Eigenvalue bounds on the pseudocodeword weight of expander codes

Four different ways of obtaining low-density parity-check codes from expander graphs are considered. For each case, lower bounds on the minimum stopping set size and the minimum pseudocodeword weight of expander (LDPC) codes are derived. These bounds are compared with the known eigenvalue-based lower bounds on the minimum distance of expander codes. Furthermore, Tanner's parity-oriented eigenvalue lower bound on the minimum distance is generalized to yield a new lower bound on the minimum pseudocodeword weight. These bounds are useful in predicting the performance of LDPC codes under graph-based iterative decoding and linear programming decoding.Comment: Journal on Advances of Mathematics of Communications, vol. 1, no. 3, pp. 287 -- 307, Aug. 200

Limit sets and commensurability of Kleinian groups

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups $G_1$ and $G_2$ of an infinite co-volume Kleinian group G \subset \Isom(\mathbf{H}^3) having $\Lambda(G_1) = \Lambda(G_2)$ are commensurable. In particular, it is proved that any finitely generated subgroup $H$ of a Kleinian group G \subset \Isom(\mathbf{H}^3) with $\Lambda(H) = \Lambda(G)$ is of finite index if and only if $H$ is not a virtually fiber subgroup.Comment: 9 page

Continuous and discrete frames generated by the evolution flow of the Schr\"odinger equation

We study a family of coherent states, called Schr\"odingerlets, both in the continuous and discrete setting. They are defined in terms of the Schr\"odinger equation of a free quantum particle and some of its invariant transformations.Comment: 20 page

Weak synchronization for isotropic flows

We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide a sufficient condition on the boundary behavior of $r$ at $0$ which guarantees that the statistical equilibrium of the flow is almost surely a singleton and its support is a weak point attractor. The condition is fulfilled in the case of negative top Lyapunov exponent, but it is also fulfilled in some cases when the top Lyapunov exponent is zero. Particular examples are isotropic Brownian flows on $S^{d-1}$ as well as isotropic Ornstein-Uhlenbeck flows on $\mathbb{R}^d$.Comment: 14 page

New Operators for Spin Net Gravity: Definitions and Consequences

Two operators for quantum gravity, angle and quasilocal energy, are briefly reviewed. The requirements to model semi-classical angles are discussed. To model semi-classical angles it is shown that the internal spins of the vertex must be very large, ~10^20.Comment: 7 pages, 2 figures, a talk at the MG9 Meeting, Rome, July 2-8, 200

An annotated bibliography on 1-planarity

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs

On the Boundedness of Positive Solutions of the Reciprocal Max-Type Difference Equation xn=max⁡{An−11xn−1,An−12xn−2,…,An−1txn−t}x_{n}=\max\left\{\frac{A^{1}_{n-1}}{x_{n-1}}, \frac{A^{2}_{n-1}}{x_{n-2}}, \ldots, \frac{A^{t}_{n-1}}{x_{n-t}}\right\} with Periodic Parameters

We investigate the boundedness of positive solutions of the reciprocal max-type difference equation $$x_{n}=\max\left\{\frac{A_{n-1}^{1}}{x_{n-1}}, \frac{A_{n-1}^{2}}{x_{n-2}}, \ldots, \frac{A_{n-1}^{t}}{x_{n-t}}\right\}, \ \ n=1, 2, \ldots,$$ where, for each value of $i$, the sequence $\{A_{n}^{i}\}_{n=0}^{\infty}$ of positive numbers is periodic with period $p_{i}$. We give both sufficient conditions on the $p_{i}$'s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded. This work essentially complements the work by Biddell and Franke, who showed that as long as every positive solution of our equation is \emph{bounded}, then every positive solution is eventually periodic, thereby leaving open the question as to when solutions are bounded.Comment: 18 page

Computation of Maxwell's equations on manifold using implicit DEC scheme

Maxwell's equations can be solved numerically in space manifold and the time by discrete exterior calculus as a kind of lattice gauge theory.Since the stable conditions of this method is very severe restriction, we combine the implicit scheme of time variable and discrete exterior calculus to derive an unconditional stable scheme. It is an generation of implicit Yee-like scheme, since it can be implemented in space manifold directly. The analysis of its unconditional stability and error is also accomplished.Comment: 9pages,4figure

Multiplicity one for certain paramodular forms of genus two

We show that certain paramodular cuspidal automorphic irreducible representations of $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$, which are not CAP, are globally generic. This implies a multiplicity one theorem for paramodular cuspidal automorphic representations. Our proof relies on a reasonable hypothesis concerning the non-vanishing of central values of automorphic $L$-series.Comment: 10 page

Gabor orthogonal bases and convexity

Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then there does not exist $S \subset {\Bbb R}^{2d}$ such that ${ \{g(x-a)e^{2 \pi i x \cdot b} \}}_{(a,b) \in S}$ is an orthonormal basis for $L^2({\Bbb R}^d)$