Combination Of Elementary Functions Research Articles

Multiobjective evolutionary discovery of equation-based analytical models for dynamical systems

In this article, an approach to modeling dynamical systems in case of unknown governing physical laws has been introduced. The systems of differential equations obtained by means of a data-driven algorithm are taken as the desired models. In this case, the problem of predicting the state of the process is solved by integrating the resulting differential equations. In contrast to classical data-driven approaches to dynamical systems representation, based on the general machine learning methods, the proposed approach is based on the principles, comparable to the analytical equation-based modeling. Models in forms of systems of differential equations, composed as combinations of elementary functions and operation with the structure, were determined by adapted multi-objective evolutionary optimization algorithm. Time-series describing the state of each element of the dynamic system are used as input data for the algorithm. To ensure the correct operation of the algorithm on data characterizing real-world processes, noise reduction mechanisms are introduced in the algorithm. The use of multicriteria optimization, held in the space of complexity and quality criteria for individual equations of the differential equation system, makes it possible to improve the diversity of proposed candidate solutions and, therefore, to improve the convergence of the algorithm to a model that best represents the dynamics of the process. The output of the algorithm is a set of Pareto-optimal solutions of the optimization problem where each individual of the set corresponds to one system of differential equations. In the course of the work, a library of data-driven modeling of dynamic systems based on differential equation systems was created. The behavior of the algorithm was studied on a synthetic validation dataset describing the state of the hunter-prey dynamic system given by the Lotka-Volterra equations. Finally, a toolset based on the solution of the generated equations was integrated into the algorithm for predicting future system states. The method is applicable to data-driven modeling of arbitrary dynamical systems (e.g. hydrometeorological systems) in cases where the processes can be described using differential equations. Models generated by the algorithm can be used as components of more complex composite models, or in an ensemble of methods as an interpretable component.

Sharp inequalities for the complete elliptic integrals of the first and second kinds

By studying the monotonicity properties of K(r), E(r) and some combinations of elementary functions and special functions, some new inequalities for the complete elliptic integrals of the first and second kinds are established. where K(r) =??/2 0 (1 ? r2 sin2?)?1/2d?, E(r)=??/2 0 (1 ? r2 sin2?)1/2d? denote the complete elliptic integrals of the first and second kind, r ? (0,1).

Electromagnetic characterization of tuneable graphene‐strips‐on‐substrate metasurface over entire THz range: Analytical regularization and natural‐mode resonance interplay

Scattering and absorption of the H-polarized plane wave by the infinite grating of flat graphene strips are considered in the environment met most frequently—on or at the surface of a dielectric-slab substrate. The full-wave meshless code is based on the analytical semi-inversion using the Riemann–Hilbert problem solution. This leads to a Fredholm second-kind matrix equation for the Floquet harmonic amplitudes that guarantees code convergence and provides easy control of computational error, which can be reduced to machine precision. The matrix elements are combinations of elementary functions, and therefore, the code is accurate and quite economical. This enables computation of the reflectance, transmittance, and absorbance as a function of the frequency in the wide band from static case to 10 THz. Numerical results show that such a metasurface with micrometre-sized strips is a composite periodic open resonator. It is highly frequency-selective thanks to the interplay of three types of natural modes—low-Q slab, moderate-Q plasmon strip, and ultra-high-Q lattice—that do not exist in the absence of the substrate. Varying the chemical potential of graphene, one can manipulate the electromagnetic characteristics of the metasurface at a fixed frequency from almost total transmission to almost total reflection.

Dynamical structures of exact soliton solutions to Burgers’ equation via the bilinear approach

The Cole–Hopf transformation is used to search for exact multiple wave solutions of Burgers’ equation. Particularly, solitary wave solutions and their interaction solutions are explicitly given, and a fission phenomenon is found, which is tough to be found by other traditional methods. Different solutions are presented with different combinations of elementary functions and a necessary condition for fission is proposed. Based on the bilinear formalism and with the aid of symbolic computation, we also determine two classes of rational solutions, rogue wave solutions and rogue–kink wave solutions, using various ansätze. To exhibit dynamics, we make diverse graphical analyses for the presented solutions, which show that the solutions are reliable in both mathematical physics and engineering.

Integrable Dynamical Systems with Dissipation on Tangent Bundles of 2D and 3D Manifolds

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 2

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in non-conservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 1

In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

A New Case of an Integrable System with Dissipation on the Tangent Bundle of a Multidimensional Sphere

The equations of motion for a dynamically symmetric n-dimensional fixed rigid body-pendulum situated in a rioricoriservative force field are studied. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of an incident medium. The complete list of transcendental first integrals expressed in terms of finite combinations of elementary functions is found.

