new homotopic factorization and symmetrization techniques for newton s and newton structured iteration we modify our earlier homotopic continuation process for iterative matrix inversion to improve the choice of the initial approximate inverse at every homotopic step this enables us to control the condition of the auxiliary matrices and to accelerate the convergence of the iteration the algorithm also extends our earlier approach where we factorized the input matrix into the product of better conditioned factors we accelerate our new algorithm in the case of a toeplitz like input comment on the extensions to the computations with other structured matrices such as cauchy pick and banded matrices and propose new techniques for the compression of the displacement of hermitian matrices

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

an optimal error estimate for the two dimensional nonlinear time fractional advection diffusion equation with smooth and non smooth solutions abstract in this paper a linearized l1 legendre galerkin spectral method for the two dimensional nonlinear time fractional advection diffusion equation is derived the numerical method is stable without the cfl conditions based on the error splitting argument technique and the discrete fractional gronwall type inequality we also consider the case of non smooth solutions by adding some correction terms the numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method

the method of particular solutions for solving nonlinear poisson problems abstract based on the recent development in the method of particular solutions we re exam three approaches using different basis functions for solving nonlinear poisson problems we further propose to simplify the solution procedure by removing the insolvency condition when the radial basis functions are augmented with high order polynomial basis functions we also specify the deficiency of some of these methods and provide necessary remedy the traditional picard method is introduced to compare with the recent proposed methods using matlab optimization toolbox solver for solving nonlinear poisson equations ranking on these three approaches are given based on the results of numerical experiment

performance of group testing algorithms with near constant tests per item we consider the nonadaptive group testing with n items of which k theta n theta are defective we study a test design in which each item appears in nearly the same number of tests for each item we independently pick l tests uniformly at random with replacement and place the item in those tests we analyze the performance of these designs with simple and practical decoding algorithms in a range of sparsity regimes and show that the performance is consistently improved in comparison with standard bernoulli designs we show that our new design requires roughly 23 fewer tests than a bernoulli design when paired with the simple decoding algorithms known as combinatorial orthogonal matching pursuit and definite defectives dd this gives the best known nonadaptive group testing performance for theta 0 43 and the best proven performance with a practical decoding algorithm for all theta in 0 1 we also give a converse result showing that the dd algorithm is optimal with respect to our randomized design when theta 1 2 we complement our theoretical results with simulations that show a notable improvement over bernoulli designs in both sparse and dense regimes

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

multi quadric quasi interpolation method coupled with fdm for the degasperis procesi equation in this paper the degasperis procesi dp equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form u p based on the equivalent formulation of dp equation a new numerical method is presented the multi quadric mq quasi interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order tvd method is used to approximate the time derivative several examples are presented to demonstrate the effectiveness of the proposed method

a quasi radial basis functions method for american options pricing abstract based on the idea of quasi interpolation and radial basis functions approximation a fast and accurate numerical method is developed for solving the black scholes equation for valuation of american options prices since the method does not require solving a resultant full matrix the ill conditioning problem resulting from using the radial basis functions as a global interpolant can be avoided the method has been shown to be effective in solving problems with free boundary condition as indicated in the numerical computation for the american option pricing an excellent approximation of the solution as well as the free optimal exercise boundary can be obtained

a note on computational algorithm for the inverse of a square matrix this short note gives some comments on computational algorithm for inverse of a square matrix considered in paper h saberi najafi m shams solary computational algorithms for computing the inverse of a square matrix quasi inverse of a non square matrix and block matrices appl math comput in press doi 10 1016 j amc 2006 05 108 and supplements the discussion on this topic

an efficient chebyshev tau method for solving the space fractional diffusion equations fractional diffusion equations fdes have recently been paid much attention finding accurate and efficient methods for solving fdes has become an active research undertaking in this paper an efficient method based on the shifted chebyshev tau idea is presented to solve an initial boundary value problem for the fdes the method is derived by expanding the required approximate solution as the elements of shifted chebyshev polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from a computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and only a small number of shifted chebyshev polynomials is needed

block ilu factorization preconditioners for a block tridiagonal h matrix abstract we propose new parallelizable block ilu incomplete lu factorization preconditioners for a nonsymmetric block tridiagonal h matrix theoretical properties of these block ilu preconditioners are compared with those of block ilu preconditioners for the corresponding comparison matrix numerical results of the bicgstab using these block preconditioners are compared with those of the bicgstab using a standard ilu factorization preconditioner to see the effectiveness of the block ilu preconditioners lastly parallel computations of the block ilu preconditioners are carried out and evaluated on the cray c90

numerical solution of the degasperis procesi equation by the cubic b spline quasi interpolation method abstract in this paper a numerical scheme is presented to solve the non dissipative degasperis procesi equation based on the u p formulation the cubic b spline quasi interpolation coupled with the finite difference method is applied to approximate the spatial derivatives and an optimal third order tvd runge kutta method to estimate the time derivative of the dependent variable the accuracy and effectiveness of the proposed method are validated by six classical problems numerical results indicate that the proposed scheme is simple easy to implement with high accuracy

finding generalized inverses by a fast and efficient numerical method in this paper a method with very high order of convergence is constructed and analyzed the method is used to compute generalized inverses the efficiency index has been employed to show its superiority numerical experiments re verify that the proposed iterative expression is more effective than the existing methods of the same type

optimal two stage algorithms for group testing problems group testing refers to the situation in which one is given a set of objects cal o an unknown subset cal p subseteq cal o and the task of determining cal p by asking queries of the type does cal p intersect cal q where cal q is a subset of cal o group testing is a basic search paradigm that occurs in a variety of situations such as quality control testing searching in storage systems multiple access communications and data compression among others group testing procedures have been recently applied in computational molecular biology where they are used for screening libraries of clones with hybridization probes and sequencing by hybridization motivated by particular features of group testing algorithms used in biological screening we study the efficiency of two stage group testing procedures our main result is the first optimal two stage algorithm that uses a number of tests of the same order as the information theoretic lower bound on the problem we also provide efficient algorithms for the case in which there is a bernoulli probability distribution on the possible sets cal p and an optimal algorithm for the case in which the outcome of tests may be unreliable because of the presence of inhibitory items in cal o our results depend on a combinatorial structure introduced in this paper we believe that it will prove useful in other contexts too

a combinatorial approach to nonlocality and contextuality so far most of the literature on quantum contextuality and the kochen specker theorem seems either to concern particular examples of contextuality or be considered as quantum logic here we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by cabello severini and winter csw in contrast to csw we explicitly include the normalization of probabilities which gives us a much finer control over the various sets of probabilistic models like classical quantum and generalized probabilistic in particular our framework specializes to quantum nonlocality in the case of bell scenarios which arise very naturally from a certain product of contextuality scenarios due to foulis and randall in the spirit of csw we find close relationships to several graph invariants the recently proposed local orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the shannon capacity of graphs our results imply that it is strictly dominated by a low level of the navascues pironio acin hierarchy of semidefinite programs which we also apply to contextuality scenarios

boolean compressed sensing and noisy group testing the fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests measurements the key contribution of this paper is in adopting a new information theoretic perspective on group testing problems we formulate the group testing problem as a channel coding decoding problem and derive a single letter characterization for the total number of tests used to identify the defective set although the focus of this paper is primarily on group testing our main result is generally applicable to other compressive sensing models

computational algorithms for computing the inverse of a square matrix quasi inverse of a non square matrix and block matrices in this paper we have written out some methods to find the inverse of a non singular matrix and also the quasi inverse of a singular matrix the corresponding algorithms have been used to solve the linear system of equations ax b as x a 1b a comparison table of the numerical results has been designed in each case to compare the implementation of the algorithms with each other the important part of the paper is presenting a block matrix method for finding the inverse matrix the block method finds the inverse matrix with a simple and easy algorithm

the maps with polynomial basis functions for solving axisymmetric time fractional equations abstract this study presents a parallel meshless approach for time fractional equations in the axisymmetric geometry in the present parallel meshless approach the method of approximate particular solutions maps in conjunction with laplace transform extended precision arithmetic epa and multiple scale technique mst is implemented in this application the laplace transform is employed to eliminate the dependence on time the maps is used to obtain the solution of time independent equation in laplace space furthermore the solution of time dependent equation is restored by stehfest s algorithm here the multiple scale technique mst and extended precision arithmetic epa are applied to alleviate the effect of an ill conditioned matrix on a numerical inverse laplace transform then multi cores parallel computing can improve effectively computing speed of the proposed meshless approach in the numerical implementation both the 2d axisymmetric equations and the original 3d equations are considered and compared

contextuality and noncommutative geometry in quantum mechanics observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values dually states can be modelled as functions from the algebra of observables to outcome values the probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality noncommuting observables which are not assumed to be jointly measurable cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state here we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical deterministic state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry by adapting the spectral presheaf of hamilton isham butterfield a formulation of quantum state space that collates contextual data we reconstruct tools of noncommutative geometry in an explicitly geometric fashion in this way we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry a la connes et al to each unital c algebra mathcal a we associate a geometric object a diagram of topological spaces collating quotient spaces of the noncommutative space underlying mathcal a that performs the role of a generalised gel fand spectrum we show how any functor f from compact hausdorff spaces to a suitable target category mathsf c can be applied directly to these geometric objects to automatically yield an extension tilde f acting on all unital c algebras this procedure is used to give a novel formulation of the operator k0 functor via a finitary variant tilde k f of the extension tilde k of the topological k functor we then delineate a c algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital c algebra the zariski topological lattice of its primitive ideal spectrum i e its lattice of closed two sided ideals we prove the von neumann algebraic analogue of this conjecture

a modified chebyshev s iterative method with at least sixth order of convergence this paper is devoted to the construction and analysis of a high order variant of the classical chebyshev method the method has order of convergence at least six for simple roots the extension to system of equations and its semilocal convergence for nonlinear equations are presented finally an application to well known algebraic riccati equation is considered

a spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion wave equations in this paper an efficient and accurate spectral numerical method is presented for solving second fourth order fractional diffusion wave equations and fractional wave equations with damping the proposed method is based on jacobi tau spectral procedure together with the jacobi operational matrix for fractional integrals described in the riemann liouville sense the main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought for spectral approximations the validity and effectiveness of the method are demonstrated by solving five numerical examples numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier an efficient numerical scheme for time fractional diffusion wave equations is proposed a time space jacobi tau approximation is developed for such equations several tau spectral methods can be achieved as special cases the validity and applicability of the proposed method are demonstrated accurate numerical results are obtained by selecting limited collocation nodes

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

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

high accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems we investigate the mechanical quadrature methods mqms for solving boundary integral equations bies of the three dimensional axisymmetric elasticity equations with dirichlet s problems the convergence of the mqms is proved based on anselone s collective compact theory moreover an asymptotic error expansion of the mqms shows that the approximation order is of o h 3 numerical examples are tested and the results verify the theoretical analysis

active lattices determine aw algebras abstract we prove that operator algebras that have enough projections are completely determined by those projections their symmetries and the action of the latter on the former this includes all von neumann algebras and all aw algebras we introduce active lattices which are formed from these three ingredients more generally we prove that the category of aw algebras is equivalent to a full subcategory of active lattices crucial ingredients are an equivalence between the category of piecewise aw algebras and that of piecewise complete boolean algebras and a refinement of the piecewise algebra structure of an aw algebra that enables recovering its total structure

approximation to the k th derivatives by multiquadric quasi interpolation method quasi interpolation is very useful in the study of approximation theory and its applications since it can yield solutions directly without the need to solve any linear system of equations based on the good performance simple to evaluate of multiquadric functions the advantages approximation order convexity and monotonlcity preserving of multiquadric quasi interpolation have been widely discussed however it is usually only for the approximation properties of the function itself and the good properties for derivatives whereas the high order derivatives have been largely ignored in this paper we go further into the approximation properties to the k th derivatives by using multiquadric quasi interpolation furthermore we develop two kinds of multiquadric quasi interpolation schemes on bounded interval x 0 x n whose derivatives converge to the corresponding derivatives of the approximated functions finally the numerical experiments are presented to confirm the accuracy of the presented scheme both theoretical results and numerical examples show this scheme provides good accuracy even if the data points in x 0 x n are irregularly distributed

solving time dependent differential equations by multiquadric trigonometric quasi interpolation multiquadric mq quasi interpolation is a popular method for the numerical solution of differential equations however mq quasi interpolation is not well suited for the equations with periodic solutions this is mainly due to the fact that its kernel the mq function is not a periodic function a reasonable way of overcoming the difficulty is to use a quasi interpolant whose kernel itself is also periodic in these cases the paper constructs such a quasi interpolant error estimates of the quasi interpolant are also provided the quasi interpolant possesses many fair properties of the mq quasi interpolant i e simplicity efficiency stability etc moreover it is more suitable than the mq quasi interpolant for periodic problems since the quasi interpolant as well as its derivatives are periodic examples of solving both linear and nonlinear partial differential equations whose solutions are periodic by the quasi interpolant and the mq quasi interpolant are compared at the end of the paper numerical results show that the quasi interpolant outperforms the mq quasi interpolant for periodic problems

stability of peakons for the degasperis procesi equation the degasperis procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the euler equations with the same asymptotic accuracy as that of the camassaholm equation in this paper we study the orbital stability problem of the peaked solitons to the degasperis procesi equation on the line by constructing a liapunov function we prove that the shapes of these peakon solitons are stable under small perturbations

hybrid chebyshev polynomial scheme for solving elliptic partial differential equations abstract in this paper we propose hybrid chebyshev polynomial scheme hcps which couples the chebyshev polynomial scheme and the method of fundamental solutions into a single matrix system this hybrid formulation requires solving only one system of equations and opens up the possibilities for solving a large class of partial differential equations in this paper we consider various boundary value problems and in particular the challenging cauchy navier equation the solution is approximated by the sum of the particular solution and the homogeneous solution chebyshev polynomials are used to approximate a particular solution of the given partial differential equation and the method of fundamental solutions is used to approximate the homogeneous solution numerical results show that our proposed approach is efficient accurate and stable

measures on projections and physical states it is shown that a finitely additive measure on the projections of a von neumann algebra withouti2 andii1 summands is the restriction of a state a definition of a physical state is proposed and it is shown that such a physical state on a simplec algebra with unit is a state

efficient algorithms for noisy group testing group testing refers to the problem of identifying with high probability a small subset of d defectives from a large set of n items via a small number of pooled tests i e tests that have a positive outcome if at least one of the items being tested in the pool is defective else have a negative outcome for ease of presentation in this paper we focus on the regime when d mathcal o n 1 delta for some delta 0 the tests may be noiseless or noisy and the testing procedure may be adaptive the pool defining a test may depend on the outcome of a previous test or non adaptive each test is performed independent of the outcome of other tests a rich body of the literature demonstrates that theta d log n tests are information theoretically necessary and sufficient for the group testing problem and provides algorithms that achieve this performance however it is only recently that reconstruction algorithms with computational complexities that are sub linear in n have started being investigated in the scenario with adaptive tests with noisy outcomes we present the first scheme that is simultaneously order optimal up to small constant factors in both the number of tests and the decoding complexity mathcal o left d log n right in both the performance metrics the total number of stages of our adaptive algorithm is small mathcal o left log d right similarly in the scenario with non adaptive tests with noisy outcomes we present the first scheme that is simultaneously near optimal in both the number of tests and the decoding complexity via an algorithm that requires mathcal o left d log d log n right tests and has a decoding complexity of mathcal o d log n log 2 d finally we present an adaptive algorithm that only requires two stages and for which both the number of tests and the decoding complexity scale as mathcal o d log n log 2 d for all three settings the probability of error of our algorithms scales as mathcal o left 1 poly d right for each of the statements mentioned earlier about the order of the number of measurements decoding complexity and probability of error we provide explicitly computed small universal factors in our theorem statements

a dispersively accurate compact finite difference method for the degasperis procesi equation in this paper we are aimed to solve the non dissipative degasperis procesi equation based on the u p formulation to resolve the computational difficulty at the wave crest where the first order derivative may diverge and the shockpeakon solution may form the first order spatial derivative term in the two step equations will be approximated in a conservative form the resulting equations will be approximated by the symplecticity preserving time stepping scheme and the spatial discretization scheme that can optimize the numerical wavenumber for the first order spatial derivative term this scheme will be developed in a three point grid stencil with the accuracy order of seventh within the combined compact finite difference framework besides the validation of numerical accuracy we will in particular address the discrete conservation of hamiltonians even when peakon collides with antipeakon and generates as a result a shockpeakon we will also demonstrate the capability of applying the proposed numerical method to sharply resolve some important features of the third order dispersive dp equation

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

fundamental kernel based method for backward space time fractional diffusion problem based on kernel based approximation technique we devise in this paper an efficient and accurate numerical scheme for solving a backward space time fractional diffusion problem bstfdp the kernels used in the approximation are the fundamental solutions of the space time fractional diffusion equation expressed in terms of inverse fourier transform of mittag leffler functions the use of inverse fast fourier transform ifft technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the bstfdp since the bstfdp is intrinsic ill posed we apply the standard tikhonov regularization technique to obtain a stable solution to the highly ill conditioned resultant system of linear equations for choosing optimal regularization parameter we combine the regularization technique with the generalized cross validation gcv method for an optimal placement of the source points in the use of fundamental solutions meanwhile the proposed algorithm also speeds up the previous method given in dou and hon 2014 several numerical examples are constructed to verify the accuracy and efficiency of the proposed method

the method of particular solutions using trigonometric basis functions abstract in this paper the method of particular solutions mps using trigonometric functions as the basis functions is proposed to solve two dimensional elliptic partial differential equations the inhomogeneous term of the governing equation is approximated by fourier series and the closed form particular solutions of trigonometric functions are derived using the method of undetermined coefficients once the particular solutions for the trigonometric basis functions are derived the standard mps can be applied for solving partial differential equations in comparing with the use of radial basis functions and polynomials in the mps our proposed approach provides another simple approach to effectively solving two dimensional elliptic partial differential equations five numerical examples are provided in this paper to validate the merits of the proposed meshless method

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

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

on the diagonal approximation of full matrices in this paper the construction of diagonal matrices in some sense approximating the inverse of a given square matrix is described the matrices are constructed using the well known computer algebra system maple the techniques we show are applicable to square matrices in general results are given for use in parallel diagonal implicit runge kutta pdirk methods for an s stage radau iia corrector we conjecture s possibilities for the diagonal matrices

conservative finite difference schemes for the degasperis procesi equation we consider the numerical integration of the degasperis procesi equation which was recently introduced as a completely integrable shallow water equation for the equation we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi hamiltonian form of the equation at the same time we also prove the unique solvability of the schemes and show some numerical examples

numerical solution of space fractional diffusion equation by the method of lines and splines abstract this paper is devoted to the application of the method of lines to solve one dimensional diffusion equation where the classical integer second derivative is replaced by a fractional derivative of the caputo type of order less than 2 as the space derivative a system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the caputo derivative the result is a numerical approach of order o x 2 t m where x and t denote spatial and temporal step sizes and 1 m 5 is an integer which is set by an ode integrator that we used the convergence and numerical stability of the method are considered and numerical tests to investigate the efficiency and feasibility of the scheme are provided

an operator splitting method for the degasperis procesi equation an operator splitting method is proposed for the degasperis procesi dp equation by which the dp equation is decomposed into the burgers equation and the benjamin bona mahony bbm equation then a second order tvd scheme is applied for the burgers equation and a linearized implicit finite difference method is used for the bbm equation furthermore the strang splitting approach is used to construct the solution in one time step the numerical solutions of the dp equation agree with exact solutions e g the multipeakon solutions very well the proposed method also captures the formation and propagation of shockpeakon solutions and reveals wave breaking phenomena with good accuracy

an iterative method for computing the approximate inverse of a square matrix and the moore penrose inverse of a non square matrix in this paper an iterative scheme is proposed to find the roots of a nonlinear equation it is shown that this iterative method has fourth order convergence in the neighborhood of the root based on this iterative scheme we propose the main contribution of this paper as a new high order computational algorithm for finding an approximate inverse of a square matrix the analytical discussions show that this algorithm has fourth order convergence as well next the iterative method will be extended by theoretical analysis to find the pseudo inverse also known as the moore penrose inverse of a singular or rectangular matrix numerical examples are also made on some practical problems to reveal the efficiency of the new algorithm for computing a robust approximate inverse of a real or complex matrix

analysis on the method of fundamental solutions for biharmonic equations abstract in this paper the error and stability analysis of the method of fundamental solution mfs is explored for biharmonic equations the bounds of errors are derived for the fundamental solutions r 2 ln r in bounded simply connected domains and the polynomial convergence rates are obtained for certain smooth solutions the bounds of condition number are also derived to show the exponential growth rates for disk domains numerical experiments are carried out to support the above analysis which is the first time to provide the rigorous analysis of the mfs using r 2 ln r for biharmonic equations

approximate iterations for structured matrices important matrix valued functions f a are e g the inverse a 1 the square root sqrt a and the sign function their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices a and f a often f a possesses this structure only approximately however intermediate matrices arising during the evaluation may lose the structure of the initial matrix this would make the computations inefficient and even infeasible however the main result of this paper is that an iterative fixed point like process for the evaluation of f a can be transformed under certain general assumptions into another process which preserves the convergence rate and benefits from the underlying structure it is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique we demonstrate our results by verifying all requirements in the case of the iterative computation of a 1 and sqrt a

an efficient computation of generalized inverse of a matrix abstract we propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop this yields a high efficiency index for that computational task the algorithm has a relatively mild numerical instability and we stabilize it at the price of adding two extra matrix multiplications per iteration loop this implies an efficiency index that exceeds the known record for numerically stable iterations for this task which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra our numerical tests cover a variety of examples in the category of generalized inverses such as drazin case rectangular case and preconditioning of linear systems the test results are in good accordance with our formal study

etale groupoids and their quantales abstract we establish close and previously unknown relations between quantales and groupoids in particular to each etale groupoid either localic or topological there is associated a unital involutive quantale we obtain a bijective correspondence between localic etale groupoids and their quantales which are given a rather simple characterization and here are called inverse quantal frames we show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids and as a consequence we obtain a non functorial correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids this generalization is entirely algebraic and it is valid in an arbitrary topos as a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen and an analogue of this holds for topological groupoids in practice we are provided with new tools for constructing localic and topological etale groupoids as well as inverse semigroups for instance via presentations of quantales by generators and relations the characterization of inverse quantal frames is to a large extent based on a new quantale operation here called a support whose properties are thoroughly investigated and which may be of independent interest

numerical simulation for the space fractional diffusion equations abstract this paper presents a novel semi analytical algorithm based on the chebyshev collocation method for the solution of space fractional diffusion equations the original fractional equation is transformed into a system of ordinary differential equations odes by the chebyshev collocation method a new semi analytical method is then used to approximate the solution of the resulting system to emphasize the reliability of the new scheme a convergence analysis is presented it is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased by presenting some numerical examples we show that the proposed method is a powerful and reliable algorithm for solving space fractional diffusion equations and can be extended to solve another space fractional partial differential equations comparison with other methods in the literature demonstrates that the proposed method is both efficient and accurate

group testing with random pools optimal two stage algorithms we study the group testing of a set of n items each of which is defective with probability p we focus on the double limit of small defect probability p 1 and large number of variables n 1 taking either p 0 after n or p 1 n with 0 1 2 in both settings the optimal number of tests which are required to identify with certainty the defectives via a two stage procedure t n p is known to scale as np log p here we determine the sharp asymptotic value of t n p np log p and construct a class of two stage algorithms over which this optimal value is attained this is done by choosing a proper bipartite regular graph of tests and variable nodes for the first stage of the detection furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree and the tests connected to a given variable are randomly chosen with uniform distribution among all tests finally we improve the existing upper and lower bounds for the optimal number of tests in the case p 1 n with 1 2 1

solving a multi order fractional differential equation using adomian decomposition abstract an algorithm has been developed to convert the multi order fractional differential equation d y t f t y t d 1 y t d n y t y k 0 c k k 0 m where m d denotes caputo fractional derivative of order into a system of fractional differential equations further adomian decomposition method has been employed to solve the system of fractional differential equations some illustrative examples are presented

the radial basis function differential quadrature method for elliptic problems in annular domains abstract we employ a radial basis function rbf differential quadrature dq method for the numerical solution of elliptic boundary value problems in annular domains with an appropriate selection of collocation points for any choice of rbf both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures these linear systems can thus be solved efficiently using matrix decomposition algorithms mdas and fast fourier transforms ffts in particular we consider problems governed by the poisson equation the inhomogeneous biharmonic equation and the inhomogeneous cauchy navier equations of elasticity in addition to its simplicity the proposed method can both achieve high accuracy and solve large scale problems the feasibility of the proposed techniques is illustrated by several numerical examples

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

an improvement on a new upper bound for moduli of eigenvalues of iterative matrices in this paper some improvements on int j comput math 80 june 2003 799 803 are presented for a bound for moduli of eigenvalues of iterative matrices and subsequently we investigate the relationships among them and show that the new bounds are better than the earlier ones int j comput math 53 1994 203 217 j comput math 7 2 1986 41 46 int j comput math 80 june 2003 799 803 under some conditions

frobenius chebyshev polynomial approximations with a priori error bounds for nonlinear initial value differential problems abstract in this paper we present a method for approximating the solution of initial value ordinary differential equations with a priori error bounds the method is based on a chebyshev perturbation of the original differential equation together with the frobenius method for solving the equation chebyshev polynomials in two variables are developed numerical results are presented

an efficient maps for solving fourth order partial differential equations using trigonometric functions abstract in this paper we apply the method of approximate particular solutions maps based on the trigonometric functions to solve the fourth order partial differential equations pdes through the use of particular solutions of two second order differential equations the derivation of the closed form particular solutions for higher order pdes is a challenge and could be tedious such task can be achieved by a simple algebraic procedure to alleviate the difficulty of the derivation of particular solutions for fourth order pdes since the closed form particular solutions for the second order pdes with constant coefficients are known the proposed solution procedure is simple and direct five numerical examples are illustrated to demonstrate the feasibility and effectiveness of the proposed method

homotopic residual correction processes we present and analyze homotopic continuation residual correction algorithms for the computation of matrix inverses for complex indefinite hermitian input matrices our homotopic methods substantially accelerate the known nonhomotopic algorithms unlike the nonhomotopic case our algorithms require no pre estimation of the smallest singular value of an input matrix furthermore we guarantee rapid convergence to the inverses of well conditioned structured matrices even where no good initial approximation is available in particular we yield the inverse of a well conditioned n x n matrix with a structure of toeplitz hankel type in o n log 3 n flops for a large class of input matrices our methods can be extended to computing numerically the generalized inverses our numerical experiments confirm the validity of our analysis and the efficiency of the presented algorithms for well conditioned input matrices and furnished us with the proper values of the parameters that define our algorithms

a two step inexact newton chebyshev like method for inverse eigenvalue problems abstract a two step inexact newton chebyshev like method is proposed to solve inverse eigenvalue problems unlike general newton type methods which need to invert the jacobian matrix that may lead to an unexpected instability the proposed method can guarantee the numerical stability by exploiting the chebyshev method to approximate the inverse of the jacobian matrix theoretically we prove that the proposed method converges cubically numerical examples are given to demonstrate the effectiveness of our method

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

the truncation error of the two variable chebyshev series expansions abstract in 1 a formula for the error bound for the truncation error of the two variable chebyshev series expansions was recently given even though the order of the error bound is correct there was a quantitative error here we find the correct formula

on the optimality of the kautz singleton construction in probabilistic group testing we consider the probabilistic group testing problem where d random defective items in a large population of n items are identified with high probability by applying binary tests it is known that the theta d log n tests are necessary and sufficient to recover the defective set with vanishing probability of error when d o n alpha for some alpha in 0 1 however to the best of our knowledge there is no explicit deterministic construction achieving theta d log n tests in general in this paper we show that a famous construction introduced by kautz and singleton for the combinatorial group testing problem which is known to be suboptimal for combinatorial group testing for moderate values of d achieves the order optimal theta d log n tests in the probabilistic group testing problem when d omega log 2 n this provides a strongly explicit construction achieving the order optimal result in the probabilistic group testing setting for a wide range of values of d to prove the order optimality of kautz and singleton s construction in the probabilistic setting we provide a novel analysis of the probability of a non defective item being covered by a random defective set directly rather than arguing from combinatorial properties of the underlying code which has been the main approach in the literature furthermore we use a recursive technique to convert this construction into one that can also be efficiently decoded with only a log log factor increase in the number of tests

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

singularity wielandt s lemma and singular values in this study some upper and lower bounds for singular values of a general complex matrix are investigated according to singularity and wielandt s lemma of matrices especially some relationships between the singular values of the matrix a and its block norm matrix are established based on these relationships one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices in addition a small error in piazza 2002 g piazza t politi an upper bound for the condition number of a matrix in spectral norm j comput appl math 143 1 2002 141 144 is also corrected some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions

a time space spectral method for the time space fractional fokker planck equation and its inverse problem abstract in the paper we consider a time space spectral method to get the numerical solution of time space fractional fokker planck initial boundary value problem the temporal discretization is constructed by jacobi polynomials and the spatial discretization is composed by legendre polynomials moreover we present the stability and convergence analysis strictly the main advantages of the provided method are spectrally accurate in time and space and high computational efficiency in addition we introduce the inverse problem based on the spectral form with high order accuracy of the direct problem for the first time the levenberg marquardt l m method is proposed to estimate the two fractional derivatives and 2 some numerical results presented are consistent with the theoretical analysis

polynomial particular solutions for solving elliptic partial differential equations abstract in the past polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms in this paper a closed form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions the newly derived particular solution is further coupled with the method of particular solutions mps for numerically solving a large class of elliptic partial differential equations in contrast to the use of chebyshev polynomial basis functions the proposed approach is more flexible in selecting the collocation points inside the domain the polynomial basis functions are well known for yielding ill conditioned systems when their order becomes large the multiple scale technique is applied to circumvent the difficulty of ill conditioning problem five numerical examples are presented to demonstrate the effectiveness of the proposed algorithm

born again group testing multiaccess communications a brief summary of the basic notions of group testing is presented together with a brief historical account one of the early papers on group testing is shown to include a description of the tree search polling algorithm of hayes the classical group testing problem is formulated including a criterion for optimality of test plans a restricted class of tests called nested testing is described and a complete description for an optimal nested strategy is given for both a finite number and an infinite number of bernoulli distributed random variables a generalization of group testing applicable to the random access communications problem is presented

a tau approach for solution of the space fractional diffusion equation fractional differentials provide more accurate models of systems under consideration in this paper approximation techniques based on the shifted legendre tau idea are presented to solve a class of initial boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain the fractional derivatives are described in the caputo sense the technique is derived by expanding the required approximate solution as the elements of shifted legendre polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from the computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use

chebyshev type methods and preconditioning techniques recently a newton s iterative method is attracting more and more attention from various fields of science and engineering this method is generally quadratically convergent in this paper some chebyshev type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system ax b theoretic analysis and numerical experiments show that chebyshev s method is more effective than newton s one in the case of constructing approximate inverse preconditioners

pure states as a dual object for c algebras we consider the set of pure states of ac algebra as a uniform space equipped with transition probabilities and orientation and show that the pure states with this structure determine thec algebra up to isomorphism

third order iterative methods under kantorovich conditions we study a class of third order iterative methods for nonlinear equations on banach spaces a characterization of the convergence under kantorovich type conditions and optimal estimates of the error are found though in general these methods are not very extended due to their computational costs we will show some examples in which they are competitive and even cheaper than other simpler methods we center our analysis in both analytic and computational aspects

a geometric construction of iterative functions of order three to solve nonlinear equations in this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations this construction can be applied to any iteration function of order two to develop an iteration function of order three some examples are given of deriving several third order iteration methods and several numerical results follow to illustrate the performance of the derived methods

stair matrices and their generalizations with applications to iterative methods i a generalization of the successive overrelaxation method stair matrices and their generalizations are introduced some properties of the matrices are presented like triangular matrices this class of matrices provides bases of matrix splittings for iterative methods a remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation in particular a generalization of the successive overrelaxation sor method is introduced the sor theory on determination of the optimal parameter is extended to the generalized method to include a wide class of matrices the asymptotic rate of convergence of the new method is derived for hermitian positive definite matrices using bounds of the eigenvalues of jacobi matrices and numerical radius finally numerical tests are presented to corroborate the obtained results

solving partial differential equation by using multiquadric quasi interpolation abstract in this paper we use a kind of univariate multiquadric mq quasi interpolation to solve partial differential equation pde we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement our numerical experiment includes two examples one is solving viscid burgers equation for initial trapezoidal conditions another is simulating the interaction of two waves travelling in opposite direction from the numerical experiment we can see that the present scheme is valid