Let $G$ be a finite group $G$, $K$ a field of characteristic $p\geq17$ and let $U$ be the group of units in $KG$. We show that if the derived length of $U$ does not exceed $4$, then $G$ must be abelian., Dishari Chaudhuri, Anupam Saikia., and Obsahuje bibliografické odkazy
Let G be a finite group. The intersection graph ΔG of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected., Xuanlong Ma., and Obsahuje seznam literatury
A positive integer n is called a square-free number if it is not divisible by a perfect square except 1. Let p be an odd prime. For n with (n, p) = 1, the smallest positive integer f such that n^{f} ≡ 1 (mod p) is called the exponent of n modulo p. If the exponent of n modulo p is p − 1, then n is called a primitive root mod p. Let A(n) be the characteristic function of the square-free primitive roots modulo p. In this paper we study the distribution \sum\limits_{n \leqslant x} {A(n)A(n + 1)} and give an asymptotic formula by using properties of character sums., Huaning Liu, Hui Dong., and Obsahuje seznam literatury
We prove that the one-parameter group of holomorphic automorphisms induced on a strictly geometrically bounded domain by a biholomorphism with a model domain is parabolic. This result is related to the Greene-Krantz conjecture and more generally to the classification of domains having a non compact automorphisms group. The proof relies on elementary estimates on the Kobayashi pseudo-metric., François Berteloot, Ninh Van Thu., and Obsahuje seznam literatury
For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive integer solution of the Pell equation $u^2-Dv^2=1,$ and let $h(4D)$ denote the class number of binary quadratic primitive forms of discriminant $4D$. If $D$ satisfies $2\nmid D$ and $v_1h(4D)\equiv0 \pmod D$, then $D$ is called a singular number. In this paper, we prove that if $(x,y,z)$ is a positive integer solution of the equation $x^y+y^x=z^z$ with $2\mid z$, then maximum $\max\{x,y,z\}<480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions $(x,y,z)$., Xiaoying Du., and Obsahuje bibliografické odkazy
We are concerned with the problem of differentiability of the derivatives of order m + 1 of solutions to the “nonlinear basic systems” of the type {\left( { - 1} \right)^m}\sum\limits_{\left| \alpha \right| = m} {{D^\alpha }{A^\alpha }\left( {{D^{\left( m \right)}}u} \right)} + \frac{{\partial u}}{{\partial t}} = 0\;in\;Q. We are able to show that {D^\alpha }u \in {L^2}\left( { - a,0,{H^\partial }\left( {B\left( \sigma \right),{\mathbb{R}^N}} \right)} \right),\;\left| \alpha \right| = m + 1, for \partial\in \left ( 0,1/2 \right )and this result suggests that more regularity is not expectable., Roberto Amato., and Obsahuje seznam literatury
For a finite group $G$, the intersection graph of $G$ which is denoted by $\Gamma(G)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H\cap K\neq1$. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of
${\rm Aut}(\Gamma(G))$., Hossein Shahsavari, Behrooz Khosravi., and Obsahuje bibliografii
For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are an(k) = (2n+n(n−1)(k−2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(an(k)ām(k), p) for k-polygonal numbers with 1 ≤ m, n ≤ p − 1, and give an interesting computational formula for it., Jing Guo, Xiaoxue Li., and Obsahuje seznam literatury
We prove and discuss some new (Hp,Lp)-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q_{k}:k\geqslant 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund means tn with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {q_{k}:k\geqslant 0} in an appropriate way., István Blahota, Lars-Erik Persson, Giorgi Tephnadze., and Obsahuje seznam literatury
A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists., Tamás Milkovszki, Zoltán Muzsnay., and Seznam literatury