Functions Of Several Variables Research Articles

Characterizations of Herglotz\u2013Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.

Post-error slowing: Large scale study in an online learning environment for practising mathematics and language.

The ability to monitor and adjust our performance is crucial for adaptive behaviour, a key component of human cognitive control. One widely studied metric of this behaviour is post‐error slowing (PES), the finding that humans tend to slow down their performance after making an error. This study is a first attempt at generalizing the effect of PES to an online adaptive learning environment where children practise mathematics and language skills. This population was of particular interest since the major development of error processing occurs during childhood. Eight million response patterns were collected from 150,000 users aged 5 to 13 years old for 6 months, across 23 different learning activities. PES could be observed in most learning activities and greater PES was associated with greater post‐error accuracy. PES also varied as a function of several variables. At the task level, PES was greater when there was less time pressure, when errors were slower, and in learning activities focusing on mathematical rather than language skills. At the individual level, students who chose the most difficult level to practise and had higher skill ability also showed greater PES. Finally, non‐linear developmental differences in error processing were found, where the PES magnitude increased from 6 to 9‐years‐old and decreased from 9 to 13. This study shows that PES underlies adaptive behaviour in an educational context for primary school students.

Synthetic Series of Electricity Generation through a Photovoltaic System by using Different Panel Temperature Models

The electric energy produced by photovoltaic systems is a function of several variables that present random behavior, such as solar irradiance, ambient temperature, wind speed, and relative humidity. Several models describe the energy produced by these systems, which reflect the various factors that influence the generation and how significant these influences are. In this way, the present work aims to compare some panel temperature models and the electrical energy produced from a photovoltaic system located in the Midwest of Brazil. The applied methodology aims to evaluate the expected averages and standard deviations of the monthly synthetic series of generated energy to identify the main random variables that influence the electric generation. The results indicate the model that best describes the physical system concerning the randomness of the variables involved, the photovoltaic technology, the location, and the measurement period.

Algorithm for Calculating the Global Minimum of a Smooth Function of Several Variables

Every year the interest of theorists and practitioners in optimisation problems is growing, and extreme problems are found in all branches of science. Local optimisation problems are well studied and there are constructive methods for their solution. However, global optimisation problems do not meet the requirements in practice; therefore, the search for the global minimum remains one of the major challenges for computational and applied mathematics. This study discusses the search for the global minimum of multidimensional and multiextremal problems with high precision. Mechanical quadrature formulas, that is, the formulas for approximate integration were applied to calculate the integrals. Of all the approximate integration formulas, the Sobolev lattice cubature formulas with a regular boundary layer were chosen. In multidimensional examples, the Sobolev formulas are optimal. Computational experiments were carried out in the most popular C++ programming language. Based on the computational experiments, a new algorithm was proposed. In three-dimensional space, the calculations of the global minimum have been described using specific examples. Computational experiments show that the proposed algorithm works for multiextremal problems with the same amount of time as for convex ones.

Hilbert’s Tenth Problem for Complex Meromorphic Functions in Several Variables

Abstract We prove that Hilbert’s Tenth Problem for meromorphic functions and for entire functions in several variables is unsolvable over the language of rings, together with constant symbols for two of the variables and a predicate for a place. This is the first result in the literature, which proves unsolvability of a diophantine problem for the ring of complex analytic functions in a number of variables. The proof rests upon an interplay between analytic geometry, analysis, number theory, and logic.

Pathway Fractional Integral Operator on Some τ-extensions of Lauricella Functions of Several Variables

Pathway Fractional Integral Operator on Some τ-extensions of Lauricella Functions of Several Variables

On additive functions with additional derivation properties

The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived which shows that if a function $f$ satisfies an addition theorem whose determining operation is derivable with respect to an additive function $d$, then the function $f$ is itself derivable with respect to $d$. As an application of this approach, new proof of a generalization of a recent result of Maksa is obtained. We also extend the result of Nishiyama and Horinouchi and formulate two open problems.

On exponential type P-functions

In this paper, we introduce and study the concept of exponential type P-function and establish Hermite-Hadamard's inequalities for this type of functions. In addition, we obtain some new Hermite-Hadamard type inequalities for functions whose first derivative in absolute value is exponential type P-function by using Hölder and power-mean integral inequalities. We also extend our initial results to functions of several variables. Next, we point out some applications of our results to give estimates for the approximation error of the integral the function in the trapezoidal formula and for some inequalities related to special means of real numbers.

Some inequalities involving two generalized beta functions in n variables

The beta and gamma functions have recently seen several developments and various extensions because of their nice properties and interesting applications. The contribution of this paper falls within this framework. After introducing a generalized gamma function and two generalized beta functions in several variables, we investigate some inequalities involving these generalized functions.

Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of Homeo([0,1]) has length 2. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of Homeo([0,1]) have length 2. We answer this question in the affirmative.We call f∈Homeo([0,1]d) singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of f∈Homeo([0,1]) has length 2 iff f is singular iff f is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem.We show that for d≥2 the graph of the generic element of Homeo([0,1]d) has infinite d-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of Homeo([0,1]d) with infinite d-dimensional Hausdorff measure is not Haar null. We show that for d≥2 the generic element of Homeo([0,1]d) is singular but not strongly singular. We also show that for d≥2 almost every element of Homeo([0,1]d) is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null).Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.

“Smooth rigidity” and Remez-type inequalities

If a \((d+1)\)-smooth function f(x) on \([-1,1]\), with \(\mathrm{max\,}_{[-1,1]}|f(x)|\ge 1,\) has \(d+1\) or more distinct zeroes on \([-1,1]\), then \(\mathrm{max\,}_{[-1,1]}|f^{(d+1)}(x)|\ge 2^{-d-1}(d+1)!\). This follows from the polynomial interpolation of f at its zeroes, with Lagrange’s remainder formula. This is one of the simplest examples of what we call “smooth rigidity”: certain geometric properties of zero sets of smooth functions f imply explicit lower bounds on the high-order derivatives of f. In dimensions greater than one, the powerful one-dimension tools such as Lagrange’s remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables (Yomdin in Proc AMS 90(4):538–542, 1984). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of Yomdin (1984), including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a pure multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that the smooth rigidity of a zeroes set Z is approximately the reciprocal Remez constant of Z.

Some results of Fekete\u2013Szeg\xf6 type for Bavrin\u2019s families of holomorphic functions in {mathbb {C}}^{n}

In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a (j, k)-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in {mathbb {C}}^{n}.

On some generalizations of the beta function in several variables

Intensive studies aiming to extend the gamma and beta functions and to establish some properties for these extensions have been recently carried out. In this paper, we first introduce a generalized gamma function in n variables. Afterwards, two generalized beta functions in several variables are introduced and their properties are discussed. Among others, we investigate recurrence relationships, Mellin transform properties, and partial differential equations involving these generalized functions. At the end, some results about partial derivatives of these extended functions are presented as well.

Non-commutative integrability, exact solvability and the Hamilton–Jacobi theory

The non-commutative integrability (NCI) is a property fulfilled by some Hamiltonian systems that ensures, among other things, the exact solvability of their corresponding equations of motion. The latter means that an “explicit formula” for the trajectories of these systems can be constructed. Such a construction rests mainly on the so-called Lie theorem on integrability by quadratures. It is worth mentioning that, in the context of Hamiltonian systems, the NCI has been for around 40 years, essentially, the unique criterium for exact solvability expressed in the terms of first integrals (containing the usual Liouville–Arnold integrability criterium as a particular case). Concretely, a Hamiltonian system with n degrees of freedom is said to be non-commutative integrable if we know a set of independent first integrals \(F_{1},\ldots ,F_{l}\) such that: the kernel of the \(l\times l\) matrix with coefficients \(\left\{ F_{i},F_{j}\right\} \), where \(\left\{ \cdot ,\cdot \right\} \) denotes the canonical Poisson bracket, has dimension \(2n-l\) (isotropy); and each bracket \(\left\{ F_{i},F_{j}\right\} \) is functionally dependent on \(F_{1},\ldots ,F_{l}\) (closure). In this paper, we develop two procedures for constructing the trajectories of a Hamiltonian system which only require isotropic first integrals (closure condition is not needed). One of them is based on an extended version of the geometric Hamilton–Jacobi theory, and does not rely on the above mentioned Lie’s theorem. We do all that in the language of functions of several variables.

Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables

Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables

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.