Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Forcing operators on MTL‐algebras

We study the forcing operators on MTL‐algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t ‐norm based logic (MTL). At logical level, they provide the notion of the forcing value of an MTL‐formula. We characterize the forcing operators in terms of some MTL‐algebras morphisms. From this result we derive the equality of the forcing value and the truth value of an MTL‐formula (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the entropy of mean curvature flow

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We also prove that a smooth selfshrinker with low entropy is a hyperplane.     

关于Finsler子流形的平均曲率

通过使用由射影球丛诱导的体积元来研究Finsler子流形几何,推导了体积泛函的第一变分公式。给出了Finsler子流形的平均曲率形式和第二基本形式的定义,该定义在Riemannian情形下与通常的概念一致．此外,通过推导射影球丛纤维上的散度公式。给出了平均曲率形式的一种非常简洁的等价表示,并得到一些关于Minkowski空间中Finsler子流形的有趣的结果．     

Creeping flow of a sphere nearby a cylinder

We present a three-dimensional solution of a sphere nearby an infinite cylinder at low Reynolds number. We utilize the Lamb’s general solution based on spherical harmonics and develop a framework based on cylindrical harmonics to solve the flow field around the sphere and outside the cylinder, respectively. The solution is solved semi-analytically by considering geometrical parameters, including sphere radius, sphere velocity, separation distance and cylinder radius. The drag force coefficients of the sphere which are dependent on the distance between the cylinder surface and the sphere, as well as the velocity contours in the vicinity of the sphere, are analyzed. We also provide an analytical formula to calculate the drag force. The analytical formula has good quantitative agreement with the semi-analytical solution when the radius of the cylinder is smaller than the sphere. Such analysis can give insights into the details of the complex interaction between the sphere and cylinder.     

On the mean value property for polyharmonic functions in the ball

We obtain the mean value property for the normal derivatives of a polyharmonic function with respect to the unit sphere. We find the values of a polyharmonic function and its Laplacians at the center of the unit ball expressed via the integrals of the normal derivatives of this function over the unit sphere.     

A positive radial product formula for the Dunkl kernel

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

     

On the smoothness of mean values of functions with summable spectral expansion

We consider spectral expansions associated with a self-adjoint extension of the Laplace operator in the n-dimensional domain. We show that if the spectral expansion of an arbitrary function at some point is summable by Riesz means, then its mean value over the sphere with center at that point has certain smoothness.     

A Mean Value Theorem for Dirichlet Series and a General Divisor Problem

 We consider the mean value formula for general Dirichlet series by applying the approximate functional equation of Ramachandra’s type, and derive the sum formula for its coefficients. The improvement of Landau’s classical results is established for the general divisor function. We also obtain the asymptotic behaviour of the mean value when the real part of s is near the abscissa of absolute convergence. Received 2 July 2001; in revised form 29 October 2001     

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m ² nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m ? 1.     

A generalization of the Poisson integral formula for the functions harmonic and biharmonic in a ball

We construct an analytic solution to the problem of extension to the unit N-dimensional ball of the potential on its values on an interior sphere. The formula generalizes the conventional Poisson formula. Bavrin’s results obtained for the two-dimensional case by methods of function theory are transferred to the N-dimensional case (N ≥ 3). We also exhibit a solution to a similar extension problem for some operator expressions depending on a potential known on an interior sphere. A connection is established between solutions to the moment problem on a segment and on a semiaxis.     

A mean value formula for the M/G/1 queues controlled by workload

We present a mean value formula for the M/G/1 queues controlled by workload (such as the D-policy queues). We first prove the formula and then demonstrate its application. This formula also works for the conventional vacation systems which are controlled by number of customers (such as the N-policy queues).     

THE POINCAR-BERTRAND FORMULA ON THE BUILDING DOMAIN OF COMPLEX BIBALLS

The Poincaré-Bertrand formula takes an important position in the study of complex singular integral.The Poincaré-Bertrand formula on the complex sphere in the multidimensional complex Euclidian spaces was given by Sheng Gong.Using the method of solid angular coefficient,the authors extend the Poincaré-Bertrand formula on the complex sphere to the building domain of the complex biballs,and obtain a more general Poincaré-Bertrand formula with the solid angular coefficients.     

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.     

Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry

Illuminating tissue with pulsed electromagnetic waves generates acoustic waves inside an object which can be measured and converted into a three‐dimensional (3d) image. This text is concerned with a two‐step reconstruction method where the acoustic pressure is measured with circular integrating detectors. In the first step, reconstruction formulas for some kind of projection of the source distribution are derived; in the second step, an inversion formula for a circular radon transform on the sphere is developed. Copyright © 2010 John Wiley & Sons, Ltd.     

On Sampling Formulas on Symmetric Spaces

We give a sampling formula using the Radon transform along a maximal geodesic subspace of the Riemannian symmetric space. For the real hyperbolic space we can get a total sampling formula. To get this formula, we prepare a sampling formula for the sphere.     

复函数微分中值公式的注记

本文给出了复函数的一个一般性的微分中值公式,获得了该公式的“中间值”的渐近性质,给出了该性质的简洁证明.     

Paneitz operator for metrics near $$S^{3}$$

We derive the first and second variation formula for the Green’s function pole’s value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the second eigenvalue and Sobolev inequality of Paneitz operator.     

On the expected value of the minimum assignment

The minimum k‐assignment of an m × n matrix X is the minimum sum of k entries of X, no two of which belong to the same row or column. Coppersmith and Sorkin conjectured that if X is generated by choosing each entry independently from the exponential distribution with mean 1, then the expected value of its minimum k‐assignment is given by an explicit formula, which has been proven only in a few cases. In this paper we describe our efforts to prove the Coppersmith–Sorkin conjecture by considering the more general situation where the entries x_ij of X are chosen independently from different distributions. In particular, we require that x_ij be chosen from the exponential distribution with mean 1/r_ic_j. We conjecture an explicit formula for the expected value of the minimum k‐assignment of such X and give evidence for this formula. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 33–58, 2002     

On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi‐Riemannian geodesic, and we compute its value in terms of the Maslov index (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)