absolutely continuous spectrum of schrodinger operators with slowly decaying and oscillating potentials the aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional schrodinger operator is essentially supported by 0 infi

bound states and the szegő condition for jacobi matrices and schrodinger operators for jacobi matrices with a n 1 1 n n b n 1 n n we study bound states and the szeg condition we provide a new proof of nevai s result that if 12 the szeg condition holds which works also if one replaces 1 n by cos n we show that if 0 0 and 12 the szeg condition fails we also show that if 1 and are small enough 2 8 2 1 24 will do then the jacobi matrix has finitely many bound states for 0 large it has infinitely many

growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

generalizations of the clustering coefficient to weighted complex networks the recent high level of interest in weighted complex networks gives rise to a need to develop new measures and to generalize existing ones to take the weights of links into account here we focus on various generalizations of the clustering coefficient which is one of the central characteristics in the complex network theory we present a comparative study of the several suggestions introduced in the literature and point out their advantages and limitations the concepts are illustrated by simple examples as well as by empirical data of the world trade and weighted coauthorship networks

absolutely continuous spectrum of the schrodinger operator with a potential representable as a sum of three functions with special properties we consider a family of schrodinger operators depending on a real parameter and prove that the absolutely continuous spectrum covers the positive half line for almost every value of the parameter the potentials in this model are representable as a sum of three terms this sum is neither fast decaying nor it rapidly changes its sign or has derivatives with respect to the angular variables

on spectral theory for schrodinger operators with operator valued potentials abstract given a complex separable hilbert space h we consider differential expressions of the type d 2 d x 2 v x with x a or x r here v denotes a bounded operator valued potential v b h such that v is weakly measurable and the operator norm v b h is locally integrable we consider self adjoint half line l 2 realizations h in l 2 a d x h associated with assuming a to be a regular endpoint necessitating a boundary condition of the type sin u a cos u a 0 indexed by the self adjoint operator b h in addition we study self adjoint full line l 2 realizations h of in l 2 r d x h in either case we treat in detail basic spectral theory associated with h and h including weyl titchmarsh theory green s function structure eigenfunction expansions diagonalization and a version of the spectral theorem

local world evolving networks with tunable clustering abstract we propose an extended local world evolving network model including a triad formation tf step in the process of network evolution random fluctuation in the number of new edges is involved we derive analytical expressions for degree distribution clustering coefficient and average path length our model can unify the generic properties of real life networks scale free degree distribution high clustering and small inter node separation moreover in our model the clustering coefficient is tunable simply by changing the expected number of tf steps after a single local preferential attachment step

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

on the absolutely continuous spectrum of multi dimensional schrodinger operators with slowly decaying potentials we consider a multi dimensional schrodinger operator v in l2 r d and find conditions on the potential v which guarantee that the absolutely continuous spectrum of this operator is essentially supported by the positive real line we prove some results which go beyond the case l1 l p with p d

distinct clusterings and characteristic path lengths in dynamic small world networks with identical limit degree distribution many real world networks belong to a particular class of structures known as small world networks that display short distance between pair of nodes in this paper we introduce a simple family of growing small world networks where both addition and deletion of edges are possible by tuning the deletion probability q t the model undergoes a transition from large worlds to small worlds by making use of analytical or numerical means we determine the degree distribution clustering coefficient and average path length of our networks surprisingly we find that two similar evolving mechanisms which provide identical degree distribution under a reciprocal scaling as t goes to infinity can lead to quite different clustering behaviors and characteristic path lengths it is also worth noting that farey graphs constitute the extreme case q t 0 of our random construction

fgsn fuzzy granular social networks model and applications abstract social network data has been modeled with several approaches including sociogram and sociomatrices which are popular and comprehensive similar to these we have developed here a novel modeling technique based on granular computing theory and fuzzy neighborhood systems which provides a uniform framework to represent social networks in this model a social network is represented with a collection of granules fuzzy sets are used for defining the granules the model is named fuzzy granular social network fgsn familiar measures of networks viz degree betweenness embeddedness and clustering coefficient are redefined in the context of this new framework two measures namely entropy of fgsn and energy of granules are defined to quantify the uncertainty involved in fgsn arising from fuzziness in the relationships of actors experimental results demonstrate the applicability of the model in two well known problems of social networks namely target set selection and community detection with comparative studies

sum rules and the szegő condition for orthogonal polynomials on the real line we study the case sum rules especially c 0 for general jacobi matrices we establish situations where the sum rule is valid applications include an extension of shohat s theorem to cases with an infinite point spectrum and a proof that if lim n a n 1 and lim nb n exist and 2 then the szeg condition fails

an omega d log n d lower bound for broadcast in radio networks we show that for any randomized broadcast protocol for radio networks there exists a network in which the expected time to broadcast a message is omega d log n d where d is the diameter of the network and n is the number of nodes this implies a tight lower bound of omega d log n for any d le n 1 varepsilon where varepsilon 0 is any constant

the degree related clustering coefficient and its application to link prediction abstract link prediction plays a significant role in explaining the evolution of networks however it is still a challenging problem that has been addressed only with topological information in recent years based on the belief that network nodes with a great number of common neighbors are more likely to be connected many similarity indices have achieved considerable accuracy and efficiency motivated by the natural assumption that the effect of missing links on the estimation of a node s clustering ability could be related to node degree in this paper we propose a degree related clustering coefficient index to quantify the clustering ability of nodes unlike the classical clustering coefficient our new coefficient is highly robust when the observed bias of links is considered furthermore we propose a degree related clustering ability path dcp index which applies the proposed coefficient to the link prediction problem experiments on 12 real world networks show that our proposed method is highly accurate and robust compared with four common neighbor based similarity indices common neighbors cn adamic adar aa resource allocation ra and preferential attachment pa and the recently introduced clustering ability ca index

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

clustering in complex directed networks many empirical networks display an inherent tendency to cluster i e to form circles of connected nodes this feature is typically measured by the clustering coefficient cc the cc originally introduced for binary undirected graphs has been recently generalized to weighted undirected networks here we extend the cc to the case of binary and weighted directed networks and we compute its expected value for random graphs we distinguish between ccs that count all directed triangles in the graph independently of the direction of their edges and ccs that only consider particular types of directed triangles e g cycles the main concepts are illustrated by employing empirical data on world trade flows

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

generating random graphs with tunable clustering coefficients abstract most real world networks exhibit a high clustering coefficient the probability that two neighbors of a node are also neighbors of each other we propose two algorithms conf and throw that take triangle and single edge degree sequences as input and generate a random graph with a target clustering coefficient we analyze them theoretically for the case of a regular graph conf generates a random graph with the input degree sequence and the clustering coefficient anticipated from the input experimental results match quite well with the anticipated clustering coefficient except for highly dense graphs in which case the experimental clustering coefficient is higher than the anticipated value for throw the degree sequence and the clustering coefficient of the generated graph varies from the input however it maintains the expected degree distribution and the clustering coefficient of the generated graph can also be predicted using analytical results experiments show that for throw the results match quite well with the analytical results typically only information about degree distribution is available we also propose an algorithm deg that takes degree sequence and clustering coefficient as input and generates a graph with the same properties experiments show results for deg that are quite similar to those for conf

absolutely continuous spectrum of schrodinger operators with potentials slowly decaying inside a cone abstract for a large class of multi dimensional schrodinger operators it is shown that the absolutely continuous spectrum is essentially supported by 0 we require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non negative bounded potential outside the cone in particular we do not require the existence of wave operators the result and method of proof extends previous work by laptev naboko and safronov

clustering coefficients of large networks abstract let g be a network with n nodes and eigenvalues 1 2 n then g is called an n d network if it is d regular and max 2 3 n it is shown that if g is an n d network and o d the average clustering coefficient c g of g satisfies c g d n for large d we show that this description also holds for strongly regular graphs and erd s renyi graphs although most real world networks are not constructed theoretically we find that many of them have c g close to d n and many close to 1 2 n d 1 d d 1 where d is the average degree of g and 2 is the average of the numbers of common neighbors over all non adjacent pairs of nodes

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

improved lower bound for deterministic broadcasting in radio networks we consider the problem of deterministic broadcasting in radio networks when the nodes have limited knowledge about the topology of the network we show that for every deterministic broadcasting protocol there exists a network of radius 2 for which the protocol takes at least w n 1 2 rounds for completing the broadcast our argument can be extended to prove a lower bound of w nd 1 2 rounds for broadcasting in radio networks of radius d this resolves one of the open problems posed in kowalski and pelc 2004 24 where the authors proved a lower bound of w n 1 4 rounds for broadcasting in constant diameter networks we prove the new lower bound for a special family of radius 2 networks each network of this family consists of o n components which are connected to each other via only the source node at the heart of the proof is a novel simulation argument which essentially says that any arbitrarily complicated strategy of the source node can be simulated by the nodes of the networks if the source node just transmits partial topological knowledge about some component instead of arbitrary complicated messages to the best of our knowledge this type of simulation argument is novel and may be useful in further improving the lower bound or may find use in other applications

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

time complexity of radio broadcasting adaptiveness vs obliviousness and randomization vs determinism we consider the time of broadcasting in ad hoc radio networks modeled as undirected graphs in such networks every node knows only its own label and a linear bound on the number of nodes but is unaware of the topology of the network or even of its own neighborhood our aim is to study to what extent the availability of two important characteristics of a broadcasting algorithm influences optimal broadcasting time these characteristics are adaptiveness and randomization our contribution is establishing upper and lower bounds on optimal broadcasting time for three classes of algorithms adaptive deterministic oblivious randomized and oblivious deterministic in two cases we present tight bounds and in one case a small gap remains we show that for deterministic adaptive algorithms time n is required even for n node networks of constant diameter this lower bound is strongest possible since linear time algorithms are known and hence establishes optimal time n for this class for oblivious randomized algorithms we show an upper bound o n min d log n and a lower bound n on optimal expected broadcasting time in n node networks of diameter d finally for oblivious deterministic algorithms we show matching upper and lower bounds n min d n on optimal broadcasting time our results imply that enforcing obliviousness has at least as strong negative impact on broadcasting time as enforcing determinism and that algorithms having both these features are strictly less efficient than those having only one of them