Functions Of Several Variables Research Articles

Local theory of stable polynomials and bounded rational functions of several variables

Local theory of stable polynomials and bounded rational functions of several variables

Partial slice regularity and Fueter's theorem in several quaternionic variables

Abstract We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions that are slice, slice regular, and circular with respect to given variables are characterized. We introduce new notions of partial spherical value and derivative for functions of several variables that extend those of one variable. We recover some of their properties as circularity, harmonicity, some relations with differential operators, and a Leibniz rule with respect to the slice product as well as studying their behavior in the context of several variables. Then, we prove our main result, which is a generalization of Fueter’s theorem for slice regular functions in several variables. This extends the link between slice regular and axially monogenic functions well known in the one variable context.

Coefficients of multiple Fourier-Haar series and variational modulus of continuity

In this paper, we introduce the concept of a variational modulus of continuity for functions of several variables, give an estimate for the sum of the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity, and prove theorems of absolute convergence of series composed of the coefficients of multiple Fourier-Haar series. In this paper, we study the issue of the absolute convergence for multiple series composed of the Fourier-Haar coefficients of functions of several variables of bounded p-variation. We estimate the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity and prove the sufficiency theorem for the condition for the absolute convergence of series composed of the Fourier-Haar coefficients of the considered function class. This paper researches the question: under what conditions, imposed on the variational modulus of continuity of the fractional order of several variables functions, there is the absolute convergence for series composed of the coefficients of multiple Fourier-Haar series.

A New Integral Function Algorithm for Global Optimization and Its Application to the Data Clustering Problem

The filled function method is an approach to finding global minimum points of multidimensional unconstrained global optimization problems. The conventional parametric filled functions have computational weaknesses when they are employed in some benchmark optimization functions. This paper proposes a new integral function algorithm based on the auxiliary function approach. The proposed method can successfully be used to find the global minimum point of a function of several variables. Some testing global optimization problems have been used to show the ability of this recommended method. The integral function algorithm is then implemented to solve the center-based data clustering problem. The results show that the proposed algorithm can solve the problem successfully.

Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain the exact-order estimates of approximation of the Nikol'skii-Besov-type classes $B^{\Omega}_{\infty,\theta}$ of periodic functions of several variables with a given function $\Omega(t)$ of a special form by using linear operators satisfying certain conditions. The approximation error is estimated in the metric of the space $L_{\infty}$. The obtained estimates of the considered approximation characteristic, in addition to independent interest, can be used to establish the lower bounds of the orthowidths of the corresponding functional classes.

Classification of Measurable Functions of Several Variables and Matrix Distributions

Classification of Measurable Functions of Several Variables and Matrix Distributions

On Universal Sampling Recovery in the Uniform Norm

It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling discretization of the uniform norm and recent results on universal sampling representation allow us to provide good universal methods of sampling recovery for anisotropic Sobolev and Nikol’skii classes of periodic functions of several variables. The sharpest results are obtained in the case of functions of two variables, where the Fibonacci point sets are used for recovery.

Bank’s Credit Risk Modeling

The article reviews the Bank’s credit risk modeling issues. The substance of the article analyzes the credit risk structure and methods for measuring its components. Credit risk is measured as a loss, that is the function of several variables. The amount of open credit risk position in case of default, expected proba-bility of credit default and recovery ratio after the default are the main variables of the given function presented in the arti-cle. These variables are reviewed as random values and meth-ods are given for its evaluation and integration as one indica-tor.The article also reviews the tasks of forming the bank’s internal credit ratings and issues related to the use of these ratings in credit risk evaluation model.

Generalized Lagrange Theorem

The present paper is devoted to possible generalizations of the classic Lagrange Mean Value Theorem. We consider a real-valued function of several variables that is only assumed to be continuous. The main concept is to replace the notion of the derivative by the so called bisequential tangent cone. We first prove Rolle and Lagrange type results and then we turn to comparing this cone with the Clarke subdifferential in the case of a Lipschitz function. We also investigate an approach using normal cones.

The General Theory of Superoscillations and Supershifts in Several Variables

In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators acting on holomorphic functions with growth conditions of exponential type. Additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for superoscillating sequences in several variables are extended to sequences of supershifts in several variables.

Multivariate Zipper Fractal Functions

A novel approach to zipper fractal interpolation theory for functions of several variables is presented. Multivariate zipper fractal functions are constructed and then perturbed through free choices of base functions, scaling functions, and a binary matrix called signature to obtain their zipper α-fractal versions. In particular, we propose a multivariate Bernstein zipper fractal function and study its coordinate-wise monotonicity which depends on the values of signature. We derive bounds for the graph of a multivariate zipper fractal function by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The box dimension result for multivariate Bernstein zipper fractal function is derived. Finally, we study some constrained approximation properties for multivariate zipper Bernstein fractal functions.

Correlation Dependence between Hydrophobicity of Modified Bitumen and Water Saturation of Asphalt Concrete

Improving the durability of asphalt concrete road surfaces by increasing their moisture resistance is an urgent task. Modified bituminous binders should be compacted into coatings with the lowest possible water saturation. The purpose of this study was to establish the effect of modifiers on the hydrophobicity of bituminous films in order to achieve minimum water saturation and to build a mathematical model of the wetting process with water. As modifiers, we used a product of amination of distillation residues of petrochemistry, waste sealing liquid (a solution of high molecular weight polyisobutylene in mineral oil), and a condensation product of polyamines and higher fatty acids. The water-repellent effect of modifiers was studied by measuring the contact angle of bituminous film with a water drop. The water saturation of asphalt concrete samples was determined by the amount of water absorbed by asphalt concrete at 20 °C. A close correlation was revealed between the hydrophobicity of modified bitumen and the water saturation of asphalt concrete. Generalized equations and a graphical representation of a function of several variables allowed for optimizing compositions by the content of modifiers to achieve the required performance properties of asphalt concrete coatings.

Normal Families of Meromorphic Functions in Several Variables

Normal Families of Meromorphic Functions in Several Variables

On Banach spaces of regulated functions of several variables. An analogue of the Riemann integral

The paper introduces the concept of a regulated function of several variables $f\colon X\to\mathbb R$, where $X\subseteq \mathbb R^n$. The definition is based on the concept of a special partition of the set $X$ and the concept of oscillation of the function $f$ on the elements of the partition. It is shown that every function defined and continuous on the closure $X$ of the open bounded set $X_0\subseteq\mathbb R^n$, is regulated (belongs to the space $\langle{\rm G(}X),\|\cdot\ |\rangle$). The completeness of the space ${\rm G}(X)$ in the $\sup$-norm $\|\cdot\|$ is proved. This is the closure of the space of step functions. In the second part of the work, the space ${\rm G}^J(X)$ is defined and studied, which differs from the space ${\rm G}(X)$ in that its definition uses $J$-partitions instead of partitions, whose elements are Jordan measurable open sets. The properties of the space ${\rm G}(X)$ listed above carry over to the space ${\rm G}^J(X)$. In the final part of the paper, the notion of $J$-integrability of functions of several variables is defined. It is proved that if $X$ is a Jordan measurable closure of an open bounded set $X_0\subseteq\mathbb R^n$, and the function $f\colon X\to\mathbb R$ is Riemann integrable, then it is $J$-integrable. In this case, the values of the integrals coincide. All functions $f\in{\rm G}^J(X)$ are $J$-integrable.

A multidimensional Hermite-Gauss sampling formula for analytic functions of several variables

A multidimensional Hermite-Gauss sampling formula for analytic functions of several variables

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

The composition H(z)=f(Φ(z)) is studied, where f is an entire function of a single complex variable and Φ is an entire function of n complex variables with a vanished gradient. Conditions are presented by the function Φ providing boundedness of the L-index in joint variables for the function H, if the function f has bounded l-index for some positive continuous function l and L(z)=l(Φ(z))(max{1,|Φz1′(z)|},…,max{1,|Φzn′(z)|}),z∈Cn. Such a constrained function L allows us to consider a function Φ with a nonempty zero set. The obtained results complement earlier published results with Φ(z)≠0. Also, we study a more general composition H(w)=G(Φ(w)), where G:Cn→C is an entire function of the bounded L-index in joint variables, Φ:Cm→Cn is a vector-valued entire function, and L:Cn→R+n is a continuous function. If the L-index of the function G equals zero, then we construct a function L˜:Cm→R+m such that the function H has bounded L˜-index in the joint variables z1,…,zn. The other group of our results concern a sum of entire functions in several variables. As a general case, a sum of functions with bounded index is not of bounded index. The same is also valid for the multidimensional case. We found simple conditions proving that f1(z1)+f2(z2) belongs to the class of functions having bounded index in joint variables z1,z2. We formulate some open problems based on the deduced results and on the usage of fractional differentiation operators in the theory of functions with bounded index.

Characteristics of the linear and nonlinear approximations of the Nikol’skii–Besov-type classes of periodic functions of several variables

Characteristics of the linear and nonlinear approximations of the Nikol’skii–Besov-type classes of periodic functions of several variables

Twisted Weyl group multiple Dirichlet series over the rational function field

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to Cr and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K, but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K=Fq(T), and describe the denominators and support of the numerators.

High-Dimensional Contact Network Epidemiology.

Contact network models are recent alternatives to equation-based models in epidemiology. In this paper, the spread of disease is modeled on contact networks using bond percolation. The weight of the edges in the contact graphs is determined as a function of several variables in which case the weight is the product of the probabilities of independent events involving each of the variables. In the first experiment, the weight of the edges is computed from a single variable involving the number of passengers on flights between two cities within the United States, and in the second experiment, the weight of the edges is computed as a function of several variables using data from 2012 Kenyan household contact networks. In addition, the paper explored the dynamics and adaptive nature of contact networks. The results from the contact network model outperform the equation-based model in estimating the spread of the 1918 Influenza virus.