probing the basins of attraction of a recurrent neural network a recurrent neural network is considered that can retrieve a collection of patterns as well as slightly perturbed versions of this pure set of patterns via fixed points of its dynamics by replacing the set of dynamical constraints i e the fixed point equations by an extended collection of fixed point like equations analytical expressions are found for the weights wij b of the net which depend on a certain parameter b this so called basin parameter b is such that for b 0 there are a priori no perturbed patterns to be recognized by the net it is shown by a numerical study via probing sets that a net constructed to recognize perturbed patterns i e with values of the connections wij b with b 0 possesses larger basins of attraction than a net made with the help of a pure set of patterns i e with connections wij b 0 the mathematical results obtained can in principle be realized by an actual biological neural net

monte carlo simulations of the ising spin glass on lattices with finite connectivity the authors have simulated the ising spin glass model on a random lattice with a finite average coordination number and also on the bethe lattice with various different boundary conditions in particular they have calculated the overlap function p q for two independent samples for the random lattice the results are consistent with a spin glass transition above which p q converges to a dirac delta function for large n number of sites and below which p q has in addition a long tail similar to previous results obtained for the infinite range model for the bethe lattice they obtain results in the interior by discarding the two outer shells of the cayley tree when calculating the thermal averages for fixed uncorrelated boundary conditions p q seems to converge to a delta function even below the spin glass transition whereas on a closed lattice correlated boundary conditions p q has a long tail similar to its behaviour in the random lattice case

generalized vector coherent state theory of sp 2n r vector operators and of sp 2n r contains implies u n reduced wigner coefficients sp 2n r contains implies u n reduced wigner coefficients coupling a positive discrete series unitary irreducible representation with the 2n dimensional non unitary irreducible representation characterized by 1 are evaluated by making use of generalized vector coherent state techniques general formulae are given in terms of the k matrix elements of vector coherent state theory and of some u n 6j or 9j type recoupling coefficients explicit analytic expressions are obtained in some important special cases such results should allow a determination of wsp 2n r approximately w n p 2n r and osp p 2n r matrix representations to be completed because both the w n generators and the osp p 2n r odd generators transform under the sp 2n r irrep 1

exact solitary and periodic wave solutions of the k 2 2 equation defocusing branch an auxiliary elliptic equation method is presented for constructing exact solitary and periodic travelling wave solutions of the k 2 2 equation defocusing branch some known results in the literature are recovered more efficiently and some new exact travelling wave solutions are obtained also new stationary wave solutions are obtained

multistate perceptrons learning rule and perceptron of maximal stability a new perceptron learning rule which works with multilayer neural networks made of multistate units is obtained and the corresponding convergence theorem is proved the definition of a perceptron of maximal stability is enlarged in order to include these new multistate perceptrons and a proof of the existence and uniqueness of such optimal solutions is outlined

coloured frt algebra and its yang baxterization leading to integrable models with non additive r matrices a colour representation of faddeev reshetikhin takhtajan frt algebra which in contrast to the standard case is related to the coloured braid group representation with generic values of q is presented explicit realizations of l or matrices occurring in this coloured variant of frt algebra are also obtained for the uq gl 2 quantized algebra though these realizations are found to depend manifestly on the colour parameters the underlying quantum group structure and associated co product are interestingly free from such dependence this allows us to perform the yang baxterization of the coloured frt algebra successfully which leads to the construction of an ancestor law operator associated with a new non additive type quantum r matrix through different realizations of this lax operator a new class of quantum integrable models representing colour generalizations of the well known models such as the lattice sine gordon model the ablowitz ladik model lattice and the derivative nonlinear schrodinger model etc is generated

learning grey toned patterns in neural networks the problem of learning multi state patterns in neural networks is investigated an analysis of the space of couplings gardner approach yields the distribution of local fields the critical storage capacity alpha c and the minimum number of errors for an overloaded network for noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width a fast converging adaptive learning algorithm is presented which finds the coupling matrix of optimal stability

the effect of correlations in neural networks the effect of correlations in neural networks is investigated by considering biased input and output patterns statistical mechanics is applied to study training times and internal potentials of the minover and adaline learning algorithms for the latter a direct extension to generalization ability is obtained comparison with computer simulations shows good agreement with theoretical predictions with biased patterns the authors find a decrease in training times and internal potentials for the minover algorithm which however does not lead to faster storage of a given information measure in adaline training characteristic times undergo a transition from order 1 to order n at any finite bias for the learning of patterns as well as for the decay of the generalization error this leads to a rescaling of the gain parameters

symmorphic space groups of finite crystal lattices the symmorphic space group of a finite n dimensional crystal lattice is studied and its factorization is presented this lattice is determined by a direct product of cyclic groups and then its translation point and space groups are defined as modular images of corresponding ones for a lattice of infinite extent a factorization of holohedries of an infinite lattice is used in order to present a symmorphic finite space group as a direct product as an example and as a very special case a space group in the case of the hypercubic lattice is studied and its irreducible representations are investigated the results obtained suggest that for vectors lying on a surface of the first brillouin zone i e for which at least one coordinate is equal to 0 or 1 2 an additional index describing their symmetry properties should be introduced this enables the authors to make a more detailed classification of states energy levels on the other hand finite symmorphic space groups can be used within e g the so called finite lattice approach

the space of interactions in neural networks gardner s computation with the cavity method gardner s 1987 1988 computation of the number of n bit patterns which can be stored in an optical neural network used as an associative memory is derived without replicas using the cavity method this allows for a unified presentation whatever the basic measure in the space of coupling constants but above all it gives the clear physical content of the assumption of replica symmetry tap equations are also derived

penalization in non classical convex programming via variational convergence a penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method proofs of convergence rely on variational convergence theory

alternating proximal algorithms for linearly constrained variational inequalities application to domain decomposition for pde s abstract let x y z be real hilbert spaces let f x r g y r be closed convex functions and let a x z b y z be linear continuous operators let us consider the constrained minimization problem p min f x g y a x b y given a sequence n which tends toward 0 as n we study the following alternating proximal algorithm a x n 1 argmin n 1 f 1 2 a b y n z 2 2 x n x 2 x y n 1 argmin n 1 g 1 2 a x n 1 b z 2 2 y n y 2 y where and are positive parameters it is shown that if the sequence n tends moderately slowly toward 0 then the iterates of a weakly converge toward a solution of p the study is extended to the setting of maximal monotone operators for which a general ergodic convergence result is obtained applications are given in the area of domain decomposition for pde s

symmetric and non symmetric waves in the osmosis k 2 2 equation we give an improved qualitative method to solve the osmosis k 2 2 equation this method combines several characteristics of other methods using this method the existence of symmetric and non symmetric wave solutions of the osmosis k 2 2 equation is studied besides abundant symmetric forms such as smooth wave solutions peaked waves cusped waves looped waves stumpons and fractal like waves this equation also admits non symmetric ones including breaking kink wave solutions breaking anti kink wave solutions and rampons as regards this equation most of those solutions either symmetric or non symmetric solutions have not appeared in the literature we also study the limiting behavior of all periodic solutions as the parameters tend to some special values

a dilution algorithm for neural networks a dilution algorithm to enlarge the storage capacity per synapse alpha eff of neural networks is proposed the algorithm is a hybrid method where hebb s rule is used to select a fraction of couplings to be removed afterwards the perceptron of optimal stability for the remaining couplings is learned the authors present an analytical calculation and the results of the numerical simulations in comparison with the fully connected or the randomly diluted perceptron the effective storage capacity alpha eff is remarkably enlarged

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

on solutions of the yang baxter equations without additivity the relationship is described between solutions found by ge and xue 1991 and by the present author 1987

distributions and size scalings for strength in a one dimensional random lattice with load redistribution to nearest and next nearest neighbors lattice and network models with elements that have random strength are useful tools in explaining various statistical features of failure in heterogeneous materials including the evolution of failure clusters and overall strength distributions and size effects models have included random fuse and spring networks where monte carlo simulation coupled to scaling analysis from percolation theory has been a common approach unfortunately severe computational demands have limited the network sizes that can be treated to gain insight at large size scales interest has returned to idealized fiber bundle models in one dimension many models can be solved exactly or asymptotically in increasing size n but at the expense of major simplification of the local stress redistribution mechanism models have typically assumed either equal load sharing among nonfailed elements or nearest neighbor local load sharing lls where a failed element redistributes its load onto its two nearest flanking survivors the present work considers a one dimensional fiber bundle model under tapered load sharing tls which assumes load redistribution to both the nearest and next nearest neighbors in a two to one ratio this rule reflects features found in a discrete mechanics model for load transfer in two dimensional fiber composites and planar lattices we assume that elements have strength 1 or 0 with probability p and q 1 ensuremath p respectively we determine the structure and probabilities for critical configurations of broken fibers which lead to bundle failure under a given load we obtain rigorous asymptotic results for the strength distribution and size effect as stackrel ensuremath rightarrow n ensuremath infty with precisely determined constants and exponents the results are a nontrivial extension of those under lls in that failure clusters are combinatorially much more complicated and contain many bridging fibers consequently certain probabilities are eigenvalues from recursive equations arising from the structure of tls next nearest neighbor effects weaken the material beyond what is predicted under lls keeping only nearest neighbor overloads our results question the validity of scaling relationships that are based largely on monte carlo simulations on networks of limited size since some failure configurations appear only in extremely large bundles the dilemma has much in common with the petersburg paradox

on the quantum symmetries associated with the two parameter free fermion model the quantum algebras associated with the r matrix obtained from the two parameter free fermion model are examined it is shown that in addition to uq s gl 1 1 there exists another algebra which is isomorphic to uq s gl 1 1 as an algebra but has a different hopf structure it does not have a classical limit and is related to up sl 2 c at a root of unity p i i2 1

sign constrained linear learning and diluting in neural networks for neural networks with predefined effects of the synapses excitatory or inhibitory the simplex algorithm is applied as a learning rule it is assumed that the given signs of the synapses can never be changed during learning the maximum possible dilution of synapses as a result of the learning is found at the maximum storage capacity of a model with only positive or randomly distributed signs for the case of infinitely many neurons a replica symmetric calculation of the free energy and the distribution of coupling strengths is presented the linear algorithm is also applied to networks with a more suitable choice of sign constraints with the result of a higher storage capacity

single peak solitary wave solutions for the ch kp 2 1 equation under boundary condition abstract the qualitative theory of differential equations is applied to the ch kp 2 1 equation u t 2 u x u 2 x x t u u x x u y y 0 our procedure shows that the ch kp 2 1 equation either has the regular peakon soliton cuspon soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u a 0 in particular mathematical analysis and numerical simulations of the ch kp 2 1 equation are provided for those peakon cuspon compacton loop soliton and smooth soliton solutions

strong asymptotic convergence of evolution equations governed by maximal monotone operators with tikhonov regularization abstract we consider the tikhonov like dynamics u t a u t e t u t where a is a maximal monotone operator on a hilbert space and the parameter function e t tends to 0 as t with 0 e t d t when a is the subdifferential of a closed proper convex function f we establish strong convergence of u t towards the least norm minimizer of f in the general case we prove strong convergence towards the least norm point in a 1 0 provided that the function e t has bounded variation and provide a counterexample when this property fails

the upper triangular solutions to the three state constant quantum yang baxter equation in this article we present all nonsingular upper triangular solutions to the constant quantum yang baxter equation rj1j2k1k2 rk1j3l1k3 rk2k3l2l3 rj2j3k2k3 rj1k3k1l3 rk1k2l1l2 in the three state case the upper triangular ansatz implies 729 equations for 45 variables fortunately many of the equations turned out to be simple allowing us to start breaking the problem into smaller ones in the end we had a total of 552 solutions but many of them were either inherited from two state solutions or subcases of others the final list contains 35 nontrivial solutions most of them new

distribution of the activities in a diluted neural network the dynamics of asymmetrically diluted neural networks can be solved exactly in the present work the distribution of the neural activities is calculated analytically for zero temperature parallel dynamics this distribution depends on the number of stored patterns and is a continuous function in the good retrieval phase the continuous part of the distribution of activities is due to the asymmetry of the synapses since it is known that networks with symmetric interactions always have a distribution of activities which is a sum of a few delta functions the expression for the distribution of activities is also given for a mixture of two patterns which have a non zero overlap

sign constrained synapses and biased patterns in neural networks the storage of biased patterns is examined in neural networks with sign constrained synapses every neuron has outgoing synapses which are either all inhibitory or all excitatory for random patterns stored in such networks it is known that the presence of a discrete gauge symmetry makes the maximal storage capacity independent of the proportion of excitatory neurons to inhibitory neurons when the stored patterns are biased however this discrete gauge symmetry is broken with the result that the maximal capacity depends on the proportion of excitatory neurons to inhibitory ones the dependence of the capacity on the fraction of excitatory neurons in the network f is calculated using the space of interactions approach it is found that the storage capacity is maximal at f 0 5 this result is true regardless of the particular value of the bias in the stored patterns the significance of this result in the neurophysiological context is discussed

approaching the solving of constrained variational inequalities via penalty term based dynamical systems abstract we investigate the existence and uniqueness of locally absolutely continuous trajectories of a penalty term based dynamical system associated to a constrained variational inequality expressed as a monotone inclusion problem relying on lyapunov analysis and on the ergodic continuous version of the celebrated opial lemma we prove weak ergodic convergence of the orbits to a solution of the constrained variational inequality under investigation if one of the operators involved satisfies stronger monotonicity properties then strong convergence of the trajectories can be shown

yang baxter matrices and differential calculi on quantum hyperplanes it is shown that any invertible matrix r that solves the yang baxter equations generates a set of quantum hyperplanes where differential calculi can be defined the number of such quantum hyperplanes is given by the number of different eigenvalues of the matrix r several examples of two dimensional quantum hyperplanes and differential calculi are presented the relations of quantum hyperplanes and differential calculi are covariant wrt the quantum groups defined by the matrix r in the generic cases the exterior differential satisfies the condition d2 0

screening and vertex operators for u n screening and vertex operators for u n are given in terms of generators of the group left action on the flag manifold u n 1 su n their matrix elements in a basis of holomorphic sections of a vector bundle are computed

learning and generalization in multineuron interacting feed forward neural networks we consider learning and generalization in the multi interacting feed forward network model recently proposed by h o carmesin 1994 with an a priori definition of the net architecture based on symmetries presented by the function to be learnt we define a generalized hebb rule extend the maximum stability learning algorithm to multi interactions and obtain training and generalization curves for rules where different orders of synapses are not correlated the results obtained for the simple perceptron concerning the hebb rule and through replica calculations in the space of couplings may be straightforwardly adapted to multi interactions through a simple renormalization of the total number of independent couplings analytical and numerical simulation results are compared and show excellent agreement

training of optimal cluster separation networks finding the optimal separation of two clusters of normalized vectors corresponds to training thresholds and weights in a neural network of maximum stability in order to achieve this two local iterative algorithms are presented which treat threshold and weights all in one avoiding the need to calculate any intermediate test quantities convergence is proved and the separation stability obtained is shown to match theoretical predictions and to be superior to existing 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

learning the unlearnable three neural network training algorithms are presented which are robust to nonlearnable problems the first algorithm converges to the gardner stability limit if the learning problem is linearly separable and otherwise finds a locally maximally stable solution the second algorithm is a robust version of rosenblatt s perceptron learning algorithm which will converge to a solution of the learning problem if one exists and otherwise will converge locally to a solution with a certain fraction of wrongly mapped patterns the third algorithm is suited most favourably to unlearnable problems it will always find a solution if the problem is learnable and otherwise it locally maximizes the number of patterns which are stored correctly the error rate of this algorithm and other known algorithms for unlearnable problems are compared for two benchmark problems proofs of the existence of solutions are given convergence is proven as well to be global in the case of learnable and local in unlearnable cases

dilution in boolean perceptrons that learn from noisy examples we investigate the effect of dilution after learning on the generalization ability of single layer boolean perceptrons that learn from noisy examples we present a thorough comparison between the relative performances of several well known learning rules in particular we show that the effect of dilution is always deleterious and that the bayes algorithm always gives the best generalization performance

two parametric solution to graded yang baxter equation and two parametric uuvgl 1 mod 1 algebra as a hopf algebra the authors discuss a two parametric solution of the graded yang baxter equation ybe and perform the yang baxterization to obtain the solution to the quantum ybe in the formalism developed previously they present the two parametric quantized superalgebra uuvgl 1 mod 1 and prove that this algebra is a hopf algebra with the hopf operations explicitly provided

time dependent fiber bundles with local load sharing fiber bundle models where fibers have random lifetimes depending on their load histories are useful tools in explaining time dependent failure in heterogeneous materials such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings various asymptotic and approximate theories have been developed for bundles with various geometries and fiber load sharing mechanisms but numerical verification has been hampered by severe computational demands in larger bundles to gain insight at large size scales interest has returned to idealized fiber bundle models in 1d such simplified models typically assume either equal load sharing els among survivors or local load sharing lls where a failed fiber redistributes its load onto its two nearest flanking survivors such models can often be solved exactly or asymptotically in increasing bundle size n yet still capture the essence of failure in real materials the present work focuses on 1d bundles under lls as in previous works a fiber has failure rate following a power law in its load level with breakdown exponent ensuremath rho surviving fibers under fixed loads have remaining lifetimes that are independent and exponentially distributed we develop both new asymptotic theories and new computational algorithms that greatly increase the bundle sizes that can be treated in large replications e g one million fibers in thousands of realizations in particular we develop an algorithm that adapts several concepts and methods that are well known among computer scientists but relatively unknown among physicists to dramatically increase the computational speed with no attendant loss of accuracy we consider various regimes of ensuremath rho that yield drastically different behavior as n increases for 1 2l ensuremath rho l 1 els and lls have remarkably similar behavior they have identical lifetime distributions at ensuremath rho 1 with approximate gaussian bundle lifetime statistics and a finite limiting mean for ensuremath rho g1 this gaussian behavior also applies to els whereas lls behavior diverges sharply showing brittle weakest volume behavior in terms of characteristic elements derived from critical cluster formation for 0l ensuremath rho l1 2 els and lls again behave similarly but the bundle lifetimes are dominated by a few long lived fibers and show characteristics of strongest link extreme value distributions which apply exactly to ensuremath rho 0

the generalization probability of a perceptron using the delta rule compares the generalization probability of a perceptron using the delta rule with other learning algorithms the numerical results indicate that the delta rule gives results which are superior to hebb s rule and comparable with the maximal stability perceptron rule minover both for randomly generated linear separable example sets and for optimal queries

spectral parameter dependent approach to quantized algebra its multiparameter deformations and their q oscillator realizations yang baxterization of faddeev reshetikhin takhtajan algebra 1989 leading to quantum ybe is considered for a class of r matrices which satisfy hecke relations the authors apply such construction of the gl n case together with its multiparameter deformations revealing the connection between frt relations and the extended trigonometric sklyanin algebra new realizations of frt algebra through q oscillator modes are presented and their potential importance in the context of quantum integrable models is discussed a multiparametrized r matrix with the inclusion of spectral parameter is constructed as a by product

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

solitons in a nonlinear model medium the periodic stationary solutions of a model nonlinear evolution equation simulating the propagation of short wave perturbations in a relaxing medium are studied solutions expressed by a multiple valued function are shown to exist a method for determining the nonlinear interaction between solitary waves is suggested an example of a collision of solitons is given

a network of oscillators a model of neuronal behaviour capable of accounting for the oscillatory plateau and rebound properties of biological neurons is derived discussed and analysed the model is based on a piecewise linear form of the fitzhugh nagumo equations but reduces to a set of maps very similar to those of the hopfield model 1982 in particular the binary description of individual neurons and the well studied form of the synaptic current sigma jijsj are preserved although the model is capable of reproducing behaviours on the slow timescales characteristic of plateau and oscillation by coupling two model cells together a mutually inhibitory or half centred oscillator and an oscillator fixed phase follower pairs are constructed

proximal point algorithm controlled by a slowly vanishing term applications to hierarchical minimization let phi 0 mathbb r n to mathbb r cup infty be a closed convex function and phi 1 mathbb r n to mathbb r be a finite convex function that are bounded from below our goal is to build an algorithm which first minimizes the map phi 0 and secondly the map phi 1 over the set s 0 rm argmin phi 0 for that purpose we define the following proximal type algorithm x n 1 x n lambda n in partial eta n phi 0 varepsilon n phi 1 x n 1 leqno mathcal a 1 where lambda n is a positive step sequence eta n is a summable error sequence and varepsilon n is a control sequence tending toward 0 partial eta denotes the eta approximate subdifferential when varepsilon n is a slow control i e sum n 0 infty varepsilon n infty we prove that under adequate conditions the sequence x n defined by mathcal a 1 tends toward an element of s 1 rm argmin s 0 phi 1 more generally given finite convex functions phi 2 ldots phi n mathbb r n to mathbb r let us define the sets s i i in 1 ldots n by the recursive relation s i rm argmin s i 1 phi i we introduce an extension of algorithm mathcal a 1 to minimize hierarchically each function phi i on the set s i 1 for i in 1 ldots n

on the unlearning procedure yielding a high performance associative memory neural network we consider a fully connected hopfield like neural network as a set of n independent perceptrons we trained these perceptrons using the so called inverse perceptron rule obtaining a matrix of synaptic couplings that make a number of spurious states unstable we numerically determine the optimum number of spurious states obtained by random shooting that must be destabilized in order to obtain an improvement in performance the unlearning procedure generated is shown to be able to give a high performance associative memory characterized by 1 an enhancement in storaging capacity 2 an enlargement in the size of attraction basins 3 a reduction in the number of spurious attractors and 4 a reliable and fast retrieval

asymptotic behavior of coupled dynamical systems with multiscale aspects abstract we study the asymptotic behavior as time variable t goes to of nonautonomous dynamical systems involving multiscale features as a benchmark case given h a general hilbert space h r and h r two closed convex functions and a function of t which tends to as t goes to we consider the differential inclusion x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled systems we show several results ranging from weak ergodic to strong convergence of the trajectories as a key ingredient we assume that for every p belonging to the range of n c 0 t p t c p t d t where is the fenchel conjugate of c is the support function of c argmin and n c x is the normal cone to c at x as a by product we revisit the system x t t x t x t 0 where t tends to zero as t goes to and 0 t d t whose asymptotic behavior can be derived from the preceding one by time rescaling applications are given in game theory optimal control variational problems and pdes

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

structure and representations of the hyperoctahedral group in this paper the main properties of the symmetry group of the n dimensional cube are reviewed and formulated with respect to possible applications in lattice theories the connection between the hyperoctahedral group wn and the orthogonal group o n is investigated by means of the canonical representation

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

solutions of the nonadditive ybe by spectral parameter factorization and re yang baxterization r matrix solutions for the nonadditive yang baxter equation are obtained in arbitrary dimension using a factorization ansatz in spectral parameters a symmetry transformation determined for the particle conserving r matrix is used to formulate a re yang baxterization scheme with multicomponent spectral parameters extending the notion of conventional yang baxterization the possible scope for such a transformation which has a striking resemblance to reshetikhin s deformation of r matrices but extended now to include spectral parameters is discussed

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

construction of coupling coefficients of su 4 in a supermultiplet basis matrix elements of the unit tensor operators of ranks 1 2 and 1 1 in the non orthonormal draayer basis of su 4 are found the evaluation algorithms for the su 4 coupling coefficients are proposed for coupling one arbitrary and one four six or ten dimensional irreducible representation of su 4 in the su 2 su 2 chain as well as in the case of the semistretched coupling the construction of the overlaps of the draayer basis states is considered

classification of the travelling waves in the nonlinear dispersive kdv equation abstract in this paper we classify the travelling wave solutions to the nonlinear dispersive kdv equation called k 2 2 equation the parameter region is specified and the parameter dependence of its solitary waves is described besides the previously known compacton solutions the equation is shown to admit more new solutions such as cuspons peakons loopons stumpons and fractal like waves furthermore by the qualitative results we give some new explicit travelling wave solutions

special exact soliton solutions for the k 2 2 equation with non zero constant pedestal abstract special exact solutions of the k 2 2 equation u t u 2 x u 2 xxx 0 are investigated by employing the qualitative theory of differential equations our procedure shows that the k 2 2 equation either has loop soliton cusped soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u 0 mathematical analysis and numerical simulations are provided for these soliton solutions of the k 2 2 equation

a disorder solution for a generalised mixed spin model on anisotropic kagome lattices a disorder solution is given for a generalised mixed spin model on an anisotropic kagome lattice because of its lower rotational symmetry the solution is obtained by decimation along four different directions in the case of a spin 1 2 and spin 1 mixed model the intra row correlations in both horizontal and vertical directions are zero when a relation imposed on the interactions is satisfied

non linear conductivity of granular superconductors a novel breakdown problem the authors study a percolating network of highly hysteretic josephson junctions in which each junction is modelled as a piecewise ohmic hysteretic device at the percolation threshold the average voltage drop across a network on a large l l grid is assumed to fall to zero as v approximately i ic l eta as the applied current is reduced to the critical current ic l the finite size monte carlo study gives the value eta 2 0 or 0 1 this value is remarkably close to the result eta 2 obtained by a scaling analysis of a continuum version of the model

matrix elements of the labelling operators for su 4 su 2 su 2 a matrix representation of the labelling operators of moshinsky and nagel 1963 and of partensky and maguin 1978 in the non orthogonal draayer basis of su 4 is derived this allows one to solve explicitly the missing label problem for the spin isospin multiplets in the arbitrary su 4 supermultiplets the simplification and degeneration of these representations are considered for two parametric irreducible representations of su 4 as well as for some special cases of irreps typical of nuclear theory the expansion of the linearly dependent states of the draayer basis is discussed

optimal basins of attraction in randomly sparse neural network models the size of the basin of attraction for randomly sparse neural networks with optimal interactions is calculated for all values of the storage ratio alpha p c 2 where p is the number of random uncorrelated patterns and c is the connectivity the basin of attraction is finite while for alpha 0 42 the basin of attraction is almost 100

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

statistical mechanics of unsupervised structure recognition a model of unsupervised learning is studied where the environment provides n dimensional input examples that are drawn from two overlapping gaussian clouds we consider the optimization of two different objective functions the search for the direction of the largest variance in the data and the largest separating gap stability between clusters of examples respectively by means of a statistical mechanics analysis we investigate how well the underlying structure is inferred from a set of examples the performances of the learning algorithms depend crucially on the actual shape of the input distribution a generic result is the existence of a critical number of examples needed for successful learning the learning strategies are compared with methods different in spirit such as the estimation of parameters in a model distribution and an information theoretical approach

fracture patterns and scaling laws abstract fracture is determined by the growth of cracks and can be described as a moving boundary problem similar to dielectric breakdown or viscous fingering the vectorial nature of the elastic medium allows however for a much larger variety of boundary conditions and growth rules than in the scalar electrostatic analogue giving rise to a number of new effects one can model this problem on a square lattice and it is found that the patterns of cracks can be fractal even without including noise due only to the interplay of anisotropy and memory the shapes of the cracks can be compared to the ones found experimentally for stress corrosion an effect that can be mimicked by a damage memory on the surface of the crack a quantity of experimental interest in the breaking characteristics force vs displacement in a model with quenched disorder and the possibility that several cracks be present in the system one finds scaling laws in the system size for the breaking characteristics just before the system breaks fully apart the distribution of local strain is multifractal this means that the volume fraction responsible for the final cracking goes to zero in the thermodynamic limit the scaling and multiscaling laws seem to be quite universal

density of states and dynamic scaling on the vicsek snowflake fractal real space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the heisenberg ferromagnet and antiferromagnet on the infinite vicsek snowflake fractal the authors have calculated the associated dynamic exponents zf and za by linearising about the respective fixed points of the ferromagnetic antiferromagnetic recursion relations and find them to satisfy the relation za zf 2 in addition using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho f omega and rho a omega have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha omega approximately bz alpha df rho lambda alpha omega near their respective fixed points alpha a f df denoting the fractal dimension finally the amplitudes galpha omega appearing in the solution to the scaling equation rho alpha omega approximately galpha omega omega dfz alpha 1 for the density of states rho alpha omega were found to be periodic functions of ln omega with period ln lambda alpha where lambda alpha is the eigenvalue associated with ferromagnetic alpha f or antiferromagnetic alpha a fixed point

nature of the largest cluster size distribution at the percolation threshold two distinct distribution functions psp m and pns m of the scaled largest cluster sizes m are obtained at the percolation threshold by numerical simulations depending on the condition whether the lattice is actually spanned or not with r pc the spanning probability the total distribution of the largest cluster is given by ptot m r pc psp m 1 r pc pns m the three distributions apparently have similar forms in three and four dimensions while in two dimensions ptot m does not exhibit a familiar form by studying the first and second cumulants of the distribution functions the different behaviour of ptot m in different dimensions may be quantified

the induced representations of brauer algebras and the clebsch gordan coefficients of induced representations of from with are discussed the induction coefficients idcs or the outer product reduction coefficients of with up to a normalization factor are derived by using the linear equation method weyl tableaux for the corresponding gel fand basis of so n are defined the assimilation method for obtaining clebsch gordan coefficients of so n in the gel fand basis for no modification rule involved couplings from idcs of brauer algebras is proposed some isoscalar factors of for the resulting irrep with are tabulated

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

the statistical mechanics of constructive algorithms the storage capacity of multilayer networks with overlapping receptive fields is studied for constructive algorithms using boolean perceptrons as their basic building block which have been investigated within a replica framework the assumption of weak coupling between subsequently constructed perceptrons is verified within a replica symmetric rs ansatz and shown to be negligible in most cases in comparison with correction due to replica symmetry breaking rsb in individual perceptrons the capacities of a tiling like and variants of the upstart algorithm are then calculated within rs and one step rsb with the quenched average taken over the individual units separately for networks with up to k 4000 and k 600 units respectively within this treatment the storage capacity seems to exhibit a power law behaviour in with an exponent n that may depend on the algorithm and the stability however due to finite size effects in k reliable estimates of n could not be extracted nevertheless the results strongly indicate that n should be strictly smaller than 1 within one step rsb whereas within rs the mitchison durbin bound is violated for finite k and n 1 may hold asymptotically

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

more on coupling coefficients for the most degenerate representations of so n we present explicit closed form expressions for the general group theoretical factor appearing in the topology of a high temperature expansion of so n symmetric lattice models this object which is closely related to 6j symbols for the most degenerate representation of so n is discussed in detail

coupling coefficients of so n and integrals involving jacobi and gegenbauer polynomials the expressions for the coupling coefficients 3j symbols for most degenerate symmetric representations of orthogonal groups so n in a canonical basis with so n restricted to so n 1 and different semicanonical or tree bases with so n restricted to so n so n n n n are considered respectively in context of integrals involving triplets of the gegenbauer and the jacobi polynomials since the directly derived triple hypergeometric series do not reveal the apparent triangle conditions of the 3j symbols they are rearranged using their relation with semistretched isofactors of the second kind for the complementary chain sp 4 su 2 su 2 and analogy with the stretched 9j coefficients of su 2 into formulae with more rich limits for summation intervals and obvious triangle conditions the isofactors of class one representations of orthogonal groups or class two representations of unitary groups and of course the related integrals involving triplets of the gegenbauer and the jacobi polynomials turn into double sums in the cases of canonical so n so n 1 or u n u n 1 and semicanonical so n so n 2 so 2 chains as well as into the 4f3 1 series under more specific conditions some ambiguities of the phase choice of the complementary group approach are adjusted as well as problems with an alternating sign parameter of so 2 representations in the so 3 so 2 and so n so n 2 so 2 chains

compacton peakon cuspons loop solutions and smooth solitons for the generalized kp mew equation in this paper we investigate all possible single traveling solitary wave solutions of the generalized kp mew equation u t u 2 x u 2 x x t x u y y 0 under boundary condition lim u a regular compacton solution of the generalized kp mew equation correspond to the case of a 0 in the case of a 0 we find new exact soliton solutions including loop solution cusped soliton peaked soliton and smooth soliton solutions the parametric conditions of existence of the loop solution cusped soliton peaked soliton smooth soliton and compacton solutions are given by using the phase portrait analytical technique mathematical analysis and numerical simulations are provided for these soliton solutions of the generalized kp mew equation we show some graphs to explain these solutions

single peak solitary wave solutions for the osmosis k 2 2 equation under inhomogeneous boundary condition abstract the qualitative theory of differential equations is applied to the k 2 2 equation with osmosis dispersion smooth peaked and cusped solitary wave solutions of the osmosis k 2 2 equation under inhomogeneous boundary condition are obtained the parametric conditions of existence of the smooth peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique asymptotic analysis and numerical simulations are provided for smooth peaked and cusped solitary wave solutions of the osmosis k 2 2 equation

mean field analysis of hierarchical associative networks with magnetisation a hopfield type neural network that can store ultrametrically organised patterns with a finite magnetisation or bias is studied both the patterns and their ancestors are remembered in one network a homogeneous field or threshold is included in the model in zero field the replica symmetric mean field equations map onto those of the hopfield model implying that transition temperatures storage capacity etc are the same by varying appropriately the field and or a constant added to all the synaptic strengths it is possible to climb up and down the hierarchical tree or to focus on a certain level in the hierarchy and thereby increase the capacity slightly

new solitary wave solutions with compact support for the kdv like k m n equations with fully nonlinear dispersion in this paper a new method different from the adomian decomposition method namely the extend sine cosine method is proposed to investigate the compacton solutions of the kdv like equations with fully nonlinear dispersion k m n equations u t u m x u n x x x 0 the new exact solitary wave solutions with compact support of the equations are found by our new method the two special cases k 2 2 and k 3 3 are chosen to illustrate the concrete scheme of our approach in k m n equations an entirely new general formulas for the solutions of k m n equations for all types where m n are established our results include not only some known results in literature as special cases but also some new exact special solutions the method presented by this paper is suitable for studying compacton solutions of some other equations

weak versus strong uniqueness of gibbs measures a regular short range example we provide an example of a nearest neighbour random model on a regular lattice which almost certainly has a unique disordered gibbs state for every boundary condition although by choosing interaction dependent boundary conditions one can obtain different gibbs states

single peak solitary wave solutions for the generalized kp mew 2 2 equation under boundary condition the qualitative theory of differential equations is applied to the kp mew 2 2 equation q t q 2 x q 2 x x t x q y y 0 our procedure shows that the kp mew 2 2 equation either has the regular peakon soliton cuspon soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x q x a 0 or possesses compacton solutions only when lim x q x a 0 in particular we present mathematical analysis and numerical simulations are provided for those peakon cuspon compacton loop soliton and smooth soliton solutions

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

random solutions from a regular density functional hamiltonian a static and dynamical theory for the structural glass transition as simple model density functional hamiltonian is shown to lead to a consistent static and dynamical theory of the structural glass transition where glassy metastable free energy states play a key role the crucial concept introduced is a probabilistic order parameter description that results when multiple solutions of the density functional theory are considered where each solution is given a canonical weight the theory suggests that there are two distinct transition temperatures or densities at the higher transition temperature an extensive number of well defined global metastable states appear for the first time at the lower transition temperature the number of relevant glassy metastable states becomes non extensive

vector coherent state theory of the semidirect sum lie algebras wsp 2n r positive discrete series irreducible representations of the semidirect sum lie algebras wsp 2n r w n sp 2n r are given in wsp 2n r contains implies sp 2n r contains implies u n bases the branching rule for the reduction of a wsp 2n r irreducible representation into a direct sum of multiplicity free sp 2n r irreducible representations is accounted for by the explicit construction of a wsp 2n r contains implies sp 2n r contains implies u n basis of vector valued holomorphic functions

the stability of solutions of vakhnenko s equation vakhnenko s equation has two families of travelling wave solutions the method of rowlands and infeld 1990 is used to investigate whether these solutions are stable to long wavelength perturbations of small amplitude the method predicts stability for both families of solutions some comments on the validity of the method are given

the isoscalar factors of o6 o6 using the tensor basis and utilising the wigner eckart theorem the authors obtain the matrix representations 00 and 10 of o6 st o3 s o3 t the isoscalar factors of o6 st o6 st o6 st o3 s o3 t are also calculated

scattering operators on fock space ii representations of the scattering operator generated by sl 2 r for pt i see ibid vol 18 p 3141 1985 a representation for unitary scattering operators acting on a symmetric fock space and invariant under an so n internal symmetry group is constructed a group of transformations commuting with so n is seen to be isomorphic to sl 2 r the representations of sl 2 r acting on the fock space are shown to come from the discrete series of representations of sl 2 r these representations are used to label the equivalent irreducible representations of so n and the partial wave amplitudes of the scattering operators are shown to be matrix elements of the discrete series of representations of sl 2 r the example of isospin internal symmetry and the pion triplet is briefly discussed

generalized solutions for the sum of two maximally monotone operators a common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions a classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations in this paper we introduce a normal problem associated with finding a zero of the sum of two maximally monotone operators if the original problem admits solutions then the normal problem returns this same set of solutions the normal problem may yield solutions when the original problem does not admit any furthermore it has attractive variational and duality properties several examples illustrate our theory

vector coherent state theory of the noncompact orthosymplectic superalgebras i general theory vector coherent states are defined for the positive discrete series irreducible representations of the non compact orthosymplectic superalgebras osp p 2n r where p 2m or 2m 1 an orthonormal bargmann berezin basis symmetry adapted to osp p 2n r so p sp 2n r so p u n is constructed and used to develop the k matrix theory for osp p 2n r a general method is provided for determining the conditions of existence of star representations and of grade star representations in the osp 2 2n r case and the branching rule for their decomposition into a direct sum of so p sp 2n r irreducible representations as a by product it also enables the matrix elements of the odd generators between basis states of lowest weight so p u n irreducible representations to be calculated in a straightforward way

fractal deterministic cracks abstract cracks are grown in an elastic medium submitted to external shear by solving the full equations of motion on a two dimensional lattice one finds that deterministic fracture patterns are in general branched and can be fractal this effect is due to the competition between the direction of global stress and the local growth direction imposed by the lattice anisotropy viscoelastic effects are also discussed