some existence and regularity results for abstract non autonomous parabolic equations abstract we study existence uniqueness and regularity of the strict classical and strong solutions u c 0 t e of the non autonomous evolution equation u t a t u t t with the initial datum u 0 x in a banach space e under the classical kato tanabe assumptions the domains of the operators a t are not needed to be dense in e we prove necessary and sufficient conditions for existence and holder regularity of the solution and its derivative

stability of time stepping methods for abstract time dependent parabolic problems we consider an abstract nonautonomous parabolic problem u t a t u t quad u t 0 u 0 where a t d t subset x to x t 0 le t le t 1 is a family of sectorial operators in a banach space x this problem is discretized in time by means of either an a theta stable runge kutta or an a theta stable linear multistep method we prove that the resulting discretization is stable under some natural assumptions on the relative total variation of a t with respect to t for strongly a theta stable runge kutta methods stability holds even for variable time step sizes our results are applicable to the analysis of time dependent parabolic problems in the lp norms

dynamics of a free boundary problem with curvature modeling electrostatic mems the dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems mems is investigated the model couples the electric potential to the deformation of the membrane the deflection of the membrane being caused by application of a voltage difference across the device more precisely the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane the gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation the main feature of the model considered herein is that unlike most previous research the small deformation assumption is dropped and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation the free boundary problem is shown to be well posed locally in time for arbitrary voltage values and globally in time for small voltages values furthermore existence of asymptotically stable steady state configurations is proved in case of small voltage values as well as non existence of steady states if the applied voltage difference is large finally convergence of solutions of the free boundary problem to the solutions of the well established small aspect ratio model is shown

abstract quasilinear parabolic equations on considere des problemes paraboliques quasi lineaires abstraits p 0 u t a t u t u t f t u t t 0 u o u 0 ou u t est une fonction a valeurs dans un espace de banach x on donne des theoremes d existence locale et d unicite pour des solutions classiques et strictes de p 0

computation of class numbers of quadratic number fields we explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields we therefore end up with a simpler and faster method for computing class numbers of quadratic number fields we also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields

global solutions to semilinear parabolic systems for small data abstract we consider semilinear parabolic systems u t au f u g u 0 u 0 a being the generator of an analytic semigroup with spectrum a in the right half plane for appropriate small data g u 0 global solutions are obtained if 0 belongs to the resolvent set and local existence is proved if 0 a the inverse function theorem is used in a class of functions which are holder continuous with respect to time t the asymptotic behaviour of the solution is given too as a first application we prove the existence of global respectively local solutions to the navier stokes equations in l spaces in bounded respectively unbounded domains second we consider the case where a is an elliptic operator of order 2 m and f has bounded growth third a quasilinear example where the nonlinearity involves derivatives of the same order as a is studied then an example of a nonautonomous nonlinearity is given and finally the convergence of solutions of the time dependent system to solutions of the stationary semilinear elliptic one is derived for 0 a

on the unnormalized solution of the filtering problem with counting process observations a general procedure to solve filtering problems for counting process observations is discussed linear nonstochastic integro differential equations describe the evolution of unnormalized conditional distribution of the state process between observation jump times while at jump times a linear updating is required final normalization is the only nonlinear operation to be implemented quite general situations may be accommodated in the present setup the state can be virtually any markov semimartingale the observation process may affect the dynamics of the state and vice versa and there is complete freedom in correlating state and observation martingale terms

asymptotic constancy for pseudo monotone dynamical systems on function spaces abstract a pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points the purpose of this paper is to present some convergence oscillation and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point some applications to neutral functional differential equations and semilinear parabolic partial differential equations with neumann boundary condition are given

parabolic evolution equations in interpolation and extrapolation spaces on etudie le probleme de cauchy pour des equations d evolution lineaires a t u f t o t t de type parabolique dans un espace de banach x 0 on considere des resultats d existence d unicite et de continuite pour a t u u f t u o t t

dynamic theory of quasilinear parabolic equations i abstracts evolution equations on etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

error estimates for semidiscrete galerkin type approximations to semilinear evolution equations with nonsmooth initial data error estimates for the semidiserete galerkin method for abstract semilinear evolution equations with non smooth initial data are given in concrete cases almost optimal order of convergence for linear finite elements results

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

primality testing and jacobi sums we present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by adleman and rumely the new algorithm performs well in practice it is the first pnmality test in existence that can routinely handle numbers of hundreds of decimal digits

global solutions of abstract quasilinear parabolic equations abstract local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied applications to quasilinear parabolic partial differential equations are also given

stability of runge kutta methods for quasilinear parabolic problems we consider a quasilinear parabolic problem u t q u t u t u t 0 u 0 d where q w d c x x w w c x is a family of sectorial operators in a banach space x with fixed domain d this problem is discretized in time by means of a strongly a stable 0 2 runge kutta method we prove that the resulting discretization is stable under some natural assumptions on the dependence of q w with respect to w our results are useful for studying in l p norms 1 p many problems arising in applications some auxiliary results for time dependent parabolic problems are also provided

bounds on the maximum modulus of solutions to quasilinear parabolic equations by means of a modified lyapunov method on obtient des bornes l aux solutions des equations paraboliques d ordre 2 de la forme u t x t u u x t u x t q ou toutes les quantites sont reelles q 0 t un ouvert borne de r n 0 t et ou l application non lineaire peut avoir une nature non locale

improvements on lower bounds for the blow up time under local nonlinear neumann conditions abstract this paper studies the heat equation u t u in a bounded domain r n n 2 with positive initial data and a local nonlinear neumann boundary condition the normal derivative u n u q on partial boundary 1 for some q 1 while u n 0 on the other part we investigate the lower bound of the blow up time t of u in several aspects first t is proved to be at least of order q 1 1 as q 1 since the existing upper bound is of order q 1 1 this result is sharp secondly if is convex and 1 denotes the surface area of 1 then t is shown to be at least of order 1 1 n 1 for n 3 and 1 1 ln 1 1 for n 2 as 1 0 while the previous result is 1 for any 1 n 1 finally we generalize the results for convex domains to the domains with only local convexity near 1

the principle of linearized stability for quasilinear parabolic evolution equations on donne des conditions pour la stabilite des points d equilibre des equations paraboliques quasilineaires avec des conditions aux limites non lineaires u a t u u u f t u 7b b t u u 0 u t 0 u 0 t 0 t

pathwise mild solutions for quasilinear stochastic partial differential equations abstract stochastic partial differential equations spdes have become a key modeling tool in applications yet there are many classes of spdes where the existence and regularity theory for solutions is not completely developed here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear cauchy problems including among others cross diffusion systems as a key application our solutions are local in time and are derived via a fixed point argument in suitable function spaces the key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations pdes with recent theory of evolution semigroups we also show how to apply our theory to the shigesada kawasaki teramoto skt model furthermore we provide examples of blow up and ill posed operators which can occur after finite time showing that solutions can only be local in time for general quasilinear spdes while they might be global in time for special subclasses of problems

instability in a geometric parabolic equation on convex domain instability on convex domain abstract we prove that any non constant smooth static solution to a geometric parabolic system is unstable provided that the domain is convex as the important applications we shall consider the landau lifshitz equation and the heat flow for harmonic map

efficient cm constructions of elliptic curves over finite fields we present an algorithm that on input of an integer n 1 together with its prime factorization constructs a finite field f and an elliptic curve e over f for which e f has order n although it is unproved that this can be done for all n a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 n log n where n is the number of distinct prime factors of n in the cryptographically relevant case where n is prime an expected run time o logn 4 e can be achieved we illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order n 10 2004 and n nextprime 102004 10 2004 4863

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

fast convolutions meet montgomery arithmetic in large ring and field extensions is an important problem of symbolic computation and it consists essentially of the combination of one multiplication and one division in the underlying ring methods are known for replacing one division by two short multiplications in the underlying ring which can be performed essentially by using convolutions however while using school book multiplication modular multiplication may be grouped into 2m r operations where m r denotes the number of operations of one multiplication in the underlying ring the short multiplication problem is an important obstruction to convolution it raises the costs in that case to 3m r in this paper we give a method for understanding and bypassing this problem thus reducing the costs of ring arithmetic to roughly 2m r when also using fast convolutions the algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper

primality testing using elliptic curves we present a primality proving algorithm a probablistic primality test that produces short certificates of primality on prime inputs we prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes as a corollary we obtain an algorithm for generating large certified primes with distribution statistically close to uniform under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size the test is shown to run in expected polynomial time for all primes yielding a las vegas primality test our test is based on a new methodology for applying group theory to the problem of prime certification and the application of this methodology using groups generated by elliptic curves over finite fields we note that our methodology and methods have been subsequently used and improved upon most notably in the primality proving algorithm of adleman and huang using hyperelliptic curves and in practical primality provers using elliptic curves

quasilinear evolution equations and parabolic systems it is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values provided the boundary conditions are time independent moreover it is shown that in the autonomous case these equations generate local semi flows on appropriate sobolev spaces our results apply in particular to the case of prescribed boundary values dirichlet boundary conditions

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

sharpening primes is in p for a large family of numbers we present algorithms that are deterministic primality tests for a large family of integers namely integers n 1 mod 4 for which an integer a is given such that the jacobi symbol a n 1 and integers n 1 mod 4 for which an integer a is given such that a n 1 a n 1 the algorithms we present run in 2 min k 2 log log n o log n 6 time where k v 2 n 1 is the exact power of 2 dividing n 1 when n 1 mod 4 and k v 2 n 1 if n 1 mod 4 the complexity of our algorithms improves up to o log n 4 when k 2 log log n we also give tests for a more general family of numbers and study their complexity

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

dynamics of a periodically pulsed bio reactor model abstract global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio reactor model in terms of principal eigenvalues of the periodic parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems abstract persistence asymptotically periodic semiflows and perturbation of global attractors

implementing the asymptotically fast version of the elliptic curve primality proving algorithm the elliptic curve primality proving ecpp algorithm is one of the current fastest practical algorithms for proving the primality of large numbers its running time cannot be proven rigorously but heuristic arguments show that it should run in time o log n 5 to prove the primality of n an asymptotically fast version of it attributed to j o shallit runs in time o log n 4 the aim of this article is to describe this version in more details leading to actual implementations able to handle numbers with several thousands of decimal digits

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

global existence and maximal regularity of solutions of gradient systems abstract in this article we use a galerkin method to prove a maximal regularity result for the following abstract gradient system u t g t e u t f t for a e t 0 t u 0 u 0 this abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients

a second order magnus type integrator for quasi linear parabolic problems in this paper we consider an explicit exponential method of classical order two for the time discretisation of quasi linear parabolic problems the numerical scheme is based on a magnus integrator and requires the evaluation of two exponentials per step our convergence analysis includes parabolic partial differential equations under a dirichlet boundary condition and provides error estimates in sobolev spaces in an abstract formulation the initial boundary value problem is written as an initial value problem on a banach space x u t a u t u t 0 t t u 0 given involving the sectorial operator a v d x with domain d c x independent of v v c x under reasonable regularity requirements on the problem we prove the stability of the numerical method and derive error estimates in the norm of certain intermediate spaces between x and d various applications and a numerical experiment illustrate the theoretical results

global existence for semilinear parabolic systems on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme u t a t u f t x u du d 2m 1 u dans t 0 b t u 0 sur t 0 u t 0 u 0 sur ou est un domaine de r n

compactness results for time dependent parabolic systems k 1 2 n where u u 8 is a function of t x e g 24 ad at q c r is a bounded domain with smooth boundary lx ak are uniformly elliptic operators and bk linear boundary operators we say that u is a regular solution of l if the function u together with its derivatives appearing in 1 is continuous in g such that eqs 1 are identically satisfied it is the object of this paper to show that under suitable conditions on a b and g the orbit t u a t b o of a bounded regular solution of 1 is relatively compact in c2 fl w results of this type are essential for the study of the asymptotic behavior of solutions u of 1 see e g 6 sect 4 61 or 17 p 3 551 we will deduce our theorems from more general results valid for abstract nonlinear evolution equations of the form