Arbitrary Number Of Variables Research Articles

Automorphic word maps and the Amit–Ashurst conjecture

Abstract In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246] using elementary techniques.

Algebraic independence of the partial derivatives of certain functions with arbitrary number of variables

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Sections 2 and 3, using the technique together with Mahler’s method, we can reduce the algebraic independence of the infinite set mentioned above to the linear independence of certain rational functions modulo the rational function field of many variables. The latter one is solved by the discussions involving a certain valuation and a generic point in Sections 3 and 4.

A systematic method of constructing weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the well-known FLIP stream cipher. In this paper, we present a systematic method of constructing weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables, which equal the direct sum of several known weightwise perfectly balanced Boolean functions. At the same time, we show two concrete constructions of weightwise almost perfectly balanced Boolean functions, whose k-weight nonlinearities and algebraic immunities are discussed at the end of this paper.

Construction of weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables

The recent FLIP cipher is an encryption scheme described by Méaux et al. at EUROCRYPT 2016. In the FLIP cipher, the input to the Boolean function used as a filter function has constant Hamming weight equal to the weight of the secret key. Weightwise perfectly balanced Boolean functions, which can be used in the FLIP cipher, only exist on 2m variables, while weightwise almost perfectly balanced Boolean functions can exist on an arbitrary number of variables. In this paper, we first introduce a class of quadratic Boolean functions on n variables, where n is a positive integer. We then propose a construction of weightwise almost perfectly balanced Boolean functions on n variables by modifying the support of the quadratic Boolean functions.

Analytic continuation of Lauricella's function F D (N) for variables close to unit near hyperplanes {z j = z l }

For the Lauricella hypergeometric function with an arbitrary number of variables , we construct formulas for analytic continuation into the vicinity of hyperplanes and their intersections providing that all variables are close to unit. Such formulas represent the function near the point as linear combinations of N–multiple hypergeometric series that are solutions of the same system of partial differential equations as . Such series are the N–dimensional analog of the Kummer solutions known for the Gauss equation. The constructed analytical continuation formulas allow one to effectively calculate the function outside the unit polydisk.

Identities of symmetry for Bernoulli polynomials and power sums

Identities of symmetry in two variables for Bernoulli polynomials and power sums had been investigated by considering suitable symmetric identities. T. Kim used a completely different tool, namely the p-adic Volkenborn integrals, to find the same identities of symmetry in two variables. Not much later, it was observed that this p-adic approach can be generalized to the case of three variables and shown that it gives some new identities of symmetry even in the case of two variables upon specializing one of the three variables. In this paper, we generalize the results in three variables to those in an arbitrary number of variables in a suitable setting and illustrate our results with some examples.

A new class of superoscillatory functions based on a generalized polar coordinate system

Is it possible for a band-limited signal to possess oscillation that is arbitrarily higher than its highest Fourier component? Common knowledge assumed that the answer is ‘No.’ Counterintuitively, it turns out that there are band-limited functions that are able to oscillate arbitrarily faster than their fastest Fourier components. These are the superoscillatory functions. Since their discovery, superoscillations have been intriguing in the world of Fourier analysis, with a vast number of applications in quantum mechanics, optics, and radar theory, among other areas. A basic aim in the literature of superoscillations is to find new types of superoscillations that will be used for such technologies. In this paper, we introduce a geometrical-based method to construct a rich class of superoscillations using the concept of directional polar coordinates, developed in this research. We investigate their basic features and show how the proposed method allows generating superoscillations with an arbitrary number of superoscillatory regions, and with an arbitrary number of variables.

On the behavior of modules of m-integrable derivations in the sense of Hasse-Schmidt under base change

We study the behavior of modules of m-integrable derivations of a commutative finitely generated algebra in the sense of Hasse-Schmidt under base change. We focus on the case of separable ring extensions over a field of positive characteristic and on the case where the extension is a polynomial ring in an arbitrary number of variables.

Analytic continuation of the Horn hypergeometric series with an arbitrary number of variables

The problem of analytic continuation is considered for the general hypergeometric Horn series with an arbitrary number of variables. An approach is proposed that allows one to find formulas for the continuation of such series into the exterior of the set of their convergence in the form of linear combinations of other hypergeometric series. These new hypergeometric series belong also to the Horn class, and are solutions of the same system of partial differential equations. Using this general result, we construct formulas for the analytic continuation for a double hypergeometric series of Horn's type, which plays an important role in the theory of the Appell function .

A classical proof that the algebraic homotopy class of a rational function is the residue pairing

Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in 1 variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

On the Complexity of Solving Restricted Word Equations

We investigate the complexity of the solvability problem for restricted classes of word equations with and without regular constraints. The solvability problem for unrestricted word equations remains [Formula: see text]-hard, even if, on both sides, between any two occurrences of the same variable no other different variable occurs; for word equations with regular constraints, the solvability problems remains [Formula: see text]-hard for equations whose two sides share no variables or with two variables, only one of which is repeated. On the other hand, word equations with only one repeated variable (but an arbitrary number of variables) and at least one non-repeated variable on each side, can be solved in polynomial-time.

Analytic continuation of the Lauricella function with arbitrary number of variables

ABSTRACTThe Lauricella function , which is a generalized hypergeometric function of N variables, and a corresponding system of partial differential equations are considered. For an arbitrary N, we give a complete collection of analytic continuation formulas of . This formulas give representation of the Lauricella function outside the polydisk in the form of a linear combination of other generalized hypergeometric series that are solutions of the same system of partial differential equations, which is also satisfied by the function . The obtained hypergeometric series are N-dimensional analogues of the Kummer solutions well known in the theory of the classical hypergeometric Gauss equation. The obtained analytic continuation formulas provide an effective algorithm for computation of the Lauricella function .

On the number of functions of k-valued logic which are polynomials modulo composite k

Abstract A function of k-valued logic is called polynomial if it may be represented by a polynomial modulo k. For any composite number k we propose a uniquely defined canonical form of polynomials for polynomial functions of k-valued logic depending on an arbitrary number of variables. This canonical form is used to find, for any composite k, a formula for the number of n-place polynomial functions of k-valued logic. As a corollary, for any composite k we find the asymptotic behaviour of the logarithm of the number of n-place polynomial functions of k-valued logic.

Variations on inversion theorems for Newton–Puiseux series

Let $f(x,y)$ be a complex irreducible formal power series without constant term. One may solve the equation $f(x,y)=0$ by choosing either $x$ or $y$ as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and 1968, Abhyankar and Zariski published proofs of an \emph{inversion theorem}, expressing the \emph{characteristic exponents} of one set of series in terms of those of the other ones. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the \emph{coefficients} of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning equations with an arbitrary number of variables.

Multi-valued Galois Shanon-Davio trees and their complexity

The idea of Shannon-Davio (S/D) trees for binary logic is a general concept that found applications in the Sum-Of-Product (SOP) minimization and the generation of new diagrams and canonical forms. Extended S/D trees are used to generate forms that include a minimum Galois Field Sum-of-Products (GFSOP) forms. Since there exist many applications of Galois field of quaternary radix especially that GF(4) is considered as an important extension of GF(2), the extension of the S/D trees to GF(4) is presented here. A general formula to calculate the number of Inclusive Forms (IFs) per variable order for an arbitrary Galois field radix and arbitrary number of variables is derived and introduced. A new fast method to count the number of IFs for an arbitrary Galois radix and functions of two variables is also introduced; the IF n,2 Triangles. The results introduced in this work can be useful for the creation of an efficient GFSOP minimizer for Galois logic and in other applications such as in reversible logic synthesis.

Moment problem in infinitely many variables

The multivariate moment problem is investigated in the general context of the polynomial algebra $\mathbb{R}[x_i \mid i \in \Omega]$ in an arbitrary number of variables $x_i$, $i\in \Omega$. The results obtained are sharpest when the index set $\Omega$ is countable. Extensions of Haviland's theorem [Amer. J. Math., 58 (1936) 164-168] and Nussbaum's theorem [Ark. Math., 6 (1965) 179-191] are proved. Lasserre's description of the support of the measure in terms of the non-negativity of the linear functional on a quadratic module of $\mathbb{R}[x_i \mid i \in \Omega]$ in [Trans. Amer. Math. Soc., 365 (2013) 2489-2504] is shown to remain valid in this more general situation. The main tool used in the paper is an extension of the localization method developed by the third author.

A solution method for autonomous first-order algebraic partial differential equations

In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.

Dimension reduction in multivariate extreme value analysis

Non-parametric assessment of extreme dependence structures between an arbitrary number of variables, though quite well-established in dimension $2$ and recently extended to moderate dimensions such as $5$, still represents a statistical challenge in larger dimensions. Here, we propose a novel approach that combines clustering techniques with angular/spectral measure analysis to find groups of variables (not necessarily disjoint) exhibiting asymptotic dependence, thereby reducing the dimension of the initial problem. A heuristic criterion is proposed to choose the threshold over which it is acceptable to consider observations as extreme and the appropriate number of clusters. When empirically evaluated through numerical experiments, the approach we promote here is found to be very efficient under some regularity constraints, even in dimension $20$. For illustration purpose, we also carry out a case study in dietary risk assessment.

Numerical Reduced Variable Optimization Methods via Implicit Functional Dependence with Applications

A systematic theoretical basis is developed that optimizes an arbitrary number of variables for (i) modeling data and (ii) the determination of stationary points of a function of several variables by the optimization of an auxiliary function of a single variable deemed the most significant on physical, experimental or mathematical grounds from which all the other optimized variables may be derived. Algorithms that focus on a reduced variable set avoid problems associated with multiple minima and maxima that arise because of the large numbers of parameters. For (i), both approximateandexact methods are presented, where the single controlling variablekof all the other variablesPkpasses through the local stationary point of the least squares metric. For (ii), an exact theory is developed whereby the solution of the optimized function of an independent variation of all parameters coincides with that due to single parameter optimization of an auxiliary function. The implicit function theorem has to be further qualified to arrive at this result. A nontrivial real world application of the above implicit methodology to rate constant and final concentration parameter determination is made to illustrate its utility. This work is more general than the reduction schemes for conditional linear parameters since it covers the nonconditional case as well and has potentially wide applicability.