comparison of two capacities in ℂ n in the past few years several capacitary set functions have been introduced in connection with the study of analytic and plurisubharmonic functions of several variables 1 1 5 9 13 there are many different such functions and the relationship between them is not at all clear we shall consider here the relative capacity of bedford and taylor and the capacity defined in terms of certain tchebycheff constants as studied by zaharjata 13 and alexander 1 see sect 2 for the definitions we show that these capacities are essentially the same our main result theorem 2 1 gives the quantitative relationship between them it happens that the relationship between the two capacities is closely connected with a result of josefson 8 on the equivalence of locally and globally pluripolar sets the quantitative estimates of theorem 2 1 allow us to give in sect 4 a new proof of josefson s lemma about normalized polynomials which are very small on the sets where a given plurisubharmonic function is nearly o e josefson s own proof is a direct construction our original motivation was to give a proof based on capacity notions it turns out that the tchebycheff polynomials themselves already do the job e1 mir 7 has obtained an extension of josefson s theorem given a function v psh on an open set he obtains a global psh function u of small growth at infinity such that u is dominated by a function h v of v his choice of h is essentially h x l o g xl thus he gets u l o g iv and so the value of the global function u is o e whenever v oe our methods give a short proof of this in fact they apply to functions h satisfying

discrete choquet integral and some of its symmetric extensions classical extensions of the choquet integral defined on 0 1 to 1 1 are the asymmetric and the symmetric choquet integral the second one being called also the sipos integral recently the balancing choquet integral was introduced as another kind of a symmetric extension of the discrete choquet integral we introduce and discuss a new type of such extension the fusion choquet integral and discuss its properties and relationship to the balancing and the symmetric choquet integral the symmetric maximum introduced by grabisch is shown to be a special case of the fusion and the balancing choquet integral several extensions of owa operators are also discussed

the nature of attractors in an asymmetric spin glass with deterministic dynamics the authors study the attractors in an infinite range ising spin glass model with deterministic dynamics where the interactions have asymmetry specified by a parameter k they find a duality relation between the attractors for models with asymmetry parameters k and 1 k the attractors are fixed points or limit cycles of short length except for k 1 at which the average cycle length diverges reminiscent of a phase transition and the model has many similarities to the random map model as well as to the infinite range symmetric spin glass in thermal equilibrium including the fact that a few attractors dominate the weight the extent of this dominance varies from sample to sample and so is given by a non trivial probability distribution pi y which they compute numerically

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

equicontinuity of families of plurisubharmonic functions with bounds on their monge ampere masses it is proved that families of solutions of the complex monge ampere equation with fixed boundary values and certain bounds on the right hand side are equicontinuous

a study of network dynamics the dynamics of a discrete time neural network model are investigated first a numerical survey of network power spectra is reported for networks of varying size with random weight matrices and initial states the steepness of the logistic function and a symmetry measure of the weight matrix are taken as control parameters summary statistics are presented to give gross measures of the model s temporal activity in parameter space second a detailed study of the dynamics of a particular network is described complex dynamical behavior is observed including hopf bifurcations the ruelle takens newhouse route to chaos showing mode locking at rational winding numbers and the destruction of an invariant torus and the period doubling route to chaos

chern forms of singular metrics on vector bundles we study singular hermitian metrics on holomorphic vector bundles following berndtsson p un previous work by raufi has shown that for such metrics it is in general not possible to define the curvature as a current with measure coefficients in this paper we show that despite this under appropriate codimension restrictions on the singular set of the metric it is still possible to define chern forms as closed currents of order 0 with locally finite mass

correlation between eigenvalue spectra and dynamics of neural networks this letter presents a study of the correlation between the eigenvalue spectra of synaptic matrices and the dynamical properties of asymmetric neural networks with associative memories for this type of neural network it was found that there are essentially two different dynamical phases the chaos phase with almost all trajectories converging to a single chaotic attractor and the memory phase with almost all trajectories being attracted toward fixed point attractors acting as memories we found that if a neural network is designed in the chaos phase the eigenvalue spectrum of its synaptic matrix behaves like that of a random matrix i e all eigenvalues lie uniformly distributed within a circle in the complex plan and if it is designed in the memory phase the eigenvalue spectrum will split into two parts one part corresponds to a random background the other part equal in number to the memory attractors the mechanism for these phenomena is discussed in this letter

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

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

from statistical mechanics to neural networks and back abstract the neural network models which have attracted so much attention in the physics community consist of a large assembly of two state elements highly interconnected by ferromagnetic and antiferromagnetic interactions methods of statistical mechanics have been applied to these models bringing new discoveries and surprises to neural network science at the same time neural network models have posed new problems and incentives to statistical mechanics one of the basic problems is that of learning which in the context of these models means a choice of the interactions for which the network dynamics leads to specific attractors the main emphasis in this paper is on the essential developments and results on this problem derived from methods of statistical mechanics

linear fuzzy game with coalition interaction and its coincident solutions abstract in this study we consider the solution concepts for fuzzy coalition games i e cooperative games with fuzzy coalition under a certain participation level in general cooperative games with fuzzy coalition are based on the assumption that all the fuzzy coalition values for different fuzzy coalitions must be represented by the same formula which may omit the coalition interaction under different participation ratios for players considering these conditions we propose the coincident fuzziness form for games with fuzzy coalition which are represented by a mapping from the characteristic function of the crisp game to that of the fuzzy coalition game the proposed fuzzy coalition games admit the differences in coalition interactions for different fuzzy coalitions where the coalition interactions are represented by the fuzzy coalition values in different ways or formulas for a fixed fuzzy coalition the maximum fuzzy coalition game is proven to be the choquet integral form on the condition that the associate crisp game is convex in order to seek appropriate solutions for the proposed games based on a certain participation level the fuzzy shapley axioms are defined and the explicit shapley value is represented by the shapley value of the associated crisp games moreover the fuzzy core of this proposed fuzzy coalition game is the stable solution set for the given fuzzy coalition which is also denoted by the crisp cores furthermore we study the relationship between the fuzzy shapley value and the fuzzy core by adding restrictions on the fuzzy core we propose a strong fuzzy core which is a more stable solution than the fuzzy core in addition the fuzzy core is equivalent to the strong fuzzy core under some condition

equilibrium properties of the linear perceptron we study the equilibrium properties of the real weights linear perceptron within the replica formalism framework focusing on the effects of the normalization of the weights on the learning and generalization capabilities of the network we also investigate the effects of static noise corrupting the training data and of dynamical noise acting on the weights during the training stage

average growth estimates for hyperplane sections of entire analytic sets suppose x is an analytic subvariety of c a basic problem in value distribution theory is to relate the growth of the volume of x to that of intersections of x with linear subspaces p of with dimx dim p n in contrast to the case where x is algebraic as described by the bezout theorem for a fixed subspace p there is no relationship between the counting functions of x and xnp it is easy to find examples where x n p is empty while x is large on the other hand an example given in 6 shows that x n p can be of infinite order even if x is of order 0 however average transcendental bezout estimates have been given by carlson 1 5 gruman 8 9 and stoll 18 gruman showed that for almost all linear subspaces p x n p has the same order of growth as x and is transcendental ifx is transcendental

domains of attraction of neural networks at finite temperature the authors consider neural networks trained by the symmetric edinburgh algorithm at finite temperature they show that while the introduction of the thermal noise affects the dynamics in a computable way it does not change the domains of attraction of stored patterns

finite morphisms to projective space and capacity theory we study conditions on a commutative ring r which are equivalent to the following requirement whenever x is a projective scheme over s spec r of fiber dimension leq d for some integer d geq 0 there is a finite morphism from x to p d s over s such that the pullbacks of coordinate hyperplanes give prescribed subschemes of x provided these subschemes satisfy certain natural conditions we use our results to define a new kind of capacity for subsets of the archimedean points of projective flat schemes x over the ring of integers of a number field this capacity can be used to generalize the converse part of the fekete szeg h o theorem

fuzzy bi cooperative games in multilinear extension form in this paper we introduce the notion of a fuzzy bi cooperative game in multilinear extension form an lg value as a possible solution concept is obtained using standard fuzzy game theoretic axioms

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

the phase space of interactions in neural networks with definite symmetry we calculate the typical fraction of the phase space of interactions which solve the problem of storing a given set of p patterns represented as n spin configurations as a function of the storage ratio a p n of the stability parameter k and of the symmetry 7 of the interaction matrices the calculation is performed for strongly diluted networks where the connectivity of each spin c is of the order of in n for each value of k and 17 there is a maximal value of a above which the volume of solutions vanishes for each value of k and a there is a typical value of 7 at which this volume is maximal the analytical studies are supplemented by numerical simulations on fully connected and diluted networks using specific learning algorithms

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 harsanyi solution for a kind of fuzzy coalition games abstract the fuzzy linear game in ref 32 is a class of fuzzy cooperative games with fuzzy coalition fuzzy coalition game for short which admits the difference of interactions for different fuzzy coalitions the fuzzy shapley value for fuzzy linear game has been studied as shapley value is just one point in the harsanyi solution it is interesting to study the explicit form of harsanyi solution in the fuzzy coalition game this paper defines the fuzzy harsanyi solution which distributes the harsanyi dividends such that the dividend shares of players in each fuzzy coalition are proportional to the corresponding players participation index when the fuzzy coalition payoff is average distributed between players in the fuzzy coalition the fuzzy harsanyi solution also coincides with the fuzzy shapley value just as their relationship in a crisp cooperative game we provide two axiomatic characterizations for the fuzzy harsanyi solution one uses axioms and the other uses distribution matrices the fuzzy harsanyi solution is a unique value on the fuzzy linear game that satisfies fuzzy efficiency fuzzy null player and fuzzy additivity property meanwhile based on probability distribution fuzzy harsanyi solution is seen as an expected marginal contribution of each player in the fuzzy coalition which provides more choices for players in the fuzzy coalition

temporal sequence storage capacity of time summating neural networks the maximum capacity for storage of temporal sequences in a time summating neural network is analysed within a statistical mechanical framework each neuron of such a network maintains an activity trace consisting of a decaying sum of all previous inputs to that neuron the maximum storage capacity is expressed as a function of the temporal correlations of the local fields of the neurons and evaluated using a perturbation expansion in powers of the decay rate

a robin formula for the fekete leja transfinite diameter we generalize the classical robin formula to higher dimensions mathematics subject classification 2000 primary 32u35 32u20 14g40 secondary 31b15 31c10 the purpose of this note is to give a formula for the fekete leja transfinite diameter on cn generalizing the classical robin formula d e e v e for the usual transfinite diameter we will disengage this from a formula for the sectional capacity proved in arithmetic intersection theory 5 theorem 1 1 p 233 first recall the definition of the fekete leja transfinite diameter for a compact set e cn see 1 16 consider the set of monomials zk z1 1 zn n in the polynomial ring c z c z1 zn let n c z be the space of polynomials of total degree at most n and let k n k zn ki 0 k1 kn n be the index set for the monomial basis of n put qn k n n n n fixing n take qn independent vector variables zi zi1 zin cn i 1 qn let k1 kqn be the indices in k n the vandermonde determinant qn z1 zqn det zj i n i j 1 work supported in part by nsf grant dms 0300784 r rumely b department of mathematics university of georgia athens ga 30602 usa e mail rr math uga edu