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A new analytical approach is presented for gravity-induced stresses in elastic half plane with a slope. The half plane is mapped onto the unit circle in ζ plane by conformal transformations. The mapping function proposed by Schwarz-Christoffel is irrational and difficult to be applied to the problem in the paper. Therefore, the explicit expression of the mapping function, which is easy to use and has high precision, is proposed through power series approximation. Based on the complex potentials with body force, a simple method for solving gravity-induced stresses which do not involve analytic continuation and Cauchy integral is established. The analytic functions are expressed as power series. Through the stress boundary condition on the ground surface, a set of linear equations with the coefficients of the power series can be directly constructed and solved. The stress results obtained by the presented analytical method agree well with the numerical solution. The stress distributions under different Poisson’s ratio and slope angle are studied.

In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of sources, we consider inhomogeneous media and use tangential components of the electric field and magnetic field at the boundary of the reference domain. The proof relies on the Fourier transform with respect to the wave numbers and the unique continuation theorems. To study the increasing stability in the source identification, we consider homogeneous media and measure the absorbing data or the tangential component of the electric field at the boundary of the reference domain as additional data. By using the Fourier transform with respect to the wave numbers, explicit bounds for analytic continuation, Huygens' principle and bounds for initial boundary value problems, increasing (with larger wave numbers intervals) stability estimate is obtained.

The main focus of this paper is on studying an order optimal regularization scheme based on the Meyer wavelets method to solve the analytic continuation problem in the high-dimensional complex domain Ω : = { x + i y ∈ C N : x ∈ R N , ‖ y ‖ ≤ ‖ y 0 ‖ , y , y 0 ∈ R + N } . This problem is exponentially ill-posed and suffers from the Hadamard's instability. Theoretically, we first provide an optimal conditional stability estimate for the proposed original problem. Applying the Meyer wavelets, an order optimal regularization scheme is then developed to stabilize the considered ill-posed problem. Some sharp error estimates of the Holder-Logarithmic type controlled by the Sobolev scale under an a-priori information are also derived. The provided error estimates are of the order optimal in the sense of Tautenhahn. Finally, some different one- and two-dimensional examples are presented to confirm the efficiency and applicability of the proposed regularization scheme. The comparison results also show that the proposed method is more accurate than the other existing methods in the literature.

We establish the existence of an internal uniform hydrostatic state of stress inside a coated non-parabolic open elastic inhomogeneity when the surrounding matrix is subjected to a concentrated couple and uniform remote in-plane stresses. The proof of existence is accomplished through analytic continuation and the introduction of a specialized form of the conformal mapping function containing an infinite number of first-order poles. The complex constants appearing in the mapping function can be uniquely determined from a resulting set of non-linear recurrence relations for given material, geometric and loading parameters. The convergence criterion for the series in the mapping function is rigorously established. A simple condition on the remote loading in the matrix is found for given elastic constants of the inhomogeneity and the matrix in order to ensure that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. The analysis indicates that the internal uniform hydrostatic stress field, the constant mean stress in the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are all independent of the specific open shapes of the two interfaces and are also independent of the existence of the nearby concentrated couple. However, the non-parabolic open shapes of the two interfaces are significantly influenced by the concentrated couple. We also accomplish the design of an uncoated non-parabolic harmonic elastic inhomogeneity in the presence of an arbitrary number of concentrated couples applied in the matrix.

We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL ( 2 , R ) due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

We employ conformal mapping and analytic continuation to study the elastic field induced by a screw dislocation interacting with a completely coated semi-infinite wedge crack under mode III loading conditions. The screw dislocation can be located either inside the matrix or in the coating itself. Explicit expressions for the local wedge stress intensity factor and the image force acting on the screw dislocation are obtained. The dislocation emission criterion from the completely coated wedge crack is established.

