set functors l fuzzy set categories and generalized terms in this work we generalize previous constructions of fuzzysetcategories introduced in 1 by consideringl fuzzysets in which the values of the characteristic functions run on a completely distri

acceleration of steady state lattice boltzmann simulations for exterior flows the simulation of a stationary fluid flow past an obstacle by the lattice boltzmann method lbm in two dimensions is discussed the combination of second order expressions for far field boundary conditions and a suitable treatment of the no slip boundary condition at the obstacle surface with the nested grid refinement technique can be applied to the lbm resulting in a highly efficient method for the treatment of exterior flows furthermore via replacing the nested time stepping by local time stepping the resolution process can be substantially accelerated a highly accurate drag coefficient was used to make an error assessment for various no slip boundary conditions imposed on the obstacle surface the analysis showed that the equilibrium method for treating the no slip conditions is superior to halfway bounce back and full way bounce back no slip conditions when the relaxation time ensuremath tau 1 also a ensuremath tau dependence test was made to evaluate the performance of different methods in the treatment of the no slip boundary conditions

on the category of lattice valued bornological vector spaces motivated by the theory of l bornological spaces of m abel and a sostak over a complete lattice l and the concept of topological system of s vickers this paper introduces the categories of l bornological vector spaces and systems and shows that the former is isomorphic to a full reflective subcategory of the latter

axisymmetric shape of gas bubbles in a uniform flow this paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method bifurcation and turning points are located by applying a special type of hermite pade approximation

similarities between powersets of terms generalisation of the foundational basis for many valued logic programming builds upon generalised terms in the form of powersets of terms a categorical approach involving set and term functors as monads allows for a study of monad compositions that provide variable substitutions and compositions thereof in this paper substitutions and unifiers appear as constructs in kleisli categories related to particular composed powerset term monads specifically we show that a frequently used similarity based approach to fuzzy unification is compatible with the categorical approach and can be adequately extended in this setting also some examples are included in order to illuminate the definitions

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

high order analysis of the limit cycle of the van der pol oscillator we have applied the lindstedt poincare method to study the limit cycle of the van der pol oscillator obtaining the numerical coefficients of the series for the period and for the amplitude to order 859 hermite pade approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy 100 digits both series have then been resummed using an approach based on pade approximants where the exact asymptotic behaviors of the period and the amplitude are taken into account our results improve drastically all previous results obtained on this subject

theory and applications of an alternative lattice boltzmann grid refinement algorithm this contribution proposes an alternative lattice boltzmann grid refinement algorithm that overcomes the drawbacks that plague existing approaches we demonstrate that this algorithm is accurate and applicable for all values of the relaxation time we also show that this algorithm can significantly speed up the flow settlement process by using a hierarchy of grid levels the stationary regime can be approached up to a thousand times faster than with a single grid resolution

lattice valued bornological systems motivated by the concept of lattice valued topological system of j t denniston a melton and s e rodabaugh which extends lattice valued topological spaces this paper introduces the notion of lattice valued bornological system as a generalization of lattice valued bornological spaces of m abel and a sostak we aim at and make the first steps towards the theory which will provide a common setting for both lattice valued point set and point free bornology in particular we show the algebraic structure of the latter

summation of the eigenvalue perturbation series by multi valued pade approximants application to resonance problems and double wells quadratic pade approximants are used to obtain energy levels both for the anharmonic oscillator x2 2 lambda x4 and for the double well x2 2 lambda x4 in the first case the complex valued energy of the resonances is reproduced by summation of the real terms of the perturbation series the second case is treated formally as an anharmonic oscillator with a purely imaginary frequency we use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function then we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants in this way we can reconstruct the real energy by summation of the complex series such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory around the minima of the double well potential incorrect

acceleration of lattice bgk schemes with grid refinement abstract the lbgk method with local grid refinement has been shown to be an efficient and accurate tool for the simulation of incompressible viscous flows over complex geometries in the present study further improvement of this concept is proposed enabling the use of smaller amount of time steps on refined grids without impairing the spatial or temporal accuracy this extension of the lbgk method has been proved by analytical and numerical investigations the gain in computational time was found to be significant

asymptotic analysis of lattice boltzmann boundary conditions in this article we use a general method for the analysis of finite difference schemes to investigate lattice boltzmann algorithms for navier stokes problems with dirichlet boundary conditions several link based boundary conditions for commonly used lattice boltzmann bgk models are considered with our method the accuracy of the algorithms can be exactly predicted moreover the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity the analysis is applicable to general geometries and instationary flows

a hermite pade perspective on the renormalization group with an application to the correlation function of lieb liniger gas renormalization group rg methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings however in cases when the function depends only on the coupling the rg technology does include a part which is related to the equation for the running coupling and which we will call the single variable rg svrg that turns out to be a purely formal method for improving the convergence properties of series expansions in fact we show that svrg is an integral hermite pade approximation preceded by a series reversion a complementary viewpoint on what svrg does to an expansion is that it performs a minimal upgrade to a self similar form the minimality being crucial in applications svrg is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable but also the exponent but not the prefactor of the scaling law for large values of the variable as a practical example we extract the scaling law prefactor for the one body density matrix of the lieb liniger gas using a new result for the fourth order term in the short distance expansion we find remarkable agreement with known quantum diffusion monte carlo numerical results

hermite pade approximation approach to mhd jeffery hamel flows in this paper we studied the problem of nonlinear flow of an incompressible conducting viscous fluid in convergent divergent channels under the influence of an externally applied homogeneous magnetic field the governing equations is obtained and solved using a special type of hermite pade approximation semi numerical approach this technique offers some advantages over solutions obtained by using traditional methods such as finite differences spectral method shooting method etc it reveals the analytical structure of the solution function and the important properties of overall flow structure including velocity field flow reversal control and bifurcations are discussed

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

a consistent grid coupling method for lattice boltzmann schemes a method of coupling grids of different mesh size is developed for classical lattice boltzmann lb algorithms on uniform grids the approach is based on an asymptotic analysis revealing suitable quantities equalized along the grid interfaces for exchanging information between the subgrids in contrast to other couplings the method works without overlap zones moreover the grid velocity mach number is not kept constant as the time step depends not linearly but quadratically on the grid spacing to illustrate the basic idea we use a simple lb algorithm solving the advection diffusion equation the proposed grid coupling is validated by numerical convergence studies indicating that the coupling does not affect the second order convergence behavior of the lb algorithm which is observed on uniform grids in order to demonstrate its principal applicability to other lb models the coupling is generalized to the standard d2p9 model for navier stokes flow and tested numerically as we use analytic tools different from the chapman enskog expansion the theoretical background material is given in two appendices in particular the results of numerical experiments are confirmed with a consistency analysis

approximants de pade hermite 2eme partie programmation in this second paper we propose an algorithm for the construction of the tables and of the pade hermite tables 3 then we present some numerical applications

advection diffusion lattice boltzmann scheme for hierarchical grids in this paper we describe an extension of a recently developed lattice boltzmann method for solving the advection diffusion equation our proposed approach allows to couple grids of different grid resolutions and includes a staggered timestepping scheme interpolations in space and time and finally a scaling step ensuring the continuity of the desired macroscopic quantities across the grid interface after validating the basic lattice boltzmann method on a uniform grid by a convergence study of analytic problems we demonstrate the consistency of our approach by solving benchmark problems and comparing results on uniform grids and multiply locally refined grids

categorical foundations of variety based topology and topological systems the paper considers a new approach to fuzzy topology based on the concept of variety and developed in the framework of topological theories resembling those of rodabaugh as a result a categorical generalization of the notion of topological system of vickers is obtained and its theory unfolded which clarifies the relations between algebra and topology we also justify the use of semi quantales as the basic underlying structure for doing lattice valued topology as well as provide a categorical framework incorporating the theory of bitopological spaces

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

numerical study of bifurcations by analytic continuation of a function defined by a power series a novel computational approach to the investigation of bifurcations relying on the use of power series in the bifurcation parameter for a particular solution branch is presented the first part of the paper is devoted to the description of a series summation technique based on the assumption that the given series is the local representation of a function algebraic in the independent variable the procedure leads to a special type of hermite pade approximant although no mathematical analysis is presented the numerical evidence suggests that the error decays faster than exponentially with the number of terms of the series used the procedure s chief merit is its ability to reveal solution branches of the underlying problem in addition to the one represented by the original series in the final part of the paper an algorithm is described for numerically generating the required power series where standard perturbation methods are inadequate thus it is shown how path following techniques may be combined

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

approximants de pade hermite 1ere partie theorie we present in this first paper a generalization of pade approximants which gives us as particular cases shafer s and baker sd log approximants first we define these approximants following an old idea of hermite then we prove some fundamental properties for their constructions

powersets of terms and composite monads composing various powerset functors with the term monad gives rise to the concept of generalized terms this in turn provides a technique for handling many valued sets of terms in a framework of variable substitutions thus being the prerequisite for categorical unification in many valued logic programming using an extended notion of terms as constructions of monads involve complicated calculations with natural transformations proofs are supported by a graphical approach that provides a useful tool for handling various conditions such as those well known for distributive laws

applications of hermite pade approximants to water waves and the harmonic oscillator on a lattice three types of hermite pade approximants are considered known respectively as quadratic integral and differential pade approximants the singularity structure of each type of approximant is described it is more complicated than that of the standard pade approximant and this property may often be used to estimate the types of singularity of a function from its power series expansion as well as evaluate it on its branch cuts two applications in different fields are described which illustrate these properties of the above types of hermite pade approximants the first concerns the characteristic values of mathieu s equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice the second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves

topological systems as a framework for institutions recently j t denniston a melton and s e rodabaugh introduced a lattice valued analogue of the concept of institution of j a goguen and r m burstall comparing it moreover with the lattice valued version of the notion of topological system of s vickers in this paper we show that a suitable generalization of topological systems provides a convenient framework for certain kinds of lattice valued institutions

sierpinski object for affine systems abstract motivated by the concept of sierpinski object for topological systems of s vickers presented recently by r noor and a k srivastava this paper introduces the sierpinski object for many valued topological systems and shows that it has three important properties of the crisp sierpinski space of general topology

fuzzy points and attachment in this paper a binary relation between l sets called attachment is defined by means of a suitable family of completely coprime filters in l examples are given using several kinds of lattice ordered algebras and a class of attachments is determined whose elements generalize pu liu s quasi coincidence relation mapping any l set on x to the set of the attached l points one gets a frame map from the l powerset l x to the powerset p s x of the set s x of l points on x this allows one to define functors from the category l top of l topological spaces either to the category top of topological spaces or to the category topsys of topological systems a comparison with the hypergraph functor is also included

categorically algebraic topology versus universal topology this paper continues to develop the theory of categorically algebraic catalg topology introduced as a common framework for the majority of the existing many valued topological settings to provide convenient means of interaction between different approaches motivated by the results of universal topology of h herrlich we show that a concrete category is fibre small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures which is definable by topological co axioms

interweaving algebra and topology lattice valued topological systems this paper is primarily dedicated to understanding the natural role that topological systems and lattice valued topological systems play in understanding the relationship between algebra and topology a relationship expressed by an interweaving of embeddings of essentially algebraic and topological categories this natural role is revealed by how various categories for systems relate to locales topological spaces and various categories for lattice valued topology

implementation of the compact interpolation within the octree based lattice boltzmann solver musubi abstract a sparse octree based parallel implementation of the lattice boltzmann method for non uniform meshes is presented in this paper to couple grids of different resolutions a second order accurate compact interpolation is employed and further extended into three dimensions the compact interpolation requires only four source elements from the adjacent level for both two and three dimensions thus improves the computational efficiency moreover a weight based domain decomposition algorithm and the level wise arrangement of elements is explained in detail the second order convergence for both velocity and strain rate are validated numerically in the taylor green vortex test case additionally the 3d flow around a sphere in the reynolds number range of 100 1000 is investigated good agreement between simulated results and those from literature is observed which provides further evidence for the accuracy of our method

asymptotic analysis of finite difference methods with this article we want to advocate the use of asymptotic methods for the analysis of finite difference schemes we present several examples to demonstrate the applicability of the approach advantages over the modified equation and truncation error analysis are pointed out