Invariant Manifolds,Nonclassical Normal Modes,and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum

It is shown that the flexible spherical pendulum undergoes purely slow motions with master and slaved components. The family of slow motions is realized as a three-dimensional invariant manifold in phase space. This manifold is computed analytically by applying the method of geometric singular perturbations. This manifold is nonlinear and for all energy and angular momentum levels is characterized precisely by three PO (proper orthogonal) modes. Its submanifold of zero angular momentum is a two-dimensional invariant manifold; it is the geometric realization of a nonclassical nonlinear normal mode. This normal mode is characterized precisely by two PO modes. The slaved slow dynamics are characterized precisely by a single PO mode. The stability of the slow invariant manifold as well as interactions between fast and slow dynamics are considered.     

Realizing Holonomic Constraints in Classical and Quantum Mechanics

We consider the problem of constraining a particle to a smooth compact submanifold Σ of configuration space using a sequence of increasing potentials. We compare the classical and quantum versions of this procedure. This leads to new results in both cases: an unbounded energy theorem in the classical case, and a quantum averaging theorem. Our two step approach, consisting of an expansion in a dilation parameter, followed by averaging in normal directions, emphasizes the role of the normal bundle of Σ, and shows when the limiting phase space will be larger (or different) than expected. Received: 16 November 2000 / Accepted: 2 February 2001     

Electroanalytical voltammetry in flowing solutions

A review is given of the principles, key developments and representative applications of small electrodes in flow-through electrochemical (ec) detectors having very low effective dead volume (<10 μl) for voltammetric and amperometric detection in flowing solutions. Emphasis is placed on factors contributing to high sensitivity, reliability and selectivity of ec detection as an integral part of larger analytical systems utilizing continuously flowing, unsegmented streams, e.g., flow-injection and liquid chromatographic analyses. Solid and mercury electrodes are considered under potentiostatic and potentiodynamic control. A review is given also of auxiliary chemical and photolytic derivatizations coupled to ec detection. The majority of the literature on the subject relates to liquid chromatography with electrochemical detection (l.c./ec); however, applications to these concepts to specific examples in flow-injection systems, as well as for on-line process control, should be obvious. Details of chromatographic separations and design of total analytical systems are not reviewed.     

Perturbation of embedded eigenvalues in the generalizedN-body problem

We discuss the perturbation of continuum eigenvalues without analyticity assumptions. Among our results, we show that generally a small perturbation removes these eigenvalues in accordance with Fermi's Golden Rule. Thus, generically (in a Baire category sense), the Schrödinger operator has no embedded non-threshold eigenvalues.Supported in part by NSF Grant DMS 8602826     

The use of a highly sulfated cyclodextrin for the simultaneous chiral separation of amphetamine-type stimulants by capillary electrophoresis

We investigated the simultaneous chiral separation of nine amphetamine type stimulants (dl-norephedrine, dl-norpseudoephedrine, dl-ephedrine, dl-pseudoephedrine, dl-amphetamine, dl-methamphetamine, dl-methylenedioxyamphetamine (MDA), dl-methylenedioxymethamphetamine (MDMA), and dl-methylenedioxyethylamphetamine (MDEA)) by capillary electrophoresis using highly sulfated gamma-cyclodextrin (SU(XIII)-gamma-CD) as a chiral selector. Three different approaches using SU(XIII)-gamma-CD with 50 mM phosphate background electrolyte were designed; (I) high CD concentration (10 mM SU(XIII)-gamma-CD) at neutral pH (pH 7.0) in the normal polarity mode, (II) low CD concentration (1.0 mM) at low pH (pH 2.6) in the normal polarity mode and (III) high CD concentration at low pH (pH 2.6) in the reversed-polarity mode. In mode (II), the effects of adding three neutral CDs (beta-CD, dimethyl-beta-CD and hydroxypropyl-beta-CD) were also investigated. The best separation was obtained after optimizing mode (III) as follows: run buffer of 10 mM SU(XIII)-gamma-CD with 50 mM phosphate background electrolyte at pH 2.6, applied voltage of -12 kV and capillary temperature of 15 degrees C.     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.     

Application of affinity capillary electrophoresis for the determination of binding and thermodynamic constants of enediynes with bovine serum albumin

The binding constants and thermodynamic properties of a series of novel enediyne compounds with bovine serum albumin (BSA) were determined. The enediynes were synthesized, characterized, and then studied by affinity capillary electrophoresis (ACE) methods to derive these recognition parameters. Change in electrophoretic mobility of BSA as a function of enediyne concentration was determined at 25 degrees C providing binding constants of 1.76 x 10(5), 1.14 x 10(5), and 0.68 x 10(5) M(-1) for enediynephenylalanine carboxylic acid, enediynephenylalanine methyl ester, and enediyne carboxylic acid, respectively. The binding constant for the enediynephenylalanine carboxylic acid was in good agreement with that obtained using conventional methodology. Binding constants for the interaction of enediynes with BSA decreased with an increase in temperature. Van't Hoff plots showed a direct correlation between intensity of the binding constant and the sign and magnitude of various thermodynamic parameters (DeltaG, DeltaS, and/or DeltaH).     

A short proof of the Zeilberger-Bressoud -Dyson theorem

We give a formal Laurent series proof of Andrews's -Dyson Conjecture, first proved by Zeilberger and Bressoud.