Residue Theorem Research Articles

Residue Theorem based soft sliding mode control for wind power generation systems

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.

MultivariateResidues: A Mathematica package for computing multivariate residues

Multivariate residues appear in many different contexts in theoretical physics and algebraic geometry. In theoretical physics, they for example give the proper definition of generalized-unitarity cuts, and they play a central role in the Grassmannian formulation of the S-matrix by Arkani-Hamed et al. In realistic cases their evaluation can be non-trivial. In this paper we provide a Mathematica package for efficient evaluation of multidimensional residues based on methods from computational algebraic geometry. The package moreover contains an implementation of the global residue theorem, which produces relations between residues at finite locations and residues at infinity. New version program summaryProgram Title:MultivariateResiduesProgram Files doi:http://dx.doi.org/10.17632/7z87sf98xs.2Licensing provisions: GNU General Public License v3Programming language: Wolfram Mathematica version 7.0 or higherSupplementary material: Additional pdf included with program filesJournal reference of previous version: Comput.Phys.Commun. 222 (2018) 250-262Does the new version supersede the previous version?: YesReasons for the new version: Bugfix and added new functionSummary of revisions: In the previous version of MultivariateResidues, the function ▪ (using the method ▪ i.e. the default method) would occasionally yield the residue with an incorrect sign, due to a numerical prefactor being inadvertently dropped from the integrand. In this version of MultivariateResidues, this prefactor is retained, and many tests have been carried out to verify that residues are now computed with correct signs.In addition, a new function ▪ has been added. This function lists the linear relations among the residues at finite locations and residues at infinity produced by the global residue theorem when the input differential form is extended to CPn. We refer to the Supplementary Material for background material on the global residue theorem.Here we give an example of the usage of the function ▪ . The syntax is [Display omitted] where ▪ denotes the numerator of ω, ▪ the denominator factors and ▪ the variables. As an example, let us consider the differential form studied in section 1.1 of the Supplementary Material, [Display omitted] There are seven linear relations that arise from the global residue theorem, [Display omitted] The relations are recorded in the form {P,{V,R}}, where P denotes the denominator partition, V the set of poles involved in the relation and R their respective residues. For the denominator partition computed in detail in section 1.1 of the Supplementary Material we have [Display omitted] It is easy to check that the residues indeed sum to zero, [Display omitted] ▪ only records non-vanishing residues and their respective poles. In cases where all the residues that appear in a global residue theorem vanish, the empty set is returned as output. [Display omitted] Nature of problem: Evaluation of multivariate complex residuesSolution method: Mathematica implementation

A general reciprocity law for symbols on arbitrary vector spaces

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov — M. Kerr and the author — together with D. Hernández Serrano — can also be deduced from this general expression.

Analysis of the Sojourn Time Distribution for M/GL/1 Queue with Bulk-Service of Exactly Size L

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.

Residual symmetries,nth Bäcklund transformation and interaction solutions for (2+1)-dimensional generalized Broer–Kaup equations

Based on the residual symmetry theorem, the residual symmetry is obtained for the (2+1)-dimensional generalized Broer–Kaup (GBK) equations. The multiple residual symmetries are presented and localized by introducing prolonged systems, and then n th Bäcklund transformation for the GBK system in terms of determinant is derived. By means of the consistent tanh expansion (CTE) method, we get various interaction solutions which describe soliton interacting with other nonlinear waves including multiple resonant soliton wave, error function wave, periodic wave and rational function wave.

Fast Frequency-Domain Algorithm for Estimating the Dynamic Wind-Induced Response of Large-Span Roofs Based on Cauchy’s Residue Theorem

Wind-induced response analysis is an important process in the design of large-span roofs. Conventional time-domain methods are computationally more expensive than frequency-domain algorithms; however, the latter are not as accurate because of the ill-treatment of the modal coupling effects. This paper revisited the derivations of the frequency-domain algorithm and proposed a fast algorithm for estimating the dynamic wind-induced response considering duly the modal coupling effects. With the wind load cross-spectra modeled by rational functions, closed-form solutions to the frequency-domain integrals can be calculated by Cauchy’s residue theorem, rather than by numerical integration, thereby reducing the truncation errors and enhancing the efficiency of computation. The algorithm is applied to the analysis of a grandstand roof and a spherical dome. Through comparison with time domain analyses results, the algorithm is proved to be reliable. A criterion of the coupling modal combination was suggested based on the cumulative modal contribution rate of over 70%.

Accurate image-charge method by the use of the residue theorem for core-shell dielectric sphere.

An accurate image-charge method (ICM) is developed for ionic interactions outside a core-shell structured dielectric sphere. Core-shell particles have wide applications for which the theoretical investigation requires efficient methods for the Green's function used to calculate pairwise interactions of ions. The ICM is based on an inverse Mellin transform from the coefficients of spherical harmonic series of the Green's function such that the polarization charge due to dielectric boundaries is represented by a series of image point charges and an image line charge. The residue theorem is used to accurately calculate the density of the line charge. Numerical results show that the ICM is promising in fast evaluation of the Green's function, and thus it is useful for theoretical investigations of core-shell particles. This routine can also be applicable for solving other problems with spherical dielectric interfaces such as multilayered media and Debye-Hückel equations.

The computation of forward speed Green function in cylindrical coordinate system

Recently, a Rankine–Kelvin hybrid method based on meshless cylinder control surface has been developed to solve the ship seakeeping with forward-speed problem. In the external domain, the integration of forward speed Green function needs to be calculated on the cylinder control surface. However, the current computation methods of forward speed Green function are most based on studies in Cartesian coordinate system and not convenient to be integrated on the cylindrical surface. The forward speed Green function and its derivatives are then studied in cylindrical coordinate system in this paper. The double Fourier integral written in tow-fold polar and wavenumber integral is analysed. The polar integral is first performed by using the theorem of residues unlike the classical formulation in which the wavenumber integral was performed before the polar integral. The resultant wavenumber integral is reformulated by removing the weak singularities and evaluated numerically. The numerical results are validated through comparison with those calculated in Cartesian coordinate system, and the wave contours due to a translating pulsating source submerged below the free surface of infinite water depth are given.

Normalized modes at selected points without normalization

As every textbook on linear algebra demonstrates, the eigenvectors for the general eigenvalue problem |K−λM|=0 involving two real, symmetric, positive definite matrices K,M satisfy some well-defined orthogonality conditions. Equally well-known is the fact that those eigenvectors can be normalized so that their modal mass μ=ϕTMϕ is unity: it suffices to divide each unscaled mode by the square root of the modal mass. Thus, the normalization is the result of an explicit calculation applied to the modes after they were obtained by some means. However, we show herein that the normalized modes are not merely convenient forms of scaling, but that they are actually intrinsic properties of the pair of matrices K,M, that is, the matrices already “know” about normalization even before the modes have been obtained. This means that we can obtain individual components of the normalized modes directly from the eigenvalue problem, and without needing to obtain either all of the modes or for that matter, any one complete mode. These results are achieved by means of the residue theorem of operational calculus, a finding that is rather remarkable inasmuch as the residues themselves do not make use of any orthogonality conditions or normalization in the first place. It appears that this obscure property connecting the general eigenvalue problem of modal analysis with the residue theorem of operational calculus may have been overlooked up until now, but which has in turn interesting theoretical implications.Á

A general closed-form solution to a Timoshenko beam on elastic foundation under moving harmonic line load

In this paper, a general closed-form solution for evaluating the dynamic behavior of a Timoshenko beam on elastic foundation under a moving harmonic line load is formulated in the frequency-wavenumber domain and in a moving coordinate system. It is found that the characteristic equation is quartic with real coefficients only, and its poles can be presented explicitly. This enables the substitution of these poles into Cauchy\'s residue theorem, leading to the general closed-form solution. The solution can be reduced to seven existing closed-form solutions to different sub-problems and a new closed-form solution to the subproblem of a Timoshenko beam on an elastic foundation subjected to a moving quasi-static line load. Two examples are included to verify the solution.

深埋公路隧洞围岩应力和位移分布的复变函数解

In order to optimize the design of highway tunnels and ensure the safety of construction, mechanical behavior of rock shall be made clear during highway tunnel excavation. With the complex variable method, firstly, the exterior domain of the tunnel was transformed into a unit-circle domain through the conformal mapping function; then 2 stress functions were derived with the Cauchy integral formula and the residue theorem; thereby, closed-form plane strain solutions of the surrounding rock were obtained for stresses and displacements. The curved-wall horseshoe-shaped cross section was adopted, and the distributions of stresses and excavation displacements along the tunnel boundary and the coordinate axes were calculated respectively with mathematical software MATLAB. To verify the accuracy of the stress and displacement distributions derived with analytical solutions, a 2D plane strain model was established with finite element software ANSYS. Comparison between the numerical results and the approximate analytical results shows good agreement. The results show that, the maximum hoop stress occurs at the arch foot, the maximum horizontal displacement occurs at the hance, the largest settlement and uplift are at the centers of the vault and the invert, respectively. Normal stresses along coordinate axes change obviously near the tunnel, and the maximum stress doesn’t always occur at the tunnel boundary. The maximum stress approaches the applied load at a distance less than 10 m from the tunnel boundary. The largest displacement occurs at the tunnel boundary, and gradually reaches zero with an increasing distance from the tunnel boundary.

Analytical solution of Fourier and non-Fourier heat transfer in longitudinal fin with internal heat generation and periodic boundary condition

In this paper, the analytical solution of Fourier and non-Fourier model of heat transfer in longitudinal fin in presence of internal heat generation has been studied under periodic boundary condition. The whole analysis is given in dimensionless form. These two mathematical models have been solved analytically, using Laplace transform technique. Temperature distribution in longitudinal fin is measured using residual theorem in complex plane for the inverse Laplace transform technique. Thermal wave nature is appeared for small value of Fo. The longitudinal fin temperature is evaluated for different value of parameters with respect to space coordinate. The effect of variability of different parameters on temperature distribution in fin is studied in detailed. It has been observed that the cooling process is faster in non-Fourier model in comparison to Fourier model.

Application of a simplified calculation for full-wave microtremor H/V spectral ratio based on the diffuse field approximation to identify underground velocity structures

Under the diffuse field approximation, the full-wave (FW) microtremor H/V spectral ratio (H/V) is modeled as the square root of the ratio of the sum of imaginary parts of the Green’s function of the horizontal components to that of the vertical one. For a given layered medium, the FW H/V can be well approximated with only surface waves (SW) H/V of the “cap-layered” medium which consists of the given layered medium and a new larger velocity half-space (cap layer) at large depth. Because the contribution of surface waves can be simply obtained by the residue theorem, the computation of SW H/V of cap-layered medium is faster than that of FW H/V evaluated by discrete wavenumber method and contour integration method. The simplified computation of SW H/V was then applied to identify the underground velocity structures at six KiK-net strong-motion stations. The inverted underground velocity structures were used to evaluate FW H/Vs which were consistent with the SW H/Vs of corresponding cap-layered media. The previous study on surface waves H/Vs proposed with the distributed surface sources assumption and a fixed Rayleigh-to-Love waves amplitude ratio for horizontal motions showed a good agreement with the SW H/Vs of our study. The consistency between observed and theoretical spectral ratios, such as the earthquake motions of H/V spectral ratio and spectral ratio of horizontal motions between surface and bottom of borehole, indicated that the underground velocity structures identified from SW H/V of cap-layered medium were well resolved by the new method.

Closed-form solution of beam on Pasternak foundation under inclined dynamic load

The dynamic response of an infinite Euler–Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution. The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction. Thus, it is more realistic than the Winkler model, which only represents compressive soil resistance. However, the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation; hence, the beam under inclined loads cannot be considered in the model. In this study, a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model, which tends to disregard the tangential interaction between the beam and the foundation. The horizontal spring reaction is assumed to be proportional to the relative tangential displacement. The governing equation can be deduced by theory of elasticity and Newton's laws, combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition. Double Fourier transformation is used to simplify the geometric equation into an algebraic equation, thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam. Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution. The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS. Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam. Results are realistic and can be used as reference for future engineering designs.

Flexural-gravity wave resistances due to a moving point source on 2-D infinite floating beam

The flexural-gravity wave responses due to a load steadily moving or suddenly accelerated along a rectilinear orbit are analytically studied within the framework of the linear potential theory. A thin viscoelastic plate model is used for a very large floating structure. The initially quiescent fluid in the ocean is assumed to be homogenous, incompressible, and inviscid, and the disturbed motion be irrotational. A moving line source on the plate surface is considered as a moving point in the two-dimensional coordinates. Under the assumptions of small-amplitude wave motion and small plate deflection, a linear fluid-plate coupling model is established. The integral solutions for the surface deflections and the wave resistances are analytically obtained by the Fourier transform method. To study the dynamic characteristics of the flexural-gravity wave response, the asymptotic representations of the wave resistances are derived by the residue theorem and the methods of stationary phase. It shows that the steady wave resistance is zero when the speed of moving load is less than the minimal phase speed. The wave resistances due to the accelerate motion consist of two parts, namely the steady and transient wave responses. Eventually the transient wave resistance declines toward zero and the wave resistance approaches the steady component as the time goes to the infinity. Furthermore, the effect of the strain relaxation time for this viscoelastic plate is studied and it exhibits more influence for a high-speed motion.

On-shell diagrams and the geometry of planar mathcal{N}

We continue the discussion of the decorated on-shell diagrammatics for planar mathcal{N}<4 Supersymmetric Yang-Mills theories started in [1]. In particular, we focus on its relation with the structure of varieties on the Grassmannian. The decoration of the on-shell diagrams, which physically keeps tracks of the helicity of the coherent states propagating along their edges, defines new on-shell functions on the Grassmannian and can introduce novel higher-order singularities, which graphically are reflected into the presence of helicity loops in the diagrams. These new structures turn out to have similar features as in the non-planar case: the related higher-codimension varieties are identified by either the vanishing of one (or more) Plücker coordinates involving at least two non-adjacent columns, or new relations among Plücker coordinates. A distinctive feature is that the functions living on these higher-codimenson varieties can be thought of distributionally as having support on derivative delta-functions. After a general discussion, we explore in some detail the structures of the on-shell functions on Gr(2, 4) and Gr(3, 6) on which the residue theorem allows to obtain a plethora of identities among them.

A Helmholtz Potential Approach to the Analysis of Guided Wave Generation During Acoustic Emission Events

This paper addresses the predictive simulation of acoustic emission (AE) guided waves that appear due to sudden energy release during incremental crack propagation. The Helmholtz decomposition approach is applied to the inhomogeneous elastodynamic Navier–Lame equations for both the displacement field and body forces. For the displacement field, we use the usual decomposition in terms of unknown scalar and vector potentials, Φ and H. For the body forces, we hypothesize that they can also be expressed in terms of excitation scalar and vector potentials, A* and B*. It is shown that these excitation potentials can be traced to the energy released during an incremental crack propagation. Thus, the inhomogeneous Navier–Lame equation has been transformed into a system of inhomogeneous wave equations in terms of known excitation potentials A* and B* and unknown potentials Φ and H. The solution is readily obtained through direct and inverse Fourier transforms and application of the residue theorem. A numerical study of the one-dimensional (1D) AE guided wave propagation in a 6 mm thick 304-stainless steel plate is conducted. A Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. The numerical studies show that the peak amplitude of A0 signal is higher than the peak amplitude of S0 signal, and the peak amplitude of bulk wave is not significant compared to S0 and A0 peak amplitudes. In addition, the effects of the source depth, higher propagating modes, and propagating distance on guided waves are also investigated.

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

• Analytical solution for dynamic response of infinite beam on foundation is developed. • The elastic foundation considered the horizontal tangential effect. • Double Fourier transformation is used for deduction and the residue theorem is employed to obtain the solution. • The analytical solution for the beam under inclined dynamic load is presented. The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.

Residues for flags of holomorphic foliations

In this work we prove a Baum–Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum–Bott type rationality conjecture in this context.

New formulations of the ship-motion Green function

The ship-motion Green function representing the potential flow generated by a time-harmonically pulsating source with unit strength advancing beneath a free surface along a straight path at a constant speed is studied in the paper. Starting with the form of a double Fourier integral with respect to wavenumber k and polar angle $$\theta $$ on the Fourier plane, one alternative formulation based on the single integral along dispersion curves for the wave component, and the integrals in polar angle after having evaluated integration in wavenumber, is first presented. Another new formulation based on performing the integral with respect to polar angle $$\theta $$ prior to the wavenumber k is developed afterward. Analytical formulations of the $$\theta $$ -integral defined as $$g_{\ell }\left( k\right) $$ are obtained by applying the theorem of residue. The behaviours of poles located in the integration contour are analyzed, and a method to efficiently evaluate the integral of $$g_{\ell }\left( k\right) $$ for different values of Brard number $$\tau $$ is outlined. Furthermore, numerical calculation is implemented. On comparing with the results obtained by means of the existing formulation, an excellent agreement is achieved with a maximum of difference in the order of $$10^{-6}$$ which shows that two formulations are intrinsically consistent. The new formulation based on the polar angle integration is shown to be suitable for the study of the spatial integration of the Green function.