An r-dynamic coloring of a graph G is proper such that every vertex in V(G) has neighbors at least $\min\{d(v),r\}$ different color classes. The chromatic number denoted as $\chi_r (G)$, the k coloring. In this paper we obtain central graph, middle total line para-line and sub-division comb $P_n\odot K_1$ by $C(P_n\odot K_1), M(P_n\odot T(P_n\odot L(P_n\odot P(P_n\odot K_1)$ $S(P_n\odot respect...

The BESIII Collaboration has observed a candidate for $c \bar c s u$ tetraquark $Z_{cs}$ at $(3982.5^{+1.8}_{-2.6} \pm 2.1)$ MeV and width $(12.8^{+5.3}_{-4.4} 3.0)$ MeV, while the LHCb in $\jpsi K^-$ channel with mass of $(4003 6 ^{+4}_{-14})$ $(131 15 26)$ MeV. In this note we examine possibility that these two states are distinct eigenstates mixing process similar to which gives rise axial-v...

‎A tree containing exactly two non-pendant vertices is called a double-star‎. ‎Let $k_1$ and $k_2$ be two positive integers‎. ‎The double-star with degree sequence $(k_1+1‎, ‎k_2+1‎, ‎1‎, ‎ldots‎, ‎1)$ is denoted by $S_{k_1‎, ‎k_2}$‎. ‎It is known that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching‎. ‎In this paper‎, ‎we study the $S_{1,2}$-decomposit...

In this work we study the strong decays of $\phi(2170)$ to final states involving kaonic resonances $K(1460)$, $K_1(1270)$ and $K_1(1400)$, on which experimental data have recently been extracted by BESIII Collaboration. The formalism developed here is based interpreting $K(1460)$ as arising from three-hadron dynamics, inspired our earlier works. For $K_1(1400)$ investigate different descriptio...

This paper studies two stochastic bulk arrivals and bulk services PH/PH/1 queue Models (A) and (B) with k_(1 ) and k_2 as the number of phases of PH arrival and PH service distributions respectively. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of th...

Steady hydromagnetic Couette flow of class-II of a viscous, incompressible and electrically conducting fluid through a porous medium in a rotating system taking Hall current into account is investigated. Heat transfer characteristics of the fluid flow are considered taking viscous and Joule dissipations into account. It is noticed that there exists flow separation at the moving plate in the sec...

Let k be a field of characteristic 0. G reductive group over the ring Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}]. We prove that has isotropic rank >=1 R iff it fractions k(x_1,...,x_n) R, and if this is case, then natural map H^1_{et}(R,G)\to H^1_{\et}(k(x_1,...,x_n),G) trivial kernel, loop reductive, i.e. contains maximal R-torus. In particular, we settle in positive conjecture V. Ch...

Abstract Let $$K_1$$ K 1 , $$K_2$$ 2 be compact Hausdorff spaces and $$E_1, E_2$$ E , Banach not containing a copy of $$c_0$$ c 0 . We establish lower estimates t...

Abstract In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$ . We also more general results where distance map $|x-y|$ replaced by function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. process, esta...