parallel and sequential computation on boolean networks abstract a model of large assemblies of computing elements is provided by the concept of boolean network f 0 1 n 0 1 n then we compare different regimes of this system namely parallel and sequential computations within the framework of discrete iterations it is shown that although the asymptotic behavior of the network as characterized by the limit cycles of the iterations may differ from one sort of regime to another it may however be largely predicted as for its spatial structure we show that for all iterations some elements never change state in any limit cycle they form the stable core we introduce the notion of forcing domain of a boolean network provide an algorithm to build it and show that it is a good approximation to the stable core this thus provides a spatial characterization of the limit cycles of the different iterations a theoretical bound on the transient length is provided through the entropy of the network

discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

chaotic simulated annealing by a neural network with a variable delay design and application in this paper we have three goals the first is to delineate the advantages of a variably delayed system the second is to find a more intuitive lyapunov function for a delayed neural network and the third is to design a delayed neural network for a quadratic cost function for delayed neural networks most researchers construct a lyapunov function based on the linear matrix inequality lmi approach however that approach is not intuitive we provide a alternative candidate lyapunov function for a delayed neural network on the other hand if we are first given a quadratic cost function we can construct a delayed neural network by suitably dividing the second order term into two parts a self feedback connection weight and a delayed connection weight to demonstrate the advantage of a variably delayed neural network we propose a transiently chaotic neural network with variable delay and show numerically that the model should possess a better searching ability than chen aihara s model wang s model and zhao s model we discuss both the chaotic and the convergent phases during the chaotic phase we simply present bifurcation diagrams for a single neuron with a constant delay and with a variable delay we show that the variably delayed model possesses the stochastic property and chaotic wandering during the convergent phase we not only provide a novel lyapunov function for neural networks with a delay the lyapunov function is independent of the lmi approach but also establish a correlation between the lyapunov function for a delayed neural network and an objective function for the traveling salesman problem

a cocycle model for topological and lie group cohomology we propose a unified framework in which the different constructions of cohomology groups for topological and lie groups can all be treated on equal footings in particular we show that the cohomology of locally continuous cochains respectively locally smooth in the case of lie groups fits into this framework which provides an easily accessible cocycle model for topological and lie group cohomology we illustrate the use of this unified framework and the relation between the different models in various applications this includes the construction of cohomology classes characterizing the string group and a direct connection to lie algebra cohomology

rotational invariance and discrete analyticity in the 2d dimer model we exploit the discrete analytic structure present in the 2d dimer model to rigorously compute the exponents of a class of two point functions in all directions this is a dimer analogue of the critical 2d ising spin spin correlation function

height fluctuations in the honeycomb dimer model we study a model of random surfaces arising in the dimer model on the honeycomb lattice for a fixed wire frame boundary condition as the lattice spacing 0 cohn kenyon and propp 3 showed the almost sure convergence of a random surface to a non random limit shape 0 in 12 okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions in particular constructing a natural conformal structure on them we show here that when 0 has no facets for a family of boundary conditions approximating the wire frame the large scale surface fluctuations height fluctuations about 0 converge as 0 to a gaussian free field for the above conformal structure we also show that the local statistics of the fluctuations near a given point x are as conjectured in 3 given by the unique ergodic gibbs measure on plane configurations whose slope is the slope of the tangent plane of 0 at x

dimers in piecewise temperleyan domains we study the large scale behavior of the height function in the dimer model on the square lattice richard kenyon has shown that the fluctuations of the height function on temperleyan discretizations of a planar domain converge in the scaling limit as the mesh size tends to zero to the gaussian free field with dirichlet boundary conditions we extend kenyon s result to a more general class of discretizations moreover we introduce a new factorization of the coupling function of the double dimer model into two discrete holomorphic functions which are similar to discrete fermions defined in smirnov proceedings of the international congress of mathematicians icm madrid spain 2006 ann math 2 172 1435 1467 2010 for temperleyan discretizations with appropriate boundary modifications the results of kenyon imply that the expectation of the double dimer height function converges to a harmonic function in the scaling limit we use the above factorization to extend this result to the class of all polygonal discretizations that are not necessarily temperleyan furthermore we show that quite surprisingly the expectation of the double dimer height function in the temperleyan case is exactly discrete harmonic for an appropriate choice of laplacian even before taking the scaling limit

the emergence of the electrostatic field as a feynman sum in random tilings with holes we consider random lozenge tilings on the triangular lattice with holes q 1 q n in some fixed position for each unit triangle not in a hole consider the average orientation of the lozenge covering it we show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole q i as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside q i this is then restated in terms of random surfaces yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is in the scaling limit a sum of helicoids

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

on the asymptotics of dimers on tori we study asymptotics of the dimer model on large toric graphs let mathbb l be a weighted mathbb z 2 periodic planar graph and let mathbb z 2 e be a large index sublattice of mathbb z 2 for mathbb l bipartite we show that the dimer partition function textsf textit z e on the quotient mathbb l mathbb z 2 e has the asymptotic expansion begin aligned textsf textit z exp a mathbf f 0 textsf fsc o 1 end aligned where a is the area of mathbb l mathbb z 2 e mathbf f 0 is the free energy density in the bulk and textsf fsc is a finite size correction term depending only on the conformal shape of the domain together with some parity type information assuming a conjectural condition on the zero locus of the dimer characteristic polynomial we show that an analogous expansion holds for mathbb l non bipartite the functional form of the finite size correction differs between the two classes but is universal within each class our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double dimer models

non fragile finite time l 2 l state estimation for discrete time markov jump neural networks with unreliable communication links this paper is concerned with the problem of finite time l 2 l non fragile state estimation for discrete time markov jump neural networks with unreliable communication links the simultaneous occurrences of packet dropouts time delays and the sensor nonlinearity stemmed from the unreliable communication links are considered the focus is on the design of non fragile state estimator such that the augmented estimation error system is mean square stochastically finite time stable with a prescribed level of l 2 l performance by employing lyapunov krasovskii approach and finite time analysis theory some sufficient conditions have been obtained for the existence of an admissible state estimator finally a numerical example is employed to demonstrate the effectiveness of our proposed approach

emergent properties in random complex automata abstract studies of large randomly assembled binary boolean node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior an important approach to characterizing these behaviors has consisted of studying ensembles of automata ensemble specifications have been based on 1 choice of boolean function regulating each node in the automaton 2 random or biased mappings of the 2 n automata states into themselves 3 the numbers of inputs k per node this article briefly reviews these approaches reports new results on the expected behavior of threshold automata and automata with a particular class of biased mappings in addition it discusses emergent highly ordered dynamical behavior in random automata rich in a specific class of canalizing boolean functions due to the crystallization of powerful subautomata called forcing structures the ordered behaviors may be of importance in biological evolution in physics and in the design of adaptive automata

a homology theory for etale groupoids etale groupoids arise naturally as models for leaf spaces of foliations for orbifolds and for orbit spaces of discrete group actions in this paper we introduce a sheaf homology theory for etale groupoids we prove its invariance under morita equivalence as well as verdier duality between haeiger cohomology and this homology we also discuss the relation to the cyclic and hochschild homologies of connes convolution algebra of the groupoid and derive some spectral sequences which serve as a tool for the computation of these homologies

delay distribution dependent exponential stability criteria for discrete time recurrent neural networks with stochastic delay this brief is concerned with the analysis problem of global exponential stability in the mean square sense for a class of linear discrete time recurrent neural networks drnns with stochastic delay different from the prior research works the effects of both variation range and probability distribution of the time delay are involved in the proposed method first a modeling method is proposed by translating the probability distribution of the time delay into parameter matrices of the transformed drnn model where the delay is characterized by a stochastic binary distributed variable based on the new method the global exponential stability in the mean square sense for the drnns with stochastic delay is investigated by using the lyapunov krasovskii functional and exploiting some new analysis techniques a numerical example is provided to show the effectiveness and the applicability of the proposed method

conformal invariance of loops in the double dimer model the dimer model is the study of random dimer covers perfect matchings of a graph a double dimer configuration on a graph mathcal g is a union of two dimer covers of mathcal g we introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes relative to a fixed set of faces of loops in the double dimer model on a planar graph as an application we prove that in the scaling limit of the uniform double dimer model on mathbb z 2 or on any other bipartite planar graph conformally approximating mathbb c the loops are conformally invariant

an index theorem for wiener hopf operators abstract we study the multivariate generalisation of the classical wiener hopf algebra which is the c algebra generated by the wiener hopf operators given by convolutions restricted to convex cones by the work of muhly and renault this c algebra is known to be isomorphic to the reduced c algebra of a certain restricted action groupoid it admits a composition series and therefore a symbol calculus using groupoid methods we obtain in the framework of kasparov s bivariant kk theory a topological expression of the index maps associated to these symbol maps in terms of geometric topological data of the underlying convex cone this generalises an index theorem by upmeier concerning wiener hopf operators on symmetric cones our result covers a wide class of cones containing polyhedral and homogeneous cones

spiking neural p systems with polarizations spiking neural p sn p systems are a class of parallel computation models inspired by neurons where the firing condition of a neuron is described by a regular expression associated with spiking rules however it is np complete to decide whether the number of spikes is in the length set of the language associated with the regular expression in this paper in order to avoid using regular expressions two major and rather natural modifications in their form and functioning are proposed the spiking rules no longer check the number of spikes in a neuron but in exchange a polarization is associated with neurons and rules one of the three electrical charges 0 surprisingly enough the computing devices obtained are still computationally complete which are able to compute all turing computable sets of natural numbers on this basis the number of neurons in a universal sn p system with polarizations is estimated several research directions are mentioned at the end of this paper

the ring structure for equivariant twisted k theory we prove under some mild conditions that the equivariant twisted k theory group of a crossed module admits a ring structure if the twisting 2 cocycle is 2multiplicative we also give an explicit construction of the transgression map t1 h a h 1 n o a for any crossed module n and prove that any element in the image is multiplicative as a consequence we prove that under some mild conditions for a crossed module n and any e s that the equivariant twisted k theory group k e n admits a ring structure as an application we prove that for a compact connected and simply connected lie group g the equivariant twisted k theory group k c g g is endowed with a canonical ring structurek c g g k j d c g g k i j d c g g where d dimg and c h gog s

the brauer group of a locally compact groupoid we define the brauer group br g of a locally compact groupoid g to be the set of morita equivalence classes of pairs a consisting of an elementary c bundle a over g 0 satisfying fell s condition and an action of g on a by isomorphisms when g is the transformation groupoid x h then br g is the equivariant brauer group br h x in addition to proving that br g is a group we prove three isomorphism results first we show that if g and h are equivalent groupoids then br g and br h are isomorphic this generalizes the result that if g and h are groups acting freely and properly on a space x say g on the left and h on the right then br g x h and br h g x are isomorphic secondly we show that the subgroup br0 g of br g consisting of classes a with a having trivial dixmier douady invariant is isomorphic to a quotient e g of the collection tw g of twists over g finally we prove that br g is isomorphic to the inductive limit ext g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of the groups e g x where x varies over all principal g spaces x and g x is the imprimitivity groupoid associated to x

loop statistics in the toroidal honeycomb dimer model the dimer model on a graph embedded in the torus can be interpreted as a collection of random self avoiding loops in this paper we consider the uniform toroidal honeycomb dimer model we prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed the winding number of the collection of loops converges in law to a two dimensional discrete gaussian distribution this is known to physicists in more generality from their analysis of toroidal two dimensional critical loop models and their mapping to the massless free field on the torus this paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model

finite time non fragile passivity control for neural networks with time varying delay abstract in this paper the problem of finite time non fragile passivity control for neural networks with time varying delay is studied we construct a new lyapunov krasovskii function with triple and four integral terms and then utilizing wirtinger type inequality technique the sufficient conditions for finite time boundedness and finite time passivity are derived furthermore a non fragile state feedback controller is designed such that the closed loop system is finite time passive moreover the proposed sufficient conditions can be simplified into the form of linear matrix inequalities lmis using matlab lmi toolbox finally three numerical examples are presented to illustrate the effectiveness of the proposed criteria

groupoid cohomology and extensions we show that haefliger s cohomology for etale groupoids moore s cohomology for locally compact groups and the brauer group of a locally compact groupoid are all particular cases of sheaf or cech cohomology for topological simplicial spaces

on extended dissipativity of discrete time neural networks with time delay in this brief the problem of extended dissipativity analysis for discrete time neural networks with time varying delay is investigated the definition of extended dissipativity of discrete time neural networks is proposed which unifies several performance measures such as the h infty performance passivity l 2 l infty performance and dissipativity by introducing a triple summable term in lyapunov function the reciprocally convex approach is utilized to bound the forward difference of the triple summable term and then the extended dissipativity criterion for discrete time neural networks with time varying delay is established the derived condition guarantees not only the extended dissipativity but also the stability of the neural networks two numerical examples are given to demonstrate the reduced conservatism and effectiveness of the obtained results

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

a pfaffian formula for monomer dimer partition functions we consider the monomer dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights with the restriction that the only non zero monomer weights are those on the boundary we prove a pfaffian formula for the corresponding partition function as a consequence of this result multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics our proof is based on the celebrated kasteleyn theorem combined with a theorem on pfaffians proved by one of the authors and a careful labeling and directing procedure of the vertices and edges of the graph

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

recurrent neural nets as dynamical boolean systems with application to associative memory discrete time discrete state recurrent neural networks are analyzed from a dynamical boolean systems point of view in order to devise new analytic and design methods for the class of both single and multilayer recurrent artificial neural networks with the proposed dynamical boolean systems analysis we are able to formulate necessary and sufficient conditions for network stability which are more general than the well known but restrictive conditions for the class of single layer networks 1 symmetric weight matrix with 2 positive diagonal and 3 asynchronous update in terms of design we use a dynamical boolean systems analysis to construct a high performance associative memory with this boolean memory we can guarantee that all fundamental memories are stored and also guarantee the size of the basin of attraction for each fundamental memory

chern weil map for principal bundles over groupoids the theory of principal g bundles over a lie groupoid is an important one unifying various types of principal g bundles including those over manifolds those over orbifolds as well as equivariant principal g bundles in this paper we study differential geometry of these objects including connections and holonomy maps we also introduce a chern weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes as an application we recover the equivariant chern weil map of bott tu we also obtain an explicit chain map between the weil model and the simplicial model of equivariant cohomology which reduces to the bott shulman map s mathfrak g g to h bg when the manifold is a point

kam theorem and quantum field theory we give a new proof of the kam theorem for analytic hamiltonians the proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale the crucial cancellations of resonances are shown to follow from the ward identities expressing the translation invariance of the corresponding field theory

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

boundary monomers in the dimer model the correlation functions of an arbitrary number of boundary monomers in the system of close packed dimers on the square lattice are computed exactly in the scaling limit the equivalence of the 2n point correlation functions with those of a complex free fermion is proved thereby reinforcing the description of the monomer dimer model by a conformal free field theory with central charge c 1 comment 15 pages 2 figures

inhomogeneous bond percolation on square triangular and hexagonal lattices the star triangle transformation is used to obtain an equivalence extending over the set of all in homogeneous bond percolation models on the square triangular and hexagonal lattices among the consequences are box crossing rsw inequalities for such models with parameter values at which the transformation is valid this is a step toward proving the universality and conformality of these processes it implies criticality of such values thereby providing a new proof of the critical point of inhomogeneous systems the proofs extend to certain isoradial models to which previous methods do not apply

dimers on surface graphs and spin structures ii partition functions for dimers on closed oriented surfaces are known to be alternating sums of pfaffians of kasteleyn matrices in this paper we obtain the formula for the coefficients in terms of discrete spin structures

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

non abelian differentiable gerbes abstract we study non abelian differentiable gerbes over stacks using the theory of lie groupoids more precisely we develop the theory of connections on lie groupoid g extensions which we call connections on gerbes and study the induced connections on various associated bundles we also prove analogues of the bianchi identities in particular we develop a cohomology theory which measures the existence of connections and curvings for g gerbes over stacks we also introduce g central extensions of groupoids generalizing the standard groupoid s 1 central extensions as an example we apply our theory to study the differential geometry of g gerbes over a manifold

relations between the coulomb gas picture and conformal invariance of two dimensional critical models partition functions of critical 2d models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit this is worked out explicitly for theo n andq state potts model forn orq integer we recover results obtained from conformal invariance but our procedure also extends to nonintegral values in the latter case the expansion on characters of the virasoro algebra involves real coefficients of either sign the operator content of both models is discussed in detail

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

observability of automata networks fixed and switching cases automata networks are a class of fully discrete dynamical systems which have received considerable interest in various different areas this brief addresses the observability of automata networks and switched automata networks in a unified framework and proposes simple necessary and sufficient conditions for observability the results are achieved by employing methods from symbolic computation and are suited for implementation using computer algebra systems several examples are presented to demonstrate the application of the results