First, we give a formula for the limiting value of the higher Mahler measures for general polynomials of one variable, which refines Biswas and Monico's result. Second, we give an analytic continuation of the zeta Mahler measures for polynomials of one variable to the whole complex plane as meromorphic functions, and we show that all of their poles are simple.

Let $F(s)=\sum_n a_n/\lambda_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $\alpha_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $\alpha_F$ and residue $\rho_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$., Comment: 11 pages

We present SpM, a sparse modeling tool for the analytic continuation of imaginary-time Green's function, licensed under GNU General Public License version 3. In quantum Monte Carlo simulation, dynamic physical quantities such as single-particle and magnetic excitation spectra can be obtained by applying analytic continuation to imaginary-time data. However, analytic continuation is an ill-conditioned inverse problem and thus sensitive to noise and statistical errors. SpM provides stable analytic continuation against noise by means of a modern regularization technique, which automatically selects bases that contain relevant information unaffected by noise. This paper details the use of this program and shows some applications., Comment: 7 pages, 4 figures

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.

Conformal mapping and analytic continuation are employed to prove the existence of an internal uniform electroelastic field inside a non-elliptical piezoelectric inhomogeneity interacting with a screw dislocation. We focus specifically on the case when the piezoelectric matrix surrounding the inhomogeneity is subjected to uniform remote anti-plane mechanical and in-plane electrical loading and a constraint is imposed between the remote loading and the screw dislocation. The constraint can be expressed in a relatively simple decoupled form by utilizing orthogonality relationships between two corresponding eigenvectors. The internal uniform electroelastic field is found to be independent of the presence of the screw dislocation; moreover, it can be expressed in decoupled form.

Using analytic continuation, we derive a new analytical solution for a screw dislocation in an N-phase laminate composed of N−2 intermediate elastic layers of equal thickness encased in two semi-infinite elastic media. The number of phases comprising the laminate can be arbitrarily large. The solution can be interpreted in terms of image screw dislocations, the magnitudes of which are determined by solving an algebraic equation of order N−2. We also obtain an elementary expression for the image force acting on the screw dislocation in terms of a single real function. Several specific examples are presented to demonstrate the solution. Analytic continuation is also applied to solve the related problem of a screw dislocation interacting with a multicoated circular elastic inhomogeneity.

We present the TRIQS / SOM analytic continuation package, an efficient implementation of the Stochastic Optimization Method proposed by Mishchenko et al. (2000). TRIQS / SOM strives to provide a high quality open source (distributed under the GNU General Public License version 3) alternative to the more widely adopted Maximum Entropy continuation programs. It supports a variety of analytic continuation problems encountered in the field of computational condensed matter physics . Those problems can be formulated in terms of response functions of imaginary time, Matsubara frequencies or in the Legendre polynomial basis representation. The application is based on the TRIQS C++/Python framework, which allows for easy interoperability with TRIQS -based quantum impurity solvers, electronic band structure codes and visualization tools. Similar to other TRIQS packages, it comes with a convenient Python interface. Program summary Program Title: TRIQS / SOM Program Files doi: http://dx.doi.org/10.17632/fcjzyhrwpw.1 Licensing provisions: GNU General Public License (GPLv3) Programming language: C++ / Python External routines/libraries: TRIQS 1.4.2 , Boost >=1.58 , cmake . Nature of problem: Quantum Monte Carlo methods (QMC) are powerful numerical techniques widely used to study quantum many-body effects and electronic structure of correlated materials. Obtaining physically relevant spectral functions from noisy QMC results in the imaginary time/Matsubara frequency domains requires solution of an ill-posed analytic continuation problem as a post-processing step. Solution method: We present an efficient C++/Python open-source implementation of the stochastic optimization method for analytic continuation. Additional comments: Project homepage http://krivenko.github.io/som/ .

We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the form of a Mellin–Barnes integral. As a corollary, we obtain an integral expression for the characteristic function. We present two approximations for the Laplace transform of the lognormal distribution, both valid in ℂ ∖ ( − ∞ , 0 ] . In the last section, we discuss how one may use our results to compute the density of a sum of independent lognormal random variables.

We propose an approach to solving and analyzing linear rational expectations models with general information frictions. Our approach is built upon policy function iterations in the frequency domain. We develop the theoretical framework of this approach using rational approximation, analytic continuation, and discrete Fourier transform. Conditional expectations, which are difficult to evaluate in the time domain, can be calculated efficiently in the frequency domain. We provide the numerical implementation accompanied by a flexible object-oriented toolbox. We demonstrate the efficiency and accuracy of our method by studying four models in macroeconomics and finance that feature asymmetric information sets, endogenous signals, and higher-order expectations.

In this paper we study some classes of generalized singular integral equations of convolution type with Cauchy kernel in the class of exponentially increasing functions. Such equations can be transformed into Riemann boundary value problems with two unknown functions on two parallel straight lines via Fourier transformation. The general solutions and the conditions of solvability are obtained by means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problem. Therefore, the classic Riemann boundary value problem is extended further.

Engineering approximations of physical systems sometimes produce models in which real-valued model-based physical-distances are added to complex-valued distances and/or (for electrical systems), real-valued current/charge image intensities are replaced with complex-valued quantities. These models are arrived at often using ad hoc approximations that allow infinite integrals or series to be approximated in closed form. Arriving at accurate ad hoc approximations in a compatible analytic form is often the difficult step in the derivation of these approximations. In this paper, we show that this difficult ad hoc step can be replaced for many classes of functions with the use of analytic continuation via Pade approximants, along with some reasonable engineering judgement. We apply our approach to several existing approximations in the electrical engineering field (overhead transmission line impedance, underground cable impedance and Green’s functions used in ground potential rise calculations) and show that these approximations can be derived elegantly, without the need for grand leaps of insight, and provide a basis for both distance parameters and current/charge intensities that are complex-valued.

We apply analytic continuation and conformal mapping techniques to investigate, in detail, the elastic interaction of a screw dislocation and subsequently an edge dislocation with a locally rough bimaterial interface. In the case of a screw dislocation, the locations and Burgers vectors of the image screw dislocations for the half-plane containing the real screw dislocation are determined in an analytical manner. The image force acting on the screw dislocation is then expediently obtained using the Peach-Koehler formula. A sufficiently accurate closed-form expression for the image force is proposed. For the case of an edge dislocation, it is assumed that the two half-planes have identical shear moduli but distinct Poisson’s ratios. An exact closed-form expression for the image force acting on the edge dislocation is obtained. The image force is found to vary with the direction of the Burgers vector of the edge dislocation due to the influence of interface roughness.

This is a continuation of our previous paper [7] , in which multiple Fibonacci zeta functions of depth 2 have been studied. In this article, we consider more general situation. In particular, we prove the meromorphic continuation of the multiple Lucas zeta functions of depth d: ∑ 0 n 1 ⋯ n d 1 U n 1 s 1 ⋯ U n d s d , where U n is the n-th Lucas number of first kind and ∑ i = j d Re ( s i ) > 0 for 1 ≤ j ≤ d . We compute a complete list of poles and their residues. We also prove that the multiple Lucas zeta values at negative integer arguments are rational.

Let r k ( n ) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoi summation formula for r k ( n ) , k ≥ 2 , proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of r k ( n ) and a product of two Bessel functions, and a series involving r k ( n ) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.

An exactly soluble model of two diabatic electronic states interacting through a coupling of strength V is used to generate data for testing the rational fraction analytic continuation technique for determining the energies and widths of metastable states of anions. By making analytical connections between the coefficients in the rational fraction and the parameters of the model, we are able to suggest how to choose the orders of the polynomials and the range of the scaling parameter, Z, within which to compute the energies for a given precision. This analysis also allows us to specify the range of Z-values to use in constructing the rational fraction in a manner that allows determination of all parameters of the model for a given precision. The constraint on the Z-value ranges can be used as a guide for constructing rational fractions of data obtained in electronic structure calculations on actual resonance states.

