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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.

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.

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/ .

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.

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.

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.

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

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.

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.

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

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.

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.

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.

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.

In this paper, we propose a numerical method for calculating Fourier transforms based on hyperfunction theory, a theory of generalized functions based on complex function theory. In the proposed method, we first obtain analytic functions giving the desired Fourier transform as a hyperfunction, and, then, we compute the Fourier transform by using continued fractions to determine the analytic continuation of these analytic functions onto the real axis. Using the proposed method, we can evaluate Fourier transforms that are difficult to evaluate using the conventional numerical integration rules. In addition, we can use the proposed method to evaluate Fourier transforms of hyperfunctions. Numerical examples show the efficiency of the presented method compared to previous methods.

We consider a two-dimensional boundary value problems for the Helmholtz equation with Dirichlet and Neumann boundary conditions on a set of arcs. This set is obtained from a closed curve by cutting out small holes situated close to each other and having a locally periodic structure. We construct the asymptotics of the scattering frequencies (poles of the analytic continuation of solutions) with small imaginary parts, which converge to the square roots of multiple eigenvalues of limit problems.

We study increasing stability in the interior inverse source problem for the Helmholtz equation from boundary Cauchy data for multiple wave numbers. By using the Fourier transform with respect to the wave number, explicit bounds for the analytic continuation, uniqueness of the continuation results, and exact observability bounds for the wave equation, a sharp uniqueness result and an increasing (with larger wave numbers intervals) stability estimate are obtained. Numerical examples in 3 spatial dimension support the theoretical prediction.

The analytic continuation of a family of Dirichlet series whose coefficients are partition functions having parts in a finite set is established. The singularities arising from this continuation are shown to be finitely many simple poles, where the residues are computable positive rational numbers. Then we give several applications. First, special values of the continuation are studied in terms of algebraic combinations of the Hurwitz zeta function. Second, a transformation formula is obtained for exponentially decaying series involving these partition functions using special values of the continuation at negative integers. Finally, asymptotic behavior and orthogonality of certain convolutions with the normalization of these partition functions are investigated in the mean value sense with precise error terms.

In scattering experiments, physicists observe so-called resonances as peaks at certain energy values in the measured scattering cross sections per solid angle. These peaks are usually associate with certain scattering processes, e.g., emission, absorption, or excitation of certain particles and systems. On the other hand, mathematicians define resonances as poles of an analytic continuation of the resolvent operator through complex dilations. A major challenge is to relate these scattering and resonance theoretical notions, e.g., to prove that the poles of the resolvent operator induce the above mentioned peaks in the scattering matrix. In the case of quantum mechanics, this problem was addressed in numerous works that culminated in Simon's seminal paper [33] in which a general solution was presented for a large class of pair potentials. However, in quantum field theory the analogous problem has been open for several decades despite the fact that scattering and resonance theories have been well-developed for many models. In certain regimes these models describe very fundamental phenomena, such as emission and absorption of photons by atoms, from which quantum mechanics originated. In this work we present a first non-perturbative formula that relates the scattering matrix to the resolvent operator in the massless Spin-Boson model. This result can be seen as a major progress compared to our previous works [13] and [12] in which we only managed to derive a perturbative formula., Comment: 26 pages, 3 figure. arXiv admin note: text overlap with arXiv:1801.04843

We use conformal mapping techniques and analytic continuation to derive an analytic solution to the problem of an edge dislocation interacting with two semi-infinite vertical cracks. Closed-form expressions for the rigid-body rotation at infinity of the left half-plane, the image force acting on the edge dislocation due to its interaction with the two cracks and mode I and mode II stress intensity factors at the two crack tips induced by the edge dislocation are obtained.

Inverse source problems for partial differential equations are of interest both in mathematics and in engineering. In this paper, we consider the problem of recovering the spatial contamination source in the diffusion model with the local measurements at fixed times. One example is constructed to show that the local measurements at one fixed time are not enough to determine the source. We propose the problem of recovering the source with the local measurements at two different fixed times. The uniqueness and conditional stability estimate are proved by the analytic continuation method.

We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the K\"all\'en-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the K\"all\'en-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularize this problem we implement an appropriate version of Tikhonov regularization supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two ---mathematically equivalent--- versions of the K\"all\'en-Lehmann spectral integral., Comment: 33 pages, 18 figures

The SternheimerGW software uses time-dependent density-functional perturbation theory to evaluate G W quasiparticle band structures and spectral functions for solids. Both the Green’s function G and the screened Coulomb interaction W are obtained by solving linear Sternheimer equations, thus overcoming the need for a summation over unoccupied states. The code targets the calculation of accurate spectral properties by convoluting G and W using a full frequency integration. The linear response approach allows users to evaluate the spectral function at arbitrary electron wavevectors, which is particularly useful for indirect band gap semiconductors and for simulations of angle-resolved photoelectron spectra. The software is parallelized efficiently, integrates with version 6.3 of Quantum Espresso, and is continuously monitored for stability using a test farm. Program summary Program Title: SternheimerGW Program Files doi: http://dx.doi.org/10.17632/7ywzjng9t5.1 Licensing provisions: GNU General Public License v3.0 Programming language: Fortran 2003 Nature of problem: The lack of the exchange–correlation discontinuity in density-functional theory (DFT) leads to a systematic underestimation of the band gap between conduction and valence states. Many-body perturbation theory in the G W approximation provides an effective solution to this problem, as well as other limitations faced by DFT in the description of electronic excitations. However, the G W method comes with its own set of limitations: (i) The perturbation of the system is typically expressed in terms of the unoccupied states, and achieving numerical convergence with respect to these states is often cumbersome. Since the underlying DFT codes rely only on the occupied states, their default behavior is often ill-suited to provide a sufficient amount of empty states. Combined with the large number of parameters that need to be converged, this limits the use of G W codes by non-expert users and automatic scripts. (ii) Currently, G W codes require that a homogeneous k -point mesh is used in the calculation. Hence, features close to the band edges are often only accessible via prohibitively expensive dense k -point meshes or interpolation techniques. (iii) To evaluate the frequency convolution of the Green’s function G and the screened Coulomb interaction W , many current G W calculations rely on approximations such as the plasmon-pole approximation, the analytic continuation, or the contour deformation. The relative merits and accuracy of the various approximations are not fully understood. Solution method: In SternheimerGW , we address (i) by replacing the summation over unoccupied states with a linear response equation. The solution depends on the occupied states only, and it employs linear response algorithms already provided by the Quantum ESPRESSO suite to compute phonons and related properties. As an additional benefit, transforming the problem in a linear response equation removes the restriction to homogeneous k -point meshes (ii), so that the G W self-energy for any arbitrary point can be evaluated. Finally (iii), we provide the possibility to perform a full frequency convolution along the real frequency axis. This feature can serve as a benchmark for approximate integrations using models or analytic continuation.

We introduce a new form of the Segal–Bargmann transform for a connected Lie group K of compact type. We show that the heat kernel ( ρ t ( x ) ) t > 0 , x ∈ K has a space-time analytic continuation to a holomorphic function ( ρ C ( τ , z ) ) Re τ > 0 , z ∈ K C , where K C is the complexification of K. The new transform is defined by the integral ( B τ f ) ( z ) = ∫ K ρ C ( τ , z k − 1 ) f ( k ) d k , z ∈ K C . If s > 0 and τ ∈ D ( s , s ) (the disk of radius s centered at s), this integral defines a holomorphic function on K C for each f ∈ L 2 ( K , ρ s ) . We construct a heat kernel density μ s , τ on K C such that, for all s , τ as above, B s , τ : = B τ | L 2 ( K , ρ s ) is an isometric isomorphism from L 2 ( K , ρ s ) onto the space of holomorphic functions in L 2 ( K C , μ s , τ ) . When τ = t = s , the transform B t , t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ = t ∈ ( 0 , 2 s ) , the transform B s , t coincides with the one introduced by the first two authors.

We consider two integrable deformations of 2d sigma models on supercosets associated with AdS n × S n . The first, the “η-deformation” (based on the Yang–Baxter sigma model), is a one-parameter generalization of the standard superstring action on AdS n × S n , while the second, the “λ-deformation” (based on the deformed gauged WZW model), is a generalization of the non-abelian T-dual of the AdS n × S n superstring. We show that the η-deformed model may be obtained from the λ-deformed one by a special scaling limit and analytic continuation in coordinates combined with a particular identification of the parameters of the two models. The relation between the couplings and deformation parameters is consistent with the interpretation of the first model as a real quantum deformation and the second as a root of unity quantum deformation. For the AdS 2 × S 2 case we then explore the effect of this limit on the supergravity background associated with the λ-deformed model. We also suggest that the two models may form a dual Poisson–Lie pair and provide direct evidence for this in the case of the integrable deformations of the coset associated with S 2 .

An ill-posed problem of synthesis of the Pierce electrodes for a cylindrical beam with a polygonal cross-section is considered. It is assumed that a beam of charged particles is extracted from a space-charge-limited planar diode and the Pierce electrodes outside of the beam ensure its zero angular divergence. A mathematical statement of the problem presumes a computation of the electrostatic potential outside of the beam that should match the Child-Langmuir 1D potential inside of the beam. An exact solution is first obtained for the potential outside of the beam near its right angle. The solution involves double analytic continuation and a numerical integration of the hypergeometric function and can be used as a benchmark for testing numerical codes. It is shown that equipotential surfaces have fractures that can be pushed away from the corner of the beam by means of smoothing the beam corners. This solution is then generalized to an angle of arbitrary magnitude., Comment: 4 pages, 2 figures

In this paper, by employing the k T factorization theorem, we made the first calculation for the space-like scalar pion form factor Q 2 F ( Q 2 ) at the leading order (LO) and the next-to-leading order (NLO) level, and then found the time-like scalar pion form factor F a , I ′ ( 1 ) by analytic continuation from the space-like one. From the analytical evaluations and the numerical results, we found the following points: (a) the NLO correction to the space-like scalar pion form factor has an opposite sign with the LO one but is very small in magnitude, can produce at most 10 % decrease to the LO result in the considered Q 2 region; (b) the NLO time-like scalar pion form factor F a , I ′ ( 1 ) describes the O ( α s 2 ) contribution to the factorizable annihilation diagrams of the considered B → π π decays, i.e. the NLO annihilation correction; (c) the NLO part of the form factor F a , I ′ ( 1 ) is very small in size, and is almost independent of the variation of cutoff scale μ 0 , but this form factor has a large strong phase around − 55 ° and may play an important role in producing large CP violation for B → π π decays; and (d) for B 0 → π + π − and π 0 π 0 decays, the newly known NLO annihilation correction can produce only a very small enhancement to their branching ratios, less than 3 % in magnitude, and therefore we could not interpret the well-known ππ-puzzle by the inclusion of this NLO correction to the factorizable annihilation diagrams.

We analyze the conditions on the Taylor coefficients of an analytic function to admit global analytic continuation, complementing a recent paper of Breuer and Simon on general conditions for natural boundaries to form. A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities, asymptotics for large values of the variable and asymptotic location of zeros.

Open string disk integrals are represented as contour integrals of a product of Beta functions using Mellin transformations. This makes the mathematical problem of computing the α ′ -expansion around the field-theory limit similar to that of the ϵ -expansion in Feynman loop integrals around the four-dimensional limit. More explicitly, the formula in Mellin space obtained directly from the standard Koba–Nielsen-like representation is valid in a region of values of α ′ that does not include α ′ = 0 . Analytic continuation is therefore needed since contours are pinched by poles as α ′ → 0 . Deforming contours that get pinched by poles generates a set of ( n − 3 ) ! multi-dimensional residues left behind which contain all the field theory information. Some analogies between the field theory formulas obtained by this method and those derived recently from using the scattering equations are commented at the end.

We use analytic continuation to derive the Euler–Lagrange equations associated to the Pfaffian in indefinite signature ( p , q ) directly from the corresponding result in the Riemannian setting. We also use analytic continuation to derive the Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. Complex metrics on the tangent bundle play a crucial role in our analysis and we obtain a version of the Chern–Gauss–Bonnet theorem in this setting for certain complex metrics.

This paper presents an efficient numerical method for approximating highly oscillatory Bessel transforms. Based on analytic continuation, we transform the integrals into the problems of integrating the forms on [ 0 , + ∞ ) with the integrand that does not oscillate and decays exponentially fast, which can be efficiently computed by using Gauss–Laguerre quadrature rule. We then derive the error of the method depending on the frequency and the node number. Moreover, we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations. Preliminary numerical results show the efficiency and accuracy of numerical approximations.

For any periodic function f : N → C with period q, we study the Dirichlet series L ( s , f ) : = ∑ n ≥ 1 f ( n ) / n s . It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1 , where it has a simple pole with residue ρ : = q − 1 ∑ 1 ≤ a ≤ q f ( a ) . Thus, the function is analytic at s = 1 when ρ = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L ( 1 , f ) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L ( 1 , f ) .

The present study recovers a time-dependent heat source H ( t ) in u t ( x , t ) = u xx ( x , t ) + H ( t ) , under measured initial heat flux, and Cauchy boundary conditions. The supplementary initial data are assumed to be analytic continuation ones being obtained by means of measurement, which are not given arbitrarily. We first transform the above problem into an inverse heat conduction problem without having the right-boundary value, and then, further transform it into another inverse heat source problem that we need to find F ( t ) in T t ( x , t ) = T xx ( x , t ) - xF ( t ) / l with measured initial and boundary conditions. By using the GL ( n , R ) Lie-group differential algebraic equations (LGDAE) method to integrate the resultant ordinary differential equations with a priori bound | F ( t ) | ⩽ F max , we can fast recover H ( t ) in real time. The accuracy and efficiency are assessed and confirmed by comparing the exact solutions with recovered results, where a large noise up to 10% or 20% is imposed on the input measured data.

We show that the analytic continuation of the exponential generating function associated to consecutive weighted pattern enumeration of permutations only has poles and no essential singularities. The proof uses the connection between permutation enumeration and functional analysis, and as well as the Laurent expansion of the associated resolvent. As a consequence, we give a partial answer to a question of Elizalde and Noy: when is the multiplicative inverse of the exponential generating function for the number permutations avoiding a single pattern an entire function? Our work implies that it is enough to verify that this function has no zeros to conclude that the inverse function is entire.