Related papers: Cylindrical Dyck paths and the Mazorchuk-Turowska …
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four…
In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of…
We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system with perturbations and two parameters $\lambda_1$ and $\lambda_2$ on locally finite graph. By…
The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a…
The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think…
Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be drawn.…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all…
The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the…
We extend the classical notion of solvability to a lambda-calculus equipped with pattern matching. We prove that solvability can be characterized by means of typability and inhabitation in an intersection type system P based on…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…
We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations $$ \Delta_{p}^2 u =…
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a…
Let $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$…
We survey the matrix product solutions of the Yang-Baxter equation obtained recently from the tetrahedron equation. They form a family of quantum $R$ matrices of generalized quantum groups interpolating the symmetric tensor representations…
A new family of skew-symmetric solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is characterized and analyzed. Such family has some remarkable properties. In first place, it is defined for…
The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear…
In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…
In this paper, we define the notion of descent for the paths in the $p$-Bratteli diagram. This leads to the definition of $p^{k}$-Eulerian polynomials, whose coefficients count the number of paths with a given number of descents. We provide…