reconstructing algebraic functions from mixed data we consider a variant of the traditional task of explicitly reconstructing algebraic functions from black box representations in the traditional setting for such problems one is given access to an unknown function f that is represented by a black box or an oracle which can be queried for the value of f at any input given a guarantee that this unknown function f is some nice algebraic function say a polynomial in its input of degree bound d the goal of the reconstruction problem is to explicitly determine the coefficients of the unknown polynomial all work on polynomial interpolation especially sparse ones are or may be presented in such a setting the work of kaltofen and trager computing with polynomials given by black boxes for their evaluations greatest common divisors factorization separation of numerators and denominators in proc 29th ann ieee symp on foundations of computer science 1988 pp 296 305 for instance highlights the utility of this setting by performing numerous manipulations on polynomials presented as black boxes the variant considered in this paper differs from the traditional setting in that our black boxes represent several algebraic functions f1 fk where at each input x the box arbitrarily chooses a subset of f1 x fk x to output and we do not know which subset it outputs we show how to reconstruct the functions f1 fk from the black box provided the black box outputs according to these functions often this allows us to group the sample points into sets such that for each set all outputs to points in the set are from the same algebraic function our methods are robust in the presence of a small fraction of arbitrary errors in the black box our model and techniques can be applied in the areas of computer vision machine learning curve fitting and polynomial approximation self correcting programs and bivariate polynomial factorization

on the kz reduction the korkine zolotareff kz reduction is one of the often used reduction strategies for lattice decoding in this paper we first investigate some important properties of kz reduced matrices specifically we present a linear upper bound on the hermit constant which is around 7 8 times of the existing sharpest linear upper bound and an upper bound on the kz constant which is polynomially smaller than the existing sharpest one we also propose upper bounds on the lengths of the columns of kz reduced matrices and an upper bound on the orthogonality defect of kz reduced matrices which are even polynomially and exponentially smaller than those of boosted kz reduced matrices respectively then we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem svp when its basis matrix is lll reduced these upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a kz reduction algorithm finally we propose a new kz reduction algorithm by modifying the commonly used schnorr euchner search strategy for solving svps and the basis expansion method proposed by zhang et al simulation results show that the new kz reduction algorithm is much faster and more numerically reliable than the kz reduction algorithm proposed by zhang et al especially when the basis matrix is ill conditioned

large gaps between consecutive prime numbers let g x denote the size of the largest gap between consecutive primes below x answering a question of erd os we show that g x f x logx log logx log log log logx log log logx 2 where f x is a function tending to innity with x our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions as such we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes

on polynomial factorial diophantine equations we study equations of the form p x n and show that for some classes of polynomials p the equation has only finitely many solutions this is the case say if p is irreducible of degree greater than 1 or has an irreducible factor of relatively large degree this is also the case if the factorization of p contains some large power s of irreducible s for example we can show that the equation x r x 1 n has only finitely many solutions for r 4 but not that this is the case for 1 r 3 although it undoubtedly should be we also study the equation p x h n where h n is one of several other highly divisible sequences proving again that for various classes of polynomials these equations have only finitely many solutions

on the turan number for the hexagon abstract a long standing conjecture of erd s and simonovits is that ex n c 2 k the maximum number of edges in an n vertex graph without a 2 k gon is asymptotically 1 2 n 1 1 k as n tends to infinity this was known almost 40 years ago in the case of quadrilaterals in this paper we construct a counterexample to the conjecture in the case of hexagons for infinitely many n we prove that ex n c 6 3 5 2 5 1 4 3 n 4 3 o n 0 5338 n 4 3 we also show that ex n c 6 n 4 3 o n 0 6272 n 4 3 if n is sufficiently large where is the real root of 16 3 4 2 3 0 this yields the best known upper bound for the number of edges in a hexagon free graph the same methods are applied to find a tight bound for the maximum size of a hexagon free 2 n n bipartite graph

distributed average consensus with dithered quantization in this paper we develop algorithms for distributed computation of averages of the node data over networks with bandwidth power constraints or large volumes of data distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted we propose a distributed algorithm in which the nodes utilize probabilistically quantized information i e dithered quantization to communicate with each other the algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely in addition we show that the expected value of the consensus is equal to the average of the original sensor data we derive an upper bound on the mean square error performance of the probabilistically quantized distributed averaging pqda moreover we show that the convergence of the pqda is monotonic by studying the evolution of the minimum length interval containing the node values we reveal that the length of this interval is a monotonically nonincreasing function with limit zero we also demonstrate that all the node values in the worst case converge to the final two quantization bins at the same rate as standard unquantized consensus finally we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios

selecting wavelengths for least squares range estimation we consider the problem of estimating the distance or range between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods more recently estimators based on the chinese remainder theorem crt and least squares have appeared recent research suggests the least squares estimator to be the most accurate in many cases the accuracy of all of these range estimators depends upon the wavelengths chosen this leads to the problem of selecting wavelengths that maximize accuracy procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described but procedures for the crt and least squares estimators are yet to be developed in this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator the algorithm minimizes an optimization criterion connected with the mean square error interesting properties of a particular class of lattices simplify the criterion allowing minimization by depth first search monte carlo simulations indicate that wavelengths that minimize the criterion can result in considerably more accurate range estimates than wavelengths selected by ad hoc means

