Irrationality of certain numbers that contain values of the di- and trilogarithm

We prove that, for z ∈ {1/2, 2/3, 3/4, 4/5}, at least one of the two numbers is irrational. Kh. Hessami Pilehrood was supported in part by Institute for Studies in Theoretical Physics and Mathematics (IPM) grant no. 83110021. T. Hessami Pilehrood was supported in part by IPM grant no. 83110020. W. Zudilin was supported in part by Russian Foundation for Basic Research grant no. 03-01-00359.     

A formulation to obtain semi-analytical planetary theories using true anomalies as temporal variables

The main methods used to obtain analytical theories of perturbed motion in celestial mechanics are based on the expansion of the disturbing function in trigonometric series of the mean anomalies (or longitudes). In this paper a new method based on the double Fourier series expansion using the true anomalies (or longitudes) is developed. The method involves a semi-analytical technique to allow the expansion of the inverse of the distance with great accuracy, and a new integration technique using a linear combination of the true anomalies based on an iterative method to integrate each term of the expansion of the Lagrange planetary equations.     

Domb's numbers and Ramanujan-Sato type series for 1/π

In this article, we construct a general series for . We indicate that Ramanujan's -series are all special cases of this general series and we end the paper with a new class of -series. Our work is motivated by series recently discovered by Takeshi Sato.     

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector by rational vectors of a lattice whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.     

Lattice ordered effect algebras

The purpose of this paper is to discuss some structural properties of lattice ordered effect algebras. We will use these structural properties to find certain lattices and classes of lattices that do not admit an effect algebra structure. Finally, using these structural properties, we will show that if L is the face lattice of a convex polytope in $ R^3 $ with more than 3 vertices, then L does not admit an effect algebra structure.Dedicated to the memory of Gian-Carlo Rota     

Continued fractions and the Markoff tree

We give here a full account of Markoff's celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey Cohn's method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers.     

An efficient algorithm for the Hurwitz zeta and related functions

A simple class of algorithms for the efficient computation of the Hurwitz zeta and related special functions is given. The algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision. A number of extensions as well as numerical examples are briefly described. The algorithms are easy to implement and compete with Euler–Maclaurin summation-based methods.     

New developments in the theory of Gröbner bases and applications to formal verification

We present foundational work on standard bases over rings and on Boolean Gröbner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Gröbner basis in the polynomial ring over Z/2ⁿ while the bit-level model leads to Boolean Gröbner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Gröbner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.     

Extensions for finite Chevalley groups III: Rational and generic cohomology

Let G be a connected reductive algebraic group and B   be a Borel subgroup defined over an algebraically closed field of characteristic p>0$p>0$. In this paper, the authors study the existence of generic G-cohomology and its stability with rational G-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of G and B  -cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of G(_Fq)$G({\mathbb{F}}_q)$ are established which generalize earlier work of Hiller, in addition to stability ranges for generic cohomology which improve on seminal work of Cline, Parshall, Scott and van der Kallen.     

Mellin transforms of Whittaker functions onGL(4,ℝ) andGL(4,ℂ)

Supported in part by NSF grant #DMS-9400324     

A note on odd unimodular Euclidean lattices

We prove that the theta series of any odd unimodular Euclidean lattice is not congruent to 1 modulo any odd prime p. Received: 18 March 2005     

A note on integral Euclidean lattices in dimension 3

We prove that the theta series of any integral Euclidean lattice of dimension 3 is not congruent to 1 modulo any odd prime. Received: 18 March 2005; revised: 19 April 2005     

Ideal bicombings for hyperbolic groups and applications

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in Monod and Shalom (Orbit equivalence rigidity and bounded cohomology, preprint, to appear) hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.     

Eisenstein series and theta functions to the septic base

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η⁴(7τ)/η⁴(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series.     

Lattice polytopes,Hecke operators,and the Ehrhart polynomial

Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.      

On Optimal Constants for Best Two-Dimensional Simultaneous Diophantine Approximations

 The main results of this paper state optimal constants for estimates of so-called successive minima in two dimensions under a constraint on the denominator. While these inequalities are known for every dimension, best possible constants within these estimates are, of course, notknown for any dimension larger than one and remain unknown for all dimensions larger than two. (Received 29 April 1998; in revised form 23 November 1998)     

Deux extensions de Théorèmes de Hamburger (portant sur l’équation fonctionnelle de la fonction zêta)

We propose two types of extensions to Hamburger’s theorems on the Dirichlet series with a functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary between the moderate meromorphic functions with the functional equation and the tempered distributions with an extended S$S$-support condition.