a topos foundation for theories of physics iv categories of systems this paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics the motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos the previous papers in this series are concerned with implementing this program for a single system in the present paper we turn to considering a collection of systems in particular we are interested in the relation between the topos representation for a composite system and the representations for its constituents we also study this problem for the disjoint sum of two systems our approach to these matters is to construct a category of systems and to find a topos representation of the entire category

trigonometric hermite wavelet approximation for the integral equations of second kind with weakly singular kernel this paper is concerned with a trigonometric hermite wavelet galerkin method for the fredholm integral equations with weakly singular kernel the kernel function of this integral equation considered here includes two parts a weakly singular kernel part and a smooth kernel part the approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric hermite wavelet constructed by e quak trigonometric wavelets for hermite interpolation math comp 65 1996 683 722 are developed the use of trigonometric hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix it is shown that we only need to compute and store o n entries for the weakly singular kernel representation matrix with dimensions n 2 which can reduce the whole computational cost and storage expense the computational schemes of the resulting matrix elements are provided for the weakly singular kernel function furthermore the convergence analysis is developed for the trigonometric wavelet method in this paper

the discrete first order system least squares the second order elliptic boundary value problem in z cai t manteuffel and s f mccormick siam j numer anal 34 1997 pp 425 454 an l2 norm version of first order system least squares fosls was developed for scalar second order elliptic partial differential equations a limitation of this approach is the requirement of sufficient smoothness of the original problem which is used for the equivalence of spaces between h1 d and h div cap h rm curl type where d 2 or 3 is the dimension by directly approximating h div cap h rm curl type space based on the helmholtz decomposition this paper develops a discrete fosls approach in two dimensions under general assumptions we establish error estimates in the l2 and h1 norms for the vector and scalar variables respectively such error estimates are optimal with respect to the required regularity of the solution a preconditioner for the algebraic system arising from this approach is also considered

piecewise trigonometric hermite interpolation based on the symmetric nonnegative and normalized basis of the trigonometric polynomial space the piecewise trigonometric hermite interpolation methods are presented the c n 1 and cn continuous piecewise trigonometric hermite interpolants of degree n are constructed and the interpolation methods are local the integral and the differential representations of the errors of the trigonometric hermite interpolants are given several examples are supplied to support the practical value of the given interpolation methods

trigonometric wavelet method for some elliptic boundary value problems abstract in this paper we apply trigonometric wavelet method to investigate the numerical solution of elliptic boundary value problem in an exterior unit disk the simple computation formulae of the entries in the stiffness matrix are obtained it shows that we only need to compute 2 2 j 1 elements of one 2 j 2 2 j 2 stiffness matrix moreover the error estimates of the approximation solutions are given and some test examples are presented

a negative norm least squares method for reissner mindlin plates in this paper a least squares method using the minus one norm developed by bramble lazarov and pasciak is introduced to approximate the solution of the reissner mindlin plate problem with small parameter t the thickness of the plate the reformulation of brezzi and fortin is employed to prevent locking taking advantage of the least squares approach we use only continuous finite elements for all the unknowns in particular we may use continuous linear finite elements the difficulty of satisfying the inf sup condition is overcome by the introduction of a stabilization term into the least squares bilinear form which is very cheap computationally it is proved that the error of the discrete solution is optimal with respect to regularity and uniform with respect to the parameter t apart from the simplicity of the elements the stability theorem gives a natural block diagonal preconditioner of the resulting least squares system for each diagonal block one only needs a preconditioner for a second order elliptic problem

self adjoint operators as functions i lattices galois connections and the spectral order observables of a quantum system described by self adjoint operators in a von neumann algebra or affiliated with it in the unbounded case form a conditionally complete lattice when equipped with the spectral order using this order theoretic structure we develop a new perspective on quantum observables

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 topos foundation for theories of physics i formal languages for physics this paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we discuss two different types of language that can be attached to a system s the first is a propositional language pl s the second is a higher order typed language l s both languages provide deductive systems with an intuitionistic logic the reason for introducing pl s is that as shown in paper ii of the series it is the easiest way of understanding and expanding on the earlier work on topos theory and quantum physics however the main thrust o

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

a comparison of two topos theoretic approaches to quantum theory the aim of this paper is to compare the two topos theoretic approaches to quantum mechanics that may be found in the literature to date the first approach which we will call the contravariant approach was originally proposed by isham and butterfield and was later extended by doring and isham the second approach which we will call the covariant approach was developed by heunen landsman and spitters

spectral order for unbounded operators abstract the spectral order a notion originated by olson for bounded operators is investigated here in the context of unbounded operators dissimilarities between bounded and unbounded cases are pointed out new criteria for two operators to be comparable are supplied a way of reducing the study of the spectral order to the case of bounded operators is proposed connections with essential selfadjointness are established integral inequalities for monotonically increasing functions are characterized in terms of distribution functions some illustrative examples are furnished

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

a topos foundation for theories of physics ii daseinisation and the liberation of quantum theory this paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we study in depth the topos representation of the propositional language pl s for the case of quantum theory in doing so we make a direct link with and clarify the earlier work on applying topos theory to quantum physics the key step is a process we term daseinisation by which a projection operator is mapped to a subobject of the spectral presheaf the topos quantum analog of a classical state space in the second part of the paper we change gear

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

finite element approximation for grad div type systems in the plane this paper studies a 3 times 3 linear system of differential equations of grad div type with appropriate boundary conditions in constructing numerical approximate solutions to this boundary val

multidimensional spectral order for selfadjoint operators abstract the notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in hilbert space it is shown that the multidimensional spectral order is preserved by transformations represented by spectral integrals of separately increasing borel functions on r in particular the dimensional spectral order is the restriction of product of spectral orders for selfadjoint operators in the context of positive tuples of pairwise commuting selfadjoint operator the relation a b holds if and only if a b for every tuples of nonnegative integer numbers

vigier s theorem for the spectral order and its applications abstract the paper mainly deals with suprema and infima of self adjoint operators in a von neumann algebra m with respect to the spectral order let m s a be the self adjoint part of m and let be the spectral order on m s a we show that a decreasing net in m s a with a lower bound has the infimum equal to the strong operator limit the similar statement is proved for an increasing net bounded above in m s a this version of vigier s theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self adjoint operators in terms of the strong operator limit and operator means as an application of our results on suprema and infima we study the order topology on m s a with respect to the spectral order we show that it is finer than the restriction of the mackey topology

the generalized marcinkiewicz zygmund inequality for trigonometric polynomials abstract the well known marcinkiewicz zygmund inequality 0 2x t x p dx 1 p a p 1 n k 0 2n t 2k 2n 1 p 1 p 1 where t is a trigonometric polynomial of degree n and ap depends on p only will be generalized to have the right hand side containing the derivative values as well the results will then be applied to prove the mean convergence of the hermite interpolation by trigonometric polynomials on equidistant nodes

trigonometric wavelets for hermite interpolation the aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials the pair of scaling functions which span the sample spaces are fundamental functions for hermite interpolation on a dyadic partition of nodes on the interval 0 2 two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions together with the corresponding dual functions these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences consequently this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment

the natural integral equations of plane elasticity problem and its wavelet methods in this paper we apply interpolatory hermite type trigonometric wavelet to investigate the numerical solution of the natural boundary integral equation of plane elasticity problem by galerkin method in our fast algorithm the computational formulae of entries of the stiffness matrix yield simple close form and for one 2 j 3x2 j 3 stiffness matrix we only need to compute 2 2 j 2 2 j 1 entries the error estimates of the approximate solution are given and the test examples are presented in the end

a topos foundation for theories of physics iii the representation of physical quantities with arrows this paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system in paper ii we studied the topos representations of the propositional language pl s for the case of quantum theory and in the present paper we do the same thing for the more extensive local language l s one of the main achievements is to find a topos representation for self adjoint operators this involves showing that for any physical quantity a there is an arrow o a r where r is the quantity value object for this theory the construction of o a is an extension of the daseinisation of projection operators that was discussed in paper ii the object r is a monoid object only in the topos setsv h op of the theory and to enhance the applicability of the formalism we apply to r a topos a

normalized b basis of the space of trigonometric polynomials and curve design a normalized b basis of the space of trigonometric polynomials of degree n is presented some interesting properties of the basis functions are given based on the basis symmetric trigonometric polynomial curves like bezier curves are constructed the trigonometric polynomial curves present the shape of their control polygons well thus the theoretics and methods are proposed for curve representation of the trigonometric polynomial space by adding additional control points the given graph examples show that the trigonometric polynomial curves are nearer their control polygons than the bezier curves for the same parametric variable and the same degree

piecewise quadratic trigonometric polynomial curves analogous to the quadratic b spline curve a piecewise quadratic trigonometric polynomial curve is presented in this paper the quadratic trigonometric polynomial curve has c2 continuity while the quadratic b spline curve has c1 continuity the quadratic trigonometric polynomial curve is closer to the given control polygon than the quadratic b spline curve

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

the least squares pseudo spectral method for navier stokes equations a spectral collocation approximation of first order system least squares for incompressible stokes equations was analyzed in kim et al 2004 12 and finite element approximations for incompressible navier stokes equations were developed in bochev et al 1998 1999 9 10 the aim of this paper is to analyze the first order system least squares pseudo spectral method for incompressible navier stokes equations the paper will be an extension of the result in kim et al 2004 12 to the navier stokes equations our least squares functional is defined by the sum of discrete spectral norms of a first order system of equations corresponding to the navier stokes equations based on legendre gauss lobatto points we show its ellipticity and continuity over an appropriate product space and spectral convergences of discretization errors are derived in the h 1 norm and the l 2 norm in each variable finally we present some numerical examples