moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

longest fault free paths in star graphs with vertex faults the star graph sn has been recognized as an attractive alternative to the hypercube since s1 s2 and s3 have trivial structures we focus our attention on sn with n 4 in this paper let fv denote the set of faulty vertices in sn we show that when fv n 5 sn with n 6 contains a fault free path of length n 2 fv 2 n 2 fv 1 between arbitrary two vertices of even odd distance since sn is bipartite with two partite sets of equal size the path is longest for the worst case scenario the situation of n 4 and fv n 5 is also discussed

panpositionable hamiltonicity and panconnectivity of the arrangement graphs abstract the arrangement graph an k is a generalization of the star graph it is more flexible in its size than the star graph there are some results concerning hamiltonicity and pancyclicity of the arrangement graphs in this paper we propose a new concept called panpositionable hamiltonicity a hamiltonian graph g is panpositionable if for any two different vertices x and y of g and for any integer l satisfying d x y l v g d x y there exists a hamiltonian cycle c of g such that the relative distance between x and y on c is l a graph g is panconnected if there exists a path of length l joining any two different vertices x and y with d x y l v g 1 we show that an k is panpositionable hamiltonian and panconnected if k 1 and n k 2

hyper hamiltonian laceability on edge fault star graph the star graph possess many nice topological properties edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes in this paper we show that the n dimensional star graph is n 3 edge fault tolerant hamiltonian laceable n 3 edge fault tolerant strongly hamiltonian laceable and n 4 edge fault tolerant hyper hamiltonian laceable all these results are optimal in a sense described in this paper

a note on fractional moments for the one dimensional continuum anderson model we give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one dimensional continuum anderson model via the fractional moments method this follows via exponential decay of fractional moments of the green function which is shown to hold at arbitrary energy and for any single site distribution with bounded compactly supported density

a fast algorithm for finding the positions of all squares in a run length encoded string squares are strings of the form ww where w is any nonempty string main and lorentz proposed an o nlogn time algorithm for finding the positions of all squares in a string of length n based on their result we show how to find the positions of all squares in a run length encoded string in time o nlogn where n is the number of runs in this string provided that we do not explicitly compute at all trivial squares occurring within runs the algorithm is optimal and its time complexity is independent of the length of the original uncompressed string

embedding hamiltonian paths in augmented cubes with a required vertex in a fixed position it is proved that there exists a path p l x y of length l if d a q n x y l 2 n 1 between any two distinct vertices x and y of aq n obviously we expect that such a path p l x y can be further extended by including the vertices not in p l x y into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle in this paper we prove that there exists a hamiltonian path r x y z l from x to z such that d r x y z l x y l for any three distinct vertices x y and z of aq n with n 2 and for any d a q n x y l 2 n 1 d a q n y z furthermore there exists a hamiltonian cycle s x y l such that d s x y l x y l for any two distinct vertices x and y and for any d a q n x y l 2 n 1

vertex pancyclicity of augmented cubes with maximal faulty edges abstract the n dimensional augmented cube denoted as aqn a variation of the hypercube possesses some properties superior to those of the hypercube in this paper we show that every vertex in aqn lies on a fault free cycle of every length from 4 to 2n even if there are up to 2n 3 link faults we also show that this result is optimal

fault tolerant cycle embedding in alternating group graphs abstract cycle embedding is an important issue in evaluating the efficiency of interconnection networks and is also an extension of the theoretical research on hamiltonicity in this paper we study the fault tolerant hamiltonicity of alternating group graphs which were recently proposed as interconnection topologies for parallel and distributed systems a cycle of length l is said to be an l cycle let f be a set of faulty vertices in an n dimensional alternating group graph ag n and let m be the number of vertices in ag n f a previous result in j m chang j s yang y l wang y cheng panconnectivity fault tolerant hamiltonicity and hamiltonian connectivity in alternating group graphs networks 44 2004 302 310 showed that for n 4 ag n f is hamiltonian i e ag n f possesses a hamiltonian cycle if f n 2 and ag n f is hamiltonian connected i e every two vertices of ag n f are connected by a hamiltonian path if f n 3 in this paper we show the following results i ag n f is pancyclic i e ag n f contains an l cycle for each l with 3 l m if f n 2 ii ag n f is vertex pancyclic i e every vertex of ag n f is contained in an l cycle for each l with 3 l m if f n 3 and iii ag n f is edge 4 pancyclic i e every edge of ag n f is contained in an l cycle for each l with 4 l m if f n 4 the first result is an improvement over the previous one and the last two results are new characterization of fault tolerant pancyclicity on ag n all the results we achieved are the best possible in the sense that the number of faulty vertices cannot be increased

conditional edge fault tolerant edge bipancyclicity of hypercubes the hypercube has been one of the most popular interconnection networks for parallel computer communication systems in this paper we assume that each node is incident with at least two fault free links under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 6 to 2 n inclusive even if it has up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated

conditional diagnosability of arrangement graphs under the pmc model abstract processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault set can t contain all the neighbors of any node in a multiprocessor system this paper considers the conditional diagnosability of an n k arrangement graph a n k a flexible interconnection network model for multiprocessor systems under the classical pmc diagnostic model and determines that the conditional diagnosability of a n k k 2 n k 2 is 4 k 4 n k 3 which is about four times its traditional diagnosability

