关于Smarandache 函数S(n)的一个猜想

对任意正整数n,著名的Smarandache 函数S(n)定义为最小的正整数m,使得n|m!.即就是S(n)=min{m∶n|m!,m∈N}.本文的主要目的是利用初等方法研究一个包含函数S(n)的猜想,并部分的得到解决.     

Homogeneity in powers of zero-dimensional first-countable spaces

A construction of L. Brian Lawrence is extended to show that the -power of every subset of the Cantor set is homogeneous via a continuous translation modulo a dense set. It follows that every zero-dimensional first-countable space has a homogeneous -power.

     

The USCI approach and elementary superposition for combinatorial enumeration

Summary The elementary superposition theorems are presented for enumerating chemical compounds that contain achiral and chiral ligands. Subduced cycle indices (SCI-CF), partial cycle indices (PCI-CF), and cycle indices (CI-CF) with chirality fittingness are defined by starting from unit subduced cycle indices with chirality fittingness (USCI-CF). All of these indices afford generating functions that are proved to be applicable to combinatorial enumeration. In addition, the concept of elementary superposition with and without chirality fittingness is proposed to provide the elementary superposition theorems. These theorems provide us with a new methodology of enumerating compounds, in which the numbers of isomers are obtained without relying on generating functions and are itemized with respect to molecular formulas (weights) and symmetries. The operation is defined on the basis of the elementary superposition. Thereby, we derive superposition theorems concerning the PCI-CFs and the CI-CFs. These are applicable to combinatorial enumeration.     

Quantenchemische Untersuchung der Elementarprozesse beim Plasmaätzen im System Fluor/Silizium

Summary Using the cluster model of the silicon (111) surface we derive by means of EHT calculations a mechanism of reactive plasma etching in the system F/Si including diffusion processes. Species SiF₂ are found to be the primary etching products at the surface. SiF₄ is formed with high probability in the gas phase.

     

Breakdown conditions and thermal instability in air

Summary Reduced electric fields (E/N) responsible for electrical breakdown in air have been calculated by solving a stationary Boltzmann equation including superelastic vibrational collisions. The results show a decrease ofE/N with increasing gas temperature. The possibility of air instability due to chemical processes producing electrons is then investigated by calculating the threshold of this instability as a function of a characteristic time for heat dissipation τ.     

Organotransition-metal chemistry: past development and future outlook

The course of the development of organometallic chemistry is reviewed from the author's personal viewpoint. The impact of the discoveries of ferrocene and the Ziegler catalyst on the later development of organotransition-metal chemistry is discussed. Through evolution of fundamental concepts, such as the insertion of olefins and carbon monoxide into metal—carbon bonds, oxidative addition and reductive elimination, and attack of nucleophiles on coordinated ligands, together with reactivities of carbene and metallacyclic complexes, various important processes have been developed. The diversity of transition-metal complexes and their unique reactivities not found in usual organic compounds warrant further developments in the various forefronts related to organometallic chemistry with almost limitless possibilities in applications as well as in fundamental studies.     

Grade 3 students’ mathematization through modeling: Situation models and solution models with mutli-digit subtraction problem solving

In considering mathematics problem solving as a model-eliciting activity ( [Lesh and Doerr, 2003], [Lesh and Harel, 2003] and [Lesh and Zawojewski, 2008]), it is important to know what students are modeling for the problems: situations or solutions. This study investigated Grade 3 students’ mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students’ modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic.     

Elementary maps on nest algebras

Let A₁, A₂ be algebras and let M:A₁→A₂, M^∗:A₂→A₁ be maps. An elementary map of A₁×A₂ is an ordered pair (M,M^∗) such that

     

Maximal line subbundles of stable bundles of rank 2 over an algebraic curve

Let E be a vector bundle of rank 2 over an algebraic curve X of genus g ≥ 2. In this paper, we prove that E is determined by its maximal line subbundles if it is general. By restudying the results of Lange and Narasimhan which relates the maximal line subbundles with the secant varieties of X, we observe that the proof can be reduced to proving some cohomological conditions satisfied by the maximal line subbundles. By noting the similarity between these conditions and the notion of very stable bundles, we get the result for the case when E has Segre invariant s(E) = g. Also by using the elementary transformation, we have the result for the case s(E) = g−1. I. Choe and J. Choy were supported by KOSEF (R01-2003-000-11634-0) and S. Park was supported by Korea Research Foundation Grant funded by Korea Government(MOEHRD, Basic Research Promotion Fund) (KRF-2005-070-C00005)