exact boundary behavior of the unique positive solution to some singular elliptic problems abstract in this paper we give an exact asymptotic of the unique solution to the following singular boundary value problem u a x g u x u 0 in u 0 here is a c 2 bounded domain in r n n 2 g c 1 0 0 is nonincreasing on 0 with lim t 0 g t 0 t d s g s c g 0 and the function a is in c l o c 0 1 satisfying 0 a 1 lim inf d x 0 a x h d x lim sup d x 0 a x h d x a 2 where h t c t exp t z s s d s 2 c 0 and z continuous on 0 for some 0 such that z 0 0 two applications of this result are also given the first concerns the boundary behavior of the unique solution of u u u 2 a x g u that vanishes on the boundary and the second concerns the behavior of u in the case where the open set is an annular and the behaviors of the function a on the interior boundary and the exterior boundary may be different

a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

asymptotic behavior of positive solutions of a semilinear dirichlet problem abstract in this paper we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem u a x u 0 x u 0 in u 0 here is a bounded c 1 1 domain 1 and the function a is in c l o c 0 1 such that there exists c 0 satisfying for each x 1 c a x x exp x z t t d t c where 2 d d i a m x d i s t x and z is a continuous function on 0 with z 0 0

design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

on a semilinear fractional dirichlet problem on a bounded domain let d be a bounded c 1 1 domain in r d d 2 0 a 2 and s 1 we are concerned with the existence the uniqueness and the asymptotic behavior of positive continuous solution for the following semilinear fractional differential equation email protected d a 2u a x u s x ind with the boundary dirichlet email protected email protected email protected x 2 au x 0 here email protected d a 2 is the fractional laplacian associated to the subordinate killed brownian motion process in d and d x d x d denotes the euclidean distance between x and d our arguments are based on potential theory associated to the fractional laplacian and some karamata regular variation theory tools

the second expansion of the solution for a singular elliptic boundary value problem abstract in this paper we analyze the second expansion of the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 g is decreasing on 0 with lim s 0 g s and g is normalised regularly varying at zero with index 1 b c is positive in may be vanishing on the boundary

the boundary behavior of the unique solution to a singular dirichlet problem abstract in this paper we analyze the boundary behavior of the unique solution to the singular dirichlet problem for a lane emden fowler equation u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim s 0 g s g is decreasing on 0 which is rapidly varying or normalized regularly varying at zero b c loc is positive in may be vanishing or singular on the boundary and belongs to the kato class k

on a family of solutions of the kadomtsev petviashvili equation which also satisfy the toda lattice hierarchy we describe the interaction pattern in the x y plane for a family of soliton solutions of the kadomtsev petviashvili kp equation the solutions considered also satisfy the finite toda lattice hierarchy we determine completely their asymptotic patterns for y and we show that all the solutions except the 1 soliton solution are of resonant type consisting of arbitrary numbers of line solitons in both asymptotics that is arbitrary n incoming solitons for y interact to form arbitrary n outgoing solitons for y we also discuss the interaction process of those solitons and show that the resonant interaction creates a web like structure having n 1 n 1 holes

exact behavior of the unique positive solution to some singular elliptic problem in exterior domains in this paper we give an exact behavior on the boundary and at infinity of the unique solution to the following singular boundary value problem u a x g u x u 0 in u 0 and lim x u x 0 here is an exterior domain in rn n 3 with compact c2c2 boundary g c1 0 0 g c1 0 0 is nonincreasing on 0 0 with limt 0g t 0tdsg s cg 0 and the function aa is in cloc 0 10 1 satisfying 0 a1 lim infd x 0a x h d x lim supd x 0a x h d x a2 and 0 b1 lim inf x a x k x lim sup x a x k x b2 where d x d x is the euclidean distance from x x to the boundary h t c1t exp t z s sds 2 2 c1 0c1 0 and zz is continuous on 0 0 for some 0 0 such that z 0 0z 0 0 and k t c2t exp 1ty s sds 2 2 c2 0c2 0 and yy is continuous on 1 1 such that limt y t 0limt y t 0

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

on clusterings good bad and spectral we motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures a simple recursive heuristic is shown to have poly logarithmic worst case guarantees under the new measure the main result of the article is the analysis of a popular spectral algorithm one variant of spectral clustering turns out to have effective worst case guarantees another finds a good clustering if one exists

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

the exact boundary behavior of the unique solution to a singular dirichlet problem with a nonlinear convection term abstract in this paper we analyze the exact boundary behavior of the unique solution to the singular nonlinear dirichlet problem u b x g u u q u 0 x u 0 where is a bounded domain with smooth boundary in r n q 0 2 0 g c 1 0 0 lim s 0 g s g is decreasing on 0 and b c loc is positive in may be vanishing or singular on the boundary we reveal that the nonlinear convection term u q does not affect the first expansion of the unique solution near the boundary to the problem

normalized graph laplacians for directed graphs abstract we consider the normalized laplace operator for directed graphs with positive and negative edge weights this generalization of the normalized laplace operator for undirected graphs is used to characterize directed acyclic graphs moreover we identify certain structural properties of the underlying graph with extremal eigenvalues of the normalized laplace operator we prove comparison theorems that establish a relationship between the eigenvalues of directed graphs and certain undirected graphs this relationship is used to derive eigenvalue estimates for directed graphs finally we introduce the concept of neighborhood graphs for directed graphs and use it to obtain further eigenvalue estimates

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

second order expansion for the solution to a singular dirichlet problem in this paper we analyze the second order expansion for the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c1 0 0 g is decreasing on 0 with lim s 0 g s and g is normalized regularly varying at zero with index 1 b c l o c 0 1 is positive in may be vanishing or singular on the boundary and belongs to the kato class k our analysis is based on the sub supersolution method and karamata regular variation theory

eigenvalues and expansion of regular graphs the spectral method is the best currently known technique to prove lower bounds on expansion ramanujan graphs which have asymptotically optimal second eigenvalue are the best known explicit expanders the spectral method yielded a lower bound of k 4 on the expansion of linear sized subsets of k regular ramanujan graphs we improve the lower bound on the expansion of ramanujan graphs to approximately k 2 moreover we construct a family of k regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k 2 this shows that k 2 is the best bound one can obtain using the second eigenvalue method we also show an upper bound of roughly 1 k 1 on the average degree of linear sized induced subgraphs of ramanujan graphs this compares positively with the classical bound 2 k 1 as a byproduct we obtain improved results on random walks on expanders and construct selection networks respectively extrovert graphs of smaller size respectively degree than was previously known

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

n soliton solutions to the dkp equation and weyl group actions we study soliton solutions to the dkp equation which is defined by the hirota bilinear form where 0 1 the functions n are given by the pfaffians of a certain skew symmetric matrix we identify a one soliton solution as an element of the weyl group of d type and discuss a general structure of the interaction patterns among the solitons soliton solutions are characterized by a 4n 4n skew symmetric constant matrix which we call the b matrix we then find that one can have m soliton solutions with m being any number from n to 2n 1 for some of the 4n 4n b matrices having only 2n nonzero entries in the upper triangular part the number of solitons obtained from those b matrices was previously expected to be just n

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

on a new kato class and positive solutions of dirichlet problems for the fractional laplacian in bounded domains abstract the purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional laplacian 2 0 2 in a bounded c 1 1 domain d in r n in particular we introduce a new kato class k d and we exploit the properties of this class to study the existence of positive solutions of some dirichlet problems for the fractional laplacian

on various solutions of the coupled kp equation the coupled kp ckp equation possesses n soliton solutions with many more degrees of freedom than the solitons for the usual kp equation have in comparison ckp solutions can therefore be expected to model far more complex interactions than their kp counterparts in this paper we present some typical solutions for the ckp system and for some of its reductions a complex coupled kdv equation and a coupled boussinesq equation and we analyse their interaction and asymptotic properties

the second expansion of the unique vanishing at infinity solution to a singular elliptic equation abstract this paper is considered with the second expansion of the unique vanishing at infinity solution to u b x g u x r n n 3 where b c r n is nonnegative and may be supercritical attenuation or critical attenuation at infinity g c 1 0 0 is non increasing on 0 with lim s 0 g s and g is normalized regularly varying at zero with index 1

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

asymptotic behavior of positive solutions of a nonlinear dirichlet problem abstract we take up the existence and the asymptotic behavior of a classical solution to the following semilinear dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a belongs to c l o c 0 1 such that there exist c 1 c 2 0 satisfying for each x c 1 x 1 exp x z 1 s s d s a x c 2 x 2 exp x z 2 s s d s where d i a m x d i s t x 1 2 2 and for i 1 2 z i is a continuous function on 0 with z i 0 0 our arguments are based on the sub supersolution method with karamata regular variation theory

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

spectral properties of hypergraph laplacian and approximation algorithms the celebrated cheeger s inequality alon and milman 1985 alon 1986 establishes a bound on the edge expansion of a graph via its spectrum this inequality is central to a rich spectral theory of graphs based on studying the eigenvalues and eigenvectors of the adjacency matrix and other related matrices of graphs it has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph in this article we introduce a new hypergraph laplacian operator generalizing the laplacian matrix of graphs in particular the operator is induced by a diffusion process on the hypergraph such that within each hyperedge measure flows from vertices having maximum weighted measure to those having minimum since the operator is nonlinear we have to exploit other properties of the diffusion process to recover the cheeger s inequality that relates hyperedge expansion with the second eigenvalue of the resulting laplacian however we show that higher order spectral properties cannot hold in general using the current framework since higher order spectral properties do not hold for the laplacian operator we instead use the concept of procedural minimizers to consider higher order cheeger like inequalities for any k n we give a polynomial time algorithm to compute an o log r approximation to the kth procedural minimizer where r is the maximum cardinality of a hyperedge we show that this approximation factor is optimal under the sse hypothesis introduced by raghavendra and steurer 2010 for constant values of k moreover using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion we show that all our results for hypergraphs extend to vertex expansion in graphs

on the complexity of approximating the independent set problem we show that for some positive constant c it is not feasible to approximate independent set for graphs of n nodes within a factor of n c provided maximum 2 satisfiability does not have a randomized polynomial time approximation scheme we also study reductions preserving the quality of approximations and exhibit complete problems

the second order expansion of solutions to a singular dirichlet boundary value problem abstract in this paper we mainly study the second order expansion of classical solutions in a neighborhood of to the singular dirichlet problem u b x g u a x f u u 0 x u 0 where is a bounded domain with smooth boundary in r n 0 the weight functions b a c loc are positive in and both may be vanishing or be singular on the boundary the function g c 1 0 0 satisfies lim t 0 g t and f c 0 0 we show that the nonlinear term a x f u does not affect the second order expansion of solutions in a neighborhood of to the problem for some kinds of functions b and a

dispersionless dkp hierarchy and the elliptic lowner equation we show that the dispersionless dkp hierarchy the dispersionless limit of the pfaff lattice admits a suggestive reformulation through elliptic functions we also consider one variable reductions of the dispersionless dkp hierarchy and show that they are described by an elliptic version of the lowner equation with a particular choice of the driving function the latter appears to be closely related to the painleve vi equation with a special choice of parameters

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

on the soliton solutions of the two dimensional toda lattice the two dimensional toda lattice 2dtl is a well known 2 1 dimensional integrable system that admits a large class of line soliton solutions we classify these soliton solutions by exploiting the structure of the casoratian expression for its tau function in general these solutions consist of unequal numbers of incoming and outgoing line solitons we classify the incoming and outgoing line solitons based on asymptotic analysis of the tau function of the 2dtl as the discrete variable tends to infinity we also identify various subclasses of solutions and characterize some of them in terms of the amplitudes and directions of the interacting solitons as a special case we obtain the reduction in the soliton solutions of the one dimensional toda lattice throughout we point out the similarities with and the differences from the corresponding results for the kadomtsev petviashvili equation

existence and estimates of solutions for singular nonlinear elliptic problems for both bounded and unbounded domain in r d 3 with smooth boundary the function 0 0 is measurable and satisfies some appropriate hypotheses detailed below the problem 1 has been studied extensively for the special nonlinearity x t p x t 0 indeed in 7 edelson proved the existence of a positive solution u c2 loc r decaying as x 2 n as x provided that p c loc r 0 0 for x 0 and satisfying the growth condition

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

multiway spectral partitioning and higher order cheeger inequalities a basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the laplacian matrix of the graph in particular the graph is disconnected if and only if there are at least two eigenvalues equal to zero cheeger s inequality and its variants provide an approximate version of the latter fact they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero it has been conjectured that an analogous characterization holds for higher multiplicities there are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets each defining a sparse cut we resolve this conjecture positively our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into rk and then apply geometric considerations to the embedding we also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size a n k and k the kth smallest eigenvalue of the normalized laplacian where n is the number of vertices in particular we show that in every graph there are at least k 2 disjoint sets one of which will have size at most 2n k each having expansion at most o k log k louis raghavendra tetali and vempala have independently proved a slightly weaker version of this last result the log k bound is tight up to constant factors for the noisy hypercube graphs

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

exact asymptotic behavior near the boundary to the solution for singular nonlinear dirichlet problems abstract we take up the existence and the exact asymptotic behavior near the boundary of the unique classical solution to a singular dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a is singular near the boundary and belongs to the kato class k

a chernoff bound for random walks on expander graphs we consider a finite random walk on a weighted graph g we show that the fraction of time spent in a set of vertices a converges to the stationary probability pi a with error probability exponentially small in the length of the random walk and the square of the size of the deviation from pi a the exponential bound is in terms of the expansion of g and improves previous results of d aldous probab engrg inform sci 1 1987 pp 33 46 l lovasz and m simonovits it random structures algorithms 4 1993 pp 359 412 m ajtai j komlos and e szemeredi deterministic simulation of logspace in proc 19th acm symp on theory of computing 1987 we show that taking the sample average from one trajectory gives a more efficient estimate of pi a than the standard method of generating independent sample points from several trajectories using this more efficient sampling method we improve the algorithms of jerrum and sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic ising system and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph

diffusion operator and spectral analysis for directed hypergraph laplacian abstract in spectral graph theory the cheeger inequality gives upper and lower bounds of edge expansion in normal graphs in terms of the second eigenvalue of the graph s laplacian operator recently this inequality has been extended to undirected hypergraphs and directed normal graphs via a non linear operator associated with a diffusion process in the underlying graph in this work we develop a unifying framework for defining a diffusion operator on a directed hypergraph with stationary vertices which is general enough for the following two applications 1 cheeger s inequality for directed hyperedge expansion 2 quadratic optimization with stationary vertices in the context of semi supervised learning despite the crucial role of the diffusion process in spectral analysis previous works have not formally established the existence of the corresponding diffusion processes in this work we give a proof framework that can indeed show that such diffusion processes are well defined in the first application we use the spectral properties of the diffusion operator to achieve the cheeger s inequality for directed hyperedge expansion in the second application the diffusion operator can be interpreted as giving a continuous analog to the subgradient method which moves the feasible solution in discrete steps towards an optimal solution

young diagrams and n soliton solutions of the kp equation we consider n soliton solutions of the kp equation an n soliton solution is a solution u x y t which has the same set of n line soliton solutions in both asymptotics y and y the n soliton solutions include all possible resonant interactions among those line solitons we then classify those n soliton solutions by defining a pair of n numbers n n with n n 1 n n n j 1 2n which labels n line solitons in the solution the classification is related to the schubert decomposition of the grassmann manifolds gr n 2n where the solution of the kp equation is defined as a torus orbit then the interaction pattern of n soliton solution can be described by the pair of young diagrams associated with n n we also show that n soliton solutions of the kdv equation obtained by the constraint u y 0 cannot have resonant interaction

asymptotic behavior of positive solutions of a singular nonlinear dirichlet problem abstract let be a c 1 1 bounded domain in r n for n 2 in this paper we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary value problem u a x u 0 in u 0 where 0 a is a nonnegative function in c loc 0 1 and there exists c 0 such that 1 c a x x k 1 m log k x k c here 2 k r is a positive constant and x dist x

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

on the subordinate killed b m in bounded domains and existence results for nonlinear fractional dirichlet problems we take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional laplacian delta d frac alpha 2 associated to the subordinate killed brownian motion process z alpha d in a bounded c 1 1 domain d our arguments are based on potential theory tools on z alpha d and properties of an appropriate kato class of functions k d

asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball abstract let m be a positive integer we investigate the existence and the asymptotic behavior of positive continuous solutions to the following semilinear polyharmonic boundary value problem in the unit ball b of r n n 2 m u a x u lim x 1 u x 1 x m 1 0 where 1 1 and a is a positive measurable function in b such that there exists c 0 satisfying for each x b 1 c a x 1 x exp 1 x z s s d s c 1 m 1 1 and z is a continuous function on 0 with z 0 0

the asymptotic behaviour of the unique solution for the singular lane emden fowler equation abstract by karamata regular variation theory and constructing comparison functions we show the exact asymptotic behaviour of the unique classical solution u c 2 c near the boundary to a singular dirichlet problem u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim t 0 g t g t for each 0 for some 0 and k c loc for some 0 1 is nonnegative on which is also singular near the boundary

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

lower bounds for matrix product in bounded depth circuits with arbitrary gates we prove superlinear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs our lower bounds are proved for all types of constant depth circuits e g constant depth arithmetic circuits and constant depth boolean circuits with arbitrary gates the bounds apply for several explicit functions and most importantly for matrix product in particular we obtain the following results we show that the number of edges in any constant depth arithmetic circuit for matrix product over any field is superlinear in m2 where m m is the size of each matrix that is the lower bound is superlinear in the number of input variables moreover if the circuit is bilinear the result applies also for the case in which the circuit gets any product of two linear functions for free we show that the number of edges in any constant depth arithmetic circuit for the trace of the product of three matrices over fields with characteristic 0 is superlinear in m2 note that the trace is a single output function we give explicit examples for n boolean functions f1 fn such that any constant depth boolean circuit with arbitrary gates for f1 fn has a superlinear number of edges the lower bound is also proved for circuits with arbitrary gates over any finite field the bound applies for matrix product over finite fields as well as for several other explicit functions

pseudorandom generators in propositional proof complexity we call a pseudorandom generator g n 0 1 n to 0 1 m hard for a propositional proof system p if p cannot efficiently prove the properly encoded statement g n x 1 ldots x n neq b for any string b in 0 1 m we consider a variety of combinatorial pseudorandom generators inspired by the nisan wigderson generator on the one hand and by the construction of tseitin tautologies on the other we prove that under certain circumstances these generators are hard for such proof systems as resolution polynomial calculus and polynomial calculus with resolution pcr

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

correlation clustering with noisy partial information in this paper we propose and study a semi random model for the correlation clustering problem on arbitrary graphs g we give two approximation algorithms for correlation clustering instances from this model the first algorithm finds a solution of value 1 opt cost o nlog 3 n with high probability where opt cost is the value of the optimal solution for every 0 the second algorithm finds the ground truth clustering with an arbitrarily small classific ation error under some additional assumptions on the instance