trading space for time in undirected s t connectivity aleliunas et al 20th annual symposium on foundations of computer science ieee computer society press los alamitos ca 1979 pp 218 223 posed the following question the reachability problem for undirected graphs can be solved in log space and o mn time m is the number of edges and n is the number of vertices by a probabilistic algorithm that simulates a random walk or in linear time and space by a conventional deterministic graph traversal algorithm is there a spectrum of time space trade offs between these extremes this question is answered in the affirmative for sparse graphs by presentation of an algorithm that is faster than the random walk by a factor essentially proportional to the size of its workspace for denser graphs this algorithm is faster than the random walk but the speed up factor is smaller

the distribution of prime ideals of imaginary quadratic fields let q x y be a primitive positive definite quadratic form with integer coefficients then for all s t e r 2 there exist m n z 2 such that q m n is prime and q m s n t q s t 0 53 1 this is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval

a general lower bound for collaborative tree exploration abstract we consider collaborative graph exploration with a set of k agents all agents start at a common vertex of an initially unknown graph with n vertices and need to collectively visit all other vertices we assume agents are deterministic moves are simultaneous and we allow agents to communicate globally for this setting we give the first non trivial lower bounds that bridge the gap between small k n and large k n teams of agents remarkably our bounds tightly connect to existing results in both domains first we significantly extend a lower bound of log k log log k by dynia et al on the competitive ratio of a collaborative tree exploration strategy to the range k n log c n for any c n second we provide a tight lower bound on the number of agents needed for any competitive exploration algorithm in particular we show that any collaborative tree exploration algorithm with k d n 1 o 1 agents has a competitive ratio of 1 while dereniowski et al gave an algorithm with k d n 1 e agents and competitive ratio o 1 for any e 0 and with d denoting the diameter of the graph lastly we show that for any exploration algorithm using k n agents there exist trees of arbitrarily large height d that require d 2 rounds and we provide a simple algorithm that matches this bound for all trees

a discrete random walk on the hypercube abstract in this paper we study the scaling for mean first passage time mfpt of random walks on the hypercube and obtain a closed form formula for the mfpt over all node pairs we also determine the exponent of scaling efficiency characterizing the random walks and compare it with those of the existing networks finally we study the random walks on the hypercube with a located trap and provide a solution of the kirchhoff index of the hypercube

cover time for random walks on arbitrary complex networks we present an analytical method for computing the mean cover time of a discrete time random walk process on arbitrary complex networks the cover time is defined as the time a random walker requires to visit every node in the network at least once this quantity is particularly important for random search processes and target localization on network structures based on the global mean first passage time of target nodes we derive a method for computing the cumulative distribution function of the cover time based on first passage time statistics our method is viable for networks on which random walks equilibrate quickly we show that it can be applied successfully to various model and real world networks our results reveal an intimate link between first passage and cover time statistics and offer a computationally efficient way for estimating cover times in network related applications

distributed averaging with linear objective maps a distributed averaging system is a linear multi agent system in which agents communicate to reach an agreement or a consensus state defined as the average of the initial states of the agents consider a more generalized situation in which each agent is given a nonnegative weight and the agreement state is defined as the weighted average of the initial conditions we characterize in this paper the weighted averages that can be evaluated in a decentralized way by agents communicating over a directed graph specifically we introduce a linear function called the objective map that defines the desired final state as a function of the initial states of the agents we then provide a complete answer to the question of whether there is a decentralized consensus dynamics over a given digraph which converges to the final state specified by an objective map in particular we characterize not only the set of objective maps that are feasible for a given digraph but also the consensus dynamics that implements the objective map in addition we present a decentralized algorithm to design the consensus dynamics

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

a gossip based algorithm for discrete consensus over heterogeneous networks quantized consensus assumes that the state of each node may only take nonnegative integer values reaching consensus under quantization is equivalent to determining a balanced assignment of identical tasks to nodes in this note we generalize this problem in two ways and denote the resulting framework discrete consensus first we consider tasks that are not identical each one is characterized by its own weight secondly we assume that nodes are not identical as well as an example in the case of task assignment that we consider as a reference problem in this framework nodes may have different speeds and should be assigned a total weight proportional to their speed we provide a gossip based distributed algorithm that aims to minimize the maximum execution time over nodes whose convergence to a bounded set is guaranteed we show that the convergence time of the proposed algorithm relies ultimately on the average meeting time between two agents performing a random walk on a graph

on approximately symmetric informationally complete positive operator valued measures and related systems of quantum states we address the problem of constructing positive operator valued measures povms in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform this is motivated by the related question of constructing symmetric informationally complete povms sic povms for which the inner products are perfectly uniform however sic povms are notoriously hard to construct and despite some success of constructing them numerically there is no analytic construction known we present two constructions of approximate versions of sic povms where a small deviation from uniformity of the inner products is allowed the first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power the second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases moreover we construct vector systems in cn which are almost orthogonal and which might turn out