A holomorphic embedding method (HEM)-based algorithm for finding Type-1 power-flow solutions is introduced whose complexity is the same as that of the HEM power-flow algorithm for calculating the high voltage (operable) power-flow solution. The algorithm is tailored to finding Type-1 solutions by using a modified embedded system and a numerical mapping from a set of integer-based boundary conditions to a floating-point-number-based reference state. The modified system can also be viewed as a homotopy whose initial point (no-load reference state) is consistent with the Type-1 solution homotopy path of the modified system, with an embedding parameter that functions simultaneously as the homotopy parameter. By using analytic continuation, starting from the initial point/reference state, the solution obtained for the modified system matches the one obtained from the original system model at the load of interest. Numerical results for three-/five-/seven-/14- and 118-bus systems are presented to demonstrate the numerical robustness.

The unified method for addressing the propagation of subsonic and transonic cracks along the interfaces of anisotropic bimaterials in Shen and Nishioka (2000) is applied in this paper for the plain strain problem of subsonic and transonic crack propagation along the interfaces of anisotropic magnetoelectroelastic (MEE) bimaterials. Using a modified Eshelby–Stroh formalism and analytic continuation, the problem here leads to a Riemann–Hilbert problem. After solving the Riemann–Hilbert equation, near-tip asymptotic fields are obtained, which shows the properties of the subsonic and transonic crack propagation in an MEE bimaterial system. Finally, the numerical results related to different singularity powers of the crack-tip are presented and discussed for barium titanate-cobalt ferrite (BaTiO3 CoFe2O4) composites. The effects of piezoelectricity and piezomagnetics are also examined.

In this paper, a simultaneous identification problem of the spacewise source term and the fractional order for a time-fractional diffusion equation is considered. Firstly, under some assumption and with two different kinds of observation data for one-dimensional and two-dimensional time-fractional diffusion equation, the unique results of the inverse problem are proven by the Laplace transformation method and analytic continuation technique. Then the inverse problems are transformed into Tikhonov type optimization problems, the existence of optimal solutions to the Tikhonov functional is proven. Finally, we adopt an alternating minimization algorithm to solve the optimization problems. The efficiency and stability of the inversion algorithm are tested by several one- and two-dimensional examples.

Divergent integrals in quantum field theory (QFT) can be given well defined existence as Lorentz covariant complex measures, which may be analyzed by means of a spectral calculus. The case of the photon self energy is considered and the spectral vacuum polarization function is shown to have very close agreement with the vacuum polarization function obtained using dimensional regularization / renormalization in the timelike domain. Using the spectral vacuum polarization function a potential function defined in the timelike domain is derived. The Uehling potential function, from which the Uehling contribution to the Lamb shift may be computed, is derived from an analytic continuation into the spacelike domain of this potential function. The spectral running coupling for QED is computed from this analytically continued potential function. The integral defining the spectral running coupling constant is shown to converge for all non-zero energies while that for the running coupling constant computed using dimensional regularization / renormalization is shown to diverge for all non-zero energies. It is seen that the spectral running coupling does not have a Landau pole and agrees both qualitatively and quantitatively with the results of scattering experiments at all energies.

We present a determination of the curvature $\kappa$ of the chiral pseudocritical line from $N_f=2+1$ lattice QCD at the physical point obtained by adopting the Taylor expansion approach. Numerical simulations performed at three lattice spacings lead to a continuum extrapolated curvature $\kappa = 0.0145(25)$, a value that is in excellent agreement with continuum limit estimates obtained via analytic continuation within the same discretization scheme, $\kappa = 0.0135(20)$. The agreement between the two calculations is a solid consistency check for both methods., Comment: Quark Matter 2018

This paper is the second part of the paper [S1] . We present the unramified computation of the local integrals, with normalized unramified data, over a p-adic field F, arising from the general Rankin–Selberg integrals for SO m × GL N , in the Cases 1b, 2 of [S1] . As in the first part, our proof is by “analytic continuation from the unramified computation in the generic case”.

We consider a circular elastic inclusion embedded in an infinite matrix each from a particular class of compressible hyperelastic materials of harmonic type. A concentrated couple is applied either inside the circular inclusion or in the matrix. Closed-form solutions of the corresponding boundary value problems are obtained using complex variable methods, in particular the principle of analytic continuation. Our analysis reveals several interesting conclusions including: the sum σ 11 + σ 22 of the normal stresses inside the inclusion remains constant when the concentrated couple is located in the surrounding matrix; the sum of the normal stresses is zero everywhere in the matrix when the concentrated couple is located inside the inclusion.

Let L be the unique even self-dual lattice of signature ( 25 , 1 ) . The automorphism group Aut ( L ) acts on the hyperbolic space H 25 . We study a Poincare series E ( z , s ) defined for z in H 25 , convergent for Re ( s ) > 25 , invariant under Aut ( L ) and having singularities along the mirrors of the reflection group of L. We compute the Fourier expansion of E ( z , s ) at a “Leech cusp” and prove that it can be meromorphically continued to Re ( s ) > 25 / 2 . Analytic continuation of Kloosterman sum zeta functions imply that the individual Fourier coefficients of E ( z , s ) have meromorphic continuation to the whole s-plane.

The holomorphic embedding load flow method (HELM) is an application for solving the power-flow problem based on a novel method developed by Dr. Trias. The advantage of the method is that it comes with a theoretical guarantee of convergence to the high-voltage (operable) solution, if it exists, provided the equations are suitably framed. While theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not; it depends on the analytic continuation algorithm chosen. Since the holomorphic embedding method (HEM) has begun to find a broader range of applications (it has been applied to nonlinear structure-preserving network reduction, weak node identification and saddle-node bifurcation point determination), examining which algorithms provide the best numerical convergence properties, which do not, why some work and not others, and what can be done to improve these methods, has become important. The numerical Achilles heel of HEM is the calculation of the Pade approximant, which is needed to provide both the theoretical convergence guarantee and accelerated numerical convergence. In the past, only two ways of obtaining Pade approximants applied to the power series resulting from power-system-type problems have been discussed in detail: the matrix method and the Viskovatov method. This paper explores several methods of accelerating the convergence of these power series and/or providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of Stahl’s theorem (i.e., those computing Pade approximants), and those that are not. For methods that are consistent with Stahl’s theoretical convergence guarantee, we identify which methods are computationally less expensive, which have better numerical performance and what remedies exist when these methods fail to converge numerically.

We consider the problem of numerical analytic continuation of an analytic function f ( z ) = f ( x + i y ) on a strip domain Ω + = { z = x + i y ∈ C | x ∈ R , 0 y y 0 } , where the data is given approximately only on the line y = 0 . This is a severely ill-posed problem. Motivated by the advantage of iterative methods for solving ill-posed problems, we propose a new modified iterative method to solve this problem under both a priori and a posteriori parameter choice rules. Moreover, some sharp error estimates between the exact solution and its approximation are proved. Some interesting numerical examples are conducted for showing that the newly-developed method works well.

In the present paper, we introduce a measure of the non-univalency of an analytic function, and we use it in order to find the best approximation of analytic function by a locally univalent normalized analytic function.

We present a numerical computation of overlaps in mass quenches in sine-Gordon quantum field theory using truncated conformal space approach (TCSA). To improve the cut-off dependence of the method, we use a novel running coupling definition which has a general applicability in free boson TCSA. The numerical results are used to confirm the validity of a previously proposed analytical Ansatz for the initial state in the sinh-Gordon quench., 13 pages, 4 pdf figures

The application of analytic continuation in quantum field theory (QFT) is juxtaposed to T-duality and mirror symmetry in string theory. Analytic continuation—a mathematical transformation that takes the time variable t to negative imaginary time— it —was initially used as a mathematical technique for solving perturbative Feynman diagrams, and was subsequently the basis for the Euclidean approaches within mainstream QFT (e.g., Wilsonian renormalization group methods, lattice gauge theories) and the Euclidean field theory program for rigorously constructing non-perturbative models of interacting QFTs. A crucial difference between theories related by duality transformations and those related by analytic continuation is that the former are judged to be physically equivalent while the latter are regarded as physically inequivalent. There are other similarities between the two cases that make comparing and contrasting them a useful exercise for clarifying the type of argument that is needed to support the conclusion that dual theories are physically equivalent. In particular, T-duality and analytic continuation in QFT share the criterion for predictive equivalence that two theories agree on the complete set of expectation values and the mass spectra and the criterion for formal equivalence that there is a “translation manual” between the physically significant algebras of observables and sets of states in the two theories. The analytic continuation case study illustrates how predictive and formal equivalence are compatible with physical inequivalence, but not in the manner of standard underdetermination cases. Arguments for the physical equivalence of dual theories must cite considerations beyond predictive and formal equivalence. The analytic continuation case study is an instance of the strategy of developing a physical theory by extending the formal or mathematical equivalence with another physical theory as far as possible. That this strategy has resulted in developments in pure mathematics as well as theoretical physics is another feature that this case study has in common with dualities in string theory.

On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the divergent series from Serre's formula can be related to these fractional multiplicities. By applying work of Avramov and Buchweitz, we positively answer Fulton's question in the context of graded rings., Comment: 8 pages, 1 figure

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings h ( Δ ) of a given discriminant Δ, over the integers, to the number of cubic rings h ˆ ( Δ ) of discriminant − 27 Δ in which every element has trace divisible by 3: (1) h ˆ ( Δ ) = { 3 h ( Δ ) if Δ > 0 h ( Δ ) if Δ 0 , where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the functions h and h ˆ ) to be put in self-reflective form. In 1998, J. Nakagawa verified (1) . We present a new proof of (1) that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity.

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

This paper is focused on a nonlinear inverse problem for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by the measured data on a single boundary point for one-dimensional case. We give the definition of a weak solution and prove its existence for the corresponding direct problem by using the Fourier method. Based on the Gronwall inequality, analytic continuation and the Laplace transformation, we obtain the uniqueness for the inverse zeroth-order coefficient problem under some simple requirements to the Neumann boundary data.

Fourier modal expansion techniques are very well suited for electromagnetic (EM) scattering in planar periodic stratified media. These techniques expand the transverse EM fields in Bloch modes in each strata unit-cell, but do not rigorously enforce EM boundary conditions at interior material interfaces which gives rise to slow convergence of the polarized S-matrix values with increasing Fourier series truncation order. We present a normal vector (NV) Fourier modal scattering approach that uses the gradient of the real permittivity in the unit-cell to generate a local NV resulting in anisotropic eigenmodes of Maxwell's equations that satisfy interior boundary conditions. The convergence of this expansion method is examine numerically for our analytic NV derived from planar primitive geometry elements for various example structures. The Fourier modal NV method is then derived for the planar cavity boundary value problem and the cavity eigenmodes and density of states is determined from the analytic continuation of the secular determinant. A complete scattering method for highly dispersive materials and sub-wavelength structures has been presented that is adaptable to planar EM cavity problems using fluctuation electrodynamics formalism.

The analytic continuation of correlation functions of composite operators from spacelike to timelike momenta is an open problem. In this talk we report on the analytic continuation of pion propagators using an effective field theory at finite temperature just below the finite temperature cross over of QCD. The spectral function is seen to be non-trivial. As a result, neither the screening nor the pole mass of the Euclidean propagators are equal to the mass in the Minkowski theory. Similarly, the “pion decay constant” computed in Euclidean is not the same as that in real time.

In this paper, we study asymptotics and fast computation of highly oscillatory Hilbert transforms ⨍ 0 + ∞ f ( x ) x − τ e i ω x H ν ( 1 ) ( ω x ) d x , where τ > 0 and f is analytic in the complex plane ℜ(z) ≥ 0, which may have an algebraic singularity at the point x = 0 . By analytic continuation, we first transform Hilbert transforms into the integrals on [ 0 , + ∞ ) with the integrands that don’t oscillate and decay exponentially, and then present the asymptotic behaviour about ω based on asymptotic expansion. For the computation of highly oscillatory Hilbert transforms, we present efficient numerical methods according to the position of τ. To be precise, we construct a Gaussian quadrature rule for it if τ = O ( 1 ) or τ ≥ 1. If 0

Individual displacements and stresses in a vertically-loaded notched finite graphite/epoxy laminated composite are determined by processing measured values of a single component of displacement with an Airy stress function in complex variables. Displacements are recorded using digital image correlation. Traction-free conditions are imposed analytically at the notch using conformal mappings and analytic continuation, and discretely on the vertical free edge. Zero shear stress is also imposed on horizontal line of symmetry. Consequences of employing different amounts and source locations of measured displacements and varying number of coefficients, as well as how displacements are differentiated to provide strains, are considered. Reliability of experimental results is demonstrated by Finite Element and force equilibrium.

We study the transient elastic field induced by an edge dislocation near a nanosized circular elastic inhomogeneity in which the effects of interface slip and diffusion are incorporated into the model of deformation. Separate Gurtin-Murdoch surface elasticities are specified on the surface of the inhomogeneity and on the adjoining surface of the surrounding matrix. In addition, rate-dependent interface slip and diffusion are assumed to occur concurrently on the inhomogeneity-matrix interface. The ensuing interaction problem is solved using a simple yet effective method based on analytic continuation and a convenient decomposition of the proposed solution. In particular, our method allows us to circumvent the second-order tangential derivative taken with respect to the interfacial normal stress, typically a source of additional complication and often an obstacle to the solution of such problems. The original problem is reduced to two coupled linear algebraic equations and a number of mutually independent sets of state-space equations, the general solutions of which can be obtained by solving the associated generalized eigenvalue problem. The image force acting on the edge dislocation is derived using the Peach-Koehler formula. Corresponding stress and displacement fields as well as the image force are found to be dependent on four size-dependent dimensionless parameters (arising from the surface elasticities) and on two size-dependent parameters (having the dimension of time) arising from the incorporation of interface slip and diffusion and they evolve with an infinite number of size-dependent relaxation times.

We propose a novel approach to solving and analyzing linear rational expectations models with general information frictions. Our approach is built upon policy function iterations in the frequency domain. We develop the theoretical framework of this approach using rational approximation, analytic continuation, and discrete Fourier transform. We provide the numerical implementation accompanied by a flexible object-oriented toolbox. We demonstrate the efficiency and accuracy of our method by studying four models in macroeconomics and finance that feature asymmetric information sets, endogenous signals, and higher-order expectations.

Well-known measures of entanglement in one-dimensional many body quantum systems, such as the entanglement entropy and the logarithmic negativity, may be expressed in terms of the correlation functions of local fields known as branch point twist fields in a replica quantum field theory. In this "replica" approach the computation of measures of entanglement generally involves a mathematically non-trivial analytic continuation in the number of replicas. In this paper we consider two-point functions of twist fields and their analytic continuation in the 1+1 dimensional massive (non-compactified) free Boson theory. This is one of the few theories for which all matrix elements of twist fields are known so that we may hope to compute correlation functions very precisely. We study two particular two-point functions which are related to the logarithmic negativity of semi-infinite disjoint intervals and to the entanglement entropy of one interval. We show that our prescription for the analytic continuation yields results which are in full agreement with conformal field theory predictions in the short-distance limit. We provide numerical estimates of universal quantities and their ratios, both in the massless (twist field structure constants) and the massive (expectation values of twist fields) theory. We find that particular ratios are given by divergent form factor expansions. We propose such divergences stem from the presence of logarithmic factors in addition to the expected power-law behaviour of two-point functions at short-distances. Surprisingly, at criticality these corrections give rise to a log(logL) correction to the entanglement entropy of one interval of length L. This hitherto overlooked result is in agreement with results by Calabrese, Cardy and Tonni and has been independently derived by Blondeau-Fournier and Doyon (in preparation)., Comment: 35 pages, 6 figures (v3 is the published version which incorporates minor improvements)

In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles.

We reconsider (namely, conclusively improve, discuss in more detail, supply with more numerical tests, at last compare with the studies of other authors) and extend our earlier works on applicability of Waterman׳s approach. In the electrostatic case which is more simple but still principally similar to the light scattering one, we discuss the condition of existence of solution to the infinite system given by this approach and of convergence of solutions to the truncated systems with a growing number of equations. The condition found earlier and depending on the particle shape only (namely, all singularities of the analytic continuation of internal field must be more distant than all singularities of the analytic continuation of “scattered” field) must include the following correction for spheroids: the approach is applicable to any particles of this shape. We discuss how this condition can be related to other light scattering methods, in particular the generalized point matching one. As illustrations we present some results of special calculations made with high accuracy for inverse spheroids. Besides showing a general agreement of the theoretical applicability condition with numerical results and the works of other authors, we note several still unclear aspects of the problem.

This paper discusses an FMM for solving waveguide problems and associated eigenvalue problems for Helmholtz’ equation in a two dimensional infinite strip with homogeneous Neumann boundary condition on the sides. Layer potentials with Green’s function for this problem are evaluated efficiently with the help of the method of images and FMM. We apply FMM to solve some boundary value problems in waveguides and associated resonance frequency problems using the Sakurai–Sugiura projection method after discussing the required analytic continuation of the solutions to complex frequencies. Some numerical examples show the accuracy and the efficiency of the proposed method.

Available Transfer Capability (ATC) is the additional power that can be securely and reliably transacted over and above the already committed transactions in a power network. In the electricity market, where every player (power producer and consumer) tries to earn higher profit, quick and accurate assessment of ATC of the power network is a prime concern. Holomorphic Embedding (HE) is a novel and non-iterative way of solving the Power Flow (PF) equations, which guarantees to find the operational High Voltage (HV) solution, in case it exists. Analytic continuation extends the domain of convergence of non-analytic functions to have voltage solutions. This paper, thus, proposes an extremely efficient and accurate approach for evaluation of ATC using the Holomorphic Embedded PF (HEPF) considering the intact system condition and N - 1 contingency situations. This approach is successfully tested on IEEE 30 bus, IEEE 118 bus, and IEEE 300 bus systems for various bilateral and multilateral transactions. The results, thus, obtained prove the efficacy of the proposed approach in comparison with those obtained from the repeated Newton-Raphson PF (NRPF), Fast Decoupled PF (FDPF) and Continuation PF (CPF) based methods.

This paper is devoted to determine the fractional order, the initial flux speed and the boundary Neumann data simultaneously in a one-dimensional time-fractional diffusion-wave equation from part boundary Cauchy observation data. We prove the uniqueness result for this inverse problem by using a new result for the Mittag-Leffler function and Laplace transform combining with analytic continuation. Then we use the iterative regularizing ensemble Kalman method in Bayesian framework to solve the inverse problem numerically. And four numerical examples are provided to show the effectiveness and stability of the proposed algorithm.