Complex Analytic Method Research Articles

Spectral deconvolution of matrix models: the additive case

Abstract We implement a complex analytic method to build an estimator of the spectrum of a matrix perturbed by the addition of a random matrix noise in the free probabilistic regime. This method, which has been previously introduced by Arizmendi, Tarrago and Vargas, involves two steps: the first step consists in a fixed point method to compute the Stieltjes transform of the desired distribution in a certain domain, and the second step is a classical deconvolution by a Cauchy distribution, whose parameter depends on the intensity of the noise. This method thus reduces the spectral deconvolution problem to a classical one. We provide explicit bounds for the mean squared error of the first step under the assumption that the distribution of the noise is unitary invariant. In the case where the unknown measure is sparse or close to a distribution with a density with enough smoothness, we prove that the resulting estimator converges to the measure in the $1$-Wasserstein distance at speed $O(1/\sqrt{N})$, where $N$ is the dimension of the matrix.

Green’s function of heat equation for heterogeneous media in 3-D

The purpose of the present paper is to study the structure of Green’s function for heat equation in several spatial dimensions and with rough heat conductivity coefficient. We take the heat conductivity coefficient to be of bounded variation in the x direction and study the dispersion in the (y, z) direction. The goal is to understand the coupling of dissipation across rough heat conductivity and the multi-dimensional dispersion in the Green’s function A series of exponential functions of path integral with coefficients over a field of complex analytic functions around imaginary axis are formulated in the Laplace and Fourier transforms variables. The Green’s function in the transformed variables is written as the sum of these integrals over random paths. The integral over a random path is rearranged through the reflection property over a variation of heat conductivity coefficient and become a simple form in terms of path phase and amplitude. The complex analytic and combinatorics method is then used to yield a precise pointwise structure of the Green’s function in the physical domain

Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of “moduli-friendly” Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method.

Retraction Note: Existence results for the general Schr\xf6dinger equations with a superlinear Neumann boundary value problem

The main goal of this paper is to study the general Schrodinger equations with a superlinear Neumann boundary value problem in domains with conical points on the boundary of the bases. First the formulation and the complex form of the problem for the equations are given, and then the existence result of solutions for the above problem is proved by the complex analytic method and the fixed point index theory, where we absorb the advantages of the methods in recent works and give some improvement and development. Finally, we are also interested in the asymptotic behavior of solutions of the mentioned equation. These results generalize some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrodinger equations or of delay Schrodinger equations.

Effective numerical evaluation of the double Hilbert transform

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.

RETRACTED ARTICLE: Existence results for the general Schr\xf6dinger\t\t\tequations with a superlinear Neumann boundary value problem

The main goal of this paper is to study the general Schrödinger\t\t\t\tequations with a superlinear Neumann boundary value problem in domains with conical\t\t\t\tpoints on the boundary of the bases. First the formulation and the complex form of\t\t\t\tthe problem for the equations are given, and then the existence result of solutions\t\t\t\tfor the above problem is proved by the complex analytic method and the fixed point\t\t\t\tindex theory, where we absorb the advantages of the methods in recent works and give\t\t\t\tsome improvement and development. Finally, we are also interested in the asymptotic\t\t\t\tbehavior of solutions of the mentioned equation. These results generalize some\t\t\t\tprevious results concerning the asymptotic behavior of solutions of non-delay\t\t\t\tsystems of Schrödinger equations or of delay Schrödinger equations.

Singularity Analysis for Heavy-Tailed Random Variables

We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindel\"of integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws $p(k) = c\exp( -k^\alpha)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^\beta)$, $\beta>2$; they cover the big jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem.

The AFD methods to compute Hilbert transform

In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive Takenaka–Malmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterized Szegö and higher order Szegö kernels. This note proposes two kinds of such analytic approximations of which one is called maximal-energy AFDs, including core AFD, Unwending AFD and Cyclic AFD; and the other is again linear combinations of Szegö kernels but generated through SVM methods. The proposed methods are based on the fact that the imaginary part of an analytic signal is the Hilbert transform of its real part. As consequence, when a sequence of rational analytic functions approximates an analytic signal, then the real parts and imaginary parts of the functions in the sequence approximate, respectively, the original real-valued signals and its Hilbert transform. The two approximations have the same errors in the energy sense due to the fact that Hilbert transformation is a unitary operator in the L2 space. This paper for the first time promotes the complex analytic method for computing Hilbert transforms. Experiments show that such computational methods are as effective as the commonly used one based on FFT.

The research of complex analytic method in spect image reconstruction

In this paper, we successfully extend Natterer’s results to the attenuated Radon transform with imaginary coefficients. And by means of the theory of complex analysis and integral transform, we study the inversion of two-dimensional weighted Radon transform and derive similar explicit inversion formulas with two different methods, which are closely related to Novikov’s inversion formula.

Computing Igusa class polynomials

We bound the running time of an algorithm that computes the genus-two class polynomials of a primitive quartic CM-field K. This is in fact the first running time bound and even the first proof of correctness of any algorithm that computes these polynomials. Essential to bounding the running time is our bound on the height of the polynomials, which is a combination of denominator bounds of Goren and Lauter and our own absolute value bounds. The absolute value bounds are obtained by combining Dupont's estimates of theta constants with an analysis of the shape of CM period lattices. The algorithm is basically the complex analytic method of Spallek and van Wamelen, and we show that it finishes in time Otilde(Delta^(7/2)), where Delta is the discriminant of K. We give a complete running time analysis of all parts of the algorithm, and a proof of correctness including a rounding error analysis. We also provide various improvements along the way.

Existence of Solutions of Inverse Problems for Elliptic Complex Equations with Degenerate Curve in Multiple Connected Domains

The present paper deals with the inverse problem for degenerate elliptic systems of first order equations in multiple connected domains with Riemann-Hilbert type map. Firstly the formulation and the complex form of the problem for the degenerate elliptic systems of first order are given, and then the coefficients of the above systems are constructed by a new complex analytic method. As an application of the above results, we can derive the corresponding results of the inverse problem for degenerate elliptic equations of second order in multiple connected domains from Dirichlet to Neumann map.

Limit Theorems for Subtree Size Profiles of Increasing Trees

Simple families of increasing trees were introduced by Bergeron, Flajolet and Salvy. They include random binary search trees, random recursive trees and random plane-oriented recursive trees (PORTs) as important special cases. In this paper, we investigate the number of subtrees of size k on the fringe of some classes of increasing trees, namely generalized PORTs and d-ary increasing trees. We use a complex-analytic method to derive precise expansions of mean value and variance as well as a central limit theorem for fixed k. Moreover, we propose an elementary approach to derive limit laws when k is growing with n. Our results have consequences for the occurrence of pattern sizes on the fringe of increasing trees.

A new process for constructing coefficients of elliptic complex equations in multiply connected domains

This article concerns the inverse problem for linear elliptic systems of first-order equations with Riemann–Hilbert-type map in multiply connected domains. First the formulation and the complex form of the problem for the systems are given, and then the coefficients of the elliptic complex equations for the above problem are constructed by a complex analytic method, where the advantage of the methods in other papers is absorbed, and the used method in this article is more simple and the obtained result is more general. As an application of the above results, we can derive the corresponding results of the inverse problem for second-order elliptic equations from Dirichlet to Neumann map in multiply connected domains.

Oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve

The present paper deals with oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point principle. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.

Computing genus 2 curves from invariants on the Hilbert moduli space

We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required.

The oblique derivative problem for second order nonlinear equations of mixed type with two degenerate lines

In [1–10], the authors posed and discussed the Tricomi problem of second order equations of mixed (elliptic-hyperbolic) type, which possesses important applications to gas dynamics, but they only consider some special equations of mixed type. In [5, 7], the authors proposed and discussed the Tricomi problem for some second order equations of mixed type with nonsmooth degenerate line under stronger conditions. The present paper deals with the oblique derivative problem for second order nonlinear equations of mixed type with two parabolic degenerate lines in a plane domain. The boundary value problem includes the Tricomi problem of Chaplygin equation as a special case. Firstly the existence of solutions of corresponding boundary value problems for degenerate elliptic and hyperbolic equations of second order are discussed, and then the solvability of the oblique derivative problem for the equations of mixed type with two degenerate lines is proved. The used method in this paper is different to those in above papers, because the equations are nonlinear, and the new notations are introduced, such that the second order equations of mixed type with degenerate lines are reduced to the first order mixed complex equations with singular coefficients, then the advantage of complex analytic method can be applied, and we can see that the method can be used to handle more general problems. Finally we mention that the authors in [21–24] discussed oblique derivative problems for the Helmholtz and Laplace equations in the general interior and exterior domains.

The inverse problem for linear elliptic systems of first order with Riemann–Hilbert-type map in multiply connected domains

The present article deals with the inverse problem for linear elliptic systems of first-order equations with Riemann–Hilbert-type map in multiply connected domains. First, the formulation and the complex form of the problem for the systems are given, and then the existence and global uniqueness of solutions for the above problem are proved by a complex analytic method.

The Inverse Problem for Elliptic Equations from Dirichlet to Neumann Map in Multiply Connected Domains

The present paper deals with the inverse problem for linear elliptic equations of second order from Dirichlet to Neumann map in multiply connected domains. Firstly the formulation and the complex form of the problem for the equations are given, and then the existence and global uniqueness of solutions for the above problem are proved by the complex analytic method, where we absorb the advantage of the methods in previous works and give some improvement and development.

Discontinuous Irregular Oblique Derivative Problems for Nonlinear Elliptic Equations of Second Order

In this paper, the discontinuous irregular oblique derivative problems (or discontinuous Poincar {\rm\acute e} boundary value problems) for nonlinear elliptic equations of second order in multiply connected domains are discussed by using a complex analytic method. Firstly the uniqueness of solutions for such boundary value problems is proved and a priori estimates of their solutions are given, and then by the Leray-Schauder theorem, the existence of solutions of the above problems is verified. As a special case the result about the continuous irregular oblique derivative problem for the nonlinear equations is derived.

Solvability of the Tricomi problem for second order equations of mixed type with degenerate curve on the sides of an angle

AbstractIn [1]–[6], the author posed and discussed the Tricomi problem of second order mixed equations, but he only consider some special mixed equations. In [3], the author discussed the uniqueness of solutions of the Tricomi problem for some second order mixed equation with nonsmooth degenerate line. The present paper deals with the Tricomi problem for general second order mixed equations with degenerate curve on the sides of an angle. I first give the formulation of the above problem, and then prove the solvability of the Tricomi problem for the mixed equations with degenerate curve on the sides of an angle, by using the existence of solutions of the mixed problem for the degenerate elliptic equations (see [11]). Here I mention that the used method in this paper is different to those in other papers or books, because I introduce the new notation (2.1) below, such that the second order equation of mixed type can be reduced to the first order complex equation of mixed type with singular coefficients, hence I can use the advantage of complex analytic method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)