multifractal behavior of linear polymers in disordered media the scaling behavior of linear polymers in disordered media modeled by self avoiding random walks saws on the backbone of two and three dimensional percolation clusters at their critical concentrations p c is studied all possible saw configurations of n steps on a single backbone configuration are enumerated exactly we find that the moments of order q of the total number of saws obtained by averaging over many backbone configurations display multifractal behavior i e different moments are dominated by different subsets of the backbone this leads to generalized coordination numbers mu q and enhancement exponents gamma q which depend on q our numerical results suggest that the relation mu 1 p c mu between the first moment mu 1 and its regular lattice counterpart mu is valid

algebraic graph theory without orientation abstract let g be an undirected graph with vertices v 1 v 2 v and edges e1 e2 e let m be the matrix whose ijth entry is 1 if ej is a link incident with vi 2 if ej is a loop at vi and 0 otherwise the matrix obtained by orienting the edges of a loopless graph g i e changing one of the 1 s to a 1 in each column of m has been studied extensively in the literature the purpose of this paper is to explore the substructures of g and the vector spaces associated with the matrix m without imposing such an orientation we describe explicitly bases for the kernel and range of the linear transformation from r to r defined by m our main results are determinantal formulas using the unoriented laplacian matrix mmt to count certain spanning substructures of g these formulas may be viewed as generalizations of the matrix tree theorem the point of view adopted in this paper also gives rise to a matroid structure on the edges of g analogous to the cycle matroid and its dual in this setting the analogue of a spanning forest can have components with one odd cycle and the analogue of an edge cut has the property that its removal creates a new bipartite component

rayleigh ritz based substructure synthesis for flexible multibody systems this paper is concerned with the modeling of flexible multibody systems by a rayleigh ritz based substructure synthesis method so that certain advantages can be accrued by using the variational approach to derive the eigenvalue problem as with the classical rayleigh ritz method if the admissible functions used to represent the motion of the substructures are not chosen properly convergence can suffer this paper presents a new substructure synthesis method with superior convergence characteristics achieved by representing the motion by means of a recently developed class of functions namely the class of quasi compari son functions this improved convergence is shown to be related to improved approximation of both the differential equations and the natural boundary conditions the theory is demonstrated by means of a numerical example i introduction t his investigation is concerned with the modeling of flexible multibody systems many structures such as fixedwing aircraft helicopters flexible spacecraft flexible robots etc can be modeled as assemblages of interacting flexible bodies a method proposed by hurty1 2 in the early 1960s and known as component mode synthesis consists of modeling the motion of the individual substructures referred to as components by means of component modes he uses three types of modes to represent the motion of a substructure rigid body modes constraint modes and normal modes to define constraint modes hurty divides the constraints into two classes statically determinate and redundant the constraint modes are equal in number to the number of redundant constraints and are defined by producing a unit displacement on each redundant constraint in turn with all other constraints fixed the normal modes represent the modes of vibration of the components with all constraints fixed an approach by craig and hampton3 differs from that of hurty 1 2 mainly in the selection of the component modes indeed craig and hampton suggest two types of component modes boundary modes providing for displacements and rotations at points along the substructure boundaries and related to the constraint modes of hurty and substructure modes corresponding to completely restrained boundaries the substructures modeled individually in refs 1 3 are made to act together as a single structure by eliminating the redundant generalized coordinates arising from the fact that displacements at points common to two adjacent substructures are included twice in the overall problem formulation once for each substructure the elimination process is based on the use of constraint equations resulting from the enforcement of compatibility conditions which amounts to saying that the displacement of a boundary point shared by two substructures is the same in another approach to the problem proposed by

cones of closed alternating walks and trails abstract consider a graph whose edges have been colored red and blue assign a nonnegative real weight to every edge so that at every vertex the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges the set of all such assignments forms a convex polyhedral cone in the edge space called the alternating cone the integral respectively 0 1 vectors in the alternating cone are sums of characteristic vectors of closed alternating walks respectively trails we study the basic properties of the alternating cone determine its dimension and extreme rays and relate its dimension to the majorization order on degree sequences we consider whether the alternating cone has integral vectors in a given box and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices the latter problem called alternating reachability is solved in a companion paper along with related results

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

self avoiding walks in quenched random environments the self avoiding walk in a quenched random environment is studied using realspace and field theoretic renormalization and flory arguments these methods indicate that the system is described for d de by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path this fixed point is characterized by scaling laws for the size of the walk l n with n the number of steps and the fluctuations in the free energy af l the bound 1 co pore and suggests that this inequality holds for d 2 and 3 although pure cannot be excluded particularly for d 2 for d d c there is a transition between strong and weak disorder phases at which pure the strong disorder fixed point for saws on percolation clusters is discussed the analogy with directed walks is emphasized

critical percolation clusters in seven dimensions and on a complete graph we study critical bond percolation on a seven dimensional hypercubic lattice with periodic boundary conditions 7d and on the complete graph cg of finite volume number of vertices v we numerically confirm that for both cases the critical number density n s v of clusters of size s obeys a scaling form n s v s n over s v d f with identical volume fractal dimension d f 2 3 and exponent 1 1 d f 5 2 we then classify occupied bonds into bridge bonds which includes branch and junction bonds and nonbridge bonds a bridge bond is a branch bond if and only if its deletion produces at least one tree deleting branch bonds from percolation configurations produces leaf free configurations whereas deleting all bridge bonds leads to bridge free configurations composed of blobs it is shown that the fraction of nonbridge biconnected bonds vanishes n cg 0 for large cgs but converges to a finite value n 7d 0 0061931 7 for the 7d hypercube further we observe that while the bridge free dimension d bf 1 3 holds for both the cg and 7d cases the volume fractal dimensions of the leaf free clusters are different d lf 7d 0 669 9 2 3 and d lf cg 0 3337 17 1 3 on the cg and in 7d the whole leaf free and bridge free clusters all have the shortest path volume fractal dimension d min 1 3 characterizing their graph diameters we also study the behavior of the number and the size distribution of leaf free and bridge free clusters for the number of clusters we numerically find the number of leaf free and bridge free clusters on the cg scale as lnv while for 7d they scale as v for the size distribution we find the behavior on the cg is governed by a modified fisher exponent 1 while for leaf free clusters in 7d it is governed by fisher exponent 5 2 the size distribution of bridge free clusters in 7d displays two scaling behavior with exponents 4 and 1 the probability distribution p c 1 v dc 1 of the largest cluster of size c 1 for whole percolation configurations is observed to follow a single variable function p over x dx with x c 1 v d f for both cg and 7d up to a rescaling factor for the variable x the probability functions for cg and 7d collapse on top of each other within the entire range of x the analytical expressions in the x 0 and x limits are further confirmed our work demonstrates that the geometric structure of high dimensional percolation clusters cannot be fully accounted for by their complete graph counterparts

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

random graph asymptotics on high dimensional tori ii volume diameter and mixing time for critical bond percolation on high dimensional torus this paper proves sharp lower bounds on the size of the largest cluster removing a logarithmic correction in the lower bound in heydenreich and van der hofstad comm math phys 270 2 335 358 2007 this improvement finally settles a conjecture by aizenman nuclear phys b 485 3 551 582 1997 about the role of boundary conditions in critical high dimensional percolation and it is a key step in deriving further properties of critical percolation on the torus indeed a criterion of nachmias and peres ann probab 36 4 1267 1286 2008 implies appropriate bounds on diameter and mixing time of the largest clusters we further prove that the volume bounds apply also to any finite number of the largest clusters finally we show that any weak limit of the largest connected component is non degenerate which can be viewed as a significant sign of critical behavior the main conclusion of the paper is that the behavior of critical percolation on the high dimensional torus is the same as for critical erd s renyi random graphs

bond and site percolation in three dimensions we simulate the bond and site percolation models on a simple cubic lattice with linear sizes up to l 512 and estimate the percolation thresholds to be p c mathrm bond 0 248 phantom rule 0 16em 0ex 811 phantom rule 0 16em 0ex 82 10 and p c mathrm site 0 311 phantom rule 0 16em 0ex 607 phantom rule 0 16em 0ex 7 2 by performing extensive simulations at these estimated critical points we then estimate the critical exponents 1 ensuremath nu 1 1410 15 ensuremath beta ensuremath nu 0 477 phantom rule 0 16em 0ex 05 15 the leading correction exponent y i ensuremath 1 2 2 and the shortest path exponent d mathrm min 1 3756 3 various universal amplitudes are also obtained including wrapping probabilities ratios associated with the cluster size distribution and the excess cluster number we observe that the leading finite size corrections in certain wrapping probabilities are governed by an exponent ensuremath approx ensuremath 2 rather than y i ensuremath approx ensuremath 1 2

recursive sampling of random walks self avoiding walks in disordered media the author presents a new algorithm for simulating random walks which is simple versatile and efficient it uses recursive function calls and can be used to obtain unbiased samples with any given length distribution this makes it particularly useful in disordered geometries where the effective connectivity constant is not known a priori when applying it to self avoiding random walks on two dimensional media with quenched randomness he finds evidence for at least two new renormalization group fixed points

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

enumeration of self avoiding walks on the square lattice we describe a new algorithm for the enumeration of self avoiding walks on the square lattice using up to 128 processors on a hp alpha server cluster we have enumerated the number of self avoiding walks on the square lattice to length 71 series for the metric properties of mean square end to end distance mean square radius of gyration and mean square distance of monomers from the end points have been derived to length 59 an analysis of the resulting series yields accurate estimates of the critical exponents and confirming predictions of their exact values likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations finally we report on an analysis giving compelling evidence that the leading non analytic correction to scaling exponent 1 3 2

geometric structure of percolation clusters we investigate the geometric properties of percolation clusters by studying square lattice bond percolation on the torus we show that the density of bridges and nonbridges both tend to 1 4 for large system sizes using monte carlo simulations we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two arm exponent we then classify bridges into two types branches and junctions a bridge is a branch iff at least one of the two clusters produced by its deletion is a tree starting from a percolation configuration and deleting the branches results in a leaf free configuration whereas deleting all bridges produces a bridge free configuration although branches account for 43 of all occupied bonds we find that the fractal dimensions of the cluster size and hull length of leaf free configurations are consistent with those for standard percolation configurations by contrast we find that the fractal dimensions of the cluster size and hull length of bridge free configurations are given by the backbone and external perimeter dimensions respectively we estimate the backbone fractal dimension to be 1 643 36 10

exact series studies of self avoiding walks on two dimensional critical percolation clusters the author presents results of exact series studies of self avoiding walks on percolation clusters on the square lattice performed very close to the percolation threshold and estimates the critical exponents nu and gamma defined by the disorder averages of the mean square radius of gyration or the end to end distance and the number of self avoiding walks respectively the self avoiding walk configurations are enumerated exactly up to 35 steps the result for nu indicates a large increase compared with the self avoiding walks on fully occupied lattice the result for gamma however indicates a behaviour similar to that on a fully occupied lattice the result for nu is completely different from that of lee nakanishi and kim 1989

low concentration series in general dimension we discuss recent work on the development and analysis of low concentration series for many models the recent breakthrough in the extremely efficient no free end method of series generation facilitates the derivation of 15th order series for multiple moments in general dimension the 15th order series have been obtained for lattice animals percolation and the edwards anderson ising spin glass in the latter cases multiple moments have been found from complete graph tables through to 13th order general dimension 13th order series have been derived for the resistive susceptibility the moments of the logarithms of the distribution of currents in resistor networks and the average transmission coefficient in the quantum percolation problem 11th order series have been found for several other systems including the crossover from animals to percolation the full resistance distribution nonlinear resistive susceptibility and current distribution in dilute resistor networks diffusion on percolation clusters the dilute ising model dilute antiferromagnet in a field and random field ising model and self avoiding walks on percolation clusters series for the dilute spin 1 2 quantum heisenberg ferromagnet are in the process of development analysis of these series gives estimates for critical thresholds amplitude ratios and critical exponents for all dimensions where comparisons are possible our series results are in good agreement with bothe expansion results near the upper critical dimension and with exact results when available in low dimensions and are competitive with other numerical approaches in intermediate realistic dimensions

dynamic critical behavior of the worm algorithm for the ising model we study the dynamic critical behavior of the worm algorithm for the two and three dimensional ising models by monte carlo simulation the autocorrelation functions exhibit an unusual three time scale behavior as a practical matter the worm algorithm is slightly more efficient than the swendsen wang algorithm for simulating the two point function of the three dimensional ising model

different self avoiding walks on percolation clusters a small cell real space renormalization group study we calculate the average number of stepsn for edge to edge normal and indefinitely growing self avoiding walks saws on two dimensional critical percolation clusters using the real space renormalization group approach with small h cells our results are of the formn al d saw b wherel is the end to end distance similarly to several deterministic fractals the fractal dimensionsd saw for these three different kinds of saws are found to be equal and the differences between them appear in the amplitudesa and in the correction termsb this behavior is atributed to the hierarchical nature of the critical percolation cluster

portal frame inertia and stiffness matrices by substructure synthesis generic expressions of mass and stiffness matrices of the portal frame are presented these are derived by means of the substructure synthesis method this method is exceptionally characterised by low order eigenvalue problems and highly accurate eigensolutions

scaling relations for logarithmic corrections multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical mechanical systems here a lee yang zero approach is used to systematically analyze the exponents of such logarithms and to propose scaling relations between them these proposed relations are then confronted with a variety of results from the literature

self consistent scaling theory for logarithmic correction exponents multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics here a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific heat critical exponent vanishes also the theory is widened to encompass the correlation function the new relations are then confronted with results from the literature and some new predictions for logarithmic corrections in certain models are made

on dynamic threshold graphs and related classes abstract this paper deals with the well known classes of threshold and difference graphs both characterized by separators i e node weight functions and thresholds we design an efficient algorithm to find the minimum separator and we show how to maintain minimum its value when the input threshold or difference graph is fully dynamic i e edges nodes are inserted removed moreover exploiting the data structure used for maintaining the minimality of the separator we study the disjoint union and the join of two threshold graphs showing that the resulting graphs are threshold signed graphs i e a superclass of both threshold and difference graphs finally we consider the complement operation on all the three introduced classes of graphs all these operations produce in output the modified graph in terms of their separator and require time linear w r t the number of different degrees we observe that recomputing from scratch the separator would run either in linear for threshold and difference graphs or quadratic for threshold signed graphs time w r t the number of nodes of the graph

where two fractals meet the scaling of a self avoiding walk on a percolation cluster the scaling properties of self avoiding walks on a d dimensional diluted lattice at the percolation threshold are analyzed by a field theoretical renormalization group approach to this end we reconsider the model of y meir and a b harris phys rev lett 63 2819 1989 and argue that via renormalization its multifractal properties are directly accessible while the former first order perturbation did not agree with the results of other methods our analytic result gives an accurate description of the available mc and exact enumeration data in a wide range of dimensions 2 ensuremath leqslant d ensuremath leqslant 6

dynamic analysis of frames by a two level finite element method abstract a two level finite element technique of constructing a frame super element is introduced to reduce the computational efforts for solving large scale frame problems the ordinary finite element method is used first to yield matrices for the beam members then the nodal displacements of all the nodes are related to those of a small number of selected joints master nodes in the frame by means of global finite element interpolating functions thus the order of the overall matrices is greatly reduced and the frame may be considered as a super element to be connected to other elements by means of the master nodes the accuracy of solution may be improved either with finer subdivision or by taking more master nodes inside each super element

flory approximant for self avoiding walks on fractals a flory approximant for the exponent describing the end to end distance of a self avoiding walk saw on fractals is derived the approximant involves the fractal dimensionalities of the backbone and of the minimal path and the exponent describing the resistance of the fractal the approximant yields values which are very close to those available from exact and numerical calculations

potts model and graph theory elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the potts model topics discussed include graphical expansions of the potts partition function and correlation functions and their relationships with the chromatic dichromatic and flow polynomials occurring in graph theory it is also shown that the potts model realization of these classic graph theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

loop cluster coupling and algorithm for classical statistical models potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin the fortuin kasteleyn fk bond or the q flow loop representation we introduce a loop cluster lc joint model of bond occupation variables interacting with q flow variables and formulate an lc algorithm that is found to be in the same dynamical universality as the celebrated swendsen wang algorithm this leads to a theoretical unification for all the representations and numerically one can apply the most efficient algorithm in one representation and measure physical quantities in others moreover by using the lc scheme we construct a hierarchy of geometric objects that contain as special cases the q flow clusters and the backbone of fk clusters the exact values of whose fractal dimensions in two dimensions remain as an open question our work not only provides a unified framework and an efficient algorithm for the potts model but also brings new insights into the rich geometric structures of the fk clusters

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

degree sequences and majorization a classical result concerning majorization is given two nonnegative integer sequences a and b such that a majorizes b a rearrangement of b can be obtained from a by a sequence of unit transformations a recent result says that a degree sequence is a threshold sequence degree sequence of a threshold graph if and only if it is not strictly majorized by any degree sequence motivated by this we define the majorization gap of a degree sequence to be the minimum number of successive reverse unit transformations required to transform it into a threshold sequence we derive a formula for the majorization gap by establishing a lower bound for it and exhibiting reverse unit transformations achieving the bound we also discuss the relationship between the majorization gap and the threshold gap introduced elsewhere and show that they are equal the degree sequences having the maximum majorization gap for a fixed number of edges or vertices are characterized

critical percolation in high dimensions we present monte carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 13 dimensions for d 6 they are preliminary for d 6 they are between 20 and 10 4 times more precise than the best previous estimates this was achieved by three ingredients i simple and fast hashing that allowed us to simulate clusters of millions of sites on computers with less than 500 mbytes memory ii a histogram method that allowed us to obtain information for several p values from a single simulation and iii a variance reduction technique that is especially efficient at high dimensions where it reduces error bars by a factor of up to 30 and more based on these data we propose a scaling law for finite cluster size corrections