generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

1 norm based regularization scheme for system identification of structures with discontinuous system parameters this paper presents a new class of regularization functions and the associated regularization scheme for structural system identification in particular 1 norm regularization functions are investigated to overcome the smearing effect of 2 norm regularization functions for the identification of discontinuous system parameters of structures the truncated singular value decomposition is employed to filter out noise polluted solution components and to impose the 1 norm regularization function on si the bilinear fitting method is proposed for selecting an optimal truncation number of the truncated singular value decomposition the validity of the proposed method is demonstrated through the identification of an inclusion in a square plate and damaged members in a two span truss copyright 2006 john wiley sons ltd

determination of an optimal regularization factor in system identification with tikhonov regularization for linear elastic continua this paper presents a geometric mean scheme gms to determine an optimal regularization factor for tikhonov regularization technique in the system identification problems of linear elastic continua the characteristics of non linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses it is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution based on this observation the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses the validity of the gms is demonstrated through two numerical examples with measurement errors and modelling errors copyright 2001 john wiley sons ltd

the critical manifolds of inhomogeneous bond percolation on bow tie and checkerboard lattices we give a conditional derivation of the inhomogeneous critical percolation manifold of the bow tie lattice with five different probabilities a problem that does not appear at first to fall into any known solvable class although our argument is mathematically rigorous only on a region of the manifold we conjecture that the formula is correct over its entire domain and we provide a non rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface we discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow tie lattices and that if it could be put on a firm foundation the negative probability portion of our method would extend this class to many further systems including f y wu s checkerboard formula for the square lattice we conclude by showing that this latter problem can in fact be proved using a recent result of grimmett and manolescu for isoradial graphs lending strong evidence in favor of our other conjectured results this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

percolation on hypergraphs with four edges we study percolation on self dual hypergraphs that contain hyperedges with four bounding vertices or four edges using three different generators each containing bonds or sites with three distinct probabilities p r and t connecting the four vertices we find explicit values of these probabilities that satisfy the self duality conditions discussed by bollobas and riordan this demonstrates that explicit solutions of the self duality conditions can be found using generators containing bonds and sites with independent probabilities these solutions also provide new examples of lattices where exact percolation critical points are known one of the generators exhibits three distinct criticality solutions p r t we carry out monte carlo simulations of two of the generators on two different hypergraphs to confirm the critical values for the case of the hypergraph and uniform generator studied by wierman et al we also determine the threshold p 0 441 374 0 000 001 which falls within the tight bounds that they derived furthermore we consider a generator in which all or none of the vertices can connect and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union jack lattice and bond percolation on the square lattice

predictions of bond percolation thresholds for the kagome and archimedean 3 12 2 lattices here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom e and 3 12 2 lattices we present two different methods one which provides an approximation to the inhomogeneous kagom e and 3 12 2 bond problems and the other which gives estimates of p c for the homogeneous kagom e 0 524 408 8 dots and 3 12 2 0 740 421 2 dots problems that respectively agree with numerical results to five and six significant figures

interdependent networks with identical degrees of mutually dependent nodes we study a problem of failure of two interdependent networks in the case of identical degrees of mutually dependent nodes we assume that both networks a and b have the same number of nodes n connected by the bidirectional dependency links establishing a one to one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links i e their degrees coincide this implies that both networks have the same degree distribution p k we call such networks correspondently coupled networks ccns we assume that the nodes in each network are randomly connected we define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes that belong to the mutual giant component remain functional we assume that initially a 1 p fraction of nodes are randomly removed because of an attack or failure and find analytically for an arbitrary p k the fraction of nodes p that belong to the mutual giant component we find that the system undergoes a percolation transition at a certain fraction p p c which is always smaller than p c for randomly coupled networks with the same p k we also find that the system undergoes a first order transition at p c 0 if p k has a finite second moment for the case of scale free networks with 2 0 finally we find that the robustness of ccn increases with the broadness of their degree distribution

