structure preserving approximation of dissipative evolution problems we present a framework for the systematic numerical approximation of nonlinear evolution problems with dissipation the approach is based on rewriting the problem in a canonical form that complies with the underlying energy dissipation structure we show that the corresponding weak formulation then allows for a dissipation preserving approximation by galerkin methods in space and discontinuous galerkin methods in time the proposed methodology is rather general and can be applied to a wide range of applications this is demonstrated by discussion of some typical examples ranging from diffusive partial differential equations to dissipative hamiltonian systems

self dual codes with an automorphism of order 11 using a method for constructing self dual codes having an automorphism of odd prime order we classify up to equivalence all binary self dual codes with an automorphism of order 11 with 6 cycles and minimum distance 12 this classification gives new 72 36 12 codes with weight enumerator that was previously not obtained as well as many 66 33 12 68 34 12 and 70 35 12 codes with new values of the parameters in their respective weight enumerators

simple calculation of lowdin s alpha function iii procedure for calculating hn 2n i lm l 2n i successively when r a 0 lowdin s function 1 r l flm a r is expressed as 1 r l flm a r 2 lm l al 1 n l imaxi l m hn 2n i lm l 2n i a2n ig 2n i a r a 2n l where a is a separation between a new center placed on the origin and the old center located at a given point on the z axis that is the origin of the coordinate system defining a function f r yml to be expanded around the new center f r is an arbitrary radial function and yml a complex spherical harmonics and r is a distance from the new center here imax min 2n 2 l l and the function g j a is expressed as g j a d dr j f ir rl 1 r a the closed form of the coefficients hn 2n i lm l is given by eq 27 in j math phys 26 3193 1985 referred to as part ii since they are expressed in terms of the different coefficients bkk lm l independent of the form of f r hn 2n i lm l constitute a set of universal constants as do bkk lm l the explicit expression for the factor lm l in front of the summa

stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

boundary constrained control of delayed nonlinear schrodinger equation this paper studies regional boundary stabilization of nonlinear schrodinger equation with state delay and bounded internal disturbance the boundary constrained control law is designed by using the backstepping method regional input to state stability of the perturbed system with time delay is established by a lyapunov function and a generalized halanay s inequality estimates on the set of initial conditions are found starting from which the solutions are exponentially attracted to a bounded set a numerical example demonstrates the efficiency of the results

stability and synchronization analysis of neural networks via halanay type inequality in this paper not only some novel sufficient criteria for stability of discrete delay neural networks but also a new discriminant method for self synchronization of hopfield neural networks are considered some novel stability criteria for discrete delay neural networks are obtained if the time average of the difference for two coefficients on some fixed lengths is lower bounded by some positive numbers moreover a new discrete halanay inequality is given which extends the existing results based on the halanay inequality a novel sufficient criterion on self synchronization of hopfield neural networks with time delay is obtained two numerical examples are given to demonstrate the effectiveness of proposed methods

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

asymptotic stability contractivity and dissipativity of one leg θ method for non autonomous delay functional differential equations abstract this paper focuses on asymptotic stability contractivity and dissipativity of non autonomous nonlinear delay functional differential equations with bounded lag and the corresponding dynamical properties of one leg method sufficient conditions for these delay functional differential equations to be dissipative asymptotically stable and contractive are established one leg method is constructed to solve such equations numerically an important result on the growth of solution of a class of difference inequalities with variable coefficients is obtained finally it is proved that the one leg method is asymptotically stable contractive and dissipative if and only if 1 numerical examples are given to confirm our theoretical results

dissipativity of θ methods for a class of nonlinear neutral delay differential equations abstract this paper is concerned with the dissipativity of theoretical solution and numerical solution of a class of nonlinear neutral delay differential equations at first a sufficient condition for the dissipativity of theoretical solution of the mentioned problem above is given then the dissipativity results are obtained for one leg and linear methods when they are applied to these problems

simple calculation of lowdin s alpha function iv procedure for evaluating the coefficients in expansion of a slater type atomic orbital by using the explicit expression for the coefficients bkk lm l eq 16 in j math phys 26 3193 1985 which is referred to hereafter as part ii a closed expression for the coefficients cij nlm l appearing in lowdin s alpha function 1 r l nlm a r for a slater type atomic orbital is obtained by manipulating its expression mathematically it was manifested that cij nlm l vanishes if i n l m l for the specific cases that each of i and j takes 0 or 1 cij nlm l take the same value from the symmetry relation on bkk lm l eq 6 in part ii it was proved that cij nlm l cji nlm l provided m l through some manipulation the recursion formula for cij nlm l in n was derived while that about m was obtained by introducing the recursion formula for bkk lm l in m eq 24 in part ii into the expression for cij nlm l for the special cases where n and m both are equal to l the expression for cij nlm l is given in a compact and symmetric form the use of this expression leads to a recursion

a lyapunov type theorem for dissipative numerical integrators with adaptive time stepping the asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time stepping can be fully investigated by a lyapunov type theorem on continuous and discrete dynamical systems respectively however once adaptive time stepping is involved such theories cease to work and usually the dynamics should be investigated from the past instead of the standard forward way such as in terms of pullback attractors in this paper we present a different approach we stick to a forward definition of limit sets and show that still we can establish a lyapunov type theorem which reveals the precise asymptotic behavior of adaptive time stepping integrators in the presence of a discrete lyapunov functional

stable attracting sets in dynamical systems and in their one step discretizations we consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set of arbitrary shape under the assumption of uniform asymptotic stability of in the sense of lyapunov we show that discretized versions of the dynamical system involving one step numerical methods have nearby attracting sets h which are also uniformly asymptotically stable our proof uses the properties of a lyapunov function which characterizes the stability of

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

generalized halanay inequalities and their applications to neural networks with unbounded time varying delays in this brief we discuss some variants of generalized halanay inequalities that are useful in the discussion of dissipativity and stability of delayed neural networks integro differential systems and volterra functional differential equations we provide some generalizations of the halanay inequality which is more accurate than the existing results as applications we discuss invariant set dissipative synchronization and global asymptotic stability for the hopfield neural networks with infinite delays we also prove that the dynamical systems with unbounded time varying delays are globally asymptotically stable

global dissipativity for a stable methods this paper concerns the discretization of the initial value problem u t f u under the three structural conditions i f bbb c n longrightarrow bbb c n re e langle f u u rangle leq a b u 2 a geq 0 b 0 for all u in bbb c n ii f bbb c n longrightarrow bbb c n re e langle f u u rangle 0 iii f w longrightarrow h re e langle f w w rangle h leq a b w h 2 a geq 0 b 0 for all w in w for complex hilbert spaces w subseteq h dahlquist s g stability theory is used to show that linear multistep and one leg methods yield dissipative discretizations for all f satisfying i if and only if the method rho sigma is a stable extensions of g stability theory are made to find necessary and sufficient conditions on rho sigma for similar properties to hold in cases ii and iii in every case conditions are found for the strict contractivity of solutions for large initial data and bounds for the asymptotic rate of decay are calculated in cases i and iii

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

a behavioral approach to delay differential systems we will study linear time invariant delay differential systems from the behavioral point of view as it was introduced for dynamical systems by willems dynam report 2 1989 pp 171 269 a ring cal h which lies between bbb r s z z 1 and bbb r s z z 1 will be presented whose elements can be interpreted as a generalized version of delay differential operators on cal c infty bbb r bbb r in this framework a behavior is the kernel of such an operator using the ring cal h an algebraic characterization of inclusion respectively equality of the behaviors under consideration is given finally controllability of the behaviors is characterized in terms of the rank of the associated matrices in the case of time delay state space systems this criterion becomes the known hautus criterion for spectral controllability

some new extremal codes of lengths 76 and 78 we find new extremal 76 38 14 and 78 39 14 binary self dual codes the code of length 76 is the first known self dual code with these parameters it is shadow extremal and the shortest known self dual code with minimal distance 14 it has an automorphism of order 19 without fixed coordinates the code of length 78 has a new weight enumerator and an automorphism of order 19 with two fixed coordinates

stability analysis of stochastic delayed systems with an application to multi agent systems this technical note is concerned with the stability problem of stochastic delayed systems by using a more general halanay type inequality with time varying coefficients a new comparison principle of the proposed halanay type inequality is developed to ensure the stability of stochastic delayed systems the p th moment stability p th moment asymptotic stability and p th moment exponential stability are investigated by means of the proposed general halanay type inequality two examples including one of the consensus of multi agent systems with time varying topologies and transmission delays are presented to demonstrate the effectiveness of the proposed results

dissipativity of linear θ methods for integro differential equations this paper is concerned with the analytic and numerical dissipativity of nonlinear integro differential equations ides a dissipativity criteria for ides is given it is shown that any a stable linear 0 method for the systems is dissipative numerical examples are given to confirm the theoretical results

dissipativity of volterra functional differential equations this paper is concerned with the dissipativity of volterra functional differential equations in a hilbert space a sufficient condition for dissipativity of one class of such equations is obtained this result is applied to delay differential equations and integro differential equations to obtain dissipativity results that are more general and deeper than related results in the previous literature

numerical dissipativity of multistep methods for delay differential equations dissipative differential equations have frequently appeared in the fields of physics engineering and biology in this paper we investigate numerical dissipativity of linear multistep and one leg methods applied to a class of dissipative delay differential equations we show that for such class of dissipative systems these numerical methods are dissipative if and only if they are a stable for ordinary differential equations one numerical experiment is given to illustrate our result

a method for constructing inequivalent self dual codes with applications to length 56 necessary and sufficient conditions are presented for two doubly even codes with an automorphism of odd prime order to be equivalent under certain assumptions for the codes as an application we obtain all extremal doubly even codes of length 56 with an automorphism of order 13 sixteen such codes are inequivalent

construction of self dual codes construction methods for self dual codes are given by using these methods some new extremal self dual 66 33 12 and 68 34 12 codes are obtained

generalized lyapunov approach for functional differential inclusions abstract this paper is concerned with the generalized lyapunov approach for functional differential inclusions fdi at first we prove a class of generalized halanay s inequalities then by applying the generalization of halanay s inequalities we investigate the uniformly ultimate boundedness and uniformly asymptotic stability of fdi

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

generalized halanay inequalities for dissipativity of volterra functional differential equations abstract this paper is concerned with the dissipativity of theoretical solutions to nonlinear volterra functional differential equations vfdes at first we give some generalizations of halanay s inequality which play an important role in study of dissipativity and stability of differential equations then by applying the generalization of halanay s inequality the dissipativity results of vfdes are obtained which provides unified theoretical foundation for the dissipativity analysis of systems in ordinary differential equations odes delay differential equations ddes integro differential equations ides volterra delay integro differential equations vdides and vfdes of other type which appear in practice

uniform ultimate boundedness of numerical solutions to nonlinear neutral delay differential equations this paper is concerned with the long time behaviour of the numerical solutions generated by runge kutta rk methods for nonlinear neutral delay differential equations nddes it is proved that the numerical solutions produced by k l algebraically stable rk methods are uniformly ultimately bounded some examples reveal that some rk methods completely preserve the long time behaviour of the exact solutions to nddes for sufficiently small time stepsize h as a comparison with the previous results a numerical example which further illustrates our theoretical results is provided

stability of θ methods for delay integro differential equations stability of methods for delay integro differential equations dides is studied on the basis of the linear equation du dt u t u t k t t u d where k are complex numbers and is a constant delay it is shown that every a stable method possesses a similar stability property to p stability i e the method preserves the delay independent stability of the exact solution under the condition that k is real and h is an integer where h is a step size it is also shown that the method does not possess the same property if h is not an integer as a result no method can possess a similar stability property to gp stability with respect to dides

simple calculation of lowdin s alpha function ii easier procedure for evaluating bk k l m l and vanishing of hn 2n i l m l for special values of i and n this paper subsequent to the one j math phys 25 1133 1984 referred to as part i presents the following new results it is found out that for m l and l 1 the coefficients bk k l m l in lowdin s function have properties other than manifested in part i the expression for bk k l m l is shown to be equivalent to the one into which sharma s expression obtained in a different manner from that in part i is simplified by rashid the use of rashid s expression leads to the recurrence formula for bk k l m l with respect to m only this formula and the expression for the bk k l m l with m l provide an easier procedure for successively evaluating bk k l m l than in part i furthermore it is proved that the coefficients hn 2n i l m l in the asymptotic form of the function vanish for i l m and for n l

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

dissipativity and asymptotic stability of nonlinear neutral delay integro differential equations this paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro differential equations ndides we first give a generalization of the halanay inequality which plays an important role in the study of dissipativity and stability of differential equations then we apply the generalization of the halanay inequality to ndides and the dissipativity and the asymptotic stability results of the theoretical solution of ndides are obtained from a numerical point of view it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions therefore the results presented in this paper provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems

stability analysis of discontinuous galerkin method for stiff volterra functional differential equations abstract this paper concerns the nonlinear stability properties of discontinuous galerkin dg method for stiff volterra functional differential equations vfdes we derive that the dg method leads to global and analogously asymptotical stability for vfdes and it is shown that the perturbations of the numerical solution are controlled by the initial perturbations this general results provide unified theoretical treatment for numerical stability analysis of volterra integro differential equations vides with constant or variable delay delay differential equations ddes ordinary differential equations odes and so on numerical examples are given to confirm our theoretical results

dissipativity of runge kutta methods for neutral delay integro differential equations this paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro differential equations the dissipativity results are obtained for algebraically stable runge kutta methods when they are applied to above problems

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

an impulsive delay differential inequality and applications an impulsive delay differential inequality is formulated in this paper an estimate of the rate of decay of solutions to this inequality is obtained it can be applied to the study of dynamical behavior of delay differential equations from the impulsive control point of view as an application we consider a class of impulsive control systems with time varying delays and establish a sufficient condition to guarantee the global exponential stability it is shown that via proper impulsive control law a linear delay differential system can be exponentially stabilized even if it is initially unstable a numerical example is given to demonstrate the effectiveness of the development method

on the numerical approximation of phase portraits near stationary points we show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one step methods or multi step methods satisfying a strong root condition this means that any continuous trajectory can be approximated by an appropriate discrete trajectory and vice versa to the correct order of convergence and uniformly on arbitrarily large time intervals in particular the stable and unstable manifolds of the discretization converge to their continuous counterparts

a 72 36 16 doubly even code does not have an automorphism of order 11 corresp we prove that there does not exist a 72 36 16 doubly even code with an automorphism of order 11 if such a code exists it can be decomposed as a direct sum of two codes one can be viewed as a self dual 12 6 4 binary code and the other a self dual 6 3 4 code over f 1024 this is shown to be impossible

dissipativity of delay functional differential equations with bounded lag delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional we first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative then we construct a one leg method to solve such dissipative equations and prove that it is dissipative if 1 one numerical example is given to confirm our theoretical result

automorphisms of codes with applications to extremal doubly even codes of length 48 general results on automorphisms of self dual binary codes are given these results are applied to the study of extremal self dual doubly even binary codes of length 48 the main theorem proved is that an extremal self dual doubly even code of length 48 with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code interesting constructions of the binary extended golay code as well as a conjecture about a possible connection between an extremal self dual doubly even code of length 72 and an extremal quaternary code of length 24 arc yielded by techniques used in the proof

asymptotic stability analysis of θ methods for functional differential equations stability analysis of methods for functional differential equations based on the test equation begin gathered y t ay t lambda by t t 0 hfill y t psi t t in left lambda 0 right hfill end gathered 0 is presented it is known thaty t 0 ast if and only if b a and we investigate whether this property is inherited by the numerical solution approximatingy

dissipativity of θ methods for nonlinear volterra delay integro differential equations this paper deals with the analytic and numerical dissipativity of nonlinear volterra delay integro differential equations vdides a dissipativity criteria for vdides is given we investigate the dissipativity properties of one leg q methods and linear q methods for the underlying systems it is shown that for q 12 1 any one leg q method and linear q method can inherit the dissipativity of the underlying systems

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen