数学物理
This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and…
We start from the classical Kadomtsev-Petviashvili hierarchy posed on formal pseudo-differential operators, and we produce two hierarchies of non-linear equations posed on non-formal pseudo-differential operators lying in the Kontsevich and…
We consider the discrete Schr\"odinger operators with potentials whose values are read along the orbits of a shift of finite type. We study a certain subset of the collection of energies at which the Lyapunov exponent is zero and prove…
In this work, we provide a treatment of scaling functional equations in a general setting involving fractals arising from sufficiently nice self-similar systems in order to analyze the tube functions, tube zeta functions, and complex…
The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing…
The Magnus expansion, introduced by Wilhelm Magnus in 1954, is an infinite Lie series employed to express solutions for first-order homogeneous linear differential equations involving a linear operator. Since its discovery it has evolved…
We consider seven-vertex two-dimensional integrable statistical model. With the help of intertwining vector method we construct its counterpart integrable model of SOS type. More general models of both types are constructed by means of…
For the scattering of plane electromagnetic waves by a general possibly anisotropic stationary linear medium in three dimensions, we give a condition on the permittivity and permeability tensors of the medium under which the first Born…
Recent works have explored relations between classical and quantum statistical physics on the one hand and Voiculescu's theory of free probability on the other. Motivated by these results, the present work focuses on the notion of effective…
In this paper we study a particular Painlev\'e V (denoted ${\rm P_{V}}$) that arises from Multi-Input-Multi-Output (MIMO) wireless communication systems. Such a $P_V$ appears through its intimate relation with the Hankel determinant that…
We study rigorously the infinite Reynolds limit of the solutions of the Landau-Lifschitz equations of fluctuating hydrodynamics for an incompressible fluid on a $d$-dimensional torus for $d\geq 2.$ These equations, which model the effects…
In this paper $4$ dimensional Riemannian (or Euclidean) vacuum general relativity is recovered from a phase transition by spontaneous symmetry breaking within a quantum field theory (all in the sense of the operator algebraic approach to…
We provide sufficient conditions such that the time evolution of a mesoscopic tight-binding open system with a local Hartree-Fock non-linearity converges to a self-consistent non-equilibrium steady state, which is independent of the initial…
We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of…
The Burgers--KdV hierarchy was introduced towards understanding intersection numbers on the moduli space of Riemann surfaces with boundaries. The goal of this paper is to derive the Dubrovin--Zhang type loop equation for the topological…
A Chern-Simons matrix model was proposed by Dorey, Tong, and Turner to describe non-Abelian fractional quantum Hall effect. In this paper we study the Hilbert space of the Chern-Simons matrix model from a geometric quantization point of…
For any local, translation-covariant quantum field theory on Minkowski spacetime, we prove that two distinct states that are invariant under the inertial time evolutions in different inertial reference frames are disjoint, i.e. neither…
We study the quasi-local masses arising in general relativity using spinors and prove their positivity property. This leads to the question of a pure quasi-local proof of the positivity of the Wang-Yau \cite{yau} quasi-local mass. More…
We consider non-twisted meromorphic connections in $\mathfrak{sl}_2(\mathbb{C})$ and the associated symplectic Hamiltonian structure. In particular, we provide explicit expressions of the Lax pair in the geometric gauge supplementing…
A mathematical model for the poroelastic materials (PEM) with the variable volume is developed in multidimensional case. Governing equations of the model are constructed using the continuity equations, which reflect the well-known physical…