Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.

A probabilistic solution for the Syracuse conjecture

We prove the veracity of the Syracuse conjecture by establishingthat starting from an arbitrary positive integer diffrent of 1 and 4, theSyracuse process will never comeback to any positive integer reachedbefore and then we conclude by using a probabilistic approach.Classification : MSC: 11A25

Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

For a prime [Formula: see text]and a positive integer[Formula: see text], let [Formula: see text] and [Formula: see text] be the extension field of [Formula: see text]. We derive a sufficient condition for the existence of a primitive element [Formula: see text] in[Formula: see text] such that [Formula: see text] is also a primitive element of [Formula: see text], a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is a primitive element of [Formula: see text], and a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is also a primitive normal element of [Formula: see text] over [Formula: see text].

A Note on the Abelian Complexity of the Rudin-Shapiro Sequence

Let {r(n)}n≥0 be the Rudin-Shapiro sequence, and let ρ(n):=max{∑j=ii+n−1r(j)∣i≥0}+1 be the abelian complexity function of the Rudin-Shapiro sequence. In this note, we show that the function ρ(n) has many similarities with the classical summatory function Sr(n):=∑i=0nr(i). In particular, we prove that for every positive integer n, 3≤ρ(n)n≤3. Moreover, the point set {ρ(n)n:n≥1} is dense in [3,3].

Power partitions and a generalized eta transformation property

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

G -Chain Mixing and G -Chain Transitivity in Metric G-Space

Firstly, we introduce the concept of G -chain mixing, G -mixing, and G -chain transitivity in metric G -space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map f has the G -shadowing property, then the map f is G -chain mixed if and only if the map f is G -mixed. (2) The map f is G -chain transitive if and only if for any positive integer k ≥ 2 , the map f k is G -chain transitive. (3) If the map f is G -pointwise chain recurrent, then the map f is G -chain transitive. (4) If there exists a nonempty open set U satisfying G U = U , U ¯ ≠ X , and f U ¯ ⊂ U , then we have that the map f is not G -chain transitive. These conclusions enrich the theory of G -chain mixing, G -mixing, and G -chain transitivity in metric G -space.

New classes of graphs with edge $ \; \delta- $ graceful labeling

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>