Non-linear circle actions on the 4-sphere and twisting spun knots

CIRCLE actions on spheres form one of the most important problems in transformation groups. The aim of this paper is to study this problem in dimension 4. We answer a question of Montgomery and Yang[7], and show that there are infinitely many non-linear circle actions on S⁴. Moreover, if the 3-dimensional Poincaré conjecture is true, these actions plus the linear ones are the only possible circle actions on S⁴. The proof of this assertion involves identifying some homotopy 4-spheres. It is closely related to the work, twisting spun knots, of Zeeman[14]. We give a different treatment of this subject. This new setting yields new proofs and substantial strengthenings of some known results. In particular, we answer two questions of Zeeman[14, pp. 493–494, Questions 3 and 4].     

On stability of non-linear differential systems

The object of this paper is to study the stability and asymptotic stability of solutions of the non-linear differential equation $\text{dx}\text{dt}\text{= A(t)x + f(t,x)}$ by using the method of equivalent inner products. This method enables one to determine a stability region without the ingenuity in constructing a Lyapunov function. It shows also that for an unstable linear system it is possible to choose a non-linear function so that the non-linear system is stable or asymptotically stable. Both global and regional stability are discussed.     

Theory of incremental motion in a body with initial elasto-plastic deformation

A theory of incremental deformation is developed for a body with initial stress and strain, valid for elastic and plastic pre-deformation. The superimposed motion is assumed hyperelastic. We present the constitutive equation, relating a measure of incremental stress to incremental strain, and the momentum equations referred to the prestrained initial configuration of the body. The latter is linearized with respect to the displacement field of the superimposed motion. For general nonlinear pre-deformation the constitutive theory of an anisotropic body may be based on the assumed strain energy that is an isotropic function of the Green strain tensor and a number of material directors of unit length. For bodies with elastoplastic prestrain and prestress, an elastic constitutive relation that depends on the plastic pre-deformation and the deformation from a relaxed state is postulated. Imposing the requirements of invariance with respect to rotations of the plastic pre-deformation and using the objectivity rule, we show that the constitutive function of stress must be an isotropic function of its arguments. This leads to a new theory of acoustoelasticity for an elasto-plastically prestrained material with anisotropic natural state by incorporating materially fixed unit directors as independent variables in the constitutive equations.

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Zusammenfassung Es wird eine Theorie der inkrementellen Deformation für einen Körper mit Vorspannung und Vorverformung entwickelt, welche sowohl für elastische wie plastische Vorverformung Gültigkeit hat. Die überlagerte Bewegung wird als hyperelastisch angenommen. Wir leiten die Materialgleichung her, welche eine inkrementelle Spannung mit einer inkrementellen Verzerrung verknüpft und geben die Impulsgleichung in einer auf die anfängliche Verformung bezogenen Konfiguration. Diese Impulsgleichung wird bezüglich des überlagerten Verschiebungsfeldes linearisiert. Für eine allgemeine nichtlineare Vorverformung kann die Materialtheorie eines anisotropen Körpers auf der Basis einer postulierten Verzerrungsenergie gegründet werden, welche eine isotrope Funktion des Greenschen Verzerrungstensors und einer Zahl materieller Direktoren mit Länge 1 darstellt. Für elasto-plastisch vorgespannte und vorverformte Körper wird eine elastische Materialgleichung postuliert, die von der plastischen Vorverformung sowie der Deformation aus einer relaxierten Konfiguration abhängt. Es wird gezeigt, daß die Materialgleichung für den Spannungstensor eine isotrope Funktion ihrer Argumente sein muß, wenn man als übergeordnetes Prinzip das Objektivitätsprinzip fordert und Invarianz des Spannungsfunktionals gegenüber Rotationen der plastischen Vordeformation verlangt. Durch Einführen von materiellen Direktoren der Einheitslänge als unabhängige Konstitutivvariablen erhält man hieraus eine neue Theorie der Akusteolastizität für elasto-plastisch vorverformte Materialien mit anisotropen natürlichen Referenzzuständen.

     

Determination of urine caffeine and its metabolites by use of high-performance liquid chromatography-tandem mass spectrometry: estimating dietary caffeine exposure and metabolic phenotyping in population studies

We have developed and validated a high-performance liquid chromatography-tandem mass spectrometric (LC-MS/MS) method for determining urine caffeine and 14 caffeine metabolites suitable for estimating caffeine exposure and metabolic phenotyping in population studies. Sample preparation consisted solely of a series of simple reagent treatments at room temperature. Stable isotope-labeled analogs were used as internal standards for all analytes. We developed rapid LC-MS/MS separations for both positive and negative ion mode electrospray ionizations to maximize measurement sensitivity. Limits of detection were 0.05–0.1 μmol/L depending on the analytes. Method imprecision, based on total coefficients of variation, was generally <7 % when analyte concentration was >1 μmol/L. Analyte recoveries were typically within 10 % of being quantitative (100 %), and good agreement was observed among analytes measured across different MS/MS transitions. We applied this method to the analysis of a convenience set of human urine samples (n?=?115) and were able to detect a majority of the analytes in ≥99 % of samples as well as calculate caffeine metabolite phenotyping ratios for cytochrome P450 1A2 and N-acetyltransferase 2. Whereas existing LC-MS/MS methods are limited in number of caffeine metabolites for which they are validated, or are designed for studies in which purposely elevated caffeine levels are expected, our method is the first of its kind designed specifically for the rapid, sensitive, accurate, and precise measurement of urine caffeine and caffeine metabolites at concentrations relevant to population studies.

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Figure The determination of caffeine and its metabolites by LC-MS/MS. Both positive and negative ion mode electrospray ionization were used to maximize measurement sensitivity and selectivity, allowing the development of a robust method suitable for estimating caffeine exposure and metabolic phenotyping in population studies

     

Markov Chains theory in the detection of research activities in physics

Conclusions The Markov Chains model was applied on a sample of citation data to predict the shifting trends of scientific inquiry from science and technology to social science subjects [7]. The validity of the findings awaits verification by future data. The same technique has recently been utilized to predict the movements of authors among nine subtopics in ethnomusicology [11]. Although a validation was attempted, unfortunately, inadequate data rendered the comparison inconclusive. In the present study, publications from the 1961–1970 Index were used to model the process of subfield shifts in a physics journal. It was assumed that authors movements among subfields of physics as shown in the subject headings assigned would reflect the subject shifts of the journal. The rate of change between the initial probability distribution and the limiting probability distribution were used to predict developmental trends. Finally, the subject distribution of publications appearing in the next ten-year Index validated the prediction. It is reasonable to conclude that the Markov Chains model appears to predict the shifts of research trends in a specialty journal.This publication was supported in part by NIH Grants RO 1-LM-04177 and KO 4-LM-00078 from the National Library of Medicine.     

On nonlinear reaction-diffusion systems

This paper presents a qualitative analysis for a coupled system of two reaction-diffusion equations under various boundary conditions which arises from a number of physical problems. The nonlinear reaction functions are classified into three basic types according to their relative quasi-monotone property. For each type of reaction functions, an existence-comparison theorem, in terms of upper and lower solutions, is established for the time-dependent system as well as some boundary value problems. Three concrete physical systems arising from epidemics, biochemistry and engineering are taken as representatives of the basic types of reacting problems. Through suitable construction of upper and lower solutions, various qualitative properties of the solution for each system are obtained. These include the existence and bounds of time-dependent solutions, asymptotic behavior of the solution, stability and instability of nontrivial steady-state solutions, estimates of stability regions, and finally the blowing-up property of the solution. Special attention is given to the homogeneous Neumann boundary condition.