An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.     

Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

An explicit formula for the square of the Riemann zeta-function in the critical strip

In this article, we prove an explicit formula for |ζ(σ + iT)|², where ζ(s) is the Riemann zeta-function and 1/2 < σ < 1, which is an analogue of Jutila’s formula. Our proof differs from that of Jutila. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 381–398, July–September, 2007.     

An explicit formula for generalized potential polynomials and its applications

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol-Bernoulli, Apostol-Euler polynomials and the polynomials involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.     

A spectral interpretation for the explicit formula in the theory of prime numbers

A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.     

An application of the Littlewood restriction formula to the Kostant-Rallis Theorem

Consider a symmetric pair of linear algebraic groups with , where and are defined as the +1 and -1 eigenspaces of the involution defining . We view the ring of polynomial functions on as a representation of . Moreover, set , where is the space of homogeneous polynomial functions on of degree . This decomposition provides a graded -module structure on . A decomposition of is provided for some classical families when is within a certain stable range.

The stable range is defined so that the spaces are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of is interpreted as a -analog of the Kostant-Rallis theorem.

     

J. Simons' type integral formula for hypersurfaces in a unit sphere

In this paper, by investigating compact rotational hypersurfaces Mⁿ in a unit sphere Sⁿ⁺¹(1), we get some integral formulas and then apply the integral formulas to characterize torus .     

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)     

Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion

The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.

     

An anomaly formula for Ray–Singer metrics on manifolds with boundary

Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a non-product metric near the boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary. Received: January 2004; Revision: February 2005; Accepted: September 2005     

An explicit formula for a Bennequin-type invariant of apparent contours

We derive an explicit formula for the computation of a Vassiliev-type invariant of an apparent contour introduced recently by Ohmoto and Aicardi using appropriate linking numbers of its Legendrian lift. Our formula does not require the construction of the Legendrian lift and only takes into account the nodes, the cusps, the extremal points and the orientation of the apparent contour. In this way the computation can be implemented into a computer program; we demonstrate its use with some examples.     

An explicit formula and its fast algorithm for a class of symmetric balanced incomplete block designs

Motivated by the field experimental designs in agriculture, the theory of block designs has been applied to several areas such as statistics, combinatorics, communication networks, distributed systems, cryptography, etc. An explicit formula and its fast computational algorithm for a class of symmetric balanced incomplete block designs are presented. Based on the formula and the careful investigation of the modulus multiplication table, the algorithm is developed. The computational costs of the algorithm is superior to those of the conventional ones.     

An invariant manifold approach to nonlinear normal modes of oscillation

Summary A method for determining the amplitude-dependent mode shapes and the corresponding modal dynamics of weakly nonlinear vibratory systems is described. The method is a combination of a Galerkin projection and invariant manifold techniques and is applied to a class of distributed parameter vibratory systems. In this paper the general theory for a class of conservative systems is outlined and applied to determine the nonlinear mode shapes and modal dynamics of a linear Euler-Bernoulli team attached to a nonlinear elastic foundation.     

An explicit formula for the determinant of a skew-symmetric pentadiagonal Toeplitz matrix

We give the explicit determinant of the general skew-symmetric pentadiagonal Toeplitz matrix in terms of the Tchebechev polynomials.     

An explicit formula for the linearization coefficients of Bessel polynomials

We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat??s formulas in their proof of the positivity results about these coefficients (Constr. Approx. 27:15?C32, 2008). As a bonus we also obtain a generalization of an integral formula of Boros and Moll (J. Comput. Appl. Math. 106:361?C368, 1999).     

An explicit formula for the Fourier coefficients of Eisenstein series attached to lattices

We prove an explicit formula for local densities of inhomogeneous quadratic forms. The formula is derived by calculating explicitly the Fourier coefficients of various types of Eisenstein series.     

An explicit formula for the action of a finite group on a commutative ring

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G, the trace (or norm) map is defined. is onto if and only if there exists an element x_H such that . We will show that the existence of x_P for every subgroup P of prime order determines the existence of x_G by exhibiting an explicit formula for x_G in terms of the x_P, where P varies over prime order subgroups. Since is onto if and only if is, where g∈G is an arbitrary element, we need to take only one P from each conjugacy class. We will also show why a formula with less factors does not exist, and show that the existence or non-existence of some of the x_P’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.     

Vibration mode of torsion balance and its response to external shock

Intrinsic vibration modes of typical torsion balances and their response to the external shock are discussed in detail. The study is important and instructive for improving measurement precision of gravitational experiments employing torsion balances, and gives a reasonable interpretation on detecting earthquake by high precision torsion balance. Project supported by the National Natural Science Foundation of China.     

An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in , . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. Received September 19, 1997