the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

homological approach to problems with jumping non linearity abstract in this paper we use a perturbed version of the rabinowitz floer homology to find solutions to pde s with jumping nonlinearities as applications we find branches for the fucik spectrum for the laplace equation and for systems on manifolds that fiber over s 1

global stability of gompertz model of three competing populations the model of three competitive populations with gompertz growth is studied the periodic solutions are ruled out by generalized dulac criteria on the basis of the analysis we obtain conditions that ensure the asymptotic behavior of the model is simple

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

testing statistical laws in complex systems the availability of large datasets requires an improved view on statistical laws in complex systems such as zipf s law of word frequencies the gutenberg richter law of earthquake magnitudes or scale free degree distribution in networks in this letter we discuss how the statistical analysis of these laws are affected by correlations present in the observations the typical scenario for data from complex systems we first show how standard maximum likelihood recipes lead to false rejections of statistical laws in the presence of correlations we then propose a conservative method based on shuffling and undersampling the data to test statistical laws and find that accounting for correlations leads to smaller rejection rates and larger confidence intervals on estimated parameters

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 periodic fucik spectrum and a superlinear sturm liouville equation in the first part of the paper a variational characterisation of the periodic eigenvalues the so called fucik spectrum of a semilinear positive homogeneous sturm liouville equation is given the proof relies on the s 1 invariance of the equation in the second part a nonlinear sturm liouville equation with typically an exponential nonlinearity is considered it is proved that under certain conditions this equation is solvable for arbitrary forcing terms the proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the fucik spectrum

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

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

a borsuk ulam theorem for orthogonal tk and zpr actions and applications abstract a version if the borsuk ulam theorem is proved for the torus tk and p torus zpr we deal with the case of a nonempty fixed point set of groups on the sphere of representation as an application a geometrical index theory is introduced for these groups the index is used to obtain the existence of multiple critical points for functions invariant under tk or zpr action

an innovation diffusion model for three competitive products abstract an innovation diffusion model is studied which describes the dynamics of three competing products in one market a complete analysis for the global stability of equilibria of the model is given by excluding the existence of periodic solutions and using the theory of three dimensional competition systems

what is the real size of a sampled network the case of the internet internet mapping projects generally consist in sampling the network from a limited set of sources by usingtraceroute probes this methodology akin to the merging of spanning trees from the different sources to a set of destinations leads necessarily to a partial incomplete map of the internet accordingly determination of internet topology characteristics from such sampled maps is in part a problem of statistical inference our contribution begins with the observation that the inference of many of the most basic topological quantities including network size and degree characteristics fromtraceroute measurements is in fact a version of the so called species problem in statistics this obse rvation has important implications since species problems are often quite challenging we focus here on the most fundamental example of atraceroute internet species the number of nodes in a network specifica lly

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

statistical properties of sampled networks we study the statistical properties of the sampled scale free networks deeply related to the proper identification of various real world networks we exploit three methods of sampling and investigate the topological properties such as degree and betweenness centrality distribution average path length assortativity and clustering coefficient of sampled networks compared with those of original networks it is found that the quantities related to those properties in sampled networks appear to be estimated quite differently for each sampling method we explain why such a biased estimation of quantities would emerge from the sampling procedure and give appropriate criteria for each sampling method to prevent the quantities from being overestimated or underestimated

efficient sampling of complex network with modified random walk strategies abstract we present two novel random walk strategies choosing seed node csn random walk and no retracing nr random walk different from the classical random walk sampling the csn and nr strategies focus on the influences of the seed node choice and path overlap respectively three random walk samplings are applied in the erdos renyi er barabasi albert ba watts strogatz ws and the weighted usair networks respectively then the major properties of sampled subnets such as sampling efficiency degree distributions average degree and average clustering coefficient are studied the similar conclusions can be reached with these three random walk strategies firstly the networks with small scales and simple structures are conducive to the sampling secondly the average degree and the average clustering coefficient of the sampled subnet tend to the corresponding values of original networks with limited steps and thirdly all the degree distributions of the subnets are slightly biased to the high degree side however the nr strategy performs better for the average clustering coefficient of the subnet in the real weighted usair networks some obvious characters like the larger clustering coefficient and the fluctuation of degree distribution are reproduced well by these random walk strategies

sampling promotes community structure in social and information networks abstract any network studied in the literature is inevitably just a sampled representative of its real world analogue additionally network sampling is lately often applied to large networks to allow for their faster and more efficient analysis nevertheless the changes in network structure introduced by sampling are still far from understood in this paper we study the presence of characteristic groups of nodes in sampled social and information networks we consider different network sampling techniques including random node and link selection network exploration and expansion we first observe that the structure of social networks reveals densely linked groups like communities while the structure of information networks is better described by modules of structurally equivalent nodes however despite these notable differences the structure of sampled networks exhibits stronger characterization by community like groups than the original networks irrespective of their type and consistently across various sampling techniques hence rich community structure commonly observed in social and information networks is to some extent merely an artifact of sampling

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

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

weak signal transmission in complex networks and its application in detecting connectivity we present a network model of coupled oscillators to study how a weak signal is transmitted in complex networks through both theoretical analysis and numerical simulations we find that the response of other nodes to the weak signal decays exponentially with their topological distance to the signal source and the coupling strength between two neighboring nodes can be figured out by the responses this finding can be conveniently used to detect the topology of unknown network such as the degree distribution clustering coefficient and community structure etc by repeatedly choosing different nodes as the signal source through four typical networks i e the regular one dimensional small world random and scale free networks we show that the features of network can be approximately given by investigating many fewer nodes than the network size thus our approach to detect the topology of unknown network may be efficient in practical situations with large network size

on the fucik spectrum of the laplacian on a torus abstract we study the fucik spectrum of the laplacian on a two dimensional torus t 2 exploiting the invariance properties of the domain t 2 with respect to translations we obtain a good description of large parts of the spectrum in particular for each eigenvalue of the laplacian we will find an explicit global curve in the fucik spectrum which passes through this eigenvalue these curves are ordered and we will show that their asymptotic limits are positive on the other hand using a topological index based on the mentioned group invariance we will obtain a variational characterization of global curves in the fucik spectrum also these curves emanate from the eigenvalues of the laplacian and we will show that they tend asymptotically to zero thus we infer that the variational and the explicit curves cannot coincide globally and that in fact many curve crossings must occur we will give a bifurcation result which partially explains these phenomena

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

revealing the efficiency of information diffusion in online social networks of microblog abstract online social networks osns such as microblogs enable people to simultaneously share information with a large number of peers and revealing the properties of the formed network structure and the corresponding impacts on information diffusion are important in this article we empirically study 10 millions user profiles from the largest chinese mircoblog sina weibo and 41 7 millions profiles from twitter by studying followers topology we find an invariant characteristic that the followers count of users obeys a power law distribution with exponent near 2 we then investigate the reason from the theoretical view and find this network structure achieves the optimal information diffusion in terms of inducing a minimal propagation times our study thus reveals the efficiency of information diffusion in osns of microblogs which is validated by experimental examination theoretical analysis and numerical simulation

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious

dancer fucik spectrum for fractional schrodinger operators with a steep potential well on rn abstract in this paper we study dancer fucik spectrum of the fractional schrodinger operators which is defined as the set of r 2 such that s u v x u u u in r n has a nontrivial solution u where the potential v has a steep potential well for sufficiently large parameter 0 it is allowed that s v has essential spectrum with finitely many eigenvalues below the infimum of e s s s v many difficulties are caused by general nonlocal operators we develop new techniques to overcome them to construct the first nontrivial curve of dancer fucik point spectrum by minimax methods to show some qualitative properties of the curve and to prove that the corresponding eigenfunctions are foliated schwartz symmetric as applications we obtain the existence of nontrivial solutions for nonlinear schrodinger equations with nonresonant nonlinearity

beyond the zipf mandelbrot law in quantitative linguistics abstract in this paper the zipf mandelbrot law is revisited in the context of linguistics despite its widespread popularity the zipf mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus in particular we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation finally it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by c tsallis

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

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

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

impact of observational incompleteness on the structural properties of protein interaction networks abstract the observed structure of protein interaction networks is corrupted by many false positive negative links this observational incompleteness is abstracted as random link removal and a specific experimentally motivated spoke link rearrangement their impact on the structural properties of gene duplication and mutation network models is studied for the degree distribution a curve collapse is found showing no sensitive dependence on the link removal rearrangement strengths and disallowing a quantitative extraction of model parameters the spoke link rearrangement process moves other structural observables like degree correlations cluster coefficient and motif frequencies closer to their counterparts extracted from the yeast data this underlines the importance to take a precise modeling of the observational incompleteness into account when network structure models are to be quantitatively compared to data

on the asymmetric may leonard model of three competing species in this paper we analyze the global asymptotic behavior of the asymmetric may leonard model of three competing species frac dx i dt x i 1 x i beta ix i 1 alpha ix i 1 x i 0 0 i 1 2 3 with x 0 x 3 x 4 x 1 under the assumption 0 b1b2b3 and p is a saddle point with one dimensional stable manifold gamma if a1a2a3 b1b2b3 then p is global asymptotically stable in int r3 ii if a1a2a3 b1b2b3 then for each initial condition x 0 not in gamma the solution v

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

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

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

what big data tells sampling the social network by communication channels big data has become the primary source of understanding the structure and dynamics of the society at large scale the network of social interactions can be considered as a multiplex where each layer corresponds to one communication channel and the aggregate of all of them constitutes the entire social network however usually one has information only about one of the channels or even a part of it which should be considered as a subset or sample of the whole here we introduce a model based on a natural bilateral communication channel selection mechanism which for one channel leads to consistent changes in the network properties for example while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one we get a monotonically decreasing distribution as observed in empirical studies of single channel data we also find that assortativity may occur or get strengthened due to the sampling method we analyze the far reaching consequences of our findings

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

systems of differential equations that are competitive or cooperative v convergence in 3 dimensional systems abstract for 3 dimensional systems that are competitive or cooperative we present new criteria for existence of equilibria and cycles convergence of all trajectories global stability and structural stability a certain type of positive feedback loop is shown to be structurally stable

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

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

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

a community based sampling method using dpl for online social networks in this paper we propose a new graph sampling method for online social networks that achieves the following first a sample graph should reflect the ratio between the number of nodes and the number of edges of the original graph second a sample graph should reflect the topology of the original graph third sample graphs should be consistent with each other when they are sampled from the same original graph the proposed method employs two techniques hierarchical community extraction and densification power law the proposed method partitions the original graph into a set of communities to preserve the topology of the original graph it also uses the densification power law which captures the ratio between the number of nodes and the number of edges in online social networks in experiments we use several real world online social networks create sample graphs using the existing methods and ours and analyze the differences between the sample graph by each sampling method and the original graph

estimating population size for capture recapture data when capture probabilities vary by time and individual animal there have been no estimators of population size associated with the capture recapture model when the capture probabilities vary by time and individual animal this work proposes a nonparametric estimation technique that is appropriate for such a model using the idea of sample coverage which is defined as the proportion of the total individual capture probabilities of the captured animals a simulation study was carried out to show the performance of the proposed estimation procedure numerical results indicate that it generally works satisfactorily when the coefficient of variation of the individual capture probabilities is relatively large an example is also given for illustration

sampling networks by nodal attributes in a social network individuals or nodes connect to other nodes by choosing one of the channels of communication at a time to re establish the existing social links since available data sets are usually restricted to a limited number of channels or layers these autonomous decision making processes by the nodes constitute the sampling of a multiplex network leading to just one though very important example of sampling bias caused by the behavior of the nodes we develop a general setting to get insight and understand the class of network sampling models where the probability of sampling a link in the original network depends on the attributes h of its adjacent nodes assuming that the nodal attributes are independently drawn from an arbitrary distribution h and that the sampling probability r h i h j for a link ij of nodal attributes h i and h j is also arbitrary we derive exact analytic expressions of the sampled network for such network characteristics as the degree distribution degree correlation and clustering spectrum the properties of the sampled network turn out to be sums of quantities for the original network topology weighted by the factors stemming from the sampling based on our analysis we find that the sampled network may have sampling induced network properties that are absent in the original network which implies the potential risk of a naive generalization of the results of the sample to the entire original network we also consider the case when neighboring nodes have correlated attributes to show how to generalize our formalism for such sampling bias and we get good agreement between the analytic results and the numerical simulations

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