Integrable Variable Dissipation Systems on the Tangent Bundle of a Multi-Dimensional Sphere and Some Applications

This paper is a survey of integrable cases in dynamics of a multi-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods to the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.

Analytic approximation for the modified Bessel function I−2/3(x)

In the present work an analytic approximation to modified Bessel function of negative fractional order I−2/3(x) is presented. The validity of the approximation is for every positive value of the independent variable. The accuracy is high in spite of the small number (4) of parameters used. The approximation is a combination of elementary functions with rational ones. Power series and assymptotic expansions are simultaneously used to obtain the approximation.

Transcendental First Integrals of Dynamical Systems on the Tangent Bundle to the Sphere

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

Integrable Motions of a Pendulum in a Two-Dimensional Plane

In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions.

Integrable Systems with Variable Dissipation on the Tangent Bundle of a Sphere

Many problems of multidimensional dynamics involve systems for which the spaces of states are spheres of finite dimension and the spaces of phases are the tangent bundles of such spheres. We study conservative systems and present nonconservative force fields such that the systems involving such forces possess a complete collection of first integrals that are expressed through a finite combination of elementary functions and, in general, are transcendental functions of their variables. Bibliography: 32 titles.

A multidimensional pendulum in a nonconservative force field under the presence of linear damping

A nonconservative force field in the dynamics of a multidimensional solid is constructed according to the results from the dynamics of real solids occurring in the force field of the action of the medium. In this case, it becomes possible to generalize the equations of motion of a multidimensional solid in a similarly constructed field of forces and to obtain a complete list of, generally speaking, transcendental first integrals expressed through a finite combination of elementary functions. In the study, the integrability in elementary functions is shown for the simultaneous equations of motion of a dynamically symmetric fixed multidimensional solid under the action of a nonconservative pair of forces in the presence of the linear damping moment (the additional dependence of the force field on the tensor of angular velocity of the solid).

Integrable nonconservative dynamical systems on the tangent bundle of the multidimensional sphere

We construct a class of nonconservative systems of differential equations on the tangent bundle of the sphere of any finite dimension. This class has a complete set of first integrals, which can be expressed as finite combinations of elementary functions. Most of these first integrals consist of transcendental functions of their phase variables. Here the property of being transcendental is understood in the sense of the theory of functions of the complex variable in which transcendental functions are functions with essentially singular points.

Integrable Cases in the Dynamics of a Multi-Dimensional Rigid Body in a Nonconservative Field in the Presence of a Tracking Force

This paper is a survey of integrable cases in the dynamics of a five-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a five-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.

Integrable systems in the dynamics on the tangent bundle of a two-dimensional sphere

A mechanical system whose phase space is the tangent bundle of a two-dimensional sphere is studied. The potential nonconservative systems describing a geodesic flow are classified. A multiparameter family of systems possessing a complete set of transcendental first integrals expressed in terms of finite combinations of elementary functions is found. Some examples illustrating the spatial dynamics of a rigid body interacting with a medium are discussed.

Performance analysis of multiple relay cooperative communication over generalized κ–μ and η–μ fading channels

In this paper, we derive the closed form expressions for symbol error rate (SER) of a multiple relay cooperative communication system using selective decode-and-forward protocol over generalized κ–μ and η–μ independent identically distributed fading channels for different modulation schemes. We also derive the exact expression of SER for non-coherent detection and an approximate expression of SER for coherent detection. At the destination, we employ the maximal ratio combining to provide diversity. The final expressions of SER are represented in terms of moment generating function which is a combination of elementary functions. We also perform the Monte Carlo simulations by assuming different channels and by varying the number of relays to validate the analytical results.

On the Inverse Curvature Problem

SummaryWe pose and investigate the inverse curvature problem. That is, we explore when elements of a certain family of monomials represent the curvature function of a curve specified by combinations of elementary functions. The main result utilizes an integration theorem of Chebyshev.

A new case of integrability in the dynamics of a multidimensional solid in a nonconservative field under the assumption of linear damping

The construction of a nonconservative force field is an important point in the dynamics of a multidimen� sional solid and is based on the results taken from the dynamics of smallsize solids in the dragforce field of the medium. It becomes possible to generalize the dynamic part of the equations to the case of the motion of a multidimensional solid in a similarly con� structed field of forces and to obtain the full list of, generally speaking, transcendental first integrals expressed through the finite combination of elemen� tary functions. The results obtained are important in the sense of the presence of a nonconservative moment in the system, while it is the potential field of forces that was used previously by other authors (see, for example, (1-3)). Moreover, when constructing the force field, an additional dependence on the angular� velocity tensor is used, the introduction of which in the system is not obvious a priori (4, 5).