lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups this feature is known as community structure and encodes precious information about the organization and the function of the nodes many algorithms have been proposed but it is not yet clear how they should be tested recently we have proposed a general class of undirected and unweighted benchmark graphs with heterogeneous distributions of node degree and community size an increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links which require suitable benchmark graphs for testing in this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built in community structure we also consider the possibility that nodes belong to more communities a feature occurring in real systems such as social networks as a practical application we show how modularity optimization performs on our benchmark

on diversity and coding gains and optimal matrix constellations for space time block codes the design of optimal space time matrix constellations for multiple input multiple output mimo wireless systems with quasistatic rayleigh flat fading channel and coherent maximum likelihood ml decoder is an open problem of great interest in 1998 tarokh et al proposed rank and determinant criteria as guidelines for designing space time codes in this paper we revisit these criteria and explore their fundamental limitations in particular we investigate how they characterize the role of the rank of the code difference matrix in the error probability of the coherent ml decoder and study the shortcomings of such a characterization motivated by the results of our study of the rank and determinant criteria we then address the problem of finding optimal space time constellations in the sense of minimizing the exact average pairwise error probability under the average energy constraint we show that for any fixed space time constellation size and block length orthogonal space time block codes ostbcs are optimal in that sense among all space time codes if the constellation size is not greater than 2k 1 where k is the number of the complex variables of the ostbc we also demonstrate that for such constellation sizes and when the number of receivers is large the optimal constellations are regular simplex furthermore we show that for any constellation size ostbcs are optimal among all linear dispersion ld codes with the same number of complex variables

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

milp formulations for the modularity density maximization problem abstract cluster analysis refers to finding subsets of vertices of a graph called clusters which are more likely to be joined pairwise than vertices in different clusters in the last years this topic has been studied by many researchers and several methods have been proposed one of the most popular is to maximize the modularity which represents the fraction of edges within clusters minus the expected fraction of such edges in a random graph with the same degree distribution however this criterion presents some issues for example the resolution limit i e the difficulty to detect clusters having small sizes in this paper we focus on a recent measure called modularity density which improves the resolution limit issue of modularity the problem of maximizing the modularity density can be described by means of a 0 1 nlp formulation we derive some properties of the optimal solution which will be used to tighten the formulation and we propose some milp reformulations which yield an improvement of the resolution time

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

eigenvalue estimates for schrodinger operators on metric trees we consider schrodinger operators on radial metric trees and prove lieb thirring and cwikel lieb rozenblum inequalities for their negative eigenvalues the validity of these inequalities depends on the volume growth of the tree we show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

efficient modularity density heuristics for large graphs abstract modularity density maximization is a community detection optimization problem which improves the resolution limit degeneracy of modularity maximization this paper presents seven scalable heuristics for modularity density and compares them with literature results from exact mixed integer linear programming and gaod imeme net hain and bmd heuristics the results are also compared with cnm and louvain which are scalable heuristics for modularity maximization the results suggest that our seven heuristics are faster than gaod imeme net hain and bmd modularity density heuristics our experiments also show that some of our heuristics surpassed the objective function value reported by imeme net hain and bmd for some real graphs our seven heuristics were tested with real graphs from the stanford large network dataset collection and the experiments show that they are scalable this feature was confirmed by an amortized complexity analysis which reveals average linear time for three of our heuristics hypothesis tests suggest that four proposed heuristics are state of the art since they are scalable for hundreds of thousands of nodes for the modularity density problem and they find the high objective value partitions for the largest instances ground truth experiments in artificial random graphs were performed and suggest that our heuristics lead to better cluster detection than both cnm and louvain

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

modularity functions maximization with nonnegative relaxation facilitates community detection in networks abstract we show here that the problem of maximizing a family of quantitative functions encompassing both the modularity q measure and modularity density d measure for community detection can be uniformly understood as a combinatoric optimization involving the trace of a matrix called modularity laplacian instead of using traditional spectral relaxation we apply additional nonnegative constraint into this graph clustering problem and design efficient algorithms to optimize the new objective with the explicit nonnegative constraint our solutions are very close to the ideal community indicator matrix and can directly assign nodes into communities the near orthogonal columns of the solution can be reformulated as the posterior probability of corresponding node belonging to each community therefore the proposed method can be exploited to identify the fuzzy or overlapping communities and thus facilitates the understanding of the intrinsic structure of networks experimental results show that our new algorithm consistently sometimes significantly outperforms the traditional spectral relaxation approaches

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

square matrix embeddable space time block codes for complex signal constellations space time block codes for providing transmit diversity in wireless communication systems are considered based on the principles of linearity and unitarity a complete classification of linear codes is given in the case when the symbol constellations are complex and the code is based on a square matrix or restriction of such by deleting columns antennas maximal rate delay optimal codes are constructed within this category the maximal rates allowed by linearity and unitarity fall off exponentially with the number of transmit antennas

nonclassical eigenvalue asymptotics abstract the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 and for x y all of which are non weyl because of infinite phase space volumes are computed along the way a general inequality on quantum partition functions computed in a kind of born oppenheimer approximation is proved

spectral determinants and zeta functions of schrodinger operators on metric graphs a derivation of the spectral determinant of the schrodinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of kostrykin and schrader to formulate the spectral determinant we first derive the spectral zeta function of the schrodinger operator using an appropriate secular equation the result obtained for the spectral determinant is along the lines of the recent conjecture texier 2010 j phys a math theor 43 425203

spectral asymptotics of the laplacian on platonic solids graphs we investigate the high energy eigenvalue asymptotics of quantum graphs consisting of the vertices and edges of the five platonic solids considering two different types of the vertex coupling one is the standard condition and the other is the preferred orientation one introduced in the work by exner and tater phys lett a 382 283 287 2018 the aim is to provide another illustration of the fact that the asymptotic properties of the latter coupling are determined by the vertex parity by showing that the octahedron graph differs in this respect from the other four for which the edges at high energies effectively disconnect and the spectrum approaches the one of the dirichlet laplacian on an interval

asymptotic behavior of jost functions near resonance points for wigner von neumann type potentials in this work are considered radial schrodinger operators v r e where v r a sin br r w r with w r bounded w r o r 2 at infinity a b real the asymptotic behavior of the jost function and the scattering matrix near the resonance point e0 b2 4 are studied if a b then this point may be an eigenvalue embedded in the continuous spectrum the leading behavior of the jost function for all values of a and b were determined somewhat surprisingly situations were found where the jost function becomes singular as e e0 even if e0 is an embedded eigenvalue moreover it was found that the scattering matrix is always discontinuous at e0 except in a few special cases it is also shown that the asymptotics for the jost function and the scattering matrix hold under weaker assumptions on w r in particular an angular momentum term l l 1 r 2 may be incorporated into w r the results were also applied to a whole line problem with a potential v x such that v x 0 for x 0 and v x of wigner von neumann

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

on the trellis complexity of certain binary linear block codes the trellis complexity s c of an n k d code c is investigated in the case where the weights of nonzero codewords in c are confined to d spl middotspl middotspl middot 2d 1 spl cup n it is shown that s c spl ges k 1 furthermore s c k 1 if the code is self complementary if the nonzero weights are confined to d spl middotspl middotspl middot 2d 3 then s c k

the free dirac operator on compact and noncompact graphs the free dirac operator defined on composite one dimensional structures consisting of finitely many half lines and intervals is investigated the influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self adjoint extensions of the dirac operator general relations between the parameters of the extensions and the eigenvalues resp the scattering coefficients are derived and then applied to the cases of a bundle of half lines a point defect a branching line and an eye shaped structure

on the sphere decoding algorithm i expected complexity the problem of finding the least squares solution to a system of linear equations where the unknown vector is comprised of integers but the matrix coefficient and given vector are comprised of real numbers arises in many applications communications cryptography gps to name a few the problem is equivalent to finding the closest lattice point to a given point and is known to be np hard in communications applications however the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known therefore in this paper rather than dwell on the worst case complexity of the integer least squares problem we study its expected complexity averaged over the noise and over the lattice for the sphere decoding algorithm of fincke and pohst we find a closed form expression for the expected complexity both for the infinite and finite lattice it is demonstrated in the second part of this paper that for a wide range of signal to noise ratios snrs and numbers of antennas the expected complexity is polynomial in fact often roughly cubic since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial this suggests that maximum likelihood decoding which was hitherto thought to be computationally intractable can in fact be implemented in real time a result with many practical implications

on agmon metrics and exponential localization for quantum graphs we investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using agmon s method to prove l2 and l infty bounds on the product of an eigenfunction with the exponential of a certain metric a generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line which are of classical liouville green wkb form examples reveal that this estimate can be the best possible but that a more rapid rate of decay is typical when the graph has additional structure in order to understand this fact we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay one of these depends on how the eigenfunction is distributed along a particular chosen path while the other applies to an average of the eigenfunction over edges at a given distance from the root point

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

inverse spectral problems for the sturm liouville operator on a d star graph abstract in this work we consider inverse spectral problems for the sturm liouville differential operator on a d star type graph with standard matching conditions in the internal vertex where the integer d 2 by using the yurko s method yurko 2008 27 yurko 2009 28 we show that 1 if the potential function q j x on a fixed edge e j is prescribed on the interval 2 then the reciprocal of d of the spectrum suffices to determine q j x on the whole interval 0 2 the 2 over d of the spectrum suffices to determine q j x on a fixed edge e j

tanner graphs for group block codes and lattices construction and complexity we develop a tanner graph tg construction for an abelian group block code l with arbitrary alphabets at different coordinates an important application of which is the representation of the label code of a lattice the construction is based on the modular linear constraints imposed on the code symbols by a set of generators for the dual code l as a necessary step toward the construction of a tg for l we devise an efficient algorithm for finding a generating set for l in the process we develop a construction for lattices based on an arbitrary abelian group block code called generalized construction a gca and explore relationships among a group code its gca lattice and their duals we also study the problem of finding low complexity tgs for abelian group block codes and lattices and derive tight lower bounds on the label code complexity of lattices it is shown that for many important lattices the minimal label codes which achieve the lower bounds cannot be supported by cycle free tanner graphs

a factorization of the scattering matrix for the schrodinger equation and for the wave equation in one dimension for the one dimensional schrodinger equation with a potential decaying at both ends of the real axis the scattering matrix of the potential is given explicitly in terms of the scattering matrices corresponding to the fragments of this potential a similar result also holds for the wave equation in a nonhomogeneous nondispersive medium where the wave speed has the same asymptotics at both ends of the real line

quantitative function for community detection we propose a quantitative function for community partition i e modularity density or d value we demonstrate that this quantitative function is superior to the widely used modularity q and also prove its equivalence with the objective function of the kernel k means both theoretical and numerical results show that optimizing the new criterion not only can resolve detailed modules that existing approaches cannot achieve but also can correctly identify the number of communities

column generation algorithms for exact modularity maximization in networks finding modules or clusters in networks currently attracts much attention in several domains the most studied criterion for doing so due to newman and girvan phys rev e 69 026113 2004 is modularity maximization many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality there are few exact algorithms prominent among which is a paper by xu et al eur phys j b 60 231 2007 modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of grotschel and wakabayashi math program 45 59 1989 applied we propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming performance of the four resulting algorithms is compared on problems from the literature instances with up to 512 entities are solved exactly moreover the computing time of previously solved problems are reduced substantially

appendix resonances on a simple graph we study the scattering problem on a graph consisting of a line with a finite length appendix the two parts are coupled through boundary conditions depending on three parameters the motion on the line is free while the appendix supports a potential the appendix bound states give rise to a ladder of resonances we construct the resolvent and solve the corresponding pole condition for a weak coupling in general the condition only admits an analytic solution in particular cases we find the pole positions numerically for a linear potential and show that the poles eventually return to the real axis when the coupling strength increases

