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We prove an analogue of Hilbert's Tenth Problem for complex meromorphic functions. More precisely, we prove that the set of integers is positive existentially definable in fields of complex meromorphic functions in several variables over…
In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in…
We present and analyze a novel sparse polynomial technique for approximating high-dimensional Hilbert-valued functions, with application to parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our…
A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…
Given the growing quantity of proposals and works of basic hypergeometric functions in the scope of $q$-calculus, it is important to introduce a systematic classification of $q$-calculus. Our aim in this article is to investigate certain…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function.…
We investigate uniqueness problems for an entire function that shares two small functions of finite order with their difference operators. In particular, we give a generalization of a result in $[2]$.
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…
Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
An explicit formula is given for a fundamental solution for a class of semielliptic operators. The fundamental solution is used to investigate properties of these operators as mappings between weighted function spaces. Necessary and…
The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite dimensional complex analysis, but this is a…
Modeling has become a widespread, useful tool in mathematics applied to diverse fields, from physics to economics to biomedicine. Practitioners of modeling may use algebraic or differential equations, to the elements of which they attribute…
Based on the success of a well-known method for solving higher order linear differential equations, a study of two of the most important mathematical features of that method, viz. the null spaces and commutativity of the product of…
Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the…
Of crucial importance to the development of quantum computing and information has been the construction of a quantum operations formalism that admits a description of quantum noise while simultaneously revealing the behavior of an open…
This work presents problems of constructing finite-difference formulas in the Hilbert space, i.e., setting problems of constructing finite-difference formulas using functional methods. The work presents a functional statement of the problem…
We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. Among all subschemes, $X$, with Hilbert function $h$, what are the possible Hilbert functions and…
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…