kolmogorov sinai entropy and lyapunov spectra of a hard sphere gas abstract the mixing behavior of a hard sphere gas has its origin in the exponential growth of small perturbations in phase space this instability is characterized by the so called lyapunov exponents in this work we compute full spectra of lyapunov exponents for the hard sphere gas for a wide range of densities and particle numbers by using a recently developed algorithm in the dilute gas regime the maximum lyapunov exponent is found to obey the krylov relation ln a formula exactly derived for the low density lorentz gas by dorfman and van beijeren we study the system size dependence and the effect of the fluid solid phase transition on the spectra in the second part of this work we describe and test a direct simulation monte carlo method dsmc for the computation of lyapunov spectra and present results for dilute hard sphere gases excellent agreement is obtained with the results of the deterministic simulations this suggests that the lyapunov instability of a hard sphere gas may be analyzed within the framework of kinetic theory

why are papers about filters on residuated structures usually trivial in this paper we introduce a notion of a t filter on residuated lattices which is a generalization of several special types of filters we provide some basic properties of t filters and show how particular results about special types of filters e g extension property triple of equivalent characteristics and quotient characteristics are uniformly covered by this simple general framework

critical storage capacity of the j 1 neural network for neural networks in which the coupling jij are allowed to take on the values jij 1 or jij 1 the authors determine numerically the critical storage capacity for random unbiased patterns as a function of the stability they use an exact enumeration scheme based on the gray code and a continuous distribution for the patterns to control finite size effects results are presented for n 25 they indicate an optimal storage capacity of c 0 82 n

fuzzy filters of mtl algebras characterizations of fuzzy filters in mtl algebras are given the fuzzy filter generated by a fuzzy set is considered the notion of boolean fuzzy filters and mv fuzzy filters are introduced and related properties are investigated a condition for a fuzzy filter to be boolean is provided a characterization of a boolean fuzzy filter is given a congruence relation on a mtl algebra induced by a fuzzy filter is established and we show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice

fuzzy filters of bl algebras the aim of this paper is to introduce the notions of fuzzy filters fuzzy prime filters and the cosets of a fuzzy filter in bl algebras and investigate some of their properties the fuzzy filter generated by a fuzzy set is discussed some characterizations of fuzzy filters and fuzzy prime filters are derived the extension theorem of fuzzy prime filters and the fuzzy prime filters theorem are established finally we prove that the algebra l f which is the set of all cosets of f is a bl algebra and is isomorphic to the bl algebra l ff 1 where ff 1 x l f x f 1 moreover each bl algebra l is a subdirect product of linearly ordered bl algebras l fi i where fi i are fuzzy prime filters of l

properties of neural networks storing spatially correlated patterns the author studies the behaviour of a feedforward neural network supplied with spatially organized data this inner structure is taken into account by a matrix cij whose coefficients equal the average correlation between two pixels i and j of the input patterns the storage capacity alpha is computed as a function of the required stability and of the eigenvalues of c the author proposes a geometrical transformation allowing an intuitive interpretation of these results numerical simulations using real and binary patterns show a very good agreement with the theory finally the author analyses the synaptic couplings correlations resulting from the training of the network with structured patterns focusing on exponentially decreasing correlations one and two dimensions the author finds that they exhibit a mexican hat profile the excitatory centre size of which depends on alpha

fuzzy boolean and positive implicative filters of bl algebras the aim of this paper is to introduce the notions of fuzzy boolean filters and fuzzy positive implicative filters in bl algebras and to investigate their properties several characterizations of fuzzy boolean filters and fuzzy positive implicative filters are derived the extension theorems of fuzzy boolean filters and fuzzy positive implicative filters are obtained the relation between fuzzy boolean filters and fuzzy positive implicative filters is investigated and it is proved that every fuzzy boolean filter is a fuzzy positive implicative filter but the converse may not be true furthermore the conditions under which a fuzzy positive implicative filter is a fuzzy boolean filter are established

optimal storage capacity of neural networks at finite temperatures gardner s analysis 1989 of the optimal storage capacity of neural networks is extended to study finite temperature effects the typical volume of the space of interactions is calculated for strongly diluted networks as a function of the storage ratio alpha temperature t and the tolerance parameter m from which the optimal storage capacity alpha c is obtained as a function of t and m at zero temperature it is found that alpha c 2 regardless of m while alpha c in general increases with the tolerance at finite temperatures the authors show how the best performance for given alpha and t is obtained which reveals a first order transition from high quality performance to a low quality one at low temperatures an approximate criterion for recalling which is valid near m 1 is also discussed

the binary perceptron and general aspects of non self averaged quantities abstract the possibility of a finite width distribution for the maximal capacity of the binary perceptron in the thermodynamic limit is discussed analytically and supported by a careful analysis of numerical simulations the results also indicate that the description of quenched random systems could take into account the possibility that in addition to non self averaged quantities other quantities such as the transition temperature might also be sample dependent

the effects of limited precision weights on the threshold adaline the effect of limited precision weights on the functional capability of a threshold adaline is examined the number of logic functions which can be implemented by the threshold adaline serves as the primary measure of functional capability closed form expressions are provided for the number of logic functions which can be implemented by a threshold adaline with four different levels of weight precision in addition realization tests and procedures based on the characterizing numbers are provided for each level of weight precision the number of realizable logic functions is compared to the capacity of a threshold adaline with limited precision weights and a relationship between the two measures is proposed

storage capacity of neural networks effect of the fluctuations of the number of active neurons per memory the storage capacity in an attractor neural network with excitatory couplings is shown to depend not only on the fraction of active neurons per pattern or coding rate but also on the fluctuations around this value in the thermodynamical limit the capacity is calculated in the case of exactly the same number of active neurons in every pattern for every coding level the capacity is increased with respect to the case of random patterns results are supported by numerical simulations done with an exhaustive search algorithm and partly solve in the sparse coding limit the paradox of the discrepancy of the capacity of the willshaw model with optimal capacity

finite size effects and bounds for perceptron models ablncl in lhis paper we consider two main aspsln of the binary perccptron problem the maximal capacity when random paltem are stored model a and its generalhation ability model b we have extended previous numerical estimates of critical capacilies and studied thermal properties of systems of small sizes to lest recent replica predictions we have also considered some simpler versions of thest models the discrete spherical vemions can be salved exactly using gardner s replica calculation for the spherical model and are shown to give a rigorous upper bound and lower bound on the capacities of models a and b respectively iby vemions of models a and b are soivcd in detail and provide information which is useful for interpreting the bnile size eeeciects present in the numerical studies of models a and b

a fractal set from the binary reflected gray code the permutation associated with the decimal expression of the binary reflected gray code with n bits is considered its cycle structure is studied considered as a set of points its self similarity is pointed out as a fractal it is shown to be the attractor of an ifs for large values of n the set is examined from the point of view of time series analysis

on filter theory of residuated lattices the aim of this paper is to develop the filter theory of general residuated lattices first we extend some particular types of filters and fuzzy filters in bl algebras and mtl algebras naturally to general residuated lattices and further enumerate some relative results obtained in bl algebras or mtl algebras which still hold in general residuated lattices next we introduce the concepts of regular filters and fuzzy regular filters to general residuated lattices which are two new types of filters and fuzzy filters and derive some of their characterizations finally we discuss the relations between fuzzy regular filters and several other special fuzzy filters and also characterize some special classes of residuated lattices by filters or fuzzy filters

two hard spheres in a spherical pore exact analytic results in two and three dimensions the partition function and the one and two body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore the equivalent problem for hard disks is addressed too we establish a relation valid for any dimension between these partition functions second virial coefficient for inhomogeneous systems in a spherical pore and third virial coefficients for polydisperse hard spheres mixtures using the established relation we were able to evaluate the cluster integral b2 v related with the second virial coefficient for the hard disc system into a circular pore finally we analyse the behaviour of the obtained expressions near the maximum density

fuzzy implicative and boolean ideals of mv algebras abstract we show how to obtain results in mv algebras by considering their fuzzy ideals in particular we prove the following theorem let be a fuzzy ideal of an mv algebra x then the folllowing are equivalent 1 x x 0 for all x x 2 x n x for all x x and all n 1 3 x x is a boolean algebra 4 is fuzzy implicative if i is an ideal of x and we take i then we obtain corresponding results for the ideal i and x we show how to extend fuzzy prime and fuzzy implicative ideals various fuzzy radicals are considered and related to the corresponding radicals of the algebra and we show how they live in the spectrum of the algebra

estimates of optimal storage conditions in neural network memories based on random matrix theory the authors formulate a method for estimating the critical conditions for storage of sets of data in neural network memory this variational method is based on random matrix theory and calculating the average spectrum of a matrix whose elements are given by overlaps of the stored patterns several generic cases of random overlap matrices are considered they investigate the cases of simply uncorrelated random patterns and spatially and semantically correlated ones they obtain bounds of the critical curve in the control parameters space which determine the stability of the stored data sets

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

vapnik chervonenkis dimension of neural networks with binary weights we investigate the vc dimension of the perceptron and simple two layer networks like the committee and the parity machine with weights restricted to values pm1 for binary inputs the vc dimension is determined by atypical pattern sets i e it cannot be found by replica analysis or numerical monte carlo sampling for small systems exhaustive enumerations yield exact results for systems that are too large for enumerations number theoretic arguments give lower bounds for the vc dimension for the ising perceptron the vc dimension is probably larger than n 2

statistical mechanics of lossy data compression using a nonmonotonic perceptron the performance of a lossy data compression scheme for uniformly biased boolean messages is investigated via methods of statistical mechanics inspired by a formal similarity to the storage capacity problem in neural network research we utilize a perceptron of which the transfer function is appropriately designed in order to compress and decode the messages employing the replica method we analytically show that our scheme can achieve the optimal performance known in the framework of lossy compression in most cases when the code length becomes infinite the validity of the obtained results is numerically confirmed

the weight space of the binary perceptron with a view to finding features of the weight space of the binary perceptron that might be instructive for training binary synapse neural networks the maximally stable perceptron having binary valued weights is compared with continuous weight perceptrons for universal choices of stored patterns the fraction of synaptic weights correctly predicted by clipping the synapses of the continuous network is calculated in the thermodynamic limit and compared with simulation results for smaller systems numerical experiments show good agreement with theory but in addition indicate that those binary synapses likely to be wrongly predicted by weight clipping are predominantly those which are weakest in the continuous synapse perceptron although not rescuing training time from growing exponentially in the system size our results suggest ways of significantly accelerating the search for successful albeit possibly imperfect neural networks with discrete valued couplings

ergodicity of classical billiard balls abstract sinai in 1970 proved the ergodicity of two discs on the 2 torus further essential progress was only reached in 1987 when chernov and sinai established the ergodicity of two billiard balls on the v torus then by applying a strategy suggested by kramli simanyi and myself in 1989 we obtained the ergodicity of three and four balls on the torus 3 in the latter case in 1991 and 1992 while recently simanyi proved that of n balls on the torus whenever n after a survey of this progress the study of toric billiards with cylindric scatterers is initiated and the k property of a general class of these is claimed

a short note on t filters i filters and extended filters on residuated lattices the notion of a t filter on residuated lattices was introduced in order to generalize a big amount of particular results about special types of filters e g implicative positive implicative fantastic etc in this short note we are going to point out the relationship between t filters and i filters and also between t filters and extended filters on residuated lattices

fuzzy t filters and their properties abstract fuzzification of special types of filters on several different algebras of many valued logics has been very popular in recent years the main aim of this paper is to point out some general principles concerning particular results about fuzzification of special types of filters we introduce the notion of a fuzzy t filter which generalizes most types of special fuzzy filters e g fuzzy implicative fuzzy boolean fuzzy fantastic etc and prove some basic properties of fuzzy t filters

some types of filters in mtl algebras the aim of this work is to introduce the concepts of imtl filters and strong mtl filters in mtl algebras and show that these filters are related to imtl algebras and strong mtl algebras respectively we also introduce the concepts of eimtl and associative filters in mtl algebras the relationships between these filters and quotient algebras that are constructed via these filters are described we clarify that eimtl imtl and fantastic filters coincide in bl algebras whereas they have different properties in mtl algebras moreover the relations between them are delineated we prove that strong filters include some current filters such as implicative positive implicative and fantastic filters

tl filters of integral residuated l monoids in this paper we introduce and discuss the concept of tl filters of integral residuated l monoids first we study some basic properties of tl filters and give the formula for calculating the tl filters generated by l subsets then we discuss some properties of tl filters under homomorphisms and study the relationship between tl filters and t congruence l relations on integral commutative residuated l monoids

the k property of four billiard balls a further step is achieved toward establishing the celebrated boltzmann sinai ergodic hypothesis for systems of four hard balls on the torus 2 it is shown that on the submanifold of the phase specified by the trivial conservation laws the system is ak flow all parts of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened the main novelties are i a refinement of the geometric algebraic methods used earlier helps us to bound the codimension of the arising implicitly given set of degeneracies even if we can not calculate their exact dimension that was possible for three billiards as a matter of fact it is this part of our arguments where further understanding and new ideas are necessary before attacking the general ergodic problem ii in the pasting part of the proof which is a sophisticated version of hopf s classical device the arguments are so general that it is hoped they work in the general case too this is achieved for four balls in particular by a version of the transversal fundamental theorem which on one hand is simpler and more suitable for applications than the previous one and on the other hand as we have discovered earlier is the main tool to prove global ergodicity of semi dispersing billiards iii the verification of the chernov sinai ansatz is essentially simplified and the new idea of the proof also promises to work in the general case

on the capacity of neural networks with binary weights the authors study the critical capacity alpha c of multilayered networks with binary couplings they show that for any network presenting a tree like architecture after the first hidden layer no fixed internal representation is required using gardner s calculations they apply statistical mechanics to the simplest network with two layers of adaptive weights following the same approach as for the binary perceptron they find from the zero entropy point a critical capacity alpha c 0 92 they discuss the validity of this result in view of exhaustive search simulations on small networks

the k property of three billiard balls sinai s strengthened version of the ergodic hypothesis is proved for three billiard balls on the v dimensional torus on connected components of the submanifold of the phase space specified by the trivial conservation laws of the energy and of the trajectory of the center of mass the system is a k flow to cope with the difficulty that in the isomorphic one particle billiard system the scatterers are not strictly convex geometric algebraic ergodic theoretic and topological methods are elaborated

soft bl algebras based on fuzzy sets by means of soft sets and q soft sets some characterizations of implicative positive implicative and fantastic filteristic soft bl algebras are investigated finally we prove that a soft set is an implicative filteristic soft bl algebra if and only if it is both a positive implicative filteristic soft bl algebra and a fantastic filteristic soft bl algebra

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

ergodicity in hard ball systems and boltzmann s entropy the ergodic property of the hard ball systems is investigated numerically by comparing the microcanonical single particle distributions and the corresponding numerical results in two and three dimensional cases a boltzmann entropy like quantity h mathit n mathit c is defined and its evolution is discussed the deviation of this entropy from the equilibrium value obeys a power decay law in the long time limit and the origin of this tail is explored heuristically textcopyright 1996 the american physical society

a perceptron with a skeletal weight space a perceptron whose space of interactions can interpolate between spherically constrained and binary valued synapses is introduced and investigated as an associative neural network memory for maximally stable storage and where the weight space remains connected the critical storage capacity alpha c is found to be reduced by a factor determined solely by the geometry of the weight space and is shown to interpolate within the replica symmetric approximation between alphac 2 in the gardner model limit and alphac 4 pi various comparisons of the synaptic weights with those of the binary perceptron show that such differences as remain between this weight space and that of the true binary perceptron are crucial to obtaining alphac or 4 pi although these differences limit the use of such models in realizing optimal binary networks they may yet provide worthwhile binary systems by simple weight clipping simulation results are presented in support of the theoretical analyses

fuzzy implicative and boolean filters of r 0 algebras the notions of fuzzy filters fuzzy implicative filters and fuzzy boolean filters are introduced in r0 algebras respectively some of their properties are investigated the relations between fuzzy implicative filters and fuzzy boolean filters are discussed the characterizations of fuzzy implicative filters are studied and the following is proved to be equivalent 1 a fuzzy filter f is implicative 2 f is a fuzzy boolean filter 3 x y z l f x z f x z y f y z 4 x y l f x y f x y y 5 x l n 1 f nx f x

optimal storage of invariant sets of patterns in neural network memories the authors investigate optimal conditions for the storage of invariant sets of patterns in neural network memories the sets of patterns that they consider are highly correlated non random and invariant with respect to some symmetry transformations they generalize gardner s method to study the fractional volume in the space of neural interactions that allow for storage of invariant patterns they demonstrate that optimal storage conditions correspond to non trivial relations between the number of stored patterns their stability and other symmetry specific characteristics of patterns

fuzzy positive implicative and fuzzy associative filters of lattice implication algebras we fuzzify the concept of positive implicative filters and associative filters in lattice implication algebras we prove that i every fuzzy positive implicative filter is a fuzzy implicative filter and ii every fuzzy associative filter is a fuzzy filter we provide equivalent conditions for both fuzzy positive implicative filter and fuzzy associative filter

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

on the capacity of threshold adalines with limited precision weights the effect of limited precision weights on the capacity of the threshold adaline is examined in particular an experimental technique is used to determine the capacity of the threshold adaline for several different levels of weight precision the results provide some insight into the manner in which the capacity of the threshold adaline increases as the number of bits per weight increases as might be expected the growth in capacity due to an additional bit per weight is greatest when the original number of bits per weight is small and decreases as the original number of bits per weight increases

adaptive genetic algorithm for the binary perceptron problem for neural networks with j couplings the perceptron problem for random unbiased patterns is considered an algorithm that uses concepts of the continuous perceptron problem as well as ideas of biological optimization is proposed and investigated the distribution of local stabilities and the critical storage capacity alpha c are determined while for n less than 50 the value of alpha c is approximately 0 83 the storage capacity goes down to alpha c 0 7 for n 255