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The article surveys the known results and conjectures about the analytic properties of dispersion relations and Fermi surfaces for periodic equations of mathematical physics and their spectral incarnations. [ABSTRACT FROM AUTHOR]

HT ht I is i B diophantine b I if for some i I > 0 i I and i I > 0 i I it belongs to (see Fig. So, we can consider it as a I submanifold i HT ht . One can smoothly embed every manifold I P i into some vector space HT ht . Obviously, the flow on HT ht equals SB I t i sb ( I i ) = I i + I t i (mod 2 I i ). [Extracted from the article]

We consider the spectral gap question for Affleck, Kennedy, Lieb, and Tasaki models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of G with a chain of n sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States approach from [Abdul-Rahman et al., Analytic Trends in Mathematical Physics, Contemporary Mathematics (American Mathematical Society, 2020), Vol. 741, p. 1.], we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy. [ABSTRACT FROM AUTHOR]

The goal of this short article is to summarize some of the recent developments in quiver Yangians and crystal meltings. This article is based on a lecture delivered by the author at International Congress on Mathematical Physics (ICMP), Geneva, 2021. [ABSTRACT FROM AUTHOR]

Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras. [ABSTRACT FROM AUTHOR]

A well-known conjecture in mathematical physics asserts that the interacting Bose gas exhibits Bose–Einstein condensation (BEC) in the thermodynamic limit. We consider the Bose gas on certain hyperbolic spaces. In this setting, one obtains a short proof of BEC in the infinite-volume limit from the existence of a volume-independent spectral gap of the Laplacian. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to face up the statistical mechanics of dense spin glasses using the well-known Ising case as a prelude for testing the methodologies we develop and then focusing on the Gaussian case as the main subject of our investigation. We tackle the problem of solving for the quenched statistical pressures of these models both at the replica symmetric level and under the first step of replica symmetry breaking by relying upon two techniques: the former is an adaptation of the celebrated Guerra's interpolation (closer to probability theory in its spirit) and the latter is an adaptation of the transport partial differential equation (closer to mathematical physics in spirit). We recover, in both assumptions, the same expression for quenched statistical pressure and self-consistency equation found with other techniques, including the well-known replica trick technique. [ABSTRACT FROM AUTHOR]

We consider the critical points (equilibria) of a planar potential generated by n Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular-restricted n-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by n + 1, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in n. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as "ring configurations," consisting of n − 1 equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as 5n − 5 equilibria. We verify analytically that the ring configuration has at least 5n − 5 equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell's problem from electrostatics and the image counting problem from gravitational lensing. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of a positive solution for a class of quasilinear Schrödinger equations involving a potential that behaves like a periodic function at infinity and the nonlinear term may exhibit critical exponential growth. In order to prove our main result, we combine minimax methods with a version of the Trudinger–Moser inequality. These equations appear naturally in mathematical physics and have been derived as models of several physical phenomena. [ABSTRACT FROM AUTHOR]

The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries. [ABSTRACT FROM AUTHOR]

Prolate spheroidal wave functions have recently attracted much attention in applied harmonic analysis, signal processing, and mathematical physics. They are eigenvectors of the sinc-kernel operator Q c : the time- and band-limiting operator. The corresponding eigenvalues play a key role, and the aim of this paper is to obtain precise non-asymptotic estimates with explicit constants related to the spectrum of Q c . This issue is rarely studied in the literature, while the asymptotic behavior of the spectrum of Q c has been well established from the 1960s. However, many recent applications require such non-asymptotic behavior. As applications of our non-asymptotic estimates, we first provide estimates for the constants appearing in the Remez- and Turàn–Nazarov-type concentration inequalities. Then, we give a non-asymptotic upper bound for the gap probability of the sinc determinantal point process. Consequently, one gets a non-asymptotic estimate for the hole probability, associated with bulk scaled asymptotics of a random matrix from the Gaussian unitary ensemble. This last result can be considered as a complement of the various and more involved asymptotic counterparts of this estimate. [ABSTRACT FROM AUTHOR]

We propose to study the Hessian metric of a functional on the space of probability measures endowed with the Wasserstein-2 metric. We name it transport Hessian metric, which contains and extends the classical Wasserstein-2 metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and mathematical physics equations are discovered. For example, the transport Hessian gradient flow, including Newton's flow, formulates a mean-field kernel Stein variational gradient flow; the transport Hessian Hamiltonian flow of Boltzmann–Shannon entropy forms the shallow water equation; and the transport Hessian gradient flow of Fisher information leads to the heat equation. Several examples and closed-form solutions for transport Hessian distances are presented. [ABSTRACT FROM AUTHOR]

This paper surveys recent progress in the analysis of nonlinear partial differential equations using Anderson localization and semi-algebraic sets method. We discuss the application of these tools from linear analysis to nonlinear equations such as the nonlinear Schrödinger equations, the nonlinear Klein–Gordon equations (nonlinear wave equations), and the nonlinear random Schrödinger equations on the lattice. We also review the related linear time-dependent problems. [ABSTRACT FROM AUTHOR]

An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrödinger operator −Δ + V(x) in R n with periodic potential near the edges of the spectrum, i.e., near extrema of the dispersion relation. A well-known and widely believed conjecture says that generically (with respect to perturbations of the periodic potential), the extrema are attained by a single branch of the dispersion relation, are isolated, and have nondegenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as the Liouville property, Green's function asymptotics, and so on hinges upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Moreover, such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist even for Schrödinger operators on simple 2D-periodic two-atomic structures, showing that the genericity fails in some discrete situations. We start with establishing in a very general situation the following natural dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. Thus, a random choice of a point in the parameter space gives the correct answer "with probability one." Noticing that the known counterexample has only two free parameters, one can suspect that this might be too tight for the genericity to hold. We thus consider the maximal Z 2 -periodic two-atomic nearest-cell interaction graph, which has nine edges per unit cell and the discrete "Laplace–Beltrami" operator on it, which has nine free parameters. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. Since the proof is non-trivial and would be much harder for more general structures, we show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters indeed cannot destroy the genericity result. This allows us to list all "bad" periodic subgraphs of the one we consider and discover that in all these cases the genericity fails for "trivial" reasons only. [ABSTRACT FROM AUTHOR]

The asymptotic iteration method is a technique for solving analytically and approximately the linear second-order differential equation, especially the eigenvalue problems that frequently appear in theoretical and mathematical physics. The analysis and mathematical justifications of the success and failure of the asymptotic iteration method are detailed in this work. A theorem explaining why the asymptotic iteration method works for the eigenvalue problem is presented. As a byproduct, a new procedure to generate unlimited classes of exactly solvable differential equations is also introduced. [ABSTRACT FROM AUTHOR]

We apply a new global dynamical system formulation to flat Robertson–Walker cosmologies with a massless and massive Yang–Mills field and a perfect-fluid with linear equation of state as the matter sources. This allows us to give proofs concerning the global dynamics of the models including asymptotic source-dominance toward the past and future time directions. For the pure massless Yang–Mills field, we also contextualize well-known explicit solutions in a global (compact) state space picture. [ABSTRACT FROM AUTHOR]

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors," usually dependent on a "time variable" t. From ladder operators [see A. Magnus, J. Comput. Appl. Math. 57(1-2), 215–237 (1995)], one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials Pn(x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by Pn(x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics, and in the special cases, they degenerate to the hypergeometric and confluent hypergeometric equations. In this paper, we look at three types of weights: the Jacobi type, the Laguerre type, and the weights deformed by the indicator function of χ(a,b)(x) and the step function θ(x). In particular, we consider the following Jacobi type weights: (1.1) xα(1 − x)βe−tx, x ∈ [0, 1], α, β, t > 0; (1.2) xα(1 − x)βe−t/x, x ∈ (0, 1], α, β, t > 0; (1.3) (1 − x 2 ) α (1 − k 2 x 2 ) β , x ∈ [ − 1 , 1 ] , α , β > 0 , k 2 ∈ (0 , 1) ; the Laguerre type weights: (2.1) xα(x + t)λe−x, x ∈ [0, ∞), t, α, λ > 0; (2.2) xαe−x−t/x, x ∈ (0, ∞), α, t > 0; and another type of deformation when the classical weights are multiplied by χ(a,b)(x) or θ(x): (3.1) e − x 2 (1 − χ (− a , a) (x)) , x ∈ R , a > 0 ; (3.2) (1 − x 2 ) α (1 − χ (− a , a) (x)) , x ∈ [ − 1 , 1 ] , a ∈ (0 , 1) , α > 0 ; (3.3) xαe−x(A + Bθ(x − t)), x ∈ [0, ∞), α, t > 0, A ≥ 0, A + B ≥ 0. The weights mentioned above were studied in a series of papers related to the deformation of "classical" weights. [ABSTRACT FROM AUTHOR]

A new expression for the Hulthén off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data. [ABSTRACT FROM AUTHOR]

This is a slightly expanded version of the plenary lecture the author gave at the International Congress on Mathematical Physics 2018 in Montreal, Canada. Reported are some work of the author and collaborators as well as related results of others on two kinds of dynamical systems: the first kind is deterministic (in the sense that nothing is left to chance) but is chaotic and unpredictable, and the second kind has a stochastic component in addition to a purely deterministic one. These two kinds of dynamical systems are compared and contrasted. The main points are that time series of observations from chaotic deterministic systems obey some of the same probabilistic limit laws as genuinely random stochastic processes, but random dynamical systems enjoy nicer properties and are technically more tractable. [ABSTRACT FROM AUTHOR]

Diffusion and heat equations are commonly investigated in mathematical physics and are solvable for potentials in polar coordinates with a separation into a radial and an angular equation. While the angular equation can be solved easily, a common method for solving the radial part consists in the application of the Neumann-Weber integral transform. The Neumann-Weber integral transform, however, has only been shown to be valid for real indices of Bessel functions. In this work, we generalize the Neumann-Weber transform to complex Bessel indices. The back transform then becomes dependent on zeros of Hankel functions, and we provide useful information for its numerical implementation. The results are relevant for solving diffusion equations and heat equations around cylindrical objects. [ABSTRACT FROM AUTHOR]

Using the Weyl-Wigner-Moyal-Groenewold description, a Moyal-deformation of a Kapustin-Witten (KW) system is obtained in this article. Starting from the known solutions of the original equations, some solutions to these deformed equations are obtained. The analytic properties of such solutions are also studied. Finally we find that in the ℏ → 0 limit, the SU(∞) KW equations are obtained. [ABSTRACT FROM AUTHOR]

An extended analysis of links between linear differential equations and the nonlinear Painlevé equation PV I is given. For linear equations, second-order equations in different forms, as well as various first-order systems, are chosen. The role of an accessory parameter is explained. The relationship to the Schlesinger system is made clear. [ABSTRACT FROM AUTHOR]

We describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and their mixed and deformed versions. The definition of the AKNS hierarchy and its deformed versions is given in the Sec. II. We also consider the continuous symmetries of the related equations and the related spectral curves. This work continues and summarises some of our previous studies dedicated to the rogue wave-like solutions associated with AKNS, nonlinear Schrödinger, and KP hierarchies. The general scheme is illustrated by the examples of small rank n, n ⩽ 7, rational or quasi-rational solutions. In particular, we consider rank-2 and rank-3 quasi-rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics, hydrodynamics, and many other branches of science. [ABSTRACT FROM AUTHOR]

In this paper, we study random lozenge tilings of non-convex polygonal regions. The interaction of the non-convexities (cuts) leads to new kernels and thus new statistics for the tiling fluctuations near these regions. This paper gives new probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts. [ABSTRACT FROM AUTHOR]

We present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. Hence our scheme extends to the quantum case a key feature of the Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action-angle variables. We work in the framework of the quantum inverse scattering method. As a first example of our approach, we give the construction of such a basis for models associated with Y(gln) and argue how it extends to their trigonometric and elliptic versions. Then we show how our general scheme applies concretely to fundamental models associated with the Y(gl2) and Y(gl3) R-matrices leading to the full characterization of their spectrum. For Y(gl2) and its trigonometric deformation, a particular case of our method reproduces Sklyanin's construction of separate variables. For Y(gl3), it gives new results, in particular, through the proper identification of the shifts acting on the separate basis. We stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including, in particular, trigonometric and elliptic spin chains, open chains with general integrable boundaries, and further higher rank cases that we will describe in forthcoming publications. [ABSTRACT FROM AUTHOR]

We investigate the persistence of spectral gaps of one-dimensional frustration free quantum lattice systems under weak perturbations and with open boundary conditions. Assuming that the interactions of the system satisfy a form of local topological quantum order, we prove explicit lower bounds on the ground state spectral gap and higher gaps for spin and fermion chains. By adapting previous methods using the spectral flow, we analyze the bulk and edge dependence of lower bounds on spectral gaps. [ABSTRACT FROM AUTHOR]

We review the authors' recent work where we obtain the uniform large s asymptotics for the Fredholm determinant D (s , γ) ≔ det (I − γ K s ↾ L 2 (− 1,1) ) , 0 ≤ γ ≤ 1. The operator Ks acts with kernel Ks(x, y) = sin(s(x − y))/(π(x − y)), and D(s, γ) appears for instance in Dyson's model of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble. [ABSTRACT FROM AUTHOR]

The complexification of the independent variables of nonlinear integrable evolution partial differential equations (PDEs) in two space dimensions, like the celebrated Kadomtsev-Petviashvili and Davey-Stewartson (DS) equations, yields nonlinear integrable equations in genuine 4 + 2, namely, in four real space dimensions (x1, x2, y1, y2) and two real time dimensions (t1, t2), as opposed to two complex space dimensions and one complex time dimension. The associated initial value problem for such equations, namely, the problem where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved via a non-local d-bar formalism. Here, the details of this formalism for the 4 + 2 DS system are presented. Furthermore, the linearised version of the 3 + 1 reduction of the 4 + 2 DS system is discussed. The construction of the nonlinear 3 + 1 reduction remains open, in spite of the fact that multi-soliton solutions for the 3 + 1 DS system already exist. [ABSTRACT FROM AUTHOR]

We define two tau functions, τ and τ ^ , on moduli spaces of spectral covers of GL(n) Hitchin systems. Analyzing the properties of τ, we express the rational divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function τ ^ is used to compute the divisor of canonical 1-forms with multiple zeros. [ABSTRACT FROM AUTHOR]

An element [ Φ ] ∈ G r n H + , F of the Grassmannian of n-dimensional subspaces of the Hardy space H + = H 2 , extended over the field F = C(x1, ..., xn), may be associated to any polynomial basis ϕ = {ϕ0, ϕ1, ⋯ } for C(x). The Plücker coordinates S λ , n ϕ ( x 1 , ... , x n ) of [Φ], labeled by partitions λ, provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system ϕ to the analog { h i (0) } of the complete symmetric functions generates a doubly infinite matrix h i (j) of symmetric polynomials that determine an element [ H ] ∈ G r n ( H + , F). This is shown to coincide with [Φ], implying a set of generalized Jacobi identities, extending a result obtained by Sergeev and Veselov [Moscow Math. J. 14, 161–168 (2014)] for the case of orthogonal polynomials. The symmetric polynomials S λ , n ϕ ( x 1 , ... , x n ) are shown to be KP (Kadomtsev-Petviashvili) τ-functions in terms of the power sums [x] of the xa's, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums ∑ λ S λ , n ϕ ([ x ]) S λ , n θ (t) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes. [ABSTRACT FROM AUTHOR]

We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function τ t associated with generic monodromy data. Using a relation of τ t to two different types of irregular c = 1 Virasoro conformal blocks and the confluence from Painlevé VI equation, connection formulas between the parameters of asymptotic expansions at 0 and i∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks. [ABSTRACT FROM AUTHOR]

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets (circular nets and planar quadrilateral nets, respectively) with edges of order ϵ. Both smooth and discrete geometries are described by integrable systems. It is shown that one can obtain an order ϵ2 approximation globally with points of the discrete nets on the smooth surface. A new simple geometric construction of principal directions of smooth surfaces is given. [ABSTRACT FROM AUTHOR]

This work constructs a well-defined and operational form factor expansion in a model having a massless spectrum of excitations. More precisely, the dynamic two-point functions in the massless regime of the XXZ spin-1/2 chain are expressed in terms of a properly regularised series of multiple integrals. These series are obtained by taking, in an appropriate way, the thermodynamic limit of the finite volume form factor expansions. The series are structured in a way allowing one to identify directly the contributions to the correlator stemming from the conformal-type excitations on the Fermi surface and those issuing from the massive excitations (deep holes, particles, and bound states). The obtained form factor series opens up the possibility of a systematic and exact study of asymptotic regimes of dynamical correlation functions in the massless regime of the XXZ spin 1/2 chain. Furthermore, the assumptions on the microscopic structure of the model's Hilbert space that are necessary so as to write down the series appear to be compatible with any model—not necessarily integrable—belonging to the Luttinger liquid universality class. Thus, the present analysis also provides the phenomenological structure of form factor expansions in massless models belonging to this universality class. [ABSTRACT FROM AUTHOR]

We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights zh, z v of the dimer model and arbitrary dimensions of the lattice m, n. We assume m is even and we show that the asymptotic expansion depends on the parity of n. We review and extend the results of Ivashkevich et al. [J. Phys. A: Math. Gen. 35, 5543 (2002)] on the full asymptotic expansion of the partition function of the dimer model, and we give a rigorous estimate of the error term in the asymptotic expansion of the partition function. [ABSTRACT FROM AUTHOR]

We study piecewise polynomial functions γk(c) that appear in the asymptotics of averages of the divisor sum in short intervals. Specifically, we express these polynomials as the inverse Fourier transform of a Hankel determinant that satisfies a Painlevé V equation. We prove that γk(c) is very smooth at its transition points and also determine the asymptotics of γk(c) in a large neighbourhood of k = c/2. Finally, we consider the coefficients that appear in the asymptotics of elliptic aliquot cycles. [ABSTRACT FROM AUTHOR]

The elliptic Calogero-Moser integrable system for an arbitrary root system has a realization as a moduli space of Higgs bundles over an Abelian variety associated with the elliptic curve and with the root system. This paper examines the Fourier-Mukai transform of this, giving an interpretation of the system on a network of elliptic curves. The rational and trigonometric versions of the systems are briefly discussed, and it is shown how they enter as degenerations in this geometric context. [ABSTRACT FROM AUTHOR]

One of the basic lessons of quantum theory is that one cannot obtain information on an unknown quantum state without disturbing it. Hence, by performing a certain measurement, we limit the other possible measurements that can be effectively implemented on the original input state. It has been recently shown by two of the authors of the present article [T. Heinosaari and T. Miyadera, Phys. Rev. A 91, 022110 (2015)] that one can implement sequentially any device, either channel or observable, which is compatible with the first measurement. In this work, we prove that this can be done, apart from some special cases, only when the succeeding device is implemented on a larger system than just the input system. This means that some part of the still available quantum information has been flown to the environment and cannot be gathered by accessing the input system only. We characterize the size of the post-measurement system by determining the class of measurements for the observable in question that allow the subsequent realization of any measurement process compatible with the said observable. We also study the class of measurements that allow the subsequent realization of any observable jointly measurable with the first one and show that these two classes coincide when the first observable is extreme. [ABSTRACT FROM AUTHOR]

Using variational methods and the potential well theory, strong instability of standing waves for a class of fractional Schrödinger-Choquard equations is established in the mass super-critical and energy sub-critical case. [ABSTRACT FROM AUTHOR]

The Bargmann-Fock-Segal space plays an important role in mathematical physics and has been extended into a number of directions. In the present paper, we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e., the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and Salomon. [ABSTRACT FROM AUTHOR]

We present numerical results for the six-vertex model with a variety of boundary conditions. Adapting an algorithm for domain wall boundary conditions, proposed in the work of Allison and Reshetikhin [Ann. Inst. Fourier 55(6), 1847–1869 (2005)], we examine some modifications of these boundary conditions. To be precise, we discuss partial domain wall boundary conditions, reflecting ends, and half turn boundary conditions (domain wall boundary conditions with half turn symmetry). Dedicated to the memory of Ludwig Faddeev [ABSTRACT FROM AUTHOR]

