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 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

the equivalence of the dla and a hydrodynamic model the authors prove that the growth process governed by the diffusion limited aggregation dla algorithm as well as by some of its natural extensions is isomorphous to the process of pushing an incompressible newtonian fluid by an inviscid fluid through a porous medium in the chamber tube ct model provided that the parameters of the ct model are properly chosen in this way they give a direct quantitative relation between hydrodynamics in random porous media and the fractal theory of stochastic dla like aggregates on the other hand this work presents an example of two equivalent dynamical systems one of them with stochastic dynamics applied to a deterministic medium and the other with deterministic dynamics applied to a random medium

the large n limit of the threshold values in mandelbrot s fractal percolation process the authors consider mandelbrot s fractal percolation process characterised by a density parameter p in 0 1 and an integer subdivision index n 1 for each n the process is known to have a percolation transition at density pc n in 0 1 they prove that lim pc n pc where pc is the threshold value of the ordinary square lattice site percolation model

phase diagram and correlation length bounds for mandelbrot aerogels the authors study a variant of the mandelbrot percolation process which is of current use as a model of aerogels the model has two parameters one of them q is the usual multiscale parameter of the mandelbrot percolation process and the other p is a bernoulli percolation parameter that is reserved for the last step of the construction they investigate the phase diagram of this model in the q p plane there are two phases a sol phase and a gel phase classified according to whether a limit of crossing probabilities vanishes or is nonzero in the sol phase they define a correlation length via the rate of decay of a rescaled connectivity function they show that this length scale diverges at the phase boundary furthermore they demonstrate that if the phase boundary is approached with q fixed and p tending up to its critical value pg q then up to logarithmic corrections the divergence is at least as fast as mod pg p mod 2dh where dh 2 mod log q log n mod can be identified as the hausdorff dimension of the background medium

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

predicting nonlinear cellular automata quickly by decomposing them into linear ones we show that a wide variety of nonlinear cellular automata cas can be decomposed into a quasidirect product of linear ones these cas can be predicted by parallel circuits of depth o log2 t using gates with binary inputs or o log t depth if sum mod p gates with an unbounded number of inputs are allowed thus these cas can be predicted by idealized parallel computers much faster than by explicit simulation even though they are nonlinear this class includes any ca whose rule when written as an algebra is a solvable group we also show that cas based on nilpotent groups can be predicted in depth o log t or o 1 by circuits with binary or sum mod p gates respectively we use these techniques to give an efficient algorithm for a ca rule which like elementary ca rule 18 has diffusing defects that annihilate in pairs this can be used to predict the motion of defects in rule 18 in o log2 t parallel time

circuit definitions of nondeterministic complexity classes we consider restictions on boolean circuits and use them to obtain new uniform circuit characterizations of nondeterministic space and time classes we also obtain characterizations of counting classes based on nondeterministic time bounded computations on the arithmetic circuit model it is shown how the notion of semiunboundedness unifies the definitions of many natural complexity classes

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

on the structure of mandelbrot s percolation process and other random cantor sets we consider generalizations of mandelbrot s percolation process for the process which we call the random sierpinski carpet we show that it passes through several different phases as its parameter increases from zero to one the final section treats the percolation phase

collision integrals for attractive potentials motivated by previous discussions of particle interactions under the manev potential u r r e r2 we construct the collision integrals for attractive potentials u r satisfying the condition u r r2 e as r 0 with e 0 for e 0 we obtain a boltzmann type integral with a collision law allowing spiral interactions and nonunique correspondence between impact parameter and scattering angle for e 0 an additional smoluchowski type coagulation integral arises all these integrals are derived and possible applications are discussed

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

the computational complexity of pattern formation the computational complexity of diffusion limited aggregation and fluid invasion in porous media is studied the time requirements on an idealized parallel computer for simulating the patterns formed by these models are investigated it is shown that these growth models are p complete these results provide strong evidence that such pattern formation processes are inherently sequential and cannot be simulated efficiently in parallel

many body problem with quadratic and or inversely quadratic potentials in one and more dimensional spaces some retrospective remarks in the context of an ambient space with an arbitrary number d of dimensions the many body problem consisting of an arbitrary number n of particles confined by a common external harmonic potential realizing a container with soft walls and interacting among themselves and with the environment with arbitrary conservative repulsive forces scaling as the inverse cube of distances displays a peculiar behaviour its effective volume oscillates isochronously without damping we recently discovered this remarkable phenomenon valid in the context of both classical and quantum mechanics and discussed its implications in the context of statistical mechanics and thermodynamics but after publishing these findings we were informed that essentially analogous results had been previously obtained by lyndell bell and lyndell bell in the present paper motivated by the need we felt to acknowledge this fact we also offer some retrospective remarks on the n body problem with quadratic and or inversely quadratic potentials in one and more dimensional space

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem we develop the cauchy theory of the spatially homogeneous inelastic boltzmann equation for hard spheres for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model we prove local in time non concentration estimates in orlicz spaces from which we deduce weak stability and existence theorem strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum for a restricted class of collision rates concerning the long time behaviour we give conditions for the cooling process to occur or not in finite time

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

sticky particles and scalar conservation laws one dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line which stick under collision with conservation of mass and momentum

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

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

parallel algorithm and dynamic exponent for diffusion limited aggregation a parallel algorithm for diffusion limited aggregation dla is described and analyzed from the perspective of computational complexity the dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random access machine pram model of parallel computation according to t sim l z where l is the cluster size t is the running time and the algorithm uses a number of processors polynomial in l it is argued that z d d 2 2 where d is the fractal dimension and d 2 is the second generalized dimension simulations of dla are carried out to measure d 2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm it is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of dla

the computational complexity of sandpiles given an initial distribution of sand in an abelian sandpile what final state does it relax to after all possible avalanches have taken place in d 3 we show that this problem is p complete so that explicit simulation of the system is almost certainly necessary we also show that the problem of determining whether a sandpile state is recurrent is p complete in d 3 and briefly discuss the problem of constructing the identity in d 1 we give two algorithms for predicting the sandpile on a lattice of size n both faster than explicit simulation a serial one that runs in time mathcal o n log n and a parallel one that runs in time mathcal o log3n i e the class nc3 the latter is based on a more general problem we call additive ranked generability this leaves the two dimensional case as an interesting open problem

on the geometric ergodicity of metropolis hastings algorithms for lattice gaussian sampling sampling from the lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper the classic metropolis hastings mh algorithm in markov chain monte carlo methods is adopted for lattice gaussian sampling two mh based algorithms are proposed which overcome the limitation of klein s algorithm the first one referred to as the independent metropolis hastings klein mhk algorithm establishes a markov chain via an independent proposal distribution we show that the markov chain arising from this independent mhk algorithm is uniformly ergodic namely it converges to the stationary distribution exponentially fast regardless of the initial state moreover the rate of convergence is analyzed in terms of the theta series leading to predictable mixing time a symmetric metropolis klein algorithm is also proposed which is proven to be geometrically ergodic

on a kinetic equation for coalescing particles existence of global weak solutions to a spatially inhomogeneous kinetic model for coalescing particles is proved each particle being identified by its mass momentum and position the large time convergence to zero is also shown the cornestone of our analysis is that for any nonnegative and convex function the associated orlicz norm is a liapunov functional existence and asymptotic behaviour then rely on weak and strong compactness methods in l1 in the spirit of the diperna lions theory for the boltzmann equation

structural and computational depth of diffusion limited aggregation diffusion limited aggregation dla is studied from the perspective of computational complexity a parallel algorithm is exhibited that requires a number of steps that scales as the depth of the tree defined by the cluster the existence of this algorithm suggests a connection between a fundamental computational and structural property of dla

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

theta function identities and the explicit formulas for theta function and their applications abstract we define two quotients of theta functions depending on two positive real parameters we then show how they are connected with two parameters of dedekind eta function and the ramanujan weber class invariants explicit formulas for determining values of the theta function is derived and several examples will be given and using them we give some complete explicit results for the complete elliptic integral of the first kind and the gaussian hypergeometric function also several new modular equations for the theta function are derived

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

majority vote cellular automata ising dynamics and p completeness we study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood these are equivalent for a restricted set of initial conditions to nonzero probability transitions in single spin flip dynamics of the ising model at zero temperature we show that in three or more dimensions these systems can simulate boolean circuits of and and or gates and are therefore p complete that is predicting their state t time steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer therefore unless a widely believed conjecture in computer science is false it is impossible even with parallel computation to predict majority vote cellular automata or zero temperature single spin flip ising dynamics qualitatively faster than by explicit simulation

the continuous coagulation fragmentation equations with diffusion existence of global weak solutions to the continuous coagulation fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition our approach relies on weak and strong compactness methods in l 1 in the spirit of the diperna lions theory for the boltzmann equation under the detailed balance condition the large time behaviour is also studied

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

average case complete problems many interesting combinatorial problems were found to be np complete since there is little hope to solve them fast in the worst case researchers look for algorithms which are fast just on average this matter is sensitive to the choice of a particular np complete problem and a probability distribution of its instances some of these tasks are easy and some not but one needs a way to distinguish the difficult on average problems such negative results could not only save positive efforts but may also be used in areas like cryptography where hardness of some problems is a frequent assumption it is shown in 1 that the tiling problem with uniform distribution of instances has no polynominal on average algorithm unless every np problem with every simple probability distribution has it

phase transitions in mandelbrot s percolation process in three dimensions we study the phase structure and transitions in three dimensional mandelbrot percolation a process which generates random fractal sets we establish the existence of three distinct phase transitions and we show that two of these transitions corresponding to percolation across the initial set by paths and sheets are discontinuous

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

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

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

parallel dynamics and computational complexity of network growth models the parallel computational complexity or depth of growing network models is investigated the networks considered are generated by preferential attachment rules where the probability of attaching a new node to an existing node is given by a power alpha of the connectivity of the existing node algorithms for generating growing networks very quickly in parallel are described and studied the sublinear and superlinear cases require distinct algorithms as a result there is a discontinuous transition in the parallel complexity of sampling these networks corresponding to the discontinuous structural transition at alpha 1 where the networks become scale free for alpha 1 networks can be generated in constant time while for 0 alpha 1 logarithmic parallel time is required the results show that these networks have little depth and embody very little history dependence despite being defined by sequential growth rules

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

coagulation fragmentation processes we study the well posedness of coagulation fragmentation models with diffusion for large systems of particles the continuous and the discrete case are considered simultaneously in the discrete situation we are concerned with a countable system of coupled reaction diffusion equations whereas the continuous case amounts to an uncountable system of such equations these problems can be handled by interpreting them as abstract vector valued parabolic evolution equations where the dependent variables take values in infinite dimensional banach spaces given suitable assumptions we prove existence and uniqueness in the class of volume preserving solutions we also derive sufficient conditions for global existence

mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation abstract this study presents the convergence analysis of the recently developed finite volume preserving scheme forestier coste and mancini 2012 for approximating a coalescence or smoluchowski equation the idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins cells the consistency of the finite volume scheme is examined thoroughly in order to prove second order convergence on uniform non uniform smooth and locally uniform grids independently of the aggregation kernel the theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels

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

on vlasov manev equations ii local existence and uniqueness we prove that the initial value problem associated with the vlasov manev system a vlasov equation in which a correction of type e r 2 is added to the newtonian or coulomb potential has a local in time classical and unique solution for sufficiently regular initial data

correlation length bounds for disordered ising ferromagnets thed dimensional nearest neighbor disordered ising ferromagnet h sum j ij sigma i sigma j is studied as a function of both temperature t and a disorder parameter which measures the size of fluctuations of couplingsjij 0 a finite size scaling correlation length f t is defined in terms of the magnetic response of finite samples this correlation length is shown to be equivalent in the scaling sense to the quenched average correlation length t defined as the asymptotic decay rate of the quenched average two point function furthermore the magnetic response criterion which defines f is shown to have a scale invariant property at the critical point the above results enable us to prove that the quenched correlation length satisfies c left log xi t right xi t geqq left t t c right 2 mathord left vphantom 2 d right kern nulldelimiterspace d which implies the boundv 2 d for the quenched correlation length exponent

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

stability and uniqueness for the spatially homogeneous boltzmann equation with long range interactions in this paper we prove some a priori stability estimates in weighted sobolev spaces for the spatially homogeneous boltzmann equation without angular cutoff covering all physical collision kernels these estimates are conditional on some regularity estimates on the solutions and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions we then prove such regularity bounds for a class of interactions including the so called non cutoff and non mollified hard potentials and moderately soft potentials in particular we obtain the first result of global existence and uniqueness for these long range interactions

the explicit formulas and evaluations of ramanujan s theta function ψ abstract we define two quotients of theta function depending on two positive real parameters we then show how they are connected with two parameters of dedekind eta function theta function and the ramanujan weber class invariants explicit formulas for determining values of the theta function are derived and several examples will be given in addition we give some applications of these parameters for the famous rogers ramanujan continued fraction r q ramanujan s cubic continued fraction g q and the modular j invariant