A Product Formula for the Twisted Lefschetz Zeta Function

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A Poisson Formula for the Schrödinger Operator

We prove a Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous article by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H ¹-critical nonlinearities are allowed.     

An Infinite Product Formula for the Elliptic Modular Function λ(τ)

We present an infinite product formula for the elliptic modular function λ(τ) of the following form:

Explicit Formula for the Hyperbolic Scattering Determinant

We prove the explicit formula for the hyperbolic scattering determinant in the case of a general subgroup F of PSL （2, R）. The class of test functions involved （not necessarily odd nor continuous） is much broader than that previously known. As an application of the technique, a new representation of the Millson-Shintani zeta function is obtained.     

A d’Alembert Formula for Hopf Hypersurfaces

A Hopf hypersurface in complex hyperbolic space ${\mathbb{C}{\rm H}^n}$ is one for which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to ambient curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit parametrization.     

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.     

A D’Alembert Formula for Flat Surfaces in the 3-Sphere

We find all the flat surfaces in the unit 3-sphere $\mathbb{S}^{3}We find all the flat surfaces in the unit 3-sphere that pass through a given regular curve of with a prescribed tangent plane distribution along this curve. The formula that solves this problem may be seen as a geometric analogue of the classical D’Alembert formula that solves the Cauchy problem for the homogeneous wave equation. We also provide several applications of this geometric D’Alembert formula, including a classification of the flat M?bius strips of  .      

On Goulden-Jackson’s Determinantal Formula for the Immanant

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially. We also present a β-extension of the theorem and a simple determinantal expression for the irreducible characters of the symmetric group.     

On the Fedosov-Hörmander Formula for Differential Operators

The Fedosov-H?rmander formula gives the Fredholm index of some pseudodifferential operators of order 0 on L ². It is well known that it can be used to calculate the index of elliptic systems under the assumption that, among other things, the coefficients are smooth and their partial derivatives of all orders satisfy specific asymptotic conditions at infinity. We prove that the formula remains valid when the coefficients are only and bounded and have vanishing oscillation at infinity. In turn, this generalization is used to obtain a nonstandard invariance property of the index as well as various sufficient conditions for the index to be 0, when the coefficients are merely continuous and bounded with vanishing oscillation.      

A Condition for the Existence of a Unique Green–Samoilenko Function for the Problem of Invariant Torus

Under the assumption that a linear homogeneous system defined on the direct product of a torus and a Euclidean space is exponentially dichotomous on the semiaxes, we obtain a condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus. We find an expression for this function in terms of projectors that determine the dichotomy on the semiaxes.     

A Stereological Version of the Gauss–Bonnet Formula

We give a stereological version of the Gauss–Bonnet formula in order to compute the Euler characteristic of a domain with boundary in a smooth orientable surface in ³, by looking at contacts with a 'sweeping' plane.     

A Reilly Formula and Eigenvalue Estimates for Differential Forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally, we also obtain, as a byproduct of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Meromorphic Functions

In the paper, a discrete limit theorem for the Matsumoto zetafunction in the space of meromorphic functions is proved.     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Analytic Functions

In the paper, a discrete distribution of the Matsumoto zetafunction is considered. It is proved that the probability measure , converges weakly as .     

Stanley’s Formula for Characters of the Symmetric Group

In his paper [9], Stanley finds a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, in [10], he gives a conjectural generalisation for any shape. Here, we will prove this formula using shifted Schur functions and Jucys-Murphy elements.     

Rate of Convergence for the Empirical Distribution Function and the Empirical Characteristic Function of Mixing Processes

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A Note on the Bloch Function in SCV

51,Intr0ductionThroughoutthispaper,Cwilldenotethecomplexfield,andC"willbethecartesianl)r0cl-uctofncopiesofClherenisanypositiveinterger.TheP0intsofC"arethusordered)I-tul>lesz=(zl,...,z`),whereeachzieC.ThainnerproductinC"is(z,w)=Zzjwj(z,weC"),j=1andtheassociatednormlzI=(z,z)1l2(zeC")makeC"intoann-dimensionalHilbertspacewhoseopenunitballwillbedenotedbyB..ForapairofpointszandwinB,,thehyperdi8tancebetweenzandwisdenotedbyp(z,w)=oplog(the),wllereh(z,w)=(lz-wl= Izwl'~lzl2It`)l2)1/2.['I1-w5l]-'…