Related papers: Matrix biorthogonal polynomials on the unit circle…
We focused on the Ablowitz--Ladik equation on a zero background, specifically considering the scenario of $N$ pairs of multiple poles. Our first goal was to establish a mapping between the initial data and the scattering data. This allowed…
Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the…
We define a dispersionless tau-symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0$/$\infty$ respectively, which extends the dispersionless 2D Toda…
Let $A_\alpha$ be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, $(A_\alpha)_{11}=\alpha$, where $\alpha\in\mathbb C$, and zero elsewhere. A basis $\{P_0,P_1,P_2,\ldots\}$ of the linear space…
This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the…
We propose a discrete analogue of null geodesics in causal sets that are approximated by a region of 2d Minkowski spacetime, in the spirit of Kronheimer and Penrose's "grids" and "beams" for an abstract causal space. The causal set…
We propose a new multi-component two-dimensional Toda lattice hierarchy (mc2dTLH) which includes two-dimensional Toda lattice equation with self-consistent sources (2dTLSCS) as the first non-trivial equation. The Lax representations for…
Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix $A\in\mathbb{F}^{N\times N}$ and a vector $b$, it is known that in the worst case $\Theta(N^2)$ operations over $\mathbb{F}$ are needed to…
This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to…
We propose a novel semi-discrete Kadomtsev--Petviashvili equation with two discrete and one continuous independent variables, which is integrable in the sense of having the standard and adjoint Lax pairs, from the direct linearisation…
We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a criterion of integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its…
A Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in\mathbb Z^+}$, $E=(e_{i,j})_{i,j\in\mathbb Z^+}$ is compact and the norms $\lVert a\rVert_{\mathcal W} = \sum_{i\in\mathbb Z}|a_i|$…
We consider solutions of the 2D Toda lattice hierarchy which are trigonometric functions of the ``zeroth'' time $t_0=x$. It is known that their poles move as particles of the trigonometric Ruijsenaars-Schneider model. We extend this…
We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is…
The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra lattices are proposed and studied by performing nonisopectral deformations on the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials without…
The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This…
This paper introduces and analyses the new grid-based tensor approach to approximate solution of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree-Fock equation over a spatial…
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how B\"acklund transformations lead to equations that can be…
Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the…