as x → ∞, where L(x) is the function L(x) = ∫ x 1 e t dt. Sarnak established this result by identifying the regulators with lengths of closed geodesics of the modular curve SL2(Z)\H, where H denotes the upper half-plane, and using the Prime Geodesic Theorem for this Riemannian surface. Actually Sarnak proved not this result but the analogue where h(O) is replaced by the class number in the narr...

A monadic formula ψ(Y) is a selector for a monadic formula φ(Y) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M. If C is a class of structures and φ is a selector for ψ in every M ∈ C, we say φ is a selector for φ over C. For a monadic formula φ(X,Y) and ordinals α ≤ ω1 and δ < ω, we decide whether there exists a monadic formula ψ(X,Y) such th...

This article is an introduction to the Petersson trace formula and Kuznetsov trace formula, both of which are now important, standard techniques in analytic number theory. To illustrate their applications to modular forms, we will explain their role in a proof of subconvexity bounds for Rankin-Selberg L-functions L(s, f ⊗ g) on the critical line σ = 1/2, where here and throughout, we write s = ...

In digital signal and image processing, digital communications, and so forth, a continuous signal is usually represented and processed by using its discrete samples. How, then, are we to reconstruct the original signal from its discrete samples? The classical Shannon sampling theorem gives the following formula for band-limited finite energy signals. For a finite energy s-band continuous signal...

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozemberg introduced the idea of predictors. A predictor of a molecule s, for a gi...

M. Saito [S] proved that the jumping numbers of a hyperplane arrangement depend only on the combinatorics of the arrangement. However, a formula in terms of the combinatorial data was still missing. In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics. We also give a combinatorial formula for part...

We study precise calculation of neutralino relic density in the minimal supergravity model. We compare the exact formula for the thermal average of the neutralino annihilation cross section times relative velocity with the expansion formula in terms of the temperature, including all the contributions to neutralino annihilation cross section analytically. We confirm that the expansion formula fa...

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the t...

We construct a stacky curve of genus 1 / 2 1/2 </inline-for...

Let S(n) denote the total number of digits ‘1’ in the binary expansions of the integers between 1 and n − 1. The Trollope-Delange formula is a classical result which provides an explicit representation for S(n) in terms of the continuous, nowhere differentiable Takagi function. Recently, connections have been established between digital sums such as S(n) and certain functional equations associa...