retrieval properties of a hopfield model with random asymmetric interactions the process of pattern retrieval in a hopfield model in which a random antisymmetric component is added to the otherwise symmetric synaptic matrix is studied by computer simulations the introduction of the antisymmetric component is found to increase the fraction of random inputs that converge to the memory states however the size of the basin of attraction of a memory state does not show any significant change when asymmetry is introduced in the synaptic matrix we show that this is due to the fact that the spurious fixed points which are destabilized by the introduction of asymmetry have very small basins of attraction the convergence time to spurious fixed point attractors increases faster than that for the memory states as the asymmetry parameter is increased the possibility of convergence to spurious fixed points is greatly reduced if a suitable upper limit is set for the convergence time this prescription works better if the synaptic matrix has an antisymmetric component

asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order abstract in this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear gradient like evolution equations of mixed first and second order the proof of convergence is based on the lojasiewicz simon inequality the construction of an appropriate lyapunov functional and some differential inequalities applications are given to nonautonomous semilinear wave and heat equations with dissipative dynamical boundary conditions a nonlinear hyperbolic parabolic partial differential equation a damped wave equation and some coupled system

decay estimates for time fractional and other non local in time subdiffusion equations in mathbb r d we prove optimal estimates for the decay in time of solutions to a rather general class of non local in time subdiffusion equations in mathbb r d an important special case is the time fractional diffusion equation which has seen much interest during the last years mostly due to its applications in the modeling of anomalous diffusion processes we follow three different approaches and techniques to study this particular case a estimates based on the fundamental solution and young s inequality b fourier multiplier methods and c the energy method it turns out that the decay behaviour is markedly different from the heat equation case in particular there occurs a critical dimension phenomenon the general subdiffusion case is treated by method b and relies on a careful estimation of the underlying relaxation function several examples of kernels including the ultraslow diffusion case illustrate our results

completely positive measures and feller semigroups on donne d abord une approche unifiee aux mesures completement positives qui jouent un role important dans la theorie des equations de volterra a travers le theoreme de bernstein puis on etudie la solution d une equation de transport liee par une mesure non negative qu elle definit a l equation qui caracterise les noyaux completement positifs on obtient une desorption complete des generateurs de semigroupes de feller induits par de telles mesures on donne une application aux equations de flux thermique lineaire dans des materiaux avec memoire

fixed points in a hopfield model with random asymmetric interactions we calculate analytically the average number of fixed points in the hopfield model of associative memory when a random antisymmetric part is added to the otherwise symmetric synaptic matrix addition of the antisymmetric part causes an exponential decrease in the total number of fixed points if the relative strength of the antisymmetric component is small then its presence does not cause any substantial degradation of the quality of retrieval when the memory loading level is low we also present results of numerical simulations which provide qualitative as well as quantitative for some aspects confirmation of the predictions of the analytic study our numerical results suggest that the analytic calculation of the average number of fixed points yields the correct value for the typical number of fixed points

a priori bounds for degenerate and singular evolutionary partial integro differential equations we study quasilinear evolutionary partial integro differential equations of second order which include time fractional p laplace equations of time order less than one by means of suitable energy estimates and de giorgi s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems we also show that a maximum principle is valid for such equations

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

a nonlocal parabolic system modelling axially symmetric thermoelastic contact of two discs abstract a system of parabolic integro differential equations modelling homogeneous and isotropic elastic thin discs subject to axially symmetric temperature fields and displacements is studied under some appropriate assumptions the questions of existence uniqueness and continuous dependence of the solutions upon the data are demonstrated the main feature of our method in this paper is that the difficulties associated with this physical problem can be seen clearly and overcome easily via the decoupling procedures and transformations

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

boundedness of weak solutions to evolutionary partial integro differential equations with discontinuous coefficients we investigate linear and quasilinear evolutionary partial integro differential equations of second order which include time fractional evolution equations of time order less than one by means of suitable energy estimates and de giorgi s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems we also show that a maximum principle holds for such equations

integro differential equations of hyperbolic type with positive definite kernels abstract the main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in hilbert spaces for kernels that decay exponentially at infinity and possess strongly positive definite primitives the exponential stability of weak solutions is obtained in the energy norm it is also shown that this theory applies to several examples of kernels with possibly variable sign and to a problem in nonlinear viscoelasticity

on the energy decay of the linear thermoelastic plate with memory abstract we analyze the longterm properties of a c 0 semigroup describing the solutions to a linear evolution system that models a thermoelastic thin plate with memory effects in the heat flux law under reasonably general assumptions on the memory kernel all single trajectories are shown to decay to zero in spite of that the semigroup is not exponentially stable

generic unfolding of the thermoelastic contact instability the classic model of a one dimensional thermoelastic rod suspended between a hot and cold wall is revisited in this model the rod is held in place at the cold end while at the hot end it is allowed to separate from or make contact with the wall when the model includes the contact and gap dependent thermal boundary condition known as the barber condition it serves to illustrate the well known thermoelastic contact instability all previous studies of this instability have focused upon the symmetric case where as a control parameter is varied the system undergoes a pitchfork bifurcation that is a new pair of linearly stable steady state solutions bifurcate symmetrically from a previously unique solution which has changed from stable to unstable here it is shown that this behavior is not generic rather for typical contact resistance functions a fold bifurcation is encountered this represents a generic unfolding of the classic pitchfork bifurcation and contains the pitchfork as a special case

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

long time behavior of nonlinear integro differential evolution equations abstract we study the long time behavior as time tends to infinity of globally bounded strong solutions to certain integro differential equations in hilbert spaces based on an appropriate new lyapunov function and the ojasiewicz simon inequality we prove that any globally bounded strong solution converges to a steady state in a real hilbert space

multidimensional contact problems in thermoelasticity we consider dynamical and quasi static thermoelastic contact problems in hbox bbb r n modeling the evolution of temperature and displacement in an elastic body that may come into contact with a rigid foundation the existence of solutions to these dynamical and quasi static nonlinear problems and the exponential stability are investigated using a penalty method interior smoothing effects in the quasi static case are also discussed

global attractors for a semilinear hyperbolic equation in viscoelasticity a semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered regarding the past history memory of the displacement as a new variable the equation is transformed into a dynamical system in a suitable hilbert space the dissipation is extremely weak and it is all contained in the memory term longtime behavior of solutions is analyzed in particular in the autonomous case the existence of a global attractor for solutions is achieved

convergence to equilibrium of solutions to a nonautonomous semilinear viscoelastic equation with finite or infinite memory abstract in this paper we consider the nonautonomous semilinear viscoelastic equation u t t u 0 k s u t s d s f x u g t in r with dirichlet boundary conditions and finite t or infinite memory here is a bounded domain in r n with smooth boundary and the nonlinearity f r r is analytic in the second variable uniformly with respect to the first one for this equation we derive an appropriate lyapunov function and we use the ojasiewicz simon inequality to show that the dissipation given by the memory term is strong enough to prove the convergence to a steady state for any global bounded solution in addition we discuss the rate of convergence to equilibrium which is polynomial or exponential depending on the ojasiewicz exponent and the decay of the time dependent right hand side g

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

lyapunov functions and convergence to steady state for differential equations of fractional order we study the asymptotic behaviour as t of bounded solutions to certain integro differential equations in finite dimensions which include differential equations of fractional order between 0 and 2 we derive appropriate lyapunov functions for these equations and prove that any global bounded solution converges to a steady state of a related equation if the nonlinear potential mathcal e occurring in the equation satisfies the ojasiewicz inequality

distribution of the number of fitness maxima in fisher s geometric model fisher s geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotypes to an n dimensional euclidean phenotype space with a nonlinear single peaked phenotype fitness map genotypes are represented by binary sequences of length l and the phenotypic effects of mutations at different sites are represented by l random vectors drawn from an isotropic gaussian distribution recent work has shown that the interplay between the genotypic and phenotypic levels gives rise to a range of different landscape topographies that can be characterised by the number of local fitness maxima extending our previous study of the mean number of local maxima here we focus on the distribution of the number of maxima when the large l limit is taken at finite n we identify the typical scale of the number of maxima for general n and determine the full scaled probability density and two point correlation function of maxima for the one dimensional case we also elaborate on the close relation of the model to the anti ferromagnetic hopfield model with n random continuous pattern vectors and show that many of our results carry over to this setting more generally we expect that our analysis can help to elucidate the fluctuation structure of metastable states in various spin glass problems

asymptotic behavior of the energy for a class of weakly dissipative second order systems with memory a class of second order abstract systems with memory and dirichlet boundary conditions is investigated by suitable liapunov functionals existence of solutions as well as asymptotic behavior are determined in particular when the memory kernel decays exponentially the polynomially decay of the solutions is proved

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity abstract we prove convergence of global bounded and smooth solutions of the wave equation with linear dissipation and analytic nonlinearity a generalization and examples of applications are given at the end of the paper

a quasistatic contact problem in thermoelasticity with a radiation condition for the temperature abstract an abstract existence theorem is proved for variational inequalities of monotone operators with a nonlinear perturbation it is used to prove the existence of a strong solution to the problem of quasistatic one dimensional contact in linearized thermoelasticity the problem after reformulation consists of the heat equation with a nonlocal and nonlinear term the uniqueness and stability of the solution are also established