Series Expansion Of Functions Research Articles

METHOD FOR GENERATING A DATA SET FOR TRAINING A NEURAL NETWORK IN A TRANSPORT CONVEYOR MODEL

The object of research is a stochastic input flow of material coming in the input of a conveyor-type transport system. Subject of research is the development of a method for generating values of the stochastic input material flow of transport conveyor to form a training data set for neural network models of the transport conveyor. The goal of the research is to develop a method for generating random values to construct implementations of the input material flow of a transport conveyor that have specified statistical characteristics calculated based on the results of previously performed experimental measurements. The article proposes a method for generating a data set for training a neural network for a model of a branched, extended transport conveyor. A method has been developed for constructing implementations of the stochastic input flow of material of a transport conveyor. Dimensionless parameters are introduced to determine similarity criteria for input material flows. The stochastic input material flow is presented as a series expansion in coordinate functions. To form statistical characteristics, a material flow implementation based on the results of experimental measurements is used. As a zero approximation for expansion coefficients, that are random variables, the normal distribution law of a random variable is used. Conclusion. It is shown that with an increase in the time interval for the implementation of the input material flow, the correlation function of the generated implementation steadily tends to the theoretically determined correlation function. The length of the time interval for the generated implementation of the input material flow was estimated.

Series expansion of some elementary functions without using mathematical analysis

We present a method to compute the power series expansions of e x , ln ⁡ ( 1 + x ) , sin ⁡ x , and cos ⁡ x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of mathematical induction. The Newton binomial formula is obtained using the expansions of the exponent and the logarithm in their power series. We build upon the approach outlined in a paper by L. P. Mironenko and O. A. Rubtsova (Approximations of Some Functions by Polynomials and the Method of Undefined Coefficients, Scientific Papers of Donetsk National Technical University. Series: Computing and Automation, No. 2 (25), p. 128–135, 2013) and complement their approach. This approach does not depend on the application of Taylor's theorem to the expansion of functions. Consequently, these expansions can be integrated into the educational process before students become acquainted with the concept of derivatives. It simplifies the study of the theory of limits, especially in the computation of standard limits such as lim x → 0 sin ⁡ ( x ) x , lim x → 0 1 − cos ⁡ ( x ) x 2 , and lim x → 0 ( 1 + x ) 1 / x .

Analytical Model of Point Spread Function under Defocused Degradation in Diffraction-Limited Systems: Confluent Hypergeometric Function

Analytical Model of Point Spread Function under Defocused Degradation in Diffraction-Limited Systems: Confluent Hypergeometric Function

Modeling Nonlinear Physical Systems in a Real-Time Operating System Environment for Control and Cyber Threat Mitigation

Nonlinearity control and interconnection between physical systems are essential for many contemporary systems. Cyber Physical systems came about because of the growing number of security concerns with network control systems. Nonlinear physical systems are the primary subject of this study's investigation into control and cyber threat mitigation. Nonlinear physical systems are the focus of this study, which also delves into their control, modelling, and real-time implementation. Here, physical systems are defined as two kinds of benchmark nonlinear robotic systems: the Segway transporter and the cruise control system. Many see the Segway as an unstable and nonlinear physical system. By factoring in kinetic and potential energy, Lagrangian dynamics has allowed for the derivation of equations of motion. Next, a state-space model of the system is obtained by linearizing the nonlinear differential equations using Taylor series expansion of functions. This model is then transformed into two transfer functions, one for the tilt angle and one for the yaw angle. The Model Predictive Controller (MPC), GA Tuned PID, and PID are the building blocks of MATLAB's effective simulation study of nonlinear control.

Solution of Laplace Equation by Modified Differential Transform Method

In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

MultiHypExp: A Mathematica package for expanding multivariate hypergeometric functions in terms of multiple polylogarithms

We present the Mathematica package MultiHypExp that allows for the expansion of multivariate hypergeometric functions (MHFs), especially those likely to appear as solutions of multi-loop, multi-scale Feynman integrals, in the dimensional regularization parameter. The series expansion of MHFs can be carried out around integer values of parameters to express the series coefficients in terms of multiple polylogarithms. The package uses a modified version of the algorithm prescribed in [1]. In the present work, we relate a given MHF to a Taylor series expandable MHF by a differential operator. The Taylor expansion of the latter MHF is found by first finding the associated partial differential equations (PDEs) from its series representation. We then bring the PDEs to the Pfaffian system and further to the canonical form, and solve them order by order in the expansion parameter using appropriate boundary conditions. The Taylor expansion so obtained and the differential operators are used to find the series expansion of the given MHF. We provide examples to demonstrate the algorithm and to describe the usage of the package, which can be found in this url. Program summaryProgram Title:MultiHypExp.wl version 1.0CPC Library link to program files:https://doi.org/10.17632/zxwkx38y7b.1Developer's repository link:https://github.com/souvik5151/MultiHypExpLicensing provisions: GNU General Public License 3Programming language: Wolfram Mathematica version 11.3 or higherExternal routines/libraries: HolonomicFunctions, CANONICA, HYPERDIRE, PolyLogTools, HPLNature of problem: Obtaining the series expansion of multivariate hypergeometric functions around integer values of Pochhammer parameters in terms of multiple polylogarithmsSolution method: Mathematica implementation of a modified version of the algorithm prescribed in Nucl. Phys. B 989 (2023) 116145.

Accurate Thermodynamic Properties of Ideal Bosons in a Highly Anisotropic 2D Harmonic Potential.

One can derive an analytic result for the issue of Bose-Einstein condensation (BEC) in anisotropic 2D harmonic traps. We find that the number of uncondensed bosons is represented by an analytic function, which includes a series expansion of q-digamma functions in mathematics. One can utilize this analytic result to evaluate various thermodynamic functions of ideal bosons in 2D anisotropic harmonic traps. The first major discovery is that the internal energy of a finite number of ideal bosons is a monotonically increasing function of anisotropy parameter p. The second major discovery is that, when p≥0.5, the changing with temperature of the heat capacity of a finite number of ideal bosons possesses the maximum value, which happens at critical temperature Tc. The third major discovery is that, when 0.1≤p<0.5, the changing with temperature of the heat capacity of a finite number of ideal bosons possesses an inflection point, but when p<0.1, the inflection point disappears. The fourth major discovery is that, in the thermodynamic limit, at Tc and when p≥0.5, the heat capacity at constant number reveals a cusp singularity, which resembles the λ-transition of liquid helium-4. The fifth major discovery is that, in comparison to 2D isotropic harmonic traps (p=1), the singular peak of the specific heat becomes very gentle when p is lowered.

Option Pricing with Fractional Stochastic Volatilities and Jumps

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&amp;P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.

Harmonic Analysis of Non-sinusoidal Currents During Heating of Transformers

The results of the analysis of currents in the transformer windings when heated by low-frequency short-circuit currents are presented, for the description of which the Fourier series expansion of piecewise exponential functions is used. This makes it possible to mathematically model the heating process and solve the issues of the ratio of the power released in the electric and magnetic circuits of the transformer.

Estimates of Coefficients for Bi-Univalent Functions in the Subclass H_∑ (n,γ,φ)

Considering that finding the bounds for the coefficients of the Taylor-Maclaurin series expansion of bi-univalent functions is one of the important subjects in geometric function theory that has attracted the attention of many researchers in the last few decades, we also take a step in this direction. Finding such bounds is the main focus or, more clearly, the main problem of our work. In this article, we study the subclass H_∑ (n,γ,φ) of bi-univalent functions which is defined in the open unit disk D. Furthermore, we obtained the upper bounds estimates for the first coefficients |a_2 | and |a_3 | of the functions in this category by using subordination method. From the main result of the article (Theorem 2.1), special cases have been derived that improve some the results of previous articles.

On the use of L-functionals in regression models

Abstract In this article, we survey and unify a large class or L L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample, Biometrika 56 (1969), no. 3, 641–650.), L L -skewness, and L L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.

Adaptive Fuzzy Control of a Four-Wheeled Mobile Robot Subject to Wheel Slip

In this paper, the adaptive fuzzy tracking control of a four-wheeled mobile robot subject to wheels slip is considered. We proposed an adaptive scheme in that fuzzy logic approximators are used to approximate the unknown system functions in designing the adaptive tracking control of a mobile robot. Fuzzy systems are expressed as a series expansion of basis functions, to adaptively compensate for the mobile robot nonlinearities. The proposed control system works online, parameter adaptation is realized in every discrete step of the control process, and a preliminary learning phase of fuzzy system parameters is not required. The stability of the algorithm is established in the Lyapunov sense, with tracking errors converging to a neighborhood of zero. Simulation results illustrate the effectiveness of the approach.

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

Non-Convex Particle-in-Cell Model for the Mathematical Study of the Microscopic Blood Flow

The field of fluid mechanics was further explored through the use of a particle-in-cell model for the mathematical study of the Stokes axisymmetric flow through a swarm of erythrocytes in a small vessel. The erythrocytes were modeled as inverted prolate spheroids encompassed by a fluid fictitious envelope. The fourth order partial differential equation governing the flow was completed with Happel-type boundary conditions which dictate no fluid slip on the inverted spheroid and a shear stress free non-permeable fictitious boundary. Through innovative means, such as the Kelvin inversion method and the R-semiseparation technique, a stream function was obtained as series expansion of Gegenbauer functions of the first and the second kinds of even order. Based on this, analytical expressions of meaningful hydrodynamic quantities, such as the velocity and the pressure field, were calculated and depicted in informative graphs. Using the first term of the stream function, the drag force exerted on the erythrocyte and the drag coefficient were calculated relative to the solid volume fraction of the cell. The results of the present research can be used for the further investigation of particle–fluid interactions.

Pre-asymptotic dispersion of active particles through a vertical pipe: the origin of hydrodynamic focusing

When motile algal cells are exposed to gyrotactic torques, their swimming directions are guided to form radial accumulation, well known as hydrodynamic focusing. The origin of hydrodynamic focusing from the effects of active swimming, ambient flow and particle anisotropy is elucidated in the present study on the pre-asymptotic dispersion of active particles through a vertical pipe. With an extension of the Galerkin method to pipe flows, time-dependent solutions directly from the Smoluchowski equation in the position and orientation space are derived by series expansions of spherical harmonics and Bessel functions. Ballistic and diffusive scaling laws are examined with the predominance of self-propelled swimming, and computation is validated against an explicit benchmark solution and Lagrangian particle simulation. In the limit of extreme shear, the competitive roles of shear dispersion and Brownian rotation are reflected concretely in the pre-asymptotic phase of hydrodynamic focusing. For flows with various shear strengths, a concentration peak in near-wall regions with a smooth transition to hydrodynamic focusing is illustrated with richer phenomena in upwelling and downwelling flows. A newly observed regime through a vertical pipe, named transient effective trapping, is revealed as a transitional mode towards hydrodynamic focusing. The pre-asymptotic approach to hydrodynamic focusing is elaborated intensively through extensive solutions of concentration moments and macroscopic transport coefficients characterised by swimming and flow Péclet numbers. The unique findings for the origin of hydrodynamic focusing could provide insight into related micro-algae reactor technology and contribute to flow control and biomass transfer in confined environments.

ϵ-expansion of multivariable hypergeometric functions appearing in Feynman integral calculus

We present a new methodology, suitable for implementation on computer, to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $\epsilon$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric functions in one, two and three variables which are typical of Feynman integral calculus.

Inertial dynamics and equilibrium correlation functions of magnetization at short times

The method of moments is developed and employed to analyze the equilibrium correlation functions of the magnetization of ferromagnetic nanoparticles in the case of inertial magnetization dynamics. The method is based on the Taylor series expansion of the correlation functions and the estimation of the expansion coefficients. This method significantly reduces the complexity of analysis of equilibrium correlation functions. Analytical expressions are derived for the first three coefficients for the longitudinal and transverse correlation functions for the uniaxial magnetocrystalline anisotropy of ferromagnetic nanoparticles with a longitudinal magnetic field. The limiting cases of very strong and negligibly weak external longitudinal fields are considered. The Gordon sum rule for inertial magnetization dynamics is discussed. In addition, we show that finite analytic series can be used as a simple and satisfactory approximation for the numerical calculation of correlation functions at short times.

An algebraic approach for the Dunkl–Killingbeck problem from the bi-confluent Heun equation

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su[Formula: see text] Lie algebra in a specific way.

The sequential power flow in complex plane for solving the VSC-MTDC hybrid AC/DC transmission grids

This paper presents new methodologies and modeling aimed at studying interwoven AC and DC subsystems of a power system in the complex plane. It is well known that the development of any former power flow application in real domain addressed to embedding FACTS devices is preceded by an arduous algebra task. To overcome this difficulty, a novel power flow solution method is proposed in this work. Thanks to the Wirtinger calculus, a Newton–Raphson method based on Taylor series expansions of nonlinear functions of complex variables and their complex conjugates is developed. Thus, the voltage-source-converter multi-terminal direct current is formulated in complex plane without any loss of accurateness and is termed as VSC-MTDC. Its performance is assessed through IEEE-test systems interconnected across a DC network prone to several scenarios, e.g., topology, loading magnitude and interchanging of active power.

Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind

Abstract In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.