Related papers: On linear equations arising in Combinatorics (Part…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to…
A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. The operational rules are described and the method is…
We show that many infinite classes of permutations over finite fields can be constructed via translators with a large choice of parameters. We first charac- terize some functions having linear translators, based on which several families of…
In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and…
This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the…
In this work we extend the range of applicability of a method recently introduced where coupled first-order nonlinear equations can be put into a linear form, and consequently be solved completely. Some general consequences of the present…
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
Given $A\in \Z^{m\times n}$ and $b\in\Z^m$, we consider the issue of existence of a nonnegative integral solution $x\in \N^n$ to the system of linear equations $Ax=b$. We provide a discrete and explicit analogue of the celebrated Farkas…
A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The…
In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule…
We extend the classical Landesman-Lazer results to the setting of second order Hamilton-Jacobi-Bellman equations. A number of new phenomena appear.
We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.
In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results…
A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between…
In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of $Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here, we determine for which of these groups the restriction of the metaplectic representation…
In [16], we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed…
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
We introduce a class of permutation polynomial over $\mathbb F_{q^n}$ that can be written in the form $\frac{L(x)}{x^{q+1}}$ or $\frac{L(x^{q+1})}x$ for some $q$-linear polynomial $L$ over $\mathbb F_{q^n}$. Specifically, we present those…