Abstract

Since the concept of Pettis integral was introduced in 1938 [10], the Pettis integrability of weakly measurable functions has been studied by many authors [5, 6, 7, 8, 9, 11]. It is known that there is a bounded function that is not Pettis integrable [10, Example 10. 8]. So it is natural to raise the question: when is a bounded function Pettis integrable\ulcorner(omitted)

Abstract

D. Dubois and H. Prade introduced the notions of fuzzy numbers and defined its basic operations [3]. R. Goetschel, W. Voxman, A. Kaufmann, M. Gupta and G. Zhang [4,5,6,9] have done much work about fuzzy numbers. Let <TEX>$\mathbb{R}$</TEX> the set of all real numbers and <TEX>$F^{*}(\mathbb{R})$</TEX> all fuzzy subsets defined on <TEX>$\mathbb{R}$</TEX>. G. Zhang [8] defined the fuzzy number <TEX>$\tilde{a}\;\in\;F^{*}(\mathbb{R})$</TEX> as follows : (omitted)

Abstract

We begin with a brief survey of some of the known results dealing with Bloch constants. Bloch's theorem asserts that there is a constant B<TEX>$\_$</TEX>1.C/(1, 0) such that if f is holomorphic in the open unit disk D and normalized by │f'(0)│<TEX>$\geq$</TEX>1, then the Riemann surface of f contains an unramified disk of radius at least B<TEX>$\_$</TEX>1.C/(1, 0) (see[7,p.14]).(omitted)

Abstract

The topos constructed in [6] is a set-like category that includes among its axioms an axiom of infinity and an axiom of choice. In its final form a topos is free from any such axioms. Set<TEX>$\^$</TEX>G/ is a topos whose object are G-set Ψ<TEX>$\sub$</TEX>s/:G<TEX>${\times}$</TEX>S\longrightarrowS and morphism f:S \longrightarrowT is an equivariants map. We already known that Set<TEX>$\^$</TEX>G/ satisfies the weak form of the axiom of choice but it does not satisfies the axiom of the choice.(omitted)

Abstract

Let R<TEX>$^n$</TEX>be n-th Euclidean space. Let be the n-th spere embeded as a subspace in R<TEX>$\^$</TEX>n+1/ centered at the origin. In this paper, we are going to consider the function space F = {f│f : S<TEX>$^n$</TEX>\longrightarrow S<TEX>$^n$</TEX>} metrized by as follow D(f,g)=d(f(<TEX>$\chi$</TEX>), g(<TEX>$\chi$</TEX>)) where f, g <TEX>$\in$</TEX> F and d is the metric in S<TEX>$^n$</TEX>. Finally we want to find certain relation these spaces.(omitted)

Abstract

We consider the linear differential equations y〃'＋ P(<TEX>$\chi$</TEX>)y'＋Q(<TEX>$\chi$</TEX>)y=0 (1)(y"＋P(<TEX>$\chi$</TEX>)y)'-Q(<TEX>$\chi$</TEX>)y =0 (2) Where (2) in the adjoint of (1) and P(<TEX>$\chi$</TEX>), Q(<TEX>$\chi$</TEX>) are continuous functions satisfying P(<TEX>$\chi$</TEX>)<TEX>$\geq$</TEX>0, Q(<TEX>$\chi$</TEX>)<TEX>$\leq$</TEX>0, P(<TEX>$\chi$</TEX>)-Q(<TEX>$\chi$</TEX>)<TEX>$\geq$</TEX>0 on [a, <TEX>${\alpha}$</TEX>). (3) In this, we show that a condition a oscillatory of(1).(omitted)

Abstract

In this paper, we introduce the almost linear spaces, a generalization of linear spaces. We prove that if the almost linear space X has a finite basis then, as in the case of a linear space, the cardinality of bases for the almost linear space X is unique. In the case X = Wx ＋ Vx, we prove that B'= {<TEX>$\chi$</TEX>'<TEX>$_1,...,x'_n</TEX>} is a basis for the algebraic dual X<TEX>$^#$</TEX> of X if B = {<TEX>$\chi$</TEX>'<TEX>$_1,...,x'_n</TEX>} is a basis for the almost linear space X. And we have an example X(<TEX>$\neq$</TEX>Wx ＋ Vx) which has no such a basis.

Abstract

Consider a solution y(t) of the nonlinear equation (E) y" ＋ f(t, y) = 0. A solution y(t) is said to be oscillatory if for every T ＞ 0 there exists <TEX>$t_{0}$</TEX> ＞ T such that y(<TEX>$t_{0}$</TEX>) = 0. Let F be the class of solutions of (E) which are indefinitely continuable to the right, i.e. y <TEX>$\in$</TEX> F implies y(t) exists as a solution to (E) on some interval of the form [t<TEX>$\sub$</TEX>y/, <TEX>$\infty$</TEX>). Equation (E) is said to be oscillatory if each solution from F is oscillatory.(omitted)

Abstract

In this paper we introduce the notion of Artinian and obtain some properties of Artinian and Noetherian BCK-algebras.

Abstract