separability of sequential isomorphisms on quantum effects in multipartite systems in the paper we study separability of sequential isomorphisms on quantum effects in multipartite systems first we obtain necessary and sufficient conditions for separable sequential isomorphisms in bipartite systems secondly in finite dimensional bipartite systems separability of sequential endomorphisms on quantum effects is discussed furthermore we extend these conclusions to the multipartite case finally applying our results an equivalent characterization of local unitary operations is given

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

linear maps preserving idempotents of tensor products of matrices abstract let f be a field of characteristic not 2 and m n the algebra of all n n matrices over f the aim of this paper is to characterize linear maps m m 1 m l m m 1 m l such that a 1 a l is an idempotent whenever a 1 a l is an idempotent

a new algorithm for n dimensional hilbert scanning there have been many applications of the hilbert curve such as image processing image compression computer hologram etc the hilbert curve is a one to one mapping between n dimensional space and one dimensional l d space which preserves point neighborhoods as much as possible there are several algorithms for n dimensional hilbert scanning such as the butz algorithm and the quinqueton algorithm the butz algorithm is a mapping function using several bit operations such as shifting exclusive or etc on the other hand the quinqueton algorithm computes all addresses of this curve using recursive functions but takes time to compute a one to one mapping correspondence both algorithms are complex to compute and both are difficult to implement in hardware in this paper we propose a new simple nonrecursive algorithm for n dimensional hilbert scanning using look up tables the merit of our algorithm is that the computation is fast and the implementation is much easier than previous ones

preservers of maximally entangled states abstract the linear structure of the real space spanned by maximally entangled states is investigated and used to completely characterize those linear maps preserving the set of maximally entangled states on m m m m where m m denotes the space of m m complex matrices aside from a degenerate rank one map such preservers are generated by a change of orthonormal basis in each tensor factor interchanging the two tensor factors and the transpose operator

hamiltonian triangular refinements and space filling curves abstract we have introduced here the concept of hamiltonian triangular refinement for any hamiltonian triangulation it is shown that there is a refinement which is also a hamiltonian triangulation and the corresponding hamiltonian path preserves the nesting condition of the corresponding space filling curve we have proved that the number of such hamiltonian triangular refinements is bounded from below and from above the relation between hamiltonian triangular refinements and space filling curves is also explored and explained

unique decompositions faces and automorphisms of separable states let sk be the set of separable states on b cm cn admitting a representation as a convex combination of k pure product states or fewer if m 1 n 1 and k max m n we show that sk admits a subset vk such that vk is dense and open in sk and such that each state in vk has a unique decomposition as a convex combination of pure product states and we describe all possible convex decompositions for a set of separable states that properly contains vk in both cases we describe the associated faces of the space of separable states which in the first case are simplexes and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras as an application of these results we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of b cm cn that preserve entanglement and separability

a competitive strategy for distance aware online shape allocation we consider the following online allocation problem given a unit square s and a sequence of numbers n i 0 1 with j 0 i n j 1 at each step i select a region c i of previously unassigned area n i in s the objective is to make these regions compact in a distance aware sense minimize the maximum normalized average manhattan distance between points from the same set c i related location problems have received a considerable amount of attention in particular the problem of determining the optimal shape of a city i e allocating a single n i has been studied we present an online strategy based on an analysis of space filling curves for continuous shapes we prove a factor of 1 8092 and 1 7848 for discrete point sets

quantum measurements and maps preserving strict convex combinations and pure states in this paper a characterization of maps between quantum states that preserve pure states and strict convex combinations is obtained based on this characterization a structure theorem for maps between multipartite quantum states that preserve separable pure states and strict convex combinations is established then these results are applied to characterize injective local quantum measurements and answer some conjectures proposed in hou and liu 2012 j phys a math theor 45 205305

a mesh partitioning algorithm for preserving spatial locality in arbitrary geometries an algorithm for partitioning computational meshes is proposed the morton order space filling curve is modified to achieve improved locality a spatial locality metric is defined to compare results with existing approaches results indicate improved performance of the algorithm in complex geometries a space filling curve sfc is a proximity preserving linear mapping of any multi dimensional space and is widely used as a clustering tool equi sized partitioning of an sfc ignores the loss in clustering quality that occurs due to inaccuracies in the mapping often this results in poor locality within partitions especially for the conceptually simple morton order curves we present a heuristic that improves partition locality in arbitrary geometries by slicing a morton order curve at points where spatial locality is sacrificed in addition we develop algorithms that evenly distribute points to the extent possible while maintaining spatial locality a metric is defined to estimate relative inter partition contact as an indicator of communication in parallel computing architectures domain partitioning tests have been conducted on geometries relevant to turbulent reactive flow simulations the results obtained highlight the performance of our method as an unsupervised and computationally inexpensive domain partitioning tool

preservers of restricted unitary orbits of projections abstract given a pure state rank 1 self adjoint projection p and a group g of unitary operators acting on a complex hilbert space let the restricted unitary orbit of p relative to g be u g p u p u 1 u g we study linear transformations t which map u g p into pure states or more stringently back into u g p a complete characterization is obtained when g is an abelian group of unitary matrices equal to its bicommutant

discontinuous galerkin finite element methods for interface problems a priori and a posteriori error estimations discontinuous galerkin dg finite element methods were studied by many researchers for second order elliptic partial differential equations and a priori error estimates were established when the solution of the underlying problem is piecewise h 3 2 epsilon smooth with epsilon 0 however elliptic interface problems with intersecting interfaces do not possess such a smoothness in this paper we establish a quasi optimal a priori error estimate for interface problems whose solutions are only h 1 alpha smooth with alpha in 0 1 and hence fill a theoretical gap of the dg method for elliptic problems with low regularity the second part of the paper deals with the design and analysis of robust residual and recovery based a posteriori error estimators theoretically we show that the residual and recovery estimators studied in this paper are robust with respect to the dg norm i e their reliability and efficiency bounds do not depend on the jump provided that the distribution of coefficients is locally quasi monotone

linear maps preserving separability of pure states abstract linear maps preserving pure states of a quantum system of any dimension are characterized this is then used to establish a structure theorem for linear maps that preserve separable pure states in multipartite systems as an application a characterization of separable pure state preserving affine maps is obtained

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

on the preservers of maximally entangled states abstract we characterize the linear maps that preserve maximally entangled states in l x y in the case where dim x divides dim y

entropy preserving maps on quantum states abstract in the paper we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations let s h be the set of all quantum states positive operators with the unit trace on a complex hilbert space h with dim h n and s the quantum entropy of the quantum state for arbitrary a surjective map s h s h we claim that preserves quantum entropy of convex combinations of quantum states i e satisfies s t 1 t s t 1 t for arbitrary s h and t 0 1 if and only if there exists a unitary or anti unitary operator u on h such that u u for all s h

quantum measurement and maps preserving convex combinations of separable states a map between convex sets is said to be convex combination preserving if for any elements x y in its domain and any number 0 t 1 there is some s with 0 s 1 such that tx 1 t y s x 1 s y every quantum measurement described by a measurement operator m introduces a map i e if that is convex combination preserving in this paper we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system h1 h2 and show that under a mild additional condition such a map is a composition of an invertible local quantum measurement i e the map of the form with s t invertible and some of the following maps the transpose the partial transpose and the swap

on the metric properties of discrete space filling curves a space filling curve is a linear traversal of a discrete finite multidimensional space in order for this traversal to be useful in many applications the curve should preserve locality we quantify locality and bound the locality of multidimensional space filling curves classic hilbert space filling curves come close to achieving optimal locality

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

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

a geometric characterization of invertible quantum measurement maps a geometric characterization is given for invertible quantum measurement maps denote by s h the convex set of all states i e trace 1 positive operators on hilbert space h with dimh and 1 2 the line segment joining two elements 1 2 in s h it is shown that a bijective map s h s h satisfies 1 2 1 2 for any 1 2 s if and only if has one of the following forms m m tr m m or m tm tr m tm where m is an invertible bounded linear operator and t is the transpose of with respect to an arbitrarily fixed orthonormal basis

compression of two dimensional data distortion free compressibility of individual pictures i e two dimensional arrays of data by finite state encoders is investigated for every individual infinite picture i a quantity rho i is defined called the compressibility of i which is shown to be the asymptotically attainable lower bound on the compression ratio that can be achieved for i by any finite state information lossless encoder this is demonstrated by means of a constructive coding theorem and its converse that apart from their asymptotic significance might also provide useful criteria for finite and practical data compression tasks the proposed picture compressibility is also shown to possess the properties that one would expect and require of a suitably defined concept of two dimensional entropy for arbitrary probabilistic ensembles of infinite pictures while the definition of rho i allows the use of different machines for different pictures the constructive coding theorem leads to a universal compression scheme that is asymptotically optimal for every picture the results are readily extendable to data arrays of any finite dimension

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

linear rank preservers of tensor products of rank one matrices abstract let n 1 n k be integers larger than or equal to 2 we characterize linear maps m n 1 n k m n 1 n k such that r a n k a 1 a k 1 whenever r a n k a 1 a k 1 for all a i m n i i 1 k applying this result we extend two recent results on linear maps that preserve the rank of special classes of matrices

linear maps preserving r potents of tensor products of matrices abstract let m n be the algebra of all n n complex matrices and r 2 a fixed integer the aim of this paper is to characterize linear maps m m 1 m l m m 1 m l such that a 1 a l is r potent whenever a 1 a l is r potent

linear maps preserving numerical radius of tensor products of matrices abstract let m n 2 be positive integers denote by m m the set of m m complex matrices and by w x the numerical radius of a square matrix x motivated by the study of operations on bipartite systems of quantum states we show that a linear map m m n m m n satisfies w a b w a b for all a m m and b m n if and only if there is a unitary matrix u m m n and a complex unit such that a b u 1 a 2 b u for all a m m and b m n where k is the identity map or the transposition map x x t for k 1 2 and the maps 1 and 2 will be of the same type if m n 3 in particular if m n 3 the map corresponds to an evolution of a closed quantum system under a fixed unitary operator possibly followed by a transposition the results are extended to multipartite systems

linear preservers on spaces of hermitian or real symmetric matrices abstract let h n denote the space of all n n hermitian matrices and l n the space of all n n real symmetric matrices let k be a fixed positive integer and let t be a linear map on h n in this paper we characterized t in each of the following cases 1 t is rank k nonincreasing and im t contains a matrix a whose rank is greater than k 2 t preserves the inertia class k 0 n k and rank t k 2 3 t is a rank k preserver in case n 2 and k 2 or k is an odd integer in 1 and 2 we assume k n in 2 we may replace h n by l n provided that the lower bound for rank t is replaced by 1 2 k k 1 in 3 we may also replace h n by l n thus obtaining a generalization of a result of lim who proved the special cases k 2 and k n odd number

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

manipulation and presentation of multidimensional image data using the peano scan a fractal curve first discussed by giuseppe peano has useful statistical properties in image processing the curve can act as a transform between a line and n dimensional space retaining some of the spatially associative properties of the space applications of the curve include multispectral image display compression in the color and spa tial domain classification and color display the technique enables the display of high quality color images on a frame store capable of displaying only 256 colors simultaneously a logically ordered and complete set of colors for false color work can be selected an im provement in compression by about 3x for color images can be ob tained by using this method in conjunction with an existing spatial compression technique a simple algorithm for constructing the curves is shown

linear maps preserving determinant of tensor products of hermitian matrices abstract let m n 2 be positive integers and let h n be the set of n n complex hermitian matrices we study linear maps h m n h m n satisfying det a b det a b for all a h m b h n the connection of the problem to quantum information science is mentioned

weak galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of weak galerkin wg finite element methods based on the second order elliptic interface problems this estimate can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator has been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator