一类超临界椭圆方程正解的存在性

本文讨论了球上半线性椭圆Ｄｉｒｉｃｈｌｅｔ问题Δｕ＋λｕ^ｑ＋ｕ^p＝０正解的存在性,其中,λ∈Ｒ,０〈ｑ〈１,ｐ〉ｐ_ｃ≡(Ｎ＋２)/(Ｎ－２)（Ｎ〉２）．在条件Ｎ≤６或Ｎ〉６,ｐ〈ｐ_Ｎ≡（Ｎ＋１－(２Ｎ－３)^1/2）／（Ｎ－３－(２Ｎ－３)^1/2）下,证明了存在唯一的λ₀,λ₀〉０,当λ＝λ₀时,有唯一的径向奇异解及无穷多个正解。     

SUPERCONVERGENCE OF THE FULL-DISCRETE F.E.M. FOR COMPRESSIBLE MISCIBLE DISPLACEMENT: THE FULL TENSOR CASE

1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｍｏｄｅｌｆｏｒｍｉｓｃｉｂｌｅｄｉｓｐｌａｃｅｍｅｎｔｏｆｏｎｅｃｏｍｐｒｅｓｓｉｂｌｅｆｌｕｉｄｂｙａｎｏｔｈｅｒｉｎａｈｏｒｉｚｏｎｔａｌｒｅｓｅｒｖｏｉｒΩ Ｒ2 ｏｆｕｎｉｔｔｈｉｃｋｎｅｓｓｄｅｓｃｒｉｂｅｄｂｙｔｈｅｎｏｎｌｉｎｅａｒｐａｒａｂｏｌｉｃｓｙｓｔｅｍ[6]ｄ(ｃ) ｐ ｔ+ ·ｕ =ｄ(ｃ) ｐ ｔ- · (ａ(ｃ) ｐ) =ｑ ,(1.1)φ ｃ ｔ+ｂ(ｃ) ｐ ｔ+ｕ· ｃ - · (Ｄ(ｕ) ｃ =(^ｃ -ｃ)ｑ ,(1.2 )ｕ·ν =(Ｄ(ｕ) ｃ-ｃｕ)…     

对“关于圆锥曲线的又一组定理”一文中错误的订正

文 [1 ]中给出如下的结论 :引理 1　对于任意的正整数ｑ ,∑ｎ-1ｋ =0ｃｏｓｑ( +2ｋπｎ ) ≡ 0引理 2∑ｎ-1ｋ=0ｃｏｓｒ( +2ｋπｎ ) =0 ,　　　ｒ:奇数ｎ2 ｒＣｒｒ2 ,　　ｒ:偶数定理 4　设圆锥曲线的焦点Ｆ ,若Ａ1 ,Ａ2 ,… ,Ａｎ 是圆锥曲线上的ｎ个点 ,且∠Ａ1 ＦＡ2 =∠Ａ2 ＦＡ3=… =∠ＡｎＦＡ1 ,则对于 ｍ ∈Ｎ ,1ＦＡ1 ｍ +1ＦＡ2 ｍ +… +1ＦＡｎ ｍ 为定值 .笔者认为 ,上述三个结论都不严密 ,现分析如下 :1　对于引理 1 ,作者显然忽视了ｑ是ｎ的倍数的情形 .因为若ｑ =ｔｎ ,则 ∑ｎ-1ｋ =0ｃｏｓｑ( +2ｋπｎ ) =∑ｎ-1ｋ=…     

Young Measure Solutions of a Class of Singular Diffusion Equations

Ｃｏｎｓｉｄｅｒｔｈｅｆｉｒｓｔｉｎｉｔｉａｌ ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ ｕ ｔ=ｄｉｖ ｑ( ｕ) ,　 (ｘ,ｔ) ∈ＱＴ,(1 )ｕ(ｘ,ｔ) =0 ,　 (ｘ,ｔ) ∈ Ω× (0 ,Ｔ) ,(2 )ｕ(ｘ,0 ) =ｕ0 (ｘ) ,　ｘ∈Ω ,(3 )ｗｈｅｒｅΩｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮｗｉｔｈｓｍｏｏｔｈｂｏｕｎｄａｒｙ Ω ,ＱＴ=Ω× (0 ,Ｔ) , ｑ = φ ,φ∈Ｃ1(ＲＮ) ,ａｎｄ φ , ｑｓａｔｉｓｆｙｔｈｅｓｔｒｕｃｔｕｒｅｃｏｎｄｉｔｉｏｎｓ(λ｜ξ｜1+δ-1 ) +≤ φ(ξ) ≤Λ｜ξ｜1+δ+ 1 ,　 ξ∈ＲＮ,(4 )｜ ｑ(ξ) …     

关于K个复相关矩阵相等的检验

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＷｅａｒｅｇｉｖｅｎｋｉｎｄｅｐｅｎｄｅｎｔＷｉｓｈａｒｔｄｅｎｓｉｔｉｅｓｏｆｔｈｅ (ｐ +ｑ)× (ｐ +ｑ)ｒａｎｄｏｍｓｙｍｍｅｔｒｉｃｐｏｓｉｔｉｖｅｄｅｆｉｎｉｔｅｍａｔｒｉｃｅｓＧ1,… ,Ｇｋｔｏｂｅｇ(Ｇｉ) =Ｋｅｘｐ -12 ｔｒＲ- 1ｉ Ｇｉ Ｇｉ12 (ｎｉ- ｑ-ｐ- 1) ,(1 )ｗｈｅｒｅｉ=1 ,… ,ｋ,ａｎｄＲｉｄｅｎｏｔｅｓｔｈｅｐｏｐｕｌａｔｉｏｎｃｏｒｒｅｌａｔｉｏｎｍａｔｒｉｘｏｆｔｈｅｉ ｔｈｐｏｐｕｌａｔｉｏｎａｎｄＫａｓａｇｅｎｅｒｉｃｌｅｔｔｅｒｄｅｎｏｔｅ…     

上期课外练习参考解答

初一年级1.50 .2 .令　Ｃ =1× 2× 3×…× 10 0 2 ,Ｄ =2× 4× 6×…× 2 0 0 4 .∵　Ａ·Ｃ =Ｂ·Ｄ ,∴　 ＡＢ =ＤＣ =2 10 0 2 .3.周长为 3× ( 43) 3 =6 4初二年级1.∵　ｐ2 +ｑ2 =ｐ2 ·ｑ2 ,　∴　 1ｐ2 +1ｑ2 =1.∴　原式 =ｐ|ｑ|- ｑ|ｐ|.当 ｐ <0 <ｑ时 ,原式 =ｐ2 +ｑ2ｑ·ｐ =ｐｑ ,当ｑ <0 <ｐ时 ,原式 =- ｐｑ .图 12 .如图 1,由于ＡＢＣＤＥＦ的各内角都是钝角 ,那么ＡＢ、ＣＤ、ＥＦ三边所在直线 ;ＢＣ、ＤＥ、ＡＦ三边所在直线 ,分别可构成△ＰＱＲ、△Ｐ′Ｑ′Ｒ′ ,而∠Ｐ =∠ＡＢＣ -∠ＰＣＢ ,∠Ｐ′ =∠ＤＥＦ -∠…     

巧化齐次式妙证不等式

若不等式两边各项的次数相等 ,不妨称之为齐次不等式 .如均值不等式中 ,ａ2 +ｂ2 ≥2ａｂ ,是齐二次不等式 ,ａ +ｂ+ｃ3 ≥ 3 ａｂｃ是齐一次不等式 ,对某些非齐次不等式的证明 ,若能结合题设条件 ,将低次项的次数适当升高 ,从而将原不等式转化为齐次不等式来处理 ,往往会产生出奇制胜的解题效果 .例 1　已知ａ、ｂ、ｃ∈Ｒ ,且ａ+ｂ +ｃ=1.求证 :ａｂ +ｂｃ+ｃａ≤ 13 .分析　所证不等式左边是二次式 ,右边是一个常数 ,即零次式 .由已知　ａ +ｂ+ｃ =1,∴　　　 (ａ+ｂ+ｃ) 2 =1,从而所证不等式可化为齐二次不等式ａｂ +ｂｃ+ｃａ≤ 13 (ａ +ｂ+ｃ) 2 ,即　ａ2 +ｂ2 +ｃ2 ≥ａｂ +ｂｃ+ｃａ .而　左边 -右边= 12 [(ａ-ｂ) 2 +(ｂ -ｃ) 2 +(ｃ -ａ) 2 ] ≥ 0 ,∴　原不等式成立 .例 2　已知 ｐ3 +ｑ3 =2 .求证 :ｐ+ｑ≤ 2 .分析　所证不等式左边为一次式 ,右边为零次式 ,考虑到已知等式是一个三次式 ,从而将所证不等式两边立方 ,得 (ｐ+ｑ) 3 ≤ 8,∵　ｐ3 +ｑ3 =2 ,　∴　 8=4( ｐ3 +ｑ...     

正n边形所有对角线与边长的2p-1次方幂和

文 [1]给出了正ｎ边形所有对角线和边长的 2 ｐ( ｐ∈Ｎ)次方幂和 ,本文将给出正边形所有对角线和边长的 2 ｐ - 1( ｐ∈Ｎ)次方幂和 .引理 1　ｓｉｎ2ｐ - 1θ =4 1-ｐ ｐ -1ｋ =0 ( - 1) ｐ- 1 ｋＣｋ2 ｐ - 1·ｓｉｎ( 2 ｐ - 1- 2ｋ)θ.证　设ｚ =ｃｏｓθ ｉｓｉｎθ , ｚ =ｃｏｓθ -ｉｓｉｎθ ,则ｓｉｎ2 ｐ - 1θ =ｚ - ｚ2ｉ2 ｐ - 1=( 2ｉ) 1- 2 ｐ 2 ｐ -1ｋ =0 ( - 1) ｋＣｋ2ｐ - 1ｚ2 ｐ - 1-ｋ ｚｋ=( 2ｉ) 1- 2 ｐ ｐ -1ｋ =0 ( - 1) ｋ　·Ｃｋ2ｐ - 1(ｚ2 ｐ - 1- 2ｋ- ｚ2 ｐ - 1- 2ｋ)(应用了Ｃｍｎ =Ｃｎ -ｍｎ …     

The Problem of Periodic Solution for Dynamical Systems（Ⅱ）

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔ：Ｈ：Ｒｎ×Ｒｎ→ＲｂｅａｓｍｏｏｔｈＨａｍｉｌｔｏｎｆｕｎｃｔｉｏｎ（ｑ，ｐ）→Ｈ（ｑ，ｐ）Ｇ：Ｒｎ×Ｒｎ→Ｒ２ｎｂｅｓｍｏｏｔｈｏｐｅｒａｔｏｒ（ｑ，ｐ）→Ｇ（ｑ，ｐ）＝（ｇ１（ｑ，ｐ），…，ｇ２ｎ（ｑ，ｐ））．　　Ｗｅｄｅｆｉｎｅｔｗｏｓｐａｃｅｓ：Ｌ＝ｓｐａｎ｛ｇｉ，｛Ｈ，ｇｉ｝，｛Ｈ，｛Ｈ，ｇｉ｝｝，…，ｉ＝１，２，…，２ｎ｝ｄＬ（ｚ）＝｛ｄｆ（ｚ）｜ｆ∈Ｌ｝　ｚ∈Ｒｎ×Ｒｎ．Ｈｅｒｅ｛，｝ｉｓｐｏｉｓｓｏｎｂｒａｃｋｅｔ．Ｔｈｒｏｕｇｈｏｕｔｔｈ…     

LOCAL CHARACTERISTIC OF HOLOMORPHIC AUTOMORPHISM ON CLASSICAL DOMAIMS

§1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　　ＬｅｔＭｂｅａｃｏｍｐｌｅｘｍａｎｉｆｏｌｄ.Ｆｏｒａｎｙｐ∈ＭａｎｄａｎｙＸ∈Ｔ1,0ｐ(Ｍ),ｔｈｅＫｏｂａｙａｓｈｉｉｎｆｉｎｉｔｅｓｉｍａｌｐｓｅｕｄｏｍｅｔｒｉｃ[1]ｏｆＸｉｓＦ(Ｘ):=ｉｎｆ｜ａ｜∈Ｃ,ｆ:ＤＭｓ.ｔ.ｆ(0)=ｐａｎｄｆ(ａｚ)=Ｘ,ｗｈｅｒｅＤｉｓｔｈｅｕｎｉｔｄｉｓｋａｎｄｆ∈Ｈｏｌ(Ｄ,Ｍ)ｉｓｔｈｅｈｏｌｏｍｏｒｐｈｉｃｍａｐｆｒｏｍＤｔｏＭ,ａｎｄｔｈｅＣａｒａｔｈｅｏｄｏｒｙｉｎｆｉｎｉｔｅｓｉｍａｌｐｅｕｓｄｏｍｅｔｒｉｃｏｆＸｉｓＥ(Ｘ):=ｓｕｐ…     

Navier-Stokes方程与Euler方程的稳定性比较

比较了Navier-Stokes方程和Euler方程的稳定性；并以它们的典型初值问题为例，分析了Navier-Stokes方程和Euler方程稳定性不同的原因．     

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.     

VISCOSITY METHOD OF A NON－HOMOGENEOUS BURGERS EQUATION

Abstract In [1], Ding et al. studied the nonhomogeneous Burgers equation ut uux = μuxx 4x.(1.1) This paper will prove that when μ → 0 the solution of (1.1) will approach the generalized solution of ut uux = 4x.(1.2) The authors notice that the equation (1.2) is beyond the scope of investigations by Oleinik O. in [2]. The solutions here are unbounded in general. The paper also studies the δ-wave phenomenon when (1.2) is jointed with some other equation.     

常系数线性差分方程的微分解法

用微分方法 ,解常系数线性差分方程     

New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author, at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by Golse and Perthame.     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

“Pexiderized” homogeneity almost everywhere

In the paper we examine Pexiderized ?-homogeneity equation almost everywhere. Assume that G and H are groups with zero, (X,G) and (Y,H) are a G- and an H-space, respectively. We prove, under some assumption on (Y,H), that if functions and satisfy Pexiderized ?-homogeneity equation

F₁(αx)=?(α)F₂(x)     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

New oscillation and nonoscillation criteria are established for the equation