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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 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.

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 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.

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.

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.

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.

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.

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.

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”.

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.

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 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.

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.

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.

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.

In this paper we establish a multivariable non-commutative generalization of Loewner's theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half-plane into itself. The non-commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples, not a priori assumed to be analytic or continuous, as a Schur complement of a positive linear matrix pencil. To achieve this we establish the abstract integral formula for these functions using only matrix convexity and LMIs that represents operator monotone and operator concave free functions as a conditional expectation of a Schur complement of a linear matrix pencil on a tensor product operator algebra. The results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so, including the Karcher mean, to obtain an explicit, closed formula for these operator means.

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.

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

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 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.

Electromagnetic problem by a plane angular sector is studied. The perfectly conducting sector is illuminated by a plane electromagnetic wave. The scattered wave field consists of several components at large distances from the vertex of the sector. In the framework of the consequent and mathematically justified procedure, based on the Watson–Bessel and Sommerfeld integral representations of the electromagnetic Debye potentials, we develop expressions for the electromagnetic diffraction coefficients of the spherical wave from the vertex. To that end, we carefully study analytic properties of the Sommerfeld transformants and give a new procedure of analytic continuation for them. Like in the acoustic problem, the singularities of the Sommerfeld transformants on the real axis are responsible for the different components in the far field asymptotics. The expressions for the diffraction coefficients are specified in the form of integrals depending on the so called spectral functions, i.e. solutions of the boundary value problem for the Laplace–Beltrami operator on the unit sphere with the cut along a segment of big circle. The spectral functions are found in terms of series of “spherical harmonics”, whereas the coefficients of the series solve a Fredholm system of linear summation equations of the second kind. The properties of the spectral functions are also discussed and their interconnections with the “nonstationary” Green’s functions for the problems for some hyperbolic operator on the unit sphere with the cut are also established. The latter Green’s functions are exploited in the formulas for the analytic continuation of the Sommerfeld transformants. The results of the work form a theoretical basis for the numerical evaluation of the diffraction coefficients and for the description of the other components in the far field asymptotics to be given in the second part of the work.

In this paper, we consider the evaluation of two kinds of oscillatory integrals with a Hankel function as kernel. We first rewrite these integrals as the integrals of Fourier-type. By analytic continuation, these Fourier-type integrals can be transformed into the integrals on 0, +∞), the integrands of which are not oscillatory, and decay exponentially fast. Consequently, the transformed integrals can be efficiently computed by using the generalized Gauss-Laguerre quadrature rule. Moreover, the error analysis for the presented methods is given. The efficiency and accuracy of the methods have been demonstrated by both numerical experiments and theoretical results.

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.

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.

Let φ be an operator monotone function from the interval ( 0 , ∞ ) into itself whose analytic continuation is univalent on the union D of the interval and upper and lower half-planes, and ψ its inverse. For a square matrix A whose all eigenvalues are in φ ( D ) , the solution X of the equation D ψ ( A ) ( X ) = Y , X = D φ ( ψ ( A ) ) ( Y ) , is given explicitly; here D φ ( A ) is the Frechet derivative of φ at A. An example related to Dyson's expansion is also discussed.

Let π and π′ be unitary automorphic cuspidal representations of GLm(QA) and GLm′(QA), respectively, where at least one of π and π′ is self-contragredient. Using the prime number theorem for Rankin–Selberg L-functions, we compute a sharper version of Selberg orthogonality that contains certain lower-order terms which depend on special values of the Rankin–Selberg L-function attached to the pair (π,π′) and a sum related to Hypothesis H. In a case by case analysis when m,m′⩽4 and Hypothesis H is known to be true, we show how the constants involved in the lower-order terms can be expressed in terms of special values of Rankin–Selberg convolutions of symmetric- and/or exterior-power L-functions. In addition to showing that these constants give arithmetic information about the representations π and π′, we demonstrate how Hypothesis H can be used to give analytic continuation of the L-functions involved in the computation of the constants.

Following ideas of Langlands and Sarnak on Beyond Endoscopy, we introduce a method to study the base change L-function of a modular form via the trace formula. We complete the analysis in the case of quadratic base change for real fields, to deduce a new proof of the analytic continuation of the base change L-function to the left of the line R(s)=1. © 2014 Elsevier Inc.

In the present paper we consider the problem of numerical analytic continuation for an analytic function f ( z ) = f ( x + iy ) on a strip domain Ω + = { z = x + iy ∈ C | x ∈ R , 0 y y 0 } . The key difference with the known methods is that a novel iteration regularization method with a priori and a posteriori parameter choice rule is presented. The comparison in numerical aspect with other methods is also discussed.

Planar central configurations of point masses that have one or more clusters of bodies are created by analytic continuation. The singularity of the gravitational interaction is removed from the continuation equations by algebraic means. The continuation splits a single body into several, and the initial mass of the single body can be nonzero. Necessary conditions are derived for these continuations that may be useful in addressing the question of finiteness of central configurations.

In this paper, the problem of numerical evaluation of a class of highly oscillatory integrals ? a b f ( x ) ( x - a ) α ( b - x ) β ? k = 1 n x - ? k γ k e i ω x dx , where α , β < 1 , γ k ≤ 1 , a < ? k < b and ω ? 1 , has been discussed. Based on analytic continuation, the integrals are transformed into the problems of integrating the forms on 0 , + ∞ ) with the integrand does not oscillate and decays exponentially fast, which can be efficiently computed by using general Gauss-Laguerre quadrature rule. Theoretical results and numerical experiments perform that the method is very efficient in obtaining very high precision approximations if ω is sufficiently large.

We prove strong spectral estimates for Ruelle transfer operators for arbitrary C 2 contact Anosov flows. As a consequence of this we obtain: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error; (c) exponential decay of correlations for Holder continuous observables with respect to any Gibbs measure.