completely bounded modules and associated extremal problems in this paper we continue our study of certain finite dimensional hilbert modules over the function algebra a cm we show that these modules are always completely bounded with the bound obtained as the matrix valued analogue of a certain scalar valued extremal problem in particular we obtain a necessary and sufficient condition for our module to be completely contractive we produce a contractive module cnm over a bm such that it is completely bounded with the complete bound equal to m that is cnm is not completely contractive

long time dynamics for semi relativistic nls and half wave in arbitrary dimension we consider the cauchy problems associated with semi relativistic nls snls and half wave hw in particular we focus on the following two main questions local global cauchy theory existence and stability instability of ground states in between other results we prove the existence and stability of ground states for snls in the l 2 supercritical regime this is in sharp contrast with the instability of ground states for the corresponding hw which is also established along the paper by showing an inflation of norms phenomenon concerning the cauchy theory we show under radial symmetry assumption the following results a local existence result in h 1 for energy subcritical nonlinearity and a global existence result in the l 2 subcritical regime

high spatial order finite element method to solve maxwell s equations in time domain this paper presents a finite element method with high spatial order for solving the maxwell equations in the time domain in the first part we provide the mathematical background of the method then we discuss the advantages of the new scheme compared to a classical finite difference time domain fdtd method several examples show the advantages of using the new method for different kinds of problems comparisons in terms of accuracy and cpu time between this method the fdtd and the finite volume time domain methods are given as well

time domain finite difference and finite element methods for maxwell equations in complex media extensions of finite difference time domain fdtd and finite element time domain fetd algorithms are reviewed for solving transient maxwell equations in complex media also provided are a few representative examples to illustrate the modeling capabilities of fdtd and fetd for complex media the term complex media refers here to media with dispersive bi anisotropic inhomogeneous and or nonlinear properties present in the constitutive tensors

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

spurious solutions in mixed finite element method for maxwell s equations dispersion analysis and new basis functions the finite element method is a well known computational technique used to obtain numerical solutions to boundary value problems including maxwell s equations this paper first presents a brief description of the mathematical structure based on the de rham diagram to discretize maxwell s equations then it uses a numerical dispersion analysis of the mixed finite element method with both electric and magnetic fields as unknowns to evaluate the presence of spurious solutions for different basis functions these unwanted spurious solutions appear when the same order of element is used for electric and magnetic fields while the system is free of spurious modes when different orders of elements are employed for electric and magnetic fields in this work finite elements in both frequency and time domain are studied and the effects of these spurious solutions in both domains are analyzed in one and three dimensional cases

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

the structure of solutions to the pseudo conformally invariant nonlinear schrodinger equation we study solutions in n of the nonlinear schrodinger equation iu t u u u where is the fixed power 4 n for this particular power these solutions satisfy the pseudo conformal conservation law and the set of solutions is invariant under a related transformation this transformation gives a correspondence between global and non global solutions if

relative yoneda cohomology for operator spaces abstract we provide two alternate presentations of the completely bounded hochschild cohomology one as a relative yoneda cohomology i e as equivalence classes of n resolutions which are relatively split and the second as a derived functor the first presentation makes clear the importance of certain relative notions of injectivity projectivity and amenability which we introduce and study we prove that every von neumann algebra is relatively injective as a bimodule over itself and consequently h n cb m m 0 for any von neumann algebra m a result obtained earlier by christensen and sinclair we prove that the relatively amenable c algebras are precisely the nuclear c algebras and hence exactly those which are amenable as banach algebras in a similar vein we prove that the only relatively projective c algebras are finite dimensional this result implies that the only c algebras that are projective as banach algebras are finite dimensional a result first obtained by selivanov and helemskii in a slightly different direction we prove that b h viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module this result is in some sense a generalization of the sz nagy foias commutant lifting theorem or of the hypoprojectivity introduced by douglas and the author

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

numerical evaluation of high order finite element time domain formulations in electromagnetics this paper compares three full wave finite element time domain fetd formulations the first is based on the vector wave equation the others on maxwell s equations viz the ebhd formulation that discretizes e rarr b rarr h rarr and drarr and the eb formulation that discretizes only e rarr and b rarr the latter two formulations use a combination of 1 and 2 form discretization to avoid an auxiliary mesh a novel method for making the ebhd formulation operational is presented conditions for finite difference time domain fdtd like explicit operation are discussed the formulations are compared numerically by solving a three dimensional cavity and a rectangular waveguide using high order field representations up to mixed fourth order the error balance between time integration and field representation is investigated difficulties in making the ebhd formulation operational which have not previously been addressed in the literature are discussed and worked around novel numerical results show that the ebhd formulation has serious performance limitations

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

operator space structure and amenability for figa talamanca herz algebras column and row operator spaces which we denote by col and row respectively over arbitrary banach spaces were introduced by the first named author for hilbert spaces these definitions coincide with the usual ones given a locally compact group g and p p 1 with 1p 1p 1 we use the operator space structure on cb col lp g to equip the figa talamanca herz algebra ap g with an operator space structure turning it into a quantized banach algebra moreover we show that for p q 2 or 2 q p and amenable g the canonical inclusion aq g ap g is completely bounded with cb norm at most kg2 where kg is grothendieck s constant as an application we show that g is amenable if and only if ap g is operator amenable for all and equivalently for one p 1 this extends a theorem by ruan

a fast higher order time domain finite element boundary integral method for 3 d electromagnetic scattering analysis a novel hybrid time domain finite element boundary integral method for analyzing three dimensional 3 d electromagnetic scattering phenomena is presented the method couples finite element and boundary integral field representations in a way that results in a sparse system matrix and solutions that are devoid of spurious modes to accurately represent the unknown fields the scheme employs higher order vector basis functions defined on curvilinear tetrahedral elements to handle problems involving electrically large objects the multilevel plane wave time domain algorithm is used to accelerate the evaluation of the boundary integrals numerical results demonstrate the accuracy and versatility of the proposed scheme

uniformly accurate finite difference schemes for p refinement the accuracy deficiency of high order conventional finite difference schemes in the high frequency range is attributed to the approximation property of a power polynomial interpolation over equispaced nodes to improve the numerical representation of high frequency solutions a new family of high order finite difference schemes is constructed by trigonometric polynomial interpolation as expected these new difference schemes can resolve a wider frequency range than conventional finite difference schemes with the same stencil in particular they remove the possibility of runge s phenomenon for uniform mesh the recently proposed optimized difference schemes are indeed types of mixed difference schemes which make a compromise between power and trigonometric polynomial interpolation

a new 3 d nonspurious discontinuous galerkin spectral element time domain dg setd method for maxwell s equations a new discontinuous galerkin spectral element time domain dg setd method for maxwell s equations based on the field variables mathbf e and mathbf b is proposed to analyze three dimensional 3 d transient electromagnetic phenomena compared to the previous setd method based on the field variables mathbf e and mathbf h the mathbf eh scheme in which different orders of interpolation polynomials for electric and magnetic field intensities are required the newly proposed method can eliminate spurious modes using basis functions with the same order interpolation for electric field intensity and magnetic flux density the mathbf eb scheme consequently it can reduce the number of unknowns and computation load domain decomposition for the mathbf eb scheme setd method is completed via the dg method in addition the mathbf eb scheme setd method is extended to the well posed time domain perfectly matched layer pml to truncate the computation domain when solving open region problems the effectiveness and advantages of the new dg setd method are validated by eigenvalue analysis and numerical results

discrete hodge operators many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms discrete differential forms provide a natural and canonical approach to their discretization however much freedom remains concerning the choice of discrete hodge operators that is discrete analogues of constitutive laws a generic discrete hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations we reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus

modeling backward wave propagation in metamaterials by the finite element time domain method in this paper we develop a leap frog type finite element method for modeling the electromagnetic wave propagation in metamaterials the metamaterial model equations are represented by integrodifferential maxwell s equations which are quite challenging to analyze and solve in that we have to solve a coupled problem with different partial differential equations given in different material regions our method is based on a mixed finite element method using edge elements which can easily handle the tangential continuity of the electric field stability analysis and optimal error estimate are carried out for the proposed scheme the scheme is implemented and confirmed to obey the proved optimal convergence rate by using a smooth analytical solution then the scheme is extended to model wave propagation in heterogeneous media composed of metamaterials and free space and extensive numerical results using a rectangular edge element a triangular edge element and mixed edge elements demonstrate the effective

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

bounded modules extremal problems and a curvature inequality abstract in this paper we study certain finite dimensional hilbert modules over the function algebra a c n these modules appear as localizations of a cowen douglas operator we show that these modules are always bounded where the bound is related to the solution of an extremal problem in particular we obtain necessary sufficient conditions for such a module to be contractive we apply the above results to produce an example of a contractive module over a b 2 which is not completely contractive

lattice electromagnetic theory from a topological viewpoint the language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice it is shown that differential forms and their discrete counterparts cochains provide a natural bridge between the continuum and the lattice versions of the theory allowing for a natural factorization of the field equations into topological field equations i e invariant under homeomorphisms and metric field equations the various potential sources of inconsistency in the discretization process are identified distinguished and discussed a rationale for a consistent extension of the lattice theory to more general situations such as to irregular lattices is considered

time domain finite element methods various time domain finite element methods for the simulation of transient electromagnetic wave phenomena are discussed detailed descriptions of test trial spaces explicit and implicit formulations nodal and edge facet element basis functions are given along with the numerical stability properties of the different methods the advantages and disadvantages of mass lumping are examined finally the various formulations are compared on the basis of their numerical dispersion performance

eb scheme based hybrid se fe dgtd method for multiscale em simulations this communication presents an eb scheme subdomainlevel discontinuous galerkin time domain dgtd method for multiscale simulations it is an extension of the previous subdomain level dgtd research by combining the degree of freedom efficiency of spectral element time domain method and the mesh flexibility of the finite element time domain method thus the multiscale problems can be solved efficiently by separating the geometrically fine and coarse parts and meshing them with hexahedrons and tetrahedrons respectively via a nonconformal mesh the implicit explicit runge kutta method is applied to the eb scheme dgtd method to obtain an efficient time integration approach

operator space approach to steering inequality in junge and palazuelos 2011 commun math phys 306 695 746 and junge et al 2010 commun math phys 300 715 39 the operator space theory was applied to study bipartite bell inequalities the aim of the paper is to follow this line of research and use the operator space technique to analyze the steering scenario we obtain a bipartite steering functional with unbounded largest violation of steering inequality as well as constructing all ingredients explicitly it turns out that the unbounded largest violation is obtained by a non maximally entangled state moreover we focus on the bipartite dichotomic case where we construct a steering functional with unbounded largest violation of steering inequality this phenomenon is different to the bell scenario where only the bounded largest violation can be obtained by any bipartite dichotomic bell functional

representations of function algebras abstract operator spaces and banach space geometry let g be the closed unit ball of some norm on cn and let a g be the closure of the polynomials in the sup norm we prove that if n 5 then there is a contractive representation of a g as operators on a hilbert space which is not completely contractive our technique involves introducing a numerical invariant x for a normed space x which measures the difference between the minimal operator space structure which can be assigned to x min x and the maximal structure max x we estimate x using banach space techniques we also prove that if x is any infinite dimensional subspace of the space of continuous functions on a compact hausdorff space then there exists a bounded linear map on x which is not completely bounded

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

minimal and maximal matrix convex sets abstract for every convex body k r d there is a minimal matrix convex set w min k and a maximal matrix convex set w max k which have k as their ground level we aim to find the optimal constant k such that w max k k w min k for example if b p d is the unit ball in r d with the l p norm then we find that b p d d 1 1 p 1 2 this constant is sharp and it is new for all p 2 moreover for some sets k we find a minimal set l for which w max k w min l in particular we obtain that a convex body k satisfies w max k w min k only if k is a simplex these problems relate to dilation theory convex geometry operator systems and completely positive maps for example our results show that every d tuple of self adjoint contractions can be dilated to a commuting family of self adjoints each of norm at most d we also introduce new explicit constructions of these and other dilations

extension properties for the space of compact operators let z be a fixed separable operator space x y general separable operator spaces and t x z a completely bounded map z is said to have the complete separable extension property csep if every such map admits a completely bounded extension to y and the mixed separable extension property msep if every such t admits a bounded extension to y finally z is said to have the complete separable complementation property cscp if z is locally reflexive and t admits a completely bounded extension to y provided y is locally reflexive and t is a complete surjective isomorphism let k denote the space of compact operators on separable hilbert space and k0 the c0 sum of mn s the space of small compact operators it is proved that k has the cscp using the second author s previous result that k0 has this property a new proof is given for the result due to e kirchberg that k0 and hence k fails the csep it remains an open question if k has the msep it is proved this is equivalent to whether k0 has this property a new banach space concept extendable local reflexivity elr is introduced to study this problem further complements and open problems are discussed

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

the similarity degree of an operator algebra ii for every integer d ge 1 there is a unital closed subalgebra a d subset b h with similarity degree equal precisely to d in the sense of our previous paper this means that for any unital homomorphism u colon a d to b h we have u cb le k u d with k 0 independent of u and the exponent d in this estimate cannot be improved the proof that the degree is larger than d 1 crucially uses an upper bound for the norms of certain gaussian random matrices due to haagerup and thorbjornsen we also include several complements to our previous publications on the same subject

limit of the solution of a nonlinear schrodinger equation at blow up time we consider the solution of the equation i v t and 0 where v is defined on 0 t rn and is a nonlinear complex valued function let us denote the function x v t x by v t we suppose that v blows up in the norm h1 at time t under suitable assumptions on and we prove the existence of a limit for v t when t t in appropriate spaces furthermore we specify the behavior of v near blow up time

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

operator spaces with few completely bounded maps we construct several examples of hilbertian operator spaces with few completely bounded maps in particular we give an example of a separable 1 hilbertian operator space x 0 such that whenever x is an infinite dimensional quotient of x 0 x is a subspace of x and t x rightarrow x is a completely bounded map then t i x s where s is compact hilbert schmidt and s 2 16 s cb s 2 moreover every infinite dimensional quotient of a subspace of x 0 fails the operator approximation property we also show that every banach space can be equipped with an operator space structure without the operator approximation property

explicit lower and upper bounds on the entangled value of multiplayer xor games the study of quantum mechanical violations of bell inequalities is motivated by the investigation and the eventual demonstration of the nonlocal properties of entanglement in recent years bell inequalities have found a fruitful re formulation using the language of multiplayer games originating from computer science this paper studies the nonlocal properties of entanglement in the context of the simplest such games called xor games when there are two players it is well known that the maximum bias the advantage over random play of players using entanglement can be at most a constant times greater than that of classical players recently perez garcia et al commun mathe phys 279 455 2008 showed that no such bound holds when there are three or more players the use of entanglement can provide an unbounded advantage and scale with the number of questions in the game their proof relies on non trivial results from operator space theory and gives a non explicit existence proof leading to a game with a very large number of questions and only a loose control over the local dimension of the players shared entanglement

computation of electromagnetic scattering with a non conforming discontinuous spectral element method in this paper we solve electromagnetic scattering problems by approximating maxwell s equations in the time domain with a high order quadrilateral discontinuous spectral element method dsem the method is a collocation form of the discontinuous galerkin method for hyperbolic systems where the solution is approximated by a tensor product legendre expansion and inner products are replaced with gauss legendre quadratures to increase flexibility of the method we use a mortar element method to couple element faces mortars provide a means for coupling element faces along which the polynomial orders differ which allows the flexibility to choose the approximation order within an element by considering only local resolution requirements mortars also permit local subdivision of a mesh by connecting element faces that do not share a full side we present evidence showing that the convergence of the non conforming approximations is spectral along with examples of their use copyright 2001 john wiley sons ltd

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces

a 3 d discontinuous spectral element time domain method for maxwell s equations a discontinuous spectral element time domain method is proposed to analyze transient electromagnetic properties of general 3 d structures this method is advantageous in that its mass matrices are block diagonal due to the gauss lobatto legendre polynomials and it allows different orders of basis functions for each subdomain the riemann solver is employed in the boundary integral terms to communicate fields between adjacent subdomains perfectly matched layers are utilized to truncate the computational domain galerkin method is used for spatial discretization and a fourth order runge kutta scheme is employed for the time integration the validity of the proposed approach is demonstrated through several numerical examples of initial value problems and scattering problems

some thoughts on ando s theorem and parrott s example in this short note we present an elementary proof of ando s theorem within a restricted class p of homomorphisms modeled after parrott s example we also show by explicit estimation that the cb norm of the contractive homomorphism of the tri disc algebra induced by the commuting triple of parrott exceeds 1 indeed we construct a polynomial p with matrix coefficients with the property p p in particular we show that there are contractive homomorphisms of the tri disc algebra which are not even 2 contractive

the origin of spurious solutions in computational electromagnetics it is commonly believed that the divergence equations in the maxwell equations are redundant for transient and timeharmonic problems therefore most of the numerical methods in computational electromagnetics solve only two first order curl equations or the second order curl curl equations this misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics by studying the div curl system this paper clarifies that the first order maxwell equations are not overdetermined and the divergence equations must always be included to maintain the ellipticity of the system in the space domain to guarantee the uniqueness of the solution and the accuracy of the numerical methods and to eliminate the infinitely degenerate eigenvalue this paper shows that the common derivation and usage of the second order curl curl equations are incorrect and that the solution of helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary the div curl method and the least squares method introduced in this paper provide rigorous derivation of the equivalent second order maxwell equations and their boundary conditions the node based least squares finite element method lsfem is recommended for solving the first order full maxwell equations directly examples of the numerical solutions by lsfem are given to demonstrate that the lsfem is free of spurious solutions