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We extend the hyperplane arrangement framework for neural network expressivity from the braid to discriminantal arrangements. Compatible piecewise linear functions are characterized by circuit relations and admit a matroidal description via…
We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs…
Special bases of orthogonal polynomials are defined, that are suited to expansions of density and potential perturbations under strict particle number conservation. Particle-hole expansions of the density response to an arbitrary…
A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
We propose a unified view of the polarity of functions, that encompasses all specific definitions, generalizes several well-known properties and provides new results. We show that bipolar sets and bipolar functions are isomorphic lattices.…
Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical…
Numerical functions, which characterize Dynkin schemes, Coxeter graphs and tame marked quivers, are considered.
The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values…
Following work of Mazur-Tate and Satoh, we extend the definition of division polynomials to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity. In analogy to the classical case of division…
Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between…
In the present paper we generate binary pseudorandom sequences using generalized polynomials. A generalized polynomial is a function in whose description we not only allow addition and product (as it is the case in usual polynomials) but…
Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in $\RR^d$. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer…
We consider the scenario in which a set of sources generate messages in a network and a receiver node demands an arbitrary linear function of these messages. We formulate an algebraic test to determine whether an arbitrary network can…
We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure…
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
Overlap functions were introduced as class of bivariate aggregation functions on [0, 1] to be applied in image processing. This paper has as main objective to present appropriates definitions of overlap functions considering the scope of…
In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a…