Mathematical Physics, Analysis and Geometry最新文献

This paper concerns the discrete version of the Painlevé identification problem, i.e., how to recognize a certain recurrence relation as a discrete Painlevé equation. Often some clues can be seen from the setting of the problem, e.g., when the recurrence is connected with some differential Painlevé equation, or from the geometry of the configuration of indeterminate points of the equation. The main message of our paper is that, in fact, this only allows us to identify the configuration space of the dynamic system, but not the dynamics themselves. The refined version of the identification problem lies in determining, up to the conjugation, the translation direction of the dynamics, which in turn requires the full power of the geometric theory of Painlevé equations. To illustrate this point, in this paper we consider two examples of such recurrences that appear in the theory of orthogonal polynomials. We choose these examples because they get regularized on the same family of Sakai surfaces, but at the same time are not equivalent, since they result in non-equivalent translation directions. In addition, we show the effectiveness of a recently proposed identification procedure for discrete Painlevé equations using Sakai’s geometric approach for answering such questions. In particular, this approach requires no a priori knowledge of a possible type of the equation.

This paper is dedicated to studying matrix solutions of the cubic Szegő equation on the real line, which is introduced in Pocovnicu [Anal PDE 4(3):379–404, 2011; Dyn Syst A 31(3):607–649, 2011] and Gérard–Pushnitski (Commun Math Phys 405:167, 2024), leading to the following cubic matrix Szegő equation on ({mathbb {R}}),

Inspired by the space-periodic case in Sun (The matrix Szegő equation, arXiv:2309.12136), we establish its Lax pair structure via double Hankel operators and Toeplitz operators. Then the explicit formula in Gérard–Pushnitski (Commun Math Phys 405:167, 2024) can be extended to two equivalent formulas in the matrix equation case, which both express every solution explicitly in terms of its initial datum and the time variable.

Motivated by the discrete-time Toda (HADT) equation and quotient-quotient-difference (QQD) scheme together with their hungry forms (hHADT equation and hQQD scheme), we derive a new class of discrete integrable systems by considering the determinant structures of bivariate orthogonal polynomials associated with the genus-two hyper-elliptic curves. The corresponding Lax pairs are expressed through the recurrence relations of this class of bivariate orthogonal polynomials. Our study emphasizes the richer structures of genus-two hyper-elliptic curves, in contrast to the genus-one curve considered in the HADT and QQD cases, as well as in the hHADT and hQQD cases.

We study the umbilicity of constant mean curvature (CMC) complete hypersurfaces immersed in an Einstein manifold satisfying appropriate curvature constraints. In this setting, we obtain new characterization results for totally umbilical hypersurfaces via suitable maximum principles which deal with the notions of convergence to zero at infinity and polynomial volume growth. Afterwards, we establish optimal estimates for the first eigenvalue of the stability operator of CMC compact hypersurfaces in such an Einstein manifold. In particular, we derive a nonexistence result concerning strongly stable CMC hypersurfaces.

We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters (p,q in (0,1]) is bounded from above and below by multiples of (|p-q|).

To any tree on n vertices we associate an n-dimensional Lotka–Volterra system with (3n-2) parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits (n-1) functionally independent integrals. We also show how each system can be reduced to an ((n-1))-dimensional system which is superintegrable and solvable by quadratures.

We prove that the magnetization is equal to the edge current in the thermodynamic limit for a large class of models of lattice fermions with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. Our result implies that edge currents in such systems are determined by bulk properties and are therefore stable against large perturbations near the boundaries. Moreover, the equality persists also after taking the derivative with respect to the chemical potential. We show that this form of bulk-edge correspondence is essentially a consequence of homogeneity in the bulk and locality of the Gibbs state. An important intermediate result is a new version of Bloch’s theorem for two-dimensional systems, stating that persistent currents vanish in the bulk.

We study infinitesimal gauge transformations of K-equivariant noncommutative principal bundles, for K a triangular Hopf algebra. They form a Lie algebra of derivations in the category of K-modules. We study Drinfeld twist deformations of these infinitesimal gauge transformations. We give several examples from abelian and Jordanian twist deformations. These include the quantum Lie algebra of gauge transformations of the instanton bundle and of the orthogonal bundle on the quantum sphere (S^4_theta ).

For each rational number (p/qin (1/2,sqrt{2}/2)) one can construct an (mathbb {S}^1)-equivariant minimal torus in (mathbb {S}^3) called Otsuki torus and denoted by (O_{p/q}). The Lawson’s bipolar surface construction applied to (O_{p/q}) gives a minimal torus (widetilde{O}_{p/q}) in (mathbb {S}^4). In this paper we give upper and lower bounds on the Morse index and the nullity of these tori for p/q close to (sqrt{2}/2). We also state a numerically assisted conjecture concerning the general case.

We discuss the sharp interface limit, leading to a mean curvature flow energy, for the rate function of the large deviation principle of a Glauber+Kawasaki process with speed change. We provide an explicit formula of the limiting functional given by the mobility and the transport coefficient.