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Firstly, the definition of p order homogeneous weighted right monogenic polynomials is given, and the hypercomplex variables are introduced in order to construct a basis of all homogeneous weighted right monogenic polynomials of degree p, then the second Taylor expansion of the weighted right monogenic functions is obtained. Secondly, the weighted left monogenic functions are constructed from continuous functions in different regions, and corresponding Taylor expansions are given. Finally, on the basis of the previous conclusions, the Laurent expansion and residue theorem of the weighted left monogenic functions are proved. [ABSTRACT FROM AUTHOR]

This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve. [ABSTRACT FROM AUTHOR]

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z = F 1 2 (1 / 2 , 1 / 2 ; 1 ; x) and z ′ = d z / d x . When a certain parameter in these series is equal to π , the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. [ABSTRACT FROM AUTHOR]

Electrolytic Manganese Residue (EMR) is a secondary material generated during the process of manganese production, poses significant environmental challenges, including land consumption and contamination threats to soil and water bodies due to its heavy metal content, soluble manganese, ammonia nitrogen, and disposal issues. This review thoroughly examines EMR, emphasizing its metallurgical principles, environmental impacts, and sustainable treatment methods. We critically analyze various approaches for EMR management, including resource recovery, utilization of construction materials, and advanced treatment techniques to mitigate its environmental challenges. Through an extensive review of recent EMR-related literature and case studies, we highlight innovative strategies for EMR valorization, such as the extraction of valuable metals, conversion into supplementary cementitious materials, and its application in environmental remediation. Our findings suggest that integrating metallurgical principles with environmental engineering practices can unlock EMR's potential as a resource, contributing to the circular economy and reducing the environmental hazards associated with its disposal. This study aims to deepen the understanding of EMR's comprehensive utilization, offering insights into future research directions and practical applications for achieving sustainable management of electrolytic manganese waste. Finally, we propose some recommendations to address the issue of EMR, intending to offer guidance for the proper disposal and effective exploitation of EMR. [ABSTRACT FROM AUTHOR]

Continuous rails are often modeled as infinite beams. In this paper, closed-form solution is newly derived by the residue theorem and Green’s function for the dynamic response of an infinite beam resting on viscoelastic foundation under a harmonic moving load. The most suitable span length will be determined for the analogic finite beam (which can be more conveniently used in practice) to best represent the deflection of the infinite beam for practical reasons. Starting from the equations of motion, closed-form solution is firstly derived for the infinite beam. Then, it is verified by the finite element method (FEM) using semi-infinite elements to simulate the infinite boundary conditions. The deflection curves of the infinite beam and analogic finite beam are compared for various parameters. It is concluded that: (1) a larger span length should be adopted for the finite beam under higher load speeds or with foundations of softer stiffness or larger damping; typically, for load speeds <90m/s (324km/h); (2) typically, a span length of 50 m can be adopted for the analogic beam for UIC rails resting on foundations with stiffness k > 3.503 × 107N/m2 and damping coefficient c < 1.7325 × 106Ns/m2; (3) the peak deflection of the infinite and analogic finite beams occurs after the moving load for foundations with larger damping; and (4) the shape and amplitude of the deflection curve vary as the load frequency and amplitude varies, but the difference between the two beams is insignificant for moving loads with high frequencies. [ABSTRACT FROM AUTHOR]

This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve.

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

The exact null distribution of the likelihood ratio statistic for simultaneous testing the equality of covariance matrices and compound symmetry of two m variate Gaussian models has been obtained and percentage points for m ≤ 7 have been computed. Inverse Mellin transform and calculus of residues have been used to derive these results. [ABSTRACT FROM AUTHOR]

A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. This approach fails only on four characteristics: 1 / 2 1 / 6 , 1 / 6 1 / 2 , 1 / 6 1 / 6 and 1 / 6 5 / 6 ; however expressions for theta derivatives with these four characteristics are found by other authors. Expressions for theta derivatives appear to be homogeneous of degree 3 with respect to theta constants. [ABSTRACT FROM AUTHOR]

We prove a duality relation for the generalized basic hypergeometric functions. It forms a q-extension of a recent result of the second- and the third-named authors and generalizes both a q-hypergeometric identity due to the third-named author (jointly with Feng and Yang) and a recent identity for the Heine's 2 ϕ 1 function due to Suzuki. We further explore various consequences of our identity leading to several presumably new multi-term relations for both terminating and non-terminating generalized basic hypergeometric series. Moreover, we give confluent versions of our results and furnish a number of explicit examples. [ABSTRACT FROM AUTHOR]

The paper is concerned with an asymptotic expansion for the oscillatory term of the solution of diffusion equation with time-like highly oscillatory forcing terms. We derive an asymptotic expansion in inverse of powers of the oscillatory parameter, for the Dirichlet case and the Neumann case. Each term of the asymptotic expansion can be computed, at a lower cost. Numerical examples are given and show that with few terms of the asymptotic expansion, we can effectively approximate the oscillatory term of the solution. [ABSTRACT FROM AUTHOR]

For positive integers p 1 , p 2 , ... , p k , q with q ≥ 2 , we define the Euler T-sums T p 1 p 2 ⋯ p k , q as the infinite series whose general terms are products of odd harmonic numbers and negative powers of half integers. Similarly to the classical Euler sums, the Euler T-sums can be evaluated by the method of contour integrals and residue computation. Using this fact, we establish explicit formulas for some T-sums. The Euler T-sums can also be written as rational linear combinations of the Hoffman t-values, which help us show that the triple Hoffman t-values are reducible to zeta values, double zeta values and double t-values. We also define the alternating Euler T-sums and another odd variant of the Euler sums, which are called the Euler S ~ -sums here. We study these sums by the contour integrals and residue computation, and find many interesting results. In particular, we show that the triple Kaneko–Tsumura T-values of even weight can be expressed in terms of single and double T-values. [ABSTRACT FROM AUTHOR]

The soliton solutions with non-zero background of KdV equation with a source are constructed by Riemann–Hilbert approach. The irregular Riemann–Hilbert problem is constructed by direct and inverse scattering transform firstly, and then it can be regularized by introducing a novel transformation. The Residue theorem is utilized to derive the multi-soliton solutions at the simple poles of the Riemann–Hilbert problem. In particular, the interaction dynamics of the two-soliton solution are illustrated by considering their evolutions at different time. [ABSTRACT FROM AUTHOR]

Telecare Medicine Information System (TMIS)'s security importance attracts a lot of attention these days. Whatever the security of TMIS improves, its application becomes wider. To address this requirement, recently, Li et al. proposed a new privacy-preserving RFID authentication protocol for TMIS. After that, Zhou et al. and also Benssalah et al. presented their scheme, which is not secure, and they presented their new authentication protocol and claim that their proposal can provide higher security for TMIS applications. In this stream, Zheng et al. proposed a novel authentication protocol with application in smart campus, including TMIS. In this paper, we present an efficient impersonation and replay attacks against Zheng et al. with the success probability of 1 and a desynchronization attack which is applicable against all of the rest three mentioned protocols with the success probability of 1-2-n, where n is the protocols parameters length. After that, we proposed a new protocol despite these protocols can resist the attacks presented in this paper and also other active and passive attacks. Our proposed protocol's security is also done both informally and formally through the Scyther tool.

Maximum power point tracking (MPPT) algorithms have become key elements in improving solar photovoltaic (PV) energy conversion systems. Numerous algorithms have been developed and implemented successfully in recent technology. This paper intended to introduce an improved incremental conductance (IC) algorithm based on the mathematical residue theorem. The major difference introduced in this paper is in considering the residual value of the IC to ensure MPPT achievement. Ensuring the minimal residue in IC improves the operation and eliminates the fluctuation around the operation points. Improved energy conversion efficiency has been achieved and the system has been proved mathematically and by simulations. One of the advantages that the system is free of parameters affect and atmospheric condition data are not required. The controller gain is based on sliding mode compensator to improve the uncertainty handling ability of the developed control approach.

Cylindrical inclusions with constant cross section in an infinite isotropic matrix are usually treated as plane elasticity problems and solved by complex potential method without considering the longitudinal eigenstrains. This paper provides a closed-form solution for the Eshelby's circular cylindrical inclusion with eigenstrains which are polynomial in transverse direction and uniform in longitudinal direction. The integrals of Green's function are decomposed into the sum of customized L-integrals. Two sets of L-integrals for the regions inside and outside the circular cylindrical inclusion are evaluated by using the residue theorem. Further, the stress and strain fields inside and outside the inclusion resulted from the polynomial eigenstrains are obtained. Circular cylinder inclusions with uniform, linear, and quadric eigenstrains are, respectively, used as examples to illustrate the proposed solution. When the cylindrical inclusion only suffers transverse eigenstrains, the solution is appropriate for the circular inclusion with polynomial eigenstrains in plane elasticity. The proposed method has convenient formulae and simplifies the integrals of Green's function with polynomial eigenstrains. [ABSTRACT FROM AUTHOR]

In this article, the exact distribution and exact percentage points for testing equality of covariance matrices of q bivariate Gaussian populations are obtained. The distribution has been derived using the inverse Mellin transformation and the residue theorem. The percentage points have been computed for q = 2(1)5. [ABSTRACT FROM AUTHOR]

To meet the challenge of compressing the vast light field data, many prediction‐based algorithms achieve satisfactory results. Although these algorithms remove most redundancy through angular prediction, they rarely develop the suitable de‐correlation solutions for the prediction residue, leading to a large gap for higher compression performance. Therefore, this letter proposes a hybrid residue transform (HRT) mechanism to fully exploit the correlations in prediction residue for a three‐layer hierarchical light field compression. Specifically, the prediction residue is hierarchically generated from the authors' three‐layer structure. Besides, considering the high‐dimensional feature of light field residue, the high‐dimensional transforms, including 3D and 4D discrete cosine transforms, are alternately utilized to make the energy of the residue more compact. Experimental results demonstrate that the proposed method outperforms the state‐of‐the‐art methods. [ABSTRACT FROM AUTHOR]

Let n > 1 be an integer, (q(e)k (n))r is a multiplicative function. In this paper, we shall study the mean value of exponential divisor function (q(e)k (n))r over cube-full number, that is where f3(n) is the characteristic function of cube-full integers, i.e. [ABSTRACT FROM AUTHOR]

Scattering problems in periodic waveguides are interesting but also challenging topics in mathematics both theoretically and numerically. As is well known, the unique solvability of these problems is not always guaranteed. To obtain a unique solution that is "physically meaningful," the limiting absorption principle (LAP) is a commonly used method. LAP assumes that the limit of a family of solutions with absorbing media converges as the absorption parameter tends to 0 and that the limit is the "physically meaningful solution." It is also called the LAP solution in this paper. It has been proved that the LAP holds for periodic waveguides in [V. Hoang, SIAM J. Appl. Math., 71 (2011), pp. 791-810]. In this paper, we consider the spectrum decomposition of periodic translation operators. With the curve integral formulation and a generalized residue theorem, the operator is explicitly described by its eigenvalues and generalized eigenfunctions, which are closely related to Bloch wave solutions. Then the LAP solution is decomposed into generalized eigenfunctions. This gives a better understanding of the structure of scattered fields. [ABSTRACT FROM AUTHOR]

We prove some combinatorial identities by an analytic method. We use the property that singular integrals of particular functions include binomial coefficients. In this paper, we prove combinatorial identities from the fact that two results of the particular function calculated as two ways using the residue theorem in the complex function theory are the same. These combinatorial identities are the generalization of a combinatorial identity that has been already obtained. [ABSTRACT FROM AUTHOR]

We find a simple expression in complex terms for homogeneous harmonic polynomials, which we use to express the Laurent series of a harmonic function around an isolated singularity. Also, we show a residue theorem and study the orientation at isolated singularities through the use of complex dilatation, focusing on those points where orientation is not preserved nor reversed, making essential the concept of exceptional set and extending it to isolated singularities. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a residue theorem based soft sliding mode control strategy for a permanent magnet synchronous generator (PMSG) based wind power generation system (WPGS), to achieve the maximum energy conversion and improved in the system dynamic performance. The main idea is to set a soft dynamic boundary for the controlled variables around a reference point. Thus the controlled variables would lie on a point inside the boundary. The convergence of the operating point is ensured by following the Forward Euler method. The proposed control has been verified via simulation and experiments, compared with conventional sliding mode control (SMC) and proportional integral (PI) control.

The stresses around tunnels with arbitrary-excavation cross sections were studied with the stress field simplified as a uniform stress field at an angle to the horizontal axis at infinity. The conformal mapping function was introduced to express the stress functions of surrounding rock with variable ζ instead of variable z. Then the Cauchy integral operator was applied to the stress boundary equation for tunnel wall. The general analytical solutions to stress functions and analytical solutions of surrounding rock stresses were obtained for tunnels with arbitrary-excavation cross sections according to the integrand analyticity and the residue theorem. Moreover, with the stress boundary conditions for elliptical tunnels substituted into the general analytical solutions, the obtained particular solutions to stress functions consisted with the results in literatures. The surrounding rock stresses were analyzed for 2-lane road tunnels based on the explicit general analytical solutions and the results accorded with engineering rules. The stress distribution of tunnel wall was analyzed with the ABAQUS finite element method, and the results coincided with the analytical solution. The analytical calculation can be expediently carried out for underground tunnels with arbitrary-excavation cross sections as long as their conformal mapping functions are presented from the unit circle exterior to the tunnel exterior. [ABSTRACT FROM AUTHOR]

Laurentiu insisted that I should give a talk. His idea was that a residue value of an old man is that he may have some stories to tell the younger generations, whence the title. I believe what he actually meant was that the only residue value an old man like me might still have is to tell stories which might be of some interest to you. Most stories took place a very long time ago, they may not be 100% accurate. [ABSTRACT FROM AUTHOR]

In this paper, we deduce the asymptotic formulas of parametric digamma function Ψ (− s ; a) at the integers and poles. Then using these identities and residue theorem, we establish a large number of formulas of double series involving parametric harmonic numbers. Some illustrative special cases as well as immediate consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper we provide a concept of fuzzy partial metric space as an extension to fuzzy setting in the sense of Kramosil and Michalek, of the concept of partial metric due to Matthews. This extension has been defined using the residuum operator associated to a continuous t-norm and without any extra condition on . Similarly, it is defined the stronger concept of GV -fuzzy partial metric (fuzzy partial metric in the sense of George and Veeramani). After defining a concept of open ball in , a topology on X deduced from P is constructed, and it is showed that is a -space. [ABSTRACT FROM AUTHOR]

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated. [ABSTRACT FROM AUTHOR]

In the present article, we introduce the second Kummer function with matrix parameters and examine its asymptotic behaviour relying on the residue theorem. Further, we provide a closed form of a solution of a Weber matrix differential equation and give a representation using the second Kummer function. [ABSTRACT FROM AUTHOR]

This paper presents a simple algorithm for computing the cumulative distribution function of the sojourn time of a random customer in an M/GL/1 queue with bulk-service of exactly size L. Both theoretical and numerical aspects related to this problem were not discussed by Chaudhry and Templeton in their monograph (1983). Our analysis is based on the roots of the so-called characteristic equation of the Laplace-Stieltjes transform (LST) of the sojourn time distribution. Using the method of partial fractions and residue theorem, we obtain a closed-form expression of sojourn time distribution, from which we can calculate the value of the distribution function for any given time t ∈ [0, + ∞). Finally, to ensure the reliability of the analytical procedure, employing the work done by Gross and Harris (1985), an effective way to validate the correctness of our results along with some numerical examples are also provided. [ABSTRACT FROM AUTHOR]

In this paper we explain how the Residue Theorem was used perhaps for the first time to determine the Laurent series development of an elliptic function. This great archievement in the history of Elliptic Functions is due to the Frech profesors Briot and Buquet. We also draw some concluisions on the role of the historical emergence of Complex Analysis, as a general theory, in the developmente of Elliptic Functions.