asymptotic behaviour of bohmian trajectories in scattering situations we study the asymptotic behaviour of bohmian trajectories in a scattering situation with short range potential v and for wavefunctions with a scattering and a bound part it is shown that the set of possible trajectories splits into trajectories whose long time behaviour is governed by the scattering part of the wavefunction scattering trajectories and trajectories whose long time behaviour is governed by the bound part of the wavefunction bound trajectories furthermore the scattering trajectories behave like trajectories in classical mechanics in the limit t as an intermediate step we show that the asymptotic velocity exists almost surely and is randomly distributed with the density where is the outgoing asymptote of the scattering part of the wavefunction

linear algebraic methods in communication complexity abstract the notion of communication complexity seeks to capture the amount of communication between different parties that is required to find the output of a boolean function when each party is provided with only part of the input different variants of the model governing the rules of this communication lead to different connections with problems in combinatorial linear algebra in particular problems arise in this context that concern the rank of a 0 1 matrix and the minimum rank of a matrix meeting a given combinatorial description this paper surveys these connections

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

amplifying the randomness of weak sources correlated with devices the problem of device independent randomness amplification against no signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the quantum devices used in the amplification procedure in this paper we relax this assumption and reconsider the original protocol of colbeck and renner using a santha vazirani sv source to do so we introduce an sv like condition for devices namely that any string of sv source bits remains weakly random conditioned upon any other bit string from the same sv source and the outputs obtained when this further string is input into the devices assuming this condition we show that a quantum device using a singlet state to violate the chained bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings for a restricted set of sv sources with 0 leq varepsilon we also study a device independent protocol that allows for correlations between the sequence of boxes used in the protocol and the sv source bits used to choose the particular box from whose output the randomness is obtained assuming the sv like condition for devices we show that the honest parties can achieve amplification of the weak source for the parameter range 0 leq varepsilon against a class of attacks given as a mixture of product box sequences made of extremal no signaling boxes with additional symmetry conditions composable security proof against this class of attacks is provided

a capacity achieving simple decoder for bias based traitor tracing schemes we investigate alternative suspicion functions for bias based traitor tracing schemes and present a practical construction of a simple decoder that attains capacity in the limit of large coalition size c we derive optimal suspicion functions in both the restricted digit model and the combined digit model these functions depend on information that is usually not available to the tracer the attack strategy or the tallies of the symbols received by the colluders we discuss how such results can be used in realistic contexts we study several combinations of coalition attack strategy versus suspicion function optimized against some attack another attack or the same in many of these combinations the usual codelength scaling ell propto c 2 changes to a lower power of c e g c 3 2 we find that the interleaving strategy is an especially powerful attack the suspicion function tailored against interleaving is the key ingredient of the capacity achieving construction

on the security of pseudorandomized information theoretically secure schemes in this paper we discuss a naive method of randomness reduction for cryptographic schemes which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator prg we propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases even against an adversary with computationally unbounded attack algorithm hence the prg based randomness reduction can be effective even for information theoretically secure cryptographic schemes especially when the amount of information received by the adversary is small in comparison to a preceding result of dubrov and ishai stoc 2006 our result removes the requirement of generalized notion of nb prgs and is effective for more general kinds of protocols we give some numerical examples to show the effectiveness of our result in practical situations and we also propose a further idea for improving the effect of the prg based randomness reduction

decoding frequency permutation arrays under chebyshev distance a frequency permutation array fpa of length n m and distance d is a set of permutations on a multiset over m symbols where each symbol appears exactly times and the distance between any two elements in the array is at least d fpa generalizes the notion of permutation array in this paper under the chebyshev distance we first prove lower and upper bounds on the size of fpa then we give several constructions of fpas and some of them come with efficient encoding and decoding capabilities moreover we show one of our designs is locally decodable i e we can decode a message bit by reading at most 1 symbols which has an interesting application to private information retrieval

on the complexity of checking semantic equivalences between pushdown processes and finite state processes simulation preorder equivalence and bisimulation equivalence are the most commonly used equivalences in concurrency theory their standard definitions are often called strong simulation bisimulation while weak simulation bisimulation abstracts from internal t actions we study the computational complexity of checking these strong and weak semantic preorders equivalences between pushdown processes and finite state processes we present a complete picture of the computational complexity of these problems and also study fixed parameter tractability in two important input parameters x the size of the finite control of the pushdown process and y the size of the finite state process all simulation problems are generally exptime complete and only become polynomial if both parameters x and y are fixed weak bisimulation equivalence is pspace complete but becomes polynomial if and only if parameter x is fixed strong bisimulation equivalence is pspace complete but becomes polynomial if either parameter x or y is fixed

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

tree algorithms for unbiased coin tossing with a biased coin we give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias we are interested in efficient algorithms where the expected number of flips is our measure of efficiency other authors have represented algorithms as lattices but by representing them instead as trees we are able to produce an algorithm more efficient than any previously appearing we also prove a conjecture of hoeffding and simons that there is no optimal algorithm further we consider generalizations where the input is a sequence of iid discrete random variables and the output is a uniform random variable with n possible outcomes in this setting we provide an algorithm significantly superior to those previously published

a simple unpredictable pseudo random number generator two closely related pseudo random sequence generators are presented the 1 p generator with input p a prime outputs the quotient digits obtained on dividing 1 by p the x 2 bmod n generato

