Functions Of Several Variables Research Articles

Mathematical modeling of minimization of electricity consumption by industrial enterprises with continuous production

Relevance. Determined by the importance of minimizing electrical power consumption in industrial enterprises with continuous production, considering the specific characteristics of their technological processes and the requirements to maintain the output volume of their products. Aim. To solve the task of minimizing electrical power consumption based on a mathematical model and gradient method under optimal planning of the production volume of the industrial enterprises with continuous production; to develop a mathematical model for optimal distribution of production over a time cycle (month, quarter, year) across departments, taking into account both simple and functional constraints, derived from the condition of ensuring minimal electrical power consumption in industrial enterprises with continuous production. Methods. When developing the mathematical model for ensuring minimal electrical power consumption while preserving the production volume, classic Lagrange optimization methods were used. To ensure sufficient calculation accuracy, iterative methods were also applied. For the task under consideration, a calculation error margin of ε=0,1 was assumed and established. It is known that the choice of calculation error margin depends on the specifics of the problem at hand and the decision-maker. To verify the adequacy of the developed model, the method of finding the relative extremum of a function of several variables was used. Results. The use of the mathematical model, which takes into account the nature of the technological process and boundary conditions in both simple and integral forms, demonstrated the feasibility of optimal planning of electrical power consumption by the enterprise. The effectiveness of the developed approaches was verified using a metallurgical enterprise as an example of an industrial enterprise with continuous production, in solving the task of minimizing electrical power consumption for products produced during the reporting period. The use of the proposed model allowed for a reduction in annual electrical power consumption by 2.5% while maintaining the same production volume. One of the classic optimization methods – the method of finding the relative extremum of functions of several variables – showed almost identical results upon verification. This serves as further evidence of the adequacy of the proposed model.

On approximation of functions, conjugate to the functions of several variables belonging to weighted Lipschitz and Zygmund classes

In the present paper, we introduce a new weighted Lipschitz class W(Lp(TN), ξ1(s1), . . . , ξN(sN)) and Zygmund class Z(Lp(TN), ξ1(s1), . . . , ξN(sN)) for N ∈ N, which generalizes the classes given in [7, 11]. We prove two theorems about
 the degree (error) of approximation of functions, conjugate to the N-variable functions (2π-periodic in each variable) belonging to these classes using the N-multiple matrix means of their N-multiple conjugate Fourier series. We improve the results of Móricz and Rhoades [6] and Móricz and Shi [7], which are given in the form of corollaries.

МАТЕМАТИЧНІ МОДЕЛІ ТА МЕТОДИ РОЗМІЩЕННЯ ОБ’ЄКТІВ І ЗОНУВАННЯ ТЕРИТОРІЙ В СИСТЕМАХ ЕКСТРЕНОЇ ЛОГІСТИКИ

The mathematical models for distribution processes related to organizing precautionary measures in the event of threats or occurrences of man-made emergencies are presented. The tasks include optimal zoning of territories with the fixing of zones by objects of social purpose for service provision. Provision is made for: the possibility of overlapping zones in case the nearest center cannot provide the service; optimal placement of a certain number of new cen-ters of emergency logistics systems with simultaneous redistribution of the load on all their structural elements; the selection of locations of structural subdivisions based on existing fa-cilities. The optimality criteria involve minimizing either the time to provide the service even to the most remote object in the given territory, or the total distance to the nearest centers from consumers that are densely distributed in the given territory, and/or the organizational costs associated with the arrangement of new centers. Mathematical models are proposed in the form of continuous problems of optimal multiplex partitioning of sets with a linear or minimax functional of quality. The latter provides such placement of centers that provides op-timal multiple coverage of the territory (with a minimum radius of multiple coverage). Meth-ods for solving the formulated problems were developed using LP-relaxation of linear prob-lems with Boolean variables, duality theory to reduce the initial problems of infinite-dimensional programming to problems of conditional optimization of a non-smooth function of several variables, and modern methods of non-differentiated optimization.

A Lagrange interpolation with preprocessing to nearly eliminate oscillations

AbstractThis work is concerned with the interpolation of a function $$\varvec{f}$$ f when using a low number of interpolation points, as required by the finite element method for solving PDEs numerically. The function $$\varvec{f}$$ f is assumed to have a jump or a steep derivative, and our goal is to minimize the oscillations produced by the Gibbs phenomenon while preserving the approximation properties for smoother functions. This is achieved by interpolating the transform $$ \varvec{\hat{f} = g \circ f} $$ f ^ = g ∘ f using Lagrange polynomials, where $$\varvec{g}$$ g is a rational transformation chosen by minimizing a suitable functional depending on the values of $$\varvec{f}$$ f . The mapping $$\varvec{g}$$ g is monotonic and constructed to possess boundary layers that remove the Gibbs phenomenon. No previous knowledge of the location of the jump is required. The extension to functions of several variables is straightforward, of which we provide several examples. Finally, we show how the interpolation fits the finite element method and compare it with known strategies.

Vowel alternation with final i offers an easy-to-learn morphological option for a sex-blind grammatical gender in French.

Like all modern Romance languages, French has a sex-based grammatical gender with two genders, feminine and masculine, and a lexicon that is highly sex-differentiated. These characteristics give rise to a number of issues, including the problematic generic use of the masculine grammatical gender, coupled with the challenge of sex categorization itself, and the epistemological difficulty of an adequate sociological description and analysis of what gender commonsense categories really are about. To remedy these concerns, several authors have proposed the creation of an additional, epicene grammatical gender. We have identified three such systematic proposals, or solutions, which specify various morphological options for new epicene nouns and gender markers on their satellite elements. These options include the use of non-standard or rarely used characters, the merging of feminine and masculine gender markers, as well as consonantal and vowel changes. In the simplest proposal, referred to as "solution I," new epicene forms are mostly derived from feminine forms by systematically replacing with an i the final e that generally differentiates feminines from their masculine counterparts in written French. Although these solutions are used in some communities, their learnability has not been addressed so far, even though it could be a determining factor in their popularity and their eventual integration into standard French. In the present study, we provide a first assessment of this aspect by means of an online translation test. For each solution, French-speaking participants were instructed that they would be trained to learn an "alien" language that does not mark sex/gender categories (these alien languages correspond to standard French where only gendered words referring to people are replaced by the new epicene forms recommended by each solution). After a short learning-by-example phase, participants were required to translate into the alien language a set of 16 standard French sentences. The translations were analyzed as a function of several variables including the participants' self-reported age and sex, the word categories and the solutions themselves. While all solutions proved quickly learnable, participants' responses with solution I achieved the highest accuracy score, in particular with regard to the production of non-standard epicene forms.

A selection theorem for essentially regulated functions of several variables

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fraňková has given a proper extension of Helly's theorem to the setting of single-variable regulated functions. We show how a similar approach yields a natural notion of regulated functions of several variables, together with an extension of the standard compactness criterion for multi-variable functions of bounded variation.

Optimization of the current spray angle of the nozzles of the adaptive distribution system of a single-support boom sprayer

One of the unsolved problems of plant protection mechanization is the uneven distribution of pesticides due to vertical vibrations of the sprayer boom. At the same time, the variation of the treatment band with each sprayer can reach 38% or more. This issue is even more relevant for single-support wheelbarrow sprayers, which are used in breeding and primary seed production. This significantly affects the quality of the technological operation. A technical solution to this problem could be the use of sprayers with variable spray geometry. Such a solution requires an analytical substantiation of the dependence of the spray pattern angle on the position angle of the sprayer boom. The purpose of the proposed research is to develop a technique for optimizing the spray angle of sprayer nozzles, taking into account the vertical vibrations of the distribution rod. Geometric and mathematical modeling, theoretical analysis of qualitative and agrotechnical parameters of spraying, substantiation of the laws of the technological process of spraying based on the known laws of physics and classical mathematics, as well as a method of nomogramming functions of several variables were used as scientific methods. It has been established that the uniformity of the application of protective equipment is determined not only by the position of the rod in the vertical plane, but also by the value of the root spray angle, the height of the installation of the sprayer and its position relative to the center of oscillation of the rod. An analytical expression is derived that relates the current width of the processing strip from one sprayer to the specified factors. The dependence of the optimal spray angle on the angle of inclination of the sprayer boom is also proposed. A nomogram has been developed that takes into account permissible deviations in the uneven distribution of the working fluid during spraying. The optimization technique is new. It is applicable to the development of software for the sprayer control complex, which is equipped with an adaptive pesticide distribution system in changing conditions.

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.