函数F(α)=(A~rB~αA~r)~((p+2r)/(α+2r))的最优单调区间

设p>0且A,B是Hilbert空间上两个正算子,Furuta给出若A>B>0,那么对任意r>0,F(α)=(ArBαAr)p+2r/α+2r 是关于α>P单调递减的,但是他指出这个结果在0<α0的条件下并不一定成立.本文给出: (1)如果-1/2-2r,那么F(α)在(-∞,-P-4r]及[p,+∞)上单调递增,并且这两个区间不能扩大; (3)如果r>0,P>0,那么[p,+∞)也是F(α)的最佳单调区间.     

函数不等式e~x≥x+1及其变形在高考压轴题中的应用

1.不等式ex≥x+1(x∈R)的证明记f(x)=ex-x-1,则f′(x)=ex-1.令f′(x)=0得x=0,当x∈(-∞,0)时,f′(x)<0,当x∈(0,+∞)时,f′(x)>0,∴f(x)在(-∞,0)单调递减,在(0,+∞)单调递增,∴f(x)在R上的最小值为f(0)=0,∴ex≥x+1(x∈R),当且仅当x=0时等号成立.     

2006年高考中的十个易错问题

一、忽视函数单调性的概念致错例1(北京卷)已知f(x)=(3a-1)x 4a,x<1logax,x≥1是(-∞, ∞)上的减函数,那么a的取值范围是()A.(0,1)B.(0,13)C.[71,31)D.[71,1)错解:因为f(x)在(-∞, ∞)上是减函数,所以f(x)在(-∞,1)和(1, ∞)上是减函数,于是3a-1<0且0相似文献   

关于多个算子的Furuta型算子单调函数

设p≥1,且A、B是Hilbert空间上两个正算子,T.Furuta给出若A≥B>0,那么对任意t∈[0,1]有,G(r,s)=A-r/2{Ar/2(A-t/2BpA-t/2)sAr/2}1-t+r/(p-t)+srA-r/2是关于r,s在r≥t及s≥1上单调递减的,我们给出该结果可以推广到多个算子的情形.     

广义加权Bloch空间与QK(p,q)空间之间的复合算子

设φ是复平面上的单位圆盘D上的全纯自映射,0＜p＜∞,q＞-2,α＞0.本文给出广义加权Bloch空间与QK(p,q)空间之间的复合算子的有界性和紧性.     

由一道高考题引发的探索与思考

1 问题出现 孰是孰非 高考结束的第二天,班里平时爱动脑筋的学生甲来问笔者:“李老师,12题怎么做?”作为最后一道选择题,此题必有含金量,笔者极为重视. 题1(2015全国理Ⅱ-12)设函数f'(x)是奇函数f(x)(x∈R)的导函数,f(-1)=0,当x＞0时,xf'(x)-f(x)＜0,则使得f(x)＞0成立的x的取值范围是() A.(-∞,-1)∪(0,1) B.(-1,0)∪(1,+∞) C.(-∞,-1)∪(-1,0) D.(0,1)∪(1,+∞) 笔者的解答是:由题意,设h(x)=f(x)/x,则h'(x)=xf'(x)-f(x)/x2.由于f(x)(x∈R)是奇函数,f(-1)=0,所以f(1)=0,f(0)=0,函数f(x)有这三个零点.显然h(x)是偶函数,由于xf'(x)-f(x)＜0,故当x＞0时,h'(x)＜0,h(x)在(0,+∞)上单调递减,因此,h(x)在(-∞,0)上单调递增.当x∈(-∞,-1)时,h(x)=f(x)/x＜0,f(x)＞0;当x∈(-1,0)时,h(x)=f(x)/x＞0,f(x)＜0;当x∈(0,1)时,h(x)=f(x)/x＞0,f(x)＞0;当x∈(1,+∞)时,h(x)=f(x)/x＜0,f(x)＜0.故综上,x∈(-∞,-1)∪(0,1),选A.     

A sharp inequality for the tail probabilities of sums of i.i.d. r.v.’s with dominatedly varying tails

Let F be a distribution function supported on (-∞, ∞) with a finite mean μ. In this note weshow that if its tail F = 1 - F is dominatedly varying, then for any γ＞ max{μ, 0}, there exist C(γ) ＞ 0 and D(γ) ＞ 0 such thatC(γ)nF(x) ≤ Fn*(x) ≤ D(γ)nF(x),for all n ≥ 1 and all x ≥γn. This inequality sharpens a classical inequality for the subexponential distributioncase.     

Maximal characterizations of Herz type Hardy spaces on homogeneous groups

In this paper, the authors establish some characterizations of Herz-type Hardy spaces HKα,p q(G) and HKq,p q(G), where 1 ＜ q ＜∞, Q(1 - 1/q) ≤α＜∞, 0 ＜ p ＜∞ and G denotes a graded homogeneous Lie group.     

正规弱空间的无限Tychonoff积

本文证明(1)如果X=∏σ∈∑Xσ是|Σ|-仿紧空间,则X是正规弱(-θ)-可加空间当且仅当 F∈[∑]＜ω,∏σ∈FXσ是正规弱(-θ)-可加空间.(2)设X=∏i∈ωXi是可数仿紧的,则下列三条等价X是正规弱(-θ)-可加的; F∈[ω]＜ω,∏i∈FXi是正规弱(-θ)-可加的; n∈ω,∏i≤nXi是正规弱(-θ)-可加的.     

一般粗糙拟微分算子的(q,r)有界性

本文考虑粗糙拟微分算子T_a的(q,r)有界性,其中振幅a∈L^pS_ρ^m(p≥1,m∈R,0≤ρ≤1).当0pS_ρ^m,T_a是L^q到L^r有界的.这个结果在多个方向推广或者改进了已知结论,并且在一般情况下r,m的范围不能改进.     

On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

On the comparison between the APV and the NPV computed via the WACC

The valuation of levered investment in the practice is made with the WACC approach, even if the superior technique of the APV is available. The paper shows that the APV can be interpreted as the arbitrage free value of the portfolio made by an investment and a supporting loan. Therefore the WACC evaluation, generally different from the APV, allows for arbitrage. We provide various conditions which completely characterize the sign of the error as a functions of all the variables entering the model.     

用常微分方程模型分析预防和隔离措施对SARS发病率的影响

通过建立常微分方程模型 ,分析了预防和隔离措施对 SARS发病率的影响 ,并把计算结果与实际统计数据进行了比较 ,结果表明 ,及时高效的预防和隔离措施能够有效地控制 SARS的传播 .     

On the Gap in the Spectrum of the Translations

The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m₁ there is no gap in the spectrum which is larger than 2m₁. In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.     

From the subalgebras of the square to the discriminator

Presented by A. F. Pixley.     

From the neutrino to the edge of the universe

Two recent findings necessitate a closer look at the existing standard models of particle physics and cosmology. These are the discovery of neutrino oscillation, and hence a non-zero mass on the one hand and, on the other, observations of distant supernovae which indicate that contrary to popular belief, the universe would continue to expand for ever, possibly accelerating in the process. In this paper it is pointed out that relatively recent studies which indicate a stochastic, quantum vacuum underpinning and a fractal structure for space time, reconcile both of the recent observations, harmoniously.     

On the dependence of the growth rate on the length of the defining relator

Let { } be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999.