diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

on the list and bounded distance decodability of reed solomon codes for an error correcting code and a distance bound the list decoding problem is to compute all the codewords within a given distance to a received message the bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance or to output the empty set if there is not obviously the bounded distance decoding problem is not as hard as the list decoding problem for a reed solomon code n k q a simple counting argument shows that for any integer 0 0 we show that the discrete logarithm problem over bf f q h can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the reed solomon code q g h q with radius q g these results show that the decoding problems for the reed solomon code are at least as hard as the discrete logarithm problem over certain finite fields for the list decoding problem of reed solomon codes although the infeasible radius that we obtain is much larger than the radius which is known to be feasible it is the first nontrivial bound our result on the bounded distance decodability of reed solomon codes is also the first of its kind the main tools for obtaining these results are an interesting connection between the problem of list decoding of reed solomon code the problem of a discrete logarithm over finite fields and a generalization of katz s theorem on representations of elements in an extension finite field by products of distinct linear factors

self similar decay in the kraichnan model of a passive scalar we study the two point correlation function of a freely decaying scalar in kraichnan s model of advection by a gaussian random velocity field that is stationary and white noise in time but fractional brownian in space with roughness exponent 0 2 appropriate to the inertial convective range of the scalar we find all self similar solutions by transforming the scaling equation to kummer s equation it is shown that only those scaling solutions with scalar energy decay exponent a d 1 are statistically realizable where d is space dimension and 2 an infinite sequence of invariants jp p 0 1 2 is pointed out where j0 is corrsin s integral invariant but the higher invariants appear to be new we show that at least one of the invariants j0 or j1 must be nonzero possibly infinite for realizable initial data initial datum with a finite nonzero invariant the first being jp converges at long times to a scaling solution p with a d p p 0 1 the latter belongs to an exceptional series of self similar solutions with stretched exponential decay in space however the domain of attraction includes many initial data with power law decay when the initial datum has all invariants zero or infinite and also it exhibits power law decay then the solution converges at long times to a nonexceptional scaling solution with the same power law decay these results support a picture of a two scale decay with breakdown of self similarity for a range of exponents d a d 2 analogous to what has recently been found in the decay of burgers turbulence

the cut metric random graphs and branching processes in this paper we study the component structure of random graphs with independence between the edges under mild assumptions we determine whether there is a giant component and find its asymptotic size when it exists we assume that the sequence of matrices of edge probabilities converges to an appropriate limit object a kernel but only in a very weak sense namely in the cut metric our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in random struct algorithms 31 3 122 2007 as well as related results of bollobas borgs chayes and riordan ann probab 38 150 183 2010 all of which involve considerably stronger assumptions we also prove corresponding results for random hypergraphs these generalize our results on the phase transition in inhomogeneous random graphs with clustering random struct algorithms 2010 to appear

the spectra of manhattan street networks the multidimensional manhattan street networks constitute a family of digraphs with many interesting properties such as vertex symmetry in fact they are cayley digraphs easy routing hamiltonicity and modular structure from the known structural properties of these digraphs we determine their spectra which always contain the spectra of hypercubes in particular in the standard two dimensional case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity

natural frequency changes of a cracked timoshenko beam by modified fourier series a new method is presented in this paper for computing the natural frequencies of a timoshenko beam with an arbitrary number of transverse open cracks the essence of this new method lies in the use of a kind of modified fourier series mfs which is developed particularly for a timoshenko beam having an arbitrary number of transverse open cracks unlike the conventional fourier series the modified fourier series can approach a function with internal geometrical discontinuities based on the modified fourier series one can treat the cracked timoshenko beam in the usual way and thus reduces the problem to a simple one by using the present method only standard linear eigenvalue equations rather than non linear algebraic equations need to be solved all the formulae are expressed in matrix form that renders the task of computer coding quite straightforward

factoring polynomials in finite fields an application of lang weil to a problem in graph theory recently chung has constructed some interesting directed graphs using finite fields and proven non trivial upper bounds for their diameters we study chung s graphs by a direct algebraic geometric method interpreting a given bound for diameter as the statement that each of a certain collection of algebraic varieties has a rational point and then using lang weil to prove the existence of the required rational points we are naturally led to showing that certain explicit two parameter families of polynomials in one indeterminate have galois group the full symmetric group

multiembedding of metric spaces metric embedding has become a common technique in the design of algorithms its applicability is often dependent on how large the embedding s distortion is for example embedding finite metric space into trees may require linear distortion as a function of the size of the metric using probabilistic metric embeddings the bound on the distortion reduces to logarithmic in the size of the metric we make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric we define multiembeddings of metric spaces in which a point is mapped onto a set of points while keeping the target metric of polynomial size and preserving the distortion of paths the distortion obtained with such multiembeddings into ultrametrics is at most o log delta log log delta where delta is the aspect ratio of the metric in particular for expander graphs we are able to obtain constant distortion embeddings into trees in contrast with the omega log n lower bound for all previous notions of embeddings we demonstrate the algorithmic application of the new embeddings for two optimization problems group steiner tree and metrical task systems

graph spectra for finite upper half planes over rings abstract graphs attached for finite analogues of the poincare upper half plane over rings z p r z are introduced for p an odd prime the spectra of these graphs for r 2 are shown to be related to those for r 1 which were studied earlier in contrast to the case r 1 we find that the graphs are not ramanujan graphs when r 2 and p 5 histograms of the eigenvalues for r 1 and 2 are also compared the graphs over rings are of interest for connections with p adic upper half planes as well as fundamental domains for congruence subgroups of the modular group sl 2 z

lower bounds of the laplacian spectrum of graphs based on diameter let g be a connected graph of order n the diameter of g is the maximum distance between any two vertices of g in the paper we will give some lower bounds for the algebraic connectivity and the laplacian spectral radius of g in terms of the diameter of g

order evaluation of products of subsets in finite groups and its applications ii in this paper we give a new estimate of the cardinality of the product of subsets ab in a finite non abelian simple group where a is normal and b is arbitrary this estimate improves the one given in j algebra 182 1996 577 603

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

on the discrete logarithm problem in finite fields of fixed characteristic for q a prime power the discrete logarithm problem dlp in mathbb f q consists in finding for any g in mathbb f q times and h in langle g rangle an integer x such that g x h we present an algorithm for computing discrete logarithms with which we prove that for each prime p there exist infinitely many explicit extension fields mathbb f p n in which the dlp can be solved in expected quasi polynomial time furthermore subject to a conjecture on the existence of irreducible polynomials of a certain form the algorithm solves the dlp in all extensions mathbb f p n in expected quasi polynomial time

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

corrigendum algorithmic enumeration of ideal classes for quaternion orders this note corrects a mathematical error in the article algorithmic enumeration of ideal classes for quaternion orders siam j comput 39 2010 pp 1714 1747

upper bounds for eigenvalues of the discrete and continuous laplace operators in this paper we are concerned with upper bounds of eigenvalues of laplace operator on compact riemannian manifolds and finite graphs while on the former the laplace operator is generated by the riemannian metric on the latter it reflects combinatorial structure of a graph respectively eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics we develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel this approach is perhaps much more general than its realization here basically we start with the following entries 1 an underlying space m with a finite measure 2 a well defined laplace operator on functions on m so that is a self adjoint operator in l m with a discrete spectrum 3 if m has a boundary then the boundary condition should be chosen so that it does not disrupt self adjointness of and is of dissipative nature 4 a distance function dist x y on m so that dist 1 for an appropriate notion of gradient

the spectra of multiplicative attribute graphs abstract a multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet and the probability of adjacency is a symmetric function t t 0 1 these graphs are a generalization of stochastic kronecker graphs and both classes have been shown to exhibit several useful real world properties we establish asymptotic bounds on the spectra of the adjacency matrix and the normalized laplacian matrix for these two families of graphs under certain mild conditions as an application we examine various properties of the stochastic kronecker graph and the multiplicative attribute graph including the diameter clustering coefficient chromatic number and bounds on low congestion routing

on the spectra of hypertrees in this paper we study the spectral properties of a family of trees characterized by two main features they are spanning subgraphs of the hypercube and their vertices bear a high degree of connectedness hierarchy such structures are here called binary hypertrees and they can be recursively defined as the so called hierarchical product of several complete graphs on two vertices ams classification 05c50 05c05