Fuzzy Perfect Mappings and Q-Compactness in Smooth Fuzzy Topological Spaces

We point out that the product of two fuzzy closed sets of smooth fuzzy topological spaces need not be fuzzy closed with respect to the the existing notion of product smooth fuzzy topology. To get this property, we introduce a new suitable product smooth fuzzy topology. We investigate whether $F_1\times F_2$ and $(F,H)$ are weakly smooth fuzzy continuity whenever $F_1$, $F_2$, $F$ and $H$ are weakly smooth fuzzy continuous. Using this new product smooth fuzzy topology, we define smooth fuzzy perfect mapping and prove that composition of two smooth fuzzy perfect mappings is smooth fuzzy perfect under some additional conditions. We also introduce two new notions of compactness called $Q$-compactness and $Q$-$\alpha$-compactness; and discuss the compactness of the image of a $Q$-compact set ($Q$-$\alpha$-compact set) under a weakly smooth fuzzy continuous function ($(\alpha ,\beta )$-weakly smooth fuzzy continuous function).     

On Bipolar Anti Fuzzy h-ideals in Hemi-rings

In this work, we present notions of bipolar anti fuzzy $h$-ideals and bipolar anti fuzzy interior $h$-ideals in hemi-rings. Investigating some of their properties, we characterize hemi-rings by means of positive anti $\beta$-cut and negative anti $\alpha$-cut. Meanwhile, some results of homomorphisms, anti images and anti pre-images are given to show the rationality of the definitions introduced in the present paper. Also, we define an equivalence relation on bipolar anti fuzzy $h$-ideals. In particular, we investigate translations, extensions and multiplications of bipolar anti fuzzy $h$-ideals. Finally, we present characterizations of h-hemi-regular and h-semi-simple hemi-rings in terms of bipolar anti fuzzy $h$-ideals.     

The Cauchy problem for the 2D magnetohydrodynamic-α equations

In this paper, we consider the Cauchy problem for the 2D incompressible magnetohydrodynamics-$\alpha$ (MHD-$\alpha$) equations. We obtain the global solution for the 2D incompressible MHD-$\alpha$ simulation model in the fractional index Sobolev space, and prove that the incompressible MHD-$\alpha$ equations reduce to the incompressible homogeneous MHD equations as $\alpha \to 0$, and the solution of MHD-$\alpha$ equations will converge to the weak solution of the corresponding MHD equations. Moreover, it is convenient to construct a numerical algorithm based on an iteration scheme in our proof.     

A fully stochastic approach to limit theorems for iterates of Bernstein operators

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f\in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k\left(n\right)$ to a polynomial $f$ when $k\left(n\right)∕n$ tends to a constant.     

Fuzzy Dot Structure of BG-algebras

In this paper, the notions of fuzzy dot subalgebras is introduced together with fuzzy normal dot subalgebras and fuzzy dot ideals of $BG$-algebras. The homomorphic image and inverse image are investigated in fuzzy dot subalgebras and fuzzy dot ideals of $BG$-algebras. Also, the notion of fuzzy relations on the family of fuzzy dot subalgebras and fuzzy dot ideals of $BG$-algebras are introduced with some related properties.     

Sur un théorème de Cauchy–Kowalewski–Nagumo global dans des espaces de Gevrey projectifs

We propose a new approach, based on a combination of the contraction principle and Picard successive approximations, for the study of a global Cauchy problem associated to partial differential operator $D_t^m?{\sum }_{(j\text{,}\alpha )\in B}{a}_{j\alpha }{D}_t^j{D}_x^{\alpha }$ with coefficients $a_{j\alpha }$ continuous or holomorphic with respect to t in projective Gevrey spaces. We extend the result of a previous Note to the case of non Kowalewskian operators. To cite this article: D. Gourdin, T. Gramchev, C. R. Acad. Sci. Paris, Ser. I 339 (2004).