Related papers: Exponential valuations on lattice polygons
We discuss two possible ways of representing tolerances: first, as a homomorphic image of some congruence; second, as the relational composition of some compatible relation with its converse. The second way is independent from the variety…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
Starting from essentially commutative exponential map $E(B|I)$ for generic tensor-valued 2-forms $B$, introduced in \cite{Akh} as direct generalization of the ordinary non-commutative $P$-exponent for 1-forms with values in matrices (i.e.…
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its…
Let $G$ be $PGL(n,F)$, $n \geq 3$, $F$ a certain non-archimedean local field; or let $G$ be $PSL(2,\mathbb{R}) \times \cdots \times PSL(2,\mathbb{R})$. Let $\Gamma$ be a lattice in $G$, and let $( \Lambda_n )$ be a sequence of lattices in…
In this note we relate the valuations of the algebras appearing in the non-commutative geometry of quantized algebras to properties of sub-lattices in some vector spaces. We consider the case of algebras with $PBW$-bases and prove that…
The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…
We introduce the notion of a categorical valuative invariant of polyhedra or matroids, in which alternating sums of numerical invariants are replaced by split exact sequences in an additive category. We provide categorical lifts of a number…
The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{Z}_{p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight…
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on…
We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a…
We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we…
All simple translation-invariant valuations on polytopes are classified. As a direct consequence the well-known conditions for translative-equidecomposability are recovered. Furthermore, a simplified proof of the classification of…
A well-known result by Lindenstrauss is that any two-dimensional normed space can be isometrically imbedded into $L_1(0,1)$. We provide an explicit form of a such an imbedding. The proof is elementary and self-contained. Applications are…
A complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence…
Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…
Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$ be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzero isotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let…
The global multiplicative properties of Laplace type operators acting on irreducible rank one symmetric spaces are considered. The explicit form of the multiplicative anomaly is derived and its corresponding value is calculated exactly, for…
We present examples of smooth lattice polytopes in dimensions 3 and higher where each coefficient of their Ehrhart polynomials that can potentially be negative is indeed negative. This answers a question by Bruns. We also discuss…
Using the entropic inequalities for Shannon and Tsallis entropies new inequalities for some classical polynomials are obtained. To this end, an invertible mapping for the irreducible unitary representation of groups $SU(2)$ and $SU(1,1)$…