Towards fuzzy differential calculus part 1: Integration of fuzzy mappings

This paper deals with fuzzy-set-valued mappings of the real line, and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line. These mappings can also be viewed as fuzzy relations. Using Zadeh's extension principle, the integral of such fuzzy mappings over a crisp interval is defined. Provided a special analytical representation of the fuzzy mapping, the practical computation of such an integral is shown to be easy. Practically speaking, it yields the fuzzy surface of a fuzzily-bounded area.     

Towards fuzzy differential calculus part 2: Integration on fuzzy intervals

The problem of integrating mappings over fuzzy domains has already received some attention in the literature. However, there is not a unique approach, but several definitions have been proposed, each carrying its own meaning. This paper tries to emphasize the relationships and the differences between the various points of view.     

On standard models of fuzzy region connection calculus

The Region Connection Calculus (RCC) is perhaps the most influential topological relation calculus. Based on the first-order logic, the RCC, however, does not fully meet the needs of applications where the vagueness of entities or relations is important and not ignorable. This paper introduces standard models for the fuzzy region connection calculus (RCC) proposed by Schockaert et al. (2008) [18]. Each of such a standard fuzzy RCC model is induced by a standard RCC model in a natural way. We prove that each standard fuzzy RCC model is canonical in the sense that any satisfiable set of fuzzy RCC8 constraints have a solution in it. A polynomial realization algorithm is also provided. As a side product, we show similar sets of fuzzy constraints have similar solutions if both are satisfiable. This allows us to approximate fuzzy RCC constraints that have arbitrary bounds by those have bounds with finite precision.     

The chain and substitution rules of interval-valued intuitionistic fuzzy calculus

The interval-valued intuitionistic fuzzy set proposed by Atanassov is the extension of intuitionistic fuzzy set. It extends the membership degree and non-membership to interval values instead of a single value. So it contains more possible values and maybe more considerate. Among all the researches, the exploration on the calculus of interval-valued intuitionistic fuzzy set is entirely new. Recently, Zhao et al. (Int J Comput Intell Syst 9:36–56, 2016) proposed the concept of interval-valued intuitionistic fuzzy function (IVIFF) and gave a calculation method of derivative and differential of IVIFF. Based on this work, in this paper, firstly, we utilize a new and easier method to express the derivative and differential of IVIFF. Secondly, we propose the chain rules of derivative and the form invariance of differential in the interval-valued intuitionistic fuzzy environment. In addition, some properties of the substation rules for interval-valued intuitionistic fuzzy indefinite integrals and definite integrals are also developed.     

Tensor calculus: unlearning vector calculus

Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can serve as a bridge for vector calculus into tensor calculus.     

Operational calculus

The literature on operational calculus and its applications, reviewed in Referativnyi Zhurnal Matematika during 1965–1977, is surveyed.Translated from Itogi Naukl i Tekhniki, Seriya Matematicheskii Analiz, Vol. 16, pp. 99–148, 1979.     

Connectivity calculus

Given a finite hypergraph H = (V, E) and, for each e ϵ E, a collection of nonempty subsets π_e of e, Möbius inversion is used to establish a recursive formula for the number of connected components of the hypergraph H = (V, ∪_eϵEπ_e). As shown elsewhere, this formula is an essential ingredient in the context of a certain divide-and-conquer strategy that allows us to define a dynamical programming scheme solving Steiner's problem for graphs in linear time (however, with a constant depending hyperexponentially on their tree width).     

Twisted calculus

A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis, the categories of twisted modules and integrable twisted differential modules are equivalent. As particular cases, one recovers classical results from the theory of finite difference equations or q-difference equations.     

Comments on some versions of intuitionistic fuzzy propositional calculus due to K. Atanassov and G. Gargov

We show that the versions of intuitionistic fuzzy propositional calculus given in Definitions 6 and 7 in Atanassov and Gargov (Fuzzy Sets and Systems 95 (1998) 39–52) do not satisfy modus ponens. Furthermore, we show that the version of intuitionistic fuzzy propositional calculus given in Definition 8 by Atanassov and Gargov is incorrect.     

Extension of the ito calculus via the malliavin calculus

The Ito formula is extended to the tempered distributions "evaluated" on the trajectories of a nondegenerate Ito process in the sense of P. Malliavin. To do this the Ito integral is extended to vector-valued adapted distributions on Wiener space. Also a Galerkin type approximation using the Skorohod integral or the divergence operator is given for the diffusion processes. At the final section we give a sufficient condition for the existence of a smooth density for the filtering of nonlinear diffusions with the help of the techniques of the Malliavin calculus and the theory of nuclear spaces.     

Convergence of domain decomposition methods via semi-classical calculus semiclassical calculus

We present a model hierarchy of hydrodynamic and quasihydrodynamic equations for plasmas consisting of electrons and ions, and give a rigorous proof of the zero-relaxation-time limits in the hydrodynamic equations. described by the Euler equations coupled with a linear or nonlinear Poisson equation. The proof is based on the high energy estimates for the Euler equations together with compactness arguments.     

Paragrassmann differential calculus

This paper significantly extends and generalizes the paragrassmann calculus of our previous paper [1]. Here we discuss explicit general constructions for paragrassmann calculus with one and many variables. For one variable, nondegenerate differentiation algebras are identified and shown to be equivalent to the algebra of (p+1)×(p+1) complex matrices. If (p+1) is a prime integer, the algebra is nondegenerate and so unique. We then give a general construction of many-variable diffeentiation algebras. Some particular examples are related to multi-parametric quantum deformations of the harmonic oscillators.Dedication This paper is in memory of Mikhail Constantinovich Polivanov. One of the authors (A.T.F.) had a privilege to be a friend of him for many and many years. He was not only a distinguished scientist but a true Russian intellectual having deep roots in Russian culture. It is a deep sorrow that we can no more have a talk with him on science, poetry, religion ...Laboratory of Theoretical Physics, JINR, Dubna. SU-101 000 Moscow, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 213–231, February, 1993.     

The Malliavin calculus

The paper presents a review of the calculus of functional derivatives introduced by Malliaving and the Malliavin technique for establishing the existence of a density for the probability law of Wiener functionals. The approach of Malliavin, Stroock and Shigekawa is compared with that of Bismut.The research was supported by the fund for the promotion of research at the Technion     

Secant tree calculus

A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.     

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence $\left(X^n\right)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $\sigma \left(B\right)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients.