maximum likelihood extracting unbiased information from complex networks the choice of free parameters in network models is subjective since it depends on what topological properties are being monitored however we show that the maximum likelihood ml principle indicates a unique statistically rigorous parameter choice associated with a well defined topological feature we then find that if the ml condition is incompatible with the built in parameter choice network models turn out to be intrinsically ill defined or biased to overcome this problem we construct a class of safely unbiased models we also propose an extension of these results that leads to the fascinating possibility to extract only from topological data the hidden variables underlying network organization making them no longer hidden we test our method on world trade web data where we recover the empirical gross domestic product using only topological information

statistical mechanics of multiplex networks entropy and overlap there is growing interest in multiplex networks where individual nodes take part in several layers of networks simultaneously this is the case for example in social networks where each individual node has different kinds of social ties or transportation systems where each location is connected to another location by different types of transport many of these multiplexes are characterized by a significant overlap of the links in different layers in this paper we introduce a statistical mechanics framework to describe multiplex ensembles a multiplex is a system formed by n nodes and m layers of interactions where each node belongs to the m layers at the same time each layer ensuremath alpha is formed by a network g ensuremath alpha here we introduce the concept of correlated multiplex ensembles in which the existence of a link in one layer is correlated with the existence of a link in another layer this implies that a typical multiplex of the ensemble can have a significant overlap of the links in the different layers moreover we characterize microcanonical and canonical multiplex ensembles satisfying respectively hard and soft constraints and we discuss how to construct multiplexes in these ensembles finally we provide the expression for the entropy of these ensembles that can be useful to address different inference problems involving multiplexes

effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

uniform concentration compactness for sobolev spaces on variable domains abstract we present a new method for proving existence results in shape optimization problems involving the eigenvalues of the dirichlet laplace operator this method brings together the convergence theory and the concentration compactness principle given a sequence of open sets a n n n in r n not necessarily bounded but of uniformly bounded measure we prove a concentration compactness result in l l 2 r n for the sequence of resolvent operators r a n n n where r a n l 2 r n h 1 0 a n r a n 1

reciprocity of networks with degree correlations and arbitrary degree sequences although most of the real networks contain a mixture of directed and bidirectional reciprocal connections the reciprocity r has received little attention as a subject of theoretical understanding we study the expected reciprocity of networks with arbitrary input and output degree sequences and given 2 node degree correlations by means of statistical ensemble approach we demonstrate that degree correlations are crucial to understand the reciprocity in real networks and a hierarchy of correlation contributions to r is revealed numerical experiments using network randomization methods show very good agreement to our analytical estimations

statistical mechanics of the international trade network analyzing real data on international trade covering the time interval 1950 2000 we show that in each year over the analyzed period the network is a typical representative of the ensemble of maximally random weighted networks whose directed connections bilateral trade volumes are only characterized by the product of the trading countries gdps it means that time evolution of this network may be considered as a continuous sequence of equilibrium states i e a quasistatic process this in turn allows one to apply the linear response theory to make and also verify simple predictions about the network in particular we show that bilateral trade fulfills a fluctuation response theorem which states that the average relative change in imports exports between two countries is a sum of the relative changes in their gdps yearly changes in trade volumes prove that the theorem is valid

hylomorphic solitons for the benjamin ono and the fractional kdv equations abstract this paper concerns with the existence of solitons namely stable solitary waves for the benjamin ono and the fractional kdv equations

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

randomizing world trade i a binary network analysis the international trade network itn has received renewed multidisciplinary interest due to recent advances in network theory however it is still unclear whether a network approach conveys additional nontrivial information with respect to traditional international economics analyses that describe world trade only in terms of local first order properties in this and in a companion paper we employ a recently proposed randomization method to assess in detail the role that local properties have in shaping higher order patterns of the itn in all its possible representations binary or weighted directed or undirected aggregated or disaggregated by commodity and across several years here we show that remarkably the properties of all binary projections of the network can be completely traced back to the degree sequence which is therefore maximally informative our results imply that explaining the observed degree sequence of the itn which has not received particular attention in economic theory should instead become one the main focuses of models of trade

gaussianization of the spectra of graphs and networks theory and applications abstract matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks such as communicability node centrality bipartivity and many more they are also intimately related to the solution of differential equations describing dynamical processes on graphs and networks here we propose a new matrix function based on the gaussianization of the adjacency matrix of a graph this function gives more weight to a selected reference eigenvalue ref which may be located in any region of the graph spectra we show here that this matrix function can be derived from physical models that consider the interactions between nearest and next nearest neighbors in the graph we first obtain a few mathematical results for the trace of this matrix function when ref 1 h 1 for simple graphs as well as for random graphs we also provide a combinatorial interpretation of this index in terms of subgraphs in the graph and in terms of the competition pressure among agents in a complex system finally we apply this index to the study of magnetic properties of molecules emerging due to spin interactions as well as to studying the temporal evolution of the international trade network in the period 1992 2002 in both cases we give a clear phenomenological interpretation of the processes described

upper bound of weak limitation for the gross pitaevskii equation abstract this paper is concerned with the blow up solutions of gross pitaevskii equation we obtain the upper bound of weak limitation for the blow up solutions by using the method of cazenave 2003 3 as well as the concentration compact principle

structural measures for multiplex networks many real world complex systems consist of a set of elementary units connected by relationships of different kinds all such systems are better described in terms of multiplex networks where the links at each layer represent a different type of interaction between the same set of nodes rather than in terms of single layer networks in this paper we present a general framework to describe and study multiplex networks whose links are either unweighted or weighted in particular we propose a series of measures to characterize the multiplexicity of the systems in terms of i basic node and link properties such as the node degree and the edge overlap and reinforcement ii local properties such as the clustering coefficient and the transitivity and iii global properties related to the navigability of the multiplex across the different layers the measures we introduce are validated on a genuinely multiplex data set of indonesian terrorists where information among 78 individuals are recorded with respect to mutual trust common operations exchanged communications and business relationships

on the lowest eigenvalue of general operators for the intersection of two domains e lieb invent math 74 1983 441 448 has proved that ifaandbare two bounded domains in rn then there exists a translation ysuch that 1 a yb 1 a 1 b where 1is the first eigenvalue of the laplacian with dirichlet boundary conditions here we extend this result to elliptic operators in divergence froml div q x c x with mixed dirichlet neumann boundary conditions ona if l a is the corresponding eigenvalue we show that there exists a translation ysuch that l a yb l a 1 b ift q x 2for allx a rn one can further improve the estimate for non isotropic operators where can be large by taking into account rotations ofb in that case a similar inequality holds ift q x 2 whereqis a mean value ofqin different directions

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

measuring the dissimilarity of multiplex networks an empirical study of international trade networks abstract in recent years multiplex networks are becoming a research focus in the domain of complex networks discovering significant correlations between layers in multiplex networks can provide an insight to their structures in this study we propose some methods to measure the dissimilarities of different layers in directed and weighted multiplex networks the dissimilarity is defined on two levels node level and layer level the node dissimilarity is computed based on the distance of the probability distribution of its link weights vectors in different layers and the layer level dissimilarity is the weighted sum of the nodes dissimilarities furthermore the dissimilarity is disintegrated into the connection based dissimilarity and the weight based dissimilarity which represent the topological structure changes and the link weight changes respectively the proposed methods are applied to international trade networks

random walk centrality in interconnected multilayer networks abstract real world complex systems exhibit multiple levels of relationships in many cases they require to be modeled as interconnected multilayer networks characterizing interactions of several types simultaneously it is of crucial importance in many fields from economics to biology and from urban planning to social sciences to identify the most or the less influent nodes in a network using centrality measures however defining the centrality of actors in interconnected complex networks is not trivial in this paper we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies and extend two well known random walk centrality measures the random walk betweenness and closeness centrality to interconnected multilayer networks for each of the measures we provide analytical expressions that completely agree with numerically results

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

enumeration and simulation methods for 0 1 matrices with given marginals data in the form of zero one matrices where conditioning on the marginals is relevant arise in diverse fields such as social networks and ecology directed graphs constitute an important special case an algorithm is given for the complete enumeration of the family of all zero one matrices with given marginals and with a prespecified set of cells with structural zero entries complete enumeration is computationally feasible only for relatively small matrices therefore a more useable monte carlo simulation method for the uniform distribution over this family is given based on unequal probability sampling and ratio estimation this method is applied to testing reciprocity of choices in social networks

an l a stable fourth order rosenbrock method with error estimator the computation of stiff systems of ordinary differential equations requires highly stable processes and this led to the development of l stable rosenbrock methods sometimes called generalized runge kutta or semi implicit runge kutta methods they are linearly implicit and require one jacobian evaluation and at least one matrix factorization per step in this paper we develop some results regarding minimum process configuration i e minimum work per step for a given order as a consequence we then develop an efficient l a stable a 89 fourth order process fifth order locally with a reference formula error estimator similar to that of fehlberg and england

extracting the multiscale backbone of complex weighted networks a large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength in recent years the study of an increasing number of large scale networks has highlighted the statistical heterogeneity of their interaction pattern with degree and weight distributions that vary over many orders of magnitude these features along with the large number of elements and links make the extraction of the truly relevant connections forming the network s backbone a very challenging problem more specifically coarse graining approaches and filtering techniques come into conflict with the multiscale nature of large scale systems here we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks preserving the edges that represent statistically significant deviations with respect to a null model for the local assignment of weights to edges an important aspect of the method is that it does not belittle small scale interactions and operates at all scales defined by the weight distribution we apply our method to real world network instances and compare the obtained results with alternative backbone extraction techniques

world trade web topological properties dynamics and evolution this paper studies the statistical properties of the web of import export relationships among world countries using a weighted network approach we analyze how the distributions of the most important network statistics measuring connectivity assortativity clustering and centrality have coevolved over time we show that all node statistic distributions and their correlation structure have remained surprisingly stable in the last 20 years and are likely to do so in the future conversely the distribution of positive link weights is slowly moving from a log normal density towards a power law we also characterize the autoregressive properties of network statistics dynamics we find that network statistics growth rates are well proxied by fat tailed densities like the laplace or the asymmetric exponential power finally we find that all our results are reasonably robust to a few alternative economically meaningful weighting schemes

decay of heat semigroups in l and applications to nonlinear parabolic problems in unbounded domains we characterize all domains of rn such that the heat semigroup decays in l l or l l1 as t namely we prove that this property is equivalent to the poincare inequality and that it is also equivalent to the solvability of u f in l for all f l in particular under mild regularity assumptions on these properties are equivalent to the geometric condition that has finite inradius next we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains first we consider the quenching problem for singular parabolic equations we prove that the solution in quenches in finite time no matter how small the nonlinearity is if and only if does not fulfill the poincare inequality second for the semilinear heat equation with a power nonlinearity we prove roughly speaking that the trivial solution is stable in l or in l1 if and only if has finite inradius

the focusing singularity of the davey stewartson equations for gravity capillary surface waves abstract we analyze the self focusing of gravity capillary surface waves modeled by the davey stewartson equations this system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two dimensional schrodinger equation with a non local cubic nonlinearity with accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical as in the usual two dimensional cubic schrodinger equation which is the deep water limit of the davey stewartson equations we also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation

semilinear equations on fractal blowups abstract the paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal where imbedding of the energy space into l p is not compact in absence of invariant transformations involved in conventional concentration compactness argument the paper develops convergence reasoning based on the fractal s self similarity

on the cauchy problem for the nonlinear schrodinger equations with time dependent linear loss gain abstract this paper is devoted to the cauchy problem for the nonlinear schrodinger equation with time dependent loss gain which reads i u t u u u i a t u 0 this equation appears in the recent studies of bose einstein condensates and optical systems we obtain some global existence and blow up results which depend on the size of the loss gain coefficient in particular we prove the global existence for the energy critical nonlinearity by scaling and compactness arguments we also discuss asymptotic profiles and concentration properties of blow up solutions

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

a minimum configuration fourth order nonautonomous explicit rosenbrock method for nonstiff differential equations abstract one class of methods for solving nonstiff ordinary differential equations is the so called explicit rosenbrock method a general formulation for these methods is proposed and a procedure for converting autonomous methods to nonautonomous form is described a fourth order method which requires the minimum computational work per step is described

the sharp threshold and limiting profile of blow up solutions for a davey stewartson system abstract the blow up solutions of the cauchy problem for the davey stewartson system which is a model equation in the theory of shallow water waves are investigated firstly the existence of the ground state for the system derives the best constant of a gagliardo nirenberg type inequality and the variational character of the ground state secondly the blow up threshold of the davey stewartson system is developed in r 3 thirdly the mass concentration is established for all the blow up solutions of the system in r 2 finally the existence of the minimal blow up solutions in r 2 is constructed by using the pseudo conformal invariance the profile of the minimal blow up solutions as t t blow up time is in detail investigated in terms of the ground state

travelling wave solutions for the discrete sine gordon equation with nonlinear pair interaction the focus of study is the nonlinear discrete sine gordon equation where the nonlinearity refers to a nonlinear interaction of neighbouring atoms the existence of travelling heteroclinic homoclinic and periodic waves is shown the asymptotic states are chosen such that the action functional is finite the proofs employ variational methods in particular a suitable concentration compactness lemma combined with direct minimisation and mountain pass arguments

vertex intrinsic fitness how to produce arbitrary scale free networks we study a recent model of random networks based on the presence of an intrinsic character of the vertices called fitness the vertex fitnesses are drawn from a given probability distribution density the edges between pairs of vertices are drawn according to a linking probability function depending on the fitnesses of the two vertices involved we study here different choices for the probability distribution densities and the linking functions we find that irrespective of the particular choices the generation of scale free networks is straightforward we then derive the general conditions under which scale free behavior appears this model could then represent a possible explanation for the ubiquity and robustness of such structures

two classes of internally s stable generalized runge kutta processes which remain consistent with an inaccurate jacobian generalized runge kutta processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a jacobian at every step however it is possible to derive processes which are internally s stable when an accurate jacobian is used but still remain consistent and highly stable if an approximate jacobian is used it is shown that these processes require at least as many function evaluations as an explicit runge kutta process of the same order and second and third order processes are developed a second class of generalized runge kutta is introduced which requires that the jacobian be evaluated accurately less than once every step a third order process of this class is developed and all three methods contain an error estimator similar to those of fehlberg or england

a comparison of some ode solvers which require jacobian evaluations a variety of third order ode solvers which have a minimum configuration i e minimum work per step have been numerically tested and the results compared they include implicit and explicit processes and share the property that a jacobian matrix must be evaluated at least once during the integration some of these processes have not been previously described in the literature

existence of nontrivial homoclinic orbits for fourth order difference equations we obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equations by using mountain pass theorem a weak convergence argument and a discrete version of lieb s lemma

on the sign of the mountain pass solution we consider a class of superlinear elliptic problems on a bounded domain in r n i t is known that besides the zero solution such superlinear problems possess at least one positive and one negative solution of mountain pass type as well as a sign changing solution moreover the least energy solutions ground states are among those one sign solutions we study the eect of the nonlinearity on the sign of such ground states and the corresponding role played by the interior geometry of and the geometry of 2001 elsevier science ltd all rights reserved

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

harnack s inequality for a nonlinear eigenvalue problem on metric spaces abstract we prove harnack s inequality for first eigenfunctions of the p laplacian in metric measure spaces the proof is based on the famous moser iteration method which has the advantage that it only requires a weak 1 p poincare inequality as a by product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains

cauchy problem for the l2 critical nonlinear schrodinger equation below h1 abstract in this paper we consider solutions of the l 2 critical focusing nonlinear schrodinger equation on r 2 for the initial data in h s s 1 mainly about the l 2 concentration phenomena for blow up solutions and global well posedness for small initial data

randomizing world trade ii a weighted network analysis based on the misleading expectation that weighted network properties always offer a more complete description than purely topological ones current economic models of the international trade network itn generally aim at explaining local weighted properties not local binary ones here we complement our analysis of the binary projections of the itn by considering its weighted representations we show that unlike the binary case all possible weighted representations of the itn directed and undirected aggregated and disaggregated cannot be traced back to local country specific properties which are therefore of limited informativeness our two papers show that traditional macroeconomic approaches systematically fail to capture the key properties of the itn in the binary case they do not focus on the degree sequence and hence cannot characterize or replicate higher order properties in the weighted case they generally focus on the strength sequence but the knowledge of the latter is not enough in order to understand or reproduce indirect effects

constrained randomization of weighted networks we propose a markov chain method to efficiently generate surrogate networks that are random under the constraint of given vertex strengths with these strength preserving surrogates and with edge weight preserving surrogates we investigate the clustering coefficient and the average shortest path length of functional networks of the human brain as well as of the international trade networks we demonstrate that surrogate networks can provide additional information about network specific characteristics and thus help interpreting empirical weighted networks

detecting spatial homogeneity in the world trade web with detrended fluctuation analysis in a spatially embedded network that is a network where nodes can be uniquely determined in a system of coordinates links weights might be affected by metric distances coupling every pair of nodes dyads in order to assess to what extent metric distances affect relationships link s weights in a spatially embedded network we propose a methodology based on dfa detrended fluctuation analysis dfa is a well developed methodology to evaluate autocorrelations and estimate long range behaviour in time series we argue it can be further extended to spatially ordered series in order to assess autocorrelations in values a scaling exponent of 0 5 uncorrelated data would thereby signal a perfect homogeneous space embedding the network we apply the proposed methodology to the world trade web wtw during the years 1949 2000 and we find in some contrast with predictions of gravity models a declining influence of distances on trading relationships

compactness via symmetrization consider two types of translation invariant functionals i and j on rm and a sequence of functions fn whose corresponding symmetric rearrangements f n are convergent we show that fn themselves converge up to translations if either limn i fn limn i fn or limn j fn limn j fn these compactness results lead to applications in variational problems and stability problems in stellar dynamics

faber krahn type inequalities and uniqueness of positive solutions on metric measure spaces abstract we consider a general class of metric measure spaces equipped with a strongly local regular dirichlet form and provide a lower bound on the hitting time probabilities of the associated hunt process using these estimates we establish i a generalization of the classical lieb s inequality and ii uniqueness of nonnegative super solutions to semi linear elliptic equations on metric measure spaces finally using heat kernel estimates we generalize the local faber krahn inequality recently obtained in 28 to local and non local dirichlet spaces

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

on minimum configuration rosenbrock methods a new structure is proposed for rosenbrock methods for solving stiff ordinary differential equations which facilitates the development of minimum configuration processes minimum computational work per step a fourth order process is described

structure and evolution of the world trade network the world trade web wtw the network defined by the international import export trade relationships has been recently shown to display some important topological properties which are tightly related to the gross domestic product of world countries while our previous analysis focused on the static undirected version of the wtw here we address its full evolving directed description this is accomplished by exploiting the peculiar reciprocity structure of the wtw to recover the directed nature of international trade channels and by studying the temporal dependence of the parameters describing the wtw topology

a variational approach to the isoperimetric inequality for the robin eigenvalue problem the isoperimetric inequality for the first eigenvalue of the laplace operator with robin boundary conditions was recently proved by daners in the context of lipschitz sets this paper introduces a new approach to the isoperimetric inequality based on the theory of special functions of bounded variation sbv we extend the notion of the first eigenvalue 1 for general domains with finite volume possibly unbounded and with irregular boundary and we prove that the balls are the unique minimizers of 1 among domains with prescribed volume

on the asymptotics of nodes ofl2 solutions of schrodinger equations in dimensions 3 let r n br wheren 3 andbr x n x r we investigate the asymptotics of real valued solutions l2 r of the schrodinger equation v e 0 wheree 0 begin gathered mathop lim inf limits r to infty r gamma int limits s r psi 2 d sigma int limits s r psi 2 d sigma 0 and hfill mathop lim inf limits r to infty ln volume d cap b r ln r geqq n 1 2 hfill end gathered results of this type are already non trivial for radial problems with satisfying non radial boundary conditions on r or for excited states of the hydrogen atom if one considers linear combinations of differentl waves

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

traveling waves for nonlinear schrodinger equations with nonzero conditions at infinity for a large class of nonlinear schrodinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity we prove the existence of nontrivial finite energy traveling waves moving with speed c in any space dimension n 3 our results are valid as well for the gross pitaevskii equation and for nls with cubic quintic nonlinearity

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

reconstructing the world trade multiplex the role of intensive and extensive biases in economic and financial networks the strength of each node has always an important economic meaning such as the size of supply and demand import and export or financial exposure constructing null models of networks matching the observed strengths of all nodes is crucial in order to either detect interesting deviations of an empirical network from economically meaningful benchmarks or reconstruct the most likely structure of an economic network when the latter is unknown however several studies have proved that real economic networks and multiplexes topologically differ from configurations inferred only from node strengths here we provide a detailed analysis of the world trade multiplex by comparing it to an enhanced null model that simultaneously reproduces the strength and the degree of each node we study several temporal snapshots and almost 100 layers commodity classes of the multiplex and find that the observed properties are systematically well reproduced by our model our formalism allows us to introduce the static concept of extensive and intensive bias defined as a measurable tendency of the network to prefer either the formation of extra links or the reinforcement of link weights with respect to a reference case where only strengths are enforced our findings complement the existing economic literature on dynamic intensive and extensive trade margins more generally they show that real world multiplexes can be strongly shaped by layer specific local constraints

a microscopic study of the fitness dependent topology of the world trade network abstract previous studies have suggested that the world trade network belongs to the class of static hidden variable models in this article we investigate the microscopic structure of the world trade network that is the hidden variable correlation matrix of the network the hidden variable is defined as a rank ordering of gross domestic products this choice significantly reduces the noise in the statistical analysis found in previous studies the hidden variable correlation matrix that expresses the probability that a trade relationship between two countries of given fitness exists suggests an attachment kernel that at least partially favours trading pairs or dissimilar fitness rather than the purely multiplicative one found previously additionally we provide an in depth look at the data source and reveal that first order results such as the degree distribution exhibit significant qualitative differences depending on the data provider furthermore we shed light on the intertemporal activity of international trade and point out that fluctuations occur mostly between countries with strong dissimilarities of fitness and connectivity

correlations in weighted networks we develop a statistical theory to characterize correlations in weighted networks we define the appropriate metrics quantifying correlations and show that strictly uncorrelated weighted networks do not exist due to the presence of structural constraints we also introduce an algorithm for generating maximally random weighted networks with arbitrary p k s to be used as null models the application of our measures to real networks reveals the importance of weights in a correct understanding and modeling of these heterogeneous systems

concentration properties of blow up solutions and instability results for zakharov equation in dimension two part ii we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right

geometric methods for nonlinear many body quantum systems abstract geometric techniques have played an important role in the seventies for the study of the spectrum of many body schrodinger operators in this paper we provide a formalism which also allows to study nonlinear systems we start by defining a weak topology on many body states which appropriately describes the physical behavior of the system in the case of lack of compactness that is when some particles are lost at infinity we provide several important properties of this topology and use them to write a simple proof of the famous hvz theorem in the repulsive case in the second step we recall the method of geometric localization in fock space as proposed by derezinski and gerard and we relate this tool to our weak topology we then provide several applications we start by studying the so called finite rank approximation which consists in imposing that the many body wavefunction can be expanded using finitely many one body functions we thereby emphasize geometric properties of hartree fock states and prove nonlinear versions of the hvz theorem in the spirit of works of friesecke in the last section we study translation invariant many body systems comprising a nonlinear term which effectively describes the interactions with a second system as an example we prove the existence of the multi polaron in the pekar tomasevich approximation for certain values of the coupling constant

influence of reciprocal edges on degree distribution and degree correlations reciprocal edges represent the lowest order cycle possible to find in directed graphs without self loops representing also a measure of feedback between vertices it is interesting to understand how reciprocal edges influence other properties of complex networks in this paper we focus on the influence of reciprocal edges on vertex degree distribution and degree correlations we show that there is a fundamental difference between properties observed on the static network compared to the properties of networks which are obtained by simple evolution mechanism driven by reciprocity we also present a way to statistically infer the portion of reciprocal edges which can be explained as a consequence of feedback process on the static network in the rest of the paper the influence of reciprocal edges on a model of growing network is also presented it is shown that our model of growing network nicely interpolates between barabasi albert ba model for undirected and the ba model for directed networks