A New Proof of Diophantine Equation （^n2） = （^m4）

By using algebraic number theory and p-adic analysis method, we give a new and simple proof of Diophantine equation （^n2） = （^m4）     

Über die Anzahl der Primfaktoren von\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)

Ohne Zusammenfassung     

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

Stability of Viscoelastic Wave Equation with Structural $\delta$-Evolution in $\mathbb{R}^{n}$

The aim of this paper is to study the Cauchy problem for the viscoelastic wave equation for structural $\delta$-evolution models. By using the energy method in the Fourier spaces, we obtain the decay estimates of the solution to considered problem.     

关于指数Diophantine方程$x^3-1=2py^2$

设p是奇素数,本文证明了:当p=48t~2 1,其中t是正整数时,方程x~3-1=2py~2无正整数解(x,y).     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

A New Proof of Diophantine Equation ■

By using algebraic number theory and p-adic analysis method,we give a new and simple proof of Diophantine equation ■.     

Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$

We study the following mean field equation$$\Delta_{g}u+\rho\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}d\mu}-\frac{1}{4\pi}\right)=0\ \ \mbox{in}\ \ \mathbb{S}^{2},$$where $\rho$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $\rho=4n(n+1)\pi$ for any odd integer $n\geq3$.     

On irreducible p,q‐representations of Lie algebras \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,1)$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,0)$\end{document}

In this paper, the irreducible p, q‐representations of the Lie algebras $\mathcal {G}(0,1)$ and $\mathcal {G}(0,0)$ are discussed. We prove two theorems that classify certain irreducible p, q‐representations of these Lie algebras and construct their one variable models in terms of p, q‐derivative and dilation operators. As an application, we derive a p, q‐special function identity based on one such model.     

Remarks on a Mean Field Equation on $\mathbb{S}^2$

In this note, we study symmetry of solutions of the elliptic equation\begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2},\end{equation*} that arises in the consideration of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.     

关于Diophantine方程(n+1)+(n+2)y=nz

设n是正整数.本文证明了:方程(n+1)+(n+2)y=nz仅当n=3时有正整数解(y,z)=(1,2).     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.     

矩阵方程$X A^*X^{-q}A=Q~(q\geq 1)$的Hermite正定解

In this paper, Hermitian positive definite solutions of the nonlinear matrix equation X ＋ A^＊X^-qA = Q （q≥1） are studied. Some new necessary and sufficient conditions for the existence of solutions are obtained. Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions, and the convergence analysis is also given. The theoretical results are illustrated by numerical examples.     

$A(n,k)$和$P(n,k)$的精确公式

设A(n,k)表示不定方程的非负整数解的个数,P(n,k)为整数n分为k个部分的无序分拆的个数,每个分部不小于1．本文给出了A(n,k)和P(n,k)的精确表达式．     

An atomless interval Boolean algebra A such that $ \mathfrak{a} (A) $ < $ \mathfrak{t} (A) $

For any Boolean algebra A, is the smallest cardinality of an infinite partition of unity in A. A tower in a Boolean algebra A is a subset X of A well-ordered by the Boolean ordering, with but with is the smallest cardinality of a tower of A. Given a linearly ordered set L with first element, the interval algebra of L is the algebra of subsets of L generated by the half-open intervals [a, b). We prove that there is an atomless interval algebra A such that . Received January 21, 2002; accepted in final form March 13, 2002.     

涉及无限族非扩张映象$\{T_n\}_{n=1}^\infty$的迭代算法$x_{n 1}=\alpha_{n 1}f(x_n) (1-\alpha_{n 1})T_{n 1}x_n$

Under the framework of uniformly smooth Banach spaces, Chang proved in 2006 that the sequence {xn} generated by the iteration xn＋1 =αn＋1f（xn） ＋ （1 - αn＋1）Tn＋1xn converges strongly to a common fixed point of a finite family of nonexpansive maps {Tn}, where f ： C → C is a contraction. However, in this paper, the author considers the iteration in more general case that {Tn} is an infinite family of nonexpansive maps, and proves that Chang＇s result holds still in the setting of reflexive Banach spaces with the weakly sequentially continuous duality mapping.     

On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.