Variable Separation Solutions in (1+1)-Dimensional and (3+1)-Dimensional Systems via Entangled Mapping Approach

In this paper, the entangled mapping approach (EMA) is applied to obtain variable separation solutions of (1 1)-dimensional and (3 1)-dimensional systems. By analysis, we firstly find that there also exists a common formula to describe suitable physical fields or potentials for these (1 1)-dimensional models such as coupled integrable dispersionless (CID) and shallow water wave equations. Moreover, we find that the variable separation solution of the (3 1)-dimensional Burgers system satisfies the completely same form as the universal quantity U1 in (2 1 )-dimensional systems. The only difference is that the function q is a solution of a constraint equation and p is an arbitrary function of three independent variables.     

On concepts of directional differentiability

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Fréchet, and Hadamard are singled out from the general framework of -directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed.     

Monitoring of yeast fermentation by ion mobility spectrometry measurement and data visualisation with Self-Organizing Maps

Ion mobility spectrometry (IMS) measurement combined with unsupervised neurocomputing is considered as a new potential method for on-line monitoring of fermentation and other processes producing volatile compounds that involve micro-organisms. This was demonstrated in a model system in which a strain of brewer’s yeast (Saccharomyces cerevisiae) was cultivated in a bench-top fermenter. Five phases of yeast growth could be detected from measurements of the exhaust gases from the fermenter, as indicated by the changes in ion mobility spectra analysed by computational methods.The data were first processed using the Self-Organizing Map (SOM) algorithm, the results showing that the phases of fermentation can be detected and identified. The cultivations were also shown by Sammon’s mapping to be comparable to a certain level of accuracy. Contaminated cultivation could be detected by its distinctive ion mobility spectrometry profile.     

Cavity search: An algorithm for the isolation and display of cavity-like binding regions

Summary A set of algorithms designed to enhance the display of protein binding cavities is presented. These algorithms, collectively entitled CAVITY SEARCH, allow the user to isolate and fully define the extent of a particular cavity. Solid modeling techniques are employed to produce a detailed cast of the active site region, which can then be color-coded to show electrostatic and steric interactions between the protein cavity and a bound ligand.     

Isolation, Mapping, DNA Sequence and RFLPs Studies of Random Single-Copy DNA Segments on Human X Chromosome

Using the total human/mouse DNA as the probe, screening has been carried out three times with in situ plaque hybridization to obtain the single-copy DNA sequence from the human X chromosome genomic library. The effective rate of screening is 1. 45%. DNAs from clones containing single-copy inserts have been analyzed by a panel of hybrid cells with or without human X chromosome. Three segments, designated by DXFD52,73,75, are mapped to the X chromosome. DXFD52 has been precisely localized on Xq12-q13 with in situ chromosomal hybridization. DXFD52 has been partially sequenced. The results indicate that DXFD52 is a new isolated single-copy segment on the X chromosome. Great progress in the RFLPs study with DXFD52 has been achieved in the population of Chongqing, Sichuan Province. The results show that the DXFD52 can be used to detect the RFLP with Hind Ⅲ, Bgl Ⅱ, and Hinf Ⅰ. DXFD52 will be a potential "landmark" for the construction of the complete linkage map of human genome and the analysis of genomic s     

Some nonlinear problems in servomechanisms

Summary A mathematical model for a hydraulic servomechanism is constructed. It is shown that the model, in general, reduces to a nonlinear third-order equation of the formxx+(₁+xx+–₂ x=p(t). Under certain conditions imposed on the constants involved, it is proved that above equation possesses a periodic solution.     

Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Crumpled laminations and manifolds of nonfinite type

Using a group-theoretic construction due to Bestvina and Brady, we build -manifolds which admit partitions into closed, connected -manifolds but which do not have finite homotopy type.

     

Derivative of the exponential mapping for infinite dimensional lie groups

It is proved that for infinite dimensional Lie groups in the sense of the differential calculus of Frölicher and Kriegl the derivative of the exponential mappings is given by the formula d(exp)(X)Y=d_exp(X)(e) ₀ ¹ Ad_exp(–tX) Y dt, where stands for the left translation ande is the neutral element.This work was supported by the Alexander von Humboldt-Stiftung.     

RECENT PROGRESS OF NANO-MECHANICAL MAPPING

 Nano-mechanical mapping by atomic force microscopy has been developed as an useful application to measure mechanical properties of soft materials at nanometer scale. To date, the Hertzian theory was used for analyzing force-distance curves as the simplest model among several contact mechanics between elastic bodies. However, the preexisting methods based on this theory do not consider the adhesive interaction in principle, which cannot be neglected in the ambient condition. A new analytical method was introduced to estimate the elasticity and the adhesive energy simultaneously by means of the JKR theory, describing adhesive contact between elastic materials. Poly(dimethylsiloxane) (PDMS) and isobutylene-co-isoprene rubber (IIR) were analyzed to verify the applicable limit of the JKR analysis. For elastic samples such as PDMS, the force-deformation plots obtained experimentally were consistent with JKR theoretical curves. Meanwhile, for viscoelastic samples, especially for IIR, the experimental plots revealed large deviations from JKR curves depending on scanning velocity and maximum loading force. Some nano-rheological arguments were employed based on the difference between these specimens.