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We define and study a generalization of the analytic Cauchy problem, that specializes to the Cauchy-Kowaleskaya-Kashiwara problem in the linear case. The main leitmotive of this text is to adapt Kashiwara's formulation of this problem both…
There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a…
In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop…
We present an alternate formulation of the partial assignment problem as matching random clique complexes, that are higher-order analogues of random graphs, designed to provide a set of invariants that better detect higher-order structure.…
In this paper, we study the notion of special ideals. We generalize the results on those as well as the algorithm obtained for finite dimensional power series rings by Mordechai Katzman and Wenliang Zhang to finite dimensional polynomial…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here,…
The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite…
When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the…
We consider several notions of genericity appearing in algebraic geometry and commutative algebra. Special emphasis is put on various stability notions which are defined in a combinatorial manner and for which a number of equivalent…
In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
Bipartite determinantal ideals are introduced by Illian and the author as a vast generalization of the classical determinantal ideals intensively studied in commutative algebra, algebraic geometry, representation theory and combinatorics.…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
Contour integral algorithms seek to compute a small number of eigenvalues located within a bounded region of the complex plane. These methods can be applied to both linear and nonlinear matrix eigenvalue problems. In the latter case, the…
Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually…
The paper addresses the problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions --…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
For any polynomial ideal $I$, let the minimal triangular set contained in the reduced Buchberger-Gr\"obner basis of $I$ with respect to the purely lexicographical term order be called the W-characteristic set of $I$. In this paper, we…
In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring…