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We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de…

New examples of matrix quasi exactly solvable Schroedinger operators are constructed. One of them constitutes a matrix generalization of the quasi exactly solvable anharmonic oscillator, the corresponding invariant vector space is…

We study distribution of zeros of a complex polynomial whose coefficients has been modified. We give a new proof of the theorem of Rubinstein, and with similar method we prove a new theorem that is not generalization of the previous…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

For every closed subset $X$ of a stratifiable [resp. metrizable] space $Y$ we construct a positive linear extension operator $T:R^{X\times X}\to R^{Y\times Y}$ preserving constant functions, bounded functions, continuous functions,…

Short integer linear programs are programs with a relatively small number of constraints. We show how recent improvements on the running-times of solvers for such programs can be used to obtain fast pseudo-polynomial time algorithms for…

Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential…

Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.

A characterization of positive operators on finite dimensional complex vector spaces based on the Routh-Hurwitz Criterion.

Let $S$ be a dense subring of the real numbers. In this paper we prove a polynomial version of Van der Waerden's theorem near zero. In fact, we prove that if $p_1,\ldots,p_m \in \mathbb{Z}[x]$ are polynomials such that $p_i(0) = 0$ and…

We consider the third order operator with periodic coefficients on the real line. This operator is used in the integration of the non-linear evolution Boussinesq equation. For the minimal smoothness of the coefficients we prove that: 1) the…

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…

In this paper, we consider periodic boundary value problems for differential equations whose coefficients are trigonometric polynomials. We construct the spaces of generalized functions, where such problems have solutions. In particular,…

We show that any finite set of linear partial differential operators with continuous coefficients is linearly dependent if and only if it is locally linearly dependent. It follows that the reflexive closure of any finite set of such…

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete- time linear systems. We mainly focus on the so-called Krasnoselskij-Mann iteration x(k+1) = ( 1 - \alpha(k) ) x(k) + \alpha(k) A…

We study multivariable spectral multipliers $F(L_1,L_2)$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_1$ and $L_2$. We prove that if $F$ satisfies H\"ormander type differentiability condition then the…

Representations of polynomial covariant type commutation relations by pairs of linear integral operators and multiplication operators on Banach spaces $L_p$ are constructed.

In this paper we prove local solvability of quasilinear pseudodifferential operators which has homogeneous principal symbol of real principal type. This generalizes Theorem A.1 in arXiv:2403.19054, which treats the case of quasilinear…

Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems…

We construct discrete analogues of the Dixmier operators, that is, commuting difference operators corresponding to a spectral curve of genus 1 whose coefficients are polynomials of the discrete variable.