一个有极限的单调减少数列

证明了{(n(4n+1)/4n-1)~(1/2)∫π/20 sin~nxdx}为严格单调减少数列,且极限为(π/2)~(1/2),因而得(π(4n-1)/2n(4n+1))~(1/2)∫π/20 sin~nxdx (π(4 n+5)/2(n+1)(4n+3))~(1/2).     

一个有趣数列的单调性

利用不等式(π4(n-1)/2n 4(n+))^(1/2)1<∫_0^(π/2)sin^nxdx<(π(4n+5)/2 (n+1)(4n+3))^(1/2),对有趣数列{(n+c)^(1/2)∫_0^(π/2)sin^nxdx}的单调性再次进行了分析论证,并对已有结论进行了改进.     

推广的Wallis数列的单调性

讨论了推广的Wallis数列{(n+c)(1/2)∫π/20sin~nxdx}(n≥1,c为非负常数)的单调性.黄永忠等(2016)证明了当0≤c≤1/2该数列严格递增;当1/2c≤1该数列对于充分大的n严格递减.本文给出了此结论的一个新的简洁证明,并对相关问题做了讨论.进一步,证明了当且仅当c2π~2-16/16-π~2=0.609945…,推广的Wallis数列为严格递减数列.     

一个与Wallis不等式有关的单调减少数列

证明了{n (64 n~3+16 n~2+72n+15)/64 n~3-16 n~2+72n-15~(1/2) integral from 0 to π/2 sin~nxdx}为严格单调减少数列,且极限为π/2~(1/2),因而得π(64 n~3-16 n~2+72n-15)/2n 64 n~3+16 n~2(+72n+15)~(1/2)integral from 0 to π/2 sin~nxdxπ(64 n~3+208 n~2+296n+167)/2 n(+1)(64 n~3+176 n~2+232n+105)~(1/2),将Wallis不等式改进为512 n~3-64 n~2+144n-15/πn (512 n~3+64 n~2+144n+15)~(1/2)2(n-1)!!/2(n)!!512 n~3+832 n~2+592n+167/(πn+0.5)(512 n~3+704 n~2+464n+105)~(1/2).     

一个有极限的单调增加数列

证明了{n(16n^2+4n+3)/16n^2-4~n+3^(1/2) integral from 0 to π/2 sin^nxdx}为严格单调增加数列,且极限为π/2^(1/2),因而得π(16n^2+36n+23)/2(n+1)(16n^2+28n+15)^(1/2)相似文献   

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

无界报酬折扣半马氏模型ε最优策略的性质

本文讨论Lippman型无界报酬折扣半马氏决策规划ε最优策略的性质,在§2中证明了:若策略π~*=(π_0~*、π_1~*,…)是ε最优的,则对任何自然数n,策略(π_0~*,π_1~*,…,π_(n+)~*)为(1-β~n)~(-1)ε最优;若策略π~*=(f_0,f_1,…,f_n,π_(n+1),…)是ε最优的,则策略f_n~∞为某ε_n最优。在§3中讨论策略的组合与分解,在§4中给出了一个策略π~*为最优的充要条件和为ε最优的充分条件。     

ON THE PROPERTIES OFε(≥0) OPTIMAL POLICIES IN DISCOUNTED UNBOUNDED RETURN MODEL

This paper investigates the properties of ε(≥0) optimal policies in the model of [2].It is shownthat,if π~*=(π_0~*,π_1~*,…,π_n~*,π_(n+1)~*,…)is a β-discounted optimal policy,then(π_0~*,π_1~*,…,π_n~*)~∞ for alln≥0 is also a β-discounted optimal policy.Under some condition we prove that stochastic stationarypolicy π_n~(*∞)corresponding to the decision rule π_n~* is also optimal for the same discounting factor β.Wehave also shown that for each β-optimal stochastic stationary policy π_0~(*∞),π_0~(*∞) can be decomposed intoseveral decision rules to which the corresponding stationary policies are also β-optimal separately;and conversely,a proper convex combination of these decision rules is identified with the former π_0~*.We have further proved that for any (ε,β)-optimal policy,say π~*=(π_0~*,π_1~*,…,π_n~*,π_(n+1)~*,…),(π_0~*,π_1~*,…,π_(n-1)~*)∞ is ((1-β~n)~(-1)ε,β)optimal for n>0.At the end of this paper we mention that the resultsabout convex combinations and de     

共轭Bochner—Riesz平均在H~p上的弱型估计

设K(x)=P(x/|x|)|x|~(-n)为一球调和核,P(x)为一m次齐次调和多项式。f(x)在R~n上的δ阶共轭Bochner-Riesz平均记为 (_(1/ε)~δf)(x)=∫_(|t|<1/ε)(t)(t)(1-|εt|~2)~δe~(iαt)dt.作者在本文中得到如下的弱型估计: |{x∈R~n:sup ε>0|(_(1/ε)~δf)(x)-_ε(x)|>λ}|≤C(‖f‖_(H~p)/λ)~p,此处δ=(n/p)-(n 2)/2,n/(n 1)≤p<1,f∈H~p(R~n),以及 _ε(x)=(2π)~(-n)∫_(|y|>ε)f(x-y)K(y)dy 。设f∈L(R~n),其δ阶的Bochner-Riesz平均为 (σ_(1/ε)~δf)(x)=∫_(|t|<1/ε)(t)(1-|εt|~2)~δe~(iαt)dt.     

THE FIRST EIGENVALUE OF AN IRREDUCIBLE HOMOGENEOUS MANIFOLD

Let M be an n-dimensional compact minimal submanifold in the unit sphere. It is shown that the diameter and volume of M satisfyd≥π/2+C(n)d~n/(d~n+V).An application is that if M is an n-dimensional compact irreducible homogeneous manifold, the first eigenvalue λ_1 of M satisfiesλ_1≥n/d~2(π/2+C(n)d~n/(d~n+V))~2.In the above two cases, C(n)'s are the same constants depending only on n.