small gaps between primes or almost primes let p n denote the n th prime goldston pintz and yildirim recently proved that li m in f pn 1 p n n log p n 0 we give an alternative proof of this result we also prove some corresponding results for numbers with two prime factors let q n denote the n th number that is a product of exactly two distinct primes we prove that lim inf qn 1 q n n 26 if an appropriate generalization of the elliott halberstam conjecture is true then the above bound can be improved to 6

detect overlapping and hierarchical community structure in networks abstract clustering and community structure is crucial for many network systems and the related dynamic processes it has been shown that communities are usually overlapping and hierarchical however previous methods investigate these two properties of community structure separately this paper proposes an algorithm eagle to detect both the overlapping and hierarchical properties of complex community structure together this algorithm deals with the set of maximal cliques and adopts an agglomerative framework the quality function of modularity is extended to evaluate the goodness of a cover the examples of application to real world networks give excellent results

anti modularity and anti community detecting in complex networks abstract many networks of interest in sciences and social research can be divided naturally into anti communities the problem of detecting and characterizing such anti community structure has attracted recent attention in this paper we first define the anti modularity as a quantitative measure of anti community partitioning on a network we also theoretically and empirically show the reliability of anti modularity as a measurement of the quality of an anti community partitioning a label propagation algorithm lpad for anti community detection is proposed experimental results on synthetic and real world networks show that our algorithm lpad can obtain higher quality anti community partitioning than other methods

two stage algorithm using influence coefficient for detecting the hierarchical non overlapping and overlapping community structure abstract community detection is one of the most important problems in complex networks many algorithms have been proposed in the last decade and most of them focus on the non overlapping community structures in the early days overlapping and hierarchical structures are another two important properties in complex networks which have attracted researchers extensive concern in recent years in this paper we proposed a two stage method which can detect the hierarchical non overlapping and overlapping community structures in complex networks in this method the cnm algorithm a fast hierarchical agglomerative algorithm proposed by clauset newman and moore is used in the first stage in the second stage a new evaluation function named as influence coefficient based on the local community structure is proposed which can get the overlapping community structures at different overlapping levels by adjusting a tunable parameter besides the proposed evaluation function can detect the wrongly classified nodes in the partition of the first stage and correct them finally the computational complexity of the algorithm is low the experimental results on both synthetic and real world network datasets show the efficiency of our method

higher correlations of divisor sums related to primes iii small gaps between primes we use divisor sums to approximate prime tuples and moments for primes in short intervals by connecting these results to classical moment problems we are able to prove that for any 0 a positive proportion of consecutive primes are within 4 times the average spacing between primes authors note this paper was written in 2004 prior to the solution in 8 of the problem considered here in 8 it is shown that 0 while the main result in theorem 1 has now been superseded we believe the method used here is both of interest and future utility in other applications in particular the work of green and tao 12 on arithmetic progressions of primes makes use of proposition 1 of this paper

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

small gaps between primes we introduce a renement of the gpy sieve method for studying prime k tuples and small gaps between primes this renement avoids previous limitations of the method and allows us to show that for each k the prime k tuples conjecture holds for a positive proportion of admissible k tuples in particular lim infn pn m pn 1 for every integer m we also show that lim inf pn 1 pn 600 and if we assume the elliott halberstam conjecture that lim infn pn 1 pn 12 and lim infn pn 2 pn 600

on limit points of the sequence of normalized prime gaps let pn denote the nth smallest prime number and let l denote the set of limit points of the sequence tppn 1 pnq logpnu 8 1 of normalized differences between consecutive primes we show that for k 9 and for any sequence of k nonnegative real numbers 1 2 k at least one of the numbers j i 1 i j k belongs to l it follows that at least 12 5 of all nonnegative real numbers belong to l

prime numbers in logarithmic intervals let x be a large parameter we will first give a new estimate for the i ntegral moments of primes in short intervals of the type p p h where p x is a prime number and h o x then we will apply this to prove that for every 1 2 there exists a positive proportion of primes p x such that the interval p p log x contains at least a prime number as a consequence we improve cheer and goldston s result on the size of real numbers 1 with the property that there is a positive proportion of int egers m x such that the interval m m log x contains no primes we also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers m x such that the interval m m log x contains at least a prime number the last application of these techniques are two theorems the fi rst one unconditional and the second one in which we assume the validity of the riemann hypothesis and of a form of the montgomery pair correlation conjecture on the positive proportion of primes p x such that the interval p p log x contains no primes ams classification 11n05 11a41

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

detection of community overlap according to belief propagation and conflict abstract most existing methods for detection of community overlap cannot balance efficiency and accuracy for large and densely overlapping networks to quickly identify overlapping communities for such networks we propose a new method that uses belief propagation and conflict pcb to occupy communities we first identify triangles with maximal clustering coefficients as seed nodes and sow a new type of belief to the seed nodes then the beliefs explore their territory by occupying nodes with high assent ability the beliefs propagate their strength along the graph to consolidate their territory and conflict with each other when they encounter the same node simultaneously finally the node membership is judged from the belief vectors the pcb time complexity is nearly linear and its space complexity is linear the algorithm was tested in extensive experiments on three real world social networks and three computer generated artificial graphs the experimental results show that pcb is very fast and highly reliable tests on real and artificial networks give excellent results compared with three newly proposed overlapping community detection algorithms

gaussian quadrature rules and numerical examples for strong extensions of mass distribution functions abstract the theory of strong moment problems has provided gaussian quadrature rules for approximate integration with respect to strong distributions in hagler ph d thesis university of colorado boulder 1997 and hagler et al lecture notes in pure and applied mathematics marcel dekker new york in press a transformation of the form v x 1 x x 0 is used to obtain strong mass distribution functions from mass distribution functions this transformation also links the systems of orthogonal polynomials and laurent polynomials and their zeros in this paper we show how the transformation method can be used to obtain the gaussian quadrature rules for strong extensions of mass distribution functions we then provide numerical examples of strong gaussian quadrature approximations to the integrals of elementary functions with respect to selected strong distributions

primes in tuples i we introduce a method for showing that there exist prime numbers which are very close together the method depends on the level of distribution of primes in arithmetic progressions assuming the elliott halberstam conjecture we prove that there are infinitely often primes differing by 16 or less even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart unconditionally we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing that is lim inf n pn 1 pn log pn log 0 we will quantify this result further in a later paper

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

inferring overlapping community structure with degree corrected block model abstract recent research has shown great interest in statistical inference methods for community detection not only in models and algorithms but also in the detectability in this paper we propose a fast community detection algorithm based on the degree corrected block model by introducing a parameter to select the candidate solutions our algorithm is able to detect overlapping communities experiments on a range of networks have achieved state of the art results moreover we show that the algorithm based on the degree corrected block model also suffers the detectability limitation which is in accord with the most recent research on the detectability threshold

a parameter free community detection method based on centrality and dispersion of nodes in complex networks abstract k means is a simple and efficient clustering algorithm to detect communities in networks however it may suffer from a bad choice of initial seeds also called centers that seriously affect the clustering accuracy and the convergence rate additionally in k means the number of communities should be specified in advance till now it is still an open problem on how to select initial seeds and how to determine the number of communities in this study a new parameter free community detection method named k rank d was proposed first based on the fact that good initial seeds usually have high importance and are dispersedly located in a network we proposed a modified pagerank centrality to evaluate the importance of a node and drew a decision graph to depict the importance and the dispersion of nodes then the initial seeds and the number of communities were selected from the decision graph actively and intuitively as the start parameter of k means experimental results on synthetic and real world networks demonstrate the superior performance of our approach over competing methods for community detection

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

primes in tuples ii we introduce a method for showing that there exist prime numbers which are very close together the method depends on the level of distribution of primes in arithmetic progressions assuming the elliott halberstam conjecture we prove that there are infinitely often primes differing by 16 or less even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart unconditionally we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing that is lim inf n 1 pnc1 pn logpn d0

exploring the structural regularities in networks in this paper we consider the problem of exploring structural regularities of networks by dividing the nodes of a network into groups such that the members of each group have similar patterns of connections to other groups specifically we propose a general statistical model to describe network structure in this model a group is viewed as a hidden or unobserved quantity and it is learned by fitting the observed network data using the expectation maximization algorithm compared with existing models the most prominent strength of our model is the high flexibility this strength enables it to possess the advantages of existing models and to overcome their shortcomings in a unified way as a result not only can broad types of structure be detected without prior knowledge of the type of intrinsic regularities existing in the target network but also the type of identified structure can be directly learned from the network moreover by differentiating outgoing edges from incoming edges our model can detect several types of structural regularities beyond competing models tests on a number of real world and artificial networks demonstrate that our model outperforms the state of the art model in shedding light on the structural regularities of networks including the overlapping community structure multipartite structure and several other types of structure which are beyond the capability of existing models

detecting one mode communities in bipartite networks by bipartite clustering triangular abstract in this paper an algorithm is proposed to detect one mode community structures in bipartite networks and to deduce which one mode community structures are weighted after analyzing the topological properties in bipartite networks bipartite clustering triangular is introduced first bipartite networks are projected into two weighted one mode networks by bipartite clustering triangular then all the maximal sub graphs from two one mode weighted networks are extracted and the maximal sub graphs are merged together using a weighted clustering threshold in addition the proposed algorithm successfully finds overlapping vertices between one mode communities experimental results using some real world network data shows that the performance of the proposed algorithm is satisfactory

community structure discovery method based on the gaussian kernel similarity matrix abstract community structure discovery in complex networks is a popular issue and overlapping community structure discovery in academic research has become one of the hot spots based on the gaussian kernel similarity matrix and spectral bisection this paper proposes a new community structure discovery method first by adjusting the gaussian kernel parameter to change the scale of similarity we can find the corresponding non overlapping community structure when the value of the modularity is the largest relatively second the changes of the gaussian kernel parameter would lead to the unstable nodes jumping off so with a slight change in method of non overlapping community discovery we can find the overlapping community nodes finally synthetic data karate club and political books datasets are used to test the proposed method comparing with some other community discovery methods to demonstrate the feasibility and effectiveness of this method

detecting and refining overlapping regions in complex networks with three way decisions the three way representation of communities as a triple of positive boundary and negative regions brings more insight into interpretation of communities the overlapping regions are further refined into 4 macro types and 8 micro types the three way decision strategies are introduced to detect the overlapping communities based on the introduced relation graph a new fitness function is devised to adapt to processing networks with overlapping the refinement of overlapping regions can reveal the different significance to impact forming or changing of communities the identification of communities is crucial to an understanding of the structural and functional properties of a complex network many methods and algorithms have been developed to detect overlapping communities a problem that has not been addressed satisfactorily is the relationship or difference between vertices in overlapping regions during the formation and growth of communities this paper investigates methods that not only detect the overlapping communities but also refine the overlapping regions we give a three way representation of a community by using interval sets and re formalize the problem of community detection as three way clustering we suggest four macro types and eight micro types of vertices to characterize members in overlapping regions for their refinement we propose an overlapping community detection algorithm by classifying the vertices into core vertices bone vertices and trivial vertices the main strategy of this algorithm is to find an initial cluster core to expand the core to a preliminary community according to a new fitness function and to merge trivial vertices based on three way decision strategies the experimental results on both real world social networks and computer generated artificial networks show the effectiveness and efficiency of the proposed methods

bounded gaps between primes it is proved that lim inf n 8 p n 1 p n 7 10 7 where p n is the n th prime our method is a refinement of the recent work of goldston pintz and yildirim on the small gaps between consecutive primes a major ingredient of the proof is a stronger version of the bombieri vinogradov theorem that is applicable when the moduli are free from large prime divisors only but it is adequate for our purpose

diophantine equations in the primes let mathfrak p mathfrak p 1 ldots mathfrak p r be a system of r polynomials with integer coefficients of degree d in n variables mathbf x x 1 ldots x n for a given r tuple of integers say mathbf s a general local to global type statement is shown via classical hardy littlewood type methods which provides sufficient conditions for the solubility of mathfrak p mathbf x mathbf s under the condition that each of the x i s is prime

uncovering the overlapping community structure of complex networks by maximal cliques abstract in this paper a unique algorithm is proposed to detect overlapping communities in the un weighted and weighted networks with considerable accuracy the maximal cliques overlapping vertex bridge vertex and isolated vertex are introduced first all the maximal cliques are extracted by the algorithm based on the deep and bread searching then two maximal cliques can be merged into a larger sub graph by some given rules in addition the proposed algorithm successfully finds overlapping vertices and bridge vertices between communities experimental results using some real world networks data show that the performance of the proposed algorithm is satisfactory

primes in intervals of bounded length in april 2013 yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs of distinct primes which differ by no more than b this is a massive breakthrough makes the twin prime conjecture look highly plausible which can be re interpreted as the conjecture that one can take b 2 and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable the original purpose of this talk was to discuss zhang s extraordinary work putting it in its context in analytic number theory and to sketch a proof of his theorem zhang had even proved the result with b 70 000 000 moreover a co operative team polymath8 collaborating only on line had been able to lower the value of b to 4680 not only had they been more careful in several difficult arguments in zhang s original paper they had also developed zhang s techniques to be both more powerful and to allow a much simpler proof indeed the proof of zhang s theorem that will be given in the write up of this talk is based on these developments in november inspired by zhang s extraordinary breakthrough james maynard dramatically slashed this bound to 600 by a substantially easier method both maynard and terry tao who had independently developed the same idea were able to extend their proofs to show that for any given integer m 1 there exists a bound bm such that there are infinitely many intervals of length bm containing at least m distinct primes we will also prove this much stronger result herein even showing that one can take bm e 8m if zhang s method is combined with the maynard tao set up then it appears that the bound can be further reduced to 576 if all of these techniques could be pushed to their limit then we would obtain b b2 12 so new ideas are still needed to have a feasible plan for proving the twin prime conjecture the article will be split into two parts the first half which appears here we will introduce the work of zhang polymath8 maynard and tao and explain their arguments that allow them to prove their spectacular results as we will discuss zhang s main novel contribution is an estimate for primes in relatively short arithmetic progressions the second half of this article sketches a proof of this result the bulletin article will contain full details of this extraordinary work part 1 primes in short intervals