internal diffusion limited aggregation parallel algorithms and complexity the computational complexity of internal diffusion limited aggregation dla is examined from both a theoretical and a practical point of view we show that for two or more dimensions the problem of predicting the cluster from a given set of paths is complete for the complexity class cc the subset of p characterized by circuits composed of comparator gates cc completeness is believed to imply that in the worst case growing a cluster of size n requires polynomial time in n even on a parallel computer a parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths and then corrects defects in the guessed cluster through a nonlocal annihilation process the parallel running time of the relaxation algorithm for two dimensional internal dla is studied by simulating it on a serial computer the numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n thus the computational resources needed to grow large clusters are significantly less on average than the worst case analysis would suggest for a parallel machine with k processors we show that random clusters in d dimensions can be generated in mathcal o n k logk n 2 d steps this is a significant speedup over explicit sequential simulation which takes mathcal o n 1 2 d time on average finally we show that in one dimension internal dla can be predicted in mathcal o logn parallel time and so is in the complexity class nc

on a system of reaction diffusion equations arising from competition with internal storage in an unstirred chemostat abstract in this paper we study a system of reaction diffusion equations arising from competition of two microbial populations for a single limited nutrient with internal storage in an unstirred chemostat the conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient the reduced system limiting system generates a strongly monotone dynamical system in its feasible domain under a partial order we construct suitable upper lower solutions to establish the existence of positive steady state solutions given the parameters of the reduced system we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors the primary conclusion is that the survival of species depends on species s intrinsic biological characteristics the external environment forces and the principal eigenvalues of some scalar partial differential equations we also lift the dynamics of the limiting system to the full system

a mathematical model of competition for two essential resources in the unstirred chemostat a mathematical model of competition between two species for two growth limiting essential complementary resources in the unstirred chemostat is considered the existence of a positive steady state solution and some of its properties are established analytically techniques include the maximum principle the fixed point index and numerical simulations the simulations also seem to indicate that there are regions in parameter space for which a globally stable positive equilibrium occurs and that there are other regions for which the model admits bistability and even multiple positive equilibria

on the computational complexity of ising spin glass models in a spin glass with ising spins the problems of computing the magnetic partition function and finding a ground state are studied in a finite two dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice in contrast to this fact the same problems are shown to belong to the class of np hard problems both in the two dimensional case within a magnetic field and in the three dimensional case np hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it

effects of random motility on microbial growth and competition in a flow reactor the authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations for pure culture a single population a sharp condition is derived which distinguishes between the two outcomes 1 washout of the population from the reactor or 2 persistence of the population and the existence of a unique single population steady state the simulations suggest that this steady state is globally attracting for the case of two populations competing for scarce nutrient they obtain sufficient conditions for the uniform persistence of the two populations for the existence of a coexistence steady state and for the ability of one population to competitively exclude a rival extensive simulations are reported which suggest that 1 all solutions approach some steady state solution 2 all possible outcomes exhibited by the classical competitive lotka volterra ode model can occur in the model and 3 the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients

a point interpolation method for simulating dissipation process of consolidation abstract a biot s consolidation problem in foundation engineering is numerically investigated using improved point interpolation method pim a weak form of biot s theory is first developed to consider the unbalanced forces at previous time step and thus guarantees the global equilibrium at current step two independent variables in the weak form displacement and excess pore water pressure are approximated using the same shape functions through pim technique the pim technique constructs its interpolation functions through a cluster of scattered points in problem domain and its shape function is of delta properties thus implementation of essential boundary conditions is as easy as in conventional finite element method crank nicholson s integration scheme is used to discretize time domain finally examples are studied and compared with finite element methods to demonstrate its capability

competition in an n vessel gradostat a mathematical model of the exploitative competition between two microbial species in a general multivessel gradostat is considered sufficient conditions for the two species to coexist in the grad

