation in the transformed basis by adiabatically eliminating the cavity mode [57], [58], [59].", "Let $\\mu ^{\\prime }=\\textrm {Tr}_{a}[\\rho ^{\\prime }]$ be the reduced density matrix of the mechanical mode.", "The cooling equation is $\\dot{\\mu }^{\\prime }&=&-i[\\omega _{m}^{\\prime }b^{\\dagger }b,\\mu ^{\\prime }]+[\\gamma (\\bar{n}_{\\textrm {th}}^{\\prime }+1)+\\Gamma _{-}]\\mathcal {D}[b]\\mu ^{\\prime }\\nonumber \\\\&+&(\\gamma \\bar{n}_{\\textrm {th}}^{\\prime }+\\Gamma _{+})\\mathcal {D}[b^{\\dagger }]\\mu ^{\\prime }$ with the rates $\\Gamma _{\\mp }=\\frac{\\kappa (G^{\\prime })^{2}}{\\kappa ^{2}/4+(\\omega _{m}^{\\prime }\\mp \\Delta _{a})^{2}}.$ The steady state of Eq.", "(REF ) is a thermal state with average phonon number $\\bar{n}_{\\textrm {eff}}^{\\prime }=\\frac{\\gamma \\bar{n}_{\\textrm {th}}^{\\prime }+\\Gamma _{+}}{\\gamma +\\Gamma },$ where $\\Gamma =\\Gamma _{-}-\\Gamma _{+}$ is the cooling rate.", "At the optimal detuning $\\Delta _{a}=\\omega _{m}^{\\prime }$ , $\\Gamma _{-}=4(G^{\\prime })^2/\\kappa $ , $\\Gamma _{+}\\approx \\kappa (G^{\\prime }/2\\omega _{m}^{\\prime })^{2}$ , and strong cooling can be achieved.", "The density matrix of the mechanical mode in the original basis $\\mu =S\\mu ^{\\prime }S^{\\dag }$ is hence a squeezed thermal state.", "The variance of the squeezed mechanical quadrature depends on the squeezing parameter $r$ and the cooling rate $\\Gamma $ , both of which are determined by the driving power." ], [ "Strong-coupling limit ", "Next, we consider the strong-coupling limit with $\\kappa \\ll G^{\\prime }\\ll \\omega ^{\\prime }_{m}$ .", "In this limit, by omitting the counter-rotating terms ($ab+a^{\\dag }b^{\\dag }$ ) in the optomechanical coupling, we can derive analytical solution for the squeezing.", "At the optimal detuning, we obtain $\\langle \\delta X^{2}\\rangle _{\\rm ss}=\\frac{2\\gamma \\bar{n}_{\\textrm {th}}+\\gamma +2\\Gamma _{\\textrm {sc}}e^{-2r}}{4(\\gamma +\\Gamma _{\\textrm {sc}})}$ with cooling rate $\\Gamma _{\\textrm {sc}}=\\frac{4(G^{\\prime })^2\\kappa }{\\kappa ^{2}+\\kappa \\gamma +4(G^{\\prime })^2}.$ The contribution of the thermal noise in $\\langle \\delta X^{2}\\rangle _{\\rm ss}$ is reduced by a factor $~\\gamma /2\\Gamma _{\\textrm {sc}}$ due to the cavity cooling.", "At zero temperature and with ultra-strong driving (when $e^{-2r}\\ll 1$ ), the squeezing will be ultimately limited by $\\langle \\delta X^{2}\\rangle _{\\rm ss}=\\frac{\\gamma }{\\gamma +4\\Gamma _{\\textrm {sc}}},$ which can be approximated as $\\langle \\delta X^{2}\\rangle _{\\rm ss}\\approx \\gamma /4\\kappa $ .", "For a typical optomechanical system with $\\gamma \\ll \\kappa $ , this indicates a strong squeezing well below the standard quantum limit.", "This analytical solution is shown in Fig.", "REF .", "It can be seen that it agrees well with that of exact numerical solution." ], [ "Detection of squeezing ", "To detect the mechanical squeezing generated in our approach, we consider an ancilla cavity mode $a_s$ (with resonant frequency $\\omega _s$ ) driven by a pump field of amplitude $\\Omega _p$ and frequency $\\omega _p$ , as depicted in Fig.", "REF .", "The frequency separation between the cavity modes $a$ and $a_s$ is much larger the frequency of the mechanical mode, i.e., $\|\\omega _a-\\omega _s\|\\gg \\omega _m$ .", "With the detection circuit included, the total Hamiltonian of this system becomes $H_{\\rm dec}=H_{t}+\\delta _{s} a_s^{\\dagger }a_s-g_s a_s^{\\dagger }a_s(b^{\\dagger }+b)+\\Omega _p(a_s^{\\dagger }+a_s),$ where $H_{t}$ is given by Eq.", "(REF ), $\\delta _s=\\omega _s-\\omega _p$ is the detuning of the ancilla mode $a_s$ , and $g_s$ is the strength of the single-photon optomechanical coupling.", "Under pumping, the ancilla mode reaches a steady-state amplitude $\\alpha _{s}$ .", "The effective Hamiltonian is then $H^{\\textrm {dec}}_{\\textrm {eff}}=H_{\\rm eff}+\\Delta _s a_s^{\\dagger }a_s-G_s(a_s+a_s^{\\dagger })(b+b^{\\dagger }),$ where $H_{\\rm eff}$ is given by Eq.", "(REF ), $\\Delta _s=\\delta _s - 2g_s\\beta $ , and $G_s = g_s\\alpha _{s}$ .", "As shown in Ref.", "[62], both the position and the momentum quadratures of the mechanical resonator in the original frame (untransformed frame) can be measured by homodyning the output field of the ancilla mode with a local oscillator.", "Effective detection of the mechanical state requires that $\\alpha _{s}\\gg 1$ while $G_{s}\\ll \\kappa _{s}$ , where $\\kappa _{s}$ is the damping rate of the ancilla cavity mode.", "Meanwhile, to reduce the detection backaction on the mechanical mode, it requires that $\\alpha _{s}\\ll \\alpha $ when the coupling constants $g_{s}\\sim g_{0}$ .", "We choose $P_s\\approx 0.1\\,\\mu \\textrm {W}$ for an ancilla cavity of $\\omega _{s}/2\\pi =1000\\,\\textrm {THz}$ , which leads to $\\alpha _{s}\\approx 50$ .", "With these parameters, the output field of the mode $a_{s}$ provides a direct measurement of the quadrature variances of the mechanical resonator.", "A weak force applied to the mechanical resonator can be detected by measuring the output field of the ancilla cavity.", "The weak impulsive force generates a displacement of the mechanical state in its phase space of the original frame, which can be detected from the output field within a finite time window shorter than the inverse of the cooling rate $\\Gamma _{s}=4G_{s}^{2}/\\kappa _{s}$ .", "Strong squeezing of the mechanical mode ensures that the detection of this force has a resolution far exceeding the standard quantum limit [16], [17]." ], [ "Discussions ", "In the previous sections, we showed that mechanical squeezing robust against thermal noise can be generated under the effective Hamiltonian $H_{\\rm eff}$ , where the nonlinear Hamiltonian $H_{\\rm nl}$ and the backaction of the detection circuit are omitted from the discussion.", "To evaluate the validity of the linearization procedure, we numerically solve the master equation that includes the nonlinear Hamiltonian and plot the steady-state variance $\\langle \\delta X^2\\rangle _{\\rm ss}$ in Fig.", "REF .", "Our results show no distinguishable difference between the solutions with and without the linearization approximation.", "Similarly, we study the influence of the detection on our squeezing scheme.", "In Fig.", "REF , the steady-state amplitudes $\|\\alpha \|$ and $\\beta $ are plotted in the presence of the detection circuit; and in Fig.", "REF , the steady-state variance $\\langle \\delta X^2\\rangle _{\\rm ss}$ is plotted.", "Our results show that detection has negligible effect on the mechanical squeezing.", "Figure: The steady-state variance 〈δX 2 〉 ss \\langle \\delta X^2\\rangle _{\\rm ss} versus the driving power PP at n ¯ th =10 2 \\bar{n}_{\\rm th}=10^2.", "Solid line: solution under the linearized Hamiltonian H eff H_{\\rm eff}; squares: with H nl H_{\\rm nl} included; asterisks: with detection circuit included.", "Here Δ a =ω m ' \\Delta _a=\\omega _m^{\\prime }, Δ s =ω m \\Delta _s=\\omega _m, and other parameters are the same as in Fig.", "." ], [ "Conclusions ", "To conclude, we presented a method to generate steady-state mechanical squeezing that is robust against thermal fluctuations.", "Our approach utilizes mechanical nonlinearity and strong driving on the cavity mode in an optomechanical system.", "The mechanical squeezing is a consequence of the joint effect of the nonlinearity-induced parametric amplification and cavity cooling.", "We showed that strong squeezing can be achieved at the optimal detuning where the cavity detuning is in resonance with the transformed mechanical frequency.", "Analytical solutions in two limiting cases are derived.", "In a wide range of driving power and thermal phonon number, squeezing well below the standard quantum limit can be achieved.", "The steady-state squeezing can be detected by measuring the output field of an ancilla cavity mode.", "XYL thanks Prof. Ying Wu and Prof. Hui Jing for valuable discussions.", "XYL and JQL are supported by the JSPS Foreign Postdoctoral Fellowships under No.", "P12204 and No.", "P12503, respectively.", "XYL is also supported by NSFC-11374116.", "LT is supported by the DARPA ORCHID program through the AFOSR, NSF-DMR-0956064, and the NSF-COINS program under No.", "NSF-EEC-0832819.", "FN is supported by the RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics, and Grant-in-Aid for Scientific Research (S)." ], [ "Generation of strong Duffing nonlinearity ", "In this appendix, we provide detailed discussion on the generation of strong mechanical nonlinearity.", "Various approaches have been studied to generate strong nonlinearity by coupling the mechanical resonator to an ancilla system [50], [51], [52], [53].", "We focus on the method in Ref.", "[52], where the nonlinearity is generated by coupling the mechanical resonator to an ancilla qubit.", "Consider an ancilla qubit with the Hamiltonian $H_{q}=(\\Delta _{q}/2)\\sigma _{x}$ , which couples to the mechanical mode via an interaction $\\lambda _{q} X\\sigma _{z}$ .", "This coupling induces an effective Duffing nonlinearity on the mechanical resonator in the form of $H_{m}^{(4)}=6\\Delta _{q} (\\lambda _{q}/\\Delta _{q})^{4}X^{4}$ , when the qubit is in an eigenstate of $\\sigma _{x}$ and under the condition $\\lambda _{q}/\\Delta _{q}\\ll 1$ .", "With $\\Delta _{q}/2\\pi =5\\,\\textrm {GHz}$ and $\\lambda _{q}=38\\,\\textrm {MHz}$ , $H_{m}^{(4)}$ gives a nonlinear amplitude $\\eta /2\\pi \\sim 0.2\\,\\textrm {kHz}$ and $\\eta /\\omega _m\\sim 10^{-4}$ , close to the parameters we used in our calculation.", "Note that the second order term induced by the qubit-resonator coupling has been absorbed into the spring constant of the mechanical resonator.", "For a typical driving power of $P=0.1\\,\\textrm {mW}$ , the dimensionless mechanical displacement in the stationary state is $X\\sim 50$ .", "The mechanical mode thus generates a backaction on the qubit in the form of $0.6\\,\\textrm {GHz}\\,\\sigma _{x}$ , the amplitude of which is much weaker than the detuning of the qubit.", "Hence, the ancilla qubit can be treated as a passive system that is not affected by the mechanical backaction." ], [ "Squeezing with cubic nonlinearity ", "In the main text, we showed that strong mechanical squeezing in the steady state can be generated for a mechanical mode with Duffing nonlinearity.", "In principle, mechanical nonlinearity in other forms can also be utilized to generate squeezing.", "In this section, we show that a cubic nonlinearity in the form of $\\eta (b+b^{\\dagger })^3$ can also be used to generate strong mechanical squeezing.", "We start with the linearization procedure for a mechanical mode with cubic nonlinearity.", "Let us denote the steady-state amplitude of the cavity (mechanical) mode as $\\alpha _{c}$ ($\\beta _{c}$ ).", "We find that these amplitudes satisfy the following nonlinear equations: $[-i(\\delta _c-2g_0\\beta _c)-\\kappa /2]\\alpha _{c}-i\\Omega _d=0, \\\\12\\eta \\beta _{\\rm c}^2+\\omega _m\\beta _{\\rm c}+3\\eta -g^2_0\|\\alpha _{\\rm c}\|^2=0,$ where we have dropped $\\gamma $ -dependent terms for $\\gamma \\ll \\kappa ,\\eta $ .", "The quantum master equation in terms of the shifted operators can be written as $\\dot{\\rho }&=&-i[H_{\\textrm {sh}}^{c},\\rho ]+\\kappa \\mathcal {D}[a]\\rho +\\gamma \\left(\\bar{n}_{\\rm th}+1\\right)\\mathcal {D}[b]\\rho \\nonumber \\\\&+&\\gamma \\bar{n}_{\\rm th}\\mathcal {D}[b^{\\dagger }]\\rho ,$ where the total Hamiltonian has the form $H_{\\textrm {sh}}^{c}=H_{\\textrm {eff}}^{c}-g_0a^{\\dagger }a(b+b^{\\dagger })+(3\\eta b^{\\dagger 2}b+\\eta b^{\\dagger 3}+h.c.", "),$ and $H_{\\textrm {eff}}^{c}$ is composed of the linear and bilinear terms with $H_{\\textrm {eff}}^{c}&=&\\Delta _{a}^{c}a^{\\dagger }a+\\tilde{\\omega }^{c}_{m}b^{\\dagger }b+\\Lambda ^{c} \\left(b^{2}+b^{\\dagger 2}\\right)\\nonumber \\\\&-&G^{c}\\left(a+a^{\\dagger }\\right)\\left(b+b^{\\dagger }\\right).$ The parameters in the above equations are $\\Delta _a^{c}&=\\delta _a-2g_0\\beta _{c},\\quad \\Lambda ^{c}=6\\eta \\beta ,\\nonumber \\\\\\tilde{\\omega }_m^{c}&=\\omega _m+2\\Lambda ^{c},\\quad G^{c}=g_{0}\|\\alpha _{c}\|.$ With $\|\\alpha _{c}\|,\\beta _c\\gg 1$ , the nonlinear terms can be neglected and $H_{\\textrm {sh}}$ can be approximated by the effective Hamiltonian $H_{\\textrm {eff}}^{c}$ .", "We want to point out that the Hamiltonian $H_{\\textrm {eff}}^{c}$ has exactly the same form as $H_{\\textrm {eff}}$ in Eq.", "(REF ) with its parameters depending on the specific form of the cubic nonlinearity.", "The squeezing of the mechanical mode can be achieved similarly as in the case of the Duffing nonlinearity." ], [ "Stability condition ", "In this appendix, we study the stability of our system by applying the Routh-Hurwitz criterion to the equations of motion (the Langevin equations) of this system.", "Based on the Hamiltonian $H_{\\textrm {eff}}$ , the equations of motion of this system can be written as $\\dot{\\textbf {R}}\\left( t\\right) =\\textbf {A}\\textbf {R}\\left( t\\right) -\\textbf {R}_{in}\\left( t\\right),$ where we introduce the operator vectors $\\textbf {R}(t) =(a^{\\dagger },a,b^{\\dagger },b)^{T}$ for the system operators and $\\textbf {R}_{in}(t) =(\\sqrt{\\kappa }a_{in}^{\\dagger },\\sqrt{\\kappa }a_{in},\\sqrt{\\gamma }b_{in}^{\\dagger },\\sqrt{\\gamma }b_{in})^{T}$ for the input noise operators, and the matrix $\\textbf {A}$ is $\\textbf {A}=\\left(\\begin{array}{cccc}i\\Delta _{a}-\\frac{\\kappa }{2} & 0 & -iG & -iG \\\\0 & -i\\Delta _{a}-\\frac{\\kappa }{2} & iG & iG \\\\-iG & -iG & i\\tilde{\\omega }_{m}-\\frac{\\gamma }{2} & 2i\\Lambda \\\\iG & iG & -2i\\Lambda & -i\\tilde{\\omega }_{m}-\\frac{\\gamma }{2}\\end{array}\\right).$ The stability for this system is determined by the eigenvalues of the matrix $\\textbf {A}$ .", "If all the eigenvalues of $\\textbf {A}$ have negative real parts, then the system is stable.", "Based on the fact that the similarity transformation does not change the eigenvalues of a matrix, below we apply a similarity transformation $\\textbf {V}=\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & \\cosh r & -\\sinh r \\\\0 & 0 & -\\sinh r & \\cosh r\\end{array}\\right)$ with $r=(1/4)\\ln (1+4\\Lambda /\\omega _{m})$ to the matrix $\\textbf {A}$ .", "The transformed matrix becomes $\\textbf {A}^{\\prime }&=\\textbf {V}^{-1}\\textbf {A}\\textbf {V}\\nonumber \\\\&=\\left(\\begin{array}{cccc}i\\Delta _{a}-\\frac{\\kappa }{2} & 0 & -iG^{\\prime } & -iG^{\\prime } \\\\0 & -i\\triangle _{a}-\\frac{\\gamma _{a}}{2} & iG^{\\prime } & iG^{\\prime } \\\\-iG^{\\prime } & -iG^{\\prime } & i\\omega _{m}^{\\prime }-\\frac{\\gamma }{2}& 0 \\\\iG^{\\prime } & iG^{\\prime } & 0 & -i\\omega _{m}^{\\prime }-\\frac{\\gamma }{2}\\end{array}\\right)$ with $G^{\\prime }=G(1+4\\Lambda /\\omega _{m})^{-1/4}$ and $\\omega _{m}^{\\prime }=\\omega _{m}\\sqrt{1+4\\Lambda /\\omega _{m}}$ .", "By calculating the eigenvalues of $\\textbf {A}^{\\prime }$ , we derive the stability condition in the red-detuned regime $\\Delta _a>0$ as $4\\omega _{m}^{\\prime }(G^{\\prime })^{2}\\Delta _{a}-\\left[(\\omega ^{\\prime }_{m})^{2}+\\frac{\\gamma ^{2}}{4}\\right]\\left(\\Delta _{a}^{2}+\\frac{\\kappa ^{2}}{4}\\right)<0.$ Converting this to the original parameters (before the squeezing transformation), the stability condition can be expressed as $16G^2<(\\omega _m+4\\Lambda )(4\\Delta _a+\\kappa ^2/\\Delta _a),$ after omitting the $\\gamma $ -dependent term as given in the main text.", "In order to generate strong squeezing, we are interested in the parameter regime of strong driving with $\|\\alpha \|, \\beta \\gg 1$ and near the optimal detuning point with $\\Delta _{a}\\sim \\omega ^{\\prime }_m$ .", "In this regime, Eq.", "(REF ) can be simplified to $g_{0}<\\sqrt{27\\omega _{m}\\eta },$ which is independent of the driving power.", "The parameter regime of interest in our scheme always satisfies this condition." ] ]	1403.0049
[ [ "The Thermodynamic Covariance Principle" ], [ "Abstract The concept of {\\it equivalent systems} from the thermodynamic point of view, originally introduced by Th.", "De Donder and I. Prigogine, is deeply investigated and revised.", "From our point of view, two systems are thermodynamically equivalent if, under transformation of the thermodynamic forces, both the entropy production and the Glansdorff-Prigogine dissipative quantity remain unaltered.", "This kind of transformations may be referred to as the {\\it Thermodynamic Coordinate Transformations} (TCT).", "The general class of transformations satisfying the TCT is determined.", "We shall see that, also in the nonlinear region ({\\it i.e.", "}, out of the Onsager region), the TCT preserve the reciprocity relations of the transformed transport matrix.", "The equivalent character of two transformations under TCT, leads to the concept of {\\it Thermodynamic Covariance Principle} (TCP) stating that all thermodynamic equations involving the thermodynamic forces and flows ({\\it e.g.", "}, the closure flux-force relations) should be covariant under TCT." ], [ "Introduction", "Onsager's theory is based on three assumptions [1]-[2]: i) The probability distribution function for the fluctuations of thermodynamic quantities (Temperature, pressure, degree of advancement of a chemical reaction etc.)", "is a Maxwellian ii) Fluctuations decay according to a linear law and iii) The principle of the detailed balance (or the microscopic reversibility) is satisfied.", "Onsager showed the equivalence of the assumptions i)-iii) with the equations $J_\\mu =\\tau _{0\\mu \\nu }X^\\nu \\qquad ;\\qquad \\frac{\\partial \\tau _{0\\mu \\nu }}{\\partial X^\\lambda }=0$ where $\\tau _{0\\mu \\nu }$ are the transport coefficients and $X^\\mu $ and $J_\\mu $ denote the thermodynamic forces and the conjugate fluxes, respectively.", "Assumption iii) allows deriving the reciprocity relations $\\tau _{0\\mu \\nu }=\\tau _{0\\nu \\mu }$ .", "Note that in Eq.", "(REF ), as well as in the sequel, the summation convention on the repeated indexes is understood.", "The Onsager theory of fluctuations starts from the Einstein formula linking the probability of a fluctuation, $\\mathcal {W}$ , with the entropy change, $\\Delta S$ , associated with the fluctuations from the state of equilibrium $\\mathcal {W}=W_0\\exp [\\Delta S/k_B]$ In Eq.", "(REF ), $k_B$ is the Bolzmann constant and $W_0$ is a normalization constant, which ensures that the sum of all probabilities equals to one.", "Prigogine generalized Eq.", "(REF ), which applies only to adiabatic or isothermal transformations, by introducing the entropy production due to fluctuations.", "Denoting by $\\xi _i$ ($i=1\\cdots m$ ) the $m$ deviations of the thermodynamic quantities from their equilibrium value, Prigogine proposed that the probability distribution of finding a state in which the values $\\xi _i$ lie between $\\xi _i$ and $\\xi _i+d\\xi _i$ is given by [3] $\\mathcal {W}=W_0\\exp [\\Delta _{\\rm I} S/k_B]\\qquad \\quad {\\rm where}\\qquad \\Delta _{\\rm I} S=\\int _E^F d_{\\rm I} s\\quad {\\rm ;}\\quad \\frac{d_{\\rm I} s}{dt}\\equiv \\int _\\Omega \\sigma dv$ $dv$ is a (spatial) volume element of the system, and the integration is over the entire space $\\Omega $ occupied by the system in question.", "$E$ and $F$ indicate the equilibrium state and the state to which a fluctuation has driven the system, respectively.", "Note that this probability distribution remains unaltered for flux-force transformations leaving invariant the entropy production.", "Concrete examples of chemical reactions, equivalent from the thermodynamic point of view, have also been analyzed in literature.", "As an example, among these, we choose the simplest of all.", "Let us consider, for example, a chemical system in which a) two isomerisations $A\\rightarrow B$ and $B\\rightarrow C$ take place [3].", "From the macroscopic point of view, the chemical changes could be equally well described by the b) two isomerisations $A\\rightarrow C$ and $B\\rightarrow C$ .", "It can be checked that, under a linear transformation of the thermodynamic forces (i.e., in this case, a linear transformation of the chemical affinities) the entropy productions, corresponding to the two chemical reactions a) and b), are equal.", "Indeed, the corresponding affinities of the reactions a) read : $A^1=\\mu _A-\\mu _B$ , and $A^2=\\mu _B-\\mu _C$ , with $A^i$ and $\\mu _i$ ($i=A,B,C$ ) denoting the affinities and the chemical potentials, respectively.", "The change per unit time of the mole numbers is given by $\\frac{dn_A}{dt}=-v_1\\ ;\\qquad \\frac{dn_B}{dt}=v_1-v_2\\ ; \\qquad \\frac{dn_C}{dt}=v_2$ with $v_i$ ($i=1,2$ ) denoting the chemical reaction rates.", "In this case the thermodynamic forces and the flows are the chemical affinities (over temperature) and the chemical reaction rates, respectively i.e., $X^\\mu =A^\\mu /T$ and $J_\\mu =v_\\mu $ .", "Hence, the corresponding entropy production reads $d_{I} S/dt=A^1/Tv_1+A^2/Tv_2>0$ .", "The affinities corresponding to reactions b) are related to the old ones by $A^{\\prime 1}=\\mu _A-\\mu _C=A^1+A^2\\ ; \\qquad A^{\\prime 2}=\\mu _B-\\mu _C=A^2$ By taking into account that $\\frac{dn_A}{dt}=-v^{\\prime }_1\\ ;\\qquad \\frac{dn_B}{dt}=-v^{\\prime }_2\\ ; \\qquad \\frac{dn_C}{dt}=v^{\\prime }_1+v^{\\prime }_2$ we get $v_1=v^{\\prime }_1\\ ;\\qquad v_2=v^{\\prime }_1+v^{\\prime }_2$ where the invariance of the entropy production is manifestly shown.", "Indeed, $\\!\\!\\!\\!", "d_{I} S/dt=(A^1/T)v_1+(A^2/T)v_2=(A^{\\prime 1}/T)v^{\\prime }_1+(A^{\\prime 2}/T)v^{\\prime }_2=d_{I} S^{\\prime }/dt$ or $J_\\mu X^\\mu = J^{\\prime }_\\mu X^{\\prime \\mu }$ (where the Einstein summation convention on the repeated indexes is adopted).", "On the basis of the above observations, Th.", "De Donder and I. Prigogine formulated, for the first time, the concept of equivalent systems from the thermodynamical point of view.", "For Th.", "De Donder and I. Prigogine, thermodynamic systems are thermodynamically equivalent if, under transformation of fluxes and forces, the bilinear form of the entropy production remains unaltered, i.e., $\\sigma =\\sigma ^{\\prime }$ .", "[4].", "However, the condition of the invariance of the entropy production is not sufficient to ensure the equivalence character of the two descriptions $(J_\\mu , X^\\mu )$ and $(J^{\\prime }_\\mu , X^{\\prime \\mu })$ .", "Indeed, we can convince ourselves that there exists a large class of transformations such that, even though they leave unaltered the expression of the entropy production, they may lead to certain paradoxes to which Verschaffelt has called attention [5]-[6].", "This obstacle can be overcome if one takes into account one of the most fundamental and general theorems valid in thermodynamics of irreversible processes : the Universal Criterion of Evolution.", "In general, Glansdorff and Prigogine have shown that : For time-independent boundary conditions, a thermodynamic system, even in strong non-equilibrium conditions, relaxes to a stable stationary state in such a way that the following Universal Criterion of Evolution is satisfied [7]-[8] : $\\int _\\Omega J_\\mu \\frac{\\partial X}{\\partial t}^\\mu \\ dV\\le 0$ Here, $\\Omega $ is the volume occupied by the system and $dV$ the volume-element, respectively.", "In addition $\\int _\\Omega J_\\mu \\frac{\\partial X}{\\partial t}^\\mu dV= 0 \\quad {\\rm At\\ the\\ steady\\ state}$ Quantity $P\\equiv \\int _\\Omega J_\\mu \\frac{\\partial X}{\\partial t}^\\mu dV$ may be referred to as the Glansdorff-Prigogine dissipative quantity.", "Let us check the validity of this theorem by considering two, very simple, examples.", "Let us consider, for instance, a closed system containing $m$ components ($i=1\\dots m$ ) among which chemical reactions are possible.", "The temperature, $T$ , and the pressure, $p$ , are supposed to be constant in time.", "The chance in the number of moles $n_i$ , of component $i$ , is $\\frac{dn_i}{dt}=\\nu _i^jv_j$ with $\\nu _i^j$ denoting the stoichiometric coefficients.", "By multiplying both members of Eq.", "(REF ) by the time derivative of the chemical potential of component $i$ , we get ${\\dot{\\mu }}^i\\frac{dn_i}{dt}=\\Bigl (\\frac{\\partial \\mu ^i}{\\partial n_\\kappa }\\Bigr )_{pT}\\frac{dn_i}{dt}\\frac{dn_\\kappa }{dt}=\\nu _i^j{\\dot{\\mu }}^iv_j$ By taking into account the De Donder law between the affinities $A^i$ and the chemical potentials i.e., $A^j=-\\nu _i^j\\mu ^i$ , we finally get $P\\equiv J_\\mu \\frac{dX^\\mu }{dt}=v_j\\frac{d}{dt}\\Bigl (\\frac{A^j}{T}\\Bigr )=-\\frac{1}{T}\\Bigl (\\frac{\\partial \\mu ^i}{\\partial n_\\kappa }\\Bigr )_{pT}\\frac{dn_i}{dt}\\frac{dn_\\kappa }{dt}\\le 0$ where the negative sign of the term on the right-hand side is due to the second law of thermodynamics.", "Hence, the Glansdorff-Prigogine dissipative quantity $P$ is always negative, and it vanishes at the stationary state.", "As a second example, we analyze the case of heat conduction in non-expanding solid.", "In this case the thermodynamic forces and the conjugate flows are the (three) components of the gradient of the inverse of the temperature, $X^\\mu =[{\\rm grad}\\ (1/T)]^\\mu $ and the (three) components of the heat flow, $J_\\mu ={\\bf J}_{q\\mu }$ (with i=1,2,3), respectively.", "Hence, $P\\equiv \\int _\\Omega J_\\mu \\frac{\\partial X^\\mu }{\\partial t}\\ dV=\\int _\\Omega {\\bf J}_{q\\mu }\\frac{\\partial }{\\partial t}[{\\rm grad}\\ (1/T)]^\\mu \\ dV$ With partial integration, and by taking into account the energy law $\\rho c_v\\frac{\\partial T}{\\partial t}=-{\\rm div}\\ {\\bf J}_q$ after simple calculations, we get $P=-\\int _\\Omega \\frac{\\rho c_v}{T^2}\\Bigl (\\frac{\\partial T}{\\partial t}\\Bigr )^2 dV\\le 0$ with $P=0$ at the steady state.", "In Eqs (REF ) and (REF ), $\\rho $ and $c_v$ are the mass density and the specific heat at volume constant, respectively.", "By summarizing, without using neither the Onsager reciprocal relations nor the assumption that the phenomenological coefficients (or linear phenomenological laws) are constant, the dissipative quantity $P$ is always a negative quantity.", "This quantity vanishes at the stationary state.", "In the two above-mentioned examples, the thermodynamic forces are the chemical affinities (over temperature) and the gradient of the inverse of temperature, respectively.", "However, we could have adopted a different choice of the thermodynamic forces.", "If we analyze, for instance, the case of heat conduction in non-expanding solid, where chemical reactions take place simultaneously, we can choose as thermodynamic forces a combination of the (dimensionless) chemical affinities (over temperature) and the (dimensionless) gradient of the inverse of temperature.", "Clearly, this representation is thermodynamically equivalent to the one where the thermodynamic forces are simply the chemical affinities (over temperature) and the gradient of the inverse of temperature, only if the transformation between the thermodynamic forces preserves the negative sign of the quantity $P$ .", "In particular, the two equations for the stationary states [the Eq.", "(REF ) and its transformed] must admit the same solutions.", "To sum up, the (admissible) thermodynamic forces should satisfy the following two conditions: 1) The entropy production, $\\sigma $ , should be invariant under transformation of the thermodynamic forces $\\lbrace X^\\mu \\rbrace \\rightarrow \\lbrace X^{\\prime \\mu }\\rbrace $ and 2) The Glansdorff-Prigogine dissipative quantity, $P$ , should also remain invariant under the forces transformations $\\lbrace X^\\mu \\rbrace \\rightarrow \\lbrace X^{\\prime \\mu }\\rbrace $.", "Condition 2) derives from 2a) A steady state should be transformed into a steady state and 2b) A stable steady state should be transformed into a stable state state, with the same \"degree\" of stability.", "Magnetically confined tokamak plasmas are a typical example of thermodynamic systems, out of Onsager's region, where the Glansdorff-Prigogine dissipative quantity remains invariant under transformation of the thermodynamic forces [9].", "Conditions 1) and 2) allow determining univocally the class of admissible thermodynamic forces [11].", "This kind of transformations may be referred to as the Thermodynamic Coordinate Transformations (TCT).", "In the light of the above, we formulate the following principle : Two systems are equivalent from the thermodynamic point of view if, under transformation $\\lbrace X^\\mu \\rbrace \\rightarrow \\lbrace X^{\\prime \\mu }\\rbrace $ , we have $\\sigma =\\sigma ^{\\prime }$ and $P=P^{\\prime }$ [11].", "The aim of this work is to derive from the admissible thermodynamic forces, the most general class of transformations (TCT) so that two thermodynamic systems are equivalent from the thermodynamic point of view.", "We shall prove that under TCT, $\\sigma $ and $P$ remain invariant and, in addition, the reciprocity relations of the transformed transport matrix are preserved.", "The thermodynamic equivalence principle leads, naturally, to the following Thermodynamic Covariance Principle (TCP) : The nonlinear closure equations (i.e.", "the flux-force relations valid out of Onsager's region) must be covariant under (TCT).", "The essence of the TCP is the following.", "The equivalent character of two representations is warranted if, and only if, the fundamental thermodynamic equations are covariant under TCT.", "In fact, a covariant formulation guarantees the writing of the fundamental laws of thermodynamics (for instance, the flux-force closure equations) in a form that is manifestly invariant under transformation between the admissible thermodynamic forces.", "This is the correct mathematical formalism to ensure the equivalence between two different representations.", "Note that the TCP is trivially satisfied by the closure equations valid in the Onsager region.", "It is worthwhile mentioning that the linear version of the TCT (see the next section) is actually largely used in a wide variety of thermodynamic processes ranging from non equilibrium chemical reactions to transport processes in tokamak plasmas (see, for examples, the papers cited in the book [10]).", "To the authors knowledge, the validity of the thermodynamic covariance principle has been verified empirically without exception in physics until now." ], [ "The Thermodynamic Covariance Principle", "Consider a thermodynamic system driven out of equilibrium by a set of $n$ independent thermodynamic forces $\\lbrace X^\\mu \\rbrace $ ($\\mu =1,\\cdots n$ ).", "It is also assumed that the system is submitted to time-independent boundary conditions.", "The set of conjugate flows, $\\lbrace J_{\\mu }\\rbrace $ , is coupled to the thermodynamic forces through the relation $J_{\\mu }=\\tau _{\\mu \\nu }(X)X^\\nu $ where the transport coefficients, $\\tau _{\\mu \\nu }(X)$ , may depend on the thermodynamic forces.", "The symmetric piece of $\\tau _{\\mu \\nu }(X)$ is denoted with $g_{\\mu \\nu }(X)$ and the skew-symmetric piece as $f_{\\mu \\nu }(X)$ : $\\tau _{\\mu \\nu }(X)=\\frac{1}{2}[\\tau _{\\mu \\nu }(X)+\\tau _{\\nu \\mu }(X)]+\\frac{1}{2}[\\tau _{\\mu \\nu }(X)-\\tau _{\\nu \\mu }(X)]=g_{\\mu \\nu }(X)+f_{\\mu \\nu }(X)$ where $&&g_{\\mu \\nu }(X)\\equiv \\frac{1}{2}[\\tau _{\\mu \\nu }(X)+\\tau _{\\nu \\mu }(X)]=g_{\\nu \\mu }(X)\\\\&&f_{\\mu \\nu }(X)\\equiv \\frac{1}{2}[\\tau _{\\mu \\nu }(X)-\\tau _{\\nu \\mu }(X)]=-f_{\\nu \\mu }(X)$ It is assumed that $g_{\\mu \\nu }(X)$ is a positive definite matrix.", "For the sake of conciseness, in the sequel we drop the symbol $(X)$ in $\\tau _{\\mu \\nu }$ , $g_{\\mu \\nu }$ and $f_{\\mu \\nu }$ , being implicitly understood that these matrices may depend on the thermodynamic forces.", "With the elements of the transport coefficients two objects are constructed: operators, which may act on thermodynamic tensorial objects and thermodynamic tensorial objects, which under coordinate (forces) transformations, obey to well specified transformation rules.", "Operators Two operators are introduced, the entropy production operator $\\sigma (X)$ and the dissipative quantity operator ${\\tilde{P}}(X)$ , acting on the thermodynamic forces in the following manner $&&\\sigma (X):\\rightarrow \\sigma (X)\\equiv X gX^T\\nonumber \\\\&& {\\tilde{P}}(X):\\rightarrow {\\tilde{P}}(X)\\equiv X\\tau \\Bigl [\\frac{dX}{d\\varsigma }\\Bigr ]^T$ In Eqs (REF ), the transport coefficients are then considered as elements of the two $n$ x $n$ matrices, $\\tau $ and $g$ .", "Symbol $T$ stands for the transpose of the matrix.", "The positive definiteness of the matrix $g_{\\mu \\nu }$ ensures the validity of the second principle of thermodynamics: $\\sigma \\ge 0$ .", "These matrices multiply the thermodynamic forces $X$ expressed as $n$ x 1 column matrices.", "We already anticipate that parameter $\\varsigma $ , defined in Eq.", "(REF ), is invariant under the thermodynamic coordinate transformations.", "Thermodynamic states $X_{s}$ such that $\\Big [{\\tilde{P}}(X)\\frac{d\\varsigma }{dt}\\Big ]_{X=X_{s}}\\!\\!\\!\\!=0$ are referred to as steady states.", "Of course, the steady states should be invariant expressions under the thermodynamic coordinate transformations.", "Eqs (REF ) should not be interpreted as the metric tensor $g_{\\mu \\nu }$ , which acts on the coordinates.", "The metric tensor acts only on elements of the tangent space (like $dX^\\mu $ , see the forthcoming sub-section) or on the thermodynamic tensorial objects.", "Transformation Rules of Entropy Production, Forces, and Flows In the previous section we stated that two systems are equivalent from the thermodynamic point of view if, under TCT, $\\sigma =\\sigma ^{\\prime }$ and ${\\tilde{P}}={\\tilde{P}}^{\\prime }$ remain unaltered.", "In mathematical terms, this implies: $\\sigma =J_\\mu X^\\mu =J^{\\prime }_\\mu X^{\\prime \\mu }$ This condition and the condition that also the dissipative quantity [cf.", "Eqs (REF )] must be an invariant expression, require that the transformed thermodynamic forces and flows satisfy the relation $&&X^{\\prime \\mu }=\\frac{\\partial X^{\\prime \\mu }}{\\partial X^\\nu } X^\\nu \\nonumber \\\\&& J^{\\prime }_\\mu =\\frac{\\partial X^{\\nu }}{\\partial X^{\\prime \\mu }}J_\\nu $ These transformations may be referred to as Thermodynamic Coordinate Transformations (TCT).", "The expression of entropy production becomes accordingly $\\sigma =J_\\mu X^\\mu =\\tau _{\\mu \\nu }X^\\mu X^\\nu =g_{\\mu \\nu }X^\\mu X^\\nu =g^{\\prime }_{\\mu \\nu }X^{\\prime \\mu }X^{\\prime \\nu }=\\sigma ^{\\prime }$ From Eqs (REF ) and (REF ) we derive $g^{\\prime }_{\\lambda \\kappa }=g_{\\mu \\nu }\\frac{\\partial X^\\mu }{\\partial X^{\\prime \\lambda }}\\frac{\\partial X^\\nu }{\\partial X^{\\prime \\kappa }}$ Moreover, inserting Eqs (REF ) and Eq.", "(REF ) into relation $J_{\\mu }=(g_{\\mu \\nu }+f_{\\mu \\nu })X^\\nu $ , we obtain $J^{\\prime }_\\lambda =\\Bigl (g^{\\prime }_{\\lambda \\kappa }+f_{\\mu \\nu }\\frac{\\partial X^\\mu }{\\partial X^{\\prime \\lambda }}\\frac{\\partial X^\\nu }{\\partial X^{\\prime \\kappa }}\\Bigr )X^{\\prime \\kappa }$ or $J^{\\prime }_\\lambda =(g^{\\prime }_{\\lambda \\kappa }+f^{\\prime }_{\\lambda \\kappa })X^{\\prime \\kappa }\\quad \\qquad {\\rm with}\\qquad f^{\\prime }_{\\lambda \\kappa }=f_{\\mu \\nu }\\frac{\\partial X^\\mu }{\\partial X^{\\prime \\lambda }}\\frac{\\partial X^\\nu }{\\partial X^{\\prime \\kappa }}$ Hence, the transport coefficients transform like a thermodynamic tensor of second order We may qualify as thermodynamic tensor or, simply thermo-tensor, (taken as a single noun) a set of quantities where only transformations Eqs (REF ) are involved.", "This is in order to qualify as a tensor, a set of quantities, which satisfies certain laws of transformation when the coordinates undergo a general transformation.", "Consequently every tensor is a thermodynamic tensor but the converse is not true..", "It is easily checked that transformations (REF ) preserve the validity of the reciprocal relations for transport coefficients i.e., if $g_{\\mu \\nu }=g_{\\nu \\mu }$ then $g^{\\prime }_{\\mu \\nu }=g^{\\prime }_{\\nu \\mu }$ (and, if $f_{\\mu \\nu }=-f_{\\nu \\mu }$ then $f^{\\prime }_{\\mu \\nu }=-f^{\\prime }_{\\nu \\mu }$ ).", "Properties of the TCT By direct inspection, it is easy to verify that the general solutions of equations (REF ) are $X^{\\prime \\mu }=X^1F^\\mu \\Bigl (\\frac{X^2}{X^1},\\ \\frac{X^3}{X^2},\\ \\cdots \\ \\frac{X^n}{X^{n-1}}\\Bigr )$ where $F^\\mu $ are arbitrary functions of variables $X^j/X^{j-1}$ with ($j=2,\\dots , n$ ).", "Eq.", "(REF ) is the most general class of transformations expressing the equivalent character of two descriptions $\\lbrace X^\\mu \\rbrace $ and $\\lbrace X^{\\prime \\mu }\\rbrace $ .", "Hence, the TCT may be highly nonlinear coordinate transformations but, in the Onsager region, we may (or we must) require that they have to reduce to $X^{\\prime \\mu }=c_\\nu ^\\mu X^\\nu $ where $c_\\nu ^\\mu $ are constant coefficients (i.e., independent of the thermodynamic forces).", "The linear and homogeneous transformations (REF ) are largely used in literature because, besides their simplicity, they are nonsingular transformations.", "We note that from Eq.", "(REF ), the following important identities are derived $X^\\nu \\frac{\\partial ^2X^{\\prime \\mu }}{\\partial X^\\nu \\partial X^\\kappa }=0\\qquad ;\\qquad X^{\\prime \\nu }\\frac{\\partial ^2X^\\mu }{\\partial X^{\\prime \\nu }\\partial X^{\\prime \\kappa }}=0$ Moreover $&&dX^{\\prime \\mu }=\\frac{\\partial X^{\\prime \\mu }}{\\partial X^\\nu } dX^\\nu \\nonumber \\\\&&\\frac{\\partial }{\\partial X^{\\prime \\mu }}= \\frac{\\partial X^{\\nu }}{\\partial X^{\\prime \\mu }}\\frac{\\partial }{\\partial X^\\nu }$ i.e., $dX^\\mu $ and $\\partial /\\partial X^{\\mu }$ transform like a thermodynamic contra-variant and a thermodynamic covariant vector, respectively.", "According to Eq.", "(REF ), thermodynamic vectors $dX^\\mu $ define the tangent space to $Ts$ .", "It also follows that the operator $P(X)$ , i.e.", "the dissipation quantity, and in particular the definition of steady states, are invariant under TCT.", "Parameter $\\varsigma $ , defined as $d\\varsigma ^2=g_{\\mu \\nu }dX^\\mu dX^\\nu $ is a scalar under TCT.", "The operator $\\mathcal {O}$ $\\mathcal {O}\\equiv X^\\mu \\frac{\\partial }{\\partial X^\\mu }=X^{\\prime \\mu }\\frac{\\partial }{\\partial X^{\\prime \\mu }}=\\mathcal {O}^{\\prime }$ is also invariant under TCT.", "This operator plays an important role in the formalism [11].", "We may now enunciate the Thermodynamic Covariance Principle (TCP) : All thermodynamic equations involving the thermodynamic forces and flows (e.g., the closure equations) should be covariant under (TCT).", "From this principle, it is possible to obtain the (nonlinear) closure equations by truncating the equations (obtained, for instance, by kinetic theory) relating the thermodynamic forces with the conjugate flows in such a way that the resulting expressions satisfy the TCP.", "A concrete example is briefly mentioned in the Conclusions." ], [ " Conclusions", "If one requires only the entropy production to be invariant, but does not impose the auxiliary condition that he Glansdorff-Prigogine dissipative quantity should also be invariant under TCT, there exists the larger class of transformations leading to the phenomenological coefficients scheme, for which the reciprocal relations are not valid [13].", "As rightly pointed out by Verschaffelt and Davies, \"to impose that only the entropy production must be invariant under flux-force transformations may lead to paradoxes or inconsistencies\" [5]-[6].", "We must also keep in mind that the original form of the entropy production, as derived from the various balance equations, is such that the Onsager relations are valid.", "The correct way to overcome this impasse is to require that both the entropy production and the Glansdorff-Prigogine dissipative quantity remain unaltered under TCT.", "The Nonlinear theory, the Thermodynamic Field Theory (TFT), reported in [11], is based on the validity of the Thermodynamic Covariance Principle.", "This assumption allows truncating, at the lowest order, the (highly nonlinear) equations relating the thermodynamic forces with the conjugate flows.", "For example, in the case of magnetically confined tokamak-plasmas, these relations are obtained by kinetic theory and are expressed by highly nonlinear integral equations [14].", "In this specific case, the generalized frictions are the thermodynamic forces whereas the conjugate flows are the Hermitian moments [10].", "The Hermitian moments are linked to the deviation of the distribution function from the (local) Maxwellian, whereas the generalized frictions are linked to the collision term.", "According to the TCP, these integral equations may be truncated in such a way that the resulting expressions are covariant under TCT.", "The derived closure equations have been recently used for estimating the electron and ion losses (particle and heat losses) as well as the electron and ion distributions functions [9]-[12].", "The theoretical predictions are fairly in line with the preliminary experimental data obtained for FTU-plasmas (FTU= Frascati Tokamak Upgrade).", "We conclude by mentioning that attempts to derive a generally covariant thermodynamic field theory (GTFT) can be found in refs [15]-[16] and [17].", "In Refs [15]-[16], the general covariance has been assumed to be valid for general transformations in the space of thermodynamic configurations, whereas in [17] it is argued that the entropy production rate should be invariant under general spatial coordinate transformations (in [17] the invariance of the Prigogine-Glansdorff dissipation is not satisfied).", "However, it is a matter of fact that a generally covariant thermodynamic theory, which is based solely on the invariance of the entropy production is, in reality, respected only by a very limited class of thermodynamic processes.", "As mentioned, a correct thermodynamic formalism must satisfy the TCP." ] ]	1403.0370
[ [ "Segmented scintillation detectors with silicon photomultiplier readout\n for measuring antiproton annihilations" ], [ "Abstract The Atomic Spectroscopy and Collisions Using Slow Antiprotons (ASACUSA) experiment at the Antiproton Decelerator (AD) facility of CERN constructed segmented scintillators to detect and track the charged pions which emerge from antiproton annihilations in a future superconducting radiofrequency Paul trap for antiprotons.", "A system of 541 cast and extruded scintillator bars were arranged in 11 detector modules which provided a spatial resolution of 17 mm.", "Green wavelength-shifting fibers were embedded in the scintillators, and read out by silicon photomultipliers which had a sensitive area of 1 x 1 mm^2.", "The photoelectron yields of various scintillator configurations were measured using a negative pion beam of momentum p ~ 1 GeV/c.", "Various fibers and silicon photomultipliers, fiber end terminations, and couplings between the fibers and scintillators were compared.", "The detectors were also tested using the antiproton beam of the AD.", "Nonlinear effects due to the saturation of the silicon photomultiplier were seen at high annihilation rates of the antiprotons." ], [ "Introduction", "The Atomic Spectroscopy and Collisions Using Slow Antiprotons (ASACUSA) experiment at the Antiproton Decelerator (AD) facility of CERN has recently developed position-sensitive detectors to measure and track charged pions emerging from antiproton annihilations.", "These segmented plastic scintillators will be used to detect annihilations that occur inside a radiofrequency Paul trap for antiprotons.", "They were also employed in a recent experiment which attempted to measure the annihilation cross-sections of low-energy (130 keV) antiprotons on various target foils [1].", "In this paper we describe the design and manufacture of these detectors, and beam tests carried out using negative pion ($\\pi ^-$ ) and antiproton beams provided by the CERN Proton Synchrotron (PS) and AD facilities.", "Depending on the atomic nucleus on which an antiproton annihilates, on average 3–4 charged pions emerge with a mean kinetic energy of a few hundred MeV [2].", "By tracking the trajectory of at least two charged pions, the initial vertex where the annihilation occurred can be reconstructed.", "This technique forms the basis of many experiments now carried out at the AD.", "Trackers based on silicon strip detectors [3], [4] or scintillation fibers [5] have been previously used to detect the formation of antihydrogen, or to count the number of antiprotons stored in traps.", "Precision laser spectroscopy of antiprotonic helium atoms have been carried out by using acrylic Cherenkov counters to detect the timing of antiproton annihilations occurring in a cryogenic helium target [6], [7].", "Recently, extruded scintillators [8], [9], [10], [11] read out by wavelength-shifting (WLS) fibers [12] and silicon photomultipliers (SiPMs) [13], [14], [15] were used as tracker detectors in long-baseline neutrino experiments [16], [17].", "They may also be used in electromagnetic calorimeters in future collider experiments [18], [19].", "Compared to conventional scintillators which are read out by photomultiplier tubes, the SiPM-based design has important advantages which include low cost, compact size, and insensitivity to magnetic fields.", "There are, however, currently numerous disadvantages such as the comparably small active area, gain, and dynamic range of the SiPMs; their high dark current and temperature sensitivity [14], [20]; and the narrow range of bias voltages in which the SiPMs can be operated in an optimal way.", "In this paper, we systematically studied the response of the scintillator against individual hits of minimum-ionizing pions.", "It was particularly important to achieve a sufficient photoelectron yield, above the backgrounds of dark current and environmental noise.", "This paper is organized in the following way.", "In Sect.", ", we describe the construction of the detector, with details of the SiPMs, scintillators, and WLS fibers.", "In Sect.", ", measurements carried out at using a 1-GeV/c pion beam provided by the CERN PS are described.", "We compared the performance of cast and extruded scintillators, fibers of various diameters, two types of SiPMs, and various methods to optically couple the scintillator and fibers.", "Sect.", "describes the tests carried out at the AD with a 70-keV antiproton beam.", "The Paul trap is housed in a 1-m-diameter cryostat (Fig.", "REF ), which has vacuum ports on its three sides comprising injection and ejection beamlines for antiprotons.", "The cryogenic equipment was located above the vacuum vessel.", "We arranged 541 scintillation bars in 11 detector modules, which together covered a solid angle of $\\sim 2$$\\pi $ steradians seen from the center of the trap.", "Figure: Schematic diagram of the scintillation detector modules surrounding the Paul trap for antiprotons.", "Two pairs of modules (blue) consisting of 17 mm -wide extruded scintillator bars reconstruct an estimated >15%>15\\% of the annihilations with 20–30 mm spatial resolution.", "Seven additional modules (orange) containing cast scintillators are optimized to detect the annihilations with a >>90 %\\% efficiency.Figure: (a) Layout of the extruded scintillator module.", "A wavelength-shifting (WLS) fiberwas placed into a hole fabricated in the center of each scintillator bar.", "(b) Layout of the cast scintillatormodule, with fibers embedded into U-shaped grooves that were machined on the surface of each scintillator bar.The fibers were coupled to SiPMs.", "Coaxial signal cables were guided from the PCB's alonga narrow channel between the outer and inner walls of the frame, and exited at the corner of the box.Figure: Detail of the connection between a SiPM and WLS fiber.", "The fiber was glued into the male part of a connector and polished (see text).", "The SiPM was placed into the female counterpart, and soldered to a printed circuit board.", "The assembly was then bolted to the aluminium box of the module, which constituted the signal ground.Seven of the modules [Fig.", "REF (b)] ranged between sensitive areas of $0.25$$\\times $$0.45$ m$^2$ and $0.8$$\\times $$0.45$ m$^2$ , and contained cast scintillator bars with a cross section of 50$\\times $ 12.7 mm$^2$ which were arranged in a XY geometry.", "They were optimized to detect annihilations that occur in the trap with a high efficiency ($>90\\%$ ).", "The four remaining modules [Fig.", "REF (a)] had larger active areas ($\\sim 1$$\\times $ 1 m$^2$ ), and contained extruded scintillators with a smaller cross section of 17$\\times $ 19 mm$^2$ which were designed for higher spatial resolution and tracking efficiency of the pions ($>15\\%$ ).", "Each module was housed in an aluminum box with 3-mm-thick walls, with support beams which held the scintillator bars in place.", "Black adhesive sheets were lined along the inside the boxes, and provided the light tightness.", "WLS fibers embedded in each scintillator bar collected the scintillation light.", "The wavelength-shifted light was subsequently guided along the fibers, and coupled to SiPMs which were connected to one end of the fibers.", "The detection and tracking efficiencies described above are optimized values, which were estimated using a Monte Carlo simulation based on the GEANT4 package [21].", "For this, a computer model of the trap was created, including its electrodes and vacuum chamber.", "We allowed antiprotons to annihilate on the electrodes according to the experimental branching ratios [2], and tracked the secondary particles for $\\sim 200$ ns, or until all the particles left the 4$\\times $ 4$\\times $ 4 m$^3$ area around the trap.", "The detection of charged pions was defined as events which deposited more than $\\sim 0.4$ MeV of energy into the scintillators." ], [ "Cast scintillators", "The cast scintillators (Saint-Gobain Crystals BC-408) with a polyvinyltoluene (PVT) base were first milled into the desired shapes.", "The surfaces were then polished using a two-blade fly-cutter.", "This consisted of an angled forecutter with a coating of polycrystalline diamond (PCD) deposited on its edge, and a flat finishing cutter with a monocrystalline diamond (DIXI Polytools type 20370 version C) edge.", "The two blades were positioned on a 85-mm diameter mount which was rotated at 2000 revolutions per minute (rpm) and fed at a speed of 250 mm/min.", "The depth of the cut during the final pass of the tool over the scintillator was adjusted to $\\sim 10$ $\\mu {\\rm m}$ , to attain a mirror-like surface.", "No additional polishing of the surfaces was carried out.", "A 1.2-mm-wide, 1.5-mm-deep U-shaped groove [Fig.", "REF (b)] was next cut along the surface of each scintillator [22], [23], [24], [25], [26], [27].", "For this a two-flute, ball-nose slot drill made of carbide with a diamond coating applied by chemical vapor deposition (CVD) was used.", "The mill was rotated at 2000 rpm and fed at a speed of 160 mm/min.", "Each pass cut the groove in increments of 200 $\\mu $ m. During machining, it was essential to constantly blow dry air on the groove to avoid immediate melting of the scintillator.", "No other lubricants were used.", "After this machining, optical grease (Eljen Technologies EJ-550) was applied to the groove, and the WLS fiber tightly pressed into it, to ensure that no bubbles formed in the contact between the fiber and scintillator.", "The bars were then wrapped in specular reflector foils (3M Vikuti), which have a reflectivity of $>98\\%$ at visible wavelengths." ], [ "Extruded scintillators", "A detector module containing extruded scintillators bars manufactured by Fermilab is shown in Fig.", "REF (a).", "Each bar with a cross section of 19$\\times $ 17 mm$^2$ contained circular, or slightly oval, holes of diameter $d=2-3.5$ mm along its center.", "During the extrusion of the scintillators, TiO$_2$ diffuse reflector coatings were deposited on their surfaces [9].", "The attenuation length of the scintillators according to Ref.", "[10] was around $\\sim 40$ –50 mm.", "We also tested extruded scintillators manufactured by CI-Kogyo K.K, with a larger cross section of 10$\\times $ 40 mm$^2$ and smaller hole diameter $1.5-2$ mm.", "These prototype bars were fabricated in 2010, by melting polystyrene (PS) pellets (Dow Chemicals No.", "679) mixed with two kinds of fluorescent dye (PPO and POPOP of respective concentrations $\\sim 1\\%$ and $\\sim 0.03\\%$ ), and extruding them as in Ref. [9].", "Some 0.3-mm-thick layers of TiO$_2$ were deposited on the surfaces by co-extrusion.", "The ends of each bar were roughly milled; polishing these surfaces or depositing layers of TiO$_2$ paint on them did not significantly increase the light yield.", "This may indicate the relatively short attenuation length of this scintillator.", "We attempted to increase the light yield by filling optical cement (Saint-Gobain Crystals BC-600) into the scintillator holes before inserting the WLS fibers.", "It was difficult to obtain a homogeneous filling, especially in the case of long ($\\sim 1$ m) scintilallator bars, although shaking the scintillator helped to reduce the forming of bubbles in the cement.", "We also attempted to coat the fibers with optical grease (Eljen Technology EJ-550) before the insertion into the hole.", "Due to the high viscosity of the grease, it was similarly difficult to achieve a homogenous filling, although the viscosity could be reduced by heating the grease to a temperature $T>35$$^\\circ $ C." ], [ "Fibers and silicon photomulipliers", "The WLS fibers (Kuraray Co., Ltd. Y-11(200)M) contained round polystyrene cores of refractive index $n=1.59$ , which were doped with 200 parts per million (ppm) of K27 dye [12].", "The peak absorption and emission wavelengths were $\\sim 420$ and $\\sim 476$ nm.", "Each core was surrounded by an inner cladding made of polymethylmethacrylate ($n=1.49$ ), and an outer core made of fluorinated polymer ($n=1.42$ ).", "These fibers of diameters $d=1$ or $1.2$ mm had numerical apertures of $\\sim 0.72$ and light attenuation lengths of $\\sim 3.5$ m. It was essential to position the SiPMs as close as possible to the cleaved ends of the fibers and center them on the fiber axis, to attain high coupling efficiencies.", "For this we used plastic connectors developed by the T2K experiment [28].", "One of the fiber ends was glued to the male part of the connector using optical cement (Fig.", "REF ).", "The fiber end was then polished, using polishing paper with diamond grain sizes that were progressively reduced from 10 $\\mu {\\rm m}$ to 1 $\\mu {\\rm m}$ .", "The other end of the fiber was left unpolished.", "To further increase the light yield, we attempted to polish the ends of the fibers opposite the SiPMs, and apply aluminum reflector layers of a few micron thickness by vacuum evaporation.", "We could not, however, attain a high-quality reflector surface that was precisely perpendicular to the fiber axis by hand-polishing.", "For 15–20$\\%$ of the fibers treated in this way, a substantial amount of light leaked out of the aluminized ends when the middle sections of the fibers were illuminated.", "We tested two types [14] of SiPMs of active area 1$\\times $ 1 mm$^2$ , i): Hamamatsu Photonics K.K.", "multi-pixel photon counters (MPPC) type S10362-11-050C, which contained 400 pixels of size 50$\\times $ 50 $\\mu $ m$^2$ , ii): S10362-11-025C containing 1600 pixels of size 25$\\times $ 25 $\\mu $ m$^2$ .", "The 400-pixel type had higher photon detection efficiency ($\\sim 50\\%$ versus $\\sim 25\\%$ ) and gain ($\\sim 8\\times 10^5$ versus $\\sim 3\\times 10^5$ ), due to the smaller dead area and higher capacitance of its larger pixels.", "These SiPMs are often used to read out, e.g., tracker scintillators which detect individual hits of minimum-ionizing particles with a high sensitivity.", "The 1600-pixel SiPMs, on the other hand, had lower dark current and higher dynamic range arising from the larger number of pixels which can sustain more simultaneous photon hits.", "These detectors are normally used to read out scintillators in calorimetric applications.", "We employed the 1600-pixel SiPMs, since many AD experiments involve an abrupt burst of pion hits which are superimposed on a less intense background of low-rate hits.", "Such a distribution may be caused by a pulsed beam of antiprotons arriving in the trap, followed by the slow annihilation of antiprotons spilling out of the trap.", "The recovery time needed for a fired pixel to recharge was around $\\sim $ 20 ns [15].", "The typical gain, dark current rate, and Geiger breakdown voltage were respectively $\\sim 3\\times 10^6$ , $\\sim $ $5\\times 10^5$ Hz, and $V_0\\sim 70\\pm 1$ V, according to the specifications of the company and Ref. [20].", "The $V_0$ -value had a temperature dependence of $dV_0/dT=60$ mV$/^\\circ $ C, and a variation of $10\\%$ [29].", "By adjusting the bias $\\sim 1$ V above $V_0$ , the optimal tradeoff between gain and noise characteristics was attained.", "The SiPMs were placed inside the 4 mm–diameter female counterparts of the plastic connector (Fig.REF ), and soldered onto double-layered printed circuit boards made of glass epoxy (Panasonic R-1705).", "Two circuit boards 16.6$\\times $ 28.5 mm$^2$ and 20$\\times $ 33 mm$^2$ were used for the two module types shown in Fig.", "REF (a) and (b), respectively.", "They supplied the bias, filtering, and readout decoupling according to the circuit diagram shown in Fig.", "REF .", "The boards were firmly grounded on the aluminum box of the detector module.", "The output signals were transmitted using 1–1.5-m-long, SMC-type coaxial cable assemblies to the outside of the module." ], [ "Pion beam measurements", "The T9 beamline of CERN provided a secondary beam of pions, muons, and electrons [30].", "It was produced by extracting protons of momentum $p=24$ GeV/c from the PS, and directing them onto a metallic target.", "The charge and momentum of the secondary particles were selected by pairs of dipole magnets and slits located downstream of the production target.", "For this experiment (Fig.", "REF ), we used negative pions and muons with $p=0.9$ –1.1 GeV/c, which is equivalent to the highest momenta of the pions emerging from antiproton annihilations [2].", "Figure: Setup of the photoelectron yield measurement at the T9 beamline of CERN.The arrival of the 1 GeV/c pions were detected by the coincidence between two scintillation counters P1P1 and P2P2.", "Two gasCherenkov detectors (C1C1 and C2C2) were used to reject the electron background.", "The SiPM was read outby a charge-sensitive preamplifier.", "The signal was amplified by a shaping amplifier with a 50-ns time constant.A peak-sensing ADC measured the signal height.Figure: (a): Pulse height distribution of BC-408 cast scintillator of 12.7 mm thickness which is struck with 1 GeV/c pions (open histogram).", "The response against a pulsed LED light source (blue filled histogram) is shown superimposed.", "Peaks corresponding tomultiple photoelectron numbers were resolved and used to convert the ADC channels to photoelectron numbers(see text).", "The solid line indicates the best fit of a convoluted Landau-Gaussian function.", "Similar histograms for (b) 19-mm-thickextruded scintillators constructed by Fermilab and (c) 10-mm-thick ones by C.I.", "Kogyo K.K.", "and embedded with 1.2-mm-diameter WLS fibers are also shown.The secondary beam was allowed to pass through two Cherenkov counters $C_1$ and $C_2$ of length 2.5 and 5 m, which were filled with nitrogen gas at a pressure of $P\\sim 1.2$ bar.", "The Cherenkov signals were used to reject the electron contamination in the beam.", "The beam then traversed two plastic scintillators $P_1$ and $P_2$ , and a third scintillator $P_3$ with a 10–mm-diameter hole which defined the size of the beam.", "The data acquisition was triggered by events for which $\\overline{C_1}\\cdot \\overline{C_2}\\cdot P_1\\cdot P_2\\cdot \\overline{P_3}$ .", "Figure: (a) Spatial distribution of photoelectron yield Γ\\Gamma of BC-408 scintillator of size 750×\\times 50×\\times 12.7 mm 3 ^3along the axis parallel to the WLS fiber.", "(b) Distribution along the axis perpendicular to the fiber.Figure: Spatial distributions of photoelectron yield of extruded scintillators constructed by Fermilab of size 1000×\\times 17×\\times 19 mm 3 ^3 along the axis parallel to the WLS fiber.", "(a) Optical grease, (b) cement, and (c) air gap were used as optical contacts between the fiber and the scintillator.The anodes of the SiPMs were biased between $-70$ and $-73$ V, corresponding to $\\sim $ 1 V above the Geiger breakdown voltage.", "The cathode signal was decoupled by a 4.7-nF capacitor, before being read out by a hybrid charge-sensitive preamplifier (Clear Pulse Co., Ltd. CS-507) with a charge-to-voltage conversion coefficient of 0.1 V/pC and integration time constant of $\\sim 1$ ms. A high-pass filter then differentiated the voltage signal with a time constant of $\\sim 50$ $\\mu {\\rm s}$ as in the design of Ref. [31].", "The preamplifier and high-pass filter were implemented on a single four-layer printed circuit board.", "The signal was transmitted by coaxial cables over a distance of $\\sim 50$ m. It then entered a shaping amplifier (Clear Pulse 4076 custom) consisting of a passive first-order differentiator and active second-order integrator with a time constant of $\\tau =50$ ns, which produced an unipolar, semi-Gaussian output pulse.", "The small shaping time in this final stage was needed to reduce the effects of integrating over the dark current of the SiPM, which had a typical rate of 0.1–1 MHz.", "The pulse amplitudes were measured by a 32-channel peak-sensing analog-to-digital converter (ADC, CAEN S.p.A. V785) with a vertical resolution of 12 bits.", "For data acquisition and analysis, the MIDAS [32] and ROOT [33] software packages were used.", "The gain of the SiPM was calibrated by measuring the light pulses produced by a light emitting diode (LED) with an emission wavelength of $\\sim 420$ nm.", "The light intensity was adjusted so that the SiPM detected an average of 3–10 wavelength-shifted photons.", "In Figs.", "REF (a)–(c), the distributions of signal amplitudes of LED light pulses measured in this way are indicated by the solid histograms.", "In SiPMs, secondary afterpulsing and optical crosstalk effects [14], [20] can cause a spurious increase in the number of discharging pixels $\\lambda _{\\rm meas}$ , relative to the number of initial photoelectrons.", "These effects are strongly dependent on gain and temperature.", "To estimate the true number of initial photoelectrons $\\lambda _{\\rm real}$ , we carried out a statistical analysis [34] of the pulse height distributions of Figs.", "REF (a)–(c).", "Assuming that the number of photons arriving from the LED follows a Poisson distribution, the photoelectron number can be estimated using the equation, $\\lambda _{\\rm real}= -\\log ( N_{\\rm zero}/N_{\\rm total})$ .", "Here $N_{\\rm total}$ denotes the total number of LED flashes, and $N_{\\rm zero}$ the number of flashes in which no photon was detected.", "In this way, the combined afterpulsing and crosstalk probability $\\varepsilon =1-\\lambda _{\\rm real}/\\lambda _{\\rm meas}$ was found to be 15–20$\\%$ at a photomultiplier gain of $\\sim 5\\times 10^6$ , and the ADC channel numbers were converted to photoelectron numbers.", "This calibration was carried out every $\\sim 10$ min during the experiment with the pion beam.", "We also estimated $\\varepsilon $ using an alternative method of measuring the peak height distributions of the dark current.", "Most of the dark counts involved the discharge of single pixels; in $\\sim 20\\%$ of the cases, however, 1–3 additional pixels fired within the $\\sim 50$ -ns shaping time of the amplifier, due to afterpulsing and crosstalk effects.", "The number of counts in these 2-pixel and 3-pixel events, relative to the total number of dark current events, provide an estimation of $\\varepsilon $ .", "The values estimated by the two methods agreed within the experimental uncertainties.", "The open histograms of Figs.", "REF (a)–(c) show the pulse height distributions of the pion events measured using various scintillators.", "Each histogram contains typically 10$^5$ pion hits.", "From these data, we obtained the photon yield $\\Gamma $ in the following way: first, we determined the most probable energy loss (i.e., the maximum of the distribution) by fitting a convoluted Landau–Gaussian function on the spectra [34], [35].", "We then corrected this value for crosstalk and afterpulsing effects, using the probabilities determined using the LED calibration measurements.", "In the examples of Fig.", "REF (a)–(c), the obtained $\\Gamma $ -values were respectively $36\\pm 3$ , $30\\pm 2$ , and $6\\pm 1$ photoelectrons.", "We repeated the measurements on at least 3 specimens of the same detector configurations.", "The experimental uncertainty on $\\Gamma $ was taken as the square root of the quadratic sum of the statistical uncertainties of the three measurements, and the systematic ones of the calibration.", "Table: Photoelectron yields Γ\\Gamma of various scintillator, WLS fiber, and silicon photomultiplierconfigurations measured using the ∼\\sim 1 GeV/c pions.", "Cast (type BC-408 manufactured by Saint Gobain Crystals)and extruded A, B (manufactured by Fermilab), and C (CI Kogyo K.K.)", "scintillators are compared.The highest photoelectron yields of $\\Gamma =35-39$ were obtained for the cast scintillators (Table REF ).", "This is presumably due to the fact that scintillators with a polyvinyltoluene base generally have a technical light yield [36], [37] which is 15–25% higher than those with a polystyrene base.", "Moreover the long ($\\sim $ 2 m) light attenuation length in the BC-408 material and the high reflectivity of the specular reflector foils should in principle allow the scintillation photons to make on average 20–30 reflections inside the bar and illuminate a larger section of the WLS fiber.", "As shown in Fig.", "REF (a), the signal is well-separated from the background of dark current counts, with a signal-to-noise ratio $>7$ .", "We next measured the spatial uniformity of $\\Gamma $ for the cast scintillator, by varying the position of the pion beam along on its surface.", "In Fig.", "REF (a), the spatial distribution of $\\Gamma $ along the horizontal (i.e, parallel to the fiber) axis of the scintillator are shown; here $x=0$ corresponds to the position of the SiPM.", "Fig.", "REF (b) shows the distribution obtained by scanning the beam along the 50-mm-width of the bar perpendicular to the fiber axis, where $y$ denotes the distance of the beam center from the fiber.", "The variations in the spatial uniformity was found to be within the experimental uncertainty on $\\Gamma $ of $\\sim $ 10%.", "The $\\Gamma $ -values for the Fermilab extruded scintillators of 19$\\times $ 17 mm$^2$ cross section (denoted as Extruded A in Table REF ) measured at a distance of $x=100$ mm from the SiPM was 34–38 photoelectrons.", "These yields were $\\sim $ 30$\\%$ less than those of cast scintillators of the same thickness, and roughly agree with the results reported by other groups [8], [17], [34].", "This provides sufficient single-to-noise ratio ($>6$ ) above the dark current for detecting minimum-ionizing particles [Fig.", "REF (b)].", "Extruded scintillators with no cement or grease in the holes retained a spatial uniformity over the 1-m length of the bar which was better than $\\sim 15\\%$ [Fig.", "REF (c)].", "Placing grease [Fig.", "REF (a)] or optical cement [Fig.", "REF (b)] in the holes increased the $\\Gamma $ -values near the SiPMs to $\\sim $ 39–41 photoelectrons.", "At the opposite ends of the bar ($x=900$ mm), however, the light yield dropped to $\\Gamma \\sim $ 25, which is similar to values obtained without any filling.", "This poor uniformity appears to indicate that it is difficult to properly distribute the cement or grease along the entire length of the bar.", "We also measured the yield of an extruded scintillator with a cross section of $40\\times 10$ mm$^2$ (denoted as Extruded B in Table REF ) manufactured by Fermilab, which resulted in a value $\\Gamma \\sim $ 14–19 depending on the SiPM.", "No obvious difference in $\\Gamma $ was observed between WLS fibers of diameters $d=1$ mm and $1.2$ mm.", "This may be due to the fact that the active area of the SiPM was smaller than the fiber diameter.", "We used $d=1.2$ mm fibers in the final detector, according to the observation in Ref.", "[38] that the alignment between the SiPM and fiber may become less critical for larger-diameter fibers.", "No significant increase in $\\Gamma $ was found for the fibers with aluminum reflectors deposited on the rear ends.", "The reason for this is not understood, but it may be due to the insufficient quality of the reflecting surfaces.", "The extruded scintillators which were manufactured by CI-Kogyo K. K. as prototypes in 2010, yielded values of $\\Gamma \\sim 7$ –9.", "This was insufficient to properly separate the signal from the dark current [Fig.", "REF (c)].", "The reason for this is not understood, but extensive efforts to reduce the impurities and moisture, and control the temperature during the extrusion process have been made by the manufacturer since 2010.", "These measurements do not represent the latest such efforts.", "As expected, the SiPMs containing 400 pixels of size 50$\\times $ 50 $\\mu $ m$^2$ provided a $\\sim $ 35% increase in the $\\Gamma $ -values, compared to the 1600-pixel ones.", "We also tested detectors (Hamamatsu Photonics K. K. S10362-33-050C) with a larger active area of 3$\\times $ 3 mm$^2$ , consisting of 3600 pixels of size 50$\\times $ 50 $\\mu $ m$^2$ , but the high dark current ($\\sim 6\\times 10^6$ Hz) was found to be a significant drawback in our application." ], [ "Antiproton beam measurements", "Fig.", "REF shows the experimental setup which was used to measure the response of the scintillators against antiproton annihilations.", "The AD provided a 300-ns long pulsed beam containing $(2-3)$$\\times $ 10$^7$ antiprotons with a kinetic energy $E=5.3 $ MeV at a repetition rate of $0.01$ Hz [39], [40].", "This beam was allowed to pass through a radiofrequency quadrupole decelerator (RFQD), which reduced the energy of some $\\sim 30\\%$ of the antiprotons to $E=70$ keV.", "The decelerated antiprotons were diverted by an achromatic momentum analyzer which was connected to the exit of the RFQD, and focused into an experimental helium target.", "The analyzer consisted of two dipole magnets which deflected the beam at an angle $\\Theta = 20^{\\circ }$ , and three 1-T solenoid magnets.", "The beam profile was measured using microwire secondary electron emission detectors [31], [41], [42].", "Some $\\sim $ 70% of the antiprotons emerged from the RFQD without being decelerated, and annihilated on the walls of the analyzer.", "Figure: Experimental setup to measure the response of the scintillator at the Antiproton Decelerator (not to scale).", "Dashed lines represent the beam envelope.", "A radiofrequency quadrupole decelerated the antiproton beam from a kinetic energy of E=E=5.3 MeV to 70 keV.A momentum analyzer transported the 70-keV antiprotons to the position of a helium gas target.", "Charged pions emerging from antiproton annihilations in the target and beamline walls were detected by an acrylic Cherenkov counter and scintillator.Figure: Output signals of (a) acrylic Cherenkov detector and (b)–(f) scintillators read out by WLS fibers and SiPMsbiased at various voltages V bias V_{\\rm bias}.", "Each waveform is an average of 14–15 antiproton pulses arriving at the experimental target.Figure: Normalized amplitudes of the output signals of the scintillator against the pulsed AD beam,plotted against the bias voltage applied to the SiPM.", "The intensity of the peaks at t=950t=950 ns (filled triangles) and 2250 ns (circles) in Fig.", "are shown superimposed.", "The signal amplitude increases rapidly around the Geiger breakdown voltage ∼69\\sim 69 V, and then saturates at higher bias voltages, presumably because the charge in the SiPM is depleted.A Cherenkov detector [2] made of UV-transparent acrylic (Mitsubishi Rayon, Acrylite000) of size 300$\\times $ 100$\\times $ 20 mm$^3$ was placed at a distance of $\\sim 1$ m from the experimental target.", "Some of the pions that emerged from the antiproton annihilations traversed the detector, and the resulting flash of Cherenkov light was detected by a fine-mesh photomultiplier (Hamamatsu R5505GX-ASSYII).", "This photomultiplier had a photocathode of diameter $17.5$ mm, and operated at a gain of $5\\times 10^4$ .", "This detector has been used for many years in precision laser spectroscopy of antiprotonic helium atoms [7], [40], and its linear behavior against high fluxes of pions have been extensively characterized [2].", "The analog waveform of the photomultiplier was recorded using a digital oscilloscope.", "In Fig.", "REF (a), the waveform of the Cherenkov signal taken as an average of 14 pulses, each containing $\\sim 6\\times 10^6$ antiprotons arriving in the target, is shown.", "The instantaneous flux of pions was so high that individual annihilations could not be resolved.", "The 300-ns-long Gaussian-shaped signal at $t\\sim 2250$ ns corresponds to the envelope of the pulsed beam of antiprotons which annihilated at the target position.", "The annihilations that occurred at other positions along the RFQD and analyzer cannot be detected, because the resulting pion flux is too low compared to the sensitivity of the detector.", "We also placed a cast scintillator of size 750$\\times $ 50$\\times $ 10 mm$^3$ , which was read out by a WLS fiber and a 1600-pixel SiPM, at the same position.", "The cathode of the SiPM was biased at $V_{\\rm bias}=65.0-73.5$ V which corresponded to a voltage between $-4$ and $+4.5$ V around the Geiger breakdown value $V_0$ .", "The anode was grounded using a 1-k$\\Omega $ resistor and 100-nF capacitor, similar to the configuration in Ref. [43].", "The anode signal was decoupled by a 100-nF capacitor, and the waveform recorded by a digital oscilloscope of input impedance 50 $\\Omega $ .", "In Fig.", "REF (b), the signal measured at a low bias of $V_{\\rm bias}\\sim 65$ V corresponding to $V_0-4$ V is shown.", "The detector operated as a conventional avalanche photodiode of low gain.", "As expected, a single peak of amplitude $\\sim 2$ mV appeared at $t\\sim 2250$ ns, as in the Cherenkov counter case [Fig.", "REF (a)].", "As $V_{\\rm bias}$ neared the breakdown voltage [Fig.", "REF (c), $V_{\\rm bias}=68.5$ V], however, two secondary peaks at $t\\sim 950$ and 1800 ns appeared, while the signal amplitude increased by an order of magnitude, to $\\sim 20$ mV.", "This was caused by the higher sensitivity and gain of the SiPM, which now detected the weak scintillation light that arose from the annihilations in the two bending magnets located upstream of the target (Fig.", "REF ).", "As we exceeded the breakdown voltage [Fig.", "REF (d), $V_{\\rm bias}=69$ V], many more structures appeared which corresponded to the small number of antiprotons that scraped the walls of the analyzer or the secondary electron emission monitors.", "The detector response against the peak at $t=2250$ ns of amplitude $\\sim 60$ mV was clearly saturated.", "Between $V_{\\rm bias}=72$ [Fig.", "REF (e)] and 73.5 V [Fig.", "REF (f)], the signal amplitude ($\\sim 400$ mV) no longer increased, presumably because the charge stored in the SiPM was depleted.", "The waveform was also deformed by the recharging time of the bias circuit and SiPM.", "When compared with the measurements in Sect.", ", we estimated that for a setting of $V_{\\rm bias}\\sim V_0+1.5\\sim 70.5$ V, which was an optimal compromise between gain and dynamic range, around $\\sim 10$ simultaneous pion hits would saturate the SiPMs.", "In Fig.REF , the amplitudes of the two intense peaks at $t=950$ ns (filled triangles) and 2250 ns (circles) normalized to the antiproton beam intensity measured by the Cherenkov counter are shown for various values of $V_{\\rm bias}$ .", "The rapid increase at $\\sim $ 68.8 V is due to the SiPM reaching Geiger breakdown.", "From these results, we decided to operate the scintillator in two modes, i): at a bias of $V_{\\rm bias}\\sim V_0-4\\sim 65$ V to measure the timing profile of the antiproton beam in a relatively linear way, ii): at $V_{\\rm bias}\\sim V_0+1.5\\sim 70.5$ V to detect single pion hits." ], [ "Discussions and conclusions", "In conclusion, we constructed a 541-channel, segmented scintillator for detecting and tracking charged pions emerging from antiproton annihilations.", "Both cast and extruded scintillators of thicknesses $t_d=12.7$ –19 mm were used, which were read out by wavelength-shifting (WLS) fibers and silicon photomultipliers (SiPMs).", "The design was optimized to attain high photoelectron yields of around $\\Gamma =35-38$ for minimum-ionizing particles.", "This was sufficient to detect antiproton annihilations in a future Paul trap with an estimated efficiency of $>90\\%$ .", "An important source of background was the dark current of the SiPMs, which reached a maximum amplitude equivalent of 3–4 simultaneous pixel discharges within the $\\sim 50$ -ns shaping time of the amplifier.", "This implied that a $\\Gamma $ -value of $>20$ was needed to achieve a sufficient signal-to-noise ratio.", "In the current detector design, it may therefore be difficult to significantly reduce the thicknesses of the scintillators (e.g., $t_d\\ll 10$ mm) and increase the spatial resolution, without sacrificing the detection efficiency.", "The response of the scintillator against the high-intensity flux of pions emerging from the annihilation of a pulsed beam of antiprotons was studied.", "At low bias voltages of the SiPM below the Geiger breakdown threshold, the detector accurately measured the envelope of the pulsed antiproton beam, but the sensitivity was too low to detect individual annihilations.", "At higher bias voltages, the detector became sensitive to individual pion hits, but a nonlinear behavior was seen.", "The scintillators were recently used to detect the small number of annihilations that occurred when an antiproton beam of kinetic energy $\\sim 130$ keV was allowed to traverse thin target foils [1].", "This technique may be used in the future to determine the total annihilation cross-sections of antiprotons on various metal targets in the low-energy region." ], [ "Acknowledgments", "We are deeply indebted to A. Pla-Dalmau for supplying the extruded scintillators used in this work and for fruitful discussions.", "We thank the ASACUSA collaboration, R. Dumps, L. Gatignon, A. László, A. Minamino, T. Nakaya, K. Niita, T. Schneider, and P. Zalán for their help.", "This work was supported by the European Research Council (ERC-Stg), European Science Foundation (EURYI), Monbukagakusho (grant no.", "20002003), and the Hungarian Research Foundation (K103917)." ] ]	1403.0451
[ [ "Fine-grained uncertainty relation under the relativistic motion" ], [ "Abstract Among various uncertainty relations, the profound fine-grained uncertainty relation is used to distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements.", "In this Letter, we explore this uncertainty relation in relativistic regime.", "For observer undergoes an uniform acceleration who is immersed in an Unruh thermal bath, we show that the uncertainty bound is dependent on the acceleration parameter and choice of Unruh modes.", "We find that the measurements in mutually unbiased bases, sharing same uncertainty bound in inertial frame, could be distinguished from each other for a noninertial observer.", "In an alternative scenario, for the observer restricted in a single rigid cavity, we show that the uncertainty bound exhibits a periodic evolution w.r.t.", "the duration of acceleration.", "With properly chosen cavity parameters, the uncertainty bounds could be protected.", "Moreover, we find that uncertainty bound can be degraded for specific quantum measurements to violate the bound exhibited in nonrelativistic limit, which can be attributed to the entanglement generation between cavity modes during particular epoch.", "Several implications of our results are discussed." ], [ "Introduction", "The distinguishability of a quantum theory from its classical counterpart is formulated by the Heisenberg uncertainty principle [1], which bounds our prediction ability for a quantum system.", "In terms of entropic measures, this uncertainty relation can be recast in a modern information-theoretical form [2], which states that $H(Q)+H(R)\\geqslant \\log _2\\frac{1}{c}$ , where $H(Q)$ and $H(R)$ denote the Shannon entropy for the probability distribution of the measurement outcomes.", "Since the overlap $c$ between observables $Q$ and $R$ does not depend on specific states to be measured, the right-hand side (RHS) of the inequality provides a fixed lower bound and a more general framework of quantifying uncertainty than the standard deviations [3], [4].", "Moreover, once using quantum memory to store information about the measured system, the entropic uncertainty bound could even be violated [5].", "All these entropic uncertainty relations play an important role in many quantum information processing.", "However, constructed from the probability distribution of a measurement taken, entropic function is still a rather coarse way of measuring the uncertainty of a set of measurements (see Ref.", "[6] for a recent review).", "For instance, one can not distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements.", "To overcome this defect, a new form of uncertainty relation, i.e., Fine-Grained Uncertainty Relation (FGUR), has been proposed recently [7].", "For a set of measurements labeled by $t$ , associating with every combination of possible outcomes $\\textbf {x}=(x^{(1)},\\ldots ,x^{(n)})\\in \\mathbf {B}^{\\times n}$ , there exist a set of inequalities $\\left\\lbrace \\sum _{t=1}^{n}p(t)p(x^{(t)}\|\\rho )\\le \\zeta _{\\textbf {x}}\\Big \|\\textbf {x}\\in \\mathbf {B}^{\\times n} \\right\\rbrace ,$ where $p(t)$ is the probability of choosing a particular measurement, $p(x^{(t)}\|\\rho )$ is the probability that one obtains the outcome $x^{(t)}$ after performing measurement $t$ on the state $\\rho $ .", "To measure the uncertainty, the maximum in function $\\zeta _{\\textbf {x}}=\\max _{\\rho }\\sum _{t=1}^n p(t)p(x^{(t)}\|\\rho )$ should be taken over all states allowed on a particular system.", "It can be proved that one can not obtain outcomes with certainty for all measurements simultaneously when $\\zeta _{\\textbf {x}}<1$ .", "On the other hand, for the so-called Maximally Certain State (MCS), the inequality (REF ) can be saturated.", "Since its introduction, many applications have been found for the FGUR.", "For instance, it was shown [8] that the FGUR could be used to discriminate among classical, quantum, and superquantum correlations involving two or three parties.", "Moreover, the uncertainty bound in (REF ) could be optimized once the measured system is assisted by a quantum memory [9].", "Dramatically, a profound link between the FGUR and the second law of thermodynamics has been found [10], which claims that a violation of uncertainty relation implies a violation of thermodynamical law.", "Other studies from various perspectives could be found in [11], [12], [13].", "While most of studies on uncertainty relations are nonrelativistic, a complete account of these relations requires one to understand them in relativistic regime, which may even shed new light on quantum gravity [14].", "In the previous work [15], we have shown that, besides the choice on the observers, the entropic uncertainty bound should also depend on the relativistic motion state of the observer who performs the measurement.", "This new character of quantum-memory-assisted Entropic Uncertainty Relation (EUR) is a direct result of entanglement generation in a relativistic system.", "In this paper, we explore the FGUR under the decoherence rooting in the relativistic motion of quantum systems, and find the uncertainty bound does depend on the motion state of the system.", "We first consider observer undergoes an uniform acceleration relative to an inertial reference.", "Since two frames differ in their description of a given quantum state due to the Unruh effect, the concept of measurement becomes observer-dependent, which implies a nontrivial relativistic modification to the FGUR.", "Without loss of generality, beyond the single-mode approximation, we show that the Unruh effect modifies the uncertainty bound for the measurements in $d=2$ Mutually Unbiased Bases (MUBs), which are intimately related to complementarity principle.", "Dramatically, we find that, for a noninertial observer, the measurements in general MUBs could be distinguished from each other, while they share same uncertainty bound in inertial frame.", "We extend these results to an alternative scenario, where, to prevent the Unruh decoherence, an observer is restricted in a single rigid cavity which undergoes an nonuniform acceleration.", "We show that the uncertainty could be degraded by the nonuniform acceleration of cavity during particular epoch, while the uncertainty bound itself exhibits a periodic evolution with respect to the duration of the acceleration.", "This phenomenon can be attributed to the entanglement generation between the field modes in single cavity that plays the role of quantum memory.", "Moreover, except the acceleration-duration time with integer periods, the measurements in different MUBs are distinguishable by the corresponding uncertainty bounds." ], [ "FGUR for an accelerating observer", "We first investigate the FGUR for an observer with uniform acceleration $a$ , who performs projective measurements on the quantum state constructed from free field modes.", "For the noninertial observer traveling in, e.g., right Rindler wedge, field modes in left Rindler wedge are unaccessible.", "The information loss associated with the acceleration horizon results in a thermal bath.", "From quantum information point of view [16], this celebrated Unruh effect induces a nontrivial evolution of quantum entanglement between field modes which plays a prominent role in most quantum-information protocols.", "For simplicity, considering fermionic field with few degrees, the most general vacuum state $\|0_U\\rangle =\\bigotimes _{k}\|0_{k,R}\\rangle \\otimes \|0_{k,L}\\rangle $ should be annihilated by Unruh operators $C_{k,U}$ [17], [18] $C_{k,U}&=&q_RC_{k,R}^\\dag +q_LC_{k,L},\\qquad q_R^2+q_L^2=1,\\nonumber \\\\C_{k,R}&=&\\cos rc_{k,\\mbox{\\tiny I}}-\\sin rd^\\dag _{k,\\mbox{\\tiny II}},\\nonumber \\\\C_{k,L}&=&\\cos rc_{k,\\mbox{\\tiny II}}-\\sin rd^\\dag _{k,\\mbox{\\tiny I}}.$ where $q_R$ and $q_L$ are real parameters, and $\\tan r=e^{-\\pi \\omega /a}$ .", "The particle and antiparticle operators $c_{k,i}$ and $d_{k,i}$ in respective Rindler wedge $\\lbrace i=\\mbox{I},\\mbox{II}\\rbrace $ satisfy the usual anti commutation relations.", "By analytic construction to whole spacetime, the proper Unruh modes are symmetric between Rindler wedges I and II.", "For particular frequency $\\Omega $ , the Unruh vacuum is [18] $\|0_{\\Omega ,U}\\rangle &=&\\cos ^2 r\|0000\\rangle -\\sin ^2 r\|1111\\rangle \\nonumber \\\\&&+\\sin r\\cos r(\|1100\\rangle -\|0011\\rangle )$ and the first excitation is $\|1_{\\Omega ,U}\\rangle &=&q_R(\\cos r\|1000\\rangle -\\sin r\|1011\\rangle )\\nonumber \\\\&&+q_L(\\sin r\|1101\\rangle +\\cos r\|0001\\rangle )$ where we introduce the notations $\|1111\\rangle =b^\\dag _{\\mbox{\\tiny I}}c^\\dag _{\\mbox{\\tiny II}}c^\\dag _{\\mbox{\\tiny I}}b^\\dag _{\\mbox{\\tiny II}}\|0_{\\Omega ,\\mbox{\\tiny I}}\\rangle ^+\|0_{\\Omega ,\\mbox{\\tiny II}}\\rangle ^-\|0_{\\Omega ,\\mbox{\\tiny I}}\\rangle ^-\|0_{\\Omega ,\\mbox{\\tiny II}}\\rangle ^+$ with particle and anti-particle vacua are denoted by $\|0\\rangle ^+$ and $\|0\\rangle ^-$ .", "It should be noted that different operator ordering in fermonic systems could lead to nonunique results in quantum information [19].", "For instance, if we rearrange operator ordering in (REF ) as $b^\\dag _{\\mbox{\\tiny I}}c^\\dag _{\\mbox{\\tiny I}}c^\\dag _{\\mbox{\\tiny II}}b^\\dag _{\\mbox{\\tiny II}}$ , then a new Fock basis is defined $\|1111\\rangle ^{\\prime }=-\|1111\\rangle $ .", "This so-called physical ordering [20], in which all region I operators appear to the left of all region II operators, was proposed to guarantee the entanglement behavior of above states would yield physical results.", "Hereafter, we adopt this particular operator ordering.", "To explore how the relativistic motion of an observer could influence the FGUR, we consider a scenario in which the state to be measured should be prepared in an inertial frame.", "After that, the observer undergoes an uniform acceleration and performs measurements.", "Since the state should be described in the corresponding Rindler frame, information would lose via Unruh effect.", "Therefore, the uncertainty obtained by the accelerated observer would be motion-dependent.", "We illustrate above insight by measurements $\\sigma _x$ and $\\sigma _z$ , behaving as the best measurement basis, where Pauli operators $\\sigma _x$ and $\\sigma _z$ with equal probability $1/2$ are chosen [7].", "Remarkably, along with $\\sigma _y$ , three set of their eigenvectors form the MUBs in Hilbert space with dimension $d=2$ , which plays central role to theoretical investigations and practical exploitations of complementarity properties [21].", "For all pure states $\|\\psi \\rangle =\\cos \\frac{\\theta }{2}\|0\\rangle +e^{i\\phi }\\sin \\frac{\\theta }{2}\|1\\rangle $ , $\\theta \\in [0,\\pi ],\\phi \\in [0,2\\pi )$ , the corresponding density matrix $\\rho =\|\\psi \\rangle \\langle \\psi \|$ should be rewritten according to the transformation (REF ) and (REF ).", "Since the field modes in Rindler wedge II is unaccessible to observer, after tracing over the modes in wedge II, the reduced density matrix is $\\rho _{red}&\\equiv &\\mbox{Tr}_{\\mbox{\\tiny II}}\|\\psi \\rangle \\langle \\psi \|\\nonumber \\\\&=&\|00\\rangle _{\\mbox{\\tiny I}}\\langle 00\|(\\cos ^2\\frac{\\theta }{2}c^4+\\sin ^2\\frac{\\theta }{2}q_L^2c^2)\\nonumber \\\\&&+\|11\\rangle _{\\mbox{\\tiny I}}\\langle 11\|(\\cos ^2\\frac{\\theta }{2}s^4+\\sin ^2\\frac{\\theta }{2}q_R^2s^2)\\nonumber \\\\&&+\|01\\rangle _{\\mbox{\\tiny I}}\\langle 01\|(\\cos ^2\\frac{\\theta }{2}s^2c^2)\\nonumber \\\\&&+\|10\\rangle _{\\mbox{\\tiny I}}\\langle 10\|(\\cos ^2\\frac{\\theta }{2}s^2c^2+\\sin ^2\\frac{\\theta }{2}q_R^2c^2+\\sin ^2\\frac{\\theta }{2}q_L^2s^2)\\nonumber \\\\&&-\|00\\rangle _{\\mbox{\\tiny I}}\\langle 01\|e^{i\\phi }\\cos \\frac{\\theta }{2}\\sin \\frac{\\theta }{2}q_Lsc^2\\nonumber \\\\&&+\|00\\rangle _{\\mbox{\\tiny I}}\\langle 10\|e^{-i\\phi }\\cos \\frac{\\theta }{2}\\sin \\frac{\\theta }{2}q_Rc^3\\nonumber \\\\&&-\|00\\rangle _{\\mbox{\\tiny I}}\\langle 11\|\\sin ^2\\frac{\\theta }{2}q_Rq_Lsc\\nonumber \\\\&&+\|01\\rangle _{\\mbox{\\tiny I}}\\langle 11\|e^{-i\\phi }\\cos \\frac{\\theta }{2}\\sin \\frac{\\theta }{2}q_Rs^2c\\nonumber \\\\&&+\|10\\rangle _{\\mbox{\\tiny I}}\\langle 11\|e^{i\\phi }\\cos \\frac{\\theta }{2}\\sin \\frac{\\theta }{2}q_Ls^3+(h.c.)_{\\mbox{\\tiny nondiag}}$ with abbreviation $c\\equiv \\cos r,\\ s\\equiv \\sin r$ .", "After performing the projective measurements $\\sigma _x$ and $\\sigma _z$ on particle sector, we have the probabilities for the outcomes $(0^x,0^z)$ $p(0^z\|\\sigma _z)_{\\rho }&\\equiv &\\mbox{tr}(\|0\\rangle ^+_{\\mbox{\\tiny I}}\\langle 0\|\\rho _{red})=c^2(\\cos ^2\\frac{\\theta }{2}+\\sin ^2\\frac{\\theta }{2}q_L^2),\\nonumber \\\\p(0^x\|\\sigma _x)_\\rho &\\equiv &\\mbox{tr}(\|+\\rangle ^+_{\\mbox{\\tiny I}}\\langle +\|\\rho _{red})\\nonumber \\\\&=&\\frac{1}{2}+\\cos \\frac{\\theta }{2}\\sin \\frac{\\theta }{2}\\cos \\phi q_Rc.$ Here, for later convenience, we chose the measurements in basis $\\lbrace \|+\\rangle _{\\mbox{\\tiny I}},\|-\\rangle _{\\mbox{\\tiny I}}\\rbrace $ and $\\lbrace \|0\\rangle _{\\mbox{\\tiny I}},\|1\\rangle _{\\mbox{\\tiny I}}\\rbrace $ which are the eigenstates of Pauli matrix $\\sigma _x$ and $\\sigma _z$ restricted in Rindler wedge I.", "Therefore, LHS of FGUR (REF ) should be $U&\\equiv &\\frac{1}{2}[p(0^z\|\\sigma _z)_\\rho +p(0^x\|\\sigma _x)_\\rho ]\\nonumber \\\\&=&\\frac{1}{4}[\\sin \\theta \\cos \\phi q_Rc+(\\cos \\theta q_R^2+q_L^2+1)c^2+1]$ For a particular Unruh mode with fixed acceleration, one can estimate that the maximum of $U$ which is $U_{max}=\\zeta $ should always be obtained with $\\theta =\\frac{\\pi }{4}$ and $\\phi =0$ [22].", "This means that the MCS which saturates the inequality (REF ) is independent of the acceleration of the observer, which indicates that once we choose the bases enabling an optimal uncertainty on average in an inertial frame, the corresponding measurements should maintain their optimality for all noninertial observer.", "This remarkable phenomenon could be useful in many real quantum process, for instance, the BB84 states $\\lbrace \|+\\rangle ,\|-\\rangle \\rbrace $ and $\\lbrace \|0\\rangle ,\|1\\rangle \\rbrace $ in quantum cryptography [23].", "Explicitly, the dependence of the fine-grained uncertainty bound with outcomes $(0^x,0^z)$ on the acceleration parameter and choice of Unruh modes could be expressed as $\\zeta _{(0^x,0^z)}&=&\\frac{1}{4}[c^2(1+q_L^2+\\frac{\\sqrt{2}}{2} q_R^2)+\\frac{\\sqrt{2}}{2}q_Rc+1]$ Above calculation can extend to any other pairs of outcomes $(0^x,1^z)$ , $(1^x,0^z)$ and $(1^x,1^z)$ , which all give the same bound $\\zeta =\\frac{1}{2}+\\frac{1}{2\\sqrt{2}}$ in inertial frame [7].", "Dramatically, we find that the nontrivial Unruh effect could distinguish these four pairs of measurements and separate them into two categories.", "For instance, we have $p(1^z\|\\sigma _z)_\\rho &\\equiv &\\mbox{tr}(\|1\\rangle _{\\mbox{\\tiny I}}\\langle 1\|\\rho _{red})\\nonumber \\\\&=&\\cos ^2\\frac{\\theta }{2}s^2+\\sin ^2\\frac{\\theta }{2}(q_R^2+q_L^2s^2),\\nonumber \\\\p(1^x\|\\sigma _x)_\\rho &\\equiv &\\mbox{tr}(\|-\\rangle _{\\mbox{\\tiny I}}\\langle -\|\\rho _{red})\\nonumber \\\\&=&\\frac{1}{2}-\\cos \\frac{\\theta }{2}\\sin \\frac{\\theta }{2}\\cos \\phi q_Rc.$ which give $\\zeta _{(1^x,1^z)}&=&\\frac{1}{4}[(1+\\frac{\\sqrt{2}}{2}c^2)q_R^2+(1+q_L^2)s^2+\\frac{\\sqrt{2}}{2}q_Rc+1]\\nonumber \\\\$ for the MCS with $\\theta =\\frac{5\\pi }{4}$ and $\\phi =0$ .", "After straightforward calculations, it can be shown that the uncertainty bound $\\zeta $ is also dependent on the direction of $z-$ axis, that measurements $(0^x,1^z)$ share the same bound (REF ) with $(1^x,1^z)$ , while $(1^x,0^z)$ has the same bound (REF ) with $(0^x,0^z)$ .", "This is definitely a new feature of fine-grained uncertainty bound triggered by noninertial relativistic motion.", "To further explore the influence of Unruh effect on FUGR, we illustrate the uncertainty bounds (REF ) and (REF ) for three different choice of Unruh modes, as depicted in Fig.", "REF .", "Figure: The value of ζ\\zeta is dependent on the acceleration parameter rr and choice of Unruh modes.", "Three set of curves correspond to the choice of Unruh modes with q L =0q_L=0 (black solid), q L =0.6q_L=0.6 (red dashed) and q L =1q_L=1 (blue dashed-double-dotted).For the case with $q_L=0$ , where the Minkowskian annihilation operator is taken to be one of the right or left moving Unruh modes, the noninertial observer would detect a single-mode state once the field is in a special superposition of Minkowski monochromatic modes from an inertial perspective [24].", "Under this Single-Mode Approximation (SMA), commonly assumed in the old literature on relativistic quantum information [16], we recover the standard result $\\zeta =\\frac{1}{2}+\\frac{1}{2\\sqrt{2}}$ for vanishing acceleration.", "As $r$ growing, the value of $\\zeta $ decreases, indicating an increment on measurement uncertainty.", "While there is no essential difference in the behavior of measurement uncertainty for outcomes $(0^x,0^z)$ and $(1^x,0^z)$ with various $q_L$ , however, the uncertainty bounds (REF ) are sensitive with the choice of Unruh modes.", "Surprisingly, as $r$ growing, for the Unruh modes with large $q_L$ , we observe a decrement on measurement uncertainty for outcomes $(0^x,1^z)$ and $(1^x,1^z)$ , which means that Unruh effect could even lift the uncertainty bound in (REF ).", "Finally, as illustrated in Fig.", "REF , we find that the distinguishability between the measurements in MUBs is a common feature for any choice of Unruh modes.", "To explain this, recall that, by definition, a set of orthonormal bases $\\lbrace \\mathcal {B}_k\\rbrace $ for a Hilbert space $\\mathcal {H}=\\mathbb {C}^d$ where $\\mathcal {B}_k=\\lbrace \|i_k\\rangle \\rbrace =\\lbrace \|0_k\\rangle ,\\cdots ,\|d-1_k\\rangle \\rbrace $ is called unbiased iff $\|\\langle i_k\|j_l\\rangle \|^2=\\frac{1}{d}$ , $\\forall \\ i,j\\in \\lbrace 0,\\cdots ,d-1\\rbrace $ holds for all basis vectors $\|i_k\\rangle $ and $\|j_l\\rangle $ belong to different bases, i.e., $\\forall \\ k\\ne l$ .", "From an inertial perspective, the MUBs are intimately related to complementarity principle [25], which indicates that the measurement of a observable reveals no information about the outcome of another one if their corresponding bases are mutually unbiased.", "However, for a noninertial observer, the bases $\\lbrace \\mathcal {B}_k\\rbrace $ should be transformed according to proper Bogoliubov transformations, which in general breaks the orthonormality.", "In other word, the MUBs in inertial frame would become non-MUBs from a noninertial perspective.", "Therefore, for the observer undergoing an uniform acceleration, we conclude that Unruh effect could distinguish measurements in MUBs." ], [ "FGUR for a nonuniform-moving cavity", "We now discuss an alternative scenario in which observer is localized in a rigid cavity.", "While the rigid boundaries of the cavity protect the inside observer from the Unruh effect, the relativistic motion of the cavity would still affect the entanglement between the free field modes inside [26], [27], [28], therefore leading to a motion-dependent uncertainty bound [15].", "For simplicity, we consider a $(1+1)$ -dimensional model, where the cavity with length $L=x_2-x_1$ imposes the Dirichlet conditions on the eigenfunctions $\\psi _n(t,x)$ of the Hamiltonian.", "A typical trajectory of nonuniform-moving cavity contains three segments referred as (I') when the cavity maintains its inertial status, then (II') begins to accelerate at $t=0$ , following the Killing vector $\\partial _\\eta $ , and finally (III') the acceleration ends at Rindler time $\\eta =\\eta _1$ , and the duration of the acceleration in proper time measured at the center of the cavity is $\\tau _1=\\frac{1}{2}(x_1+x_2)\\eta _1$ .", "The Dirac field can be expanded in quantized eigenfunctions as $\\psi =\\sum _{n\\geqslant 0}a_n\\psi _n+\\sum _{n\\leqslant 0}b^\\dag _n\\psi _n$ in segment I', and similarly be expressed by $\\hat{\\psi }_n$ in segment II' and $\\tilde{\\psi }_n$ in segment III'.", "The nonvanishing anticommutators $\\lbrace c_m,c^\\dag _n\\rbrace =\\lbrace d_m,d_n^\\dag \\rbrace =\\delta _{mn}$ define the vacuum $c_n\|0\\rangle =d_n\|0\\rangle =0$ .", "Any two field modes in distinct regions can be related by Bogoliubov transformations like $\\hat{\\psi }_m=\\sum _{n}A_{mn}\\psi _n$ and $\\tilde{\\psi }_m=\\sum _{n}\\mathbb {A}_{mn}\\psi _n$ , where the coefficients can be calculated perturbatively in the limit of small cavity acceleration [26].", "More specifically, by introducing the dimensionless parameter $h=2L/(x_1+x_2)$ , which is the product of the cavity's length and the acceleration at the center of the cavity, the coefficients can be expanded in a Maclaurin series to $h^2$ order, $A=A^{(0)}+A^{(1)}+A^{(2)}+\\mathcal {O}(h^3)$ , and similarly $\\mathbb {A}=\\mathbb {A}^{(0)}+\\mathbb {A}^{(1)}+\\mathbb {A}^{(2)}+\\mathcal {O}(h^3)$ .", "We start from a pure state $\|\\psi _k\\rangle =\\cos \\frac{\\theta }{2}\|0_k\\rangle +e^{i\\phi }\\sin \\frac{\\theta }{2}\|1_k\\rangle ^+$ in segment I.", "After the uniform acceleration, we can express this state in segment III' by means of the Bogoliubov transformations, which contains modes within all frequency.", "Throughout the process, we assume that the observer can only be sensitive to modes in particular frequency $k$ .", "Therefore all other modes with frequency $k^{\\prime }\\ne k$ should be traced out in the density matrix $\\tilde{\\rho }=\|\\tilde{\\psi }_k\\rangle \\langle \\tilde{\\psi }_k\|$ , which leads to $\\tilde{\\rho }_{red}&=&\\mbox{tr}_{\\lnot k}\|\\tilde{\\psi }_k\\rangle \\langle \\tilde{\\psi }_k\|\\nonumber \\\\&=&\|\\tilde{0}_k\\rangle \\langle \\tilde{0}_k\|(\\cos ^2\\frac{\\theta }{2}-\\cos ^2\\frac{\\theta }{2}f^-_k+\\sin ^2\\frac{\\theta }{2}f^+_k)\\nonumber \\\\&&+\|\\tilde{1}_k\\rangle ^{++}\\langle \\tilde{1}_k\|(\\sin ^2\\frac{\\theta }{2}+\\cos ^2\\frac{\\theta }{2}f^-_k-\\sin ^2\\frac{\\theta }{2}f^+_k)\\nonumber \\\\&&+\|\\tilde{0}_k\\rangle ^+\\langle \\tilde{1}_k\|\\frac{1}{2}\\sin \\theta e^{-i\\phi }(G_k+\\mathbb {A}^{(2)}_{kk})\\nonumber \\\\&&+\|\\tilde{1}_k\\rangle ^+\\langle \\tilde{0}_k\|\\frac{1}{2}\\sin \\theta e^{i\\phi }(G_k+\\mathbb {A}^{(2)}_{kk})^*$ where the coefficients are $f^+_k\\equiv \\sum _{p\\geqslant 0}\|\\mathbb {A}^{(1)}_{pk}\|^2$ and $f^-_k\\equiv \\sum _{p<0}\|\\mathbb {A}^{(1)}_{pk}\|^2$ .", "The probability of measurements $(0^x,0^z)$ are $p(0^z\|\\sigma _z)_{\\tilde{\\rho }}&\\equiv &\\mbox{tr}(\|\\tilde{0}_k\\rangle \\langle \\tilde{0}_k\|\\tilde{\\rho }_{red})\\nonumber \\\\&=&\\cos ^2\\frac{\\theta }{2}-\\cos ^2\\frac{\\theta }{2}f^-_k+\\sin ^2\\frac{\\theta }{2}f^+_k,\\nonumber \\\\p(0^x\|\\sigma _x)_{\\tilde{\\rho }}&\\equiv &\\mbox{tr}(\|+\\rangle \\langle +\|\\tilde{\\rho }_{red})\\nonumber \\\\&=&\\frac{1}{2}\\lbrace 1+\\sin \\theta \\mbox{Re}[e^{-i\\phi }(G_k+\\mathbb {A}^{(2)}_{kk})]\\rbrace .$ The uncertainty bound should be the maximum of LHS of FGUR (REF ), $U\\equiv \\frac{1}{2}[p(0^z\|\\sigma _z)_{\\tilde{\\rho }}+p(0^x\|\\sigma _x)_{\\tilde{\\rho }}]$ .", "Along the analysis before, we know that the acceleration of cavity would not change the MCS with parameters $\\theta =\\frac{\\pi }{4}$ and $\\phi =0$ .", "Therefore, we obtain the uncertainty bound for the cavity system $\\tilde{\\zeta }_{(0^x,0^z)}=\\frac{1}{4}[2+\\frac{\\sqrt{2}}{2}(1-F_+)+F_-+\\frac{\\sqrt{2}}{2}\\mbox{Re}(G_k+\\mathbb {A}^{(2)}_{kk})]\\nonumber \\\\$ The coefficients in the bound has been given in [26], which are $F_+&=&\\sum _{p=-\\infty }^{\\infty }\|E_1^{k-p}-1\|^2\|A^{(1)}_{kp}\|^2\\nonumber \\\\&=&\\frac{4h^2}{\\pi ^4}[4(k+s)^2(Q_6(1)-Q_6(E_1))+Q_4(1)-Q_4(E_1)]\\nonumber $ and [15] $F_-&\\equiv & f^+_k-f^-_k=\\bigg (\\sum _{p\\geqslant 0}-\\sum _{p<0}\\bigg )\|E_1^{k-p}-1\|^2\|A^{(1)}_{kp}\|^2\\nonumber \\\\&=&\\frac{16h^2}{\\pi ^4}2(k+s)[Q_5(1)-Q_5(E_1)]+P(k,s,E_1)\\nonumber $ with $s\\in [0,1)$ characterizing the self-adjoint extension of the Hamiltonian.", "Here we use the notation $Q_\\alpha (\\beta )\\equiv \\mbox{Re}\\big [\\mbox{Li}_\\alpha (\\beta )-\\frac{1}{2^\\alpha }\\mbox{Li}_\\alpha (\\beta ^2)\\big ]$ , Li is the polylogarithm and $E_1\\equiv \\exp (\\frac{i\\pi \\eta _1}{\\ln (x_2/x_1)})=\\exp (\\frac{i\\pi h\\tau _1}{2Lx_1\\tanh (h/2)})$ .", "$P$ is a polynomial summing for all terms with odd number $\\sum _{odd\\;m=1}^{k}\\frac{4h^2}{\\pi ^4}\\big (1-\\mbox{Re}(E_1^{m})\\big )\\bigg [4(k+s)\\bigg (\\frac{k+s}{m}-1\\bigg )+\\frac{1}{m^4}\\bigg ]$ and $\\mbox{Re}(G_k+\\mathbb {A}^{(2)}_{kk})&=&1-h^2\\bigg \\lbrace \\bigg (\\frac{1}{48}+\\frac{\\pi ^2(k+s)^2}{120}\\bigg )\\nonumber \\\\&&-\\frac{2}{\\pi ^4}\\big [4(k+s)^2Q_6(E_1)+Q_4(E_1)\\big ]\\bigg \\rbrace \\nonumber $ In previous section, we show that an uniformly-accelerating observer can distinguish the measurements in MUBs which share the same uncertainty bounds in an inertial frame.", "Here we generalize this to the scenario with rigid cavity.", "To proceed, we calculate the probabilities of measurements $(1^x,1^z)$ $p(1^z\|\\sigma _z)_{\\tilde{\\rho }}&\\equiv &\\mbox{tr}(\|\\tilde{1}_k\\rangle ^{++}\\langle \\tilde{1}_k\|\\tilde{\\rho }_{red})\\nonumber \\\\&=&\\sin ^2\\frac{\\theta }{2}+\\cos ^2\\frac{\\theta }{2}f^-_k-\\sin ^2\\frac{\\theta }{2}f^+_k,\\nonumber \\\\p(1^x\|\\sigma _x)_{\\tilde{\\rho }}&\\equiv &\\mbox{tr}(\|+\\rangle \\langle +\|\\tilde{\\rho }_{red})\\nonumber \\\\&=&\\frac{1}{2}\\lbrace 1-\\sin \\theta \\mbox{Re}[e^{-i\\phi }(G_k+\\mathbb {A}^{(2)}_{kk})]\\rbrace .$ which give the uncertainty bound $\\tilde{\\zeta }_{(1^x,1^z)}$ for MCS with $\\theta =\\frac{5\\pi }{4},\\ \\phi =0$ $\\tilde{\\zeta }_{(1^x,1^z)}=\\frac{1}{4}[2+\\frac{\\sqrt{2}}{2}(1-F_+)-F_-+\\frac{\\sqrt{2}}{2}\\mbox{Re}(G_k+\\mathbb {A}^{(2)}_{kk})]\\nonumber \\\\$ We depict above uncertainty bounds for measurements performed within cavity in Fig.", "REF .", "We find that both uncertainty bounds (REF ) and (REF ) are now periodic in time $\\tau _1$ , which measures the duration of the cavity acceleration, with the period $T=4Lx_1\\tanh (h/2)/h$ .", "The evolution of the uncertainty bound can be interpreted as a result of the entanglement generation between the field modes in the single rigid cavity that plays the role of quantum memory [9].", "By properly choosing the parameters to ensure that $\\tau _1=nT$ with $n\\in \\mathbb {N}$ , the uncertainty bounds are protected [15], recovering the value $\\frac{1}{2}+\\frac{1}{2\\sqrt{2}}$ as in inertial case.", "While the uncertainty change is very small due to the low acceleration approximation $\|k\|h\\ll 1$ imposed, our results could provide a novel way to detect the relativistic effect by future quantum metrology.", "Figure: The value of ζ\\zeta depends on the duration time of acceleration of rigid cavity.", "We choose k=1k=1.", "For each pair of measurements ((0 x ,0 z )(0^x,0^z) and (1 x ,0 z )(1^x,0^z), (1 x ,1 z )(1^x,1^z) and (0 x ,1 z )(0^x,1^z)), three cures from top to bottom correspond to parameters s=0,0.3,0.6s=0,\\ 0.3,\\ 0.6.", "The parameter u=η 1 /(2ln(x 2 /x 1 ))=hτ 1 /[4Lx 1 tanhh/2]u=\\eta _1/(2\\ln (x_2/x_1))=h\\tau _1/[4Lx_1\\tanh {h/2}] characterizes the duration time of cavity acceleration.", "To demonstrate the low acceleration approximation, the uncertainty is estimated under h=0.1h=0.1.On the other hand, for arbitrary acceleration duration $\\tau _1\\ne nT$ , measurements with outcomes $(0^x,0^z)$ and $(1^x,1^z)$ can be distinguished from each other by the corresponding uncertainty bounds (REF ) and (REF ).", "By a straightforward calculation, it can be shown that measurements $(0^x,1^z)$ share the same bound (REF ) with $(1^x,1^z)$ , while $(1^x,0^z)$ has the same bound (REF ) with $(0^x,0^z)$ .", "Therefore, same as Unruh effect, we conclude that the relativistic motion of a rigid cavity can provoke the distinguishability between the measurements in MUBs that share the same bound $\\frac{1}{2}+\\frac{1}{2\\sqrt{2}}$ for an inertial observer." ], [ "Discussions", "In this paper, we have explorer the nontrivial relativistic modification to the FGUR.", "We have shown that, for the observer undergoes a large acceleration, the associated Unruh effect could increase or reduce the fine-grained uncertainty bounds, depending on the choice of Unruh modes.", "Nevertheless, the MCS still could be independent of the acceleration, which indicates that once we choose the bases enabling an optimal uncertainty on average in an inertial frame, the corresponding measurements should maintain their optimality for all noninertial observer.", "Moreover, we have shown that the measurements in MUBs, sharing same uncertainty bound in inertial frame, could be distinguished from each other when the observer undergoes a nonvanishing acceleration.", "In an alternative scenario, we have investigated the FGUR for the measurements restricted in a single rigid cavity, where the uncertainty bound itself exhibits a periodic evolution with respect to the duration of the acceleration.", "This phenomenon could be understand by the entanglement generation in a single rigid cavity that plays the role of quantum memory.", "Our results provide a novel way to investigate the relativistic effect from a quantum-information perspective, that could be experimental tested by quantum metrology [29].", "Our results could be linked to many interesting issues.", "For instance, we can generalize above analysis to fundamental mutually unbaised bases in higher dimensional Hilbert spaces [30], [31], where a $n$ -qubit can be truncated from free scalar field modes with infinite levels.", "On the other hand, the fascinating link between FGUR and the second thermodynamical law has been explored in [10], which proved that a deviation of the FGUR implies a violation of the second law of thermodynamics.", "In this spirit, by investigating the influence of the relativistic motion of observer on a thermodynamical cycle, one could relate the relativistic effect to thermodynamics in an information-theoretic way.", "In particular, one can investigate the uncertainty relations in general dynamical spacetimes [32], e.g., cosmological background, where the entanglement generation under the evolution of spacetime is expected to play a dramatical role in measurements [33].", "With a proper designed thermodynamical cycle, such investigation might provide us a new perspective on quantum gravity." ], [ "ACKNOWLEDGEMENT", "This work is supported by the Australian Research Council through DP110103434.", "J. F. thanks Li-Hang Ren for stimulating discussions.", "H. F. acknowledges the support of NSFC, 973 program through 2010CB922904." ] ]	1403.0313
[ [ "Visualization of Skewed Data: A Tool in R" ], [ "Abstract In this work we present a visualization tool specifically tailored to deal with skewed data.", "The technique is based upon the use of two types of notched boxplots (the usual one, and one which is tuned for the skewness of the data), the violin plot, the histogram and a nonparametric estimate of the density.", "The data is assumed to lie on the same line, so the plots are compatible.", "We show that a good deal of information can be extracted from the inspection of this tool; in particular, we apply the technique to analyze data from synthetic aperture radar images.", "We provide the implementation in R." ], [ "Introduction", "TukeyEDA work set the basis for the Exploratory Data Analysis, which is the art of seeking for relevant information within the data with the least possible distributional assumptions about the underlying process.", "Such quest is frequently based upon graphical representations.", "Among the schematic plots which survived or emerged since the advent of powerful personal computers, one should mention, for univariate data, the scatterplot, the histogram Pearson1895Histogramdefined and discussed in, the boxplot , the beanplot , the shifting boxplot , the violin plot , and their many variations.", "The histogram and the boxplot are the most used plots which convey information about the shape of the underlying distribution.", "They work in the same fashion: they extract and display key quantifiers from the data.", "These quantifiers can be tuned for specific situations as, for instance, the choice of the bins in the histograms (the Feedman-Diaconis, Sturges, and Scott options in the hist function available in R).", "A key point when using more than a single graphical presentation of the same data set is to clearly convey the same or complementary information.", "A common mistake is simply showing side-by-side several summaries, but the precision and extent of enhancement such juxtaposition provides is arguable.", "Since no single plot is able to provide all the relevant information in every conceivable case, a possible solution for this problem consists of presenting the plots with clear visual clues of their same origin: the data set.", "Visualization techniques are often used to drive important decisions.", "If the information conveyed by the graphical summaries is hindered, decisions may be biased or completely wrong.", "For instance, AutomatedNonGaussianClusteringPolSAR present a segmentation procedure for polarimetric synthetic aperture radar (PolSAR) imagery which, albeit automatic, exhibits the quality of the product at each iteration in the form of histograms overlapped with fitted densities; the closer the fit, the better the result.", "When the data is overly asymmetric, the automatic presentation is hard to grasp as the abscissas span a huge interval.", "In this article we present a tool for the visual display of skewed data developed and freely available in R. The tool is based on the integration and coordination of several graphical representations, some of them tailored to this kind of data.", "We test the tool on PolSAR data, which exhibit intense asymmetry.", "Section presents the graphical summaries that will be integrated in our visualization tool.", "Section presents the data: different types of land cover as retrieved by Synthetic Aperture Radar – SAR sensors.", "This kind of data is prone to presenting extreme deviations from the Gaussian hypothesis, as they are heavily skewed.", "Finally, section concludes the paper with further suggestions.", "The Appendix provides details about the implementation and instructions to obtain the code and the data." ], [ "The summaries and their coordination", "In the following we define and comment advantages and disadvantages of some commonly used graphical summaries of data.", "The use of these summaries is illustrated with the same data set: a sample from a PolSAR image of the Niigata area; see Fig.", "REF .", "As presented in Table , the VV polarization is the one with strongest asymmetry, so the data henceforth presented come from this band.", "More information about this and other images is given in Section ." ], [ "Histograms and kernels", "In Data Exploratory Analysis the probability density estimation is one of the main tools to extract data information about the distribution of the underlying population.", "The estimated density may help revealing patterns and features representative of the targeted object for data modeling, analysis and decision management.", "Generally speaking, the problem of density estimation can be defined as the estimating processes of the unknown distribution by means of the also unknown density function $f$ on the set of attributes of data $\\mathcal {X}$ using the information of the observations of the random sample of size $n$ , namely $\\mathbb {X}=\\lbrace X_1, \\ldots , X_n \\rbrace \\subset \\mathcal {X}$ draw from the target function $f$ .", "The Histogram is a basic form of nonparametric density estimator where the region covered by $\\mathcal {X}$ is usually divided into equal-sized bins whose height is proportional to the count of hits within that bin.", "This estimator is depends on the choice of bin width $h$ and the starting points of the bin too.", "These two values will determine how the data will be grouped i.e.", "to which bin the data will belong to.", "The number of bins $k$ should be related to the bin width $h$ as, for instance, $ k = {\\rm range}(\\mathbb {X})/h$ .", "Different rules to choose $h$ are available.", "For example, Scott79 proposed to $h= 3.5 \\widehat{\\sigma }/n^3$ with $\\widehat{\\sigma }$ the sample standard deviation, while Sturges26 proposed $k = 1 + \\log _2(n)$ .", "Freedman81 proposed to use $h=2 IQR(\\mathbb {X})/n^3$ where $IQR(\\cdot )$ stands for the interquartile range of the data set.", "These methods usually affect strongly the histogram by the start end points of the bins and their width.", "Additionally, the histogram present the inconvenient feature of non-smoothness .", "Thus, rosenblatt1956 and parzen1962 developed the kernel density estimator that is smoth, controls bin boundary effects and (under very mild conditions) also converges to the true density, but faster than the histogram.", "A “kernel” is any smooth function (generally, a symmetric probability density) that depends on the bandwith parameter $h$ which controls both the spread and the orientation.", "Just like in histograms, $h$ determines the smoothness of the estimation.", "In practice, the choice of kernel is less important than tat of $h$ .", "For example, a small value of $h$ will lead to under-smoothing and masking important features of the data, such as skewness and multimodality.", "On the other hand, rough curves produced by larger values of $h$ yield smoother estimates but might dodge significant peaks or other important structure estimate .", "Figure REF shows the histogram along with several density estimates using different bandwidths for the intensity VV of Niigata data set.", "We see how sensitive the estimate of $\\widehat{f}$ in relation to $h$ .", "Note that the graphs reveal the strong data asymmetry.", "In particular, due to the combined effect of asymmetry and large data spread, the bulk of the information is confined to a small region, approximately in the interval $[0,.7]$ , while there is little to visualize in the remaining area of the plot which spans in $[.7,2.5]$ .", "Figure: Histogram with kernel density estimation for the intensity VV of Niigata data set.", "The bandwith hh was chosen for different values.R produces high-quality and fully customizable histograms with the hist function." ], [ "Boxplots and variants", "TukeyEDA introduced the boxplot to analyze univariate data sets by graphically displaying important core statistics of the data.", "The plot extracts a few descriptive parameters and shows information about location, spread, skewness as well as the tails of the data.", "A boxplot shows the data distribution in terms of its quartiles, labelled $Q_1, Q_2, Q_3$ (the first, second and third quartiles).", "Define the interquartile range as $IQR=1.5(Q_3 - Q_1)$ the boxplot is comprised of the following elements: a box, with horizontal lines at $Q_1,$ $Q_2$ (the median) and $Q_3$ ; vertical lines at ${W}_{L} = Q_1 - IQR$ and ${W}_{U} = Q_1 + IQR$ (the “whiskers”, omitted inside the box); and individual observations: all observations outside the $({W}_{L}, {W}_{U})$ range (outliers), plus two observations on either en which just fall inside this range.", "A variation of the boxplot is the notched boxplot which is useful for determining whether two samples were drawn from the same population in terms of their median values.", "The notch displays a confidence interval around the median based on the Gaussian hypothesis: $Q_2 \\pm 1.57\\cdot IQR/\\sqrt{n}$ .", "According to chambers1983gmd, although not a formal test, if two boxes notches do not overlap there is “strong evidence” (95% confidence) that their medians differ.", "R's default graphical tools include the boxplot function which has the option notch=TRUE to add a notch to the box.", "In many situations, mainly when with skewed data, the boxplot may erroneously identify values which exceed the whiskers as outliers.", "To correct this distortion, AdjustedBoxplotSkewed proposed an adjusted boxplot for skewed distributions.", "The main idea is the inclusion of the medcouple introduced by Brys as a robust measure of skewness in the determination of the whiskers.", "Also, the adjusted boxplot can be useful as a fast tool for automatic outlier detection, without making any assumption about the distribution of the data.", "The function adjbox of the R package robustbase can be used to produce an adjusted boxplot.", "Figure REF illustrates these three types of boxplot with the Niigata VV data set.", "Notice that the classical and notched boxplots look alike; cf.", "figures REF and REF , resp.", "This is typical of large samples, $n=4446$ in this case, for which the width of the notch becomes negligible.", "Figure: Adjusted boxplotThe difference between both classical and notched boxplots and the adjusted boxplot shown in Fig.", "REF is noticeable.", "While the two former identify numerous observations as outliers, the latter only considers a few as surprising data.", "This last graphical representation is more adequate and may lead to better informed decisions than the former.", "Warren proposed the box-percentile plot as a variant of the boxplot which allows the sides of the plot to convey more information, presenting details about the distribution of the data.", "The width of the box is not fixed, but proportional the the number of data.", "In this way, the box-percentile plot summarizes more than the histogram, but shows more details than the boxplot.", "The HMisc package in R computes and displays this graphical summary with the function bpplot().", "Generally speaking, boxplots are useful for small or moderate-sized data sets.", "The classical boxplot can be expected to label an increasing number of observations as outliers as the sample size grows.", "To improve the boxplot in this direction, Letter proposed the letter-value boxplot by displaying more detailed information over the tails of distribution using letter values, but only out to the depths where the letter values are reliable estimates.", "In this form the “outliers” can be defined as a function of the most extreme letter value shown.", "The function LVboxplot, implemented in R in the appendix of Letter, can be used to produce letter-valued boxplots.", "Recently, Marmolejo provided a comprehensive literature review on boxplot graphs, and proposed an important variant, the Shifting boxplot.", "This graphical summary incorporates the mean as basilar information instead of the median.", "The methodology supports conducting parametric tests.", "The code that produces is can be obtained directly from the authors.", "Figure REF presents these last three graphical representations of the Niigata VV data set.", "Although the visual complexity is somewhat increased from the plots presented in Fig.", "REF , the amount of visual information is also enhanced.", "Figure: Shifting boxplotBeanplot introduced the beanplot, an alternative to the boxplot for visual comparison of univariate data between groups.", "In this plot, individual observations are shown as small lines in a one-dimensional scatter plot, and then a density estimator of data distribution and the mean are showed.", "An interesting feature of this plot is that bimodalities and possible duplicated measurements are easily exposed.", "R has the beanplot package which includes a function for this purpose.", "Finally, ViolinPlot presented the violin plot: a combination of a boxplot and a (doubled) kernel density plot.", "The violin plot does not include the individual points, but it displays the median and a box indicating the interquartile range.", "It is useful when comparing multiple groups and with large dataset.", "Overlaid on this boxplot is a kernel density estimation plot.", "There is also a vioplot package in R. Fig.", "REF presents the bean plot and the violin plot for the Niigata VV data set.", "Figure: Violin plot" ], [ "The proposed tool", "We discussed in the previous section a number of graphical summaries.", "Each conveys an important aspect of the properties of the sample.", "Our proposal consists of the simultaneous use of them in a synchronized fashion, for an enhanced visualization of those different aspects.", "In order to assess more adequately data whit highly asymmetry and to extract the largest possible amount of information and features from the data data set, we propose the simultaneous use of the histogram enhanced with density estimation, the boxplot with notches (B-N) , the violin plot (V-P), the shifting boxplot (S-P), the adjusted boxplot (A-B), and the box-percentile plot (B-P).", "All graphical summaries are synchronized with respect to a location parameter estimate.", "The proposed tool follows the guidelines proposed by Tufte01 for quality visual display of quantitative information, in particular, the ratio information/ink was maximized by showing only graphical elements which convey essential information.", "The median is shown by as a vertical line connecting all plots A rug plot is displayed in the middle of the graphical summaries to reinforce the position of the data, and to reinforce the visual notion that the data is common to all plots.", "Fig.", "REF presents the result of computing the proposed visualization on the Niigata VV data set.", "Figure: Synchronized graphs in the median of intensity VV of Niigata data set.SAR sensors provide relevant and complementary information about the target.", "Their use has proven valuable from as diverse applications as the mapping of the surface of Venus by the Magellan and Venera missions , the unearthing of lost Maya ruins , and the 4-D (space and time) monitoring of the environment .", "The main characteristics that make valuable the data provided by SAR sensors are (i) their ability to provide images with high spatial resolution independent from daylight, cloud coverage and weather and environmental conditions as fog, smoke, smog, rain etc., and (ii) the fact that their return is the result of complex interactions between the incident signal and the target, which complement the information available in the visible and near-visible spectrum.", "The recent tutorial by TutorialSAR is an excellent starting point for the reader interested in this field.", "The simplest form SAR data adopt is the intensity.", "MejailJacoboFreryBustos:IJRS presented evidence that the $\\mathcal {G}_I^0$ model is able to describe many types of target textures, from textureless (such as crops) to extremely textured (as, for instance, urban areas), but including areas with moderate texture (e.g.", "forests).", "This model had been proposed by frery96, and was later extended to the full polarimetric case by FreitasFreryCorreia:Environmetrics:03.", "The $\\mathcal {G}_I^0$ distribution has positive support, and it is indexed by three parameters: the number of looks, which describes the signal-to-noise ratio, the scale, which amounts for the energy incident in the receiving antenna relative to the emitted power, and the texture.", "Depending on a relationship between the two last parameters, the $r$ -th order moment of this this distribution is infinite.", "The visualization of both the density and data from this distribution can be demanding, since extreme observations are expected.", "Among the works which present qualitative analyses of SAR data, FreryCorreiaFreitas:ClassifMultifrequency:IEEE:2007 and AutomatedNonGaussianClusteringPolSAR make critical decisions based on such information.", "The former decides which joint distribution will be used for the classification, while the latter forms a stopping rule for the iterative segmentation.", "In the following we present the data that will be analyzed with the proposed technique.", "Fig.", "REF presents the color composites of three images from different PolSAR sensors and areas.", "Fig.", "REF is from the San Francisco area, and the area under analysis is highlighted in yellow; it is a textureless sample from the sea.", "Fig.", "REF is from the Death Valley, and the sample in red has moderate texture.", "Fig.", "REF is from Niigata, and the sample in yellow has extreme texture since it is from an urban area.", "Figure: Niigata, sample in yellowTable presents quantitative summary information about these data sets.", "As expected, regardless the range, the image and the polarization of the data, there is intense skewness and kurtosis in all cases.", "[hbt] Summary statistics.", "Table: NO_CAPTIONFigures , and show the results of applying the proposed technique to each polarization of the samples from figures REF , REF and REF , respectively.", "As can be seen from the values presented in Table , these data sets are comparable within each image, but the ranges differ widely among images.", "[!hbt] Intensity HHFigure: NO_CAPTION Intensity HVFigure: NO_CAPTION Intensity VVFigure: NO_CAPTION Synchronized graphs in the median of intensity bands of San Francisco data set.", "The samples presented in Fig.", "look alike when comparing the histograms and fitted densities, but important differences arise in the violin and shifting boxplots.", "In these graphical summaries is clearer that the HV band has more spread than the other two.", "The extent can be visually quantified by the shifting boxplots.", "[!hbt] Intensity HHFigure: NO_CAPTION Intensity HVFigure: NO_CAPTION Intensity VVFigure: NO_CAPTION Synchronized graphs in the median of intensity bands of Death Valley data set.", "The data from the Death Valley image, summarized in Fig.", ", are the most symmetric; this confirms the values presented in Table .", "Nevertheless, one observes in the boxplots that there are outliers to the right of the three samples.", "If asymmetry is assumed, the adjusted boxplot also detects ouliers to the left of two of the three polarizations, namely in figures REF and REF .", "[!hbt] Intensity HHFigure: NO_CAPTION Intensity HVFigure: NO_CAPTION Intensity VVFigure: NO_CAPTION Synchronized graphs in the median of intensity bands of Niigata data set.", "Fig.", "shows at a glance the different behavior among polarizations, with the HV channel exhibiting more spread than the other two; cf.", "the box-percentile and violin plots Although HH and VV polarizations behave alike, the adjusted boxplot reveals that the former has more outliers than the latter." ], [ "Conclusions and future work", "As presented in the examples, the use of synchronized graphical summaries promoted the discovery of information conveyed by the data.", "Had only one type of plot been used, some of these features would have not been identified.", "Synchronization is essential for retaining the ability to compare graphical representations of the same data set.", "A loose presentation of two or more of the plots would not allow the discovery of such information.", "More customizable options are being added to the tool as, for instance, the ability to choose interactively the order in which the plots appear.", "In its current version, the function does not return any object.", "This can be easily customized, for instance using lists." ], [ " Acknowledgments", "The authors wish to thank Dr. Fernando Marmolejo for providing the R codes of the shifting boxplot.", "The study was supported partially by CNPq and Fapeal grants, from Brazil." ], [ "Implementation", "The tool for producing synchronized plots was written in the R programming language .", "The code involves functions from other freely available packages: Hmisc, robustbase, vioplot, bootstrap, MASS, lfstat, graphics, gplots, beanplot, and some new implementations for horizontal histogram, lines density estimates in violin plots, and Box-plot percentile.", "The proposed new R function, termed SincronizedPlot, joins in synchronized form the histogram enhanced with a density estimation, the boxplot with notches (B-N), the violin plot (V-P), the shifting boxplot (S-P), the adjusted boxplot (A-B), and the box-percentile plot (B-P).", "The main argument that must be supplied is data, a numeric vector or an R object which is coercible to one by 'as.vector(x, \"numeric\")' The time required to produce an output is negligible.", "The code and data used in this work are freely available from http://www.de.ufpe.br/~raydonal/SynchronizedPlots/SynchronizedPlots.zip.", "In order to try the tool, the user must load the R scripts, and issue the following R commands: # Data x=rgamma(100, shape=0.5) # Plot SincronizedPlot(x)" ] ]	1403.0532
[ [ "Thermodynamically consistent modeling for dissolution/growth of bubbles\n in an incompressible solvent" ], [ "Abstract We derive mathematical models of the elementary process of dissolution/growth of bubbles in a liquid under pressure control.", "The modeling starts with a fully compressible version, both for the liquid and the gas phase so that the entropy principle can be easily evaluated.", "This yields a full PDE system for a compressible two-phase fluid with mass transfer of the gaseous species.", "Then the passage to an incompressible solvent in the liquid phase is discussed, where a carefully chosen equation of state for the liquid mixture pressure allows for a limit in which the solvent density is constant.", "We finally provide a simplification of the PDE system in case of a dilute solution." ], [ "Introduction", "The process of dissolution or growth of gas bubbles in an ambient liquid phase is very common in many situations.", "In everyday life, we often see bubbles in carbonated mineral water, beer, champagne etc.", "In particular the dissolution of gases is of huge technological and industrial importance in the context of gas scrubbing.", "This is, for instance, relevant for CO$_2$ disposal, where gas from a combustion process is injected into a reactive liquid medium.", "Such processes are usually run under pressure control instead of volume control.", "Note that the latter is much more common in the mathematical analysis of such mass transfer problems, since it allows for a fixed domain in which the mathematical model–usually in the form of a system of partial differential equations–holds.", "The massive impact of the external pressure is known from the above mentioned everyday life examples, but also can be seen in the medical context.", "This is the case with decompression sickness or caisson disease, where severe symptoms can be caused by bubble generation in the blood after a fast change of the ambient pressure.", "There is a large literature on experiments and numerical computation of dissolution/growth of bubbles in a liquid, e.g.", "Liger-Belair et al.", "[10], Sauzade and Cubaud [13], Takemura and Yabe [16].", "A rigorous mathematical model is necessary for possible theoretical investigations and mathematical analysis on this topic.", "Based on Continuum Physics, we derive a mathematical model of a two-phase fluid system of type liquid/gas, where both gas and liquid phases are composed of molecularly miscible constituents and the pressure is controlled via a free (upper) surface $\\Gamma (t)$ .", "The system consists of chemical components $A_1,\\ldots ,A_N$ .", "The gas phase is denoted by $\\Omega ^+(t)$ , the liquid phase by $\\Omega ^-(t)$ and the movable free interface by $\\Sigma (t)$ .", "See the Figure 1 below.", "Figure: NO_CAPTION Figure 1.", "The two phase system under pressure control.", "In common mathematical models for mass transfer from or to gas bubbles in a liquid phase, the transferred gas is treated as a dilute component in both phases.", "This allows to use a two-phase Navier-Stokes system together with advection-diffusion equations for passive scalars.", "If the bubble is composed of a pure gas, this is no longer possible since the dissolution then significantly changes the bubble volume.", "In this case a much more elaborate modeling is required for both of the bulk phases and the transmission condition at the interface.", "In particular, the two one-sided limits of the bulk velocities at the interface and the interface's own velocity need to be distinguished.", "Since such a more rigorous model accounts for the mass and volume of individual constituents, an incompressible model for the liquid phase will still lead to non-zero divergence of the barycentric velocity field.", "Moreover, a thermodynamically rigorous model needs to be developed for compressible bulk phases in the first place.", "Only then, an incompressible version may be derived as a limit, where the latter depends on the notion of incompressibility which is neither a priori clear nor unique in the mixture context.", "The novel aspect in the present paper is the idea of an incompressible solvent (associated to $A_N$ ) carrying dissolved gas components which add their partial pressure to the total one like being ideal gases.", "The underlying mixture is supposed to be described by an equation of state according to $p=p^R_N+K(\\frac{\\rho _N}{\\rho _N^R}-1)+\\sum _{k=1}^{N-1}\\frac{\\rho _k}{M_k}RT,$ where $p^R_N$ is a reference pressure and $\\rho _N^R$ a reference density for the solvent, while $K$ is the solvent bulk modulus.", "The incompressible limit will be attained (formally) by letting $K$ tend to infinity.", "This leads to the constraint $\\rho _N\\equiv \\rho _N^R,$ i.e.", "to a constant solvent density.", "Since the continuity equation for the solvent then reduces to $\\nabla \\cdot v_N=0,$ it makes sense to employ the solvent momentum balance instead of the one for the mixture.", "This is attractive, because it leads to a standard incompressible Navier-Stokes equation for the bulk liquid.", "Only the diffusive fluxes, which rely on the relative velocity to the barycentric one, become slightly more intricate, but only involving a simple linear relation.", "The obtained PDE systems still comprise of a compressible gas phase model.", "Low Mach number approximation seems possible and will be given in a forthcoming paper.", "Note that the gas phase density in the incompressible limit will still be a function of time, determined by the dynamical mass transfer process." ], [ "Balance Equations", "(i) Mass balance For simplicity, we assume that there are no chemical reactions (which could be easily added) and that there is no absorbed mass at the interface, i.e.", "$\\rho _i^\\Sigma \\equiv 0$ for all $i=1,\\ldots ,N$ .", "The partial mass balance in its integral form for a fixed control volume $V$ with the outer normal $n$ reads as $&& \\frac{d}{dt}\\int _V \\rho _i\\, dx=-\\int _{\\partial V} \\rho _i v_i \\cdot n\\,do.$ Using the two-phase transport and divergence theorems (see the appendix), this implies $\\int _{V\\setminus \\Sigma }\\partial _t\\rho _i\\,dx-\\int _{\\Sigma _V}[\\!", "[\\rho _i]\\!]", "v^\\Sigma \\cdot n_\\Sigma \\,do=-\\int _{V\\setminus \\Sigma }\\nabla \\cdot (\\rho _iv_i)\\,dx-\\int _{\\Sigma _V}[\\!", "[\\rho _iv_i\\cdot n_\\Sigma ]\\!", "]\\,do$ with $\\Sigma _V:=\\Sigma (t)\\cap V$ , the surface velocity $v^\\Sigma $ and the surface unit normal $n_\\Sigma $ pointing toward $\\Omega ^-$ .", "Comparison of bulk and interface terms yields the local form $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t\\rho _i+\\nabla \\cdot (\\rho _iv_i)=0\\,\\,\\,\\,\\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&[\\!", "[\\rho _i(v_i-v^\\Sigma )\\cdot n_\\Sigma ]\\!", "]=0\\,\\,\\,\\,\\mbox{on $\\Sigma (t)$}.\\end{array}\\right.$ Above, the bracket $[\\![\\,\\,]\\!", "]$ denotes the jump of a quantity across the interface (crossing $\\Sigma $ in the direction opposite to $n_\\Sigma $ ).", "The mixture is described by the total density $\\rho $ and the barycentric velocity $v$ , given by $\\rho :=\\sum _{i=1}^N\\rho _i,\\quad \\rho v:=\\sum _{i=1}^N\\rho _iv_i.$ As a consequence of (REF ), the mixture obeys the continuity equation $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t\\rho +\\nabla \\cdot (\\rho v)=0\\,\\,\\,\\,\\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&[\\!", "[\\rho (v-v^\\Sigma )\\cdot n_\\Sigma ]\\!", "]=0\\,\\,\\,\\,\\mbox{on $\\Sigma (t)$}.\\end{array}\\right.$ Let $\\dot{m}^\\pm $ denote the one-sided limits $\\rho ^\\pm (v^\\pm -v^\\Sigma )\\cdot n_\\Sigma $ on $\\Sigma (t)$ .", "Then the second equation in (REF ) becomes $\\dot{m}^-=\\dot{m}^+$ , and hence $\\dot{m}:=\\dot{m}^-=\\dot{m}^+$ is well-defined.", "Similarly, we introduce $\\dot{m}_i:=\\dot{m}_i^\\pm =\\rho _i^\\pm (v_i^\\pm -v^\\Sigma )\\cdot n_\\Sigma $ .", "We define diffusion velocities $u_i:=v_i-v$ , mass fractions $y_i:=\\rho _i/\\rho $ and diffusion mass fluxes $j_i:=\\rho _iu_i=\\rho _i(v_i-v)$ .", "Then we have the following equivalent form of the equations (REF ): $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t\\rho _i+\\nabla \\cdot (\\rho _iv+j_i)=0\\,\\,\\,\\,\\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&[\\!", "[j_i\\cdot n_\\Sigma ]\\!]+[\\!", "[\\rho _i(v-v^\\Sigma )\\cdot n_\\Sigma ]\\!", "]=0\\,\\,\\,\\,\\mbox{at $\\Sigma (t)$},\\end{array}\\right.$ or $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\rho (\\partial _ty_i+v\\cdot \\nabla y_i)+\\nabla \\cdot j_i=0\\,\\,\\,\\,\\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&[\\!", "[j_i\\cdot n_\\Sigma ]\\!]+\\dot{m}[\\![y_i]\\!", "]=0\\,\\,\\,\\,\\mbox{at $\\Sigma (t)$}.\\end{array}\\right.$ In the common models for mass transfer, the jump condition in (REF ) or (REF ) is simplified to read $[\\!", "[ j_i\\cdot n_\\Sigma ]\\!", "]=0$ , assuming $\\dot{m}=0$ which means that the total phase change effect of the mass transfer is neglected; cf.", "Bothe and Fleckenstein [4] for an assessment of this approximation.", "(ii) Momentum balance The mixture is to be described by a so-called class-I model, where we consider only a single (common) momentum balance.", "The integral form is $\\frac{d}{dt}\\int _V\\rho v \\,dx&=&-\\int _{\\partial V}\\rho v(v\\cdot n)\\,do+\\int _{\\partial V}Sn\\,do+\\int _V\\rho b\\,dx+\\int _{\\partial \\Sigma _V}S^\\Sigma \\nu \\,ds$ with the bulk stress tensor $S$ , the surface stress tensor $S^\\Sigma $ and the body force $\\rho b$ .", "Note that $\\rho b=\\sum _{k=1}^N\\rho _kb_k$ with (possibly) individual body forces $b_k$ , for instance due to forces in an electrical field.", "Here $\\nu $ is the outer unit normal of the bounding curve $\\partial \\Sigma _V$ of $\\Sigma _V$ , being tangential to $\\Sigma $ .", "The transport and (surface) divergence theorems yield the local form $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t(\\rho v)+\\nabla \\cdot (\\rho v\\otimes v- S)=\\rho b\\,\\,\\,\\,\\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&\\dot{m}[\\![v]\\!]-[\\!", "[Sn_\\Sigma ]\\!", "]=\\nabla _\\Sigma \\cdot S^\\Sigma \\,\\,\\,\\,\\mbox{at $\\Sigma (t)$}.\\end{array}\\right.$ We assume non-polar fluids, for which the balance of angular momentum has a simple form without body couples or surface couples.", "This is equivalent to the assumptions $S=S^{\\sf T}, \\quad S^\\Sigma =(S^\\Sigma )^{\\sf T}.$ This is a constitutive assumption which is made right away.", "(iii) Energy balance The integral form of the total energy balance is $&&\\frac{d}{dt}\\left[ \\int _V \\rho (e+\\frac{v^2}{2})\\,dx+\\int _{\\Sigma _V}u^\\Sigma \\,do \\right]=-\\int _{\\partial V}\\rho (e+\\frac{v^2}{2})v\\cdot n\\, do-\\int _{\\partial \\Sigma _V}u^\\Sigma v^\\Sigma \\cdot \\nu \\,ds\\\\&&\\qquad -\\int _{\\partial V}q\\cdot n \\,do -\\int _{\\partial \\Sigma _V}q^\\Sigma \\cdot \\nu \\,ds + \\int _{\\partial V} v\\cdot Sn\\,do+\\int _{\\partial \\Sigma _V}v^\\Sigma \\cdot S^\\Sigma \\nu \\, ds \\\\&&\\qquad +\\int _{V} v\\cdot \\rho b\\,dx+\\int _V \\sum _{k=1}^N j_k\\cdot b_k\\,dx$ with the specific internal energy of the bulk $e$ and the internal energy density of the surface $u^\\Sigma $ .", "After straightforward computations, the local form turns out as $\\,\\,\\,\\,\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t\\left(\\rho (e+\\frac{v^2}{2})\\right)+\\nabla \\cdot \\left(\\rho (e+\\frac{v^2}{2})v+q\\right)=\\medskip \\\\&\\displaystyle \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\nabla \\cdot (Sv)+\\rho v\\cdot b+\\sum _{k=1}^N j_k\\cdot b_k \\quad \\mbox{ in $\\Omega ^+(t) \\cup \\Omega ^-(t)$,}\\medskip \\\\&\\displaystyle \\frac{D^\\Sigma u^\\Sigma }{Dt}+u^\\Sigma \\nabla _\\Sigma \\cdot v^\\Sigma +[\\!", "[\\rho (e+\\frac{v^2}{2})(v-v^\\Sigma )\\cdot n_\\Sigma ]\\!]+[\\!", "[q\\cdot n_\\Sigma ]\\!", "]+\\nabla _\\Sigma \\cdot q^\\Sigma =\\medskip \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad [\\!", "[Sv\\cdot n_\\Sigma ]\\!", "]+\\nabla _\\Sigma \\cdot (S^\\Sigma v^\\Sigma )\\,\\,\\,\\,\\mbox{ at $\\Sigma (t)$}.\\end{array}\\right.$ Subtracting the balance of kinetic energy derived from (REF ), one obtains the balance of internal energy as $\\,\\,\\,\\,\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t(\\rho e)+\\nabla \\cdot (\\rho e v+q)=\\nabla v:S +\\sum _{k=1}^N j_k\\cdot b_k\\quad \\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&\\displaystyle \\frac{D^\\Sigma u^\\Sigma }{Dt}+u^\\Sigma \\nabla _\\Sigma \\cdot v^\\Sigma +\\dot{m}[\\!", "[e+\\frac{(v-v^\\Sigma )^2}{2}-\\frac{1}{\\rho }n_\\Sigma \\cdot Sn_\\Sigma ]\\!]+[\\!", "[q\\cdot n_\\Sigma ]\\!", "]\\\\& \\qquad \\qquad \\qquad \\qquad +\\nabla _\\Sigma \\cdot q^\\Sigma - [\\!", "[(v-v^\\Sigma )_\\parallel \\cdot Sn_\\Sigma ]\\!", "]=\\nabla _\\Sigma v^\\Sigma :S^\\Sigma \\,\\,\\,\\,\\mbox{at $\\Sigma (t)$},\\end{array}\\right.$ where $(v-v^\\Sigma )_\\parallel $ stands for the tangential projection of $(v-v^\\Sigma )$ onto the local tangent plane to $\\Sigma $ , i.e.", "$(v-v^\\Sigma )_\\parallel =P_\\Sigma (v-v^\\Sigma )$ with the projection tensor $P_\\Sigma =I-n_\\Sigma \\otimes n_\\Sigma $ .", "Later, we will use the constitutive relation $S^\\Sigma =\\gamma ^\\Sigma P_\\Sigma $ with a scalar $\\gamma ^\\Sigma $ .", "Then we have $\\nabla _\\Sigma v^\\Sigma :S^\\Sigma =\\gamma ^\\Sigma \\nabla _\\Sigma \\cdot v^\\Sigma $ .", "(iv) Entropy balance Let $\\rho s$ denote the density of entropy in the bulk (i.e.", "$s$ is the specific entropy) and $\\eta ^\\Sigma $ the area-density of interfacial entropy.", "The integral form of the entropy balance is $\\frac{d}{dt} \\left[ \\int _V\\rho s\\,dx + \\int _{\\Sigma _V}\\eta ^\\Sigma \\,do \\right]&=&-\\int _{\\partial V}(\\rho s v +\\Phi )\\cdot n\\,do - \\int _{\\partial \\Sigma _V} (\\eta ^\\Sigma v^\\Sigma +\\Phi ^\\Sigma )\\cdot \\nu \\,ds \\\\&& + \\int _{V}\\xi \\,dx +\\int _{\\Sigma _V} \\xi ^\\Sigma \\,do$ with the bulk entropy flux $\\Phi $ and the interfacial entropy flux $\\Phi ^\\Sigma $ .", "Hence we obtain the local form $\\,\\,\\,\\,\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t(\\rho s)+\\nabla \\cdot (\\rho s v+\\Phi )=\\xi \\quad \\mbox{in $\\Omega ^+(t)\\cup \\Omega ^-(t)$,}\\medskip \\\\&\\displaystyle \\frac{D^\\Sigma \\eta ^\\Sigma }{Dt}+\\eta ^\\Sigma \\nabla _\\Sigma \\cdot v^\\Sigma +\\dot{m}[\\![s]\\!", "]+\\nabla _\\Sigma \\cdot \\Phi ^\\Sigma +[\\!", "[\\Phi \\cdot n_\\Sigma ]\\!", "]=\\xi ^\\Sigma \\,\\,\\,\\,\\mbox{at $\\Sigma (t)$}.\\end{array}\\right.$" ], [ "Entropy Principle", "If the entropy fluxes $\\Phi $ and $\\Phi ^\\Sigma $ in (REF ) are related to the primitive variables via constitutive relations in such a way that the following entropy principle holds, we speak of a thermodynamically consistent model.", "Entropy principle.", "The entropy flux ($\\Phi ,\\Phi ^\\Sigma $ ) is such that The entropy production is a sum of binary products of “fluxes” times “driving force”, i.e.", "$\\displaystyle \\xi =\\sum _m F_m D_m$ and $\\displaystyle \\xi ^\\Sigma =\\sum _{m^{\\prime }} F^\\Sigma _{m^{\\prime }} D_{m^{\\prime }}^\\Sigma $ .", "$\\xi \\ge 0$ , $\\xi ^\\Sigma \\ge 0$ for any thermodynamical process.", "$\\xi \\equiv 0$ and $\\xi ^\\Sigma \\equiv 0$ characterizes equilibria of the system.", "This is a condensed form of the full entropy principle.", "For more details see Bothe and Dreyer [3], as well as Dreyer [7].", "We consider the simplest class of isotropic fluids without mesoscopic forces.", "This corresponds to the choice of certain primitive variables in modeling the entropy of the material.", "We assume $\\rho s=h(\\rho e,\\rho _1,\\ldots ,\\rho _N),\\quad \\eta ^\\Sigma =h^\\Sigma (u^\\Sigma ),$ where $h$ and $h^\\Sigma $ are concave functions.", "The concavity is required for thermodynamic stability properties of the mixture.", "Then we define the (absolute) temperature $T$ , respectively $T^\\Sigma $ of bulk and interface, as well as the bulk chemical potentials $\\mu _i$ via $\\frac{1}{T}:=\\frac{\\partial h}{\\partial (\\rho e)},\\,\\,\\,\\,\\,-\\frac{\\mu _i}{T}:=\\frac{\\partial h}{\\partial \\rho _i},\\,\\,\\,\\,\\,\\frac{1}{T^\\Sigma }:=\\frac{\\partial h^\\Sigma }{\\partial u^\\Sigma }.$ Next, we compute $\\xi $ and $\\xi ^\\Sigma $ from (REF ), (REF ), (REF ), where we eliminate the derivatives of $\\rho _i$ , $\\rho e$ , $u^\\Sigma $ by means of the balance equations in (REF ), (REF ).", "This yields the following results.", "(i) Bulk entropy production $\\xi &=& \\nabla \\cdot (\\Phi - \\frac{q}{T}+\\sum _{k=1}^N\\frac{\\mu _kj_k}{T})-\\frac{1}{T}(\\rho e -\\rho s T-\\sum _{k=1}^N \\rho _k \\mu _k)\\nabla \\cdot v\\\\&&+\\frac{1}{T}\\nabla v:S+q\\cdot \\nabla \\frac{1}{T}-\\sum _{k=1}^Nj_k\\cdot \\left(\\nabla \\frac{\\mu _k}{T}-\\frac{b_k}{T}\\right).$ We choose the entropy flux as $\\Phi =\\frac{q}{T}-\\sum _{k=1}^N\\frac{\\mu _kj_k}{T}$ and determine further constitutive relations so that the entropy principle holds.", "We decompose the stress tensor $S$ as $S=-PI+S^\\circ $ with the traceless part $S^\\circ $ of $S$ and $P=-\\frac{1}{3}\\mbox{tr}(S)$ .", "We decompose the pressure $P$ as $P=p+\\Pi $ , where $\\Pi $ vanishes in equilibrium.", "This is important, since $\\Pi $ can depend on $\\nabla \\cdot v$ , while $p$ cannot.", "Hence $S$ is rewritten as $S=-(p+\\Pi )I+S^\\circ .$ Introducing the Helmholtz free energy $\\rho \\psi =\\rho \\psi (T,\\rho _1,\\ldots ,\\rho _N)=\\rho e-\\rho sT,$ we change from $\\rho e$ as a primitive variable to $T$ (via Legendre transform with $\\frac{\\partial (\\rho e)}{\\partial (\\rho s)}=T$ ).", "Then $\\xi $ becomes $\\xi =-\\frac{1}{T}\\left( \\rho \\psi + p-\\sum _{k=1}^N\\rho _k\\mu _k \\right)\\nabla \\cdot v - \\frac{\\Pi }{T}\\nabla \\cdot v+\\frac{1}{T}\\nabla v :S^\\circ + q \\cdot \\nabla \\frac{1}{T}-\\sum _{k=1}^Nj_k\\cdot \\left(\\nabla \\frac{\\mu _k}{T}-\\frac{b_k}{T}\\right).$ Now, $\\xi \\ge 0$ for any thermodynamical process implies the Gibbs-Duhem relation $p&=&-\\rho \\psi +\\sum _{i=k}^N\\rho _k\\mu _k.$ Thus the entropy production in the bulk reduces to read $\\xi &=&-\\frac{\\Pi }{T}\\nabla \\cdot v +\\frac{1}{T}\\nabla v: S^\\circ +q\\cdot \\nabla \\frac{1}{T}-\\sum _{k=1}^Nj_k\\cdot \\left(\\nabla \\frac{\\mu _k}{T}-\\frac{b_k}{T}\\right).$ Since $\\xi \\ge 0$ is required, the simplest closure is linear in the driving forces and such that a quadratic form is obtained.", "Note that the constraint $\\sum _{k=1}^N j_k=0$ has to be accounted for.", "Hence we eliminate $j_N$ , which is chosen as $-\\sum _{k=1}^{N-1} j_k$ .", "For $D:=\\frac{1}{2}(\\nabla v+(\\nabla v)^{\\sf T})$ and its traceless part $D^\\circ $ we have $\\nabla v:S^\\circ =D^\\circ :S^\\circ $ and $\\mbox{tr}D=\\nabla \\cdot v$ .", "Then $\\xi $ becomes $\\xi =-\\frac{\\Pi }{T} \\nabla \\cdot v +\\frac{1}{T}D^\\circ :S^\\circ +q\\cdot \\nabla \\frac{1}{T}-\\sum _{k=1}^{N-1} j_k\\cdot \\left(\\nabla \\frac{\\mu _k-\\mu _N}{T}-\\frac{b_k-b_N}{T}\\right).$ Note that the viscous entropy production can be written as $\\frac{1}{T}D:S^{irr}$ , if we let $S^{irr}:=-\\Pi I+S^\\circ $ , i.e.", "$S^{irr}$ is the irreversible part of $S$ which produces entropy.", "(ii) Interfacial entropy production We do not consider viscous surface dissipation, hence $S^\\Sigma =\\gamma ^\\Sigma P_\\Sigma $ .", "Then it follows from the second equation in (REF ) and the other balance equations that $\\xi ^\\Sigma &=&\\frac{1}{T^\\Sigma }(\\gamma ^\\Sigma -u^\\Sigma +T^\\Sigma \\eta ^\\Sigma )\\nabla _\\Sigma \\cdot v^\\Sigma +\\nabla _\\Sigma \\cdot (\\Phi ^\\Sigma -\\frac{q^\\Sigma }{T^\\Sigma })+q^\\Sigma \\cdot \\nabla _\\Sigma \\frac{1}{T^\\Sigma }\\\\&&+[\\!", "[ (\\frac{1}{T}-\\frac{1}{T^\\Sigma })(\\dot{m}sT+q\\cdot n_\\Sigma ) ]\\!", "]+\\frac{1}{T^\\Sigma }[\\!", "[ (v-v^\\Sigma )_\\parallel \\cdot (S^{irr}n_\\Sigma ) ]\\!", "]- \\sum _{k=1}^N[\\!", "[ \\frac{\\mu _kj_k\\cdot n_\\Sigma }{T} ]\\!", "]\\\\&&-\\frac{\\dot{m}}{T^\\Sigma }[\\!", "[ \\sum _{k=1}^Ny_k\\mu _k+\\frac{(v-v^\\Sigma )^2}{2}-\\frac{1}{\\rho }n_\\Sigma \\cdot S^{irr}n_\\Sigma ]\\!", "].$ We choose the entropy flux as $\\Phi ^\\Sigma =q^\\Sigma /T^\\Sigma $ and obtain the surface Gibbs-Duhem equation $\\gamma ^\\Sigma =u^\\Sigma -T^\\Sigma \\eta ^\\Sigma ,$ which shows that $\\gamma ^\\Sigma $ is the interfacial free energy.", "For simplification, we assume from here on that there is no temperature jump at $\\Sigma (t)$ , i.e.", "$[\\![T]\\!", "]=0,\\quad T\|_{\\Sigma }=T^\\Sigma .$ Then, with (REF ), we see that $\\xi ^\\Sigma $ becomes $\\xi ^\\Sigma &=&q^\\Sigma \\cdot \\nabla _\\Sigma \\frac{1}{T}+\\frac{1}{T}[\\!", "[ (v-v^\\Sigma )_\\parallel \\cdot (S^{irr} n_\\Sigma ) ]\\!", "]\\\\\\nonumber &&\\qquad \\qquad \\qquad \\qquad -\\frac{1}{T}\\sum _{k=1}^N\\dot{m}_k[\\!", "[ \\mu _k+\\frac{(v-v^\\Sigma )^2}{2}-\\frac{1}{\\rho }n_\\Sigma \\cdot S^{irr}n_\\Sigma ]\\!", "],$ where $\\dot{m}_i$ satisfies $[\\![\\dot{m}_i]\\!", "]=0$ for all $i=1,\\ldots ,N$ .", "In the next section, we further determine appropriate constitutive relations such that the entropy principle holds.", "In addition, one needs a constitutive modeling for the Helmholtz free energy $\\rho \\psi $ .", "This will be constructed from an equation of state for the pressure $p$ and from the chemical potentials $\\mu _i$ ." ], [ "Constitutive Modeling", "Constitutive relations can be derived from the entropy principle in (REF ) and (REF ).", "The standard closure is as follows (cf.", "de Groot and Mazur [8]; Slattery [15]; Hutter and Jöhnk [9]).", "(i) Bulk (B1) $\\Pi =-\\lambda \\nabla \\cdot v, \\,\\,\\,\\lambda =\\lambda (T,\\rho _i)\\ge 0$ the bulk viscosity, (B2) $S^\\circ =2\\eta D^\\circ ,\\,\\,\\,\\eta =\\eta (T,\\rho _i)\\ge 0$ the dynamic viscosity (Newton's law), (B3) $q=\\alpha \\nabla \\frac{1}{T},\\,\\,\\,\\alpha =\\alpha (T,\\rho _i)\\ge 0$ the heat conductivity (Fourier's law), (B4) $j_i=-\\sum _{k=1}^{N-1} L_{ik}\\left(\\nabla \\frac{\\mu _k-\\mu _N}{T}-\\frac{b_k-b_N}{T}\\right)$ with a positive (semi-)definite matrix $[L_{ik}]=[L_{ik}(T,\\rho _1,\\ldots ,\\rho _N)]$ of mobilities (Fick's law for multi-component mixture).", "(ii) Interface (B5) $q^\\Sigma =\\alpha ^\\Sigma \\nabla _\\Sigma \\frac{1}{T}$ , $\\alpha ^\\Sigma =\\alpha ^\\Sigma (T)\\ge 0$ the interfacial heat conductivity, (B6) $[\\!", "[ v_\\parallel ]\\!", "]=0$ , $v_\\parallel ^\\pm =v_\\parallel ^\\Sigma $ , i.e.", "continuous tangential velocities, (B7) If $i\\in I^\\pm :=\\lbrace i\\,\|\\,A_i \\mbox{ is only in }\\Omega ^\\pm \\rbrace $ , then $\\dot{m}_i=0$ (no transfer) and otherwise $ [\\!", "[ \\mu _i]\\!]=[\\!", "[ \\frac{1}{\\rho }n_\\Sigma \\cdot S^{irr}n_\\Sigma -\\frac{(v-v^\\Sigma )^2}{2}]\\!", "],$ or, more general but still neglecting mass transfer cross-effects, (B7') $\\displaystyle \\dot{m}_i=-\\beta _i[\\!", "[ \\mu _i+\\frac{(v-v^\\Sigma )^2}{2}-\\frac{1}{\\rho }n_\\Sigma \\cdot S^{irr}n_\\Sigma ]\\!", "]$ , $\\beta _i=\\beta _i(T)\\ge 0$ .", "Now we model the Helmholtz free energy $\\rho \\psi $ , where we follow Example 2 in Bothe and Dreyer [3].", "The free energy can be constructed from an equation of state for the pressure $p$ and from relations for the “chemical part” of the chemical potential $\\mu _i$ .", "We consider the gas phase as an ideal mixture of ideal gases and the liquid phase as a solution with $A_N$ as the solvent and $A_1,\\ldots ,A_{N-1}$ the solutes (i.e.", "dissolved components).", "We introduce the following notation: $c_i:=\\frac{\\rho _i}{M_i}\\mbox{ (molar density)},\\,\\,\\,c:=\\sum _{i=1}^Nc_i,\\,\\,\\,x_i:=\\frac{c_i}{c}\\mbox{ (molar fraction)},\\,\\,\\,x^{\\prime }:=(x_1,\\ldots ,x_{N-1}),$ where $\\sum _{k=1}^{N}x_k=1$ .", "We use $(\\rho ,x^{\\prime })$ as a set of primitive variables as well as $(\\rho _1,\\ldots ,\\rho _N)$ .", "Note that $(\\rho ,x^{\\prime })\\mapsto (\\rho _1,\\ldots ,\\rho _N)$ is one-to-one with the relations above and $\\rho _i=\\rho _i(\\rho ,x^{\\prime }):=\\frac{\\rho M_ix_i}{M(x^{\\prime })},\\quad M(x^{\\prime }):=\\sum _{k=1}^NM_kx_k,\\quad x_N:=1-\\sum _{k=1}^{N-1}x_k.$ Each thermodynamic quantity $f$ is represented as $f=f(T,\\rho _1,\\ldots ,\\rho _N)=\\tilde{f}(T,\\rho ,x^{\\prime }),$ where we always suppose the above relations among $(T,\\rho ,x^{\\prime })$ , $x_N$ and $(T,\\rho _1,\\ldots ,\\rho _N)$ .", "Now we model the pressure.", "In the gas phase $\\Omega ^+(t)$ , we assume $p=\\sum _{k=1}^Np_k$ with partial pressures $p_i$ according to the ideal gas law $p_i=\\frac{\\rho _i}{M_i}RT$ , namely $p=p(T,\\rho _1,\\ldots ,\\rho _N)=\\sum _{k=1}^N\\frac{\\rho _i}{M_i}RT=\\tilde{p}(T,\\rho ,x^{\\prime })=\\frac{\\rho R T}{M(x^{\\prime })},$ where $\\rho _i=0$ means that $A_i$ does not exist in $\\Omega ^+(t)$ .", "In the liquid phase $\\Omega ^-(t)$ , for the later passage to the incompressible case, we use $p_N=p_N^R+K(\\frac{\\rho _N}{\\rho _N^R}-1)$ with a bulk modulus $K=\\partial _{\\rho _N}p_N(\\rho _N^R)\\rho _N^R>0$ and reference quantities $p_N^R$ and $\\rho _N^R$ .", "Later we let $K\\rightarrow \\infty $ , which leads to $\\rho _N\\equiv \\rho _N^R$ .", "Note that the “1” in (REF ) can be generalized to an appropriate function of temperature and composition, but then $\\rho _N$ will not become constant in the incompressible limit.", "For all other species in the liquid, we assume that they behave as ideal gas components (in the solvent “matrix” instead of a gas volume), namely $p_i=\\frac{\\rho _i}{M_i} RT$ for all $i<N$ .", "Hence we have $p&=&p(T,\\rho _1,\\ldots ,\\rho _N)=p_N^R+K(\\frac{\\rho _N}{\\rho _N^R}-1) + \\sum _{k=1}^{N-1}\\frac{\\rho _k}{M_k}RT\\\\\\nonumber &=& \\tilde{p}(T,\\rho ,x^{\\prime })= p_N^R+K\\left(\\frac{\\rho M_N x_N}{\\rho _N^R M(x^{\\prime })}-1\\right) + \\frac{\\rho R T}{M(x^{\\prime })}\\sum _{k=1}^{N-1}x_k,$ where $\\rho _i=0$ ($i<N$ ) means that $A_i$ does not exist in $\\Omega ^-(t)$ .", "The full chemical potential cannot be modeled directly, but needs to be computed from a Helmholtz free energy function $\\rho \\psi $ .", "The modeling of $\\psi $ follows the concept laid out in Section 13 of Bothe and Dreyer [3] and employs a decomposition of $\\psi $ into an “elastic” part $\\psi ^{el}$ , which takes into account the mechanical (pressure) work, and a “thermal” part $\\psi ^{th}$ which accounts for the entropy of mixing.", "We start with a fixed temperature $T$ and a reference pressure $p^R$ .", "We have a reference density function $\\rho ^\\ast =\\rho ^\\ast (T,x^{\\prime })$ through the equation $\\tilde{p}(T,\\rho ^\\ast ,x^{\\prime })=p^R.$ We then define $\\psi ^{th}(T,x^{\\prime }):=\\tilde{\\psi }(T,\\rho ^\\ast (T,x^{\\prime }),x^{\\prime }), \\,\\,\\,\\,\\, \\psi ^{el}(T,\\rho ,x^{\\prime }):=\\tilde{\\psi }(T,\\rho ,x^{\\prime })-\\psi ^{th}(T,x^{\\prime }).$ Note that $\\psi ^{el}(T,\\rho ^\\ast ,x^{\\prime })=0$ .", "From the Gibbs-Duhem relation, we obtain $\\rho ^\\ast \\psi ^{th}(T,x^{\\prime })=p^R+\\sum _{k=1}^N\\rho _k(\\rho ^\\ast ,x^{\\prime })\\mu _k^{th}(T,x^{\\prime }),\\,\\,\\,\\mu _k^{th}(T,x^{\\prime }):=\\tilde{\\mu }_k(T,\\rho ^\\ast (T,x^{\\prime }),x^{\\prime }).$ The thermal part of the chemical potential needs to be modeled, where we only consider the case of ideal mixtures (only containing entropy of mixing), namely $\\mu ^{th}_i(T,x^{\\prime })=g_i(T,p^R)+\\frac{RT}{M_i} \\ln x_i,\\,\\,\\,i=1,\\ldots ,N,$ where $g_i$ denotes the Gibbs free energy of the pure component $A_i$ in the respective phase.", "Next we compute $\\psi ^{el}$ through the relation $\\frac{\\partial }{\\partial \\rho }\\psi ^{el}(T,\\rho ,x^{\\prime })=\\frac{\\tilde{p}(T,\\rho ,x^{\\prime })}{\\rho ^2},$ inserting $\\tilde{p}$ modeled in (REF ) and (REF ), respectively.", "For the gas phase, we obtain $\\psi ^{el}(T,\\rho ,x^{\\prime })=\\int ^\\rho _{\\rho ^\\ast }\\frac{\\tilde{p}(T,\\tilde{\\rho },x^{\\prime })}{\\tilde{\\rho }^2}\\,d\\tilde{\\rho }= \\frac{RT}{M(x^{\\prime })}\\ln \\frac{\\rho }{\\rho ^\\ast }.$ Hence we have $\\rho \\psi =\\rho \\tilde{\\psi }(T,\\rho ,x^{\\prime })=-p^R\\frac{\\rho }{\\rho ^\\ast }+\\frac{\\rho }{\\rho ^\\ast }\\sum _{k=1}^N\\rho _k(\\rho ^\\ast ,x^{\\prime })\\left\\lbrace g_k(T,p^R)+\\frac{RT}{M_k}\\ln x_k\\right\\rbrace +\\frac{\\rho RT}{M(x^{\\prime })}\\ln \\frac{\\rho }{\\rho ^\\ast }.$ In oder to compute $\\rho \\psi (T,\\rho _1\\ldots ,\\rho _N)=\\rho \\tilde{\\psi }(T,\\rho ,x^{\\prime }(\\rho _1,\\ldots ,\\rho _N))$ , we observe the following relations: $\\nonumber &&\\frac{\\tilde{p}(T,\\rho ,x^{\\prime })}{\\tilde{p}(T,\\rho ^\\ast ,x^{\\prime })}=\\frac{\\rho ^\\ast RT/M(x^{\\prime })}{\\rho RT/M(x^{\\prime })}=\\frac{\\rho ^\\ast }{\\rho }=\\frac{p}{p^R}=\\frac{RT}{p^R}\\sum _{k=1}^N\\frac{\\rho _k}{M_k},\\\\&& M(x^{\\prime }(\\rho _1\\ldots ,\\rho _N))=\\frac{\\rho }{c},\\,\\,\\,\\,\\,x_i=\\frac{\\rho _i/M_i}{ \\sum _{k=1}^N\\rho _k/M_k},\\\\&&\\rho _i(\\rho ^\\ast (T,x^{\\prime }(\\tilde{\\rho }_1\\ldots ,\\tilde{\\rho }_N)),x^{\\prime }(\\tilde{\\rho }_1\\ldots ,\\tilde{\\rho }_N))=\\rho ^\\ast \\frac{M_ix_i}{M(x^{\\prime })}=\\rho ^\\ast \\frac{\\tilde{\\rho }_i}{\\tilde{\\rho }}.$ Direct calculation yields $\\rho \\psi &=&\\rho \\psi (T,\\rho _1\\ldots ,\\rho _N)=\\rho \\tilde{\\psi }(T,\\rho ,x^{\\prime }(\\rho _1,\\ldots ,\\rho _N))\\\\&=&-RT\\sum _{k=1}^N\\frac{\\rho _k}{M_k}+ \\sum _{k=1}^N\\rho _k\\left(g_k(T,p^R)+\\frac{RT}{M_k}\\ln \\frac{\\rho _k}{M_k}\\right) + RT\\left(\\sum _{k=1}^N\\frac{\\rho _k}{M_k}\\right)\\ln \\frac{RT}{p^R}.$ Hence we obtain for $i=1,\\ldots , N$ the chemical potentials as $\\mu _i=\\mu _i(T,\\rho _1,\\ldots ,\\rho _N):=\\frac{\\partial (\\rho \\psi (T,\\rho _1,\\ldots ,\\rho _N))}{\\partial \\rho _i}=g_i(T,p^R)+\\frac{RT}{M_i}\\ln \\left( \\frac{\\rho _iRT}{p^RM_i} \\right).$ With the relation $\\rho _iRT/M_i=(RT/M_i)(\\rho M_ix_i/M(x^{\\prime }))=\\tilde{p}(x,\\rho ,x^{\\prime })x_i$ , we also obtain $\\mu _i=\\tilde{\\mu }_i(T,\\rho ,x^{\\prime })=g_i(T,p^R)+\\frac{RT}{M_i}\\ln \\frac{\\tilde{p}(T,\\rho ,x^{\\prime })}{p^R}+\\frac{RT}{M_i}\\ln x_i\\mbox{ \\,\\,\\,\\,for $i=1,\\ldots ,N$}.$ This reproduces the formulas known from the thermodynamical literature; see, e.g., Müller [11].", "For the liquid phase, we obtain $\\psi ^{el}(T,\\rho ,x^{\\prime })&=&\\int ^\\rho _{\\rho ^\\ast }\\frac{\\tilde{p}(T,\\tilde{\\rho },x^{\\prime })}{\\tilde{\\rho }^2}d\\tilde{\\rho }\\\\&=& -(p^R_N-K)\\left( \\frac{1}{\\rho }-\\frac{1}{\\rho ^\\ast }\\right)+\\frac{KM_Nx_N}{\\rho ^R_NM(x^{\\prime })}\\ln \\frac{\\rho }{\\rho ^\\ast }+\\frac{RT}{M(x^{\\prime })}\\left(\\sum _{k=1}^{N-1}x_k\\right)\\ln \\frac{\\rho }{\\rho ^\\ast }.$ Hence we have $\\rho \\psi &=&\\rho \\tilde{\\psi }(T,\\rho ,x^{\\prime })=-p^R\\frac{\\rho }{\\rho ^\\ast }+\\frac{\\rho }{\\rho ^\\ast }\\sum _{k=1}^N\\rho _k(\\rho ^\\ast ,x^{\\prime })\\left(g_k(T,p^R)+\\frac{RT}{M_k}\\ln x_k\\right)\\\\&&+(p^R_N-K)\\left( \\frac{\\rho }{\\rho ^\\ast }-1\\right)+\\frac{K\\rho M_Nx_N}{\\rho ^R_NM(x^{\\prime })}\\ln \\frac{\\rho }{\\rho ^\\ast }+\\frac{\\rho RT}{M(x^{\\prime })}\\left(\\sum _{k=1}^{N-1}x_k\\right)\\ln \\frac{\\rho }{\\rho ^\\ast }.$ Solving $\\tilde{p}(T,\\rho ^\\ast ,x^{\\prime })=p^R$ with (REF ), we get $\\frac{\\rho }{\\rho ^\\ast }=\\frac{\\rho }{\\rho ^\\ast (T,x^{\\prime }(\\rho _1,\\ldots ,\\rho _N))}=\\left(K\\frac{\\rho _N}{\\rho _N^R}+RT\\sum _{k=1}^{N-1}\\frac{\\rho _i}{M_i} \\right)(p^R-p^R_N+K)^{-1}.$ Straightforward computation with (REF ) and () yields $\\rho \\psi &=&\\rho \\psi (T,\\rho _1\\ldots ,\\rho _N)=\\rho \\tilde{\\psi }(T,\\rho ,x^{\\prime }(\\rho _1,\\ldots ,\\rho _N))\\\\&=&\\left( RT\\sum _{k=1}^{N-1}\\frac{\\rho _k}{M_k}+K\\frac{\\rho _N}{\\rho _N^R} \\right)\\left( \\ln \\frac{\\rho }{\\rho ^\\ast } -1\\right)+ K-p^R_N+\\sum _{k=1}^N\\rho _k\\left(g_k(T,p^R)+\\frac{RT}{M_k}\\ln \\frac{\\rho _k/M_k}{c}\\right),$ where the above $\\rho /\\rho ^\\ast $ and $c$ still have to be plugged in.", "Therefore we obtain, for $i=1,\\ldots , N-1$ , $\\mu _i&=&\\mu _i(T,\\rho _1,\\ldots ,\\rho _N):=\\frac{\\partial (\\rho \\psi (T,\\rho _1,\\ldots ,\\rho _N))}{\\partial \\rho _i}\\\\\\nonumber &=&g_i(T,p^R)+\\frac{RT}{M_i}\\ln \\frac{\\rho _i/M_i}{c} + \\frac{RT}{M_i}\\ln \\left(\\Big (K\\frac{\\rho _N}{\\rho _N^R}+RT\\sum _{k=1}^{N-1}\\frac{\\rho _k}{M_k} \\Big )\\frac{1}{p^R-p^R_N+K}\\right).$ For $i=N$ , we obtain $\\quad \\mu _N&=&\\mu _N(T,\\rho _1,\\ldots ,\\rho _N):=\\frac{\\partial (\\rho \\psi (T,\\rho _1,\\ldots ,\\rho _N))}{\\partial \\rho _N}\\\\\\nonumber &=&g_N(T,p^R)+\\frac{RT}{M_N}\\ln \\frac{\\rho _N/M_N}{c} + \\frac{K}{\\rho ^R_N}\\ln \\left(\\Big (K\\frac{\\rho _N}{\\rho _N^R}+RT\\sum _{k=1}^{N-1}\\frac{\\rho _i}{M_i} \\Big )\\frac{1}{p^R-p^R_N+K}\\right).$ Let us sum up: Up to here, the balance equations (REF ), (REF ) and (REF ) with constitutive relations (B1) to (B7), where the chemical potentials are modeled via (REF ), (REF ) and (REF ), form – up to boundary and initial conditions – a thermodynamically consistent full PDE system for a two-phase gas/liquid multicomponent system with compressible liquid and gas phases and mass transfer.", "For the non-isothermal case, the temperature dependencies need to be specified and the internal energy balance is usually transformed into a temperature form, i.e.", "of heat equation type.", "In the isothermal case, it can be dropped." ], [ "Incompressible Limit", "We discuss the passage to an incompressible limit for the liquid solvent.", "As $K\\rightarrow \\infty $ , assuming that the pressure stays bounded, we get $\\rho _N/\\rho ^R_N\\rightarrow 1$ .", "After a (formal) computation, the passage $K\\rightarrow \\infty $ yields $\\mu _i\\rightarrow \\mu _i^\\infty $ , where $\\mu _i^\\infty &=&g_i(T,p^R)+\\frac{RT}{M_i}\\ln x_i\\mbox{\\,\\,\\,\\,\\, for $i<N$},\\\\\\mu _N^\\infty &=&g_N(T,p^R)+\\frac{p-p^R}{\\rho ^R_N}+\\frac{RT}{M_N}\\ln x_N.$ Note that for an incompressible pure substance $A_N$ , the Gibbs free energy satisfies $g_N(T,p)=g_N(T,p^R)+\\frac{p-p^R}{\\rho _N}.$ Hence we have $\\mu _N^\\infty =g_N(T,p)+\\frac{RT}{M_N}\\ln x_N.$ Therefore, we obtain the usual formulas for the chemical potential in the limit of $K\\rightarrow \\infty $ , except for the fact that the chemical potentials of the solutes do not depend on the pressure.", "This is not a priori clear.", "Below, the superscript “$\\infty $ ” is dropped.", "Note that $\\rho _N$ is constant and $p=p_N+\\sum _{k=1}^{N-1}\\frac{\\rho _k}{M_k}RT$ with $p_N$ a free primitive variable.", "In fact, $p_N$ acts as a Lagrange multiplier in the liquid phase to account for the constraint $\\nabla \\cdot v_N=0$ which results from (REF ) for $i=N$ .", "As mentioned in the introduction, we employ the solvent momentum balance in the liquid phase and couple it to the barycentric momentum balance in the gas phase.", "For this purpose we use the relation $v_N=v+u_N=v+\\frac{j_N}{\\rho _N}=v-\\frac{1}{\\rho _N}\\sum _{k=1}^{N-1}j_k.$ Then each mass balance equation in (REF ) is rewritten with $v_N$ , instead of $v$ , in the liquid phase.", "In particular, the mass transfer transmission conditions $[\\!", "[ \\dot{m}_i]\\!", "]=0$ become, for $i<N$ , $\\left( j^+_i+\\rho ^+_i(v^+-v^\\Sigma ) \\right)\\cdot n_\\Sigma =\\left( J^-_i+\\rho ^-_i(v^-_N-v^\\Sigma ) \\right)\\cdot n_\\Sigma \\mbox{\\quad on $\\Sigma (t)$} ,$ where $J_i:=\\rho _i(v_i-v_N)=j_i+\\frac{\\rho _i}{\\rho _N}\\sum _{k=1}^{N-1}j_k$ is the diffusion flux relative to the solvent.", "For $i=N$ , the transfer condition is rewritten to become a substitution for the second equation in (REF ) and reads as $j^+_N\\cdot n_\\Sigma + \\dot{m}y^+_N \\cdot n_\\Sigma =\\rho ^-_N(v^-_N-v^\\Sigma )\\cdot n_\\Sigma \\mbox{\\quad on $\\Sigma (t)$}.$ If the solvent evaporation is neglected, i.e.", "$\\dot{m}_N=0$ and $\\rho ^+_N=0$ , then (REF ) simplifies to $v^\\Sigma \\cdot n_\\Sigma =v^-_N\\cdot n_\\Sigma \\mbox{\\quad on $\\Sigma (t)$}.$ As for the momentum balance, the standard approach would be to employ the barycentric momentum balance (REF ).", "However, this would lead to a velocity field $v$ of non-zero divergence.", "As an interesting alternative which leads to a divergence free velocity field, we make use of the partial momentum balance for $A_N$ .", "According to Bothe and Dreyer [3], the partial momentum balance for $A_i$ reads as $\\rho _i(\\partial _tv_i+v_i\\cdot \\nabla v_i)=-\\rho _i\\nabla \\mu _i+\\nabla \\cdot S_i^{irr}+\\rho _ib_i-T\\sum _{k=1}^{N}f_{ik}\\rho _i\\rho _k(v_i-v_k),$ where $S_i^{irr}=p_iI+S_i$ is the irreversible part of $S_i$ and $f_{ik}=f_{ki}>0$ are friction coefficients governing the exchange of momentum between the constituents.", "Comparing (REF ) to the barycentric momentum balance in dimensionless form, it turns out that the difference of $\\partial _tv_i+v_i\\cdot \\nabla v_i$ to the mixture acceleration $\\partial _tv+v\\cdot \\nabla v$ is negligible against the remaining terms, if the characteristic speed of diffusion is small compared to $\\sqrt{p/\\rho }$ which is about the speed of sound in a gas.", "The latter is assumed to hold, in which case (REF ) can be replaced by $\\rho _i(\\partial _tv_i+v_i\\cdot \\nabla v_i)=-y_i\\nabla p+y_i\\lambda \\nabla (\\nabla \\cdot v)+y_i\\nabla \\cdot S^\\circ +\\rho _i b.$ Applied to the solvent ($i=N$ ), we obtain $\\rho ^-(\\partial _tv_N^-+v_N^-\\cdot \\nabla v_N^-)=-\\nabla p_N^--RT\\sum _{k=1}^{N-1}\\nabla c_k^-+\\lambda \\nabla (\\nabla \\cdot v^-)+\\nabla \\cdot S^\\circ {}^-+\\rho ^-b^-$ with the standard constraint $\\nabla \\cdot v_N^-\\equiv 0$ in the incompressible limit, where the superscript “$-$ ” indicates that a quantity refers to the liquid phase.", "For the momentum transmission, the jump condition in (REF ) is rewritten with $v_N$ and $j_k$ , namely $\\frac{\\rho ^-}{\\rho _N^-}(\\dot{m}_N^--j_N^-\\cdot n_\\Sigma )(v_N^--\\frac{j_N^-}{\\rho _N^-})-S^-n_\\Sigma -(\\dot{m}^+v^+-S^+n_\\Sigma )=\\nabla _\\Sigma \\cdot S^\\Sigma \\mbox{\\quad on $\\Sigma (t)$}.$ In (REF ) and (REF ), $v^-$ , $S^\\circ {}^-=2\\eta D^\\circ {}^-$ and $S^-=-p^-I+\\lambda (\\nabla \\cdot v^-)I+2\\eta D^\\circ {}^-$ are to be rewritten with $v_N$ and $j_k$ instead of $v$ by means of (REF ).", "In order to obtain more detailed information about the diffusive fluxes, we first compute $\\nabla (\\mu _i/T)$ .", "Since we are finally interested in the isothermal case, we consider constant $T$ from here on.", "In the gas phase, with $\\mu _i^+=g_i^+(T,p^R)+\\frac{RT}{M_i}\\ln \\frac{x_i^+p^+}{p^R}=g_i^+(T,p^R)+\\frac{RT}{M_i}\\ln \\frac{c^+_iRT}{p^R}$ for the assumed ideal gas mixture, we obtain the result $\\nabla \\frac{\\mu _i^+}{T}=\\frac{R}{M_i}\\frac{\\nabla c_i^+}{c_i^+}=\\frac{R}{c_i^+}\\nabla \\rho _i^+,$ where the superscript “$+$ ” indicates that a quantity refers to the gas phase.", "In the liquid phase, we obtain $\\nabla (\\mu _i^-/T)$ for $i<N$ and $i=N$ as $\\nabla \\frac{\\mu _i^-}{T}=\\frac{R}{M_i}\\frac{\\nabla x_i^-}{x_i^-},\\quad \\nabla \\frac{\\mu _N^-}{T}=\\frac{\\nabla p^-}{T\\rho _N^R}+\\frac{R}{M_N}\\frac{\\nabla x_N^-}{x_N^-}=\\frac{\\nabla p_N^-}{T\\rho _N^R}+\\frac{R}{\\rho ^R_N}\\sum _{k=1}^{N-1}\\nabla c_k^-+\\frac{R}{M_N}\\frac{\\nabla x_N^-}{x_N^-}.$ If these are inserted into the Fickean form of the diffusive mass fluxes, the (molar) mass densities in the denominator only cancel, if the dependence of the phenomenological coefficients $L_{ik}$ on $\\rho _1,\\ldots ,\\rho _N$ has a special structure.", "To incorporate such structural information, while keeping the derivation as rigorous as possible, we prefer to use the generalized Maxwell-Stefan equations as constitutive relations determining the diffusion fluxes.", "The Maxwell-Stefan equations read $-\\sum _{k=1}^N\\frac{x_kj_i^m-x_ij_k^m}{{ik}}=\\frac{\\rho _i}{RT}\\nabla \\mu _i-\\frac{y_i}{RT}\\nabla p -\\frac{\\rho _i}{RT}(b_i-b)$ with an individual body force $b_i$ for $A_i$ and the molar mass fluxes $j_i^m:=\\frac{j_i}{M_i}=c_i(v_i-v).$ For a rigorous derivation of (REF ) see Bothe and Dreyer [3].", "There you also find the additional contribution $\\nabla \\cdot S_i-y_i\\nabla \\cdot S$ in the right-hand side of (REF ).", "The latter is not included in the classical form of the Maxwell-Stefan equations as given in, e.g., Taylor and Krishna [17] and Bird et al.", "[1].", "For simplicity, we also neglect the effect of diffusion driven by viscous stress.", "In (REF ), the ${ik}$ are the so-called Maxwell-Stefan diffusivities, which are symmetric (cf.", "[3]).", "From measurements, one knows that the ${ik}$ depend only weakly on the composition (often as affine functions), in contrast to the Fickean diffusivities.", "We assume the ${ik}$ to be constant with ${ik}={ki}>0$ .", "Note that the Maxwell-Stefan equations sum up to zero, and hence the N equations are not independent.", "Concerning the inversion of this equation system, see Bothe [2].", "From here on, we assume equal body forces $b_k\\equiv b$ for all components.", "Insertion of the chemical potential gradients yields for the gas phase $-\\sum _{k\\ne i}\\frac{x_k^+j_i^m{}^+-x_i^+j_k^m{}^+}{{ik}^+}=\\nabla c_i^+-\\frac{y_i^+}{RT}\\nabla p^+ =\\nabla c_i^+-y_i^+\\nabla c^+.$ In the liquid phase, we obtain for $i<N$ $-\\sum _{k\\ne i}\\frac{x_k^-j_i^m{}^--x_i^-j_k^m{}^-}{{ik}^-}=c^-\\nabla x_i^-- \\frac{y_i^-}{RT}\\nabla p^-.$ For $i=N$ , we obtain $-\\sum _{k\\ne N}\\frac{x_k^-j_N^m{}^--x_N^-j_k^m{}^-}{{Nk}^-}=\\frac{\\nabla p^-}{RT}+c^-\\nabla x_N^-- \\frac{y_N^-}{RT}\\nabla p^-.$ We simplify the jump conditions of the chemical potential.", "Neglecting the viscous and the kinetic effect in (B7), we assume $[\\!", "[\\mu _i]\\!", "]=0.$ See Bothe and Fleckenstein [4] for an assessment of this approximation.", "For $i<N$ , we have $\\mu _i^+(T,\\rho ^+,x^{\\prime }{}^+)&=&g_i^+(T,p^R)+\\frac{RT}{M_i}\\ln \\frac{p^+x_i^+}{p^R}=g_i^+(T,p^R)+\\frac{RT}{M_i}\\ln \\frac{p_i^+}{p^R},\\\\\\mu _i^-(T,\\rho ^-,x^{\\prime }{}^-)&=&g_i^-(T,p^R)+\\frac{RT}{M_i}\\ln x_i^-.$ For given $T$ , choose $p^R=p_i^R(T)$ so that $g_i^+(T,p_i^R(T))=g_i^-(T,p_i^R(T))$ holds for each $i$ and for a planar interface.", "Then, neglecting curvature effects via the pressure jump, we obtain $\\mu _i^+(T,\\rho ^+,x^{\\prime }{}^+)=\\mu ^-_i(T,\\rho ^-,x^{\\prime }{}^-)\\Leftrightarrow \\ln x_i^-=\\ln \\frac{p^+_i}{p^R_i(T)}=\\ln \\frac{\\rho ^+_i RT}{M_ip_i^R(T)}\\Leftrightarrow x_i^-p^R_i(T)=c_i^+RT.$ This is a version of Henry's law.", "Thus we obtain the following PDE system for incompressible solvent and compressible gas phase.", "Non-dilute solution with incompressible solvent: Gas phase: $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t c_i+ \\nabla \\cdot (c_iv+j_i^m)=0,\\qquad i=1,\\ldots ,N, \\medskip \\\\&\\displaystyle -\\sum _{k\\ne i}\\frac{x_kj_i^m-x_ij_k^m}{{ik}}=\\nabla c_i-y_i\\nabla c,\\qquad i=1,\\ldots ,N, \\medskip \\\\&\\displaystyle \\rho (\\partial _tv+v\\cdot \\nabla v)+\\nabla p =\\lambda \\nabla (\\nabla \\cdot v)+\\eta \\Delta v + \\rho b,\\medskip \\\\&\\displaystyle p=cRT=RT\\sum _{k=1}^Nc_k,\\,\\,\\,\\rho =\\sum _{k=1}^NM_kc_k.\\end{array}\\right.$ Liquid phase: $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t c_i+ \\nabla \\cdot (c_iv_N+J_i^m)=0,\\qquad i=1,\\ldots ,N-1, \\medskip \\\\&\\displaystyle -\\sum _{k\\ne i}\\frac{x_kj_i^m-x_ij_k^m}{{ik}}=c\\nabla x_i-\\frac{y_i}{RT}\\nabla p,\\qquad i=1,\\ldots ,N-1, \\medskip \\\\&\\displaystyle -\\sum _{k\\ne N}\\frac{x_kj_N^m-x_Nj_k^m}{{Nk}}=c\\nabla x_N+\\frac{1-y_N}{RT}\\nabla p, \\medskip \\\\&\\displaystyle J_i^m=j_i^m+\\frac{\\rho _i}{\\rho _N}\\sum _{k=1}^{N-1}j^m_k,\\,\\,\\,j_i^m=\\frac{j_i}{M_i},\\,\\,\\,p=p_N+\\sum _{k=1}^{N-1}c_kRT,\\medskip \\\\&\\displaystyle \\rho (\\partial _tv_N+v_N\\cdot \\nabla v_N)+\\nabla p_N = \\lambda \\nabla (\\nabla \\cdot v) +\\nabla \\cdot S^\\circ + \\rho b -RT\\sum _{k=1}^{N-1}\\nabla c_k, \\medskip \\\\&\\displaystyle \\nabla \\cdot v_N\\equiv 0,\\,\\,\\,\\rho _N\\equiv \\rho _N^R,\\end{array}\\right.$ where $v$ and $S^\\circ =2\\eta D^\\circ =\\eta \\big (\\nabla v+(\\nabla v)^{\\sf T}-\\frac{1}{3}(\\nabla \\cdot v)I\\big )$ are to be rewritten with $v_N$ and $j_k$ through (REF ).", "Interface: $\\left\\lbrace \\begin{array}{lll}&\\displaystyle (j_i^m{}^++c_i^+(v^+-v^\\Sigma ))\\cdot n_\\Sigma =(j_i^m{}^-+c_i^-(v_N^--v^\\Sigma ))\\cdot n_\\Sigma , \\qquad i=1,\\ldots ,N-1, \\medskip \\\\&\\displaystyle (j^+_N\\cdot n_\\Sigma +y_N^+\\rho ^+(v^+-v^\\Sigma ))\\cdot n_\\Sigma =\\rho ^-_N(v^-_N-v^\\Sigma )\\cdot n_\\Sigma \\medskip \\\\& \\mbox{(or $v^\\Sigma \\cdot n=v^-_N\\cdot n_\\Sigma $ in case of negligible evaporation of $A_N$)}, \\medskip \\\\&\\displaystyle x_i^-p_i^R(T)=c_i^+RT,\\qquad i=1,\\ldots ,N-1,\\medskip \\\\&\\displaystyle \\frac{\\rho ^-}{\\rho _N^-}(\\dot{m}_N^--j_N^-\\cdot n_\\Sigma )(v_N^--\\frac{j_N^-}{\\rho _N^-})-S^-n_\\Sigma -(\\dot{m}^+v^+-S^+n_\\Sigma )=\\gamma ^\\Sigma \\kappa _\\Sigma n_\\Sigma +\\nabla _\\Sigma \\gamma ^\\Sigma ,\\end{array}\\right.$ where $\\kappa _\\Sigma =\\nabla _\\Sigma \\cdot (-n_\\Sigma )$ is the curvature, $S^\\pm =-p^\\pm I+\\lambda (\\nabla \\cdot v)^\\pm I+2\\eta D^\\circ {}^\\pm $ with $D^\\circ =\\frac{1}{2}\\big (\\nabla v+(\\nabla v)^{\\sf T}-\\frac{1}{3}(\\nabla \\cdot v)I\\big )$ and $S^-$ is to be rewritten with $v_N$ and $j_k$ through (REF )." ], [ "Dilute Solution with Incompressible Solvent", "We ignore bulk viscosities in both phases and assume $\\dot{m}_N=0$ .", "Note that in the dilute case ($x_i\\ll 1$ for $i<N$ ), we have $J_i^m\\approx j_i^m$ , $\\rho \\approx \\rho _N$ and $S\\approx S_N$ in the liquid phase.", "We obtain a simple Fick's law for $i<N$ , namely $j_i^m{}^-= -{Ni}^-\\nabla c_i^-.$ We may approximate $c^-\\approx c_N^-$ .", "Then Henry's law becomes $\\frac{c^-_i}{c_i^+}=\\frac{c_N^-RT}{p^R_i(T)}=:H_i.$ Thus we obtain the following PDE system for a dilute solution with incompressible solvent and compressible gas phase.", "Dilute solution with incompressible solvent: Gas phase: $\\left\\lbrace \\begin{array}{lll}&\\displaystyle \\partial _t c_i+ \\nabla \\cdot (c_iv+j_i^m)=0,\\qquad i=1,\\ldots ,N,\\medskip \\\\&\\displaystyle -\\sum _{k\\ne i}\\frac{x_kj_i^m-x_ij_k^m}{{ik}}=\\nabla c_i-y_i\\nabla c,,\\qquad i=1,\\ldots ,N, \\medskip \\\\&\\displaystyle \\rho (\\partial _tv+v\\cdot \\nabla v)+RT\\nabla c =\\eta \\Delta v + \\rho b,\\medskip \\\\&\\displaystyle p=cRT=RT\\sum _{k=1}^Nc_k,\\quad \\rho =\\sum _{k=1}^NM_kc_k.\\end{array}\\right.$ Liquid phase: $\\left\\lbrace \\begin{array}{lll}& \\partial _t c_i+ \\nabla \\cdot (c_iv_N+j_i^m)=0,\\qquad i=1,\\ldots ,N-1,\\medskip \\\\&j_i^m=-{iN}\\nabla c_i,\\qquad i=1,\\ldots ,N-1,\\medskip \\\\&\\displaystyle \\rho _N(\\partial _tv_N+v_N\\cdot \\nabla v_N)+\\nabla p_N =\\eta _N\\Delta v_N + \\rho _N b-RT\\sum _{k=1}^{N-1}\\nabla c_k,\\medskip \\\\&\\nabla \\cdot v_N\\equiv 0,\\,\\,\\,\\rho _N\\equiv \\rho _N^R.\\end{array}\\right.$ Interface: $\\left\\lbrace \\begin{array}{lll}&(j_i^m{}^++c_i^+(v^+-v^\\Sigma ))\\cdot n_\\Sigma =j_i^m{}^-\\cdot n_\\Sigma ,\\qquad i=1,\\ldots ,N-1,\\medskip \\\\&\\displaystyle v^\\Sigma \\cdot n=v^-_N\\cdot n_\\Sigma ,\\medskip \\\\&\\displaystyle \\frac{c_i^-}{c_i^+}=H_i,\\medskip \\\\&\\displaystyle (-j_N^-\\cdot n_\\Sigma )(v_N^--\\frac{j_N^-}{\\rho _N^-})-S_N^-n_\\Sigma -(\\dot{m}^+v^+-S^+n_\\Sigma )=\\gamma ^\\Sigma \\kappa _\\Sigma n_\\Sigma +\\nabla _\\Sigma \\gamma ^\\Sigma ,\\end{array}\\right.$ where $S^+=-p^+I+2\\eta D^\\circ {}^+$ and $S^-_N=-p_N^-I+2\\eta _N D^\\circ {}^-=-p_N^-I+\\eta ^-_N(\\nabla v_N+(\\nabla v_N)^{\\sf T})^-$ ." ], [ "Boundary Conditions", "The mathematical model is to be complemented by appropriate boundary conditions at the fixed walls, called $\\partial \\Omega $ , and at the free upper surface $\\Gamma (t)$ .", "Since the derivation of physically sound boundary conditions is a topic on its own (cf., e.g.", "Bothe, Köhne and Prüss [5]), we rest content with the simplest reasonable choice.", "(i) Boundary conditions at fixed walls The fixed walls are impermeable.", "Hence $\\mbox{ $v\\cdot n_w=0$ \\,\\,and\\,\\, $j_i\\cdot n_w=0$ \\,\\,at $\\partial \\Omega $,}$ where $n_w$ is the unit outer normal to the walls.", "This also implies $\\mbox{$v_N\\cdot n_w=0$\\,\\, at $\\partial \\Omega $.", "}$ In order to allow for a movable upper surface, the tangential velocities $v_\\parallel $ and $v_N{}_\\parallel $ shall not be assumed to vanish.", "Instead, we assume a Navier slip condition of the form $\\mbox{$v_\\parallel +a(Sn_w)_\\parallel =0$, \\,\\,$v_N{}_\\parallel +a_N(S_Nn_w)_\\parallel =0$ \\,\\, at $\\partial \\Omega $ with $a,a_N\\ge 0$.", "}$ In the non-isothermal case, we add a Robin-condition for the temperature, i.e.", "$\\mbox{$T+\\beta \\nabla T\\cdot n_w=T_{ext}$ \\,\\, at $\\partial \\Omega $}$ with $\\beta \\ge 0$ .", "(ii) Boundary conditions at the free upper surface The Robin condition for the temperature can also be applied at the free surface.", "The other conditions are $\\mbox{$v\\cdot n_\\Gamma =V_\\Gamma $\\, and\\, $(p_{ext}-p)n_\\Gamma +S^{irr}n_\\Gamma =\\gamma ^\\Gamma \\kappa _\\Gamma n_\\Gamma +\\nabla _\\Gamma \\gamma ^\\Gamma $ \\,\\, on $\\Gamma (t)$}$ with the outer unit normal $n_\\Gamma $ on $\\Gamma (t)$ and the curvature $\\kappa _\\Gamma =\\nabla _\\Gamma \\cdot (-n_\\Gamma )$ .", "Let us note that, in the dilute solution limit and for a constant surface tension $\\gamma ^\\Gamma $ , the condition (REF ) becomes $\\mbox{$v_N\\cdot n_\\Gamma =V_\\Gamma $,\\,\\,\\, $p_{ext}-p_N+n_\\Gamma \\cdot S^{irr}_Nn_\\Gamma =\\gamma _\\Gamma \\kappa _\\Gamma $ \\,\\,and\\,\\, $n_\\Gamma \\times S_N^{irr}n_\\Gamma =0$ \\,\\, on $\\Gamma (t)$.", "}$ We assume the mixture composition to be given at $\\Gamma (t)$ due to local chemical equilibrium with a large and well-mixed external gas phase.", "Hence $\\mbox{$x_i^-=x_i^\\Gamma $\\,\\,\\, on $\\Gamma (t)$ for $i<N$ with $x_i^\\Gamma \\ge 0$,}$ where we assume $\\sum _{i=1}^{N-1}x_i^\\Gamma <1$ .", "(iii) Condition at the contact line The free surface $\\Gamma (t)$ touches the fixed wall in a set of points which forms the so-called contact line $C$ .", "The modeling of dynamic contact lines is, again, a topic on its own and we refer to Shikhmurzaev [14] and the reference therein for detailed information.", "Here, in order to close the system in the simplest possible manner, we assume a fixed contact angle of $\\pi /2$ , i.e.", "$\\mbox{$n_\\Gamma \\perp n_w$ \\,\\, on $C$.", "}$" ], [ "The derivation of the balance equations is based on standard two-phase transport and divergence theorems: Let $V$ denote an arbitrary fixed control volume with outer normal $n$ .", "Then $\\frac{d}{dt}\\int _V\\phi dx = \\int _{V\\setminus \\Sigma } \\partial _t \\phi dx -\\int _{\\Sigma _V}[\\!", "[\\phi ]\\!", "]v^\\Sigma \\cdot n_\\Sigma do$ with $\\Sigma _V:=\\Sigma (t)\\cap V$ , the surface velocity (including tangential part) $v^\\Sigma $ and the surface unit normal $n_\\Sigma $ .", "Here $[\\!", "[\\phi ]\\!", "]:=\\lim _{h\\rightarrow 0+}(\\phi (x+hn_\\Sigma )-\\phi (x-hn_\\Sigma ) )$ defined for $x\\in \\Sigma $ .", "We also have $\\int _{\\partial V}f \\cdot n do=\\int _{V\\setminus \\Sigma } \\nabla \\cdot f dx+\\int _{\\Sigma _V}[\\!", "[ f\\cdot n_{\\Sigma }]\\!", "]do.$ Since the internal energy and the entropy have surface contributions, we also need the surface transport theorem for $\\phi ^\\Sigma $ defined on $\\Sigma $ .", "It states that $\\frac{d}{dt}\\int _{\\Sigma _V}\\phi ^\\Sigma do =\\int _{\\Sigma _V}\\left( \\frac{D^\\Sigma \\phi ^\\Sigma }{Dt}+\\phi ^\\Sigma \\nabla _\\Sigma \\cdot v^\\Sigma \\right)do -\\int _{\\partial \\Sigma _V} \\phi ^\\Sigma v^\\Sigma \\cdot \\nu ds,$ which – in this simple form – holds for all fixed $V$ such that its outer normal $n$ satisfies $n\\perp n_\\Sigma $ on $\\Sigma _V$ , and hence $n=\\nu $ , where $\\nu $ is tangential to $\\Sigma $ and normal to the bounding curve $\\partial \\Sigma _V$ .", "We always choose such control volumes in the integral balances above.", "For more details and mathematical proofs see, e.g., Slattery [15], Romano and Marasco [12] or the appendix in Bothe, Prüss and Simonett [6]." ] ]	1403.0248
[ [ "Planar infinite groups" ], [ "Abstract We will determine all infinite $2$-locally finite groups as well as infinite $2$-groups with planar subgroup graph and show that infinite groups satisfying the chain conditions containing an involution do not have planar embeddings.", "Also, all connected outer-planar groups and outer-planar groups satisfying the chain conditions are presented.", "As a result, all planar groups which are direct product of connected groups are obtained." ], [ "Introduction", "Embedding of graphs on surfaces is a central problem in the theory of graphs particularly when the surface has low genus.", "A lattice can be visualized as a graph in which there is an edge between two elements if one of the elements covers the other.", "Hence it is natural to ask which lattices can be embedded, as a graph, on a specific surface, say the plane or sphere.", "A Kuratowski's like theorem for planar lattices is given by Kelly and Rival in [4] by providing a list of forbidden sublattices.", "An important family of lattices can be constructed from the subgroups of a group, the subgroup lattice, which is studied from various points of view.", "Drawing of subgroups of a group according to the natural subgroup partial order appears in several contexts, hence it is interesting to know which groups have planar lattices of subgroups.", "For convenience, the subgroup graph of a group $G$ is defined as the graph of its lattice of subgroups, that is, the graph whose vertices are the subgroups of $G$ such that two subgroups $H$ and $K$ are adjacent if one of $H$ or $K$ is maximal in the other.", "The subgroup graph of $G$ is denoted by $\\mathcal {L}(G)$ .", "If $\\mathcal {P}$ is a graph theoretical property, then we say that a group has property $\\mathcal {P}$ or it is a $\\mathcal {P}$ -group if the lattice of its subgroups, as a graph, has property $\\mathcal {P}$ .", "A group $G$ is planar, or $G$ is a planar group if $\\mathcal {L}(G)$ is planar.", "We note that subgroups and quotients of planar groups are planar.", "Starr and Turner III [9] were the first to study groups $G$ with $\\mathcal {L}(G)$ planar and classified all planar abelian groups.", "Also, Schmidt [8] and Bohanon and Reid [2] simultaneously classified all finite planar groups.", "In this paper, we shall study the planarity of infinite non-abelian groups and show that planarity of the lattice of subgroups of a group imposes many restrictions on the structure of the given group.", "We show that if $G$ is a group such that any two elements of $G$ generate a finite subgroup, then $G$ is planar if and only if it is a finite group or an infinite abelian group as described in Theorems REF and REF , respectively.", "As a result, it follows that there are no planar infinite solvable groups as well as FC-groups other than planar infinite abelian groups and that all Engel groups satisfying the ascending chain condition are finite.", "Remind that an FC-group is a group in which all conjugacy classes are finite.", "Also, we prove that a planar infinite group satisfying the ascending and descending chain conditions, henceforth chain conditions, is generated by two elements and it has neither involutions nor elements whose orders are a product of three distinct primes.", "In contrast to chain conditions, a planar infinite group that is the union of finite pairwise comparable subgroups is abelian (see Theorem REF ).", "Therefore, by utilizing the fact that infinite 2-groups satisfy the normalizer condition on finite subgroups, it can be shown that every planar infinite 2-groups is the union of finite pairwise comparable subgroups and hence is ableian.", "In the remainder of this paper, we shall use the notion of outer-planar graphs.", "A graph is outer-planar if it has a planar embedding in which all vertices lie on the outer (unbounded) region.", "We classify all outer-planar groups satisfying the chain conditions.", "Finally, by using the structure of connected outer-planar groups, we show that a planar group that is a product of two nontrivial groups with connected subgroup graphs is isomorphic to one of the groups $\\mathbb {Z}_{p^mq}$ , $\\mathbb {Z}_{p^mqr}$ or $\\mathbb {Z}_{p^m}\\times \\mathbb {Z}_p$ for distinct primes $p,q$ and $r$ .", "The following theorem of Schmidt, Bohanon and Reid is crucial in our proofs.", "We note that a group $G$ is called Hasse planar if its subgroup graph can be drawn in the $xy$ -plane such that whenever $H$ is a maximal subgroup of $K$ for subgroups $H$ and $K$ of $G$ , then the $y$ -coordinate of $H$ is less than that of $K$ .", "Theorem 1.1 ([2], [8]) Up to isomorphism, the only finite planar groups are the trivial group and (1) $\\mathbb {Z}_{p^m}$ , $\\mathbb {Z}_{p^mq^n}$ , $\\mathbb {Z}_{p^mqr}$ , $\\mathbb {Z}_{p^m}\\times \\mathbb {Z}_p$ , (2) $Q_8=\\langle a,b:a^4=1,b^2=a^2,bab^{-1}=a^{-1}\\rangle $ , (3) $Q_{16}=\\langle a,b:a^8=1, b^2=a^4, bab^{-1}=a^{-1}\\rangle $ , (4) $QD_{16}=\\langle a,b:a^8=b^2=1, bab^{-1}=a^3\\rangle $ , (5) $M_{p^m}=\\langle a,b:a^{p^{m-1}}=b^p=1, bab^{-1}=a^{p^{m-2}+1}\\rangle $ , (6) $\\mathbb {Z}_p\\rtimes \\mathbb {Z}_{q^n}=\\langle a,b:a^p=b^{q^n}=1, bab^{-1}=a^i\\rangle $ , where $q\\big {\|}(p-1)$ and $\\mathrm {ord}_p(i)=q$ , (7) $(\\mathbb {Z}_p\\times \\mathbb {Z}_p)\\rtimes \\mathbb {Z}_q=\\langle a,b,c:a^p=b^p=c^q=1,ab=ba, cac^{-1}=a^ib^j,cbc^{-1}=a^kb^l\\rangle $ , where ${\\tiny \\left[\\begin{array}{cc}i&j\\\\k&l\\end{array}\\right]}$ is an element of order $q>2$ in $GL_2(p)$ and $q\\big {\|}(p+1)$ , where $p,q$ and $r$ are distinct primes.", "The groups $\\mathbb {Z}_{p^mqr}$ and $QD_{16}$ are the only planar groups that are not Hasse planar.", "We note that in Theorem REF , $M_8\\cong D_8$ and that the groups in part (6) include all dihedral groups of order $2p$ , where $p$ is a prime.", "The following theorem of Starr and Turner III will be used in the next section.", "Theorem 1.2 ([9]) An infinite abelian group is planar if and only if it is isomorphic to one of the groups $\\mathbb {Z}_{p^{\\infty }}$ , $\\mathbb {Z}_{p^{\\infty }}\\times \\mathbb {Z}_p$ , $\\mathbb {Z}_{p^{\\infty }}\\times \\mathbb {Z}_{q^m}$ , $\\mathbb {Z}_{p^{\\infty }}\\times \\mathbb {Z}_{q^{\\infty }}$ or $\\mathbb {Z}_{p^{\\infty }}\\times \\mathbb {Z}_q\\times \\mathbb {Z}_r$ , where $p$ , $q$ and $r$ are distinct primes and $m$ is a positive integer." ], [ "Locally finite groups", "Locally finite groups are the groups which are most closely related to finite groups and are the best candidates for our investigation of planar groups.", "Indeed we study a more general case.", "We need the following simple lemma for the proof of Theorem REF .", "Lemma 2.1 Let $G$ be a planar group and $H,K$ be non-abelian finite subgroups of $G$ such that $H<K$ .", "Then either $K\\cong Q_{16}$ and $H\\cong Q_8$ , or $K\\cong QD_{16}$ and $H\\cong D_8$ or $Q_8$ .", "The result follows from the fact that the maximal subgroups of groups of types (5), (6) and (7) in Theorem REF are abelian and that $Q_{16}$ has subgroups isomorphic to $Q_8$ and $QD_{16}$ has subgroups isomorphic to $D_8$ and $Q_8$ .", "Definition An $n$ -locally finite group is a group such that the subgroups generated by every subset with at most $n$ elements is finite.", "A group that is $n$ -locally finite for all $n\\ge 1$ is called a locally finite group.", "It is clear that all subgroups and quotients of $n$ -locally finite groups are $n$ -locally finite and that all $(n+1)$ -locally finite groups (locally finite groups) are $n$ -locally finite for all $n\\ge 1$ .", "In what follows, $\\omega (G)$ stands for the set of all orders of elements of a group $G$ and $\\pi (G)$ denotes the set of all primes in $\\omega (G)$ .", "Also, a Sylow $p$ -subgroup of a group $G$ is denoted by $S_p(G)$ .", "Theorem 2.2 A planar 2-locally finite group is locally finite.", "Let $\\pi ^(G)=\\lbrace p\\in \\pi (G):p^2\\in \\omega (G)\\rbrace $ .", "We claim that $\\langle x\\rangle \\cap \\langle y\\rangle \\ne 1$ for all $p$ -elements $x,y\\in G$ of orders $>p$ .", "Suppose on the contrary that $\\langle x\\rangle \\cap \\langle y\\rangle =1$ .", "If $xy=yx$ then $\\langle x,y\\rangle \\cong \\langle x\\rangle \\times \\langle y\\rangle $ , which contradicts Theorem REF .", "Thus $H=\\langle x,y\\rangle $ is a non-abelian finite group with $p$ -elements $x,y$ of order $>p$ and, by Theorem REF , we have the following cases: (i) $H\\cong Q_8$ , $Q_{16}$ , $QD_{16}$ or $M_{p^k}$ is a $p$ -group.", "(ii) $H\\cong \\mathbb {Z}_q\\rtimes \\mathbb {Z}_{p^k}$ .", "If (i) holds, then $H$ has a maximal cyclic subgroup $\\langle z\\rangle $ , which implies that $x^p,y^p\\in \\langle z\\rangle $ .", "Hence $\\langle x^p\\rangle \\cap \\langle y^p\\rangle \\ne 1$ and consequently $\\langle x\\rangle \\cap \\langle y\\rangle \\ne 1$ , which is a contradiction.", "Assume that (ii) holds.", "Then $[S_p(H):C_{S_p(H)}(S_q(H))]=p$ , which implies that $x^p,y^p\\in C_{S_p(H)}(S_q(H))$ .", "Hence $\\langle x^p\\rangle \\cap \\langle y^p\\rangle \\ne 1$ so that $\\langle x\\rangle \\cap \\langle y\\rangle \\ne 1$ , which is a contradiction.", "Having eliminated (i) and (ii) we have $\\langle x\\rangle \\cap \\langle y\\rangle \\ne 1$ , from which it follows that $\\Theta _p(G)=\\bigcap _{\|x\|=p^m>p}\\langle x\\rangle $ is a non-trivial normal cyclic subgroup of $G$ for all $p\\in \\pi ^(G)$ .", "Let $\\Theta (G)=\\langle \\Theta _p(G):p\\in \\pi ^(G)\\rangle $ .", "Then $\\Theta (G)$ is a finite cyclic normal subgroup of $G$ , which implies that $\|\\pi ^(G)\|\\le 3$ .", "Let $\\Theta _0(G)=1$ and define $\\Theta _i(G)$ inductively by $\\Theta _i(G)/\\Theta _{i-1}(G)=\\Theta (G/\\Theta _{i-1}(G))$ for all $i\\ge 1$ .", "Since $\\Theta _i(G)/\\Theta _{i-1}(G)$ is a cyclic group for all $i\\ge 1$ , $T=\\bigcup \\Theta _i(G)$ is a locally finite group.", "If $\\overline{G}=G/T$ is finite, then $G$ is locally finite and we are done.", "Thus we may assume that $\\overline{G}$ is infinite.", "Since $\\Theta (\\overline{G})=1$ , the elements of $\\overline{G}$ have square-free order.", "If $\\overline{x},\\overline{y}$ are non-commuting elements of $\\overline{G}$ , then by Theorem REF , $\\langle \\overline{x},\\overline{y}\\rangle \\cong \\mathbb {Z}_p\\rtimes \\mathbb {Z}_q$ or $(\\mathbb {Z}_p\\times \\mathbb {Z}_p)\\rtimes \\mathbb {Z}_q$ , which implies that $\\overline{x},\\overline{y}$ have prime orders.", "Thus every element with composite order is central.", "By Theorem REF , $Z_\\infty (\\overline{G})$ is finite and $\\overline{G}/Z_\\infty (\\overline{G})$ is a centerless group.", "Thus we may assume without loss of generality that $\\overline{G}$ is a centerless group with nontrivial elements of prime orders.", "If the number of $p$ -elements of $\\overline{G}$ is finite for some $p\\in \\pi (\\overline{G})$ , then $\\overline{G}$ has a finite Sylow $p$ -subgroup $\\overline{S}$ with finitely many conjugates.", "Since $N_{\\overline{G}}(\\overline{S})/C_{\\overline{G}}(\\overline{S})\\le \\mathrm {Aut}(\\overline{S})$ and $C_{\\overline{G}}(\\overline{S})\\le \\overline{S}$ , it follows that $N_{\\overline{G}}(\\overline{S})$ is finite.", "On the other hand, $[\\overline{G}:N_{\\overline{G}}(\\overline{S})]$ is finite, which implies that $\\overline{G}$ is finite, a contradiction.", "Hence $\\overline{G}$ has infinitely many $p$ -elements for all primes $p\\in \\pi (\\overline{G})$ .", "Clearly, $\\overline{G}$ is not a $p$ -group.", "Let $p=\\min \\pi (\\overline{G})$ and $q\\in \\pi (\\overline{G})\\setminus \\lbrace p\\rbrace $ , $\\overline{x}$ be a $p$ -element and $\\overline{Y}$ be the set of all $q$ -elements of $\\overline{G}$ .", "Then $\\langle \\overline{x},\\overline{y}\\rangle =\\langle \\overline{y}\\rangle \\rtimes \\langle \\overline{x}\\rangle $ or $\\langle \\overline{x},\\overline{x}^{\\overline{y}}\\rangle \\rtimes \\langle \\overline{y}\\rangle $ for all $\\overline{y}\\in \\overline{Y}$ .", "If $\\langle \\overline{x},\\overline{y}\\rangle =\\langle \\overline{y}\\rangle \\rtimes \\langle \\overline{x}\\rangle $ and $\\langle \\overline{x},\\overline{z}\\rangle =\\langle \\overline{z}\\rangle \\rtimes \\langle \\overline{x}\\rangle $ for some $\\overline{y},\\overline{z}\\in \\overline{Y}$ such that $\\langle \\overline{y}\\rangle \\ne \\langle \\overline{z}\\rangle $ , then $\\overline{x}\\notin \\langle \\overline{y},\\overline{z}\\rangle $ for otherwise $\\overline{y}\\overline{z}\\ne \\overline{z}\\overline{y}$ and $\\langle \\overline{y},\\overline{z}\\rangle =\\langle \\overline{x},\\overline{x}^{\\overline{y}}\\rangle \\rtimes \\langle \\overline{y}\\rangle =\\langle \\overline{x},\\overline{y}\\rangle =\\langle \\overline{y}\\rangle \\rtimes \\langle \\overline{x}\\rangle $ , which is a contradiction.", "Therefore $\\langle \\overline{x},\\overline{y},\\overline{z}\\rangle $ is a non-abelian finite subgroup of $\\overline{G}$ properly containing $\\langle \\overline{y},\\overline{z}\\rangle $ , which is impossible by Theorem REF .", "Thus $\\langle \\overline{x},\\overline{y}\\rangle =\\langle \\overline{x},\\overline{x}^{\\overline{y}}\\rangle \\rtimes \\langle \\overline{y}\\rangle $ for infinitely many elements $\\overline{y}\\in \\overline{Y}$ , from which it follows that $C_{\\overline{G}}(\\overline{x})$ is infinite.", "On the other hand, $C_{\\overline{G}}(\\overline{x})$ is a $p$ -group so that $C_{\\overline{G}}(\\overline{x})$ is an infinite elementary abelian $p$ -group leading us to a contradiction.", "The proof is complete.", "Theorem 2.3 A planar infinite locally finite group is abelian.", "If $G$ is not abelian, then it has two non-abelian finite subgroups $H,K$ such that $H<K$ and $\|H\|>16$ contradicting Lemma REF .", "Therefore $G$ is abelian.", "Corollary 2.4 A planar infinite soluble group is abelian.", "Corollary 2.5 A planar infinite FC-group is abelian.", "The result follows from [7].", "Corollary 2.6 A planar Engel group satisfying the ascending chain condition is finite.", "The result follows from [7].", "We conclude this section by a partial result on left Engel elements.", "Proposition 2.7 Let $x$ be a left Engel element of a planar group $G$ which satisfies the ascending chain condition.", "Then $x$ has finitely many conjugates.", "In particular, one of the following conditions hold: (1) $\\langle x\\rangle \\trianglelefteq G$ and $\|x^G\|\\le \\varphi (\|x\|)$ , (2) $\\langle x\\rangle \\lnot \\trianglelefteq G$ , $\|x\|=p^k\\ne 4$ ($k>1$ ) and $\|x^G\|\\le p^k(p-1)$ , (3) $\\langle x\\rangle \\lnot \\trianglelefteq G$ , $\|x\|=p\\ne 2$ and $\|x^G\|\\le p^2-1$ , (4) $\\langle x\\rangle \\lnot \\trianglelefteq G$ , $\|x\|=2$ and $\|x^G\|\\le 5$ , or (5) $\\langle x\\rangle \\lnot \\trianglelefteq G$ , $\|x\|=4$ and $\|x^G\|\\le 6$ , where $\\varphi $ denotes the Euler totient function.", "First suppose that $x^G$ is infinite.", "By [7], $x\\in F(G)$ , the Fitting subgroup of $G$ , and hence $F(G)$ is an infinite nilpotent subgroup of $G$ .", "Since $F(G)$ is a locally finite group, by Theorem REF , $F(G)$ is abelian, which contradicts Theorem REF .", "Now, suppose that $x^G$ is finite.", "Since $F(G)$ is nilpotent, $\\langle x^G\\rangle $ is nilpotent and by Theorem REF , it is isomorphic to $\\mathbb {Z}_{p^m}$ , $\\mathbb {Z}_{p^mq^n}$ , $\\mathbb {Z}_{p^mqr}$ , $\\mathbb {Z}_{p^m}\\times \\mathbb {Z}_p$ , $Q_8$ , $Q_{16}$ , $QD_{16}$ or $M_{p^n}$ .", "A simple verification shows that the size of $x^G$ is bounded above by the numbers given in the proposition and we are done." ], [ "Groups satisfying chain conditions", "In this section, we shall study planar infinite groups satisfying the chain conditions.", "We note that, by Theorem REF , such groups are non-abelian.", "It is worth noting that the class of groups satisfying the chain conditions contains some important infinite simple groups, for instance Tarski monsters.", "A Tarski group is a group in which every nontrivial proper subgroup has a fixed prime order.", "It is well-known that Tarski groups exist for all primes $>10^{75}$ (see [5]).", "Clearly, the subgroup graph of a Tarski group is an infinite bipartite graph $K_{2,\\infty }$ , where the infinite part is countable.", "Tarski groups are planar groups satisfying the chain conditions, hence a complete classification of all planar groups satisfying the chain conditions might be very difficult in general.", "Nevertheless, we are able to give some general properties of such groups.", "Our first result gives a criterion for the order of elements of such groups whose proof uses the following famous theorem of Platt.", "Theorem 3.1 (Platt [6]) A lattice $L$ is Hasse planar if and only if the graph obtained from joining the minimum and maximum elements is planar.", "Theorem 3.2 Let $G$ be a planar infinite group satisfying the chain conditions.", "Then $G$ has no elements of order $pqr$ , for distinct primes $p$ , $q$ and $r$ .", "Suppose on the contrary that $G$ has an element $x$ of order $pqr$ for some distinct primes $p$ , $q$ and $r$ .", "If there is a path in $\\mathcal {L}(G)$ from 1 to $\\langle x\\rangle $ disjoint from $\\mathcal {L}(\\langle x\\rangle )$ , then by Platt's theorem and Theorem REF , $\\mathcal {L}(G)$ is not planar contradicting the hypothesis.", "Thus $\\langle x\\rangle \\cap \\langle y\\rangle \\ne 1$ for all $y\\in G$ , otherwise refinements of $1\\subset \\langle y\\rangle \\subset \\langle x,y\\rangle $ and $\\langle x\\rangle \\subset \\langle x,y\\rangle $ give rise to a path from 1 to $\\langle x\\rangle $ disjoint from $\\mathcal {L}(\\langle x\\rangle )$ .", "Now, if $y$ is a $p$ -element, a $q$ -element, or a $r$ -element, then $\\langle x^{qr}\\rangle \\le \\langle y\\rangle $ , $\\langle x^{pr}\\rangle \\le \\langle y\\rangle $ , or $\\langle x^{pq}\\rangle \\le \\langle y\\rangle $ , respectively, from which it follows that $\\langle x^{qr}\\rangle $ , $\\langle x^{pr}\\rangle $ and $\\langle x^{pq}\\rangle $ are normal subgroups of $G$ .", "Hence $\\langle x\\rangle \\trianglelefteq G$ and $G/C_G(x)$ is isomorphic to a subgroup of $\\mathrm {Aut}(\\langle x\\rangle )$ .", "Since $\\mathrm {Aut}(\\langle x\\rangle )$ is finite, $C_G(x)$ is infinite.", "Let $H=C_G(x)$ .", "Then $x\\in Z(H)$ and $\\pi (H)=\\lbrace p,q,r\\rbrace $ .", "For every element $g\\in H$ and $s\\in \\pi (H)$ , let $g=g_sg_{s^{\\prime }}$ , where, $g_s$ is a $s$ -element, $g_{s^{\\prime }}$ is a $s^{\\prime }$ -element and $g_sg_{s^{\\prime }}=g_{s^{\\prime }}g_s$ .", "Let $x_0=x$ , $h_0=x_s$ and $H_1=H/\\langle h_0\\rangle $ .", "If $s\\in \\pi (H_1)$ , then there exists $h_1\\langle h_0\\rangle \\in H_1$ such that $\|h_1\\langle h_0\\rangle \|=s$ .", "Let $x_1=h_1x_{s^{\\prime }}$ .", "Since $\|x_1\\langle h_0\\rangle \|=pqr$ , by the same argument as above $\\langle x_1\\langle h_0\\rangle \\rangle \\trianglelefteq H_1$ which implies that $\\langle h_1\\langle h_0\\rangle \\rangle \\trianglelefteq H_1$ .", "Let $H_2=H/\\langle h_0,h_1\\rangle $ .", "Continuing this way, we obtain a sequence of groups $\\lbrace H_k\\rbrace $ such that $H_{k+1}=H_k/\\langle h_0,h_1,...,h_k\\rangle $ and $\|h_k\\langle h_0,\\ldots ,h_{k-1}\\rangle \|=s$ for each $k$ .", "Since $G$ satisfies the chain conditions $\\lbrace H_k\\rbrace $ is finite, which implies that $s\\notin \\pi (H_{k+1})$ for some $k$ .", "Hence $S_s=\\langle h_0,h_1,\\ldots ,h_k\\rangle $ is a normal Sylow $s$ -subgroup of $H$ for all $s\\in \\pi (H)$ .", "Clearly, $S_p$ , $S_q$ and $S_r$ are finite and hence $H=S_pS_qS_r$ is finite, which is a contradiction.", "In what follows, we will provide all the tools we need to prove that a planar infinite group satisfying the chain conditions has no involutions.", "In the following lemmas and corollaries, $G$ stands for a planar group satisfying the chain conditions.", "Also, the set of all involutions of a group $G$ is denoted by $I(G)$ .", "The following result will be used in the sequel.", "Lemma 3.3 If $H=\\langle x,y,z\\rangle $ is a subgroup of $G$ properly containing $\\langle x,y\\rangle $ , $\\langle y,z\\rangle $ , $\\langle z,x\\rangle $ and $\\langle x,yz\\rangle $ , then $\\langle x,yz\\rangle =\\langle xyz\\rangle $ .", "In particular, $H=\\langle y,xyz\\rangle $ is 2-generated.", "Suppose on the contrary that $\\langle x,yz\\rangle \\ne \\langle xyz\\rangle $ .", "Then $\\mathcal {L}(G)$ has a subdivision of $K_{3,3}$ drawn in Figure 1, which is a contradiction.", "Note that dashed lines indicate paths (maximal chains of subgroups) between the corresponding vertices.", "[scale=1.25] circle,fill=white,inner sep=1pt] (D) at (2cos(30)+2.3,2sin(30)) ; circle,fill=black,inner sep=1pt,label=below:$\\langle y,z\\rangle $ ] (A) at (2cos(270),2sin(270)) ; circle,fill=black,inner sep=1pt,label=315:$\\langle x,y\\rangle \\cap \\langle y,z\\rangle $ ] (B) at (sqrt(3)cos(300),sqrt(3)sin(300)) ; circle,fill=black,inner sep=1pt,label=right:$\\langle x,y\\rangle $ ] (C) at (2cos(330),2sin(330)) ; circle,fill=black,inner sep=1pt,label=right:$\\langle x,y,z\\rangle $ ] (D) at (2cos(30),2sin(30)) ; circle,fill=black,inner sep=1pt,label=above:$\\langle x,z\\rangle $ ] (E) at (2cos(90),2sin(90)) ; circle,fill=black,inner sep=1pt,label=left:$\\langle x,y\\rangle \\cap \\langle x,z\\rangle $ ] (F) at (2cos(150),2sin(150)) ; circle,fill=black,inner sep=1pt,label=left:$\\langle x,y\\rangle \\cap \\langle x,z\\rangle \\cap \\langle xyz\\rangle $ ] (G) at (sqrt(3)/cos(10)cos(170),sqrt(3)/cos(10)sin(170)) ; circle,fill=black,inner sep=1pt,label=left:$\\langle xyz\\rangle $ ] (H) at (sqrt(3)/cos(10)cos(190),sqrt(3)/cos(10)sin(190)) ; circle,fill=black,inner sep=1pt,label=left:$\\langle x,yz\\rangle $ ] (I) at (2cos(210),2sin(210)) ; circle,fill=black,inner sep=1pt,label=225:$\\langle y,z\\rangle \\cap \\langle x,yz\\rangle $ ] (J) at (sqrt(3)cos(240),sqrt(3)sin(240)) ; circle,fill=black,inner sep=1pt,label=right:$\\langle x,z\\rangle \\cap \\langle y,z\\rangle $ ] (K) at (cos(270),sin(270)) ; [dashed] (A)–(B)–(C)–(D)–(E)–(F)–(G)–(H)–(I)–(J)–(A)–(E); [dashed] (C)–(F); [dashed] (D)–(I); Figure 1 The above lemma has the following interesting consequences.", "Corollary 3.4 The group $G$ is generated by two elements.", "Since $G$ is finitely generated, the result follows by Lemma REF .", "Corollary 3.5 If $x\\in G$ such that $\\langle x,y\\rangle \\subset G$ for all $y\\in G$ , then $x\\in \\Phi (G)$ .", "Suppose on the contrary that $x\\notin \\Phi (G)$ .", "Then $G=\\langle x,X\\rangle \\ne \\langle X\\rangle $ for some subset $X$ of $G$ .", "By Lemma REF , $\\langle X\\rangle =\\langle y,z\\rangle $ for some $y,z\\in G$ .", "Hence $G=\\langle x,y,z\\rangle $ and by Lemma REF , we obtain $\\langle x,yz\\rangle =\\langle xyz\\rangle $ is abelian so that $[x,yz]=1$ .", "Since $\\langle X\\rangle =\\langle y^{-1},yz\\rangle $ a same argument shows that $[x,z]=[x,y^{-1}\\cdot yz]=1$ , which implies that $x\\in Z(G)$ .", "Now let $N=\\langle x\\rangle \\cap \\langle X\\rangle $ .", "Then $G/N\\cong \\langle x\\rangle /N\\times \\langle X\\rangle /N$ and by Corollary REF of the last section, $\\langle X\\rangle /N=\\langle gN\\rangle $ for some $g\\in G$ .", "Hence, $G=\\langle x,g\\rangle $ contradicting the assumption on $x$ .", "Corollary 3.6 We have (1) $Z(G)\\le \\Phi (G)$ if $G$ is non-abelian, and (2) $Z_\\infty (G)\\le \\Phi (G)$ if $G$ is infinite.", "(1) It is clear by Corollary REF .", "(2) We proceed by induction on $n$ to show that $Z_n(G)\\subseteq \\Phi (G)$ for all $n\\ge 1$ .", "Since $G$ is non-abelian, by Corollary REF , $Z_1(G)\\subseteq \\Phi (G)$ .", "Now, suppose that $Z_n(G)\\subseteq \\Phi (G)$ .", "Then $\\frac{Z_{n+1}(G)}{Z_n(G)}=Z\\left(\\frac{G}{Z_n(G)}\\right)\\subseteq \\Phi \\left(\\frac{G}{Z_n(G)}\\right)=\\frac{\\Phi (G)}{Z_n(G)},$ which implies that $Z_{n+1}(G)\\subseteq \\Phi (G)$ .", "Therefore $Z_{\\infty }(G)=\\bigcup _{n=1}^{\\infty }Z_n(G)\\le \\Phi (G)$ , as required.", "Corollary 3.7 If $H$ is a non-abelian subgroup of $G$ , then $C_G(H)=Z(H)$ .", "Suppose on the contrary that $C_G(H)\\ne Z(H)$ .", "Then there exists $z\\in C_G(H)$ such that $z\\notin H$ .", "Let $K=\\langle H,z\\rangle $ .", "Then $z\\in Z(K)$ but $z\\notin \\Phi (K)$ contradicting Corollary REF (1).", "Corollary 3.8 $Z(G)=\\bigcap Z(H)$ , where the intersection is taken over all non-abelian subgroups $H$ of $G$ .", "In the sequel, we assume that the groups under consideration have some involutions and try to reach to a contradiction.", "Lemma 3.9 If $G$ has a Klein 4-subgroup $K$ , then $C_K(x)\\cong \\mathbb {Z}_2$ and $K^x=K$ for all $x\\in I(G)\\setminus K$ .", "Let $K=\\langle a,b\\rangle $ , $x\\in I(G)\\setminus K$ and $H=\\langle K,x\\rangle $ .", "By Lemma REF , $H=\\langle k,x\\rangle $ for some $k\\in K\\setminus \\lbrace 1\\rbrace $ for otherwise $\\langle xab\\rangle =\\langle x,ab\\rangle $ is a dihedral group, which is impossible.", "Hence, by Theorem REF , $H\\cong D_8$ .", "Therefore, $K^x=K$ and $K\\cap Z(H)\\ne 1$ , from which the result follows.", "In the following two lemmas, $G$ is assumed to be a planar infinite group satisfying chain conditions.", "Lemma 3.10 $G$ has no Klein 4-subgroups.", "Let $J=\\langle I(G)\\rangle $ and $K$ be a Klein 4-subgroup of $G$ .", "If $J$ is finite, then $C_G(J)$ is infinite for $J\\trianglelefteq G$ and $N_G(J)/C_G(J)$ is isomorphic to a subgroup of $\\mathrm {Aut}(J)$ .", "By [7], $C_G(J)$ is not a 2-group and hence $C_G(J)$ has a nontrivial element $x$ of odd order.", "Then, by Theorem REF , $\\langle K,x\\rangle \\cong K\\times \\langle x\\rangle $ is not planar, which is a contradiction.", "Therefore, $J$ is infinite.", "By Lemma REF , $K\\trianglelefteq J$ .", "So $J/C_J(K)$ is isomorphic to a subgroup of $\\mathrm {Aut}(K)\\cong S_3$ .", "Thus $C_J(K)$ is infinite and, by using the same argument as above, we reach to a contradiction.", "Lemma 3.11 If $G$ has an involution, then it has an involution with infinite centralizer.", "If $I(G)$ is finite then we are done.", "Thus we may assume that $I(G)$ is infinite.", "Let $a,b\\in I(G)$ be two distinct involutions.", "We have two cases: (i) $b^c\\in \\langle a,b\\rangle $ for all $c\\in I(G)\\setminus I(\\langle a,b\\rangle )$ .", "Since $I(\\langle a,b\\rangle )$ is finite, there exist distinct elements $c,c_1,c_2,\\ldots \\in I(G)\\setminus I(\\langle a,b\\rangle )$ such that $b^c=b^{c_1}=b^{c_2}=\\cdots $ .", "Thus $cc_i\\in C_G(b)$ for all $i$ , which implies that $C_G(b)$ is infinite.", "(ii) There exists $c\\in I(G)\\setminus I(\\langle a,b\\rangle )$ such that $b^c\\notin \\langle a,b\\rangle $ .", "Let $H=\\langle a,b,c\\rangle $ .", "If $H$ is a dihedral group, then by Theorem REF and Lemma REF , $H\\cong D_{2p}$ for some prime $p$ .", "But then $\\langle a,b\\rangle =H$ and $c\\in \\langle a,b\\rangle $ , which is a contradiction.", "Now, it is easy to see that $\\langle a,b\\rangle $ , $\\langle a,c\\rangle $ , $\\langle b,c\\rangle $ , $\\langle a,b^c\\rangle $ are distinct and maximal chains from $\\langle a,c\\rangle $ , $\\langle a,b\\rangle $ and $\\langle a,b^c\\rangle $ to $H$ are disjoint.", "Therefore, the subgraph induced by 1, $\\langle a\\rangle $ , $\\langle b\\rangle $ , $\\langle c\\rangle $ , $\\langle b^c\\rangle $ , $\\langle a,b\\rangle $ ,$\\langle a,c\\rangle $ , $\\langle b,c\\rangle $ , $\\langle a,b^c\\rangle $ and $\\mathcal {L}(H)$ contains a subdividion of $K_{3,3}$ , which is a contradiction (see Figure 2).", "[scale=1] circle,fill=black,inner sep=1pt] (A) at (2,0) ; t (2+0.4,0) $\\langle a,b\\rangle $; circle,fill=black,inner sep=1pt] (B) at (2cos(60),2sin(60)) ; t ((2+0.25)cos(60),(2+0.25)sin(60)) $\\langle b\\rangle $; circle,fill=black,inner sep=1pt] (C) at (2cos(120),2sin(120)) ; t ((2+0.25)cos(120),(2+0.25)sin(120)) 1; circle,fill=black,inner sep=1pt] (D) at (2cos(180),2sin(180)) ; t ((2+0.25)cos(180),(2+0.25)sin(180)) $\\langle a\\rangle $; circle,fill=black,inner sep=1pt] (E) at (2cos(240),2sin(240)) ; t ((2+0.25)cos(240),(2+0.25)sin(240)) $\\langle a,c\\rangle $; circle,fill=black,inner sep=1pt] (F) at (2cos(300),2sin(300)) ; t ((2+0.25)cos(300),(2+0.25)sin(300)) $H$; circle,fill=black,inner sep=1pt] (G) at (1cos(60),1sin(60)) ; t ((1+0.25)cos(45),(1+0.25)sin(45)) $\\langle b,c\\rangle $; circle,fill=black,inner sep=1pt] (H) at (1cos(120),1sin(120)) ; t ((1+0.25)cos(135),(1+0.25)sin(135)) $\\langle b^c\\rangle $; circle,fill=black,inner sep=1pt] (I) at (1cos(240),1sin(240)) ; t ((1+0.15)cos(228),(1+0.15)sin(228)) $\\langle c\\rangle $; circle,fill=black,inner sep=1pt] (J) at (1cos(300),1sin(300)) ; t ((1+0.3)cos(318),(1+0.3)sin(318)) $\\langle a,b^c\\rangle $; (C)–(H)–(J); (B)–(G)–(I)–(E)–(D)–(C)–(B)–(A)–(D); [dashed] (A)–(F)–(E); [dashed] (F)–(J); Figure 2 Theorem 3.12 A planar infinite group satisfying the chain conditions has no involutions.", "Suppose on the contrary that $G$ has an involution.", "By Lemma REF , $G$ has an involution $x$ such that $C_G(x)$ is infinite.", "Since $G$ satisfies the ascending chain condition, by Lemma REF , we may assume that $C_G(x)/\\langle x\\rangle $ has no involutions for otherwise we may construct an infinite series of subgroups $\\langle x_1\\rangle \\subset \\langle x_1,x_2\\rangle \\subset \\cdots $ such that $x_i\\langle x_1,\\ldots ,x_{i-1}\\rangle $ is involution and $\\frac{G_i}{\\langle x_1,\\ldots ,x_{i-1}\\rangle }=C_{\\frac{G_{i-1}}{\\langle x_1,\\ldots ,x_{i-1}\\rangle }}(x_i\\langle x_1,\\ldots ,x_{i-1}\\rangle )$ is infinite for all $i\\ge 1$ , where $G_0=G$ .", "If $H/\\langle x\\rangle $ is a finite subgroup of $C_G(x)/\\langle x\\rangle $ , then $\|H/\\langle x\\rangle \|$ is odd and there exists a subgroup $K$ of $C_G(x)$ such that $H=\\langle x\\rangle \\times K$ .", "By Theorems REF and REF , $K\\cong \\mathbb {Z}_{p^m}$ is a cyclic $p$ -group.", "Hence every finite subgroup of $C_G(x)/\\langle x\\rangle $ is a cyclic $p$ -group for some prime $p$ .", "If $y,z\\in C_G(x)$ are elements of odd orders such that $\\langle y\\rangle \\cap \\langle z\\rangle =\\langle z\\rangle \\cap \\langle yz\\rangle =\\langle yz\\rangle \\cap \\langle y\\rangle =1$ , then as it is drawn in Figure 3, $\\mathcal {L}(C_G(x))$ has a subdivision of $K_{3,3}$ , which is a contradiction.", "As before, dashed lines indicate paths (maximal chains of subgroups) between the corresponding vertices.", "Let $C_1=C_G(x)$ .", "If there exist two distinct elements $y,z$ of odd prime orders such that $\\langle y\\rangle \\ne \\langle z\\rangle $ , then $\\langle y\\rangle \\cap \\langle z\\rangle =\\langle z\\rangle \\cap \\langle yz\\rangle =\\langle yz\\rangle \\cap \\langle y\\rangle =1$ .", "Therefore, by the same reason as above, $C_1$ has a subgraph isomorphic to a subdivision of $K_{3,3}$ , which is a contradiction.", "Hence $C_1/\\langle x\\rangle $ is a $p$ -group and $C_1$ contains a unique cycle $\\langle x_1\\rangle $ of order $p$ .", "Let $C_2=C_1/\\langle x_1\\rangle $ .", "The same argument shows that there exists a unique cycle $\\langle x_2\\langle x_1\\rangle \\rangle $ in $C_2$ such that $\|x_2\\langle x_1\\rangle \|=p$ .", "By a repetitive argument, there exist elements $x_1,x_2,\\ldots ,x_n,\\ldots $ such that $\|x_i\\langle x_1,x_2,x_3,\\ldots ,x_{i-1}\\rangle \|=p$ for all $i\\ge 1$ .", "Therefore $G$ has an infinite ascending chain of subgroups $\\langle x_1\\rangle \\subset \\langle x_1,x_2\\rangle \\subset \\cdots \\subset \\langle x_1,\\ldots ,x_n\\rangle \\subset \\cdots ,$ which contradicts the assumption.", "The proof is complete.", "[scale=1] circle,fill=black,inner sep=1pt] (A) at (2,0) ; t (2+0.25,0) $\\langle x\\rangle $; circle,fill=black,inner sep=1pt] (B) at (2cos(60),2sin(60)) ; t ((2+0.25)cos(60),(2+0.25)sin(60)) $\\langle x,z\\rangle $; circle,fill=black,inner sep=1pt] (C) at (sqrt(3)/cos(10)cos(90),sqrt(3)/cos(10)sin(90)) ; t ((sqrt(3)/cos(10)+0.25)cos(90),(sqrt(3)/cos(10)+0.25)sin(90)) $\\langle z\\rangle $; circle,fill=black,inner sep=1pt] (D) at (2cos(120),2sin(120)) ; t ((2+0.25)cos(120),(2+0.25)sin(120)) 1; circle,fill=black,inner sep=1pt] (E) at (sqrt(3)/cos(10)cos(150),sqrt(3)/cos(10)sin(150)) ; t ((sqrt(3)/cos(10)+0.25)cos(150),(sqrt(3)/cos(10)+0.25)sin(150)) $\\langle y\\rangle $; circle,fill=black,inner sep=1pt] (F) at (2cos(180),2sin(180)) ; t ((2+0.4)cos(180),(2+0.4)sin(180)) $\\langle x,y\\rangle $; circle,fill=black,inner sep=1pt] (G) at (2cos(240),2sin(240)) ; t ((2+0.25)cos(240),(2+0.25)sin(240)) $\\langle x,y,z\\rangle $; circle,fill=black,inner sep=1pt] (H) at (2cos(300),2sin(300)) ; t ((2+0.25)cos(300),(2+0.25)sin(300)) $\\langle x,(yz)^2\\rangle $; circle,fill=black,inner sep=1pt] (I) at (1cos(300),1sin(300)) ; t (0.85cos(270),0.85sin(270)) $\\langle (yz)^2\\rangle $; (B)–(C); (E)–(F); (H)–(I); [dashed] (C)–(D)–(E); [dashed] (D)–(I); [dashed] (H)–(A)–(F)–(G)–(B)–(A); [dashed] (G)–(H); Figure 3 Regardless of chain conditions, it is yet possible to decide on the planarity of a group with involutions whenever it is a 2-group.", "Recall that for a $p$ -group $G$ , $\\Omega _i(G)$ denotes the subgroup of $G$ generated by all elements of orders at most $p^i$ for all $i\\ge 0$ .", "Theorem 3.13 A planar infinite 2-group is abelian.", "First we show that all finite subgroups of $G$ are abelian.", "Suppose on the contrary that $G$ has a finite non-abelian subgroup $H$ .", "Then, by [7], we may assume that $\|H\|>16$ .", "Hence, by Theorem REF , $H\\cong M_{2^m}$ for some $m$ .", "With the same reason, there exists a finite non-abelian subgroup $K\\cong M_{2^n}$ properly containing $H$ , which contradicts Lemma REF .", "Now, if $x,y\\in G$ are two involutions, then $\\langle x,y\\rangle $ is a finite dihedral group, which implies that $\\langle x,y\\rangle $ is abelian, that is, $xy=yx$ .", "Thus $\\Omega _1(G)$ is an elementary abelian 2-group, from which by Theorem REF , it follows that $\\Omega _1(G)\\cong \\mathbb {Z}_2$ or $\\mathbb {Z}_2\\times \\mathbb {Z}_2$ .", "Now, let $A_0=1$ and $A_i$ be a subgroup of $G$ defined inductively by $A_i/A_{i-1}=\\Omega _1(G/A_{i-1})$ for all $i\\ge 1$ .", "Then $A_i$ 's are finite abelian subgroups of $G$ such that $G=\\bigcup A_i$ .", "Therefore $G$ is abelian, as required.", "Albeit the classification of all planar groups satisfying the chain conditions seems to be a difficult task, the situation is much more convenient for a stronger notion of planarity known as outer-planarity.", "By a result of Chartrand and Harary [3], a finite connected graph is outer-planar if and only if it has no subdivisions of $K_4$ and $K_{2,3}$ as subgraphs.", "Therefore, a graph with a subdivision of $K_4$ or $K_{2,3}$ is never an outer-planar graph.", "Lemma 3.14 A finite group is outer-planar if and only if it is isomorphic to $\\mathbb {Z}_{p^m}$ or $\\mathbb {Z}_{p^mq}$ for some distinct primes $p,q$ .", "Theorem 3.15 Every outer-planar group satisfying the chain conditions is finite (and hence is isomorphic to $\\mathbb {Z}_{p^m}$ or $\\mathbb {Z}_{p^mq}$ , where $p,q$ are distinct primes).", "Suppose $G$ is a non-abelian group.", "Since $G$ satisfies the chain conditions, there exist an integer $n$ such that $Z(\\overline{G})=1$ , where $\\overline{G}=G/Z_n(G)$ .", "By Corollary REF , $\\overline{G}=\\langle \\overline{x},\\overline{y}\\rangle $ for some elements $\\overline{x},\\overline{y}\\in \\overline{G}$ , which implies that $\\overline{G}=\\langle \\overline{x},\\overline{x}\\overline{y}\\rangle =\\langle \\overline{y},\\overline{x}\\overline{y}\\rangle $ .", "It is easy to see that $\\langle \\overline{x}\\rangle \\cap \\langle \\overline{y}\\rangle =\\langle \\overline{x}\\rangle \\cap \\langle \\overline{x}\\overline{y}\\rangle =\\langle \\overline{y}\\rangle \\cap \\langle \\overline{x}\\overline{y}\\rangle =1$ .", "Therefore $\\overline{G},\\langle \\overline{x}\\rangle ,\\langle \\overline{y}\\rangle ,\\langle \\overline{x}\\overline{y}\\rangle ,1$ along with subgroups in maximal chains connecting 1 and $G$ to $\\langle \\overline{x}\\rangle $ , $\\langle \\overline{y}\\rangle $ and $\\langle \\overline{x}\\overline{y}\\rangle $ give rise to a subdivision of $K_{2,3}$ , which is a contradiction.", "Therefore, $G$ is abelian and by Theorem REF , $G$ is a finite group.", "Now, the result follows by Theorem REF ." ], [ "Product of groups", "A theorem of Behzad and Mahmoodian [1] states that the Cartesian product of two finite connected graphs is planar if they are both paths, one of them is a path and the other is a cycle, or one of them is an outer-planar graph and the other is a single edge.", "Hence the classification of such graphs relies on the classification of outer-planar graphs.", "To end this, we first determine all connected outer-planar groups.", "Theorem 4.1 Let $G$ be a connected outer-planar group.", "Then $G$ is isomorphic to one of the groups $\\mathbb {Z}_{p^m}$ or $\\mathbb {Z}_{p^mq}$ for some distinct primes $p$ and $q$ .", "First we observe that if $G$ is finite, then by Theorem REF , $G$ is a cyclic group of order $p^m$ or $p^mq$ for some primes $p$ and $q$ .", "Thus we may assume that $G$ is infinite.", "Since $\\mathcal {L}(G)$ is connected, $G$ has an infinite subgroup $H$ with a maximal finite cyclic subgroup $\\langle x\\rangle $ .", "Clearly, $\|x\|=p^m$ or $p^mq$ .", "We claim that $\\langle x\\rangle $ has infinitely many conjugates in $H$ .", "If not then either $\\langle x\\rangle \\subset N_H(\\langle x\\rangle )\\subseteq H$ or $N_H(\\langle x\\rangle )=H$ , that is, $\\langle x\\rangle \\trianglelefteq H$ and hence $\\langle x\\rangle \\subset \\langle x,y\\rangle \\subset H$ for all $y\\in H\\setminus \\langle x\\rangle $ , which contradict maximality of $\\langle x\\rangle $ in $H$ .", "Now, a simple verification shows that there exist three elements $a,b,c\\in H$ such that $\\langle x\\rangle ^a$ , $\\langle x\\rangle ^b$ and $\\langle x\\rangle ^c$ are distinct and their pairwise intersections are equal to a fix subgroup $N$ .", "Therefore, $\\mathcal {L}(G)$ has a subgraph isomorphic to a subdivision of $K_{2,3}$ , which is a contradiction.", "The proof is complete.", "Corollary 4.2 Let $G=H\\times K$ be a planar group, where $H$ and $K$ are non-trivial groups with connected subgroup graph.", "Then $G$ is isomorphic to one of the groups $\\mathbb {Z}_{p^mq}$ , $\\mathbb {Z}_{p^mqr}$ or $\\mathbb {Z}_{p^m}\\times \\mathbb {Z}_p$ for distinct primes $p,q$ and $r$ .", "Since $\\mathcal {L}(G)$ has a subgraph isomorphic to $\\mathcal {L}(H)\\times \\mathcal {L}(K)$ , the Cartesian product of $\\mathcal {L}(H)$ and $\\mathcal {L}(K)$ , $\\mathcal {L}(H)\\times \\mathcal {L}(K)$ is planar and by [1], it follows that $\\mathcal {L}(H)$ and $\\mathcal {L}(K)$ are outer-planar.", "Hence, in all cases, $H,K$ are cyclic groups of orders $p^m$ or $p^mq$ for some primes $p$ and $q$ , from which the result follows.", "Corollary 4.3 Let $G$ be an outer-planar infinite group.", "Then the infinite subgroups of $G$ have no maximal finite subgroups.", "Corollary 4.4 If $G$ is an outer-planar infinite group the girth of its subgroup graph is greater than four, then the finite subgroups of $G$ induce a tree as a connected component.", "Acknowledgment The authors would like to thank the referee for careful reading of the paper and giving some helpful comments and pointing out a gap in the proof of Lemma REF ." ] ]	1403.0331
[ [ "Model-Free Sure Screening via Maximum Correlation" ], [ "Abstract We consider the problem of screening features in an ultrahigh-dimensional setting.", "Using maximum correlation, we develop a novel procedure called MC-SIS for feature screening, and show that MC-SIS possesses the sure screen property without imposing model or distributional assumptions on the response and predictor variables.", "Therefore, MC-SIS is a model-free sure independence screening method as in contrast with some other existing model-based sure independence screening methods in the literature.", "Simulation examples and a real data application are used to demonstrate the performance of MC-SIS as well as to compare MC-SIS with other existing sure screening methods.", "The results show that MC-SIS outperforms those methods when their model assumptions are violated, and it remains competitive when the model assumptions hold." ], [ "Introduction", "With rapid development of modern technology, various types of high-dimensional data are collected in a variety of areas such as next-generation sequencing and biomedical imaging data in bioinformatics, high-frequency time series data in quantitative finance, and spatial-temporal data in environmental studies.", "In those types of high-dimensional data, the number of variables p can be much larger than the sample size n, which is referred to the `large $p$ small $n$ ' scenario.", "To deal with this scenario, a commonly adopted approach is to impose the sparsity assumption that the number of important variables is small relative to $p$ .", "Based on the sparsity assumption, a variety of regularization procedures have been proposed for high-dimensional regression analysis such as the lasso [29], the smoothly clipped absolute deviation method [7], and elastic net [33].", "All these methods work when $p$ is moderate.", "However, when applied to analyze ultrahigh-dimensional data where dimensionality grows exponentially with sample size (e.g., $p = \\exp (n^{\\alpha })$ with $\\alpha >0$ ), their performances will deteriorate in terms of computational expediency, statistical accuracy and algorithmic stability [10].", "To address the challenges of ultrahigh dimensionality, a number of marginal screening procedures have been proposed under different model assumptions.", "They all share the same goal that is to reduce dimensionality from ultrahigh to high while retaining all truly important variables.", "When a screening procedure achieves this goal, it is said to have the sure screening property in the literature.", "[8] proposed to use Pearson correlation for feature screening and showed that the resulting procedure possesses the sure screening property under the linear model assumption.", "They refer to the procedure as the Sure Independence Screening (SIS) procedure.", "[11] extended SIS from linear models to generalized linear models by using maximum marginal likelihood values.", "[6] developed a Nonparametric Independence Screening (NIS) procedure and proved that NIS has the sure screening property under the additive model.", "[21] proposed to use distance correlation to rank the predictor variables, and showed that the resulting procedure, denoted as DC-SIS, has the sure screening property without imposing any specific model assumption.", "Compared with the other screening procedures discussed previously, DC-SIS is thus model-free.", "Distance correlation was introduced in [28], which uses joint and marginal characteristic functions to measure the dependence between two random variables.", "From the review above, it is clear that the standard approach to developing a valid screening procedure consists of two steps.", "First, a proper dependence measure between the response and predictor variables needs to be defined and further used to rank-order all the predictor variables; and second, the sure screening property needs to be established for the screening procedure based on the dependence measure.", "The screening methods discussed in the previous paragraph differ from each other in these two steps.", "For example, SIS uses Pearson correlation as the dependence measure and possesses the sure screening property under linear models, whereas NIS uses the goodness of fit measure of the nonparametric regression between the response and predictor variable as the dependence measure and possesses the sure screening property under additive models.", "For the purpose of screening in an ultrahigh dimensional setting, we argue that an effective screening procedure should employ a sensitive dependence measure and satisfy the sure screening requirement without model specifications.", "The goal of screening is not to precisely select the true predictors, instead, it is to reduce the number of predictor variables from ultrahigh to high while retaining the true predictor variables.", "Therefore, false positives or selections can be tolerated to a large degree, and sensitive dependence measures are more preferred than insensitive measures.", "In ultrahigh dimensional data, there usually does not exist information about the relationship between the response and predictor variables, and it is extremely difficult to explore the possible relationship due to the presence of a large number of predictors.", "Therefore, model assumptions should be avoided as much as possible in ultrahigh dimensional screening, and we should prefer screening procedures that possess the sure screening property without model specifications.", "In other words, model-free sure screening procedures are more preferable.", "Among the existing screening procedures discussed previously, only DC-SIS does not require restrictive model assumption and therefore is model-free.", "However, the distance correlation measure used by DC-SIS may not be sensitive especially when sample size is small, because empirical characteristic functions are employed to estimate distance correlations.", "A more sensitive dependence measure between the response and a predictor variable is maximum correlation, which was originally proposed by [13] and studied by [22] as a general dependence measure between two random variables.", "[22] gave a list of seven fundamental properties a reasonable dependence measure must have, and maximum correlation is one of a few measures that satisfy this requirement.", "The definition and estimation of maximum correlation involve maximizations over functions (see Section REF ), and thus it is fairly sensitive even when sample size is small.", "Recently, there have been some revived interests in using maximum correlation as a proper dependence measure in high-dimensional data analysis [1], [15], [23], [25].", "In this paper, we propose to use maximum correlation as a dependence measure for ultrahigh dimensional screening, and prove that the resulting procedure has the sure screening property without imposing model specifications (see Theorem REF in Section REF ).", "We adopt the B-spline functions-based estimation method [4] to estimate maximum correlation.", "We refer to our proposed procedure as the maximum correlation-based sure independence screening procedure, or in short, the MC-SIS procedure.", "Numerical results show that MC-SIS is competitive to other existing model-based screening procedures, and is more sensitive and robust than DC-SIS when sample size is small or the distributions of the predictor variables have heavy tails.", "The rest of the paper is organized as follows.", "In Section , we introduce maximum correlation and the B-spline functions-based method for estimating maximum correlation, propose the MC-SIS procedure, and establish the sure screening property for MC-SIS.", "In Section , we develop a three-step procedure for selecting tuning parameters for MC-SIS in practice.", "Section presents results from simulation study and a real life screening application.", "Section concludes the paper with additional remarks and future research.", "The proofs of the theorems are given in the Appendix." ], [ "Maximum correlation and optimal transformation", "Let $Y$ denote the response variable and $ \\mathbf {X} = (X_1,\\ldots , X_p)$ be the vector of predictor variables.", "We assume the supports of $Y$ and $X_j$ $(j= 1,\\ldots , p)$ are compact, and they are further assumed to be [0,1] without loss of generality.", "For any given $j$ , consider a pair of random variables ($X_j,Y$ ).", "The maximum correlation coefficient between $X_j$ and $Y$ , denoted as $\\rho _j^$ , is defined as follows.", "$\\rho _j^(X_j,Y) =\\underset{\\theta , \\phi }{{sup}}\\lbrace \\rho \\left( \\theta (Y),\\phi (X_j)\\right): 0 < {E} \\lbrace \\theta ^2(Y) \\rbrace < \\infty , 0 < {E}\\lbrace \\phi ^2(X_j) \\rbrace < \\infty \\rbrace ,$ where $\\rho $ is the Pearson correlation, and $\\theta $ and $\\phi $ are Borel-measurable functions of $Y$ and $X_j$ .", "We further denote $\\theta _j^$ and $\\phi _j^$ as the optimal transformations that attain the maximum correlation.", "Maximum correlation coefficient enjoys the following properties [22]: (a) $0 \\le \\rho _j^(X_j,Y) \\le 1$ ; (b) $\\rho _j^(X_j,Y) = 0$ if and only if $X_j$ and $Y$ are independent; (c) $\\rho _j^(X_j,Y) = 1$ if there exist Borel-measurable functions $\\theta ^$ and $\\phi ^$ such that $\\theta ^(Y) = \\phi ^(X_j)$ ; and (d) if $X_j$ and $Y$ are jointly Gaussian, then $\\rho _j^(X_j,Y) = \\mathopen \|\\rho (X_j,Y)\\mathclose \|$ .", "Some other properties of maximum correlation coefficient are discussed in [27], [5], [3], and [31].", "Due to Property (d), it is clear that maximum correlation is a natural extension of Pearson correlation.", "Note that Pearson correlation does not possess Properties (b) and (c).", "For property (c), there are cases that Pearson correlation coefficient can be as low as zero when $Y$ is functionally determined by $X_j$ .", "For example, if $Y = X_1^2$ where $X_1 \\sim \\mathcal {N}(0,1)$ , the Pearson correlation between $Y$ and $X_1$ is zero, whereas the maximum correlation is one.", "Therefore, maximum correlation is a more proper measure of the dependence between two random variables than Pearson correlation.", "[22] established the existence of maximum correlation under certain sufficient conditions, and a different set of sufficient conditions are given in [2].", "[2] also showed that optimal transformations $\\theta ^_j$ and $\\phi ^_j$ can be obtained via the following minimization problem.", "$\\begin{aligned}& \\underset{\\theta _j, \\phi _j \\in L_2(P)}{\\text{min}}& & e_j^{2} = {E}[\\lbrace \\theta _j(Y) - \\phi _j(X_j)\\rbrace ^2], \\\\& \\text{subject to}& & {E} \\lbrace \\theta _j(Y)\\rbrace = {E}\\lbrace \\phi _j(X_j)\\rbrace = 0; \\\\&&& {E} \\lbrace \\theta ^2_j(Y)\\rbrace = 1.\\end{aligned}$ Here, P denotes the joint distribution of ($X_j$ ,$Y$ ) and $L_2(P)$ is the class of square integrable functions under the measure P. Let $e_j^{2}$ be the minimum of $e_j^2$ .", "[2] derived two critical connections between $e_j^{2}$ , squared maximum correlation $\\rho _j^{2}$ , and optimal transformation $\\phi _j^$ , which we state as Fact 0 below.", "$\\mbox{\\textit {Fact 0.}}", "\\hspace{158.99377pt} e_j^{2} = 1 & - \\rho _j^{2};\\hspace{130.08621pt}\\\\{E} ( \\phi _j^{2} ) & = \\rho _j^{2}.$ Fact 0 suggests that the minimization problem (REF ) is equivalent to the optimization problem (REF ).", "Furthermore, the squared maximum correlation coefficient is equal to the expectation of the squared optimal transformation $\\phi ^_j$ .", "Various algorithms have been proposed in the literature to compute maximum correlation, including Alternating Conditional Expectations (ACE) in [2], B-spline approximation in [4], and polynomial approximation in [1] and [15].", "Equation () indicates that maximum correlation coefficient $\\rho _j^$ can be calculated through the optimal transformation $\\phi _j^$ .", "In this paper, we apply Burman's approach to first estimate $\\phi ^_j$ , and then estimate $\\rho _j^$ , which will be further used in screening." ], [ "B-spline estimation of optimal transformations", "Let $\\mathcal {S}_n$ be the space of polynomial splines of degree $\\ell \\ge 1$ and $\\lbrace B_{jm}, m = 1,\\ldots , d_n \\rbrace $ denote a normalized B-spline basis with $\\mathopen \| \\mathopen \| B_{jm} \\mathopen \| \\mathopen \|_{{sup}} \\le 1$ , where $\\mathopen \| \\mathopen \|\\cdot \\mathopen \| \\mathopen \|_{{sup}}$ is the sup-norm.", "We have $\\theta _{nj}(Y) = {\\alpha }_j^T \\mathbf {B}_{j}(Y) $ , $\\phi _{nj}(X_j) = {\\beta }_j^T \\mathbf {B}_{j}(X_j) $ for any $\\theta _{nj}(Y), \\phi _{nj}(X_j) \\in \\mathcal {S}_n$ , where $\\mathbf {B}_j(\\cdot ) = (B_{j1}(\\cdot ),\\ldots , B_{jd_n}(\\cdot ))^T$ denotes the vector of $d_n$ basis functions.", "Additionally, we let $k$ be the number of knots where $k = d_n - \\ell $ .", "The population version of B-spline approximation to the minimization problem (REF ) can be written as follows.", "$\\begin{aligned}& \\underset{\\theta _{nj}, \\phi _{nj} \\in \\mathcal {S}_n}{\\text{min}}& & {E}[\\lbrace \\theta _{nj}(Y) - \\phi _{nj}(X_j)\\rbrace ^2], \\\\& \\text{subject to}& & {E} \\lbrace \\theta _{nj}(Y) \\rbrace = {E} \\lbrace \\phi _{nj}(X_j) \\rbrace = 0; \\\\&&& {E} \\lbrace \\theta ^2_{nj}(Y)\\rbrace = 1.\\end{aligned}$ [4] applied a technique to remove the first constraint ${E} \\lbrace \\theta _{nj}(Y) \\rbrace = \\\\ {E} \\lbrace \\phi _{nj}(X_j) \\rbrace = 0$ in the optimization problem above as follows.", "First, let $\\mathbf {z}_1,\\ldots , \\mathbf {z}_{d_n - 1} $ ($\\mathbf {z}_i = (z_{i1}, \\ldots , z_{id_n})^T$ for $i = 1, \\ldots , d_n -1$ ) be $d_n$ -dimensional vectors which are orthogonal to each other, orthogonal to the vector of 1's and $\\mathbf {z}_i^{T} \\mathbf {z}_i = 1$ for $i = 1,\\ldots , d_n-1$ .", "Second, obtain matrix $\\mathbf {D}_j$ with the $(s,m)$ -entry $\\mathbf {D}_{j,sm} = z_{sm}/(kb_{jm})$ where $b_{jm} = {E} \\lbrace B_{jm}(X_{j}) \\rbrace $ , for $s=1,\\ldots , d_n-1$ and $m = 1,\\ldots , d_n$ .", "Third, let $\\phi _{nj}(X_j) = {\\eta }_j^T \\psi _j(X_j)$ where $\\psi _j(X_j) = \\mathbf {D}_j \\mathbf {B}_j(X_j) $ .", "With this construction, it is easy to verify that ${E} \\lbrace \\phi _{nj}(X_j) \\rbrace = 0$ , and the minimization of $ {E}[\\lbrace \\theta _{nj}(Y) - \\phi _{nj}(X_j) \\rbrace ^2]$ subject to ${E} \\lbrace \\theta ^2_{nj}(Y)\\rbrace = 1$ ensures that ${E} \\lbrace \\theta _{nj}(Y) \\rbrace = 0$ .", "[4] showed the equivalence between the optimization problem (REF ) and the one stated below.", "$\\begin{aligned}& \\underset{\\theta _{nj}, \\phi _{nj} \\in \\mathcal {S}_n}{\\text{min}}& & {E}[\\lbrace \\theta _{nj}(Y) - \\phi _{nj}(X_j)\\rbrace ^2], \\\\& \\text{subject to}&& {E} \\lbrace \\theta ^2_{nj}(Y)\\rbrace = 1.\\end{aligned}$ For fixed $\\theta _{nj}(Y)$ (i.e.", "fixed ${\\alpha }_j$ ), the minimizer of (REF ) with respect to ${\\eta }_j$ and $\\phi _{nj}(X_j)$ are $\\begin{aligned}& {\\eta }_j = [ {E} \\lbrace {\\psi }_j(X_j) {\\psi }_{j}^T(X_j) \\rbrace ] ^{-1} {E} \\lbrace {\\psi }_j(X_j) \\mathbf {B}_{j}^T(Y) \\rbrace {\\alpha }_j, \\\\& \\phi _{nj}(X_j) = {\\psi }_j^T(X_j) [ {E} \\lbrace {\\psi }_j(X_j) {\\psi }_{j}^T(X_j) \\rbrace ] ^{-1} {E} \\lbrace {\\psi }_j(X_j) \\mathbf {B}_{j}^T(Y) \\rbrace {\\alpha }_j.\\end{aligned}$ By plugging $\\phi _{nj}(X_j)$ back in (REF ), we obtain the following maximization problem, $\\begin{aligned}& \\underset{{\\alpha }_j \\in \\mathbb {R}^{d_n}}{\\text{max}}& & {\\alpha }_j^T {E} \\lbrace \\mathbf {B}_{j} (Y) {\\psi }_j^T(X_j) \\rbrace [ {E} \\lbrace {\\psi }_j(X_j) {\\psi }_{j}^T(X_j)\\rbrace ]^{-1} {E} \\lbrace {\\psi }_j(X_j) \\mathbf {B}_{j}^T(Y) \\rbrace {\\alpha }_j, \\\\& \\text{subject to}& & {\\alpha }_j^T {E} \\lbrace \\mathbf {B}_{j}(Y) \\mathbf {B}_{j}^T(Y) \\rbrace {\\alpha }_j = 1.\\end{aligned}$ Following the notation in [4], we denote $\\begin{aligned}\\mathbf {A}_{j00} = {E} \\lbrace \\mathbf {B}_{j}(Y) \\mathbf {B}_{j}^T(Y) \\rbrace , \\qquad \\mathbf {A}_{jXX} = {E} \\lbrace \\psi _{j}(X_j) \\psi _{j}^T(X_j) \\rbrace , \\\\\\mathbf {A}_{jX0} = {E} \\lbrace \\psi _{j}(X_j) \\mathbf {B}_{j}^T(Y) \\rbrace , \\quad \\mbox{and} \\quad \\mathbf {A}_{j0X} = \\mathbf {A}_{jX0}^T.\\end{aligned}$ It is clear that (REF ) is a generalized eigenvalue problem, which can be solved by the largest eigenvalue and its corresponding eigenvector of $\\mathbf {A}_{j00}^{-1/2} \\mathbf {A}_{j0X} \\mathbf {A}_{jXX}^{-1} \\mathbf {A}_{jX0} \\mathbf {A}_{j00}^{-1/2}$ .", "We denote the largest eigenvalue by $\\lambda _{j1}^$ , which is equal to $\|\| \\mathbf {A}_{j00}^{-1/2} \\mathbf {A}_{j0X} \\mathbf {A}_{jXX}^{-1} \\mathbf {A}_{jX0} \\mathbf {A}_{j00}^{-1/2}\|\|$ , where $\|\|\\cdot \|\|$ is the operator norm, and further denote the corresponding eigenvector by ${\\alpha }^_j$ .", "Let $\\phi ^_{nj}(X_j) = {\\psi }_j^T(X_j) [ {E} \\lbrace {\\psi }_j(X_j) {\\psi }_{j}^T(X_j) \\rbrace ] ^{-1} {E} \\lbrace {\\psi }_j(X_j) \\mathbf {B}_{j}^T(Y) \\rbrace {\\alpha }^_j$ .", "$\\phi ^_{nj}$ can be considered the spline approximation to the optimal transformation $\\phi ^_j$ defined previously.", "Note that the target function in (REF ) is ${E}(\\phi _{nj}^{2})$ , and we also have $ {E}(\\phi _{nj}^{2}) = \\lambda _{j1}^$ .", "Given the data $\\lbrace Y_u\\rbrace _{u=1}^n$ and $\\lbrace X_{uj}\\rbrace _{u=1}^n$ , we estimate $\\mathbf {A}_{j00}$ , $\\mathbf {A}_{jXX}$ , $\\mathbf {A}_{jX0}$ , and $\\mathbf {A}_{j0X}$ as follows.", "$\\begin{aligned}\\widehat{\\mathbf {A}_{j00}} = n^{-1} \\sum _{u=1}^{n} \\mathbf {B}_j(Y_u)\\mathbf {B}_j^T(Y_u), \\qquad \\widehat{\\mathbf {A}_{jXX}} = n^{-1} \\sum _{u=1}^{n} \\widehat{\\psi }_j(X_{uj}) \\widehat{\\psi }_j^T(X_{uj}), \\\\\\widehat{\\mathbf {A}_{jX0}} = n^{-1} \\sum _{u=1}^{n} \\widehat{\\psi }_j(X_{uj}) \\mathbf {B}_j^T(Y_u), \\quad \\mbox{and} \\quad \\widehat{\\mathbf {A}_{j0X}} = \\widehat{\\mathbf {A}_{jX0}}^T,\\end{aligned}$ where $\\widehat{\\psi }_j(X_{uj}) = \\widehat{\\mathbf {D}_j} \\mathbf {B}_j(X_{uj})$ , the ($s,m$ )-entry of $\\widehat{\\mathbf {D}_j}$ is $\\widehat{\\mathbf {D}}_{j,sm} = z_{sm}/(k \\widehat{b_{jm}})$ , and $\\widehat{b_{jm}} = n^{-1} \\sum _{u=1}^{n} B_{jm}(X_{uj})$ , for $s = 1,\\ldots , d_n - 1$ and $m = 1,\\ldots , d_n$ .", "Then, $\\lambda _{j1}^$ is estimated by $\\widehat{\\lambda _{j1}^} = \|\| \\widehat{\\mathbf {A}_{j00}}^{-1/2} \\widehat{\\mathbf {A}_{j0X}} \\widehat{\\mathbf {A}_{jXX}}^{-1} \\widehat{\\mathbf {A}_{j0X}}^T \\widehat{\\mathbf {A}_{j00}}^{-1/2}\|\|$ , and ${\\alpha }^_j$ is estimated by the eigenvector of $\\widehat{\\mathbf {A}_{j00}}^{-1/2} \\widehat{\\mathbf {A}_{j0X}} \\widehat{\\mathbf {A}_{jXX}}^{-1} \\widehat{\\mathbf {A}_{j0X}}^T \\widehat{\\mathbf {A}_{j00}}^{-1/2}$ corresponding to $\\widehat{\\lambda _{j1}^}$ , which we denote as $\\widehat{{\\alpha }^_j}$ .", "Therefore, the optimal transformation of $Y$ is estimated by $\\widehat{\\theta ^_{nj}} = \\widehat{{\\alpha }^_j}^T B_j(Y)$ .", "Furthermore, based on (REF ), the optimal transformation of $X_j$ can be obtained by $\\widehat{\\phi ^_{nj}} = \\widehat{{\\eta }^_j}^T {\\psi }_j(X_j)$ with $\\widehat{{\\eta }^_j} = \\widehat{\\mathbf {A}_{jXX}}^{-1} \\widehat{\\mathbf {A}_{jX0}} \\widehat{{\\alpha }^_j} $ .", "Based on the two relationships ($i$ ) ${E} ( \\phi _j^{2} ) = ( \\rho _j^* )^2$ and ($ii$ ) $ {E}(\\phi _{nj}^{2}) = \\lambda _{j1}^$ , and the fact that $\\phi _{nj}^$ is the optimal spline approximation to $\\phi _{j}^$ , we propose to screen important variables using the magnitudes of $\\widehat{\\lambda _{j1}^}$ for $1\\le j \\le p$ ." ], [ "MC-SIS procedure", "Let $\\nu _n$ be a pre-specified threshold, and $\\widehat{\\mathcal {D}_{\\nu _n}} $ the collection of selected important variables.", "Then our proposed screening procedure can be defined as $\\widehat{\\mathcal {D}_{\\nu _n}} =\\lbrace 1 \\le j \\le p \\colon \\widehat{\\lambda _{j1}^} \\ge \\nu _n \\rbrace .$ $\\left[ a \\right] $ denotes the integer part of $a$ .", "Since $\\widehat{\\lambda _{j1}^}$ is the estimate of $\\lambda _{j1}^$ , which is an approximation to the squared maximum correlation coefficient $\\rho _j^{2}$ , we refer to the procedure as the MC-SIS procedure.", "The sure screening property of the procedure will be discussed in next section." ], [ "Sure Screening Property", "Adopting notations from [21], we use $F(Y\|\\mathbf {X})$ to denote the conditional distribution of $Y$ given $\\mathbf {X}$ and $\\Psi _Y$ the support for $Y$ .", "We define $\\mathcal {D} = \\lbrace j : F(y\|\\mathbf {X})$ functionally depends on $X_j$$\\rbrace $ , and $\\mathcal {I} = \\lbrace j : F(y\|\\mathbf {X})$ does not functionally depends on $X_j$$\\rbrace $ .", "Let $\\mathbf {X}_{\\mathcal {D}} = \\lbrace X_j$ : $j \\in \\mathcal {D} \\rbrace $ and $\\mathbf {X}_{\\mathcal {I}} = \\lbrace X_j$ : $j \\in \\mathcal {I} \\rbrace $ , which are referred to the active and inactive sets, respectively.", "Furthermore, we refer the variables in the active set and inactive set as active predictor variables and inactive predictor variables, respectively.", "Ideally, the goal of a screening procedure is to retain $\\mathcal {D}$ after screening, which is referred to as the sure screening property.", "We have established the sure screening property of the MC-SIS procedure under certain conditions.", "Before stating the theorem, we first list the conditions below.", "(C1) If the transformations $\\theta _j$ and $\\phi _j$ with zero means and finite variances satisfy $\\theta _j(Y) + \\phi _j(X_j) = 0 \\mbox{ a.s., then each of them is zero a.s.}$ (C2) The conditional expectation operators $ {E}\\lbrace \\phi _j(X_j) \\mid Y \\rbrace : H_2(X_j) \\rightarrow H_2(Y)$ and ${E}\\lbrace \\theta _j(Y) \\mid X_j \\rbrace : H_2(Y) \\rightarrow H_2(X_j)$ are all compact operators.", "$H_2(Y)$ and $H_2(X_j)$ are Hilbert spaces of all measurable functions with zero mean, finite variance and usual inner product.", "(C3) The optimal transformations $\\lbrace \\theta ^_j, \\phi ^_j \\rbrace _{j = 1}^p$ belong to a class of functions $\\mathcal {F}$ , whose $\\textit {r}$ th derivative $\\mathit {f}^{(r)}$ exists and is Lipschitz of order $\\alpha _1$ , that is, $\\mathcal {F} = \\lbrace \\mathit {f} : \|\\mathit {f}^{(r)}(s) - \\mathit {f}^{(r)}(t)\| \\le K\|s-t\|^{\\alpha _1} \\mbox{ for all } s,t \\rbrace \\mbox{ for some positive constant \\textit {K}}$ , where $\\textit {r}$ is a nonnegative integer and $\\alpha _1 \\in (0,1]$ such that $d = \\textit {r} + \\alpha _1 > 0.5$ .", "(C4) The joint density of $Y \\mbox{ and } X_j$ $(j = 1, \\ldots , p)$ is bounded and the marginal densities of $Y$ and $X_j$ are bounded away from zero.", "(C6) There exist positive constant $C_1$ and constant $\\xi \\in (0,1)$ such that $d_n^{-d-1} \\le c_1 (1-\\xi ) n^{-2\\kappa }/C_1$ .", "Conditions (C1) and (C2) are adopted from [2], which ensure that the optimal transformations exist.", "Conditions (C3) and (C4) are from [4], but modified for our two-variable scenario.", "Condition (C5) above is similar to Condition 3 in [8], Condition C in [6], and Condition (C2) in [21], which all require that the dependence between the response and active predictor variables cannot be too weak.", "We note that this condition is necessary, since a marginal screening procedure will fail when the marginal dependence between the response and an active predictor variable is too weak.", "The following lemma shows that the maximum correlations achieved by B-spline-based transformations are at the same level as the original maximum correlations.", "Lemma 1 Under conditions (C3) – (C6), we have $ \\mathop {\\min }\\limits _{j \\in \\mathcal {D}} \\lambda _{j1}^ \\ge c_1 \\xi d_n n^{-2\\kappa }$ .", "Based on condition (C1) – (C6), we establish the following sure screening properties for MC-SIS.", "Theorem 1 (a) Under conditions (C1) – (C4), for any $c_2 > 0$ , there exist positive constants $c_3$ and $c_4$ such that $P( \\mathop {\\max }\\limits _{1\\le j \\le p} \| \\widehat{\\lambda _{j1}^} - \\lambda _{j1}^ \| \\ge c_2 d_n n^{-2 \\kappa } ) \\le \\mathcal {O} \\left( p \\zeta (d_n, n) \\right).$ where $\\zeta (d_n, n) = d_n^2 \\exp ( - c_3 n^{1-4\\kappa } d_n^{-4} ) + d_n \\exp ( -c_4 n d_n^{-7}) $ .", "(b) Additionally, if conditions (C5) and (C6) hold, by taking $\\nu _n = c_5 d_n n^{-\\kappa }$ with $c_5 \\le c_1 \\xi /2$ , we have that $P ( \\mathcal {D} \\subseteq \\widehat{\\mathcal {D}_{\\nu _n}}) \\ge 1- \\mathcal {O} \\left( s \\zeta (d_n, n)\\right),$ where $s$ is the cardinality of $\\mathcal {D}$ .", "Note that Theorem REF is stated for fixed number of predictor variables $p$ .", "In fact, the same theorem holds for divergent number of predictor variables $p_n$ .", "As long as $p_n \\zeta (d_n, n)$ goes to zero asymptotically, MC-SIS can possess the sure screening property.", "And we remark that the number of basis functions $d_n$ affects the final performance of MC-SIS.", "To obtain the sure screening property, an upper bound of $d_n$ is $o(n^{1/7})$ .", "Since $d_n$ is determined by the choices of the degree of B-spline basis functions and the number of knots, different combinations of degree and the number of knots can lead to different screening results.", "Additionally, knots placement can further affect the behavior of B-spline functions, and in practice, knots are usually equally spaced or placed at sample quantiles.", "In next section, we will propose a data-driven three-step procedure for determine $d_n$ for MC-SIS in practice.", "The optimal choice of $d_n$ and knots placement are beyond the scope of this paper and can be an interesting topic for future research.", "The sure screening property from Theorem REF guarantees that MC-SIS retains the active set.", "The size of the selected set can be much larger than the size of the active set.", "Therefore, it is of interest to assess the size of the selected set, similar to [6].", "The next theorem is such a result for MC-SIS.", "Theorem 2 Under Conditions (C1) – (C6), we have that for any $\\nu _n = c_5 d_n n^{-\\kappa }$ , there exist positive constants $c_3$ and $c_4$ such that $P [ \|\\widehat{\\mathcal {D}_{\\nu _n}}\| \\le \\mathcal {O}\\lbrace n^{2\\kappa } \\lambda _{max}(\\mathbf {\\Sigma }) \\rbrace ] \\ge 1- \\mathcal {O} \\left( p_n \\zeta (d_n, n)\\right),$ where $\|\\widehat{\\mathcal {D}_{\\nu _n}}\|$ is the cardinality of $\\widehat{\\mathcal {D}_{\\nu _n}}$ , , $\\mathbf {\\Sigma } = E \\lbrace {\\psi } {\\psi }^T\\rbrace $ , $ {\\psi } = ( {\\psi }^T_1, \\ldots , {\\psi }^T_{p_n} )^T$ , $p_n$ is the divergent number of predictor variables, and $\\zeta (d_n, n)$ is defined in Theorem REF .", "From Theorem REF , we have that when $\\lambda _{max}(\\mathbf {\\Sigma }) = \\mathcal {O}(n^{\\tau })$ , the cardinality of the selected set by MC-SIS will be of order $\\mathcal {O}(n^{2\\kappa +\\tau })$ .", "Thus, by applying MC-SIS, we can reduce dimensionality from the original exponential order to a polynomial order, while retaining the entire active set." ], [ "Numerical Results", "We illustrate the MC-SIS procedure by studying its performance under different model settings and distributional assumptions of the predictor variables.", "For all examples, we compare MC-SIS with SIS, NIS, and DC-SIS.", "As mentioned at the end of Section REF , the ACE algorithm in [2] can also be used to calculate maximum correlation coefficient.", "Therefore, the ACE algorithm can also be used to perform maximum correlation-based screening, and we refer to the resulting procedure as the ACE-based MC-SIS procedure.", "We also include the ACE-based MC-SIS procedure in our simulation study.", "To avoid confusion, we refer to our proposed procedure as the B-spline-based MC-SIS procedure in this section.", "For each simulation example, we set $p = 1000$ and choose $n \\in \\lbrace 200, 300, 400 \\rbrace $ .", "Following [8] and [6], we measure the effectiveness of MC-SIS using minimum model size (MMS) and robust estimate of its standard deviation (RSD).", "MMS is defined as the minimum number of selected variables, i.e., the size of the selected set, that is required to include the entire active set.", "RSD is defined as IQR/1.34, where IQR is the interquartile range.", "When constructing B-spline basis functions, Example 1 (1.a): $Y = {\\beta }^{T} \\mathbf {X} + \\varepsilon $ , with the first s components of $\\beta ^{\\ast }$ taking values $\\pm 1$ alternatively and the remaining being 0, where $s = 3, 6 \\mbox{ or } 12$ ; $ X_k $ are independent and identically distributed as $N\\left( 0,1\\right)$ for $1 \\le k \\le 950$ ; $ X_k = \\sum _{j=1}^{s}X_{j}\\left( -1\\right) ^{j+1} / 5+ ( 1- s \\varepsilon _{k}/25)^{1/2}$ where $\\varepsilon _k$ are independent and identically distributed as $N\\left( 0,1\\right)$ for $k = 951, \\ldots , 1000$ ; and $\\varepsilon \\sim N\\left( 0,3\\right)$ .", "Here, $\\mathcal {D} = \\lbrace 1,2, \\ldots , s\\rbrace .$ (1.b): $Y = X_1 + X_2 + X_3 + \\varepsilon $ , where $ X_k$ are independent and identically distributed as $N\\left( 0,1\\right)$ for $k=1, \\mbox{and } 3 \\le k \\le 1000$ ; $X_2 = \\dfrac{1}{3} X_1^3 + \\tilde{\\varepsilon }$ , and $\\tilde{\\varepsilon } \\sim N\\left( 0,1\\right)$ ; and $\\varepsilon \\sim N\\left( 0,3\\right)$ .", "Here, $\\mathcal {D} = \\lbrace 1, 2, 3\\rbrace $ .", "The first example is from [6] and the simulation results are presented in Table REF .", "Under model (1.a), SIS demonstrates the best performance across all cases, which is expected since SIS is specifically developed for linear models.", "Under the models (1.a) with $s = 3 \\mbox{ or } 6$ , when $n = 200$ , MC-SIS under-performs all other methods.", "However, when sample size increases to 300 or 400, MC-SIS becomes comparable to others.", "For the case with $s = 12$ , MC-SIS under-performs other methods for all choices of $n$ .", "The cause for the relatively poor performance of MC-SIS is due to the weak signal.", "With $s=12$ , it requires more samples for MC-SIS to estimate maximum correlation coefficient, without taking advantages of linearity assumptions.", "In model (1.b), SIS fails as there exists a nonlinear relationship between $X_1$ and $X_2$ .", "NIS demonstrates the best performance as NIS is designed for dealing with nonparametric additive models.", "The ACE-based MC-SIS procedure demonstrates the second best performance.", "The B-spline-based MC-SIS procedure performs better than DC-SIS.", "Table: MMS and RSD (in parenthesis) for Example Example 2 (2.a): $Y = X_1X_2 + X_3X_4 + \\varepsilon $ ; $\\mathcal {D} = \\lbrace 1,2,3,4 \\rbrace $ ; (2.b): $Y = X_1^2 + X_2^3 + X_3^2X_4 + \\varepsilon $ ; $\\mathcal {D} = \\lbrace 1,2,3,4\\rbrace $ ; (2.c): $Y = X_1\\sin (X_2) + X_2\\sin (X_1) + \\varepsilon $ ; $\\mathcal {D} = \\lbrace 1,2 \\rbrace $ ; (2.d): $Y = X_1\\exp (X_2) + \\varepsilon $ ; $\\mathcal {D} = \\lbrace 1,2\\rbrace $ ; (2.e): $Y = X_1\\log (\|c_0 + X_2\|) + \\varepsilon $ ; $\\mathcal {D} = \\lbrace 1,2\\rbrace $ ; (2.f): $Y = X_1/(c_0 + X_2) + \\varepsilon $ ; $\\mathcal {D} = \\lbrace 1,2\\rbrace $ .", "Here $X_{1},\\ldots ,X_{1000}$ and $\\epsilon $ are generated independently from $N(0,1)$ , and $c_0=10^{-4}$ .", "The eight models considered in this example are non-additive, and the simulation results are presented in Table REF .", "Due to the presence of non-additive structures, we notice that SIS and NIS fail in all models, and increasing sample size does not help improve the performances of SIS and NIS for most models.", "Both MC-SIS and DC-SIS work well in this example, but MC-SIS outperforms DC-SIS for almost all the models in terms of MMS.", "Even when the sample size is as small as 200, MC-SIS can effectively retain the active set under models (2.c), (2.e) and (2.f).", "This example demonstrates the advantages of MC-SIS and DC-SIS over SIS and NIS for non-additive models as well as the effectiveness of MC-SIS over DC-SIS.", "Table: MMS and RSD (in parenthesis) for Example Example 3 The models considered in this example are modifications of the models considered in Example REF .", "First, the error term $\\epsilon $ in each original model is removed; and second, the predictor variables $X_1, X_2, \\ldots , X_p$ are drawn independently from $Cauchy(0,1)$ instead of $N(0,1)$ .", "The resulting models are denoted as (3.a)-(3.f), correspondingly.", "Simulation results based on these models are presented in Table REF .", "Intuitively, the absence of the error terms in the models is expected to help the screening methods, but the use of heavy-tailed distributions for the predictor variables is expected to hinder the methods.", "The exact performance of a screening method in this example depends on the trade-off between those two changes.", "Comparing Table REF with Table REF , we can see that the performances of SIS and NIS have improved, though they are still far from being satisfactory.", "The performance of DC-SIS has improved in models (3.a) and (3.c), but has much deteriorated in the other models, which indicates that DC-SIS is susceptible to heavy-tailed distributions.", "In the presence of heavy tails, Condition (C1) in [21] is violated, and DC-SIS may not have the sure screening property.", "The performances of ACE-based and B-spline-based MC-SIS are better over DC-SIS in most models, which indicates the robustness of MC-SIS towards heavy-tailed distributions.", "Table: MMS and RSD (in parenthesis) for Example Example 4 In this example, we consider a real data set that contains the expression levels of 6319 genes and the expression levels of a G protein-coupled receptor (Ro1) in 30 mice [24].", "The same data set has been analyzed in [14] and in [21] using DC-SIS.", "The goal is to identify the most influential genes for Ro1.", "We apply SIS, NIS, DC-SIS, ACE-based MC-SIS and B-spline-based MC-SIS to select the top two most important genes, separately.", "Additionally, we note that almost all of the procedures considered here, including B-spline-based MC-SIS, consistently ranked Msa.741.0, Msa.2134.0 and Msa.2877.0 among the top ranked genes.", "The top-ranked two genes by individual procedures are reported in Table REF .", "To further compare the performances of the screening procedures, we fit regression models for the response, which is the expression level of Ro1, using the top two genes selected by the procedures.", "Three different models are considered, which are the linear regression model $ Y = \\beta _0 + \\beta _{1} X_{1} + \\beta _{2} X_{2} + \\varepsilon $ , the additive model $ Y = \\ell _{1}(X_{1}) + \\ell _{2}(X_{2}) + \\varepsilon $ , and the optimal transformation model $ \\theta ^(Y) = \\phi ^_{1}(X_{1}) + \\phi _{2}^ (X_{2}) + \\varepsilon $ , where $\\theta ^$ , $\\phi _1^$ and $\\phi _2^$ are the optimal transformations [2].", "For each procedure, all three models are fitted using the top ranked gene as well as using the top ranked two genes, and the resulting adjusted $R^2$ values are reported in Table REF .", "Table: Top ranked genes for Example Table: Adjusted R 2 R^2 (in percentage) of fitting 3 different models for Example Under the linear model, as expected, SIS achieves the largest adjusted $R^2$ values, whereas the adjusted $R^2$ values of ACE-based MC-SIS are rather poor.", "The major cause for the difference between SIS and ACE-based MC-SIS is that the former is specifically developed for screening under the linear model, whereas the latter is for screening under the optimal transformation model.", "Under the additive model, when the top one gene is used, NIS achieves the largest adjusted $R^2$ value; and when the top two genes are used, DC-SIS achieves the largest adjusted $R^2$ value.", "Under the optimal transformation model, MC-SIS (both ACE-based and B-spline-based) methods achieve the largest adjusted $R^2$ values with both the top one gene and top two genes.", "When plotting the expression levels of Ro1 against the expression levels of various selected genes, different patterns including linear and nonlinear patterns emerge for different screening methods.", "In practice, we believe that the top ranked genes by different methods are all worth further investigation." ], [ "Discussion", " The performances and results of B-spline-based MC-SIS depend on the choice of degree and the number of knots for B-splines.", "In this paper, we have developed a data-driven three-step procedure to construct B-spline basis functions for MC-SIS in practice.", "The proposed procedure demonstrates satisfactory performance in simulation study as well as real data application.", "We hope to investigate and characterize the theoretical property of the procedure in the future.", "Similar to other existing screening procedures, MC-SIS fail to retain active predictor variables that are marginally independent with the response variable.", "Under the linear regression model, [8] proposed an iterative procedure to recover such predictor variables.", "Similarly, we have developed an iterative version of MC-SIS with the hope to recover active predictor variables missed by MC-SIS.", "Currently, we are investigating the empirical performance and theoretical property of this iterative version and hope to report the results in a future publication.", "Most existing marginal screening procedures under nonparametric model assumptions, including MC-SIS, make use of independent measures, whose estimation typically involves nonparametric model fitting and tuning parameter selection.", "Nonparametric methods are known to be sensitive to tuning parameter selection.", "Therefore, this can also become a drawback for those screening procedures.", "On the other hand, there are various independence measures that are based on cumulative distribution functions, and the estimation of those measures does not involve nonparametric fitting and tuning parameter selection.", "Two examples include Hoeffding's test [18] and Heller-Heller-Gorfine tests [17].", "It will be of interest to explore the application of these measures for screening and the potential of using these methods for variable selection after screening.", "Appendix" ], [ "Notation", " $n$ : sample size $p$ : dimension size $\\ell $ : degree of polynomial spline $k$ : number of knots $d_n$ : dimension of B-spline basis $\\mathcal {D}$ : active set $\\mathcal {I}$ : inactive set $\\theta _j$ : transformation of response $Y$ for pair $\\left(X_j,Y \\right)$ , $j = 1,2,\\ldots ,p$ $\\phi _j$ : transformation of $X_j$ for pair $\\left(X_j,Y \\right)$ $\\rho _j$ : Pearson correlation of pair $\\left(X_j,Y \\right)$ $e_j^2$ : squared error by regressing $\\phi _j$ on $\\rho _j$ $\\theta _j^$ : optimal transformation of response $Y$ for pair $\\left(X_j,Y \\right)$ $\\phi _j^$ : transformation of $X_j$ for pair $\\left(X_j,Y \\right)$ $\\rho _j^$ : maximum correlation of pair $\\left(X_j,Y \\right)$ $e_j^{2}$ : squared error by regressing $\\phi _j^$ on $\\theta _j^$ $\\theta _{nj}^$ : spline approximation to optimal transformation $\\theta _j^$ $\\phi _{nj}^$ : spline approximation to optimal transformation $\\phi _j^$ $s$ : cardinality of active set $\\mathcal {D}$ $\\mathopen \| \\mathopen \| \\cdot \\mathopen \| \\mathopen \|$ : operator norm $\\mathopen \| \\mathopen \| \\cdot \\mathopen \| \\mathopen \|_{{sup}}$ : sup norm" ], [ "Bernstein's inequality and four facts", "Lemma 2 (Bernstein's inequality, Lemma 2.2.9, [30]) For independent random variables $Y_1, \\ldots , Y_n$ with bounded ranges $\\left[-M,M\\right]$ and 0 means, $P \\left( \| Y_1 + \\ldots + Y_n \| > x \\right)\\le 2 \\exp [ -x^2 / \\lbrace 2(v + Mx/3) \\rbrace ]$ for $v \\ge var( Y_1 + \\ldots + Y_n)$ .", "Under conditions (C3) and (C4), the following four facts hold when $\\ell \\ge d$ .", "Fact 1.", "There exists a positive constant $C_1$ such that [4] ${E} \\lbrace (\\phi ^_{j} - \\phi ^_{nj} )^2 \\rbrace \\le C_1 k^{-d}$ Fact 2.", "There exists a positive constant $C_2$ such that [26], [19] ${E} \\lbrace B_{jm}^2(\\cdot ) \\rbrace \\le C_2 d_n^{-1}$ Fact 3.", "There exist positive constants $c_{11}$ , $c_{12}$ such that [4], [32] $\\begin{aligned}c_{11} d_n^{-1} \\le \\lambda _{min}\\left( {E} \\lbrace \\mathbf {B}_j(\\cdot ) \\mathbf {B}_j^T(\\cdot ) \\rbrace \\right) &\\le \\lambda _{max}\\left( {E} \\lbrace \\mathbf {B}_j(\\cdot ) \\mathbf {B}_j^T(\\cdot ) \\rbrace \\right) \\le c_{12} d_n^{-1} \\\\c_{11} k^{-1} \\le \\lambda _{min}\\left( {E} \\lbrace {\\psi }_j(X_j) {\\psi }_j^T(X_j)\\rbrace \\right)&\\le \\lambda _{max}\\left( {E} \\lbrace {\\psi }_j(X_j) {\\psi }_j^T(X_j)\\rbrace \\right) \\le c_{12} k^{-1}\\end{aligned}$ Fact 4.", "There exists a positive constant $C_3 $ such that [4], [12] $\\begin{aligned}C_3 k^{-1} \\le b_{jm} \\le 1, \\quad \\quad 0 \\le \\widehat{b_{jm}} \\le 1\\end{aligned}$ Remark 1 The choice of knots plays a role in establishing the sure screening property.", "When the knots of the B-splines are placed at the sample quantiles, $\\widehat{b_{jm}}$ is positive.", "When knots are uniform placed, $\\widehat{b_{jm}}$ can be zero with a small probability.", "According to [4], when the marginal density $f_{X_j}(x) > \\gamma _0 > 0$ by condition (C4) for each $X_j$ , we have $ P ( \\widehat{b_{jm}}= 0 \\mbox{ for some } m = 1,\\ldots , d_n ) \\le k \\exp ( - \\gamma _0 n/k )$ .", "The results in [4] are based on equally spaced knots, and our proof for MC-SIS use the same choice of knots, as the probability of $\\widehat{b_{jm}}$ being zero is a small probability, we just acknowledge $\\widehat{b_{jm}} > 0$ in the proof.", "In fact, sure screening property still hold when the event $\\widehat{b_{jm}}= 0$ is included.", "Remark 2 With $\\ell $ fixed, $k$ and $d_n$ are of the same order, we replace $k$ with $d_n$ in the following proof for convenience." ], [ "Proof of Lemma 1", "By Cauchy-Schwarz inequality, we have ${E} (\\phi ^{2}_{j}) \\le 2 {E} \\lbrace (\\phi ^_{j} - \\phi ^_{nj} )^2\\rbrace + 2 {E} (\\phi ^{2}_{nj})$ Therefore, ${E} (\\phi ^{2}_{nj}) \\ge \\frac{1}{2} {E} (\\phi ^{2}_{j}) - {E} \\lbrace (\\phi ^_{j} - \\phi ^_{nj} )^2 \\rbrace $ Lemma 1 follows from condition (C5) together with $ {E} (\\phi ^{2}_{nj}) = \\lambda _{j1}^$ ." ], [ "Proof of eight basic results", "We list and prove eight results (R1) – (R8) which together form the major parts in proving sure screening property of MC-SIS.", "For the rest of the paper, we use ${P}_n$ to denote the sample average.", "R1.", "With $c_{11}$ in Fact 3, we have that, $\|\|\\mathbf {A}_{j00}^{-1/2}\|\| \\le c_{11}^{-1/2} d_n ^{1/2}$ $\|\|\\mathbf {A}_{j00}^{-1/2}\|\| = \\lambda _{min}^{-1/2}(\\mathbf {A}_{j00})$ , result follows by Fact 3.", "R2.", "There exist positive constant $c_{13}$ such that $\|\|\\mathbf {A}_{j0X}\|\| \\le c_{13} d_n ^{-1/2}$ Let $\\mathbf {u} = (u_1, \\ldots , u_{d_n})^T \\in R^{d_n}$ with $\\sum _{m=1}^{d_n} u_m^2 = 1$ .", "$\\begin{aligned}\\mathbf {u}^T {E} \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(Y) \\rbrace & {E} \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \\mathbf {u} = \\sum \\limits _{i=1}^{d_n} \\left[ \\int \\lbrace \\sum \\limits _{m=1}^{d_n} u_m B_{jm}(X_j)\\rbrace B_{ji}(Y) dF \\right]^2 \\\\&\\le \\int \\lbrace \\sum \\limits _{m=1}^{d_n} u_m B_{jm}(X_j) \\rbrace ^2 dF \\times \\sum \\limits _{i=1}^{d_n} \\lbrace \\int B_{ji}^2(Y) dF \\rbrace \\\\&\\le \\lambda _{max} [{E} \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace ] \\times d_n \\max \\limits _{i} {E} \\lbrace B_{ji}^2(Y)\\rbrace \\end{aligned}$ Then, $\|\| {E} \\lbrace \\mathbf {B}_{j}(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\le ( c_{12} C_2/ d_n) ^{1/2} $ by Fact 2 and Fact 3.", "It can be easily shown that, for $\\mathbf {u} \\in R^{d_n-1}$ with $\\sum _{i=1}^{d_n-1} u_i^2 = 1$ , $\\begin{aligned}\\mathbf {u}^T \\mathbf {D}_j \\mathbf {D}_j^T \\mathbf {u} = \\sum \\limits _{m=1}^{d_n} \\frac{1}{k^2 b_{jm}^2} \\left( \\sum \\limits _{i=1}^{d_n - 1} u_i z_{im} \\right)^2 \\le C_3^{-2} \\sum \\limits _{m=1}^{d_n} \\left( \\sum \\limits _{i=1}^{d_n - 1} u_i z_{im} \\right)^2 \\le C_3^{-2}\\end{aligned}$ which indicates $\|\|\\mathbf {D}_j^T\|\| \\le C_3^{-1}$ .", "Then, $\|\|\\mathbf {A}_{j0X}\|\| \\le \|\| {E} \\lbrace \\mathbf {B}_{j}(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\|\\mathbf {D}_j^T\|\| \\le c_{13} d_n^{-1/2}$ with $c_{13}=(c_{12} C_2)^{1/2} C_{3}^{-1}$ .", "R3.", "For any given constant $c_4$ , there exists a positive constant $c_8$ such that $P\\lbrace \|\|\\widehat{\\mathbf {A}_{j00}}^{-1/2}\|\| \\ge \\left( (c_8 +1)c_{11}^{-1} d_n \\right)^{1/2} \\rbrace \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} )$ Since $\|\|\\widehat{\\mathbf {A}_{j00}}^{-1/2}\|\| = \\sqrt{\|\|[{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(Y)\\rbrace ]^{-1}\|\|}$ .", "R3 can be obtained via equation (26) in [6], which is $P\\lbrace \|\|[{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(Y)\\rbrace ]^{-1}\|\| \\ge (c_8 +1) c_{11}^{-1} d_n \\rbrace \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} )$ .", "R4.", "There exist some positive constants $c_6$ , $c_{7}$ such that, $P\\lbrace \|\|\\widehat{\\mathbf {A}_{j0X}} \|\| \\ge c_{6} d_n^{-1/2} \\rbrace \\le 4 d_n^2 \\exp ( -c_7 n d_n^{-2} )$ As $\|\|\\widehat{\\mathbf {A}_{j0X}} \|\| = \|\| {P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \\widehat{\\mathbf {D}_j}^T \|\| \\le \|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j)\\rbrace \|\| \\hspace{3.61371pt} \|\|\\widehat{\\mathbf {D}_j}^T\|\|$ , we firstly deal with $\|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\|$ .", "For any square matrix $\\mathbf {A}$ and $\\mathbf {B}$ , $\|\|\\mathbf {A} + \\mathbf {B}\|\| \\le \|\|\\mathbf {A}\|\| +\|\|\\mathbf {B}\|\|$ .", "We have $\\begin{aligned}\|\|\\mathbf {A}\|\| - \|\|\\mathbf {B}\|\| \\le \|\|\\mathbf {A} - \\mathbf {B}\|\|\\mbox{\\qquad and \\quad }\|\|\\mathbf {B}\|\| -\|\|\\mathbf {A}\|\|\\le \|\|\\mathbf {B} - \\mathbf {A}\|\|\\end{aligned}$ Then, $\\mid \|\|\\mathbf {A}\|\| -\|\|\\mathbf {B}\|\|\\mid \\le \|\|\\mathbf {A} - \\mathbf {B}\|\|$ Let $\\mathbf {V}_j = {P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace - {E} \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j)\\rbrace $ .", "It follows that, $\\mid \|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| -\|\|{E}\\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\mid \\le \|\|\\mathbf {V}_j\|\|$ It is easy to verify that, $\\mid \|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| - \|\| {E}\\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\|\\mid \\le d_n \|\|\\mathbf {V}_j\|\|_{sup}$ Since $\|\|B_{jm}(\\cdot )\|\|_{sup} \\le 1$ and using Fact 2, we have $\\mbox{var}( B_{jm_1}(Y) B_{jm_2}(X_j)) \\le {E} \\lbrace B_{jm_1}^2(Y) B_{jm_2}^2(X_j) \\rbrace \\le {E} \\lbrace B_{jm_1}^2(Y)\\rbrace \\le C_2d_n^{-1}$ By Bernstein's inequality, for any $\\delta > 0$ , $P\\lbrace \| ({P}_n - {E}) \\lbrace B_{jm_1}(Y) B_{jm_2}(X_j) \\rbrace \| \\ge \\delta /n \\rbrace \\le 2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace $ Therefore, $P\\lbrace \\mid \|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace )\|\| - \|\|{E}\\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\|\\mid \\ge d_n \\delta /n \\rbrace \\le 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace $ Recalling R2, we have, $P\\lbrace \|\| {P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\ge d_n \\delta /n + (c_{12} C_2/d_n)^{1/2} \\rbrace \\le 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace $ By taking $\\delta = c_8 (c_{12} C_2)^{1/2} n d_n^{-3/2}$ , we obtain that for some positive constant $c_4$ , $P\\lbrace \|\|({P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace )\|\| \\ge (c_8 +1) (c_{12} C_2/d_n)^{1/2} \\rbrace \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-2} )$ Next we deal with $\|\| \\widehat{\\mathbf {D}_j}^T \|\|$ .", "Using Bernstein's inequality, we obtain that, $P\\lbrace \|\\widehat{b_{jm}} - b_{jm}\| \\ge \\delta /n \\rbrace \\le 2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace $ Since $ b_{jm} \\ge C_3 k^{-1}$ , by taking $\\delta = C_3 w_1 n d_n ^{-1}$ for $w_1 \\in (0,1)$ , we have that there exists some positive constant $c_5$ such that $P\\lbrace \\widehat{b_{jm}} \\le C_3 (1-w_1) d_n^{-1} \\rbrace \\le 2 \\exp ( -c_5 n d_n^{-1} )$ For $\\mathbf {u} = (u_1, \\ldots , u_{d_n-1})^T \\in R^{d_n-1}$ with $\\sum _{i=1}^{d_n-1} u_i^2 = 1$ , $\\begin{aligned}\\mathbf {u}^T \\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T \\mathbf {u} = \\sum \\limits _{m=1}^{d_n} \\frac{1}{k^2 \\widehat{b_{jm}}^2} \\left( \\sum \\limits _{i=1}^{d_n - 1} u_i z_{im} \\right)^2 \\le \\max \\limits _{m} \\frac{1}{k^2 \\widehat{b_{jm}}^2}\\end{aligned}$ Combing (REF ), (REF ) and (REF ), we have that $\\begin{aligned}P\\lbrace \|\| \\widehat{\\mathbf {D}_j}^T \|\| \\ge C_3^{-1} (1-w_1)^{-1} \\rbrace & \\le P\\lbrace \\max \\limits _{m} \\frac{1}{k \\widehat{b_{jm}}} \\ge C_3^{-1} (1-w_1)^{-1} \\rbrace \\\\& \\le P\\lbrace \\min \\limits _{m} \\widehat{b_{jm}} \\le C_3 (1-w_1) k^{-1} \\rbrace \\\\&\\le 2 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ Combining (REF ), (REF ), and $\|\|\\widehat{\\mathbf {A}_{j0X}} \|\| \\le \|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\|\\widehat{\\mathbf {D}_j}^T\|\|$ , we have $\\begin{aligned}& \\qquad P\\lbrace \|\|\\widehat{\\mathbf {A}_{j0X}} \|\| \\ge (c_8 +1)(c_{12} C_2)^{1/2} d_n^{-1/2} C_3^{-1} (1-w_1)^{-1} \\rbrace \\\\& \\le P\\lbrace \|\|{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\ge (c_8 +1) (c_{12} C_2)^{1/2} d_n^{-1/2} \\rbrace + P\\lbrace \|\| \\widehat{\\mathbf {D}_j}^T \|\| \\ge C_3^{-1} (1-w_1)^{-1} \\rbrace \\\\& \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-2} ) + 2 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ Result in R4 follows by choosing $c_6$ , $c_7$ accordingly.", "R5.", "There exist some positive constants $c_{9}$ , $c_{10}$ such that, for any $\\delta > 0$ , $\\begin{aligned}P\\lbrace \|\|\\widehat{\\mathbf {A}_{j0X}} & - \\mathbf {A}_{j0X}\|\| \\ge c_{9} d_n^2 \\delta ^2/n^2 + c_{10} d_n \\delta /n \\rbrace \\\\ &\\le 8 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + 4 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ It is easy to derive $\\begin{aligned}& \|\|\\widehat{\\mathbf {A}_{j0X}} - \\mathbf {A}_{j0X}\|\| = \|\| {P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \\widehat{\\mathbf {D}_j}^T - {E}\\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace {\\mathbf {D}_j}^T \|\|\\\\& \\le \|\|({P}_n -{E}) \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\| \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j^T\|\| + \|\|{E}\\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\| \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j^T\|\|\\\\& \\qquad \\qquad + \|\|({P}_n -{E}) \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\| {\\mathbf {D}_j}^T \|\|\\end{aligned}$ It is proved in R2 that $\|\| {E} \\lbrace \\mathbf {B}_{j}(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\le (c_{12} C_2/ d_n)^{1/2}$ and that $\|\| {\\mathbf {D}_j}^T \|\| \\le C_3^{-1}$ .", "Combining (REF ) and the fact that $\|\|({P}_n - {E}) \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\le d_n \|\|({P}_n - {E}) \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\|_{sup},$ we have that, $P\\lbrace \|\|({P}_n -{E}) \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\ge d_n \\delta /n \\rbrace \\le 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace .$ For $\\mathbf {u} \\in R^{d_n-1}$ with $\\sum _{i=1}^{d_n-1} u_i^2 = 1$ , $\\begin{aligned}\\mathbf {u}^T (\\widehat{\\mathbf {D}_j} - \\mathbf {D}_j) (\\widehat{\\mathbf {D}_j} - \\mathbf {D}_j) ^T \\mathbf {u} & = \\sum \\limits _{m=1}^{d_n} \\left( \\frac{1}{k \\widehat{b_{jm}}} - \\frac{1}{k b_{jm}} \\right)^2 \\left( \\sum \\limits _{i=1}^{d_n - 1} u_i z_{im} \\right)^2\\\\& \\le C_3^{-2} \\mathop {\\max }\\limits _{m} \\frac{(\\widehat{b_{jm}} - b_{jm})^2}{\\widehat{b_{jm}}^2}\\end{aligned}$ From (REF ), (REF ) and (REF ), we have that, $\\begin{aligned}& P\\lbrace \|\| \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j^T\|\| \\ge C_3^{-2} (1-w_1)^{-1} d_n \\delta /n \\rbrace \\\\& \\le P\\lbrace C_3^{-1} \\mathop {\\max }\\limits _{m} \\frac{\|\\widehat{b_{jm}} - b_{jm}\|}{\\widehat{b_{jm}}}\\ge C_3^{-1} \\frac{\\delta /n}{ C_3 (1-w_1) d_n^{-1}} \\rbrace \\\\& \\le P\\lbrace \\mathop {\\max }\\limits _{m} \|\\widehat{b_{jm}} - b_{jm}\| \\ge \\delta /n \\rbrace + P\\lbrace \\mathop {\\min }\\limits _{m} \\widehat{b_{jm}} \\le C_3 (1-w_1) d_n^{-1} \\rbrace \\\\& \\le 2 d_n \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + 2 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ Therefore, together with (REF ), (REF ), (REF ) and union bound of probability, we have $\\begin{aligned}& P\\lbrace \|\|\\widehat{\\mathbf {A}_{j0X}} - \\mathbf {A}_{j0X}\|\| \\ge \\frac{d_n^2 \\delta ^2/n^2}{C_3^2 (1-w_1) } + \\frac{(c_{12} C_2)^{1/2} d_n^{1/2} \\delta /n}{C_3^2 (1-w_1) } + C_3^{-1} d_n \\delta /n \\rbrace \\\\& \\le 4 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + 4 d_n \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + 4 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ Result in R5 can be obtained by adjusting the values of $c_{9}$ and $c_{10}$ .", "R6.", "For given $c_4$ and $c_5$ , there exist positive constants $c_{15}$ and $ c_{16}$ such that, $\\begin{aligned}P\\lbrace \|\|& \\widehat{\\mathbf {A}_{jXX}}^{-1}\|\| \\ge c_{16} d_n \\rbrace \\\\ & \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} ) + 2 d_n^3 \\exp (-c_{15} n d_n^{-7} ) + 2 d_n^3 \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ Follow the proof in Lemma 5 of [6], we have that, $\|\\lambda _{min}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) - \\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T) \| \\le d_n \|\|\\mathbf {V}_j\|\|_{sup}, \\mbox{ where } \\mathbf {V}_j = \\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j \\mathbf {D}_j^T$ The $(s,m)$ -entry of $\\mathbf {V}_j$ is $\\begin{aligned}\\left(\\mathbf {V}_j \\right) ^{(s,m)} &= \|\\sum \\limits _{i=1}^{d_n} \\frac{z_{si} z_{mi}}{k^2} \\left( \\frac{1}{ \\widehat{b_{ji}}^2} - \\frac{1}{b_{ji}^2} \\right)\| = \|\\sum \\limits _{i=1}^{d_n} \\frac{z_{si} z_{mi}}{k^2 b_{ji}^2} \\left( \\frac{b_{ji}^2 - \\widehat{b_{ji}}^2}{ \\widehat{b_{ji}}^2} \\right)\|\\\\&\\le C_3^{-2} d_n \\max \\limits _{i} \| \\frac{b_{ji}^2 - \\widehat{b_{ji}}^2}{ \\widehat{b_{ji}}^2}\| \\le 2 C_3^{-2} d_n \\max \\limits _{i} \| \\frac{ b_{ji} - \\widehat{b_{ji}}}{ \\widehat{b_{ji}}^2}\|\\end{aligned}$ It is clear that $\|\|\\mathbf {V}_j\|\|_{sup} \\le 2 C_3^{-2} d_n \\max \\limits _{i} \| ( b_{ji} - \\widehat{b_{ji}})/{ \\widehat{b_{ji}}^2}\|$ .", "Together with (REF ) and (REF ) , we have $\\begin{aligned}& P\\lbrace \|\\lambda _{min}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) - \\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T) \| \\ge 2 C_3^{-4}(1-w_1)^{-2} d_n^4 \\delta /n \\rbrace \\\\& \\le P\\lbrace 2 C_3^{-2} d_n^2 \\max \\limits _{i} \| \\frac{ b_{ji} - \\widehat{b_{ji}}}{ \\widehat{b_{ji}}^2}\| \\ge 2 C_3^{-2} d_n^2 \\delta /n \\times {C_3^{-2} (1-w_1)^{-2} d_n^{2}} \\rbrace \\\\& \\le P\\lbrace \\mathop {\\max }\\limits _{m} \|\\widehat{b_{jm}} - b_{jm}\| \\ge \\delta /n \\rbrace + P\\lbrace \\mathop {\\min }\\limits _{m} \\widehat{b_{jm}} \\le C_3 (1-w_1) d_n^{-1} \\rbrace \\\\& \\le 2 d_n \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + 2 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ which indicates that there exists a positive constant $c_{14}$ , $\\begin{aligned}P\\lbrace \|\\lambda _{min}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) - &\\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T) \| \\ge c_{14} d_n^4 \\delta /n \\rbrace \\\\& \\le 2 d_n \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + 2 d_n \\exp (-c_5 n d_n^{-1} )\\end{aligned}$ Due to the facts that $\\begin{aligned}c_{11} k^{-1} \\le \\lambda _{min}(\\mathbf {D}_j {E} \\lbrace & \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \\mathbf {D}_j^T ) \\le \\\\& \\lambda _{max}({E} \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace ) \\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T) \\le c_{12} k^{-1} \\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T)\\end{aligned}$ and that $\\begin{aligned}c_{11} k^{-1} \\lambda _{max}(\\mathbf {D}_j \\mathbf {D}_j^T) \\le \\lambda _{min}({E} \\lbrace \\mathbf {B}_j(X_j)& \\mathbf {B}_j^T(X_j) \\rbrace ) \\lambda _{max}(\\mathbf {D}_j \\mathbf {D}_j^T) \\le \\\\ & \\lambda _{max}(\\mathbf {D}_j {E} \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \\mathbf {D}_j^T ) \\le c_{12} k^{-1}\\end{aligned}$ we have $\\begin{aligned}\\frac{c_{11}}{c_{12}} \\le \\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T) \\le \\lambda _{max}(\\mathbf {D}_j \\mathbf {D}_j^T) \\le \\frac{c_{12}}{c_{11}}\\end{aligned}$ By taking $\\delta = w_2/c_{14} n d_n^{-4} \\times c_{11}/c_{12}$ in (REF ) for any $w_2 \\in (0,1)$ , there exists a positive constant $c_{15}$ such that, $\\begin{aligned}P\\lbrace \|\\lambda _{min}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) - \\lambda _{min}(\\mathbf {D}_j& \\mathbf {D}_j^T) \| \\ge w_2 \\lambda _{min}(\\mathbf {D}_j \\mathbf {D}_j^T) \\rbrace \\\\ & \\le 2 d_n \\exp (-c_{15} n d_n^{-7} ) + 2 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ By following a similar argument in proving inequality (26) in NIS [6], we have, $P\\lbrace \\lambda _{min}^{-1}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) \\ge (c_8 +1)c_{12}/c_{11} \\rbrace \\le 2 d_n \\exp (-c_{15} n d_n^{-7} ) + 2 d_n \\exp ( -c_5 n d_n^{-1} )$ Similarly, it is easy to obtain $\\begin{aligned}P\\lbrace \\lambda _{min}^{-1}({P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace ) \\ge (c_8 +1)c_{11}^{-1} d_n \\rbrace \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} )\\end{aligned}$ Due to the fact that $\\lambda _{max}({\\mathbf {H}}^{-1}) = \\lambda _{min}^{-1}({\\mathbf {H}})$ , we have $\|\|\\widehat{\\mathbf {A}_{jXX}}^{-1}\|\| = \\lambda _{min}^{-1}(\\widehat{\\mathbf {A}_{jXX}}) \\le \\lambda _{min}^{-1}({P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace ) \\hspace{3.61371pt} \\lambda _{min}^{-1}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T)$ Together with (REF ) and (REF ), we can obtain that $\\begin{aligned}& P\\lbrace \|\| \\widehat{\\mathbf {A}_{jXX}}^{-1}\|\| \\ge (c_8 +1)^2 c_{12} c_{11}^{-2} d_n \\rbrace \\\\& \\le P\\lbrace \\lambda _{min}^{-1}({P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace ) \\hspace{3.61371pt} \\lambda _{min}^{-1}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) \\ge (c_8 +1)^2 c_{12} c_{11}^{-2} d_n \\rbrace \\\\& \\le P\\lbrace \\lambda _{min}^{-1}({P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace ) \\ge (c_8 +1)c_{12}/c_{11} \\rbrace + P\\lbrace \\lambda _{min}^{-1}(\\widehat{\\mathbf {D}_j} \\widehat{\\mathbf {D}_j}^T) \\ge (c_8 +1)c_{11}^{-1} d_n \\rbrace \\\\& \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} ) + 2 d_n \\exp (-c_{15} n d_n^{-7} ) + 2 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ Therefore, R6 follows by choosing $c_{16} = (c_8 +1)^2 c_{12} c_{11}^{-2}$ .", "R7.", "For any $\\delta > 0$ , given positive constant $c_4$ , there exists a positive constant $c_{17}$ such that, $P\\lbrace \|\| \\widehat{\\mathbf {A}_{j00}}^{-1/2} - \\mathbf {A}_{j00}^{-1/2}\|\|\\ge c_{17} d_n^{5/2} \\delta /n \\rbrace \\le 2 d_n^2 \\exp ( -c_{4} n d_n^{-3} ) + 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace $ Using perturbation theory [20], it is proved [4] that for some $c_{18} > 0$ , $\|\| \\widehat{\\mathbf {A}_{j00}}^{-1/2} - \\mathbf {A}_{j00}^{-1/2}\|\| \\le c_{18} \\tilde{\\gamma }^{-3/2} \|\|\\widehat{\\mathbf {A}_{j00}} - \\mathbf {A}_{j00}\|\|$ where $\\tilde{\\gamma }$ is the minimum of the smallest eigenvalues of $\\widehat{\\mathbf {A}_{j00}}$ and $\\mathbf {A}_{j00}$ .", "$\\tilde{\\gamma }$ is positive by definition.", "Therefore, $\\tilde{\\gamma }^{-1} = \\max \\lbrace \\lambda _{min}^{-1} (\\widehat{\\mathbf {A}_{j00}} ), \\lambda _{min}^{-1} (\\mathbf {A}_{j00} ) \\rbrace = \\max \\lbrace \|\|\\widehat{\\mathbf {A}_{j00}}^{-1}\|\|, \|\|\\mathbf {A}_{j00}^{-1}\|\| \\rbrace $ From Fact 3 and R3, we have, $& c_{12}^{-1} d_n \\le \|\|\\mathbf {A}_{j00}^{-1}\|\| \\le c_{11}^{-1} d_n\\\\& P\\lbrace \|\|[{P}_n \\lbrace \\mathbf {B}_j(Y) \\mathbf {B}_j^T(Y) \\rbrace ]^{-1}\|\| \\ge (c_8 +1) c_{11}^{-1} d_n \\rbrace \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} )$ Combining (REF ) and () yields $P\\lbrace \\tilde{\\gamma }^{-1} \\ge \\max \\left( (c_8 +1) c_{11}^{-1} d_n, c_{11}^{-1} d_n \\right) \\rbrace \\le 2 d_n^2 \\exp (-c_{4} n d_n^{-3} )$ which is, $P \\lbrace \\tilde{\\gamma }^{-1} \\ge (c_8 +1) c_{11}^{-1} d_n \\rbrace \\le 2 d_n^2 \\exp ( -c_{4} n d_n^{-3} )$ Additionally, as proved in equation (33) in [6], we have large deviation bound for $\|\|({P}_n - {E}) \\lbrace \\mathbf {B}_j(Y)\\mathbf {B}_j^T(Y) \\rbrace \|\|$ , $\\begin{aligned}P\\lbrace \|\|({P}_n - {E}) \\lbrace \\mathbf {B}_j(Y)\\mathbf {B}_j^T(Y) \\rbrace \|\| \\ge d_n \\delta /n \\rbrace \\le 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace \\end{aligned}$ By (REF ), (REF ), (REF ) and under the union bound of probability, we have that, $\\begin{aligned}& P\\lbrace \|\| \\widehat{\\mathbf {A}_{j00}}^{-1/2} - \\mathbf {A}_{j00}^{-1/2}\|\| \\ge c_{18} (c_8 +1)^{3/2} c_{11}^{-3/2} d_n^{5/2}\\delta /n \\rbrace \\\\& \\le P\\lbrace c_{18} \\tilde{\\gamma }^{-3/2} \|\|\\widehat{\\mathbf {A}_{j00}} - \\mathbf {A}_{j00}\|\| \\ge c_{18} (c_8 +1)^{3/2} c_{11}^{-3/2} d_n^{3/2} \\hspace{3.61371pt} d_n\\delta /n \\rbrace \\\\&\\le P\\lbrace \\tilde{\\gamma }^{-1} \\ge (c_8 +1) c_{11}^{-1} d_n \\rbrace + P\\lbrace \|\| \\widehat{ \\mathbf {A}_{j00}} - \\mathbf {A}_{j00}\|\| \\ge d_n\\delta /n \\rbrace \\\\& \\le 2 d_n^2 \\exp ( -c_{4} n d_n^{-3} ) + 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace \\end{aligned}$ Therefore, R7 follows by choosing $c_{17} = c_{18} (c_8 +1)^{3/2} c_{11}^{-3/2}$ .", "R8.", "For any $\\delta > 0$ , given positive constant $c_4$ , there exist a positive constant $c_{19}$ such that, $\\begin{aligned}P\\lbrace \|\| \\widehat{\\mathbf {A}_{jXX}}^{-1} - \\mathbf {A}_{jXX}^{-1}\|\|& \\ge c_{19} (d_n^5 \\delta ^3/n^3 + d_n^3 \\delta /n) \\rbrace \\le 8 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace + \\\\ & 4 d_n^2 \\exp ( -c_{4} n d_n^{-3} ) +2 d_n \\exp (-c_{15} n d_n^{-7} ) + 6 d_n \\exp ( -c_5 n d_n^{-1} )\\end{aligned}$ It's obvious that $\\begin{aligned}\|\|\\widehat{\\mathbf {A}_{jXX}}^{-1} - \\mathbf {A}_{jXX}^{-1}\|\| \\le \|\| \\mathbf {A}_{jXX}^{-1}\|\| \\hspace{3.61371pt} \|\|\\mathbf {A}_{jXX} - \\widehat{\\mathbf {A}_{jXX}} \|\| \\hspace{3.61371pt} \|\|\\widehat{\\mathbf {A}_{jXX}}^{-1}\|\|\\end{aligned}$ and that $\\begin{aligned}&\|\|\\widehat{\\mathbf {A}_{jXX}} - \\mathbf {A}_{jXX}\|\| = \|\| \\widehat{\\mathbf {D}_j} {P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j {E} \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \\mathbf {D}_j^T \|\|\\\\& \\le \|\|\\widehat{\\mathbf {D}_j} - \\mathbf {D}_j\|\| \\hspace{3.61371pt} \|\|({P}_n -{E}) \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\| \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j^T\|\| + 2 \|\|{P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\times \\\\& \\qquad \|\| \\widehat{\\mathbf {D}_j}^T - \\mathbf {D}_j^T\|\| + \|\| \\mathbf {D}_j^T \|\| \\hspace{3.61371pt} \|\|({P}_n -{E}) \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\hspace{3.61371pt} \|\| \\mathbf {D}_j \|\|\\end{aligned}$ From the similar reasoning in proving (REF ) and (REF ), it is easy to obtain that $P\\lbrace \|\|{P}_n \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\ge (c_8 +1)c_{13} d_n^{-1} \\rbrace \\le 2 d_n^2 \\exp ( -c_4 n d_n^{-3} )$ $P\\left(\|\|({P}_n -{E}) \\lbrace \\mathbf {B}_j(X_j) \\mathbf {B}_j^T(X_j) \\rbrace \|\| \\ge d_n \\delta /n \\right) \\le 2 d_n^2 \\exp \\lbrace - \\frac{\\delta ^2}{2(C_2 n d_n^{-1} + 2\\delta /3)} \\rbrace $ With $c_{19}$ chosen properly, results in R8 follows by combining Fact 3, (REF ), (REF ), (REF ), (REF ), (REF ), (REF ) and the fact $\|\|\\mathbf {D}_j^T\|\| < C_3^{-1}$ ." ], [ "Proof of Theorem 1", "Proof of Theorem 1.", "Recall that $\\lambda _{j1}^ = \|\| \\mathbf {A}_{j00}^{-1/2} \\mathbf {A}_{j0X} \\mathbf {A}_{jXX}^{-1} \\mathbf {A}_{jX0} \\mathbf {A}_{j00}^{-1/2}\|\|$ and that $\\widehat{\\lambda _{j1}^} = \|\| \\widehat{\\mathbf {A}_{j00}}^{-1/2} \\widehat{\\mathbf {A}_{j0X}} \\widehat{\\mathbf {A}_{jXX}}^{-1} \\widehat{\\mathbf {A}_{j0X}}^T \\widehat{\\mathbf {A}_{j00}}^{-1/2}\|\|$ Let $\\mathbf {a} = \\mathbf {A}_{j00}^{-1/2}$ , $\\mathbf {b} = \\mathbf {A}_{j0X}$ , $\\mathbf {H}= \\mathbf {A}_{jXX}^{-1}$ , $\\mathbf {a_n} = \\widehat{\\mathbf {A}_{j00}}^{-1/2}$ , $\\mathbf {b_n} = \\widehat{\\mathbf {A}_{j0X}} $ , $\\mathbf {H_n} =\\widehat{\\mathbf {A}_{jXX}}^{-1}$ , $\\begin{aligned}& \\widehat{\\lambda _{j1}^} - \\lambda _{j1}^* = \|\|\\mathbf {a_n}^T \\mathbf {b_n}^T \\mathbf {H_n} \\mathbf {b_n} \\mathbf {a_n}\|\| - \|\|\\mathbf {a}^T \\mathbf {b}^T \\mathbf {H} \\mathbf {b} \\mathbf {a}\|\|\\\\& \\le \|\|(\\mathbf {a}_n - \\mathbf {a})^T \\mathbf {b}_n^T \\mathbf {H}_n \\mathbf {b}_n (\\mathbf {a}_n - \\mathbf {a})\|\| + 2 \|\| (\\mathbf {a}_n - \\mathbf {a})^T \\mathbf {b}_n^T \\mathbf {H}_n \\mathbf {b}_n \\mathbf {a} \|\| + \|\|\\mathbf {a}^T (\\mathbf {b}_n^T \\mathbf {H}_n \\mathbf {b}_n - \\mathbf {b}^T \\mathbf {H} \\mathbf {b}) \\mathbf {a}\|\| \\\\& \\qquad \\triangleq S_1 + S_2 + S_3\\end{aligned}$ We denote the terms in r.h.s as $S_1$ , $S_2$ and $S_3$ respectively.", "Furthermore, we let the r.h.s of inequalities (REF ),(REF ),(REF ),(REF ),(REF ) as $Q_4$ , $Q_5$ , $Q_6$ , $Q_7$ , $Q_8$ .", "Note that $S_1 \\le \|\|\\mathbf {a}_n - \\mathbf {a}\|\|^2 \\hspace{3.61371pt} \|\|\\mathbf {b}_n\|\|^2 \\hspace{3.61371pt} \|\|\\mathbf {H}_n\|\|$ By (REF ),(REF ),(REF ), we have that there exist a positive constant $c_{20}$ such that, $\\begin{aligned}P\\lbrace S_1 \\ge c_{20} d_n^5 \\delta ^2/n^2\\rbrace \\le Q_4 + Q_6 + Q_7\\end{aligned}$ As to $S_2$ , $S_2 \\le \|\|\\mathbf {a}_n - \\mathbf {a}\|\| \\hspace{3.61371pt} \|\|\\mathbf {b}_n\|\|^2 \\hspace{3.61371pt} \|\|\\mathbf {H}_n\|\| \\hspace{3.61371pt} \|\|\\mathbf {a}\|\|$ By (REF ),(REF ),(REF ),(REF ), we have that there exist a positive constant $c_{21}$ such that, $\\begin{aligned}P\\lbrace S_2 \\ge c_{21} d_n^3 \\delta /n \\rbrace \\le Q_4 + Q_6 + Q_7\\end{aligned}$ As to $S_3$ , $\\begin{aligned}S_3 &\\le \|\|\\mathbf {a}\|\|^2 \\hspace{3.61371pt} \|\|\\mathbf {b}_n^T \\mathbf {H}_n B_n - \\mathbf {b}^T \\mathbf {H} \\mathbf {b}\|\|\\\\& \\le \|\|\\mathbf {a}\|\|^2 (\|\|(\\mathbf {b}_n - \\mathbf {b})^T \\mathbf {H}_n (\\mathbf {b}_n - \\mathbf {b})\|\| + 2 \|\| (\\mathbf {b}_n - \\mathbf {b})^T \\mathbf {H}_n \\mathbf {b} \|\| + \|\|\\mathbf {b}^T (\\mathbf {H}_n - \\mathbf {H}) \\mathbf {b}\|\|)\\\\& \\qquad \\triangleq \|\|\\mathbf {a}\|\|^2 (S_{31} + 2S_{32} + S_{33})\\end{aligned}$ Note that $S_{31} \\le \|\|\\mathbf {b}_n - \\mathbf {b}\|\|^2 \\hspace{3.61371pt} \|\|\\mathbf {H}_n\|\|$ By (REF ),(REF ), we have that there exist a positive constant $c_{22}$ such that, $\\begin{aligned}P\\lbrace S_{31} \\ge c_{22} d_n^5 (\\delta ^2/n^2 +\\delta /n)^2\\rbrace \\le Q_5 + Q_6\\end{aligned}$ As to $S_{32}$ , $S_{32} \\le \|\|\\mathbf {b}_n - \\mathbf {b}\|\| \\hspace{3.61371pt} \|\|\\mathbf {H}_n\|\| \\hspace{3.61371pt} \|\|\\mathbf {b}\|\|$ By (REF ),(REF ),(REF ),(REF ), we have that there exist a positive constant $c_{23}$ such that, $\\begin{aligned}P\\lbrace S_{32} \\ge c_{23} d_n^{5/2} (\\delta ^2/n^2 +\\delta /n) \\rbrace \\le Q_5 + Q_6\\end{aligned}$ As to $S_{33}$ , $S_{33} \\le \|\|\\mathbf {b}\|\|^2 \\hspace{3.61371pt} \|\|\\mathbf {H}_n - \\mathbf {H}\|\|$ By (REF ),(REF ), we have that there exist a positive constant $c_{24}$ such that, $\\begin{aligned}P\\lbrace S_{33} \\ge c_{24}(d_n^4 \\delta ^3/n^3 + d_n^2 \\delta /n) \\rbrace \\le Q_8\\end{aligned}$ Combining (REF ),(REF ),(REF ),(REF ),(REF ), we have $\\begin{aligned}P\\lbrace S_3 \\ge c_{22} & d_n^6 (\\delta ^2/n^2 +\\delta /n)^2 + c_{23} d_n^{7/2}(\\delta ^2/n^2 +\\delta /n) + c_{24}(d_n^5 \\delta ^3/n^3 + d_n^3 \\delta /n)\\rbrace \\\\& \\le 2 Q_5 + 2 Q_6 + Q_8\\end{aligned}$ Define $\\varsigma (d_n, \\delta ) = c_{20} d_n^5 \\delta ^2/n^2 + c_{21} d_n^3 \\delta /n + c_{22} d_n^6 (\\delta ^2/n^2 +\\delta /n)^2 + c_{23} d_n^{7/2} (\\delta ^2/n^2 +\\delta /n) + c_{24}(d_n^5 \\delta ^3/n^3 + d_n^{3} \\delta /n)$ .", "Then from (REF ),(REF ),(REF ),(REF ), we have that due to symmetry, $\\begin{aligned}P\\lbrace \|\\widehat{\\lambda _{j1}^} - \\lambda _{j1}^\| \\ge \\varsigma (d_n, \\delta ) \\rbrace \\le 4 Q_4 + 4 Q_5 + 8 Q_6 + 4 Q_7 + 2 Q_8\\end{aligned}$ By properly choosing the value of $\\delta $ (i.e., taking $\\delta = {c_2}{(c_{22} +c_{23})^{-1}} d_n^{-5/2} n ^{1-2\\kappa }$ ), we can make $\\varsigma (d_n, \\delta ) = c_2 d_n n^{-2\\kappa }$ , for any $c_2>0$ .", "Then, we have $\\begin{aligned}P( \|\\widehat{\\lambda _{j1}^} - \\lambda _{j1}^\| \\ge c_2 d_n n^{-2\\kappa } ) \\le \\mathcal {O} \\left( d_n^2 \\exp ( - c_3 n^{1-4\\kappa } d_n^{-4} ) + d_n \\exp ( -c_4 n d_n^{-7} ) \\right)\\end{aligned}$ The first part of Theorem 1 follows via the union bound of probability.", "To prove the second part, we define a event $\\mathcal {A}_n \\equiv \\lbrace \\mathop {max}\\limits _{j \\in \\mathcal {D}}\|\\widehat{\\lambda _{j1}^} - \\lambda _{j1}^\|\\le c_1 \\xi d_n n^{-2\\kappa } /2 \\rbrace $ By Lemma REF , we have $\\widehat{\\lambda _{j1}^} \\ge c_1 \\xi d_n n^{-2\\kappa } /2 , \\forall j \\in \\mathcal {D}$ Thus, by choosing $\\nu _n = c_5 d_n n^{-2\\kappa }$ with $c_5 \\le c_1 \\xi /2$ .", "We have that $\\mathcal {D} \\subseteq \\widehat{\\mathcal {D}_{\\nu _n}}$ .", "Therefore, $P (\\mathcal {A}_n^c) \\le \\mathcal {O} \\left( s \\lbrace d_n^2 \\exp \\lbrace - c_3 n^{1-4\\kappa } d_n^{-4} \\rbrace + d_n \\exp ( -c_4 n d_n^{-7} ) \\rbrace \\right)$ Then the probability bound for the second part of Theorem 1 is attained." ], [ "Proof sketch of Theorem 2", "Proof of Theorem 2.", "From subsection REF , we have that $ \\lambda _{j1}^ = {E} (\\phi ^{2}_{nj}) $ and $ \\widehat{\\lambda _{j1}^} = {P_n} (\\phi ^{2}_{nj}) $ .", "From equation (REF ), after obtaining $\\theta ^_{nj}$ where $\\mbox{Var}(\\theta ^{}_{nj}) = 1$ , $\\phi ^_{nj}$ can be obtained via the following optimization problem.", "$\\begin{aligned}& \\underset{ \\phi _{nj} \\in \\mathcal {S}_n}{\\text{arg min}}& & {E}[\\lbrace \\theta ^_{nj}(Y) - \\phi _{nj}(X_j)\\rbrace ^2], \\mbox{ where $\\phi _{nj}(X_j) = {\\eta }_j^T \\psi _j(X_j)$.", "}\\end{aligned}$ Therefore, $\\phi ^_{nj} = \\psi _j^T E \\lbrace {\\psi _j} {\\psi _j}^T\\rbrace ^{-1} E\\psi _j \\theta ^{}_{nj} $ .", "We notice that the only difference between our proof and the proof of Theorem 2 in [6] is the role of $Y$ .", "As MC-SIS essentially uses transformation of $Y$ , we can not deal directly with $Y$ .", "However, from the formulation above, $\\theta _{nj}^$ here plays the same role as $Y$ in [6].", "With this connection, our proof follows immediately by replacing $Y$ in the proof of Theorem 2 in [6] with $\\theta _{nj}^*$ ." ] ]	1403.0048
[ [ "The honeycomb lattice with multi-orbital structure: topological and\n quantum anomalous Hall insulators with large gaps" ], [ "Abstract We construct a minimal four-band model for the two-dimensional (2D) topological insulators and quantum anomalous Hall insulators based on the $p_x$- and $p_y$-orbital bands in the honeycomb lattice.", "The multiorbital structure allows the atomic spin-orbit coupling which lifts the degeneracy between two sets of on-site Kramers doublets $j_z=\\pm\\frac{3}{2}$ and $j_z=\\pm\\frac{1}{2}$.", "Because of the orbital angular momentum structure of Bloch-wave states at $\\Gamma$ and $K(K^\\prime)$ points, topological gaps are equal to the atomic spin-orbit coupling strengths, which are much larger than those based on the mechanism of the $s$-$p$ band inversion.", "In the weak and intermediate regime of spin-orbit coupling strength, topological gaps are the global gap.", "The energy spectra and eigen wave functions are solved analytically based on Clifford algebra.", "The competition among spin-orbit coupling $\\lambda$, sublattice asymmetry $m$ and the N\\'eel exchange field $n$ results in band crossings at $\\Gamma$ and $K (K^\\prime)$ points, which leads to various topological band structure transitions.", "The quantum anomalous Hall state is reached under the condition that three gap parameters $\\lambda$, $m$, and $n$ satisfy the triangle inequality.", "Flat bands also naturally arise which allow a local construction of eigenstates.", "The above mechanism is related to several classes of solid state semiconducting materials." ], [ "Introduction", "The two-dimensional (2D) quantum Hall effect [1] is among the early examples of topological states of matter whose magnetic band structure is characterized by the first Chern number [2], [3], [4], [5].", "Later on, quantum anomalous Hall (QAH) insulators were proposed with Bloch band structures [5].", "Insulators with nontrivial band topology were also generalized into time-reversal (TR) invariant systems, termed topological insulators (TIs) in both 2D and 3D, which have become a major research focus in contemporary condensed matter physics [6], [7], [8].", "The topological index of TR invariant TIs is no longer just integer valued, but $\\mathbb {Z}_2$ valued, in both 2D and 3D [9], [10], [11], [12], [13], [14], [15].", "In 4D, it is the integer-valued second Chern number [16], [12].", "Various 2D and 3D TI materials were predicted theoretically and observed experimentally [11], [17], [18], [19], [20], [21].", "They exhibit gapless helical 1D edge modes and 2D surface modes through transport and spectroscopic measurements.", "Solid state materials with the honeycomb lattice structure (e.g., graphene) are another important topic of condensed matter physics [22], [23], [24].", "There are several proposals of QAH model in the honeycomb lattice[25], [26].", "As a TR invariant doublet of Haldane's QAH model [9], [27], the celebrated 2D Kane-Mele model was originally proposed in the context of graphene-like systems with the $p_z$ band.", "However, the atomic level spin-orbit (SO) coupling in graphene does not directly contribute to opening the topological band gap [28].", "Because of the single band structure and the lattice symmetry, the band structure SO coupling is at the level of a high-order perturbation theory and thus is tiny.", "Recently, the $p_x$ - and $p_y$ -orbital physics in the honeycomb lattice has been systematically investigated in the context of ultracold-atom optical lattices [29], [30], [31], [32], [33], [34], [35].", "The optical potential around each lattice potential minimum is locally harmonic.", "The $s$ - and $p$ -orbital bands are separated by a large band gap, and thus the hybridization between them is very small.", "The $p_z$ -orbital band can also be tuned to high energy by imposing strong laser beams along the $z$ direction.", "Consequently, we can have an ideal $p_x$ - and $p_y$ -orbital system in the artificial honeycomb optical lattice.", "Such an orbitally active system provides a great opportunity to investigate the interplay between nontrivial band topology and strong correlations, which is fundamentally different from graphene [29], [31], [32].", "Its band structure includes not only Dirac cones but also two additional narrow bands which are exactly flat in the limit of vanishing $\\pi $ bonding.", "Inside the flat bands, due to the vanishing kinetic energy scale, nonperturbative strong correlation effects appear, such as the Wigner crystallization of spinless fermions [29], [31] and ferromagnetism [33] of spinful fermions as exact solutions.", "Very recently, the honeycomb lattice for polaritons has been fabricated [36].", "Both the Dirac cone and the flat dispersion for the $p_x/p_y$ orbital bands have been experimentally observed.", "The band structure can be further rendered topologically nontrivial by utilizing the existing experimental technique of the on-site rotation around each trap center [37].", "This provides a natural way to realize the QAH effect (QAHE) as proposed in Refs.", "wu2008 and zhang2011a, and the topological gaps are just the rotation angular velocity [32], [35].", "In the Mott-insulating states, the frustrated orbital exchange can be described by a novel quantum 120$^\\circ $ model [30], whose classic ground states map to all the possible loop configurations in the honeycomb lattice.", "The $p_x$ - and $p_y$ -orbital structure also enables unconventional $f$ -wave Cooper pairing even with conventional interactions exhibiting flat bands of zero energy Majorana edge modes along boundaries parallel to gap nodal directions [34].", "The $p_x$ - and $p_y$ -orbital structures have also been studied very recently in several classes of solid state semiconducting materials including fluoridated tin film [38], [39], [26], functionalized germanene systems [40], Bi$X$ /Sb$X$ ($X$ =H,F,Cl,Br) systems [41], [42], and in organic materials [43], [44], [45].", "All these materials share the common feature of the active $p_x$ and $p_y$ orbitals in the honeycomb lattice, enabling a variety of rich structures of topological band physics.", "The most striking property is the prediction of the large topological band gap which can even exceed room temperature.", "In the literature, a common mechanism giving rise to topological band gaps is the band inversion, which typically applies for two bands with different orbital characters, say, the $s$ -$p$ bands.", "However, although band inversion typically occurs in systems with strong SO coupling, the SO coupling does not directly contribute to the value of the gap.", "The band inversion would lead to gap closing at finite momenta in the absence of the $s$ -$p$ hybridization, and the $s$ -$p$ hybridization reopens the gap whose nature becomes topological.", "The strength of the hybridization around the $\\Gamma $ point linearly depends on the magnitude of the momenta, in the spirit of the $k\\cdot p$ perturbation theory, which is typically small.", "This is why in usual topological insulators based on band inversion, in spite of considerable SO coupling strengths, the topological gap values are typically small.", "On the other hand, as for the single band systems in the honeycomb lattice such as graphene, the effect from the atomic level SO coupling to the band structure is also tiny, as a result of the high-order perturbation theory.", "In the model presented in this paper, here are only $p$ orbitals.", "The two-sublattice structure and the $p_x/p_y$ -orbital configuration together greatly enhance the effect of SO coupling, as illustrated in Fig.", "REF .", "The atomic-scale SO coupling directly contributes to the opening of the topological gap at the $K(K^{\\prime })$ point between bands 2 and 3, and that at the $\\Gamma $ point between bands 1 and 2.", "Since the atomic SO coupling can be very large, the topological band gap can even reach the level of $0.3\\,eV$ according to the estimation in Ref. si2014a.", "In this article, we construct a minimal four-band model to analyze the topological properties based on the $p_x$ - and $p_y$ -orbital structure in the honeycomb lattice.", "The eigen energy spectra and wave functions can be analytically solved with the help of Clifford $\\Gamma $ matrices.", "The atomic SO coupling lifts the degeneracy between two on-site Kramers pairs with $j_z=\\pm \\frac{3}{2}$ and $j_z=\\pm \\frac{1}{2}$ .", "As explained in the preceding paragraph, the topological gap in this class of systems is extraordinary large.", "In the weak and intermediate regime of spin-orbit coupling strength, the topological gaps are the global gap.", "The lattice asymmetry and the SO coupling provide two different gap opening mechanisms, and their competition leads to a variety of topological band structures.", "With the introduction of both the sublattice anisotropy and the Néel exchange field, the system can become a large gap QAH insulator.", "The article is organized as follows.", "The four-band model for the $p_x$ - and $p_y$ -orbital system in the honeycomb lattice is constructed in Sec. .", "The symmetry analysis is presented in Sec. .", "In Sec.", ", the analytic solutions of energy spectra and eigen wave functions are presented.", "The study of band topology and band crossing is presented in Sec. .", "Effective two-band models are constructed around high-symmetry points near band crossings in Sec. .", "The mechanism of large topological band gap is explained in Sec. .", "We add the Néel exchange field term in Sec.", ", and investigate how to get a large gap QAH insulator.", "Conclusions are presented in Sec.", "." ], [ "The $p_x$ and {{formula:20f1460d-8a08-43d8-9b9d-98064afa5e89}} band Hamiltonian", "The two sublattices of the honeycomb lattice are denoted $A$ and $B$ .", "The bonding part of the Hamiltonian is $H_0&=&t_{\\parallel } \\sum _{ \\vec{r} \\in A,s} \\big \\lbrace p^\\dagger _{i,s}(\\vec{r}) p_{i,s}(\\vec{r} +a \\hat{e}_i)+{\\rm H.c.} \\big \\rbrace \\nonumber \\\\&-&t_{\\perp } \\sum _{ \\vec{r} \\in A,s} \\big \\lbrace p^{\\prime \\dagger }_{i,s}(\\vec{r})p^\\prime _{i,s}(\\vec{r}+a\\hat{e}_i) + {\\rm H.c.} \\big \\rbrace ,$ where $s=\\uparrow ,\\downarrow $ represents two eigenstates of spin $s_z$ ; $\\hat{e}_{1,2}=\\pm \\frac{\\sqrt{3}}{2}\\hat{e}_x+\\frac{1}{2}\\hat{e}_y$ and $\\hat{e}_3=-\\hat{e}_y$ are three unit vectors from one $A$ site to its three neighboring $B$ sites; $a$ is the nearest neighbor bond length; $p_i\\equiv (p_x\\hat{e}_x+p_y\\hat{e}_y)\\cdot \\hat{e}_i$ and $p^\\prime _i\\equiv (-p_x\\hat{e}_y+p_y\\hat{e}_x)\\cdot \\hat{e}_i$ are the projections of the $p$ orbitals parallel and perpendicular to the bond direction $\\hat{e}_i$ for $i=1,\\cdots ,3$ , respectively; $t_\\parallel $ and $t_\\perp $ are the corresponding $\\sigma $ - and $\\pi $ -bonding strengths, respectively.", "Typically speaking, $t_\\perp $ is much smaller than $t_\\parallel $ .", "The signs of the $\\sigma $ - and $\\pi $ -bonding terms are opposite to each other because of the odd parity of $p$ -orbitals.", "The $p_z$ orbital is inactive because it forms $\\sigma $ bonding with halogen atoms or the hydrogen atom.", "There exists the atomic SO coupling $\\vec{s} \\cdot \\vec{L}$ on each site.", "However, under the projection into the $p_x$ - and $p_y$ -orbital states, there are only four on-site single-particle states.", "They can be classified into two sets of Kramers doublets: $p^\\dagger _{+,\\uparrow }\|0\\rangle $ and $p_{-,\\downarrow }^\\dagger \|0\\rangle $ with $j_z=\\pm \\frac{3}{2}$ , and $p_{+,\\downarrow }^\\dagger \|0\\rangle $ and $p_{-,\\uparrow }^\\dagger \|0\\rangle $ with $j_z=\\pm \\frac{1}{2}$ , where $p^\\dagger _{\\pm ,s}=\\frac{1}{\\sqrt{2}} (p^\\dagger _{x,s}\\pm i p^\\dagger _{y,s})$ are the orbital angular momentum $L_z$ eigenstates and $j_z$ is the $z$ component of total angular momentum.", "These four states cannot be mixed under $j_z$ conservation, and thus only the $s_z L_z$ term survives which splits the degeneracy between the two sets of Kramers doublets.", "The SO coupling is modeled as $H_{so}=-\\lambda \\sum _{\\vec{r},\\sigma ,s} \\sigma \\,s \\,p^\\dagger _{\\sigma ,s}(\\vec{r})p_{\\sigma ,s} (\\vec{r}),$ where $\\sigma =\\pm $ refers to the orbital angular momentum number $L_z$ , $s=\\pm $ corresponds to the eigenvalues of $s_z=\\uparrow ,\\downarrow $ , and $\\lambda $ is the SO coupling strength.", "For completeness, we also add the sublattice asymmetry term $H_m&=&m\\Big \\lbrace \\sum _{\\vec{r} \\in A,\\sigma ,s}p^\\dagger _{\\sigma ,s}(\\vec{r}) p_{\\sigma ,s} (\\vec{r})-\\sum _{\\vec{r} \\in B,\\sigma ,s}p^\\dagger _{\\sigma ,s}(\\vec{r}) p_{\\sigma ,s} (\\vec{r})\\Big \\rbrace .\\nonumber \\\\$ In Sec.", ", we will consider the QAH state based on this system by adding the following time-reversal (TR) symmetry breaking Néel exchange term $H_n&=&n\\Big \\lbrace \\sum _{\\vec{r} \\in A,\\sigma ,s}s\\,p^\\dagger _{\\sigma ,s}(\\vec{r}) p_{\\sigma ,s} (\\vec{r}) \\nonumber \\\\&-&\\sum _{\\vec{r} \\in B,\\sigma ,s} s\\,p^\\dagger _{\\sigma ,s}(\\vec{r}) p_{\\sigma ,s} (\\vec{r}) \\Big \\rbrace .$ where $n$ is the Néel exchange field strength.", "Before Sec.", ", we only consider the Hamiltonian $H_0+H_{so}+H_{m}$ without the Néel exchange term." ], [ "Symmetry properties", "One key observation is that electron spin $s_z$ is conserved for the total Hamiltonian $H_0+H_{so}+H_m$ .", "We will analyze the band structure in the sector with $s=\\uparrow $ , and that with $s=\\downarrow $ can be obtained by performing time-reversal (TR) transformation.", "$H_0+H_{so}$ is a TR doubled version of the QAH model proposed in ultracold fermion systems in honeycomb optical lattices [32].", "In the sector with $s=\\uparrow $ , we introduce the four-component spinor representation in momentum space defined as $\\psi _{\\uparrow \\tau \\sigma }(\\vec{k})&=&(\\psi _{\\uparrow ,A,+}(\\vec{k}),\\psi _{\\uparrow ,B,+}(\\vec{k}), \\nonumber \\\\&&\\psi _{\\uparrow ,A,-}(\\vec{k}), \\psi _{\\uparrow ,B,-}(\\vec{k}))^T,$ where two sublattice components are denoted $A$ and $B$ .", "The doublet of orbital angular momentum and that of the sublattice structure are considered as two independent pseudospin degrees of freedom, which are denoted by two sets of Pauli matrices as $\\sigma _{1,2,3}$ and $\\tau _{1,2,3}$ , respectively.", "Unlike $s_z$ , these two pseudospins are not conserved.", "The nearest neighbor hopping connects $A$ -$B$ sublattices, which does not conserve the orbital angular momentum due to orbital anisotropy in lattice systems.", "The Hamiltonian $H_{\\uparrow }(\\vec{k})$ can be conveniently represented as $H_{\\uparrow }(\\vec{k})&=&h_{03} 1_\\tau \\otimes \\sigma _3+ h_{30} \\tau _3 \\otimes 1_\\sigma +h_{10}(\\vec{k})\\tau _1 \\otimes 1_\\sigma \\nonumber \\\\&+&h_{20} (\\vec{k}) \\tau _2 \\otimes 1_\\sigma +h_{11} (\\vec{k}) \\tau _1\\otimes \\sigma _1 + h_{22} (\\vec{k})\\tau _2\\otimes \\sigma _2 \\nonumber \\\\&+&h_{21} (\\vec{k}) \\tau _2\\otimes \\sigma _1+ h_{12} (\\vec{k}) \\tau _1\\otimes \\sigma _2,$ with the expressions of $h_{03}&=&-\\lambda , \\ \\ \\, \\ \\ \\, h_{30}=m, \\nonumber \\\\h_{10}&=& t_1\\sum _{i=1}^3 \\cos (\\vec{k} \\cdot \\hat{e}_i), \\nonumber \\\\h_{20}&=& -t_1\\sum _{i=1}^3 \\sin (\\vec{k} \\cdot \\hat{e}_i), \\nonumber \\\\h_{11}&=& t_2\\sum _{i=1}^3 \\cos (\\vec{k} \\cdot \\hat{e}_i) \\cos 2\\theta _i, \\nonumber \\\\h_{22}&=& -t_2\\sum _{i=1}^3 \\sin (\\vec{k} \\cdot \\hat{e}_i) \\sin 2\\theta _i, \\nonumber \\\\h_{21}&=& -t_2\\sum _{i=1}^3 \\sin (\\vec{k} \\cdot \\hat{e}_i) \\cos 2\\theta _i, \\nonumber \\\\h_{12}&=& t_2\\sum _{i=1}^3 \\cos (\\vec{k} \\cdot \\hat{e}_i) \\sin 2\\theta _i,$ where $t_{1,2}=\\frac{1}{2}(t_\\parallel \\pm t_\\perp )$ and $\\theta _i=\\frac{1}{6}\\pi ,\\frac{5}{6}\\pi $ , $\\frac{3}{2}\\pi $ are the azimuthal angles of the bond orientation $\\hat{e}_{i}$ for $i=1,2$ and 3, respectively.", "For the sector with $s=\\downarrow $ , the four-component spinors $\\psi _{\\downarrow }$ are constructed as $\\psi _{\\downarrow \\tau \\sigma }(\\vec{k})=(\\psi _{\\downarrow ,A,+}(\\vec{k}),\\psi _{\\downarrow ,B,+}(\\vec{k}), \\psi _{\\downarrow ,A,-}(\\vec{k}),\\psi _{\\downarrow ,B,-}(\\vec{k}))^T$ .", "Under this basis, $H_{\\downarrow }(\\vec{k})$ has the same matrix form as that of $H_{\\uparrow }(\\vec{k})$ except we flip the sign of $\\lambda $ in the $h_{03}$ term.", "Next we discuss the symmetry properties of $H_{\\uparrow }(\\vec{k})$ .", "We first consider the case of $m=0$ , i.e., in the absence of the lattice asymmetry.", "$H_{\\uparrow }(\\vec{k})$ satisfies the parity symmetry defined as $P H_\\uparrow (\\vec{k}) P^{-1}= H_\\uparrow (-\\vec{k}),$ with $P=\\tau _1\\otimes 1_\\sigma $ .", "$H_{\\uparrow }(\\vec{k})$ also possesses the particle-hole symmetry $C^\\prime H_\\uparrow (\\vec{k}) (C^{\\prime })^{-1}=-H_\\uparrow ^(-\\vec{k}),$ where $C^\\prime =\\tau _3\\otimes \\sigma _1$ , satisfying $(C^{\\prime })^2=1$ , and $$ represents complex conjugation.", "$C^\\prime $ is the operation of $p_{\\uparrow , A, \\sigma }\\rightarrow p_{\\uparrow , A, \\sigma }$ and $p_{\\uparrow , B, \\sigma } \\rightarrow -p_{\\uparrow , B, \\sigma }$ combined with switching eigenstates of $L_z$ .", "Furthermore, when combining two sectors of $s=\\uparrow $ and $\\downarrow $ together, the system satisfies the TR symmetry defined as $T=is_2 \\otimes 1_\\tau \\otimes \\sigma _1\\otimes K$ with $T^2=-1$ , where $K$ is the complex conjugation.", "Due to the above symmetry proprieties, our system is in the DIII class [46] in the absence of lattice asymmetry.", "However, in the presence of lattice asymmetry, the particle-hole symmetry $C^\\prime $ is broken, and only the TR symmetry exists.", "In that case, the system is the in sympletic class AII.", "In both cases, the topological index is $\\mathbb {Z}_2$ .", "Nevertheless, in the presence of sublattice asymmetry $m$ , the product of parity and particle-hole transformations remains a valid symmetry as $C H_{\\uparrow }(\\vec{k}) C^{-1}= - H^_\\uparrow (\\vec{k}),$ where $C=i\\tau _2\\otimes \\sigma _1$ satisfying $C^2=-1$ .", "This symmetry ensures the energy levels, for each $\\vec{k}$ , appear symmetric with respect to the zero energy.", "Without loss of generality, we choose the convention that $m>0$ and $\\lambda >0$ throughout the rest of this article.", "The case of $m<0$ can be obtained through a parity transformation that flips the $A$ and $B$ sublattices as $H_{m<0}(\\vec{k})= (\\tau _1 \\otimes 1_\\sigma ) H_{m>0} (-\\vec{k})(\\tau _1 \\otimes 1_\\sigma )^{-1}.$ The case of $\\lambda <0$ can be obtained through a partial TR transformation only within each spin sector but without flipping electron spin: $H_{\\lambda <0}(\\vec{k})= (1_\\tau \\otimes \\sigma _1) H^_{\\lambda >0}(-\\vec{k})(1_\\tau \\otimes \\sigma _1)^{-1}.$" ], [ "Energy spectra and eigenfunctions", "In this section, we provide solutions to the Hamiltonian of $p_x$ - and $p_y$ -orbital bands in honeycomb lattices.", "Based on the properties of $\\Gamma $ matrices, most results can be expressed analytically." ], [ "Analytic solution to eigen energies", "Due to Eq.", "(REF ), the spectra of $H_\\uparrow (\\vec{k})$ are symmetric with respect to the zero energy.", "Consequently, they can be analytically solved as follows.", "The square of $H_\\uparrow (\\vec{k})$ can be represented in the standard $\\Gamma $ -matrix representation as $H^2(\\vec{k})= g_0 (\\vec{k})+ 2\\sum _{i=1}^5 g_i(\\vec{k}) \\Gamma _i,$ with the $g_i$ 's expressed as $g_0&=&\\lambda ^2+ m^2 + 3(t_1^2 +t_2^2)+(2t_2^2-t_1^2)\\sum _{j=1}^3 \\cos \\vec{k} \\cdot \\vec{b}_j, \\nonumber \\\\g_1&=&-t_1t_2 \\sum _{j=1}^3 \\cos \\vec{k} \\cdot \\vec{b}_j \\sin \\theta _i, \\ \\ \\,g_5= -t_1t_2 \\sum _{j=1}^3 \\cos \\vec{k} \\cdot \\vec{b}_j \\cos \\theta _i,\\nonumber \\\\g_2 &=& -\\lambda t_2 \\sum _{j=1}^3 \\cos \\vec{k} \\cdot \\vec{a}_j, \\ \\ \\, \\ \\ \\,g_3= -\\lambda t_2 \\sum _{j=1}^3 \\sin \\vec{k} \\cdot \\vec{a}_j, \\nonumber \\\\g_4&=&\\frac{\\sqrt{3}}{2}t_1^2 \\sum _{j=1}^3 \\sin \\vec{k} \\cdot \\vec{b}_j-m\\lambda , \\ \\ \\, \\ \\ \\,$ where $\\vec{b}_1=\\hat{e}_2-\\hat{e}_3$ , $\\vec{b}_2=\\hat{e}_3-\\hat{e}_1$ , and $\\vec{b}_3=\\hat{e}_1-\\hat{e}_2$ .", "The $\\Gamma $ matrices satisfy the anticommutation relation as $\\lbrace \\Gamma _i, \\Gamma _j\\rbrace =2\\delta _{ij}$ .", "They are defined here as $\\Gamma _1=1_\\tau \\otimes \\sigma _1, \\ \\ \\,\\Gamma _{2,3,4}=\\tau _{1,2,3}\\otimes \\sigma _3, \\ \\ \\,\\Gamma _{5}=1_\\tau \\otimes \\sigma _2.", "\\ \\ \\ $ The spectra are solved as $E^2(\\vec{k})=g_0\\pm 2 (\\sum _{i=1}^5 g_i^2)^{\\frac{1}{2}}$ .", "In the case of neglecting the $\\pi $ bonding, i.e., $t_1=t_2=\\frac{1}{2}t_\\parallel $ , the spectra can be expressed as $E_{1,4}(\\vec{k})&=& \\pm \\sqrt{f_1(\\vec{k})+\\sqrt{f_2(\\vec{k})} },\\nonumber \\\\E_{2,3}(\\vec{k})&=& \\pm \\sqrt{ f_1(\\vec{k})-\\sqrt{f_2(\\vec{k})} },$ where $f_1(\\vec{k}) &=&\\lambda ^2+m^2+\\frac{3}{2}t_\\parallel ^2 +\\frac{1}{4}t_\\parallel ^2\\eta _c(\\vec{k}), \\nonumber \\\\f_2(\\vec{k}) &=&\\big \\lbrace \\frac{t^2_\\parallel }{4}[3-\\eta _c(\\vec{k}) ]-4\\lambda ^2 \\big \\rbrace ^2 \\nonumber \\\\&+& \\lambda ^2(9t_\\parallel ^2 -16 \\lambda ^2+4m^2)-\\frac{\\sqrt{3}}{4}t_\\parallel ^2m\\lambda \\eta _s(\\vec{k}), \\ \\ \\ $ and the expressions for $\\eta _c$ , $\\eta _s$ are defined as $\\eta _c(\\vec{k})&=& \\sum _{j=1}^3 \\cos \\vec{k} \\cdot \\vec{b}_j, \\ \\ \\,\\eta _s(\\vec{k})=\\sum _{j=1}^3 \\sin \\vec{k} \\cdot \\vec{b}_j.$" ], [ "Solution to eigen wave functions", "Eigen-wave functions $\\psi _i(\\vec{k})$ for the band index $i=1,\\cdots , 4$ can be obtained by applying two steps of projection operators successively.", "The first projection is based on $H^2(\\vec{k})$ which separates the subspace spanned by $\\psi _{1,4}(\\vec{k})$ from that by $\\psi _{2,3}(\\vec{k})$ .", "We define $P_{14}(\\vec{k})&=&\\frac{1}{2} \\big [1+\\sum _{i=1}^5g^\\prime _i(\\vec{k}) \\Gamma _i\\big ],\\nonumber \\\\P_{23}(\\vec{k})&=&\\frac{1}{2} \\big [1- \\sum _{i=1}^5g^\\prime _i(\\vec{k}) \\Gamma _i\\big ],$ where $g^\\prime _i$ is normalized according to $g_i^\\prime (\\vec{k})=g_i(\\vec{k})/\\sqrt{f_2(\\vec{k})}$ such that $\\sum _i g^{\\prime ,2}_i=1$ .", "In each subspace, we can further distinguish the positive and negative energy states by applying $P_i(\\vec{k}) =\\frac{1}{2}\\big \\lbrace 1+ \\frac{1}{E_i}H_\\uparrow (\\vec{k})\\big \\rbrace .$ for each band $i=1,\\cdots , 4$ .", "In other words, starting from an arbitrary state vector $\\psi (\\vec{k})$ , we can decompose it into $\\psi (\\vec{k})= \\sum _{i=1}^4 \\phi _i(\\vec{k})$ according to $\\phi _{1,4}(\\vec{k})&=&P_{1,4} (\\vec{k}) P_{14}(\\vec{k})\\psi , \\nonumber \\\\\\phi _{2,3}(\\vec{k})&=&P_{2,3} (\\vec{k}) P_{23}(\\vec{k})\\psi .$ which satisfy $H\\phi _i(\\vec{k})=E_i \\phi _i(\\vec{k})$ .", "Nevertheless, the concrete expressions of eigen wave functions $\\psi _i (i=1,\\cdots , 4)$ after normalization are rather complicated and thus we will not present their detailed forms." ], [ "A new set of bases", "Below we present a simplified case in the absence of SO coupling, i.e., $\\lambda =0$ , in which the two-step diagonalizations can be constructed explicitly.", "This also serves as a set of convenient bases for further studying the band topology after turning on SO coupling.", "We introduce a new set of orthonormal bases denoted as $\|A_1(\\vec{k})\\rangle &=&\\frac{1}{\\sqrt{2N_k}}\\left(\\begin{array}{c}\\gamma ^_{1-}(\\vec{k})\\\\0\\\\\\gamma ^_{1+}(\\vec{k}) \\\\0\\\\\\end{array}\\right), \\nonumber \\\\\|B_1(\\vec{k})\\rangle &=&\\frac{1}{\\sqrt{2N_k}}\\left(\\begin{array}{c}0\\\\\\gamma _{1+}(\\vec{k})\\\\0\\\\\\gamma _{1-}(\\vec{k})\\\\\\end{array}\\right),$ and $\|A_2(\\vec{k})\\rangle &=&\\frac{1}{\\sqrt{2N_k}}\\left(\\begin{array}{c}\\gamma _{2-}(\\vec{k})\\\\0\\\\\\gamma _{2+}(\\vec{k})\\\\0\\end{array}\\right), \\nonumber \\\\\|B_2(\\vec{k})\\rangle &=&\\frac{1}{\\sqrt{2N_k}}\\left(\\begin{array}{c}0\\\\\\gamma _{2+}^(\\vec{k})\\\\0\\\\\\gamma _{2-}^(\\vec{k})\\end{array}\\right)$ where $\\gamma _{1\\pm }(\\vec{k})&=&\\sum _{i=1}^3 e^{i\\vec{k} \\cdot \\hat{e}_i \\pm 2 i\\theta _i}, \\ \\ \\,\\gamma _{2\\pm }(\\vec{k})=\\sum _{i=1}^3 e^{i\\vec{k} \\cdot \\hat{e}_i \\pm i\\theta _i}, \\nonumber \\\\N(\\vec{k})&=&3-\\eta _c(\\vec{k}).$ In terms of this set of new bases, $H_\\uparrow (\\vec{k})$ is represented as $H_\\uparrow (\\vec{k})=\\left[\\begin{array}{cccc}m-n(\\vec{k})& -\\frac{3}{2} t_\\parallel & h(\\vec{k})& 0 \\\\-\\frac{3}{2}t_\\parallel & -m +n(\\vec{k}) &0 & h(-\\vec{k})\\\\h^(\\vec{k})& 0 & m +n(\\vec{k})& -\\frac{1}{2} t_\\parallel l^(\\vec{k})\\\\0& h^(-\\vec{k}) & -\\frac{1}{2} t_\\parallel l(\\vec{k})& -m - n(\\vec{k})\\end{array}\\right],$ where for simplicity $t_\\perp $ is set to 0; $n(\\vec{k})$ , $l(\\vec{k})$ , and $h(\\vec{k})$ are expressed as $n(\\vec{k})&=&\\frac{\\sqrt{3} \\lambda }{N_k} \\eta _s(\\vec{k}), \\ \\ \\, \\ \\ \\,l(\\vec{k})= \\sum _i e^{i\\vec{k} \\cdot \\hat{e}_i}, \\nonumber \\\\h(\\vec{k})&=&\\frac{i\\lambda }{N_k}\\Big \\lbrace (\\sum _i e^{i\\vec{k} \\cdot \\hat{e}_i})^2-3(\\sum _i e^{-i\\vec{k} \\cdot \\hat{e}_i})\\Big \\rbrace .$ In the absence of SO coupling, $h(\\vec{k})=n(\\vec{k})=0$ , the above matrix of $H_\\uparrow (\\vec{k})$ is already block diagonalized.", "The left-up block represents the Hamiltonian matrix in the subspace spanned by the bottom band $\|\\phi _1(\\vec{k})\\rangle $ and top band $\|\\phi _4(\\vec{k})\\rangle $ , and the right-bottom block represents that in the subspace spanned by the middle two bands $\|\\phi _{2,3}(\\vec{k})\\rangle $ .", "Apparently, the bottom and top bands are flat as $E_{1,4}=\\pm \\sqrt{(\\frac{3}{2}t_\\parallel )^2+m^2},$ whose eigen wave functions are solved as $\\left[\\begin{array}{c}\|\\phi _1(\\vec{k})\\rangle \\\\\|\\phi _4(\\vec{k})\\rangle \\end{array}\\right]=\\left[ \\begin{array}{cc}\\sin \\frac{\\alpha }{2} & \\cos \\frac{\\alpha }{2} \\\\\\cos \\frac{\\alpha }{2} & -\\sin \\frac{\\alpha }{2}\\end{array}\\right]\\left[\\begin{array}{c}\|A_1(\\vec{k})\\rangle \\\\\|B_1 (\\vec{k}) \\rangle \\end{array}\\right],$ where $\\alpha =\\arctan \\frac{3t_\\parallel }{2m}$ .", "As for the middle two bands, the spectra can be easily diagonalized as $E_{2,3}(\\vec{k})=\\pm \\sqrt{\\frac{1}{4}t^2_\\parallel \\eta ^2_c(\\vec{k})+m^2}.$ The spectrum is the same as that in graphene at $m=0$ .", "The eigen wave functions are enriched by orbital structures which can be solved as $\\left[\\begin{array}{c}\|\\phi _2(\\vec{k})\\rangle \\\\\|\\phi _3(\\vec{k})\\rangle \\end{array}\\right]=\\left[ \\begin{array}{cc}\\sin \\frac{\\beta }{2} & \\cos \\frac{\\beta }{2} e^{i\\phi } \\\\\\cos \\frac{\\beta }{2} e^{-i\\phi } & -\\sin \\frac{\\beta }{2}\\end{array}\\right]\\left[\\begin{array}{c}\|A_2(\\vec{k})\\rangle \\\\\|B_2 (\\vec{k}) \\rangle \\end{array}\\right], \\nonumber \\\\$ where $\\beta (\\vec{k})=\\arctan [\\frac{t_\\parallel }{2m} l(\\vec{k})]$ and $\\phi (\\vec{k})=\\arg {l(\\vec{k})}$ ." ], [ "Appearance of flat bands", "According to the analytical solution of spectra Eq.", "(REF ), flat bands appear in two different situations: (i) In the absence of SO coupling such that the bottom and top bands are flat with the eigen energies described by Eq.", "(REF ); (ii) in the presence of SO coupling, at $\\lambda =\\frac{3}{4}t_\\parallel $ , the two middle bands are flat with the energies $E_{2,3}(\\vec{k})=\\pm \\frac{3}{4}t_\\parallel $ .", "In both cases, the band flatness implies that we can construct eigenstates localized in a single hexagon plaquette.", "The localized eigenstates for the case of $\\lambda =0$ are constructed in Ref.", "wu2007, and those for the case of $\\lambda =\\frac{3}{4}t_\\parallel $ were presented in Ref. zhang2011a.", "Since the kinetic energy is suppressed in the flat bands, interaction effects are nonperturbative.", "Wigner crystallization [29] and ferromagnetism [33] have been studied in the flat band at $\\lambda =0$ ." ], [ "Band topology and band crossings", "In this section, we study the topology of band structures after SO coupling $\\lambda $ is turned on.", "Due to the $s_z$ conservation, the $Z_2$ topological class is augmented to the spin Chern class.", "Without loss of generality, we only use the pattern of Chern numbers of the sector $s=\\uparrow $ to characterize the band topology, and that of the $s=\\downarrow $ sector is just with an opposite sign.", "The Berry curvature for the $i$ -th band is defined as $F_i(\\vec{k})=\\partial _{k_x} A_y(\\vec{k}) -\\partial _{k_y} A_x(\\vec{k})$ in which the Berry connection is defined as $\\vec{A}_i(\\vec{k})=-i \\langle \\phi _i(\\vec{k})\|\\vec{\\nabla }_k\|\\phi _i(\\vec{k})\\rangle $ .", "The spin Chern number of band $i$ can be obtained through the integral over the entire first Brillouin zone as $C_{s,i}=\\frac{1}{2\\pi }\\int _{FBZ} d k_x dk_y F_i(\\vec{k}_x, \\vec{k}_y).$ Figure: The spectra along the cut of K ' K^\\prime -Γ\\Gamma -KK in Brillouinzone.The spectra evolution is shown with fixed m/t ∥ =0.3m/t_{\\parallel }=0.3 and increasing λ\\lambda from 0.1 (a) to 2.8 (g), which passes phases A 1 A_1, B 1 B_1, A 2 A_2,and C 1 C_1.The pattern of spin-Chern numbers in the gapped states are marked.Parameters of (b), (d), and (f) are located at phase boundariesand gaps are closed at K ' K^\\prime , Γ\\Gamma , and KK points for(b), (d), and (f), respectively.Please note the appearance of single Dirac cones forthe sector of spin-↑\\uparrow , which is possiblein 2D when two masses from sublattice asymmetryand SO coupling compete.Figure: The same plot as in Fig.", "but for theevolution with fixed m/t ∥ =0.5m/t_{\\parallel }=0.5 and increasing λ\\lambda from 0.2 (a) to 2 (g), which passes phases A 1 A_1, B 1 B_1, and B 2 B_2and C 1 C_1.Gaps are closed at K ' K^\\prime , KK, and Γ\\Gamma points in(b), (d), and (f), respectively." ], [ "Band crossings at $\\Gamma $ , {{formula:99d63331-9efe-4dcc-9bbf-286c8024b96e}} and {{formula:d94e569b-3729-4e5e-a610-e022d21af628}}", "We have performed the numerical integration for spin Chern numbers $(C_{s,1},C_{s,2},C_{s,3}, C_{s,4})$ for $H_\\uparrow (\\vec{k})$ as presented in Fig.", "REF based on Eq.", "(REF ).", "The phase boundary lines $L_{1,2,3}$ are associated with band touching, which occurs at high symmetry points $\\Gamma $ , $K$ , and $K^\\prime $ , respectively.", "The momenta of these points are defined as $(0,0)$ , $(\\pm \\frac{4 \\pi }{3\\sqrt{3}}, 0)$ .", "Since the dispersions of $H_\\uparrow (\\vec{k})$ are symmetric with respect to zero energy, the band crossing occurs either between bands 2 and 3 at zero energy, or between 1 and 2, 3 and 4 symmetrically with respect to zero energy.", "We first check the crossing at the $\\Gamma $ -point.", "According to Eq.", "(REF ), the energies of the two middle levels are $E_{2,3}(\\Gamma )=\\pm \\left(\\lambda -\\sqrt{m^2+\\big (\\frac{3}{2}t_{\\parallel }\\big )^2}\\right).$ The level crossing can only occur at zero energy with the hyperbolic condition $\\lambda ^2=m^2+\\Big (\\frac{3}{2}t_\\parallel \\Big )^2,$ which corresponds to line $L_1$ in Fig.", "REF .", "The sublattice asymmetry parameter $m$ and SO coupling $\\lambda $ are different mass generation mechanisms.", "The former breaks parity and contributes equally at $K$ and $K^\\prime $ , while the latter exhibits opposite signs.", "Their total effects superpose constructively or destructively at $K$ and $K^\\prime $ , respectively, as shown in the spectra of the two lower energy levels at $K$ and $K^\\prime $ .", "At $K^\\prime =(-\\frac{4\\pi }{3\\sqrt{3}},0)$ , they are $E_{2,3}(K^{\\prime })=\\pm (\\lambda -m),$ and those at $K=(\\frac{4\\pi }{3\\sqrt{3}},0)$ are $E_{1,4}(K)&=&\\mp \\sqrt{(m-\\lambda )^2+\\Big (\\frac{3}{2}t_\\parallel \\Big )^2}, \\nonumber \\\\E_{2,3}(K)&=&\\mp (m+\\lambda ).$ Thus the level crossing at $K^\\prime $ occurs at zero energy with the relation $\\lambda =m,$ which is line $L_2$ in Fig.", "REF .", "Similarly, the level crossing at $K$ occurs when $E_2(K)=E_1(K)$ leading to the condition $\\lambda \\, m=\\Big (\\frac{3}{4}t_\\parallel \\Big )^2,$ which is line $L_3$ in Fig.", "REF .", "Figure: The same plot as in Fig.", "but for theevolution fixing m/t ∥ =1m/t_{\\parallel }=1 and increasing λ\\lambda from 0.2 (a) to 2 (g), which passes phases A 1 A_1, C 2 C_2, and B 2 B_2and C 1 C_1.Gaps are closed at K ' K^\\prime , KK, and Γ\\Gamma points in(b), (d), and (f), respectively." ], [ "Evolution of the topological band structures", "The lattice asymmetry term $m$ by itself can open a gap at $K$ and $K^\\prime $ in the absence of SO coupling.", "In this case, the gap value is $m$ at both $K$ and $K^\\prime $ .", "The lower two bands remain touched at the $\\Gamma $ point with quadratic band touching.", "Nevertheless, the overall band structure remains nontopological.", "The SO coupling $\\lambda $ brings nontrivial band topology.", "Its competition with the lattice asymmetry results in a rich structure of band structure topology presented in Fig.", "REF , which are characterized by their pattern of spin Chern numbers.", "There are two phases characterized by the same spin Chern number pattern $(1,-1,1-1)$ marked as $A_1$ and $A_2$ , respectively; two phases characterized by $(1,0,0,-1)$ marked as $B_1$ and $B_2$ ; and two trivial phases denoted as $C_1$ and $C_2$ $(0,0,0,0)$ .", "Even an infinitesimal value of $\\lambda $ removes the quadratic band touching between the band 1 and 2, and brings nontrivial band topology.", "The line of $m=0$ corresponds to the situation investigated in the QAH insulator based on the $p_x$ - and $p_y$ -orbital bands in the honeycomb lattice [32], [35].", "The current situation is a 2D topological insulator with $s_z$ conserved, which is just a double copy of the previous QAH model.", "At small values of $\\lambda $ , the system is in the $B_1$ phase.", "It enters the $A_2$ phase after crossing the line $L_1$ at $\\lambda =\\frac{3}{2} t_\\parallel $ .", "If the system begins with a nonzero lattice asymmetry parameter $m$ , it first enters the $A_1$ .", "If we increase SO coupling strength $\\lambda $ by fixing $m$ at different values, different band topology transitions appear.", "To further clarify these transitions, we plot the spectra evolutions with increasing $\\lambda $ while fixing $m=0.3,0.5$ , and 1 in Figs.", "REF , REF , REF , respectively.", "Only the spectra along the line cut from $K^\\prime $ to $\\Gamma $ to $K$ in the Brillouin zone are plotted.", "At small values of $m$ as shown in Fig.", "REF , the gap first closes at $K^\\prime $ , and then at $\\Gamma $ , and finally at $K$ with increasing $\\lambda $ .", "The sequence of phase transitions is $A_1\\rightarrow B_1 \\rightarrow A_2 \\rightarrow C_1$ .", "At intermediate values of $m$ shown in Fig.", "REF , the gap first closes at $K^\\prime $ , then at $K$ , and finally at $\\Gamma $ leading to a sequence of phase transitions $A_1\\rightarrow B_1\\rightarrow B_2 \\rightarrow C_1$ .", "At large values of $m$ as shown in Fig.", "REF , the gap first closes at $K^\\prime $ , then at $K$ , and finally at $\\Gamma $ .", "The sequence of phases is $A_1\\rightarrow C_2 \\rightarrow B \\rightarrow C_1$ ." ], [ "Reduced two-band models around band crossings", "In order to further clarify topological band transitions, we derive the effective two-band Hamiltonians around the gap closing points ($\\Gamma $ , $K$ , and $K^{\\prime }$ ) respectively in this section.", "Since the crossing at the $\\Gamma $ point occurs at zero energy, we consider the middle two states.", "We construct the two bases as $\|\\phi _2(\\vec{k})\\rangle &=& \\cos \\frac{\\alpha }{2}\|\\psi _{A,-}(\\vec{k})\\rangle +\\sin \\frac{\\alpha }{2}\|\\psi _{B,-}(\\vec{k})\\rangle \\nonumber \\\\\|\\phi _3(\\vec{k})\\rangle &=& -\\sin \\frac{\\alpha }{2}\|\\psi _{A,+}(\\vec{k})\\rangle +\\cos \\frac{\\alpha }{2}\|\\psi _{B,+}(\\vec{k})\\rangle ,$ where $\\alpha =\\arctan \\frac{3t_\\parallel }{2m}$ .", "Right at the $\\Gamma $ point, these two bases are the eigenvectors of the middle two bands with energies are $E_{2,3}(\\Gamma )= \\mp (\\sqrt{m^2+(\\frac{3}{2}t_\\parallel )^2}-\\lambda )$ , respectively.", "As $\\lambda \\rightarrow \\sqrt{m^2+ (\\frac{3}{2}t_\\parallel )^2}$ , we construct the low-energy Hamiltonian for $\\vec{k}$ around the $\\Gamma $ point by using $\|\\phi _{2,3}(\\vec{k})\\rangle $ as bases: $&&\\left[\\begin{array}{cc}\\langle \\phi _2\|H\|\\phi _2\\rangle &\\langle \\phi _2\|H\|\\phi _3\\rangle \\\\\\langle \\phi _3\|H\|\\phi _2\\rangle &\\langle \\phi _3\|H\|\\phi _3\\rangle \\end{array}\\right]\\nonumber \\\\&=&\\left[\\begin{array}{cc}-\\lambda +\\sqrt{m^2+(\\frac{3}{2}t_\\parallel )^2}& \\frac{3}{4}t_\\parallel (k_x+ik_y)\\\\\\frac{3}{4}t_\\parallel (k_x-ik_y)& \\lambda -\\sqrt{m^2+(\\frac{3}{2}t_\\parallel )^2}\\end{array}\\right], \\ \\ \\ $ which describes the band crossing of line $L_1$ in Fig.", "REF .", "The two-band effective model for the crossing at the $K^\\prime $ point is just what we have constructed in Eq.", "(REF ).", "It describes the crossing at zero energy represented by line $L_2$ in Fig.", "REF .", "As for the band crossing at the $K$ point, it occurs between band 1 and 2, and between 3 and 4 symmetrically with respect to zero energy ($B_2$ , $C_1$ , and $C_2$ phases).", "For simplicity, we only consider the effective two-band model at small values of $m$ .", "In this case, the band crossing is described by line $L_3$ in Fig.", "REF occurring at large values of $\\lambda \\gg m$ .", "The on-site energy level splitting between the states of $(p_+,\\uparrow )$ and $(p_-,\\uparrow )$ is larger than the hopping integral $t_\\parallel $ , and each of them will develop a single band in the honeycomb lattice.", "The bands of $p_{\\pm }$ orbitals lie symmetrically with respect to zero energy.", "Nevertheless, as shown in Refs.", "wu2008,zhang2011a, the interband coupling at the second-order perturbation level effectively generates the complex-valued next-nearest-neighbor hopping as in Haldane's QAH model [5].", "Our current situation is a TR double copy and thus it gives rise to the Kane-Mele model.", "To describe the above physics, we only keep the $p_{+}$ orbitals on each site in the case of large values of $\\lambda $ .", "Then the terms of $h_{11}$ , $h_{22}$ , $h_{21}$ , and $h_{12}$ in Eq.", "(REF ) become perturbations.", "By the second-order perturbation theory, we derive the low-energy Hamiltonian of $(p_{A,+}(\\vec{k}), p_{B,+}(\\vec{k}))$ bands as $&&\\left[\\begin{array}{cc}\\langle \\psi _{A+}\|H\|\\psi _{A+}\\rangle &\\langle \\psi _{A+}\|H\|\\psi _{B_+}\\rangle \\\\\\langle \\psi _{B+}\|H\|\\psi _{A+}\\rangle &\\langle \\psi _{B+}\|H\|\\psi _{B_+}\\rangle \\end{array}\\right]\\nonumber \\\\&=&\\left[\\begin{array}{cc}m+m_H(\\vec{k}) &-\\frac{t_\\parallel }{2}l^(\\vec{k})\\nonumber \\\\-\\frac{t_\\parallel }{2}l(\\vec{k})& -m-m_H(\\vec{k})\\end{array}\\right],$ where $m_H(\\vec{k})=\\frac{\\sqrt{3}}{8}\\frac{t^2_\\parallel }{\\lambda }\\sin \\eta _s(\\vec{k}).$ Around the $K$ point, $m_H(K)=-\\frac{9}{16}\\frac{t_\\parallel ^2}{\\lambda }$ .", "The band crossing occurs when $m+m_H(\\vec{k})$ switches sign, which gives rise to line $L_3$ in Fig.", "REF .", "The topological gap opens at the $K^\\prime $ point between bands 2 and 3.", "According to Eq.", "(REF ), we only need to keep the right-bottom block for the construction of the low-energy two-band model.", "By expanding around the $K^\\prime $ point, we have $&&\\left[\\begin{array}{cc}\\langle A_2\|H\|A_2\\rangle &\\langle A_2\|H\|B_2\\rangle \\\\\\langle B_2\|H\|A_2\\rangle &\\langle B_2\|H\|B_2\\rangle \\end{array}\\right]\\nonumber \\\\&=&\\left[\\begin{array}{cc}m-\\lambda & -\\frac{3}{4}t_\\parallel (\\delta k_x+i \\delta k_y)\\\\-\\frac{3}{4}t_\\parallel (\\delta k_x-i\\delta k_y)& -m+\\lambda \\end{array}\\right],$ where $\\delta \\vec{k}= \\vec{k}-\\vec{K}^\\prime $ , and thus the mass term is controlled by $m-\\lambda $ .", "For completeness, we also derive the effective two-band Hamiltonian for bands 2 and 3 around the $K$ point similarly, which yields the gap value $m+\\lambda $ .", "In the absence of lattice asymmetry, the gap values at $K$ and $K^\\prime $ are both the SO coupling strength.", "Now let us look more carefully at the eigen wave functions of the effective two-band Hamiltonian for bands 2 and 3 at $K^\\prime $ and $K$ points and check their orbital angular momenta.", "The eigenstates are just $\|A_2(K^\\prime )\\rangle $ and $\|B_2(K^\\prime )\\rangle $ at $K^\\prime $ , and $\|A_2(K)\\rangle $ , and $\|B_2(K)\\rangle $ at $K$ .", "In the bases of Eq.", "(REF ), we express $&&\|A_2(K^\\prime )\\rangle =\\left[\\begin{array}{c}1\\\\0\\\\0\\\\0\\end{array}\\right], \\ \\ \\,\|B_2(K^\\prime )\\rangle =\\left[\\begin{array}{c}0\\\\0\\\\0\\\\1\\end{array}\\right],\\nonumber \\\\&&\|A_2(K)\\rangle =\\left[\\begin{array}{c}0\\\\0\\\\1\\\\0\\end{array}\\right], \\ \\ \\,\|B_2(K)\\rangle =\\left[\\begin{array}{c}0\\\\1\\\\0\\\\0\\end{array}\\right].$ All of them are the orbital angular momentum eigenstates with $L_z=\\pm 1$ .", "Considering this is the sector with $s=\\uparrow $ , the gap is just the atomic SO coupling strength $\\lambda $ in the absence of the lattice asymmetry term $m$ ." ], [ "Large topological band gaps", "The most striking feature of the these $p_x$ -$p_y$ systems is the large topological band gap at $K^{\\prime }$ , $K$ , and $\\Gamma $ points.", "In this section, we analyze the origin of large topological band gaps at these $k$ points in the $B_1$ phase (quantum spin Hall (QSH) phase, $\\lambda >m$ ).", "For the case of a single-component fermion QAH model studied in Ref.", "wu2008, it has been analyzed that the gap values at the $\\Gamma $ , $K$ , and $K^\\prime $ points are just the on-site rotation angular velocity $\\Omega $ in the absence of the lattice asymmetry term.", "The situation in this paper is a TR invariant double copy of the previously single component case, and thus the role of of $\\Omega $ is replaced by the on-site atomic SO coupling strength $\\lambda $ .", "At the $K^{\\prime }$ point, according to Eq.", "(REF ), the eigenstates for the bands 2,3 are orbital angular momentum eigenstates with $L_z=\\pm 1$ .", "The energy and corresponding eigenstates for bands 2 and 3 are $E_2(K^{\\prime })&=& m-\\lambda , \\quad \|\\phi _2(K^{\\prime })\\rangle = \|\\psi _{A,+}(K^{\\prime })\\rangle , \\nonumber \\\\E_3(K^{\\prime })&=& \\lambda -m, \\quad \|\\phi _3(K^{\\prime })\\rangle = \|\\psi _{B,-}(K^{\\prime })\\rangle , \\nonumber \\\\\\Delta _{K^{\\prime }}&=& 2 (\\lambda - m).$ As shown in Fig.", "REF , the eigenstate for band 2 has $L_z=+1$ with the energy $m-\\lambda $ , which is of $p_x+i p_y$ type, and its wave function is totally on the $A$ sublattice.", "In contrast, the eigenstate for band 3 has $L_z=-1$ with the energy $\\lambda -m$ .", "It is of the $p_x-i p_y$ type whose wave function completely distributes on the $B$ sublattice.", "The topological band gap is thus $2(\\lambda -m)$ .", "If the sublattice asymmetry term vanishes, i.e., $m=0$ , the band gap is just $2\\lambda $ .", "Obviously, the atomic on-site SO coupling strength $\\lambda $ directly contributes to the topological band gap, leading to a large band splitting.", "It is because at the $K^{\\prime }$ point, the eigenstates of the system are also $L_z$ eigenstates, which means the topological band gap is the eigenenergy difference between the SO coupling term $s_z L_z$ for $L_z=\\pm 1$ .", "It is easy to generalize the analysis to the $K$ point similarly.", "At the $\\Gamma $ point, the Hamiltonian $H(\\vec{k})$ preserves all the rotation symmetries of the system, and thus the SO coupling term $s_z L_z$ commutes with $H(\\vec{k})$ .", "The eigenstates simultaneously diagonalize the SO coupling term and $H(\\vec{k})$ .", "The energy and corresponding eigenstates for bands 1 and 2 at the $\\Gamma $ point are $E_1(\\Gamma ) &=& -\\lambda -\\sqrt{m^2+(\\frac{3}{2}t_{\\parallel })^2}, \\nonumber \\\\\|\\phi _1(\\Gamma )\\rangle &=& \\sin \\frac{\\alpha }{2}\|\\psi _{A,+}(\\Gamma )\\rangle +\\cos \\frac{\\alpha }{2}\|\\psi _{B,+}(\\Gamma )\\rangle , \\nonumber \\\\E_2(\\Gamma )&=& \\lambda -\\sqrt{m^2+(\\frac{3}{2}t_{\\parallel })^2}, \\nonumber \\\\\|\\phi _2(\\Gamma )\\rangle &=& \\cos \\frac{\\alpha }{2}\|\\psi _{A,-}(\\Gamma )\\rangle +\\sin \\frac{\\alpha }{2}\|\\psi _{B,-}(\\Gamma )\\rangle , \\nonumber \\\\\\Delta _{\\Gamma }&=& 2\\lambda .$ The eigenstates for bands 1,2 are the superpositions of wave functions on both the $A$ and $B$ sublattices.", "However, for band 1, the eigenstate is an $L_z=-1$ eigenstate, and the eigenstate for band 2 is an $L_z=1$ eigenstate (see Fig.", "REF ).", "As a result, the topological band gap $\\Delta _{\\Gamma }$ is the energy difference of the SO coupling term $s_z L_z$ , which is $2\\lambda $ .", "We discuss the dependence of the topological gap values on SO coupling strength in the $B_1$ phase.", "Let us first consider the gap between the lowest two bands.", "For the case without lattice asymmetry, i.e., $m/t_{\\parallel }=0$ , in the weak and intermediate regimes of the SO coupling strength $0< \\lambda /t_{\\parallel }<3/(4\\sqrt{2})$ , the minimal gap is located at the $\\Gamma $ point as shown in Fig.", "REF (b).", "In typical solid state systems, $\\lambda $ lies in these regimes, and thus, typically, the topological gap can approach up to $2\\lambda =3/(2\\sqrt{2}) t_{\\parallel }$ , which is a very large gap.", "If $\\lambda $ further increases, then the minimal gap shifts from the $\\Gamma $ point to the $K$ points, and the value of the gap shrinks as $\\lambda $ increases.", "Similarly, consider the topological gap between the middle two bands, and for parameters $m/t_{\\parallel }=0$ : as long as the SO coupling strength is in the $B_1$ phase, the minimal gap is located at the $K(K^{\\prime })$ point, which can approach up to $2\\lambda =3\\, t_{\\parallel }$ ." ], [ "Quantum Anomalous Hall State", "In this section, we add the Néel antiferromagnetic exchange field term [Eq.", "(REF )] to the Hamiltonian.", "This term gives rise to another mass generation mechanism.", "Together with the atomic SO coupling term of $L_z \\sigma _z$ , and the sublattice asymmetry term [Eq.", "(REF )] discussed before, we can drive the system to a QAH state.", "A similar mechanism was also presented in the single-orbital honeycomb lattice [47], and here we generalize it to the $p_x$ -$p_y$ -orbital systems.", "We consider the gap opening at the $K$ and $K^\\prime $ points, and assume that bands 1 and 2 are filled.", "In the absence of the Néel term [Eq.", "(REF )], the system is in the trivially gapped phase $A_1$ at $m> \\lambda $ , and in the QSH phase $B_1$ at $\\lambda > m$ .", "Let us start with the QSH phase $B_1$ with $\\lambda >m>0$ , and gradually turn on the Néel exchange magnitude $n>0$ .", "The energy levels for different spin sectors at the $K^{\\prime }$ and $K$ points for the middle two bands are $E_{2,3, \\uparrow }(K^{\\prime })&=& \\mp (\\lambda -m-n), \\nonumber \\\\E_{2,3, \\downarrow }(K^{\\prime })&=& \\mp (\\lambda +m-n), \\nonumber \\\\E_{2,3,\\uparrow }(K)&=& \\mp (\\lambda +m+n),\\nonumber \\\\E_{2,3,\\downarrow }(K)&=& \\mp (\\lambda -m+n).$ The gap will not close for both spin-$\\uparrow $ and spin-$\\downarrow $ sectors at the $K$ point with increasing $n$ , and thus we focus on the band crossing at the $K^\\prime $ point.", "At this point, the first band crossing occurs in the spin-$\\uparrow $ sector at $n=\\lambda -m$ , which changes the spin-$\\uparrow $ sector into the topologically trivial regime.", "Meanwhile, the spin-$\\downarrow $ sector remains topologically nontrivial, and thus the system becomes a QAH state.", "If we further increase $n$ , the second band crossing occurs in the spin-$\\downarrow $ sector at $n=\\lambda +m$ , at which the spin-$\\downarrow $ sector also becomes topologically trivial.", "In this case, the entire system is a trivial band insulator.", "The QAH state can be realized for $\\lambda -m<n<\\lambda +m$ .", "The band crossing diagrams are shown in Fig.", "REF (a).", "Similarly, we start from the $A_1$ trivially gapped phase ($0<\\lambda <m$ ), and gradually turn on the Néel exchange field $n$ .", "The middle two energy levels for both spin sectors at the $K^{\\prime }$ and $K$ points are $E_{2,3, \\uparrow }(K^{\\prime })&=& \\mp ( m-\\lambda +n), \\nonumber \\\\E_{2,3,\\downarrow }(K^{\\prime })&=& \\mp (m+\\lambda -n) , \\nonumber \\\\E_{2,3,\\uparrow }(K)&=& \\mp (m+ \\lambda +n) ,\\nonumber \\\\E_{2,3,\\downarrow }(K)&=& \\mp (m-\\lambda -n).$ In this case, the spin-$\\uparrow $ sector remains in the trivially gapped phase with increasing $n$ , since there is no band inversion in this sector [see Fig.", "REF (b)].", "The first band crossing occurs in the spin-$\\downarrow $ sector at the $K$ point when $n=m-\\lambda $ , rendering this sector topologically nontrivial, and then the whole system goes into a QAH phase.", "The second band inversion occurs at the $K^{\\prime }$ point also in the spin-$\\downarrow $ sector at $n=\\lambda +m$ .", "Now the spin-$\\downarrow $ sector is back into a topologically trivial phase, and the whole system is a trivial band insulator for $n>\\lambda +m$ .", "Similarly to the previous case, the QAH phase is realized at $-\\lambda +m<n<\\lambda +m$ .", "Figure: (Color online)Diagrams of the energy level crossing with increasingantiferromagnetic exchange field strength nn for two parameter regimes(a) 0<m<λ0<m<\\lambda and (b) 0<λ<m0<\\lambda <m.Red wavy lines indicate the range of nn for thesystem to be in a QAH phase (\|λ-m\|<n<λ+m\|\\lambda -m\|<n<\\lambda +m).There are three gap parameters in our model, the spin-orbit coupling $\\lambda $ , the sublattice asymmetry term $m$ , and the Néel exchange field $n$ .", "Combining the two situations discussed above, we summarize the condition for the appearance of the QAH state as follows: $\|\\lambda -m\|<n<\\lambda +m,$ which is also equivalent to $\|m-n\|<\\lambda <m+n$ , or $\|\\lambda -n\|<m<\\lambda +n$ .", "In other words, the three gap parameters $\\lambda $ , $m$ , and $n$ can form a triangle.", "For the buckled honeycomb lattices, the A and B sublattices are at different heights.", "The Néel exchange field $n$ can be generated by attaching two ferromagnetic substrates with opposite magnetizations to the two surfaces, and the sublattice asymmetry term $m$ can also be generated if the contacts with these two substrates are asymmetric.", "In the parameter regime for the QAH state, it is easy to check that the maximal topological gap is the minimum of $\\lambda $ and $m$ ." ], [ "Conclusions and Outlooks", "In summary, we have presented a minimal model to describe the 2D topological insulator states in the honeycomb lattice which have been recently proposed in the literature.", "The $p_x$ and $p_y$ orbitals are the key, and thus their properties are dramatically different from those in graphene.", "The atomic level SO coupling directly contributes to the topological gap opening, and thus the gap can be large.", "Due to the conservation of $s_z$ , the band structures are a TR invariant doublet of the previously investigated QAHE based on the $p$ orbital in the honeycomb lattice.", "The band topology is described by the spin Chern numbers.", "Both sublattice asymmetry and the on-site SO coupling can open the gap, and their competition leads to a rich structure of topological band insulating phases.", "Due to the underlying structure of Clifford algebra, the energy spectra and eigen wave functions can be obtained analytically.", "Also, the transition lines among different topological insulators are also analytically obtained.", "Low-energy two-band models are constructed around band crossings.", "Furthermore, with the help of the Néel antiferromagnetic exchange field, the model can enter into a QAH phase.", "This work provides a useful platform for further exploring interaction and topological properties in such systems.", "In addition to a class of solid state materials, the model constructed in this article can, in principle, also be realized in ultracold-atom optical lattices.", "For example, in previous papers by one of the authors and his collaborators (Refs.", "wu2008b and zhang2011a), the quantum anomalous Hall models were proposed for spinless fermions of the $p_x/p_y$ bands in the honeycomb optical lattices.", "By this technique, each optical site is rotating around its own center, which can be modeled as an orbital Zeeman term.", "The quantum spin Hall model of Eqs.", "(REF ) and (REF ) is a time-reversal invariant double of the anomalous quantum Hall model, which in principle can be realized by the spin-dependent on-site rotations of the honeycomb lattice, i.e., the rotation angular velocities for spin-$\\uparrow $ and spin-$\\downarrow $ fermions are opposite to each other.", "This is essentially a spin-orbit coupling term $L_z\\, S_z$ and the rotation angular velocity plays the role of spin-orbit coupling strength.", "In order to observe the topological phase, we need the fermions population to fill the $p$ -orbital bands.", "Then, the phase diagram will be the same as in Fig.REF , by replacing the spin-orbit coupling strength with the magnitude of the angular velocity.", "Note added Near the completion of this work, we became aware of the work Ref.", "liu2014 in which the low energy effective model of the 2D topological insulators on honeycomb lattice are also constructed.", "G. F. Z. and C. W. are supported by the NSF DMR-1410375 and AFOSR FA9550-11-1-0067(YIP).", "Y. L. thanks the Inamori Fellowship and the support at the Princeton Center for Theoretical Science.", "C.W.", "acknowledges financial support from the National Natural Science Foundation of China (11328403)." ] ]	1403.0563
[ [ "On vanishing theorems for local systems associated to Laurent\n polynomials" ], [ "Abstract We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials.", "In particular, we extend one of the results of Gelfand-Kapranov-Zelevinsky into various directions." ], [ "Introduction", "The study of the cohomology groups of local systems is an important subject in algebraic geometry, hyperplane arrangements, topology and hypergeometric functions of several variables.", "Many mathematicians are interested in the conditions for which we have their concentrations in the middle degrees (for a review of this subject, see for example [4]).", "Here let us consider this problem in the following situation.", "Let $B=\\lbrace b(1), b(2), \\ldots ,b(N)\\rbrace \\subset \\mathbb {Z}^{n-1}$ be a finite subset of the lattice $\\mathbb {Z}^{n-1}$ .", "Assume that the affine lattice generated by $B$ in $\\mathbb {Z}^{n-1}$ coincides with $\\mathbb {Z}^{n-1}$ .", "For $z=(z_1, \\ldots , z_N) \\in \\mathbb {C}^N$ we consider Laurent polynomials $P(x)$ on the algebraic torus $T_0=( \\mathbb {C}^)^{n-1}$ defined by $P(x)=\\sum _{j=1}^N z_j x^{b(j)}$ ($x=(x_1, \\ldots , x_{n-1})\\in T_0=( \\mathbb {C}^)^{n-1}$ ).", "Then for $c=(c_1, \\ldots , c_n) \\in \\mathbb {C}^n$ we obtain a possibly multi-valued function $P(x)^{-c_n}x_1^{c_1-1} \\cdots x_{n-1}^{c_{n-1}-1}$ on $W=T_0 \\setminus P^{-1}(0)$ .", "It generates the rank one local system $\\mathcal {L}= \\mathbb {C}_{W}P(x)^{-c_n}x_1^{c_1-1} \\cdots x_{n-1}^{c_{n-1}-1}$ on $W$ .", "Under the nonresonance condition (see Definition REF ) on $c \\in \\mathbb {C}^n$ , Gelfand-Kapranov-Zelevinsky [10] proved that we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-1)$ for non-degenerate Laurent polynomials $P(x)$ .", "This result was obtained as a byproduct of their study on the integral representations of $A$ -hypergeometric functions in [10].", "Since their proof of this concentration heavily relies on the framework of the $-moduletheory, it is desirable toprove it more directly.In this paper, by applying thetwisted Morse theory to perverse sheaveswe extend the result ofGelfand-Kapranov-Zelevinskyto various directions.First in Theorem \\ref {VTM}we relax the non-degeneracycondition on $ P(x)$ by replacing itwith a weaker one (see Definition\\ref {WND}).", "We thusextend the result of \\cite {G-K-Z-2}to the case where the hypersurface $ P-1(0) T0$ may have isolatedsingular points in $ T0$.In fact, in Theorem \\ref {VTM}we relax also the condition that$ B$ generates $ Zn-1$ to a weaker one thatthe dimension of the convexhull $ Rn-1$ of$ B$ in $ Rn-1$ is $ n-1$.", "In Theorem \\ref {MVTM}we extend these resultsto more general local systemsassociated to several Laurent polynomials.Namely we obtain a vanishing theorem forarrangements of toric hypersurfaceswith isolated singular points.", "Our proofsof Theorems \\ref {VTM} and \\ref {MVTM}are very natural and obtained by taking(possibly singular) ``minimal\" toriccompactifications of $ T0$.", "In order to workon such singular varieties, we useour previous idea in the proof of\\cite [Lemma 4.2]{E-T-2}.See Section \\ref {sec:3} for the details.Moreover in Theorem \\ref {NTM}(assuming the non-degeneracy ofGelfand-Kapranov-Zelevinsky \\cite {G-K-Z-2}for Laurent polynomials) we relaxthe nonresonance condition of $ c Cn$in Theorem \\ref {MVTM} by replacing it with themuch weaker one $ c Zn$.To prove Theorem \\ref {NTM}, we first perturbLaurent polynomials by multiplyingmonomials.", "Then we apply thetwisted Morse theory to thereal-valued functionsassociated to themby using some standard propertiesof vanishing cycles of perversesheaves.", "See Sections \\ref {sec:4}and \\ref {sec:5} for the details.In the course of the proof of Theorem \\ref {NTM},we obtain also the following result whichmight be of independent interest.Let $ Q1, ..., Ql$ beLaurent polynomials on $ T=( C)n$and for $ 1 i l$ denote by$ i Rn$ the Newtonpolytope $ NP(Qi)$ of $ Qi$.Set $ = 1 + + l$.$ Theorem 1.1 Let $\\mathcal {L}$ be a non-trivial local system of rank one on $T=( \\mathbb {C}^)^{n}$ .", "Assume that for any $1 \\le i \\le l$ we have ${\\rm dim}\\Delta _i =n$ and the subvariety $Z_i = \\lbrace x \\in T \\ \| \\ Q_1(x)= \\cdots = Q_i(x)=0 \\rbrace \\subset T$ of $T$ is a non-degenerate complete intersection.", "Then for any $1 \\le i \\le l$ we have the concentration $H^j(Z_i ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-i).$ Moreover we have ${\\rm dim}H^{n-i} (Z_i ; \\mathcal {L}) =\\displaystyle \\sum _{{{\\scriptstyle \\begin{matrix}m_1,\\ldots ,m_i \\ge 1\\\\m_1+\\cdots +m_i=n\\end{matrix}}}}{\\rm Vol}_{\\mathbb {Z}}(\\underbrace{\\Delta _1,\\ldots ,\\Delta _1}_{\\text{$m_1$-times}},\\ldots ,\\underbrace{\\Delta _i,\\ldots ,\\Delta _i}_{\\text{$m_i$-times}}),$ where ${\\rm Vol}_{\\mathbb {Z}}(\\underbrace{\\Delta _1,\\ldots ,\\Delta _1}_{\\text{$m_1$-times}},\\ldots ,\\underbrace{\\Delta _i,\\ldots ,\\Delta _i}_{\\text{$m_i$-times}})\\in \\mathbb {Z}$ is the normalized $n$ -dimensional mixed volume with respect to the lattice $\\mathbb {Z}^n\\subset \\mathbb {R}^n$ Note that this result can be considered as a refinement of the classical Bernstein-Khovanskii-Kushnirenko theorem (see [13]).", "On the other hand, Matusevich-Miller-Walther [20] and Saito-Sturmfels-Takayama [25] studied the condition on the parameter vector $c \\in \\mathbb {C}^n$ for which the corresponding local system of $A$ -hypergeometric functions is non-rank-jumping.", "They also relaxed the nonresonance condition of $c \\in \\mathbb {C}^n$ .", "It would be an interesting problem to study the relationship between Theorem REF and their results.", "Acknowledgement: We express our hearty gratitude to Professor N. Takayama for drawing our attention to this problem.", "Moreover some discussions with Professor M. Yoshinaga were very useful during the preparation of this paper." ], [ "Preliminary results", "In this section, we recall basic notions and results which will be used in this paper.", "In this paper, we essentially follow the terminology of [4], [12] etc.", "For example, for a topological space $X$ we denote by ${\\bf D}^{b}(X)$ the derived category whose objects are bounded complexes of sheaves of $\\mathbb {C}_X$ -modules on $X$ .", "Denote by ${\\bf D}_{c}^{b}(X)$ the full subcategory of ${\\bf D}^{b}(X)$ consisting of constructible objects.", "Let $\\Delta \\subset \\mathbb {R}^n$ be a lattice polytope in $\\mathbb {R}^n$ .", "For an element $u \\in \\mathbb {R}^n$ of (the dual vector space of) $\\mathbb {R}^n$ we define the supporting face $\\gamma _u \\prec \\Delta $ of $u$ in $ \\Delta $ by $\\gamma _u = \\left\\lbrace v \\in \\Delta \\ \| \\ \\langle u , v \\rangle =\\min _{w \\in \\Delta }\\langle u ,w \\rangle \\right\\rbrace ,$ where for $u=(u_1,\\ldots ,u_n)$ and $v=(v_1,\\ldots , v_n)$ we set $\\langle u,v\\rangle =\\sum _{i=1}^n u_iv_i$ .", "For a face $\\gamma $ of $\\Delta $ set $\\sigma (\\gamma ) = \\overline{ \\lbrace u \\in \\mathbb {R}^n \\ \| \\ \\gamma _u = \\gamma \\rbrace } \\subset \\mathbb {R}^n .$ Then $\\sigma (\\gamma )$ is an $(n- {\\rm dim}\\gamma )$ -dimensional rational convex polyhedral cone in $\\mathbb {R}^n$ .", "Moreover the family $\\lbrace \\sigma (\\gamma ) \\ \| \\ \\gamma \\prec \\Delta \\rbrace $ of cones in $\\mathbb {R}^n$ thus obtained is a subdivision of $\\mathbb {R}^n$ .", "We call it the dual subdivision of $\\mathbb {R}^n$ by $\\Delta $ .", "If ${\\rm dim}\\Delta =n$ it satisfies the axiom of fans (see [8] and [21] etc.).", "We call it the dual fan of $\\Delta $ .", "Let $\\Delta _1, \\ldots , \\Delta _p\\subset \\mathbb {R}^n$ be lattice polytopes in $\\mathbb {R}^n$ and $\\Delta =\\Delta _1 + \\cdots + \\Delta _p\\subset \\mathbb {R}^n$ their Minkowski sum.", "For a face $\\gamma \\prec \\Delta $ of $\\Delta $ , by taking a point $u \\in \\mathbb {R}^n$ in the relative interior of its dual cone $\\sigma (\\gamma )$ we define the supporting face $\\gamma _i \\prec \\Delta _i$ of $u$ in $\\Delta _i$ .", "Then it is easy to see that $\\gamma = \\gamma _1 + \\cdots + \\gamma _p$ .", "Now we recall Bernstein-Khovanskii-Kushnirenko's theorem [13].", "Definition 2.1 Let $g(x)=\\sum _{v \\in \\mathbb {Z}^n} c_vx^v$ be a Laurent polynomial on the algebraic torus $T=(\\mathbb {C}^)^n$ ($c_v\\in \\mathbb {C}$ ).", "We call the convex hull of ${\\rm supp}(g):=\\lbrace v\\in \\mathbb {Z}^n \\ \| \\ c_v\\ne 0\\rbrace \\subset \\mathbb {Z}^n \\subset \\mathbb {R}^n$ in $\\mathbb {R}^n$ the Newton polytope of $g$ and denote it by $NP(g)$ .", "For a face $\\gamma \\prec NP(g)$ of $NP(g)$ , we define the $\\gamma $ -part $g^{\\gamma }$ of $g$ by $g^{\\gamma }(x):=\\sum _{v \\in \\gamma } c_vx^v$ .", "Definition 2.2 (see [22] etc.)", "Let $g_1, g_2, \\ldots , g_p$ be Laurent polynomials on $T=(\\mathbb {C}^)^n$ .", "Set $\\Delta _i=NP(g_i)$ $(i=1,\\ldots , p)$ and $\\Delta = \\Delta _1 + \\cdots + \\Delta _p$ .", "Then we say that the subvariety $Z=\\lbrace x\\in T=(\\mathbb {C}^)^n \\ \| \\ g_1(x)=g_2(x)=\\cdots =g_p(x)=0 \\rbrace $ of $T=(\\mathbb {C}^)^n$ is a non-degenerate complete intersection if for any face $\\gamma \\prec \\Delta $ of $\\Delta $ the $p$ -form $dg_1^{\\gamma _1} \\wedge dg_2^{\\gamma _2} \\wedge \\cdots \\wedge dg_p^{\\gamma _p}$ does not vanish on $\\lbrace x\\in T=(\\mathbb {C}^)^n \\ \| \\ g_1^{\\gamma _1}(x)= \\cdots =g_p^{\\gamma _p}(x)=0 \\rbrace $ .", "Theorem 2.3 [[13]] Let $g_1, g_2, \\ldots , g_p$ be Laurent polynomials on $T=(\\mathbb {C}^)^n$ .", "Assume that the subvariety $Z=\\lbrace x\\in T=(\\mathbb {C}^)^n \\ \| \\ g_1(x)=g_2(x)=\\cdots =g_p(x)=0 \\rbrace $ of $T=(\\mathbb {C}^)^n$ is a non-degenerate complete intersection.", "Set $\\Delta _i=NP(g_i)$ $(i=1,\\ldots , p)$ .", "Then we have $\\chi (Z)=(-1)^{n-p}\\displaystyle \\sum _{{{\\scriptstyle \\begin{matrix}m_1,\\ldots ,m_p \\ge 1\\\\ m_1+\\cdots +m_p=n\\end{matrix}}}}{\\rm Vol}_{\\mathbb {Z}}(\\underbrace{\\Delta _1,\\ldots ,\\Delta _1}_{\\text{$m_1$-times}},\\ldots ,\\underbrace{\\Delta _p,\\ldots ,\\Delta _p}_{\\text{$m_p$-times}}),$ where ${\\rm Vol}_{\\mathbb {Z}}(\\underbrace{\\Delta _1,\\ldots ,\\Delta _1}_{\\text{$m_1$-times}},\\ldots ,\\underbrace{\\Delta _p,\\ldots ,\\Delta _p}_{\\text{$m_p$-times}})\\in \\mathbb {Z}$ is the normalized $n$ -dimensional mixed volume with respect to the lattice $\\mathbb {Z}^n\\subset \\mathbb {R}^n$ $($ see the remark below$)$ .", "Remark 2.4 Let $\\Delta _1,\\ldots ,\\Delta _n$ be lattice polytopes in $\\mathbb {R}^n$ .", "Then their normalized $n$ -dimensional mixed volume ${\\rm Vol}_{\\mathbb {Z}}( \\Delta _1,\\ldots ,\\Delta _n)\\in \\mathbb {Z}$ is defined by the formula ${\\rm Vol}_{\\mathbb {Z}}( \\Delta _1, \\ldots , \\Delta _n)=\\frac{1}{n!", "}\\displaystyle \\sum _{k=1}^n (-1)^{n-k}\\sum _{{{\\scriptstyle \\begin{matrix}I\\subset \\lbrace 1,\\ldots ,n\\rbrace \\\\ \\sharp I=k\\end{matrix}}}}{\\rm Vol}_{\\mathbb {Z}}\\left(\\displaystyle \\sum _{i\\in I} \\Delta _i \\right)$ where ${\\rm Vol}_{\\mathbb {Z}}(\\ \\cdot \\ )= n!", "{\\rm Vol}(\\ \\cdot \\ ) \\in \\mathbb {Z}$ is the normalized $n$ -dimensional volume with respect to the lattice $\\mathbb {Z}^n\\subset \\mathbb {R}^n$ ." ], [ "A vanishing theorem\nfor local systems", "Let $B=\\lbrace b(1), b(2), \\ldots ,b(N)\\rbrace \\subset \\mathbb {Z}^{n-1}$ be a finite subset of the lattice $\\mathbb {Z}^{n-1}$ .", "Let $\\Delta \\subset \\mathbb {R}^{n-1}$ be the convex hull of $B$ in $\\mathbb {R}^{n-1}$ .", "Assume that ${\\rm dim}\\Delta =n-1$ .", "For $z=(z_1, \\ldots , z_N) \\in \\mathbb {C}^N$ we define a Laurent polynomial $P(x)$ on $T_0=( \\mathbb {C}^)^{n-1}$ by $P(x)=\\sum _{j=1}^N z_j x^{b(j)}$ ($x=(x_1, \\ldots , x_{n-1})\\in T_0=( \\mathbb {C}^)^{n-1}$ ).", "Then for $c=(c_1, \\ldots , c_n) \\in \\mathbb {C}^n$ the possibly multi-valued function $P(x)^{-c_n}x_1^{c_1-1} \\cdots x_{n-1}^{c_{n-1}-1}$ on $W=T_0 \\setminus P^{-1}(0)$ generates the local system $\\mathcal {L}= \\mathbb {C}_{W}P(x)^{-c_n}x_1^{c_1-1} \\cdots x_{n-1}^{c_{n-1}-1}.$ Set $a(j)=(b(j), 1) \\in \\mathbb {Z}^n$ ($1 \\le j \\le N$ ) and $A=\\lbrace a(1), a(2), \\ldots ,a(N)\\rbrace \\subset \\mathbb {Z}^{n}$ .", "Then $K= \\mathbb {R}_+ A \\subset \\mathbb {R}^n$ is an $n$ -dimensional closed convex polyhedral cone in $\\mathbb {R}^n$ .", "For a face $\\Gamma \\prec K$ of $K$ let ${\\rm Lin}(\\Gamma ) \\simeq \\mathbb {C}^{{\\rm dim}\\Gamma } \\subset \\mathbb {C}^n$ be the $\\mathbb {C}$ -linear subspace of $\\mathbb {C}^n$ generated by $\\Gamma $ .", "Definition 3.1 (Gelfand-Kapranov-Zelevinsky [10]) We say that the parameter vector $c \\in \\mathbb {C}^n$ is nonresonant (with respect to $A$ ) if for any face $\\Gamma \\prec K$ of $K$ such that ${\\rm dim}\\Gamma =n-1$ we have $c \\notin \\lbrace \\mathbb {Z}^n+{\\rm Lin}(\\Gamma ) \\rbrace $ .", "Definition 3.2 (see [22] etc.)", "We say that the Laurent polynomial $P(x)= \\sum _{j=1}^N z_j x^{b(j)}$ is “weakly\" non-degenerate if for any face $\\gamma $ of $\\Delta $ such that ${\\rm dim}\\gamma < {\\rm dim}\\Delta =n-1$ the hypersurface $\\lbrace x \\in T_0=(\\mathbb {C}^)^{n-1} \\ \| \\ P^{\\gamma }(x)= \\sum _{j: b(j) \\in \\gamma }z_j x^{b(j)}=0 \\rbrace \\subset T_0$ is smooth and reduced.", "Let $\\iota : W=T_0 \\setminus P^{-1}(0)\\hookrightarrow T_0$ be the inclusion map and set $\\mathcal {M}= R \\iota _ \\mathcal {L}\\in {\\bf D}_{c}^{b}(T_0)$ .", "Then the following theorem generalizes one of the results in Gelfand-Kapranov-Zelevinsky [10] to the case where the hypersurface $P^{-1}(0) \\subset T_0$ may have isolated singular points.", "Theorem 3.3 Assume that ${\\rm dim}\\Delta =n-1$ , the parameter vector $c \\in \\mathbb {C}^n$ is nonresonant and the Laurent polynomial $P(x)$ is weakly non-degenerate.", "Then there exists an isomorphism $H^j_c(T_0; \\mathcal {M}) \\simeq H^j(T_0; \\mathcal {M})\\simeq H^j(W ; \\mathcal {L})$ for any $j \\in \\mathbb {Z}$ .", "Moreover we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-1).$ Let $\\Sigma _0$ be the dual fan of $\\Delta $ in $\\mathbb {R}^{n-1}$ and $X$ the (possibly singular) toric variety associated to it.", "Then there exists a natural action of $T_0$ on $X$ whose orbits are parametrized by the faces of $\\Delta $ .", "For a face $\\gamma $ of $\\Delta $ denote by $X_{\\gamma } \\simeq (\\mathbb {C}^)^{{\\rm dim}\\gamma }$ the $T_0$ -orbit associated to $\\gamma $ .", "Note that $X_{\\Delta } \\simeq T_0$ is the unique open dense $T_0$ -orbit in $X$ and its complement $X \\setminus X_{\\Delta }$ is the union of $X_{\\gamma }$ for $\\gamma \\prec \\Delta $ such that ${\\rm dim}\\gamma <n-1$ .", "Let $i :X_{\\Delta } \\simeq T_0 \\hookrightarrow X$ be the inclusion map.", "Then by the weak non-degeneracy of $P(x)$ , the closure $S= \\overline{i (P^{-1}(0))}\\subset X$ of the hypersurface $i( P^{-1}(0)) \\subset i (T_0)$ in $X$ intersects $T_0$ -orbits $X_{\\gamma }$ in $X \\setminus X_{\\Delta }$ transversally.", "Moreover by the nonresonance of $c \\in \\mathbb {C}^n$ , for any $\\gamma \\prec \\Delta $ such that ${\\rm dim}\\gamma =n-2$ the monodromy of the local system $\\mathcal {L}$ around the codimension-one $T_0$ -orbit $X_{\\gamma } \\subset X$ in $X$ is non-trivial.", "Indeed, let $\\gamma \\prec \\Delta $ be such a facet of $\\Delta $ .", "We denote by $\\Gamma $ the facet of the cone $K= \\mathbb {R}_+ A$ generated by $\\gamma \\times \\lbrace 1 \\rbrace \\subset K$ .", "Let $\\nu \\in \\mathbb {Z}^{n-1}\\setminus \\lbrace 0 \\rbrace $ be the primitive inner conormal vector of the facet $\\gamma $ of $\\Delta \\subset \\mathbb {R}^{n-1}$ and set $m= \\min _{v \\in \\Delta }\\langle \\nu , v \\rangle =\\min _{v \\in \\gamma }\\langle \\nu , v \\rangle \\in \\mathbb {Z}.$ Then the primitive inner conormal vector $\\widetilde{\\nu } \\in \\mathbb {Z}^{n}\\setminus \\lbrace 0 \\rbrace $ of the facet $\\Gamma $ of $K \\subset \\mathbb {R}^{n}$ is explicitly given by the formula $\\widetilde{\\nu } =\\left( \\begin{array}{c}\\nu \\\\-m\\end{array} \\right) \\in \\mathbb {Z}^{n}\\setminus \\lbrace 0 \\rbrace .$ and the condition $c=(c_1, \\ldots , c_{n-1}, c_n) \\notin \\lbrace \\mathbb {Z}^n+ {\\rm Lin}(\\Gamma ) \\rbrace $ is equivalent to the one $m( \\gamma ):=\\biggl \\langle \\nu , \\quad \\left( \\begin{array}{c}c_1-1 \\\\\\vdots \\\\c_{n-1}-1\\end{array} \\right) \\biggr \\rangle - m \\cdot c_n\\quad \\notin \\mathbb {Z}.$ We can easily see that the order of the (multi-valued) function $P(x)^{-c_n}x_1^{c_1-1} \\cdots x_{n-1}^{c_{n-1}-1}$ along the codimension-one $T_0$ -orbit $X_{\\gamma } \\subset X$ in $X$ is equal to $m( \\gamma ) \\notin \\mathbb {Z}$ .", "Then by constructing suitable distance functions as in the proof of [7], we can show that for the open embedding $i: T_0 \\hookrightarrow X$ we have $(Ri_ \\mathcal {M})_p \\simeq 0\\qquad \\text{for any} \\ p \\in X \\setminus i(T_0)$ as follows.", "Let us first assume that the point $p \\in X \\setminus i(T_0)$ lies in a 0-dimensional $T_0$ -orbit $X_{\\gamma }$ .", "Let $U_{\\gamma }\\subset X$ be an $(n-1)$ -dimensional affine toric variety containing $p=X_{\\gamma }$ and regard it as a subvariety of $\\mathbb {C}^l$ for some $l$ .", "Then as in the proof of [7] we can construct a distance function on $\\mathbb {C}^l$ to prove the isomorphism $(Ri_* \\mathcal {M})_p \\simeq 0 $ .", "When the point $p \\in X \\setminus i(T_0)$ lies in a $T_0$ -orbit $X_{\\gamma }$ such that ${\\rm dim}X_{\\gamma } = {\\rm dim}\\gamma >0$ , by taking a normal slice of $X_{\\gamma }$ in $X$ we can reduce the problem to the case where ${\\rm dim}X_{\\gamma } =0$ .", "We thus obtain an isomorphism $i_!", "\\mathcal {M}\\simeq Ri_* \\mathcal {M}$ in ${\\bf D}_{c}^{b}(X)$ .", "Applying the functor $R \\Gamma _c(X; \\cdot ) =R \\Gamma (X; \\cdot )$ to it we obtain the desired isomorphisms $H^j_c(T_0; \\mathcal {M}) \\simeq H^j(T_0; \\mathcal {M})\\simeq H^j(W ; \\mathcal {L})$ for $j \\in \\mathbb {Z}$ .", "Now recall that $T_0$ is an affine variety and $\\mathcal {M}\\in {\\bf D}_{c}^{b}(T_0)$ is a perverse sheaf on it (up to some shift).", "Then by Artin's vanishing theorem for perverse sheaves over affine varieties (see [4] etc.)", "we have $H^j_c(T_0; \\mathcal {M}) \\simeq 0 \\quad \\text{for} \\ j< {\\rm dim}T_0 =n-1$ and $H^j(T_0; \\mathcal {M}) \\simeq 0 \\quad \\text{for} \\ j> {\\rm dim}T_0 =n-1,$ from which the last assertion immediately follows.", "This completes the proof.", "By Theorem REF we obtain the following corollary of Theorem REF .", "Corollary 3.4 In the situation of Theorem REF , let $p_1, \\ldots , p_r \\in P^{-1}(0)$ be the (isolated) singular points of $P^{-1}(0) \\subset T_0$ and for $1 \\le i \\le r$ let $\\mu _i>0$ be the Milnor number of $P^{-1}(0)$ at $p_i$ .", "Then we have ${\\rm dim}H^{n-1}(W ; \\mathcal {L}) ={\\rm Vol}_{\\mathbb {Z}}( \\Delta ) -\\sum _{i=1}^r \\mu _i.$ We can generalize Theorem REF to the case where the hypersurface $S= \\overline{i (P^{-1}(0))}\\subset X$ has (stratified) isolated singular points $p$ also in $T_0$ -orbits $X_{\\gamma }\\subset X \\setminus i(T_0)$ as follows.", "For such a point $p \\in S \\cap X_{\\gamma }$ of $S$ let us show that we have the vanishing $(Ri_* \\mathcal {M})_p\\simeq 0$ in general.", "First consider the case where the codimension of $X_{\\gamma }$ in $X$ is one.", "The question being local, it suffices to consider the case where $X= \\mathbb {C}^{n-1}_y \\supset X_{\\gamma }= \\lbrace y_{n-1}=0 \\rbrace $ , $S= \\lbrace f(y)=0 \\rbrace \\ni p=0$ , $T_0= \\mathbb {C}^{n-1} \\setminus \\lbrace y_{n-1}=0 \\rbrace $ , $i: \\mathbb {C}^{n-1} \\setminus \\lbrace y_{n-1}=0 \\rbrace \\hookrightarrow \\mathbb {C}^{n-1}$ and $\\mathcal {L}= \\mathbb {C}_{\\mathbb {C}^{n-1} \\setminus \\lbrace f(y) \\cdot y_{n-1}=0 \\rbrace }f(y)^{\\alpha } y_{n-1}^{\\beta }$ for $\\alpha = -c_n$ and some $\\beta \\in \\mathbb {C}$ (by the notation in the proof of Theorem REF we have $\\beta = m ( \\gamma )$ ).", "Here $f(y)$ is a polynomial on $\\mathbb {C}^{n-1}$ such that $S=f^{-1}(0)$ has a (stratified) isolated singular point at $p=0 \\in S \\cap X_{\\gamma }$ .", "Moreover for the inclusion map $\\iota : \\mathbb {C}^{n-1}\\setminus \\lbrace f(y) \\cdot y_{n-1}=0 \\rbrace \\hookrightarrow \\mathbb {C}^{n-1}\\setminus \\lbrace y_{n-1}=0 \\rbrace $ we have $\\mathcal {M}\\simeq R \\iota _* \\mathcal {L}$ .", "By the nonresonance of $c \\in \\mathbb {C}^n$ we have $\\beta = m ( \\gamma )\\notin \\mathbb {Z}$ and there exists an isomorphism $i_!", "( \\mathbb {C}_{\\mathbb {C}^{n-1}\\setminus \\lbrace y_{n-1}=0 \\rbrace }y_{n-1}^{\\beta } )\\overset{\\sim }{\\longrightarrow }Ri_* ( \\mathbb {C}_{\\mathbb {C}^{n-1}\\setminus \\lbrace y_{n-1}=0 \\rbrace }y_{n-1}^{\\beta } ) .$ Set $\\mathcal {N}= i_!", "( \\mathbb {C}_{\\mathbb {C}^{n-1}\\setminus \\lbrace y_{n-1}=0 \\rbrace }y_{n-1}^{\\beta } )$ .", "Then $\\mathcal {N}$ is a perverse sheaf (up to some shift) on $X= \\mathbb {C}^{n-1}$ and satisfies the condition $\\psi _{f}( \\mathcal {N})_p\\simeq \\phi _{f}( \\mathcal {N})_p$ (use (REF )), where $\\psi _f, \\phi _f : {\\bf D}_{c}^{b}(X)\\longrightarrow {\\bf D}_{c}^{b}( \\lbrace f=0 \\rbrace )$ are the nearby and vanishing cycle functors associated to $f$ respectively (see [4] etc.).", "By the $t$ -exactness of the functor $\\phi _{f}$ the constructible sheaf $\\phi _{f}( \\mathcal {N})$ on $S=f^{-1}(0)$ is perverse (up to some shift).", "Moreover by our assumption its support is contained in the point $\\lbrace p \\rbrace = \\lbrace 0 \\rbrace \\subset X = \\mathbb {C}^{n-1}$ .", "This implies that we have the concentration $H^j \\psi _{f}( \\mathcal {N})_p\\simeq H^j \\phi _{f}( \\mathcal {N})_p\\simeq 0 \\qquad (j \\ne n-2).$ Hence in order to show the vanishing $(Ri_* \\mathcal {M})_p \\simeq 0$ it suffices to prove that the monodromy operator $\\Phi : H^{n-2} \\psi _{f}( \\mathcal {N})_p \\overset{\\sim }{\\longrightarrow }H^{n-2} \\psi _{f}( \\mathcal {N})_p$ does not have the eigenvalue $\\exp (- 2 \\pi i \\alpha )$ .", "For this purpose, we shall use the results in [17].", "Let $\\Gamma _+(f)\\subset \\mathbb {R}_+^{n-1}$ be the convex hull of $\\cup _{v \\in {\\rm supp}(f)} (v+ \\mathbb {R}^{n-1}_+)$ in $\\mathbb {R}^{n-1}_+$ .", "We call it the Newton polyhedron of $f$ at the origin $p=0 \\in \\mathbb {C}^{n-1}$ .", "Definition 3.5 (see [22] etc.)", "We say that $f$ is Newton non-degenerate at the origin $p=0 \\in \\mathbb {C}^{n-1}$ if for any compact face $\\gamma \\prec \\Gamma _+(f)$ of $\\Gamma _+(f)$ the hypersurface $\\lbrace y \\in (\\mathbb {C}^)^{n-1} \\ \| \\ f^{\\gamma }(y)=0 \\rbrace $ of $(\\mathbb {C}^)^{n-1}$ is smooth and reduced.", "For each subset $I \\subset \\lbrace 1,2, \\ldots , n-1 \\rbrace $ we set $\\mathbb {R}_+^I= \\lbrace v=(v_1, \\ldots , v_{n-1})\\in \\mathbb {R}_+^{n-1}\\ \| \\ v_i=0 \\ \\text{for any} \\ i \\notin I \\rbrace \\simeq \\mathbb {R}_+^{\\sharp I}.$ Let $\\gamma _1^I,\\ldots , \\gamma _{n(I)}^I \\prec \\Gamma _+(f) \\cap \\mathbb {R}_+^I$ be the compact facets of $\\Gamma _+(f) \\cap \\mathbb {R}_+^I$ .", "For $1 \\le i \\le n(I)$ denote by $d_i^I \\in \\mathbb {Z}_{>0}$ the lattice distance of $\\gamma _i^I$ from the origin $0 \\in \\mathbb {R}^{I}_+$ and let $u_i^I=(u_{i,1}^I, \\ldots ,u_{i,n-1}^I) \\in \\mathbb {R}_+^I \\cap \\mathbb {Z}^{n-1}$ be the unique (non-zero) primitive vector which takes its minimum exactly on $\\gamma _i^I$ .", "For simplicity we set $\\delta _i^I:=u_{i,n-1}^I$ .", "Finally we define a finite subset $E_p \\subset \\mathbb {C}$ of $\\mathbb {C}$ by $E_p= \\bigcup _{I: I \\ni n-1}\\bigcup _{i=1}^{n(I)}\\lbrace \\lambda \\in \\mathbb {C}\\ \|\\ \\lambda ^{d_i^I} =\\exp (2 \\pi \\sqrt{-1}\\beta \\cdot \\delta _i^I) \\rbrace .$ Then the following result is a special case of [17].", "Proposition 3.6 In the above situation, assume moreover that $f$ is Newton non-degenerate at the origin $p=0 \\in \\mathbb {C}^{n-1}$ .", "Then the set of the eigenvalues of the monodromy operator $\\Phi : H^{n-2}\\psi _{f}( \\mathcal {N})_p \\overset{\\sim }{\\longrightarrow }H^{n-2} \\psi _{f}( \\mathcal {N})_p$ is contained in $E_p$ .", "Corollary 3.7 Assume that ${\\rm dim}\\Delta =n-1$ , $c \\in \\mathbb {C}^n$ is nonresonant, $\\exp (- 2 \\pi \\sqrt{-1} \\alpha )= \\exp ( 2 \\pi \\sqrt{-1} c_n )\\notin E_p$ and $f$ is Newton non-degenerate at the origin $p=0 \\in \\mathbb {C}^{n-1}$ .", "Then we have $(Ri_* \\mathcal {M})_p \\simeq 0$ .", "In fact, by [17] we can generalize this corollary to the case where the codimension of the $T_0$ -orbit $X_{\\gamma }$ in $X_{\\gamma }\\subset X \\setminus i(T_0)$ containing the (stratified) isolated singular point $p$ of $S$ is larger than one.", "We leave the precise formulation to the reader and omit the details here.", "In this way, our Theorem REF can be generalized to the case where $S$ has (stratified) isolated singular points $p$ also in $T_0$ -orbits $X_{\\gamma }\\subset X \\setminus i(T_0)$ .", "In particular we have the following result.", "For a face $\\gamma $ of $\\Delta $ let $L_{\\gamma } \\simeq \\mathbb {R}^{{\\rm dim}\\gamma }$ be the linear subspace of $\\mathbb {R}^{n-1}$ parallel to the affice span of $\\gamma $ in $\\mathbb {R}^{n-1}$ and consider the $\\gamma $ -part $P^{\\gamma }$ of $P$ as a function on $T_{\\gamma }= {\\rm Spec}( \\mathbb {C}[ L_{\\gamma } \\cap \\mathbb {Z}^{n-1} ] ) \\simeq ( \\mathbb {C}^)^{{\\rm dim}\\gamma }$ .", "Theorem 3.8 Assume that ${\\rm dim}\\Delta =n-1$ and for any face $\\gamma $ of $\\Delta $ the hypersurface $(P^{\\gamma })^{-1}(0)\\subset T_{\\gamma }$ of $T_{\\gamma }$ has only isolated singular points.", "Then for generic parameter vectors $c \\in \\mathbb {C}^n$ we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-1).$ From now, let us generalize Theorem REF to the following more general situation.", "For $0<k<n$ let $B_i=\\lbrace b_i(1), b_i(2), \\ldots ,b_i(N_i)\\rbrace \\subset \\mathbb {Z}^{n-k}$ ($1 \\le i \\le k$ ) be $k$ finite subsets of the lattice $\\mathbb {Z}^{n-k}$ and set $N=N_1 + N_2 + \\cdots +N_k$ .", "For $1 \\le i \\le k$ and $(z_{i1}, \\ldots , z_{iN_i}) \\in \\mathbb {C}^{N_i}$ we define a Laurent polynomial $P_i(x)$ on $T_0=( \\mathbb {C}^)^{n-k}$ by $P_i(x)=\\sum _{j=1}^{N_i} z_{ij} x^{b_i(j)}$ ($x=(x_1, \\ldots , x_{n-k}) \\in T_0=( \\mathbb {C}^)^{n-k}$ ).", "Let us set $W=T_0 \\setminus \\cup _{i=1}^k P_i^{-1}(0)$ .", "Then for $c=(c_1, \\ldots , c_{n-k}, \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\in \\mathbb {C}^n$ the possibly multi-valued function $P_1(x)^{-\\widetilde{c_1}} \\cdots P_k(x)^{-\\widetilde{c_k}}x_1^{c_1-1} \\cdots x_{n-k}^{c_{n-k}-1}$ on $W$ generates the local system $\\mathcal {L}= \\mathbb {C}_{W}P_1(x)^{-\\widetilde{c_1}} \\cdots P_k(x)^{-\\widetilde{c_k}}x_1^{c_1-1} \\cdots x_{n-k}^{c_{n-k}-1}.$ Let $e_i=(0,0, \\ldots , 0,1,0, \\ldots , 0) \\in \\mathbb {Z}^k$ ($1 \\le i \\le k$ ) be the standard basis of $\\mathbb {Z}^k$ and set $a_i(j)=(b_i(j), e_i) \\in \\mathbb {Z}^{n-k}\\times \\mathbb {Z}^k=\\mathbb {Z}^n$ ($1 \\le i \\le k$ , $1 \\le j \\le N_i$ ) and $A=\\lbrace a_1(1), \\ldots , a_1(N_1),\\ldots \\ldots ,a_k(1), \\ldots , a_k(N_k)\\rbrace \\subset \\mathbb {Z}^{n}.$ For $1 \\le i \\le k$ let $\\Delta _i \\subset \\mathbb {R}^{n-k}$ be the convex hull of $B_i$ in $\\mathbb {R}^{n-k}$ .", "Denote by $\\Delta \\subset \\mathbb {R}^{n-k}$ their Minkowski sum $\\Delta _1 + \\cdots +\\Delta _k$ .", "Assume that ${\\rm dim}\\Delta =n-k$ .", "Then by using the $n$ -dimensional closed convex polyhedral cone $K= \\mathbb {R}_+ A \\subset \\mathbb {R}^n$ generated by $A$ in $\\mathbb {R}^n$ we can define the nonresonance of the parameter $c \\in \\mathbb {C}^n$ as in Definition REF .", "For a face $\\gamma \\prec \\Delta $ of $\\Delta $ let $\\gamma _i \\prec \\Delta _i$ be the faces of $\\Delta _i$ ($1 \\le i \\le k$ ) canonically associated to $\\gamma $ such that $\\gamma = \\gamma _1 + \\cdots +\\gamma _k$ .", "Definition 3.9 (see [22] etc.)", "We say that the $k$ -tuple of the Laurent polynomials $(P_1, \\ldots , P_k)$ is “weakly\" (resp.", "“strongly\") non-degenerate if for any face $\\gamma $ of $\\Delta $ such that ${\\rm dim}\\gamma < {\\rm dim}\\Delta = n-k$ (resp.", "${\\rm dim}\\gamma \\le {\\rm dim}\\Delta = n-k$ ) and non-empty subset $J \\subset \\lbrace 1,2, \\ldots , k \\rbrace $ the subvariety $\\lbrace x \\in T_0=(\\mathbb {C}^)^{n-k} \\ \| \\ P_i^{\\gamma _i}(x)=0 \\ \\ (i \\in J) \\rbrace \\subset T_0$ is a non-degenerate complete intersection.", "Remark 3.10 Denote the convex hull of $\\cup _{i=1}^k (\\Delta _i \\times \\lbrace e_i \\rbrace ) \\subset \\mathbb {R}^{n-k} \\times \\mathbb {R}^{k} =\\mathbb {R}^{n}$ in $\\mathbb {R}^{n}$ by $\\Delta _1 * \\cdots * \\Delta _k$ .", "Then $\\Delta _1 * \\cdots * \\Delta _k$ is naturally identified with the Newton polytope of the Laurent polynomial $R(x,t)=\\sum _{i=1}^k P_i(x)t_i $ on $\\widetilde{T_0}:= T_0 \\times (\\mathbb {C}^)_t^k \\simeq (\\mathbb {C}^)_{x,t}^{n}$ .", "In [10] the authors considered the condition that for any face $\\gamma $ of $\\Delta _1 * \\cdots * \\Delta _k$ the hypersurface $\\lbrace (x,t) \\in \\widetilde{T_0} \\ \| \\ R^{\\gamma }(x,t)=0 \\rbrace \\subset \\widetilde{T_0}$ of $\\widetilde{T_0}$ is smooth and reduced.", "It is easy to see that our strong non-degeneracy of the $k$ -tuple $(P_1, \\ldots , P_k)$ in Definition REF is equivalent to their condition.", "Let $\\iota :W=T_0 \\setminus \\cup _{i=1}^k P_i^{-1}(0)\\longrightarrow T_0$ be the inclusion map and set $\\mathcal {M}= R \\iota _* \\mathcal {L}\\in {\\bf D}_{c}^{b}(T_0)$ .", "Theorem 3.11 Assume that ${\\rm dim}\\Delta = n-k$ , the parameter vector $c \\in \\mathbb {C}^n$ is nonresonant and $(P_1, \\ldots , P_k)$ is weakly non-degenerate.", "Then there exists an isomorphism $H^j_c(T_0; \\mathcal {M}) \\simeq H^j(T_0; \\mathcal {M})\\simeq H^j(W ; \\mathcal {L})$ for any $j \\in \\mathbb {Z}$ .", "Moreover we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-k).$ The proof is similar to that of Theorem REF .", "Let $\\Sigma _0$ be the dual fan of $\\Delta $ in $\\mathbb {R}^{n-k}$ and $X$ the (possibly singular) toric variety associated to it.", "For a face $\\gamma $ of $\\Delta $ we denote by $X_{\\gamma } \\simeq (\\mathbb {C}^)^{{\\rm dim}\\gamma }$ the $T_0$ -orbit associated to $\\gamma $ .", "Let $i :X_{\\Delta } \\simeq T_0 \\hookrightarrow X$ be the inclusion map.", "Then by the weak non-degeneracy of the $k$ -tuple $(P_1, \\ldots , P_k)$ , for any $T_0$ -orbits $X_{\\gamma }$ in $X \\setminus X_{\\Delta }$ and the closure $S= \\overline{i (\\cup _{i=1}^k P_i^{-1}(0))}\\subset X$ of the hypersurface $i(\\cup _{i=1}^kP_i^{-1}(0)) \\subset i (T_0)$ in $X$ their intersection $S \\cap X_{\\gamma }\\subset X_{\\gamma }$ is a normal crossing divisor in $X_{\\gamma }$ .", "In fact $S$ itself is normal crossing on a neighborhood of such $X_{\\gamma }$ and any irreducible component of it intersects $X_{\\gamma }$ transversally.", "Moreover by the nonresonance of $c \\in \\mathbb {C}^n$ , for any $\\gamma \\prec \\Delta $ such that ${\\rm dim}\\gamma =n-k-1$ the monodromy of the local system $\\mathcal {L}$ around the codimension-one $T_0$ -orbit $X_{\\gamma } \\subset X$ in $X$ is non-trivial.", "Indeed, let $\\gamma \\prec \\Delta $ be such a facet of $\\Delta $ and $\\gamma _i \\prec \\Delta _i$ the faces of $\\Delta _i$ ($1 \\le i \\le k$ ) associated to $\\gamma $ such that $\\gamma = \\gamma _1 + \\cdots +\\gamma _k$ .", "We denote the convex hull of $\\cup _{i=1}^k (\\Delta _i \\times \\lbrace e_i \\rbrace )$ (resp.", "$\\cup _{i=1}^k (\\gamma _i \\times \\lbrace e_i \\rbrace )$ ) $\\subset \\mathbb {R}^{n-k} \\times \\mathbb {R}^{k} = \\mathbb {R}^{n}$ in $\\mathbb {R}^{n}$ by $\\Delta _1 \\cdots * \\Delta _k$ (resp.", "$\\gamma _1 * \\cdots * \\gamma _k$ ).", "Then $\\Delta _1 * \\cdots * \\Delta _k$ is the join of $\\Delta _1, \\ldots , \\Delta _k$ and $\\gamma _1 * \\cdots * \\gamma _k$ is its facet.", "We denote by $\\Gamma $ the facet of the cone $K= \\mathbb {R}_+ A$ generated by $\\gamma _1 * \\cdots * \\gamma _k \\subset K$ .", "Let $\\nu \\in \\mathbb {Z}^{n-k}\\setminus \\lbrace 0 \\rbrace $ be the primitive inner conormal vector of the facet $\\gamma $ of $\\Delta \\subset \\mathbb {R}^{n-k}$ and for $1 \\le i \\le k$ set $m_i= \\min _{v \\in \\Delta _i}\\langle \\nu , v \\rangle =\\min _{v \\in \\gamma _i}\\langle \\nu , v \\rangle \\in \\mathbb {Z}.$ Then the primitive inner conormal vector $\\widetilde{\\nu } \\in \\mathbb {Z}^{n}\\setminus \\lbrace 0 \\rbrace $ of the facet $\\Gamma $ of $K \\subset \\mathbb {R}^{n}$ is explicitly given by the formula $\\widetilde{\\nu } =\\left( \\begin{array}{c}\\nu \\\\-m_1 \\\\\\vdots \\\\-m_k\\end{array} \\right) \\in \\mathbb {Z}^{n}\\setminus \\lbrace 0 \\rbrace .$ and the condition $c=(c_1, \\ldots , c_{n-k}, \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\notin \\lbrace \\mathbb {Z}^n+ {\\rm Lin}(\\Gamma ) \\rbrace $ is equivalent to the one $m( \\gamma ):=\\biggl \\langle \\nu , \\quad \\left( \\begin{array}{c}c_1-1 \\\\\\vdots \\\\c_{n-k}-1\\end{array} \\right) \\biggr \\rangle - \\sum _{i=1}^k m_i \\cdot \\widetilde{c_i}\\quad \\notin \\mathbb {Z}.$ Moreover we can easily see that the order of the (multi-valued) function $P_1(x)^{-\\widetilde{c_1}} \\cdots P_k(x)^{-\\widetilde{c_k}}x_1^{c_1-1} \\cdots x_{n-k}^{c_{n-k}-1}$ along the codimension-one $T_0$ -orbit $X_{\\gamma } \\subset X$ in $X$ is equal to $m( \\gamma ) \\notin \\mathbb {Z}$ .", "Finally, by constructing suitable distance functions as in the proof of [7], we can show that $(Ri_* \\mathcal {M})_p \\simeq 0 \\qquad \\text{for any} \\ p \\in X \\setminus T_0.$ Namely there exists an isomophism $i_!", "\\mathcal {M}\\simeq Ri_* \\mathcal {M}$ in ${\\bf D}_{c}^{b}(X)$ .", "Applying the functor $R \\Gamma _c(X; \\cdot ) =R \\Gamma (X; \\cdot )$ to it we obtain the desired isomorphisms $H^j_c(T_0; \\mathcal {M}) \\simeq H^j(T_0; \\mathcal {M})\\simeq H^j(W ; \\mathcal {L})$ for $j \\in \\mathbb {Z}$ .", "Then the remaining assertion can be proved as in the proof of Theorem REF .", "This completes the proof.", "As in the case where $k=1$ we have the following result.", "For a face $\\gamma $ of $\\Delta $ let $L_{\\gamma } \\simeq \\mathbb {R}^{{\\rm dim}\\gamma }$ be the linear subspace of $\\mathbb {R}^{n-k}$ parallel to the affice span of $\\gamma $ in $\\mathbb {R}^{n-k}$ and for $1 \\le i \\le k$ consider the $\\gamma _i$ -part $P_i^{\\gamma _i}$ of $P_i$ as a function on $T_{\\gamma }= {\\rm Spec}( \\mathbb {C}[ L_{\\gamma } \\cap \\mathbb {Z}^{n-k} ] ) \\simeq ( \\mathbb {C}^)^{{\\rm dim}\\gamma }$ .", "Theorem 3.12 Assume that ${\\rm dim}\\Delta = n-k$ and for any $1 \\le i \\le k$ the hypersurface $P_i^{-1}(0) \\subset T_0$ of $T_0$ has only isolated singular points.", "Assume moreover that for any face $\\gamma $ of $\\Delta $ such that ${\\rm dim}\\gamma <{\\rm dim}\\Delta = n-k$ and non-empty subset $J \\subset \\lbrace 1,2, \\ldots , k \\rbrace $ the $k$ -tuple of the Laurent polynomials $(P_1, \\ldots , P_k)$ satisfies the following condition: If $J=\\lbrace i \\rbrace $ for some $1 \\le i \\le k$ and ${\\rm dim}\\gamma _i= {\\rm dim}\\gamma ={\\rm dim}\\Delta -1= n-k-1$ the hypersurface $(P_i^{\\gamma _i})^{-1}(0)\\subset T_{\\gamma }$ of $T_{\\gamma }$ has only isolated singular points.", "Otherwise, the subvariety $\\lbrace x \\in T_0=(\\mathbb {C}^)^{n-k} \\ \| \\ P_i^{\\gamma _i}(x)=0 \\ \\ (i \\in J) \\rbrace \\subset T_0$ of $T_0$ is a non-degenerate complete intersection.", "Then for generic parameter vectors $c \\in \\mathbb {C}^n$ we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-k).$ Let $\\Sigma _0$ be the dual fan of $\\Delta $ in $\\mathbb {R}^{n-k}$ and $X$ the (possibly singular) toric variety associated to it.", "Then our assumptions imply that for any $1 \\le i \\le k$ the hypersurface $S_i= \\overline{i ( P_i^{-1}(0))}\\subset X$ has only stratified isolated singular points in $X$ and we can prove the assertion following the proofs of Theorems REF and REF .", "For a face $\\gamma $ of $\\Delta $ and $1 \\le i \\le k$ such that ${\\rm dim}\\gamma _i <{\\rm dim}\\gamma \\le n-k-1$ the hypersurface $(P_i^{\\gamma _i})^{-1}(0)\\subset T_{\\gamma }$ of $T_{\\gamma }$ is smooth or has non-isolated singularities.", "In the latter case, we cannot prove the concentration in Theorem REF by our methods.", "This is the reason why we do not allow such cases in our assumptions of Theorem REF .", "However, in the very special case where the Newton polytopes $\\Delta _1, \\ldots , \\Delta _k$ are similar each other, we do not have this problem and obtain the following simpler result.", "Theorem 3.13 Assume that ${\\rm dim}\\Delta = n-k$ , the Newton polytopes $\\Delta _1, \\ldots , \\Delta _k$ are similar each other and for any face $\\gamma $ of $\\Delta $ and $1 \\le i \\le k$ the hypersurface $(P_i^{\\gamma _i})^{-1}(0)\\subset T_{\\gamma }$ of $T_{\\gamma }$ has only isolated singular points.", "Assume moreover that for any face $\\gamma $ of $\\Delta $ such that ${\\rm dim}\\gamma <{\\rm dim}\\Delta = n-k$ and any subset $J \\subset \\lbrace 1,2, \\ldots , k \\rbrace $ such that $\\sharp J \\ge 2$ the subvariety $\\lbrace x \\in T_0=(\\mathbb {C}^)^{n-k} \\ \| \\ P_i^{\\gamma _i}(x)=0 \\ \\ (i \\in J) \\rbrace \\subset T_0$ of $T_0$ is a non-degenerate complete intersection.", "Then for generic parameter vectors $c \\in \\mathbb {C}^n$ we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-k).$" ], [ "Some results on the twisted\nMorse theory", "In this section, we prepare some auxiliary results on the twisted Morse theory which will be used in Section .", "The following proposition is a refinement of the results in [5].", "See also [7].", "Proposition 4.1 Let $T$ be an algebraic torus $( \\mathbb {C}^)^n_x$ and $T= \\sqcup _{\\alpha } Z_{\\alpha }$ its algebraic stratification.", "In particular we assume that each stratum $Z_{\\alpha }$ in it is smooth.", "Let $h(x)$ be a Laurent polynomial on $T=( \\mathbb {C}^)^n_x$ such that the hypersurface $\\lbrace h=0 \\rbrace \\subset T$ intersects $Z_{\\alpha }$ transversally for any $\\alpha $ .", "For $a \\in \\mathbb {C}^n$ consider the (possibly multi-valued) function $g_a(x):=h(x) x^{-a}$ on $T$ .", "Then there exists a non-empty Zariski open subset $\\Omega \\subset \\mathbb {C}^n$ of $\\mathbb {C}^n$ such that the restriction $g_a\|_{Z_{\\alpha }}:Z_{\\alpha } \\longrightarrow \\mathbb {C}$ of $g_a$ to $Z_{\\alpha }$ has only isolated non-degenerate (i.e.", "Morse type) critical points for any $a \\in \\Omega \\subset \\mathbb {C}^n$ and $\\alpha $ .", "We may assume that each stratum $Z_{\\alpha }$ is connected.", "We fix a stratum $Z_{\\alpha }$ and set $k= {\\rm dim}Z_{\\alpha }$ .", "For a subset $I \\subset \\lbrace 1,2, \\ldots , n \\rbrace $ such that $\|I\|=k= {\\rm dim}Z_{\\alpha }$ denote by $\\pi _I : T=( \\mathbb {C}^)^n_x\\longrightarrow ( \\mathbb {C}^)^k$ the projection associated to $I$ .", "We also denote by $Z_{\\alpha , I} \\subset Z_{\\alpha }$ the maximal Zariski open subset of $Z_{\\alpha }$ such that the restriction of $\\pi _I$ to it is locally biholomorphic.", "By the implicit function theorem, the variety $Z_{\\alpha }$ is covered by such open subsets $Z_{\\alpha , I}$ .", "For simplicity, let us consider the case where $I= \\lbrace 1,2, \\ldots ,k \\rbrace \\subset \\lbrace 1,2, \\ldots , n \\rbrace $ .", "Then we may regard $g_a\|_{Z_{\\alpha }}$ locally as a function $g_{a, \\alpha , I}(x_1, \\ldots , x_k)$ on the Zariski open subset $\\pi _I (Z_{\\alpha , I})\\subset ( \\mathbb {C}^)^k$ of the form $g_{a, \\alpha , I}(x_1, \\ldots , x_k) =\\frac{h_{a, \\alpha , I}(x_1, \\ldots ,x_k)}{x_1^{a_1} \\cdots x_k^{a_k}}.$ By our assumption, the hypersurface $\\lbrace h_{a, \\alpha , I}=0 \\rbrace \\subset \\pi _I (Z_{\\alpha , I}) \\subset ( \\mathbb {C}^)^k$ is smooth.", "Then as in the proof of [7] we can show that there exists a non-empty Zariski open subset $\\Omega _{\\alpha , I} \\subset \\mathbb {C}^n$ such that the (possibly multi-valued) function $g_{a, \\alpha , I}(x_1, \\ldots , x_k)$ on $\\pi _I (Z_{\\alpha , I}) \\subset ( \\mathbb {C}^)^k$ has only isolated non-degenerate (i.e.", "Morse type) critical points for any $a \\in \\Omega _{\\alpha , I}\\subset \\mathbb {C}^n$ .", "This completes the proof.", "Corollary 4.2 In the situation of Proposition REF , assume moreover that for the Newton polytope $NP(h) \\subset \\mathbb {R}^n$ of $h$ we have ${\\rm dim}NP(h)=n$ .", "Then there exists $a \\in {\\rm Int}NP(h)$ such that the restriction $g_a\|_{Z_{\\alpha }}: Z_{\\alpha } \\longrightarrow \\mathbb {C}$ of $g_a$ to $Z_{\\alpha }$ has only isolated non-degenerate (i.e.", "Morse type) critical points for any $\\alpha $ .", "Now let $Q_1, \\ldots , Q_l$ be Laurent polynomials on $T=( \\mathbb {C}^)^{n}$ and for $1 \\le i \\le l$ denote by $\\Delta _i \\subset \\mathbb {R}^{n}$ the Newton polytope $NP(Q_i)$ of $Q_i$ .", "Set $\\Delta = \\Delta _1 + \\cdots + \\Delta _l$ .", "Then by Corollary REF we obtain the following result which might be of independent interest.", "Theorem 4.3 Let $\\mathcal {L}$ be a non-trivial local system of rank one on $T=( \\mathbb {C}^)^{n}$ .", "Assume that for any $1 \\le i \\le l$ we have ${\\rm dim}\\Delta _i =n$ and the subvariety $Z_i = \\lbrace x \\in T \\ \| \\ Q_1(x)= \\cdots = Q_i(x)=0 \\rbrace \\subset T$ of $T$ is a non-degenerate complete intersection.", "Then for any $1 \\le i \\le l$ we have the concentration $H^j(Z_i ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-i).$ Moreover we have ${\\rm dim}H^{n-i} (Z_i ; \\mathcal {L}) =\\displaystyle \\sum _{{{\\scriptstyle \\begin{matrix}m_1,\\ldots ,m_i \\ge 1\\\\m_1+\\cdots +m_i=n\\end{matrix}}}}{\\rm Vol}_{\\mathbb {Z}}(\\underbrace{\\Delta _1,\\ldots ,\\Delta _1}_{\\text{$m_1$-times}},\\ldots ,\\underbrace{\\Delta _i,\\ldots ,\\Delta _i}_{\\text{$m_i$-times}}).$ We prove the assertion by induction on $i$ .", "For $i=0$ we have $Z_i=T$ and the assertion is obvious.", "Since $Z_i \\subset T$ is affine, by Artin's vanishing theorem we have the concentration $H^j(Z_i ; \\mathcal {L}) \\simeq 0 \\qquad (j > n-i= {\\rm dim}Z_i).$ On the other hand, by Corollary REF there exists $a_i \\in {\\rm Int}NP(Q_i) \\subset \\mathbb {R}^n$ such that the real-valued function $g_i: Z_{i-1} \\longrightarrow \\mathbb {R}, \\qquad x \\longmapsto \| Q_i(x) x^{-a_i} \|$ has only isolated non-degenerate (Morse type) critical points.", "Note that the Morse index of $g_i$ at each critical point is ${\\rm dim}Z_{i-1} = n-i+1$ .", "Let $\\Sigma _0$ be the dual fan of the $n$ -dimensional polytope $\\Delta $ in $\\mathbb {R}^n$ and $\\Sigma $ its smooth subdivision.", "We denote by $X_{\\Sigma }$ the toric variety associated to $\\Sigma $ .", "Then $X_{\\Sigma }$ is a smooth compactification of $T$ such that $D= X_{\\Sigma } \\setminus T$ is a normal crossing divisor in it.", "By our assumption, the closures $\\overline{Z_{i-1}}, \\overline{Z_{i}}\\subset X_{\\Sigma }$ of $Z_{i-1}, Z_i$ in $X_{\\Sigma }$ are smooth.", "Moreover they intersect $D$ etc.", "transversally.", "Let $U$ be a sufficiently small tubular neighborhood of $\\overline{Z_i}\\cap D$ in $\\overline{Z_{i-1}}$ .", "Then by [27] (see also [15]), for any $t \\in \\mathbb {R}_+$ there exist isomorphisms $H^j( \\lbrace g_i<t \\rbrace ; \\mathcal {L}) \\simeq H^j( \\lbrace g_i<t \\rbrace \\setminus U ; \\mathcal {L})\\qquad (j \\in \\mathbb {Z}).$ Moreover the level set $g_i^{-1}(t) \\cap (Z_{i-1} \\setminus U)$ of $g_i$ in $Z_{i-1} \\setminus U$ is compact in $Z_{i-1}$ and intersects $\\partial U$ transversally for any $t \\in \\mathbb {R}_+$ .", "Hence for $t \\gg 0$ we have isomorphisms $H^j( \\lbrace g_i<t \\rbrace ; \\mathcal {L})\\simeq H^j(Z_{i-1} ; \\mathcal {L})\\qquad (j \\in \\mathbb {Z}).$ Moreover for $0 < t \\ll 1$ we have isomorphisms $H^j( \\lbrace g_i<t \\rbrace ; \\mathcal {L})\\simeq H^j(Z_{i} ; \\mathcal {L})\\qquad (j \\in \\mathbb {Z}).$ When $t \\in \\mathbb {R}$ decreases passing through one of the critical values of $g_i$ , only the dimensions of $H^{n-i+1}( \\lbrace g_i<t \\rbrace ; \\mathcal {L})$ and $H^{n-i}( \\lbrace g_i<t \\rbrace ; \\mathcal {L})$ may change and the other cohomology groups $H^j( \\lbrace g_i<t \\rbrace ; \\mathcal {L})$ $(j \\ne n-i+1, n-i)$ remain the same.", "Then by our induction hypothesis for $i-1$ and (REF ) we obtain the desired concentration $H^j(Z_i ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-i).$ Moreover the last assertion follows from Theorem REF .", "This completes the proof.", "From now on, assume also that the $l$ -tuple $(Q_1, \\ldots , Q_l)$ is strongly non-degenerate and ${\\rm dim}\\Delta _l=n$ .", "Let $T= \\sqcup _{\\alpha }Z_{\\alpha }$ be the algebraic stratification of $T$ associated to the hypersurface $S= \\cup _{i=1}^{l-1} Q_i^{-1}(0) \\subset T$ and set $M=T \\setminus S$ .", "Then by Corollary REF there exists $a \\in {\\rm Int}( \\Delta _l)$ such that the restriction of the (possibly multi-valued) function $Q_l(x)x^{-a}$ to $Z_{\\alpha }$ has only isolated non-degenerate (i.e.", "Morse type) critical points for any $\\alpha $ .", "In particular, it has only stratified isolated singular points.", "We fix such $a \\in {\\rm Int}( \\Delta _l)$ and define a real-valued function $g: T \\longrightarrow \\mathbb {R}_+$ by $g(x)=\| Q_l(x)x^{-a} \|$ .", "For $t \\in \\mathbb {R}_+$ we set also $M_t= \\lbrace x \\in M=T \\setminus S \\ \| \\ g(x)<t \\rbrace \\subset M.$ Then we have the following result.", "Lemma 4.4 Let $\\mathcal {L}$ be a local system on $M=T \\setminus S$ .", "Then for any $c >0$ there exists a sufficiently small $0 < \\varepsilon \\ll 1$ such that we have the concentration $H^j( M_{c+ \\varepsilon }, M_{c- \\varepsilon } ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n).$ Let $\\Sigma _0$ be the dual fan of the $n$ -dimensional polytope $\\Delta $ in $\\mathbb {R}^n$ and $\\Sigma $ its smooth subdivision.", "We denote by $X_{\\Sigma }$ the toric variety associated to $\\Sigma $ .", "Then $X_{\\Sigma }$ is a smooth compactification of $T$ such that $D= X_{\\Sigma } \\setminus T$ is a normal crossing divisor in it.", "By the strong non-degeneracy of $(Q_1, \\ldots , Q_l)$ , the hypersurface $\\overline{Q_l^{-1}(0)} \\subset X_{\\Sigma }$ intersects $D$ etc.", "transversally.", "Let $U$ be a sufficiently small tubular neighborhood of $\\overline{Q_l^{-1}(0)}\\cap D$ in $X_{\\Sigma }$ and for $t \\in \\mathbb {R}_+$ set $M_t^{\\prime }=M_t \\setminus U$ .", "Then by [27], for any $t \\in \\mathbb {R}_+$ there exist isomorphisms $H^j( M_t ; \\mathcal {L}) \\simeq H^j( M_t^{\\prime } ; \\mathcal {L})\\qquad (j \\in \\mathbb {Z}).$ Moreover the level set $g^{-1}(t) \\cap (T \\setminus U)$ of $g$ in $T \\setminus U$ is compact in $T$ and intersects $\\partial U$ transversally for any $t \\in \\mathbb {R}_+$ .", "For $c>0$ let $p_1, \\ldots , p_r\\in T \\setminus g^{-1}(0)=T \\setminus Q_l^{-1}(0)$ be the stratified isolated singular points of the function $h(x)=Q_l(x)x^{-a}$ in $T$ such that $g(p_i)=\|h(p_i)\|=c$ .", "Note that we have $g(x)=\|h(x)\|= \\exp [ {\\rm Re}\\lbrace \\log h(x) \\rbrace ].$ Then there exist small open balls $B_i$ centered at $p_i$ in $T$ and $0 < \\varepsilon \\ll 1$ such that we have isomorphisms $H^j( M_{c+ \\varepsilon }^{\\prime },M_{c- \\varepsilon }^{\\prime } ; \\mathcal {L})\\simeq \\bigoplus _{i=1}^rH^j( B_i \\cap M_{c+ \\varepsilon },B_i \\cap M_{c- \\varepsilon } ; \\mathcal {L})\\qquad (j \\in \\mathbb {Z}).$ For $1 \\le i \\le r$ by taking a local branch $\\log h$ of the logarithm of the function $h \\ne 0$ on a neighborhood of $p_i \\in T \\setminus h^{-1}(0)$ we set $f_i= \\log h - \\log h(p_i)$ .", "Then $f_i$ has also a stratified isolated singular point at $p_i$ .", "Let $F_i \\subset B_i$ be the Milnor fiber of $f_i$ at $p_i \\in f_i^{-1}(0)$ .", "Then for any $1 \\le i \\le r$ by shrinking $B_i$ if necessary we can easily prove the isomorphisms $H^j( B_i \\cap M_{c+ \\varepsilon },B_i \\cap M_{c- \\varepsilon } ; \\mathcal {L})\\simeq H^j( B_i \\setminus S,F_i \\setminus S ; \\mathcal {L})\\qquad (j \\in \\mathbb {Z}).$ Let $j : M=T \\setminus S \\hookrightarrow T$ be the inclusion.", "Since the Milnor fibers $F_i \\subset B_i$ intersect each stratum $Z_{\\alpha }$ transversally, we have also isomorphisms $H^j( B_i \\setminus S,F_i \\setminus S ; \\mathcal {L})\\simeq H^{j-1} \\phi _{f_i}( Rj_* \\mathcal {L})_{p_i}\\qquad (j \\in \\mathbb {Z}),$ where $\\phi _{f_i}$ are Deligne's vanishing cycle functors.", "Hence by (the proof of) [4] the assertion follows from ${\\rm supp} \\ \\phi _{f_i}( Rj_* \\mathcal {L}) \\subset \\lbrace p_i \\rbrace \\qquad (1 \\le i \\le r)$ and the fact that $Rj_* \\mathcal {L}$ and $\\phi _{f_i}( Rj_* \\mathcal {L})$ are perverse sheaves (up to some shifts).", "This completes the proof." ], [ "A new vanishing theorem", "Now let $P_1, \\ldots , P_k$ be Laurent polynomials on $T_0=( \\mathbb {C}^)^{n-k}$ and for $1 \\le i \\le k$ denote by $\\Delta _i \\subset \\mathbb {R}^{n-k}$ the Newton polytope $NP(P_i)$ of $P_i$ .", "Set $\\Delta = \\Delta _1 + \\cdots + \\Delta _k$ .", "Let us set $W=T_0 \\setminus \\cup _{i=1}^k P_i^{-1}(0)$ and for $(c, \\widetilde{c} )=(c_1, \\ldots , c_{n-k}, \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\in \\mathbb {C}^n$ consider the local system $\\mathcal {L}= \\mathbb {C}_{W}P_1(x)^{\\widetilde{c_1}} \\cdots P_k(x)^{\\widetilde{c_k}}x_1^{c_1} \\cdots x_{n-k}^{c_{n-k}}$ on $W$ .", "Theorem 5.1 Assume that the $k$ -tuple of the Laurent polynomials $(P_1, \\ldots , P_k)$ is strongly non-degenerate, $(c, \\widetilde{c} ) =(c_1, \\ldots , c_{n-k}, \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\notin \\mathbb {Z}^n$ and for any $1 \\le i \\le k$ we have ${\\rm dim}\\Delta _i =n-k$ .", "Then we have the concentration $H^j(W ; \\mathcal {L}) \\simeq 0 \\qquad (j \\ne n-k).$ Set $T=T_0 \\times ( \\mathbb {C}^)^k_{t_1, \\ldots , t_k}\\simeq ( \\mathbb {C}^)^n_{x,t}$ and consider the Laurent polynomials $\\widetilde{P_i}(x,t)=t_i -P_i(x)\\qquad (1 \\le i \\le k)$ on $T$ .", "For $1 \\le i \\le k$ we set also $Z_i = \\lbrace (x,t) \\in T\\ \| \\ \\widetilde{P_1}(x,t)= \\cdots = \\widetilde{P_i}(x,t)=0 \\rbrace .$ We define a local system $\\widetilde{\\mathcal {L}}$ on $T$ by $\\widetilde{\\mathcal {L}} =\\mathbb {C}_T x_1^{c_1}\\cdots x_{n-k}^{c_{n-k}}t_1^{\\widetilde{c_1}} \\cdots t_k^{\\widetilde{c_k}}.$ Then $Z_k \\simeq W$ and we have isomorphisms $H^j(W ; \\mathcal {L}) \\simeq H^j (Z_k ; \\widetilde{\\mathcal {L}} )\\qquad (j \\in \\mathbb {Z}).$ First let us consider the case where $ \\widetilde{c} = ( \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\notin \\mathbb {Z}^k$ .", "In this case, without loss of generality we may assume that $\\widetilde{c_k} \\notin \\mathbb {Z}$ .", "Then by the Künneth formula, for $i=1,2, \\ldots , k-1$ we have the vanishings $H^j (Z_i ; \\widetilde{\\mathcal {L}} )\\simeq 0 \\qquad (j \\in \\mathbb {Z}).$ Moreover we can naturally identify $Z_{k-1}\\subset T$ with $(T_0 \\setminus \\cup _{i=1}^{k-1} P_i^{-1}(0)) \\times \\mathbb {C}^_{t_k}$ .", "Consider $\\widetilde{P_k}$ as a Laurent polynomial on $T_1=T_0 \\times \\mathbb {C}^_{t_k} \\simeq ( \\mathbb {C}^)^{n-k+1}$ .", "Note that we have ${\\rm dim}NP(\\widetilde{P_k})=n-k+1= {\\rm dim}T_1$ .", "By taking a sufficiently generic $( a_1, \\ldots , a_{n-k},a_{n-k+1} ) \\in {\\rm Int}NP(\\widetilde{P_k})\\subset \\mathbb {R}^{n-k+1}$ we define a real-valued function $g$ on $T_1=T_0 \\times \\mathbb {C}^_{t_k}$ by $g(x, t_k)= \\left\|\\widetilde{P_k} (x, t_k) \\times x_1^{- a_1} \\cdots x_{n-k}^{- a_{n-k}}t_k^{- a_{n-k+1}} \\right\|.$ Then by applying Lemma REF to the Morse function $g: T_1=T_0 \\times \\mathbb {C}^\\longrightarrow \\mathbb {R}$ and arguing as the proof of Theorem REF we obtain the desired concentration $H^j (Z_k ; \\widetilde{\\mathcal {L}} )\\simeq 0 \\qquad (j \\ne n-k).$ The proof for the remaining case where $(c, \\widetilde{c} )=(c_1, \\ldots , c_{n-k}, \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\notin \\mathbb {Z}^n$ and $ \\widetilde{c} = ( \\widetilde{c_1},\\ldots , \\widetilde{c_k}) \\in \\mathbb {Z}^k$ is similar.", "In this case, $Z_1 \\subset T$ is isomorphic to the product $Z_1^{\\prime } \\times ( \\mathbb {C}^)^{k-1}$ for a hypersurface $Z_1^{\\prime }$ in $T_0 \\times \\mathbb {C}^_{t_1}$ and $\\widetilde{\\mathcal {L}}$ is isomoprhic to the pull-back of a local system on $T_0\\times \\mathbb {C}^*_{t_1}$ .", "Hence by the Künneth formula and the proof of Theorem REF we obtain the concentration $H^j (Z_1 ; \\widetilde{\\mathcal {L}} )\\simeq 0 \\qquad (j \\ne n-k, \\ldots , n-1).$ Repeating this argument with the help of Lemma REF and the proof of Theorem REF we obtain also $H^j(W ; \\mathcal {L}) \\simeq H^j (Z_k ; \\widetilde{\\mathcal {L}} )\\simeq 0 \\qquad (j \\ne n-k, \\ldots , n-1).$ Then the assertion is obtained by applying Artin's vanishing theorem to the $(n-k)$ -dimensional affine variety $Z_k \\subset T$ .", "This completes the proof." ] ]	1403.0103
[ [ "Approximation of the inertial manifold for a nonlocal dynamical system" ], [ "Abstract We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-\\Delta)^{\\frac{\\alpha}{2}}$, with $0<\\alpha<2$.", "The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect.", "We first establish the existence of an inertial manifold and highlight the influence of the parameter $\\alpha$.", "Then we approximate the inertial manifold when a small normal diffusion $\\varepsilon \\Delta$ (with $\\varepsilon \\in (0, 1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and without normal diffusion." ], [ "Introduction", "Nonlocal operators appear in complex systems, such as anomalous diffusion and geophysical flows [5], [21], [22], a thin obstacle problem [30], finance [8], and stratified materials [28].", "A special but important nonlocal operator is the fractional Laplacian operator arising in non-Gaussian stochastic systems.", "For a stochastic differential system with a symmetric $\\alpha $ -stable Lévy motion (a non-Gaussian stochastic process) $L_t^\\alpha $ for $\\alpha \\in (0, 2)$ , $ dX_t =b(X_t)dt + dL^\\alpha _t, \\;\\; X_0 =x, $ the corresponding Fokker-Planck equation contains the fractional Laplacian operator $(-\\Delta )^{\\frac{\\alpha }{2}}$ .", "When the drift term $b$ in the above stochastic differential system depends on the probability distribution of the system state, the Fokker-Planck equation becomes a nonlinear, nonlocal partial differential equation [1].", "For nonlocal partial differential equations with the fractional Laplacian operator $(-\\Delta )^{\\frac{\\alpha }{2}}$ , $\\alpha \\in (0, 2)$ , there are recent works about modeling techniques, well-posedness and regularity of solutions; see, for example [12], [30], [3], [4], [5], [6], [31], [7], [10].", "It is desirable to further study dynamical behaviors of such nonlocal systems.", "In the present paper, we consider the inertial manifold and its approximation for a system described by a nonlocal partial differential equation.", "An inertial manifold is a Lipschitz manifold that captures asymptotic long time dynamics of the system evolution [33], [27], [35].", "The Lyapunov-Perron method is often used to study an inertial manifold.", "With this method, the system is converted into an integral equation and the inertial manifold is constructed as the graph of a unique fixed point of a corresponding mapping.", "The method is also used to construct the inertial manifolds of stochastic partial differential equations [11].", "A spectral gap condition is sufficient to guarantee the existence of the fixed point.", "This spectral gap condition may be understood as a relationship of the gap between eigenvalues of the linear operator in the system, with the Lipschitz constant of the nonlinearity.", "See the inequality (REF ) in the next section.", "We consider a nonlocal dynamical system containing anomalous diffusion $(-\\Delta )^{\\frac{\\alpha }{2}}$ , with or without small normal diffusion $ \\varepsilon \\Delta $ for $\\varepsilon \\in (0, 1)$ .", "We prove the existence of an inertial manifold and then consider its approximation when $\\varepsilon $ is sufficiently small.", "The paper is arranged as follows.", "In section 2, we review a basic theory of inertial manifolds.", "In section 3, we prove the existence of the inertial manifold of a nonlocal system with both anomalous and normal diffusion.", "We point out that the interaction between the normal diffusion and the anomalous diffusion in the nonlocal operator plays a significant role in the existence of the inertial manifold.", "Section 4 is devoted to the existence of the inertial manifold when the normal diffusion is absent; however, in this case we do not have the existence of the inertial manifold when the parameter $\\alpha $ in the anomalous diffusion is less than 1, since the spectral gap condition does not hold.", "Finally, in section 5, we derive an asymptotic approximation of the inertial manifold when the normal diffusion is small enough." ], [ "Preliminaries", "In this section, we recall basic facts about inertial manifolds of infinite dimensional dynamical systems [33], [27], [17].", "We consider an evolution equation in a Hilbert space $H$ with norm $\|\\cdot \|$ and scalar product $\\langle \\cdot ,\\cdot \\rangle $ $\\frac{du}{dt}+Au = f(u),~u\\in H,~u\|_{t=0}=u_0,$ where $A$ is a closed linear operator on $H$ and $f$ is a nonlinear mapping.", "We make the following assumptions.", "Assumption I.", "The linear operator $A$ is a positive definite, self-adjoint operator with discrete spectrum.", "More specifically, the eigenvalues may be arranged as follows $0 < \\lambda _1 < \\lambda _2 \\le \\cdot \\cdot \\cdot \\le \\lambda _j\\le \\cdot \\cdot \\cdot ,~~~\\lambda _j \\rightarrow + \\infty ~\\mathrm {as}~j \\rightarrow \\infty .$ For example, if a linear closed operator $A$ is positive and self-adjoint with the compact inverse, then it has discrete spectrum.", "Let $V$ be a subspace of $H$ with norm $\\Vert \\cdot \\Vert $ and scalar product $\\ll \\cdot ,\\cdot \\gg $ .", "Assumption II.", "The nonlinear mapping $f: V\\longrightarrow H$ is globally Lipschitz continuous $\|f(u)-f(v)\|\\le l_f\\Vert u-v\\Vert ,~\\forall ~ u,v\\in V,$ where $l_f$ is the Lipschitz constant.", "It is well-known that if the operator $-A$ is the infinitesimal generator of a $C_0-$ semigroup, the problem (REF ) is well-posed, that is for every $u_0\\in H$ , there exists a unique global solution $u(t)$ , and, in fact, $u\\in C(0,T;H)\\cap L^2(0,T;H^1_0(\\Omega ))$ for all $T>0$ ; see [18].", "The solution of the system (REF ) can be defined by the formula of variation of constant $u(t)=e^{-At}u_0 + \\int ^t_0e^{-A(t-s)}f(u(s))ds.$ If we define $T(t):u_0\\longrightarrow u(t)$ , then $T(t)$ is continuous and satisfies semigroup properties: $T(0)=I \\textrm {(identity~operator)}$ ; $T(t)T(s)=T(t+s)$ , $t,s\\ge 0$ .", "Let $P_n$ a projection operator with rank $n$ from $H$ to $P_nH$ , and then we define $Q_n=I-P_n$ .", "Definition 2.1 We say that a manifold $\\mathcal {M}$ in space $H$ is an inertial manifold for the system (REF ) or the corresponding semigroup $T(t)$ if the manifold $\\mathcal {M}$ is invariant under $T(t)$ , i.e., $T(t)\\mathcal {M}=\\mathcal {M}$ , $t\\ge 0$ ; it can be represented as the graph of a Lipschitz continuous function $\\Phi :P_nH\\longrightarrow Q_nH$ , i.e., $\\mathcal {M}=\\lbrace u+\\Phi (u),u\\in P_nH\\rbrace ;$ it possesses exponential tracking property, i.e., there exist positive constants $\\eta $ and $\\beta $ such that, for every $u\\in H$ , there is a $v\\in \\mathcal {M}$ such that $\|T(t)u-T(t)v\|\\le \\eta e^{-\\beta t}\|u-v\|.$ Under certain further assumptions, for a large enough $N$ , such that $\\lambda _{N+1}-\\lambda _N>2l_f$ , an inertial manifold for system (REF ) can be realized as the graph of a function $\\Phi :P_N H\\longrightarrow Q_N H$ .", "The inertial form $\\frac{dp}{dt}+ Ap=P_N f(p+\\Phi (p)),~p\\in P_N H,$ captures all long-time behaviors of (REF ).", "The dimension of $\\mathcal {M}$ is the dimension of $P_N H$ , i.e., $dim(\\mathcal {M})=dim(P_NH).$ We assume that there exists an exponential dichotomy (see [29]): $\\Vert e^{-tA}P_n\\Vert _{\\mathcal {L}(H,H)}\\le K_1e^{-\\lambda _nt},~\\forall t\\le 0$ , $\\Vert e^{-tA}P_n\\Vert _{\\mathcal {L}(V,H)}\\le K_1\\lambda _n^se^{-\\lambda _nt},~\\forall t\\le 0$ , $\\Vert e^{-tA}(I-P_n)\\Vert _{\\mathcal {L}(H,H)}\\le K_2e^{-\\lambda _{n+1}t},~\\forall t\\ge 0$ , $\\Vert e^{-tA}(I-P_n)\\Vert _{\\mathcal {L}(V,H)}\\le K_2(t^{-s}+\\lambda _{n+1}^s)e^{-\\lambda _{n+1}t},~\\forall t\\ge 0$ , where constants $K_1,~K_2\\ge 1,$ $0\\le s<1$ , and $\\Vert \\cdot \\Vert _{\\mathcal {L}(V,H)}$ is operator norm.", "If a spectral gap condition holds, i.e., there is an $N\\in \\mathbb {N}$ such that $\\lambda _{N+1}-\\lambda _N>2l_f, $ then, we can choose a $\\sigma $ such that $\\lambda _N + 2l_fK_1<\\sigma <\\lambda _{N+1}-2l_fK_1K_2.$ This $\\sigma $ is used to define the Banach space $\\mathcal {F}_\\sigma =\\lbrace \\varphi \\in C((-\\infty ,0],H)~\|~\\Vert \\varphi \\Vert _{\\mathcal {F}_\\sigma }=\\sup _{t\\le 0}e^{\\sigma t}\|\\varphi (t)\|<\\infty \\rbrace .$ A trajectory on the inertial manifold can be found as the fixed point $\\varphi =\\varphi (p)$ of a mapping $J(\\cdot ,p):\\mathcal {F}_\\sigma \\longrightarrow \\mathcal {F}_\\sigma $ defined by $J(\\varphi ,p)(t)=e^{-tA}p-\\int ^0_te^{-(t-s)A}P_Nf(\\varphi (s))ds+\\int ^t_{-\\infty }e^{-(t-s)A}(I-P_N)f(\\varphi (s))ds.$ The inertial manifold $\\mathcal {M}$ is the graph of $\\Phi :P_N H\\longrightarrow (I-P_N)H$ , which is defined in terms of the fixed point $\\varphi $ of (REF ) as follows $\\Phi (p)=(I-P_N)\\varphi (p)(0)&=(I-P_N)(\\int ^0_{-\\infty }e^{sA}(I-P_N)f(\\varphi (s))ds)\\nonumber \\\\&=\\int ^0_{-\\infty }e^{sA}(I-P_N)f(\\varphi (s))ds,~\\forall ~p\\in P_N H.$ Note that $\\varphi (p)(0)=p+\\Phi (p)$ .", "The spectral gap condition is used to ensure not only that $J$ has a fixed point by contraction mapping principle, but also that the resulting manifold is exponentially tracking, i.e., there exist positive constants $\\eta $ and $\\beta $ such that, for $u\\in H$ , there is $v_0\\in \\mathcal {M}$ such that $\|T(t)u-T(t)v_0\|\\le \\eta e^{-\\beta t}\|u-v_0\|.$ The conclusion mentioned above is summarized in the following theorem (see [35], [33]) Theorem 2.1 Let the Assumptions I and II hold.", "If there is an $N\\in \\mathbb {N}$ , such that the following spectral gap condition is satisfied $\\lambda _{N+1}-\\lambda _N>2l_f,$ then there exists an $N$ -dimensional inertial manifold $\\mathcal {M}$ , which is the graph of a Lipschitz continuous function $\\Phi $ satisfying (REF ).", "Remark 2.1 If $f$ is $C^{k}$ , then $\\mathcal {M}$ is $C^{k}$ .", "Remark 2.2 If the system (REF ) is dissipative, i.e., it possesses an absorbing set, then $f$ only needs to be locally Lipschitz continuous." ], [ "Inertial manifold for a nonlocal system with both anomalous and normal diffusion", "Now we consider the existence of the inertial manifold for a nonlocal evolution equation with both fractional and usual Laplacian (i.e., both anomalous and normal diffusion).", "We recall the definition of the fractional Laplacian operator.", "Definition 3.1 For $u\\in C^\\infty _0(\\mathbb {R}^n)$ and $\\alpha \\in (0,2)$ , define $(-\\Delta )^{\\frac{\\alpha }{2}} u=C(n,\\alpha )P.V.\\int _{\\mathbb {R}^n}\\frac{u(x)-u(y)}{\|x-y\|^{n+2\\alpha }}dy,$ where the principle value (P.V.)", "is taken as the limit of the integral over $\\mathbb {R}^n\\backslash B_\\epsilon (x)$ as $\\epsilon \\longrightarrow 0$ , with $B_\\epsilon (x)$ the ball of radius $\\epsilon $ centered at $x$ , and $C(n,\\alpha )=\\frac{\\alpha }{2^{1-\\alpha }\\pi ^{\\frac{n}{2}}}\\frac{\\Gamma (\\frac{n+\\alpha }{2})}{\\Gamma (1-\\frac{\\alpha }{2})}.$ Here $\\Gamma $ is the Gamma function defined by $\\Gamma (\\lambda )=\\int ^\\infty _0t^{\\lambda -1}e^{-t}dt$ for every $\\lambda >0$ ; for more information see[15], [6].", "In this paper, $n=1$ and the usual local Laplacian operator is $\\Delta = \\partial _{xx}$ .", "We consider the existence of the inertial manifold $\\mathcal {M}_\\varepsilon $ for the following system ${\\left\\lbrace \\begin{array}{ll}\\dfrac{du}{dt} - \\varepsilon \\Delta u +(-\\Delta )^{\\frac{\\alpha }{2}} u+f(u)=g(x),&in\\ \\Omega =(-\\pi ,\\pi ).\\\\u\\big \|_{\\Omega ^c}=0.\\\\u(x,0)=u_0, &\\ x\\in \\Omega ,\\end{array}\\right.", "}$ where $0<\\varepsilon <<1$ , and $\\Omega ^c=\\mathbb {R}\\backslash \\Omega $ .", "Let $H=L^2(\\Omega )$ .", "The nonlocal Laplacian operator $(-\\Delta )^{\\frac{\\alpha }{2}}$ is defined on $H$ .", "Assume that the nonlinear function $f:H\\longrightarrow H$ is locally Lipschitz, i.e., for every $B\\subset H$ $\|f(u)-f(v)\|\\le l_f\|u-v\|,~\\forall ~ u,v\\in B.$ Assumption III.", "The nonlinear function $f$ satisfies the following condition $C\|s\|^{p}-C\\le f(s)s\\le C\|s\|^{p}+C,\\;\\; s \\in \\mathbb {R}, $ for some $p\\ge 2$ .", "We recall the eigenvalues of the nonlocal operator $(-\\Delta )^{\\frac{\\alpha }{2}}$ in $H=L^2(\\Omega )$ .", "Lemma 3.1 ([20]) The eigenvalues of the following spectral problem $(-\\Delta )^{\\frac{\\alpha }{2}}\\varphi (x)=\\lambda \\varphi (x),~x\\in \\Omega ,$ where $\\varphi (x)\\in L^2(\\Omega )$ is extended to $\\mathbb {R}$ by 0, are $\\lambda _n=\\Big (\\frac{n}{2}-\\frac{(2-\\alpha )}{8}\\Big )^\\alpha +o(\\frac{1}{n}).$ Moreover, $0 < \\lambda _1 < \\lambda _2 \\le \\cdot \\cdot \\cdot \\le \\lambda _n\\le \\cdot \\cdot \\cdot , \\; \\mbox{for}~ n=1, 2, \\cdots .$ Moreover, the corresponding eigenfunctions $\\varphi _n$ form a complete orthonormal basis in $L^2(\\Omega )$ .", "Then we have the following result on the well-posedness for the system (REF ).", "For some related results, see [16].", "Denote by $C$ a general positive constant which may be different in different places.", "Theorem 3.1 (Well-posedness) Assume that $g$ is in $L^2(\\Omega )$ , $f$ satisfies the condition (REF ), and that Assumption III hold.", "Then there exists a unique solution $u(t)\\in C(0,T;H)$ (for every $T>0$ ) for the system (REF ).", "The solution is given by the formula of variation of constants $u(t)=e^{-A_\\varepsilon t}u_0 - \\int ^t_0e^{-A_\\varepsilon (t-s)}(f(u(s))-g(x))ds,$ where $A_\\varepsilon =- \\varepsilon \\Delta +(-\\Delta )^{\\frac{\\alpha }{2}} $ .", "Recall that the eigenvalues of the local Laplacian operator $- \\Delta $ in $L^2(\\Omega )$ with domain $H^2(\\Omega )\\cap H^1_0(\\Omega )$ are $ n^2$ , $n\\in \\mathbb {N}$ , and the corresponding eigenfunctions are $\\omega _n(x)=\\sin (nx)$ which form a complete orthonormal basis of $L^2(\\Omega )$ .", "By Lemma REF and [2], we know that the eigenvalues of $A_\\varepsilon $ are $\\lambda _n=\\varepsilon n^2+(\\frac{n}{2}-\\frac{(2-\\alpha )}{8})^\\alpha +o(\\frac{1}{n})$ $n\\in \\mathbb {Z}$ , $n\\ge 1$ , and they satisfy $0 < \\lambda _1 < \\lambda _2 \\le \\cdot \\cdot \\cdot \\le \\lambda _n\\le \\cdot \\cdot \\cdot $ .", "The corresponding eigenfunctions form a complete orthonormal basis in $L^2(\\Omega )$ .", "Hence, we conclude that the operator $-A_\\varepsilon =\\varepsilon \\Delta -(-\\Delta )^{\\frac{\\alpha }{2}} $ is dissipative in $L^2(\\Omega )$ , that is, $<-A_\\varepsilon u,u>\\le 0$ .", "Thus, $-A_\\varepsilon $ is a infinitesimal generator of an analytic semigroup [25].", "As $f$ is local Lipschitz continuous, we obtain the existence and uniqueness of solution $u(t)\\in C(0,T;H)\\cap L^2(0,\\tau (u_0);H^1_0(\\Omega ))$ for some $\\tau =\\tau (u_0)>0$ ; see [18].", "In fact, the solution is given by the formula of variation of constants.", "Now, we prove that the solution exists globally.", "Multiplying $u$ on both sides of the equation (REF ) and integrating over $\\Omega $ , we have $\\frac{1}{2}\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+\\varepsilon \\Vert u\\Vert ^2 +\\int _\\Omega f(u)uds&=-&\\int _\\Omega u(-\\Delta )^{\\frac{\\alpha }{2}}u\\textrm {d}x+\\int _\\Omega g(x)u \\textrm {d}x.$ Furthermore, $\\frac{1}{2}\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+\\varepsilon \\Vert u\\Vert ^2 +C\\int _\\Omega \|u\|^pdx\\le -\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x+C\\int _\\Omega u^2 \\textrm {d}x+C\|g(x)\|^2 +C\|\\Omega \|,$ where $\|\\Omega \|$ denote the measure of $\\Omega $ .", "Using the nonlocal Poincaré $\|u\|^2\\le C\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x$ (see [13]), and local Poincaré inequality, we get $\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+C\|u\|^2 \\le C\|g(x)\|^2+C\|\\Omega \|.$ By uniform Gronwall's inequality, we have $\|u\|^2 \\le \|u_0\|^2\\textrm {e}^{-Ct}+C\|g(x)\|^2+C\|\\Omega \|.$ which implies that $\\displaystyle {\\sup _{0\\le t<+\\infty }\|u(x,\\,t)\|<+\\infty }$ .", "Thus the solution exists for all time.", "The proof is complete.", "By Theorem REF , we can define a semigroup by $T(t):u_0\\longrightarrow u(t)$ , where $u(t)$ is the solution of (REF ) and the operator $-A_\\varepsilon $ is the infinitesimal generator of $T(t)$ .", "Furthermore, for the semigroup $\\lbrace T(t)\\rbrace _{t\\ge 0}$ , there exists a bounded absorbing set $B_\\varepsilon $ in $H$ , i.e., for every bounded set $B\\subset H$ , we can find a constant $t_0=t_0(B)>0$ , such that when $t\\ge t_0$ , $T(t)B\\subset B_\\varepsilon $ .", "We are ready to present the result on the existence of the inertial manifold $\\mathcal {M}_\\varepsilon $ .", "Theorem 3.2 (Inertial manifold $\\mathcal {M}_\\varepsilon $ ) Assume that $g$ belongs to $L^2(\\Omega )$ , the nonlinearity $f$ is local Lipschitz continuous, and Assumption III holds.", "Then there exists an $N-$ dimensional inertial manifold $\\mathcal {M}_\\varepsilon $ for the system (REF ), as a graph of a Lipschitz continuous function $\\Phi $ from $P_NH$ to $(I-P_N)H$ .", "Both $-\\Delta $ and $(-\\Delta )^{\\frac{\\alpha }{2}}$ , $\\alpha \\in (0,2)$ are positive, seif-adjoint operators in $L^2(\\Omega )$ .", "By Lemma REF and [2], the eigenvalues of the operator $A_\\varepsilon =- \\varepsilon \\Delta +(-\\Delta )^{\\frac{\\alpha }{2}}$ are $\\lambda _n=\\varepsilon n^2+\\displaystyle {\\left(\\frac{n}{2}-\\frac{(2-\\alpha )}{8}\\right)}^\\alpha +o(\\frac{1}{n}),$ which satisfy $0 < \\lambda _1 < \\lambda _2 \\le \\cdot \\cdot \\cdot \\le \\lambda _n\\le \\cdot \\cdot \\cdot ,$ and the corresponding eigenfunctions $\\varphi _n$ form a complete orthonormal basis of $L^2(\\Omega )$ .", "The assumptions I and II are all satisfied.", "Hence, by Theorem REF , we only need to verify the spectral gap condition.", "We do this in the following cases: Case 1, $\\alpha =1$ : $\\lambda _{n+1}-\\lambda _{n}=\\varepsilon (2n+1)+\\frac{1}{2}+o(\\frac{1}{n}).$ So the spectral gap condition is satisfied for fixed $\\varepsilon $ , i.e., we can find some $N$ such that $\\lambda _{N+1}-\\lambda _{N}\\ge 2l_f$ , while $n\\ge N$ , we obtain $\\lambda _{n+1}-\\lambda _{n}\\ge \\lambda _{N+1}-\\lambda _{N}\\ge 2l_f$ .", "Next, suppose that $G(n)=\\lambda _{n+1}-\\lambda _{n}=\\varepsilon (2n+1)+ (\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^\\alpha -(\\frac{n}{2}-\\frac{2-\\alpha }{8})^\\alpha +o(\\frac{1}{n})$ .", "Then $G^{^{\\prime }}(n) = 2\\varepsilon +\\frac{\\alpha }{2}\\Big [\\big (\\frac{n+1}{2}-\\frac{2-\\alpha }{8}\\big )^{\\alpha -1}-\\big (\\frac{n}{2}-\\frac{2-\\alpha }{8}\\big )^{\\alpha -1}\\Big ]+\\gamma o(\\frac{1}{n^2}).$ If $G^{^{\\prime }}(n)>0$ , then the spectral gap becomes larger and larger, hence for $2l_f$ , we can find suitable $N$ such that spectral gap condition hold.", "Case 2, $1<\\alpha <2$ : In this case, the term $\\frac{\\alpha }{2}[(\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}-(\\frac{n}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}]>0$ , and $\\gamma o(\\frac{1}{n^2})$ becomes very small when $n$ large enough whenever the sign of $\\gamma $ .", "Hence we can choose big $n$ , such that $G^{^{\\prime }}(n)>0$ , so the gap becomes larger and larger.", "Hence for $2l_f$ , we can choose big $N$ , such that $n\\ge N$ , we obtain $\\lambda _{n+1}-\\lambda _{n}\\ge \\lambda _{N+1}-\\lambda _{N}\\ge 2l_f$ .", "Case 3, $0<\\alpha <1$ : From (REF ), we know that in this situation $\\frac{\\alpha }{2}[(\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}-(\\frac{n}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}]<0.$ Our aim is to obtain $\\lambda _{n+1}-\\lambda _{n}$ larger and larger, so necessary, we need $G^{^{\\prime }}(n)>0$ .", "By above analysis, if $\\gamma o(\\frac{1}{n^2})>0$ , then we choose $-\\frac{\\alpha }{4}[(\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}-(\\frac{n}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}]-\\frac{\\gamma }{2} o(\\frac{1}{n^2})<\\varepsilon <<1$ , such that $G^{^{\\prime }}(n)>0$ , i.e., there exist $N$ , such that $n\\ge N$ , we obtain $\\lambda _{n+1}-\\lambda _{n}\\ge \\lambda _{N+1}-\\lambda _{N}\\ge 2l_f$ .", "If $\\gamma o(\\frac{1}{n^2})<0$ , then we choose $-\\frac{\\alpha }{4}[(\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}-(\\frac{n}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}]+\\frac{\\gamma }{2} o(\\frac{1}{n^2})<\\varepsilon <<1$ , such that $G^{^{\\prime }}(n)>0$ , i.e., there exists $N$ , such that $n\\ge N$ , we obtain $\\lambda _{n+1}-\\lambda _{n}\\ge \\lambda _{N+1}-\\lambda _{N}\\ge 2l_f$ .", "In conclusion, we see that in cases 1 and 2, for arbitrary $\\varepsilon >0$ , there exists inertial manifold, but in case 3, the existence of inertial manifold under the choice of $\\varepsilon $ .", "The proof is complete." ], [ "Inertial manifold for a nonlocal system with only anomalous diffusion ", "In this section, we consider the existence of the inertial manifold $\\mathcal {M}_0$ of following equation with only anomalous diffusion ${\\left\\lbrace \\begin{array}{ll}\\dfrac{du}{dt} +(-\\Delta )^{\\frac{\\alpha }{2}}+f(u)=g(x),&in\\ \\Omega =(-\\pi ,\\pi ).\\\\u\\big \|_{\\Omega ^c}=0.\\\\u(x,0)=u_0(x)\\in H^{\\frac{\\alpha }{2}}_0(\\Omega ),&in\\ x\\in \\Omega ,\\end{array}\\right.", "}$ where $\\Omega ^c=\\mathbb {R}\\backslash \\Omega $ .", "Assume that for any $B\\subset H^{\\frac{\\alpha }{2}}_0(\\Omega )$ function $f:B\\longrightarrow H^{\\frac{\\alpha }{2}}_0(\\Omega )$ is locally Lipschitz continuous $\\Vert f(u)-f(v)\\Vert _{H^{\\frac{\\alpha }{2}}_0}\\le l_f\\Vert u-v\\Vert _{H^{\\frac{\\alpha }{2}}_0},~\\forall ~ u,v\\in B.$ We will prove the following Theorem.", "Theorem 4.1 (Inertial manifold $\\mathcal {M}_0$ ) Assume that $g$ belongs to $L^2(\\Omega )$ and the nonlinearity of problem (REF ) be local Lipschitz continuous (REF ).", "Also suppose that Assumption $III$ hold.", "Then there exists an $N-$ dimensional inertial manifold $\\mathcal {M}_0$ which is a graph of Lipschitz continuous function $\\Phi $ from $P_NH^{\\frac{\\alpha }{2}}(\\Omega )$ to $(I-P_N)H^{\\frac{\\alpha }{2}}(\\Omega )$ .", "First, we begin with the existence and uniqueness of problem (REF ), we will use semigroup method ([25]).", "Denote $A_\\alpha =-(-\\Delta )^{\\frac{\\alpha }{2}}$ and $\\Vert \\ \\Vert $ norm of space or operator." ], [ "Some estimates on the nonlocal Laplacian", "Definition 4.1 $A$ is a sectorial operator, if $A$ is dense defined for some $\\phi \\in (0,\\,\\frac{\\pi }{2})$ , $M\\ge 1$ , and $a\\in \\mathbb {R}$ , $S_{a,\\,\\phi }=\\lbrace \\lambda \|~\\phi \\le \|\\mbox{\\textrm {arg}}(\\lambda -a)\|\\le \\pi ,~\\lambda \\ne a\\rbrace \\subset \\rho (A)$ and $\\Vert (\\lambda I-A)^{-1}\\Vert \\le M/\|\\lambda -a\|$ .", "Lemma 4.1 The nonlocal Laplacian operator $A_{\\alpha }$ is sectorial one, satisfying the estimates as follows $\\Vert \\textrm {e}^{tA_{\\alpha }}\\Vert _{L^2(\\Omega )}\\le C\\textrm {e}^{-\\delta t},~~~~\\Vert A_{\\alpha }\\textrm {e}^{tA_{\\alpha }}\\Vert _{L^2(\\Omega )}\\le \\frac{C}{t}\\textrm {e}^{-\\delta t},$ where $C,\\,\\delta >0$ are constants independent of $t$ .", "By Lemma REF , $A_{\\alpha }$ is sectorial can be proved by definition.", "Set $\\mu =\\lambda t$ , $\\lambda >0$ , $\\Vert \\textrm {e}^{tA_{\\alpha }}\\Vert _{L^2(\\Omega )}=\\displaystyle {\\left\\Vert \\frac{1}{2\\pi i}\\int _{\\Gamma }\\textrm {e}^{\\mu }(\\frac{\\mu }{t}-A_{\\alpha })^{-1}\\frac{\\textrm {d}\\mu }{t}\\right\\Vert }_{L^2(\\Omega )} \\le \\frac{M}{2\\pi }\\int _{\\Gamma }\\mid \\textrm {e}^{\\mu }\\mid \\frac{\\mid \\textrm {d}\\mu \\mid }{\\mid \\mu \\mid } \\le C\\textrm {e}^{-\\delta t}.$ $\\Vert A_{\\alpha }\\textrm {e}^{tA_{\\alpha }}\\Vert _{L^2(\\Omega )}=\\displaystyle {\\left\\Vert \\frac{1}{2\\pi i}A_{\\alpha }\\int _{\\Gamma }\\textrm {e}^{\\mu }(\\frac{\\mu }{t}-A_{\\alpha })^{-1}\\frac{\\textrm {d}\\mu }{t}\\right\\Vert }_{L^2(\\Omega )} \\le \\frac{1}{2\\pi }\\frac{M}{\\delta }\\int _{\\Gamma }\\mid \\textrm {e}^{\\mu }\\mid \\frac{\\mid \\textrm {d}\\mu \\mid }{\\mid \\mu \\mid }\\frac{1}{t} \\le \\frac{C}{t}\\textrm {e}^{-\\delta t}.$ The proof is complete.", "Definition 4.2 $A$ is a sectorial operator.", "if $\\Re \\sigma (-A)>0$ , then for every $\\beta >0$ , $A^{-\\beta }=\\displaystyle {\\frac{1}{\\Gamma (\\beta )}\\int _0^{\\infty }t^{\\beta -1}\\textrm {e}^{tA}\\textrm {d}t}$ .", "Moreover, $A^{\\beta }=(A^{-\\beta })^{-1}$ and $A^0=I$ .", "Lemma 4.2 $A$ is a sectorial operator, $\\Re \\sigma (-A)>\\delta >0$ .", "$\\forall \\,\\beta \\ge 0$ , such that $\\exists \\,C(\\beta )<\\infty $ , $\\forall \\,t>0$ , $\\Vert A^{\\beta }\\textrm {e}^{tA}\\Vert \\le C(\\beta ) t^{-\\beta }\\textrm {e}^{-\\delta t}.$ $\\forall $ $m=1,\\,2,\\,\\cdots $ , $\\Vert A^m\\textrm {e}^{tA}\\Vert =\\left\\Vert \\left(A\\textrm {e}^{\\frac{tA}{m}}\\right)^m\\right\\Vert \\le (Cm)^m t^{-m}\\textrm {e}^{-\\delta t}.$ For $0<\\beta <1$ , $t>0$ $\\Vert A^{\\beta }\\textrm {e}^{tA}\\Vert &=&\\Vert A^{\\beta -1}A\\textrm {e}^{tA}\\Vert =\\Vert A^{-(1-\\beta )}A\\textrm {e}^{tA}\\Vert \\\\&=&\\displaystyle {\\left\\Vert \\frac{1}{\\Gamma (1-\\beta )}\\int _0^{\\infty }\\tau ^{1-\\beta -1}A\\textrm {e}^{-A(t+\\tau )}\\textrm {d}\\tau \\right\\Vert }\\le \\frac{1}{\\Gamma (1-\\beta )}\\int _0^{\\infty }\\tau ^{-\\beta }\\Vert A\\textrm {e}^{-A(t+\\tau )}\\Vert \\textrm {d}\\tau \\\\&\\le & \\frac{C}{\\Gamma (1-\\beta )}\\int _0^{\\infty }\\tau ^{-\\beta }(t+\\tau )^{-1}\\textrm {e}^{-\\delta (t+\\tau )}\\textrm {d}\\tau =C\\Gamma (\\beta )t^{-\\beta }\\textrm {e}^{-\\delta t}.$ Hence, $\\forall $ $\\beta \\ge 0$ , $\\Vert A^{\\beta }\\textrm {e}^{tA}\\Vert \\le C(\\beta ) t^{-\\beta }\\textrm {e}^{-\\delta t}$ .", "The proof is complete.", "Lemma 4.3 $\\textrm {Dom}(A_{\\alpha }) \\hookrightarrow H_0^{\\frac{\\alpha }{2}}(\\Omega )$ .", "In paper [7], we know $\\textrm {Dom}(A_{\\alpha }) \\subset H_0^{\\frac{\\alpha }{2}}(\\Omega )$ .", "According to the embedding results and the nonlocal calculus in paper [13], we have by the Hölder inequality $\\Vert u\\Vert ^2_{H^{\\frac{\\alpha }{2}}(\\Omega )}&=&\\Vert u\\Vert ^2_{L^2(\\Omega )}+\|u\|^2_{H^{\\frac{\\alpha }{2}}(\\Omega )} \\\\&\\le & \\Vert u\\Vert ^2_{L^2(\\Omega )}+\\frac{1}{2}\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u)(x,\\,y))^2 \\textrm {d}x \\textrm {d}y+C\\varepsilon ^{-(1+\\alpha )}\\Vert u\\Vert ^2_{L^2(D)}\\\\&=& C(\\Omega ,\\,\\varepsilon ,\\,\\alpha )\\Vert u\\Vert ^2_{L^2(\\Omega )}-\\frac{1}{2} \\langle A_{\\alpha }u,\\,u \\rangle \\le C\\Vert u\\Vert ^2_{L^2(\\Omega )}+C\\Vert A_{\\alpha }u\\Vert _{L^2(\\Omega )}\\Vert u\\Vert _{L^2(\\Omega )} \\\\&=& C\\Vert u\\Vert _{L^2(\\Omega )}(\\Vert u\\Vert _{L^2(\\Omega )}+\\Vert A_{\\alpha }u\\Vert _{L^2(\\Omega )}).$ Through the nonlocal Poincarè inequality[13], we get $\\Vert u\\Vert ^2_{H^{\\frac{\\alpha }{2}}(\\Omega )}&\\le & C\\Vert u\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )}(\\Vert u\\Vert _{L^2(\\Omega )}+\\Vert A_{\\alpha }u\\Vert _{L^2(\\Omega )}).$ Therefore, $\\Vert u\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )}\\le C(\\Vert u\\Vert _{L^2(\\Omega )}+\\Vert A_{\\alpha }u\\Vert _{L^2(\\Omega )})$ .", "The proof is complete.", "Lemma 4.4 $\\textrm {Dom}(A_{\\alpha }^{\\beta }) \\hookrightarrow H^{\\frac{\\alpha }{2}}(\\Omega )$ , $\\frac{1}{2}<\\beta <1$ .", "Through the Lemma $\\ref {embetoHs}$ , we have $\\textrm {Dom}(A_{\\alpha }^{\\beta })\\hookrightarrow \\textrm {Dom}(A_{\\alpha }) \\hookrightarrow H^{\\frac{\\alpha }{2}}(\\Omega )$ .", "The proof is complete." ], [ "The local and global solution of problem (", "The existence proof of the local solution is a standard contraction argument.", "With numbers $T>0$ and $R>0$ to be fixed below, in the Banach space $X=C^0([0,\\,T],\\,H_0^{\\frac{\\alpha }{2}}(\\Omega ))$ , we consider the closed set $S=\\lbrace u\\in X:~\\Vert u-u_0\\Vert _X\\le R\\rbrace .$ It follows that the map $u=\\textrm {e}^{tA_{\\alpha }}u_0-\\int _0^t \\textrm {e}^{(t-\\tau )A_{\\alpha }}(f(u(\\tau ))-g(x))\\textrm {d}\\tau :=\\Theta (u)$ is contraction from $S$ into itself.", "Since $\\textrm {e}^{tA_{\\alpha }}$ is a strongly continuous semigroup, we can choose $T_1$ such that $\\Vert \\textrm {e}^{tA_{\\alpha }}u_0-u_0\\Vert _{H^{\\frac{\\alpha }{2}}(D)}\\le R/2$ for $t\\in [0,\\,T_1]$ .", "Denote $F(x,u)=-f(u)+g(x)$ .", "If $u\\in X$ , because $f$ is Lipschitz continuous from bounded subsets of $L^{6}(\\Omega )$ to $L^{2}(\\Omega )$ , then we have a bound $\\Vert F(x,u)\\Vert _X\\le K_3 R$ , where $K_3>0$ is a constant.", "Thus, using Lemmas $\\ref {semiest}$ and $\\ref {fracembe}$ , we have $&&\\int _0^t \\Vert \\textrm {e}^{(t-\\tau )A_{\\alpha }}(F(\\tau ,u(\\tau )))\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&\\le & C\\int _0^t \\Vert \\textrm {e}^{(t-\\tau )A_{\\alpha }}(F(\\tau ,u(\\tau )))\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau +C\\int _0^t \\Vert A^{\\beta }_{\\alpha }\\textrm {e}^{(t-\\tau )A_{\\alpha }}(F(\\tau ,u(\\tau )))\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&\\le & C\\int _0^t\\Vert \\textrm {e}^{(t-\\tau )A_{\\alpha }}\\Vert _{L^2(\\Omega )}\\Vert F(\\tau ,u(\\tau ))\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau +C\\int _0^t \\Vert A^{\\beta }_{\\alpha }\\textrm {e}^{(t-\\tau )A_{\\alpha }}\\Vert _{L^2(\\Omega )}\\Vert F(\\tau ,u(\\tau ))\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&\\le & CK_3R\\int _0^t\\textrm {e}^{-\\delta (t-\\tau )}\\textrm {d}\\tau +CK_3R\\int _0^t\\frac{\\textrm {e}^{-\\delta (t-\\tau )}}{(t-\\tau )^{\\beta }}\\textrm {d}\\tau \\nonumber \\\\&\\le & \\frac{C K_3R}{\\delta }(1-\\textrm {e}^{-\\delta T_2})+\\frac{C K_3R}{1-\\beta }T_2^{1-\\beta },$ where $\\frac{1}{2}<\\beta <1$ .", "If we pick up $T_2\\le T_1$ small enough, then such that $\\frac{C K_3R}{\\delta }(1-\\textrm {e}^{-\\delta T_2})+\\frac{C K_3R}{1-\\beta }T_2^{1-\\beta }\\le R/2$ for $t\\in [0,\\,T_2]$ .", "Therefore $\\Theta :~S\\rightarrow S$ when provided $T\\le T_2$ .", "To arrange that $\\Theta $ be a contraction mapping, we also use the Lipschitz continuous properties of $F(\\tau ,u(\\tau ))$ for $u,\\,\\bar{u}\\in X$ .", "Hence, for $t\\in [0,T_2]$ , through Lemmas $\\ref {semiest}$ and $\\ref {embetoHs}$ , we have $&&\\Vert \\Theta (u)-\\Theta (\\bar{u})\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )}\\le \\int _0^t \\Vert \\textrm {e}^{(t-\\tau )A_{\\alpha }}(F(\\tau ,u(\\tau ))-F(\\tau ,\\bar{u}(\\tau )))\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&\\le & C\\int _0^t \\Vert \\textrm {e}^{(t-\\tau )A_{\\alpha }}(F(\\tau ,u(\\tau ))-F(\\tau ,\\bar{u}(\\tau )))\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&&+C\\int _0^t \\Vert A^{\\beta }_{\\alpha }\\textrm {e}^{(t-\\tau )A_{\\alpha }}(F(\\tau ,u(\\tau ))-F(\\tau ,\\bar{u}(\\tau )))\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&\\le & Cl_f(R)\\int _0^t\\Vert \\textrm {e}^{(t-\\tau )A_{\\alpha }}\\Vert _{L^2(\\Omega )}\\Vert u-\\bar{u}\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&&\\!+C l_f(R)\\!\\!\\!\\int _0^t \\Vert A^{\\beta }_{\\alpha }\\textrm {e}^{(t-\\tau )A_{\\alpha }}\\Vert _{L^2(\\Omega )}\\Vert u-\\bar{u}\\Vert _{L^2(\\Omega )}\\textrm {d}\\tau \\nonumber \\\\&\\le & \\frac{C l_f(R)}{\\delta }(1-\\textrm {e}^{-\\delta T})\\Vert u-\\bar{u}\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )}+\\frac{C l_f(R)}{1-\\beta }T^{1-\\beta }\\Vert u-\\bar{u}\\Vert _{H^{\\frac{\\alpha }{2}}(\\Omega )},$ where $l_f(R)$ denote the Lipschitz constant and $\\frac{1}{2}<\\beta <1$ ; now if $T\\le T_2$ is choosen small enough, then we get $\\Vert \\Theta (u)-\\Theta (\\bar{u})\\Vert _X\\le L\\Vert u-\\bar{u}\\Vert _X$ , $L<1$ , making $\\Phi $ a contraction mapping from $S$ into itself.", "Thus $\\Phi $ has a unique fixed point $u$ in $S$ , solving ($\\ref {solsemi}$ ).", "We have proved the following result Theorem 4.2 If $f$ is Lipschitz continuous locally, then problem (REF ) has a unique solution $u\\in C^0([0,\\,T],\\,H_0^{\\alpha /2}(\\Omega ))$ , where $T>0$ is chosen above.", "Following, we prove the global solution to the nonlocal semi-linear equations basing on the result of the local existence.", "Theorem 4.3 Let $g$ belongs to $ L^2(\\Omega )$ and Assumption $III$ hold.", "Then the solution of problem (REF ) exists globally in the space $C^0(\\mathbb {R}^+,\\,H_0^{\\frac{\\alpha }{2}}(\\Omega ))$ , $\\mathbb {R}^+=\\lbrace t\\in \\mathbb {R}~\|~t\\ge 0\\rbrace $ .", "It's enough to prove that $\\displaystyle {\\sup _{0\\le t<+\\infty }\\Vert u(x,\\,t)\\Vert _{H^{\\frac{\\alpha }{2}}_0(\\Omega )}<+\\infty }$ by standard energy estimates.", "Multiplying $u$ on the both side of the Equation (REF ) and integrating over $\\Omega $ , we have $\\frac{1}{2}\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+\\int _\\Omega f(u)u \\textrm {d}x&=&\\int _\\Omega uA_{\\alpha }u\\textrm {d}x+\\int _\\Omega g(x)u \\textrm {d}x \\nonumber \\\\&\\le & -\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x+C\\int _\\Omega u^2 \\textrm {d}x+C\|g(x)\|^2 .$ Using the nonlocal Poincaré inequality $\|u\|^2\\le C\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x$ , we get $\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+C\|u\|^2+C\\int _\\Omega \|u\|^pdx+2\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x \\le C\|g(x)\|^2+C\|\\Omega \|,$ where $\|\\Omega \|$ denote the measure of $\\Omega $ .", "Hence we have $\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+C\|u\|^2\\le C\|g(x)\|^2+C\|\\Omega \|.$ By uniform Gronwall inequality, we have $\|u\|^2 \\le \|u_0\|^2\\textrm {e}^{-Ct}+C\|g(x)\|^2+C\|\\Omega \|.$ which implies $\\displaystyle {\\sup _{0\\le t<+\\infty }\|u(x,\\,t)\|<+\\infty }$ .", "Integrating (REF ) between $t$ and $t+1$ , we get $\|u(t+1)\|^2-\|u(t)\|^2+C\\int _t^{t+1}\|u(\\tau )\|^2 \\textrm {d}\\tau &+\\int _t^{t+1}\\int _\\Omega \|u(\\tau )\|^2d\\tau \\nonumber \\\\&+2\\int _t^{t+1}\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u(\\tau )))^2\\textrm {d}y \\textrm {d}x \\textrm {d}\\tau \\le C\|g(x)\|^2+C\|\\Omega \|.$ By ($\\ref {L2estsemi1}$ ), we obtain $\\int _t^{t+1}\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u(\\tau )))^2\\textrm {d}y \\textrm {d}x \\textrm {d}\\tau \\le C\|g(x)\|^2+\\frac{1}{2}\|u(t)\|^2+C\|\\Omega \|\\le C(\|g(x)\|^2,\|\\Omega \|,\\,\|u_0\|^2).$ At the same time, let $F(s) = \\int ^s_0f(s)ds$ .", "By Assumption $III$ , we obtain $C\|s\|^{p} - C\|\\Omega \| \\le F(s)\\le C\|s\|^{p} +C\|\\Omega \|.$ Therefore $C\\int _{\\Omega }\|u\|^p - C\|\\Omega \| \\le \\int _{\\Omega }F(u)dx\\le C\\int _{\\Omega }\|u\|^p + C\|\\Omega \|.$ Combining with (REF ) and (REF ), we have $\\frac{\\textrm {d}}{\\textrm {d}t}\|u\|^2+C\|u\|^2+\\int _{\\Omega }F(u)dx+\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x \\le C\|g(x)\|^2+C\|\\Omega \|.$ Multiplying $\\frac{du}{dt}$ on the both side of the equation (REF ) and integrating over the domain $\\Omega $ , we have $\\frac{1}{2}\|\\frac{du}{dt}\|^2+\\frac{1}{2}\\frac{d}{dt}\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x+\\frac{d}{dt}\\int _\\Omega F(u)dx\\le \\frac{1}{2}\|g(x)\|^2,$ here we also use H$\\ddot{o}$ lder inequality and Cauchy inequality.", "Combining with (REF ) and (REF ), we have $\\frac{d}{dt}(\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x+\\int _\\Omega F(u)dx)+C(\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x+\\int _\\Omega F(u)dx)\\le C\|g(x)\|^2+C\|\\Omega \|.$ Applying uniform Gronwall inequality, we deduce from (REF ) that $\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u))^2\\textrm {d}y \\textrm {d}x+\\int _\\Omega F(u)dx\\le e^{-Ct}(\\int _\\Omega \\int _\\Omega (\\mathcal {D}^{\\ast }(u_0))^2\\textrm {d}y \\textrm {d}x+\\int _\\Omega F(u_0)dx)+C\|g(x)\|^2+C\|\\Omega \|,$ which implies $\\displaystyle {\\sup _{0\\le t<+\\infty }\\Vert u(x,\\,t)\\Vert _{H^{\\frac{\\alpha }{2}}_0(\\Omega )}<+\\infty }$ .", "The proof is complete.", "By Theorem REF , the semigroup $T(t)$ corresponding to problem (REF ) can be defined by $T(t):u_0\\longrightarrow u(t)$ , $u(t)$ is the solution of problem (REF ).", "Also, the semigroup $T(t)$ possesses a bounded absorbing set $B_0$ , namely, for any $B\\subset H^{\\frac{\\alpha }{2}}_0(\\Omega )$ , there exists a time $t_0=t_0(B)$ such that when $t\\ge t_0$ , $T(t)B\\subset B_0$ ." ], [ "Proof of Theorem ", "Now we ready to prove Theorem REF .", "Proof of Theorem REF Thanks to an embedding theorem, we obtain the well-posedness of problem (REF ) in space $H^{\\frac{\\alpha }{2}}(\\Omega )$ .", "By Lemma REF , the assumptions $I$ and $II$ are all satisfied.", "Hence by Theorem REF , next we only need to verify spectral gap condition which similar to the proof of Theorem REF , we also decompose it into three cases: Case 1, $\\alpha =1$ : $\\lambda _{n+1}-\\lambda _{n}=\\frac{1}{2}+o(\\frac{1}{n}).$ While $n$ large enough, we have $\\lambda _{n+1}-\\lambda _{n}\\ge \\frac{1}{2}.$ Thus the Lipschitz constant of nonlinearity do not bigger than $\\frac{1}{2}$ , the spectral gap condition be satisfied.", "Supposing that $G(n)=\\lambda _{n+1}-\\lambda _{n}= (\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^\\alpha -(\\frac{n}{2}-\\frac{2-\\alpha }{8})^\\alpha +o(\\frac{1}{n})$ .", "Then $G^{^{\\prime }}(n) = \\frac{\\alpha }{2}\\Big [\\big (\\frac{n+1}{2}-\\frac{2-\\alpha }{8}\\big )^{\\alpha -1}-\\big (\\frac{n}{2}-\\frac{2-\\alpha }{8}\\big )^{\\alpha -1}\\Big ]+\\gamma o(\\frac{1}{n^2}).$ Our aim is to obtain $G^{^{\\prime }}(n)>0$ , the spectral gap becomes larger and larger, hence for $2l_f$ , we can find suitable $N$ such that spectral gap condition hold.", "Case 2, $1<\\alpha <2$ : In this case, the term $\\frac{\\alpha }{2}[(\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}-(\\frac{n}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}]>0$ , and $\\gamma o(\\frac{1}{n^2})$ becomes very small when $n$ large enough whenever the sign of $\\gamma $ .", "Hence we can choose a big enough $N$ , such that $G^{^{\\prime }}(N)>0$ , so the spectral gap condition hold.", "Case 3, $0<\\alpha <1$ : From (REF ), we know that in this situation $\\frac{\\alpha }{2}[(\\frac{n+1}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}-(\\frac{n}{2}-\\frac{2-\\alpha }{8})^{\\alpha -1}]<0.$ While $n$ large enough, we have $G^{^{\\prime }}(n)<0$ .", "Thus we can not fine suitable $N$ such that the spectral gap condition hold.", "In conclusion, we see that in cases 1 and 2, there exists inertial manifold, but in case 3, there is not exists inertial manifold for problem (REF ).", "The proof is complete.$\\Box $" ], [ "Asymptotic approximation of inertial manifold when normal diffusion is sufficiently small", "In this section, we approximate the inertial manifold $\\mathcal {M}_\\varepsilon $ , when the normal diffusion $\\varepsilon $ is sufficiently small [32].", "We will see the relationship between $\\mathcal {M}_\\varepsilon $ and $\\mathcal {M}_0$ , as normal diffusion $\\varepsilon $ convergent to $0^+$ .", "We know, from section 3, that the inertial manifold $\\mathcal {M}_\\varepsilon $ is the graph of a Lipschitz continuous mapping $\\Phi ^\\varepsilon (p)=(I-P_N)\\varphi (p)(0)=-\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)(f(\\varphi (s))-g(x))ds,~\\forall ~p\\in P_N H,$ where $A_\\varepsilon =- \\varepsilon \\Delta +(-\\Delta )^{\\frac{\\alpha }{2}}.$ That is, $\\mathcal {M}_\\varepsilon =\\lbrace p+\\Phi ^\\varepsilon (p)~\|~p\\in P_N H\\rbrace .$ Similarly, from Section 4, the inertial manifold $\\mathcal {M}_0$ is the graph of a Lipschitz continuous mapping $\\Phi ^0(p)=(I-P_N)\\varphi (p)(0)=-\\int ^0_{-\\infty }e^{-sA_\\alpha }(I-P_N)(f(\\varphi (s))-g(x))ds,~\\forall ~p\\in P_N H,$ where $A_\\alpha $ is the nonlocal operator.", "That is, $\\mathcal {M}_0=\\lbrace p+\\Phi ^0(p)~\|~p\\in P_N H\\rbrace .$ We expand $\\Phi ^\\varepsilon (p)$ as follows.", "For $p\\in P_N H$ , set $\\Phi ^\\varepsilon (p)=\\Phi ^0(p)+\\varepsilon \\Phi ^1(p)+\\varepsilon ^2\\Phi ^2(p)+\\cdot \\cdot \\cdot +\\varepsilon ^k\\Phi ^k(p)+\\cdot \\cdot \\cdot .$ We write the solution of problem (REF ) in the form $u(t)=u_0(t)+\\varepsilon u_1(t)+\\varepsilon ^2u_2(t)+\\cdot \\cdot \\cdot +\\varepsilon ^ku_k(t)+\\cdot \\cdot \\cdot ,$ with the initial condition $u(0)=p+\\Phi ^\\varepsilon (p)=p+\\Phi ^0(p)+\\varepsilon \\Phi ^1(p)+\\varepsilon ^2\\Phi ^2(p)+\\cdot \\cdot \\cdot +\\varepsilon ^k\\Phi ^k(p)+\\cdot \\cdot \\cdot .$ At $\\varepsilon =0$ , we expand $f(u(t))$ (which depends on $\\varepsilon $ ) by Taylor expansion, $f(u)=f(u_0(t))+\\varepsilon f^{^{\\prime }}(u_0(t))u_1(t)+\\frac{f^{^{\\prime \\prime }}(u_0(t))}{2!", "}u_2(t)\\varepsilon ^2+\\cdot \\cdot \\cdot +\\frac{f^{(k)}(u_0(t))}{k!", "}u_k(t)\\varepsilon ^k+\\cdot \\cdot \\cdot ,$ where $f^{(k)}$ denote the $kth$ Fréchet derivative of $f$ .", "Substituting (REF ) and (REF ) into problem (REF ), we have ${\\left\\lbrace \\begin{array}{ll}\\dfrac{du_0(t)}{dt} -A_\\alpha u_0(t)+f(u_0(t))=g(x),&in\\ \\Omega .\\\\u_0(t)\\big \|_{\\Omega ^c}=0.\\\\u_0(x,0)=p+\\Phi ^0(p),\\end{array}\\right.", "}$ ${\\left\\lbrace \\begin{array}{ll}\\dfrac{du_1(t)}{dt} +A_\\varepsilon u_1(t)+f^{^{\\prime }}(u_0(t))u_1(t)=0,&in\\ \\Omega .\\\\u_1(t)\\big \|_{\\Omega ^c}=0.\\\\u_1(x,0)=\\Phi ^{1}(p).\\end{array}\\right.", "}$ $\\cdot \\cdot \\cdot \\cdot \\cdot $ ${\\left\\lbrace \\begin{array}{ll}\\dfrac{du_k(t)}{dt} +A_\\varepsilon u_k(t)+\\frac{f^{(k)}(u_0(t))}{k!", "}u_k(t)=0,&in\\ \\Omega .\\\\u_k(t)\\big \|_{\\Omega ^c}=0.\\\\u_k(x,0)=\\Phi ^{k}(p).\\end{array}\\right.", "}$ and so on.", "Solving the above problems, we obtain $u_0(t)=e^{A_\\alpha t}u_0(0) - \\int ^t_0e^{A_\\alpha (t-s)}(f(u_0(s))-g(x))ds.$ $u_1(t)=e^{-A_\\varepsilon t}u_1(0) - \\int ^t_0e^{-A_\\varepsilon (t-s)}f^{^{\\prime }}(u_0(s))u_1(s)ds.$ $\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot $ $u_k(t)=e^{-A_\\varepsilon t}u_k(0) - \\int ^t_0e^{-A_\\varepsilon (t-s)}\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds,$ and so forth.", "The right hand side of (REF ) can be represented as $&-\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)(f(\\varphi (s))-g(x))ds\\nonumber \\\\&=-\\int ^0_{-\\infty }e^{-sA_\\alpha }(I-P_N)(f(u_0(s))-g(x))ds \\nonumber \\\\&- \\varepsilon \\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)f^{^{\\prime }}(u_0(s))u_1(s)ds\\nonumber \\\\&\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\nonumber \\\\&-\\varepsilon ^k\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds\\nonumber \\\\&-\\cdot \\cdot \\cdot \\cdot \\cdot \\nonumber \\\\&=I_0+\\varepsilon I_1+\\cdot \\cdot \\cdot +\\varepsilon ^k I_k+\\cdot \\cdot \\cdot ,$ where $I_0=-\\int ^0_{-\\infty }e^{-sA_\\alpha }(I-P_N)(f(u_0(s))-g(x))ds$ , $I_k=-\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds$ , $k\\ge 1.$ By (REF ) and (REF ), we infer from (REF ) that $\\Phi ^0(p)+\\varepsilon \\Phi ^1(p)+\\varepsilon ^2\\Phi ^2(p)+\\cdot \\cdot \\cdot +\\varepsilon ^k\\Phi ^k(p)+\\cdot \\cdot \\cdot =I_0+\\varepsilon I_1+\\cdot \\cdot \\cdot +\\varepsilon ^k I_k+\\cdot \\cdot \\cdot .$ Matching the powers of $\\varepsilon $ , we obtain $\\Phi ^0(p)=-\\int ^0_{-\\infty }e^{-sA_\\alpha }(I-P_N)(f(u_0(s))-g(x))ds,$ and $\\Phi ^1(p)=-\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)f^{^{\\prime }}(u_0(s))u_1(s)ds,$ $\\cdot \\cdot \\cdot \\cdot \\cdot $ $\\Phi ^k(p)=-\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds,$ and so on.", "Thus, we see that if the inertial manifold $\\mathcal {M}_\\varepsilon $ of (REF ) exists and is a graph of a sufficiently smooth function of $\\varepsilon $ , then $\\Phi ^0(p)$ and $\\Phi ^k(p)$ , $p\\in P_N H$ , $k\\ge 1$ as obtained above are well defined.", "Theorem 5.1 Let $\\mathcal {M}_\\varepsilon $ be the inertial manifold for the system (REF ).", "Assume that the following conditions hold: Nonlinear function $f$ is sufficiently smooth, There exists an $N\\in \\mathbb {N}$ such that $\\lambda _{N+1}-\\lambda _N>2l_f.$ Then for a sufficiently small $\\varepsilon $ , the inertial manifold $\\mathcal {M}_\\varepsilon $ can be represented as $\\mathcal {M}_\\varepsilon =\\lbrace p+\\Phi ^0(p)+\\varepsilon \\Phi ^1(p)+\\varepsilon ^2\\Phi ^2(p)+\\cdot \\cdot \\cdot +\\varepsilon ^k\\Phi ^k(p)+\\cdot \\cdot \\cdot ~\|~p\\in P_N H\\rbrace ,$ where $\\Phi ^0(p)$ , $\\Phi ^k(p)$ , $k\\ge 1$ , are in (REF ) and (REF ), respectively.", "By the above analysis, we only need to show that (REF ) and (REF ) are well defined.", "According to section 4, the existence and uniqueness of $\\Phi ^0(p)$ obviously.", "Next, we will verify (REF ) well defined.", "Thanks to (REF ), (REF ) and (REF ), for problem (REF ) we have $u(t,p)=J(u,p)(t)=&e^{-tA_\\varepsilon }p+\\int ^0_te^{-(t-s)A_\\varepsilon }P_N(f(u(s))-g(x))ds\\nonumber \\\\&-\\int ^t_{-\\infty }e^{-(t-s)A_\\varepsilon }(I-P_N)(f(u(s))-g(x))ds.$ Equating the terms with the same power of $\\varepsilon $ , we get $u_k(t,p)=J(u_k,p)(t)=&\\int ^0_te^{-(t-s)A_\\varepsilon }P_N\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds\\nonumber \\\\&-\\int ^t_{-\\infty }e^{-(t-s)A_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds.$ Under further assumptions, for a large enough $N$ , such that $\\lambda _{N+1}-\\lambda _N>2l_f$ , one can choose $\\sigma $ such that $\\lambda _N + 2l_fK_1<\\sigma <\\lambda _{N+1}-2l_fK_1K_2.$ This $\\sigma $ is used to define the Banach space $\\mathcal {F}_\\sigma =\\lbrace \\varphi \\in C((-\\infty ,0],H)~\|~\\Vert \\varphi \\Vert _{\\mathcal {F}_\\sigma }=\\sup _{t\\le 0}e^{\\sigma t}\|\\varphi (t)\|<\\infty \\rbrace .$ By section 2, we know that $u_k(t,p)$ is a well defined mapping from $\\mathcal {F}_\\sigma \\times P_N H\\longrightarrow \\mathcal {F}_\\sigma .$ Indeed, for $\|\\frac{f^{(k)}(u_0(s))}{k!", "}\|\\le 2l_f$ , we have $\|J(u_k,p)(t)\|\\le 2l_f\\max \\lbrace \\int ^0_t\|e^{-(t-s)A_\\varepsilon }P_N\|\|u_k(s)\|ds,\\int ^t_{-\\infty }\|e^{-(t-s)A_\\varepsilon }(I-P_N)\|\|u_k(s)\|ds\\rbrace .$ By exponential dichotomy properties, we have $\\Vert J(u_k,p)(t)\\Vert _{\\mathcal {F}_\\sigma }&\\le 2l_f\\sup _{t\\le 0}\\max \\lbrace \\int ^0_tK_1e^{(\\sigma -\\lambda _N)(t-s)}ds,\\int ^t_{-\\infty } K_2e^{(\\sigma -\\lambda _{N+1})(t-s)}ds\\rbrace \\Vert u_k(s)\\Vert _{\\mathcal {F}_\\sigma }\\nonumber \\\\&\\le \\max \\lbrace \\frac{2l_fK_1}{\\sigma -\\lambda _N},\\frac{2l_fK_2}{\\lambda _{N+1}-\\sigma }\\rbrace \\Vert u_k(s)\\Vert _{\\mathcal {F}_\\sigma }<\\infty .$ Next we prove that $J(u_k,p)(t)$ is a contraction mapping on $\\mathcal {F}_\\sigma $ .", "$\\Vert J(u^1_k,p)(t)-J(u^2_k,p)(t)\\Vert _{\\mathcal {F}_\\sigma }&=\\sup _{t\\le 0}e^{\\sigma t}\\lbrace \|\\int ^0_te^{-(t-s)A_\\varepsilon }P_N\\frac{f^{(k)}(u_0(s))}{k!", "}(u_k^1(s)-u_k^2(s))ds+\\nonumber \\\\&\\int ^t_{-\\infty }e^{-(t-s)A_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}(u_k^1(s)-u_k^2(s))ds\|\\rbrace \\nonumber \\\\&\\le \\sup _{t\\le 0}e^{\\sigma t}\\max \\lbrace \\int ^0_t\|e^{-(t-s)A_\\varepsilon }P_N\\frac{f^{(k)}(u_0(s))}{k!", "}(u_k^1(s)-u_k^2(s))\|ds,\\nonumber \\\\&\\int ^t_{-\\infty }\|e^{-(t-s)A_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}(u_k^1(s)-u_k^2(s))\|ds\\rbrace .$ By exponential dichotomy condition which was presented in section 2, and $\|\\frac{f^{(k)}(u_0(s))}{k!", "}\|\\le 2l_f$ , we have $&\\Vert J(u^1_k,p)(t)-J(u^2_k,p)(t)\\Vert _{\\mathcal {F}_\\sigma }\\le \\sup _{t\\le 0}e^{\\sigma t}\\max \\lbrace \\int ^0_t\|K_1l_fe^{-\\lambda _N(t-s)}\|(u_k^1(s)-u_k^2(s))\|ds,\\nonumber \\\\&\\int ^t_{-\\infty }K_2l_fe^{-\\lambda _{N+1}(t-s)}\|(u_k^1(s)-u_k^2(s))\|ds\\rbrace .\\nonumber \\\\&\\le \\sup _{t\\le 0}\\Vert (u_k^1(s)-u_k^2(s))\\Vert _{\\mathcal {F}_\\sigma }\\max \\lbrace K_12l_f\\int ^0_te^{(\\sigma -\\lambda _N)(t-s)}ds\\nonumber \\\\&,K_2l_f\\int ^t_{-\\infty }e^{(\\sigma -\\lambda _{N+1})(t-s)}ds\\rbrace \\nonumber \\\\&\\le \\max \\lbrace \\frac{2l_fK_1}{\\sigma -\\lambda _N},\\frac{2l_fK_2}{\\lambda _{N+1}-\\sigma }\\rbrace \\Vert (u_k^1(s)-u_k^2(s))\\Vert _{\\mathcal {F}_\\sigma }.$ Since by spectral condition, $\\frac{2l_fK_1}{\\sigma -\\lambda _N}<1$ and $\\frac{2l_fK_2}{\\lambda _{N+1}-\\sigma }<1$ , hence $J(u_k,p)(t)$ is a contraction mapping on $\\mathcal {F}_\\sigma $ .", "Using the contraction mapping principle, there exists unique $u_k(t,p)$ satisfy (REF ).", "Let $\\Phi _k(p)=(I-P_N)u_k(p)(0),~\\forall ~p\\in P_N H.$ We have $\\Phi ^k(p)=-\\int ^0_{-\\infty }e^{sA_\\varepsilon }(I-P_N)\\frac{f^{(k)}(u_0(s))}{k!", "}u_k(s)ds,$ which satisfy (REF ).", "The proof is complete.", "Following, we consider the asymptotic behavior between $\\mathcal {M}_\\varepsilon $ and $\\mathcal {M}_0$ when $\\varepsilon \\longrightarrow 0^+$ .", "We recall that the inertial manifold for problem (REF ) is $\\mathcal {M}_\\varepsilon =\\lbrace p+\\Phi ^0(p)+\\varepsilon \\Phi ^1(p)+\\varepsilon ^2\\Phi ^2(p)+\\cdot \\cdot \\cdot +\\varepsilon ^k\\Phi ^k(p)+\\cdot \\cdot \\cdot ~\|~p\\in P_N H\\rbrace ,$ where $\\Phi ^0(p)$ , $\\Phi ^k(p)$ are as (REF ) and (REF ) respectively.", "The inertial manifold for problem (REF ) is $\\mathcal {M}_0=\\lbrace p+\\Phi ^0(p)~\|~p\\in P_N H\\rbrace ,$ where $\\Phi ^0(p)$ as in (REF ).", "In proof of Theorem REF , we know that $u_k(t,p)\\in \\mathcal {F}_\\sigma $ , $p\\in P_NH$ , hence we have $\|\\Phi ^k(p)\|<\\infty $ , $k\\ge 1$ .", "Thus, we have following main result in this section Theorem 5.2 Let $\\mathcal {M}_\\varepsilon $ and $\\mathcal {M}_0$ be inertial manifolds for problems (REF ) and (REF ) respectively.", "Then $\\mathcal {M}_\\varepsilon $ convergence to $\\mathcal {M}_0$ in the norm of $H$ as $\\varepsilon \\longrightarrow 0^+,$ that is $~\\forall ~0<\\varepsilon <<1$ $dist_H(\\mathcal {M}_\\varepsilon ,\\mathcal {M}_0)\\le o(\\varepsilon ),~as~\\varepsilon \\longrightarrow 0^+,$ where $dist_H(\\cdot ,\\cdot )$ denote Hausdorff semi-distance between two sets in space $H$ .", "Supposing that in construction of $\\mathcal {M}_\\varepsilon $ and $\\mathcal {M}_0$ , we choose same initial data $p\\in P_N H$ .", "Then by the expression of $\\mathcal {M}_\\varepsilon $ and $\\mathcal {M}_0$ we get $dist_H(\\mathcal {M}_\\varepsilon ,\\mathcal {M}_0)\\le \\varepsilon \|\\Phi ^1(p)\|+\\varepsilon ^2\|\\Phi ^2(p)\|+\\cdot \\cdot \\cdot +\\varepsilon ^k\|\\Phi ^k(p)\|+\\cdot \\cdot \\cdot .$ Since $\|\\Phi ^k(p)\|<\\infty $ , $k\\ge 1$ , we obtain the result.", "The proof is complete." ] ]	1403.0165
[ [ "Universal quantum gates on microwave photons assisted by circuit quantum\n electrodynamics" ], [ "Abstract Based on a microwave-photon quantum processor with two superconducting resonators coupled to one transmon qutrit, we construct the controlled-phase (c-phase) gate on microwave-photon-resonator qudits, by combination of the photon-number-dependent frequency-shift effect on the transmon qutrit by the first resonator and the resonant operation between the qutrit and the second resonator.", "This distinct feature provides us a useful way to achieve the c-phase gate on the two resonator qudits with a higher fidelity and a shorter operation time, compared with the previous proposals.", "The fidelity of our c-phase gate can reach 99.51% within 93 ns.", "Moreover, our device can be extended easily to construct the three-qudit gates on three resonator qudits, far different from the existing proposals.", "Our controlled-controlled-phase gate on three resonator qudits is accomplished with the assistance of a transmon qutrit and its fidelity can reach 92.92% within 124.64 ns." ], [ "Introduction", "Quantum computation has attracted much attention in recent years [1].", "Many important schemes have been proposed for quantum computation by using different quantum systems, such as photonic systems [2], [3], [4], [5], [6], nuclear magnetic resonance [7], [8], quantum dots [9], [10], [11], [12], and diamond nitrogen-vacancy (NV) centers [13], [14], [15].", "Universal quantum gates are the key elements in constructing a universal quantum computer.", "Moreover, they can be used to produce the entanglement of multipartite quantum systems.", "The controlled-phase (c-phase) gate is one of the important universal two-qubit gates.", "It has the same role as the controlled-not gate in quantum computation.", "The controlled-controlled-phase (cc-phase) gate is an important three-qubit gate which can play the same role as the three-qubit Toffoli gate which can be used to construct a universal quantum computation with single-qubit Hadamard operations [1].", "Circuit quantum electrodynamics (QED), which combines superconducting circuits and cavity QED, provides a good platform for quantum computation [17], [16], [18], [19], [20].", "A superconducting Josephson junction can act as a perfect qubit and it has some good features, such as the large scale integration [21], a relatively long coherence time of about $0.1$ ms [22], the versatility in its energy-level structure with $\\Xi $ , $\\Lambda $ , $V$ , and even $\\Delta $ types [23] which cannot be found in atom systems, and the superconducting qubit with tunable coupling strength [24], [25], [26].", "All these characteristics have attracted much attention focused on quantum information processing on superconducting qubits in circuit QED.", "Some interesting proposals for quantum information processing on qubits have been presented, such as the reset of a superconducting qubit [27], [28], [29], universal quantum gates and entanglement generation [30], [31], [32], [33], [34], [35], and single-shot individual qubit measurement and the joint qubit readout [36], [37], [38].", "A superconducting coplanar resonator whose quality factor $Q$ can be increased to be $10^{6}$ [39], [40], [41], [42], can act as a qudit because it contains some microwave photons whose lifetimes are much longer than that of a superconducting qubit [42], [43], [44].", "The coupling strength between a resonator and a transmission line is tunable [45].", "Moreover, the strong and even ultrastrong coupling [16], [46], [47], [48], [49], [50], [51] in circuit QED affords a strong nonlinear interaction between a superconducting qubit and a microwave-photon qudit.", "These good features make resonators a powerful platform for quantum computation as well.", "There are some interesting studies on resonator qudits.", "For example, Moon and Girvin [52] studied theoretically the parametric down-conversion and squeezing of microwaves inside a transmission line resonator, resorting to circuit QED in 2005.", "In 2007, Marquardt [53] presented an efficient scheme for the generation of microwave photon pairs by parametric down-conversion in a superconducting resonator coupled to a superconducting qubit.", "In 2008, Hofheinz et al.", "[54] demonstrated the preparation of pure Fock states with a microwave resonator, resorting to a superconducting phase qubit.", "In the next year, they synthesized arbitrary quantum states in a superconducting resonator [55].", "In 2009, Rebić et al.", "[56] introduced a scheme for generating giant Kerr nonlinearities in circuit QED.", "In 2010, Bergeal et al.", "[57] proposed a practical microwave device for achieving parametric amplification.", "In this year, Johnson et al.", "[58] demonstrated a quantum nondemolition detection scheme that measures the number of photons inside a high-quality-factor microwave cavity on a chip and Strauch et al.", "[59] presented an effective method to synthesize an arbitrary quantum state of two superconducting resonators.", "In 2011, Mariantoni et al.", "[60] used a three-resonator circuit to shuffle one- and two-photon Fock states between the three resonators and demonstrated qubit-mediated vacuum Rabi swaps between two resonators.", "In addition, there are some interesting works to generate the entanglement between the resonator qudits [61], [62], [63], [64], [65], [66], [67], [68].", "To realize the quantum computation based on resonator qudits, people should construct the universal quantum gates on qudits.", "In 2007, Schuster et al.", "[69] proposed the effect of the number-state-dependent interaction between a superconducting qubit and resonator qudits, which provides an interesting way to achieve the state-selective qubit rotation.", "Based on this effect, Strauch [70] presented an interesting scheme to construct the c-phase gate on two superconducting resonator qudits in 2011.", "In his work, each of two resonators (A and B) is coupled to an auxiliary three-level transmon or phase qutrit (a and b), and each qutrit is coupled to each other directly.", "The operation time of the c-phase gate on two resonator qudits is 150 ns.", "In 2012, Wu et al.", "[71] gave an effective scheme for the construction of the c-phase gate on two resonators by using the number-state-dependent interaction between a two-energy-level charge qubit and two resonator-qudits for one-way quantum computation, and the operation time of the c-phase gate was 125 ns.", "In this paper, we give a microwave-photon quantum processor with two resonators which are coupled to just one transmon qutrit which has the characteristics of a lesser anharmonicity energy level and a long coherence time [72], and we construct an effective c-phase gate on two resonator qudits, resorting to the combination of the number-state-dependent interaction between the qutrit and one resonator-qudit subsystem and the simple resonant operation between the qutrit and another resonator-qudit subsystem.", "This different physical mechanism provides us a faster way to achieve a higher-fidelity c-phase gate on the two resonator qudits without increasing the difficulty of its implementation, compared with the previous proposals [70], [71].", "The fidelity of our c-phase gate is $99.51\\%$ within the operation time of 93 ns.", "Moreover, our device can be extended easily to construct the three-qudit cc-phase gate on three resonator qudits, by using a resonator to complete the simple resonant operation and the other two resonators to achieve the number-state-dependent interaction on the transmon qutrit, far different from the existing proposals.", "Its fidelity is $92.92\\%$ within the operation time of 124.64 ns.", "Figure: (Color online) Schematic diagram for our c-phase gate ontwo microwave-photon-resonator qudits, by combination of thenumber-state-dependent interaction between the transmon qutritand the left resonator (r 1 r_1) and the simple resonantoperation between the transmon qutrit and the right resonator (r 2 r_2).", "Thetwo resonators are capacitively coupled to thequtrit whose transition frequency can be tuned by anexternal flux." ], [ "Controlled-phase gate on two resonators in circuit QED", "Let us first consider a system composed of a perfect two-level superconducting qubit $q$ and a resonator ($r_1$ ), whose schematic diagram is the same as that shown in the dashed-line box in Fig.", "REF (by replacing the three-energy-level qutrit with a two-energy-level qubit).", "The Hamiltonian for this system under the rotating-wave approximation ($\\hbar =1$ ) is $ H_1=\\omega _{r_1}a^{+}a+\\omega _q \\sigma ^{+}\\sigma ^{-}+g\\left(a\\sigma ^{+}+a^{+}\\sigma ^{-}\\right), $ where $\\sigma ^{+}=\\vert 1\\rangle \\langle 0\\vert $ and $a^{+}$ are the creation operators of the superconducting qubit $q$ and the resonator $r_1$ , respectively.", "$g$ is the coupling strength between the qubit and the resonator.", "$\\omega _{r_1}$ and $\\omega _q$ are the transition frequencies of the resonator $r_1$ and the qubit, respectively.", "In the dispersive regime ($\\frac{g^{2}}{\\Delta }$ $\\le 1$ ) in circuit QED, by making the unitary transformation $U=\\exp \\left[ \\frac{g}{\\Delta }\\left(a\\sigma ^{+}-a^{+}\\sigma ^{-}\\right)\\right]$ , the Hamiltonian $ H_1$ becomes [16] $ H^{\\prime }_1=UH_{1}U^{+}\\approx \\omega _{r_1}a^{+}a+\\frac{1}{2}\\left[\\omega _{q}+\\frac{g^{2}}{ \\Delta }(2a^{+}a+1)\\right]\\sigma _{z}.$ The effect of the photon-number-dependent transition frequency of the qubit can be described as $ {\\omega ^{\\prime }}^{n}_{q}=\\omega _{q}+\\frac{g^{2}}{\\Delta }(2n+1).$ Here $\\Delta =\\omega _{r_1}-\\omega _{q}$ .", "$n=a^{+}a$ is the photon number in a resonator.", "${\\omega ^{\\prime }}^{n}_{q}$ is the changed transition frequency of the qubit due to the different photon numbers in the resonator.", "In the dispersive strong regime ($\\frac{g^{2}}{\\Delta } \\ll 1$ ) [69], the photon-number-dependent transition frequency of the qubit is too small to distinguish the different transition frequencies of the qubit due to the different photon numbers in the resonator.", "By keeping $\\frac{g}{\\Delta }$ to be a small value to make Eq.", "(REF ) work well, one can increase the coupling strength to make the transition frequency of the qubit depend on largely the photon number, shown in Eq.", "(REF ).", "That is, if we apply a drive field with the frequency equivalent to the transition frequency of the qubit when $n=1$ , and take the proper amplitude $ \\left\|\\Omega \\right\|\\ll \\frac{g^{2}}{\\Delta }$ to suppress the error generated by off-resonant transitions sufficiently, the field will flip the qubit only if there is one microwave photon in the resonator.", "On the other hand, if we apply a drive field with the frequency equivalent to the transition frequency of the qubit when $n=0$ , and take the proper amplitude, the field will flip the qubit only if there is no microwave photon in the resonator.", "To describe this effect, we consider a system with the resonator $r_{1}$ coupled to a practical transmon qutrit [73], whose Hamiltonian is (under the rotating-wave approximation) $ H_{2} &=& \\sum _{l=g,e,f}E_{l}\\left\|l\\right\\rangle _{q}\\left\\langle l\\right\|+\\omega _{r_1}a_{1}^{+}a_{1}+g_{r_1}^{g,e}(a_{1}^{+}\\sigma _{g,e}^{-}+a_{1}\\sigma _{g,e}^{+})\\nonumber \\\\&&+g_{r_1}^{e,f}(a_{1}^{+}\\sigma _{e,f}^{-}+a_{1}\\sigma _{e,f}^{+}),$ where $\\left\|g\\right\\rangle _{q}$ , $\\left\|e\\right\\rangle _{q}$ , and $\\left\|f\\right\\rangle _{q}$ are the first three lower-energy levels of the qutrit.", "$\\sigma _{g,e}^{+}$ and $\\sigma _{e,f}^{+}$ are the creation operators for the transitions $\\left\|g\\right\\rangle _{q}\\rightarrow \\left\|e\\right\\rangle _{q}$ and $\\left\|e\\right\\rangle _{q}\\rightarrow \\left\|f\\right\\rangle _{q}$ of the qutrit $q$ , respectively.", "$a_{1}^{+}$ is the creation operator of the resonator $r_1$ .", "The energy for the level $l$ of $q$ is $E_{l}$ , and $ \\omega _{r_1}$ is the transition frequency of $r_{1}$ .", "$g_{1}^{g,e}$ and $g_{1}^{e,f}$ are the coupling strengths between these two transitions of $q$ and $r_{1}$ .", "A microwave drive field $H_{d}=\\Omega (\\left\|f\\right\\rangle _{q}\\left\\langle e\\right\|e^{-i\\omega _{d}t}+\\left\|e\\right\\rangle _{q}\\left\\langle f\\right\|e^{i\\omega _{d}t})$ with a proper amplitude $\\Omega $ is applied to interact with the qutrit, and here the frequency $\\omega _{d}$ is chosen to be equivalent to the transition frequency ($\\left\|e\\right\\rangle _{q}\\leftrightarrow \\left\|f\\right\\rangle _{q}$ ) of the qutrit $q$ when there is no microwave photon in the resonator.", "Due to the realistic quantum Rabi oscillation (ROT) occurring between the dress states of the system [74], we simulate the expectation value of ROT$_0^{e,f}$ $(\\left\|0\\right\\rangle _{r_1}\\left\|e\\right\\rangle _{q})_{dress}\\leftrightarrow (\\left\|0\\right\\rangle _{r_1}\\left\|f\\right\\rangle _{q})_{dress}$ and ROT$_1^{e,f}$ $(\\left\|1\\right\\rangle _{r_1}\\left\|e\\right\\rangle _{q})_{dress}\\leftrightarrow (\\left\|1\\right\\rangle _{r_1}\\left\|f\\right\\rangle _{q})_{dress}$ , shown in Fig.", "REF .", "The transition frequencies of $\\left\|g\\right\\rangle _{q}\\leftrightarrow \\left\|e\\right\\rangle _{q}$ and $ \\left\|e\\right\\rangle _{q}\\leftrightarrow \\left\|f\\right\\rangle _{q}$ of the qutrit are chosen to be $\\omega _{g,e}/(2\\pi )=E_e-E_g=8.7GHz$ and $\\omega _{e,f}/(2\\pi )=E_f-E_e=8.0GHz$ , respectively.", "$\\omega _{r_1}/(2\\pi )=7.5GHz$ .", "The coupling strengths between two transitions of the qutrit and $r_{1}$ are taken in convenience with $g_{1}^{g,e}/(2\\pi )=g_{1}^{e,f}/(2\\pi )=0.2GHz$ .", "The frequency and amplitude of the drive field are $\\omega _{d}/(2\\pi )=8.043GHz$ and $ \\Omega =0.0115GHz$ , respectively.", "Figure: (Color online) The expectation values of the probabilitydistributions of the quantum Rabi oscillations ROT 0 g,e _0^{g,e},ROT 0 e,f _0^{e,f}, ROT 1 g,e _1^{g,e}, and ROT 1 e,f _1^{e,f}.", "ROT 0 g,e _0^{g,e} andROT 0 e,f _0^{e,f} represent the oscillations of (0 r 1 g q ) dress ↔(0 r 1 e q ) dress (\\left\|0\\right\\rangle _{r_1}\\left\|g\\right\\rangle _{q})_{dress}\\leftrightarrow (\\left\|0\\right\\rangle _{r_1}\\left\|e\\right\\rangle _{q})_{dress} and (0 r 1 e q ) dress ↔(0 r 1 f q ) dress (\\left\|0\\right\\rangle _{r_1}\\left\|e\\right\\rangle _{q})_{dress}\\leftrightarrow (\\left\|0\\right\\rangle _{r_1}\\left\|f\\right\\rangle _{q})_{dress}, respectively.ROT 1 g,e _1^{g,e} and ROT 1 e,f _1^{e,f} represent the oscillations of(1 r 1 g q ) dress ↔(1 r 1 e q ) dress (\\left\|1\\right\\rangle _{r_1}\\left\|g\\right\\rangle _{q})_{dress}\\leftrightarrow (\\left\|1\\right\\rangle _{r_1}\\left\|e\\right\\rangle _{q})_{dress} and (1 r 1 e q ) dress ↔(1 r 1 f q ) dress (\\left\|1\\right\\rangle _{r_1}\\left\|e\\right\\rangle _{q})_{dress}\\leftrightarrow (\\left\|1\\right\\rangle _{r_1}\\left\|f\\right\\rangle _{q})_{dress}, respectivelyAs shown in Fig.", "REF , the maximal probability of ROT$_{0}^{e,f}$ can reach $100\\%$ .", "After a period of ROT$_{0}^{e,f}$ , a $\\pi $ phase shift can be generated in the state $ (\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress}$ , and ROT$_{1}^{e,f}$ and the other oscillations take place with a very small probability, which indicates that the final state of the system composed of $r_{1}$ and $q$ becomes $ \\left\|\\phi _{f}\\right\\rangle &=& \\frac{1}{2}[(\\left\|0\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}-(\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|1\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress}]$ after the state-selective qubit rotation if the initial state of the system is $ \\left\|\\phi _{0}\\right\\rangle &=& \\frac{1}{2}[(\\left\|0\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|1\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress}].$ This is just the outcome of a hybrid c-phase gate on $r_1$ and $q$ by using $r_1$ as the control qubit and $q$ as the target qubit.", "At the beginning and the end of the algorithm, one can turn on and off the coupling between $r_1$ and $q$ to evolve the dress states into the computational states [74].", "On one hand, one can tune the transition frequency of the resonator or the qutrit to make them resonate or largely detune with each other, in order to turn on or off the interaction between $q$ and a non-computational resonator.", "On the other hand, one can tune on or off the coupling between the qutrit and the noncomputational resonator [75], [76].", "The principle of our c-phase gate on two resonator qudits based on the state-selective qubit rotation is shown in Fig.", "REF .", "The matrix representation of the c-phase gate can be written as $ \\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & -1\\end{array}\\right)$ in the basis of a two-resonator-qudit system $\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}$ , $\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}$ , $\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}$ , $\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}$ }.", "The Hamiltonian of the system composed of the resonators $r_1$ , $r_2$ , and $q$ can be written as (under the rotating-wave approximation) $ H_2 &=&\\sum \\limits _{l=g,e,f}E_{l}\\left\|l\\right\\rangle _{q}\\left\\langle l\\right\|+ \\sum _{i=1,2}[\\omega _{r_i}a_{i}^{+}a_{i}+g_{i}^{g,e}(a_{i}^{+}\\sigma _{g,e}^{-}+a_{i}\\sigma _{g,e}^{+})\\nonumber \\\\&&+g_{i}^{e,f}(a_{i}^{+}\\sigma _{e,f}^{-}+a_{i}\\sigma _{e,f}^{+})].$ Suppose that the initial state of the system is $ \\left\|\\psi _{0}\\right\\rangle = \\frac{1}{2}(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}+\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2})\\otimes \\left\|g\\right\\rangle _{q}.$ Our c-phase gate on the two resonators can be accomplished with three steps as follows.", "Figure: (Color online) (a) The density operator (ρ 0 \\rho _0) of theinitial state ψ 0 \\left\|\\psi _{0}\\right\\rangle of the systemcomposed of the two resonators and the qutrit in our c-phase gate.", "Panels(b) and (c) are the real part (Real[ρ 3 \\rho _3]) and the imaginary part(Imag[ρ 3 \\rho _3]) of the final state ψ f \\left\|\\psi _{f}\\right\\rangle of the system, respectively.First, by resonating $r_{2}$ and $q$ with $g^{g,e}_{2}t=\\frac{\\pi }{2}$ , and turning off the interaction between $r_{1}$ and $q$ , the system evolves from the initial state $\\left\|\\psi _{0}\\right\\rangle $ to the state $ \\left\|\\psi _{1}\\right\\rangle = \\frac{1}{2}(\\left\|0\\right\\rangle _{1}\\left\|g\\right\\rangle _{q}-i\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q}\\nonumber \\\\+\\left\|1\\right\\rangle _{1}\\left\|g\\right\\rangle _{q}-i\\left\|1\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})\\otimes \\left\|0\\right\\rangle _{2}.$ Second, by turning on the coupling between $r_{1}$ and $q$ , and turning off the coupling between $r_{2}$ and $q$ , the state of the system becomes $ \\left\|\\psi _{1}^{\\prime }\\right\\rangle &=&\\frac{1}{2}[(\\left\|0\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}-i(\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&& +(\\left\|1\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}-i(\\left\|1\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress}]\\otimes \\left\|0\\right\\rangle _{2}.$ By applying a drive field $H_{d}=\\Omega (\\left\|f\\right\\rangle _{q}\\left\\langle e\\right\|e^{-i\\omega _{d}t}+\\left\|e\\right\\rangle _{q}\\left\\langle f\\right\|e^{i\\omega _{d}t})$ with the frequency equivalent to the transition frequency ($\\left\|e\\right\\rangle _{q}\\leftrightarrow \\left\|f\\right\\rangle _{q}$ ) of the qutrit when there is no microwave photon in the resonator $r_1$ , after an operation time of $\\Omega t=\\pi $ , the state of the system is changed to be $ \\left\|\\psi _{2}\\right\\rangle &=& \\frac{1}{2}[(\\left\|0\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&& +(\\left\|1\\right\\rangle _{1}\\left\|g\\right\\rangle _{q})_{dress}-i(\\left\|1\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})_{dress}]\\otimes \\left\|0\\right\\rangle _{2}.$ Third, by turning off the coupling between $r_{1}$ and $q$ , the state of the system evolves from $\\left\|\\psi _{2}\\right\\rangle $ into $ \\left\|\\psi _{2}^{\\prime }\\right\\rangle =\\frac{1}{2}(\\left\|0\\right\\rangle _{1}\\left\|g\\right\\rangle _{q}+i\\left\|0\\right\\rangle _{1}\\left\|e\\right\\rangle _{q}+\\left\|1\\right\\rangle _{1}\\left\|g\\right\\rangle _{q}-i\\left\|1\\right\\rangle _{1}\\left\|e\\right\\rangle _{q})\\otimes \\left\|0\\right\\rangle _{2}.$ By resonating $r_{2}$ and $q$ with $g^{g,e}_{2}t=\\frac{\\pi }{2}$ again, and turning off the interaction between $r_{1}$ and $q$ , the state of the system becomes $ \\left\|\\psi _{f}\\right\\rangle =\\frac{1}{2}(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\!+\\!\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2} \\!+\\!\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2} \\!-\\!\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2})\\!\\otimes \\!\\left\|g\\right\\rangle _{q}.", "$ This is just the result of the c-phase gate on $r_{1}$ and $r_{2}$ by using $r_{1}$ as the control qubit and $r_{2}$ as the target qubit.", "The reduced density operators of the two-resonator system in the initial state $\\vert \\psi _0\\rangle $ in Eq.", "(REF ) and the final state $\\vert \\psi _f\\rangle $ in Eq.", "(REF ) are shown in Fig.", "REF .", "One can see that the fidelity of our c-phase gate on two microwave-photon qudits is about $99.51\\%$ within about 93 ns.", "Here the fidelity is defined as $F=Tr(\\left\|\\sqrt{\\rho _{f}}\\,\\rho _{ideal}\\sqrt{\\rho _{f}}\\right\|)$ [77].", "$\\rho _{f}$ is the density operator of the final state of the two-microwave-photon-qudit system $\\left\|\\psi _{f}\\right\\rangle $ and $\\rho _{ideal}$ is the density operator of the final state of the system after an ideal c-phase gate operation is performed with the initial state $ {\\left\|\\psi _{0}\\right\\rangle }$ .", "Figure: (Color online) (a) The schematic diagram for ourcc-phase gate on a three-qudit microwave-photonsystem.", "(b) The schematic diagram for the coupling between atransmon qutrit and a microwave-photon resonator.", "Here qqrepresents a transmon qutrit.", "r 1 r_1, r 2 r_2, and r 3 r_3 arethree microwave-photon resonators which have the same structure asthose shown in Fig.." ], [ "Controlled-controlled-phase gate on three resonators", "The principle of our cc-phase gate on a three-resonator system is shown in Fig.REF .", "Here the resonator $r_1$ has the same role as $r_2$ and they both are used to provide the effect of the photon-number-dependent transition frequency of the $\\Xi $ -type three-level qutrit to accomplish the state-selective qubit rotation, different from the resonator $r_3$ .", "The photon-number-dependent transition frequency between $ \\left\|e\\right\\rangle _{q}$ and $\\left\|f\\right\\rangle _{q}$ of the qutrit can be written as [59], [74], [71] $ {\\omega ^{\\prime }}^{n_{1},\\,n_{2}}_{\\,e,\\,f}\\approx \\omega _{e,\\,f}+\\frac{(g_{1}^{e,\\,f})^{2}}{\\omega _{e,\\,f}-\\omega _{r_1}}(2n_{1}+1)+\\frac{(g_{2}^{e,\\,f})^{2}}{\\omega _{e,\\,f}-\\omega _{r_2}} (2n_{2}+1).", "$ Here $n_1$ and $n_2$ are the photon numbers in the resonators $r_1$ and $r_2$ , respectively.", "The photon-number-dependent transition frequency of $q$ depends on the relationship of the photon numbers in two resonators.", "That is, one can afford a drive field with the frequency equivalent to the changed transition frequency of the qutrit to achieve the state-selective qubit rotation with different relations between $n_{1}$ and $n_{2}$ .", "Suppose that $\\frac{3(g_{1}^{e,\\,f})^{2}}{ \\omega _{e,f}-\\omega _{r_1}} =\\frac{(g_{2}^{e,\\,f})^{2}}{\\omega _{e,f}-\\omega _{r_2}}$ , one can obtain the relation ${\\omega ^{\\prime }}^{n_{1},\\,n_{2}}_{\\,e,\\,f}=\\omega _{e,\\,f}+\\frac{(g_{1}^{e,\\,f})^{2}}{\\omega _{e,\\,f}-\\omega _{r_1}}(N+4)$ , where $N=2n_{1}+6n_{2}$ .", "The transition frequency of $q$ can be divided into four groups, according to the photon-number relations between $r_{1}$ and $r_{2}$ .", "That is, $\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}$ , $\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}$ , $\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}$ , and $\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}$ with $N=0,2,6$ , and 8, respectively.", "Considering $N=8$ , a drive field with the frequency $\\omega _{d}=\\omega _{e,f}+\\frac{12(g_{1}^{e,\\,f})^{2}}{\\omega _{e,f}-\\omega _{r_1}}$ can flip the qutrit between $\\left\|e\\right\\rangle _{q}$ and $\\left\|f\\right\\rangle _{q}$ only if there is one microwave photon in each of the two resonators $r_1$ and $r_2$ .", "If we take the initial state of the hybrid system composed of $r_1$ , $r_2$ , and $q$ as $ \\left\|\\Phi _{0}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}[(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress}], $ applying a drive field to complete a state-selective qubit rotation operation on the transition between $\\left\|e\\right\\rangle _{q}$ and $\\left\|f\\right\\rangle _{q}$ when there is one microwave photon in each of the two resonators $r_{1}$ and $r_{2}$ , the state of the hybrid system becomes $ \\left\|\\Phi _{f}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}[(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}-(\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress}] $ after the operation time $t=\\frac{\\pi }{ \\Omega }$ .", "This is just the result of a hybrid cc-phase gate on the system composed of $r_{1}$ , $r_{2}$ , and $q$ , by using $r_{1}$ and $r_{2}$ as the control qudits and $q$ as the target qubit.", "With the hybrid cc-phase gate above, we can construct the cc-phase gate on three resonator qudits, shown in Fig.REF .", "The Hamiltonian of the hybrid system composed of the three resonators $r_{1}$ , $r_{2}$ , and $r_{3}$ and the transmon qutrit $q$ is $ H_3 &=& \\sum \\limits _{l=g,e,f}E_{l}\\left\|l\\right\\rangle _{q}\\left\\langle l \\,\\right\|+ \\sum _{i=1,2,3}[\\omega _{i}^{r}a_{i}^{+}a_{i}+g_{i}^{g,e}(a_{i}^{+}\\sigma _{g,e}^{-}+a_{i}\\sigma _{g,e}^{+})\\nonumber \\\\&&+g_{i}^{e,f}(a_{i}^{+}\\sigma _{e,f}^{-}+a_{i}\\sigma _{e,f}^{+})].$ Suppose that the initial state of the system is $ \\left\|\\Psi _{0}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|1\\right\\rangle _{3} \\nonumber \\\\&&+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|1\\right\\rangle _{3} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|1\\right\\rangle _{3} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|1\\right\\rangle _{3})\\otimes \\left\|g\\right\\rangle _{q}.", "$ The cc-phase gate can be achieved with three steps as follows.", "First, we turn off the interaction between the two resonators $r_{1}r_{2}$ and $q$ , and then resonate $r_3$ and the qutrit $q$ in the transition between $\\left\|g\\right\\rangle _{q}$ and $\\left\|e\\right\\rangle _{q}$ .", "After the operation time $t=\\frac{\\pi }{2g_3^{g,e}}$ , the state of the system evolves from $\\left\|\\Psi _{0}\\right\\rangle $ into $ \\left\|\\Psi _{1}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q} \\nonumber \\\\&&+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})\\otimes \\left\|0\\right\\rangle _{3}.$ Second, we turn off the interaction between $r_3$ and $q$ , and turn on the interactions between $r_1$ and $q$ and between $r_2$ and $q$ .", "The state $\\left\|\\Psi _{1}\\right\\rangle $ is changed to be $\\left\|\\Psi ^{\\prime } _{1}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}[(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1 \\!\\right\\rangle _{1}\\!\\left\|1 \\!\\right\\rangle _{2} \\!\\left\|g\\right\\rangle _{q})_{dress}\\!+\\!i(\\left\|1\\!\\right\\rangle _{1}\\left\|1 \\!\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress}] \\!\\otimes \\!", "\\left\|0\\right\\rangle _{3}.\\;\\;\\;\\;\\;\\;\\; $ By taking the hybrid cc-phase gate on $r_{1}$ , $r_{2}$ , and $q$ , we can get $\\left\|\\Psi _{2}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}[(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q})_{dress}+i(\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q})_{dress} \\nonumber \\\\&&+(\\left\|1 \\!\\right\\rangle _{1} \\!\\left\|1 \\!\\right\\rangle _{2} \\!\\left\|g\\right\\rangle _{q})_{dress}\\!-\\!i(\\left\|1\\!\\right\\rangle _{1} \\!\\left\|1 \\!\\right\\rangle _{2}\\!\\left\|e\\right\\rangle _{q})_{dress}]\\!\\otimes \\!", "\\left\|0\\right\\rangle _{3}.\\;\\;\\;\\;\\;\\;\\; $ Third, we turn off the coupling between $r_{1} r_{2}$ and $q$ , the state of the system becomes $\\left\|\\Psi _{2}^{\\prime }\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}[\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q} \\nonumber \\\\&&+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}+i\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|e\\right\\rangle _{q} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|g\\right\\rangle _{q}-i\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|e\\right\\rangle _{q}]\\otimes \\left\|0\\right\\rangle _{3}.", "$ By resonating $r_3$ and $q$ , we can get the final state of the system as $\\left\|\\Psi _{f}\\right\\rangle &=&\\frac{1}{2\\sqrt{2}}[\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|0\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|1\\right\\rangle _{3} \\nonumber \\\\&&+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|0\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|1\\right\\rangle _{3} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}+\\left\|1\\right\\rangle _{1}\\left\|0\\right\\rangle _{2}\\left\|1\\right\\rangle _{3} \\nonumber \\\\&&+\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|0\\right\\rangle _{3}-\\left\|1\\right\\rangle _{1}\\left\|1\\right\\rangle _{2}\\left\|1\\right\\rangle _{3}]\\otimes \\left\|g\\right\\rangle _{q}.", "\\;\\;\\;\\;\\;\\;\\;\\;$ This is just the outcome of the cc-phase gate operation on $r_{1}$ , $r_{2}$ , and $r_3$ , by using $r_{1}$ and $r_{2}$ as the control qudits and $r_3$ as the target qudit.", "Figure: (Color online) The real part (a) and the imaginary part (b)of the final state Ψ f \\left\|\\Psi _{f}\\right\\rangle of the systemcomposed of the three microwave-photon resonators and thesuperconducting qutrit in our cc-phase gate, respectively.We simulate the evolution for the density operator of the system with the initial state $\\left\|\\Psi _{0}\\right\\rangle $ , and the reduced density operator of the final state $\\left\|\\Psi _{f}\\right\\rangle $ shown in Eq.", "(REF ) is shown in Fig.REF .", "Here, $\\omega ^{r_1}/(2\\pi )=6.5GHz$ , $\\omega ^{r_2}/(2\\pi )=7.5GHz$ , $\\omega ^{r_3}/(2\\pi )=7.5GHz$ , $\\omega ^{g,e}/(2\\pi )=E_e-E_g=8.7GHz$ , $\\omega ^{e,f}/(2\\pi )=E_f-E_e=8.0GHz$ , $g^{g,e}_1/(2\\pi )=g^{e,f}_1/(2\\pi )=0.2GHz$ , $g^{g,e}_2/(2\\pi )=g^{e,f}_2/(2\\pi )=0.2GHz$ , $g^{g,e}_3/(2\\pi )=g^{e,f}_3/(2\\pi )=0.12GHz$ , $\\omega ^{d}/(2\\pi )=8.1768GHz$ , and $\\Omega =0.0266GHz$ .", "From Fig.REF , one can see that the fidelity of our cc-phase gate can reach $92.92\\%$ within about 124.64 ns, without considering the decoherence and leakage of the resonators." ], [ "Discussion and summary", "The deterministic approaches to realize the nonlinear interaction between two photons for quantum computation are usually based on the Kerr effect.", "Here we constructed the local c-phase and cc-phase gates on the resonator qudits in a microwave-photon quantum processor assisted by only one transmon qutrit, resorting to the combination of the number-state-dependent interactions between the transmon qutrit and the resonator qudits and the simple resonant interaction between the qutrit and one resonator qudit.", "Usually, the processor on microwave-photon systems needs a tunable coupling superconducting qubit or some tunable resonators [59], [70], [71], [74].", "The experiments showed that a tunable coupling strength between a superconducting qubit and a superconducting resonator is feasible [24], [25], [26].", "Some recent experiments were demonstrated for tuning the frequency of a resonator [78], [79], [80].", "In order to avoid shortening the relaxation time of the qutrit, the processor needs some high-$Q$ resonators.", "That is, the present c-phase and cc-phase gates are feasible, similar to those in Refs.", "[59], [71], [67], [63] In our calculation, the parameters of the transmon qutrit are chosen as the same as those in Ref.", "[73].", "Actually, the coupling strength for the two different transitions of a transmon qutrit and a microwave-photon resonator is asymptotically increased as $(E_{J}/E_{C})^{1/4}$ (for a transmon qubit, $20<E_{J}/E_{C}<5\\times 10^{4}$ ) [72].", "That is, it is reasonable to use the same coupling strength for the two different transitions of a transmon qutrit and a resonator for convenience.", "The amplitudes of the drive fields for constructing the c-phase and cc-phase gates are too small (compared with the anharmonicity between the two transitions of the qutrit) to induce the influences coming from the higher excited energy levels of the transmon qutrit.", "The coherence time of a transmon qubit approaches 0.1 ms [22] and the life time of microwave photons contained in resonators are always longer than that of a qutrit [42], which means our gates can be operated several hundreds of times within the life time of the processor.", "In order to evolve the systems from the dress states to the computational states in our schemes for constructing the gates, one needs to tune on or off the interaction between the resonators and the qutrit, the same as in Refs.", "[59], [74].", "The quantum error coming from this method in experiment is determined by the technique of the tunable transition frequency of a superconducting resonator or the tunable coupling strength between the qutrit and the resonators.", "In our calculation, we don't consider the error coming from the preparation of the initial states shown in Eqs.", "(REF ) and (REF ).", "To prepare the states shown in Eqs.", "(REF ) and (REF ) from the state of the system composed of multiple resonators coupled to a qutrit $\\otimes _i\|0\\rangle _i\|g\\rangle _q$ , one needs to take the single-qubit gate by appling a $\\frac{\\pi }{2}$ pulse on the qutrit, and the state of the system is changed into $\\otimes _i\\frac{1}{\\sqrt{2}}\|0\\rangle _i(\|g\\rangle _q-i\|e\\rangle _q)$ .", "By resonating the qutrit and the resonator $j$ with the time $t=\\frac{3\\pi }{2g_j^{g,e}}$ , one can obtain the state $\\otimes _{i,i\\ne j}\\frac{1}{\\sqrt{2}}(\|0\\rangle _j+\|1\\rangle _j)\|0\\rangle _i\|g\\rangle _q$ [71].", "By repeating the steps to the rest resonators, one can obtain the states shown in Eqs.", "(REF ) and (REF ) $\\otimes _i\\frac{1}{\\sqrt{2^i}}(\|0\\rangle _i+\|1\\rangle _i)\|g\\rangle _q$ .", "The single-qubit gate can be realized with the quantum error of $0.007 \\pm 0.005$ [81] and even smaller than 0.0009 [35], which is too small to influence the fidelity of our gates.", "By using a qubit to read out the state of the resonator [54], [55], [58], one can complete fully the tomography of the resonator logic gates [70].", "Now, let us compare our c-phase gate on two resonators with the one constructed in Ref.", "[70].", "In Ref.", "[70], Strauch presented an interesting scheme to construct the c-phase gate on two superconducting resonator qudits.", "In his work, each of two resonators (A and B) is coupled to an auxiliary three-level transmon or phase qutrit (a and b), and each qutrit should be coupled to each other directly.", "Moreover, the c-phase gate on two resonators based on the Fock states $\|0\\rangle $ and $\|1\\rangle $ is constructed by first using the number-state-dependent interactions between a superconducting qutrit and a resonator qudit (A to a and B to b) twice, and then turning on the interaction between two superconducting qutrits.", "Finally, the gate is completed by repeating the first step.", "The operation time of this gate is 150 ns.", "In our work, the c-phase gate is accomplished with two resonators which are coupled to just one transmon qutrit.", "It is easier to extend our two-resonator c-phase gate to a three-resonator processor assisted by a transmon qutrit.", "The effects used for constructing our c-phase gate on two resonators include the number-state-dependent interaction between the qubit and one resonator-qudit subsystem, and the simple resonant operation between the qubit and another resonator-qudit subsystem.", "These different characteristics make us get a higher fidelity c-phase gate with a faster operation time.", "The fidelity of our gate is 99.51% within the operation time of 93 ns.", "Different from the effective c-phase gate constructed by Wu et al.", "[71] in which both two resonators are coupled to a two-energy-level charge qubit by the number-state-dependent interactions and it is completed without considering the existence of the third energy level of the charge qubit (the operation time of this gate is 125 ns), our c-phase gate on the two resonators (1 and 2) is accomplished by combination of the photon-number-dependent frequency-shift effect on the transmon qutrit by the first resonator and the simple resonant operation between the qutrit and the second resonator.", "That is, resonator 2 is used to resonate with the qutrit and resonator 1 is used to complete the selective rotation on the qutrit by using the effect that the transition frequency of the qutrit is determinated by the photon number in only resonator 1, which is simpler than the effect used in Ref.", "[71].", "This different physical mechanism makes us obtain the higher fidelity and faster c-phase gate on the two resonators.", "Moreover, there are no works about the construction of the cc-phase gate on three microwave-photon-resonator qudits.", "The different devices in our work make it possible to construct the cc-phase gate on three resonators, far different from the previous proposals [70], [71].", "The fidelity of our cc-phase gate is $92.92\\%$ within the operation time of 124.64 ns.", "In summary, we have constructed two universal quantum gates, i.e., the c-phase and cc-phase gates in a microwave-photon quantum processor which contains multiple superconducting microwave-photon-resonator qudits coupled to a $\\Xi $ -type transmon qutrit.", "Our gates are based on the combination of the number-state-dependent interaction between a transmon qutrit and a resonator-qudit subsystem and the simple resonant operation between the qutrit and another resonator-qudit subsystem, and they have a high fidelity in a short operation time.", "The algorithms of our gates are based on the Fock states of the resonators, and the microwave photon number in each resonator is limited to none or just one.", "Our universal quantum processor can deal with the quantum computation with microwave photons in resonators.", "It is worth pointing out that the techniques for catching and releasing microwave-photon states from a resonator to the transmission line [45] and the single-photon router in the microwave regime [73] have been realized in experiments.", "The microwave-photon quantum processor can act as an important platform for quantum communication as well." ], [ "Acknowledgments", "This work is supported by the National Natural Science Foundation of China under Grant No.", "11174039 and NECT-11-0031." ] ]	1403.0031

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