Fine topology is one of the several well-known topologies of physical and mathematical relevance. In the present paper, it is obtained that the nonempty open sets of different dimensional Minkowski spaces with the fine topology are not homeomorphic. This leads to the introduction of a new class of manifolds. It turns out that the technique developed here is also applicable to some other topologies, namely, the s-topology, space topology, f-topology, and A-topology. [ABSTRACT FROM AUTHOR]

We note that large classes of contractions of algebras that arise in physics can be understood purely algebraically via identifying appropriate Z m -gradings (and their generalizations) on the parent algebra. This includes various types of flat space/Carroll limits of finite and infinite dimensional (A)dS algebras, as well as Galilean and Galilean conformal algebras. Our observations can be regarded as providing a natural context for the Grassmann approach of Krishnan et al. [J. High Energy Phys. 2014(3), 36]. We also introduce a related notion, which we call partial grading, that arises naturally in this context. [ABSTRACT FROM AUTHOR]

A set of new tensors of purely geometric origin have been investigated, which form a hierarchy. A tensor of a lower rank plays the role of the potential for the tensor of one rank higher. The tensors have interesting mathematical and physical properties. The highest rank tensor of the hierarchy possesses all the geometrical properties of the Weyl tensor. [ABSTRACT FROM AUTHOR]

The blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations is investigated. More precisely, we consider a functional F(t) associated with the momentum and weighted by a general test function f and show that if F(0) is sufficiently large, then the finite time blowup of the solutions of the non-isentropic compressible Euler equations occurs. As the test function f is a general function with only mild conditions imposed, a class of blowup conditions is established. [ABSTRACT FROM AUTHOR]

We study the existence of ground state sign-changing solutions for the fractional Kirchhoff problem. Under mild assumptions on the nonlinearity, by using some new analytical skills and the non-Nehari manifold method, we prove that the fractional Kirchhoff problem possesses a ground state sign-changing solution ub. Moreover, we show that the energy of ub is strictly larger than twice that of the ground state solutions of Nehari-type. Finally, we establish the convergence property of ub as the parameter b ↘ 0. Our results generalize some results obtained by Shuai [J. Differ. Equations 259, 1256 (2015)] and Tang and Cheng [J. Differ. Equations 261, 2384 (2016)]. [ABSTRACT FROM AUTHOR]

We start with Fannes' type andWinter's type tight (uniform) continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying Hilbert space is finite and obtain Fannes' type tight continuity bounds for the Holevo quantity in this case. In the general case, conditions for local continuity of the Holevo quantity for discrete and continuous ensembles are found. Winter's type tight continuity bound for the Holevo quantity under constraint on the average energy of ensembles is obtained and applied to the system of quantum oscillators. The above results are used to obtain tight and close-to-tight continuity bounds for basic capacities of finite-dimensional channels (significantly refining the Leung-Smith continuity bounds). [ABSTRACT FROM AUTHOR]

The Lax representation for different matrix generalizations of Short Pulse Equations (SPEs) is considered. The four-dimensional Lax representations of four-component Matsuno, Feng, and Dimakis-Müller-Hoissen-Matsuno equations are obtained. The four-component Feng system is defined by generalization of the two-dimensional Lax representation to the four-component case. This system reduces to the original Feng equation, to the two-component Matsuno equation, or to the Yao-Zang equation. The three-component version of the Feng equation is presented. The fourcomponent version of the Matsuno equation with its Lax representation is given. This equation reduces the new two-component Feng system. The two-component Dimakis-Müller-Hoissen-Matsuno equations are generalized to the four-parameter family of the four-component SPE. The bi-Hamiltonian structure of this generalization, for special values of parameters, is defined. This four-component SPE in special cases reduces to the new two-component SPE. [ABSTRACT FROM AUTHOR]

The diameter of the apparent horizon, defined by the distance between furthest points on the horizon, in spacetimes with a positive cosmological constant Λ has been investigated. It is established that the diameter of the apparent horizon on the totally umbilic partial Cauchy surface cannot exceed 2π/√3Λ. Then, it is argued that arbitrarily long black strings cannot be formed in our universe. [ABSTRACT FROM AUTHOR]