dynamical systems defining jacobi s ϑ constants we propose a system of equations that defines weierstrass jacobi s eta and theta constant series in a differentially closed way this system is shown to have a direct relationship to a little known dynamical system obtained by jacobi the classically known differential equations by darboux halphen chazy and ramanujan are the differential consequences or reductions of these systems the proposed system is shown to admit the lagrangian hamiltonian and nambu formulations we explicitly construct a pencil of nonlinear poisson brackets and complete set of involutive conserved quantities as byproducts of the theory we exemplify conserved quantities for the ramamani dynamical system and quadratic system of halphen brioschi

linearly degenerate hierarchies of quasiclassical sdym type we demonstrate that sdym self dual yang mills equations for the lie algebra of one dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy we define the reduction in terms of wave functions and introduce a generating relation lax sato equations and the dressing scheme for the reduced hierarchy a multidimensional case is also discussed

bifurcation analysis on a two neuron system with distributed delays abstract a general two neuron model with distributed delays is studied in this paper its local linear stability is analyzed by using the routh hurwitz criterion if the mean delay is used as a bifurcation parameter we prove that hopf bifurcation occurs for a weak kernel this means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value the direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem some numerical simulations for justifying the theoretical analysis are also given

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

the lax pair by dimensional reduction of chern simons gauge theory we show that the nonlinear schrodinger equation and the related lax pair in 1 1 dimensions can be derived from 2 1 dimensional chern simons topological gauge theory the spectral parameter a main object for the loop algebra structure and the inverse spectral transform has to appear as a homogeneous part condensate of the statistical gauge field connected with the compactified extra space coordinate in terms of solitons a natural interpretation for the one dimensional analog of chern simons gauss law is given

global hopf bifurcation in a neural netlet sufficient conditions are obtained for the global hopf bifurcation of periodic solutions in the dynamics of a neural netlet of activation and inhibition with continuously distributed delays the orbital asymptotic stability of the bifurcating periodic solutions is investigated the results are illustrated with numerical simulations a few problems are formulated in the form of conjectures

convergence under dynamical thresholds with delays necessary and sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium of a class of delay differential equations modeling the action of a neuron with dynamical threshold effects

analysis of a pseudostress based mixed finite element method for the brinkman model of porous media flow in this paper we introduce and analyze a new mixed finite element method for the two dimensional brinkman model of porous media flow with mixed boundary conditions we use a dual mixed formulation in which the main unknown is given by the pseudostress the original velocity and pressure unknowns are easily recovered through a simple postprocessing in addition since the neumann boundary condition becomes essential we impose it in a weak sense which yields the introduction of the trace of the fluid velocity over the neumann boundary as the associated lagrange multiplier we apply the babuska brezzi theory to establish sufficient conditions for the well posedness of the resulting continuous and discrete formulations in particular a feasible choice of finite element subspaces is given by raviart thomas elements of order k ge 0 k 0 for the pseudostress and continuous piecewise polynomials of degree k 1 k 1 for the lagrange multiplier we also derive a reliable and efficient residual based a posteriori error estimator for this problem suitable auxiliary problems the continuous inf sup conditions satisfied by the bilinear forms involved a discrete helmholtz decomposition and the local approximation properties of the raviart thomas and clement interpolation operators are the main tools for proving the reliability then helmholtz s decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are employed to show the efficiency finally several numerical results illustrating the performance and the robustness of the method confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithm are provided

hopf bifurcation of an n 1 neuron bidirectional associative memory neural network model with delays recent studies on hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two three four five or six neurons it is well known that neural networks are complex and large scale nonlinear dynamical systems so the dynamics of the delayed neural networks are very rich and complicated although discussing the dynamics of networks with a few neurons may help us to understand large scale networks there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large scale networks in this paper a general delayed bidirectional associative memory neural network model with n 1 neurons is considered by analyzing the associated characteristic equation the local stability of the trivial steady state is examined and then the existence of the hopf bifurcation at the trivial steady state is established by applying the normal form theory and the center manifold reduction explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution furthermore the paper highlights situations where the hopf bifurcations are particularly critical in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections it is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons numerical simulations are carried out to illustrate the main results

global existence of periodic solutions in a tri neuron network model with delays we consider a delayed differential system that models a network of three neurons with memory using a global hopf bifurcation theorem for fde due to j wu and a bendixson s criterion for high dimensional ode due to li and muldowney we obtain a group of sufficient conditions for the system to have multiple periodic solutions when the sum of delays is sufficiently large

a priori and a posteriori error analysis of an augmented mixed fem for the navier stokes brinkman problem abstract we introduce and analyze an augmented mixed finite element method for the navier stokes brinkman problem with nonsolenoidal velocity we employ a technique previously applied to the stationary navier stokes equation which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the convective term and propose an augmented pseudostress velocity formulation for the model problem the resulting augmented scheme is then written equivalently as a fixed point equation so that the well known banach fixed point theorem combined with the lax milgram lemma are applied to prove the unique solvability of the continuous and discrete systems we point out that no discrete inf sup conditions are required for the solvability analysis and hence in particular for the galerkin scheme arbitrary finite element subspaces of the respective continuous spaces can be utilized for instance given an integer k 0 the raviart thomas spaces of order k and continuous piecewise polynomials of degree k 1 constitute feasible choices of discrete spaces for the pseudostress and the velocity respectively yielding optimal convergence we also emphasize that since the dirichlet boundary condition becomes a natural condition the analysis for both the continuous an discrete problems can be derived without introducing any lifting of the velocity boundary datum in addition we derive a reliable and efficient residual based a posteriori error estimator for the augmented mixed method the proof of reliability makes use of a global inf sup condition a helmholtz decomposition and local approximation properties of the clement interpolant and raviart thomas operator on the other hand inverse inequalities the localization technique based on element bubble and edge bubble functions approximation properties of the l 2 orthogonal projector and known results from previous works are the main tools for proving the efficiency of the estimator finally some numerical results illustrating the performance of the augmented mixed method confirming the theoretical rate of convergence and properties of the estimator and showing the behavior of the associated adaptive algorithms are reported

the 2 1 dimensional nonisospectral relativistic toda hierarchy related to the generalized discrete painleve hierarchy in this paper we will concentrate on the topic of integrable discrete hierarchies in 2 1 dimensions and their connection with discrete painleve hierarchies by considering a 2 1 dimensional nonisospectral discrete linear problem two new 2 1 dimensional nonisospectral integrable lattice hierarchies the 2 1 nonisospectral relativistic toda lattice hierarchy and the 2 1 nonisospectral negative relativistic toda lattice hierarchy are constructed it is shown that the reductions of the two new 2 1 nonisospectral lattice hierarchies lead to the 2 1 dimensional nonisospectral volterra lattice hierarchy and the 2 1 dimensional nonisospectral negative volterra lattice hierarchy we also obtain two new 1 1 dimensional nonisospectral integrable lattice hierarchies and two new ordinary difference hierarchies which are direct reductions of the two 2 1 nonisospectral integrable lattice hierarchies one of the two difference hierarchies yields our previously obtained generalized discrete first painleve dpi hierarchy and another one yields a generalized alternative discrete second painleve alt dpii hierarchy

a priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems abstract in this paper we introduce and analyze new augmented mixed finite element methods allowing much more flexibility in the choice of the corresponding finite dimensional subspaces for a class of nonlinear elasticity problems arising in hyperelasticity the starting mixed method is based on the incorporation of the strain tensor as an auxiliary unknown which together with the usual stress displacement rotation approach employed in linear elasticity yields a nonlinear twofold saddle point operator equation as the resulting weak formulation we first extend known results on the well posedness of the associated galerkin scheme with peers of order k 0 to the case k 1 then the augmented schemes are obtained by adding consistent galerkin type terms arising first from the constitutive equation and then from the equilibrium equation and the relations defining the rotation in terms of the displacement and the strain tensor as independent unknown all of them multiplied by suitably chosen stabilization parameters we apply classical results on the solvability analysis of nonlinear saddle point and strongly monotone operator equations to prove that the corresponding continuous and discrete augmented schemes are well posed more precisely and as the main advantages of our present approach we show that the well posedness of a partially augmented galerkin scheme is ensured by any finite element subspace for the strain tensor together with the peers space of order k 0 for the remaining unknowns whereas any finite element subspace of the whole continuous space will do in the case of a fully augmented scheme then we derive reliable and efficient residual based a posteriori error estimators for all the schemes finally we provide several numerical results illustrating the good performance of the mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

delay induced periodicity in a neural netlet of excitation and inhibition abstract the dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated it is shown that in the absence of delay the netlet relaxes to the trivial resting state if the delay is of sufficient magnitude the network is excited to a temporally periodic cyclic behaviour the analytical mechanism for the onset of cyclic behaviour is through a hopf type bifurcation approximate solusions to the periodic output of the netlet is calculated stability of the temporally periodic cycle is investigated it is shown that the bifurcation is supercritical a related discrete version of the continuous time system is formulated it is found that the discrete system also displays a cyclic behaviour results of a number of computer simulations are displayed graphically the article concludes with a brief neurobiological discussion

a residual based a posteriori error estimator for a fully mixed formulation of the stokes darcy coupled problem abstract in this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2d flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we consider dual mixed formulations in both media which yields the pseudostress and the velocity in the fluid together with the velocity and the pressure in the porous medium and the traces of the porous media pressure and the fluid velocity on the interface as the resulting unknowns the set of feasible finite element subspaces includes raviart thomas elements of lowest order and piecewise constants for the velocities and pressures respectively in both domains together with continuous piecewise linear elements for the traces we derive a reliable and efficient residual based a posteriori error estimator for the coupled problem the proof of reliability makes use of the global inf sup condition helmholtz decompositions in both media and local approximation properties of the clement interpolant and raviart thomas operator on the other hand inverse inequalities the localization technique based on element bubble and edge bubble functions and known results from previous works are the main tools for proving the efficiency of the estimator finally some numerical results confirming the theoretical properties of this estimator and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution are reported

some reductions of the self dual yang mills equations to integrable systems in 2 1 dimensions a reduction of the self dual yang mills sdym equations is studied by imposing two space time symmetries and by requiring that the connection one form belongs to a lie algebra of formal matrix valued differential operators in an auxiliary variable in this article the scalar case and the canonical cases for 2 2 matrices are examined in the scalar case it is shown that the field equations can be reduced to the forced burgers equation in the matrix case several well known 2 1 integrable equations are obtained also examined are certain transformation properties between the solutions of some of these 2 1 equations

stability and bifurcation in a neural network model with two delays abstract a simple neural network model with two delays is considered linear stability of the model is investigated by analyzing the associated characteristic transcendental equation for the case without self connection it is found that the hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem an example is given and numerical simulations are performed to illustrate the obtained results

adler kostant symes construction bi hamiltonian manifolds and kdv equations this paper focuses a relation between adler kostant symes aks theory applied to fordy kulish scheme and bi hamiltonian manifolds the spirit of this paper is closely related to casati magri pedroni work on hamiltonian formulation of the kp equation here the kdv equation is deduced via the superposition of the fordy kulish scheme and aks construction on the underlying current algebra c s1 g c this method is different from the drinfeld sokolov reduction method it is known that aks construction is endowed with bi hamiltonian structure in this paper we show that if one applies the fordy kulish construction in the adler kostant symes scheme to construct an integrable equation associated with symmetric spaces then this superposition method becomes closer to casati magri pedroni s bi hamiltonian method of the kp equation we also add a self contained appendix where we establish a direct relation between aks scheme and bi hamiltonian methods

two 2 1 dimensional expanding dynamical systems associated to the mkp hierarchy an isospectral problem with a parameter is presented for which a modified kdv mkdv hierarchy is easily derived from the tu scheme then two different expanding integrable models are obtained respectively the main purpose for this focuses on deriving a 2 1 dimensional mkdv hierarchy called the mkp hierarchy and the corresponding different 2 1 dimensional expanding models including the linear and nonlinear whose hamiltonian structures are obtained by employing an identity developed by us in the paper as the reduced consequences the linear and nonlinear 2 1 dimensional expanding models of the mkp equation are generated

two 2 1 dimensional hierarchies of evolution equations and their hamiltonian structures two kinds of appropriate isospectral problems are introduced by using a lie algebra with the help of the tah scheme we generate two new 2 1 dimensional hierarchies of evolution equations whose hamiltonian structures are derived from the trace identity proposed by tu guizhang andrushkiw r i and huang x c finally we propose some problems worth thinking about

an augmented mixed finite element method for the vorticity velocity pressure formulation of the stokes equations this paper deals with the numerical approximation of the stationary two dimensional stokes equations formulated in terms of vorticity velocity and pressure with non standard boundary conditions here by introducing a galerkin least squares term we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem we propose two families of mixed finite elements to solve the discrete problem in the first family the unknowns are approximated by piecewise continuous and quadratic elements brezzi douglas marini and piecewise constant finite elements respectively while in the second family the unknowns are approximated by piecewise linear and continuous raviart thomas and piecewise constant finite elements respectively the wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the babuska brezzi theory we establish a priori error estimates in the natural norms and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes additional examples of cases not covered by our theory are also presented

on an adaptive stabilized mixed finite element method for the oseen problem with mixed boundary conditions abstract we consider the oseen problem with nonhomogeneous dirichlet boundary conditions on a part of the boundary and a neumann type boundary condition on the remaining part suitable least squares terms that arise from the constitutive law the momentum equation and the dirichlet boundary condition are added to a dual mixed formulation based on the pseudostress velocity variables we prove that the new augmented variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas elements and the velocity is approximated by continuous piecewise polynomials we develop an a posteriori error analysis based on a helmholtz type decomposition and derive a posteriori error indicators that consist of two residual terms per element except on those elements with a side on the dirichlet boundary where they both have two additional terms we prove that these a posteriori error indicators are reliable and locally efficient finally we provide several numerical experiments that support the theoretical results

2 1 dimensional integrable lattice hierarchies related to discrete fourth order nonisospectral problems in this paper we consider the 2 1 dimensional discrete fourth order nonisospectral problem by using the lax technique three new 2 1 dimensional nonisospectral four field integrable lattice hierarchies are constructed their reductions yield three 1 1 dimensional isospectral four field integrable lattice hierarchies due to mlaszak marciniak we make a comparison between the 1 1 dimensional discrete fourth order nonisospectral problem and the third order nonisospectral problem we found that the integrable lattice hierarchies related to the discrete fourth order nonisospectral problem have new characteristics

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

augmented mixed finite element methods for the stationary stokes equations in this paper we introduce and analyze two augmented mixed finite element methods for a velocity pressure stress formulation of the stationary stokes equations our approach which extends analog results for linear elasticity problems is based on the introduction of the galerkin least squares type terms arising from the constitutive and equilibrium equations and the dirichlet boundary condition for the velocity all of them multiplied by suitable stabilization parameters we show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms whence the associated galerkin schemes become well posed for any choice of finite element subspaces in particular we can use continuous piecewise linear velocities piecewise constant pressures and raviart thomas elements for the stresses thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation alternatively the above factor reduces to 4 when a second augmented variational formulation involving only the velocity and the stress as unknowns is employed next we derive a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element schemes finally extensive numerical experiments illustrating the performance of the augmented mixed finite element methods confirming the properties of the a posteriori estimators and showing the behavior of the associated adaptive algorithms are reported

aks hierarchy and bi hamiltonian geometry of gelfand zakharevich type a bi hamiltonian system is a system of differential equations which can be written in hamiltonian form in two distinct ways the applications of gelfand zakharevich bi hamiltonian structure which is an extension of a poisson nijenhuis structure on phase space has been extensively explored by falqui magri and pedroni in the context of separation of variables it is well known that the integrable hamiltonian systems defined by the adler kostant symes aks scheme contains bi hamiltonian structure in this paper we unveil the connection between adler kostant symes formalism applied to loop algebra and the gelfand zakharevich bi hamiltonian structure by superposition the results of fordy and kulish in the aks scheme we also study the commuting flows of the aks hierachy and its connection to the zakharov shabat hierarchy

analysis of a velocity pressure pseudostress formulation for the stationary stokes equations abstract we consider a non standard mixed approach for the stokes problem in which the velocity the pressure and the pseudostress are the main unknowns alternatively the pressure can be eliminated from the original equations thus yielding an equivalent formulation with only two unknowns in this paper we develop a priori and a posteriori error analyses of both approaches we first apply the babuska brezzi theory to prove the well posedness of the continuous and discrete formulations in particular we show that raviart thomas elements of order k 0 for the pseudostresses and piecewise polynomials of degree k for the velocities and the pressures ensure unique solvability and stability of the associated galerkin schemes then we derive reliable and efficient residual based a posteriori error estimators for both schemes without and with pressure unknown finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

stability in a scalar differential equation with multiple distributed time delays abstract we consider a linear scalar delay differential equation dde consisting of two arbitrary distributed time delays we formulate necessary conditions for stability of the trivial solution which are independent of the distributions for the case of one discrete and one gamma distributed delay we give an explicit description of the region of stability of the trivial solution and discuss how this depends on the model parameters

dynamics of simple electronic neural networks abstract we examine systems of one and two nonlinear threshold switching elements neurons of the kind used in electronic neural networks characteristics of these systems which deviate from standard ideal models are found to induce complex dynamics when the neurons possess a finite frequency response or a transfer characteristic with a time delay underdamped transients and instability leading to oscillation can occur inertia in the neuron connections is found to cause ringing about fixed points convoluted basin boundaries instability and spontaneous oscillation and chaotic behavior when driven furthermore the collective behavior of a network of multiple neurons can be underdamped even when the individual connections are overdamped these results imply that care should be exercised in implementing networks with electronic devices or when adding inertia to enhance the performance of optimizing networks

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

global attractivity in delayed hopfield neural network models two different approaches are employed to investigate the global attractivity of delayed hopfield neural network models without assuming the monotonicity and differentiability of the activation functions liapunov functionals and functions combined with the razumikhin technique are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays in the case of monotone and smooth activation functions the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model such criteria depend on the magnitude of delays and show that self inhibitory connections can contribute to the global convergence

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

a priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows abstract in this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem to the case of incompressible fluid flows with symmetric stress tensor similarly as before the present approach is based on the introduction of the galerkin least squares type terms arising from the constitutive and equilibrium equations and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement all of them multiplied by stabilization parameters we show that these parameters can be suitably chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form whence the associated galerkin scheme becomes well posed for any choice of finite element subspaces next we present a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element scheme finally several numerical results confirming the theoretical properties of this estimator and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution are reported

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

the reduced self dual yang mills equation binary and infinitesimal darboux transformations the self dual yang mills equation in the 2 1 dimensional space time is considered binary darboux transformation of a new kind is applied to obtain the infinite hierarchy of solutions expressed through ones of corresponding lax pairs on an initial solution of the equation studied sufficient conditions are given for the solutions built to take values in the class of unimodular positive definite matrices a new infinitesimal darboux transformation is introduced and a particular solution of the linearization of the self dual yang mills equation is given two hierarchies of infinitesimal symmetries depending on a solution of the nonlinear equation are extracted

on the stability of bdms and peers elements using fortin operators we give a new proof of stability for stenberg s family of bdms elements in linear elasticity our approach yields the inf sup condition with respect to the standard norms which is indispensable for a posteriori error analysis furthermore our technique allows the construction of another family of finite elements strongly related to the classical peers element the given estimates are robust for nearly incompressible materials

natural symmetries of the yang mills equations a natural generalized symmetry of the yang mills equations is defined as an infinitesimal transformation of the yang mills field built in a local gauge invariant and poincare invariant fashion from the yang mills field strength and its derivatives to any order which maps solutions of the field equations to other solutions on the jet bundle of yang mills connections a spinorial coordinate system is introduced that is adapted to the solution subspace defined by the yang mills equations in terms of this coordinate system the complete classification of natural symmetries is carried out in a straightforward manner it is found that all natural symmetries of the yang mills equations stem from the gauge transformations admitted by the equations

an augmented mixed finite element method with lagrange multipliers a priori and a posteriori error analyses in this paper we provide a priori and a posteriori error analyses of an augmented mixed finite element method with lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions the augmented scheme is obtained by including the galerkin least squares terms arising from the constitutive and equilibrium equations we use the classical babuska brezzi theory to show that the resulting dual mixed variational formulation and its galerkin scheme defined with raviart thomas spaces are well posed and also to derive the corresponding a priori error estimates and rates of convergence then we develop a reliable and efficient residual based a posteriori error estimate and a reliable and quasi efficient ritz projection based one as well finally several numerical results illustrating the performance of the augmented scheme and the associated adaptive algorithms are reported

a priori and a posteriori error analyses of a velocity pseudostress formulation for a class of quasi newtonian stokes flows abstract in this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear stokes models arising in quasi newtonian fluids the methods are based on a non standard mixed approach in which the velocity the pressure and the pseudostress are the original unknowns however we use the incompressibility condition to eliminate the pressure and set the velocity gradient as an auxiliary unknown which yields a twofold saddle point operator equation as the resulting dual mixed variational formulation in addition a suitable augmented version of the latter showing a single saddle point structure is also considered we apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well posed in particular we show that raviart thomas elements of order k 0 for the pseudostress and piecewise polynomials of degree k for the velocity and its gradient ensure the well posedness of the associated galerkin schemes moreover we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation then we derive a reliable and efficient residual based a posteriori error estimator for each scheme finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms

a simple construction of recursion operators for multidimensional dispersionless integrable systems abstract we present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect to the latter new examples of recursion operators obtained using our technique include inter alia those for the general heavenly equation which describes a class of anti self dual solutions of the vacuum einstein equations and a six dimensional equation resulting from a system of ferapontov and khusnutdinova

mixed methods for stationary navier stokes equations based on pseudostress pressure velocity formulation in this paper we develop and analyze mixed finite element methods for the stokes and navier stokes equations our mixed method is based on the pseudostress pressure velocity formulation the pseudostress is approximated by the raviart thomas brezzi douglas marini or brezzi douglasfortin marini elements the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree it is shown that these sets of finite elements are stable and yield optimal accuracy for the stokes problem for the pseudostress pressure velocity formulation of the stationary navier stokes equations the well posedness and error estimation results are established by eliminating the pseudostress variables in the resulting algebraic system we obtain cell centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum

global exponential stability for a class of retarded functional differential equations with applications in neural networks in this paper the global exponential stability and asymptotic stability of retarded functional differential equations with multiple time varying delays are studied by employing several lyapunov functionals a number of sufficient conditions for these types of stability are presented our results show that these conditions are milder and more general than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity furthermore the results obtained for neural networks with time varying delays do not assume symmetry of the connection matrix

robust stability for interval hopfield neural networks with time delay the conventional hopfield neural network with time delay is intervalized to consider the bounded effect of deviation of network parameters and perturbations yielding a novel interval dynamic hopfield neural network idhnn model a sufficient condition related to the existence of unique equilibrium point and its robust stability is derived

hidden symmetries and integrable hierarchy of the mathcal n 4 supersymmetric yang mills equations we describe an infinite dimensional algebra of hidden symmetries of mathcal n 4 supersymmetric yang mills sym theory our derivation is based on a generalization of the supertwistor correspondence using the latter we construct an infinite sequence of flows on the solution space of the mathcal n 4 sym equations the dependence of the sym fields on the parameters along the flows can be recovered by solving the equations of the hierarchy we embed the mathcal n 4 sym equations in the infinite system of the hierarchy equations and show that this sym hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations presumably the existence of such nonlocal symmetries underlies the observed integrable structures in quantum mathcal n 4 sym theory

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

a self dual yang mills hierarchy and its reductions to integrable systems in 1 1 and 2 1 dimensions the self dual yang mills equations play a central role in the study of integrable systems in this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self dual yang mills flow as the first member we show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1 1 and 2 1 dimensions and can be considered as a universal integrable hierarchy prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators symmetries and conservation laws are discussed

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

bifurcation analysis of a chemostat model with a distributed delay a chemostat model of a single species feeding on a limiting nutrient supplied at a constant rate is proposed the model incorporates a general nutrient uptake function and a distributed delay the delay indicates that the growth of the species depends on the past concentration of nutrient using the average time delay as a bifurcation parameter it is proven that the model undergoes a sequence of hopf bifurcations stability criteria for the bifurcating periodic solutions are derived it is also found that the periodic solutions become unstable if the dilution rate is increased computer simulations illustrate the results

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

the symmetry structure of the anti self dual einstein hierarchy an important example of a multi dimensional integrable system is the anti self dual einstein equations by studying the symmetries of these equations a recursion operator is found and the associated hierarchy constructed owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them in addition the lax pair for this hierarchy is constructed

a priori and a posteriori error analyses of a pseudostress based mixed formulation for linear elasticity in this paper we present the a priori and a posteriori error analyses of a non standard mixed finite element method for the linear elasticity problem with non homogeneous dirichlet boundary conditions more precisely the approach introduced here is based on a simplified interpretation of the pseudostress displacement formulation originally proposed in arnold and falk 1988 which does not require symmetric tensor spaces in the finite element discretization in addition physical quantities such as the stress the strain tensor of small deformations and the rotation are computed through a simple postprocessing in terms of the pseudostress variable furthermore we also introduce a second element by element postprocessing formula for the stress which yields an optimally convergent approximation of this unknown with respect to the broken h div norm we apply the classical babuska brezzi theory to prove that the corresponding continuous and discrete schemes are well posed in particular raviart thomas spaces of order k 0 for the pseudostress and piecewise polynomials of degree k for the displacement can be utilized moreover we remark that in the 3d case the number of unknowns behaves approximately as 9 times the number of elements tetrahedra of the triangulation when k 0 this factor increases to 12 5 when one uses the classical peers next we derive a reliable and efficient residual based a posteriori error estimator for the mixed finite element scheme finally several numerical results illustrating the performance of the method confirming the theoretical properties of the estimator and showing the expected behaviour of the associated adaptive algorithm are provided

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

asymptotic stability criteria for a two neuron network with different time delays in this paper the asymptotic stability of a two neuron system with different time delays has been investigated some criteria for determining the global asymptotically stability of equilibrium are derived from the theory of monotonic dynamical system and the approach of lyapunov functional for local asymptotic stability some elegant criteria are also obtained by the nyquist criteria we find that one of them depends on the length of delays while the other ones do not in the latter case the delays are sometimes called harmless delays the results obtained have leading significance in the study of neural networks composed of a large number of neurons with different time delays

hopf bifurcation and stability for a neural network model with mixed delays abstract a generalized model of the two neuron network with mixed delays is studied the main purpose of this paper is to explore the linear stability of the trivial solution and hopf bifurcation of a two neuron network with continuous and discrete delays the general formula of the direction the estimation formula of period and stability of bifurcated periodic solutions are also studied finally the numerical simulations are given to illustrate the theoretical analysis

formulation and analysis of fully mixed methods for stress assisted diffusion problems abstract this paper is devoted to the mathematical and numerical analysis of a mixed mixed pde system describing the stress assisted diffusion of a solute into an elastic material the equations of elastostatics are written in mixed form using stress rotation and displacements whereas the diffusion equation is also set in a mixed three field form solving for the solute concentration for its gradient and for the diffusive flux this setting simplifies the treatment of the nonlinearity in the stress assisted diffusion term the analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of schauder and banach fixed point theorems together with the babuska brezzi and lax milgram theories concerning numerical discretization we propose two families of finite element methods based on either peers or arnold falk winther elements for elasticity and a raviart thomas and piecewise polynomial triplet approximating the mixed diffusion equation we prove the well posedness of the discrete problems and derive optimal error bounds using a strang inequality we further confirm the accuracy and performance of our methods through computational tests

on a new hierarchy of symmetries for the kadomtsev petviashvili equation abstract we present a new hierarchy of symmetries for the kadomtsev petviashvili equation these new symmetries depend on the space and time variables explicitly together with the previously known classical symmetries they constitute an infinite dimensional lie algebra

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed

new fully mixed finite element methods for the stokes darcy coupling abstract in this paper we introduce and analyze two new fully mixed variational formulations for the coupling of fluid flow with porous media flow flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we first extend recent related results involving a pseudostress velocity based formulation in the fluid and consider a fully mixed formulation in which the main unknowns are given now by the stress the vorticity and the velocity all them in the fluid together with the velocity and the pressure in the porous medium the aforementioned formulation is then partially augmented by introducing galerkin least squares type terms arising from the constitutive and equilibrium equations of the stokes equation and from the relation defining the vorticity in terms of the free fluid velocity these three terms are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well posed the classical babuska brezzi theory is applied to provide sufficient conditions for the well posedness of the continuous and discrete formulations of both approaches next we derive a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element scheme the proof of reliability makes use of the global inf sup condition helmholtz decomposition and local approximation properties of the clement interpolant and raviart thomas operator in turn inverse inequalities the localization technique based on element bubble and edge bubble functions and known results from previous works are the main tools to prove the efficiency of the estimator finally several numerical results illustrating the good performance of both methods confirming the aforementioned properties of the estimator and showing the behavior of the associated adaptive algorithm are provided