一类非线性方程存在多个极限环的一个充分条件

关于方程(1)的极限环唯多性问题,不少人作了研究,[1]引进“n重互相包含”曲线类,,经[2]的改进获得较好的结果,但这类曲线要求的条件较多。本文不从[1]的观点出发,不包含[1]的全部条件,独立地用①11。’,LtnnoB变换折叠相平面,据轨线与折叠线交点的位置,对(1)的等价方程按线性方程积分,将积分的正负项作成比值进行比较,得出一组充分条件。     

Alternative Approach to the Concept of Shape Invariance in Quantum Mechanics

It is shown that in the case of shapeinvariance, with the use of the mixing functionformalism, exact solvability of the Schrodinger equationcan be obtained without recourse to the technique ofsupersymmetrization. Some consequences of the method are outlinedand illustrated by specific examples.     

Synthesis and nuclear magnetic resonance analysis of starch‐g‐poly(1,4‐dioxan‐2‐one) copolymers

A new biodegradable starch graft copolymer, starch‐g‐poly(1,4‐dioxan‐2‐one), was synthesized through the ring‐opening graft polymerization of 1,4‐dioxan‐2‐one onto a starch backbone. The grafting reactions were conducted with various 1,4‐dioxan‐2‐one/starch feed ratios to obtain starch‐g‐poly(1,4‐dioxan‐2‐one) copolymers with various poly(1,4‐dioxan‐2‐one) graft structures. The microstructure of starch‐g‐poly(1,4‐dioxan‐2‐one) was characterized in detail with one‐ and two‐dimensional NMR spectroscopy. The effect of the feed composition on the resulting microstructure of starch‐g‐poly(1,4‐dioxan‐2‐one) was investigated. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 3417–3422, 2004     

Hyperfine structure and isotope shift investigations in202–222Rn for the study of nuclear structure beyond Z=82

The hyperfine structure (hfs) and isotope shift (IS) in the isotopic chain of the radioactive element radon have been studied for the first time. The measurements were carried out by collinear fast-beam laser spectroscopy at the mass separator facility ISOLDE at CERN. The IS between 16 isotopes in the mass range 202A222 and the hfs of 7 odd-A isotopes were determined in the transitions 7s [3/2]₂-7p [5/2]₃ (745 nm) of Rn I. The nuclear spins and moments, as well as the observed inversion of the odd-even staggering for^218–222Rn, can be associated with the effects of octupole instability around N=134.This work was supported by the Bundesministerium für Forschung und Technologie and the Deutsche Forschungsgemeinschaft.     

Alkylation of ethylenethiourea with alcohols: a convenient synthesis of S-alkyl-isothioureas without toxic alkylating agents

The alkylation of ethylenethiourea with alcohols and aqueous acids (HCl, HBr, and HI) allows the synthesis of the respective S-alkyl-isothioureas in high yield and purity. Consistently high yields (91-98%) were obtained with 56% HI, the yields for 48% HBr (48-93%) and 37% HCl (36-85%) were lower and varied with the type of alcohol. The method is a convenient low-cost alternative to the use of alkyl iodides and an easy access to the S-tert-butyl isothiourea.     

Some Properties of Solutions to the Novikov Equation with Weak Dissipation Terms

In this paper, we investigate the Novikov equation with weak dissipation terms. First, we give the local well-posedness and the blow-up scenario. Then, we discuss the global existence of the solutions under certain conditions. After that, on condition that the compactly supported initial data keeps its sign, we prove the infinite propagation speed of our solutions, and establish the large time behavior. Finally, we also elaborate the persistence property of our solutions in weighted Sobolev space.     

On a class of graphs with prescribed mean curvature

We study a class of quasilinear elliptic equations on the unit ball of ℝ ⁿ in the divergence form ∑ _j=1 ⁿ D _j{G(|x|²,|Du|²)D _j u} =H(|x|) and get estimates on the boundary by using a modified barrier-function technique of Bernstein. We establish a maximum principle for the gradients of solutions and get a global gradient estimate. We prove that solutions with constant boundary condition must be radial. Finally, we apply these results to graphs {(x,u(x)):x∈H ⁿ } whereu:H ⁿ →ℝ is a smooth map of then-hyperbolic spaceH ⁿ =B(0,1) with the metric to get the existence of graphs with radial prescribed mean curvature.     

Alkylation of ethyl bromomalonate by alkylcobaloximes

Allylcobaloximes react under very mild conditions with ethyl bromomalonate to yield allyl-substituted ethylmalonates in good yield. In the case of crotyl-, 3,3 dimethyl, allyl- and cinnamyl-cobaloximes, the substitution occurs with total rearrangement of the allyl groups. Similar rearrangements are observed during the reactions of propargyl- and allenylcobaloximes with BrCH(CO₂Et)₂ yielding allenyl- and propargyl-malonic esters respectively.