Related papers: On linear dependence over complete differential al…
The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…
The continuous dependence of solutions to certain (non-autonomous, partial, integro-differential-algebraic, evolutionary) equations on the coefficients is addressed. We give criteria that guarantee that convergence of the coefficients in…
The quest for complete observables in general relativity has been a longstanding open problem. We employ methods from descriptive set theory to show that no complete observable on rich enough collections of spacetimes is Borel definable. In…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain…
Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact…
This is an introduction to the theory of disconjugacy for a second order linear differential equation. We give new proofs of some of basic results and obtain new sufficient conditions for disconjugacy (in particular, on the whole real…
We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no…
An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
Let $G=\{e^{tA}:t\in\mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on $\mathbb{R}^{n}$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
We show that there are no non-trivial linear dependencies among p-norms of vectors in finite dimensions that hold for all p. The proof is by complex analytic continuation.
This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a…
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…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…
A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there…
The study of linear independence of $L(k, \chi)$ for a fixed integer $k>1$ and varying $\chi$ depends critically on the parity of $k$ vis-\`a-vis $\chi$. This has been investigated by a number of authors for Dirichlet characters $\chi$ of a…
The paper is devoted to two new results concerning varieties of Leibnitz algebras over a field of the zero characteristic. Here is proved the sufficient condition for finiteness colength of variety of Leibnitz algebras. Here is also defined…