entropy satisfying relaxation method with large time steps for euler ibvps this paper could have been given the title how to positively and implicitly solve euler equations using only linear scalar advections the new relaxation method we propose is able to solve euler like systems as well as initial and boundary value problems with real state laws at very low cost using a hybrid explicit implicit time integration associated with the arbitrary lagrangian eulerian formalism furthermore it possesses many attractive properties such as i the preservation of positivity for densities ii the guarantee of min max principle for mass fractions iii the satisfaction of entropy inequality under an expressible bound on the cfl ratio the main feature that will be emphasized is the design of this optimal time step which takes into account data not only from the inner domain but also from the boundary conditions

numerical approximations of the 10 moment gaussian closure we propose a numerical scheme to approximate the weak solutions of the 10 moment gaussian closure the moment gaussian closure for gas dynamics is governed by a conservative hyperbolic system supplemented by entropy inequalities whose solutions satisfy positiveness of density and tensorial pressure we consider a suliciu type relaxation numerical scheme to approximate the solutions these methods are proved to satisfy all the expected positiveness properties and all the discrete entropy inequalities the scheme is illustrated by several numerical experiments

communication lower bounds using directional derivatives we study the set disjointness problem in the most powerful model of bounded error communication the k party randomized number on the forehead model we show that set disjointness requires n 2kk bits of communication where n is the size of the universe our lower bound generalizes to quantum communication where it is essentially optimal proving this bound was a longstanding open problem even in restricted settings such as one way classical protocols with k 4 parties wigderson 1997 the proof contributes a novel technique for lower bounds on multiparty communication based on directional derivatives of protocols over the reals

fast relaxation solvers for hyperbolic elliptic phase transition problems phase transition problems in compressible media can be modelled by mixed hyperbolic elliptic systems of conservation laws within this approach phase boundaries are understood as shock waves that satisfy additional constraints sometimes called kinetic relations for numerical approximation tracking type algorithms have been suggested the core piece of these algorithms is the usage of exact riemann solvers incorporating the kinetic relation however exact riemann solvers are computationally expensive or even not available in this paper we present a class of approximate riemann solvers for hyperbolic elliptic models that relies on a generalized relaxation procedure it preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions in combination with a novel subiteration procedure the approximate riemann solvers are used in the tracking algorithms the efficiency of the approach is validated on a barotropic system with linear kin

robust parent identifying codes and combinatorial arrays an n word y y1 yn over a finite alphabet of cardinality q is called a descendant of a set of t words x1 xt if every coordinate yi i 1 n is contained in the set x1i xti a code c x1 xm is said to have the t ipp property if for any n word y that is a descendant of at most t parents belonging to the code it is possible to identify at least one of them from earlier works it is known that t ipp codes of positive rate exist if and only if t q 1 we introduce a robust version of ipp codes which allows error free identification of parents in the presence of a certain number of mutations i e coordinates in y that can break away from the descent rule taking arbitrary values from the alphabet or becoming completely unreadable we show existence of robust t ipp codes for all t q 1 and some positive proportion of such coordinates we uncover a relation between the hash distance of codes and the ipp property and use it to find the exact proportion of mutant coordinates that permits identification of pirates with zero probability of error in the case of size 2 coalitions

a note on approximation of uniform distributions from variable to fixed length codes in this correspondence we prove that the probability distribution induced on the leaves of a tunstall parse tree for a given source is a unique lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves and ordered according to the majorization partial order we apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin

approximation of boolean functions by combinatorial rectangles this paper deals with the number of monochromatic combinatorial rectangles required to approximate a boolean function on a constant fraction of all inputs where each rectangle may use its own partition of the input variables the main result of the paper is that the number of rectangles required for the approximation of boolean functions in this model is very sensitive to the allowed error there is an explicitly defined sequence of boolean functions fn on n variables such that fn has rectangle approximations with a constant number of rectangles and one sided error 13 o 1 or two sided error 14 o 1 but on the other hand fn requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant as applications of this result the following separation results for read once branching programs are obtained the functions from the main result require only linear size for nondeterministic read once branching programs and randomized read once branching programs with two sided error 13 o 1 while randomized read once branching programs with constant two sided error smaller than 13 and unambiguous nondeterministic read once branching programs require exponential size

2 server pir with subpolynomial communication a 2 server private information retrieval pir scheme allows a user to retrieve the ith bit of an n bit database replicated among two noncommunicating servers while not revealing any information about i to either server in this work we construct a 2 server pir scheme with total communication cost no log log n log n this improves over current 2 server protocols which all require n1 3 communication our construction circumvents the n1 3 barrier of razborov and yekhanin 2007 which holds for the restricted model of bilinear group based schemes covering all previous 2 server schemes the improvement comes from reducing the number of servers in existing protocols based on matching vector codes from 3 or 4 servers to 2 this is achieved by viewing these protocols in an algebraic way using polynomial interpolation and extending them using partial derivatives

on the quality of random sequences produced with a combined random bit generator in this paper we prove that a combined binary generator which combines modulo 2 bit streams produced by independent generators or by generators that can be grouped into independent pairs of dependent generators can provide unbiased and uncorrelated bit streams this result was obtained under the realistic assumption that each source generator produces a sequence of correlated and biased bits the proposed theorems and formulas are useful in designing random bit generators with satisfactory properties that are robust to external manipulations or to environmental changes up to a certain degree

relaxation approximation of the euler equations the aim of this paper is to show how solutions to the one dimensional compressible euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term the enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate riemann solver from a theoretical point of view the convergence of solutions to the enlarged system towards solutions to the euler equations is proved for local in time smooth solutions we also show that arbitrarily large shock waves for the euler equations admit smooth shock profiles for the enlarged relaxation system in the end we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system we test it on various cases

a robust high order lagrange projection like scheme with large time steps for the isentropic euler equations we present an extension to high order of a first order lagrange projection like method for the approximation of the euler equations introduced in coquel et al math comput 79 1493 1533 2010 the method is based on a decomposition between acoustic and transport operators associated to an implicit explicit time integration thus relaxing the constraint of acoustic waves on the time step we propose here to use a discontinuous galerkin method for the space approximation considering the isentropic euler equations we derive conditions to keep positivity of the mean value of density and to satisfy a discrete entropy inequality in each element of the mesh at any approximation order in space these results allow to design limiting procedures to restore these properties at nodal values within elements numerical experiments support the conclusions of the analysis and highlight stability and robustness of the present method while it allows the use of large time steps

on the global existence of bohmian mechanics we show that the particle motion in bohmian mechanics given by the solution of an ordinary differential equation exists globally for a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values a substantial part of the analysis is based on the probabilistic significance of the quantum flux we elucidate the connection between the conditions necessary for global existence and the self adjointness of the schrodinger hamiltonian

maximum principle on the entropy and second order kinetic schemes we consider kinetic schemes for the multidimensional inviscid gas dynamics equations compressible euler equations we prove that the discrete maximum principle holds for the specific entropy this fixes the choice of the equilibrium functions necessary for kinetic schemes we use this property to perform a second order oscillation free scheme where only one slope limitation for three conserved quantities in 1d is necessary numerical results exhibit stability and strong convergence of the scheme introduction we consider the gas dynamics equations in one or two space dimensions atp div pu 0 1 atpu div puju a p 0 j 1 2 9 e div e p u 0 where x xl x2 u ul u2 and the total energy e piu12 2 pt y 1 is related to the pressure by the relation p pt 1 oh and in 11d godunov and lax friedrichs schemes preserve this property at the discretized level because they solve exactly the system 1 a reason why 4 should hold is that s satisfies the meaningless equation ats u vxs 0 received by the editor march 10 1992 and in revised form january 7 1993 1991 mathematics subject classification primary 35l65 76n10 65m06 76po5

the complexity of counting quantifiers on equality languages an equality language is a relational structure with infinite domain whose relations are first order definable in equality we classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers in doing this we find ourselves in various worlds in which dichotomies or trichotomies subsist

a relaxation method for two phase flow models with hydrodynamic closure law this paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two phase flows in pipelines the pdes are closed by two highly nonlinear algebraic relations namely a pressure law and a hydrodynamic one the severe nonlinearities encoded in these laws make the classical approximate riemann solvers virtually intractable at a reasonable cost of evaluation we propose a strategy for relaxing solely these two nonlinearities the relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate such a property not only makes it trivial to solve the riemann problem but also enables us to enforce some further stability requirements in addition to those coming from a chapman enskog analysis the new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions extensive numerical evidences are provided

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

randomizing functions simulation of a discrete probability distribution using a source of unknown distribution in this paper we characterize functions that simulate independent unbiased coin flips from independent coin flips of unknown bias we call such functions randomizing our characterization of randomizing functions enables us to identify the functions that generate the largest average number of fair coin flips from a fixed number of biased coin flips we show that these optimal functions are efficiently computable then we generalize the characterization and we present a method to simulate an arbitrary rational probability distribution optimally in terms of the average number of output digits and efficiently in terms of computational complexity from outputs of many faced dice of unknown distribution we also study randomizing functions on exhaustive prefix free sets

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

extractors for a constant number of polynomially small min entropy independent sources we consider the problem of randomness extraction from independent sources we construct an extractor that can extract from a constant number of independent sources of length n each of which have min entropy n gamma for an arbitrarily small constant gamma 0 our extractor is obtained by composing seeded extractors in simple ways we introduce a new technique to condense independent somewhere random sources which looks like a useful way to manipulate independent sources our techniques are different from those used in recent work b barak r impagliazzo and a wigderson siam j comput 36 2006 pp 1095 1118 b barak g kindler r shaltiel b sudakov and a wigderson simulating independence new constructions of condensers ramsey graphs dispersers and extractors in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 1 10 r raz extractors with weak random seeds in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 11 20 j bourgain int j number theory 1 2005 pp 1 32 for this problem in the sense that they do not rely on any results from arithmetic combinatorics using an extractor of bourgain s int j number theory 1 2005 pp 1 32 as a black box we obtain a new extractor for two independent block sources with few blocks even when the min entropy is as small as operatorname polylog n we also show how to modify the 2 source disperser for linear min entropy of barak et al simulating independence new constructions of condensers ramsey graphs dispersers and extractors in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 1 10 and the three source extractor of raz extractors with weak random seeds in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 11 20 to get dispersers extractors with exponentially small error and linear output length where previously both were constant

uniform random sampling not recommended for large graph size estimation abstract the norm of data size estimation is to use uniform random samples whenever possible there have been tremendous efforts in obtaining uniform random samples using methods such as metropolis hasting random walk or importance sampling 2 this paper shows that on the contrary to the common practice uniform random sampling should be avoided when pps probability proportional to size sampling is available for large data to develop intuition of the sampling process we discuss the sampling and estimation problem in the context of graph the size is the number of nodes in the graph uniform random sampling corresponds to uniform random node rn sampling and pps sampling is approximated by random edge re sampling in this setting we show that for large graphs re sampling outperforms rn sampling with a ratio proportional to the normalized graph degree variance this result is particularly important in the era of big data when data are typically large and scale free 3 resulting in large degree variance we derive the result by giving the variances of rn and re estimators each step of the derivation is supported and demonstrated by simulation studies assuming power law distributions then we use 18 real world networks to verify the result furthermore we show that the performance of random walk rw sampling is data dependent and can be significantly worse than rn and re more specifically rw can estimate online social networks but not web graphs due to the difference of the graph conductance

