Cohomological Hasse principle and motivic cohomology for arithmetic schemes

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme X. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.     

A trace formula for the scalar product of Hecke series and its applications

A trace formula expressing the mean values of the form (k=2,3,...)

A generating function for Blattner's formula

Let be a connected, semisimple Lie group with finite center and let be a maximal compact subgroup. We investigate a method to compute multiplicities of -types in the discrete series using a rational expression for a generating function obtained from Blattner's formula. This expression involves a product with a character of an irreducible finite-dimensional representation of and is valid for any discrete series system. Other results include a new proof of a symmetry of Blattner's formula, and a positivity result for certain low rank examples. We consider in detail the situation for of type split . The motivation for this work came from an attempt to understand pictures coming from Blattner's formula, some of which we include in the paper.

     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given.     

A trace formula for isometric pairs

It is well known that for every isometry , This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries , namely

where is the complete anti-symmetric sum and is the Fredholm index of the pair . The major tool is what we call the fringe operator. Two examples are considered.

     

A trace formula for Hankel operators

We show that if is an operator valued analytic function in the open right half plane such that the Hankel operator with symbol is of trace-class, then has continuous extension to the imaginary axis,

exists in the trace-class norm, and .

     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given. This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562.     

A symmetric formula for hypergeometric series

The Ramanujan Journal - In terms of Dougall’s $$_2H_2$$ series identity and the series rearrangement method, we establish a symmetric formula for hypergeometric series. Then it is utilized to...     

A trace formula for canonical differential expressions

We prove a trace formula for pairs of self-adjoint operators associated to canonical differential expressions. An important role is played by the associated Weyl function.     

A factorization for formal laurent series and lattice path enumeration

If f = Σ_n=?∞^∞a_ntⁿ is a formal Laurent series with certain restrictions on the a_n, then f = f_?f₀f₊, where f_? contains only negative powers of t, f₊ contains only positive powers of t, and f₀ is independent of t. Applications include Lagrange's formula for series reversion, the problem of counting lattice paths below a diagonal, and a theorem of Furstenberg that the diagonal of a rational power series in two variables is algebraic.     

A closed formula for the generating function of p-Bernoulli numbers

Abstract

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials.     

A recursive formula for even order harmonic series

A useful recursive formula for obtaining the infinite sums of even order harmonic series Σ^∞_n=1 (1/n^2k), k = 1, 2, …, is derived by an application of Fourier series expansion of some periodic functions. Since the formula does not contain the Bernoulli numbers, infinite sums of even order harmonic series may be calculated by the formula without the Bernoulli numbers. Infinite sums of a few even order harmonic series, which are calculated using the recursive formula, are tabulated for easy reference.     

A trace formula for Weyl transforms with radial symbols

A trace formula for Weyl transforms onL ² () with radial symbols is given.This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562.     

A trace formula for the forcing relation of braids

The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.