We discuss relationships among T-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral set T that contains 0, the asymptotic T-coloring ratio R(T ) is equal to the fractional chromatic number of the distance graph G(Z, D), where D=T&[0]. This fact is then used to study the distance ...

In this article, we give some equality and basic theorems about quaternion numbers, and some special operations. the notation and terminology for this paper. In this paper z 1 , z 2 , z 3 , z 4 , z are quaternion numbers. The following propositions are true: (1) (z 1 · z 2) = (z 2 · z 1). (2) If z is a real number, then z + z 3 = (z) + (z 3) + 1 (z 3) · i + 2 (z 3) · j + 3 (z 3) · k. (4) If z i...

For simplicity, we use the following convention: n denotes a natural number, i denotes an integer, p, x, x0, y denote real numbers, q denotes a rational number, and f denotes a partial function from R to R. Let q be an integer. The functor qZ yields a function from R into R and is defined as follows: (Def. 1) For every real number x holds (qZ)(x) = x q Z. Next we state a number of propositions:...

In this paper we examine the polynomials An(z) and the rational numbers An = An(0) defined by means of oo exzx2(ex x I)'1 =22 An(z)x"/n\. n=0 We prove that the numbers An are related to the Stirling numbers and associated Stirling numbers of the second kind, and we show that this relationship appears to be a logical extension of a simUar relationship involving Bernoulli and Stirling numbers. Ot...

After Zadeh introduced the concept of z-number scientists in various fields have shown keen interest applying this applications. In applications z-numbers, to compare two a ranking procedure is essential. While few functions been already proposed literature there need evolve some more good functions. paper, novel function for z-numbers proposed- "the Momentum Ranking Function"(MRF). Also, game ...

We prove that for a dense set of irrational numbers $\alpha$, the analytic centraliser map $e^{2\pi i \alpha} z+ z^2$ near $0$ is trivial. also some circle diffeomorphisms in Arnold family, with rotation numbers, have trivial centralisers. These provide first examples such maps

Z-numbers have been used to address environmental monitoring issues. A computational experiment based on a combination of odd and even numbers was also performed their comparative analysis considered. Results experimental research using Z-assessment in obscure conclusion systems applied, this approach has issues the case poorly formed process.

That is, W0 is the set of all finite products of the generators, allowing repetitions. Let W signify the larger multiplicative semigroup generated by W0 together with {1/2}. We call W the wild semigroup and W0 the Wooley semigroup. The question we consider is: Which integers belong to these semigroups? The sets of integer elements W(Z) := W ∩ Z and W0(Z) := W0 ∩ Z themselves form multiplicative...

This paper presents a hybrid fuzzy multi criteria decision making model for z-numbers using intuitive vectorial centroid. There are two novelty discuss here: 1) development of intuitive vectorial centroid defuzzification and; 2) development of hybrid fuzzy multi criteria decision making model based on consistent fuzzy preference relations and fuzzy technique for order performance by similarity ...