on 4 manifolds with singular torus actions this paper studies the homeomorphism classification problem for closed oriented 4 manifolds which support effective codimension two torus actions with suitable restrictions on the actions a classification is obtained in terms of numerical invariants computable from orbit data these invariants are related to the fundamental groups and the cohomology rings of the underlying manifolds they do not distinguish among all 4 manifolds with actions as above and so the general classification problem remains open

convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled data systems under dwell time constraints stability analysis and control of linear impulsive systems is addressed in a hybrid framework through the use of continuous time time varying discontinuous lyapunov functions necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas extensions to the stability of aperiodic systems under minimum maximum and ranged dwell times are then derived by exploiting further the particular structure of the stability conditions the results are non conservatively extended to quadratic stability analysis of linear uncertain impulsive systems these stability criteria are in turn losslessly extended to stabilization using a particular yet broad enough class of state feedback controllers providing then a convex solution to the open problem of robust dwell time stabilization of impulsive systems using hybrid stability criteria relying finally on the representability of sampled data systems as impulsive systems the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled data systems are straightforwardly solved using the same ideas several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach

work by robert kalaba on automated sensitivity analysis this paper reviews work by robert kalaba on the development of a program nasa for the automated sensitivity analysis of parametrized nonlinear systems over parameter intervals nasa incorporates a fast and efficient algorithm feed for higher order partial derivative evaluation as well as an adaptive homotopy continuation procedure for obtaining all required initial conditions applications are envisioned for fields such as economics and the biological sciences where models tend to be complex and closed form solutions are difficult to obtain

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

retracted asymptotic constancy for a differential equation with multiple state dependent delays in this paper we investigate the asymptotic behavior of solutions to a differential equation with multiple state dependent delays it is shown that every bounded solution of such an equation tends to a constant as t our results improve and extend some corresponding ones already known

retracted asymptotic behavior of solutions to a system of differential equations with state dependent delays in this paper we investigate the asymptotic behavior of solutions to a system of differential equations with state dependent delays it is shown that every bounded solution of such a system tends to a constant vector as t our results improve and extend some corresponding ones already known

automatic differentiation of functions of derivatives in a recent paper an algorithm feed was introduced for the systematic exact evaluation of higher order partial derivatives of functions of many variables the present paper demonstrates that feed can be applied to a much broader class of functions than envisioned previously specifically feed can be used to evaluate the higher order partial derivatives of functions which are defined in terms of the derivatives of other functions a task required in many applications

existence of periodic solutions for delay differential equations with state dependent delay abstract in this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state a fixed point problem related to a poincare operator is constructed and solved using an ejective fixed point theorem

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

boundary layer phenomena for differential delay equations with state dependent time lags i in this paper we begin a study of the differential delay equation mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0jc9yq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpeps0he9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiabew7aljqadi hagaqbaiaacicacawg0bgaaiykaiabg2da9iabgkhitiaadihacagg oagaamidaiaacmcacqghrawkcawgmbgaaiikaiaadihacaggoagaam idaiabgkhitiaadkhacaggpagaaiykaiaacycacaqggagaaeiiaiaa dkhacqgh9aqpcawgybgaaiikaiaadihacaggoagaamidaiaacmcaca ggpaaaaa 5192 varepsilon x t x t f x t r r r x t we prove the existence of periodic solutions for 0 0 where 0 is an optimal positive number we investigate regularity and monotonicity properties of solutions x t which are defined for all t and of associated functions like t t r x t we begin the development of a poincare bendixson theory and phase plane analysis for such equations in a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as approaches zero

solving nonlinear equations by adaptive homotopy continuation standard homotopy continuation methods for solving systems of nonlinear equations require the continuation parameter to move from 0 to 1 along the real line difficulties can occur however if a point of singularity is encountered during the course of the integration to ameliorate these difficulties this paper proposes extending the continuation parameter to complex values and adaptively computing a continuation path in the complex plane that avoids points giving rise to singularities specifically it is proposed that the continuation parameter move from 0 0i to 1 0i along a spider web grid centered at 1 0i in the complex plane the actual path through the grid is determined step by step in accordance with two objectives short path length and avoidance of singular points a two phase homotopy continuation is used to study the implementation of this idea numerical examples are presented which indicate the effectiveness of the approach

complete comparative static differential equations equations having form 1 arise as first order conditions in microeconomic constrained optimization models as defining characterizations for general equilibrium and macroeconomic models and as steady state solution characterizations for descriptive and optimal growth models see silberberg 12 hansen 7 cass and shell 3 and brock l if y r r is continuously differentiable over a neighborhood of a point x0 a in r n l for which y xo lx 0 and iy x0 or 0 then the implicit function theorem guarantees the existence of a continuously differentiable function x n cr r over some open neighborhood n cro of cx in r such that dx z ix yx x x a lyjx a x g e n cp 2

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

automatic computation of derivatives with the use of the multilevel differentiating technique 1 algorithmic basis abstract a multilevel technique for derivative computation is introduced which rests upon a multilevel modular unification of the back and the forward methods with an increased flexibility for accurate differrentiation this method provides in many cases a reasonable compromise between time and memory the structure of the method is in a good agreement with the conventional modular principle of programming which facilitates its software realization

an ada library for automatic evaluation of derivatives requirements for the evaluation of high order partial derivatives arise in many applications including optimal control system identification and the numerical solution of nonlinear algebraic equations two point boundary value problems and integral equations these evaluations have long been shunned by numerical analysts because of the difficulty in forming symbolic expressions for the derivatives this paper describes in detail an implementation in ada of a set of simple procedures for evaluating derivatives exactly without the requirement to form their symbolic expressions and without resorting to finite difference methods an expression whose derivatives are desired is written in the usual way using standard mathematical operators and intrinsic functions and values of all desired derivatives are returned automatically two examples of parameter identification are shown one from economics and one from guidance and control a complete listing of all ada code is included

slowly oscillating periodic solutions of autonomous state dependent delay equations more than a dozen years since the peak of activity on the question of the existence of periodic slowly oscillating solutions of autonomous delay differential equations following the early work of jones l wright 2 and grafton 3 the work of nussbaum 4 51 is to be specially noted for providing several new fixed point results and a global bifurcation theorem which are particularly useful for proving the existence of periodic solutions other important works include those of hadeler and tomiuk 6 kaplan and yorke 7 chow 8 chow and hale 9 alt lo 111 and walther 12 131 see hale s book 14 for an overview to the best of our knowledge only nussbaum 5 and alt 1 l considered the question of the existence of periodic solutions of autonomous state dependent delay differential equations nussbaum considered a special equation 0 3 below in 5 but did not prove a general result alt 1 l considered more complicated integral threshold type state dependent delays which have arisen in various models in epidemiology and structured population models alt obtained a theorem for a general class of state dependent equations on which we comment below the lack of results on periodic solutions for state dependent equations is probably explained by the fact that it is not clear what kinds of state dependent delays are interesting or natural and on a lack of compelling examples of such equations arising from plausible mathematical models however some recent papers of belair and mackey 15 and belair 16 contain some interesting state dependent delay equations arising in economics and population biology state dependent delays of threshold type arise naturally in structured population models see lo 111 the motivation for this paper stems from consideration of the innocent looking equation x t kx t r x t 0 1 where r x is bell shaped e g r x 01 emxz 1 01 0 i 01 5 1 if k 7r 2 and r o 1 then the zero solution of 0 1 is unstable a formal linearization would yield z t kz t 1 which is known to be unstable for

a finite algorithm for the exact evaluation of higher order partial derivatives of functions of many variables abstract a formal algorithm is given for the systematic exact evaluation of higher order partial derivatives of functions of many variables the algorithm improves upon wengert s method in two key respects applications are envisioned wherever gradients jacobians hessians and power series expansions could be employed

periodic solutions for differential equations with state dependent delay and positive feedback existence of a nontrivial periodic orbit and a homoclinic orbit connecting 0 to the periodic orbit is shown for the differential equation x t x t f x t r r r x t with state dependent delay and positive feedback here 0 f and r are smooth real functions with r 0 1 and f 0

nonlocal automated sensitivity analysis previously a complete ordinary differential equation ode system was developed for tracking the solution x of a parameterized system of nonlinear equations 0 x over an interval 0 this paper develops a two phase complex homotopy continuation method for obtaining the required initial conditions at t an initial short artificial continuation is followed by a continuation which essentially proceeds through the physically meaningful function which can ameliorate the problem of artificially induced singularities also the path in the complex plane followed by the continuation parameter in each phase evolves sequentially in an attempt to keep the path both short minimal number of integration steps and numerically stable avoidance of singular points a fortran program nasa has been developed for solving the complete ode system starting with a two phase initialization the program incorporates a fast and efficient procedure for the automatic evaluation of partial derivatives numerical experiments are reported which illustrate the program s effectiveness in ref 1 an analytically complete system of ordinary differential equations odes was developed for finding a solution vector x for a parameterized system of nonlinear equations 0 p x the well known ode dx d x was supplemented with odes for the adjoint a and the determinant 6 of the jacobian matrix vx x this differential system in principle permits nonlocal sensitivity analysis i e the tracking of the solution vector x and the sensitivity vector dx d together with the adjoint a and the determinant i over any closed interval 0 where the determinant i remains nonzero a fortran program was used successfully in ref 1 to solve this differential system for several illustrative examples subsequently the program was modified to incorporate a fast and efficient algorithm feed for the automatic evaluation of higher order partial derivatives 2 nevertheless to make the program a truly practical tool a simple and reliable procedure was still needed for obtaining the initial conditions x a and c required by the differential system at the initial parameter point ct this paper develops a simple but effective homotopy continuation method for obtaining these initial conditions which differs in two principal respects from standard homotopy continuation methods first the proposed continuation essentially proceeds through the function rather than through an artificial construct letting f x d x t a standard continuation method for finding a solu tion vector x x to the system of equations 0 f x would generally involve solving a system of equations of the form

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

local and nonlocal comparative static analysis of economic systems a complete differential equation system has been developed for the local and nonlocal comparative static analysis of general parametrized economic systems j x a 0 the differential equations allow both qualitative and quantitative sensitivity studies at parameter points or over parameter intervals a fortran program comstat is now available for the numerical integration of the differential equations the only input requirements are j and initial conditions initialization is rendered routine by a two stage application of comstat derivatives are automatically evaluated via a generalized wengert method freeing the user from ever having to form analytical derivatives comstat is applied to a problem in optimal economic growth to show the sensitivity of the steady state capital labor ratio to changes in the time preference parameter

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence