Q ℒ,q-Scales in a Hyperbolic Setting

Q_p -scales arise in complex analysis as an interpolation scale between BMO, Bloch and Dirichlet spaces. They were generalized to the n-dimensional case by means of the conformal group of the unit ball and a modified fundamental solution of the Laplacian; however, this operator is no longer invariant under the action of group in consideration. In this talk we propose an approach to Q _ℒ,q-scales for homogeneous hyperbolic manifolds using a fundamental solution for the (α -homogeneous) hyperbolic Dirac operator based on a spherical Radon transform. We present also some properties of this scales. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Recognizing ℙn in Classical and Modern Setting

In classical projective algebraic geometry, ?ⁿ was seen mainly as a linear subspace. The modern setting has produced in the last 40 years several remarkable abstract characterizations of projective space. We survey some interaction between these two points of view.     

Lattice Algorithms for Multivariate L ∞ Approximation in the Worst-Case Setting

We approximate d-variate functions from weighted Korobov spaces with the error of approximation defined in the L _∞ sense. We study lattice algorithms and consider the worst-case setting in which the error is defined by its worst-case behavior over the unit ball of the space of functions. A lattice algorithm is specified by a generating (integer) vector. We propose three choices of such vectors, each corresponding to a different search criterion in the component-by-component construction. We present worst-case error bounds that go to zero polynomially with n ^?1, where n is the number of function values used by the lattice algorithm. Under some assumptions on the weights of the function space, the worst-case error bounds are also polynomial in d, in which case we have (polynomial) tractability, or even independent of d, in which case we have strong (polynomial) tractability. We discuss the exponents of n ^?1 and stress that we do not know if these exponents can be improved.     

The Method of Virtual Components in the Multivariate Setting

We describe the so-called method of virtual components for tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. Two examples of the virtual components for tight wavelet frames based on bivariate box splines on three or four direction mesh are given. As a byproduct, a new construction of tight wavelet frames based on box splines under the quincunx dilation matrix is presented.     

Krein–Langer Factorization and Related Topics in the Slice Hyperholomorphic Setting

We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling–Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results is the fact that the right spectrum of a quaternionic linear operator and the point S-spectrum coincide. Finally, we study the Krein–Langer factorization for slice hyperholomorphic generalized Schur functions. Both the Beurling–Lax type theorem and the Krein–Langer factorization are far-reaching results which have not been proved in the quaternionic setting using notions of hyperholomorphy other than slice hyperholomorphy.     

Convergence and Smoothness of Nonlinear Lane–Riesenfeld Algorithms in the Functional Setting

We investigate the Lane–Riesenfeld subdivision algorithm for uniform B-splines, when the arithmetic mean in the various steps of the algorithm is replaced by nonlinear, symmetric, binary averaging rules. The averaging rules may be different in different steps of the algorithm. We review the notion of a symmetric binary averaging rule, and we derive some of its relevant properties. We then provide sufficient conditions on the nonlinear binary averaging rules used in the Lane–Riesenfeld algorithm that ensure the convergence of the algorithm to a continuous function. We also show that, when the averaging rules are C ² with uniformly bounded second derivatives, then the limit is a C ¹ function. A canonical family of nonlinear, symmetric averaging rules, the p-averages, is presented, and the Lane–Riesenfeld algorithm with these averages is investigated.     

Test Functions, Schur–Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

Given a collection of test functions, one defines the associated Schur–Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur–Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with ${{\mathbb C}^{N}}$ of the scalar-valued version.     

A Remark on the Convergence of the Douglas–Rachford Iteration in a Non-convex Setting

Using a known construction of a Lyapunov function, it is shown that the Douglas–Rachford iteration with respect to a sphere and a line in a Hilbert space converges to the intersection point in a fashion which is stronger than uniform convergence on compact sets.     

Stability Properties for Feynman’s Operational Calculus in the Combined Continuous/Discrete Setting

We establish some stability theorems for Feynman’s operational calculus in the setting where the time-ordering measures are allowed to have both continuous and discrete parts. In particular, we investigate stability in a number of special cases of this blended approach to the operational calculus.Mathematics Subject Classifications (2000) Primary: 46J15, 47A56, 47A60, 60B10; secondary: 46N50, 47N50.     

Why and How Much Brouwer’s Fixed Point Theorem Fails in Noncompact Setting?

The celebrated Brouwer’s Fixed Point Theorem is dated in 1912. Its extension to compact set setting in Banach spaces due to Schauder appeared in 1930. Immediately it raised the question whether the Theorem can be extended to noncompact setting. The works of Kakutani, Klee, Benyamini and Sterfeld, Sternfeld and Lim solved the qualitative part of the problem. Lack of compactness makes the statement of the theorem false. However, there are some quantitative aspects of the question. The two basic are called minimal displacement problem, and optimal retraction problem. The aim of this article is to present the historical back ground and possibly, up to date state of investigations in this field. A list of open problems with comments will be discussed.     

Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.     

Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time-Dependent Setting

We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution \((\mathbf{u},\mathbf{w},\mathbf{b}) (t)\) in its maximal interval \([0,T^{*})\) provided that \(T^{*}<\infty\), which show for \(\delta\in(0,1)\) that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-(\delta s)/(1+2\delta)}\) for \(s\geq1/2+\delta\). In particular, by choosing a suitable \(\delta\), one concludes that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-s/4}\), and \((T^{*}-t)^{1/4-s/2}\) for \(s\geq1\), and \(1/2< s<3/2\), respectively. We also show that \((T^{*}-t)^{-s/3}\) is a lower rate for \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) if \(s>3/2\).     

Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices

We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.     

化“隐”为“显”“圆”来完“美”

1 问题提出 在江苏高考中,“圆”作为8个C级要求的知识点之一,是高考必考的知识点.纵观2008年至今的江苏高考方案,有关圆的试题的呈现时明时隐,有时明隐难辨.具体表现为:2008年13题(隐)、18题(明),2009年18题(明),2010年9题(明),2011年14题(明),2012年12题(明),2013年17题(2)(隐),2014年9题(明),2015年10题(明),2016年18题(明).对圆“显性”的考查,学生在求解时难度不大,若题目中“隐性”存在圆,如果不能充分挖掘题中隐含的信息,将圆化“隐”为“显”,则计算往往会非常繁琐,以致难以求解.笔者对圆的定义、性质、方程等方面展开阐述.     

化“错”为“思” 设“停”促“悟”

面对学生解决问题时形形色色的错误,应注重分析、引导,启发学生自主思考,还应巧妙地设计一些“停留”,通过深刻解析与错误题型密切相关的知识点促使学生“悟”出解题之道,积累问题解决的方法、经验.这种“停留”的设计往往起因于一类基本问题解决的不得法,从学生犯错的起因开始分析该如何设计恰当的“停留”,包括该联系哪些知识点、具体有哪些纠错策略、该用什么方式对学生进行引导等.     

Setting gates for activities in the stochastic project scheduling problem through the cross entropy methodology

This paper addresses the problem of scheduling activities in projects with stochastic activity durations. The aim is to determine for each activity a gate—a time before it the activity cannot begin. Setting these gates is analogous to setting inventory levels in the news vendor problem. The resources required for each activity are scheduled to arrive according to its gate. Since activities’ durations are stochastic, the start and finish time of each activity is uncertain. This fact may lead to one of two outcomes: (1) an activity is ready to start its processing as all its predecessors have finished, but it cannot start because the resources required for it were scheduled to arrive at a later time. (2) The resources required for the activity have arrived and are ready to be used but the activity is not ready to start because of precedence constraints. In the first case we will incur a “holding” cost while in the second case, we will incur a “shortage” cost. Our objective is to set gates so as to minimize the sum of the expected holding and shortage costs. We employ the Cross-Entropy method to solve the problem. The paper describes the implementation of the method, compares its results to various heuristic methods and provides some insights towards actual applications.     

“亲子”不是“仇敌”

像尼克这样和父母赌气离家出走的孩子,在我们的现实生活中并不少见。甚至还有孩子会做出自杀等更极端的举动。虽然尼克回来了．但贝卡他们内心的思考并未停止。父母和孩子,、明明是世上互相依靠、彼此深爱的人。却究竟为什么常常互相指责、互相伤害呢？     

“等”与“不等”

在数学中,“等”和“不等”是既对立又统一的,并且具有很多相似的性质,相互关联着．充分理解并运用“等”与“不等”的辩证关系,能大大地拓展学生的思维层次,提升学生的解题素养．     

“现值”与“终值”

“现值”与“终值”是利息计算中两个非常重要的基本概念 ,掌握好这两个概念 ,对于顺利解决有关金融中的数学问题以及理解各种不同的算法都是十分有益的 .所谓“现值”是指在 n期末的金额 A,把它扣除利息后 ,折合成现时的值 .而“终值”是指 n期后的本利和 .它们计算的基点分别     

“多思”与“统一”

一首旋律优美的音乐百听不厌,余音绕梁,叫人永世难忘;一道好题优雅的解法简捷明快.一针见血,令人叹为观止.我们在解决很多数学难题时,经常是百思不得其解,但往往当你把其中几个不同的事物转化、统一成同一