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

convergence properties of the softassign quadratic assignment algorithm the softassign quadratic assignment algorithm is a discrete time continuous state synchronous updating optimizing neural network while its effectiveness has been shown in the traveling salesman problem graph matching and graph partitioning in thousands of simulations its convergence properties have not been studied here we construct discrete time lyapunov functions for the cases of exact and approximate doubly stochastic constraint satisfaction which show convergence to a fixed point the combination of good convergence properties and experimental success makes the softassign algorithm an excellent choice for neural quadratic assignment optimization

theory of neuromata a finite automaton the so called neuromaton realized by a finite discrete recurrent neural network working in parallel computation mode is considered both the size of neuromata i e the number of neurons and their descriptional complexity i e the number of bits in the neuromaton representation are studied it is proved that a constraint time delay of the neuromaton output does not play a role within a polynomial descriptional complexity it is shown that any regular language given by a regular expression of length n is recognized by a neuromaton with n neurons further it is proved that this network size is in the worst case optimal on the other hand generally there is not an equivalent polynomial length regular expression for a given neuromaton then two specialized constructions of neural acceptors of the optimal descriptional complexity n for a single n bit string recognition are described they both require o n1 2 neurons and either o n connections with constant weights or o n1 2 edges with weights of the o 2 n size furthermore the concept of hopfield languages is introduced by means of so called hopfield neuromata i e of neural networks with symmetric weights it is proved that the class of hopfield languages is strictly contained in the class of regular languages the necessary and sufficient so called hopfield condition stating when a regular language is a hopfield language is formulated a construction of a hopfield neuromaton is presented for a regular language satisfying the hopfield condition the class of hopfield languages is shown to be closed under union intersection concatenation and complement and it is not closed under iteration finally the problem whether a regular language given by a neuromaton or by a hopfield acceptor is nonempty is proved to be pspace complete as a consequence the same result for a neuromaton equivalence problem is achieved

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

boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains in a uniform domain we investigate the boundary behavior of positive superharmonic functions u satisfying the nonlinear inequality u x 0 and p 0 where is the laplacian and x is the distance from a point x to the boundary of in particular we present a fatou type theorem concerning the existence of nontangential limits and a littlewood type theorem concerning the nonexistence of tangential limits

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

thin sets at the boundary in his letter 18 hayman raised the following questions if u is negative subharmonic in the unit disk then for almost all u z has a limit as z tends to e nontangentially and outside a near thin set 17 chapter 7 there ought to be a global result i e not in an angle outside a suitable exceptional set what should it be it might be easier for green potentials where the limit is always zero of course the questions also exist in higher dimensions the purpose of this paper is to give an answer to hayman s questions we shall show that there are exceptional sets suitable for the description of especially tangential boundary behavior of superharmonic functions as a result we shall prove that the above nontangential restriction can be relaxed see corollary 1 1 let us first recall the notion of minimal thinness this notion can be defined in a very general framework let d be a martin space with martin boundary md a set e d is called minimally thin at a minimal boundary point x if the reduced function over e of the martin kernel at x is a potential the fatou naim doob theorem describes the boundary behavior of nonnegative superharmonic functions or equivalently nonpositive subharmonic functions with minimally thin sets

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

quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions we show that ii a set e is dispersely decomposed into subsets then the capacity of e is comparable to the summation of tile capacities of the subsets from this fact it is derived that the lebesgue measure of a certain expanded set is estimated by the ca

exponential transients in continuous time liapunov systems we consider the convergence behavior of a class of continuous time dynamical systems corresponding to so called symmetric hopfield nets studied in neural networks theory we prove that such systems may have transient times that are exponential in the system dimension i e number of neurons despite the fact that their dynamics are controlled by liapunov functions this result stands in contrast to many proposed uses of such systems in e g combinatorial optimization applications in which it is often implicitly assumed that their convergence is rapid an additional interesting observation is that our example of an exponential transient continuous time system a simulated binary counter in fact converges more slowly than any discrete time hopfield system of the same representation size this suggests that continuous time systems may be worth investigating for gains in descriptional efficiency as compared to their discrete time counterparts

positive neural networks in discrete time implement monotone regular behaviors we study the expressive power of positive neural networks the model uses positive connection weights and multiple input neurons different behaviors can be expressed by varying the connection weights we show that in discrete time and in the absence of noise the class of positive neural networks captures the so called monotone regular behaviors which are based on regular languages a finer picture emerges if one takes into account the delay by which a monotone regular behavior is implemented each monotone regular behavior can be implemented by a positive neural network with a delay of one time unit some monotone regular behaviors can be implemented with zero delay and interestingly some simple monotone regular behaviors cannot be implemented with zero delay

discrete time analogs for a class of continuous time recurrent neural networks this paper is concerned with the problem of local and global asymptotic stability for a class of discrete time recurrent neural networks which provide discrete time analogs to their continuous time counterparts i e continuous time recurrent neural networks with distributed delay some stability criteria which include some existing results as their special cases are derived a discussion about the dynamical consistence of discrete time neural networks versus their continuous time counterparts is provided an unconventional finite difference method is proposed and an example is also given to show the effectiveness of the method

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

efficient simulation of finite automata by neural nets let k m denote the smallest number with the property that every m state finite automaton can be built as a neural net using k m or fewer neurons a counting argument shows that k m is at least o m log m 1 3 and a construction shows that k m is at most o m 3 4 the counting argument and the construction allow neural nets with arbitrarily complex local structure and thus may require neurons that themselves amount to complicated networks mild and in practical situations almost necessary constraints on the local structure of the network give again by a counting argument and a construction lower and upper bounds for k m that are both linear in m

discrete time convergence theory and updating rules for neural networks with energy functions we present convergence theorems for neural networks with arbitrary energy functions and discrete time dynamics for both discrete and continuous neuronal input output functions we discuss systematically how the neuronal updating rule should be extracted once an energy function is constructed for a given application in order to guarantee the descent and minimization of the energy function as the network updates we explain why the existing theory may lead to inaccurate results and oscillatory behaviors in the convergence process we also point out the reason for and the side effects of using hysteresis neurons to suppress these oscillatory behaviors

the computational power of discrete hopfield nets with hidden units we prove that polynomial size discrete hopfield networks with hidden units compute exactly the class of boolean functions pspace poly i e the same functions as are computed by polynomial space bounded nonuniform turing machines as a corollary to the construction we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions p poly i e the class computed by polynomial time bounded nonuniform turing machines

a finite element formulation for the coupled thermoviscoelastic response of solid fluid systems the coupled thermoviscoelastic response of a solid fluid system is computed by a finite element formulation which involves only temperature and displacement fields the laplace transform with respect to time is applied to the coupled equations this results in a global matrix in the transform domain that is symmetric and banded and the time response is obtained by numerical inversion of the transformed solution also this formulation permits the use of fractional derivatives to model viscoelastic material behaviour

convergence analysis of a discrete time recurrent neural network to perform quadratic real optimization with bound constraints presents a model of a discrete time recurrent neural network designed to perform quadratic real optimization with bound constraints the network iteratively improves the estimate of the solution always maintaining it inside of the feasible region several neuron updating rules which assure global convergence of the net to the desired minimum have been obtained some of them also assure exponential convergence and maximize a lower bound for the convergence degree simulation results are presented to show the net performance

on the computational complexity of binary and analog symmetric hopfield nets we investigate the computational properties of finite binary and analog state discrete time symmetric hopfield nets for binary networks we obtain a simulation of convergent asymmetric networks by symmetric networks with only a linear increase in network size and computation time then we analyze the convergence time of hopfield nets in terms of the length of their bit representations here we construct an analog symmetric network whose convergence time exceeds the convergence time of any binary hopfield net with the same representation length further we prove that the min energy problem for analog hopfield nets is np hard and provide a polynomial time approximation algorithm for this problem in the case of binary nets finally we show that symmetric analog nets with an external clock are computationally turing universal

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

champagne subdomains with unavoidable bubbles abstract a champagne subdomain of a connected open set u 0 in r d d 2 is obtained by omitting pairwise disjoint closed balls b x r x x x the bubbles where x is an infinite locally finite set in u the union a of these balls may be unavoidable that is brownian motion starting in u a and killed when leaving u may hit a almost surely or equivalently a may have harmonic measure 1 for u a recent publications by gardiner and ghergu d 3 and by pres d 2 give rather sharp answers to the question of how small such a set a may be when u is the unit ball in this paper using a totally different approach optimal results are obtained which hold also for arbitrary connected open sets u

continuous time symmetric hopfield nets are computationally universal we establish a fundamental result in the theory of computation by continuous time dynamical systems by showing that systems corresponding to so called continuous time symmetric hopfield nets are capable of general computation as is well known such networks have very constrained lyapunov function controlled dynamics nevertheless we show that they are universal and efficient computational devices in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discrete time binary neurons with in general asymmetric coupling weights can be simulated by a symmetric continuous time hopfield net containing only 18n 7 units employing the saturated linear activation function moreover if the asymmetric network has maximum integer weight size wmax and converges in discrete time t then the corresponding hopfield net can be designed to operate in continuous time t e for any e 0 such that wmax212n e21 e in terms of standard discrete computation models our result implies that any polynomially space bounded turing machine can be simulated by a family of polynomial size continuous time symmetric hopfield nets

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

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 the design of bsb neural associative memories using semidefinite programming this article is concerned with the reliable search for optimally performing bsb brain state in a box neural associative memories given a set of prototype patterns to be stored as stable equilibrium points by converting and or modifying the nonlinear constraints of a known formulation for the synthesis of bsb based associative memories into linear matrix inequalities we recast the synthesis into semidefinite programming problems and solve them by recently developed interior point methods the validity of this approach is illustrated by a design example

quasi periodic β expansions and cut languages abstract motivated by the analysis of neural net models between integer and rational weights we introduce a so called cut language over a real digit alphabet which contains finite expansions i e base representations of the numbers less than a given threshold we say that an infinite expansion is eventually quasi periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits contains an infinite constant subsequence we prove that a cut language is regular iff its threshold is a quasi periodic number whose all expansions are eventually quasi periodic by showing that altogether they have a finite number of tail values for algebraic bases we prove that there is an eventually quasi periodic expansion with an infinite number of tail values iff there is a conjugate of on the unit circle for transcendental combined with algebraic digits a expansion is eventually quasi periodic iff it has a finite number of tail values for a pisot base and digits from the smallest field extension q over rational numbers including we show that any number from q is quasi periodic in addition we achieve a dichotomy that a cut language is either regular or non context free and we show that any cut language with rational parameters is context sensitive

unavoidable collections of balls for censored stable processes abstract we study avoidability of collections of balls in bounded c 1 1 opens sets for censored stable processes 1 2 the results are analogous to the ones obtained for brownian motion in gardiner and ghergu 2010 11 on the way we derive a wiener aikawa type criterion for minimal thinness with respect to the censored stable processes

interpolating sequences for besov spaces abstract we characterise the interpolating sequences for the besov spaces b p and for their multiplier spaces we also construct linear operators of interpolation

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

dynamics of discrete time continuous state hopfield networks the dynamics of discrete time continuous state hopfield networks is driven by an energy function in this paper we use this tool to prove under mild hypotheses that any trajectory converges to a fixed point for the sequential iteration and to a cycle of length 2 or a fixed point for the parallel iteration perhaps surprisingly it seems that no rigorous proof of these results was published before