postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

dissecting the qutrit to visualize a higher dimensional object it is convenient to consider its two dimensional cross sections the set of quantum states for a three level system has eight dimensions we supplement a recent paper by goyal et al arxiv 1111 4427 by considering the set of all possible two dimensional cross sections of the qutrit each such cross section is bounded by a plane cubic curve

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

probability measure generated by the superfidelity we study the probability measure on the set of density matrices induced by the metric defined by using superfidelity we give the formula for the probability density of eigenvalues we also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure

higher rank numerical ranges and compression problems we consider higher rank versions of the standard numerical range for matrices a central motivation for this investigation comes from quantum error correction we develop the basic structure theory for the higher rank numerical ranges and give a complete description in the hermitian case we also consider associated projection compression problems

convergence and superconvergence analysis of a nonconforming finite element method for solving the signorini problem abstract in this paper we present the carey nonconforming finite element approximation of the variational inequality resulting from the signorini problem firstly we show that if the displacement field is of h 2 regularity the optimal convergence rate of o h can be obtained with respect to the energy norm secondly if stronger but reasonable h 5 2 regularity is available the superconvergence rate of o h 3 2 can be derived through the interpolated postprocessing technique finally numerical experiments are given which are consistent with our theoretical analysis

the generalized numerical range of a set of matrices abstract for a given set of n n matrices f we study the union of the c numerical ranges of the matrices in the set f denoted by w c f we obtain basic algebraic and topological properties of w c f and show that there are connections between the geometric properties of w c f and the algebraic properties of c and the matrices in f furthermore we consider the starshapedness and convexity of the set w c f in particular we show that if f is the convex hull of two matrices such that w c a and w c b are convex then the set w c f is star shaped we also investigate the extensions of the results to the joint c numerical range of an m tuple of matrices

numerical range for random matrices abstract we analyze the numerical range of high dimensional random matrices obtaining limit results and corresponding quantitative estimates in the non limit case for a large class of random matrices their numerical range is shown to converge to a disc in particular numerical range of complex ginibre matrix almost surely converges to the disk of radius 2 since the spectrum of non hermitian random matrices from the ginibre ensemble lives asymptotically in a neighborhood of the unit disk it follows that the outer belt of width 2 1 containing no eigenvalues can be seen as a quantification the non normality of the complex ginibre random matrix we also show that the numerical range of upper triangular gaussian matrices converges to the same disk of radius 2 while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to 2 e

discriminant of symmetric matrices as a sum of squares and the orthogonal group it is proved that the discriminant of n n real symmetric matrices can be written as a sum of squares where the number of summands equals the dimension of the space of n variable spherical harmonics of degree n the representation theory of the orthogonal group is applied to express the discriminant of 3 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares it is proved that the discriminant of 4 4 real symmetric matrices can be written as a sum of seven squares these improve results of kummer from 1843 and borchardt from 1846 2010 wiley periodicals inc

flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices abstract let t be an n n matrix the davis wielandt shell of t is defined as the set w t t t x tx x t tx x c r x x 1 in this paper we discuss the existence of flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices

a least squares petrov galerkin finite element method for the stationary navier stokes equations in this paper a galerkin least squares type finite element method is proposed and analyzed for the stationary navier stokes equations the method is consistent and stable for any combination of discrete velocity and pressure spaces without requiring a babuska brezzi stability condition the existence uniqueness and convergence at optimal rate of the discrete solution is proved in the case of sufficient viscosity or small data

real numerical shadow and generalized b splines abstract restricted numerical shadow p a x z of an operator a of order n is a probability distribution supported on the numerical range w x a restricted to a certain subset x of the set of all pure states normalized one dimensional vectors in c n its value at point z c equals the probability that the inner product u a u is equal to z where u stands for a random complex vector from the set x distributed according to the natural measure on this set induced by the unitarily invariant fubini study measure for a hermitian operator a of order n we derive an explicit formula for its shadow restricted to real states p a r x show relation of this density to the dirichlet distribution and demonstrate that it forms a generalization of the b spline furthermore for operators acting on a space with tensor product structure h a h b we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order n 4

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes

classification of joint numerical ranges of three hermitian matrices of size three abstract the joint numerical range w f of three hermitian 3 by 3 matrices f f 1 f 2 f 3 is a convex and compact subset in r 3 generically we find that w f is a three dimensional oval assuming dim w f 3 every one or two dimensional face of w f is a segment or a filled ellipse we prove that only ten configurations of these segments and ellipses are possible we identify a triple f for each class and illustrate w f using random matrices and dual varieties

superconvergent analysis of a nonconforming mixed finite element method for non stationary conduction convection problem abstract in this paper a backward euler mixed finite element mfe approximation scheme is presented for the non stationary conduction convection problem in which the constrained nonconforming rotated cnr q 1 element is used to approximate the velocity u the temperature t and the q 0 element to the pressure p based on the characters of these elements and some special skills i e mean value skill and a new transforming skill with respect to the superclose results of order o h 2 for u t in the broken h 1 norm and p in l 2 norm are deduced respectively here h is the subdivision parameter and the time step furthermore the global superconvergent results are obtained through the interpolation postprocessing technique finally numerical results are provided to confirm the theoretical analysis

a least squares galerkin petrov nonconforming mixed finite element method for the stationary conduction convection problem abstract in this paper a least squares galerkin petrov nonconforming mixed finite element method lsgpnmfm is proposed and analyzed for the stationary conduction convection problem we use p 2 nonconforming as approximation space for the velocity the linear element for the pressure space and the quadratic element for the temperature space the mixed finite element spaces x h and m h need not satisfy inf sup condition the existence uniqueness and convergence of the discrete solution are presented and error estimates of optimal order are derived in the case of sufficient viscosity

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

linear images of joint unitary orbits of hermitian matrices abstract let h n be the set of all n n hermitian matrices and h n m be the set of all m tuples of n n hermitian matrices for a a 1 a m h n m and for any linear map l h n m r l we define the l numerical range and the l numerical radius of a by w l a l u a 1 u u a m u u c n n u u i n and r l a max x x t x w l a respectively they can be regarded as a generalization of the c numerical range and the c numerical radius respectively of n n complex matrices in this paper interplay between the geometric properties of w l a and algebraic properties of linear maps l h n m r l and a h n m will be studied we shall give a characterization of l such that r l defines a norm on h n m furthermore we shall also characterize linear maps l h n m r l and a h n m in which w l a is a singleton set more generally the characterization of linear maps l h n m r l and a h n m in which w l a lies inside a k dimensional affine subspace of r l will be given

induced operators on symmetry classes of tensors let v be an n dimensional hilbert space suppose h is a subgroup of the symmetric group of degree m and x h c is a character of degree 1 on h consider the symmetrizer on the tensor space m v s v 1 v m 1 h v 1 1 x v 1 m defined by h and x the vector space m v m x h s v is a subspace of m v called the symmetry class of tensors over v associated with h and x the elements in v m x h of the form s v 1 v m are called decomposable tensors and are denoted by v 1 v m for any linear operator t acting on v there is a unique induced operator k t acting on v m x h satisfying k t v 1 v m tv 1 tv m in this paper several basic problems on induced operators are studied

numerical shadows measures and densities on the numerical range abstract for any operator m acting on an n dimensional hilbert space h n we introduce its numerical shadow which is a probability measure on the complex plane supported by the numerical range of m the shadow of m at point z is defined as the probability that the inner product mu u is equal to z where u stands for a random complex vector from h n satisfying u 1 in the case of n 2 the numerical shadow of a non normal operator can be interpreted as a shadow of a hollow sphere projected on a plane a similar interpretation is provided also for higher dimensions for a hermitian m its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional b spline in the case of a normal m the numerical shadow corresponds to a shadow of a transparent solid simplex in r n 1 onto the complex plane numerical shadow is found explicitly for jordan matrices j n direct sums of matrices and in all cases where the shadow is rotation invariant results concerning the moments of shadow measures play an important role a general technique to study numerical shadow via the cartesian decomposition is described and a link of the numerical shadow of an operator to its higher rank numerical range is emphasized

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

almost certain convergence in double arrays for double arrays of constants ani 1 i k n n 1 and i i d random variables x x i i 1 conditions are given under which the row sums sum limits i 1 k n a ni x i xrightarrow text a text c text 0 both cases k n and k n are treated in general the hypotheses involve a trade off between moment requirements on x and the magnitude of the itani a marcinkiewicz zygmund type strong law is obtained for the special case a ni a i mathord left vphantom a i left sum limits j 1 n a j p right 1 p right kern nulldelimiterspace left sum limits j 1 n a j p right 1 p a i 0 text sum limits text 1 n a j p to infty text 0 p 2

