Infinite-dimensional homology and multibump solutions

We start by introducing a Čech homology with compact supports which we then use in order to construct an infinite-dimensional homology theory. Next we show that under appropriate conditions on the nonlinearity there exists a ground state solution for a semilinear Schr?dinger equation with strongly indefinite linear part. To this solution there corresponds a nontrivial critical group, defined in terms of the infinite-dimensional homology mentioned above. Finally, we employ this fact in order to construct solutions of multibump type. Although our main purpose is to survey certain homological methods in critical point theory, we also include some new results. Dedicated to Felix Browder on the occasion of his 80th birthday     

Fractional and Hypersingular Operators in Variable Exponent Spaces on Metric Measure Spaces

We prove the continuity of potential type operators and hypersingular operators in variable Lebesgue and Sobolev spaces on a metric measure space (Χ, d, μ). Two variants of such operators are considered, according to the regularity admitted on the measure μ.     

Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.     

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras H^∞(E), which we call Hardy algebras, and which are noncommutative generalizations of classical H^∞, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which “reproduces” the values, on , of the “functions” coming from H^∞(E). We present results that are natural generalizations of the Poisson integral formula. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the “model space” for the canonical model that can be attached to a point in the disc . We also connect our Poisson kernel to various “point evaluations” and to the idea of curvature. The first named author was supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation. The second named author was supported in part by the U.S.-Israel Binational Science Foundation and by the B. and G. Greenberg Research Fund (Ottawa).     

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set . A polynomial of this class describes the volume of the set V + tB ⁿ as a function of t, where t is a positive number and B ⁿ denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of into for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

基于网络学习共同体的教师专业发展研究

教师专业发展被视为应对教育挑战的有效方法,而教师学习共同体则是教师发展的重要途径之一。研究和实践表明,网络学习共同体的有效建立和应用能够促进教师的专业发展。经过多年的研究与实践,提炼出一套教师网络学习共同体的建构策略。基于网络学习共同体的教师专业发展模式主要包括基于网络的自主探究模式、基于问题的协作学习模式、基于课题的专家引领模式等。     

基于“教、学、做”一体化的FMS课程开发与实施

基于自动化生产过程的专业和课程改革探讨在FMS技术专业应用中有诸多难点,该文结合作者所在单位自动化专业改革的实践,论述了"教学做"一体化教学的必要性,通过"教学做"一体化的教学模式,从专业培养目标定位、岗位能力分析、学习领域设计、学习情境设计等几方面进行FMS课程建设改革的探讨。     

关注幼小数学课程目标的衔接——基于全美《州共同核心数学标准》K-1年级的内容解读

幼小数学课程目标的衔接是整个幼小数学课程衔接的关键环节,解读全美《州共同核心数学标准》K-1年级部分的内容,认为其在儿童心理与内容制定角度上较好地衔接了幼小数学课程目标。目前,我国幼小数学课程目标的衔接存在一定问题,需要制定幼(学前一年)小统一的数学课程标准;将数学作为学前教育中独立的学习领域;兼顾行为性与展开性的数学课程目标取向;注重幼小数学课程目标内容的衔接。     

PBL教学法在制药工程专业《工程制图》教学中的应用初探

为改进《工程制图》的教学效果,在制药工程专业的《工程制图》教学中引入了PBL教学法,探讨了PBL教学模式在该课程教学中的应用方法及效果,指出了PBL教学应用于该课程教学的优势。实践证明,PBL教学模式适当地引进在该课程的教学中,有助于发挥学生的主动性和积极性,更好的激发学生的学习兴趣。