On α-Relation and Transitivity Conditions of α

The main tools in the theory of hyperstructurs are the fundamental relations. The fundamental relation on hyperring was introduced by Vougiouklis at the fourth AHA congress (1990). The fundamental relation on a hyperring is defined as the smallest equivalence relation so that the quotient would be the ring. Note that, generally, the commutativity in the ring are not assumed. In this article, we introduce a new strongly regular equivalence relation on hyperring so that the quotient is a commutative ring. Also we state the condition that is equivalent with the transitivity of this relation and finally we characterize the complete hyperring (with the fundamental relation as commutative).     

On α-accessible domains

The class A _ρ ^α of domains D ? ?ⁿ which are α-accessible with respect to the origin, α ∈ [0, 1], and have the property \(\rho = \mathop {\min }\limits_{\rho \in \partial D} ||p||\), where ρ ∈ (0, +∞) is a fixed number, is considered. Such domains satisfy the so-called cone condition, i.e., are conically accessible from the interior. The maximal set of points a such that all domains D ∈ A _ρ ^α are β-accessible with respect to a, 0 ≤ β ≤ α, is found, which solves a problem posed by Professor S.I. Dudov. This set is proved to be the open ball of radius ρ centered at 0 if α = 1 and β = 0 and the closed ball of radius \(\rho \sin \frac{{\left( {\alpha - \beta } \right)\pi }}{2}|\) centered at 0 otherwise. An answer to a question of Professor S. R. Nasyrov is also given. For domains D ? ? ⁿ α-accessible with respect to the origin, α ∈ (0, 1], a sharp upper bound for \(\rho = \mathop {\max }\limits_{\rho \in \partial D} ||p||/\rho = \mathop {\min }\limits_{\rho \in \partial D} ||p||\) is found.     

On α-quasi-irresolute functions

Joseph and Kwack [9] introduced the notion of (θ,s)-continuous functions in order to investigateS-closed spaces due to Thompson [32]. In [26], the present authors investigated further properties of (θ,s)-continuous functions. In this paper, we introduce a new class of functions called α-quasi-irresolute functions which is weaker than (θ,s)-continuous and improve some results established in [26].     

On α-close-to-convex functions

LetP(α) denote the class of functionsf analytic in the unit discE, withf(0)=0,f(z)≠0 (0<|z|<1) andf′(z)≠0 inE, satisfying the condition $$\int\limits_{\theta _1 }^{\theta _2 } {\operatorname{Re} } \left\{ {a\left( {1 + \frac{{zf''\left( z \right)}}{{f'\left( z \right)}}} \right) + \left( {1 - a} \right)\frac{{zf'\left( z \right)}}{{f\left( z \right)}}} \right\}d\theta > - \pi $$ whenever 0≤θ₁≤θ₂≤θ₁+2π,z=re^iθ r<1 and α is any positive real number. The functions inP(α) unify the classes of close-to-starlike (α=0) and close-to-convex (α=1) functions. We callf∈P(α) and α-close-to-convex function. In this paper we investigate certain properties of the classP(α).     

On Weighted α-Besov Spaces and α-Bloch Spaces of Quaternion-Valued Functions

In this paper, we give the definitions of weighted α-Besov-type spaces and α-Bloch spaces of quaternion-valued functions, then we obtain characterizations of these quaternion α-Bloch spaces by quaternion α-Besov-type spaces. Relations between Q _p norms and weighted α-Besov norms are also considered. The role of ρ?^α sequences in securing non-Bloch functions is highlighted in quaternion sense.     

More on Weak α-Openness

In [6] the authors, introduced the notion of weak -openness and investigated its fundamental properties. In this paper we investigate some more properties of this type of openness. In this connection, we obtain a new decomposition of -openness and it is also shown that weakly -open bijections preserve clopen sets.     

On Open and α-Covering Mappings

We prove that an open mapping of compact metric spaces can be made α-covering if the metric of the image is changed. An example is given in which this cannot be achieved by changing the metric in the departure space.     

Normal functions and α-Normal Functions

This paper has studied two open questions about normal functions due to Lappan, and obtained two corresponding results for α-normal functions. Received April 5, 1999, Accepted June 9, 1999     

Finding the αn-th largest element

We describe an algorithm for selecting the n-th largest element (where 0<<1), from a totally ordered set ofn elements, using at most (1+(1+o(1))H())·n comparisons whereH() is the binary entropy function and theo(1) stands for a function that tends to 0 as tends to 0. For small values of this is almost the best possible as there is a lower bound of about (1+H())·n comparisons. The algorithm obtained beats the global 3n upper bound of Schönhage, Paterson and Pippenger for <1/3.     

Rodabaugh α-闭包的推广

对LF拓扑空间的分明集及模糊格L中的元素α,Rodabaugh提出了α-闭包的概念.本文对此做了推广，对LF拓扑空间中的任一LF集，定义了HFα-闭包．并顺便引入了HFα-闭集的概念.文中讨论了这两个概念的基本性质。     

BOUNDARY LIMITS OF α-HARMONIC FUNCTIONS

In this paper, the existence of weighted boundary limits of α-harmonic functions is discussed.     

Multivariate Normal α-Stable Exponential Families

Mediterranean Journal of Mathematics - The normal inverse Gaussian distributions are used to introduce the class of multivariate normal α-stable distributions. Some fundamental properties of...     

Upper Bounds for α-Domination Parameters

We provide a new upper bound for the α-domination number in terms of a parameter α, 0 < α ≤ 1, and graph vertex degrees. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. Using a different probabilistic construction, we prove similar upper bounds for the α-rate domination number, which combines the concepts of α-domination and k-tuple domination.     

λ-STATISTICAL CONVERGENCE OF ORDER α

In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong(V,λ)-summability of order α are given.