fault diagnosability of arrangement graphs the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and to replace the faulty processors to maintain a system s high reliability the fault diagnosis is the process of identifying faulty processors in a system through testing in this paper we show that the largest connected component of the survival graph contains almost all of the remaining vertices in the n k arrangement graph a n k when the number of moved faulty vertices is up to twice or three times the traditional connectivity using this fault resiliency we establish the conditional diagnosability of a n k under the comparison model and prove that the conditional diagnosability of a n k is 3k 2 n k 3 for k 4 n k 2 and the conditional diagnosability of a n n 1 is 3n 7 for n 5

fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs the arrangement graph a sub n k is a generalization of the star graph there are some results concerning fault hamiltonicity and fault hamiltonian connectivity of the arrangement graph however these results are restricted in some particular cases and thus are less completed we improve these results and obtain a stronger and simpler statement let n k spl ges 2 and f spl sube v a sub n k spl cup e a sub n k we prove that a sub n k f is hamiltonian if f spl les k n k 2 and a sub n k f is hamiltonian connected if f spl les k n k 3 these results are optimal

hamiltonian cycle embedding for fault tolerance in balanced hypercubes abstract the balanced hypercube bh n defined by wu and huang is a variant of the hypercube network yang proposed that fault tolerance of balanced hypercube bh n is an important issue in parallel computing which needs further study yang 2010 24 in this paper we prove that there exists a fault free hamiltonian path between any two adjacent vertices in bh n with 2 n 2 faulty edges as a corollary we derive that for any fault free edge e there exists a fault free hamiltonian cycle containing e in bh n with 2 n 2 faulty edges which is optimal in the sense of the number of faulty edges

embedding of meshes in mobius cubes the hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement mobius cubes form a class of hypercube variants that give better performance with the same number of edges and vertices in this paper we consider embedding of meshes in mobius cubes the main results obtained in this paper are 1 for n 1 there exists a 2x2 n 1 mesh that can be embedded in the n dimensional mobius cube with dilation 1 and expansion 1 2 for n 4 there exists a 4x2 n 2 mesh that can be embedded in the n dimensional mobius cube with dilation 2 and expansion 1 3 for n 4 there are two disjoint 4x2 n 3 meshes that can be embedded in the 0 type n dimensional mobius cube with dilation 1 4 for n 4 there are two disjoint 4x2 n 3 meshes that can be embedded in the 1 type n dimensional mobius cube with dilation 2 results of 1 and 3 are optimal in the sense that the dilations of the embeddings are 1

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

conditional matching preclusion for the arrangement graphs the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex recently the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex it is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings in this paper we find this number and classify all optimal sets for the arrangement graphs one of the most popular interconnection networks

geodesic pancyclicity of twisted cubes the hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement an n dimensional twisted cube tq n is an important variation of the hypercube q n and preserves many of its desirable properties the problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years the geodesic cycle embedding problem is for any two distinct vertices to find all the possible lengths of cycles that include a shortest path joining them in this paper we prove that tq n is geodesic 2 pancyclic for each odd integer n 3 this result implies that tq n is edge pancyclic for each odd integer n 3 moreover tq nxk 2 is also demonstrated to be geodesic 4 pancyclic

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

bipanconnectivity of faulty hypercubes with minimum degree abstract in this paper we consider the conditionally faulty hypercube q n with n 2 where each vertex of q n is incident with at least m fault free edges 2 m n 1 we shall generalize the limitation m 2 in all previous results of edge bipancyclicity we also propose a new edge fault tolerant bipanconnectivity called k edge fault tolerant bipanconnectivity a bipartite graph is k edge fault tolerant bipanconnected if g f remains bipanconnected for any f e g with f k for every integer m under the same hypothesis we show that q n is n 2 edge fault tolerant edge bipancyclic and bipanconnected and the results are optimal with respect to the number of edge faults tolerated this not only improves some known results on edge bipancyclicity and bipanconnectivity of hypercubes but also simplifies the proof

distinct squares in run length encoded strings squares are strings of the form ww where w is any nonempty string two squares ww and w w are of different types if and only if w w fraenkel and simpson avieri s fraenkel jamie simpson how many squares can a string contain journal of combinatorial theory series a 82 1998 112 120 proved that the number of square types contained in a string of length n is bounded by o n the set of all different square types contained in a string is called the vocabulary of the string if a square can be obtained by a series of successive right rotations from another square then we say the latter covers the former a square is called a c square if no square with a smaller index can cover it and it is not a trivial square the set containing all c squares is called the covering set note that every string has a unique covering set furthermore the vocabulary of the covering set are called c vocabulary in this paper we prove that the cardinality of c vocabulary in a string is less than 143n where n is the number of runs in this string

edge fault tolerant pancyclicity of arrangement graphs the arrangement graph a n k is a well known interconnection network day and tripathi proved that a n k is pancyclic for n k 2 in this paper we improve this result and we demonstrate that a n k is also pancyclic even if it has no more than k n k 2 faulty edges for n k 2 our result is optimal concerning the edge fault tolerance

embedding meshes into twisted cubes the n dimensional twisted cube tn n is a variation of the hypercube in this paper we study embedding of meshes into tn n we prove three major results in this paper 1 for any integer n 1 a 2x2 n 1 mesh can be embedded into tn n with dilation 1 and expansion 1 2 for any integer n 4 an mxk m 3 k 3 mesh cannot be embedded into tn n with dilation 1 3 for any integer n 4 two node disjoint 4x2 n 3 meshes can be embedded into tn n with dilation 2 and expansion 1

geodesic pancyclicity and fault tolerant panconnectivity of augmented cubes abstract choudum and sunitha s a choudum v sunitha augmented cubes networks 40 2002 71 84 proposed the class of augmented cubes as a variation of hypercubes and showed that augmented cubes possess several embedding properties that the hypercubes and other variations do not possess recently hsu et al h c hsu p l lai c h tsai geodesic pancyclicity and balanced pancyclicity of augmented cubes information processing letters 101 2007 227 232 showed that the n dimensional augmented cube aq n n 2 is weakly geodesic pancyclic i e for each pair of vertices u v aq n and for each integer l satisfying max 2 d u v 3 l 2 n where d u v denotes the distance between u and v in aq n there is a cycle of length l that contains a u v geodesic in this paper we obtain a stronger result by proving that aq n n 2 is indeed geodesic pancyclic i e for each pair of vertices u v aq n and for each integer l satisfying max 2 d u v 3 l 2 n every u v geodesic lies on a cycle of length l to achieve the result we first show that aq n f n 3 remains panconnected and thus is also edge pancyclic if f aq n is any faulty vertex the result of fault tolerant panconnectivity is the best possible in the sense that the number of faulty vertices in aq n n 3 cannot be increased

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

an extension of the kunz souillard approach to localization in one dimension and applications to almost periodic schrodinger operators abstract we generalize the approach to localization in one dimension introduced by kunz souillard and refined by delyon kunz souillard and simon in the early 1980 s in such a way that certain correlations are allowed several applications of this generalized kunz souillard method to almost periodic schrodinger operators are presented on the one hand we show that the schrodinger operators on l 2 z with limit periodic potential that have pure point spectrum form a dense subset in the space of all limit periodic schrodinger operators on l 2 z more generally for any bounded potential one can find an arbitrarily small limit periodic perturbation so that the resulting operator has pure point spectrum our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit periodic schrodinger operators on l 2 z on the other hand we show that schrodinger operators on l 2 z with arbitrarily small one frequency quasi periodic potential may have pure point spectrum for some phases this was previously known only for one frequency quasi periodic potentials with norm exceeding 2 namely the super critical almost mathieu operator with a typical frequency and phase moreover this phenomenon can occur for any frequency whereas no previous quasi periodic potential with liouville frequency was known that may admit eigenvalues for any phase

embedding binary trees into crossed cubes the recently introduced interconnection network crossed cube has attracted much attention in the parallel processing area due to its many attractive features like the ordinary hypercube the n dimensional crossed cube is a regular graph with 2 sup n vertices and n2 sup n 1 edges the diameter of the crossed cube is approximately half that of the ordinary hypercube these advantages of the crossed cube motivated the study of how well it can simulate other networks such as the complete binary tree we show that the 2 sup n 1 node complete binary tree can be embedded into the n dimensional crossed cube with dilation 1

on embedding rings into a star related network abstract it is shown that in the star graph it is always possible to form a hamiltonian path between two nodes as long as the distance between them is odd this result leads to the proof of hamiltonicity for the clustered star graph the results can readily be used in optimal embedding of a linear array of processors or a ring in the star graph the embedding problem is also solved when one or more processors are faulty in the network the length of the linear array and the ring in this case is shown to be n m where m is the dimension of the smallest star graph which contains all the faulty processors

embedding of cycles in arrangement graphs arrangement graphs have been proposed as an attractive interconnection topology for large multiprocessor systems the authors study these graphs by proving the existence of hamiltonian cycles in any arrangement graph they also prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph they show that an arrangement graph can be decomposed into node disjoint cycles in many different ways

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

brief paper quantized consensus in hamiltonian graphs the main contribution of this paper is an algorithm to solve an extended version of the quantized consensus problem over networks represented by hamiltonian graphs i e graphs containing a hamiltonian cycle which we assume to be known in advance given a network of agents we assume that a certain number of tokens should be assigned to the agents so that the total number of tokens weighted by their sizes is the same for all the agents the algorithm is proved to converge almost surely to a finite set containing the optimal solution a worst case study of the expected convergence time is carried out thus proving the efficiency of the algorithm with respect to other solutions recently presented in the literature moreover the algorithm has a decentralized stop criterion once the convergence set is reached

a note on the number of squares in a word fraenkel and simpson a s fraenkel j simpson how many squares can a string contain j combin theory ser a 82 1998 112 120 proved that the number of squares in a word of length n is bounded by 2n in this note we improve this bound to 2n q logn based on the numerical evidence the conjectured bound is n

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard