Addition Theorem Research Articles

On Bäcklund transformations to stationary equations in hierarchy of the second Painlevé equation

The analytical properties of solutions to stationary equations of the second and fourth orders in the hierarchy of the second Painlevé equation are considered. For the second-order equation, it is shown that the Bäcklund transformation in the general case determines the formula of the addition theorem for the Weierstrass elliptic function. For the fourth and sixth-order equations, Bäcklund transformations and special classes of solutions are constructed. It has been established that, for a certain relationship between the parameters, the set of solutions to the first term of the stationary hierarchy is a subset of solutions to the second term and the set of solutions to the second term of the hierarchy is a subset of solutions of the six-order equation of the stationary hierarchy of the second Painlevé equation.

The usage of stochastic matrices while learning the topic “Eigenvalues and eigenvectors of a matrix” in the course of Higher Mathematics

Abstract The article looks into teaching students mathematical modeling during the mastering of the topic “Eigenvalues and eigenvectors of a matrix” in the Linear Algebra. The modeling is based on the usage of matrices built based on Markov chains. The analysis of the researchers’ papers shows that the usage of matrices of this type can be implemented while teaching students of different university specialties. The analysis of the prerequisites for the involvement of stochastic matrices in the consideration of students has helped to determine additional concepts and theorems that should be considered while learning “Eigenvalues and eigenvectors of a matrix”. The theorem on the presence of stochastic vectors in a stochastic matrix can serve as a formula for systematizing problems that can present applications of eigenvalues and eigenvectors of the matrix in real-life situations. The results of the experiment conducted using the developed system of problems for students of different University specialties show the improvement of skills provided by mastering the topic “Eigenvalues and eigenvectors of a matrix”.

Some Aspects of Neutrosophic Set Theory

This article aims to introduce my work on constructing neutrosophic set theory. Neutrosophy is a new branch of mathematical systems proposed by Smarandache in 1995 as an extension of an intuitionistic fuzzy set according to three-degree membership functions. The new perspective of neutrosophic sets consists of two paths, the first path depends on the three degrees including the degree of truth-membership, degree of indeterminacy-membership, and degree of false-non-membership respectively, while the second path depends on the generated classical set by three types of neutrosophic sets and study the concepts of neutrosophic theory. In previous work, I Presented the concepts of neutrosophic sets including universal, empty, compliment, and subsets. I also explored neutrosophic operations and their properties, such as neutrosophic unions and neutrosophic intersections. In this article, I shall delve into additional materials and theorems related to these concepts and discuss neutrosophic and symmetric differences, including their properties. Furthermore, I shall present a generalization of De Morgan's theorem.

A new entropy on metric spaces with respect to Bourbaki-bounded subsets

In this paper, we define a new entropy for every self-map on metric spaces, which is referred to as the Bourbaki entropy. We show, by means of an example, that the metric entropy is not necessarily equal to the Bourbaki entropy. Finally, the basic properties of the Bourbaki entropy are studied. The obtained results include the logarithmic law, invariance under conjugation, the weak addition theorem, and the completion theorem.

Computation of acoustic scattered fields and derived radiation force and torque for axisymmetric objects at arbitrary orientations.

This study presents a theoretical framework for calculating acoustic scattering fields, as well as radiation force and torque resulting from the interaction between an incident wave and an axisymmetric object positioned at arbitrary orientations. Grounded in the partial-wave expansion method, it formulates scattering products using beam-shape and scalar scattering coefficients. The incorporation of geometric features into the scalar scattering coefficients is achieved through the conformal transformation approach. Notably, its applicability is restricted to scenarios where the object is positioned at its standard orientation, a limitation circumvented by employing rotational transformations to extend the model to non-standard orientations. A rotational transformation tunes the original frame (observation coordinate system) into a reference frame (computation coordinate system), for any deviated orientation and facilitating solution of scattering products. While the non-intuitive nature of rotational transformations disrupts the inheritability of the partial-wave expressions for the scattering products, an alternative approach is provided based on rotation addition theorem. This method directly incorporates object orientations into the beam-shape and scalar scattering coefficients, bypassing rotational transformations and preserving the partial-wave format. Comparative analysis with full three-dimensional numerical simulations shows theoretical methods are computationally more efficient while ensuring substantial consistency.

