On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

On the Superlinear Lazer–McKenna Conjecture: Part II

ABSTRACT

We prove that certain weakly nonlinear elliptic equations have many solutions when a paragraph is large. The nonlinearity grows superlinearly for y positive but grows linearly for y negative.     

Monte Carlo simulation of cutaneous reflectance and fluorescence measurements--the effect of melanin contents and localization

Melanin content and distribution in skin were studied by examining a patient with white, brown and blue skin tones expressed on skin affected by vitiligo. Both diffuse reflectance and autofluorescence spectra of the three distinction skin sites were measured and compared. Monte Carlo simulations were then performed to help explain the measured spectral differences. The modeling is based on a six-layer skin optical model established from published skin optical parameters and by adding melanin content into different locations in the model skin. Both the reflectance and fluorescence spectra calculated by Monte Carlo (MC) simulation were approximately in agreement with experimental results. The study suggests that: (1) trichrome vitiligo skin may be an ideal in vivo model for studying the effect of skin melanin content and distribution on skin spectroscopy properties. (2) Based on the skin optical model and MC simulation, the content and distribution of melanin in skin, or other component of skin could be simulated and predicted. (3) Both reflectance and fluorescence spectra provided information about superficial skin structures but fluorescence spectra are capable of providing information from deeper cutaneous structures. (4) The research method, including the spectral ratio method, the method of adding and modifying the melanin content in skin optical models, and MC simulation could be applied in other non-invasive optical studies of the skin.     

Infinitely many positive solutions for an elliptic problem with critical or supercritical growth

We prove that for some supercritical exponents and for some smooth domains D in RN there are infinitely many (distinct) positive solutions to the following Lane–Emden–Fowler equation: This seems to be the first result for such type of equations.     

Substituent Effect on 31P Chemical Shifts and Conformation of some Cyclic Phosphonates

Abstract

A series of 2-alkyl-2-oxo-1,2,3-dioxaphosphorinane (1) and some 2-alkyl-2-oxo-1,2,3-dioxaphosphepane (II) were synthesized and their NMR, IR as well as MS were reported. The substituent effect on ³¹P chemical shifts and conformations of these compounds were investigated by the molecule, mechanics calculations using Allinger's MM2 program.     

Molecular Mechanics Study of Organophosphorus Compounds

Abstract

Molecular Mechanics calculation (Allinger's force field and MM2 1985 program) was successfully applied for the structure-reactivity studies of organophosphorus compounds in our laboratory.     

色散、群速与群折射率

本文讨论了群折射率存在的物理意义,指出群折射率不仅包含了折射率概念而且还体现了折射率的色散性质,同样能作为材料本身的光学性质参量.接着指出在折射率的白光干涉法测量中存在着忽视色散作用而导致的将群折射率误认为折射率的情况.将群折射率与折射率区别开来,不仅可以消除折射率测量中所存在的错误,而且可以突出色散现象在光传播时间特性上的意义.为此建立了折射率与群折射率的相互变换关系,通过对于典型物质的折射率与群折射率变换关系的计算,说明二者在数值上的差别甚至对于气体介质来说也是不能忽略的.     

紫激光辐射人肺组织的光学特性

应用光纤探测系统和自行研制的光纤探针深度针,成功实现了Ｋｒ＾＋紫激光辐射新鲜离体人肺组织体表面漫反射率和内部光能流率分布的精确测定,并利用漫射理论和加倍（ＡｄｄｉｎｇＤｏｕｂｌｉｎｇ）方法的部分结论,获得了紫光从肺癌组织和正常组织的主要光学特性参数,与蒙特卡罗法拟合实验的光分布曲线得出的结果一致秋临床实际应用提供了重要的原始参数。     

基于神经网络的反射式表面粗糙度光纤传感器

本文提出了一种新型表面粗糙度光纤传感器,阐述了它的结构、工作原理,给出了测 试结果。利用神经网络很强的输入-输出映射能力,将实验数据交给神经网络处理,减小了测量误差,提高了测量精度,扩大了测量范围。本系统能准确地识别Ra分别为0.012μm, 0.025μm,0.5μm,0.1μm的被测表面。