fuzzy n cell numbers and the differential of fuzzy n cell number value mappings in this paper we give the definition of a special kind of n dimension fuzzy numbers fuzzy n cell numbers discuss their operations and representation theorems define a complete metric on the fuzzy n cell number space and prove that the metric is equivalent to the supremum metric derived by the hausdorff metric between the level sets of the n dimension fuzzy numbers and obtain an embedding theorem of the fuzzy n cell number space isometrically embeds it into a concrete banach space we also consider the differential of the fuzzy mappings from an interval into the fuzzy n cell number space by using the embedding theorem

the logarithmic cotangent complex we define the cotangent complex of a morphism of fine log schemes prove that it is functorial and construct under certain restrictions a transitivity triangle we also discuss its relationship with deformation theory

on several definitions of the differential of a fuzzy mapping abstract several definitions of the differential of a fuzzy function are discussed in the restricted framework of fuzzy mappings from r to r

umbral calculus and the theory of multispecies non ideal gases abstract relations in the cluster expansion theory of non ideal gases are presented using the recently developed formalism of umbral calculus this paper provides an overview of the mathematical theory and explicitly derives the virial expansion and relations between the partition function and reducible and irreducible cluster coefficients for single and multispecies non ideal gases it is shown that these results are obtained remarkably easily using the methods of umbral calculus

the theoretical fundamentals of learning theory based on fuzzy complex random samples statistical learning theory based on real valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning the key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory in this paper we discuss a statistical learning theory based on fuzzy complex random samples firstly the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated secondly we carry out further research focused on a special type of fuzzy complex number namely rectangular fuzzy complex number and establish some properties and develop important theorems we also prove the strong law of large numbers based on fuzzy complex random variables thirdly the definitions of the fuzzy complex expected risk functional the fuzzy complex empirical risk functional the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed finally the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed

generalized lebesgue integrals of fuzzy complex valued functions in this paper on the basis of the theory of fuzzy complex numbers introduced by buckley we define fuzzy complex valued measurable functions fuzzy complex valued fuzzy measures and corresponding generalized lebesgue integrals of the fuzzy complex valued functions furthermore some basic properties and convergence theorems with respect to this kind of lebesgue integrals will be discussed

mathbb a 1 connected varieties of rank one over nonclosed fields in this paper we prove two results regarding the arithmetics of separably mathbb a 1 connected varieties of rank one first we prove that over a large field there is an mathbb a 1 curve through any rational point of the boundary if the boundary divisor is smooth and separably rationally connected secondly we generalize a theorem of hassett tschinkel for the zariski density of integral points over function fields of curves

on the restudy of fuzzy complex analysis part 1 the sequence and series of fuzzy complex numbers and their convergences this paper studies the convergence of the sequence and series of fuzzy complex numbers we define the metric convergence and levelwise convergence of sequences on the metric space of fuzzy complex numbers and discuss the relationship between two convergences all this may be a foundation for researching fuzzy complex analysis

barnes type degenerate euler polynomials in this paper we consider the barnes type degenerate euler polynomials we present several explicit formulas and recurrence relations for these polynomials also we establish a connection between our polynomials and several known families of polynomials

fuzzy complex analysis ii integration abstract this paper continues the author s research in fuzzy complex analysis where we now define and study the basic properties of a fuzzy contour integral for fuzzy mappings which map a rectifiable curve in the complex plane into fuzzy complex numbers

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

umbral calculus and sheffer sequences of polynomials in this paper we investigate some properties of sheffer sequences of polynomials arising from umbral calculus from these properties we derive new and interesting identities between sheffer sequences of polynomials an application to normal ordering is presented

an umbral calculus for polynomials characterizing u n tensor operators after the change of variables i i i and xi i 1 i i 1 we show that the invariant polynomials g n q i xi i 1 characterizing u n tensor operators p q q 0 0 become an integral linear combination of schur functions s in the symbol where denotes the difference of the two sets of variables 1 n and 1 n we obtain a similar result for the yet more general bisymmetric polynomials m g n q 1 n 1 m making use of properties of skew schur functions s and s we put together an umbral calculus for m g n q that is working entirely with polynomials we uniquely determine m g n q from m g n q 1 and combinatorial rules involving ferrers diagrams i e partitions provided that n 1 q this restriction does not interfere with writing the general case of m g n q as a linear combination of s as an application we deduce conjugation symmetry for n g n q from transposition symmetry by showing that these two symmetries are equivalent

on fuzzy n cell numbers and n dimension fuzzy vectors fuzzy n cell numbers are a kind of special n dimension fuzzy numbers whose sets of cuts are all hyperrectangles n dimension fuzzy vectors are the cartesian products of n one dimension fuzzy numbers in this paper we show that the fuzzy n cell number space and the n dimension fuzzy vector space are identical and obtain the relations of the joint membership function and the edge membership functions of a fuzzy n cell number i e a n dimension fuzzy vector using fuzzy vector form representations of fuzzy n cell numbers we study the metric of two fuzzy n cell numbers the supremum of a set of fuzzy n cell numbers and operational rules of addition scalar multiplication and multiplication of fuzzy n cell numbers and get the fuzzy vector form characterizations of the differentiability measurability and integrability of fuzzy n cell number valued fuzzy mappings

ancfis a neurofuzzy architecture employing complex fuzzy sets complex fuzzy sets cfss are an extension of type 1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane although there has been considerable progress made in determining the properties of cfss and complex fuzzy logic there has yet to be any practical application of this concept we present the adaptive neurocomplex fuzzy inferential system ancfis which is the first neurofuzzy system architecture to implement complex fuzzy rules and in particular the signature property of rule interference we have applied this neurofuzzy system to the domain of time series forecasting which is an important machine learning problem we find that ancfis performs well in one synthetic and five real world forecasting problems and is also very parsimonious experimental comparisons show that ancfis is comparable with existing approaches on our five datasets this work demonstrates the utility of complex fuzzy logic on real world problems

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

on starshaped fuzzy sets in this paper we firstly define some new types of fuzzy starshapedness discuss the relationships among the different types of fuzzy starshapedness and then present some basic properties of them finally we develop several important results on the shadows of starshaped fuzzy sets

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

complex fuzzy logic a novel framework for logical reasoning termed complex fuzzy logic is presented in this paper complex fuzzy logic is a generalization of traditional fuzzy logic based on complex fuzzy sets in complex fuzzy logic inference rules are constructed and fired in a manner that closely parallels traditional fuzzy logic the novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets characterized by complex valued membership functions the range of these membership functions is extended from the traditional fuzzy range of 0 1 to the unit circle in the complex plane thus providing a method for describing membership in a set in terms of a complex number several mathematical properties of complex fuzzy sets which serve as a basis for the derivation of complex fuzzy logic are reviewed in this paper these properties include basic set theoretic operations on complex fuzzy sets namely complex fuzzy union and intersection complex fuzzy relations and their composition and a novel form of set aggregation vector aggregation complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic while benefiting from the properties of complex numbers and complex fuzzy sets the introduction of complex valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties and considerable potential for further research and application

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

more on the umbral calculus with emphasis on the q umbral calculus abstract the interrelationship between distinct umbral calculi is studied these ideas are applied in particular to q umbral calculus which shows how andrews q theory relates to the q theory discussed in the previous paper by the author the q hermite polynomials and basic hypergeometric series are briefly discussed

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

stable logarithmic maps to deligne faltings pairs i we introduce a new compactication of the space of relative stable maps this approach uses logarithmic geoemetry in the sense of kato fontaineillusie without taking expansions of the target the underlying structures of the stable logarithmic maps are stable in the usual sense

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

heisenberg algebra umbral calculus and orthogonal polynomials umbral calculus can be viewed as an abstract theory of the heisenberg commutation relation p m 1 in ordinary quantum mechanics p is the derivative and m the coordinate operator here we shall realize p as a second order differential operator and m as a first order integral one we show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials related to laguerre polynomials these polynomials are particularly well suited for describing the so called flatenned beams in laser theory

a method for multiple periodic factor prediction problems using complex fuzzy sets multiple periodic factor prediction mpfp problems exist widely in multisensor data fusion applications development of an effective prediction method should integrate information for multiple periodically changing factors because the uncertainty and periodicity coexist in the information used the prediction method should be able to handle them simultaneously in this study complex fuzzy sets are used to represent the information with uncertainty and periodicity a product sum aggregation operator psao is developed for a set of complex fuzzy sets which is used to integrate information with uncertainty and periodicity and a psao based prediction psaop method is then proposed to generate a solution of mpfp problems this study illustrates the details of the psaop method through two real applications in annual sunspot number prediction and bushfire danger rating prediction experiments indicate that the proposed psaop method effectively handles the uncertainty and periodicity in the information of multiple periodic factors simultaneously and can generate accurate predictions for mpfp problems

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

log fano varieties over function fields of curves consider a smooth log fano variety over the function field of a curve suppose that the boundary has positive normal bundle choose an integral model over the curve then integral points are zariski dense after removing an explicit finite set of points on the base curve

degenerate mittag leffler polynomials we present several explicit formulas and recurrence relations for the degenerate mittag leffler polynomials we also give several connections between mittag leffler polynomials and other known families of polynomials