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

density length profiles and trellis complexity of lattices the density length profile dcp of a lattice spl lambda is analogous to the dimension length profile of a linear code the dlp is a geometrical invariant of spl lambda that includes the coding gain of spl lambda duality results analogous to those of linear block codes are derived for lattices bounds on the dlp may be derived from bounds on hermite s constants these hold with equality for many dense lattices in turn the dlp lowerbounds the state complexity profile of a minimal trellis diagram for spl lambda in any coordinate system it is shown that this bound can be met for the e sub 8 lattice by a laminated lattice construction with a novel trellis diagram bounds and constructions for other important low dimensional lattices are given two laminated lattice constructions of the leech lattice yield trellis diagrams with maximum state space sizes 1024 and 972

on the number of states bound by one dimensional finite periodic potentials bound states and zero energy resonances of one dimensional finite periodic potentials are investigated by means of levinson s theorem for finite range potentials supporting no bound states a lower bound for the reduced time delay at threshold is derived

good lattice constellations for both rayleigh fading and gaussian channels recent work on lattices matched to the rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency we present a new family of lattice constellations based on complex algebraic number fields which have good performance on rayleigh fading channels some of these lattices also present a reasonable packing density and thus may be used at the same time over a gaussian channel conversely we show that particular versions of the best lattice packings d sub 4 e sub 6 e sub 8 k sub 12 spl lambda sub 16 spl lambda sub 24 constructed from totally complex algebraic cyclotomic fields present better performance over the rayleigh fading channel the practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results

super orthogonal space time trellis codes we introduce a new class of space time codes called super orthogonal space time trellis codes these codes combine set partitioning and a super set of orthogonal space time block codes in a systematic way to provide full diversity and improved coding gain over earlier space time trellis code constructions we also study the optimality of our set partitioning and provide coding gain analysis codes operating at different rates up to the highest theoretically possible rate for different number of states can be designed by using our optimal set partitioning super orthogonal space time trellis codes can provide a tradeoff between rate and coding gain simulation results show more than 2 db improvements over the codes presented in the literature while providing a systematic design methodology

fading resilient super orthogonal space time signal sets can good constellations survive in fading in this correspondence first tier indirect direct discernible constellation expansions are defined for generalized orthogonal designs the expanded signal constellation leading to so called super orthogonal codes allows the achievement of coding gains in addition to diversity gains enabled by orthogonal designs conditions that allow the shape of an expanded multidimensional constellation to be preserved at the channel output on an instantaneous basis are derived this property referred to as fading resilience is first discussed in the context of quasi static fading then generalized to coordinate interleaved transmissions for rapid fading in both cases and despite of piecing together in the latter case a constellation point as a result of coordinate deinterleaving the channel alters neither the relative distances nor the angles between signal points in a fading resilient signal constellation it is shown that nonorthogonal mimo constellations having a geometric structure that is resilient to fading do exist and their further study is warranted

mellin transforms and asymptotics harmonic sums this survey presents a unified and essentially self contained approach to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics discrete probabilistic models and the average case analysis of algorithms it relies on the mellin transform a close relative of the integral transforms of laplace and fourier the method applies to harmonic sums that are superpositions of rather arbitrary harmonics of a common base function its principle is a precise correspondence between individual terms in the asymptotic expansion of an original function and singularities of the transformed function the main applications are in the area of digital data structures probabilistic algorithms and communication theory

weighting links based on edge centrality for community detection abstract link weights have the equally important position as links in complex networks and they are closely associated with each other for the emergence of communities how to assign link weights to make a clear distinction between internal links of communities and external links connecting communities is of vital importance for community detection edge centralities provide a powerful approach for distinguishing internal links from external ones here we first use edge centralities such as betweenness information centrality and edge clustering coefficient to weight links of networks respectively to transform unweighted networks into weighted ones and then a weighted function that both considers links and link weights is adopted on the weighted networks for community detection we evaluate the performance of our approach on random networks as well as real world networks better results are achieved on weighted networks with stronger weights of internal links of communities and the results on unweighted networks outperform that of weighted networks with weaker weights of internal links of communities the availability of our findings is also well supported by the study of granovetter that the weak links maintain the global integrity of the network while the strong links maintain the communities especially in the karate club network all the nodes are correctly classified when we weight links by edge betweenness the results also give us a more comprehensive understanding on the correlation between links and link weights for community detection

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

local modularity measure for network clusterizations many complex networks have an underlying modular structure i e structural subunits communities or clusters characterized by highly interconnected nodes the modularity q has been introduced as a measure to assess the quality of clusterizations q has a global view while in many real world networks clusters are linked mainly locally among each other local cluster connectivity here we introduce a measure of localized modularity lq which reflects local cluster structure optimization of q and lq on the clusterization of two biological networks shows that the localized modularity identifies more cohesive clusters yielding a complementary view of higher granularity

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

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

siso app searches in lattices with tanner graphs an efficient low complexity soft output detector for general lattices is presented based on their tanner graph tg representations closest point searches in lattices can be performed as nonbinary belief propagation on associated tgs soft information output is naturally generated in the process the algorithm requires no backtrack cf classic sphere decoding and extracts extrinsic information a lattice s coding gain enables equivalence relations between lattice points which can be thereby partitioned in cosets total and extrinsic a posteriori probabilities at the detector s output further enable the use of soft detection information in iterative schemes the algorithm is illustrated via two scenarios that transmit a 32 point uncoded super orthogonal so constellation for multiple input multiple output mimo channels carved from an 8 dimensional nonorthogonal lattice d4 d4 it achieves maximum likelihood performance in quasistatic fading and performs close to interference free transmission and identically to list sphere decoding in independent fading with coordinate interleaving and iterative equalization and detection latter scenario outperforms former despite absence of forward error correction coding because the inherent lattice coding gain allows for the refining of extrinsic information the lattice constellation is the same as the one employed in the so space time trellis codes first introduced for 2 2 mimo by ionescu et al then independently by jafarkhani and seshadri algorithmic complexity is log linear in lattice dimensionality versus cubic in classic sphere decoders

on the absence of absolutely continuous spectra for schrodinger operators on radial tree graphs the subject of the paper is schrodinger operators on tree graphs which are radial having the branching number bn at all the vertices at the distance tn from the root we consider a family of coupling conditions at the vertices characterized by bn 1 2 4 real parameters we prove that if the graph is sparse so that there is a subsequence of tn 1 tn growing to infinity in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings but on the the other hand there are cases when the spectrum of such a schrodinger operator can be purely absolutely continuous

regular and pathological eigenvalue behavior for the equation λu vuon the semiaxis we study the behavior of the eigenvalue distribution functionn for the equation u vuon r u 0 0 typically n o 1 2 and moreover 1 2n 1 v dxas 0 we obtain criteria for each of these relations examples are presented where lim 0 1 2n does exist but is not equal to 1 v dx

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

application of the wittrick williams algorithm to the sturm liouville problem on homogeneous trees a structural mechanics analogy the wittrickwilliams ww algorithm was developed over 30 years ago and has been applied with increasing sophistication to problems in structural mechanics ever since much wider applications to a

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

loops and multiple edges in modularity maximization of networks the modularity maximization model proposed by newman and girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges however the applications usually correspond to simple graphs these graphs are compared to a null model where the degree distribution is maintained but edges are placed at random therefore in this null model there will be loops and possibly multiple edges sharp bounds on the expected number of loops and their impact on the modularity are derived then building upon the work of massen and doye but using algebra rather than simulation we propose modified null models associated with graphs without loops but with multiple edges graphs with loops but without multiple edges and graphs without loops nor multiple edges we validate our models by using the exact algorithm for clique partitioning of grotschel and wakabayashi

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

systematic and optimal cyclotomic lattices and diagonal space time block code designs in this correspondence a new and systematic design of cyclotomic lattices with full diversity is proposed by using some algebraic number theory this design provides infinitely many full diversity cyclotomic lattices for a given lattice size based on the packing theory and the concrete form of the design optimal cyclotomic lattices are presented by minimizing the mean transmission signal power for a given minimum diversity product or equivalently maximizing the minimum product for a given mean transmission signal power the newly proposed cyclotomic lattices can be applied to both space time code designs for multi antenna systems and linear precode design for signal space diversity in single antenna systems over fast rayleigh fading channels although there are some cyclotomic lattices space time codes existing in the literature most of them are not optimal

scattering on graphs and one dimensional approximations to n dimensional schrodinger operators in the present article we develop the spectral analysis of schrodinger operators on lattice type graphs for the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices using an explicit expression for the resolvent of lattice type operator we develop in the second section appropriate lippmann schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix it serves as a base for the approximation of the multi dimensional schrodinger operator by a one dimensional operator on the graph in the third section of the paper for given n dimensional schrodinger operators with rapidly decreasing potential we construct a lattice type operator on a cubic graph embedded into rn and show that the original n dimensional scattering problem can be approximated in a p

low density parity check lattices construction and decoding analysis low density parity check codes ldpc can have an impressive performance under iterative decoding algorithms in this paper we introduce a method to construct high coding gain lattices with low decoding complexity based on ldpc codes to construct such lattices we apply construction d due to bos conway and sloane to a set of parity checks defining a family of nested ldpc codes for the decoding algorithm we generalize the application of max sum algorithm to the tanner graph of lattices bounds on the decoding complexity are derived and our analysis shows that using ldpc codes results in low decoding complexity for the proposed lattices the progressive edge growth peg algorithm is then extended to construct a class of nested regular ldpc codes which are in turn used to generate low density parity check lattices using this approach a class of two level lattices is constructed the performance of this class improves when the dimension increases and is within 3 db of the shannon limit for error probabilities of about 10 6 this is while the decoding complexity is still quite manageable even for dimensions of a few thousands

asymptotically good codes have infinite trellis complexity the trellis complexity s c of a block code c is defined as the logarithm of the maximum number of states in the minimal trellis realization of the code the parameter s c governs the complexity of maximum likelihood decoding and is considered a fundamental descriptive characteristic of the code in a number of recent works we derive a new lower bound on s c which implies that asymptotically good codes have infinite trellis complexity more precisely for i spl ges 1 let c sub i be a code over an alphabet of size q of length n sub i rate r sub i and minimum distance d sub i the infinite sequence of codes c sub c sub 2spl middotspl middotspl middot such that n sub ispl rarrspl infin when i spl rarrspl infin is said to be asymptotically good if both r sub i and d sub in sub i are bounded away from zero as i spl rarrspl infin we prove that the complexity s c sub i increases linearly with n sub i in any asymptotically good sequence of codes

modular structure of functional networks in olfactory memory graph theory enables the study of systems by describing those systems as a set of nodes and edges graph theory has been widely applied to characterize the overall structure of data sets in the social technological and biological sciences including neuroscience modular structure decomposition enables the definition of sub networks whose components are gathered in the same module and work together closely while working weakly with components from other modules this processing is of interest for studying memory a cognitive process that is widely distributed we propose a new method to identify modular structure in task related functional magnetic resonance imaging fmri networks the modular structure was obtained directly from correlation coefficients and thus retained information about both signs and weights the method was applied to functional data acquired during a yes no odor recognition memory task performed by young and elderly adults four response categories were explored correct hit and incorrect false alarm fa recognition and correct and incorrect rejection we extracted time series data for 36 areas as a function of response categories and age groups and calculated condition based weighted correlation matrices overall condition based modular partitions were more homogeneous in young than elderly subjects using partition similarity based statistics and a posteriori statistical analyses we demonstrated that several areas including the hippocampus caudate nucleus and anterior cingulate gyrus belonged to the same module more frequently during hit than during all other conditions modularity values were negatively correlated with memory scores in the hit condition and positively correlated with bias scores liberal conservative attitude in the hit and fa conditions we further demonstrated that the proportion of positive and negative links between areas of different modules i e the proportion of correlated and anti correlated areas accounted for most of the observed differences in signed modularity taken together our results provided some evidence that the neural networks involved in odor recognition memory are organized into modules and that these modular partitions are linked to behavioral performance and individual strategies

two body scattering on a graph and application to simple nanoelectronic devices the two body system on a graph with one junction is considered the effective three body scattering problem turns out to be exactly solvable for pointwise interactions additional degrees of freedom corresponding to a dynamics of some structure e g an atomic cluster located in the junction point of common contact of three thin electrodes are considered these degrees of freedom bring effective energy dependent interaction into the effective schrodinger equation in the scattering channel the wave function of the system is constructed in the explicit form using the extension theory methods the obtained results are applied to the qualitative description of a simple three electrode nanoelectronic device the perturbation theory approach based on the analysis of the liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

locally optimal heuristic for modularity maximization of networks community detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics hybrids and in a few papers exact algorithms we consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way this heuristic is compared with the spectral based hierarchical divisive heuristic of newman proc natl acad sci usa 103 8577 2006 and with the hierarchical agglomerative heuristic of clauset newman and moore phys rev e 70 066111 2004 computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges they show that the proposed divisive heuristic gives better results than the divisive heuristic of newman and than the agglomerative heuristic of clauset et al

localization for anderson models on metric and discrete tree graphs we establish spectral and dynamical localization for several anderson models on metric and discrete radial trees the localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies all results are proved under the minimal hypothesis on the type of disorder the random variables generating the trees assume at least two distinct values this level of generality in particular allows us to treat radial trees with disordered geometry as well as schrodinger operators with bernoulli type singular potentials our methods are based on an interplay between graph theoretical properties of radial trees and spectral analysis of the associated random differential and difference operators on the half line

a universal lattice code decoder for fading channels we present a maximum likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver the decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point by judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment

schrodinger operators on regular metric trees with long range potentials weak coupling behavior consider a regular d dimensional metric tree with root o define the schrodinger operator v where v is a non negative symmetric potential on with neumann boundary conditions at o pro

on the trellis complexity of root lattices and their duals trellis complexity of root lattices a sub n d sub n e sub n and their duals is investigated using n the number of distinct paths in a trellis as the measure of trellis complexity for lattices a trellis is called minimal if it minimizes n it is proved that the previously discovered trellis diagrams of some of the above lattices d sub n n odd a sub n 4 spl les n spl les 9 a sub 4 a sub 5 a sub 6 a sub 9 e sub 6 e sub 6 and e sub 7 are minimal we also obtain minimal trellises for a sub 7 and a sub 8 it is known that the complexity n of any trellis of an n dimensional lattice with coding gain spl gamma satisfies n spl ges spl gamma sup n 2 here this lower bound is improved for many of the root lattices and their duals for a sub n and a sub n lattices we also propose simple constructions for low complexity trellises in an arbitrary dimension n and derive tight upper bounds on the complexity of the constructed trellises in some dimensions the constructed trellises are minimal while for some other values of n they have lower complexity than previously known trellises