Turán's famous tetrahedron problem is to compute the Turán density of K 4 3 $K_4^3$ . This equivalent determining maximum ℓ 1 $\ell _1$ -norm codegree vector a -free n $n$ -vertex 3-uniform hypergraph. We introduce new way for measuring extremality hypergraphs and determine asymptotically extremal function in our notion. The squared sum, co 2 ( G ) $\mbox{co}_2(G)$ , hypergraph $G$ sum codegree...

Let $$G=(V,E)$$ be a finite graph (here V and E denote the set of vertices edges G respectively) $$M_G$$ centered Hardy–Littlewood maximal operator defined there. We find optimal value $$\mathbf{{C}}_{G,p}$$ such that inequality $$\begin{aligned} \mathrm{Var\,}_{p}M_{G}f\le \mathbf{C}_{G,p}\mathrm{Var\,}_{p}f \end{aligned}$$ holds for every $$f:V\rightarrow {\mathbb {R}},$$ where $$\mathrm{Var\...

A graph $G$ is a non-separating planar if there drawing $D$ of on the plane such that (1) no two edges cross each other in and (2) for any cycle $C$ $D$, vertices not are same side $D$.
 Non-separating graphs closed under taking minors subclass superclass outerplanar graphs.
 In this paper, we show only it does contain $K_1 \cup K_4$ or K_{2,3}$ $K_{1,1,3}$ as minor.
 Furthermore...

Given a finite graph G, the maximum length of sequence $$(v_1,\ldots ,v_k)$$ vertices in G such that each $$v_i$$ dominates vertex is not dominated by any $$\{v_1,\ldots ,v_{i-1}\}$$ called Grundy domination number, $$\gamma _\mathrm{gr}(G)$$ , G. A small modification definition yields Z-Grundy which dual invariant well-known zero forcing number. In this paper, we prove _\mathrm{gr}(G) \ge \fra...

We study a dynamic coordination process in which agents are uncertain about the actions of their fellow agents, and anticipate strategically relevant information. Because of the uncertainty and learning, (ir)reversibility of actions has important strategic consequences. We find that the reversibility option can either enhance or hamper efficient coordination, and we characterize the direction o...

We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hh-immune, and indicate how to generalise these results to n-c.e. sets, n > 2.

For optical studies of boiling heat transfer process, it is essential that the refractive index and its dependence on temperature for the test liquid be known. In this work a simple set-up was built to measure the refractive index of Fluorinert Liquid PF-5060 with temperature compensation and temperature control. The pinhole laser light beam was used for determining the refraction angle. The re...

Weprovide a simple set of sufficient conditions for theweak convergence of discrete-time, discrete-state Galton–Watson branching processes with immigration to continuous-time, continuous-state branching processes with immigration.

We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand we know that there are intervals which contain no hypersimple set.