algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws abstract in this work we modify a continuous galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property preserving low order scheme the addition of a second order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains validity of local maximum principles and entropy stability the new methodology combines the advantages of modern entropy stable entropy conservative schemes and their local extremum diminishing counterparts the process of algebraic flux correction is based on inequality constraints which provably provide the desired properties no free parameters are involved the proposed algebraic fixes are readily applicable to unstructured meshes finite element methods general time discretizations and steady state residuals numerical studies of explicit entropy constrained schemes are performed for linear and nonlinear test problems

orthogonal bases for vertex mapped pyramids discontinuous galerkin dg methods discretized under the method of lines must handle the inverse of a block diagonal mass matrix at each time step efficient implementations of the dg method hinge upon inexpensive and low memory techniques for the inversion of each dense mass matrix block we propose an efficient time explicit dg method on meshes of pyramidal elements based on the construction of a seminodal high order basis which is orthogonal for a class of transformations of the reference pyramid despite the nonaffine nature of the mapping we give numerical results confirming both expected convergence rates and discuss efficiency of dg methods under such a basis

interaction in quantum communication in some scenarios there are ways of conveying information with many fewer even exponentially fewer qubits than possible classically moreover some of these methods have a very simple structure they involve only few message exchanges between the communicating parties it is therefore natural to ask whether every classical protocol may be transformed to a simpler quantum protocol one that has similar efficiency but uses fewer message exchanges we show that for any constant k there is a problem such that its k 1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity this in particular proves a round hierarchy theorem for quantum communication complexity and implies via a simple reduction an omega n1k lower bound for k message quantum protocols for set disjointness for constant k enroute we prove information theoretic lemmas and define a related measure of correlation the informational distance that we believe may be of significance in other contexts as well

the space complexity of recognizing well parenthesized expressions in the streaming model the index function revisited we show an p n t lower bound for the space required by any unidirectional constanterror randomized t pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well parenthesized this proves a conjecture due to magniez mathieu and nayak 2009 and rigorously establishes the peculiar power of bi directional streams over unidirectional ones observed in the algorithms they present the lower bound is obtained by analysing the information that is necessarily revealed by the players about their respective inputs in a two party communication protocol for a variant of the index function

quantum automata and quantum grammars abstract to study quantum computation it might be helpful to generalize structures from language and automata theory to the quantum case to that end we propose quantum versions of finite state and push down automata and regular and context free grammars we find analogs of several classical theorems including pumping lemmas closure properties rational and algebraic generating functions and greibach normal form we also show that there are quantum context free languages that are not context free so qcfl cfl

matrix free subcell residual distribution for bernstein finite element discretizations of linear advection equations abstract in this work we introduce a new residual distribution rd framework for the design of bound preserving high resolution finite element schemes the continuous and discontinuous galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing led approximations to that end we perform mass lumping and redistribute the element residuals in a manner which guarantees the led property the hierarchical correction procedure for high order bernstein finite element discretizations involves localization to subcells and definition of bound preserving weights for subcell contributions using strong stability preserving ssp runge kutta methods for time integration we prove the validity of discrete maximum principles under cfl like time step restrictions the low order version of our method has roughly the same accuracy as the one derived from a piecewise multi linear approximation on a submesh with the same nodal points in high order extensions we use an element based flux corrected transport fct algorithm which can be interpreted as a nonlinear rd scheme the proposed led corrections are tailor made for matrix free implementations which avoid the rapidly growing cost of matrix assembly for high order bernstein elements the results for 1d 2d and 3d test problems compare favorably to those obtained with the best matrix based approaches

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

multiplicity results of periodic solutions for a class of second order delay differential systems abstract this paper is concerned with the periodic boundary value problem 1 u t u t f t u t a e t 0 2 u 0 u 2 u 0 u 2 0 where 0 is a given constant r is a parameter the variational principle is given and some multiplicity results of periodic solutions of 1 are obtained directly via variational methods

the learnability of quantum states traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n but using ideas from computational learning theory we show that one can do exponentially better in a statistical setting in particular to predict the outcomes of most measurements drawn from an arbitrary probability distribution one needs only a number of sample measurements that grows linearly with n this theorem has the conceptual implication that quantum states despite being exponentially long vectors are nevertheless reasonable in a learning theory sense the theorem also has two applications to quantum computing first a new simulation of quantum one way communication protocols and second the use of trusted classical advice to verify untrusted quantum advice

quantum versus deterministic counter automata this paper focuses on quantum analogues of various models of counter automata and almost completely proves the relation between the classes of languages recognizable by bounded error quantum ones and classical deterministic ones in every model of counter automata it is proved that i there are languages that can be recognized by two way quantum one counter automata with bounded error but cannot be recognized by two way deterministic one counter automata ii under some reasonable restriction every language that can be recognized by two way deterministic one counter automata can also be recognized by two way reversible one counter automata and hence by bounded error two way quantum one counter automata and iii for any fixed k quantum ones and deterministic ones are incomparable in one way k counter automata

non uniform order mixed fem approximation implementation post processing computable error bound and adaptivity the present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form the work contains details of the discretisation using non uniform order mixed finite elements of arbitrarily high order a new local post processing scheme for the primary variable the use of the post processing scheme in the derivation of new fully computable bounds for the error in the flux variable and an hp adaptive refinement strategy based on the a posteriori error estimator numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten pole electric motor with singularities arising from both geometric features and discontinuities in material properties the procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns

characterizations of one way general quantum finite automata generally unitary transformations limit the computational power of quantum finite automata qfa in this paper we study a generalized model named one way general quantum finite automata 1gqfa in which each symbol in the input alphabet induces a trace preserving quantum operation instead of a unitary transformation two different kinds of 1gqfa will be studied measure once one way general quantum finite automata mo 1gqfa where a measurement deciding to accept or reject is performed at the end of a computation and measure many one way general quantum finite automata mm 1gqfa where a similar measurement is performed after each trace preserving quantum operation on reading each input symbol we characterize the measure once model from three aspects the closure property the language recognition power and the equivalence problem we prove that mo 1gqfa recognize with bounded error precisely the set of all regular languages our results imply that some models of quantum finite automata proposed in the literature which were expected to be more powerful still cannot recognize non regular languages we prove that mm 1gqfa also recognize only regular languages with bounded error thus mm 1gqfa and mo 1gqfa have the same language recognition power in sharp contrast with traditional mo 1qfa and mm 1qfa the former being strictly less powerful than the latter finally we present a necessary and sufficient condition for two mm 1gqfa to be equivalent

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

state succinctness of two way finite automata with quantum and classical states two way finite automata with quantum and classical states 2qcfa were introduced by ambainis and watrous in 2002 in this paper we study state succinctness of 2qcfa for any email protected z and any e 1 2 we show that 1 there is a promise problem a e q m which can be solved by a 2qcfa with one sided error e in a polynomial expected running time with a constant number that depends neither on m nor on e of quantum states and o email protected classical states whereas the sizes of the corresponding deterministic finite automata dfa two way nondeterministic finite automata 2nfa and polynomial expected running time two way probabilistic finite automata 2pfa are at least 2m 2 logm and logm b3 respectively 2 there exists a language l t w i n m wcw email protected a b w m over the alphabet s a b c which can be recognized by a 2qcfa with one sided error e in an exponential expected running time with a constant number of quantum states and o email protected classical states whereas the sizes of the corresponding dfa 2nfa and polynomial expected running time 2pfa are at least 2 m m and m b3 respectively where b is a constant

low complexity finite element algorithms for the de rham complex on simplices we combine recently developed finite element algorithms based on bernstein polynomials m ainsworth g andriamaro and o davydov siam j sci comput 33 2011 pp 3087 3109 r c kirby numer math 117 2011 pp 631 652 with the explicit basis construction of the finite element exterior calculus d n arnold r s falk and r winther comput methods appl mech engrg 198 2009 pp 1660 1672 to give a family of algorithms for the de rham complex on simplices that achieves stiffness matrix construction and matrix free action in optimal complexity these algorithms are based on realizing the exterior calculus bases as short combinations of bernstein polynomials numerical results confirm optimal scaling of the algorithms and favorable comparison with fenics at high polynomial order as well additional empirical studies show that very high accuracy is achieved in the mixed discretization of the poisson equation and the maxwell eigenvalue problem as the polynomial degree increases

bernstein bezier bases for tetrahedral finite elements abstract we present a new set of basis functions for h curl conforming h div conforming and l 2 conforming finite elements of arbitrary order on a tetrahedron the basis functions are expressed in terms of bernstein polynomials and augment the natural h 1 conforming bernstein basis the basis functions respect the differential operators namely the gradients of the high order h 1 conforming bernstein bezier basis functions form part of the h curl conforming basis and the curl of the high order non gradients h curl conforming basis functions form part of the h div conforming basis and the divergence of the high order non curl h div conforming basis functions form part of the l 2 conforming basis procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for piecewise constant coefficients on affine tetrahedra numerical results are presented to illustrate the use of the basis to approximate representative problems

flux correction tools for finite elements flux correction in the finite element context is addressed criteria for positivity of the numerical solution are formulated and the low order transport operator is constructed from the discrete high order operator by adding modulated dissipation so as to eliminate negative off diagonal entries the corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes rather than element contributions which represent bilateral mass exchange between individual nodes thereby essentially one dimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes the proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit fct schemes based on a unified limiting strategy numerical results for a number of benchmark problems illustrate the performance of the algorithm

a maximum principle preserving finite element method for scalar conservation equations this paper introduces a first order viscosity method for the explicit approximation of scalar conservation equations with lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension provided the lumped mass matrix is positive definite the method is shown to satisfy the local maximum principle under a usual cfl condition the method is independent of the cell type for instance the mesh can be a combination of tetrahedra hexahedra and prisms in three space dimensions

on the homogeneous algebraic graphs of large girth and their applications abstract families of finite graphs of large girth were introduced in classical extremal graph theory one important theoretical result here is the upper bound on the maximal size of the graph with girth 2 d established in even circuit theorem by p erdos we consider some results on such algebraic graphs over any field the upper bound on the dimension of variety of edges for algebraic graphs of girth 2 d is established getting the lower bound we use the family of bipartite graphs d n k with n 2 over a field k whose partition sets are two copies of the vector space k n we consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory finally we present a very short survey on applications of finite homogeneous algebraic graphs to coding theory and cryptography

entanglement sampling and applications a natural measure for the amount of quantum information that a physical system e holds about another system a a 1 ldots a n is given by the min entropy rm h min a e in particular the min entropy measures the amount of entanglement between e and a and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography decoupling used in channel coding to physical processes such as thermalization or the thermodynamic work cost or gain of erasing a quantum system as such it is a central question to determine the behavior of the min entropy after some process mathcal m is applied to the system a here we introduce a new generic tool relating the resulting min entropy to the original one and apply it to several settings of interest a simple example of such a process is the one of sampling where a subset s of the systems a 1 ldots a n is selected at random our tool allows us to quantify the entanglement that e has with the selected systems a s i e rm h min a s es as a function of the original rm h min a e we give two applications of this result first it directly provides the first local quantum to classical randomness extractors for use in quantum cryptography as well as decoupling operations acting on only a small fraction a s of the input a moreover it gives lower bounds on the dimension of k out of n fully quantum random access encodings another natural example of such a process is a measurement in e g bb84 bases commonly used in quantum cryptography we establish the first entropic uncertainty relations with quantum side information that are nontrivial whenever e is not maximally entangled with a as a consequence we are able to prove optimality of quantum cryptographic schemes in the noisy storage model this model allows for the secure implementation of two party cryptographic primitives under the assumption that the adversary cannot store quantum information perfectly a special case is the bounded quantum storage model bqsm which assumes that the adversary s quantum memory device is noise free but limited in size ever since the inception of the bqsm it has been a vexing open question to determine whether the security is possible as long as the adversary can only store strictly less than the number of qubits n transmitted during the protocol here we show that security is even possible as long as the adversary s device is not larger than n o log 2 n qubits which finally settles the fundamental limits of the bqsm

a comparison of finite element and finite difference solutions of the one and two dimensional burgers equations abstract solutions to the one and two dimensional burgers equations with moderate to severe internal and boundary gradients have been used to compare minimum truncation error three five and seven point finite difference schemes with linear quadratic and cubic rectangular finite element schemes the various schemes demonstrate the theoretically predicted convergence rates if the mesh is sufficiently refined in one dimension the quadratic finite element scheme and the five point finite difference scheme are computationally the most efficient in multidimensions the finite element method is less economical than the finite difference method even if the group representation is used for the convective terms in two dimensions the linear group finite element representation and the three point finite difference scheme are computationally the most efficient on a coarse mesh or if a severe gradient is present

a short note on a bernstein bezier basis for the pyramid we introduce a bernstein bezier basis for the pyramid whose restriction to the face reduces to the bernstein bezier basis on the triangle or quadrilateral the basis satisfies the standard positivity and partition of unity properties common to bernstein polynomials and spans the same space as nonpolynomial pyramid bases in the literature procedures for differentiation and integration of these basis functions are also discussed

quantum inductive inference by finite automata freivalds and smith r freivalds c h smith memory limited inductive inference machines springer lecture notes in computer science 621 1992 19 29 proved that probabilistic limited memory inductive inference machines can learn with probability 1 certain classes of total recursive functions which cannot be learned by deterministic limited memory inductive inference machines we introduce quantum limited memory inductive inference machines as quantum finite automata acting as inductive inference machines these machines we show can learn classes of total recursive functions not learnable by any deterministic nor even by probabilistic limited memory inductive inference machines

the group finite element formulation abstract the introduction of trial solutions for groups of variables within a finite element formulation is shown to provide a significant economy with no loss of accuracy for steady solutions of the two dimensional burgers equations with linear rectangular elements the application of the group finite element to the convective term produces a threefold reduction in execution time with a small improvement in accuracy operation count estimates indicate even larger economies in three dimensions or if higher order nonlinearities occur or if higher order shape functions are used applications of the group formulation are described for inviscid and viscous laminar and turbulent separated and unseparated and incompressible and compressible flows

determining the equivalence for one way quantum finite automata two quantum finite automata are equivalent if for any input string x the two automata accept x with equal probability in this paper we first focus on determining the equivalence for one way quantum finite automata with control language cl 1qfas defined by bertoni et al and then as an application we address the equivalence problem for measure many one way quantum finite automata mm 1qfas introduced by kondacs and watrous more specifically we obtain that i two cl 1qfas a 1 and a 2 with control languages regular languages l 1 and l 2 respectively are equivalent if and only if they are c 1n 1 2 c 2n 2 2 1 equivalent where n 1 and n 2 are the numbers of states in a 1 and a 2 respectively and c 1 and c 2 are the numbers of states in the minimal dfas that recognize l 1 and l 2 respectively furthermore if l 1 and l 2 are given in the form of dfas with m 1 and m 2 states respectively then there exists a polynomial time algorithm running in time o m 1n 1 2 m 2n 2 2 4 that takes as input a 1 and a 2 and determines whether they are equivalent ii as an application of item i two mm 1qfas a 1 and a 2 with n 1 and n 2 states respectively are equivalent if and only if they are 3n 1 2 3n 2 2 1 equivalent furthermore there is a polynomial time algorithm running in time o 3n 1 2 3n 2 2 4 that takes as input a 1 and a 2 and determines whether a 1 and a 2 are equivalent

existence of periodic solutions for a class of first order delay differential equations dealing with vectors abstract by the weak linking theorem and the local linking theorem we study the existence of periodic solutions for the following delay non autonomous systems 1 u t a t u t f t u t r 0 where r 0 0 r f c r r n r n satisfies f t r x f t x for all x r n and a t a i j t n n is an anti symmetric matrix valued function with a i j t c r r which is r periodic in t for all i j 1 2 n the variational principle is given some existence theorems of 4 r periodic solutions of 1 are obtained

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

fast simplicial finite element algorithms using bernstein polynomials fast algorithms for applying finite element mass and stiffness operators to the b form of polynomials over d dimensional simplices are derived these rely on special properties of the bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor product techniques in rectangular domains first special structure leading to fast application of mass matrices is developed then by factoring stiffness matrices into products of sparse derivative matrices with mass matrices fast algorithms are also obtained for stiffness matrices

monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws abstract using the theoretical framework of algebraic flux correction and invariant domain preserving schemes we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations nonlinear scalar conservation laws and hyperbolic systems in contrast to flux corrected transport fct algorithms that apply limited antidiffusive corrections to bound preserving low order solutions our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds there is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately moreover the steady state limit of the nonlinear discrete problem is well defined and independent of the pseudo time step in the case study for the euler equations the components of the bar states are constrained sequentially to satisfy local maximum principles for the density velocity and specific total energy in addition to positivity preservation for the density and pressure the results of numerical experiments for standard test problems illustrate the ability of built in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner

invariant domains and second order continuous finite element approximation for scalar conservation equations we propose a technique for approximating nonlinear scalar conservation equations that uses continuous finite elements and is formally at least second order accurate in space and maximum principle preserving the method is explicit in time uses unstructured continuous finite elements in space and works in any space dimension the stability and accuracy of the method are achieved by adapting the artificial viscosity

bernstein bezier weight adjusted discontinuous galerkin methods for wave propagation in heterogeneous media abstract this paper presents an efficient discontinuous galerkin method to simulate wave propagation in heterogeneous media with sub cell variations this method is based on a weight adjusted discontinuous galerkin method wadg which achieves high order accuracy for arbitrary heterogeneous media 1 however the computational cost of wadg grows rapidly with the order of approximation in this work we propose a bernstein bezier weight adjusted discontinuous galerkin method bbwadg to address this cost by approximating sub cell heterogeneities by a fixed degree polynomial the main steps of wadg can be expressed as polynomial multiplication and l 2 projection which we carry out using fast bernstein algorithms the proposed approach reduces the overall computational complexity from o n 2 d to o n d 1 in d dimensions numerical experiments illustrate the accuracy of the proposed approach and computational experiments for a gpu implementation of bbwadg verify that this theoretical complexity is achieved in practice

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

a lowest order composite finite element exact sequence on pyramids abstract composite basis functions for pyramidal elements on the spaces h 1 h curl h div and l 2 are presented in particular we construct the lowest order composite pyramidal elements and show that they respect the de rham diagram i e we have an exact sequence and satisfy the commuting property moreover the finite elements are fully compatible with the standard finite elements for the lowest order raviart thomas nedelec sequence on tetrahedral and hexahedral elements that is to say the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements and the basis functions are conforming in the sense that they maintain the required level of continuity full tangential component normal component etc across the interface furthermore we study the approximation properties of the spaces as an initial partition consisting of tetrahedra hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same optimal order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions

a second order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations this paper proposes an explicit at least second order maximum principle sat isfying lagrange finite element method for solving nonlinear scalar conservation equations the technique is based on a new viscous bilinear form introduced in guermond and nazarov com put methods appl mech engrg 272 2014 pp 198 213 a high order entropy viscosity method and the boris book zalesak flux correction technique the algorithm works for arbitrary meshes in any space dimension and for all lipschitz fluxes the formal second order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems

pyramid algorithms for bernstein bezier finite elements of high nonuniform order in any dimension the archetypal pyramid algorithm is the de casteljau algorithm which is a standard tool for the evaluation of bezier curves and surfaces pyramid algorithms replace an operation on a single high order polynomial by a recursive sequence of self similar affine combinations and are ubiquitous in computer aided geometric design for computations involving high order curves and surfaces pyramid algorithms have received no attention whatsoever from the high or low order finite element community we develop and analyze pyramid algorithms for the efficient handling of all of the basic finite element building blocks including the assembly of the element load vectors and element stiffness matrices the complexity of the algorithm for generating the element stiffness matrix is optimal a new nonuniform order variant of the de casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm the work provides the

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

new two dimensional slope limiters for discontinuous galerkin methods on arbitrary meshes summary in this paper we introduce an extension of van leer s slope limiter for two dimensional discontinuous galerkin dg methods on arbitrary unstructured quadrangular or triangular grids the aim is to construct a non oscillatory shock capturing dg method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion unlike some splitting techniques that are limited to linear approximations on rectangular grids in this work the solution is approximated by means of piecewise quadratic functions the main idea of this new reconstructing and limiting technique follows a well known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters copyright 2004 john wiley sons ltd

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

high order composite finite element exact sequences based on tetrahedral hexahedral prismatic pyramidal partitions abstract the combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces this paper presents high order exact sequences of finite element approximations in h 1 h curl h div and l 2 based on such kind of three dimensional mesh configurations the approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra the traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid in order to maintain conformity with shared neighboring hexahedron or prism two classes of composite exact sequences are constructed one using classic nedelec spaces of first kind and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces projection based interpolants satisfying the commuting diagram property are presented in a general form for each type of element the interpolants are expressed as the sum of linearly independent contributions associated with vertices edges faces and volume according to the kind of traces appropriate to the space under consideration furthermore we study applications to the mixed formulation of darcy s problems based on compatible pairs of approximations in h div l 2 for such tetrahedral hexahedral prismatic pyramidal meshes an error analysis is outlined showing same optimal orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied the predicted convergence orders are verified for some test problems

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

high order exactly divergence free hybrid discontinuous galerkin methods for unsteady incompressible flows abstract in this paper we present an efficient discretization method for the solution of the unsteady incompressible navier stokes equations based on a high order hybrid discontinuous galerkin formulation the crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non linear parts with respect to their temporal and spatial treatment exploiting the flexibility of operator splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub problems a corresponding hyperbolic transport problem and an unsteady stokes problem for the hyperbolic transport problem a spatial discretization with an upwind discontinuous galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation the treatment of the stokes part involves the solution of linear systems in this case a discretization with hybrid discontinuous galerkin methods is better suited we consider such a discretization for the stokes part with two important features h div conforming finite elements to guarantee exactly divergence free velocity solutions and a projection operator which reduces the number of globally coupled unknowns we present the method discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems

flux corrected transport algorithms for continuous galerkin methods based on high order bernstein finite elements abstract this work extends the flux corrected transport fct methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes using bernstein polynomials as local basis functions we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the bezier net the design of accuracy preserving fct schemes for high order bernstein bezier finite elements requires the development of new algorithms and or generalization of limiting techniques tailored for linear and multilinear lagrange elements in this paper we propose i a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils ii high order variational stabilization based on the difference between two gradient approximations and iii new localized limiting techniques for antidiffusive element contributions the optional use of a smoothness indicator based on a second derivative test makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions the accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1d and 2d

improved constructions of quantum automata we present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata our automata use email protected states to recognize a language that requires p states classically the construction is both substantially simpler and achieves a better constant in the front of logp than the previously known construction our construction is by a probabilistic argument we consider the possibility to derandomize it and present some results in this direction

edge based nonlinear diffusion for finite element approximations of convection diffusion equations and its relation to algebraic flux correction schemes for the case of approximation of convection diffusion equations using piecewise affine continuous finite elements a new edge based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle the diffusion operator is shown to be lipschitz continuous and linearity preserving using these properties we provide a full stability and error analysis which in the diffusion dominated regime shows existence uniqueness and optimal convergence then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters the performance of the method is illustrated on some numerical test cases in two space dimensions

analysis of algebraic flux correction schemes a family of algebraic flux correction afc schemes for linear boundary value problems in any space dimension is studied these methods main feature is that they limit the fluxes along each one of the edges of the triangulation and we suppose that the limiters used are symmetric for an abstract problem the existence of a solution existence and uniqueness of the solution of a linearized problem and an a priori error estimate are proved under rather general assumptions on the limiters for a particular but standard in practice choice of the limiters it is shown that a local discrete maximum principle holds the theory developed for the abstract problem is applied to convection diffusion reaction equations where in particular an error estimate is derived numerical studies show its sharpness

dense quantum coding and quantum finite automata we consider the possibility of encoding m classical bits into many fewer n quantum bits qubits so that an arbitrary bit from the original m bits can be recovered with good probability we show that nontrivial quantum codes exist that have no classical counterparts on the other hand we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding the proof is based on an entropy coalescence principle that is obtained by viewing holevo s theorem from a new perspective in the existing implementations of quantum computing qubits are a very expensive resource moreover it is difficult to reinitialize existing bits during the computation in particular reinitialization is impossible in nmr quantum computing which is perhaps the most advanced implementation of quantum computing at the moment this motivates the study of quantum computation with restricted memory and no reinitialization that is of quantum finite automata it was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata here we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata

an o p3 hp version fem in two dimensions preconditioning and post processing abstract the bernstein polynomials have been known for over a century and are widely used in the spline literature computer aided geometric design and computer graphics however the realization that the bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development for instance it is known ainsworth et al 2011 that the bernstein basis can be exploited to compute all of the entries in the load vector in o p 3 operations even in the case of non linear problems on curvilinear elements for a degree p approximation moreover the element matrices can be assembled in o 1 operations per entry we show that properties of the bernstein polynomials can also be exploited to obtain o p 3 complexity procedures for all of the main components needed to implement a high order finite element code including computation of the residuals needed for an iterative solution method evaluating the action of a preconditioner for the global mass matrices and visualization and post processing of the resulting finite element approximations the construction of a preconditioner for the mass matrix whose condition number does not degenerate with the order p at a cost of o p 3 operations is one of the main contributions of the present work the preconditioner is based on an abstract additive schwarz method recently developed by the authors the preconditioner can be implemented at a cost of o p 3 operations by exploiting properties of the bernstein polynomials in particular we present an algorithm which allows one to invert the interior block of the element mass matrix in o p 3 operations numerical examples are provided to illustrate the applicability of the bernstein basis to challenging non linear reaction diffusion problems non linear wave propagation of solitons and to robust approximation of problems exhibiting boundary layers

bernstein bezier finite elements on tetrahedral hexahedral pyramidal partitions abstract a construction for high order continuous finite elements on partitions consisting of tetrahedra hexahedra and pyramids based on polynomial bernstein bezier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity o 1 per entry

recursive parameter free explicitly defined interpolation nodes for simplices a rule for constructing interpolation nodes for n th degree polynomials on the simplex is presented these nodes are simple to define recursively from families of 1d node sets such as the lobatto

a property of quantum relative entropy with an application to privacy in quantum communication we prove the following information theoretic property about quantum states substate theorem let and be quantum states in the same hilbert space with relative entropy s c tr log log c then for all 0 there is a state such that the trace distance tr tr 2 and 2o c 2 it states that if the relative entropy of and is small then there is a state close to i e with small trace distance tr that when scaled down by a factor 2o c sits inside or becomes a substate of this result has several applications in quantum communication complexity and cryptography using the substate theorem we derive a privacy trade off for the set membership problem in the two party quantum communication model here alice is given a subset a n bob an input i i n and they need to determine if i i a privacy trade off for set membership in any two party quantum communication protocol for the set membership problem if bob reveals only k bits of information about his input then alice must reveal at least n 2o k bits of information about her input we also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem

some remarks on spanning families and weights for high order whitney spaces on simplices abstract high order whitney finite element spaces generally lack natural choices of bases but they do have spanning families in these pages we recall such a family on simplicial meshes and we prove theoretically its effectiveness we also comment on some aspects of a new set of degrees of freedom the so called weights on the small simplices to represent discrete functions in these spaces

fast simplicial quadrature based finite element operators using bernstein polynomials we derive low complexity matrix free finite element algorithms for simplicial bernstein polynomials on simplices our techniques based on a sparse representation of differentiation and special block structure in the matrices evaluating b form polynomials at warped gauss points apply to variable coefficient problems as well as constant coefficient ones thus extending our results in kirby numer math 2011 in press

monotonicity preserving finite element schemes based on differentiable nonlinear stabilization in this work we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping the method relies on an artificial diffusion method based on a graph laplacian operator it is nonlinear since it depends on a shock detector further the resulting method is linearity preserving the same shock detector is used to gradually lump the mass matrix the resulting method is led positivity preserving and also satisfies a global dmp lipschitz continuity has also been proved however the resulting scheme is highly nonlinear leading to very poor nonlinear convergence rates we propose a smooth version of the scheme which leads to twice differentiable nonlinear stabilization schemes it allows one to straightforwardly use newton s method and obtain quadratic convergence in the numerical experiments steady and transient linear transport and transient burgers equation have been considered in 2d using the newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of anderson acceleration with the original non smooth scheme in any case these properties are only true for the converged solution but not for iterates in this sense we have also proposed the concept of projected nonlinear solvers where a projection step is performed at the end of every nonlinear iterations onto a fe space of admissible solutions the space of admissible solutions is the one that satisfies the desired monotonic properties maximum principle or positivity

a partition of unity approach to adaptivity and limiting in continuous finite element methods abstract the partition of unity finite element method pufem proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner the lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures in contrast to traditional h p adaptivity for finite elements the proposed approach preserves discrete conservation properties and the continuity of traces at common boundaries of adjacent mesh cells in the context of space discretizations a continuous blending function is used to combine finite element bases corresponding to high order polynomials and piecewise linear approximations based on the same set of nodes in a similar vein spatially partitioned time discretizations can be designed using weights that depend continuously on the space variable the design of blending functions may be based on a priori knowledge e g in applications to problems with singularities or boundary layers local error estimates smoothness indicators and or discrete maximum principles in adaptive methods changes of the finite element approximation exhibit continuous dependence on the data the presented numerical examples illustrate the typical behavior of local h 1 and l 2 errors

embedded discontinuous galerkin transport schemes with localised limiters motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction we introduce locally bounded transport schemes for partially continuous finite element spaces the underlying high order transport scheme is constructed by injecting the partially continuous field into an embedding discontinuous finite element space applying a stable upwind discontinuous galerkin dg scheme and projecting back into the partially continuous space we call this an embedded dg transport scheme we prove that this scheme is stable in l 2 provided that the underlying upwind dg scheme is we then provide a framework for applying limiters for embedded dg transport schemes standard dg limiters are applied during the underlying dg scheme we introduce a new localised form of element based flux correction which we apply to limiting the projection back into the partially continuous space so that the whole transport scheme is bounded we provide details in the specific case of tensor product finite element spaces on wedge elements that are discontinuous p1 q1 in the horizontal and continuous p2 in the vertical the framework is illustrated with numerical tests

high order local maximum principle preserving mpp discontinuous galerkin finite element method for the transport equation abstract in this work we present a fct like maximum principle preserving mpp method to solve the transport equation we use high order polynomial spaces in particular we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension the method combines the concepts of positive basis functions for discontinuous galerkin finite element spatial discretization locally defined solution bounds element based flux correction and non linear local mass redistribution we consider a simple 1d problem with non smooth initial data to explain and understand the behavior of different parts of the method convergence tests in space indicate that high order accuracy is achieved numerical results from several benchmarks in two and three dimensions are also reported

invariant domains and first order continuous finite element approximation for hyperbolic systems we propose a numerical method for solving general hyperbolic systems in any space dimension using forward euler time stepping and continuous finite elements on nonuniform grids the properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant set containing the initial data is an invariant domain for the method the invariant domain property is proved for any hyperbolic system provided a cfl condition holds the solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system the method is formally first order accurate in space and can be made high order in time by using strong stability preserving algorithms this technique extends to continuous finite elements the work of d hoff math comp 33 1979 pp 1171 1193 d hoff trans amer math soc 289 1985 pp 591 610 and h frid arch ration mech anal 160 2001 pp 245 269

second order invariant domain preserving approximation of the euler equations using convex limiting a new second order method for approximating the compressible euler equations is introduced the method preserves all the known invariant domains of the euler system positivity of the density posi

a note on quantum sequential machines quantum sequential machines qsms are a quantum version of stochastic sequential machines ssms recently we showed that two qsms m 1 and m 2 with n 1 and n 2 states respectively are equivalent iff they are n 1 n 2 2 equivalent l z li d w qiu determination of equivalence between quantum sequential machines theoretical computer science 358 2006 65 74 however using this result to check the equivalence is likely to need exponential expected time in this note we consider the time complexity of deciding the equivalence between qsms and related problems the main results are as follows 1 we present a polynomial time algorithm for deciding the equivalence between qsms and if two qsms are not equivalent this algorithm will produce an input output pair with length not more than n 1 n 2 2 2 we improve the bound for the equivalence between qsms from n 1 n 2 2 to n 1 2 n 2 2 1 by employing moore and crutchfield s method c moore j p crutchfield quantum automata and quantum grammars theoretical computer science 237 2000 275 306 also quant ph 9707031 1997 in addition by viewing mo 1qfas as a special case of qsms we briefly discuss the equivalence between mo 1qfas where the method used and the result obtained are slightly different from those given by koshiba t koshiba polynomial time algorithms for the equivalence for one way quantum finite automata in proceedings of the 12th international symposium on algorithms and computation isaac 2001 christchurch new zealand in lecture notes in computer science vol 2223 springer berlin 2001 pp 268 278

determination of equivalence between quantum sequential machines quantum sequential machines qsms that may be viewed as a quantum variant of stochastic sequential machines ssms are one of the important quantum computing models a very crucial result on ssms is that two ssms with n and n states respectively and the same input and output alphabets are equivalent if and only if they are n n 1 equivalent therefore gudder asked whether or not it holds for qsms as an open problem in this direction since the further study is closely related to this result qiu demonstrated that in qsms this result does not hold and therefore answered in part the problem however as qiu indicated the sufficient and necessary conditions for the equivalence between two qsms are still not discovered in this paper we show that if the condition of n n 1 equivalence is appropriately relaxed then we can give a sufficient and necessary condition for justifying the equivalence between two qsms more precisely we show that any two qsms m and m are equivalent if and only if they are n n 2 equivalent where n and n are respectively the numbers of states in m and m we therefore solve the open problem suggested by gudder and provide a basic result for further developing qsms as well we discuss strongly factorizable qsms and present the conditions for the equivalence between two strongly factorizable qsms

geometric decompositions and local bases for spaces of finite element differential forms abstract we study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions these spaces are generalizations of the classical finite element spaces for vector fields frequently referred to as raviart thomas brezzi douglas marini and nedelec spaces in the present paper we derive geometric decompositions of these spaces which lead directly to explicit local bases for them generalizing the bernstein basis for ordinary lagrange finite elements the approach applies to both families of finite element spaces for arbitrary polynomial degree arbitrary order of the differential forms and an arbitrary simplicial triangulation in any number of space dimensions a prominent role in the construction is played by the notion of a consistent family of extension operators which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis

subcell flux limiting for high order bernstein finite element discretizations of scalar hyperbolic conservation laws abstract this work extends the concepts of algebraic flux correction and convex limiting to continuous high order bernstein finite element discretizations of scalar hyperbolic problems using an array of adjustable diffusive fluxes the standard galerkin approximation is transformed into a nonlinear high resolution scheme which has the compact sparsity pattern of the piecewise linear or multilinear subcell discretization the representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under cfl like conditions in contrast to predictor corrector approaches based on the flux corrected transport methodology the proposed flux limiting strategy is monolithic i e limited antidiffusive terms are incorporated into the well defined residual of a nonlinear semi discrete problem a stabilized high order galerkin discretization is recovered if no limiting is performed in the limited version the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors a formal proof of sparsity is provided for simplicial and box elements the involved element contributions can be calculated efficiently making use of matrix free algorithms and precomputed element matrices of the reference element numerical studies for q 2 discretizations of linear and nonlinear two dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions

bernstein bezier finite elements of arbitrary order and optimal assembly procedures algorithms are presented that enable the element matrices for the standard finite element space consisting of continuous piecewise polynomials of degree n on simplicial elements in mathbb r d to be computed in optimal complexity mathcal o n 2d the algorithms i take into account numerical quadrature ii are applicable to nonlinear problems and iii do not rely on precomputed arrays containing values of one dimensional basis functions at quadrature points although these can be used if desired the elements are based on bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements

index coding with side information motivated by a problem of transmitting supplemental data over broadcast channels birk and kol infocom 1998 we study the following coding problem a sender communicates with n receivers r1 rn he holds an input x 0 01l n and wishes to broadcast a single message so that each receiver ri can recover the bit xi each ri has prior side information about x induced by a directed graph grain nodes ri knows the bits of a in the positions j i j is an edge of g g is known to the sender and to the receivers we call encoding schemes that achieve this goal indexcodes for 0 1 n with side information graph g in this paper we identify a measure on graphs the minrank which exactly characterizes the minimum length of linear and certain types of nonlinear index codes we show that for natural classes of side information graphs including directed acyclic graphs perfect graphs odd holes and odd anti holes minrank is the optimal length of arbitrary index codes for arbitrary index codes and arbitrary graphs we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph this bound holds even for randomized codes but has been shown not to be tight

orientation embedded high order shape functions for the exact sequence elements of all shapes a unified construction of high order shape functions is given for all four classical energy spaces h1 h curl h div and l2 and for elements of all shapes segment quadrilateral triangle hexahedron tetrahedron triangular prism and pyramid the discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element the shape functions are conforming hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape the polynomial order is allowed to differ for each separate topological entity vertex edge face or interior in the mesh so the shape functions can be used to implement local p adaptive finite element methods each topological entity may have its own orientation and the shape functions can have that orientation embedded by a simple permutation of arguments