Related papers: On the dimension of systems of algebraic differenc…
A differential category is an additive symmetric monoidal category, that is, a symmetric monoidal category enriched over commutative monoids, with an algebra modality, axiomatizing smooth functions, and a deriving transformation on this…
Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite…
We define the probability of an equation in a finite algebra as the proportion of tuples in its domain that satisfy it. We call the probabilistic spectrum of an algebra the set of probability values obtained when the equation varies. We…
Given finitely many consecutive terms of an infinite sequence, we discuss the construction of a polynomial difference equation that the sequence may satisfy. We also present a method to seek a candidate polynomial differential equation for…
One often distinguishes between a line and a plane by saying that the former is one-dimensional while the latter is two. But, what does it mean for an object to have $d-$dimensions? Can we define a consistent notion of dimension rigorously…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations…
We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a…
A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every…
The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic…
We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with…
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.
Algebraic and combinatorial properties of a monomial ideal and its radical are compared.
We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
To allow for Division By Zero, we develop a new algebraic structure containing addition and multiplication called an S-Extension of a Field. This unique structure extends a Field so that the equation $0\cdot s=x$ has exactly one solution…
We present a null witness of the dimension of a quantum system, discriminating real, complex and classical spaces, based on equality due to linear independence. The witness involves only a single measurement with sufficiently many outcomes…
We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope.…
We consider a rational system of first order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of…
In this paper we study the monomial dynamical systems of dimension one over finite fields from the viewpoints of arithmetic and graph theory. We give formulas for the number of periodic points with period r and cycles with length r. Then we…