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Rotation symmetric Boolean functions represent an interesting class of Boolean functions as they are relatively rare compared to general Boolean functions. At the same time, the functions in this class can have excellent properties, making…
Lucas polynomials are polynomials in $s_1$ and $s_2$ defined recursively by $\{0\}=0$, $\{1\}=1$, and $\{m\}=s_1\{m-1\}+s_2\{m-2\}$ for $m \geq 2$. We generalize Lucas polynomials from 2-variable polynomials to multivariable polynomials.…
As the amount of textual data has been rapidly increasing over the past decade, efficient similarity search methods have become a crucial component of large-scale information retrieval systems. A popular strategy is to represent original…
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…
Chebyshev polynomials and their modifications are attributes of various fields of mathematics. In particular, they are generating functions of the rows elements of certain Riordan matrices. In paper, we give a selection of some…
We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a…
We propose a new iterative algorithm for generating a subset of eigenvalues and eigenvectors of large matrices which generalizes the method of optimal relaxations. We also give convergence criteria for the iterative process, investigate its…
We describe a procedure for the generation of functional digraphs up to isomorphism; these are digraphs with uniform outdegree 1, also called mapping patterns, finite endofunctions, or finite discrete-time dynamical systems. This procedure…
This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…
In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables.…
A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
This paper presents preliminary work on using deep neural networks to guide general-purpose heuristic algorithms for performing utilitarian combinatorial assignment. In more detail, we use deep learning in an attempt to produce heuristics…
We analyze and partially solve system of recurrences that can be derived from the properties of martingale orthogonal polynomials that characterize quadratic harnesses (QH). We also specify conditions for the existence of moments of one…
A hash function is constructed based on a three-layer neural network. The three neuron-layers are used to realize data confusion, diffusion and compression respectively, and the multi-block hash mode is presented to support the plaintext…
Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules…
We present a hierarchy of semidefinite programs (SDPs) for the problem of fitting a shape-constrained (multivariate) polynomial to noisy evaluations of an unknown shape-constrained function. These shape constraints include convexity or…
Probabilistic circuits compute multilinear polynomials that represent multivariate probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in…