ergodicity of probabilistic cellular automata a constructive criterion we give a sequence of criteria of increasing complexity for the exponential ergodicity of discrete time interacting particle systems each criterion involves estimating the dependence on initial conditions of the process on finite space time volumes it generalizes and improves the existing single site condition and is the analog of the dobrushin shlosmanc v condition in equilibrium statistical mechanics our dynamic criterion may also be used to prove the uniqueness of gibbs state in situations where thec v condition fails as a converse we prove that if there is a certain form of convergence to the stationary measure faster thann d wheren is the time andd is the dimension of the lattice then our condition holds for some space time volumes and hence the convergence must be exponentially fast

when is an interacting particle system ergodic we consider stochastic flip dynamics for an infinite number of ising spins on the lattice d we find a sequence of constructive criteria for the system to be exponentially ergodic the main idea is to approximate the continuous time process with discrete time processes itseuler polygon and to use an improved version of previous results ms about constructive ergodicity of discrete time processes

on the exponential decay of the n point correlation functions and the analyticity of the pressure the goal of this paper is to provide estimates leading to a direct proof of the exponential decay of the n point correlation functions for certain unbounded models of kac type the methods are based on estimating higher order derivatives of the solution of the witten laplacian equation on 1 forms associated with the hamiltonian of the system we also provide a formula for the taylor coefficients of the pressure that is suitable for a direct proof of the analyticity

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

diffusion in energy conserving coupled maps we consider a dynamical system consisting of subsystems indexed by a lattice each subsystem has one conserved degree of freedom energy the rest being uniformly hyperbolic the subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved we prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit

poincare and transportation inequalities for gibbs measures under the dobrushin uniqueness condition in in this paper we establish an explicit and sharp estimate of the spectral gap poincare inequality and the transportation inequality for gibbs measures under the dobrushin uniqueness condition moreover we give a generalization of the liggett s m e theorem for interacting particle systems

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

data driven fuzzy modeling for takagi sugeno kang fuzzy system this paper presents a systematic approach to design first order tagaki sugeno kang tsk fuzzy systems this approach attempts to obtain the fuzzy rules without any assumption about the structure of the data the structure identification and parameter optimization steps in this approach are carried out automatically and are capable of finding the optimal number of the rules with an acceptable accuracy starting with an initial structure the system first tries to improve the structure and then as soon as an improved structure is found it fine tunes its rules parameters then it goes back to improve the structure again to find a better structure and re fine tune the rules parameters this loop continues until a satisfactory solution tsk model is found the proposed approach has successfully been applied to well known benchmark datasets and real world problems the obtained results are compared with those obtained with other methods from the literature experimental studies demonstrate that the predicted properties have a good agreement with the measured data by using the elicited fuzzy model with a small number of rules finally as a case study the proposed approach is applied to the desulfurization process of a real steel industry comparing the proposed approach with some other fuzzy systems and neural networks it is shown that the developed tsk fuzzy system exhibits better results with higher accuracy and smaller size of architecture

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

some numerical results on the block spin transformation for the 2d ising model at the critical point we study the block spin transformation for the 2d ising model at the critical temperaturetc we consider the model with the constraint that the total spin in each block is zero an old argument by cassandro and gallavotti strongly supports the gibbsianness of the transformed measure provided that such model has a critical temperaturet c lower thantc after describing a possible rigorous approach to the problem we present numerical evidence that indeedt c tc and study the dobrushin shlosman uniqueness condition

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

measure concentration for euclidean distance in the case of dependent random variables let qn be a continuous density function in n dimensional euclidean space we think of qn as the density function of some random sequence xn with values in bbb r n for i 1 n let xi denote the collection of coordinates xi i i and let widebar x i denote the collection of coordinates xi i i we denote by q i x i bar x i the joint conditional density function of xi given widebar x i we prove measure concentration for qn in the case when for an appropriate class of sets i i the conditional densities q i x i bar x i as functions of xi uniformly satisfy a logarithmic sobolev inequality and ii these conditional densities also satisfy a contractivity condition related to dobrushin and shlosman s strong mixing condition

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

quasi factorization of the entropy and logarithmic sobolev inequalities for gibbs random fields we show that the entropy functional exhibits a quasi factorization property with respect to a pair of weakly dependent algebras as an application we give a simple proof that the dobrushin and shlosmans complete analyticity condition for a gibbs specification with finite range summable interaction implies uniform logarithmic sobolev inequalities this result has been previously proven using several different techniques the advantage of our approach is that it relies almost entirely on a general property of the entropy while very little is assumed on the dirichlet form no topology is introduced on the single spin space thus discrete and continuous spins can be treated in the same way

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

coarse graining schemes for stochastic lattice systems with short and long range interactions we develop coarse graining schemes for stochastic many particle microscopic models with competing short and long range interactions on a dimensional lattice we focus on the coarse graining of equilibrium gibbs states and by using cluster expansions we analyze the corresponding renormalization group map we quantify the approximation properties of the coarse grained terms arising from different types of interactions and present a hierarchy of correction terms we derive semi analytical numerical coarse graining schemes that are accompanied by a posteriori error estimates for lattice systems with short and long range interactions

constructive fractional moment criteria for localization in random operators abstract we present a family of finite volume criteria which cover the regime of exponential decay for the fractional moments of green functions of operators with random potentials such decay is a technically convenient characterization of localization for it is known to imply spectral localization absence of level repulsion dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases the constructive criteria also preclude fast power law decay of the green functions at mobility edges

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

random walks in space time mixing environments we prove that random walks in random environments that are exponentially mixing in space and time are almost surely diffusive in the sense that their scaling limit is given by the wiener measure

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

the log sobolev inequality with quadratic interactions we assume one site measures without a boundary e x dx z that satisfies a log sobolev inequality we prove that if these measures are perturbed with quadratic interactions then the associated infinite dimensional gibbs measure on the lattice always satisfies a log sobolev inequality furthermore we present examples of measures that satisfy the inequality with a phase that goes beyond convexity at infinity

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

occurrence repetition and matching of patterns in the low temperature ising model we continue our study of the exponential law for occurrences and returns of patterns in the context of gibbsian random fields for the low temperature plus phase of the ising model we prove exponential laws with error bounds for occurrence return waiting and matching times moreover we obtain a poisson law for the number of occurrences of large cylindrical events and a gumbel law for the maximal overlap between two independent copies as a by product we derive precise fluctuation results for the logarithm of waiting and return times the main technical tool we use in order to control mixing is disagreement percolation

phase diagram of the two dimensional ising antiferromagnet computer assisted proof we study the phase diagram of the ising antiferromagnet on a square lattice in a neighbourhood of ground state critical pointsh 4 t 0 it leads to a question about the valuea c of the critical activity of the hard square lattice gas using a constructive criterion of uniqueness we prove thata c 1 and that the phase diagram of the antiferromagnet does not bulge near mentioned critical points it is a specific feature of this work that the proof was completed with the help of a computer

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

cluster expansion for d dimensional lattice systems and finite volume factorization properties we consider classical lattice systems with finite range interactions ind dimensions by means of a block decimation procedure we transform our original system into a polymer system whose activity is small provided a suitable factorization property of finite volume partition functions holds in this way we extend a result of olivieri

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

isoperimetry for gibbs measures we show that a strong mixing condition implies a bakry bobkov ledoux inequality for a probability measure on infinite dimensional space

perturbative analysis of disordered ising models close to criticality we consider a two dimensional ising model with random i i d nearest neighbor ferromagnetic couplings and no external magnetic field we show that if the probability of supercritical couplings is small enough the system admits a convergent cluster expansion with probability one the associated polymers are defined on a sequence of increasing scales in particular the convergence of the above expansion is compatible with the infinite differentiability of the free energy but does not imply its analyticity the basic tools in the proof are a general theory of graded cluster expansions and a stochastic domination of the disorder

complete analyticity of the 2d potts model above the critical temperature we investigate the complete analyticity ca of the two dimensional q state potts model for large values of q we are able to prove it for every temperature provided we restrict ourselves to nice subsets their niceness depending on the temperature t contrary to this restricted complete analyticity rca the full ca is known to fail for some values of the temperature above

renormalization group transformations near the critical point some rigorous results we consider renormalization group rg transformations for classical ising type lattice spin systems in the infinite volume limit formally the rg maps a hamiltonian h into a renormalized hamiltonian h exp h t exp h where t denotes a specific rg probability kernel t 1 for every configuration with the help of the dobrushin uniqueness condition and standard results on the polymer expansion haller and kennedy gave a sufficient condition for the existence of the renormalized hamiltonian in a neighborhood of the critical point by a more complicated but reasonably straightforward application of the cluster expansion machinery the present investigation shows that their condition would further imply a band structure on the matrix of partial derivatives of the renormalized interaction with respect to the original interaction this in turn gives an upper bound for the rg linearization

exponential distribution for the occurrence of rare patterns in gibbsian random fields we study the distribution of the occurrence of rare patterns in sufficiently mixing gibbs random fields on the lattice d d 2 a typical example is the high temperature ising model this distribution is shown to converge to an exponential law as the size of the pattern diverges our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns a similar result holds for the repetition of a rare pattern we apply these results to the fluctuation properties of occurrence and repetition of patterns we prove a central limit theorem and a large deviation principle

absence of phase transition for antiferromagnetic potts models via the dobrushin uniqueness theorem we prove that theq state potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures including zero temperature wheneverq 2r we also prove slightly better bounds for several two dimensional lattices square lattice exponential decay forq 7 triangular lattice q 11 hexagonal lattice q 4 and kagome lattice q 6 the proofs are based on the dobrushin uniqueness theorem

on the ergodic properties of glauber dynamics we show that if there is an infinite volume gibbs measure which satisfies a logarithmic sobolev inequality with local coefficients of moderate growth then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

completely analytic interactions with infinite values in this note we extend the notion of completely analytic interactions of gibbs random fields that is known for finite interactions with finite range to interactions that can have infinite values too we formulate a set of ten conditions on such interactions in terms of analyticity properties of the partition functions or correlation decay the main theorem states that all these conditions are equivalent therefore an interaction is called a completely analytic interaction if it satisfies one of these conditions

a modified pnn algorithm with optimal pd modeling using the orthogonal least squares method in this paper a modified algorithm is suggested for developing polynomial neural network pnn models optimal partial description pd modeling is introduced at each layer of the pnn expansion a task accomplished using the orthogonal least squares ols method based on the initial pd models determined by the polynomial order and the number of pd inputs ols selects the most significant regressor terms reducing the output error variance the method produces pnn models exhibiting a high level of accuracy and superior generalization capabilities additionally parsimonious models are obtained comprising a considerably smaller number of parameters compared to the ones generated by means of the conventional pnn algorithm three benchmark examples are elaborated including modeling of the gas furnace process as well as the iris and wine classification problems extensive simulation results and comparison with other methods in the literature demonstrate the effectiveness of the suggested modeling approach

ill defined block spin transformations at arbitrarily high temperatures examples are presented of block spin transformations which map the gibbs measures of the ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non gibbsian measures in this way we provide the first example of this kind of pathology for very high temperatures and as a corollary also the first example of such a pathology happening at a critical point

witten laplacian method for the decay of correlations the aim of this paper is to apply direct methods to the study of integrals that appear naturally in statistical mechanics and euclidean field theory we provide weighted estimates leading to the exponential decay of the two point correlation functions for certain classical convex unbounded models the methods involve the study of the solutions of the witten laplacian equations associated with the hamiltonian of the system

a fuzzy asymmetric garch model applied to stock markets in this paper we derive a new class of flexible threshold asymmetric generalized autoregression conditional heteroskedasticity garch models we use this tool for analysis and modeling of the properties that are apparent in many financial time series in general the transmission of volatility in the stock market is time varying nonlinear and asymmetric with respect to both positive and negative results given this fact we adopt the method of fuzzy logic systems to modify the threshold values for an asymmetric garch model our simulations use stock market data from the taiwan weighted index taiwan the nikkei 225 index japan and the hang seng index hong kong to illustrate the performance of our proposed method from the simulation results we have determined that the forecasting of volatility performance is significantly improved if the leverage effect of clustering is considered along with the use of expert knowledge enabled by the garch model

trace estimates for stable processes in this paper we study the behaviour in time of the trace the partition function of the heat semigroup associated with symmetric stable processes in domains of rd in particular we show that for domains with the so called r smoothness property the second terms in the asymptotic as t 0 involves the surface area of the domain just as in the case of brownian motion

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

relaxed criteria of the dobrushin shlosman mixing condition an interacting particle system glauber dynamics which evolves on a finite subset in thed dimensional integer lattice is considered it is known that a mixing property of the gibbs state in the sense of dobrushin and shlosman is equivalent to several very strong estimates in terms of the glauber dynamics we show that similar but seemingly much milder estimates are again equivalent to the dobrushin shlosman mixing condition hence to the original ones found by stroock and zegarlinski this may be understood as the absence of intermediate speed of convergence to equilibrium

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

on the asymptotics of the heat equation and bounds on traces associated with the dirichlet laplacian we use the feynman kac formula and a decomposition of the brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel we also compute bounds on trace et d where d is the dirichlet laplacian for i a bounded region d in rn with smooth boundary or ii an unbounded region d in r2 with finite area

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

identification of fuzzy models using a successive tuning method with a variant identification ratio the paper is concerned with the identification of fuzzy inference systems fuzzy models realized with the aid of the successive tuning method using a variant identification ratio that is based on hierarchical fair competition based parallel genetic algorithms hfc pga and information granulation the hfc pga is a certain multi population version of parallel genetic algorithms pga which is suitable for a simultaneous optimization of both the structure of the fuzzy model as well as its parameters the granulation of information is realized with the aid of the c means clustering algorithm information granules formed in this way become essential at further stages of the construction of the fuzzy models by forming the centers modal values of the fuzzy sets constituting individual rules of the inference schemes further optimization of the fuzzy model deals with an adjustment of a suite of parameters such as the number of input variables to be used in the model a collection of specific subsets of the input variables and the number of membership functions being used by these variables and the order and parameters of the polynomial occurring in the conclusions of the corresponding rules an iterative development of the fuzzy model deals with its structural as well as parametric optimization via hfc pga the c means algorithm and a standard least square estimation method to evaluate the performance of the proposed model we exploit well known and commonly used data sets such as gas furnace mackey glass time series as well as medical imaging system a comparative analysis demonstrates that the proposed model leads to the superior performance when compared with other fuzzy models reported in the literature

a combinatorial proof of tree decay of semi invariants we consider finite range gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi invariants supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions this hypothesis holds for completely analytical gibbs fields in this context the tree decay of semi invariants has been proven via analyticity arguments however the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called griffiths phase when analyticity arguments fail

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

pathological behavior of renormalization group maps at high fields and above the transition temperature we show that decimation transformations applied to high q potts models result in non gibbsian measures even for temperatures higher than the transition temperature we also show that majority transformations applied to the ising model in a very strong field at low temperatures produce non gibbsian measures this shows that pathological behavior of renormalization group transformations is even more widespread than previous examples already suggested

a fixed point equation for the high temperature phase of discrete lattice spin systems a fixed point equation on an infinite dimensional space is proposed as an alternative to the usual definition of the infinite volume limit in discrete lattice spin systems in the high temperature phase it is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities a theorem is then proved that says that if a certain finite volume condition is satisfied then this fixed point equation has a solution and the resulting free energy is analytic in the parameters in the hamiltonian for particular values of the temperature this finite volume condition may be checked with the help of a computer the two dimensional ising model is considered as a test case and it is shown that the finite volume condition is satisfied for 0 77 critical

some remarks on pathologies of renormalization group transformations for the ising model the results recently obtained by van enter fernandez and sokal on non gibbsianness of the measurev t b h arising from the application of a single decimation transformationt b with spacingb to the gibbs measure h of the ising model for suitably chosen large inverse temperature and nonzero external fieldh are critically analyzed in particular we show that if keeping fixed the same values of h andb one iterates a sufficiently large number of timesn the transformationt b one obtains a new measurev t b n h which is gibbsian and moreover very weakly coupled

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

coexistence of partially disordered ordered phases in an extended potts model we consider a generalization of the standard potts model in which there are q r s states with an interaction that distinguishes the two subspecies we develop a graphical representation of the fk type for the system and show that this representation may be incorporated directly into reflection positivity arguments using combinations of these techniques we establish detailed properties of the phase diagram including the existence of sharp triple points whenever relevant the phases are characterized by the percolation properties of the underlying representation

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

cluster centric fuzzy modeling in this study we propose a cluster oriented development of fuzzy models an overall design process is focused on an efficient usage of fuzzy clustering fuzzy c means fcm in particular to form information granules clusters that are used in the construction of the fuzzy model fuzzy models are regarded as mappings from information granules expressed in the input and output spaces this position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules this study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages the augmented fcm method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and this way the performance of the resulting fuzzy model is optimized it is shown that the cluster oriented modeling gives rise to the mamdani like fuzzy rules and a zero order takagi sugeno model under a certain decoding scheme we identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding decoding or granulation degranulation encountered in processing fuzzy sets and granular computing in general furthermore refinements of zero order fuzzy models are investigated leading to first order fuzzy models with linear functions standing in the conclusions of the rules a series of experiments is reported where we used synthetic and real world data using which an issue of generalization capabilities is elaborated in detail

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

fuzzy radial basis function neural networks with information granulation and its parallel genetic optimization fuzzy modeling of complex systems is a challenging task which involves important problems of dimensionality reduction and calls for various ways of improving the accuracy of modeling the ig frbfnn a hybrid architecture of the ig fis fuzzy inference system and frbfnn fuzzy radial basis function neural networks is proposed to address these problems the paper is concerned with the analysis and design of ig frbfnns and their optimization by means of the hierarchical fair competition based parallel genetic algorithm hfc pga in the proposed network the membership functions of the premise part of the fuzzy rules of the ig based frbfnn model directly rely on the computation of the relevant distance between data points and the use of four types of polynomials such as constant linear quadratic and modified quadratic are considered for the consequent part of fuzzy rules moreover the weighted least square wls learning is exploited to estimate the coefficients of the polynomial forming the conclusion part of the rules since the performance of the ig rbfnn model is affected by some key design parameters such as a specific subset of input variables the fuzzification coefficient of the fcm the number of rules and the order of polynomial of the consequent part of fuzzy rules it becomes beneficial to carry out both structural as well as parametric optimization of the network in this study the hfc pga is used as a comprehensive optimization vehicle the performance of the proposed model is illustrated by means of several representative numerical examples

self organizing polynomial neural networks based on polynomial and fuzzy polynomial neurons analysis and design abstract in this study we introduce and investigate a class of neural architectures of polynomial neural networks pnns develop a comprehensive design methodology and carry out a series of numeric experiments pnn is a flexible neural architecture whose structure topology is developed through learning in particular the number of layers of the pnn is not fixed in advance but is generated on the fly in this sense pnn is a self organizing network we distinguish between two kinds of sopnn architectures that is a polynomial neuron pn based and b fuzzy polynomial neuron fpn based self organizing networks this taxonomy is based on the character of each neuron structure in the network each of them comes with two structures referred here to as basic and the modified topology moreover for each topology of the sopnn we identify two types that is a generic and advanced type the essence of the design procedure of self organizing polynomial neural networks sopnn dwells on the group method of data handling gmdh ieee trans systems man and cybernet 12 1971 364 each node of the pn based sopnn exhibits a high level of flexibility and realizes a polynomial type of mapping linear quadratic and cubic between input and output variables fpn based sopnn dwells on the ideas of fuzzy rule based computing and neural networks especially in fpn based sopnn the generic rules in the system assume the form if a then y p x where a is a fuzzy relation in the condition space while p x is a polynomial forming a conclusion part of the rule each fpn processing element consists of a series of the nonlinear inference rules the conclusion part of the rules especially the regression polynomial uses several types of high order polynomials such as linear quadratic modified quadratic and cubic as the premise part of the rules both triangular and gaussian like membership functions are studied simulations involve a series of synthetic as well as experimental data used across various neurofuzzy systems a detailed comparative analysis is included as well

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

hybrid identification in fuzzy neural networks this paper introduces an identification method for nonlinear models in the form of fuzzy neural networks fnn in this model we use two forms of the fuzzy inference methods a simplified and linear fuzzy inference and exploit a standard error back propagation learning algorithm the fnn modeling and identification environment realizes parameter identification through a synergistic usage of clustering techniques genetic optimization and a complex search method we use a hard c means hcm clustering algorithm to determine initial apexes of the membership functions of the information granules used in this fuzzy model the parameters such as apexes of membership functions learning rates and momentum coefficients are then adjusted using hybrid algorithm the proposed hybrid identification algorithm is carried out by combining both genetic optimization genetic algorithm ga and the improved complex method an aggregate objective function performance index with a weighting factor is introduced to achieve a sound balance between approximation and generalization of the model according to the selection and adjustment of the weighting factor of this objective function we reveal how to design a model with sound approximation and generalization abilities the proposed model is experimented with using several time series data gas furnace sewage treatment process and nox emission process data of gas turbine power plant

renormalization group at criticality and complete analyticity of constrained models a numerical study we study the majority rule transformation applied to the gibbs measure for the 2d ising model at the critical point the aim is to show that the renormalized hamiltonian is well defined in the sense that the renormalized measure is gibbsian we analyze the validity of dobrushin shlosman uniqueness dsu finite size condition for the constrained models corresponding to different configurations of the image system it is known that dsu implies in our 2d case complete analyticity from which as recently shown by haller and kennedy gibbsianness follows we introduce a monte carlo algorithm to compute an upper bound to vasserstein distance appearing in dsu between finite volume gibbs measures with different boundary conditions we get strong numerical evidence that indeed the dsu condition is verified for a large enough volumev for all constrained models

identification of fuzzy systems by means of an auto tuning algorithm and its application to nonlinear systems abstract the study concerns a design procedure of rule based systems the proposed rule based fuzzy modeling implements system structure and parameter identification in the efficient form of if then statements and exploits the theory of system optimization and fuzzy implication rules two types of methods for rule based fuzzy modeling are studied this classification concerns the form of the conclusion part of the rules that can be either constant or formed by some linear functions both triangular and gaussian like membership function are studied the optimization hinges on an auto tuning algorithm that covers a modified constrained optimization method known as a complex method the study introduces a weighted performance index objective function that helps achieve a sound balance between the quality of results produced for the training and testing set this methodology sheds light on the role and impact of different parameters of the model on its performance in particular the mapping and predicting capabilities of the rule based computing the study is illustrated with the aid of several representative numerical examples

heat content estimates over sets of finite perimeter abstract this paper studies by means of standard analytic tools the small time behavior of the heat content over a bounded lebesgue measurable set of finite perimeter by working with the set covariance function and by imposing conditions on the heat kernels applications concerning the heat kernels of rotational invariant stable processes are given

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field

phase diagram of ising systems with additional long range forces we consider ferromagnetic ising systems where the interaction is given by the sum of a fixed reference potential and a kac potential of intensity 0 and scaling parameter 0 in the lebowitz penrose limit 0 the phase diagram in the t positive quadrant is described by a critical curve mf t which separates the regions with one and two phases respectively below and above the curve we prove that if mf t i e above the curve there are at least two gibbs states for small values of if instead mf t and if the reference gibbs state i e without the kac potential satisfies a mixing condition at the temperature t then at the same temperature the full interaction i e with also the kac potential satisfies the dobrushin shlosman uniqueness condition for small values of so that there is a unique gibbs state

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc