short communication some remarks on the category fuz h of m eytan the aim of this paper is to disprove a result asserted by m eytan in his paper fuzzy sets a topos logical point of view in this journal vol 5 pp 47 67 namely that the category fuz h is a topos for any complete heyting algebra h it is proved that this result can be true only when h is a boolean algebra

improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

h fuzzy relations i a topological universe viewpoint abstract we introduce the subcategory rel r h of rel h consisting of fuzzy reflexive relational space on sets and we study structures of rel r h in a viewpoint of the topological universe introduced by l d nel we show that rel r h is a topological universe over set also we introduce the subcategories rel pr h rel p h rel e h and rel po h of rel r h we will show that rel pr h resp rel p h and rel e h is a topological universe over set

short communication the categories set h and fuz h this paper follows up d ponasse s paper some remarks on the category fuz h of m eytan in this journal 4 in particular it gives a new proof that fuz h cannot have a subobject classifier if h is a complete chain with at least three elements

one point newton type iterative methods in this paper a unified point of view that includes the most of one point newton type iterative methods for solving nonlinear equations is introduced a simple idea to design iterative methods with quadratic or cubic convergence is also described this idea is extended to construct one point iterative methods of order four in addition several numerical examples are given to illustrate and compare different known methods and some introduced by using this unifying idea

a survey of fuzzy set and topos theory abstract this paper is a comparison and contrast of approaches to many valued mathematics offered by fuzzy set theory and topos theory it gives a survey of the categorical foundations of fuzzy set theory and related topoi topoi are not a basis for fuzzy set theory butthey do suggest appropriate directions to go and questions to ask for a synthesis which does provide a foundation one possible structure which has a lopos like internal logic and the rich variety of loggical connectives used in fuzzy sets is included

study on gas kinetic unified algorithm for flows from rarefied transition to continuum the modified bgk equation adapted to various flow regimes can be presented by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium the discrete velocity ordinate method is developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space and then the velocity distribution function equation is cast into hyperbolic conservation law form with nonlinear source terms based on the unsteady time splitting method and the non oscillatory containing no free parameters and dissipative nnd scheme the gas kinetic finite difference second order scheme is constructed for the computation of the discrete velocity distribution functions the mathematical model on the interaction of molecules with solid surface is studied and used in the numerical method four types of numerical quadrature rules such as the modified gauss hermite formula the composite newton cotes integration method the gauss legendre numerical quadrature rule and the golden section number theoretic integral method are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space as a result a unified simplified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime based on analyzing the inner parallel degree of the unified algorithm the parallel strategy adapted to the gas kinetic numerical algorithm is studied and then the hpf parallel processing software for the unified algorithm is developed to test the present method the one dimensional shock tube problems the flows past two dimensional circular cylinder and the flows around three dimensional sphere and spacecraft shape with various knudsen numbers are simulated the computational results are found in high resolution of the flow fields and good agreement with the theoretical dsmc n s and experimental results

variants of newton s method using fifth order quadrature formulas abstract some variants of newton s method are developed in this work in order to solve nonlinear equations depending on one or several variables based in rules of quadrature of fifth order we prove the third or fifth order of convergence of these methods for dimension one and the second or third order in several variables depending on the behaviour of the second derivative moreover different numeric tests confirm or improve theoretic results and allow us to compare these variants with newton s classical method

an extension of gander s result for quadratic equations in the study of iterative methods with high order of convergence gander provides a general expression for iterative methods with order of convergence at least three in the scalar case taking into account an extension of this result we define a family of iterations in banach spaces with r order of convergence at least four for quadratic equations

improved iterative methods for solving nonlinear equations in this paper we use the decomposition technique of noor m aslam noor numerical analysis and optimization lecture notes comsats institute of information technology islamabad pakistan 2006 to suggest and analyze some new iterative methods for solving nonlinear equations our method of developing these iterative methods is very simple as compared with other methods several numerical examples are given to illustrate the efficiency and performance of the new method our method can be considered as an improvement of the existing methods

reconstruction possibilities for fuzzy sets towards a new level of understanding abstract using the methodological approach of rational reconstruction some possible explications of the notion of fuzzy set are critically discussed with respect to their intuitive meaning plausibility and consistency together with a historical survey we try to provide a better understanding of the development of various concepts within traditional fuzzy set theory based on the approaches of 39 and 18 we show that zadeh s original intuition of a fuzzy set and its formalization as generalized membership function can be reinterpreted in a plausible and consistent way and thus are supported not only on a formal level by mathematical results but can also be justified in principle on a semantical level hence both the operational as well as the traditional valuational viewpoint have their own rights in their corresponding areas of modeling reality and it is shown how the latter can be given a sound foundation based upon the former

efficient iterative methods applied to the solution of transonic flows we investigate the use of an inexact newton s method to solve the potential equations in the transonic regime as a test case we solve the two dimensional steady transonic small disturbance equation approximate factorization adi techniques have traditionally been employed for implicit solutions of this nonlinear equation instead we apply newton s method using an exact analytical determination of the jacobian with preconditioned conjugate gradient like iterative solvers for solution of the linear systems in each newton iteration two iterative solvers are tested a block s step version of the classical orthomin k algorithm called orthogonal s step orthomin osomin and the well known gmres method the preconditioner is a vectorizable and parallelizable version of incomplete lu ilu factorization efficiency of the newton iterative method on vector and parallel computer architectures is the main issue addressed in vectorized tests on a single processor of the cray c 90 the performance of newton osomin is superior to newton gmres and a more traditional monotone af adi method maf for a variety of transonic mach numbers and mesh sizes newton gmres is superior to maf for some cases the parallel performance of the newton method is also found to be very good on multiple processors of the cray c 90 and on the massively parallel thinking machine cm 5 where very fast execution rates up to 9 gflops are found for large problems

fuzzy sets a topos logical point of view abstract benabou deduction categories are defined with a set of additional assumptions that define categories with formal finite limits resp formal regular categories formal logoi formal topoi they are shown to be generalized structures in which higher order many sorted languages can be realized the corresponding gentzen type higher order calculus of sequents is explicited and the soundness theorem is formulated a construction is given which associates to each deduction category with formal properties a real category with the corresponding real properties in a universal way the corresponding sounddess and completeness properties are formulated for the real categories thus obtained fuzzy sets as generalized by goguen are introduced considered as the objects of a category fuz h which turns out to be the real category associated to a very simple formal topos and thus to be itself a topos furthermore this is proved to be a grothendieck topos which is a strictly full epireflective subcategory of higgs category of h valued sets topoi are proposed as generalized fuzzy sets and deductio0 categories as generalized 2 fuzzy sets some related topics such as arbib manes fuzzy theories probability many valued and fuzzy logics intensional logic are very briefly touched upon

fuzzy sets do not form a topos abstract the relationship between fuzzy sets and h valued sets is briefly examined and a claim of eytan s that the former comprise the objects of a topos is corrected

a new iteration method with cubic convergence to solve nonlinear algebraic equations abstract in this paper a new iteration scheme is proposed to solve the roots of an algebraic equation f x 0 given an initial guess x0 the roots of the equation can be obtained using the following iteration scheme x n 1 x n f x n f 2 x n 2 f x n f x n f x n this iteration scheme has unique convergence characteristics different from the well known newton s method it is shown that this iteration method has cubic local convergence in the neighborhood of the root using this scheme real or complex roots for specific algebraic equations can be found because there are two iteration directions for a given initial guess two solutions can be found for certain algebraic equations with multiple roots examples are presented and compared with other methods

on a family of halley like methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation here we introduce a family of halley like methods and show that euler chebyshev and bsc are just members of the family we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction

fuzzy sets and toposes towards higher order fuzzy logic abstract a series of paper which have appeared in fuzzy sets and systems over the last three years are reviewed in an expository manner the claims and counter claims made in them are dealt with some of the problems of representing knowledge in the first order predicate calculus are discussed briefly and a case is made for exposing the relationship between topoi and fuzzy mathematics with a view to making a true fuzzy logic both consistent complete and respectable the paper is an attempt to present the controversy to the non specialist and especially to practitioners in expert systems and artificial intelligence

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

a new iterative method to compute nonlinear equations the aim of this paper is to construct a new efficient iterative method to solve nonlinear equations the new method is based on the proposals of abbasbandy on improving the order of accuracy of newton raphson method s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method applied mathematics and computation 145 2003 887 893 and on the proposals of babolian and biazar on improving the order of accuracy of adomian s decomposition method e babolian j biazar on the order of convergence of adomian method applied mathematics and computation 130 2002 383 387 the convergence of the new scheme is proved and at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of this new method also it is shown in this paper that the modified adomian s method developed by babolian and biazar to solve nonlinear equations e babolian j biazar solution of nonlinear equations by modified adomian decomposition method applied mathematics and computation 132 2002 167 172 should be slightly modified due to the fact that convergence of adomian s method does not ensure convergence of the modified method an example illustrates this fact which unlike what is claimed by the authors does not converge with their method but with a simple different choice of the zero component becomes convergent

a family of new newton like methods abstract in this paper we present a family of new newton like methods which removes the severe condition f x 0 in a neighborhood of the required root imposed on newton s method the new methods are quadratically convergent and moreover have the same error equation as newton s method the detailed analysis of convergence is supplied the numerical results show that the new methods have definite practical utility

on explicit expressions in additive cellular automata theory abstract in the present paper the problem of additive cellular automata is posed as a second order linear difference equation in two indices time step t and position i subject to a periodicity condition for all t and an initial condition for t 0 an explicit solution of this mathematical problem is obtained by using the method of generating function this explicit formula enables us to compute the configuration at any time without any explicit simulation the explicit formula has been derived for all important additive rules with the help of these many salient characteristics of the transition may be obtained with surprising ease the explicit formula method appears to be very efficient and has possibilities to be applied to diverse situations in cellular automata theory

categorical properties of f set theory abstract in the first part of the paper we show that the interpretation of a fuzzy set as a bundle of sets over a base space makes the category of fuzzy sets into a topos f set v i e to paraphrase pitts 5 fuzzy sets do form a topos the second part exhibit close connections to operations proposed for fuzzy set theory from an intuitive and operational viewpoint by toth 7 these two results together seem to indicate that there are two essentially different ways to understand the concept of a fuzzy set either as an evaluation over a universe or as a collection of evaluated objects the latter one being a set theoretical interpretation and henceforth leading to a topos

modified families of newton halley and chebyshev methods this paper presents new families of newton type iterative methods newton halley and chebyshev methods for finding simple zero of univariate non linear equation permitting f x 0 in the vicinity of the root newton type iterative methods have well known geometric interpretation and admit their geometric derivation from a parabola these algorithms are comparable to the well known powerful classical methods of newton halley and chebyshev respectively and these can be seen as special cases of these families the efficiency of the presented methods is demonstrated by numerical examples

preordered sets valued in a gl monoid let m l be a gl monoid an m valued preordered set is an l subset endowed with a reflexive and m transitive l relation it is essentially a category enriched in a quantaloid generated by m this paper presents a study of m valued preordered sets with emphasis on symmetrization and the cauchy completion the main result states that symmetrization and the cauchy completion of m valued preordered sets commute up to a natural isomorphism

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

categorial studies of fuzzy sets abstract this paper presents some recent studies on the categorial aspects of fuzzy sets and sets with a valued equality in order to suitably develop a theory fuzzy sets it seems useful to construct a category such an approach is very helpful when a classical concept has to be generalized for performing a choice between all the apparently possible definitions such a category has besides to look as much as possible like the usual category of sets one can wish it would be a topos or at least something similar

on a two step optimal steffensen type method relaxed local and semi local convergence analysis and dynamical stability abstract in this paper we study on the local and semi local convergence and dynamic of an iterative optimal two step steffensen s method we study local convergence by using lipschitz conditions and continuity of differentiable function f we also obtain semi local convergence by using majorizing sequences by this approach we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions moreover we analyze the stability of the method via julia sets and parameter planes that is a novelty for steffensen type methods we extend this method for obtaining the sign function of a matrix numerical examples are given to confirm our theoretical results

categories of fuzzy sets and weak topos in this paper definitions of middle object in a category and concept of weak topos are given a weak topos is a cartesian closed category with a middle object it is proved that category of functors from a small category to category fuz of fuzzy sets is a weak topos