on the o spektrum of regular operators and the spectrum of measures the banach algebra m g of all bounded regular borel measures on a locally compact group g is often represented on lz g by the mapping p t 2 t zf f fel2 g this representation has the advantage of bringing the hilbert space structure into play thus the involution is preserved and the spectrum of tu 2 is easy to compute if g is abelian or compact there is nevertheless an inconvenience the spectrum of in the banach algebra m g does not coincide with the spectrum of the operator tu z in general this could be overcome by considering li g instead of lz g another possibility which conserves the hilbert space structure makes use of the fact that the operators t 2 are regular operators on the banach lattice l2 g i e linear combinations of positive operators and indeed the representation of m g in the smaller banach algebra l2 g of all regular operators on l2 g behaves nicely if g is amenable it is an algebraic and lattice isomorphism of m g onto the full subalgebra and sublattice of those regular operators which commute with the right translations consequently the spectrum of in m g is the same as the spectrum of t 2 in the banach algebra r l2 g

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

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

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

subordinated discrete semigroups of operators given a power bounded linear operator t in a banach space and a probability f on the non negative integers one can form a subordinated operator s k 0 f k t k we obtain asymptotic properties of the subordinated discrete semigroup s n n 1 2 under certain conditions on f in particular we study probabilities f with the property that s satisfies the ritt resolvent condition whenever t is power bounded examples and counterexamples of this property are discussed the hypothesis of power boundedness of t can sometimes be replaced by the weaker kreiss resolvent condition

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

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

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

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

probabilistic low rank matrix completion from quantized measurements we consider the recovery of a low rank real valued matrix m given a subset of noisy discrete or quantized measurements such problems arise in several applications such as collaborative filtering learning and content analytics and sensor network localization we consider constrained maximum likelihood estimation of m under a constraint on the entry wise infinity norm of m and an exact rank constraint we provide upper bounds on the frobenius norm of matrix estimation error under this model previous theoretical investigations have focused on binary 1 bit quantizers and been based on convex relaxation of the rank compared to the existing binary results our performance upper bound has faster convergence rate with matrix dimensions when the fraction of revealed observations is fixed we also propose a globally convergent optimization algorithm based on low rank factorization of m and validate the method on synthetic and real data with improved performance over previous methods

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

computing the partition function ensemble averages and density of states for lattice spin systems by sampling the mean an algorithm to approximately calculate the partition function and subsequently ensemble averages and density of states of lattice spin systems through non monte carlo random sampling is developed this algorithm called the sampling the mean algorithm can be applied to models where the up or down spins at lattice nodes interact to change the spin states of other lattice nodes especially non ising like models with long range interactions such as the biological model considered here because it is based on the central limit theorem of probability the sampling the mean algorithm also gives estimates of the error in the partition function ensemble averages and density of states easily implemented parallelization strategies and error minimizing sampling strategies are discussed the sampling the mean method works especially well for relatively small systems systems with a density of energy states that contains sharp spikes or oscillations or systems with little a priori knowledge of the density of states

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

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

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

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

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 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

almost everywhere convergence of powers of some positive lp contractions abstract we extend the solution of burkholder s conjecture for products of conditional expectations obtained by delyon and delyon for l 2 and by cohen for l p 1 p to the context of badea and lyubich let t be a finite convex combination of operators t j which are products of finitely many conditional expectations then t n f converges a e for every f l p 1 p with sup n t n f l p the proof uses the work of le merdy and xu on positive l p contractions satisfying ritt s resolvent condition as another application of the work of le merdy and xu we extend a result of bellow jones and rosenblatt proving that if a probability a k k z has bounded angular ratio then for every positive invertible isometry s of an l p space 1 p the operator t k z a k s k is a positive l p contraction such that for every f l p t n f converges a e and sup n t n f l p if a k is supported on n the same result is true when s is only a positive contraction of l p similar results are obtained for averages of bounded continuous representations of a compact lca group by positive operators in one l p space 1 p for a positive contraction t on l p which satisfies ritt s condition and f i t l p 0 1 we prove that n t n f 0 a e and sup n n t n f l p

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

on the adaptive finite element analysis of the kohn sham equations methods algorithms and implementation in this paper details of an implementation of a numerical code for computing the kohn sham equations are presented and discussed a fully self consistent method of solving the quantum many body problem within the context of density functional theory using a real space method based on finite element discretisation of realspace is considered various numerical issues are explored such as i initial mesh motion aimed at coaligning ions and vertices ii a priori and a posteriori optimization of the mesh based on kelly s error estimate iii the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy additionally iv explicit implicit and gaussian approaches to treat the ionic potential are compared a quadrupole expansion is employed to provide boundary conditions for the poisson problem to exemplify the soundness of our method accurate computations are performed for hydrogen helium lithium carbon oxygen neon the hydrogen molecule ion and the carbon monoxide molecule our methods algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment copyright 2015 john wiley sons ltd

resolvent representations for functions of sectorial operators abstract we obtain integral representations for the resolvent of a where is a holomorphic function mapping the right half plane and the right half axis into themselves and a is a sectorial operator on a banach space as a corollary for a wide class of functions we show that the operator a generates a sectorially bounded holomorphic c 0 semigroup on a banach space whenever a does and the sectorial angle of a is preserved when is a bernstein function this was recently proved by gomilko and tomilov but the proof here is more direct moreover we prove that such a permanence property for a can be described at least on hilbert spaces in terms of the existence of a bounded h calculus for a as byproducts of our approach we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for ritt operators

the ritt property of subordinated operators in the group case abstract let g be a locally compact abelian group let be a regular probability measure on g let x be a banach space let g b x be a bounded strongly continuous representation consider the average or subordinated operator s g t d t x x we show that if x is a umd banach lattice and has bounded angular ratio then s is a ritt operator with a bounded h functional calculus next we show that if is the square of a symmetric probability measure and x is k convex then s is a ritt operator we further show that this assertion is false on any non k convex space x

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

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

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

convergence of iterates of averages of certain operator representations and of convolution powers abstract 1 1 let g be a locally compact compact group and let t t be a continuous representation of g by isometries in a uniformly convex banach space b let be an adapted strictly aperiodic and spread out probability on g and let px t t x d t then pn converges strongly 2 2 if g is not compact and is as above then n 0 for every continuous vanishing at infinity

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

analyticity and discrete maximal regularity on lp spaces abstract in the first part of this paper we give the following interpolation result on the analyticity i e the property t i tn c n for all n n of an operator t on lp if t is powerbounded on lp and lq as well as analytic on lp then t is powerbounded and analytic on lr for all r strictly between p and q this is a discrete analogue of the well known corresponding result for analytic semigroups eta as recently shown by the author the analyticity of t is a necessary condition for the maximal regularity of the discrete time evolution equation un 1 tun fn for all n z u0 0 in the second part of this paper we establish the following two sufficient conditions for its maximal regularity t is a subpositive analytic contraction or t is an integral operator satisfying certain poisson bounds these results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u t au t f t for all t r u 0 0 due to lamberton weis coulhon and duong and hieber and pruss for the poisson bound result of coulhon and duong and hieber and pruss we give a slight improvement and a short proof

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

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

a max norm constrained minimization approach to 1 bit matrix completion we consider in this paper the problem of noisy 1 bit matrix completion under a general non uniform sampling distribution using the max norm as a convex relaxation for the rank a max norm constrained maximum likelihood estimate is introduced and studied the rate of convergence for the estimate is obtained information theoretical methods are used to establish a minimax lower bound under the general sampling model the minimax upper and lower bounds together yield the optimal rate of convergence for the frobenius norm loss computational algorithms and numerical performance are also discussed

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 subordination of holomorphic semigroups abstract we prove that for any bernstein function the operator a generates a bounded holomorphic c 0 semigroup e t a t 0 on a banach space whenever a does this answers a question posed by kishimoto and robinson moreover giving a positive answer to a question by berg boyadzhiev and de laubenfels we show that e t a t 0 is holomorphic in the holomorphy sector of e t a t 0 and if e t a t 0 is sectorially bounded in this sector then e t a t 0 has the same property we also obtain new sufficient conditions on in order that for every banach space x the semigroup e t a t 0 on x is holomorphic whenever e t a t 0 is a bounded c 0 semigroup on x these conditions improve and generalize well known results by carasso kato and fujita

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

on functional calculus properties of ritt operators we compare various functional calculus properties of ritt operators we show the existence of a ritt operator t x x on some banach space x with the following property t has a bounded h functional calculus with respect to the unit disc that is t is polynomially bounded but t does not have any bounded h functional calculus with respect to a stolz domain of with vertex at 1 also we show that for an r ritt operator the unconditional ritt condition of kalton and portal is equivalent to the existence of a bounded h functional calculus with respect to such a stolz domain