Analytic Coefficients Research Articles

Reduction with degenerate Gram matrix for one-loop integrals

An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the new method is a self-completed method in [3]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences.

Growth of Solutions to Linear Differential Equations with Analytic Coefficients in a Punctured Disc

Growth of Solutions to Linear Differential Equations with Analytic Coefficients in a Punctured Disc

DESCRIPTION OF GROWTH AND OSCILLATION OF SOLUTIONS OF COMPLEX LDE’S

AbstractIt is known that, in the unit disc as well as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc ,A_{k-2}$ of $$ \begin{align*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geqslant 2, \end{align*} $$ determines, under certain growth restrictions, not only the growth but also the oscillation of the equation’s nontrivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$ , $0<R\leqslant \infty $ , by using several measures for growth that are more flexible than those in the existing literature, and therefore permit more detailed analysis. In particular, the results obtained are not restricted to cases where the solutions are of finite (iterated) order of growth in the classical sense. The new findings are based on an accurate integrated estimate for logarithmic derivatives of meromorphic functions, which preserves generality in terms of three free parameters.

On the Continuation of Solutions of Linear Equations with Analytic Coefficients

On the Continuation of Solutions of Linear Equations with Analytic Coefficients

Elliptic Differential Operators with Analytic Coefficients and Linear Degeneracy

Elliptic Differential Operators with Analytic Coefficients and Linear Degeneracy

Analytical solution for stress and displacement of a finite thickness soil layer subjected to strip footing

Consider an elastic soil layer with a shallow-buried strip footing at the upper boundary and bedrock at the lower boundary, the displacement along the contact surface between the soil layer and the bedrock can be considered as zero when the bedrock is sufficiently rigid relative to the soil layer. The stress and displacement fields in the soil layer are solved on the premise that the displacement on the contact surface between the soil layer and the strip footing is given. The soil layer with the finite thickness on the physical plane is mapped onto the inner domain of the unit circle on the image plane based on the conformal transformation. Then, the complex variable method is applied to give the stress and displacement fields in the soil layer by two analytic functions, the coefficients of which can be determined by the stress and displacement boundary conditions on the upper and lower boundaries of the soil layer. In calculation, the power series method is introduced to establish a set of linear equations containing the analytic function coefficients, and then the coefficients are determined by solving the linear equations. This paper presents the analytical solutions of stress and displacement in the soil layer under two different displacement boundary conditions on the contact surface between the soil layer and the footing. The obtained results are compared with the results of the numerical method based on ANSYS. Finally, the influence of different displacement boundary conditions and soil thickness on the stresses on the surface and internal region of the soil layer is analyzed.

Double order of growth of solutions to linear differential equations with analytic coefficients

In this paper, we investigate the [p,q]-order of growth of solutions to certain linear differential equations with entire coefficients and analytic coefficients in the unit disc by using the Nevanlinna theory of meromorphic functions. This work is an improvement and generalization of some previous results by the second author.

Global unique continuation from the boundary for a system of viscoelasticity with analytic coefficients and a memory term

<p style='text-indent:20px;'>A global unique continuation property (UCP) from the boundary for solutions to a viscoelastic system with a memory term is presented. The density and elasticity tensors are assumed to be real analytic. The tensors can be anisotropic and satisfy physically natural conditions such as full symmetry and strong convexity. The global UCP is given in terms of the travel time of the slowest wave of the viscoelastic system, which is the optimal description for the global UCP in our setup.</p>

Inverse, ill-posed problems, essentially different solutions and explicit formulas for solutions

A request for an inverse problem, as well as for an incorrect problem produces tens of millions of answers in the Internet. In the past few decades, hundreds of international conferences on these topics have been held annually. Problems of this kind are quite involved, and their numerical analysis requires the development of special methods and numerical algorithms. Explicit formulas provide the main tool for testing these methods and numerical algorithms. The Cauchy problem for an elliptic equation is a classical ill-posed problem, which serves as a model for many inverse and incorrect problems. In the present paper we give a numerically realizable explicit formula for solving the Cauchy problem in a two-dimensional domain for a general second-order linear elliptic equation with analytic coefficients and the Cauchy analytic data on the analytic boundary.

Analytic tadpole coefficients of one-loop integrals

One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.

BLUES function method applied to partial differential equations and analytic approximants for interface growth under shear

An iteration sequence based on the BLUES (beyond linear use of equation superposition) function method is presented for calculating analytic approximants to solutions of nonlinear partial differential equations. This extends previous work using this method for nonlinear ordinary differential equations with an external source term. Now, the initial condition plays the role of the source. The method is tested on three examples: a reaction-diffusion-convection equation, the porous medium equation with growth or decay, and the nonlinear Black-Scholes equation. A comparison is made with three other methods: the Adomian decomposition method (ADM), the variational iteration method (VIM), and the variational iteration method with Green function (GVIM). As a physical application, a deterministic differential equation is proposed for interface growth under shear, combining Burgers and Kardar-Parisi-Zhang nonlinearities. Thermal noise is neglected. This model is studied with Gaussian and space-periodic initial conditions. A detailed Fourier analysis is performed and the analytic coefficients are compared with those of ADM, VIM, GVIM, and standard perturbation theory. The BLUES method turns out to be a worthwhile alternative to the other methods. The advantages that it offers ensue from the freedom of choosing judiciously the linear part, with associated Green function, and the residual containing the nonlinear part of the differential operator at hand.

Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

We prove that the homogeneously polyanalytic functions of total order $m$, defined by the system of equations $\overline{D}^{(k_1,\ldots,k_n)} f=0$ with $k_1+\cdots+k_n=m$, can be written as polynomials of total degree $<m$ in variables $\overline{z_1},\ldots,\overline{z_n}$, with some analytic coefficients. We establish a weighted mean value property for such functions, using a reproducing property of Jacobi polynomials. After that, we give a general recipe to transform a reproducing kernel by a weighted change of variables. Applying these tools, we compute the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in $\mathbb{C}^n$ and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (1977), Pessoa (2014), Hachadi and Youssfi (2019).

Fast growing solutions of linear differential equations with analytic coefficients in the unit disc

In this paper, we investigate the growth of solutions of higher order linear differential equations with analytic coefficients of ϕ-order in the unit disc. We introduce new definitions of the lower order and the type related to the ϕ-order concepts to generalise and extend previous results due to Chyzhykov-Semochko [6], Semochko [14], Belaïdi [1,2,3], Hu-Zheng [12].

Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity

In this article, we are interested in the Gevrey properties of the formal power series solution in time of the partial differential equations with a polynomial semilinearity and with analytic coefficients at the origin of $${\mathbb {C}}^{n+1}$$ . We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any $$s\ge s_c$$ , where $$s_c$$ is a nonnegative rational number fully determined by the Newton polygon of the associated linear PDE. In the opposite case $$s<s_c$$ , we show that the solution is generically $$s_c$$ -Gevrey while the inhomogeneity is s-Gevrey, and we give an explicit example in which the solution is $$s'$$ -Gevrey for no $$s'<s_c$$ .

Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients

We are interested in the Gevrey properties of the formal power series solution in time of the inhomogeneous semilinear heat equation with a power-law nonlinearity in $1$-dimensional time variable $t\in\mathbb{C}$ and $n$-dimensional spatial variable $x\in\mathbb{C}^n$ and with analytic initial condition and analytic coefficients at the origin $x=0$. We prove in particular that the inhomogeneity of the equation and the formal solution are together $s$-Gevrey for any $s\geq1$. In the opposite case $s<1$, we show that the solution is $1$-Gevrey at most while the inhomogeneity is $s$-Gevrey, and we give an explicit example in which the solution is $s'$-Gevrey for no $s'<1$.

Analytic hypoellipticity of Keldysh operators

We consider Keldysh-type operators, $ P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q (x, D_{x'} ) $, $ x = ( x_1, x') $ with analytic coefficients, and with $ Q ( x, D_{x'} ) $ second order, principally real and elliptic in $ D_{x'} $ for $ x $ near zero. We show that if $ P u =f $, $ u \in C^\infty $, and $ f $ is analytic in a neighbourhood of $ 0 $ then $ u $ is analytic in a neighbourhood of $ 0 $. This is a consequence of a microlocal result valid for operators of any order with Lagrangian radial sets. Our result proves a generalized version of a conjecture made by the second author and Lebeau and has applications to scattering theory.

Next-to-Next-to-Leading Order Calculation of Quasiparton Distribution Functions.

We present the next-to-next-to-leading order (NNLO) calculation of quark quasiparton distribution functions (PDFs) in the large momentum effective theory. The nontrivial factorization at this order is established explicitly and the full analytic matching coefficients between the quasidistribution and the light-cone distribution are derived. We demonstrate that the NNLO numerical contributions can improve the behavior of the extracted PDFs sizably. With the unprecedented precision study of nucleon tomography at the planned electron-ion collider, high precision lattice QCD simulations with our NNLO results implemented will enable to test the QCD theory and more precise results on the PDFs of nucleons will be obtained.

Diagonal symmetrizers for hyperbolic operators with triple characteristics

Symmetrizers for hyperbolic operators are obtained by diagonalizing the Bézoutian matrix of the principal symbols and its derivatives. Such diagonal symmetrizers are applied to the Cauchy problem for hyperbolic operators with triple characteristics. In particular, the Ivrii’s conjecture concerned with strongly hyperbolic operators with triple effectively hyperbolic characteristics is proved for differential operators with time dependent coefficients, also for third order differential operators with two independent variables with analytic coefficients.

An Inverse Problem for LTV ODE’s

It is well known that any linear combination of products of polynomials and exponentials (a.k.a. Bohl functions) are annihilated by LTI-ODE’s. In this paper, we consider the more general classes of piecewise Bohl functions and of real analytic functions and present the realization of the latter as solutions of linear differential polynomials with real analytic coefficients. In particular, we show that the minimal order depends solely on the multiplicity of the zeros of the given function. We extend the problem further to meromorphic functions without poles on the real line. We briefly comment on piecewise-defined functions and the nonhomogeneous problem.

Linear Differential Equations with Analytic Coefficients Having the Same Order Near a Singular Point

In this paper, we investigate the growth of solutions of the differential equation: $$\begin{aligned} f^{\left( k\right) }+A_{k-1}\left( z\right) \exp \left\{ \frac{a_{k-1}}{ \left( z_{0}-z\right) ^{n}}\right\} f^{\left( k-1\right) }+\cdots +A_{0}\left( z\right) \exp \left\{ \frac{a_{0}}{\left( z_{0}-z\right) ^{n}}\right\} f=0, \end{aligned}$$ where $$A_{j}\left( z\right) $$ are analytic functions in the closed complex plane except at $$z_{0}$$ and $$a_{j}\left( j=0,\ldots ,k-1\right) $$ are distinct complex numbers. Other linked cases have been studied.