The π-calculus is a process algebra for modelling concurrent systems in which the pattern of communication between processes may change over time. This paper describes the results of preliminary work on a definitional formal theory of the π-calculus in higher order logic using the HOL theorem prover. The ultimate goal of this work is to provide practical mechanized support for reasoning with th...

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GLn×GLn′ . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s,π× π̃), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GLn is determined by a finite number ...

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GLn×GLn′ . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s, π × π̃), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GLn is determined by a finite numbe...

We find, in the form of a continued fraction, the generating function for the number of (132)avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern. We find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan. the electronic journa...

d -descents are permutation statistics that generalize the notions of descents and inversions. It is known that the distribution of d -descents of permutations of length n satisfies a central limit theorem as n goes to infinity. We provide an explicit formula for the mean and variance of these statistics and obtain bounds on the rate of convergence using Stein’s method. Introduction For π ∈ Sn,...

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the ¯ ∂-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth projective varieties of positive dimension and f : Y − → X is a holomorphic immersion with ample normal bundle, then the image of π 1 (Y) in π 1 (X) is of finite in...

In this technical report we present a complete proof of the Marcinkiewicz-Zygmund inequality on the bi-Sphere from [1, Theorem 4.5]. We use the same notation as in [1]. The Theorem 4.5 of [1] reads as follows: Theorem 1. Let N ∈ N2 and R be a admissible decomposition of S2 × S2 according to a sampling set X such that ‖R‖ ≤ η 21 Ñ , (1) where η ∈ (0, 1) is arbitrarily fixed. Then for r = 1 or r ...

Given a complex semisimple Lie algebra g = k+ ik, we consider the converse question of Kostant’s convexity theorem for a normal x ∈ g. Let π : g → h be the orthogonal projection under the Killing form onto the Cartan subalgebra h := t+it where t is a maximal abelian subalgebra of k. If π(Ad(K)x) is convex, then there is k ∈ K such that each simple component of Ad(k)x can be rotated into the cor...

The so-called Ambarzumyan theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator − d2 dx2 +q with an integrable real-valued potential q on [0,π] are {n2 : n 0} , then q = 0 for almost all x ∈ [0,π] . In this work, the classical Ambarzumyan theorem is extended to star graphs with Dirac operators on its edges. We prove that if the spectrum of Dirac operator on star graphs ...