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.     

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     

零国王勇战食数兽

一天,数字王国突然闯进来一个三只角的怪兽,吓得数字王国的公民纷纷躲避走,怪兽张开血盆大口,一口吞下数36,接着它又吞吃了-6,数2吓得脚软,奇怪的是,怪兽看也没看它     

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。     

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     

Entrainment by Chaos

A new phenomenon, the entrainment of limit cycles by chaos, which results from the appearance of cyclic irregular behavior, is discussed. In this study, sensitivity is considered as the main ingredient of chaos to be captured, and the period-doubling cascade is chosen for extension. Theoretical results are supported by simulations and discussions regarding Chua’s oscillators, entrainment of toroidal attractors by chaos, synchronization, and controlling problems. It is demonstrated that the entrainment cannot be considered as generalized synchronization of chaotic systems.     

由斜坡函数激发的神经网络算子逼近

首先,引入一种由斜坡函数激发的神经网络算子,建立了其对连续函数逼近的正、逆定理,给出了其本质逼近阶.其次,引入这种神经网络算子的线性组合以提高逼近阶,并且研究了这种组合的同时逼近问题.最后,利用Steklov函数构造了一种新的神经网络算子,建立了其在L~p[a,b]空间逼近的正、逆定理.     

Implementation by mediated equilibrium

Implementation theory tackles the following problem given a social choice correspondence (SCC), find a decentralized mechanism such that for every constellation of the individuals’ preferences, the set of outcomes in equilibrium is exactly the set of socially optimal alternatives (as specified by the correspondence). In this paper we are concerned with implementation by mediated equilibrium; under such an equilibrium, the players’ strategies can be coordinated in a way that discourages deviation. Our main result is a complete characterization of SCCs that are implementable by mediated strong equilibrium. This characterization, in addition to being strikingly concise, implies that some important SCCs that are not implementable by strong equilibrium are in fact implementable by mediated strong equilibrium.     

Sublattices Generated by Polars

In an algebraic frame L, the complete sublattice CP(L) generated by the polars of L is studied, in comparison with FP(L), the subframe generated by the polars. It is shown, by an example from the theory of ?-groups, that these are distinct, in general. The relationship between FP(L), CP(L), and other established constructs, closely related to the boolean algebra of polars, is also studied.     

Integration by Parts Formula and Applications for SDEs Driven by Fractional Brownian Motions

By using coupling by change of measures, the Driver-type integration by parts formula is established for a class of stochastic differential equations driven by fractional Brownian motions. As applications, (log) shift Harnack inequalities and estimates on the distribution density of the solutions are presented.     

Induction heating of cylindrical nonmagnetic ingots by rotation in static magnetic field generated by permanent magnets

Induction heating of cylindrical nonmagnetic billets by their rotation in static magnetic field is modeled. The magnetic field is produced by a system of appropriately arranged permanent magnets. The numerical model is solved by our own full adaptive higher-order finite element method in a monolithic formulation, i.e., both magnetic and temperature fields are solved simultaneously, respecting their mutual interaction. All principal nonlinearities are included in the model (permeability of ferromagnetic parts of the system as well as temperature dependences of physical parameters of the heated metal). The methodology is illustrated by two examples whose results are discussed.     

On loops covered by subloops

The topic of this paper is the investigation of coverings of a loop by subloops. A loop has a covering by subloops if it is the set-theoretic union of proper subloops. If the set of subloops is finite, the covering is called finite. Coverings of groups by subgroups have been widely investigated and key results are detailed in the introduction. Various analogues for loops of the results for groups are obtained. An example of an infinite loop which is the union of three proper commutative subloops, but has no finite homomorphic images and has a trivial center, shows that the results for loops cannot be as general as for groups, justifying additional assumptions on the loops or the coverings.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

Scoring by usage

This paper aims to discover whether the predictive accuracy of a new applicant scoring model for a credit card can be improved by estimating separate scoring models for applicants who are predicted to have high or low usage of the card. Two models are estimated. First we estimate a model to explain the desired usage of a card, and second we estimate separately two further scoring models, one for those applicants whose usage is predicted to be high, and one for those for whom it is predicted to be low. The desired usage model is a two-stage Heckman model to take into account the fact that the observed usage of accepted applicants is constrained by their credit limit. Thus a model of the determinants of the credit limit, and one of usage, are both estimated using Heckman's ML estimator. We find a large number of variables to be correlated with desired usage. We also find that the two stage scoring methodology gives only very marginal improvements over a single stage scoring model, that we are able to predict a greater percentage of bad payers for low users than for high users and a greater percentage of good payers for high users than for low users.     

Diffraction by a Dielectric Wedge

Approximate expressions are obtained for the field producedwhen an electromagnetic source field is diffracted by a dielectricwedge. The boundary value problem, of the diffraction of an E-or H-pokarizedelectromagnetic line source by an arbitrary angled dielectricwedgeis formulated, and its solution is given in the form ofa Fredholm integral equation. The solution of the integral equationis obtained by a standard perturbation technique. The perturbationparameter is dependent on the refractive index of the dielectricwedge. The right angle dielectric wedge, which is illuminatedby an E-polarized plane wate, and whose refractive index (=n)is such that 1<n < 2, is considered in detail.