graph optimisation problems and the potts glass the np complete problems of partitioning and colouring of random graphs with p partitions and colours respectively are mapped onto the statistical mechanical problem of p state potts glasses an estimate of the cost functions of these optimisation problems has been derived using the potts glass mean field theory this estimate applies to dense graphs in the thermodynamic limit an exact expression for the cost function in the large p limit is given

correlation structure of the landscape of the graph bipartitioning problem the relaxation of random walks and the autocorrelation function on the landscape of the graph bipartitioning problem are calculated

statistical mechanics of the knapsack problem the knapsack problem is an np complete combinatorial optimization problem with inequality constraints using the replica method of statistical physics we study the space of its solutions for a large problem size it turns out that this problem is closely related to the theory of the binary perceptron

two combinatorial models with identical statics yet different dynamics motivated by the problem of sorting we introduce two simple combinatorial models with distinct hamiltonians yet identical spectra and hence partition function and show that the local dynamics of these models is very different after a deep quench one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima

bipartitioning of random graphs of fixed extensive valence the procedure of fu and anderson ibid vol 19 p 1605 1986 for the application of statistical mechanics to the problem of bipartitioning random graphs is extended to graphs of fixed extensive valence the cost function is shown to be independent of whether the valence is locally or globally constrained

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

quenched random graphs spin models on quenched random graphs are related to many important optimization problems we give a new derivation of their mean field equations that elucidates the role of the natural order parameter in these models

a monte carlo study of scaling and universality in 2d directed percolation monte carlo data for the pair correlation function is obtained on five different lattices for the directed percolation problem in two dimensions scaling is tested and found to hold beyond 20 lattice spacings for all models evidence is found to support the universality of the scaling function the values of the exponents gamma v sub and vperpendicular to v sub are extracted from the data the results for all models are summarised by gamma v sub 1 305 or 0 007 and vperpendicular to v sub 0 629 or 0 010 comparison is made with estimates derived by other means

on the stability of the replica symmetric theory of the matching problem the longitudinal sector a replica symmetric theory of the matching problem was given by orland 1985 and by mezard and paris 1985 here the authors investigate the stability of this solution in the zero temperature limit they find that the longitudinal spectrum is continuous and lies in the interval 0 lambda 2 they conclude that the solution of mezard and parisi is stable to longitudinal fluctuation they give the explicit form of the longitudinal eigenvectors the method used lends itself to dealing with replica symmetry broken solutions where necessary

monte carlo simulations of the ising spin glass on lattices with finite connectivity the authors have simulated the ising spin glass model on a random lattice with a finite average coordination number and also on the bethe lattice with various different boundary conditions in particular they have calculated the overlap function p q for two independent samples for the random lattice the results are consistent with a spin glass transition above which p q converges to a dirac delta function for large n number of sites and below which p q has in addition a long tail similar to previous results obtained for the infinite range model for the bethe lattice they obtain results in the interior by discarding the two outer shells of the cayley tree when calculating the thermal averages for fixed uncorrelated boundary conditions p q seems to converge to a delta function even below the spin glass transition whereas on a closed lattice correlated boundary conditions p q has a long tail similar to its behaviour in the random lattice case

dynamics of an ising spin glass on the bethe lattice we study the dynamical low temperature behaviour of the ising spin glass on the bethe lattice starting from glauber dynamics we propose a cavity such as an ansatz that allows for the treatment of the slow low temperature part of dynamics assuming a continuous phase transition and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature the theory is formulated in terms of correlation response functions of arbitrary order they can however be broken down completely to products of pair functions depending on two time arguments only for binary couplings j i a spin glass solution is found which approaches the corresponding solution for the sk model in the limit of high connectivity for more general distributions p j no stable or marginal solution of this type appears to exist the nature of the low temperature phase in this more general case is unclear

disappearance of spurious states in analog associative memories we show that symmetric n mixture states when they exist are almost never stable in autoassociative networks with threshold linear units only with a binary coding scheme we could find a limited region of the parameter space in which either 2 mixture or 3 mixture states are stable attractors of the dynamics

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

the random quadratic assignment problem the quadratic assignment problem qap is one of the most difficult of all com binatorial optimization problems here we use an abbreviated application of the statistical mechanics replica method to study the asymptotic behavior of instances in which the en tries of at least one of the two matrices that specify the problem are chosen from a random distributionp s urprisingly the qap has not been studied before using the replica method despite the fact that the qap was first proposed over 50 years ago and the replica method was developed over 30 years ago we find simple forms forc min andc max t he costs of the minimal and maximum solutions respectively notable features of our results are the sym metry of the results forc min andc max and their dependence onponly through its mean and standard deviation independent of the details ofp

data clustering using a model granular magnet we present a new approach to clustering based on the physical properties of an inhomogeneous ferromagnet no assumption is made regarding the underlying distribution of the data we assign a potts spin to each data point and introduce an interaction between neighboring points whose strength is a decreasing function of the distance between the neighbors this magnetic system exhibits three phases at very low temperatures it is completely ordered all spins are aligned at very high temperatures the system does not exhibit any ordering and in an intermediate regime clusters of relatively strongly coupled spins become ordered whereas different clusters remain uncorrelated this intermediate phase is identified by a jump in the order parameters the spin spin correlation function is used to partition the spins and the corresponding data points into clusters we demonstrate on three synthetic and three real data sets how the method works detailed comparison to the performance of other techniques clearly indicates the relative success of our method

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

analysis of parallel finite element algorithm based on three linearization methods for the steady incompressible mhd flow abstract based on full domain partition parallel finite element algorithm based on three linearizations is designed to solve the stationary incompressible magnetohydrodynamics mhd in every algorithm each processor is in charge of a subproblem which is linearized by three options stokes type newton s and picard s iterations each subproblem is defined in the whole domain with vast majority of the degrees of freedom associated with the particular subdomain the error estimates of these proposed algorithms with respect to iterative step and small mesh sizes are analyzed finally some numerical experiments are provided to show the validity and high efficiency of our algorithms

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

a statistical mechanics analysis of the set covering problem the dependence of the optimal solution average cost of the set covering problem on the density of 1 s of the incidence matrix and on the number of constraints p is investigated in the limit where the number of items n goes to infinity the annealed approximation is employed to study two stochastic models the constant density model where the elements of the incidence matrix are statistically independent random variables and the karp model where the rows of the incidence matrix possess the same number of 1 s lower bounds for are presented in the case that p scales with ln n and is of order 1 as well as in the case that p scales linearly with n and is of order 1 n it is shown that in the case that p scales with exp n and is of order 1 the annealed approximation yields exact results for both models

the travelling salesman problem on a randomly diluted lattice the authors study the problem of a travelling salesman who must visit a randomly chosen subset of sites of a d dimensional lattice the average length of the shortest path per chosen site is alpha q where 1 q is the density of chosen sites for a triangular lattice they show that alpha q differs from 1 only by terms of order q5 for the square lattice they show that to first order in q optimal paths can be found from the dynamics of a model of a one dimensional gas of kinks and antikinks the authors find alpha q or l 3 1 q 1 2 as q tends to 1

an expandable local and parallel two grid finite element scheme an expandable local and parallel two grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of poisson equation compared with the usual local and parallel finite element schemes the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of cpus convergence results based on h 1 and l 2 a priori error estimation of the scheme are obtained which show that the scheme can reach the optimal convergence orders within ln h 2 or ln h two grid iterations if the coarse mesh size h and the fine mesh size h are properly configured in 2 d or 3 d case respectively some numerical results are presented at the end of the paper to support our analysis

on network bipartivity systems with two types of agents with a preference for heterophilous interaction produces networks that are more or less close to bipartite we propose two measures quantifying the notion of bipartivity the two measures one well known and natural but computationally intractable one computationally less complex but also less intuitive are examined on model networks that continuously interpolates between bipartite graphs and graphs with many odd circuits we find that the bipartivity measures increase as we tune the control parameters of the test networks to intuitively increase the bipartivity and thus conclude that the measures are quite relevant we also measure and discuss the values of our bipartivity measures for empirical social networks constructed from professional collaborations internet communities and field surveys here we find as expected that networks arising from romantic online interaction have high and professional collaboration networks have low bipartivity values in some other cases probably due to low average degree of the network the bipartivity measures cannot distinguish between romantic and friendship oriented interaction

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

postprocessing fourier galerkin method for the navier stokes equations a full discrete two level postprocessing fourier galerkin scheme for the unsteady navier stokes equations with a periodic boundary condition is proposed in this paper by defining a new projection the interaction between the large and small eddies is reflected by the associated space splitting to some extent therefore a weakly coupled system of the large and small eddies is obtained stability and error estimates for the weakly coupled two level scheme are established in the paper the proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse level subspace by solving the coarse level standard galerkin equation and the matrix of the linear algebraic equations arising in each fine level time stepping is a sparse matrix especially for large scale computations as a result the new scheme saves a lot of cpu time and memory in comparison with the standard fourier galerkin method for deriving an approximation of prescribed accuracy

towards the solution of the graph bipartitioning problem the author presents the replica symmetric solution of the graph bipartitioning problem he points out the possibility of many solutions even with the replica symmetry assumption he also comments on the possibility of constructing a model with an arbitrary finite number of transition temperatures

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

exact solution of potts random energy models with weak connectivity and of the many states potts spin glass the author generalises derrida s random energy model to the potts spin glass with p spin interactions in the limit of large p he solves the model exactly for the cases of strong and weak connectivity in the latter case he shows how to solve the model using replicas and obtains a solution for the value of the set of order parameters qr1 rsalpha 1 alpha s he then derives the large q limit of the ordinary q state potts spin glass on a lattice with finite connectivity

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

cost distributions in large combinatorial optimisation problems the authors have examined the cost distribution of locally optimal solutions in certain combinatorial optimisation problems this distribution is found to be peaked about a value characteristics of the algorithm involved with a width that decreases with the system size n it is shown that the distribution of costs epsilon is of the form exp ng epsilon and that g epsilon is self averaging consequently estimation of the optimal cost is best achieved by curve fitting to g epsilon possible forms for g epsilon are proposed

local and parallel finite element post processing scheme for the stokes problem abstract a local and parallel finite element post processing scheme based on partition of unity method is proposed and analyzed in this paper for the stokes problem firstly a standard galerkin finite element method on a relatively coarse grid is used to obtain the approximation of the lower frequency components secondly the higher frequency components are computed on fine grids by some local and parallel procedure to post process the standard galerkin approximation the motivation of the proposed local and parallel finite element post processing scheme is based on the superposition principle finally to eliminate the effect of the dirichlet boundary conditions which are imposed on the internal artificial boundaries a global coarse grid correction is done to improve the l 2 accuracy of the approximation

neutrality in fitness landscapes the interplay of ruggedness and neutrality in fitness landscapes plays an important role in explaining the dynamics of evolutionary adaptation while various measures of ruggedness correlation functions adaptive walks or the density of local optima are reasonably well understood and models for constructing landscapes with a desired degree of ruggedness are readily available very little is known about neutrality we introduce the notion of additive random landscapes as a framework for tuning both neutrality and ruggedness at once and we develop a formalism that allows the explicit computation of the most salient parameters that are associated with neutrality in landscapes of this type

finite connectivity attractor neural networks we study a family of diluted attractor neural networks with a finite average number of symmetric connections per neuron as in finite connectivity spin glasses their equilibrium properties are described by order parameter functions for which we derive an integral equation in replica symmetric approximation a bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin glass transition lines in the phase diagram the line separating the retrieval phase from the spin glass phase is calculated at zero temperature all phase transitions are found to be continuous

optimization of output load in a junction model an optimization problem of minimizing the output load of a junction model with multiple input and output channels is solved by the replica method simulation results show that for the case of extensive connectivity the replica solution obtained by assuming a vanishing solution space is satisfactory however for the case of intensive connectivity the degeneracy of the solution space causes the simulation results to deviate from the replica solution of vanishing volume

exact value for the average optimal cost of the bipartite traveling salesman and two factor problems in two dimensions we show that the average cost for the traveling salesman problem in two dimensions which is the archetypal problem in combinatorial optimization in the bipartite case is simply related to the average cost of the assignment problem with the same euclidean increasing convex weights in this way we extend a result already known in one dimension where exact solutions are avalaible the recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction overline e n frac 1 pi log n for large number of points 2n as a byproduct of our analysis also the loop covering problem has the same optimal average cost we also explain why this result cannot be extended at higher dimensions we numerically check the exact predictions

when are networks truly modular abstract the study of cluster or community structure of complex networks contributes to the understanding of networks at a functional level in many networks latent classes of nodes are suspected which manifest themselves as communities i e groups of nodes with a high link density among the nodes of the same class and low link density between nodes of different classes community detection algorithms are used to infer these classes e g by finding a partition of the network which maximizes a quality function such as the network modularity q m newman m girvan finding and evaluating community structure in networks phys rev e 69 2004 026113 however it is known from numerical experiments that even purely random networks display intrinsic modularity and may be partitioned yielding high values of q extending on our earlier results j reichardt s bornholdt statistical mechanics of community detection phys rev e 74 2006 016110 the mapping of the community detection problem onto finding the ground state of a spin glass is exploited in order to derive analytical expressions for the expected modularity in random graphs and assess the theoretical limits to community detection the results are independent of any specific community detection algorithm and allow for differentiation between modularity arising purely due to the search process in the large configuration space of possible partitionings on the one hand or due to the actual presence of different classes of nodes on the other hand

series expansion studies of directed percolation estimates of the correlation length exponents series expansion methods have been used to estimate the longitudinal and transverse correlation length exponents for directed bond percolation on the square lattice the central estimate for the ratio of these exponents phi 1 579 or 0 006 is in good agreement with the value obtained by the phenomenological scaling renormalisation group method

potts models on feynman diagrams we investigate numerically and analytically potts models on thin random graphs generic feynman diagrams using the idea that such models may be expressed as the limit of a matrix model the thin random graphs in this limit are locally tree like in distinction to the fat random graphs that appear in the planar feynman diagram limit more familiar from discretized models of two dimensional gravity the interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree like structures such as the bethe lattice q state potts models display a first order transition in the mean field for q 2 so the thin graph potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation such discontinuous transitions also appear in multiple ising models on thin graphs and may have implications for the use of the replica trick in spin glass models on random graphs

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

statistical mechanics of an np complete problem subset sum we study the statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics the problem is a generalization of the number partitioning problem which is also an npcomplete problem and has been studied in the physics literature the asymptotic expressions for the number of solutions are obtained these results applied to the number partitioning problem as a special case are compared with those which were previously obtained by a different method we discuss the limit of applicability of the techniques of statistical mechanics to the present problem

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

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

maximum independent sets on random regular graphs we determine the asymptotics of the independence number of the random d regular graph for all d geq d 0 it is highly concentrated with constant order fluctuations around n alpha c log n for explicit constants alpha d and c d our proof rigorously confirms the one step replica symmetry breaking heuristics for this problem and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs

tangent space correction method for the galerkin approximation based on two grid finite element abstract a novel fully discrete two grid finite element method for certain semi linear parabolic equations is presented in this paper the new scheme is based on two different finite element subspaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h h nonlinearity is treated in the coarse grid subspace by solving the standard galerkin equation while in the fine grid subspace only a linear equation has to be solved differing from the usual two grid method the splitting of the coarse grid subspace and its related fine grid incremental subspace is based on a new projection in which sense the incremental subspace is closely identified with the tangent space of certain operator on each time step with linear finite element discretization the stability and error estimate results for such new scheme are derived the results show that the difference between this new approximation and the fine grid standard galerkin approximation in h 1 norm is of the order h 4 d 2 where d 1 2 3 is the space dimension

detecting network communities by propagating labels under constraints we investigate the recently proposed label propagation algorithm lpa for identifying network communities we reformulate the lpa as an equivalent optimization problem giving an objective function whose maxima correspond to community solutions by considering properties of the objective function we identify conceptual and practical drawbacks of the label propagation approach most importantly the disparity between increasing the value of the objective function and improving the quality of communities found to address the drawbacks we modify the objective function in the optimization problem producing a variety of algorithms that propagate labels subject to constraints of particular interest is a variant that maximizes the modularity measure of community quality performance properties and implementation details of the proposed algorithms are discussed bipartite as well as unipartite networks are considered

replica symmetric solutions of the graph bipartitioning problem with fixed finite valence using the replica method the author presents the replica symmetric solutions of the graph bipartitioning problem with fixed finite valence with the constraint sigma ni 1 si 0 strictly enforced the author is able to find another solution which gives a lower cost function than that given by the spin glass solution

local and parallel finite element algorithms for the stokes problem based on two grid discretizations some local and parallel finite element algorithms for the stokes problem are proposed and analyzed in this paper these algorithms are motivated by the observation that for a solution to the stokes problem low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure one technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape regular grids

complexity and ultradiffusion the authors present the exact solution to the problem of ultradiffusion in an arbitrary hierarchical space they derive rigorous upper and lower bounds for the dynamic exponent describing the decay of the autocorrelation function they show that the upper bound is saturated by both uniformly and randomly multifurcating hierarchical trees and identify a class of highly unbalanced trees that saturate the lower bound they conclude that the speed of relaxation is a measure of the complexity or lack of self similarity of the underlying tree they point out that complexity may be revealed by the temperature dependence of the dynamic exponent and in particular by the nature of the transition from exponential to power law decay

directed percolation mean field theory and series expansions for some two dimensional lattices the shape of percolation clusters in the directed percolation problem is studied within mean field theory low density series expansions are obtained for bond and site problems on the square and triangular lattices these are used to test a scaling formula for the pair connectedness which involves two lengths xi sub rho and xi perpendicular to rho determination of the exponents nu sub and nu perpendicular to for these lengths shows that all four problems are in the same universality class and the values support the hyperscaling relation beta 1 2 d nu perpendicular to nu sub gamma an independent argument is given for this relation

a statistical analysis of the knapsack problem we investigate the dependence of the multi knapsack objective function on the knapsack capacities and on the number of capacity constraints p in the case when all n objects are assigned the same profit value and the weights are uniformly distributed over the unit interval a rigorous upper bound to the optimal profit is obtained employing the annealed approximation and then compared with the exact value obtained through the lagrangian relaxation method the analysis is restricted to the regime where n goes to infinity and p remains finite

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

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

replica symmetry breaking of the ising spin glass with finite connectivity the authors consider the ising spin glass on a lattice with finite connectivity m 1 using the recent method of large connectivity expansions the free energies at finite temperatures are evaluated numerically at first step replica symmetry breaking rsb the 1 m expansion at finite temperature diverges as t0 as in the replica symmetric case but is well behaved for a larger range the 1 m expansion at zero temperature is also calculated at higher steps of rsb an expression for the free energy expansion is derived for an arbitrary step of rsb up to order 1 m explicit numerical values at second step rsb are obtained from the results they speculate that the divergence in the finite temperature expansion might disappear for the exact infinite step rsb solution they also compare their results with the simulation results of graph bipartitioning

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

parallel dynamics of disordered ising spin systems on finitely connected random graphs we study the dynamics of bond disordered ising spin systems on random graphs with finite connectivity using generating functional analysis rather than disorder averaged correlation and response functions as for fully connected systems the dynamic order parameter is here a measure which represents the disorder averaged single spin path probabilities given external perturbation field paths in the limit of completely asymmetric graphs our macroscopic laws close already in terms of the single spin path probabilities at zero external field for the general case of arbitrary graph symmetry we calculate the first few time steps of the dynamics exactly and we work out numerical and analytical procedures for constructing approximate stationary solutions of our equations simulation results support our theoretical predictions

cut size statistics of graph bisection heuristics we investigate the statistical properties of cut sizes generated by heuristic algorithms which solve the graph bisection problem approximately on an ensemble of sparse random graphs we find empirically that the distribution of the cut sizes found by local algorithms becomes peaked as the number of vertices in the graphs becomes large evidence is given that this distribution tends toward a gaussian whose mean and variance scales linearly with the number of vertices of the graphs given the distribution of cut sizes associated with each heuristic we provide a ranking procedure that takes into account both the quality of the solutions and the speed of the algorithms this procedure is demonstrated for a selection of local graph bisection heuristics

correlation length isotropy and meta stable states abstract a landscape is rugged if it has many local optima if it gives rise to short adaptive walks and if it exhibits a rapidly decreasing pair correlation function and hence if it has a short correlation length the correlation length conjecture allows to estimate the number of meta stable states from the correlation length provided the landscape is typical isotropy originally introduced as a geometrical condition on the covariance matrix of a random field can be re interpreted as a maximum entropy condition that lends a precise meaning to the notion of a typical landscape the xy hamiltonian which violates isotropy only to a relatively small extent is an ideal model for investigating the influence of anisotropies numerical estimates for the number of local optima and predictions obtained from the correlation length conjecture indeed show deviations that increase with the extent of anisotropies in the model

adaptive variational multiscale methods based on a posteriori error estimation energy norm estimates for elliptic problems we develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution the fine scale is approximated by a sum of solutions to decoupled localized problems which are solved numerically on a fine grid partition of a patch of coarse grid elements the sizes of the patches of elements may be increased to control the error caused by localization we derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters the coarse grid mesh size the fine grid mesh size and the sizes of the patches based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters finally we show how the method works in practice by presenting various numerical examples

application of statistical mechanics to combinatorial optimization problems the chromatic number problem and q partitioning of a graph methods of statistical mechanics are applied to two important np complete combinatorial optimization problems the first is the chromatic number problem which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color the second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections both models are mapped into a frustrated potts model which is related to theq state potts spin glass for the first problem we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation the quenched model is also discussed for the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq 3 and 4 potts spin glass using parisi s replica symmetry breaking we also perform some numerical simulations to test the theoretical result and obtain very good agreement

replica symmetry breaking in finite connectivity systems a large connectivity expansion at finite and zero temperature parisi replica symmetry breaking is extended to random bond systems with finite fixed connectivity m 1 the free energy f is explicitly calculated for large m within the first stage of symmetry breaking at finite temperature t the 1 m expansion is found to diverge as t to o indeed direct evaluation at zero temperature shows that the large m expansion is in powers of 1 m introduction of symmetry breaking brings the value of f close approximately 1 for 10 m 20 to numerical estimates of banavar et al the same techniques apply to systems with average finite connectivity

on the survey propagation equations in random constraint satisfiability problems in this paper we study the existence of a solution of the survey propagation equations for a given instance of a random problem in the framework of constraint satisfiability we consider the concrete examples of k satisfiability and coloring we conjecture that when the number of variables goes to infinity the solution of the survey propagation equations for a given instance can be obtained by finding the supposed unique solution of the corresponding equations on an infinite tree we conjecture that the survey propagation equations on a random infinite tree have a unique solution in the suitable range of parameters we also present analytic arguments that indicate that the survey propagation equations do have solutions in the satisfiable phase for simplicity of notation the argument is presented in the case of the coloring problem the same results extend to other optimization problems where exist configurations that have cost zero i e in the satisfiable phase on a random graph the solutions of th

an aim and one step newton method for the navier stokes equations abstract in this paper we investigate a two level finite element approximation to the navier stokes equations by means of a new approximate inertial manifold aim then we construct a new aim based numerical scheme and show that the convergence rate of the new approximation obtained by this aim method is better than the double of the convergence rate of the standard galerkin finite element solution in addition we also show that the new aim scheme is equivalent to a one step newton iterative scheme for the navier stokes equations in a suitable hilbert space

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

q partitioning of graphs with finite coordination number the np complete optimisation problem of q partitioning of graphs with finite connectivity is discussed using the effective local field method the authors obtain the local field distributions with and without the continuous part and use them to calculate the ground state energies of the three state potts spin glass on lattices with finite coordination number equal to three this in turn gives analytic results for the optimal cost of the 3 partitioning of graphs they perform simulations of actual 3 partitions of random graphs to compare with their theoretical results and obtain very good agreement ways for further improvement of the estimates are discussed

a boundary multiplier fictitious domain method for the steady incompressible navier stokes equations we analyze the error of a fictitious domain method with boundary lagrange multiplier it is applied to solve a non homogeneous steady incompressible navier stokes problem in a domain with a multiply connected boundary the interior mesh in the fictitious domain and the boundary mesh are independent up to a mesh length ratio

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

monte carlo structural optimization in discrete variables with annealing algorithm the paper describes the basic ideas of monte carlo annealing algorithms for structural optimization with discrete design parameters the algorithm generates randomly a set of design parameters with probability depending on the objective function and given by the boltzmann gibbs distribution in this model the search for the global minimum is simulated by a relaxation process of the statistical mechanical system with the hamiltonian proportional to the objective function the rate of the convergence of the method and its dependence upon the annealing probability are discussed numerical implementation of the method for the weight optimization of the ten bar planar cantilever truss is presented the results of numerical simulation are compared with those obtained by the dual methods the principal conjecture is that the method is fairly efficient and has great potential for applicaton in engineering design

graph optimization problems on a bethe lattice thep partitioning andp coloring problems on a bethe lattice of coordination numberz are analyzed it is shown that these two np complete optimization problems turn out to be equivalent to finding the ground state energy ofp state potts models with frustration numerical calculation of the cost function of both problems are carried out for several values ofz andp in the case ofp 2 the results are identical to those obtained by mezard and parisi for the case of the bipartitioning problem a numerical upper bound to the chromatic number is found for several values ofz

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

partitioning and modularity of graphs with arbitrary degree distribution we solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method we find the cut size to scale universally with k in contrast earlier results studying the problem in graphs with a poissonian degree distribution had found a scaling with kfu and anderson j phys a 19 1605 1986 our results also generalize to the problem of q partitioning they can be used to find the expected modularity q newman and girvan phys rev e 69 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms given a graph or network gn m of n nodes and m edges the problem of graph partitioning is finding a partition of the nodes into q equal sized parts such that the number of edges connecting different parts the cut size is minimal the solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing very large scale integrated chip design vlsi and data mining 1 due to the problem being np complete 2 i e no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph only heuristics exist to approximate a solution furthermore one cannot determine in polynomial time whether a solution found is indeed optimal the only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size

cooperative updating in the hopfield model we propose a new method for updating units in the hopfield model with this method two or more units change at the same time so as to become the lowest energy state among all possible states since this updating algorithm is based on the detailed balance equation convergence to the boltzmann distribution is guaranteed if our algorithm is applied to finding the minimum energy in constraint satisfaction and combinatorial optimization problems then there is a faster convergence than those with the usual algorithm in the neural network this is shown by experiments with the travelling salesman problem the four color problem the n queen problem and the graph bi partitioning problem in constraint satisfaction problems for which earlier neural networks are effective in some cases our updating scheme works fine even though we still encounter the problem of ending up in local minima our updating scheme has a great advantage compared with the usual updating scheme used in combinatorial optimization problems also we discuss parallel computing using our updating algorithm

probabilistic analysis of the number partitioning problem given a sequence of n positive real numbers the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized in the case in which the s are statistically independent random variables uniformly distributed in the unit interval this np complete problem is equivalent to the problem of finding the ground state of an infinite range random antiferromagnetic ising model we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large n limit furthermore we calculate analytically the fraction of metastable states i e states that are stable against all single spin flips and found that it vanishes like