quadratic forms on graphs we introduce a new graph parameter called the grothendieck constant of a graph g v e which is defined as the least constant k such that for every a e sup f v to s v 1 sum u v in e a u v cdot langle f u f v rangle le k sup varphi v to 1 1 sum u v in e a u v cdot varphi u varphi v the classical grothendieck inequality corresponds to the case of bipartite graphs but the case of general graphs is shown to have various algorithmic applications indeed our work is motivated by the algorithmic problem of maximizing the quadratic form u v e a u v u v over all v 1 1 which arises in the study of correlation clustering and in the investigation of the spin glass model we give upper and lower estimates for the integrality gap of this program we show that the integrality gap is o log vartheta overline g where vartheta overline g is the lovasz theta function of the complement of g which is always smaller than the chromatic number of g this yields an efficient constant factor approximation algorithm for the above maximization problem for a wide range of graphs g we also show that the maximum possible integrality gap is always at least log g where g is the clique number of g in particular it follows that the maximum possible integrality gap for the complete graph on n vertices with no loops is logn more generally the maximum possible integrality gap for any perfect graph with chromatic number n is logn the lower bound for the complete graph improves a result of kashin and szarek on gram matrices of uniformly bounded functions and settles a problem of megretski and of charikar and wirth

coexistence and stability of an unstirred chemostat model with beddington deangelis function this paper deals with an unstirred chemostat model with the beddington deangelis functional response first a sufficient condition to the existence of positive steady state solutions is established second the effect of the parameter b 1 in the beddington deangelis functional response which models mutual interference between species u is considered the result shows that if b 1 is sufficiently large the solution of this model is determined by a limiting equation the main tool used here includes the fixed point index theory the perturbation technique and the bifurcation theory

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

coexistence in the unstirred chemostat we consider a rather general system of reaction diffusion equations containing multiple parameters and prove existence of positive solutions for some sets of parameters the problem models long term coexistence of competing microorganisms in an unstirred chemostat with diffusion and flow the proof as in previous work depends on a bifurcation argument using a crandall rabinowitz theorem since stability of these steady states as well as the more general question of persistence is unresolved we include some numerical results which shed light on these questions

the geometry of algorithms using hierarchical tensors in this paper the differential geometry of the novel hierarchical tucker format for tensors is derived the set ht k of tensors with fixed tree t and hierarchical rank k is shown to be a smooth quotient manifold namely the set of orbits of a lie group action corresponding to the non unique basis representation of these hierarchical tensors explicit characterizations of the quotient manifold its tangent space and the tangent space of ht k are derived suitable for high dimensional problems the usefulness of a complete geometric description is demonstrated by two typical applications first new convergence results for the nonlinear gauss seidel method on ht k are given notably and in contrast to earlier works on this subject the task of minimizing the rayleigh quotient is also addressed second evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of ht k in addition a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions 2013 elsevier inc

asymptotic behavior of an unstirred chemostat model with internal inhibitor this paper deals with a chemostat model with an internal inhibitor first the elementary stability and asymptotic behavior of solutions of the system are determined second the effects of the inhibitor are considered it turns out that the parameter which measures the effect of the inhibitor plays a very important role in deciding the stability and longtime behavior of solutions of the system the results show that if is sufficiently large this model has no coexistence solution and one of the semitrivial equilibria is a global attractor when the maximal growth rate a of the species u lies in certain range but when a belongs to another range all positive solutions of this model are governed by a limit problem and two semitrivial equilibria are bistable the main tools used here include monotone system theory degree theory bifurcation theory and perturbation technique

globall estimates for a class of reaction diffusion systems abstract we show that a priori l 1 estimates dissipativity imply l estimates dissipativity for a class of reaction diffusion systems this generalizes the results obtained in biological models such as unmixed bioreactors plug flow models and lotka volterra systems

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

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 nonlinear boundary value problem arising from competition in the chemostat a boundary value problem with two parameters for two coupled second order nonlinear ordinary differential equations is considered this problem arises in a study of chemostats without a well mixing hypothesis the existence of a solution in the interior of the positive cone is established via two bifurcations some open questions are pointed out

positive solutions of a competition model for two resources in the unstirred chemostat this paper deals with a competition model between two species for two growth limiting and perfectly complementary resources in the unstirred chemostat the main purpose is to determine the exact range of the parameters of two species so that the system possesses positive solutions and to investigate multiple positive steady states of the system the main tools used here include the monotone methods and the topological fixed point theory developed by amann

