Application of coupling methods to the Neumann eigenvalue problem

Summary By using coupling methods, some lower bounds are obtained for the first Neumann eigenvalue on Riemannian manifolds. This method is new and the results improve some known estimates. An example shows that our estimates can be sharp.Research supported in part by the Foundation of Beijing Normal University     

Function spaces over GO-spaces: Part I

We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained.     

基团共轭效应研究Ⅱ：^13C核磁共振的化学位移与取代基的共轭效应强度

引言在定量和定性研究有机化合物的结构与性质的关系中,最广泛使用的是Hammett等所提出的σ_m、σ_p等取代基常数以及基团的诱导(场)效应和共轭效应常数。在建立基团的共轭效应常数方面,人们做了大量的工作,但多数工作是基于对基团的Hammett常数σ的人为分解上,建立的模型和分解方式(系数因子)不同,则得到不同的结果。另一方面纯理论计算工作大多引入较多参数,使计算复杂和使用不便。作者曾在建立表征诱导效应大小的     

Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

Synthesis of first nonannulated 1,2,3,4-tetrazine 1,3-dioxides

Thermolysis of o-diazidobenzotetrazine 1,3-dioxides was accompanied by cleavage of the C-C bond of the benzene ring to give nonannulated 1,2,3,4-tetrazine 1,3-dioxides. The structures of these first representatives of nonfused 1,2,3,4-tetrazines were confirmed by ¹³C and ¹⁴N NMR spectroscopy. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1589–1594, September, 2006.     

Cyclizations via Palladium-Catalyzed Allylic Alkylations [New Synthetic Methods (79)]

The history of ring systems in organic chemistry parallels their synthetic accessibility. Transition-metal-catalyzed cyclizations offer a new opportunity to create carbo- and heterocyclic compounds with great facility. Among these methods, allylic alkylations catalyzed by palladium have proven unusually productive because of the extraordinary chemo-, regio-, and diastereoselectivity and the continuing possibility for the development of enantioselectivity. The rules for ring closure differ from those for non-transition-metal-catalyzed reactions. A major benefit is the ability to generate medium (eight-, nine-, ten-, and eleven-membered) and large rings in preference to normal (five-, six- and seven-membered) rings. With the appropriate substrate, efficient macrocyclizations are possible under conditions of normal concentrations. A second major benefit derives from the complementary stereochemistry of the metalcatalyzed substitution (net retention of configuration) compared to non-metal-catalyzed reactions (inversion of configuration). Further, the requirement for the substrate to conform to the transition-metal template may impose a stereochemical preference in the intermediate that ultimately translates into the thermodynamically less stable organic product regardless of the stereochemistry of the starting material. While more work has focused on carbocyclic synthesis, the possibilities for heterocyclic synthesis are just beginning to be tapped. In addition to forming heterocycles by C? C bond formation, use of a heteroatom as a nucleophile has already proven effective for oxygen and nitrogen, with other nucleophiles awaiting investigation. New dimensions for cyclization via allylic alkylation arise by generating the requisite π-allylpalladium intermediates by methods other than palladium(0)-initiated allylic ionizations. In addition, metals other than palladium will clearly expand the possibilities, but as yet remain untapped.     

A new general-purpose isothermal microcalorimeter for use at temperatures up to 200 °C

The design and some properties of a new general-purpose isothermal microcalorimeter are reported. The instrument is a twin thermopile heat conduction calorimeter, which is designed for use up to 200 °C. The calorimetric units and surrounding heat sink are suspended inside a hollow aluminium construction, which is thermostated. Above that unit a second thermostated block is positioned and the whole assembly is suspended inside a Dewar vessel. When the instrument is used at room temperature and below, the thermostated units are cooled by use of an insertion Peltier effect cooler. The instrument can be used with a wide range of different reaction vessels (diameter 14 mm). Baseline experiments have been conducted in the temperature range 15-200 °C. Typical values obtained during 10 h periods at 200 °C are ±3 and ±10 nW for the baseline drift and baseline fluctuations, respectively. The heat detection limit, determined by release of electrical energy, is about 2 μJ. Preliminary stability measurements have been conducted at 100 °C on samples of stabilised and non-stabilised polyamide film.     

Local structure of graphs with λ=μ=2,a 2=4

Several properties of graphs with ==2,a ₂=4 are studied. It is proved that such graphs are locally unions of triangles, hexagons or heptagons. As a consequence, a distance regular graph with intersection array (13,10,7;1,2,7) does not exist.     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.