krein milman type problems for compact matricially convex sets abstract matricially convex sets are convex sets of matrices in which matrix valued convex coefficients are admitted along with the usual scalar valued convex coefficients accordingly extremal elements of these sets are defined with respect to the larger class of coefficients as in scalar valued convexity theory a krein milman theorem is highly desirable in this paper krein milman problems are formulated for compact matricially convex sets and are solved in the special case where the matricially convex sets are generated by normals

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

convergence analysis of moving finite element methods for space fractional differential equations most existent papers have focused on the fixed mesh methods for solving space fractional differential equations however since some classes of space fractional differential equations may have singular or even finite time blowup solutions it is highly needed to develop adaptive mesh methods to solve these problems in this paper the moving finite element methods are studied for a class of time dependent space fractional differential equations the convergence theories of the methods are derived with l 2 norm and numerical examples are provided to support the theoretical results to simplify the analysis a fractional ritz projection operator is introduced and the error estimation of the projection is derived under the moving mesh setting

analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations this work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations the minimization of the tracking functional subject to implicit parabolic equations is examined existence of an optimal solution is proved and an optimality system of equations is derived semi discrete in space error estimates for the finite element approximations of the optimality system are presented these estimates are symmetric and applicable for higher order discretizations finally fully discrete error estimates of arbitrarily high order are presented based on a discontinuous galerkin in time and conforming in space scheme two examples related to the lagrangian moving mesh galerkin formulation for the convection diffusion equation are described

conformal actions of nilpotent groups on pseudo riemannian manifolds we study conformal actions of connected nilpotent lie groups on compact pseudo riemannian manifolds we prove that if a type p q compact manifold m supports a conformal action of a connected nilpotent group h then the degree of nilpotence of h is at most 2p 1 assuming p q further if this maximal degree is attained then m is conformally equivalent to the universal type p q compact conformally flat space up to finite or cyclic covers the proofs make use of the canonical cartan geometry associated to a pseudo riemannian conformal structure

the geometric structure of generalized state spaces the facial structure of the generalized state space of a c algebra a is investigated the norm closed and w closed faces are characterized in terms of projections in a mn matricially convex faces are introduced and shown to correspond to the faces of the state space of a

an l stability analysis for the finite difference solution of one dimensional linear convection diffusion equations on moving meshes the stability of three moving mesh finite difference schemes is studied in the l norm for one dimensional linear convection diffusion equations these schemes use central finite differences for spatial discretization and the q method for temporal discretization and they are based on conservative and non conservative forms of transformed partial differential equations the stability conditions obtained consist of the cfl condition and the mesh speed related conditions the cfl condition is independent of the mesh speed and has the same form as that for fixed meshes the mesh speed related conditions restrict how fast the mesh can move the conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine illustrative numerical results are presented

the identity component of the isometry group of a compact lorentz manifold this result may be compared with a theorem of e ghys ghy see also bel asserting a similar conclusion but assuming that m has dimension 3 and that the action is just volume preserving and locally free the statement there is that the action of ag may be extended to an action of a finite cover of psl 2 r or to an action of the solvable 3 dimensional lie group sol here we have another motivation we want to understand the structure of lie groups acting isometrically on compact lorentz manifold the first results in the subject are due to zim and gro a final result is due to a s and zeg1 independently necessary and sufficient conditions were given in order that a lie group acts isometrically and locally faithfully on a compact lorentz manifold note however that if a group acts in such a fashion then its subgroups also act in the same way for instance all known examples of isometric actions of ag are obtained by viewing it as a subgroup of sl 2 r so a natural question is what are the maximal connected lie groups acting isometrically on a compact lorentz manifold equivalently

some remarks on c convexity abstract recently the study of completely positive maps has become important to the results of brown douglas and fillmore on ext a c algebra attempts to solve questions related to ext have often turned into questions about the matrix algebras mn in this paper we wish to discuss a notion of c convexity related to completely positive linear maps to state some facts about c convexity and to ask some questions about c convexity to a large degree the tone of this paper is expository

isotropy of semisimple group actions on manifolds with geometric structure we prove results concerning isotropy of noncompact semisimple lie group actions that preserve pseudo riemannian or affine structures on compact or finite volume manifolds for example if a lie group g acts locally faithfully on a connected compact complete affine manifold m and preserves the affine structure then the connected component of the identity of each isotropy subgroup is a compact extension of a solvable group this is a consequence of the result that nontrivial affine actions of connected groups locally isomorphic to sl 2 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i on such manifolds cannot have fixed points we also prove that if g is connected semisimple without compact factors and acts isometrically and topologically transitively on a connected finite volume pseudo riemannian m then the action must be everywhere locally free this answers affirmatively a special case of a conjecture by g d ambra and m gromov with only the assumption that no factor of g acts trivially local freeness is also assured when m is compact and complete

on the convex structure of process positive operator valued measures measurements on quantum channels are described by so called process positive operator valued measures or process povms we study implementing schemes of extremal process povms as it turns out the corresponding measurement must satisfy certain extremality property which is stronger than the usual extremality given by the convex structure this property motivates the introduction and investigation of the a convex structure of povms which generalizes both the usual convex and c convex structures we show that extremal points and faces of the set of process povms are closely related to a extremal points and a faces of povms for a certain subalgebra a we also give a characterization of a extremal and a pure povms

on the completely positive and positive semidefinite preserving cones abstract the cone cp n q of completely positive linear transformations from m n c m n to m q is shown to be isometrically isomorphic to p nq the cone of nq by nq positive semidefinite matrices generalizations of scalar and matrix results to cp n q hp n q l m n m q where hp n q represents the hermitian preserving linear transformations are discussed relationships among the completely positives the set of positive semidefinite preservers p n and its dual m n are given left and right facial ideals of cp are characterized properties of the joint angular field of values of a finite sequence of hermitian matrices h 1 h m are studied leading to a characterization of p q p n

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

on lorentz dynamics from group actions to warped products via homogeneous spaces we show a geometric rigidity of isometric actions of non compact semisimple lie groups on lorentz manifolds namely we show that the manifold has a warped product structure of a lorentz manifold with constant curvature by a riemannian manifold

the essentially commutative dilations of dynamical semigroups on m n for identity and trace preserving one parameter semigroups tt t 0 on then n matricesmn we obtain a complete description of their essentially commutative dilations i e dilations which can be constructed on a tensor product ofmn by a commutativew algebra

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

on the extreme points of quantum channels abstract let l m n denote the convex set of completely positive trace preserving operators from c m m to c n n i e quantum channels we give a necessary condition for l l m n to be an extreme point we show that generically this condition is also sufficient we characterize completely the extreme points of l 2 2 and l 3 2 i e quantum channels from qubits to qubits and from qutrits to qubits

noncompact simple automorphism groups of lorentz manifolds and other geometric manifolds when such a g does act zimmer has proven that the full automorphism group of the lorentz manifold must be locally isomorphic to sl 2 r x k where k is a compact lie group m gromov has shown 2 that the universal cover m must then be lorentz isometric to a warped product sl 2 r f x s where sl 2 ir is the universal cover of sl 2 ii with the cartan killing metric s is a riemannian manifold and f s 0 o oo is a warping function this paper is mostly concerned with the following situation a noncompact simple r algebraic group g acting on a manifold m preserving a lorentz metric we do not assume m compact or finite volume or even complete in contrast the theorems above require compactness and use it in an essential way in their proofs but as we show strong analogues of these theorems hold even for m noncompact our first result is the following

analysis of some moving space time finite element methods two space time finite element methods for solving time dependent partial differential equations are defined and analyzed the methods are based on the use of isoparametric finite elements to implicitly define the time discretization on a moving mesh in the space dimensions one method allows for adding and deleting knots in a continuous fashion while the other allows for discontinuous changes in the mesh static rezone a detailed convergence analysis for a model parabolic equation with a possibly large convection term is presented here the authors obtain symmetric best approximation error estimates similar to those obtained by dupont math comp 39 1982 pp 85 107 for the semidiscrete case

pure matrix states on operator systems abstract an operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital c algebra a matrix state on an operator system v is a unital completely positive linear map of v into a full matrix algebra pure matrix states are studied and a new and somewhat simplified proof of a krein milman type theorem of webster and winkler is given if v is 3 dimensional then the matrix state space of v is matrix affinely homeomorphic to the matricial range of some hilbert space operator with the aid of this representation pure matrix states on 3 dimensional operator systems are examined and in some cases completely determined

faces of sets of operators with the numerical range in a prescribed polyhedron abstract sets of operators with numerical range contained in a given closed subset k of the complex plane are considered a characterization of faces and in particular extreme points of these sets of operators is obtained when the underlying hilbert space is finite dimensional and k is a closed polyhedron

a symmetric error estimate for galerkin approximations of time dependent navier stokes equations in two dimensions a symmetric error estimate for galerkin approximations of solutions of the navier stokes equations in two space dimensions plus time is given the finite dimensional function spaces are taken to be divergence free and time is left continuous the estimate is similar to known results for scalar parabolic equations an application of the result is given for mixed method formulations a short discussion of examples is included finally there are some remarks about a partial extension to three space dimensions

two party locc convertibility of quadpartite states and kraus cirac number of two qubit unitaries non local properties globalness of a non separable unitary determine how the unitary affects the entanglement properties of a quantum state we apply a given two qubit unitary in a quadpartite system including two reference systems and analyze its local operations and classical communication locc partial invertibility under two party locc a decomposition given by kraus and cirac for two qubit unitaries shows that globalness is completely characterized by three parameters our analysis shows that the number of non zero parameters the kraus cirac number has operational significance when converting entanglement properties of multipartite states all two qubit unitaries have a kraus cirac number of at most 3 while those with at most 1 or 2 are equivalent up to local unitaries to a controlled unitary or matchgate respectively the presented operational framework distinguishes the unitaries with a kraus cirac number of 2 and 3 which was not possible by the known measure of the operator schmidt decomposition we also analyze how the kraus cirac number changes when two or more two qubit unitaries are applied sequentially this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

c convexity of norm unit balls abstract we characterize the norms on b h whose unit balls are c convex we call such norms m norms and investigate their dual norms named l norms we show that the class of l norms greater than a given norm enjoys a minimum element and the class of m norms less than a given norm enjoys a maximum element these minimum and maximum elements will be determined in some cases finally we give a constructive result to obtain m norms and l norms on b h

isometry groups of lorentz manifolds soit v une variete de lorentz a n dimensions si v g est compacte analytique reelle simplement convexe alors le groupe d isometrie is v g est compact

on birkhoff s theorem for doubly stochastic completely positive maps of matrix algebras abstract a study is made of the extreme points of the convex set of doubly stochastic completely positive maps of the matrix algebra m n if n 2 the extreme points are precisely the unitary maps but if n 3 there are nonunitary extreme points examples of which are exhibited a tilde operation is defined on the linear maps of m n and used to give an elementary derivation of a result of kummerer and maassen

symmetric error estimates for moving mesh mixed methods for advection diffusion equations a mixed method allowing a general class of mesh movements is proposed for an advection diffusion equation in either conservative or nonconservative form several symmetric error estimates are derived for the method under certain conditions in one space dimension optimal order l2 convergence and superconvergence are proved as corollaries of the symmetric estimates

compact lorentz manifolds with local symmetry we prove a structure theorem for compact aspherical lorentz manifolds with abundant local symmetry if m is a compact aspherical real analytic complete lorentz manifold such that the isometry group of the universal cover has semisimple identity component then the local isometry orbits in m are roughly fibers of a fiber bundle a corollary is that if m has an open dense locally homogeneous subset then m is locally homogeneous acknowledgements i thank my dissertation advisor benson farb for his guidance and encouragement i also thank abdelghani zeghib with whom i had helpful conversations while working on this project in loving memory of laura sue jung kang

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

on the structure of positive maps ii low dimensional matrix algebras we use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on m2 c and m3 c our main tool stems from classical grothendieck theorem on tensor product of banach spaces and is an older and more general version of choi jamiolkowski isomorphism between positive maps and block positive choi matrices it takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps in this setting we show that in m2 c case a large class of nice positive maps can be generated from the small set of maps represented by self adjoint unitaries 2px with x maximally entangled vector and p 1 with p rank 1 projector we indicate problems with passing this result to m3 c case among similarities in both m2 c and m3 c cases any unital positive map represented by self adjoint unitary is unitarily equivalent to the transposition map consequently

ein operatorwertiger hahn banach satz abstract we generalize arveson s extension theorem for completely positive mappings 1 to a hahn banach principle for matricial sublinear functionals with values in an injective c algebra or an ideal in b h we characterize injective w algebras by a matricial order condition we illustrate the matricial hahn banach principle by three applications 1 let a b b be unital c algebras b a subalgebra of a and b b injective if a b is a completely bounded self adjoint b bihomomorphism then it can be expressed as the difference of two completely positive b bihomomorphism 2 let m be a w algebra containing 1 h on a hilbert space h if m is finite and hyperfinite there exists an invariant expectation mapping p of b h onto m p is an extension of the center trace 3 combes 7 proved that a lower semicontinuous scalar weight on a c algebra is the upper envelope of bounded positive functionals we generalize this result to unbounded completely positive mappings with values in an injective w algebra

error estimates for the discontinuous galerkin methods for parabolic equations the classical discontinuous galerkin method for a general parabolic equation is analyzed symmetric error estimates for schemes of arbitrary order are presented the ideas developed below relax many assumptions required in previous work for example different discrete spaces may be used at each time step and the spatial operator need not be self adjoint or independent of time our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular