The degree of the eigenvalues of generalized Moore geometries

AbstractUsing elementary methods it is proved that the eigenvalues of generalized Moore geometries of type GMm(s, t, c) are of degree at most 3 with respect to the field of rational numbers, if st > 1

Impact of geometry on many-body localization

The impact of geometry on many body localization is studied on simple, exemplary systems amenable to exact diagonalization treatment. The crossover between ergodic and MBL phase for uniform as well as quasi-random disorder is analyzed using statistics of energy levels. It is observed that the transition to many-body localized phase is correlated with the number of nearest coupled neighbors. The crossover from extended to localized systems is approximately described by the so called plasma model.Comment: 8pp. comments welcom

Largest regular multigraphs with three distinct eigenvalues

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\leq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.Comment: 9 pages, no figur

Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion

We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces result in less stringent CFL restrictions than equivalent piecewise continuous or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L2 n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods

Chip-firing may be much faster than you think

A new bound (Theorem \ref{thm:main}) for the duration of the chip-firing game with $N$ chips on a $n$-vertex graph is obtained, by a careful analysis of the pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is expressed in terms of the entries of the pseudo-inverse. It is shown (Section 5) to be always better than the classic bound due to Bj{\"o}rner, Lov\'{a}sz and Shor. In some cases the improvement is dramatic. For instance: for strongly regular graphs the classic and the new bounds reduce to $O(nN)$ and $O(n+N)$, respectively. For dense regular graphs - $d=(\frac{1}{2}+\epsilon)n$ - the classic and the new bounds reduce to $O(N)$ and $O(n)$, respectively. This is a snapshot of a work in progress, so further results in this vein are in the works

Enhanced Symmetries in Multiparameter Flux Vacua

We give a construction of type IIB flux vacua with discrete R-symmetries and vanishing superpotential for hypersurfaces in weighted projective space with any number of moduli. We find that the existence of such vacua for a given space depends on properties of the modular group, and for Fermat models can be determined solely by the weights of the projective space. The periods of the geometry do not in general have arithmetic properties, but live in a vector space whose properties are vital to the construction.Comment: 32 pages, LaTeX. v2: references adde

Matrix Models, Topological Strings, and Supersymmetric Gauge Theories

We show that B-model topological strings on local Calabi-Yau threefolds are large N duals of matrix models, which in the planar limit naturally give rise to special geometry. These matrix models directly compute F-terms in an associated N=1 supersymmetric gauge theory, obtained by deforming N=2 theories by a superpotential term that can be directly identified with the potential of the matrix model. Moreover by tuning some of the parameters of the geometry in a double scaling limit we recover (p,q) conformal minimal models coupled to 2d gravity, thereby relating non-critical string theories to type II superstrings on Calabi-Yau backgrounds.Comment: 22 pages, minor correction