competition in the unstirred chemostat with periodic input and washout the model of an unstirred chemostat is generalized to that of a chemostat with time dependent input washout rates the novelty of the new model is that time periodicity appears in the boundary conditions the asymptotic dynamics of the competition between two microbial populations is determined in terms of the corresponding period map which is shown to preserve the standard competitive ordering it is shown that the dynamics of competition is similar to that of a chemostat with constant boundary conditions simple criteria for coexistence versus competitive exclusion are presented 1 introduction the chemostat represents a basic model of an open system in microbial ecology in its simplest form it consists of three vessels the rst called the feed bottle contains medium with all of the nutrients needed for growth in sur plus except one which hereafter is simply called the nutrient the contents of the feed bottle are pumped at a constant rate into the second vessel called the culture vessel or bioreactor the culture vessel is charged with one or more populations of microorganisms the contents of the culture vessel are pumped into the remaining vessel called the over ow vessel at a constant rate keeping the volume of the reactor constant the organisms compete for the nutrient in a purely exploitative manner basic assumptions include that the vessel is well mixed and that all other parame ters ph temperature etc are strictly controlled the ow rate is assumed to be sucient to preclude wall growth or the accumulation of metabolic products let s t denote the concentration of the nutrient in the culture vessel and xi t i 1 2 denote the concentration of the competitors lets 0 denote the concentration of the input nutrient and let d denote the dilution rate ow rate volume if growth is assumed to be proportional to consumption then the basic equations take the form

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

a reaction advection diffusion system modeling the competition for two complementary resources with seasonality in a flowing habitat abstract in this current paper we investigate a periodic bio reactor model where microorganisms compete for two essential nutrient resource s needed for growth for the single population growth model we show that when the trivial solution is locally asymptotically stable then the single population will be washed out when the trivial solution is unstable there is a unique periodic positive solution which attracts all solutions with nonzero initial data for the two species model we prove that the existence of a periodic coexistence state is possible if each species can invade the semi trivial periodic state established by the other species more precisely if the semi trivial periodic solutions are both unstable there exists at least one periodic coexistence solution numerical work indicates conditions for persistence depend on the flow characteristics advection and diffusivity from our numerical simulations competitive exclusion bistability and coexistence are all observed

approximate inference on planar graphs using loop calculus and belief propagation we introduce novel results for approximate inference on planar graphical models using the loop calculus framework the loop calculus chertkov and chernyak 2006a allows to express the exact partition function of a graphical model as a finite sum of terms that can be evaluated once the belief propagation bp solution is known in general full summation over all correction terms is intractable we develop an algorithm for the approach presented in chertkov et al 2008 which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of pfaffians of matrices we analyze the performance of the algorithm for models with binary variables and pairwise interactions on grids and other planar graphs we study in detail both the loop series and the equivalent pfaffian series and show that the first term of the pfaffian series for the general intractable planar model can provide very accurate approximations the algorithm outperforms previous truncation schemes of the loop series and is competitive with other state of the art methods for approximate inference

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

a point interpolation method for two dimensional solids a point interpolation method pim is presented for stress analysis for two dimensional solids in the pim the problem domain is represented by properly scattered points a technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points the pim equations are then derived using variational principles in the pim the essential boundary conditions can be implemented with ease as in the conventional finite element methods the present pim has been coded in fortran the validity and efficiency of the present pim formulation are demonstrated through example problems it is found that the present pim is very easy to implement and very flexible for obtained displacements and stresses of desired accuracy in solids as the elements are not used for meshing the problem domain the present pim opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures copyright 2001 john wiley sons ltd

optimizing transformations of structural balance in signed networks with potential relationships abstract a signed network includes positive edges negative edges and 0 edges the last of which denote potential relationships however 0 edges are commonly ignored in transformations of unbalanced networks in this paper we take 0 edges into account and solve the optimization problem in a more comprehensive way we transform the unbalanced network by not only changing signs of edges but also changing edges into potential the experimental results show that our method can solve this problem efficiently and that our solutions are cost saving

bound entanglement in tree graphs in this paper we discuss the entanglement properties of graph diagonal states with particular emphasis on calculating the threshold for the transition between the presence and absence of entanglement i e the separability point special consideration is made of the thermal states of trees including the linear cluster state we characterize the type of entanglement present and describe the optimal entanglement witnesses and their implementation on a quantum computer up to an additive approximation in the case of general graphs we invoke a relation with the partition function of the classical ising model thereby intimating a connection to computational complexity theoretic tasks finally we show that the entanglement is robust to some classes of local perturbations

systems of ordinary differential equations which generate an order preserving flow a survey of results this article consists of a survey of results concerning the qualitative behavior of solutions of systems of ordinary differential equations which generate an order preserving flow we restrict our consideration to partial orderings on r n induced by any one of its orthants a flow preserves ordering if any two solutions x t and y t are ordered x t leqq y t for all t 0 whenever x 0 leqq y 0 many of the important results for such systems have only recently been obtained principally by m w hirsch who pointed out the tendency of their solutions to converge to equilibrium less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria a particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibrium one of our main goals is to show that by allowing partial orderings on r n generated by orthants

j ising model on durer lattices abstract this paper tackles the problem of finding analytical expressions describing the ground state properties of durer lattices over which a generalized edwards anderson model j ising model is defined a local frustration analysis is performed based on representative cells for these lattices the concentration of ferromagnetic f bonds x is used as the independent variable in the analysis 1 x is the concentration for antiferromagnetic a bonds where x spans the range 0 00 1 00 the presence of a bonds brings frustration whose clear manifestation is when bonds around the minimum possible circuit of bonds plaquette cannot be simultaneously satisfied the distribution of curved frustrated plaquettes within the representative cell is determinant for the evaluation of the parameters of interest such as average frustration segment energy per bond and fractional content of non frustrated bonds an analytical method is developed to cope with this analysis based on the direct probability of a plaquette being curved exact numerical simulations on a large number of randomly generated samples associated to durer lattices allow to validate previously described theoretical analysis this analytical method gives an excellent description for the most of the range for x a particular discussion for the point x 0 50 belonging to spin glass phase also sheds light on the general trends of the properties described here

elasto plastic analysis of reinforced soils using mesh free method in this paper the application of the mesh free method mfm for the elasto plastic analysis of reinforced soils is studied for the first time the applied mesh free method is called the radial point interpolation method rpim in mfm unlike the finite element method fem there is no need of mesh in the traditional sense and the shape functions are based on nodes in the present study the reinforced soil is divided into three separate parts of soil reinforcements and interface layers the displacement field in each part is constructed by rpim the final system of equations is derived by the substitution of the displacement field into the weak form of the governing equation the elasto plastic behaviors of soil reinforcements and interface layers are considered there is also the ability of slippage modeling between the soil and reinforcement based on the derived equations a computer code has been developed and its validity investigated by solving some examples at the end of the paper

a reaction diffusion model of harmful algae and zooplankton in an ecosystem abstract this paper is devoted to the investigation of an unstirred chemostat system modeling the interactions of two essential nutrients i e nitrogen and phosphorus harmful algae i e p parvum and cyanobacteria and a small bodied zooplankton in an ecosystem to obtain a weakly repelling property of a compact and invariant set on the boundary we introduce an associated elliptic eigenvalue problem it turns out that the model system admits a coexistence steady state and is uniformly persistent provided that the trivial steady state two semi trivial steady states and a global attractor on the boundary are all weak repellers

a food chain model for two resources in un stirred chemostat this paper deals with a multiple food chain model in un stirred chemostat where the prey feeds on two growth limiting nonreproducing resources the conditions for existence of positive steady states are established the global bifurcation of positive solutions is considered also furthermore we obtain the conditions under which the predator possesses uniformly strong persistence or goes to extinction the techniques used in this paper include the degree theory the global bifurcation theory the semigroup theory and the maximum principle

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

tighter linear and semidefinite relaxations for max cut based on the lovasz schrijver lift and project procedure we study how the lift and project method introduced by lovasz and schrijver siam j optim 1 1991 pp 166 190 applies to the cut polytope we show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no k5 minor therefore for a graph g with n ge 4 nodes with stability number alpha g n 4 iterations suffice instead of the m number of edges iterations required in general and under some assumption n alpha g 3 iterations suffice the exact number of needed iterations is determined for small n le 7 by a detailed analysis of the new relaxations if positive semidefiniteness is added to the construction then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by anjos and wolkowicz discrete appl math to appear we also show how the lovasz schrijver relaxations for the stable set polytope of g can be strengthened using the corresponding relaxations for the cut polytope of the graph g nabla obtained from g by adding a node adjacent to all nodes of g

coexistence solutions for a reaction diffusion system of un stirred chemostat model in this paper we study a reaction diffusion system of multiple food chain model where two predators feed on a single prey growing in an un stirred chemostat the conditions for the coexistence of steady states are determined the main technique used here is the degree theory in cones

algorithms for estimating the partition function of restricted boltzmann machines abstract accurate estimates of the normalization constants partition functions of energy based probabilistic models markov random fields are highly important for example for assessing the performance of models monitoring training progress and conducting likelihood ratio tests several algorithms for estimating the partition function in relation to a reference distribution have been introduced including annealed importance sampling ais and bennett s acceptance ratio method bar however their conceptual similarities and differences have not been worked out so far and systematic comparisons of their behavior in practice have been missing we devise a unifying theoretical framework for these algorithms which comprises existing variants and suggests new approaches it is based on a generalized form of crooks equality linking the expectation over a distribution of samples generated by a transition operator to the expectation over the distribution induced by the reversed operator the framework covers different ways of generating samples such as parallel tempering and path sampling an empirical comparison revealed the differences between the methods when estimating the partition function of restricted boltzmann machines and ising models in our experiments bar using parallel tempering worked well with a small number of bridging distributions while path sampling based ais performed best when many bridging distributions were available because bar gave the overall best results we favor it over ais furthermore the experiments showed the importance of choosing a proper reference distribution

constrained codes as networks of relations we address the well known problem of determining the capacity of constrained coding systems while the one dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity in contrast computing the exact capacity of a two dimensional constrained coding system is still an elusive research challenge the only known exception in the two dimensional case is an exact however not rigorous solution to the run length limited rll system on the hexagonal lattice furthermore only exponential time algorithms are known for the related problem of counting the exact number of constrained two dimensional information arrays we present the first known rigorous technique that yields an exact capacity of a two dimensional constrained coding system in addition we devise an efficient polynomial time algorithm for counting the exact number of constrained arrays of any given size our approach is a composition of a number of ideas and techniques describing the capacity problem as a solution to a counting problem in networks of relations graph theoretic tools originally developed in the field of statistical mechanics techniques for efficiently simulating quantum circuits as well as ideas from the theory related to the spectral distribution of toeplitz matrices using our technique we derive a closed form solution to the capacity related to the path cover constraint in a two dimensional triangular array the resulting calculated capacity is path cover is a generalization of the well known one dimensional rll constraint for which the capacity is known to be

reaction diffusion equations of two species competing for two complementary resources with internal storage abstract this paper examines a system of reaction diffusion equations arising from a mathematical model of two microbial species competing for two complementary resources with internal storage in an unstirred chemostat the governing system can be reduced to a limiting system based on two uncoupled conservation principles one of main technical difficulties in our analysis is the singularities in the reaction terms conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems maximum principle and the theory of monotone dynamical systems

from near to eternity spin glass planting tiling puzzles and constraint satisfaction problems we present a methodology for generating ising hamiltonians of tunable complexity and with a priori known ground states based on a decomposition of the model graph into edge disjoint subgraphs the idea is illustrated with a spin glass model defined on a cubic lattice where subproblems whose couplers are restricted to the two values 1 1 are specified on unit cubes and are parametrized by their local degeneracy the construction is shown to be equivalent to a type of three dimensional constraint satisfaction problem known as the tiling puzzle by varying the proportions of subproblem types the hamiltonian can span a dramatic range of typical computational complexity from fairly easy to many orders of magnitude more difficult than prototypical bimodal and gaussian spin glasses in three space dimensions we corroborate this behavior via experiments with different algorithms and discuss generalizations and extensions to different types of graphs

tensor network contractions for sat the computational cost of counting the number of solutions satisfying a boolean formula which is a problem instance of sat has proven subtle to quantify even when finding individual satisfying solutions is computationally easy e g 2 sat which is in mathsf p determining the number of solutions can be mathsf p hard recently computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics inspired techniques by these methods we give an algorithm using an axiomatic tensor contraction language for n variable sat instances with complexity o g cd o 1 2 c where c is the number of copy tensors g is the number of gates and d is the maximal degree of any copy tensor thus n variable counting problems can be solved efficiently when their tensor network expression has at most o log n copy tensors and polynomial fan out this framework also admits an intuitive proof of a variant of the tovey conjecture the r 1 sat instance of the dubois tovey theorem this study increases the theory expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science

fast approximate maximum a posteriori restoration of multicolour images we propose a new algorithm for the approximation of the maximum a posteriori map restoration of noisy images the image restoration problem is considered in a bayesian setting we assume as prior distribution multicolour markov random fields on a graph whose main restriction is the presence of only pairwise site interactions the noise is modelled as a bernoulli field computing the mode of the posterior distribution is np complete i e can very likely be done only in a time exponential in the number of sites of the underlying graph our algorithm runs in polynomial time and is based on the coding of the colours it produces an image with the following property either a pixel is coloured with one of the possible colours or it is left blank in the first case we prove that this is the colour of the site in the exact map restoration the quality of the approximation is then measured by the number of sites being left blank we assess the performance of the new algorithm by numerical experiments on the simple three colour potts model more rigorously we present a probabilistic analysis of the algorithm the results indicate that the approximation is quite often sufficiently good for the interpretation of the image

quantum computational complexity of the n representability problem qma complete we study the computational complexity of the n representability problem in quantum chemistry we show that this problem is quantum merlin arthur complete which is the quantum generalization of nondeterministic polynomial time complete our proof uses a simple mapping from spin systems to fermionic systems as well as a convex optimization technique that reduces the problem of finding ground states to n representability

the material point method in large strain engineering problems abstract the material point method is a variant of the finite element method formulated in an arbitrary lagrangian eulerian description of motion two kinds of spatial discretization are utilized in the method the motion of material points representing subregions of the analysed continuum is traced against a background of the computational element mesh the mesh can be chosen arbitrarily so that the main disadvantage of the finite element method related to mesh distortion is avoided application of the method to some large strain industrial problems as granular flow in a silo as well as geomechanical and plastic forming problems are shown

a system of partial differential equations modeling the competition for two complementary resources in flowing habitats abstract this paper examines a system of reaction diffusion equations arising from a flowing water habitat in this habitat one or two microorganisms grow while consuming two growth limiting complementary essential resources for the single population model the existence and uniqueness of a positive steady state solution is proved furthermore the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values mathematical analysis for the two competing populations is carried out more precisely the long time behavior is determined by using the monotone dynamical system theory when the semi trivial solutions are both unstable it is also shown that coexistence solutions exist by using the fixed point index theory when the semi trivial solutions are both asymptotically stable

the ugc hardness threshold of the lp grothendieck problem for p 2 we consider the problem of given an n n matrix a aij whose diagonal entries vanish approximating in polynomial time the number rm opt p a max left sum i j 1 na ij x ix j x 1 ldots x n in mathbb r n wedge left sum i 1 n x i p right 1 p le 1 right when p 2 this is simply the problem of computing the maximum eigenvalue of a whereas for p actually it suffices to take p log n it is the grothendieck problem on the complete graph which was shown to have a o log n approximation algorithm in nemirovski et al nemirovski a c roos t terlaky 1999 on maximization of quadratic form over intersection of ellipsoids with common center math program ser a86 3 463 473 megretski megretski a 2001 relaxations of quadratic programs in operator theory and system analysis systems approximation singular integral operators and related topics bordeaux 2000 vol 129 operator theory advances and applications birkhauser basel 365 392 charikar and wirth charikar m a wirth 2004 maximizing quadratic programs extending grothendieck s inequality proc 45th annual sympos foundations comput sci ieee computer society 54 60 and was used in the work of charikar and wirth noted above to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering thus the problem of approximating optp a interpolates between the spectral p 2 case and the correlation clustering p case from a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard wall potential well we design a polynomial time algorithm which given p 2 and an n n matrix a aij with zeros on the diagonal computes optp a up to a factor p e 30 log p on the other hand assuming the unique games conjecture ugc we show that it is np hard to approximate optp a up to a factor smaller than p e hence as p the ugc hardness threshold for computing optp a is exactly p e 1 o 1

a system of resource based growth models with two resources in the unstirred chemostat abstract models of single species growth in the unstirred chemostat on two growth limiting nonreproducing resources are considered for the case of two complementary resources the existence and uniqueness of a positive steady state solution is established it is also proved that the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values for the case of two substitutable resources the existence of a positive steady state solution is determined for a range of the parameter m n techniques include the maximum principle monotone method and global bifurcation theory the longtime behavior of the corresponding limiting system is given for a range of m n in the special case of m n the uniqueness and global attractivity of the positive steady state solution of the original system is established

steady state solutions of a reaction diffusion system arising from intraguild predation and internal storage abstract intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat at first we established the sharp a priori estimates for nonnegative solutions of the system which assure that all of nonnegative solutions belong to a special cone the selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions secondly existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone it turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive

a competition model for two resources in un stirred chemostat abstract this paper studies a un stirred chemostat with two species competing for two growth limiting non reproducing resources we determine the conditions for positive steady states of the two species and then consider the global attractors of the model in addition we obtain the conditions under which the two populations uniformly strongly persist or go to extinction since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species in particular it is found that both instability and bi stability subcases of the two semitrivial steady states are included in the coexistence region the two populations will go to extinction when both possess large diffusion rates if just one of them spreads faster with the other one diffusing slower then the related semitrivial steady state will be globally attracting the techniques used for the above results consist of the degree theory the semigroup theory and the maximum principle

percolation of unsatisfiability in finite dimensions the optimization of two dimensional boolean formulas is studied using percolation theory rare region arguments and boundary effects in contrast with mean field results there is no satisfiability transition as the constraint density is varied although there is a logical connectivity transition in the disconnected phase there is a transition in the solution time the thermodynamic ground state for this np hard optimization problem is unique local solutions can be adjoined to find the global ground state these results have implications for the computational study of disordered materials

fast computing global structural balance in signed networks based on memetic algorithm abstract structural balance is a large area of study in signed networks and it is intrinsically a global property of the whole network computing global structural balance in signed networks which has attracted some attention in recent years is to measure how unbalanced a signed network is and it is a nondeterministic polynomial time hard problem many approaches are developed to compute global balance however the results obtained by them are partial and unsatisfactory in this study the computation of global structural balance is solved as an optimization problem by using the memetic algorithm the optimization algorithm named meme sb is proposed to optimize an evaluation function energy function which is used to compute a distance to exact balance our proposed algorithm combines genetic algorithm and a greedy strategy as the local search procedure experiments on social and biological networks show the excellent effectiveness and efficiency of the proposed method

the computational complexity of the self avoiding walk on random lattices it is proved that finding a minimum energy n step self avoiding walk on a 2d square lattice with assigned site energies is an np complete problem however if the fraction of lowest energy sites exceeds the percolation threshold then it is argued the problem can almost always be solved in polynomial time

on the approximability of the ground state problem for ising spin glasses by quantum devices a lower bound result about the approximability of the energy function of ising spin glasses in the quantum context is determined the ground state problem for the three dimensional ising spin glasses represented on two level grids such that the vertical interactions are at most p n where n is the number of spins set on the grid is considered we prove that unless np bqp there is a constant 0 such that every quantum approximate polynomial time algorithm finds a solution with absolute error greater than p n with high probability infinitely often

exact ground states of large two dimensional planar ising spin glasses studying spin glass physics through analyzing their ground state properties has a long history although there exist polynomial time algorithms for the two dimensional planar case where the problem of finding ground states is transformed to a minimum weight perfect matching problem the reachable system sizes have been limited both by the needed cpu time and by memory requirements in this work we present an algorithm for the calculation of exact ground states for two dimensional ising spin glasses with free boundary conditions in at least one direction the algorithmic foundations of the method date back to the work of kasteleyn from the 1960s for computing the complete partition function of the ising model using kasteleyn cities we calculate exact ground states for huge two dimensional planar ising spin glass lattices up to 3000 2 spins within reasonable time according to our knowledge these are the largest sizes currently available kasteleyn cities were recently also used by thomas and middleton in the context of extended ground states on the torus moreover they show that the method can also be used for computing ground states of planar graphs furthermore we point out that the correctness of heuristically computed ground states can easily be verified finally we evaluate the solution quality of heuristic variants of the l bieche et al approach

on a system of reaction diffusion equations arising from competition in an unstirred chemostat a system of reaction diffusion equations modeling two species competition in an unstirred chemostat is considered the asymptotic behavior of solutions is given as a function of the parameters and it is determined when neither one or both competing populations survive techniques include the maximum principle theory of uniform persistence in infinite dimensional dynamical systems and the theory of strongly order preserving semidynamical system

j ising model on homogeneous archimedean lattices abstract we tackle the problem of finding analytical expressions describing the ground state properties of homogeneous archimedean lattices over which a generalized edwards anderson model j ising model is defined a local frustration analysis is performed based on representative cells for square lattices triangular lattices and honeycomb lattices the concentration of ferromagnetic f bonds x is used as the independent variable in the analysis 1 x is the concentration for antiferromagnetic a bonds where x spans the range 0 0 1 0 the presence of a bonds brings frustration whose clear manifestation is when bonds around the minimum possible circuit of bonds plaquette cannot be simultaneously satisfied the distribution of curved frustrated plaquettes within the representative cell is determinant for the evaluation of the parameters of interest such as average frustration segment energy per bond and fractional content of unfrustrated bonds two methods are developed to cope with this analysis one based on the direct probability of a plaquette being curved the other one is based on the consideration of the different ways bonds contribute to the particular plaquette configuration exact numerical simulations on a large number of randomly generated samples allow to validate previously described theoretical analysis it is found that the second method presents slight advantages over the first one however both methods give an excellent description for most of the range for x the small deviations at specific intervals of x for each lattice have to do with the self imposed limitations of both methods due to practical reasons a particular discussion for the point x 0 5 for each one of the lattices also shines light on the general trends of the properties described here

complexity of tile rotation problems in this paper we introduce tile rotation problems the instances or initial configurations are tile assignments on a n x n lattice board and the question to be answered is the following does there exist any configuration obtained horn the initial one by tile rotations only whose cost is less than a given bound notice that a zero cost configuration corresponds to a perfect tiling we prove here the np completeness for both the zero cost problem for a particular set of 5 tiles and the minimization problem for a particular set of 2 tiles finally by showing the polynomiality of some subproblems we establish complexity border results

more really is different abstract in 1972 p w anderson suggested that more is different meaning that complex physical systems may exhibit behavior that cannot be understood only in terms of the laws governing their microscopic constituents we strengthen this claim by proving that many macroscopic observable properties of a simple class of physical systems the infinite periodic ising lattice cannot in general be derived from a microscopic description this provides evidence that emergent behavior occurs in such systems and indicates that even if a theory of everything governing all microscopic interactions were discovered the understanding of macroscopic order is likely to require additional insights

the swendsen wang process does not always mix rapidly the swendsen wang process provides one possible dynamics for the q state potts model computer simulations of this process are widely used to estimate the expectations of various observables random variables of a potts system in the equilibrium or gibbs distribution the legitimacy of such simulations depends on the rate of convergence of the process to equilibrium as measured by the mixing time empirical observations suggest that the mixing time of the swendsen wang process is short in many instances of practical interest although proofs of this desirable behavior are known only for some very special cases nevertheless we show that there are occasions when the mixing time of the swendsen wang process is exponential in the size of the system this undesirable behavior is related to the phenomenon of first order phase transitions in potts systems with q 2 states

complexity of two dimensional patterns in dynamical systems such as cellular automata and iterated maps it is often useful to look at a language or set of symbol sequences produced by the system there are well established classification schemes such as the chomsky hierarchy with which we can measure the complexity of these sets of sequences and thus the complexity of the systems which produce them in this paper we look at the first few levels of a hierarchy of complexity for two or more dimensional patterns we show that several definitions of regular language or local rule that are equivalent in d 1 lead to distinct classes in d 2 we explore the closure properties and computational complexity of these classes including undecidability and l nl and np completeness results we apply these classes to cellular automata in particular to their sets of fixed and periodic points finite time images and limit sets we show that it is undecidable whether a ca in d 2 has a periodic point of a given period and that certain local lattice languages are not finite time images or limit sets of any ca we also show that the entropy of a d dimensional ca s finite time image cannot decrease faster than t d unless it maps every initial condition to a single homogeneous state

consistent inference of a general model using the pseudolikelihood method recently a maximum pseudolikelihood mpl inference method has been successfully applied to statistical physics models with intractable likelihoods we use information theory to derive a relation between the pseudolikelihood and likelihood functions furthermore we show the consistency of the pseudolikelihood method for a general model

making classical ground state spin computing fault tolerant we examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation this model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions in its most primitive form systems constructed in this model cannot compute in an error free manner when working at nonzero temperature however by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free we achieve this by using techniques in fault tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature we further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class merlin arthur this provides an interesting connection between the physical behavior of certain many body spin systems and computational complexity