Functions Of Several Variables Research Articles

Оптимальное управление группой беспилотных планирующих летательных аппаратов

The paper focuses on an algorithm for controlling a group of unmanned gliding aircraft which are notable for the absence of a propulsion system. The flight is accomplished by using the full mechanical energy reserve received by the unmanned gliding aircraft at the time of the air launch. The task of a group flight with each gliding device being autonomous is currently relevant. Developing a group control algorithm and forming the trajectory of each unmanned gliding aircraft make it possible to solve a wide range of practical problems. This paper states the problem implying the maximization of the flight range with boundary conditions set at the ends of the trajectory. The trajectory of each of the group unmanned gliding aircraft is formed by specifying a reference function for each phase coordinate (xg, yg, zg). The flight range of the group is maximized as a result of solving a boundary value problem by the Ritz --- Galerkin method based on optimization of a function of several variables. In contrast to classical optimization methods, it allows solving the considered optimization problem with sufficient accuracy in a practical sense, without requiring large computational resources for its implementation. This method was used to optimize flight trajectories; however, in this work, it is applied with a number of significant differences both in terms of the problem statement and in terms of requirements for the dynamic capabilities of the aircraft

A (pedagogical) note on the log-linearization of functions of several variables

Dynamic stochastic general equilibrium (DSGE) models are analytically intractable and numerical methods must be used to approximate a solution. A key input shared by many solution methods is log-linearization. While the basics of the procedure have been extensively documented, applying the methodology to complicated functions of model variables remains uncharted territory, often resulting in cumbersome and error-prone calculations. This paper offers a procedure – the log-linear product approach – that automates and simplifies this task, as I show with a full working example. The procedure relies on the basic fact that the product of second order terms is zero when dealing with a linear expansion.

Comparison of Classical and Robust Estimates of Threshold Auto-regression Parameters

The exponential auto-regression model is a discrete analog of the second-order nonlinear differential equations of the type of Duffing and van der Pol oscillators. It is used to describe nonlinear stochastic processes with discrete time, such as vehicle vibrations, ship roll, electrical signals in the cerebral cortex. When applying the model in practice, one of the important tasks is its identification, in particular, an estimate of the model parameters from observations of the stochastic process it described. A traditional technique to estimate autoregressive parameters is the nonlinear least squares method. Its disadvantage is high sensitivity to the measurement errors of the process observed. The M-estimate method largely has no such a drawback. The M-estimates are based on the minimization procedure of a non-convex function of several variables. The paper studies the effectiveness of several well-known minimization methods to find the M-estimates of the parameters of an exponential autoregressive model. The paper demonstrates that the sequential quadratic programming algorithm, the active set algorithm, and the interior-point algorithm have shown the best and approximately the same accuracy. The quasi-Newton algorithm is inferior to them in accuracy a little bit, but is not inferior in time. These algorithms had approximately the same speed and were one and a half times faster than the Nelder-Mead algorithm and 14 times faster than the genetic algorithm. The Nelder-Mead algorithm and the genetic algorithm have shown the worst accuracy. It was found that all the algorithms are sensitive to initial conditions. The estimate of parameters, on which the autoregressive equation linearly depends, is by an order of magnitude more accurate than that of the parameter on which the auto-regression equation depends in a nonlinear way.

Lie Group Analysis of the Time-delayed Inviscid Burgers' Equation

In this paper, we discuss group analysis of rst-order delay partial di erential equations and use it to obtain symmetries of the Invis- cid Burgers' equation with delay, its kernel and extensions of the kernel. We obtain a Lie type invariance condition for rst-order delay partial di erential equations by using Taylor's theorem for a function of several variables. We obtain the symmetries admitted by this delay partial di er- ential equation. Further, we obtain representations of analytic solutions and the reduced equations from the symmetries.

Biomonitoring selenium, mercury, and selenium:mercury molar ratios in selected species in Northeastern US estuaries: risk to biota and humans.

The mutual mitigation of selenium and mercury toxicity is particularly interesting, especially for humans. Mercury is widely recognized as a pantoxic element; all forms are toxic to all organisms. Less well known is that selenium in excess is toxic as well. The high affinity between these elements influences their bioavailability and toxicity. In this paper, we use selected species from Barnegat and Delaware Bays in New Jersey to examine variations in levels of selenium and mercury, and selenium:mercury molar ratios between and within species. We report on species ranging from horseshoe crab eggs (Limulus polyphemus), a keystone species of the food chain, to several fish species, to fish-eating birds. Sampling began in the 1970s for some species and in the 1990s for others. We found no clear time trends in mercury levels in horseshoe crab eggs, but selenium levels declined at first, then remained steady after the mid1990s. Concentrations of mercury and selenium in blood of migrant shorebirds directly reflected levels in horseshoe crab eggs (their food at stopover). Levels of mercury in eggs of common terns (Sterna hirundo) varied over time, and may have declined slightly since the mid2000s; selenium levels also varied temporally, and declined somewhat. There were variations in mercury and selenium levels in commercial, recreational, and subsistence fish as a function of species, season, and size (a surrogate for age). Selenium:mercury molar ratios also varied as a function of species, year, season, and size in fish. While mercury levels increased with size within individual fish species, selenium levels remained the same or declined. Thus selenium:mercury molar ratios declined with size in fish, reducing the potential of selenium to ameliorate mercury toxicity in consumers. Mercury levels in fish examined were higher in early summer and late fall, and lower in the summer, while selenium stayed relatively similar; thus selenium:mercury molar ratios were lower in early summer and late fall than in midsummer. We discuss the importance of temporal trends in biomonitoring projects, variations in levels of mercury, selenium, and the molar ratios as a function of several variables, and the influence of these on risks to predators and humans eating the fish, and the eggs of gulls, terns. Our data suggests that variability limits the utility of the selenium:mercury molar ratio for fish consumption advisories and for risk management.

Branched Continued Fraction Expansions of Horn’s Hypergeometric Function H3 Ratios

The paper deals with the problem of construction and investigation of branched continued fraction expansions of special functions of several variables. We give some recurrence relations of Horn hypergeometric functions H3. By these relations the branched continued fraction expansions of Horn’s hypergeometric function H3 ratios have been constructed. We have established some convergence criteria for the above-mentioned branched continued fractions with elements in R2 and C2. In addition, it is proved that the branched continued fraction expansions converges to the functions which are an analytic continuation of the above-mentioned ratios in some domain (here domain is an open connected set). Application for some system of partial differential equations is considered.

Hypergeometric Functions of Several Variables and Evaluation of Error Probability in Fading Multichannel System

A method is proposed of evaluation of symbol and/or bit error probabilities for coherent diversity receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables.

Gypsum polymer materials in construction

The modern level of technological development involves the use of traditional materials modified with additives of various types and functional purposes, as well as composite materials allowing to obtain a product with improved properties. Expanding the area of application of products based on gypsum for facade systems involves the creation of weather-resistant, and, first of all, waterproof materials based on gypsum polymers. The purpose of the experiment, the results of which are presented in the article, was to assess the possibility of using polycondensation polymers as a component of gypsum polymer, to model the properties of the material and to evaluate its characteristics as a result of climatic and humidity influences. The modeling and optimization of gypsum polymer properties were based on statistical methods as well as methods of mathematical analysis of functions of several variables. The assessment of the water resistance of gypsum polymer samples was carried out under test conditions in an open reservoir with an almost unlimited reaction capacity of the medium. The weather resistance was checked according to the results of tests in a climatic chamber. Experiments have shown that the strength of samples with 20% modified melamine-formaldehyde resin in compression and in bending for 80 days of storage in air increases by 30% and 25%, respectively. The compressive strength is 60 MPa, and the flexural strength is 12 MPa. Gypsum polymer has high frost resistance up to 150 cycles of alternate freezing and thawing. The result of the research was the confirmation of the possibility of using polycondensation resins and the foundations of the method for selecting the composition of the gypsum polymer were developed. The results obtained can be used in the development of the technology of gypsum polymer products, and, in particular, piece products (building cladding tiles).

An algorithm determining the set of lengths of polynomial cycles in $Z_K^N$

We give a finitary procedure of finding the set of lengths of cycles for polynomial mappings in several variables over discrete valuation domains. As a consequence, we obtain a procedure of determining the set of cycle-lengths of $N\ge 2$ variables over $

On an equation characterizing multi-Jensen-quartic mappings and its stability

In this paper, we define and investigate the mappings of several variables which are quartic in each variable. We show that such mappings can be unified as an equation, say the multi-quartic functional equation. We also establish the Hyers-Ulam stability of a such functional equation by a fixed point theorem in non-Archimedean normed spaces. Moreover, we generalize some known stability and hyperstability results.

Students' Conception on Multiple Integrals of a Function of Several Variables: A Case of Adama Science and Technology University

The study was conducted at Adama Science and Technology University to investigate students' conceptual understanding in learning Applied Mathematics II in general and multiple integrals in particular. A case study research design was employed on a Mechanical engineering group one student. This group was randomly selected through simple random sampling techniques. The number of students involved in this study was 50. Qualitative data were collected through reasoning part of the multiple choice items of the pre-test and interview items of the post-test were analyzed using APOS analysis based on proposed genetic decompositions. These tools were intended to investigate the conceptual understanding of students and the way they justify their answers. The study shows that the majority of the students' conception of multiple integrals could be categorized under action level whereas a few students were categorized under process conception. Students' conceptual understanding on multiple integrals of a function of two variables is a straight forward as that of a function of a single variable, which reveals that students have not developed a new schema for a function of two variables, as different from a function of a single variable. The majority of the respondents was poor at extending previous concepts to the new concept and had difficulty to represent multiple integrals using graph. Thus; the researchers recommended the utilization of an appropriate instructional approach in order to scaffold students' conceptual understanding of multiple integrals.

Numerical calculation of multidimensional integrals depended on input information about the function in mathematical modelling of technical and economic processes

The object of study is a numerical integration of the functions of several variables using new information operators. The cubature formulas of the approximate calculation of double and triple integrals for different information operators are presented. Information about function is given not only by the values of a function in nodes, but also as a set of traces of a function on planes or as a set of traces of a function on lines. Attention is paid to the application of such approximate calculation of integrals in various fields of science.

Аппарат производных чисел и возможности применения

The article develops the apparatus of derived numbers, the use of which allows one to study the behavior of functions of several variables without requiring their differentiability. In addition, the application of this apparatus to the problem of integrability of the field of planes tangent to a differential manifold allows one to generalize the Frobenius theorem and expand its scope by weakening the restrictions on the degree of smoothness of the differential manifolds under consideration. Conditions and criteria for using the apparatus of partial and external derivatives of numbers are proposed.

Observer-based adaptive control for HIV infection therapy using the Baskakov operator

This paper presents an adaptive observer-based robust controller design for HIV infection therapy using the Baskakov operators as a Universal approximator. The purpose of the treatment is to reduce, asymptotically, the number of infected CD4+T cells to zero using the assumption of accessibility to the total number of CD4+T cells. The lumped uncertainties including un-modeled dynamics and parameter uncertainties are modeled with the Baskakov operators. All of the aforementioned uncertainties are assumed as a function of time instead a function of several variables. The Baskakov operators’ coefficients are then adaptively adjusted. It must be noted that Baskakov operators’ have been used for function approximation usually in an offline manner in mathematics. However, the contribution of this paper is using Baskakov operators’ in an online manner for an engineering problem. Also, considering the nonlinearities originated from antiretroviral drugs' dosage limitation is another important novelty of this paper which has not been considered in previous related works. Uniformly ultimately boundedness of the observer estimation error and the systems’ states are guaranteed through Lyapunov stability concept. The performance is evaluated analytically to confirm that the observer transient response performs satisfactorily. Simulation results and a comparison with observer/controller based on radial basis functions neural network (RBFNN) verify that the proposed Baskakov operator-based observer/controller achieves reasonable performance for reducing the number of infected CD4+ T cells even in the presence of un-modeled dynamics and parametric uncertainties.

On the convergence of multidimensional S-fractions with independent variables

The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

Vector-valued holomorphic functions in several variables

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

PREDICTIVE CONTROL METHODS IN TASKS OF SEARCHING SADDLE POINTS

The article presents new methods for searching critical points of a function of several variables, including saddle points. Such problems are found in various fields of theoretical and practical science, for example, saddle-point construction lens design, machine and deep learning, problems of convex optimization and nonlinear programming (necessary and sufficient conditions for the solution are formulated using saddle points of the Lagrange function and proved in the Kuhn-Tucker theorem. When training neural networks, it is necessary to repeat the training process on large clusters and check the network's trainability at different loss functions and different network depth. Which means that thousands of new calculations are run, where each time the loss function is optimized on large amounts of data. So any acceleration in the process of finding critical points is a major advantage and saves computing resources. Many modern methods of searching saddle points are based on calculating the Hessian matrix, inverting this matrix, the scalar product of the gradient vector and the current vector, finding the full Lagrangian, etc. However, all these operations are computationally “expensive” and it would make sense to bypass such complex calculations. The idea of modifying the standard gradient methods used in the article is to apply fixed-point search schemes for nonlinear discrete dynamical systems for gradient descent problems. It is assumed that these fixed points correspond to unstable equilibrium positions, and there are large units among the multipliers of each equilibrium position. The averaged predictive control methods are used. Results of numerical modeling and visualization are presented in the form of two tables, which indicate basins of attraction for each critical point in each scheme, and statistical data by the convergence rates.

Estimates of the best approximations of the functions of the Nikol’skii–Besov class in the generalized space of Lorentz

In this paper, we consider the generalized Lorentz space of periodic functions of several variables and the Nikol’skii–Besov space of functions. The article establishes a sufficient condition for a function to belong from one generalized Lorentz space to another space in terms of the difference of the partial sums of the Fourier series of a given function. Exact in order estimates of the best approximation by trigonometric polynomials of functions of the Nikol’skii–Besov class are obtained.

Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

Application of Hypergeometric Functions of Several Variables in the Mathematical Theory of Communication: Evaluation of Error Probability in Fading Singlechannel System

A method is proposed of evaluation of symbol and/or bit error probabilities for coherent receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables.