Spectral properties of geometric-arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the geometric-arithmetic index GA(1) from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices. Moreover, using this matrix, we define a GA Laplacian matrix which determines the geometric-arithmetic index of a graph and satisfies properties similar to the ones of the classical Laplacian matrix. (C) 2015 Elsevier Inc. All rights reserved.This research was supported in part by a Grant from Ministerio de Economía y Competitividad (MTM 2013-46374-P), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry

In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus $g\geq 2$. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology and the cohomology of the cone of the local monodromy $N$ at arithmetic infinity as introduced by the first author of this paper.Comment: 68 pages, 10pt LaTeX, xy-pic (v2: to appear in Selecta Mathematica

Archimedean cohomology revisited

Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger's approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic infinity and Givental's homological geometry on loop spaces.Comment: 28 pages LaTeX 3 eps figure

Continuum and Emission-Line Properties of Broad Absorption Line Quasars

We investigate the continuum and emission-line properties of 224 broad absorption line quasars (BALQSOs) with 0.9<z<4.4 drawn from the Sloan Digital Sky Survey (SDSS) Early Data Release (EDR), which contains 3814 bona fide quasars. We find that low-ionization BALQSOs (LoBALs) are significantly reddened as compared to normal quasars, in agreement with previous work. High-ionization BALQSOs (HiBALs) are also more reddened than the average nonBALQSO. Assuming SMC-like dust reddening at the quasar redshift, the amount of reddening needed to explain HiBALs is E(B-V)~0.023 and LoBALs is E(B-V)~0.077 (compared to the ensemble average of the entire quasar sample). We find that there are differences in the emission-line properties between the average HiBAL, LoBAL, and nonBAL quasar. These differences, along with differences in the absorption line troughs, may be related to intrinsic quasar properties such as the slope of the intrinsic (unreddened) continuum; more extreme absorption properties are correlated with bluer intrinsic continua. Despite the differences among BALQSO sub-types and nonBALQSOs, BALQSOs appear to be drawn from the same parent population as nonBALQSOs when both are selected by their UV/optical properties. We find that the overall fraction of traditionally defined BALQSOs, after correcting for color-dependent selection effects due to different SEDs of BALQSO and nonBALQSOs, is 13.4+/-1.2% and shows no significant redshift dependence for 1.7<z<3.45. After a rough completeness correction for the effects of dust extinction, we find that approximately one in every six quasars is a BALQSO.Comment: 35 pages, 11 figures (1 color), 1 table; accepted by A

Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwiller's semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N(E), which in turn is successfully applied in a quantization condition.Comment: 32 pages, compressed with gzip / uuencod

The notion of dimension in geometry and algebra

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.Comment: 29 pages, a revie