covering problems for brownian motion on spheres bounds are given on the mean time taken by a strong markov process to visit all of a finite collection of subsets of its state space these bounds are specialized to brownian motion on the surface of the unit sphere sigma p in r p this leads to bounds on the mean time taken by this brownian motion to come within a distance varepsilon of every point on the sphere and bounds on the mean time taken to come within varepsilon of every point or its opposite the second case is related to the grand tour a technique of multivariate data analysis that involves a search of low dimensional projections in both cases the bounds are asymptotically tight as varepsilon rightarrow 0 on sigma p for p geq 4

small gaps between prime numbers the work of goldston pintz yildirim in early 2005 dan goldston janos pintz and cem yildirim 12 made a spectacular breakthrough in the study of prime numbers resolving a long standing open problem they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes before their work it was known only that there were infinitely many gaps which were about a quarter the size of the average gap the new result may be viewed as a step towards the famous twin prime conjecture that there are infinitely many prime pairs p and p 2 the gap here being 2 the smallest possible gap between primes perhaps most excitingly their work reveals a connection between the distribution of primes in arithmetic progressions and small gaps between primes assuming certain admittedly difficult conjectures on the distribution of primes in arithmetic progressions they are able to prove the existence of infinitely many prime pairs that differ by at most 16 the aim of this article is to explain some of the ideas involved in their work let us begin by explaining the main question in a little more detail the number of primes up to x denoted by x is roughly x log x for large values of x this is the celebrated prime number theorem therefore if we randomly choose an integer near x then it has about a 1 in log x chance of being prime in other words as we look at primes around size x the average gap between consecutive primes is about log x as x increases the primes get sparser and the gap between consecutive primes tends to increase here are some natural questions about these gaps between prime numbers do the gaps always remain roughly about size log x or do we sometimes get unexpectedly large gaps and sometimes surprisingly small gaps can we say something about the statistical distribution of these gaps that is can we quantify how often the gap is between say log x and log x given 0 except for the primes 2 and 3 clearly the gap between consecutive primes must be even does every even number occur infinitely often as a gap between consecutive primes for example the twin prime conjecture says that the

the difference between consecutive primes ii with enough effort the value of x0 could be determined effectively the paper has much in common with 1 in particular we use the sieve method of harman 4 5 we no longer use zero density estimates however but rather mean value results on dirichlet polynomials similar to those that give rise to such estimates compare for example work of iwaniec and pintz 9 and baker harman and pintz 2 much of the improvement over 1 arises from the use of watt s theorem 11 on a particular kind of mean value more accurate estimates for six dimensional integrals are also used to good effect there is in addition a device which uses a two dimensional sieve to get an asymptotic formula for a one dimensionally sieved set see lemmas 16 17 unfortunately these lemmas which would be of great signi cance for v 0 53 are not very numerically signi cant when v drops to 0 525 the same applies to the roale reversals discussed below let us introduce enough notation to permit an outline of the proof when e is a nite sequence of positive integers counted with multiplicity we write jej for the number of terms of e and ed fm dm 2 eg let p z y

error correction in polynomial remainder codes with non pairwise coprime moduli and robust chinese remainder theorem for polynomials this paper investigates polynomial remainder codes with non pairwise coprime moduli we first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small namely the robust chinese remainder theorem crt for polynomials it basically says that a polynomial can be reconstructed from its erroneous residues such that the degree of the reconstruction error is upper bounded by tau whenever the degrees of all residue errors are upper bounded by tau where a sufficient condition for tau and a reconstruction algorithm are obtained by relaxing the constraint that all residue errors have small degrees another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues we finally obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result some errors with small degrees can be also corrected in the residues with this newly obtained result improvements in uncorrected error probability and burst error correction capability in data transmission are illustrated

consensus in time delayed multi agent systems with quantized dwell times abstract this paper investigates the effects of quantized dwell times in solving consensus problems in time delayed multi agent systems with the asynchronous property of unpredictable quantization jumps in inter agent communications the quantized dwell times can be introduced on purpose or be inherent from quantizing devices they beneficially avoid undesirable fast chattering of quantizers outputs and mathematical difficulties in convergence analysis the main result shows that the largest eigenvalue of the interaction graph laplacian well characterizes the dependence of maximum allowable time delay on quantized dwell times and quantizer parameters under the assumption of heterogeneous time varying transmission delays simulations are presented to show the effectiveness of the main result

tight bounds for the cover time of multiple random walks we study the cover time of multiple random walks on undirected graphs g v e we consider k parallel independent random walks that start from the same vertex the speed up is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks recently alon et al 2008 5 derived several upper bounds on the cover time which imply a speed up of w k for several graphs however for many of them k has to be bounded by o logn they also conjectured that for any 1 2 our bounds are tight up to logarithmic factors our findings also reveal a surprisingly sharp threshold behaviour for certain graphs e g the d dimensional torus with d 2 and hypercubes there is a value t such that the speed up is approximately min t k for any 1

convex subclass of harmonic starlike functions complex valued harmonic functions that are univalent and sense preserving in the unit disk u can be written in the form f h g where h and g are analytic in u we define and investigate a convex subclass of harmonic starlike functions of order a 0 a 1 we obtain coefficient conditions extreme points distortion bounds convolution conditions and convex combination for the above class of harmonic functions

the distribution of integers with a divisor in a given interval we determine the order of magnitude of h x y z the number of integers n x having a divisor in y z for all x y and z we also study hr x y z the number of integers n x having exactly r divisors in y z whenr 1 we establish the order of magnitude ofh1 x y z for allx y z satisfying z x 1 2 for every r 2 c 1 and 0 we determine the order of magnitude of hr x y z uniformly for y large and y y logy log 4 1 z min y c x 1 2 as a consequence of these bounds we settle a 1960 conjecture of erd os and some conjectures of tenenbaum one key element of the proofs is a new result on the distribution of uniform order statistics

neighborhoods of certain p valently analytic functions with negative coefficients in this investigation by making use of the familiar concept of neighborhoods of p valently analytic functions the authors prove coefficient bounds distortion inequalities associated inclusion relations for the n neighborhoods of a family of p valently analytic functions and its certain derivatives which is defined by means of a certain non homogenous differential equation

a distributed algorithm for finding hamiltonian cycles in random graphs in o logn time abstract it is known for some time that a random graph g n p contains w h p a hamiltonian cycle if p is larger than the critical value p c r i t log n log log n n n the determination of a concrete hamiltonian cycle for g n p is a nontrivial task even when p is much larger than p c r i t in this paper we consider random graphs g n p with p in 1 n where hides poly logarithmic factors in n for this range of p we present a distributed algorithm a hc that finds w h p a hamiltonian cycle in o log n rounds the algorithm works in the synchronous model and uses messages of size o log n and o log n memory per node

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

harmonic functions starlike in the unit disk complex valued harmonic functions that are univalent and sense preserving in the unit disk can be written in the form f h where h and g are analytic in we give univalence criteria and sufficient coefficient conditions for normalized harmonic functions that are starlike of order 0 1 these coefficient conditions are also shown to be necessary when h has negative and g has positive coefficients these lead to extreme points and distortion bounds

sparse stable systems abstract in the design of decentralized networked systems it is important to know whether a given network topology can sustain stable dynamics we consider a basic version of this problem here given a vector space of sparse real matrices does it contain a stable hurwitz matrix said differently is a feedback channel corresponding to a non zero entry necessary for stabilization or can it be done without we provide in this paper a set of necessary conditions and a set of sufficient conditions for the existence of stable matrices in a vector space of sparse matrices we further prove some properties of the set of sparse matrix spaces that contain hurwitz matrices the conditions we exhibit are most easily stated in the language of graph theory which we thus adopt in this paper

neighborhoods of a new class of harmonic multivalent functions abstract we introduce and investigate a new subclass of harmonic multivalent functions defined by using a differential operator we obtain coefficient conditions distortion bounds extreme points convex combination for the above class of harmonic multivalent functions we also derive inclusion relationships involving the neighborhoods of harmonic multivalent functions

radius of convexity of partial sums of functions in the close to convex family abstract let f denote the class of all normalized analytic functions f that are locally univalent in the unit disk z 1 satisfying the condition re 1 z f z f z 1 2 for z 1 functions in f are known to be close to convex univalent in the unit disk this class plays a crucial role in the discussion on certain extremal problems for the class of complex valued and sense preserving harmonic convex functions and some other related problems in determining univalence criteria for sense preserving harmonic mappings in this article we show that every section of a function in the class f is convex in the disk z 1 6 the radius 1 6 is best possible we conjecture that every section of functions in the family f is univalent and close to convex in the disk z 1 3

distributed random walks performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science including distributed computing in this article we focus on the problem of sampling random walks efficiently in a distributed network and its applications given bandwidth constraints the goal is to minimize the number of rounds required to obtain random walk samples all previous algorithms that compute a random walk sample of length e as a subroutine always do so naively that is in o e rounds the main contribution of this article is a fast distributed algorithm for performing random walks we present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk our algorithm performs a random walk of length e in o ed rounds o hides polylog n factors where n is the number of nodes in the network with high probability on an undirected network where d is the diameter of the network for small diameter graphs this is a significant improvement over the naive o e bound furthermore our algorithm is optimal within a poly logarithmic factor as there exists a matching lower bound nanongkai et al 2011 we further extend our algorithms to efficiently perform k independent random walks in o ked k rounds we also show that our algorithm can be applied to speedup the more general metropolis hastings sampling our random walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine we present two main applications first we give a fast distributed algorithm for computing a random spanning tree rst in an arbitrary undirected unweighted network which runs in o md rounds with high probability m is the number of edges our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network our algorithm is fully decentralized and can serve as a building block in the design of topologically aware networks

injectivity of sections of univalent harmonic mappings abstract if s denotes the class of functions h z z n 2 a n z n which are analytic and univalent in the unit disk z 1 then the classical result of szego shows that every section s n h z k 1 n a k z k of h is univalent in z 1 4 exact largest radius of univalence r n of s n h remains an open problem although the corresponding results for sections of various geometric subclasses of s have been obtained which include those h s for which re h z 0 holds in the unit disk however no attempt has been made to derive harmonic analog of these results the central object in this case is the class s h 0 of sense preserving harmonic univalent mappings f h g defined on the unit disk normalized so that h 0 g 0 h 0 1 g 0 0 our primary objective in this paper is to solve the univalency of every section of a harmonic function in the class p h 0 where p h 0 denotes the class of normalized univalent harmonic mapping f h g in the unit disk z 1 satisfying the condition re h z g z for z 1 where g 0 0 and 0 1 in this paper we first present sharp bounds for the moduli of the coefficients a n b n of f p h 0 and then determine the value of r so that the partial sums of f p h 0 are close to convex in z r

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

tight analyses of two local load balancing algorithms this paper presents an analysis of the following load balancing algorithm at each step each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d 1 fewer tokens where d is the maximum degree of any node in the network we show that within o delta alpha steps the algorithm reduces the maximum difference in tokens between any two nodes to at most o d 2 log n alpha where delta is the global imbalance in tokens i e the maximum difference between the number of tokens at any node initially and the average number of tokens n is the number of nodes in the network and alpha is the edge expansion of the network the time bound is tight in the sense that for any graph with edge expansion alpha and for any value delta there exists an initial distribution of tokens with imbalance delta for which the time to reduce the imbalance to even delta 2 is at least omega delta alpha the bound on the final imbalance is tight in the sense that there exists a class of networks that can be locally balanced everywhere i e the maximum difference in tokens between any two neighbors is at most 2d while the global imbalance remains omega d 2 log n alpha furthermore we show that upon reaching a state with a global imbalance of o d 2 log n alpha the time for this algorithm to locally balance the network can be as large as omega n 1 2 we extend our analysis to a variant of this algorithm for dynamic and asynchronous networks we also present tight bounds for a randomized algorithm in which each node sends at most one token in each step

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

precise coefficient estimates for close to convex harmonic univalent mappings the class sh consists of harmonic univalent and sense preserving functions f in the open unit disk u z z 1 such that f h g where h z z n 2 anzn and g z n 1 a nzn let s0h ch and c0h denote the subclass of sh with a 1 0 the subclass of sh with f being a close to convex mapping and the intersection of s0h and ch respectively in this paper for f c0h and f ch we prove that the harmonic analogue of the bieberbach conjecture and the generalization of the bieberbach conjecture are true

harmonic univalent functions with negative coefficients we give sufficient coefficient conditions for normalized harmonic functions to map onto either starlike or convex regions these conditions are also shown to be necessary when the coefficients are negative this leads to distortion bounds and extreme points

convergence time for unbiased quantized consensus over static and dynamic networks in this paper the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs and some strong results are obtained the paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks it is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk using the theory of harmonic functions for reversible markov chains following this and using electric resistance analogy of the reversible markov chains the paper provides an upper bound of o nmd log n for the expected convergence time to consensus where n m and d denote respectively the number of nodes the number of edges and the diameter of the network moreover the paper shows that the above upper bound is tight up to a factor of log n for some simple graphs such as line graph and cycle finally the results are extended to bound the expected convergence time of the underlying dynamics in time varying networks modeling such dynamics as the evolution of a time inhomogeneous markov chain the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks this upper bound is significantly better than earlier ones given for different quantized consensus protocols over time varying graphs

fault tolerant convolution via chinese remainder codes constructed from non coprime moduli this paper develops a framework for performing fault tolerant convolution via error correcting codes based on the chinese remainder theorem crt with non coprime moduli in contrast to convolution that is protected through crt codes with coprime moduli our scheme allows errors to be detected and located in a highly parallel manner and has the advantage of being robust to faults that occur during the error handling stage in addition for certain important classes of errors the codes developed in this paper require less redundancy than crt codes constructed from coprime moduli we demonstrate the applicability and advantages of our codes by comparing them to both crt codes with coprime moduli and repetition codes we focus on codes over polynomial rings but our constructions apply naturally to integer codes we discuss this extension and show that it generalizes the distance properties and error correction strategies associated with previously developed approaches for codes based on non coprime integer moduli

a robust chinese remainder theorem with its applications in frequency estimation from undersampled waveforms the chinese remainder theorem crt allows to reconstruct a large integer from its remainders modulo several moduli in this paper we propose a robust reconstruction algorithm called robust crt when the remainders have errors we show that using the proposed robust crt the reconstruction error is upper bounded by the maximal remainder error range named remainder error bound if the remainder error bound is less than one quarter of the greatest common divisor gcd of all the moduli we then apply the robust crt to estimate frequencies when the signal waveforms are undersampled multiple times it shows that with the robust crt the sampling frequencies can be significantly reduced

compact sequences of co primes and their applications to the security of crt based threshold schemes crt based threshold secret sharing schemes use sequences of pairwise co prime positive integers in their construction if these sequences are not compact then the crt based threshold secret sharing schemes may have a weak security in the sense of a massive loss of entropy or an arbitrarily large information rate in this paper compact sequences of co primes are introduced and their applications to the security of crt based threshold secret sharing schemes is throughout investigated it is shown that all the results regarding the security of crt based threshold secret sharing schemes that use sequences of consecutive primes also hold for threshold secret sharing schemes that use compact sequences of co primes moreover compact sequences of co primes may be significantly denser than sequences of consecutive primes of the same length and their use in the construction of crt based threshold secret sharing schemes may lead to better security properties

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

difference set codes codes with squared euclidean distance of six for partial response channels we present a new construction of block codes for the 1 d pr partial response channel the codewords in the code correspond to constant sum subsets of a difference set it is shown that at the output of a noiseless 1 d pr channel the minimum squared euclidean distance of such a code is at least six compared to two for the uncoded system this construction yields larger code rates than previously known codes with the same minimum distance for large code lengths the construction technique also imposes upper bounds on the decoding complexity of the codes

a modular approach to key safeguarding a method is proposed for a key safeguarding scheme threshold scheme in which the shadows are congruence classes of a number associated with the original key a variation of this scheme provides efficient error detection and even exposes deliberate tampering certain underlying similarities of this scheme with shamir s interpolation method make it possible to incorporate these protective features in that method as well

diophantine approximation with arithmetic functions i we prove a strong simultaneous diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes

row stochastic matrices with a common left fixed vector abstract we consider the convex polytope s n x that consist of those n n row stochastic matrices having a common nonnegative left fixed vector xt we examine the 1 skeleton of s n x and show how to construct all extreme points adjacent to a given one as vertices of the 1 skeleton connections with transportation polytopes are discussed further we give a formula for the degree of an extreme point in the 1 skeleton of s n x find its maximum and minimum values and determine when all degrees are equal an explicit description of the 1 skeleton is given for n 3

neighborhoods of certain multivalent functions with negative coefficients by means of a differential operator we introduce and investigate two new subclasses of p valently analytic functions of complex order the various results obtained here for each of these function classes include coefficient inequalities and the consequent inclusion relationships involving the neighborhoods of the p valently analytic functions

optimal estimates of two common remainders for a robust generalized chinese remainder theorem estimation of multiple common remainders from a sequence of erroneous residue sets is an important step for the robust generalized chinese remainder theorem crt this paper considers the problem of how to estimate the two common remainders from their residue sets modulo a set of moduli to measure the errors properly under the modular operation we introduce two type circular distances based on these circular distances two estimation methods are proposed by properly grouping the erroneous remainders into two ordered clusters both of the two methods perform better with lower computational complexities than the existing methods in this paper theoretical analysis as well as analytical results for the two proposed methods are obtained for the first method the two optimal estimates are proved to be in a finite set with no more than l candidates where l is the number of the given moduli the second method have closed forms simulation results show that the two proposed methods have nearly the same performance these optimal estimates can improve the performance of the robust generalized crt significantly

neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator in the present investigation by making use of the familiar concept of neighborhoods of analytic and multivalent functions we derive coefficient bounds and distortion inequalities associated inclusion relations for the n d neighborhoods of subclasses of analytic and multivalent functions with negative coefficients which are defined by means of a certain nonhomogenous differential equation several special cases of the main results are mentioned

pluriharmonic mappings in cn and complex banach spaces abstract in this paper we obtain a sufficient condition for pluriharmonic mappings on the euclidean unit ball b n to be univalent sense preserving quasiconformal and bi lipschitz diffeomorphisms on b n and to have linearly connected images also we give a sufficient condition for pluriharmonic mappings on b n to have quasiconformal extensions to c n next we generalize the harmonic schwarz lemma to pluriharmonic mappings of the unit ball b x of a complex banach space x into the unit ball b n of c n with respect to an arbitrary norm further we obtain a generalization of the harmonic schwarz pick lemma to the case of pluriharmonic mappings of the homogeneous unit ball b x of a complex banach space x into the unit ball b n we also obtain a version of the holomorphic schwarz pick lemma for the jacobian determinant on the euclidean unit ball b n to the case of pluriharmonic mappings of the homogeneous unit ball b x into b n in the case that b x is an open subset of c n finally we obtain the landau and the bloch theorems for pluriharmonic or holomorphic mappings on finite dimensional homogeneous unit balls

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

reaching a consensus in a dynamically changing environment a graphical approach this paper presents new graph theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments the concepts of rooted strongly rooted and neighbor shared are defined and conditions are derived for compositions of sequences of directed graphs to be of these types the graph of a stochastic matrix is defined and it is shown that under certain conditions the graph of a sarymsakov matrix and a rooted graph are one and the same as an illustration of the use of the concepts developed in this paper graph theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied vicsek consensus problem

connections between various subclasses of planar harmonic mappings involving hypergeometric functions the main purpose of this paper is to establish connections between various subclasses of harmonic mappings in the plane by applying certain convolution operators involving hypergeometric functions to be more precise we investigate such connections with harmonic k uniformly starlike and harmonic k uniformly convex mappings in the plane

the security of all rsa and discrete log bits we study the security of individual bits in an rsa encrypted message en x we show that given en x predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy is through a polynomial time reduction as hard as breaking rsa moreover we prove that blocks of o log log n bits of x are computationally indistinguishable from random bits the results carry over to the rabin encryption scheme considering the discrete exponentiation function gx modulo p with probability 1 o 1 over random choices of the prime p the analog results are demonstrated the results do not rely on group representation and therefore applies to general cyclic groups as well finally we prove that the bits of ax b modulo p give hard core predicates for any one way function f all our results follow from a general result on the chosen multiplier hidden number problem given an integer n and access to an algorithm px that on input a random a i zn returns a guess of the ith bit of ax mod n recover x we show that for any i if px has at least a nonnegligible advantage in predicting the ith bit we either recover x or obtain a nontrivial factor of n in polynomial time the result also extends to prove the results about simultaneous security of blocks of o log log n bits

quantized consensus over directed networks with switching topologies abstract this paper studies the quantized consensus problem for a group of agents over directed networks with switching topologies we propose an effective distributed protocol with an adaptive finite level uniform quantized strategy under which consensus among agents is guaranteed with weaker communication conditions in particular we analytically prove that each agent sending 5 level quantized information to each of its neighbors together with 3 level quantized information to itself at each time step which suffices for attaining consensus with an exponential convergence rate as long as the duration of all link failures in the directed network is bounded by dropping the typical common left eigenvector requirement for the existence of common quadratic lyapunov function we conduct the convergence analysis based on the notion of input to output stability the proposed quantized protocol has favorable merits of requiring little communication overhead and increasing robustness to link unreliability and it fits well into the digital network framework

disk of convexity of sections of univalent harmonic functions abstract one of the classical results from szego shows that if h z z n 2 a n z n is analytic and univalent in the unit disk d z c z 1 then the section s n h z k 1 n a k z k of h is univalent in z 1 4 the exact largest radius of the univalence r n of s n h remains an open problem on the other hand not much is known in the case of harmonic univalent functions it is then natural to consider the class p h 0 of normalized harmonic mappings f h g in the unit disk d satisfying the condition re h z g z for z d where g 0 0 functions in p h 0 are known to be univalent and close to convex in d in this paper we first show that each f p h 0 is convex in the disk z 2 1 and then determine the value of r so that the partial sums of f p h 0 are convex in z r

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

chinese remaindering with errors the chinese remainder theorem states that a positive integer m is uniquely specified by its remainder module k relatively prime integers p sub 1 spl middot spl middot spl middot p sub k provided m spl pi sub i 1 sup k p sub i thus the residues of m module relatively prime integers p sub 1 p sub 2 spl middot spl middot spl middot p sub n form a redundant representation of m if m spl pi sub i 1 sup k p sub i and k n this gives a number theoretic construction of an error correcting code that has been considered often in the past in this code a message integer m spl pi sub i 1 sup k p sub i is encoded by the list of its residues module p sub 1 spl middot spl middot spl middot p sub n by the chinese remainder theorem if a codeword is corrupted in e n k 2 coordinates then there exists a unique integer m whose corresponding codesword differs from the corrupted word in at most e places furthermore mandelbaum 1976 1978 shows how m can be recovered efficiently given the corrupted word provided that the p sub i s are very close to one another to deal with arbitrary p sub i s we present a variant of his algorithm that runs in almost linear time and recovers from e log p sub 1 log p sub 1 log p sub n spl middot n k errors our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than n k 2 specifically given n residues r sub 1 spl middot spl middot spl middot r sub n and an agreement parameter t we find a list of all integers m spl pi sub i 1 sup k p sub i such that m mod p sub i r sub i for at least t values of i spl isin 1 spl middot spl middot spl middot n provided t spl omega spl radic kn log p sub n log p sub 1 for n spl gt k and p sub n spl les p sub 1 sup o 1 the fraction of error corrected by the algorithm is almost twice that corrected by the previous work more significantly the algorithm recovers the message even when the amount of agreement between the received word and the codeword is much smaller than the number of errors

the cover time of deterministic random walks for general transition probabilities abstract a deterministic random walk is a deterministic process analogous to a random walk while there are some results on the cover time of the rotor router model which is a deterministic random walk corresponding to a simple random walk nothing is known about the cover time of deterministic random walks emulating general transition probabilities this paper is concerned with the shortest remaining time srt router model with multiple tokens which is a deterministic process coping with general transition probabilities possibly containing irrational numbers for the model we give an upper bound of the cover time which is the first result on the cover time of deterministic random walks for general transition probabilities our upper bound also improves the existing bounds for the rotor router model in some cases

on the time taken by random walks on finite groups to visit every state consider a random walk on a finite group g suppose that the time taken for the transition probabilities to approach the uniform distribution is small compared with g then the time taken for the random walk to visit every state is approximately r g log g where r is the mean number of returns to the initial state in the short term

non linear protocols for optimal distributed consensus in networks of dynamic agents we consider stationary consensus protocols for networks of dynamic agents with fixed topologies at each time instant each agent knows only its and its neighbors state but must reach consensus on a group decision value that is function of all the agents initial state we show that the agents can reach consensus if the value of such a function is time invariant when computed over the agents state trajectories we use this basic result to introduce a non linear protocol design rule allowing consensus on a quite general set of values such a set includes e g any generalized mean of order p of the agents initial states as a second contribution we show that our protocol design is the solution of individual optimizations performed by the agents this notion suggests a game theoretic interpretation of consensus problems as mechanism design problems under this perspective a supervisor entails the agents to reach a consensus by imposing individual objectives we prove that such objectives can be chosen so that rational agents have a unique optimal protocol and asymptotically reach consensus on a desired group decision value we use a lyapunov approach to prove that the asymptotical consensus can be reached when the communication links between nearby agents define a time invariant undirected network finally we perform a simulation study concerning the vertical alignment maneuver of a team of unmanned air vehicles

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

distributive weighted threshold secret sharing schemes the concept of distributive weighted threshold access structure is introduced which is an weighted threshold access structure where the participants are distributed on levels the participants on the same level are assigned the same weight and the threshold of the access structure is 1 the weight of the participants on the ith level is of the form 1 ki and therefore the ith level induces a standard threshold access structure with threshold ki we propose a crt based realization of distributive weighted threshold access structures which is asymptotically perfect and perfect zero knowledge we also show that distributive weighted threshold access structures do not generally have ideal realizations in case of just one level our scheme can be viewed as an asymptotically perfect and perfect zero knowledge variation of the asmuth bloom secret sharing scheme

lattice based threshold changeability for standard shamir secret sharing schemes we consider the problem of increasing the threshold parameter of a secret sharing scheme after the setup share distribution phase without further communication between the dealer and the shareholders previous solutions to this problem require one to start off with a nonstandard scheme designed specifically for this purpose or to have communication between shareholders in contrast we show how to increase the threshold parameter of the standard shamir secret sharing scheme without communication between the shareholders our technique can thus be applied to existing shamir schemes even if they were set up without consideration to future threshold increases our method is a new positive cryptographic application for lattice reduction algorithms inspired by recent work on lattice based list decoding of reed solomon codes with noise bounded in the lee norm we use fundamental results from the theory of lattices geometry of numbers to prove quantitative statements about the information theoretic security of our construction these lattice based security proof techniques may be of independent interest

irreducible algebraic integers in short intervals we study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field k exceeds 2 every interval of the form x x x with a fixed 1 2 contains absolute value of the norm of an irreducible algebraic integer from k provided x x 0 the constant x 0 effectively depends on k and

bounded distance 1 soft decision reed solomon decoding we present a new reed solomon decoding algorithm which embodies several refinements of an earlier algorithm some portions of this new decoding algorithm operate on symbols of length lgq bits other portions operate on somewhat longer symbols in the worst case the total number of calculations required by the new decoding algorithm is proportional to nr where n is the code s block length and r is its redundancy this worst case workload is very similar to prior algorithms but in many applications average case workload and error correcting performance are both much better the input to the new algorithm consists of n received symbols from gf q and n nonnegative real numbers each of which is the reliability of the corresponding received symbol any conceivable errata pattern has a score equal to the sum of the reliabilities of its locations with nonzero errata values a max likelihood decoder would find the minimum score over all possible errata patterns our new decoding algorithm finds the minimum score only over a subset of these possible errata patterns the errata within any candidate errata pattern may be partitioned into errors and erasures depending on whether the corresponding reliabilities are above or below an erasure threshold different candidate errata patterns may have different thresholds each chosen to minimize its corresponding errata count which is defined as 2 spl middot number of errors number of erasures the new algorithm finds an errata pattern with minimum score among all errata patterns for which errata count spl les r 1 where r is the redundancy of the rs code conventional algorithms also require that the erasure threshold be set a priori the new algorithm obtains the best answer over all possible settings of the erasure threshold conventional cyclic rs codes have length n q 1 and their locations correspond to the nonzero elements of gf q the new algorithm also applies very naturally to rs codes which have been doubly extended by the inclusion of 0 and spl infin as additional locations

sections of stable harmonic convex functions abstract in this article we consider the class shc of normalized stable harmonic convex mappings f h g in the unit disk and determine r such that every section of f is close to convex or convex in the disk z r also we show that the convolution f f shc whenever f shc in addition we prove that the harmonic convolution f 1 f 2 is stable harmonic close to convex whenever f 1 shc and f 2 is either a slanted half plane mapping or an asymmetric vertical strip mapping or a mapping in the family p h 0 which is contained in the family of harmonic close to convex functions in z 1

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

decompositions into spanning rainbow structures a subgraph of an edge coloured graph is called rainbow if all its edges have distinct colours the study of rainbow subgraphs goes back more than two hundred years to the work of euler on latin squares and has been the focus of extensive research ever since euler posed a problem equivalent to finding properly n edge coloured complete bipartite graphs kn n which can be decomposed into rainbow perfect matchings while there are proper edge colourings of kn n without even a single rainbow perfect matching the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings in particular we prove that if some fraction of the colour classes have at most 1 o 1 n edges then one can nearly decompose the edges of kn n into edge disjoint perfect rainbow matchings as an application of this we establish in a very strong form a conjecture of akbari and alipour and asymptotically prove a conjecture of barat and nagy both these conjectures concern rainbow perfect matchings in edge colourings of kn n with quadratically many colours using our techniques we also prove a number of results on near decompositions of graphs into other rainbow structures like hamiltonian cycles and spanning trees most notably we prove that any properly coloured complete graph can be nearly decomposed into spanning rainbow trees this asymptotically proves the brualdi hollingsworth and kaneko kano suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions