A maximum matching of a graph G is a matching of G with the largest number of edges. The matching number of a graph G, denoted by {\alpha }'(G), is the number of edges in a maximum matching of G. In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture is true for every connected graph G with {\alpha }'(G)\leq 3., Fuyuan Chen., and Obsahuje seznam literatury
Let Q = (qn)n=1∞ be a sequence of bases with qi ≥ 2. In the case when the qi are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose Q-Cantor series expansion is both Q-normal and Q-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of Q, and from this construction we can provide computable constructions of numbers with atypical normality properties., Dylan Airey, Bill Mance, Joseph Vandehey., and Obsahuje seznam literatury
A graph G is a k-tree if either G is the complete graph on k + 1 vertices, or G has a vertex v whose neighborhood is a clique of order k and the graph obtained by removing v from G is also a k-tree. Clearly, a k-tree has at least k + 1 vertices, and G is a 1-tree (usual tree) if and only if it is a 1-connected graph and has no K_{3} -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of k-trees as follows: if G is a graph with at least k + 1 vertices, then G is a k-tree if and only if G has no K_{k+2} -minor, G does not contain any chordless cycle of length at least 4 and G is k-connected., De-Yan Zeng, Jian-Hua Yin., and Obsahuje seznam literatury
A theorem of Burnside asserts that a finite group G is p-nilpotent if for some prime p a Sylow p-subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of |G|, the order of G. Let P \in Syl_{p} (G). As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p-subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P \not\cong \left\langle {a,b;{a^{{p^{n - 1}}}} = 1,{b^2} = 1,{b^{ - 1}}ab = {a^{1 + {p^{n - 2}}}}} \right\rangle, where n\geq 3 for p > 2 and n\geq 4 for p = 2, then G is p-nilpotent or p-closed., Jiangtao Shi., and Obsahuje seznam literatury
We deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the Q-transform employed previously by Varshamov and Garakov (1969) as well as by Meyn (1990) for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper these conditions are removed. We construct infinite sequences of irreducible polynomials of nondecreasing degree starting from any irreducible polynomial., Simone Ugolini., and Obsahuje seznam literatury
A general theorem (principle of a priori boundedness) on solvability of the boundary value problem ${\rm d} x={\rm d} A(t)\cdot f(t,x),\quad h(x)=0$ is established, where $f\colon[a,b]\times\mathbb{R}^n\to\mathbb{R}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A\colon[a,b]\to\mathbb{R}^{n\times n}$ with bounded total variation components, and $h\colon\operatorname{BV}_s([a,b],\mathbb{R}^n)\to\mathbb{R}^n$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x(t_1(x))=\mathcal{B}(x)\cdot x(t_2(x))+c_0,$ where $t_i\colon\operatorname{BV}_s([a,b],\mathbb{R}^n)\to[a,b]$ $(i=1,2)$ and $\mathcal{B}\colon\operatorname{BV}_s([a,b],\mathbb{R}^n)\to\mathbb{R}^n$ are continuous operators, and $c_0\in\mathbb{R}^n$., Malkhaz Ashordia., and Obsahuje bibliografické odkazy
We consider the annihilator of certain local cohomology modules. Moreover, some results on vanishing of these modules will be considered., Ahmad Khojali., and Obsahuje bibliografii
We give a simple proof that critical values of any Artin L-function attached to a representation ̺ with character χ̺ are stable under twisting by a totally even character χ, up to the dim̺-th power of the Gauss sum related to χ and an element in the field generated by the values of χ̺ and χ over Q. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward., Peng-Jie Wong., and Seznam literatury
Let $G$ be a finite group. A normal subgroup $N$ of $G$ is a union of several $G$-conjugacy classes, and it is called $n$-decomposable in $G$ if it is a union of $n$ distinct $G$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5., Ruifang Chen, Xianhe Zhao., and Obsahuje bibliografické odkazy