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
Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $\|p'\|_{[-1,1]}\leq\frac12\|p\|_{[-1,1]}$ for a constrained polynomial $p$ of degree at most $n$, initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(-1,1)$ and establish a new asymptotically sharp inequality., Lai-Yi Zhu, Da-Peng Zhou., and Obsahuje bibliografii
We examine the q-Pell sequences and their applications to weighted partition theorems and values of L-functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous authors noted in the introduction. Showing that simple combinatorial manipulations give rise to an identity published by the present author, a weighted form of a Lebesgue partition theorem is given as the main application to partitions. The conclusion of the paper summarizes some directions for further research, pointing out that certain conditions on the q-polynomial would be desired, and also possibly looking at the operator equation in the present paper from the position of using modular forms., Alexander E. Patkowski., and Obsahuje bibliografii
We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are introduced: quantum group of gauge transformations, Pontryagin dual of a quantum group, and Galois-Hopf-algebra of an algebra extension., Maysam Maysami Sadr., and Obsahuje bibliografii
In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators M+ and M−. More precisely, we prove that M+ and M− map W¹,p(R) → W1,p(R) with 1 < p < nekonečno, boundedly and continuously. In addition, we show that the discrete versions M+ and M− map BV(ℤ) → BV(ℤ) boundedly and map l¹(ℤ) → BV(ℤ) continuously. Specially, we obtain the sharp variation inequalities of M+ and M−, that is Var(M+(f))<Var(f) and Var(M−(f))<Var(f) if f ∈ BV(ℤ), where Var(f) is the total variation of f on ℤ and BV(ℤ) is the set of all functions f: ℤ → R satisfying Var(f) < nekonečno., Feng Liu, Suzhen Mao., and Obsahuje bibliografii