Related papers: Partial difference equations over compact Abelian …
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a subgroup H of a group G must be determined from a quantum state y uniformly supported…
We introduce a pair of novel difference equations, whose solutions are expressed in terms of Racah or Wilson polynomials depending on the nature of the finite-difference step. A number of special cases and limit relations are also examined,…
It is well known that every modular form~$f$ on a discrete subgroup $\Gamma\leqslant \textrm{SL}(2, \mathbb R)$ satisfies a third-order nonlinear ODE that expresses algebraic dependence of the functions~$f$, $f'$, $f''$ and~$f'''$. These…
The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial…
In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to…
Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the unit disk $\ID$ and satisfying the normalization $f(0)=0= f'(0)-1$. Let $\mathcal{S}$ denote the subclass of ${\mathcal A}$ consisting of univalent functions in…
We construct a simplified resolution for the trivial G-module Z, where G is a finite abelian group, and compare it with the standard resolution. We use it to calculate cohomologies of irreducible G-lattices and their duals.
The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and…
We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the…
We give an effective procedure to explicitly find the decomposition of a polarized abelian variety into its simple factors if a period matrix is known. Since finding this datum is not easy, we also provide two methods to compute the period…
We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the…
Let k be an algebraically closed field, let R be an associative k-algebra, and let F = {M_a: a in I} be a family of orthogonal points in R-Mod such that End_R(M_a) = k for all a in I. Then Mod(F), the minimal full sub-category of R-Mod…
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the…
The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state…
This paper investigates connections between discrete and continuous approaches for decomposable submodular function minimization. We provide improved running time estimates for the state-of-the-art continuous algorithms for the problem…
This paper works out the structure of singular points of p-adic differential equations (i.e. differential modules over the ring of functions analytic in some annulus with external radius 1). Surprisingly results look like in the formal case…
Let $G$ be a topological group. We investigate relations between two classes of "polynomial like" continuous functions on $G$ defined, respectively, by the conditions (1) $\Delta_h^{n+1}f=0$ for every $h \in G$, and (2) $\Delta_{h_{n+1}}…
A normalized analytic function f is shown to be univalent in the open unit disk D if its second coefficient is sufficiently small and relates to its Schwarzian derivative through a certain inequality. New criteria for analytic functions to…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions,…