We use the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator δ constructed from an elliptic family of operators with base S. We show that the regularized values η(δt, 0) and tζ(δt, 0) have smooth limits as t → 0, and we identify the limits with the holonomy of the determinant bundle, respectively wi...

We announce misère-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misère quotients—commutative monoids that encode the additive structure of specific misère-play games. We also introduce several advances in the structure theory of misère quotients, including a connection between the combinatorial structure of normal and misère play.

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral Teichmüller character representations Bernoulli polynomials, we give reciprocity law these These sums their generalized some classical Dedekind law. It be noted that laws, a fine study existing symmetry relations between finite sums, considered in our study, symmetries through permutations...

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and – we feel – simpler proofs of theorems of L. Claborn and C.R. Leedham-Green. Luther Claborn received his PhD from U. Mi...

Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, w...

Many operations exist for constructing Scott-domains. This paper presents Dedekind completion as a new operation for constructing such domains and outlines an application of the operation. Dedekind complete Scott domains are of particular interest when modeling versions of λ-calculus that allow quantification over sets of arbitrary cardinality. Hence, it is of interest when constructing models ...

We study the asymptotics of the heat trace Tr{fPe 2 } where P is an operator of Dirac type, where f is an auxiliary smooth smearing function which is used to localize the problem, and where we impose spectral boundary conditions. Using functorial techniques and special case calculations, the boundary part of the leading coefficients in the asymptotic expansion is found.

We give explicit, polynomial–time computable formulas for the number of integer points in any two– dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind–Rademacher sums, which are polynomial–time computable finite Fourier series. As a by–product we rederive a reciprocity law for these...