Narrow Results

Let $K_n$ denote a complete graph on $n$ vertices and $S_k$ denote a complete bipartite graph $K_{1,k}$. A Bowtie $B_l$ is a graph formed by the union of two cycles $C_n$ and $C_m$ intersecting at a common vertex. A decomposition of a graph $G$ is a collection of edge-disjoint subgraphs $H$, such that every edge of G belongs to exactly one $H$. Given non-isomorphic subgraphs $H_1$ and $H_2$ of $G$, a $(H_1,H_2)$ — multi-decomposition of $G$ is the decomposition of $G$ into $a$ copies of $H_1$ and $b$ copies of $H_2$, such that $a{H}_1\oplus b{H}_2=G$, for some integers $a,b\ge 0$. In this paper, the multi-decomposition of $K_n$ into $S_k$ and $B_l$ has been investigated and obtained a necessary and sufficient condition when $k=l=6$. It is proved that for a given positive integer $n$, $K_n$ can be decomposed into $a$ copies of $S_6$ and $b$ copies of $B_6$ for some pair of non-negative integers $(a,b)$ if and only if $6(a+b)=\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$, for all $n\ge 9$.

A regular language $L$ is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each $n\ge 4$, there exists a ternary witness of state complexity $n$ that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has ${(n-1)}^n$ elements and requires at least $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

The spherically symmetric thin shells of the barotropic fluids with the linear equation of state are considered within the frameworks of general relativity. We study several aspects of the shells as completely relativistic models of stars, first of all the neutron stars and white dwarfs, and circumstellar shells. The exact equations of motion of the shells are obtained. Also we calculate the parameters of the equilibrium configurations, including the radii of static shells. Finally, we study the stability of the equilibrium shells against radial perturbations.

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For a set ${𝒦}$ of connected graphs, a spanning subgraph $H$ of $G$ is called a ${𝒦}$-factor if each component of $H$ is isomorphic to a member of ${𝒦}$. In this paper, some sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the $\{K_{1,j}:1\le j\le 2k\}$-factor and $\{P_2,P_5\}$-factor for any graph are obtained.

The $\{K_{1,1},K_{1,2},\dots ,K_{1,k},{𝒯}(2k+1)\}$-factor and $\{K_{1,2},K_{1,3},K_5\}$-factor of a graph are a spanning subgraph whose each component is an element of $\{K_{1,1},K_{1,2},\dots ,K_{1,k},{𝒯}(2k+1)\}$ and $\{K_{1,2},K_{1,3},K_5\}$, respectively, where ${𝒯}(2k+1)$ is a special family of trees. In this paper, we obtain a sufficient condition in terms of tight toughness, isolated toughness and binding number bounds to guarantee the existence of a $\{K_{1,1},K_{1,2},\dots ,K_{1,k},{𝒯}(2k+1)\}$-factor and $\{K_{1,2},K_{1,3},K_5\}$-factor for any graph.

For a set ${\mathscr{H}}$ of connected graphs, a spanning subgraph $H$ of a graph $G$ is an ${\mathscr{H}}$-factor if every component of $H$ is isomorphic to some member of ${\mathscr{H}}$. In this paper, we give a criterion for the existence of tight toughness, isolated toughness and binding number bounds in a graph of a strong ${𝒮}$-star factor, $\{1,3,\dots ,2n-1\}$-factor and $f$-star factor. Moreover, we show that the bounds of the sufficient conditions are sharp.

For two given graphs G₁ and G₂, the Ramsey number R(G₁, G₂) is the smallest positive integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let S_n denote a star of order n and W_m a wheel of order m + 1. In this paper we show that R(S_n, W₈) = 2n + 1 for n ≥ 5 and n ≡ 1 (mod 2).

Recent results of pulsar observations at Nanshan are present in this paper. In total 47 glitches were detected in the young pulsars, including a giant glitch in PSR B2334+61. We also investigated the scintillation parameters with the quasi-continuous observations of PSR B0329+54 spanning twenty days. Mainly using the same dataset, consecutive mode-switching events were analyzed in pulse profiles of PSR B0329+54.