multistability of second order competitive neural networks with nondecreasing saturated activation functions in this paper second order interactions are introduced into competitive neural networks nns and the multistability is discussed for second order competitive nns socnns with nondecreasing saturated activation functions firstly based on decomposition of state space cauchy convergence principle and inequality technique some sufficient conditions ensuring the local exponential stability of 2n equilibrium points are derived secondly some conditions are obtained for ascertaining equilibrium points to be locally exponentially stable and to be located in any designated region thirdly the theory is extended to more general saturated activation functions with 2r corner points and a sufficient criterion is given under which the socnns can have r 1 n locally exponentially stable equilibrium points even if there is no second order interactions the obtained results are less restrictive than those in some recent works finally three examples with their simulations are presented to verify the theoretical analysis

multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks we show that an n neuron cellular neural network with time varying delay can have 2n periodic orbits located in saturation regions and these periodic orbits are locally exponentially attractive in addition we give some conditions for ascertaining periodic orbits to be locally or globally exponentially attractive and allow them to locate in any designated region as a special case of exponential periodicity exponential stability of delayed cellular neural networks is also characterized these conditions improve and extend the existing results in the literature to illustrate and compare the results simulation results are discussed in three numerical examples

multistability analysis of delayed quaternion valued neural networks with nonmonotonic piecewise nonlinear activation functions abstract this paper deals with the multistability problem of the quaternion valued neural networks qvnns with nonmonotonic piecewise nonlinear activation functions and unbounded time varying delays by virtue of the non commutativity of quaternion multiplication resulting from hamilton rules the qvnns can be separated into four real valued systems by using the fixed point theorem and other analytical tools some novel algebraic criteria are established to guarantee that the qvnns can have 54n equilibrium points 34n of which are locally stable some criteria that guarantee the multiple exponential stability multiple power stability multiple log stability multiple log log stability are also derived as special cases the obtained results reveal that the introduced qvnns in this paper can have larger storage capacity than the complex valued ones finally one numerical example is presented to clarify the validity of the theoretical results

multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions in this paper we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions it is proved that under some conditions such n neuron neural networks can have at least 5 n equilibrium points 3 n of which are locally stable and the others are unstable based on the contraction mapping theorem and the theory of strict diagonal dominance matrix the investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous mexican hat type activation function or discontinuous two level activation functions an illustrative example with computer simulations is presented to verify the theoretical analysis

effects of the computational time step on numerical solutions of turbulent flow effects of large computational time steps on the computed turbulence were investigated using a fully implicit method in turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined turbulence fluctuations could not be sustained if the computational time step was near or larger than the kolmogorov time scale

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

multistability and convergence in delayed neural networks abstract we present the existence of 2 n stable stationary solutions for a general n dimensional delayed neural networks with several classes of activation functions the theory is obtained through formulating parameter conditions motivated by a geometrical observation positively invariant regions for the flows generated by the system and basins of attraction for these stationary solutions are established the theory is also extended to the existence of 2 n limit cycles for the n dimensional delayed neural networks with time periodic inputs it is further confirmed that quasiconvergence is generic for the networks through justifying the strongly order preserving property as the self feedback time lags are small for the neurons with negative self connection weights our theory on existence of multiple equilibria is then incorporated into this quasiconvergence for the network four numerical simulations are presented to illustrate our theory

multistability and multiperiodicity of delayed cohen grossberg neural networks with a general class of activation functions abstract in this paper by using analysis approach and decomposition of state space the multistability and multiperiodicity issues are discussed for cohen grossberg neural networks cgnns with time varying delays and a general class of activation functions where the general class of activation functions consist of nondecreasing functions with saturation s including piecewise linear functions with two corner points and standard activation functions as its special case based on the cauchy convergence principle some sufficient conditions are obtained for checking the existence and uniqueness of equilibrium points of the n neuron cgnns it is shown that the n neuron cgnns can have 2n locally exponentially stable equilibrium points located in saturation regions also some conditions are derived for ascertaining equilibrium points to be locally exponentially stable or globally exponentially attractive and to be located in any designated region as an extension of multistability some similar results are presented for ascertaining multiple periodic orbits when external inputs of the n neuron cgnns are periodic finally three examples are given to illustrate the effectiveness of the obtained results

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

multiple periodic solutions in impulsive hybrid neural networks with delays abstract the existence of multiple periodic solutions and their exponential stability are investigated for impulsive hybrid hopfield type neural networks with both time dependent and distributed delays using the leray schauder fixed point theorem and lyapunov functionals the criteria given are easily verifiable possess many adjustable parameters and depend on impulses providing flexibility for the analysis and design of delayed neural networks with impulse effects examples are given

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

multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions abstract in this paper the multistability and multiperiodicity issues are discussed for 2 n dimensional delayed bidirectional associative memory neural networks with r level discontinuous activation functions sufficient conditions are established to ensure the existence of r n locally exponentially stable equilibria as an extension of multistability conditions are established to ensure the existence of r n locally exponentially stable periodic solutions when external inputs are periodic finally two examples are given to illustrate the effectiveness of the obtained results

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

solving systems of linear equations via gradient systems with discontinuous righthand sides application to ls svm a gradient system with discontinuous righthand side that solves an underdetermined system of linear equations in the l sub 1 norm is presented an upper bound estimate for finite time convergence to a solution set of the system of linear equations is shown by means of the persidskii form of the gradient system and the corresponding nonsmooth diagonal type lyapunov function this class of systems can be interpreted as a recurrent neural network and an application devoted to solving least squares support vector machines ls svm is used as an example

on attracting basins of multiple equilibria of a class of cellular neural networks in this paper we study the distribution of attraction basins of multiple equilibrium points of cellular neural networks cnns under several conditions the boundaries of the attracting basins of the stable equilibria of a completely stable cnn system are composed of the closures of the stable manifolds of unstable equilibria of n 1 dimensions as demonstrations of this idea under the conditions proposed in the literature which depicts stable and unstable equilibria we identify the attraction basin of each stable equilibrium of which the boundary is composed of the stable manifolds of the unstable equilibria precisely we also investigate the attracting basins of a simple class of symmetric 1 d cnns via identifying the unstable equilibria of which the stable manifold is n 1 dimensional and the completely stable asymmetric cnns with stable equilibria less than 2n

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks this technical note proposes to study the activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks the concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons keeps invariant all the time conditions are obtained for locating activity invariant sets under some conditions it shows that an activity invariant set can have one equilibrium point which attracts exponentially all trajectories starting in the set since the attractors are located in activity invariant sets each attractor has binary pattern and also carries analog information such results can provide new perspective to apply attractor networks for applications such as group winner take all associative memory etc

monostability and multistability for almost periodic solutions of fractional order neural networks with unsaturating piecewise linear activation functions since the unsaturating activation function is unbounded more complex dynamics may exist in neural networks with this kind of activation function in this article monostability and multistability results of almost periodic solutions are developed for fractional order neural networks with unsaturating piecewise linear activation functions some globally mittag leffler attractive sets are given and the existence of globally mittag leffler stable almost periodic solution is demonstrated by using ascoli arzela theorem in particular some sufficient conditions are provided to ascertain the multistability of almost periodic solutions based on locally positively invariant set it shows that there exists an almost periodic solution in each positively invariant set and all trajectories converge to this periodic trajectory in that rectangular area two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria

fast construction of fejer and clenshaw curtis rules for general weight functions the main purpose of this paper is to compute the weights of clenshaw curtis and fejer type quadrature rules via dct and dst for general weight functions w the approach is different from that used by waldvogel 2006 in 10 where the author considered the computation of these sets only for the legendre weight w 1 using dft arguments

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

multistability and new attraction basins of almost periodic solutions of delayed neural networks in this paper we investigate multistability of almost periodic solutions of recurrently connected neural networks with delays simply called delayed neural networks we will reveal that under some conditions the space rn can be divided into 2n subsets and in each subset the delayed n neuron neural network has a locally stable almost periodic solution furthermore we also investigate the attraction basins of these almost periodic solutions we reveal that the attraction basin of almost periodic trajectory is larger than the subset where the corresponding almost periodic trajectory is located in addition several numerical simulations are presented to corroborate the theoretical results

multistability of discrete time recurrent neural networks with unsaturating piecewise linear activation functions this paper studies the multistability of a class of discrete time recurrent neural networks with unsaturating piecewise linear activation functions it addresses the nondivergence global attractivity and complete stability of the networks using the local inhibition conditions for nondivergence are derived which not only guarantee nondivergence but also allow for the existence of multiequilibrium points under these nondivergence conditions global attractive compact sets are obtained complete stability is studied via constructing novel energy functions and using the well known cauchy convergence principle examples and simulation results are used to illustrate the theory

multiple almost periodic solutions in nonautonomous delayed neural networks a general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed we consider a general class of nonautonomous neural networks with delays and various activation functions a geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed we further derive criteria under which the n neuron network admits 2n exponentially stable sets in addition we establish the existence of 2n exponentially stable almost periodic solutions for the system when the connection strengths time lags and external bias are almost periodic functions of time through applying the contraction mapping principle finally three numerical simulations are presented to illustrate our theory

discrete orthogonal function expansions for non uniform grids using the fast fourier transform abstract discrete orthogonal function expansions are obtained for non uniformly spaced data by combined use of mapping functions and the fast fourier transform the result yields alias free differentiation and integration operators the method is applicable to periodic zero or zero derivative boundary conditions or combinations thereof truncation is required to avoid aliasing and or singularity but the truncation error is explicit quantitatively expressible and generally small this approach is demonstrated in an infinite physical domain with application to linear convection and diffusion the resulting errors are orders of magnitude smaller than those generated by standard finite difference methods difficulties arising from image fields are an important phenomena the method of differentiation as well as the boundary conditions imply artificial image fields a vortex pairing problem is presented which shows that image flows can totally alter the solution the new scheme avoids this undesirable effect by keeping image flows infinitely far away

numerical simulation of two dimensional flows over a circular cylinder using the immersed boundary method in this work a virtual boundary method is applied to the numerical simulation of a uniform flow over a cylinder the force source term added to the two dimensional navier stokes equations guarantees the imposition of the no slip boundary condition over the body fluid interface these equations are discretized using the finite differences method the immersed boundary is represented with a finite number of lagrangian points distributed over the solid fluid interface a cartesian grid is used to solve the fluid flow equations the key idea is to propose a method to calculate the interfacial force without ad hoc constants that should usually be adjusted for the type of flow and the type of the numerical method when this kind of model is used in the present work this force is calculated using the navier stokes equations applied to the lagrangian points and then distributed over the eulerian grid the main advantage of this approach is that it enables calculation of this force field even if the interface is moving or deforming it is unnecessary to locate the eulerian grid points near this immersed boundary the lift and drag coefficients and the strouhal number calculated for an immersed cylinder are compared with previous experimental and numerical results for different reynolds numbers

multistability of neural networks with mexican hat type activation functions in this paper we are concerned with a class of neural networks with mexican hat type activation functions due to the different structure from neural networks with saturated activation functions a set of new sufficient conditions are presented to study the multistability including the total number of equilibrium points their locations and stability furthermore the attraction basins of stable equilibrium points are investigated for two neuron neural networks the investigation shows that the stable manifolds of unstable equilibrium points constitute the boundaries of attraction basins of stable equilibrium points several illustrative examples are given to verify the effectiveness of our results

high order compact schemes for incompressible flows a simple and efficient method with quasi spectral accuracy in this paper a finite difference code for direct and large eddy simulation dns les of incompressible flows is presented this code is an intermediate tool between fully spectral navier stokes solvers limited to academic geometry through fourier or chebyshev representation and more versatile codes based on standard numerical schemes typically only second order accurate the interest of high order schemes is discussed in terms of implementation easiness computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations the equivalence rules between operations in physical and spectral spaces are efficiently used to solve the poisson equation introduced by the projection method it is shown that for the pressure treatment an accurate fourier representation can be used for more flexible boundary conditions than periodicity or free slip using the concept of the modified wave number the incompressibility can be enforced up to the machine accuracy the benefit offered by this alternative method is found to be very satisfactory even when a formal second order error is introduced locally by boundary conditions that are neither periodic nor symmetric the usefulness of high order schemes combined with an immersed boundary method ibm is also demonstrated despite the second order accuracy introduced by this wall modelling strategy in particular the interest of a partially staggered mesh is exhibited in this specific context three dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high order schemes to reduce the computational cost for a given accuracy the main conclusion of this paper is that finite difference schemes with quasi spectral accuracy can be very efficient for dns les of incompressible flows while allowing flexibility for the boundary conditions and easiness in the code development therefore this compromise fits particularly well for very high resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments

an analysis of a class of neural networks for solving linear programming problems a class of neural networks that solve linear programming problems is analyzed the neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact nondifferentiable penalty functions it is proved that for a given linear programming problem and sufficiently large penalty parameters any trajectory of the neural network converges in finite time to its solution set for the analysis lyapunov type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets the results are illustrated via numerical simulation examples

multistability of recurrent neural networks with nonmonotonic activation functions and unbounded time varying delays this paper is concerned with the coexistence of multiple equilibrium points and dynamical behaviors of recurrent neural networks with nonmonotonic activation functions and unbounded time varying delays based on a state space partition by using the geometrical properties of the activation functions it is revealed that an n neuron neural network can exhibit prod i 1 n 2k i 1 equilibrium points with k i geq 0 in particular several sufficient criteria are proposed to ascertain the asymptotical stability of prod i 1 n k i 1 equilibrium points for recurrent neural networks these theoretical results cover both monostability and multistability furthermore the attraction basins of asymptotically stable equilibrium points are estimated it is shown that the attraction basins of the stable equilibrium points can be larger than their originally partitioned subsets finally the results are illustrated by using the simulation results of four examples

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

generalized lyapunov approach for convergence of neural networks with discontinuous or non lipschitz activations the paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non lipschitz functions some tools are developed which enable us to apply a lyapunov like approach to differential equations with discontinuous right hand side modeling the neural network dynamics the tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable lyapunov function and a comparison principle for this time derivative which yields conditions for exponential convergence or convergence in finite time by means of the lyapunov like approach a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions moreover new results on global convergence in finite time are established which are applicable to neuron activations with jump discontinuities or neuron activations modeled by means of continuous non lipschitz holder functions

complete stability in multistable delayed neural networks we investigate the complete stability for multistable delayed neural networks a new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property an iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity the existence of 3n equilibria and 2n positively invariant sets for the n neuron system remains valid under the new formulation the theory is demonstrated by a numerical illustration

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

numerical simulation of unsteady incompressible flow em re protect boldmath leq 9500 on the curvilinear half staggered mesh in this paper we continue our efforts toward numerical simulation of high re 1000 leq re leq 9500 unsteady incompressible flows with the finite difference method on the curvilinear half staggered mesh the finite difference scheme and the projection method for the numerical solution of the incompressible navier stokes equations in primitive variables was proposed and verified in l c huang j comput math to appear here we derive the discrete poisson equation on the curvilinear half staggered mesh and discuss its properties for unsteady complex flow simulation rapid solution of this equation is crucial the multilevel one way dissection methods offer effective solutions numerical simulation of high re flows around circular and aerofoil shaped cylinders show our method to be efficient and robust

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