Related papers: Explicit solutions of division problems for matric…
Solving the holography equation has long been a numerical task. While effective, the numeric approach has its own set of limitations. Relying solely on numerical approaches often obscures the intricate interplay and influence of the…
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…
We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…
We describe the images of multilinear polynomials of degree up to four on the upper triangular matrix algebra.
We introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Then, among other things, by using the formula about representing one lambda-Sheffer polynomial in terms of other lambda-Sheffer…
In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…
It is well-known that any solution of the Laplace equation is a real or imaginary part of a complex holomorphic function. In this paper, in some sense, we extend this property into four order hyperbolic and elliptic type PDEs. To be more…
Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with positive real part.…
We consider the class of polynomial optimization problems $\inf \{f(x):x\in K\}$ for which the quadratic module generated by the polynomials that define $K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such problems,…
We present explicit representations in terms of hypergeometric functions for the scaling functions in the $C^0$ orthogonal multiresolution analyses associated with piecewise continuous polynomials. Closed formulas for the Mellin transform…
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation…
We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a…
Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute…
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector…