quantum reverse hypercontractivity its tensorization and application to strong converses in this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log sobolev inequalities next we prove a generalization of the stroock varopoulos inequality in the non commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2 log sobolev and 1 log sobolev inequalities respectively we then prove some tensorization type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers finally as an application of these results we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting we prove strong converse bounds for the problems of quantum hypothesis testing and classical quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

a least squares finite element formulation for unsteady incompressible flows with improved velocity pressure coupling in the weak form galerkin formulation for incompressible flows the pressure has a well understood role at all times it may be interpreted as a lagrange multiplier that enforces the divergence free constraint on the velocity field this is not the case in least squares formulations for incompressible flows where the divergence free constraint is enforced in a least squares sense in a variational setting of residual minimization thus the role of the pressure in a least squares formulation is rather vague we find that this lack of velocity pressure coupling in least squares formulations may induce spurious temporal pressure oscillations when using the non stationary form of the equations we present a least squares formulation with improved velocity pressure coupling based on the use of a regularized divergence free constraint a first order system least squares fosls approach based on velocity pressure and vorticity is used to allow the use of practical c0 element expansions in the finite element model we use high order spectral element expansions in space and second and third order time stepping schemes excellent conservation of mass and accuracy of computed pressure metrics are demonstrated in the numerical results

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

on mass conserving least squares methods least squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations including symmetric positive definite discrete operators and a sharp error measure one of the potential drawbacks especially in three dimensions is that mass conservation is achieved only in a least squares sense and underresolved solutions are especially vulnerable to poor conservation for the stationary navier stokes equations which are typically rewritten as a larger system of first order equations the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used a new first order system reformulation of the navier stokes equations is presented that admits a wider range of mass conserving boundary conditions this new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems a square channel backward facing step and branching tubes with two generations of bifurcations

efficiency based adaptive local refinement for first order system least squares formulations in this paper we propose new adaptive local refinement alr strategies for first order system least squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration the goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements to accomplish this the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost two efficiency measures are discussed predicted error reduction and predicted computational cost these methods are first applied to a two dimensional 2d poisson problem with steep gradients and the results are compared with the threshold based methods described in w dorfler siam j numer anal 33 1996 pp 1106 1124 next these methods are applied to a 2d reduced model of the incompressible resistive magnetohydrodynamic equations these equations are used to simulate instabilities in a large aspect ratio tokamak we show that by using the new alr strategies on this system we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance

multiary α resolution principle for a lattice valued logic this paper focuses on resolution based automated reasoning theory in a lattice valued logic system with truth values that are defined in a lattice valued logical algebraic structure lattice implication algebras lias which essentially aims to extend the classical logic to handle automated deduction under an uncertain environment concretely we investigate a generalization of the known conjunctive normal form cnf in the classical logic representation that we call the generalized conjunctive normal form gcnf which aims to characterize the constants and implication connectives that are essentially different from the ones in classical logic we then extend the established resolution principle at a certain truth value level called resolution in this lattice valued logic to a more general form i e from binary resolution to multiary resolution the extension to multiary resolution starts from lattice valued propositional logic lp x while its theorems of both soundness and completeness are proved multiary resolution principle is then further established in the corresponding lattice valued first order logic lf x along with its soundness theorem lifting lemma and completeness theorem meanwhile an important result that multiary resolution principle in lf x can be equivalently transformed into that in lp x to some extent is obtained all these works will theoretically support the establishment of an automated reasoning algorithm and its implementation with further applications into automated deduction and decision making problems under uncertainty

on the refutational completeness of signed binary resolution and hyperresolution signed binary resolution and hyperresolution belong to the basic resolution proof methods both the resolution methods are refutation sound and under some finitary restrictions refutation complete our aim is to investigate their refutational completeness in a more general case we shall assume that clausal theories may be countable sets and the set of truth values an arbitrary one there are unsatisfiable countable clausal theories for which there do not exist refutations by signed binary resolution and hyperresolution we propose a criterion on the existence of a refutation of an unsatisfiable countable clausal theory by the investigated resolution methods two important applications of the achieved results to automated deduction in signed logic herbrand s theorem and a signed davis putnam logemann loveland dpll procedure will be discussed

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

hp adaptive least squares spectral element method for hyperbolic partial differential equations this paper describes a hp adaptive spectral element formulation which is used to discretize the weak formulation obtained by minimizing the residuals in the l 2 norm the least squares error indicator will be briefly discussed refinement of the numerical approximation is based on an estimate of the regularity of the underlying exact solution if the underlying exact solution is sufficiently smooth polynomial enrichment is employed in areas with limited regularity h refinement is used for this purpose the sobolev regularity is estimated functionally and geometrically non conforming neighbouring elements are patched together using so called mortar elements results of this approach are compared to uniform h and p refinement for a linear advection equation

binary resolution over complete residuated stone lattices this paper extends previous results obtained for binary resolution proof method over complete boolean lattices we shall assume that truth values form a complete residuated stone lattice in addition we investigate binary resolution over residuated lattices with strict negation the refutational soundness and completeness of the generalised resolution proof method will be proved finally we discuss the boundaries of the refutational completeness over arbitrary residuated lattices

least squares finite element method with high continuity nurbs basis for incompressible navier stokes equations modern least squares finite element method lsfem has advantage over mixed finite element method for non self adjoint problem like navier stokes equations but has problem to be norm equivalent and mass conservative when using c 0 type basis in this paper lsfem with non uniform b splines nurbs is proposed for navier stokes equations high order continuity nurbs is used to construct the finite dimensional spaces for both velocity and pressure variational form is derived from the governing equations with primitive variables and the dofs due to additional variables are not necessary there is a novel k refinement which has spectral convergence of least squares functional the method also has same advantages as isogeometric analysis like automatic mesh generation and exact geometry representation several benchmark problems are solved using the proposed method the results agree well with the benchmark solutions available in literature the results also show good performance in mass conservation

conservation of mass and momentum of the least squares spectral collocation scheme for the stokes problem from the literature it is known that spectral least squares schemes perform poorly with respect to mass conservation and compensate this lack by a superior conservation of momentum this should be revised since the here presented new least squares spectral collocation scheme leads to an outstanding performance with respect to conservation of momentum and mass the reasons can be found in using only a few elements each with high polynomial degree avoiding normal equations for solving the overdetermined linear systems of equations and by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with our least squares spectral collocation scheme to discretize the internal flow problems

spectral element formulations on non conforming grids a comparative study of pointwise matching and integral projection methods pointwise matching pm and integral projection ip methods are two widely used techniques to extend the classical weak formulations to include non conforming grids we present spectral element formulations on polynomial p type and geometric h type non conforming grids using both the pm also known as the constrained approximation and ip also known as the mortar element methods we systematically compare the convergence characteristics of pm and ip methods for diffusion convection and convection diffusion equations consistency errors due to the non conforming formulations of the diffusion equation result in convergence problems for the pm method using the maximum rule both non conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values causing convergence problems due to the consistency errors however small physical diffusion in the convection diffusion equation eliminates these problems resulting in spectral convergence for both methods encouraged by this we present spectral element formulations for incompressible navier stokes equations using pm and ip methods on p type and h type non conforming grids and demonstrate spectral convergence for unsteady and steady test cases results for two dimensional lid driven cavity flow at re 1000 are also presented

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

a collocation penalty least squares finite element formulation for incompressible flows abstract we present a penalty least squares finite element formulation for incompressible flows the pressure degree of freedom is eliminated using the classical penalty approach and the least squares model is formed in terms of velocity vorticity and dilatation an iterative penalization is implemented which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix the h convergence is verified using the exact solution of kovasznay s flow numerical results are presented for a number of benchmark problems e g steady flow past a large circular cylinder in a channel transient flow over a backward facing step unsteady flow past a circular cylinder and buoyancy driven natural convection in a square cavity for all numerical examples the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters 10 100 these problems are also solved on coarse meshes to show that good mass conservation is achieved

a least squares finite element method for the navier stokes equations the navier stokes equations for flow in a plane are reformulated as a first order system in terms of stress and stream functions solutions of this system are obtained by the least squares finite element method a feature of this approach is that the linearised system gives rise to a symmetric and positive definite linear algebra problem at each newton iteration care over handling the incompressibility term is needed to ensure good results are obtained

spectral hp penalty least squares finite element formulation for the steady incompressible navier stokes equations in this paper we present spectral hp penalty least squares finite element formulation for the numerical solution of steady incompressible navier stokes equations the continuity equation is treated as a constraint on velocity field and the constraint is enforced using the penalty method which is equivalent to replacing the pressure variable p with p n p n email protected u where u is the velocity field c is the penalty parameter and superscript n is non linear iteration number the modified navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity dilatation or stresses as additional independent variables and the least squares method is used to develop the finite element model high order basis functions are used to construct the discrete model unlike other penalty finite element formulations equal order integration is used for all terms of the coefficient matrix schur complement method is used and all interior element degrees of freedom are condensed out and interface degrees of freedom are solved either by direct solver banded cholesky factorization or iterative solver preconditioned conjugate gradient present formulation produces very accurate results for even very low penalty parameters 10 40 in this way present formulation and implementation fixes perennial problem of bad conditioning of coefficient matrix associated with penalty formulations proposed formulation is verified using the exact solution of kovasznay flow numerical results are also presented for a number of two dimensional benchmark problems e g 2d lid driven cavity flow flow over a backward facing step steady flow past a circular cylinder and 2d flow past a large circular cylinder in a channel finally the proposed formulation is extended to coupled velocity temperature problems and buoyant flow inside a square enclosure is solved to test the penalty least squares finite element model developed herein

binary resolution over boolean lattices abstract we propose a binary resolution system over boolean lattices we shall assume that truth values form a complete boolean lattice l where its supremum operator is infinitely distributive over the infimum one binary l resolution will operate on graded clauses a graded clause is a pair of the form l 1 l n c where l i are literals and c is a truth degree in l the refutational soundness and completeness of the proposed resolution proof method will be proved

least squares spectral element solution of incompressible navier stokes equations with adaptive refinement least squares spectral element solution of steady two dimensional incompressible flows are obtained by approximating velocity pressure and vorticity variable set on gauss lobatto legendre nodes constrained approximation method is used for h and p type nonconforming interfaces of quadrilateral elements adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient effective use of p refinement to overcome poor mass conservation drawback of least squares formulation and successful use of h and p refinement together to solve problems with geometric singularities are demonstrated capabilities and limitations of the developed code are presented using kovasznay flow flow past a circular cylinder in a channel and backward facing step flow

determination of α resolution in lattice valued first order logic lf x key issues for resolution based automated reasoning in lattice valued first order logic lf x are investigated with truth values in a lattice valued logical algebraic structure lattice implication algebra lia the determination of resolution at a certain truth value level called a resolution in lf x is proved to be equivalently transformed into the determination of a resolution in lattice valued propositional logic lp x based on lia the determination of a resolution of any quasi regular generalized literals and constants under various cases in lp x is further analyzed specified and subsequently verified hence the determination of a resolution in lf x can be accordingly solved to a very broad extent which not only lays a foundation for the practical implementation of automated reasoning algorithms in lf x but also provides a key support for a resolution based automated reasoning approaches and algorithms in lia based linguistic truth valued logics

dynamic refinement algorithms for spectral element methods abstract a numerical method is presented that combines high order domain decomposition and adaptive mesh refinement the spatial discretization is based on a nonconforming spectral element method meshes are generated by adaptive refinement of a quadtree using either local solution gradients or the local polynomial spectrum as a refinement criteria the method is applied to solve scalar test problems and to simulate the flow in a lid driven cavity at moderate reynolds numbers

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

adaptive refinement for least squares finite elements with element by element conjugate gradient solution an error analysis and an adaptive refinement scheme are developed for use with a class of least squares finite element methods the error indicators for the refinement procedure are based on element residuals which are calculated as part of the least squares method the solution scheme used in the supporting numerical studies employs an elment by element conjugate gradient algorithm consequently storage and computation are not influenced by nodal numbering