Heat Kernel Expansion for Operators of the Type of the Square Root of the Laplace Operator

A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG) coefficients for the operator $\sqrt{-\nabla^2 + V(x)}$ basing on a generalization of the pseudodifferential operator technique. The lowest DWSG coefficients for the operator $\sqrt{-\nabla^2} + V(x)$ are calculated by using the method proposed. It is shown that the method admits a generalization to the case of operators of the type $(-\nabla^2 + V(X))^{1/{\rm m}}$, where m is an arbitrary rational number. A more simple method is proposed for the calculation of the DWSG coefficients for the case of strictly positive operators under the sign of root. By using this method, it is shown that the problem of the calculation of the DWSG coefficients for such operators is exactly solvable. Namely, an explicit formula expressing the DWSG coefficients for operators with root through the DWSG coefficients for operators without root is deduced.Comment: 17 pages, LaTeX, no figure

A new and efficient method for the computation of Legendre coefficients

An efficient procedure for the computation of the coefficients of Legendre expansions is here presented. We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of an Abel-type transform of f(x). The computation of N Legendre coefficients can then be performed in O(N log N) operations with a single Fast Fourier Transform of the Abel-type transform of f(x).Comment: 5 page

The New Algorithm for the Determination of the Williams Asymptotic Expansion Coefficients for Notched Semidiscs Using the Photoelasticity Method and Finite Element Method

The study proposes the algorithm for the determination of the coefficients of the Williams series expansion in notched semidisks with different angles of the notch. The algorithm is based on the experimental procedure of the photoelasticity method and the finite element analysis. The large series of experiments for semidiscs was performed. Digital photoelasticity method is used to analyse experimentally the complete Williams series expansion of the stress and displacement fields in the vicinity of the crack tip in isotropic linear elastic plates under Mixed Mode loading. The distribution of the isochromatic fridge patterns is employed for obtaining the stress field near the crack tip by the use of the complete Williams asymptotic expansion for various classes of the experimental specimens (plates with two collinear cracks under tensile loading and under mixed mode loading conditions). The higher order terms of the Williams series expansion are taken into account and the coefficients of the higher order terms are experimentally obtained. The stress field equation of Williams up to fifty terms in each in mode I and mode II has been considered. The comparison of the experimental results and the calculations performed with finite element analysis has shown the importance and significant advantages of photoelastic observations for the multi-parameter description of the stress field in the neighborhood of the crack tip.Financial support from the Russian Foundation of Basic Research (project No. 19-01-00631) is gratefully acknowledged

Non-perturbative determination of anisotropy coefficients and pressure gap at the deconfining transition of QCD

We propose a new non-perturbative method to compute derivatives of gauge coupling constants with respect to anisotropic lattice spacings (anisotropy coefficients). Our method is based on a precise measurement of the finite temperature deconfining transition curve in the lattice coupling parameter space extended to anisotropic lattices by applying the spectral density method. We determine the anisotropy coefficients for the cases of SU(2) and SU(3) gauge theories. A longstanding problem, when one uses the perturbative anisotropy coefficients, is a non-vanishing pressure gap at the deconfining transition point in the SU(3) gauge theory. Using our non-perturbative anisotropy coefficients, we find that this problem is completely resolved.Comment: LATTICE98(hightemp