generalized cell dual cell transformation and exact thresholds for percolation suggested by scullard s recent star triangle relation for correlated bond systems we propose a general cell dual cell transformation which allows in principle an infinite variety of lattices with exact percolation thresholds to be generated we directly verify scullard s site percolation thresholds and derive the bond thresholds for his martini lattice p c 1 sqrt 2 and the a lattice p c 0 625457 dots solution to p 5 ensuremath 4 p 4 3 p 3 2 p 2 ensuremath 1 0 we also present a precise monte carlo test of the site threshold for the a lattice

equality of bond percolation critical exponents for pairs of dual lattices for a certain class of two dimensional lattices lattice dual pairs are shown to have the same bond percolation critical exponents a computational proof is given for the martini lattice and its dual to illustrate the method the result is generalized to a class of lattices that allows the equality of bond percolation critical exponents for lattice dual pairs to be concluded without performing the computations the proof uses the substitution method which involves stochastic ordering of probability measures on partially ordered sets as a consequence there is an infinite collection of infinite sets of two dimensional lattices such that all lattices in a set have the same critical exponents

currents and flat chains associated to varifolds with an application to mean curvature flow we prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer multiplicity rectifiable currents the convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow

a computationally efficient approach for inverse material characterization combining gappy pod with direct inversion abstract an approach for computationally efficient inverse material characterization from partial field response measurements that combines the gappy proper orthogonal decomposition pod machine learning technique with a physics based direct inversion strategy is presented and evaluated gappy pod is used to derive a data reconstruction tool from a set of potential system response fields that are generated from available a priori information regarding the potential distribution of the unknown material properties then the gappy pod technique is applied to reconstruct the full spatial distribution of the system response from whatever portion of the response field has been measured with the chosen system testing method lastly a direct inversion strategy is presented that is derived from the equations governing the system response i e physics of the system which utilizes the full field response reconstructed by gappy pod to produce an estimate of the spatial distribution of the unknown material properties the direct inversion technique is a particularly computationally efficient inversion technique requiring a cost equivalent to a single numerical analysis therefore the majority of the computational expense of the presented approach is the one time potential response generation for the gappy pod technique which leads to an approach that is substantially computationally efficient overall two numerically simulated examples are shown in which the elastic modulus distribution was characterized based on partial field displacement response measurements both static and dynamic the inversion procedure was shown to have the capability to efficiently provide accurate estimates to material property distributions from partial field response measurements the direct inversion with gappy pod response estimation was also shown to be substantially tolerant to noise in comparison to the direct inversion given measured full field response lastly a physical example regarding elastography of an arterial construct from ultrasound imaging response measurements is shown to validate the practical applicability of the direct inversion approach with gappy pod response reconstruction

percolation on shopping and cashback electronic commerce networks abstract many realistic networks live in the form of multiple networks including interacting networks and interdependent networks here we study percolation properties of a special kind of interacting networks namely shopping and cashback electronic commerce networks scecns we investigate two actual scecns to extract their structural properties and develop a mathematical framework based on generating functions for analyzing directed interacting networks then we derive the necessary and sufficient condition for the absence of the system wide giant in and out component and propose arithmetic to calculate the corresponding structural measures in the sub critical and supercritical regimes we apply our mathematical framework and arithmetic to those two actual scecns to observe its accuracy and give some explanations on the discrepancies we show those structural measures based on our mathematical framework and arithmetic are useful to appraise the status of scecns we also find that the supercritical regime of the whole network is maintained mainly by hyperlinks between different kinds of websites while those hyperlinks between the same kinds of websites can only enlarge the sizes of in components and out components

parameter identification with weightless regularization although the regularization increased the popularity of parameter identification due to its capability of deriving a stable solution the significant problem is that the solution depends upon the regularization parameters chosen this paper presents a technique for deriving solutions without the use of the parameters and further an optimization method which can work efficiently for problems of concern numerical examples show that the technique can efficiently search for appropriate solutions copyright 2001 john wiley sons ltd

clique percolation in random networks the notion of k clique percolation in random graphs is introduced where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated for the erdos renyi graph of n vertices we obtain that the percolation transition of k cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p c k k 1 n 1 k 1 at the transition point the scaling of the giant component with n is highly nontrivial and depends on k we discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks

critical percolation of virtually free groups and other tree like graphs this article presents a method for finding the critical probability p c for the bernoulli bond percolation on graphs with the so called tree like structure such a graph can be decomposed into a tree of pieces each of which has finitely many isomorphism classes this class of graphs includes the cayley graphs of amalgamated products hnn extensions or general groups acting on trees it also includes all transitive graphs with more than one end the idea of the method is to find a multi type galton watson branching process with a parameter p which has finite expected population size if and only if the expected percolation cluster size is finite this provides sufficient information about p c in particular if the pairwise intersections of pieces are finite then p c is the smallest positive p such that det m 1 0 where m is the first moment matrix of the branching process if the pieces of the tree like structure are finite then p c is an algebraic number and we give an algorithm computing p c as a root of some algebraic function we show that any cayley graph of a virtually free group i e a group acting on a tree with finite vertex stabilizers with respect to any finite generating set has a tree like structure with finite pieces in particular we show how to compute p c for the cayley graph of a free group with respect to any finite generating set

universality for bond percolation in two dimensions all in homogeneous bond percolation models on the square triangular and hexagonal lattices belong to the same universality class in the sense that they have identical critical exponents at the critical point assuming the exponents exist this is proved using the star triangle transformation and the box crossing property the exponents in question are the one arm exponent the 2j alternating arms exponents 2j for j 1 the volume exponent and the connectivity exponent by earlier results of kesten this implies universality also for the near critical exponents assuming these exist for any of these models that satisfy a certain additional hypothesis such as the homogeneous bond percolation models on these three lattices

exact site percolation thresholds using a site to bond transformation and the star triangle transformation i construct a two dimensional lattice on which the inhomogeneous site percolation threshold is exactly calculable and use this result to find two more lattices on which the site thresholds can be determined the primary lattice studied here the martini lattice is a hexagonal lattice with every second site transformed into a triangle the site threshold of this lattice is found to be 0 764826 i e the solution to p4 3p3 1 0 while the others have square root 5 1 2 the inverse of the golden ratio and 1 square root 2 this last solution suggests a possible approach to establishing the bound for the hexagonal site threshold pc 1 square root 2 to derive these results i solve a correlated bond problem on the hexagonal lattice by use of the star triangle transformation and then by a particular choice of correlations derived from a site to bond transformation solve the site problem on the martini lattice

tight bounds for the bond percolation threshold of the 3 122 lattice improved mathematically rigorous upper and lower bounds for the bond percolation threshold are established using the substitution method the lower bound is greater than 0 739 830 which was conjectured by tsallis to be the exact value of the percolation threshold together the bounds determine the first three digits of the threshold the results are obtained using computational improvements to the substitution method in which stochastic ordering is checked using a network flow algorithm

some exact results on bond percolation we present some exact results on bond percolation we derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice by l bonds connecting the same adjacent vertices thereby yielding the lattice l this relation is used to calculate the bond percolation threshold on l we show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d 2 but changes it on a one dimensional lattice and quasi one dimensional infinite length strips we also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the n limits of several families of n vertex graphs finally we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as n

multi objective parameter identification of euler bernoulli beams under axial load abstract identification of physical parameters of the partial differential equation describing transverse vibrations of an axially loaded euler bernoulli beam ebb is proposed via a multi objective optimization formulation and solved by a genetic algorithm conflicting objectives such as performance and stability are specifically formulated and optimized simultaneously stability is quantified in terms of the solution s time growth factor physical parameter sets in the resulting pareto front approximation represent best trade offs with respect to the multiple objectives to compute output error performance objectives the ebb equation is discretized via finite differences in space and time and reformulated to a state space system identifiability is verified by checking regularity of the so called fisher information matrix the identification methodology is capable of determining material parameters including damping as well as the axial load from few spatially concentrated measurements its features are demonstrated and successfully validated on specific simulation data and measurement data obtained from a laboratory testbed

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

exact bond percolation thresholds in two dimensions recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices here we show how these results can be extended to other classes of graphs significantly increasing the number and variety of solved problems any graph that can be decomposed into a certain arrangement of triangles which we call self dual gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle triangle transformation we use this method to generalize wierman s solution of the bow tie lattice to yield several new solutions we also give another example of a self dual arrangement of triangles that leads to a further class of solvable problems there are certainly many more such classes

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

transfer matrix computation of generalized critical polynomials in percolation percolation thresholds have recently been studied by means of a graph polynomial pb p henceforth referred to as the critical polynomial that may be defined on any periodic lattice the polynomial depends on a finite subgraph b called the basis and the way in which the basis is tiled to form the lattice the unique root of pb p in 0 1 either gives the exact percolation threshold for the lattice or provides an approximation that becomes more accurate with appropriately increasing size of b initially pb p was defined by a contraction deletion identity similar to that satisfied by the tutte polynomial here we give an alternative probabilistic definition of pb p which allows for much more efficient computations by using the transfer matrix than was previously possible with contraction deletion we present bond percolation polynomials for the 4 82 kagome and 3 122 lattices for bases of up to respectively 96 162 and 243 edges much larger than the previous limit of 36 edges using contraction deletion we discuss in detail the role of the symmetries and the embedding of b for the largest bases we obtain the thresholds pc 4 82 0 676 803 329 pc kagome 0 524 404 998 pc 3 122 0 740 420 798 comparable to the best simulation results we also show that the alternative definition of pb p can be applied to study site percolation problems this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

percolation thresholds on two dimensional voronoi networks and delaunay triangulations the site percolation threshold for the random voronoi network is determined numerically with the result pc 0 714 10 0 000 02 using monte carlo simulation on periodic systems of up to 40 000 sites the result is very close to the recent theoretical estimate pc approximately 0 7151 of neher for the bond threshold on the voronoi network we find pc 0 666 931 0 000 005 implying that for its dual the delaunay triangulation pc 0 333 069 0 000 005 these results rule out the conjecture by hsu and huang that the bond thresholds are 2 3 and 1 3 respectively but support the conjecture of wierman that for fully triangulated lattices other than the regular triangular lattice the bond threshold is less than 2 sin pi 18 approximately 0 3473

random cluster models on the triangular lattice we study percolation and the random cluster model on the triangular lattice with 3 body interactions starting with percolation we generalize the star triangle transformation we introduce a new parameter the 3 body term and identify configurations on the triangles solely by their connectivity in this new setup necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non percolation next we apply this set of ideas to the q 1 random cluster model we derive duality relations for the suitable random cluster measures prove necessary and sufficient conditions for them to have positive correlations and finally prove some rigorous theorems concerning phase transitions

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

embedded self similar shrinkers of genus 0 we conrm a well known conjecture that the round sphere is the only compact embedded self similar shrinking solution of mean curvature ow in r 3 with genus 0 more generally we show that the only properly embedded self similar shrinkers in r 3 with vanishing intersection form are the sphere the cylinder and the plane this answers two questions posed by t ilmanen

width of percolation transition in complex networks it is known that the critical probability for the percolation transition is not a sharp threshold actually it is a region of nonzero width deltap c for systems of finite size here we present evidence that for complex networks deltap c approximately p c l where l approximately nnu opt where is the average length of the percolation cluster and n is the number of nodes in the network for erdos renyi graphs nu opt 1 3 while for scale free networks with a degree distribution p k approximately k lambda and 3 lambda 4 nu opt lambda 3 lambda 1 we show analytically and numerically that the survivability s p l which is the probability of a cluster to survive l chemical shells at probability p behaves near criticality as s p l s p c l exp p p c l p c thus for probabilities inside the region p p c p c l the behavior of the system is indistinguishable from that of the critical point

a bond percolation critical probability determination based on the star triangle transformation the bond percolation critical probability of a planar graph with square and triangular faces obtained by inserting a diagonal in every other face of the square lattice is the root of 1 p 6p2 6p3 p5 0 in 0 1 which is approximately 0 404 518 the proof uses the star triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

robust structural damage identification based on multi objective optimization structural damage identification is an inverse problem which often is formulated as an optimization problem the design variables are degrees of damage in computational model and the objective is the discrepancies between the computed responses and the measured responses conventional single objective optimization approach defines the objective function by combining multiple error terms into a single one which leads to a weaker constraint in solving the identification problem an alternative approach explained in this paper is a multi objective approach that simultaneously minimizes multiple error terms a stronger constraint from multiple objectives promotes the solutions to converge to the correct solution numerical examples based on static testing are provided to illustrate the multi objective approach also conceptual explanations are given to extend the approach to include both the static testing and the dynamic testing expected challenges will also be explained

co evolutionary algorithm for structural damage identification using minimal physical testing the problem of damage identification using minimum test data is studied in this work data sparsity in damage identification applications commonly results in inverse problems that are mathematically ill posed e g non unique solutions although solution non uniqueness may be addressed by performing multiple tests on a structure it is not trivial to decide which tests to carry out given that actual physical testing is costly this problem is addressed in this work through a new co evolutionary algorithm that interactively searches for damage scenarios and optimum physical tests the algorithm is composed of two stages the estimation phase which searches for damage scenarios that can predict current physical tests and the exploration phase which searches for tests that increase the level of information about the damaged system the feasibility of the methodology is demonstrated using numerical examples copyright 2006 john wiley sons ltd

potts and percolation models on bowtie lattices we give the exact critical frontier of the potts model on bowtie lattices for the case of q 1 the critical frontier yields the thresholds of bond percolation on these lattices which are exactly consistent with the results given by ziff et al j phys a 39 15083 2006 for the q 2 potts model on a bowtie a lattice the critical point is in agreement with that of the ising model on this lattice which has been exactly solved furthermore we do extensive monte carlo simulations of the potts model on a bowtie a lattice with noninteger q our numerical results which are accurate up to seven significant digits are consistent with the theoretical predictions we also simulate the site percolation on a bowtie a lattice and the threshold is s c 0 5479148 7 in the simulations of bond percolation and site percolation we find that the shape dependent properties of the percolation model on a bowtie a lattice are somewhat different from those of an isotropic lattice which may be caused by the anisotropy of the lattice

crossover from isotropic to directed percolation we generalize the directed percolation dp model by relaxing the strict directionality of dp such that propagation can occur in either direction but with anisotropic probabilities we denote the probabilities as p p pd and p p 1 pd with p representing the average occupation probability and pd controlling the anisotropy the leath alexandrowicz method is used to grow a cluster from an active seed site we call this model with two main growth directions biased directed percolation bdp standard isotropic percolation ip and dp are the two limiting cases of the bdp model corresponding to pd 1 2 and pd 0 1 respectively in this work besides ip and dp we also consider the 1 2 pd 1 region extensive monte carlo simulations are carried out on the square and the simple cubic lattices and the numerical data are analyzed by finite size scaling we locate the percolation thresholds of the bdp model for pd 0 6 and 0 8 and determine various critical exponents these exponents are found to be consistent with those for standard dp we also determine the renormalization exponent associated with the asymmetric perturbation due to pd 1 2 6 0 near ip and confirm that such an asymmetric scaling field is relevant at ip