dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

nonlinear resonance in systems of conservation laws the riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed the inhomogeneity is modeled by a linearly degenerate field the solution of the riemann problem determines the nature of wave interactions and thus the riemann problem serves as a canonical form for nonlinear systems of conservation laws generic conditions on the fluxes are stated and it is proved that under these conditions the solution of the riemann problem exists is unique and has a fixed structure this demonstrates that in the above sense resonant inhomogeneous systems generically have the same canonical form the wave curves for these systems are only lipschitz continuous in a neighborhood of the states where the wave speeds coincide and so in contrast to strictly hyperbolic systems the implicit function theorem cannot be applied directly to obtain existence and uniqueness here we show that existence

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

multistability of neural networks with time varying delays and concave convex characteristics in this paper stability of multiple equilibria of neural networks with time varying delays and concave convex characteristics is formulated and studied some sufficient conditions are obtained to ensure that an n neuron neural network with concave convex characteristics can have a fixed point located in the appointed region by means of an appropriate partition of the n dimensional state space when nonlinear activation functions of an n neuron neural network are concave or convex in 2k 2m 1 intervals this neural network can have 2k 2m 1 n equilibrium points this result can be applied to the multiobjective optimal control and associative memory in particular several succinct criteria are given to ascertain multistability of cellular neural networks these stability conditions are the improvement and extension of the existing stability results in the literature a numerical example is given to illustrate the theoretical findings via computer simulations

why many theories of shock waves are necessary kinetic relations for non conservative systems for a class of nonconservative hyperbolic systems of partial dif ferential equations endowed with a strictly convex mathematical entropy we formulate the initial value problem by supplementing the equations with a ki netic relation prescribing the rate of entropy dissipation across shock waves our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by abeyaratne knowles and truskinovsky for subsonic phase transitions and by lefloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems the proposed kinetic relation for non conservative systems turns out to be equivalent for the class of systems under consideration at least to dal maso lefloch and murat s definition based on a prescribed family of lipschitz continuous paths in agreement with previous theories the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconserva tive system we illustrate with several examples that nonconservative systems arising in the applications fit in our framework and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems abstract in this paper we establish a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one space dimension first of all we introduce a new notion of admissible solutions based on prescribed sets of admissible discontinuities and admissible speeds our definition unifies in a single framework the various notions of entropy solutions known for hyperbolic systems of conservation laws as well as for systems in nonconservative form for instance it covers the nonclassical undercompressive shock waves generated by a vanishing diffusion dispersion regularization and characterized via a kinetic relation it also covers dal maso lefloch and murat s definition of weak solutions of nonconservative systems under certain natural assumptions on the prescribed sets and and assuming the existence of a l 1 continuous semi group of admissible solutions we prove that for each cauchy datum at t 0 there exists at most one admissible solution to the cauchy problem depending l 1 continuously upon the initial data in particular our result shows the uniqueness of the l 1 continuous semi group of admissible solutions in short this paper proves that supplementing a hyperbolic system with the dynamics of elementary discontinuities characterizes at most one l 1 continuous and admissible solution

multistability of two kinds of recurrent neural networks with activation functions symmetrical about the origin on the phase plane in this paper we investigate multistability of two kinds of recurrent neural networks with time varying delays and activation functions symmetrical about the origin on the phase plane one kind of activation function is with zero slope at the origin on the phase plane while the other is with nonzero slope at the origin on the phase plane we derive sufficient conditions under which these two kinds of n dimensional recurrent neural networks are guaranteed to have 2m 1 n equilibrium points with m 1 n of them being locally exponentially stable these new conditions improve and extend the existing multistability results for recurrent neural networks finally the validity and performance of the theoretical results are demonstrated through two numerical examples

on the solution to the riemann problem for the compressible duct flow the quasi one dimensional euler equations in a duct of variable cross section form probably one of the simplest nonconservative systems we consider the riemann problem for it and discuss its properties in particular for some initial conditions the solution to the riemann problem appears to be nonunique in order to rule out the nonphysical solutions we provide two dimensional computations of the euler equations in a duct of corresponding geometry and compare them with the one dimensional 1d results then the physically relevant 1d solutions satisfy a kind of entropy rate admissibility criterion

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

nonlinear resonance for quasilinear hyperbolic equation the purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system for the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source analytical understanding is used to design a numerical scheme based on the random choice method an important physical example is transonic gas flow through a nozzle this analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow including such phenomena as nonlinear stability instability and changing types of waves

stochastic integral representation theorem for quantum semimartingales abstract the quantum stochastic integral of ito type formulated by hudson and parthasarathy is extended to a wider class of adapted quantum stochastic processes on boson fock space an ito formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and wick exponentials of the basic martingales in quantum stochastic calculus

propagating phase boundaries formulation of the problem and existence via the glimm method this paper treats the hyperbolic elliptic system of two conservation laws which describes the dynamics of an elastic material having a non monotone strain stress function followingabeyaratne knowles we propose a notion of admissible weak solution for this system in the class of functions of bounded variation the formulation includes an entropy inequality a kinetic relation imposed along any subsonic phase boundary and an initiation criterion for the appearance of new phase boundaries we prove thel1 continuous dependence of the solution to the riemann problem our main result yields the existence and the stability of propagating phase boundaries the proofs are based onglimm s scheme and in particular on the techniques ofglimm andlax in order to deal with the kinetic relation we prove a result of pointwise convergence of the phase boundary

multistability of recurrent neural networks with time varying delays and the piecewise linear activation function in this brief stability of multiple equilibria of recurrent neural networks with time varying delays and the piecewise linear activation function is studied a sufficient condition is obtained to ensure that n neuron recurrent neural networks can have 4k 1 n equilibrium points and 2k n of them are locally exponentially stable this condition improves and extends the existing stability results in the literature simulation results are also discussed in one illustrative example

hyperbolic two pressure models for two phase flow abstract for some time it has been known that many of the two phase flow models lead to ill posed cauchy problems because they have complex characteristic values a necessary condition at least in the linear case for the cauchy problem to be well posed is that it be stable in the sense of von neumann for systems of partial differential equations of first order stability in the sense of von neumann is essentially equivalent to the condition that the model be hyperbolic all real characteristic values and complete set of characteristic vectors herein models are developed which have real characteristic values for all physically acceptable states state space and except for a set of measure zero have a complete set of characteristic vectors in state space therefore these models are hyperbolic a e almost everywhere in state space also they are stable in the sense of von neumann a e in state space even without inclusion of viscosity terms the models discussed herein are developed for the case of two phase separated planar flow and include transverse momentum considerations these models are referred to as two pressure models because each phase is assumed to exist at an average pressure different from the average pressure in the other phase the pressure fields are related through momentum considerations numerical results on a steady state problem show good agreement with existing steady state results numerical results on a transient problem agree with a single pressure model until the onset of numerical instability in the single pressure model compared to the single pressure hydrostatic model the two pressure model approximates additional physical features and is shown to be a viable approach for the case of separated flow

absolute continuity of the integrated density of states of the quantum lorentz gas for a class of repulsive potentials for a class of repulsive potentials for instance of the type phi x mod x mod kappa 1 x2 lambda kappa lambda positive in a certain range in the random schrodinger operators h omega p2 v x omega p2 sigma j phi x xj acting in l2 rd with poisson distributed xjs the quantum lorentz gas we show that the integrated density of states n e is absolutely continuous for e zeta rho phi here is phi the integral of phi over rd rho the averaged density of points xj and zeta 0 depends on phi and d in the above example zeta d kappa 2 our method makes use of a fock space representation for the poisson random system recently developed by maassen and the author 1993 within this fock space formalism the mourre commutator method is then employed to obtain the announced result

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

analysis of cyclic dynamics for networks of linear threshold neurons the network of neurons with linear threshold lt transfer functions is a prominent model to emulate the behavior of cortical neurons the analysis of dynamic properties for lt networks has attracted growing interest such as multistability and boundedness however not much is known about how the connection strength and external inputs are related to oscillatory behaviors periodic oscillation is an important characteristic that relates to nondivergence which shows that the network is still bounded although unstable modes exist by concentrating on a general parameterized two cell network theoretical results for geometrical properties and existence of periodic orbits are presented although it is restricted to two dimensional systems the analysis can provide a useful contribution to analyze cyclic dynamics of some specific lt networks of high dimension as an application it is extended to an important class of biologically motivated networks of large scale the winner take all model using local excitation and global inhibition

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

a fock space representation for the quantum lorentz gas a fock space representation is given for the quantum lorentz gas i e for random schrodinger operators of the form h p2 v p2 x xj acting in h l2 rd with poisson distributed xjs an operator h is defined in k h p h l2 p d l2 p d h by the action of h on its fibers in a direct integral decomposition the stationarity of the poisson process allows a unitarily equivalent description in terms of a new family h k k rd where each h k acts in p a tip j math phys 35 113 1994 the space p is then unitarily mapped upon the symmetric fock space over l2 rd dx with the intensity of the poisson process the average number of points xj per unit volume the scatterer density and the equivalent of h k is determined averages now become vacuum expectation values and a further unitary transformation removing in dx is made which leaves the former invariant the resulting operator hf k has an interesting structure on the nth fock layer we encounter a single particle mo

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

random time evolution and direct integrals constants of the motion and the mass operator the members h p2 v x of ergodic families of random operators defined in h l2 rd are considered as acting in fibers in a direct integral decomposition of k l2 p d h p being the underlying probability space such a formulation may turn out to be useful in providing a rigorous background for theoretical physical methods employed in treating random systems as an example a rigorous definition of the mass operator z is presented and the formulation of transport equations in a functional analytic setting is briefly indicated in case there is translational invariance in the probability law the operator h is shown to be unitarily equivalent to an operator h which can be decomposed h h k in a way that diagonalizes p thus problems about averaged time evolution can be formulated entirely in terms of h k in l2 p

a quantum nonadapted ito formula and stochastic analysis in fock scale abstract a generalized definition of quantum stochastic qs integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of malliavin derivative on a projective fock space and their uniform continuity with respect to the inductive limite convergence is proved a new form of qs calculus based on an inductive algebraic structure in an indefinite space is developed and a nonadaptive generalization of the qs ito formula for its representation in fock space is derived 1 the problem of solution of general qs evolution equations in a hilbert space is solved in terms of the constructed operator representation of chronological products defined in the indefinite space and the isometry and homomorphism property respectively for operators and maps of these solutions corresponding to the peseudounitary and homomorphism property of the qs integrable generators is proved

annihilation derivative creation derivative and representation of quantum martingales on the basis of the quantum white noise theory we introduce the notion of creation and annihilation derivatives of fock space operators and study the differentiability of white noise operators we define the hitsuda skorohod quantum stochastic integrals by the adjoint actions of quantum stochastic gradients and show explicit formulas for their creation and annihilation derivatives as an application we derive direct formulas for the integrands in the quantum stochastic integral representation of a regular quantum martingale

cohomology of power sets with applications in quantum probability square integrable wiener functionals may be represented as sums of multiple ito integrals this leads to an identification of such functionals with square integrable functions on the symmetric measure space of the lebesgue spacer when the pointwise product of wiener functionals is thus carried over the product takes a pleasing form cf wick s theorem and various non commutative perturbations of this wiener product have been considered here we employ cohomological arguments to analyse deformations of an abstract wiener product this leads to the construction of levy fields which are neither bosonic nor fermionic and also gives rise to homotopies between quasi free boson and fermion fields finally we unify existence and uniqueness results for quantum stochastic differential equations by treating mixed noise differential equations

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

gaussian hypercontractivity revisited abstract recent work on convolution type products on fock space allow an elementary treatment of hypercontractivity for second quantised contractions in the fermi case new results are obtained lending further support to a conjecture of l gross

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

a new comment on the computation of non conservative products using roe type path conservative schemes abstract we are interested in the numerical approximation of the discontinuous solutions of non conservative hyperbolic systems we more precisely consider a non conservative formulation of the usual gas dynamics equations and show how to slightly modify the so called roe type path conservative schemes to properly capture the underlying shock discontinuities numerical evidences are proposed the present note follows a first comment on the computation of non conservative products in 1

a martingale characterization of canonical commutation and anticommutation relations using a martingale condition and some restrictions on moments up to fourth order the characterisation problem of boson fermion and classical brownian motions is studied from a unified point of view entirely within the framework of elementary operator theory global commutation and anticommutation rules turn out to be consequences of corresponding commutation and anticommutation rules between past and future observables

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

quantum random walk for uq su 2 and a new example of quantum noise we derive the quantum version of the central limit theorem for sample sums of q independent q identically distributed quantum variables for q r 1 in particular we consider uq su 2 for which we obtain in the limit a state on the algebra generated by q commuting q oscillators indexed by pairs of real numbers r s where 0 r s giving a new example of non additive quantum noise

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

multistability of delayed recurrent neural networks with mexican hat activation functions this letter studies the multistability analysis of delayed recurrent neural networks with mexican hat activation function some sufficient conditions are obtained to ensure that an dimensional recurrent neural network can have equilibrium points with and of them are locally exponentially stable furthermore the attraction basins of these stable equilibrium points are estimated we show that the attraction basins of these stable equilibrium points can be larger than their originally partitioned subsets the results of this letter improve and extend the existing stability results in the literature finally a numerical example containing different cases is given to illustrate the theoretical results

nontrivial global attractors in 2 d multistable attractor neural networks attractor dynamics is a crucial problem for attractor neural networks as it is the underling computational mechanism for memory storage and retrieval in neural systems this brief studies a class of attractor network consisting of linearized threshold neurons and analyzes global attractors based on a parameterized 2 d model on the basis of previous results on nondegenerate and degenerate equilibria in mathematics we further elucidate all possible nontrivial global attractors our theoretical result provides precise descriptions on how the changes of network parameters affect the attractors distribution and landscape and it may give a feasible solution towards specifying attractors by specifying weights simulations are presented to illustrate the theoretical results