Functions Of Several Variables Research Articles

Construction of solutions to PDEs using holomorphic functions of several variables

The work develops Humbert's method of solving PDEs. It applies method for constructing solutions of the three-dimensional Laplace, Helmholtz, and Poisson equations in the form of components of holomorphic functions of several variables (complex and hypercomplex). For more information see https://ejde.math.txstate.edu/Volumes/2024/71/abstr.html

Research on the convergence of some types of functional branched continued fractions

An analysis of research on the problem of convergence of various types of functional branched continued fractions has been carried out. Branched continued fractions with $N$ branching branches and branched continued fractions with independent variables are considered. The definition and, in our opinion, characteristic criteria of convergence of multidimensional generalizations of C-, S-, g-, J-fractions are given, both for branched continued fractions of the general form with $N$ branching branches and branched continued fractions with independent variables. Such multidimensional generalizations of continued fractions arise, in particular, in the development of various classes of hypergeometric functions of several variables, in particular, the functions of Appel, Lauricella, Horn, etc.

Assessing the Probability of Winning a Point in Men’s Padel: A Comprehensive Analysis

Background—The number of studies on padel has grown significantly in recent years, reflecting the growing importance of the sport. However, more research is still needed on a comprehensive analysis of performance indicators. This study had a double objective: on the one hand, to analyze the probability of winning a point as a function of several variables and, on the other hand, to develop a match analysis. Methods—A total of 980 points from the Menorca 2020 World Padel Tour Master Final Men’s category were examined using observational methodology. The participants were the eight pairs who competed in the final rounds of the tournament. To obtain the results, various analytical techniques were used, such as descriptive analysis and the chi-square test, with a significance level of p < 0.05. Results—The results indicated that most points were initiated on the first serve (88.5%) and were most often won by the serving pair (59.3%). Short rallies predominated (42.2%), with the serving pair most likely to win the point (71%). As the rally became longer, the probability of winning the point decreased, reaching 57% for medium rallies and 47% for long rallies. Almost half of the points were completed in less than 10 s, with an average point duration of 15 s. Most points were finished from the middle area and near the net (41.4% and 36%, respectively), mainly by smashes (25.9%) or volleys (28.9%), with the cross-court trajectory being the most effective. Conclusions—Short rallies (0–6 shots) were the most common, with an average point duration of approximately 15 s. Most points ended with a shot from the middle of the court, using a volley or a smash with a cross-court trajectory. There is a clear relationship between proximity to the net on the final shot and an increased probability of winning the point.

Approximation of classes of Poisson integrals by rectangular Fejér means

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane.

Students' conception of partial derivatives of a function of several variables

The study aimed to explore and assess students' conceptual understanding of partial derivatives in functions of several variables. A case study research design was employed, with four classes randomly selected for participation. Qualitative data were collected through interviews and open-ended questions developed by the researchers. The analysis was conducted using APOS theory, based on the proposed genetic decomposition. The findings reveal that the majority of students’ understanding falls under action conception, interpreting partial derivatives similarly to single-variable derivatives. Additionally, many students struggled to extend their knowledge from single-variable derivatives to partial derivatives and faced difficulties in converting between algebraic/symbolic and graphical representations. Based on these results, the researchers recommend that educators adopt instructional strategies that foster deeper conceptual understanding, rather than relying on traditional teaching methods. Keywords: APOS Theory; conceptual understanding; function of several variables; partial derivatives

Photometric Completeness Modelled with Neural Networks

In almost any study involving optical/near-infrared photometry, understanding the completeness of detection and recovery is an essential part of the work. The recovery fraction is, in general, a function of several variables including magnitude, color, background sky noise, and crowding. We explore how completeness can be modeled, with the use of artificial-star tests, in a way that includes all of these parameters simultaneously within a neural network (NN) framework. The method is able to manage common issues including asymmetric completeness functions and the bilinear dependence of the detection limit on color index. We test the method with two sample Hubble Space Telescope data sets: the first involves photometry of the star cluster population around the giant Perseus galaxy NGC 1275, and the second involves the halo-star population in the nearby elliptical galaxy NGC 3377. The NN-based method achieves a classification accuracy of > 94% and produces results entirely consistent with more traditional techniques for determining completeness. Additional advantages of the method are that none of the issues arising from the binning of the data are present and that a recovery probability can be assigned to every individual star in real photometry. Our data, models, and code (called COINTOSS) can be found online on Zenodo at the following link: https://doi.org/10.5281/zenodo.8306488.

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency

In this paper, for a wide class of nonparametric regression models, new local linear kernel estimators are proposed that are uniformly consistent under close-to-minimal and visual conditions on design points. These estimators are universal in the sense that their designs can be either fixed and not necessarily satisfying the traditional regularity conditions, or random, while not necessarily consisting of independent or weakly dependent random variables. With regard to the design elements, only dense filling of the regression function domain with the design points without any specification of their correlation is assumed. This study extends the dense data methodology and main results of the authors’ previous work for the case of regression functions of several variables.

On Banach spaces of regulated functions of several variables. Analogue of the Riemann–Stieltjes integral

In the previous work of the authors, the concept of a regulated function of several variables $f\colon X\to\mathbb R$ was introduced, where $X\subseteq \mathbb R^n.$ The definition is based on the concept of a special partition of the set $X$ and the concept oscillation of the function $f$ on the elements of the partition. The space ${\rm G}(X)$ of such functions is Banach in the $\sup$-norm and is the closure of the space of step functions. In this paper, the space ${\rm G}^F(X)$ is defined and studied, which differs from ${\rm G}(X)$ in that here, in defining regulated functions of several variables, instead of special partitions, $F$-partitions are used: their elements are non-empty open sets measurable by the generalized Jordan measure (by the measure $m_{_{\!F}}$). (Symbol $F$ denotes the function generating the measure $m_{_{\!F}}.$) In the second part of the work, the concept of $F$-integrability of functions of several variables is defined. It is proved that if $X$ is the closure of a non-empty open bounded set $X_0\subseteq {\mathbb R}^n,$ measurable with respect to measure $m_{_{\!F}},$ and the function $f\colon X\to {\mathbb R}$ is integrable in the Riemann–Stieltjes sense with respect to the measure $m_{_{\!F}}$, then it is $F$-integrable. In this case, the values of the multiple integrals coincide. All functions from the space ${\rm G}^F(X)$ are $F$-integrable. The main properties of the Riemann–Stieltjes $F$-integral are proved.

Analytical Solution for Seismic Stability of 3D Rock Slope Reinforced with Prestressed Anchor Cables

Currently, the study of analytical solutions for the seismic stability of slopes under anchorage conditions is one of the hottest subjects in engineering. In this paper, an analytical solution for the seismic safety of the three-dimensional (3D) two-stage rock slope reinforced with prestressed anchor cables governed by the nonlinear Hoek–Brown criterion was deduced, in which the analyses of seismicity were performed by the latest modified pseudo-dynamic method. This method supplements the consideration of the damping effect of the rock medium on seismic waves, which is more in line with the real seismic situation. A mathematical geometric model was developed for calculating the external forces work and internal energy dissipation acting on 3D rotating rock masses reinforced with prestressed anchor cables, in which the seismic work rate was calculated using a new layer-by-layer superposition summation method. The analytical solution of the safety factor could be collated as an explicit function of several variables, and then, the optimal value was obtained by the Particle Swarm Optimization (PSO) algorithm. To corroborate the accuracy of new analytical solutions, the results were contrasted with those of the pertinent literature. The results of the two comparisons were very close. Ultimately, the sensitivity analyses and coupling effects of seismic pseudo-dynamic factors and prestressing anchorage factors were carried out. It was found that even small seismic intensities had a large effect on the stability of rock slopes with developed joints. Increasing the number of steps and prestressing anchors can effectively improve the stability of rock slopes under seismic effects. The conclusions have significant implications for the anchorage design of the 3D two-stage rock slope as seismic events occur.

On application of Gaussian kernels and Laplace functions combined with Kolmogorov's theorem for approximation of functions of several variables

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace functions). It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

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.

Positive Matrix Representations of Rational Positive Real Functions of Several Variables

Positive Matrix Representations of Rational Positive Real Functions of Several Variables

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.

On Grothendieck-type duality for spaces of holomorphic functions of several variables

We describe the strong dual space $({\mathcal O} (D))^*$ of the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables in a bounded domain $D$ with Lipschitz boundary and connected complement (as usual, ${\mathcal O} (D)$ is endowed with the topology of local uniform convergence in $D$). We identify the dual space with the closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, whose elements vanish at the point at infinity and satisfy the Cauchy-Riemann tangential conditions on $\partial D$. In particular, we generalize classical Grothendieck-Köthe-Sebastião e Silva duality for holomorphic functions of one variable to the multivariate situation. We prove that the duality we produce holds if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of Sobolev-class holomorphic functions in $D$ is dense in ${\mathcal O} (D)$. Bibliography: 35 titles.

EXPANSIONS OF KAMP ́E DE F ́ERIET HYPERGEOMETRIC FUNCTIONS

When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called H-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these H-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kampe de Feriet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kampe de Feriet functions of the superior order. Some Kampe de Feriet functions are expanded in terms of the Appell and Humbert functions as illustrative examples