partial orders on b h in this paper we characterize the sets of all b b h such that a b and the sets of all b b h such that b a where a b h is given and

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

geographic coarse graining analysis of the railway network of china abstract we investigate the detailed empirical analysis of the statistical properties of the railway network of china rnc in space l and space g constructed by geographic coarse graining process the rnc exhibits similar properties in the cumulative distributions of degree and strength in two spaces and it presents the hierarchical structure small world behavior and assortativity areciprocal connection both in space l and space g we also investigate the path length that every train runs the distribution of the railroad length per degree and the optimal distribution of stations

automorphisms of b h with respect to minus partial order abstract let h be an infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h we discuss possible extensions of the concept of the minus partial order from matrix to operator algebras in particular we show that mitra s unified theory of matrix partial orders based on generalized inverses can be modified in such a way that we get a unified approach to partial orders on b h or even more general algebras and rings then we choose the most natural among possible definitions of minus partial order on b h and describe the structure of corresponding automorphisms

consensus dynamics on random rectangular graphs abstract a random rectangular graph rrg is a generalization of the random geometric graph rgg in which the nodes are embedded into a rectangle with side lengths a and b 1 a instead of on a unit square 0 1 2 two nodes are then connected if and only if they are separated at a euclidean distance smaller than or equal to a certain threshold radius r when a 1 the rrg is identical to the rgg here we apply the consensus dynamics model to the rrg our main result is a lower bound for the time of consensus i e the time at which the network reaches a global consensus state to prove this result we need first to find an upper bound for the algebraic connectivity of the rrg i e the second smallest eigenvalue of the combinatorial laplacian of the graph this bound is based on a tight lower bound found for the graph diameter our results prove that as the rectangle in which the nodes are embedded becomes more elongated the rrg becomes a large world i e the diameter grows to infinity and a poorly connected graph i e the algebraic connectivity decays to zero the main consequence of these findings is the proof that the time of consensus in rrgs grows to infinity as the rectangle becomes more elongated in closing consensus dynamics in rrgs strongly depend on the geometric characteristics of the embedding space and reaching the consensus state becomes more difficult as the rectangle is more elongated

emergence of scale free behavior in networks from limited horizon linking and cost trade offs abstract we study network growth from a fixed set of initially isolated nodes placed at random on the surface of a sphere the growth mechanism we use adds edges to the network depending on strictly local gain and cost criteria only nodes that are not too far apart on the sphere may be considered for being joined by an edge given two such nodes the joining occurs only if the gain of doing it surpasses the cost our model is based on a multiplicative parameter that regulates in a function of node degrees the maximum geodesic distance that is allowed between nodes for them to be considered for joining for n nodes distributed uniformly on the sphere and for n within limits that depend on cost related parameters we have found that our growth mechanism gives rise to power law distributions of node degree that are invariant for constant n we also study connectivity and distance related properties of the networks

a duality approach to the symmetry of bernstein sato polynomials of free divisors abstract in this paper we prove that the bernstein sato polynomial of any free divisor for which the d s module d s h s admits a spencer logarithmic resolution satisfies the symmetry property b s 2 b s this applies in particular to locally quasi homogeneous free divisors for instance to free hyperplane arrangements or more generally to free divisors of linear jacobian type we also prove that the bernstein sato polynomial of an integrable logarithmic connection e and of its dual e with respect to a free divisor of linear jacobian type are related by the equality b e s b e s 2 our results are based on the behaviour of the modules d s h s and d s e s h s under duality

range limited centrality measures in complex networks here we present a range limited approach to centrality measures in both nonweighted and weighted directed complex networks we introduce an efficient method that generates for every node and every edge its betweenness centrality based on shortest paths of lengths not longer than ensuremath ell 1 l in the case of nonweighted networks and for weighted networks the corresponding quantities based on minimum weight paths with path weights not larger than w ensuremath ell ensuremath ell ensuremath delta ensuremath ell 1 2 l r ensuremath delta these measures provide a systematic description on the positioning importance of a node edge with respect to its network neighborhoods one step out two steps out etc up to and including the whole network they are more informative than traditional centrality measures as network transport typically happens on all length scales from transport to nearest neighbors to the farthest reaches of the network we show that range limited centralities obey universal scaling laws for large nonweighted networks as the computation of traditional centrality measures is costly this scaling behavior can be exploited to efficiently estimate centralities of nodes and edges for all ranges including the traditional ones the scaling behavior can also be exploited to show that the ranking top list of nodes edges based on their range limited centralities quickly freezes as a function of the range and hence the diameter range top list can be efficiently predicted we also show how to estimate the typical largest node to node distance for a network of n nodes exploiting the afore mentioned scaling behavior these observations were made on model networks and on a large social network inferred from cell phone trace logs ensuremath sim 5 5 ifmmode times else texttimes fi 10 6 nodes and ensuremath sim 2 7 ifmmode times else texttimes fi 10 7 edges finally we apply these concepts to efficiently detect the vulnerability backbone of a network defined as the smallest percolating cluster of the highest betweenness nodes and edges and illustrate the importance of weight based centrality measures in weighted networks in detecting such backbones

corrected mean field model for random sequential adsorption on random geometric graphs a notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non overlapping congruent spheres in the d dimensional euclidean space with d ge 2 spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres due to spatial correlations characterizing the fraction of accepted spheres remains largely intractable we study this fraction by taking a novel approach that compares random sequential adsorption in euclidean space to the nearest neighbor blocking on a sequence of clustered random graphs this random network model can be thought of as a corrected mean field model for the interaction graph between the attempted spheres using functional limit theorems we characterize the fraction of accepted spheres and its fluctuations

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

differential forms on free and almost free divisors we introduce a variant of the usual kahler forms on singular free divisors and show that it enjoys the same depth properties as kahler forms on isolated hypersurface singularities using these forms it is possible to describe analytically the vanishing cohomology and the gauss manin connection in families of free divisors in precise analogy with the classical description for the milnor fibration of an isolated complete intersection singularity due to brieskorn and greuel this applies in particular to the family formula of discriminants of a versal deformation formula of a singularity of a mapping

star partial order on b h abstract let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h in the paper the equivalent definition of the star partial order on b h using selfadjoint idempotent operators is introduced also some properties of the generalized concept of order relations on b h defined with the help of idempotent operators are investigated

random walks on fractals and stretched exponential relaxation stretched exponential relaxation exp t tau beta k is observed in a large variety of systems but has not been explained so far studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7 we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters thus we find that in each dimension the decay exponent beta k is related to well known exponents of the percolation theory in the corresponding flat space we suggest that the stretched exponential behavior observed in many complex systems polymers colloids glasses is due to the fractal character of their configuration space

subgraphs and network motifs in geometric networks many real world networks describe systems in which interactions decay with the distance between nodes examples include systems constrained in real space such as transportation and communication networks as well as systems constrained in abstract spaces such as multivariate biological or economic data sets and models of social networks these networks often display network motifs subgraphs that recur in the network much more often than in randomized networks to understand the origin of the network motifs in these networks it is important to study the subgraphs and network motifs that arise solely from geometric constraints to address this we analyze geometric network models in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes we present analytical solutions for the numbers of all three and four node subgraphs in both directed and nondirected geometric networks we also analyze geometric networks with arbitrary degree sequences and models with a bias for directed edges in one direction scaling rules for scaling of subgraph numbers with system size lattice dimension and interaction range are given several invariant measures are found such as the ratio of feedback and feed forward loops which do not depend on system size dimension or connectivity function we find that network motifs in many real world networks including social networks and neuronal networks are not captured solely by these geometric models this is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information processing functions

on families of square matrices in this paper we classify families of square matrices up to the following natural equivalence thinking of these families as germs of smooth mappings from a manifold to the space of square matrices we allow arbitrary smooth changes of co ordinates in the source and pre and post multiply our family of matrices by generally distinct families of invertible matrices all dependent on the same variables we obtain a list of all the corresponding simple mappings that is those that do not involve adjacent moduli this is a non linear generalisation of the classical notion of linear systems of matrices we also make a start on an understanding of the associated geometry 2000 mathematics subject classification 58k40 58k50 58k60 32s25

extremal optimization for graph partitioning extremal optimization is a new general purpose method for approximating solutions to hard optimization problems we study the method in detail by way of the computationally hard np hard graph partitioning problem we discuss the scaling behavior of extremal optimization focusing on the convergence of the average run as a function of run time and system size the method has a single free parameter which we determine numerically and justify using a simple argument on random graphs our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes with an approximation error decreasing over run time roughly as a power law t 20 4 on geometrically structured graphs the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials but when only the best runs are considered results consistent with theoretical arguments are recovered

adaptive walk on complex networks we investigate the properties of adaptive walks on an uncorrelated fitness landscape which is established in sequence spaces of complex structure in particular we perform numerical simulations of adaptive walks on random graphs and scale free networks for the former we also derive some analytical approximations for the density of local optima of the fitness landscape and the mean length walk we compare our results with those obtained for regular lattices we obtain that the density of local optima decreases as 1 z where z is the mean connectivity for all networks we have investigated in random graphs the mean length walk overline l reaches the asymptotic value e ensuremath 1 for large z which corresponds to the result for regular networks although we could not find an exact estimate we derive an underestimated value for overline l unlike random graphs scale free networks show an upper asymptotic value of overline l

coverage properties of the target area in wireless sensor networks an analytical approximation is developed for the probability of sensing coverage in a wireless sensor network with randomly deployed sensor nodes each having an isotropic sensing area this approximate probability is obtained by considering the properties of the geometric graph in which an edge exists between any two vertices representing sensor nodes with overlapping sensing areas the principal result is an approximation to the proportion of the sensing area that is covered by at least one sensing node given the expected number of nodes per unit area in a two dimensional poisson process the probability of a specified region being completely covered is also approximated simulation results corroborate the probabilistic analysis with low error for any node density the relationship between this approximation and noncoverage by the sensors is also examined these results will have applications in planning and design tools for wireless sensor networks and studies of coverage employing computational geometry

an analytical calculation of neighbourhood order probabilities for high dimensional poissonian processes and mean field models consider that the coordinates of n points are randomly generated along the edges of a d dimensional hypercube random point problem the probability p d n m n that an arbitrary point is the mth nearest neighbour to its own nth nearest neighbour cox probabilities plays an important role in spatial statistics also it has been useful in the description of physical processes in disordered media here we propose a simpler derivation of cox probabilities where we stress the role played by the system dimensionality d in the limit d the distances between pair of points become independent random link model and closed analytical forms for the neighbourhood probabilities are obtained both for the thermodynamic limit and finite size system breaking the distance symmetry constraint drives us to the random map model for which the cox probabilities are obtained for two cases whether a point is its own nearest neighbour or not

relationships between dmp relation and some partial orders let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h the equivalent definition of the dmp relation on b h using operator block matrices is introduced we present a new interpretation of this relation which allows to generalize many known results for matrices to general hilbert spaces regarding the dmp relation and its relationship with the star order and minus partial order also some properties of dmp relation defined with the help of idempotent operators are investigated

finding metastable states in real world time series with recurrence networks in the framework of time series analysis with recurrence networks we introduce a self adaptive method that determines the elusive recurrence threshold and identifies metastable states in complex real world time series as initial step we introduce a way to set the embedding parameters used to reconstruct the state space from the time series we set them as the ones giving the maximum shannon entropy of the diagonal line length distribution for the first simultaneous minima of recurrence rate and shannon entropy to identify metastable states as well as the transitions between them we use a soft partitioning algorithm for module finding which is specifically developed for the case in which a system shows metastability we illustrate our method with a complex time series example finally we show the robustness of our method for identifying metastable states our results suggest that our method is robust for identifying metastable states in complex time series even when introducing considerable levels of noise and missing data points

networks based on collisions among mobile agents we investigate in detail a recent model of colliding mobile agents phys rev lett 96 088702 used as an alternative approach to construct evolving networks of interactions formed by the collisions governed by suitable dynamical rules the system of mobile agents evolves towards a quasi stationary state which is apart small fluctuations well characterized by the de nsity of the system and the residence time of the agents the residence time defines a collision rate and by varying the collision ra te the system percolates at a critical value with the emergence of a giant cluster whose critical exponents are the ones of t wodimensional percolation further the degree and clusteri ng coefficient distributions and the average path length sho w that the network associated with such a system presents non trivial features which depending on the collision rule enables one not only to recover the main properties of standard networks such as exponential random and scale free networks but also to obtain other topological structures namely we show a specific example where the obtained structure has topological fe atures which characterize accurately the structure and evolution of social networks in different contexts ranging from networks of acquaintances to networks of sexual contacts

properties of the star supremum for arbitrary hilbert space operators abstract we give a simple necessary and sufficient condition for the existence of star supremum for two arbitrary operators on a hilbert space it is shown that the results of hartwig 1979 15 and janowitz 1983 18 when applied to the ring of bounded operators on a hilbert space require much simpler conditions we also give a complete answer to a question stated by hartwig and drazin 1982 in 16 regarding the extremal values of range and null space of the star infimum

on mappings preserving the sharp and star orders abstract the present paper is devoted to the study of linear maps preserving certain relations such as the sharp partial order and the star partial order in semisimple banach algebras and c algebras

the minus order and range additivity abstract we study the minus order on the algebra of bounded linear operators on a hilbert space by giving a characterization in terms of range additivity we show that the intrinsic nature of the minus order is algebraic applications to generalized inverses of the sum of two operators to systems of operator equations and to optimization problems are also presented

naming games in two dimensional and small world connected random geometric networks we investigate a prototypical agent based model the naming game on two dimensional random geometric networks the naming game a baronchelli et al j stat mech theory exp 2006 p06014 is a minimal model employing local communications that captures the emergence of shared communication schemes languages in a population of autonomous semiotic agents implementing the naming games with local broadcasts on random geometric graphs serves as a model for agreement dynamics in large scale autonomously operating wireless sensor networks further it captures essential features of the scaling properties of the agreement process for spatiallyembedded autonomous agents among the relevant observables capturing the temporal properties of the agreement process we investigate the cluster size distribution and the distribution of the agreement times both exhibiting dynamic scaling we also present results for the case when a small density of long range communication links are added on top of the random geometric graph resulting in a small world like network and yielding a significantly reduced time to reach global agreement we construct a finite size scaling analysis for the agreement times in this case

coverage processes on spheres and condition numbers for linear programming this paper has two agendas firstly we exhibit new results for coverage processes let p n m alpha be the probability that n spherical caps of angular radius alpha in s m do not cover the whole sphere s m we give an exact formula for p n m alpha in the case alpha in pi 2 pi and an upper bound for p n m alpha in the case alpha in 0 pi 2 which tends to p n m pi 2 when alpha to pi 2 in the case alpha in 0 pi 2 this yields upper bounds for the expected number of spherical caps of radius alpha that are needed to cover s m secondly we study the condition number mathscr c a of the linear programming feasibility problem exists x in mathbb r m 1 ax le0 x ne0 where a in mathbb r n times m 1 is randomly chosen according to the standard normal distribution we exactly determine the distribution of mathscr c a conditioned to a being feasible and provide an upper bound on the distribution function in the infeasible case using these results we show that mathbf e ln mathscr c a le2 ln m 1 3 31 for all n m the sharpest bound for this expectancy as of today both agendas are related through a result which translates between coverage and condition

poisson point processes cascades and random coverings ofr n the generalized random energy model grem is formulated in terms of hierarchies of poisson point processes this allows one to relate the high temperature region with a random covering ofr n

random spherical graphs this work addresses a modification of the random geometric graph rgg model by considering a set of points uniformly and independently distributed on the surface of a d 1 sphere with radius r in a d dimensional euclidean space instead of on an unit hypercube 0 1 d then two vertices are connected by a link if their great circle distance is at most s in the case of d 3 the topological properties of the random spherical graphs rsgs generated by this model are studied as a function of s we obtain analytical expressions for the average degree degree distribution connectivity average path length diameter and clustering coefficient for rsgs by setting r sqrt pi 2 pi we also show the differences between the topological properties of rsgs and those of two dimensional rggs and random rectangular graphs rrgs surprisingly in terms of the average clustering coefficient rsgs look more similar to the analytical estimation for rggs than rggs themselves when their boundary effects are considered we support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for rsgs

generic structure of two dimensional dimages under gaussian blurring for medical imaging problems computer scientists pizer and eberly introduced the core as the analogue for grayscale images of the medial axis it is defined for two dimensional images as the ridge of a medial function defined on 2 1 dimensional scale space in this paper we extend methods from singularity theory to analyze the generic properties of ridges and cores for two dimensional images including the generic transitions which occur for sequences of images varying with a parameter such as time we do so by placing the ridge construction into a larger relative critical set structure which captures properties undetected by the disjoint ridge core segments the genericity properties are established using transversality conditions for associated mappings we introduce a method for establishing transversality conditions for medial functions obtained by gaussian blurring by using a basis of local solutions to the heat equation the method applies to many other properties besides those we consid

cohomology of the complement of a free divisor we prove that if d is a strongly quasihomogeneous free divisor in the stein manifold x and u is its complement then the de rham cohomology of u can be computed as the cohomology of the complex of meromorphic differential forms on x with logarithmic poles along d with exterior derivative the class of strongly quasihomogeneous free divisors introduced here includes free hyperplane arrangements and the discriminants of stable mappings in mather s nice dimensions and in particular the discriminants of coxeter groups

enhancing robustness and immunization in geographical networks we find that different geographical structures of networks lead to varied percolation thresholds although these networks may have similar abstract topological structures thus strategies for enhancing robustness and immunization of a geographical network are proposed using the generating function formalism we obtain an explicit form of the percolation threshold q c for networks containing arbitrary order cycles for three cycles the dependence of q c on the clustering coefficients is ascertained the analysis substantiates the validity of the strategies with analytical evidence

morphology of technological levels in an innovation propagation model we study the dynamical properties of the propagation of innovation on a two dimensional lattice random network scale free network and cayley tree in order to investigate the diversity of technological level we study the scaling property of width w n t which represents the root mean square of the technological level of agents here n is the total number of agents from the numerical simulations we find that the steady state value of w n t w sat n scales as w sat n n 1 2 when the system is in a flat ordered phase for d 2 in the flat ordered phase most of the agents have the same technological level on the other hand when the system is in a smooth disordered phase the value of w sat n does not depend on n these behaviors are completely different from those observed on a one dimensional 1d lattice by considering the effect of the underlying topology on the propagation dynamics for d 2 we also provide a mean field analysis for w sat n which agrees very well with the observed behaviors of w sat n this directly shows that the morphological properties in order disorder transition on a 1d lattice is completely different from that on higher dimensions it also provides an evidence that the upper critical dimension for the roughening transition of the propagation of innovation is d u 2

on the legacy of free divisors discriminants and morse type singularities we investigate when the freeness of a divisor v 0 is inherited by the discriminants for the versal deformations of nonlinear sections of v we introduce morse type singularities for sections and give a criterion for freeness of the discriminant in terms of v 0 generically having morse type singularities this criterion is applied to determine when the bifurcation sets of mappings and smoothings of space curves and complete intersections are free it also explains the failure of freeness for discriminantal arrangements of hyperplanes

deformation of singular lagrangian subvarieties we investigate deformations of lagrangian manifolds with singularities we introduce a complex similar to the de rham complex whose cohomology calculates deformation spaces this cohomology turns out to be constructible in many cases examples of singular lagrangian varieties are presented and deformations are calculated explicitly

nature s way of optimizing we propose a general purpose method for finding high quality solutions to hard optimization problems inspired by self organizing processes often found in nature the method called extremal optimization successively eliminates extremely undesirable components of sub optimal solutions drawing upon models used to simulate far from equilibrium dynamics it complements approximation methods inspired by equilibrium statistical physics such as simulated annealing with only one adjustable parameter its performance proves competitive with and often superior to more elaborate stochastic optimization procedures we demonstrate it here on two classic hard optimization problems graph partitioning and the traveling salesman problem 2000 elsevier science b v all rights reserved

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

complex network view of evolving manifolds we study complex networks formed by triangulations and higher dimensional simplicial complexes representing closed evolving manifolds in particular for triangulations the set of possible transformations of these networks is restricted by the condition that at each step all the faces must be triangles stochastic application of these operations leads to random networks with different architectures we perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties this characterization includes the hausdorff and spectral dimensions of the resulting networks their degree distributions and various structural correlations our results reveal a rich zoo of architectures and geometries of these networks some of which appear to be small worlds while others are finite dimensional with hausdorff dimension equal or higher than the original dimensionality of their simplices the range of spectral dimensions of the evolving triangulations turns out to be from about 1 4 to infinity our models include simplicial complexes representing manifolds with evolving topologies for example an h holed torus with a progressively growing number of holes this evolving graph demonstrates features of a small world network and has a particularly heavy tailed degree distribution

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

growing networks under geographical constraints inspired by the structure of technological weblike systems we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks thus we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors these are the dependence of the spatial interaction range of vertices on the vertex attractiveness or importance within the network and on the inhomogeneous distribution of vertices in space we propose and analyze numerically a simple model based on these generating mechanisms which seems for instance to be able to reproduce known structural features of the internet

spatially embedded random networks many real world networks analyzed in modern network theory have a natural spatial element e g the internet social networks neural networks etc yet aside from a comparatively small number of somewhat specialized and domain specific studies the spatial element is mostly ignored and in particular its relation to network structure disregarded in this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding specifically we model connectivity as dependent on the distance between network nodes our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real world networks but rather to gain intuition for the effects of spatial embedding on network structure nevertheless we are able to demonstrate in a quite general setting some constraints of spatial embedding on connectivity such as the effects of spatial symmetry conditions for scale free degree distributions and the existence of small world spatial networks we also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples

gradient mechanism in a communication network we study the efficiency of the gradient mechanism of message transfer in a two dimensional communication network of regular nodes and randomly distributed hubs each hub on the network is assigned some randomly chosen capacity and hubs with lower capacities are connected to the hubs with maximum capacity the average travel times of single messages traveling on the lattice decrease rapidly as the number of hubs increase the functional dependence of the average travel times on the hub density shows q exponential behavior with a power law tail we also study the relaxation behavior of the network when a large number of messages are created simultaneously at random locations and travel on the network toward their designated destinations for this situation in the absence of the gradient mechanism the network can show congestion effects due to the formation of transport traps we show that if hubs of high betweenness centrality are connected by the gradient mechanism efficient decongestion can be achieved the gradient mechanism is less prone to the formation of traps than other decongestion schemes we also study the spatial configurations of transport traps and propose minimal strategies for their elimination

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

degree correlations in random geometric graphs spatially embedded networks are important in several disciplines the prototypical spatial network we assume is the random geometric graph of which many properties are known here we present new results for the two point degree correlation function in terms of the clustering coefficient of the graphs for two dimensional space in particular with extensions to arbitrary finite dimensions

transversality in families of mappings by a classical result of thom 20 if n and p are smooth manifolds and q is a smooth submanifold of p the set of maps f n p transverse to q is residual in c n p if q is a closed subset of p the set of transverse maps is also open however if u is a further smooth manifold and fu a family of smooth mappings we cannot expect to be able to deform the family to make each fu transverse to q this situation was considered by bruce 2 who showed that for a residual set of maps f n p u we have a for each u u the set of points in n at which fu is not transverse to q is discrete and b at each such point the class of the contact of fu n with q has finite codimension this is the result he established the formal statement given was more explicit but weaker this result is not entirely satisfactory we may hope also for example that each contact class presented is universally unfolded in the family we address this problem and show that this hope is justified under a hypothesis of nice dimensions after recalling mather s theory of topological stability we establish a result valid in all dimensions with topologically versal unfoldings since the proofs of these results involve numerous translations between equivalent versality conditions we begin with a section developing a notation and approach to facilitate such proofs we then consider the corresponding question with the submanifold q replaced by a stratified subset of p this requires an extensive study of theories of contact equivalence relative to a subset of the target good algebraic conditions and criteria are only known under an analyticity hypothesis for some results this needs to be strengthened to a holonomic condition with these preliminaries we obtain a parallel to the theory of the earlier section however a full account of a topological analogue remains beyond our reach

new heuristic percolation criterion for continuum systems a heuristic criterion is introduced and applied for derivation of percolation thresholds in continuum systems the criterion is comprehensive in the sense that it applies to systems with diverse bonding criteria as well as to systems with a correlated distribution of sites its usefulness is manifested by the excellent quantitative agreement between the values obtained and available monte carlo data

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

evaluating transport in irregular pore networks a general approach for investigating transport phenomena in porous media is presented this approach has the capacity to represent various kinds of irregularity in porous media without the need for excessive detail or computational effort the overall method combines a generalized effective medium approximation ema with a macroscopic continuum model in order to derive a transport equation with explicit analytical expressions for the transport coefficients the proposed form of the ema is an anisotropic and heterogeneous extension of kirkpatrick s ema which allows the overall model to account for microscopic alterations in connectivity with the locations of the pores and the orientation and length of the throat as well as macroscopic variations in transport properties a comparison to numerical results for randomly generated networks with different properties is given indicating the potential for this methodology to handle cases that would pose significant difficulties to many other analytical models

distance functions and geodesics on submanifolds of r d and point clouds a theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of bbb r d given by point clouds is introduced and developed in this paper

a and the vanishing topology of discriminants suppose thatf n 0 p 0 is finitely a determined withn p we define a milnor fiber for the discriminant off it is the discriminant of a stabilization off we prove that this discriminant milnor fiber has the homotopy type of a wedge of spheres of dimensionp 1 whose number we denote by f one of the main theorems of the paper is a type result if n p is in the range of nice dimensions in the sense of mather then mu delta f geqq a e codium with equality iff is weighted homogeneous outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization these results are further extended to nonlinear sections of free divisors

the computation of the logarithmic cohomology for plane curves we will give algorithms of computing bases of logarithmic cohomology groups for square free polynomials in two variables

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

star order on jbw algebras abstract the goal of the paper is to extend the star order from associative algebras to non associative jordan banach structures let a be a jbw algebra we define a relation on a as the set of all pairs a b a a such that the range projections of a and b a are orthogonal we show that this relation defines a partial order on a which in the case of the self adjoint part of a von neumann algebra gives the star order after showing basic properties of this order we shall prove the following preserver theorem let a be a jbw algebra without type i 2 direct summand and let be a continuous map from a to b preserving the star order in both directions if for each scalar one has 1 f z where f is a continuous function and z is a central invertible element then there is a unique jordan isomorphism a b such that a f a z moreover we show that if a is a type i n factor where n 2 then the equation above holds for all continuous maps preserving the star order in both directions

topological versus spectral properties of random geometric graphs in this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs rggs a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space rggs g n l consist of n vertices uniformly and independently distributed on the unit square where two vertices are connected by an edge if their euclidian distance is less than or equal to the connection radius l 0 sqrt 2 to evaluate the topological properties of rggs we chose two well known topological indices the randic index r g and the harmonic index h g we characterize the spectral and eigenvector properties of the corresponding randomly weighted adjacency matrices by the use of random matrix theory measures the ratio between consecutive eigenvalue spacings the inverse participation ratios and the information or shannon entropies s g first we review the scaling properties of the averaged measures topological and spectral on rggs then we show that i the averaged scaled indices r g and h g are highly correlated with the average number of nonisolated vertices v g and ii surprisingly the averaged scaled shannon entropy s g is also highly correlated with v g therefore we suggest that very reliable predictions of eigenvector properties of rggs could be made by computing topological indices

signatures of chaotic and stochastic dynamics uncovered with recurrence networks an old and important problem in the field of nonlinear time series analysis entails the distinction between chaotic and stochastic dynamics recently recurrence networks have been proposed as a t

random geometric graphs in high dimension many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks these proximity relations define a so called geometric graph where two nodes are linked if they are sufficiently close to each other random geometric graphs where the positions of nodes are randomly generated in a subset of r d offer a null model to study typical properties of data sets and of machine learning algorithms up to now most of the literature focused on the characterization of low dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high dimensional spaces d 10 2 in this work we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k e g the average number of k cliques this analysis highlights that local observables display different behaviors depending on the chosen ensemble soft random geometric graphs with continuous activation functions converge to the naive infinite dimensional limit provided by erdos renyi graphs whereas hard random geometric graphs can show systematic deviations from it we present numerical evidence that our analytical results exact in infinite dimensions provide a good approximation also for dimension d 10

localization at hyperplane arrangements combinatorics and d modules abstract we describe an algorithm deciding if the annihilating ideal of the meromorphic function 1 f where f 0 defines an arrangement of hyperplanes is generated by linear differential operators of order 1 the algorithm is based on the comparison of two characteristic cycles and uses a combinatorial description of the characteristic cycle of the d module of meromorphic functions with respect to f due to alvarez montaner garcia lopez and zarzuela

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

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

core inverse and core partial order of hilbert space operators abstract the core inverse of matrix is generalized inverse which is in some sense in between the group and moore penrose inverse in this paper a generalization of core inverse and core partial order to hilbert space operator case is presented some properties are generalized and some new ones are proved connections with other generalized inverses are obtained the useful matrix representations of operator and its core inverse are given it is shown that a is less than b under the core partial order if and only if they have specific kind of simultaneous diagonalization induced by appropriate decompositions of hilbert space the relation is also characterized by the inclusion of appropriate sets of generalized inverses the spectral properties of core inverse are also obtained

fluctuations of principal eigenvalues and random scales we investigate the principal dirichlet eigenvalue of the laplacian with soft poissonian obstacles in large boxes of d 2 with the help of our recent version of the method of enlargement of obstacles 18 we derive quantitative confidence intervals for these eigenvalues we also provide less quantitative estimates which however point out the correct size of fluctuations and indicate a stiffness in their behavior in the two dimensional case we derive geometric controls which relate these eigenvalues to certain empty circular droplets our results also have natural applications to the study of the location of minima of certain intermittent random variational problems motivated by 13 17

stable mappings and logarithmic relative symplectic forms let d be the image of a stable weighted homogeneous map f mathbb c n to mathbb c n 1 with rm dim mathbb c rm ker df 0 1 and which is not a trivial deformation of a lower dimensional map by proving a variant of the buchsbaum eisenbud structure theorem for grade 3 gorenstein quotients we show the existence of a form omega in omega 2 log d which restrictst to a non degenerate holomorphic 2 form on the milnor fibres of d experiments with the computer algebra programme macaulay suggest this restriction is closed and is thus a holomorphic symplectic form

analytical framework for recurrence network analysis of time series recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems while there are already many successful applications ranging from medicine to paleoclimatology a solid theoretical foundation of the method has still been missing so far here we interpret an ensuremath varepsilon recurrence network as a discrete subnetwork of a continuous graph with uncountably many vertices and edges corresponding to the system s attractor this step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ensuremath varepsilon in particular we introduce local measures such as the ensuremath varepsilon clustering coefficient mesoscopic measures such as ensuremath varepsilon motif density path based measures such as ensuremath varepsilon betweennesses and global measures such as ensuremath varepsilon efficiency this new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ensuremath varepsilon via a percolation threshold and it shows that estimation can be improved by so called node splitting invariant versions of the measures we finally illustrate the framework for a number of archetypical chaotic attractors such as those of the bernoulli and logistic maps periodic and two dimensional quasiperiodic motions and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments more generally the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints which appear frequently in disciplines ranging from biology to climate science

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

star order automorphisms on the poset of type 1 operators abstract let h be a complex infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h the star partial order is defined by a b if and only if a a a b and a a a b for any a and b in b h we give a type decomposition of operators with respect to star order for any a b h there are unique type 1 operator a 1 which is 0 or the supremum of those rank 1 operators less than a 1 and type 2 operator a 2 which is not greater than any rank 1 operator under star order such that a i a i 1 2 and a a 1 a 2 moreover we determine all automorphisms on the poset of type 1 operators as a consequence we characterize continuous automorphisms on b h

characterizing normal crossing hypersurfaces the objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds it is shown that a divisor hypersurface has normal crossings if and only if it is a free divisor has a radical jacobian ideal and a smooth normalization using k saito s theory of free divisors also a characterization in terms of logarithmic differential forms and vector fields is found finally we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of m granger and m schulze

concentric network symmetry we developed a new node center symmetry measurement for complex networks the symmetry values were compared among six network models as well as four real world networks by using principal component analysis we obtained distinctive signatures of symmetry for each considered network quantification of symmetries in complex networks is typically done globally in terms of automorphisms extending previous methods to locally assess the symmetry of nodes is not straightforward here we present a new framework to quantify the symmetries around nodes which we call connectivity patterns we develop two topological transformations that allow a concise characterization of the different types of symmetry appearing on networks and apply these concepts to six network models namely the erd s renyi barabasi albert random geometric graph waxman voronoi and rewired voronoi real world networks namely the scientific areas of wikipedia the world wide airport network and the street networks of oldenburg and san joaquin are also analyzed in terms of the proposed symmetry measurements several interesting results emerge from this analysis including the high symmetry exhibited by the erd s renyi model additionally we found that the proposed measurements present low correlation with other traditional metrics such as node degree and betweenness centrality principal component analysis is used to combine all the results revealing that the concepts presented here have substantial potential to also characterize networks at a global scale we also provide a real world application to the financial market network

evolution of cooperation on spatially embedded networks in this work we study the behavior of classical two person two strategies evolutionary games on networks embedded in a euclidean two dimensional space with different kinds of degree distributions and topologies going from regular to random and to scale free ones using several imitative microscopic dynamics we study the evolution of global cooperation on the above network classes and find that specific topologies having a hierarchical structure and an inhomogeneous degree distribution such as apollonian and grid based networks are very conducive to cooperation spatial scale free networks are still good for cooperation but to a lesser degree both classes of networks enhance average cooperation in all games with respect to standard random geometric graphs and regular grids by shifting the boundaries between cooperative and defective regions these findings might be useful in the design of interaction structures that maintain cooperation when the agents are constrained to live in physical two dimensional space

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well