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

joint numerical range and its generating hypersurface in this paper we study the joint numerical range of m tuples of hermitian matrices via their generating hypersurfaces an example is presented which shows the invalidity of an analogous kippenhahn theorem for the joint numerical range of three hermitian matrices

continuity of the maximum entropy inference convex geometry and numerical ranges approach we study the continuity of an abstract generalization of the maximum entropy inference a maximizer it is defined as a right inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body using convex geometry we prove amongst others the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations further we use numerical range methods in the case of quantum inference which refers to two observables one result is a complete characterization of points of discontinuity for 3 3 matrices

joint numerical ranges quantum maps and joint numerical shadows abstract we associate with a k tuple of hermitian n n matrices a probability measure on r k supported on their joint numerical range the joint numerical shadow of these matrices when k 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices we apply this material to the theory of quantum information thus we show that quantum maps on the set of quantum states defined by kraus operators satisfying the identity resolution assumption shrink joint numerical ranges

a defect correction method for unsteady conduction convection problems ii time discretization in this paper a fully discrete defect correction mixed finite element method mfem for solving the non stationary conduction convection problems in two dimension which is leaded by combining the back euler time discretization with the two step defect correction in space is presented in this method we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect correction technique the stability and the error analysis are derived the theory analysis shows that our method is stable and has a good convergence property some numerical results are also given which show that this method is highly efficient for the unsteady conduction convection problems

unitarily achievable zero patterns and traces of words in a and a abstract our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices which 0 patterns may be achieved for any given n by n complex matrix via some unitary similarity of it to this end certain restrictions on achievable 0 patterns are mentioned both positional and more important on the maximum number of achievable 0 s prior results fitting this general question are mentioned as well as the first unresolved pattern for 3 by 3 matrices in the process a recent question is answered a closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of specht s theorem about which pairs of complex matrices are unitarily similar which seems not widely known to matrix theorists in the process we mention the number of words necessary for small size matrices

error estimates on a new nonlinear galerkin method based on two grid finite elements a new nonlinear galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations the new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time with linear finite element discretizations it is proved that the difference between the new nonlinear galerkin solution and the standard galerkin solution in h 1 omega norm is of the order of h 3

numerical measure of a complex matrix we introduce a natural probability measure over the numerical range of a complex matrix a mn input amssym bbb c this numerical measure a can be defined as the law of the random variable ax x input amssym bbb c when the vector x input amssym bbb c n is uniformly distributed on the unit sphere if the matrix a is normal we show that a has a piecewise polynomial density fa which can be identified with a multivariate b spline in the general nonnormal case we relate the radon transform of a to the spectrum of a family of hermitian matrices and we deduce an explicit representation formula for the numerical density that is appropriate for theoretical and computational purposes as an application we show that the density fa is polynomial in some regions of the complex plane that can be characterized geometrically and we recover some known results about lacunae of symmetric hyperbolic systems in 2 1 dimensions finally we prove under general assumptions that the numerical measure of a matrix a mn input amssym bbb c concentrates to a dirac mass as the size n goes to infinity 2011 wiley periodicals inc

the quantum canonical ensemble the phase space of quantum mechanics can be viewed as the complex projective space cpn endowed with a kahlerian structure given by the fubini study metric and an associated symplectic form we can then interpret the schrodinger equation as generating a hamiltonian dynamics on based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles the resulting density matrix for the canonical ensemble differs from the density matrix of the conventional approach as an illustration the results are applied to the case of a spin one half particle in a heat bath with an applied magnetic field

complementation in the face lattice of a proper cone abstract barker proved that the lattice of faces f k of a finite dimensional proper cone k is always complemented the proof however contained a gap in this paper we offer a correct proof of the result at the same time new characterizations of lattices of finite length which are section complemented or relatively complemented are found it is also proved that the lattice of exposed faces of a proper cone k is complemented finally we show that the class of proper cones with a relatively complemented face lattice is strictly contained in the class of proper cones with a dual atomistic face lattice for a few natural related questions counterexamples are given it turns out that one of our examples also answers in the negative an open problem posed recently by the second author

error analysis of a fully discrete finite element variational multiscale method for the natural convection problem abstract for the natural convection problem we propose a new projection based finite element variational multiscale method by defining the stabilization terms via two local gauss integrations at the element level based on the implicit backward euler and implicit crank nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization we establish two numerical schemes for the natural convection problem unconditional stabilities of the two numerical schemes are proved we derive error bounds of the fully discrete solution which are first and second order in time respectively the optimal error estimates in space could be achieved for velocity and temperature in the h 1 semi norm and for pressure in the l 2 norm with the proper choosing of stabilized parameters however the error estimates in space are suboptimal for velocity and temperature in the l 2 norm the derived theoretical results are supported by two numerical examples

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

positive maps positive polynomials and entanglement witnesses we link the study of positive quantum maps block positive operators and entanglement witnesses with problems related to multivariate polynomials for instance we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares although the general problem of describing the set of positive maps remains open in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity

numerical ranges of tensors abstract the main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms we show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges we make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained also we consider tensor numerical ranges based on inner products however they may not be convex in general

a small eddy correction method for nonlinear dissipative evolutionary equations considering the interaction between the large and small eddy components of solutions and using the idea of the newton iteration a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative pdes of parabolic type in particular the navier stokes equations nse we assume that the large eddy approximation to the solution is known formally applying the newton iterative procedure to the small eddy equation we then generate approximate systems it is shown that the first two steps in fact lead to the standard galerkin method sgm and the so called optimum nonlinear galerkin method ong and therefore the small eddy correction method is actually a certain generalization of sgm and ong the boundedness and convergence analysis are presented in the framework of the two dimensional nse the results show that the small eddy correction method can greatly improve the accuracy of sgm approximate solutions

product numerical range in a space with tensor product structure we study operators acting on a tensor product hilbert space and investigate their product numerical range product numerical radius and separable numerical range concrete bounds for the product numerical range for hermitian operators are derived product numerical range of a non hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum while the latter set is convex the product range needs not to be convex nor simply connected the product numerical range of a tensor product is equal to the minkowski product of numerical ranges of individual factors

3 3 matrices with a flat portion on the boundary of the numerical range the set of unitarily irreducible complex 3 3 matrices whose numerical range has a flat portion is explicitly described it is proved that the set is connected and real analytic properties of the set are studied connectedness of the sets of unitarily irreducible 3 3 matrices having two other possible shapes of their numerical ranges is proved as well

pauli group classification and joint higher rank numerical range abstract in mousavi and salemi 2015 17 authors showed that joint higher rank numerical range of triple of pauli group elements is often convex and characterized it exactly in the case that it is convex in the present paper those results are derived in a different manner also n qubit pauli group using simultaneously unitary similarity are classified which may generally be used to characterize joint higher rank numerical range of any hermitian m tuple in the n qubit pauli group besides some more results about joint numerical range are here derived

the automorphism group of separable states in quantum information theory we show that the linear group of automorphism of hermitian matrices which preserves the set of separable states is generated by natural automorphisms change of an orthonormal basis in each tensor factor partial transpose in each tensor factor and interchanging two tensor factors of the same dimension we apply our results to the preservers of the product numerical range

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

on the dimension of subspaces with bounded schmidt rank we consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states this is motivated in part by results of hayden et al e print arxiv quant ph 0407049 commun math phys 265 95 2006 which show that in large d d dimensional systems there exist random subspaces of dimension almost d2 all of whose states have entropy of entanglement at least logd o 1 it is also a generalization of results on the dimension of completely entangled subspaces which have connections with the construction of unextendible product bases here we take as entanglement measure the schmidt rank and determine for every pair of local dimensions da and db and every r the largest dimension of a subspace consisting only of entangled states of schmidt rank r or larger this exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in hayden et al we also determine the converse the l

spectral theory for one parameter groups of isometries abstract if u t t r is a given unitary one parameter group of operators in hilbert space there are well known l p integrability conditions p 1 on the imaginary part of the resolvent operators which imply absolute continuity of the spectral resolution for u t although one parameter groups of isometrics in banach space do not in general have spectral resolutions they still have an attractive spectral theory we shall display resolvent type conditions which imply spectral continuity for such one parameter groups thus generalizing the hilbert space results the results are applied to the case of one parameter groups of automorphisms of c algebras in the last section

generic local distinguishability and completely entangled subspaces a subspace of a multipartite hilbert space is completely entangled if it contains no product states such subspaces can be large with a known maximum size smax approaching the full dimension of the system d we show that almost all subspaces with dimension s smax are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n d smax this condition holds for almost all sets of states of all multipartite systems and reveals something surprising the criterion is identical for separable and nonseparable states entanglement makes no difference

singular points of cyclic weighted shift matrices abstract let s be an n by n cyclic weighted shift matrix and f s t x y det t i x s y s be a ternary form associated with s we investigate the number of singular points of the curve f s t x y 0 and show that the number of singular points of f s t x y 0 associated with a cyclic weighted shift matrix whose weights are neither 1 periodic nor 2 periodic is less than or equal to n n 3 2 furthermore we verify the upper bound n n 3 2 is sharp for 4 n 7

linear preservers of tensor product of unitary orbits and product numerical range abstract it is shown that the linear group of automorphism of hermitian matrices which preserves the tensor product of unitary orbits is generated by natural automorphisms change of an orthonormal basis in each tensor factor partial transpose in each tensor factor and interchanging two tensor factors of the same dimension the result is then applied to show that automorphisms of the product numerical ranges have the same structure

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

joint ranges of hermitian matrices and simultaneous diagonalization abstract let a a 1 a k be a k tuple of heritian operators on an n diensional inner product space x with unit sphere s u x u u 1 with 1 k as the corresponding quadratic forms i e j x x a j x for all x x j 1 k we study ranges of over various subsets t of x the interplay between geometric properties of t and algebraic properties of a is our main concern in particular we characterize those a for which s is identical with the convex hull of the joint spectrum of a defined by a c k a j u j u for some u s the closely related problem of diagonalizing several hermitian operators simultaneously is also studied

convexity of the joint numerical range topological and differential geometric viewpoints abstract we investigate the convexity of the joint numerical range of m tuples of n n hermitian matrices the methods come from differential geometry and the differential and algebraic topology our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n modulo a mild technical assumption this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the m 1 dimensional unit sphere the condition is that the eigenvalue has constant multiplicity we show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges as a byproduct of our main result we obtain a new proof of the celebrated toeplitz hausdorff theorem and of its extension the numerical range of any triple of hermitian n n matrices is convex if n 2

ergodic properties of one parameter automorphism groups of operator algebras we shall consider the following two types of dynamical systems employed in the foundations of quantum theory c systems and w systems it is widely believed that quantum dynamics is suitably described by one of the two types of systems c for quantum spin systems with short range interaction and w for long range 5 8 17 25 27 301 a c system is a pair a where g is a c algebra always assumed to contain an identity and x is a strongly continuous one parameter group of automorphism the continuity requirement is 11 a a a ii 0 for t 0 a e 0 a w system is a pair m yi where m is a w algebra and yi is a point ultraweakly continuous one parameter group of automorphisms of m the continuity requirement here is 01 y x x 0 for t 0 x e m and u e m are the a m m continuous ultraweakly linear functionals on m the cone of positive elements in m is denoted by mf and consists of the normal positive functionals on m states when normalized both of the stated continuity requirements are well known to be natural for the respective systems from the point of view of both mathematics 4 14 231 and physics l 5 27 291 a mathematical scattering theory for the two types of abstract quantum systems does not exist the closest such seems to be rigorous conditions for return to equilibrium 28 in this paper we look into some of the foundational problems associated to such a non commutative scattering theory related results have earlier been obtained by arveson 3 4 muhly and co workers 20 22 kawamura and tomiyama 1151 and zsido 37 common to these results is the focus on finite systems i e the special case where traces are present in applications 10 12 131 it is known however that the actual systems are type iii and in this paper we therefore turn to the infinite case 354 0022 247x 82 060354 19 02 00 0

sets of matrices with given joint numerical range abstract the following question is addressed to what extent the n tuple of m m hermitian matrices is determined by its joint numerical range the cases m 2 n arbitrary and m n 3 are considered in detail

differential topology of numerical range abstract the numerical range of an n n matrix also known as its field of values is reformulated as the image of a smooth quadratic mapping from the n 1 dimensional complex projective space to the complex plane this paper investigates the numerical range from the perspective of differential topology morse theory more specifically the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values more importantly the map is shown to have additional critical value curves in the interior of the numerical range these additional curves are shown to have such singularity phenomena as cusps and swallow tails to be the caustic envelopes of families of lines and to exhibit the so called normal bifurcation when an eigenvalue becomes unitarily decoupled

numerical analysis of the unconditionally stable discontinuous galerkin schemes for the nonstationary conduction convection problem abstract this work aims at employing the discontinuous galerkin dg methods for solving the nonstationary conduction convection model which is motivated by advantages of dg methods the discontinuous element pair p k p k 1 p k in the space discretization and backward euler scheme in time are applied as well as an explicit upwind scheme based on picard iteration is introduced for the coupled term existence and uniqueness of the approximation solutions are proved along with preserving the unconditional stability of the numerical schemes the priori error estimates for velocity pressure and temperature are also derived numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the dg methods in simulating practical problems

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

a new low order least squares nonconforming characteristics mixed finite element method for burgers equation in this paper a new low order least squares nonconforming characteristics mixed finite element method mfem is considered for two dimensional burgers equation by use of two typical characters of the elements for approximating the velocity and flux variables a the consistency errors of the elements can be estimated as order o h 2 with respect to the mesh size h in the broken h 1 norm one order higher than their interpolation errors b the elements interpolation operators satisfy certain orthogonalities which lead to o h 2 order error estimates in l 2 norm and some superclose results the accuracy of corresponding variables in l 2 norm is improved by one order compared with other mfems in the existing literature it seems that the results provided herein have never been derived before