Hardy spaces H~p over non-homogeneous metric measure spaces and their applications

Let(X,d,)be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈(1,∞),0p≤1≤q≤∞,p≠q,γ∈[1,∞)and ∈∈(0,∞).In this paper,the authors introduce the atomic Hardy space Hp,q,γ atb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ∈(μ)via the discrete coefficient K(ρ),p B,S,and prove that the Calder′on-Zygmund operator is bounded from Hp,q,γ,δmb,ρ(μ)(or Hp,q,γatb,ρ(μ))into Lp(μ),and from Hp,q,γ+1atb,ρ(ρ+1)(μ)into H p,q,γ,12(δ-νp+ν)mb,ρ(μ).The boundedness of the generalized fractional integral Tβ(β∈(0,1))from Hp1,q,γ,θmb,ρ(μ)(or Hp1,q,γatb,ρ(μ))into Lp2(μ)with 1/p2=1/p1-β is also established.The authors also introduce theρ-weakly doubling condition,withρ∈(1,∞),of the measure and construct a non-doubling measure satisfying this condition.If isρ-weakly doubling,the authors further introduce the Campanato space Eα,qρ,η,γ(μ)and show that Eα,qρ,η,γ(μ)is independent of the choices ofρ,η,γand q;the authors then introduce the atomic Hardy space Hp,q,γatb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ(μ),which coincide with each other;the authors finally prove that Hp,q,γatb,ρ(μ)is the predual of E1/p-1,1ρ,ρ,1(μ).Moreover,if is doubling,the authors show that Eα,qρ,η,γ(μ)and the Lipschitz space Lipα,q(μ)(q∈[1,∞)),or Hp,q,γatb,ρ(μ)and the atomic Hardy space Hp,q at(μ)(q∈(1,∞])of Coifman and Weiss coincide.Finally,if(X,d,)is an RD-space(reverse doubling space)with(X)=∞,the authors prove that Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ)and Hp,q at(μ)coincide for any q∈(1,2].In particular,when(X,d,):=(RD,||,dx)with dx being the D-dimensional Lebesgue measure,the authors show that spaces Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ),Hp,q,γatb,ρ(μ)and Hp,q,γ,mb,ρ(μ)all coincide with Hp(RD)for any q∈(1,∞).     

有得有失 有加有减

正林泽宇说,他的爸爸妈妈经常吵架,他很羡(xiàn)慕(mù)其他同学有温暖和谐的家庭,其实,很多家庭都有不如意的事情,就拿我家来说,我爸爸是做生意的,他经常在全国各地飞来飞去,很少有时间陪我,为此我经常闷闷不乐。昨天早上,爸爸匆匆忙忙地整理好行李就去机场了,谁知过了半天,他又回来了,原来,由于雾太大,航班取消了。     

一类随机最优控制问题的单调控制解

讨论了一类可允许控制策略满足单调非降条件的随机最优控制问题,给出了值函数v(t,x,y,)满足一类受梯度限制的Hamilton-Jacobi-Bellman(HJB)方程:max{Lv(t,x,y), v(t,x,y)/ y}=0,其中Lv(t,x,y)= v/ t b(t,x,y,) v/ x 1/2σ2(t,x,y) 2v/ x2 f(t,x,y).借助粘性解的思想,定义了该类HJB方程的粘性解并在此意义下证明了v(t,x,y)是唯一粘性解,这类方程在随机控制,金融数学等领域内有重要应用.     

一道“得票高低”竞赛试题的另解

<正>文[1]中有一道关于"得票高低"的初二竞赛试题(简称题[1]),其题目与解答如下:题目学生会选举有四个候选人A,B,C,D,已知D得票比B得票多,A、B得票之和超过C、D得票之和,A、C得票之和与B、D得票之和相等,则四人得票数由高到低的排列次序是().(A)A,D,C,B(B)D,B,A,C(C)D,A,B,C(D)A,D,B,C答:(D).解用圆形图表示,因A,C得票之和与B,D得票之和相等,作一直径分圆为两半,上部分B,D;下部为A,C;已知D得票比B得票多,画得扇形D大于扇形B;由A,B得票之和超过C,D得     

对一道中考模拟题的剖析

2011年,北京石景山区二模第22题,其中蕴含的数学知识非常值得探究,题目如下:(1)已知:如图1,在四边形ABCD中,E是AD上一点,EC//AB,EB//CD,若S△DEC=1,S△ABE=3,则S△BCE=;若S=S,S=S,S=S,请直     

例说数学思维的困境及其摆脱

思维受挫就是在解题时,按一般的思路介入后,或者浅尝辄止,或者眼花缭乱,或者误入歧途,一错再错,或者前途渺茫,无法推进,就算使尽了浑身解数,左冲右突,终觉“山重水复疑无路”.笔者以为,要修复受挫的思维,只要仔细读题审题,寻找蛛丝马迹,类比联想相关知识点,分析可能的思想方法,探索恰当的思维路径,就会“峰回路转,柳暗花明”.下面举例说明,供同学们参考.     

怎一个“又”字了得

拐角和问题,图形变化多端,神秘莫测,破解之法是挖掘图形中的一个又字,这个又字时大时小,时正时偏,扑朔迷离,然连接又字撇、捺的末端,好似变8字,构造规则图形,则会揭开其神秘的面纱.析解几例,以飨读者.     

例谈“以退求进”法解题策略

著名数学家华罗庚说过:善于退,足够的退,退到最原始而不失去重要的地方,是学好数学的一个诀窍.这里所谓的退,当然不是逃跑,而是养精蓄锐,蓄势待发,是在为进寻求途径,即以退为进.它的实质是借助转化的数学思想,把复杂的问题简单化,运动的问题静止化,高维问题低维化,变量问题常量化,抽象问题具体化,代数问     

突破直线与圆锥曲线综合问题的多种策略研究

湖北八校(华中师大一附中,黄冈中学,黄石二中,鄂南高中,荆州中学,襄阳四中,襄阳五中,孝感高中)2012届高三第一次联考结束后,笔者发现此次联考理科解析几何题虽然不是特别难,但得分率仍然普遍很低,随即进行了追踪调查(数据分析,学生访谈,教师访谈,调查问卷,质量分析会),大部分     

珠心算可以防止老人痴呆症

犹记得在24年前。仲夏之际,母亲在三军总医院住院一个月才把如小玉西瓜大的肿瘤切除,当母亲身体逐渐康复,正准备隔天要出院时,父亲却因为太劳累,脑溢血倒下去,住进特护病房、重病房插管不醒人世,二周后与我们永别,与世长辞。世事如此无常,对我们的打击犹如晴天霹雳,母亲也变得郁郁寡欢,精神恍惚,思念父亲溢于言表。     

两类带有确定潜伏期的SEIS传染病模型的分析

通过研究两类带有确定潜伏期的SEIS传染病模型,发现对种群的常数输入和指数输入会使疾病的传播过程产生本质的差异．对于带有常数输入的情形,找到了地方病平衡点存在及局部渐近稳定的阈值,证明了地方病平衡点存在时一定局部渐近稳定,并且疾病一致持续存在．对于带有指数输入的情形,发现地方病平衡点当潜伏期充分小时是局部渐近稳定的,当潜伏期充分大时是不稳定的．     

Milnor方图中的w-凝聚性

整环R称为ω-凝聚整环,是指R的每个有限型理想是有限表现型的.本文证明了ω-凝聚整环是v-凝聚整环,且若(RDTF,M)是Milnor方图,则在Ⅰ型情形,R是ω-凝聚整环当且仅当D和T都是ω-整环,且T_M是赋值环;对于Ⅱ-型情形,R是ω-凝聚整环当且仅当D是域,[F:D]<∞,M是R的有限型理想,T是ω-凝聚整环,且R_M是凝聚整环.     

Compromise solutions and estimation of the noninferior set

A general framework is presented in which the relation of the set of noninferior points and the set of compromise solutions is studied. It is shown that the set of compromise solutions is dense in the set of noninferior points and that each compromise solution is properly noninferior. Also, under convexity of the criteria space, a characterization of the properly noninferior points in terms of the compromise solutions is presented. In this characterization, the compromise solutions depend continuously on the weights. Use of the maximum norm is studied also. It is shown that a subset of these max-norm solutions, obtained by taking certain limits of compromise solutions, is dense and contained in the closure of the set of noninferior points.     

Logic optimality for multi-objective optimization

Pareto dominance is one of the most basic concepts in multi-objective optimization. However, it is inefficient when the number of objectives is large because in this case it leads to an unmanageable number of Pareto solutions. In order to solve this problem, a new concept of logic dominance is defined by considering the number of improved objectives and the quantity of improvement simultaneously, where probabilistic logic is applied to measure the quantity of improvement. Based on logic dominance, the corresponding logic nondominated solution is defined as a feasible solution which is not dominated by other ones based on this new relationship, and it is proved that each logic nondominated solution is also a Pareto solution. Essentially, logic dominance is an extension of Pareto dominance. Since there are already several extensions for Pareto dominance, some comparisons are given in terms of numerical examples, which indicates that logic dominance is more efficient. As an application of logic dominance, a house choice problem with five objectives is considered.     

Chaotic synchronization with filter based on wavelet transformation

A kind of chaotic synchronization method is presented in the paper. In the transmitter, part signals are transformed by wavelet and the detail information is removed. In the receiver, the component with low frequency is reconstructed and discrete feedback is used, we show that synchronization of two identical structure chaotic systems is attained. The effect of feedback on chaotic synchronization is discussed. Using the synchronous method, the transmitting signal is transported in compressible way, system resource is saved, the component with high frequency is filtered and the effect of disturbance on synchronization is reduced. The synchronization method is illustrated by numerical simulation experiment.     

Polynomial-time algorithms for regular set-covering and threshold synthesis

A set-covering problem is called regular if a cover always remains a cover when any column in it is replaced by an earlier column. From the input of the problem - the coefficient matrix of the set-covering inequalities - it is possible to check in polynomial time whether the problem is regular or can be made regular by permuting the columns. If it is, then all the minimal covers are generated in polynomial time, and one of them is an optimal solution. The algorithm also yields an explicit bound for the number of minimal covers. These results can be used to check in polynomial time whether a given set-covering problem is equivalent to some knapsack problem without additional variables, or equivalently to recognize positive threshold functions in polynomial time. However, the problem of recognizing when an arbitrary Boolean function is threshold is NP-complete. It is also shown that the list of maximal non-covers is essentially the most compact input possible, even if it is known in advance that the problem is regular.     

油田注水系统拓扑布局优化的混合遗传算法

以投资最小为目标函数,建立了注水系统拓扑布局优化数学模型.根据模型特点,将优化问题分为两层,分别采用遗传算法和非线性优化方法进行求解.并对遗传算法的操作过程进行了改进,调整了适应函数,改进了交叉和变异操作,结合了模拟退火算法,在操作过程中使约束条件得到满足,减少了不可行解的产生,使遗传算法的优化性能得到了提高.优化算例说明了该方法的有效性.     

An Improved Branch & Bound Method for the Uncapacitated Competitive Location Problem

In this paper, the problem of locating new facilities in a competitive environment is considered. The problem is formulated as the firm expected profit maximization and a set of nodes is selected in a graph representing the geographical zone. Profit depends on fixed and deterministic location costs and, since customers are independent decision-makers, on the expected market share. The problem is an instance of nonlinear integer programming, because the objective function is concave and submodular. Due to this complexity a branch & bound method is developed for solving small size problems (that is, when the number of nodes is less than 50), while a heuristic is necessary for larger problems. The branch & bound is called data-correcting method, while the approximate solutions are obtained using the heuristic-concentration method.     

On the density of the minimal subspaces generated by discrete linear Hamiltonian systems

This paper focuses on the density of the minimal subspaces generated by a class of discrete linear Hamiltonian systems. It is shown that the minimal subspace is densely defined if and only if the maximal subspace is an operator; that is, it is single valued. In addition, it is shown that, if the interval on which the systems are defined is bounded from below or above, then the minimal subspace is non-densely defined in any non-trivial case.     

沪深股市收益的相关性

以概率作为相关度量指标,分整体相关性和尾部相关性对沪深两市收益进行考察。整体相关性采用概率方法中的变化协调形成的相关性作为度量,结果表明沪深两市收益在整体上具有一定的正相关性。对于尾部相关性,先用t分布分别拟事两市收益底分布,然后用蒙特卡洛模拟确定尾部的最优门限,进而求得尾部相关性,结果显示当市场剧烈波动时两市收益具有正的相关性,且比整体相关性强,尤其在暴跌的时候,两市具有很强的正相关性。