simulating fair dice with biased coins this paper is concerned with simulating a fair die with a bounded number of coin flips and a bounded number of possibly biased coins as the main results this paper shows that for anyn 2 a set ofh n coins is sufficient to simulate a fairn sided die withind lgn coin flips whereh n is the number of 1 s of the binary representation of an integern and that for anyn 2d 1 d 3 a set ofd h n coins is necessary and sufficient to simulate a fairn sided die withind lgn coin flips

green s theorem and isolation in planar graphs we show a simple application of email protected s theorem from multivariable calculus to the isolation problem in planar graphs in particular we give a log space construction of a skew symmetric polynomially bounded edge weight function for directed planar graphs such that the weight of any simple cycle in the graph is non zero with respect to this weight function as a direct consequence of the above weight function we are able to isolate a directed path between two fixed vertices in a directed planar graph we also show that given a bipartite planar graph we can obtain an edge weight function using the above function in log space which isolates a perfect matching in the given graph earlier this was known to be true only for grid graphs which is a proper subclass of planar graphs we also look at the problem of obtaining a straight line embedding of a planar graph in log space although we do not quite achieve this goal we give a piecewise straight line embedding of the given planar graph in log space

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

lower bounds for one way probabilistic communication complexity and their application to space complexity abstract we prove three different types of complexity lower bounds for the one way unbounded error and bounded error error probabilistic communication protocols for boolean functions the lower bounds are proved in terms of the deterministic communication complexity of functions and in terms of the notion probabilistic communication characteristic that we define we present boolean functions with the different probabilistic communication characteristics which demonstrates that each of these lower bounds can be more precise than the others depending on the probabilistic communication characteristics of a function our lower bounds are good enough for proving that proper hierarchy for one way probabilistic communication complexity classes depends on a measure of bounded error as the application of lower bounds for probabilistic communication complexity we prove two different types of complexity lower bounds for the one way bounded error error probabilistic space complexity our lower bounds are good enough for proving proper hierarchies for different one way probabilistic space communication complexity classes inside space n namely for bounded error probabilistic computation and for errors of probabilistic computation

post challenge leakage in public key encryption when an adversary can measure the physical memory storing the decryption key decryption functionality often comes in handy halevi and lin tcc 11 studied after the fact or post challenge leakage in public key encryption pke in which an adversary can make leakage queries from a split state after seeing the challenge ciphertext but left security against chosen ciphertext attacks cca as a future work in this paper we follow their work and formulate the definition of entropic leakage resilient cca secure pke which we show can be realized by the naor yung double encryption paradigm stoc 90 we then leverage it to get a cca secure key encapsulation mechanism in the presence of post challenge leakage in the same model of bounded memory leakage from a split state finally we prove that the hybrid encryption framework is still applicable by presenting a construction of cca secure pke in the presence of post challenge leakage as additional results we extend these concepts to the identity based setting where many identity based secret keys can be leaked after the adversary got the challenge and give a construction of identity based encryption in the presence of post challenge leakage in the split state model which can be instantiated by the identity based hash proof systems of alwen et al eurocrypt 10 and chow et al ccs 10

navier stokes equations with several independent pressure laws and explicit predictor corrector schemes this work is concerned with the numerical capture of stiff viscous shock solutions of navier stokes equations for complex compressible materials in the regime of large reynolds numbers after 2 and 6 a relevant numerical capture is known to require the satisfaction of an extended set of non classical rankine hugoniot conditions due to the non conservation form of the governing pde model here we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate riemann solutions a relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method

large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms we propose large time step and asymptotic preserving schemes for the gas dynamics equations with external forces and friction terms by asymptotic preserving we mean that the numerical scheme is able to reproduce at the discrete level the parabolic type asymptotic behavior satisfied by the continuous equations by large time step we mean that the scheme is stable under a cfl stability condition driven by the slow material waves and not by the fast acoustic waves customary in godunov type schemes numerical evidence is proposed and shows a gain of several orders of magnitude in both accuracy and efficiency

quantum equilibrium and the origin of absolute uncertainty what is randomness probability certainty knowledge these are old and difficult questions and we shall not focus on them here nonetheless we shall obtain sharp striking conclusions concerning the relationship between these concepts

lagrangian systems of conservation laws we study the mathematical structure of 1d systems of conservation laws written in the lagrange variable modifying the symmetrization proof of systems of conservation laws with three hypothesis we prove that these models have a canonical formalism these hypothesis are i the entropy flux is zero ii galilean invariance iii reversibility for smooth solutions then we study a family of numerical schemes for the solution of these systems we prove that they are entropy consistent we also prove from general considerations the symmetry of the spectrum of the jacobian matrix

simple proof for global existence of bohmian trajectories we address the question whether bohmian trajectories exist for all times bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the schrodinger or the dirac equation some trajectories may end in finite time for example by running into a node of the wavefunction where the law of motion is ill defined the aim is to show under suitable assumptions on the initial wavefunction and the potential global existence of almost all solutions we provide an alternative proof of the known global existence result for spinless schrodinger particles and extend the result to particles with spin to the presence of magnetic fields and to dirac wavefunctions our main new result is conditions on the current vector field on configuration space time which are sufficient for almost sure global existence

on the locality of codeword symbols consider a linear n k d q code c we say that the ith coordinate of c has locality r if the value at this coordinate can be recovered from accessing some other r coordinates of c data storage applications require codes with small redundancy low locality for information coordinates large distance and low locality for parity coordinates in this paper we carry out an in depth study of the relations between these parameters we establish a tight bound for the redundancy n k in terms of the message length the distance and the locality of information coordinates we refer to codes attaining the bound as optimal we prove some structure theorems about optimal codes which are particularly strong for small distances this gives a fairly complete picture of the tradeoffs between codewords length worst case distance and locality of information symbols we then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes using our structure theorem we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings we prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance

the sign rank of ac 0 the sign rank of a matrix a a ij with pm1 entries is the least rank of a real matrix b b ij with a ij b ij 0 for all i j we obtain the first exponential lower bound on the sign rank of a function in mathsf ac 0 namely let f x y bigwedge i 1 dots m bigvee j 1 dots m 2 x ij wedge y ij we show that the matrix f x y x y has sign rank exp omega m this in particular implies that sigma 2 cc not subseteq mathsf upp cc which solves a longstanding open problem in communication complexity posed by babai frankl and simon proceedings of the 27th symposium on foundations of computer science focs 1986 pp 337 347 our result additionally implies a lower bound in learning theory specifically let phi 1 dots phi r 0 1 n to mathbb r be functions such that every dnf formula f 0 1 n to 1 1 of polynomial size has the representation f equiv mathrm sgn a 1 phi 1 dots a r phi r for some reals a 1 dots a r we prove that then r geqslant exp omega n 1 3 which essentially matches an upper bound of exp tilde o n 1 3 due to klivans and servedio j comput system sci 68 2004 pp 303 318 finally our work yields the first exponential lower bound on the size of threshold of majority circuits computing a function in mathsf ac 0 this substantially generalizes and strengthens the results of krause and pudlak theoret comput sci 174 1997 pp 137 156

new methods for digital generation and postprocessing of random data a new method for digital true random number generation based on asynchronous logic circuits with feedback is introduced in particular a concrete technique using the so called galois and fibonacci ring oscillators is developed and analyzed both theoretically and experimentally the generated random binary sequences may have a very high speed and a higher and more robust entropy rate in comparison with previous proposals for digital random number generators a new method for digital postprocessing of random data based on irregularly clocked nonautonomous synchronous logic circuits with feedback is also introduced and a concrete technique using a self clock controlled linear feedback shift register is proposed the postprocessing can provide both randomness extraction and computationally secure speed increase of input random data

semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation we investigate totally linearly degenerate hyperbolic systems with relaxation we aim to study their semilinear behavior which means that the local smooth solutions cannot develop shocks and the global existence is controlled by the supremum bound of the solution in this paper we study two specific examples the it suliciu type and the it kerr debye type models for the suliciu model which arises from the numerical approximation of isentropic flows the semilinear behavior is obtained using pointwise estimates of the gradient for the kerr debye systems which arise in nonlinear optics we show the semilinear behavior via energy methods for the original kerr debye model thanks to the special form of the interaction terms we can show the global existence of smooth solutions

computing pure bayesian nash equilibria in games with finite actions and continuous types we extend the well known fictitious play fp algorithm to compute pure strategy bayesian nash equilibria in private value games of incomplete information with finite actions and continuous types g facts we prove that if the frequency distribution of actions fictitious play beliefs converges then there exists a pure strategy equilibrium strategy that is consistent with it we furthermore develop an algorithm to convert the converged distribution of actions into an equilibrium strategy for a wide class of games where utility functions are linear in type this algorithm can also be used to compute pure e nash equilibria when distributions are not fully converged we then apply our algorithm to find equilibria in an important and previously unsolved game simultaneous sealed bid second price auctions where various types of items e g substitutes or complements are sold finally we provide an analytical characterisation of equilibria in games with linear utilities specifically we show how equilibria can be found by solving a system of polynomial equations for a special case of simultaneous auctions we also solve the equations confirming the results obtained numerically

how to turn loaded dice into fair coins we present a new technique for simulating fair coin flips using a biased stationary source of randomness sequences of random numbers are of pervasive importance in cryptography and vital to many other computing applications many sources of randomness such as radioactive or quantum mechanical sources possess the property of stationarity in other words they produce independent outputs over fixed probability distributions the output of such sources may be viewed as the result of rolling a biased or loaded die while a biased die may be a good source of entropy many applications require input in the form of unbiased bits rather than biased ones for this reason von neumann 1951 presented a now well known and extensively investigated technique for using a biased coin to simulate a fair coin we describe a new generalization of von neumann s algorithm distinguished by its high level of practicality and amenability to analysis in contrast to previous efforts we are able to prove our algorithm optimally efficient in the sense that it simulates the maximum possible number of fair coin flips for a given number of die rolls in fact we are able to prove that in an asymptotic sense our algorithm extracts the full entropy of its input moreover we demonstrate experimentally that our algorithm achieves a high level of computational and output efficiency in a practical setting

towards 3 query locally decodable codes of subexponential length a q query locally decodable code ldc encodes an n bit message x as an n bit codeword c x such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword c x even after some constant fraction of codeword bits has been corrupted we give new constructions of three query ldcs of vastly shorter length than that of previous constructions specifically given any mersenne prime p e 2t 1 we design three query ldcs of length n e exp o n1 t for every n based on the largest known mersenne prime this translates to a length of less than exp o n10 7 compared to exp o n1 2 in the previous constructions it has often been conjectured that there are infinitely many mersenne primes under this conjecture our constructions yield three query locally decodable codes of length n e exp no 1 log log n for infinitely many n we also obtain analogous improvements for private information retrieval pir schemes we give 3 server pir schemes with communication complexity of o n10 7 to access an n bit database compared to the previous best scheme with complexity o n1 5 25 assuming again that there are infinitely many mersenne primes we get 3 server pir schemes of communication complexity no 1 log logn for infinitely many n previous families of ldcs and pir schemes were based on the properties of low degree multivariate polynomials over finite fields our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set

simulating independence new constructions of condensers ramsey graphs dispersers and extractors we present new explicit constructions of deterministic randomness extractors dispersers and related objects we say that a distribution x on binary strings of length n is a source if x assigns probability at most 2 n to any string of length n for every 0 we construct the following poly n time computable functions 2 source disperser d 0 1 n 2 0 1 such that for any two independent sources x1 x2 we have that the support of d x1 x2 is 0 1 bipartite ramsey graph let ne2n a corollary is that the function d is a 2 coloring of the edges of kn n the complete bipartite graph over two sets of n vertices such that any induced subgraph of size n by n is not monochromatic 3 source extractor e 0 1 n 3 0 1 such that for any three independent sources x1 x2 x3 we have that e x1 x2 x3 is o 1 close to being an unbiased random bit no previous explicit construction was known for either of these for any a component in these results is a new construction of condensers that may be of independent interest this is a function c 0 1 n 0 1 n c d where c and d are constants that depend only on such that for every source x one of the output blocks of c x is exponentially close to a 0 9 source this result was obtained independently by ran raz the constructions are quite involved and use as building blocks other new and known objects a recurring theme in these constructions is that objects that were designed to work with independent inputs sometimes perform well enough with correlated high entropy inputs the construction of the disperser is based on a new technique which we call the challenge response mechanism that in some sense allows identifying high entropy regions in a given pair of sources using only one sample from the two sources

error correcting data structures we study data structures in the presence of adversarial noise we want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object even if the data structure has been corrupted by a constant fraction of errors we measure the efficiency of a data structure in terms of its length the number of bits in its representation and query answering time measured by the number of bit probes to the possibly corrupted representation the main issue is the trade off between these two this new model is the common generalization of static data structures and locally decodable error correcting codes ldcs we prove a number of upper and lower bounds on various natural error correcting data structure problems in particular we show that the optimal length of t probe error correcting data structures for the membership problem where we want to store subsets of size s from a universe of size n such that membership queries can be answered effic

combinatorial properties of frameproof and traceability codes in order to protect copyrighted material codes may be embedded in the content or codes may be associated with the keys used to recover the content codes can offer protection by providing some form of traceability ta for pirated data several researchers have studied different notions of ta and related concepts in previous years strong versions of ta allow at least one member of a coalition that constructs a pirate decoder to be traced weaker versions of this concept ensure that no coalition can frame a disjoint user or group of users all these concepts can be formulated as codes having certain combinatorial properties we study the relationships between the various notions and we discuss equivalent formulations using structures such as perfect hash families we use methods from combinatorics and coding theory to provide bounds necessary conditions and constructions sufficient conditions for the objects of interest

kernel and fast algorithm for dense triplet inconsistency we study the parameterized complexity of inferring supertrees from sets of rooted triplets an important problem in phylogenetics for a set l of labels and a dense set t of triplets distinctly leaf labeled by 3 subsets of l we seek a tree distinctly leaf labeled by l and containing all but at most k triplets from t as homeomorphic subtree our results are the first polynomial kernel for this problem with o k 2 labels and a subexponential fixed parameter algorithm running in time o l 4 2 o k 1 3 l o g k

divisors of mersenne numbers we add to the heuristic and empirical evidence for a conjecture of gillies about the distribution of the prime divisors of mersenne numbers we list some large prime divisors of mersenne numbers mp in the range 17000 p 105

on the existence of pseudorandom generators pseudorandom generators suggested and developed by blum and micali and yao are efficient deterministic programs that expand a randomly selected k bit seed into a much longer pseudorandom bit sequ

the flux across surfaces theorem for short range potentials and wave functions without energy cutoffs the quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time the relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the flux across surfaces theorem which was conjectured by combes newton and shtokhamer phys rev d 11 366 372 1975 we prove the flux across surfaces theorem for short range potentials and wave functions without energy cutoffs the proof is based on the free flux across surfaces theorem daumer et al lett math phys 38 103 116 1996 and on smoothness properties of generalized eigenfunctions it is shown that if the potential v x decays like x at infinity with n n then the generalized eigenfunctions of the corresponding hamiltonian 1 2 v are n 2 times continuously differentiable with respect to the momentum variable

on the fingerprinting capacity under the marking assumption we address the maximum attainable rate of fingerprinting codes under the marking assumption studying lower and upper bounds on the value of the rate for various sizes of the attacker coalition lower bounds are obtained by considering typical coalitions which represents a new idea in the area of fingerprinting and enables us to improve the previously known lower bounds for coalitions of size two and three for upper bounds the fingerprinting problem is modeled as a communications problem it is shown that the maximum code rate is bounded above by the capacity of a certain class of channels which are similar to the multiple access channel mac converse coding theorems proved in the paper provide new upper bounds on fingerprinting capacity it is proved that capacity for fingerprinting against coalitions of size two and three over the binary alphabet satisfies and respectively for coalitions of an arbitrary fixed size we derive an upper bound on fingerprinting capacity in the binary case finally for general alphabets we establish upper bounds on the fingerprinting capacity involving only single letter mutual information quantities

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

computing pure nash and strong equilibria in bottleneck congestion games bottleneck congestion games properly model the properties of many real world network routing applications they are known to possess strong equilibria a strengthening of nash equilibrium to resilience against coalitional deviations in this paper we study the computational complexity of pure nash and strong equilibria in these games we provide a generic centralized algorithm to compute strong equilibria which has polynomial running time for many interesting classes of games such as e g matroid or single commodity bottleneck congestion games in addition we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics using unilateral improvement dynamics in matroid games pure nash equilibria can be reached efficiently in contrast computing even a single coalitional improvement move in matroid and single commodity games is strongly np hard in addition we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics they continue to hold even for convergence to approximate equilibria

high rate locally correctable and locally testable codes with sub polynomial query complexity locally correctable codes lccs and locally testable codes ltcs are error correcting codes that admit local algorithms for correction and detection of errors those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword the fundamental question about lccs and ltcs is to determine the optimal tradeoff among their rate distance and query complexity in this work we construct the first lccs and ltcs with constant rate constant relative distance and sub polynomial query complexity specifically we show that there exist lccs and ltcs with block length n constant rate which can even be taken arbitrarily close to 1 and constant relative distance whose query complexity is exp o slog n for lccs and log n o log log n for ltcs in addition to having small query complexity our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by lccs or ltcs specifically over large but constant size alphabet our codes approach the singleton bound that is they have almost the best possible relationship between their rate and distance over the binary alphabet our codes meet the zyablov bound such tradeoffs between the rate and the relative distance were previously not known for any o n query complexity our results on lccs also immediately give locally decodable codes with the same parameters

a new comment on the computation of non conservative products using roe type path conservative schemes abstract we are interested in the numerical approximation of the discontinuous solutions of non conservative hyperbolic systems we more precisely consider a non conservative formulation of the usual gas dynamics equations and show how to slightly modify the so called roe type path conservative schemes to properly capture the underlying shock discontinuities numerical evidences are proposed the present note follows a first comment on the computation of non conservative products in 1

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

high rate codes with sublinear time decoding locally decodable codes are error correcting codes that admit efficient decoding algorithms any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword the tradeoff between the rate of a code and the locality efficiency of its decoding algorithms has been well studied and it has widely been suspected that nontrivial locality must come at the price of low rate a particular setting of potential interest in practice is codes of constant rate for such codes decoding algorithms with locality o k were known only for codes of rate 1 where k is the length of the message furthermore for codes of rate 1 2 no nontrivial locality had been achieved in this article we construct a new family of locally decodable codes that have very efficient local decoding algorithms and at the same time have rate approaching 1 we show that for every 0 and 0 for infinitely many k there exists a code c which encodes messages of length k with rate 1 and is locally decodable from a constant fraction of errors using o k queries and time these codes which we call multiplicity codes are based on evaluating multivariate polynomials and their derivatives multiplicity codes extend traditional multivariate polynomial codes they inherit the local decodability of these codes and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance

fast mixed searching and related problems on graphs abstract in this paper we introduce the fast mixed search model which is a combination of the fast search model and the mixed search model given a graph g in which a fugitive hides on vertices or along edges the fast mixed search problem is to find the minimum number of searchers to capture the fugitive in the fast mixed search model we establish a relation between the fast mixed search problem and the induced path cover problem we also consider relations between the fast mixed search problem and other graph search problems we prove that fast mixed searching from leaves is np complete and it remains np complete for planar graphs with maximum degree 3 we also prove that fast mixed searching between leaves is np complete and it remains np complete for graphs with maximum degree 4 we present linear time algorithms for computing the fast mixed search number and optimal search strategies of some classes of graphs including trees cacti and proper interval graphs

extraction of optimally unbiased bits from a biased source we explore the problem of transforming n independent and identically biased 1 1 valued random variables x sub 1 x sub n into a single 1 1 random variable f x sub 1 x sub n so that this result is as unbiased as possible in general no function f produces a completely unbiased result we perform the first study of the relationship between the bias b of these x sub i and the rate at which f x sub 1 x sub n can converge to an unbiased 1 1 random variable as n spl rarr spl infin a 1 1 random variable has bias b if e x sub i b fixing a bias b we explore the rate at which the output bias e f x sub 1 x sub n can tend to zero for a function f 1 1 spl rarr 1 1 this is accomplished by classifying the behavior of the natural normalized quantity spl xi b spl delta inf sub f lim sub n spl rarr spl infin n spl radic e f x sub 1 x sub n this infimum taken over all such f we show that for rational b spl xi b 1 s where 1 b 2 r s r and s relatively prime developing the theory of uniform distribution of sequences to suit our problem we then explore the case where b is irrational we prove a new metrical theorem concerning multidimensional diophantine approximation type from which we show that for lebesgue almost all biases b spl xi b 0 finally we show that algebraic biases exhibit curious boundary behavior falling into two classes class 1 those algebraics b for which spl xi b 0 and furthermore c sub 1 spl les spl xi b spl les c sub 2 where c sub 1 and c sub 2 are positive constants depending only on b s algebraic characteristics class 2 those algebraics b for which there exist n 0 and f 1 1 sup n spl rarr 1 1 so that e f x sub 1 x sub n 0 notice that this classification excludes the possibility that n spl radic e f x sub 1 x sub n limits to zero for algebraics for rational and algebraic biases we also study the computational problem by restricting f to be a polynomial time computable function finally we discuss natural extensions where output distributions other than the uniform distribution on 1 1 are sought

two query pcp with subconstant error we show that the np complete language 3sat has a pcp verifier that makes two queries to a proof of almost linear size and achieves subconstant probability of error eeo 1 the verifier performs only projection tests meaning that the answer to the first query determines at most one accepting answer to the second query the number of bits representing a symbol in the proof depends only on the error e previously by the parallel repetition theorem there were pcp theorems with two query projection tests but only arbitrarily small constant error and polynomial size there were also pcp theorems with subconstant error and almost linear size but a constant number of queries that is larger than 2 as a corollary we obtain a host of new results in particular our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem a partial list includes the following 1 3sat cannot be efficiently approximated to within a factor of 7 8po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 7 8pe for any constant e 0 under polynomial reductions hastad 2 3lin cannot be efficiently approximated to within a factor of 1 2po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 1 2pe for any constant e 0 under polynomial reductions hastad 3 a pcp theorem with amortized query complexity 1 p o 1 and amortized free bit complexity o 1 previously the best known amortized query complexity and free bit complexity were 1pe and e respectively for any constant e 0 samorodnitsky and trevisan one of the new ideas that we use is a new technique for doing the composition step in the classical proof of the pcp theorem without increasing the number of queries to the proof we formalize this as a composition of new objects that we call locally decode reject codes ldrc the notion of ldrc was implicit in several previous works and we make it explicit in this work we believe that the formulation of ldrcs and their construction are of independent interest

kernels for below upper bound parameterizations of the hitting set and directed dominating set problems in the hitting set problem we are given a collection f of subsets of a ground set v and an integer p and asked whether v has a p element subset that intersects each set in f we consider two parameterizations of hitting set below tight upper bounds p m k and p n k in both cases k is the parameter we prove that the first parameterization is fixed parameter tractable but has no polynomial kernel unless conp np poly the second parameterization is w 1 complete but the introduction of an additional parameter the degeneracy of the hypergraph h v f makes the problem not only fixed parameter tractable but also one with a linear kernel here the degeneracy of h v f is the minimum integer d such that for each x v the hypergraph with vertex set v x and edge set containing all edges of f without vertices in x has a vertex of degree at most d in nonblocker directed nonblocker we are given an undirected graph a directed graph g on n vertices and an integer k and asked whether g has a set x of n k vertices such that for each vertex y x there is an edge arc from a vertex in x to y nonblocker can be viewed as a special case of directed nonblocker replace an undirected graph by a symmetric digraph dehne et al proc sofsem 2006 proved that nonblocker has a linear order kernel we obtain a linear order kernel for directed nonblocker

local correctability of expander codes in this work we present the first local decoding algorithm for expander codes this yields a new family of constant rate codes that can recover from a constant fraction of errors in the codeword symbols and where any symbol of the codeword can be recovered with high probability by reading n e symbols from the corrupted codeword where n is the block length of the code expander codes introduced by sipser and spielman are formed from an expander graph g v e of degree d and an inner code of block length d over an alphabet each edge of the expander graph is associated with a symbol in a string in e will be a codeword if for each vertex in v the symbols on the adjacent edges form a codeword in the inner code we show that if the inner code has a smooth reconstruction algorithm in the noiseless setting then the corresponding expander code has an efficient local correction algorithm in the noisy setting instantiating our construction with inner codes based on finite geometries we obtain novel locally decodable codes with rate approaching one this provides an alternative to the multiplicity codes of kopparty saraf and yekhanin stoc 11 and the lifted codes of guo kopparty and sudan itcs 13

upper bounds on matching families in bbz_ pq n matching families are one of the major ingredients in the construction of locally decodable codes ldcs and the best known constructions of ldcs with a constant number of queries are based on matching families the determination of the largest size of any matching family in zmn where zm is the ring of integers modulo m is an interesting problem in this paper we show an upper bound of o pq 0 625n 0 125 for the size of any matching family in zpqn where p and q are two distinct primes our bound is valid when n is a constant p and p q 1 our result improves an upper bound of dvir and coworkers

optimal algorithms for universal random number generation from finite memory sources we study random number generators rngs both in the fixed to variable length fvr and the variable to fixed length vfr regimes in a universal setting in which the input is a finite memory source of arbitrary order and unknown parameters with arbitrary input and output finite alphabet sizes applying the method of types we characterize essentially unique optimal universal rngs that maximize the expected output respectively minimize the expected input length in the fvr respectively vfr case for the fvr case the rng studied is a generalization of elias s scheme while in the vfr case the general scheme is new we precisely characterize up to an additive constant the corresponding expected lengths which include second order terms similar to those encountered in universal data compression and universal simulation furthermore in the fvr case we consider also a twice universal setting in which the markov order k of the input source is also unknown

average and randomized communication complexity the communication complexity of a two variable function f x y is the number of information bits two communicators need to exchange to compute f when initially each knows only one of the variables there are several communication complexity measures corresponding to whether 1 the worst case or average number of bits is considered 2 computation errors are allowed and 3 randomization is allowed tight bounds are provided for the typical behavior of all bounded error communication complexity measures of boolean functions in the present work the authors formally define the deterministic model they describe randomized protocols and compare them to deterministic ones they both survey previous work and describe original results

space lower bounds for online pattern matching we present space lower bounds for online pattern matching under a number of different distance measures given a pattern of length m and a text that arrives one character at a time the online pattern matching problem is to report the distance between the pattern and a sliding window of the text as soon as the new character arrives we require that the correct answer is given at each position with constant probability we give w m bit space lower bounds for l 1 l 2 l hamming edit and swap distances as well as for any algorithm that computes the cross correlation convolution we then show a dichotomy between distance functions that have wildcard like properties and those that do not in the former case which includes as an example pattern matching with character classes we give w m bit space lower bounds for other distance functions we show that there exist space bounds of w logm and o log 2m bits finally we discuss space lower bounds for non binary inputs and show how in some cases they can be improved

optimal probabilistic fingerprint codes we construct binary codes for fingerprinting digital documents our codes for n users that are e secure against c pirates have length o c2log n e this improves the codes proposed by boneh and shaw l1998r whose length is approximately the square of this length the improvement carries over to works using the boneh shaw code as a primitive for example to the dynamic traitor tracing scheme of tassa l2005r by proving matching lower bounds we establish that the length of our codes is best within a constant factor for reasonable error probabilities this lower bound generalizes the bound found independently by peikert et al l2003r that applies to a limited class of codes our results also imply that randomized fingerprint codes over a binary alphabet are as powerful as over an arbitrary alphabet and the equal strength of two distinct models for fingerprinting