Algebraic Method for Group Classification of (1+1)-Dimensional Linear Schrödinger Equations

In the present paper, we study the existence of solutions for some nonlocal problems involving the \(p(x)\)-Laplacian operator. The approach is based on a new sub-supersolution method.     

Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.     

Universal Associative Envelopes of (n+ 1)-Dimensional n-Lie Algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Gröbner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.     

Обобшенные проиэводные и граничные своиства проиэводных интеграла Пуассона для полупространства R + k+1 (k>1)

Реэуме. Вводятся понятия обобшеннои частнои проиэводнои и обобшенного оератора Лапласа для функции многих переменных. Докаэана теоремы типа теоремы Фату о граничных своиствах частхых проиэводных интегралал Пуассона для полупространства, когда плотностя интеграла имеет обобшеннуу частнуу проиэводнуу и обобшенныи оператор Лапласа.     

(一)近似公式(1+α)~n≈1+nα的条件

高中代数教材介绍了如下的近似公式: “当α的绝对值与1相比很小时,(1+a)~n的近似值可用公式(1+α)~n≈1+nα来计算。”其实,在仅有α的绝对值与1相比很小的条件下,运用(1+α)~n≈l+nα来计算,往往得到不可思议的结果。如计算(1-0.001)~2000。这里-0.001的绝对值与1相比很小,若按上述公式计算:(1-0.001)~2000=1-2000×0.001=-1。显然(1-0.001)~2000是个正值,而经过公式近似成了一个负值,荒谬。由此可见,仅有α的绝对值与1相比很小还不行,得须加上条件:nα的绝对值与1相比也很小。     

Lump Solutions,Interaction Solutions and Breather Solutions of Generalized (3+1)-Dimensional KdV Equations北大核心CSCD

Based on the bilinear form of the generalized (3+1)-dimensional KdV equation, the lump solution, the interaction solution and the breather solution of the equation were obtained. The obtained lump solutions were proved to be rationally localized in all directions of the space, then the “fusion” and “fission” phenomena were observed during the interaction between the lump soliton wave and the one-stripe soliton. Finally, the breather solution of the equation was obtained. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

不等式k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)的几种证法

为什么要证明不等式k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)下面通过实例来说明,高中数学第三册P.147.3(4)题:求证1/1~(1/2)+1/2~(1/2)+…+1/n~(1/2)>n~(1/2)(n>1)。我们用数学归纳法来证明。 (1)当n=2时不等式左边=1/1~(1/2)+1/2~(1/2)=(2+2~(1/2))/2右边=2~(1/2)=(2~(1/2)+2~(1/2))/2,显然不等式成立。 (2)假设当n=k(k>1)时不等式成立,     

由k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)的证明所想到的

本刊1984年第8期《不等式k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)的几种证法》一文(以下简称文〔1〕中,对不等式k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)作出了六种证法。其实,这个不等式还有一种更简单的证法,现补充如下,我们姑且叫它为〔1〕的第七个证法吧: g) 间接证法:     

Iteration xn+1=αn+1f(xn) + (1 - αn+1)Tn+1xn for an Infinite Family of Nonexpansive Maps {Tn}∞n=1

Under the framework of uniformly smooth Banach spaces, Chang[1] 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.     

(1+1)维Burgers方程新的行波解

通过采用新的exp(-ρ(ξ))展式法,得到了(1+1)维Burgers方程形如u(ξ)=αm(exp(-ρ(ξ)))m+αm-1(exp(-ρ(ξ)))m-1+…的新行波解.该方法也可以应用于求解其它许多的非线性演化方程.     

Spectral Geometry of Closed Minimal Real Hypersurfaces in CP~(n+1) and QP~(n+1)

ＳｐｅｃｔｒａｌＧｅｏｍｅｔｒｙｏｆＣｌｏｓｅｄＭｉｎｉｍａｌＲｅａｌＨｙｐｅｒｓｕｒｆａｃｅｓｉｎＣＰｎ＋１ａｎｄＱＰｎ＋１＊）ＱｕＣｈｅｎｇｑｉｎ（瞿成勤）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｎａ...     

关于ξ(1/2+it)

<正> 曾经从事于求上极限Q使得(?)成立的这个问题有 Van der Corpur,Koksma,Walfisz,Tichmarsh,Phillips Titchmarsh 和闵词鹤,他们的结果是     

A Characterization of l + 1(Γ)

In this article we give necessary and sufficient conditions in order a closed cone of a Banach space E to be isomorphic to the positive cone l ₁ ⁺(Γ) of an l ₁(Γ) space. This problem has applications in the theory of Banach spaces as in the characterization of reflexive spaces and also on the problem of the embeddability of 1in Banach spaces.     

y=f(x+1)与y=f~(-1)(x+1)互为反函数吗?

从图象上看　ｙ =ｆ(ｘ + 1)与 ｙ =ｆ- 1(ｘ + 1)的图象是分别将 ｙ =ｆ(ｘ)与 ｙ =ｆ- 1(ｘ)的图象向左平移一个单位所得 ,因 ｙ =ｆ(ｘ)与 ｙ =ｆ- 1(ｘ)图象关于 ｙ =ｘ对称 ,将 ｙ =ｘ向左平移一个单位得 ｙ =ｘ + 1,所以函数 ｙ =ｆ(ｘ + 1)与 ｙ =ｆ- 1(ｘ + 1)的图象关于 ｙ =ｘ + 1对称 ,因而 ｙ =ｆ(ｘ + 1)与 ｙ =ｆ- 1(ｘ + 1)不一定互为反函数 .从求反函数过程看　由 ｙ =ｆ(ｘ + 1)有ｘ + 1=ｆ- 1(ｙ)即ｘ =ｆ- 1(ｙ) - 1,互换ｘ ,ｙ ,有 ｙ =ｆ- 1(ｘ)- 1,所以 ｙ =ｆ(ｘ + 1)的反函数为 ｙ =ｆ- 1(ｘ) - 1.记号 ｙ =ｆ- 1(…     

y′=(a_0x~n+a_1x~(n-1)y+…+a_ny~n)/(b_0x~n+b_1x~(n-1)y+…+b_ny~n)的积分曲綫的拓扑結构

<正> 設Y_n=a_ox~n+a_1x~(n-1)y+…+a_ny~n,X_n=b_ox~n+b_1x~(n-1)y+…+b_ny~n.其中Y_n,X_n无公因子.微分方程 y′=Y_n/X_n(1)只有一个奇点(0,0).当n=1时,Poincare决定了(1)的积分曲綫的拓扑結构.当n=2时,决定了(1)的积分曲綫的結构.当n=3时,张棣决定了(1)的积分曲綫