Related papers: A note on multiplicative-inverse chaining in finit…

Thanks to a new construction of the so-called Chudnovsky-Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation…

In this paper we prove some results on the possible multiplicative orders of $\alpha + \alpha^{-1}$ when $\alpha$ is a non-zero element of a finite field of characteristic 2. The results of the paper rely on a previous investigation on the…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

We discuss the properties and the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over finite fields. The results in this paper unify and generalize several recently discovered results on…

Inversion of function sinc(x) is studied. New series and integral representations of branches of inverse function are obtained using Fourier analysis.

In this note, we solve an inverse spectral problem for a class of finite band symmetric matrices. We provide necessary and sufficient conditions for a matrix valued function to be a spectral function of the operator corresponding to a…

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations…

Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to…

We investigate the permutation property of polynomials of the form $x^{r}(x^{s} -a)^{t}$, and give the expressions of their inverses. In particular, explicit expressions of inverses of permutation polynomials $x(x^3 -a)^2$ and $x(x^2 -a)^3$…

We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the $2$-reverse pass…

We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis.

We describe several families of permutation polynomials obtained using functions with linear translators.

We study Ising chains with arbitrary multispin finite-range couplings, providing an explicit solution of the associated inverse Ising problem, i.e. the problem of inferring the values of the coupling constants from the correlation…

In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. We extend its application to the case of formal power series over a field of arbitrary characteristic and illustrate the proposed approach…

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the…

The double-direction orthogonalization algorithm is applied to construct sequences of polynomials, which are orthogonal over the interval [0,1]with the weighting function 1. Functional and recurrent relations are derived for the sequences…

Knutson introduced two families of reverse juggling Markov chains (single and multispecies) motivated by the study of random semi-infinite matrices over $\mathbb{F}_q$. We present natural generalizations of both chains by placing generic…

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing,…

Let $f_1(x),\ldots,f_n(x)$ be some polynomials. The upper bound on the number of $x\in\mathbb F_p$ such that $f_1(x),\ldots,f_n(x)$ are roots of unit of order $t$ is obtained. This bound generalize the bound of the paper \cite{V-S} to the…

We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a…