Functions Of Several Variables Research Articles

ОБ УНИФИКАЦИИ ИЗЛОЖЕНИЯ КРАТНЫХ, КРИВОЛИНЕЙНЫХ И ПОВЕРХНОСТНЫХ ИНТЕГРАЛОВ НА ИНЖЕНЕРНЫХ НАПРАВЛЕНИЯХ БАКАЛАВРИАТОВ

In modern world, the up-to-date trends in reforming of higher education lead to a significant reduction of classroom activities in the educational process, which extremely negatively affects the study of disciplines of the mathematical cycle due to their abstractness and often complexity of material presentation. Thus, there is a necessity to increase the efficiency of contact work and to widen the role of unsupervised work of students. This problem is especially evident in the study of the topic “Integral calculus for functions of several variables” in the engineering directions of undergraduate studies. One of the factors that contribute to increasing the efficiency of contact and unsupervised work is the unification of the educational material presentation related to this topic. In this article, we suggest a new approach to the study of multiple, curvilinear and surface integrals in the engineering directions of undergraduate studies, which include a unified parallel consideration of these integrals when using: 1) a preliminary revision of the topic “Defined integral” and the study of new material on this basis, 2) interdisciplinary connections, 3) information technologies. It is advisable to use appropriate methodical support to improve the effectiveness of this approach. As the latter, it is appropriate to use tutorials, problem books and individual tasks, posters and 3d models, interactive guidelines integrated with applications, written in high-level languages, generators of typical tasks, etc.

Primjenom otpadne drvne biomase i Miscanthusa do održivog sektora termoenergetike u vremenu energetske tranzicije / Sustainable Energy Sector Using Waste Woody Biomass and Miscanthus in the Energy Transition

The thermal energy sector is very important at this time of global energy transition. From the aspect of stable energy independence, this is especially important in countries whose energy system is based on fossil fuels, especially coal. This is why decarbonisation of the energy sector is necessary, with a partial and gradual substitution of coal with renewable fuels. However, the use of these fuels is not always possible in existing plants. This refers to the establishment of energy, economic and environmentally acceptable proportion of the co-firing of these fuels, i.e. a mixture of coal and renewable fuels. The success of the establishment of this process is a function of several variables, the dominant of which are the aggregate properties of the fuel, basic and their mixtures, including the ash properties of those fuels. With the motive of contributing to a more successful implementation of the energy transition in the thermal energy sector, and with the aim of obtaining new scientific knowledge about the characteristics of the combustion of lignite and brown coal with different renewable fuels, laboratory research was carried out. In particular, different mixtures of lignite and brown coal, waste woody biomass and Miscanthus as a fast-growing energy crop were subjected to co-firing with variable process conditions. In addition to changing the composition of fuel mixtures, the test regimes included a significant change in process temperature (1250-1450 °C) and the primary measure of staged air supply to the furnace. In these combustion conditions, the emission of undesirable and harmful components into the environment, the efficiency of combustion, and the tendency of the ash from the fuel mixture to possible soiling of the heating surfaces in the furnace were evaluated - an analysis of the characteristics of the ash samples from the reaction zone and samples of the slag and ash to the furnace. Here, in both cases of co-firing, lignite and brown coal with different types of biomass, it was shown that in real conditions it is possible to establish a sustainable primary energy conversion process from fuel with a low unburnt carbon content in the slag (unburnt carbon content, UBC<1%) as well as low CO emission, below 350 mg/mn 3. At the same time, the NOx emission is below 320 mg/mn 3 during the co-firing of lignite mixtures at 1250 °C, and in the case of brown coal mixtures below 740 mg/mn 3 at 1450 °C. In both cases, the net CO2 emission decreases in proportion to the proportion of biomass in the mixture, while the SO2 emission is still high, at a level of up to 2500 mg/mn 3 for lignite mixtures and up to 6400 mg/mn 3 for brown coal mixtures. None of the treated types of biomass, up to the level of applicable content in the mixture with lignite and brown coal, does not worsen the progress of the process from the aspect of possible slagging/fouling heating surfaces in the boiler.

One-sided differentiability: a challenge for computer algebra systems

<abstract><p>Computer Algebra Systems (CASs) are extremely powerful and widely used digital tools. Focusing on differentiation, CASs include a command that computes the derivative of functions in one variable (and also the partial derivative of functions in several variables). We will focus in this article on real-valued functions of one real variable. Since CASs usually compute the derivative of real-valued functions as a whole, the value of the computed derivative at points where the left derivative and the right derivative are different (that we will call conflicting points) should be something like "undefined", although this isn't always the case: the output could strongly differ depending on the chosen CAS. We have analysed and compared in this article how some well-known CASs behave when addressing differentiation at the conflicting points of five different functions chosen by the authors. Finally, the ability for calculating one-sided limits of CASs allows to directly compute the result in these cumbersome cases using the formal definition of one-sided derivative, which we have also analysed and compared for the selected CASs. Regarding teaching, this is an important issue, as it is a topic of Secondary Education and nowadays the use of CASs as an auxiliary digital tool for teaching mathematics is very common.</p></abstract>

On multivariable Plancherel–Pólya inequality and truncation error upper bounds in irregular sampling

In the note is shown that for the d-dimensional Bernstein functions class B_{varvec{sigma },d}^p,, p>0 the Plancherel–Pólya inequality holds with the constant which equals to the product of the constants occuring in the one-dimensional cases. Related truncation error upper bounds are precised in the irregular sampling restoration of functions in several variables.

Some New Results for the Kampé de Fériet Function with an Application

The generalized hypergeometric functions in one and several variables and their natural generalizations appear in many mathematical problems and their applications. The theory of generalized hypergeometric functions in several variables comes from the fact that the solutions of the partial differential equations appearing in a large number of applied problems of mathematical physics have been expressed in terms of such generalized hypergeometric functions. In particular, the Kampé de Fériet function (in two variables) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics, statistics, and mathematical physics. In this context, in a very recent paper, Progri successfully calculated the 2F2 generalized hypergeometric function for a particular set of parameters and expressed the result in terms of the difference between two Kampé de Fériet functions. Inspired by his work, in the present paper, we obtain three results for a terminating 3F2 series of arguments 1 and 2, together with a transformation formula of a 3F2(z) generalized hypergeometric function in terms of the difference between two Kampé de Fériet functions. One application of this result is also provided. The paper concludes with six reduction formulas for the Kampé de Fériet function. Of note, symmetry occurs naturally in the generalized hypergeometric functions pFq and the Kampé de Fériet function involving two variables, which are the two most important functions discussed in this paper.

Trigonometric $$F^{mn}$$-transform of multi-variable functions and its application to the partial differential equations and image processing

In this study, we focus on the extension of the trigonometric F-transform for functions in one variable to: (i) a larger domain, (ii) a higher degree of the $$F^m$$ -transform, and (iii) many-variable functions to improve its approximation properties over the entire domain and especially at its boundaries. In addition, the properties of approximation and convergence of direct and inverse extended and multidimensional trigonometric $$F^{m}$$ -transforms are discussed. Then, direct formulas for partial derivatives of functions of several variables are obtained in terms of trigonometric $$F^{m}$$ -transforms, which are used to solve the Cauchy problem for the transport equation. A new image compression method is proposed and compared with well-established compression methods such as JPEG, JPEG 2000 and their less complex variations JPEG(APDCBT), JPEG(APUBT3), APUBT3-NUP, and JPEG-FT. We have shown that this $$^t\bar{F}^ {11 }$$ -transform image compression method has high accuracy and reasonably low (irreducible) complexity.

Data-driven statistical nonlinearization technique based on information entropy

For a given nonlinear stochastic system, an exact stationary probability density function, which is important for system response prediction and control, is not always available. Owing to the development of various sensors and measurement systems, data acquisition and processing have become increasingly convenient. Inspired by traditional statistical nonlinearization techniques and measured data, a data-driven statistical nonlinearization technique was developed in this study based on information entropy. First, a class of appropriate equivalent systems with exact solutions was selected. Subsequently, the optimal parameters of the equivalent systems were determined based on the minimization of the Shannon information entropy and some statistical moments of the system state from the measured data. The derivation of the optimal parameters of the equivalent systems was then converted into an extremum problem of the entropy function of several variables. Finally, three typical nonlinear stochastic systems were presented to illustrate and validate the effectiveness and accuracy of the proposed procedure. The proposed data-driven statistical nonlinearization technique can be used to provide more comprehensive information on the system response, which is important for system response evaluation and control.

ON ESTIMATES OF M-TERM APPROXIMATIONS ON CLASSES OF FUNCTIONS WITH BOUNDED MIXED DERIVATIVE IN THE LORENTZ SPACE

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space $$L_{q, \tau }(\mathrm {T}^{m})$$ , the class of functions with bounded mixed fractional derivative $$W_{q, \tau }^{\overline{r}}$$ , $$1< q, \tau < \infty$$ , and studies the order of the best M-term approximation of a function $$f \in L_{p, \tau }(\mathrm {T}^{m})$$ by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class $$W_{q, \tau _{1}}^{\overline{r}}$$ in the norm of the space $$L_{p, \tau _{2 }}(\mathrm {T}^{m})$$ for various relations between the parameters $$p, q, \tau _{1}, \tau _{2}$$ .

Generalized Approach to Differentiability

In the traditional approach to differentiability, found in almost all university textbooks, this notion is considered only for interior points of the domain of function or for functions with an open domain. This approach leads to the fact that differentiability has usually been considered only for functions with an open domain in Rn, which severely limits the possibility of applying the potential techniques and tools of differential calculus to a broader class of functions. Although there is a great need for generalization of the notion of differentiability of a function in various problems of mathematical analysis and other mathematical branches, the notion of differentiability of a function at the non-interior points of its domain has almost not been considered or successfully defined. In this paper, we have generalized the differentiability of scalar and vector functions of several variables by defining it at non-interior points of the domain of the function, which include not only boundary points but also all points at which the notion of linearization is meaningful (points admitting nbd rays). This generalization allows applications in all areas where standard differentiability can be applied. With this generalized approach to differentiability, some unexpected phenomena may occur, such as a function discontinuity at a point where a function is differentiable, the non-uniqueness of differentials… However, if one reduces this theory only to points with some special properties (points admitting a linearization space with dimension equal to the dimension of the ambient Euclidean space of the domain and admitting a raylike neighborhood, which includes the interior points of a domain), then all properties and theorems belonging to the known theory of differentiability remain valid in this extended theory. For generalized differentiability, the corresponding calculus (differentiation techniques) is also provided by matrices—representatives of differentials at points. In this calculus the role of partial derivatives (which in general cannot exist for differentiable functions at some points) is taken by directional derivatives.

Precisely controlled polydopamine-mediated antibacterial system: mathematical model of polymerization, prediction of antibacterial capacity, and promotion of wound healing

In this work, the polydopamine (PDA)-mediated antibacterial system is synthesized for in vitro and in vivo antimicrobial activities. To precisely control the surface modification of nanodiamonds (NDs), a mathematical kinetics model of PDA deposition is established, and the conditions of synthesis reaction are discussed including the concentrations of dopamine, the reaction time, and the kinetic constant k 1, which is a function of several variables associated with the reaction temperature, light irradiance, pH value and concentration of dissolved O2 in the solution. A simulation reveals that the deposition of PDA is positively correlated with these factors and will be terminated if dissolved O2 levels are inadequate. Then, the precisely controlled thickness of PDA layers can regulate the formation of AgNPs, resulting in a rise in the intensity of Raman peaks and a predicted antibacterial activity against E. coli in vitro. An optimized antibacterial hydrogel incorporating NDs-PDA/Ag is prepared and characterized by the Fourier transform infrared spectroscopy and field emission scanning electron microscopy. Finally, the antibacterial tests are performed to promote in vivo wound healing, as confirmed by pathological and immunohistochemical-stained tissue sections. This work provides a theoretical foundation for predicting the PDA-assisted surface modification of NDs, resulting in a divinable antibacterial effect and promoting in vivo wound healing.

Hölder–Zygmund classes on smooth curves

We prove that a function in several variables is in the local Zygmund class $\mathcal{Z}^{m,1}$ if and only if its composite with every smooth curve is of class $\mathcal{Z}^{m,1}$. This complements the well-known analogous result for local Hölder–Lipschitz classes $\mathcal{C}^{m,\alpha}$, which we reprove along the way. We demonstrate that these results generalize to mappings between Banach spaces and use them to study the regularity of the superposition operator $f\_\colon g \mapsto f \circ g$ acting on the global Zygmund space $\Lambda\_{m+1}(\mathbb{R}^d)$. We prove that, for all integers $m,k\ge 1$, the map $f\_\colon \Lambda\_{m+1}(\mathbb{R}^d) \to \Lambda\_{m+1}(\mathbb{R}^d)$ is of Lipschitz class $\mathcal{C}^{k-1,1}$ if and only if $f \in \mathcal{Z}^{m+k,1}(\mathbb{R})$.

Singularities of rational inner functions in higher dimensions

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

Fekete–Szegö problem for Bavrin’s functions and close-to-starlike mappings in {mathbb {C}}^{n}

The paper is devoted to the study of a family of complex-valued holomorphic functions and a family of holomorphic mappings in {mathbb {C}}^{n}. More precisely, the article concerns a Bavrin’s family of functions defined on a bounded complete n-circular domain {mathcal {G}} of {mathbb {C}}^{n} and a family of biholomorphic mappings on the Euclidean open unit ball in {mathbb {C}}^{n}. The presented results include some estimates of a combination of the Fréchet differentials at the point z=0, of the first and second order for Bavrin’s functions, also of the second and third order for biholomorphic close-to-starlike mappings in {mathbb {C}}^{n}, respectively. These bounds give a generalization of the Fekete–Szegö coefficients problem for holomorphic functions of a complex variable on the case of holomorphic functions and mappings of several variables.

On superoscillations and supershifts in several variables

The aim of this paper is to study a class of superoscillatory functions in several variables, removing some restrictions on the functions that we introduced in a previous paper. Since the tools that we used with our approach are not common knowledge we will give detailed proof for the case of two variables. The results proved for superoscillatory functions in several variables can be further extended to supershifts in several variables.

Polynomial inequalities, o-minimality and Denjoy–Carleman classes

We study several intimately related problems in the theory of multivariate polynomial inequalities. Firstly, given a map h in certain quasianalytic Denjoy–Carleman classes, we show how to decide whether the image under h of a set satisfying Markov's (resp. Nikolskii's) inequality satisfies Markov's (resp. Nikolskii's) inequality. Our approach relies heavily on the theory of o-minimal structures, particularly on the work of Rolin, Speissegger and Wilkie. Secondly, we establish the relation between Markov's inequality and Nikolskii's inequality (both in general setting and in o-minimal setting). Thirdly, we prove that each compact, definable (in an o-minimal structure) set satisfying Markov's inequality is fat. In particular, this solves in the o-minimal category the well-known problem of nonpluripolarity of sets satisfying Markov's inequality. And lastly, we develop a unified method of polynomial approximation for various classes of smooth functions of several variables.

On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field

AbstractLet $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.

Generalized difference-property for functions of several variables

Generalized difference-property for functions of several variables

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ^*-algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on {mathbb {R}}^{2^n}. One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set.

The level sets of functions with bounded criticalsets and bounded Hess+ complements

"We denote by ${\rm Hess}^+(f)$ the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we prove several properties of real-valued functions of several variables by showing the connectedness of their level sets for sufficiently high levels, under the boundedness assumption on the critical set. In the case of three variables we also prove the convexity of the levels surfaces for sufficiently high levels, under the additional boundedness assumption on the ${\rm Hess}^+$ complement. The selection of the {\em a priori} convex levels, among the connected regular ones, is done through the positivity of the Gauss curvature function which ensure an ovaloidal shape of the levels to be selected. The ovaloidal shape of a level set makes a diffeomorphism out of the associated Gauss map. This outcome Gauss map diffeomorphism is then extended to a smooth homeomorphism which is used afterwards to construct one-parameter families of smooth homeomorphisms of Loewner chain flavor. "

Dirichlet Averages of Generalized Mittag-Leffler Type Function

Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and its various extensions (referred to as Mittag-Leffler type functions) have been researched and applied to a wide range of problems in physics, biology, chemistry, and engineering. In the context of fractional calculus, Mittag-Leffler type functions have been widely studied. Since Carlson established the notion of Dirichlet average and its different variations, these averages have been explored and used in a variety of fields. This paper aims to investigate the Dirichlet and modified Dirichlet averages of the R-function (an extended Mittag-Leffler type function), which are provided in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Principal findings in this article are (possibly) applicable. This article concludes by addressing an open problem.