the hille yosida space of an arbitrary operator abstract let a be an arbitrary banach space operator with resolvent defined for all 0 we define a linear manifold z in the given space and a norm on z majorizing the given norm such that z is a banach space and the restriction of a to z generates a strongly continuous semigroup of contractions in z this so called hille yosida space z is maximal unique in a suitable sense

ill posed problems associated with sectorial operators in banach space abstract ill posed problems of the form u t a u h 0 t t u 0 are studied in banach space under two related scenarios both of which consider a well posed logarithmic approximation in the first scenario a generates a bounded holomorphic semigroup of angle 2 and so a is sectorial of angle 0 in the second where may be relaxed to any value in 0 2 we impose an additional assumption that e a is sectorial regularization results for the original ill posed problem are then established in the linear case where h h t and subsequently extended to nonlinear problems where h is replaced by h t u t we compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations both linear and nonlinear in l p r n 1 p

invariant conserving finite difference algorithms for the nonlinear klein gordon equation a formalism for systematicaiiy deriving second order accurate finite difference algorithms which conserve certain invariant quantities in the original nonlinear pdes is presented three algorithms are derived for the nonlinear klein gordon equation nlkge based on the proposed formalism the local conservation laws of the nlkge form the basic starting point in our derivation which hinges essentially on the commutativity of certain finite difference operators such commutativity in the discrete approximations allows a preservation of the derivation properties of the continuous counterparts at the pde level with appropriate boundary conditions the proposed algorithms preserve in the discrete sense either the total system energy or the system s linear momentum several variants of the present algorithms and their relation to previously proposed algorithms are discussed an analysis of the accuracy and stability is conducted to compare the different variants of the proposed algorithms the preservation of energy of the present algorithms for the nlkge can also be viewed as providing a method of stabilization for conditionally stable algo thms for the linear wave equation the computer impiementation of the proposed algorithms with the treatment of the boundary conditions is presented in detail numerical examples are given concerning soliton collisions in the sine gordon equation the double sine gordon equation and the cb phi four equation the numerical results demonstrate that the present algorithms can preserve accurately up to 10 decimal digits the total system energy for a very coarse grid reliable algorithms for josephson junction models which contain dissipation damping mechanisms and driving bias current are obtained as direct by products of the proposed invariant conserving algorithms for the nlkge even though presented mainly for the 1 d case the proposed algorithms are generalizable to the 2 d and 3 d cases and to the case of complex valued nlkge

pointwise functional calculi supposeais a possibly unbounded closed linear operator on a banach spacex x x and f is a banach algebra of functions we introduce apointwiseffunctional calculus for a at x this is a bounded linear map from f intox with the properties that one would expect from a mapf f a x ifahad a f functional calculus howeveramay not have such a functional calculus we show that the existence of a pointwise f functional calculus is equivalent to the existence of a continuously embedded banach subspace on whichahas a global f functional calculus we characterize being pointwise generalized scalar atxand give simple sufficient conditions we also discuss the relationship between pointwise functional calculi and the many physical problems that may be modelled as an abstract cauchy problem

unbounded holomorphic functional calculus and abstract cauchy problems for operators with polynomially bounded resolvents abstract we consider linear unbounded operators b on a banach space x whose spectrum is contained in an open set v such that the resolvent w b 1 exists and is o 1 w for w v for some 1 we will call this an operator of type v for such operators we construct a functional calculus b defined for functions holomorphic on v whose real part is bounded above such that b generates an exponentially bounded b m semigroup where m 2 v this map is a continuous algebra homomorphism in a sense that is made precise later we give best possible spectral mapping theorems applications of this construction include the following we characterize subsets v of the complex plane with the property that whenever a is densely defined and of type v then the abstract cauchy problem has a solution for all initial data in a dense set we show that when b is of a type v then there exists a ranach subspace y such that d b m y x and there exists an h v functional calculus for b y on this interpolating space the imaginary powers of b form a c 0 group and we have spectral decompositions of b when v is disconnected our construction produces solutions for all initial data in dense sets of wide classes of first and second order cauchy problems including the schrodinger equation the backwards heat equation and the laplace equation on l p r n or l p i p where is a region in r n with smooth boundary we also consider incomplete cauchy problems of arbitrarily high order we emphasize that in our construction of b we do not assume that is meromorphic at this allows us to define exponentials cosines and logarithms of b directly for large classes of operators b for example we may define the exponential e 5 of an operator b with real spectrum

nonlinear structure of some classical quasi banach spaces and f spaces we investigate the lipschitz structure of lp and lp for 0 p 1 as quasi banach spaces and as metric spaces with the metric induced by the p norm and show that they are not lipschitz isomorphic we prove that the f space l0 is not uniformly homeomorphic to any other lp space that the lp spaces for 0 p 1 embed isometrically into one another and reduce the problem of the uniform equivalence amongst lp spaces to their lipschitz equivalence as metric spaces

on the second order cauchy problem associated with a linear operator abstract let a be a linear operator in a banach space x with the resolvent defined for all 0 we introduce a subspace of initial values x in x for which both the first and the second order abstract cauchy problem for a have a solution our results complete those of s kantorovitz 5 on the hille yosida subspace of a

approximation properties and schauder decompositions in lipschitz free spaces abstract we prove that the lipschitz free space over a doubling metric space has the bounded approximation property we also show that the lipschitz free spaces over l 1 n or l 1 have monotone finite dimensional schauder decompositions

h functional calculus and models of nagy foias type for sectorial operators we prove that a sectorial operator admits an h functional calculus if and only if it has a functional model of nagy foias type furthermore we give a concrete formula for the characteristic function in a generalized sense of such an operator more generally this approach applies to any sectorial operator by passing to a different norm the mcintosh square function norm we also show that this quadratic norm is close to the original one in the sense that there is only a logarithmic gap between them

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

spectral representations for unbounded operators with real spectrum soit t un operateur agissant dans un espace de banach reflexif x avec domaine d t et spectre reel alors il existe une mesure spectrale e sur z la variete de semisimplicite pour t telle que 1 pour chaque b r e commute avec chaque u b x qui commute avec t 2 d t z x z r se ds x existe et appartient a z et 3 tx r se ds x pour tout x d t z

quasi distribution semigroups and integrated semigroups abstract suppose x is a banach space we introduce the concept of quasi distribution semigroups on x as a generalization of the concept of distribution semigroups introduced by lions in 19 in our approach the generator a of a quasi distribution semigroup may not be densely defined also introduced is a functional calculus for a in terms of the fourier transform for fixed r 0 k n 0 define order r k for a quasi distribution semigroup and define an f r k functional calculus we prove that a generates a k 1 times integrated semigroup of exponential type r with local lipschitz continuity if and only if a generates a quasi distribution semigroup of order r k if and only if a has an f r k functional calculus when a is densely defined a quasi distribution semigroup reduces to a distribution semigroup in the sense of lions

on schauder bases in lipschitz free spaces abstract the main result implies that the lipschitz free spaces f l 1 and f r n have a schauder basis this improves in a special case on the previous work of godefroy and kalton who showed that f x has a bounded approximation property if and only if the banach space x does

conservation properties of a time fe method part iii mechanical systems with holonomic constraints a galerkin based discretization method for index 3 differential algebraic equations pertaining to finite dimensional mechanical systems with holonomic constraints is proposed in particular the mixed galerkin mg methodis introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained hamiltonian system namely total energy and angular momentum in addition to that the constraints on the configuration level and on the velocity momentum level are fulfilled exactly the application of the mg method to specific mechanical systems such as the pendulum rigid body dynamics and the coupled motion of rigid and flexible bodies is presented related numerical examples are investigated to evaluate the numerical performance of the mg 1 and mg 2 method

conservative spline methods for second order ivps of odes abstract one of the most fundamental characteristics in physics are the conservation laws 1 therefore the numerical methods in computational physics should be conservative unfortunately the most popular numerical methods like runge kutta multistep and taylor expansion methods when they are applied to the second order nonlinear newtonian initial value problems do not preserve fundamental physical invariants like energy and momentum when these are present in 2 4 a family of spline methods proposed for the initial value problems of order n a variant of these methods are energy conserving approximations of order three four and five for the second order newtonian initial value problems furthermore it seems to be easy to extend for equations containing the first derivative of the unknown function even for coupled equations some applications and numerical results are presented 1999 elsevier science ltd all rights reserved

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

energy preserving schemes for nonlinear hamiltonian systems of wave equations application to the vibrating piano string the problem of the vibration of a string is well known in its linear form describing the transversal motion of a string nevertheless this description does not explain all the observations well enough nonlinear coupling between longitudinal and transversal modes seams to better model the piano string as does for instance the geometrically exact model gem this report introduces a general class of nonlinear systems nonlinear hamiltonian systems of wave equations in which fits the gem mathematical study of these systems is lead in a first part showing central properties energy preservation existence and unicity of a global smooth solution finite propagation velocity ldots space discretization is made in a classical way variational formulation and time discretization aims at numerical stability using an energy technique a definition of preserving schemes is introduced and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is linear a general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear hamiltonian systems of wave equations class

embeddability of lp and bases in lipschitz free p spaces for 0 p 1 abstract our goal in this paper is to continue the study initiated by the authors in 4 of the geometry of the lipschitz free p spaces over quasimetric spaces for 0 p 1 denoted f p m here we develop new techniques to show that by analogy with the case p 1 the space l p embeds isomorphically in f p m for 0 p 1 going further we see that despite the fact that unlike the case p 1 this embedding need not be complemented in general complementability of l p in a lipschitz free p space can still be attained by imposing certain natural restrictions to m as a by product of our discussion on bases in f p 0 1 we obtain examples of p banach spaces for p 1 that are not based on a trivial modification of banach spaces which possess a basis but fail to have an unconditional basis