A district-level building electricity use profile simulation model based on probability distribution inferences

District-level building energy systems play a significant role in urban energy networks in the future. Understanding the key distributive features of district electricity use profiles is essential for the optimal planning and design of energy networks. Due to the diversity of building electricity use characteristics, the district-level electricity use profile exhibits a prominent “peak staggering effect.” Current physics-based and statistical models cannot fully represent realistic distributions and the uncertainties of district profiles. Thus, it is critical to quantitatively investigate the changing patterns and distributive features of electricity use profiles at various district levels. This paper proposes a novel approach for district building electricity use profile simulation. Probability distribution inference methods integrating Gaussian Mixture Model (GMM)/lognorm distribution fitting, singular value decomposition (SVD)-based feature transformation, and distribution addition theorems have been proposed to generate the feature parameters of electricity use profiles at various district scales, thus generating simulated district electricity use profiles. The performance of the proposed model was validated using engineering-informed metrics, including peak demands, load duration curves, and standard deviations of the load parameters. The results of the case study suggest that the average relative error of the 99 % peak demand is reduced from 17.60 % in the baseline model to 3.48 % in the proposed model, the average relative error of the duration of 2Qm reduced from 40.82 % in the baseline model to 0.99 % in the proposed model, and the average relative error of the standard deviation of load parameters was reduced from >100 % in the baseline model to <35 % in the proposed model. The results indicate a better quantification of district electricity use distributions and uncertainties, providing practical tools to support the capacity design and optimization of integrated district energy systems.

The dynamic performance of semi-infinite piezoelectric strip containing irregular boundaries based on the boundary element method

In this article, the problem of a semi-infinite piezoelectric strip with irregular boundaries is studied by boundary element method, and SH guided wave and line source loads are considered as external loads acting on piezoelectric strip. The superiority of the boundary element method is induced by two different arithmetic examples. Firstly, when the irregular boundary is not subjected to line source loads, the dynamic characteristics are analyzed in the first example, employing the image method and Graf addition theorem. Then, the Green’s identities are introduced and the Green’s function in infinite three-dimensional space is solved. When irregular boundaries are subjected to line source loads, the dynamic characteristics are investigated using the boundary element method in the second example. Finally, results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Borehole radiation wavefield for multipole logging while drilling with an eccentric collar

Abstract This study investigates the radiation characteristics of the borehole-source in acoustic logging while drilling (ALWD) with an eccentric collar. Understanding these characteristics is crucial for the development of acoustic remote detection technology, which has shown great potential in ALWD. Considering the frequent occurrence of tool eccentricity during drilling, it is essential to take into account its impact on the acoustic wavefield outside the borehole. In this paper, we establish an ALWD model with an eccentric collar in a dual-cylindrical coordinate system. By applying the translational Bessel addition theorem and the steepest descent method, we obtain the far-field asymptotic solution of the problem and model the radiation wavefield. Our findings indicate that the asymmetrical radiation pattern resulting from drill collar eccentricity can effectively resolve ambiguities in determining the reflector azimuth.

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique.

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

Analytical solution for acoustic radiation force and torque on a spheroid near a rigid or free planar boundary.

An analytical solution is developed for the acoustic radiation force and torque caused by an arbitrary sound field that is incident on a compressible spheroid of any size near a planar boundary that is either rigid or pressure release. The analysis is an extension of a recent solution for a compressible sphere near a planar boundary [Simon and Hamilton, J. Acoust. Soc. Am. 153, 627-642 (2023)]. Approximations that account for a boundary formed by a two-fluid interface may be incorporated as in the previous analysis for a sphere. The present solution is based on expansions of the total acoustic pressure field in spheroidal wave functions and the use of addition theorems. Verification of the solution is accomplished by comparison with a finite element model. Examples are presented for incident fields that are either plane or spherical waves. Effects resulting from the presence of the boundary are studied by comparing the full theory with a simplified model in which multiple scattering is neglected. Numerical implementation of the proposed solution is also discussed.

Propagation attenuation of elastic waves in multi-row infinitely periodic pile barriers: A closed-form analytical solution

Periodically arranged media exhibit attenuation zones for elastic waves, and the related periodic structures play a crucial role in the design of seismic isolation and vibration mitigation. From the perspective of wave theory, this paper presents a closed-form analytical solution for the propagation attenuation of elastic waves through infinitely periodic, multi-row pile barriers. The innovative approach begins by deriving the extended Graf’s addition theorem to transform wave potentials across arbitrary coordinate systems, overcoming the constraints of traditional formulas that limit pile centers to linear configurations. Subsequently, it completes the wave function expansion by utilizing the phase differences in the frequency domain of wave fields around different periodic elements, thus skillfully integrating the effects of incident and scattered wave fields. For the first time, this study delivers exact expressions for the scattering of various types of plane waves (SH, SV, and P) by multi-row pile barriers with an infinite periodic array, addressing the complexities of large pile numbers in previous theoretical studies. The proposed solution increases computational efficiency by tens of times and markedly reduces the majority of resource usage, improving the analysis of complex vibration isolation systems involving numerous piles. Building on the accuracy verification and convergence analysis, several numerical examples are conducted to explore the effects of pile row, pile spacing, and pile arrangement on the propagation attenuation. The findings elucidate the mechanisms driving vibration reduction design using periodic wave barrier, furthering its application in engineering structures.

Efficient Analysis of Unlined Elliptic Tunnels: An Approximate Method for Dynamic Response to Incident Plane SH Waves

ABSTRACT Dynamic analysis of underground structures, such as tunnels, is crucial to ensure their safety and stability during seismic loading. This paper proposes a more general method of large elliptic arc assumption-based wave function expansion, which is attributed to the important role of the large circular arc assumption-based wave function expansion method in the elastic wave scattering problem. The study is conducted within elliptic coordinate systems, employing the Mathieu function and Mathieu function addition theorem. The applicability and limitations of this method are analysed by comparing it with the existing exact analytical solutions obtained through the mirror-based wave function expansion method.

Near- and far-field radiated acoustic pressures from a vibrating three-dimensional cylindrical shell in an underwater acoustic waveguide

The vibroacoustic behavior of a vibrating infinitely-long thin three-dimensional cylindrical shell immersed in a perfect underwater acoustic waveguide is predicted using an analytical method. Vibration is induced by a point force excitation on the surface of the shell that is ingested into the equations of motion modeled using Flügge’s shell theory. Radiated acoustic pressures are then investigated in relation to an influence from the acoustic boundaries of an underwater acoustic waveguide. The waveguide, which simulates a simplified shallow water environment, is composed of an upper free surface and a lower rigid floor. Its influence is modeled with the image-source method. Graf’s addition theorem is employed to reconcile the coordinate systems of the infinite image sources in the fluid–structure coupling analytical expressions. Predictions of the near-field acoustic pressure are obtained by a direct numerical evaluation of the inverse Fourier integral, while the far-field acoustic pressure uses the stationary phase method. Axial directivities are of interest as the longitudinal length of the 3D shell imposes an additional dimension to the most commonly used 2D models in the literature to which radiated acoustic waves can dissipate energy. The far-field sound propagation from the shell in different fluid domain configurations is also shown.

Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one

Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one

On the Weihrauch degree of the additive Ramsey theorem

We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals and of the natural numbers. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals respectively natural numbers satisfying properties determined by some additional algebraic structure on the set of colours. In the context of reverse mathematics, most of the principles we study are equivalent to Σ 2 0 -induction over RCA 0 . The associated problems in the Weihrauch lattice are related to TC N ∗ , ( LPO ′ ) ∗ or their product, depending on their precise formalizations.

Anti‐plane stress analysis for V‐notch in a piezomagnetic half space under SH wave

AbstractIn this paper, the anti‐plane stress analysis of a V‐notch with complex boundary conditions in a piezomagnetic half space is studied. Firstly, SH wave is considered as an external load acting on piezomagnetic half space, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the boundary conditions on the boundary of the half space. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V‐notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green's function method is applied, the half space is divided into two parts along the vertical interface, a pair of in‐plane magnetic field and out‐plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Rayleigh Scattering from a Sphere Located Near a Planar Rigid Boundary

Rayleigh scattering from a spherical object located near a planar rigid boundary at distances smaller than the wavelength is calculated. Low frequency analysis reduces a scattering problem to a sequence of potential problems. An analytical solution based on expansion in spherical solid harmonics and the use of addition theorem is presented. Analytical perturbation approach is validated by comparison with numerical calculations. The velocity of the center of the particle and the scattering amplitude are determined. In the lowest order in wavenumber, the scattering amplitude is expressed in terms of the monopole and dipole components. In contrast to the behavior of a bubble, under the same conditions, dipole oscillations of the particle in the direction normal to the boundary are not excited and the monopole component of the scattering amplitude does not depend on the location of the particle relative to the boundary.

Nonlinear susceptibilities for weakly turbulent magnetized plasma: Electromagnetic formalism

This is a companion paper to the previous work [P. H. Yoon, Phys. Plasmas 31, 032309 (2024)] in which the nonlinear susceptibilities of weakly turbulent magnetized plasma are derived under a simplifying assumption of electrostatic interaction. The present paper extends the analysis to a general situation of electromagnetic interaction. The main novelty of the previous and present papers is that by employing the Bessel function addition theorem, the mathematical definitions for the susceptibilities are substantially simplified, a procedure that has not been discussed in the existing literature. In the present paper, a full set of Maxwell’s equations are considered in conjunction with the nonlinear Vlasov equation, which is solved by a perturbative method. The result is a fully general nonlinear susceptibility, given in tensorial form, which is applicable for weakly turbulent magnetized plasmas.

Scattering from a semi‐elliptical cavity with arbitrary orientation placed in a perfect electric conductor plane

AbstractA semi‐analytic solution for scattering from a semi‐elliptical cavity with arbitrary orientation placed in a perfect electric conductor plane is suggested. By imposing the boundary conditions at an elliptical border, three independent equations containing the Mathieu functions series are extracted. Finally, the solution of the problem is reduced to the solution of a simple system of linear equations by using Mathieu function addition theorem and the orthogonal properties of the angular Mathieu functions. Some examples are presented to demonstrate the reliability and the accuracy of the proposed solution.

Nonlinear susceptibilities for weakly turbulent magnetized plasma: Electrostatic approximation

The plasma weak turbulence theory is a perturbative nonlinear theory, which has been proven to be quite valid in a number of applications. However, the standard weak turbulence theory found in the literature is fully developed for highly idealized unmagnetized plasmas. As many plasmas found in nature and laboratory are immersed in a background static magnetic field, it is necessary to extend the existing discussions to include the effects of ambient magnetic field. Such a task is quite formidable, however, which has prevented fundamental and significant progresses in the subject matter. The central difficulty lies in the formulation of the complete nonlinear response functions for magnetized plasmas. The present paper derives the nonlinear susceptibilities for weakly turbulent magnetized plasmas up to the third order nonlinearity, but in doing so, a substantial reduction in mathematical complexity is achieved by the use of Bessel function addition theorem (or sum rule). The present paper also constructs the weak turbulence wave kinetic equation in a formal sense. For the sake of simplicity, however, the present paper assumes the electrostatic interaction among plasma particles. Fully electromagnetic generalization is a subject of a subsequent paper.