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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]

Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.

In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function x M (x + 1) n λ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu' results. [ABSTRACT FROM AUTHOR]

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo methods. Assessment of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis. [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.

For a function field in one variable F/K and a discrete valuation v of K with perfect residue field k, we bound the number of discrete valuations on F extending v whose residue fields are non-ruled function fields in one variable over k. Assuming that K is relatively algebraically closed in F, we find that the number of non-ruled residually transcendental extensions of v to F is bounded by g + 1 where g is the genus of F/K. An application to sums of squares in function fields of curves over R ((t)) is outlined. [ABSTRACT FROM AUTHOR]

Based on the energy approach to the process of brittle fracture, a new energy criterion for fracture was obtained in the case when an arbitrary normal symmetrical load acts on the surface of an isolated crack. [ABSTRACT FROM AUTHOR]

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]

In this study, the Fourier series expansions of the Apostol-Frobenius type of Tangent and Genocchi polynomials of higher order are derived using the Cauchy residue theorem. Some novel and intriguing results are obtained by applying the Fourier series expansion of these types of polynomials. [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]

Let λ s y m 2 f (n) be the n th normalized Fourier coefficient of symmetric square L-function. In this paper, we will establish an asymptotic formula for ∑ (a , b , c , d) ∈ Z 4 a 2 + b 2 + c 2 + d 2 ≤ x λ s y m 2 f θ (a 2 + b 2 + c 2 + d 2) where θ = 3 , 4 and x ≥ x 0 (sufficiently large). [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function x M (x + 1) n λ (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu' results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1 + 1 -dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this associated hyperbolic secant Fourier spectrum can be represented by a convergent infinite series, which can be achieved using Mittag–Leffler's expansion theorem. Conversely, given the expression of the series of the spectrum, we recover its closed form by employing Cauchy's residue theorem for summation. We further confirm that the fundamental soliton indeed satisfies essential characteristics such as Parseval's relation and the stretch-bandwidth reciprocity relationship. The fundamental bright soliton finds rich applications in nonlinear fiber optics and optical telecommunication systems. [ABSTRACT FROM AUTHOR]

To resolve several challenging applications in many scientific domains, general formulas of improper integrals are provided and established for use in this article. The suggested theorems can be considered generators for new improper integrals with precise solutions, without requiring complex computations. New criteria for handling improper integrals are illustrated in tables to simplify the usage and the applications of the obtained outcomes. The results of this research are compared with those obtained by I.S. Gradshteyn and I.M. Ryzhik in the classical table of integrations. Some well-known theorems on improper integrals are considered to be simple cases in the context of our work. Some applications related to finding Green's function, one-dimensional vibrating string problems, wave motion in elastic solids, and computing Fourier transforms are presented. [ABSTRACT FROM AUTHOR]

Many formulas of improper integrals are shown every day and need to be solved in different areas of science and engineering. Some of them can be solved, and others require approximate solutions or computer software. The main purpose of this research is to present new fundamental theorems of improper integrals that generate new formulas and tables of integrals. We present six main theorems with associated remarks that can be viewed as generalizations of Cauchy's results and I.S. Gradshteyn integral tables. Applications to difficult problems are presented that cannot be solved with the usual techniques of residue or contour theorems. The solutions of these applications can be obtained directly, depending on the proposed theorems with an appropriate choice of functions and parameters. [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]

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order α are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type Frobenius–Euler polynomials resulting in their expansions and integral representations. [ABSTRACT FROM AUTHOR]

Many improper integrals appear in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik. It is a challenge for some researchers to determine the method in which these integrations are formed or solved. In this article, we present some new theorems to solve different families of improper integrals. In addition, we establish new formulas of integrations that cannot be solved by mathematical software such as Mathematica or Maple. In this article, we present three main theorems that are essential in generating new formulas for solving improper integrals. To show the efficiency and the simplicity of the presented techniques, we present some applications and examples on integrations that cannot be solved by regular methods. Furthermore, we acquire new results for integrations and compare them to that obtained in the classical table of integrations. Some previous results, become special cases of our outcomes or generalizations to acquire new integrals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the polynomials B n , α (k) (x ; q) generated by a function including Jackson q-Bessel functions J α (k) (x ; q) (k = 1 , 2 , 3) , α > - 1 . The cases α = ± 1 2 are the q-analogs of Bernoulli and Euler , s polynomials introduced by Ismail and Mansour for (k = 1 , 2) , Mansour and Al-Towalib for (k = 3) . We study the main properties of these polynomials, their large n degree asymptotics and give their connection coefficients with the q-Laguerre polynomials and little q-Legendre polynomials. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by constructing contour integral and using Cauchy’s residue theorem, we provide a novel proof of Chu’s two partial fraction decompositions.

We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this associated hyperbolic secant Fourier spectrum can be represented by a convergent infinite series, which can be achieved using Mittag–Leffler’s expansion theorem. Conversely, given the expression of the series of the spectrum, we recover its closed form by employing Cauchy’s residue theorem for summation. We further confirm that the fundamental soliton indeed satisfies essential characteristics such as Parseval’s relation and the stretch-bandwidth reciprocity relationship. The fundamental bright soliton finds rich applications in nonlinear fiber optics and optical telecommunication systems.

To resolve several challenging applications in many scientific domains, general formulas of improper integrals are provided and established for use in this article. The suggested theorems can be considered generators for new improper integrals with precise solutions, without requiring complex computations. New criteria for handling improper integrals are illustrated in tables to simplify the usage and the applications of the obtained outcomes. The results of this research are compared with those obtained by I.S. Gradshteyn and I.M. Ryzhik in the classical table of integrations. Some well-known theorems on improper integrals are considered to be simple cases in the context of our work. Some applications related to finding Green’s function, one-dimensional vibrating string problems, wave motion in elastic solids, and computing Fourier transforms are presented.

Many formulas of improper integrals are shown every day and need to be solved in different areas of science and engineering. Some of them can be solved, and others require approximate solutions or computer software. The main purpose of this research is to present new fundamental theorems of improper integrals that generate new formulas and tables of integrals. We present six main theorems with associated remarks that can be viewed as generalizations of Cauchy’s results and I.S. Gradshteyn integral tables. Applications to difficult problems are presented that cannot be solved with the usual techniques of residue or contour theorems. The solutions of these applications can be obtained directly, depending on the proposed theorems with an appropriate choice of functions and parameters.

In this paper, by constructing contour integral and using Cauchy's residue theorem, we provide a novel proof of Chu's two partial fraction decompositions. [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.

Many improper integrals appear in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik. It is a challenge for some researchers to determine the method in which these integrations are formed or solved. In this article, we present some new theorems to solve different families of improper integrals. In addition, we establish new formulas of integrations that cannot be solved by mathematical software such as Mathematica or Maple. In this article, we present three main theorems that are essential in generating new formulas for solving improper integrals. To show the efficiency and the simplicity of the presented techniques, we present some applications and examples on integrations that cannot be solved by regular methods. Furthermore, we acquire new results for integrations and compare them to that obtained in the classical table of integrations. Some previous results, become special cases of our outcomes or generalizations to acquire new integrals.

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]