Forced lattice vibrations: Part I

This is the First part of a two-part series on forced lattice vibrations in which a semi-infinite lattice of one-dimensional particles {x_n}_n≧1 is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion, _x0(_t) = _at + _h(_yt), where a and γ are constants and h(·) has period 2π. For a wide variety of restoring forces F (i.e., F′ > 0), numerical calculations indicate the existence of a sequence of thresholds γ₁ = γ₁(a, h, F) > γ₂ = γ₂(a,h,F) > … > γ_k = γ_k(a,h,F) > …, γ_k → 0, as k → ∞. If γ_k > γ > γ_k+1, a k-phase wave that is well described by the wave form, emerges and travels through the lattice. The goal of this series is to describe the emergence and calculate some properties of these wave forms. In Part I the authors first consider the case where F(x) = e^x (i.e., Toda forces) but h is arbitrary, and show how to compute a basic diagnostic (see J(λ), formula (1.26)) for the system in terms of the solution of an associated scalar Riemann-Hilbert problem, once a certain finite set of numbers is known. In another direction, the authors consider the case where F(x) is restoring but arbitrary, and h is small. Here the authors prove a general result, asserting that if there exists a sufficiently ample family of traveling-wave solutions of the doubly infinite lattice, then it is possible to construct time-periodic k-phase wave solutions with asymptotics in n of type (iii) for the driven system (i). In Part II, the authors prove that sufficiently ample families of traveling-wave solutions of the system (iv) exist in the cases γ > γ₁ and γ₁ > γ > γ₂ for general restoring forces F. In the case with Toda forces, F(x) = e^x, the authors prove that sufficiently ample families of traveling-wave solutions.     

Investigations into the parallel kinetic resolution of 2-phenylpropanoyl chloride using quasi-enantiomeric oxazolidinones

The resolution of 2-phenylpropanoyl chloride using an equimolar combination of quasi-enantiomeric oxazolidinones is discussed. The levels of diastereoselectivity were found to be dependent upon the structural nature of the metallated oxazolidinone, temperature and metal counter-ion.     

Unified approach to KdV modulations

We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial‐value problem for the zero‐dispersion KdV as the steepest descent for the scalar Riemann‐Hilbert problem [6] and on the method of generating differentials for the KdV‐Whitham hierarchy [9]. By assuming the hyperbolicity of the zero‐dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)‐plane. The resulting system effectively solves the zero‐dispersion KdV with an arbitrary initial datum. © 2001 John Wiley & Sons, Inc.     

Efficient coupling of low boiling point alkynes and 5-iodonucleosides

The coupling of unprotected 5-iodonucleosides and low boiling point alkynes has been achieved in a high yield for the first time. The coupling is highly dependent on concentration and can be carried out at low temperature and pressure conditions.     

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface ${\mathcal {R} = \mathcal {R}(x,t)}The initial value problem for an integrable system, such as the Nonlinear Schr?dinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface R = R(x,t){\mathcal {R} = \mathcal {R}(x,t)} in the spectral variable, where the space-time variables (x, t) play the role of external parameters. The curves in the x, t plane, separating regions of different genuses of R(x,t){\mathcal {R}(x,t)}, are called breaking curves or nonlinear caustics. The genus of R(x,t){\mathcal {R}(x,t)} is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. The evolution theorem ([10]) guarantees continuous evolution of the asymptotic solution in the space-time away from the breaking curves. In the case of the analytic scattering data f(z; x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function á h(z;x,t){{\Im\,h(z;x,t)}}, which is closely related to the g function. Namely, a break can be caused ([10]) either through the change of topology of zero level curves of á h(z;x,t){\Im\,h(z;x,t)} (regular break), or through the interaction of zero level curves of á h(z;x,t){{\Im\,h(z;x,t)}} with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in the region across the curve.     

Chemoselectivity and Conformational Analysis of the Reaction of 2‐Acetylcycloalkanones with Benzohydrazide: Synthesis and Reduction of 1‐Aroylcycloalkapyrazole Derivatives

From the reaction of 2‐acetylcyclopentanone and 2‐acetyl‐2‐methylcyclopentanone with benzohydrazide, the 1‐benzoyl‐6a‐hydroxycyclopentapyrazole derivatives 2a and 2b were obtained as the only reaction products, whereas from the reaction of 2‐acetylcyclohexanone an epimeric cis/trans mixture of the 2‐benzoyl‐3‐hydroxy‐2H‐indazole derivative 3c was formed. The dehydration of the isolated compounds 2a and 3c , as well as the NaBH₄ and NaBH₃CN reduction products of 2a were studied. The structural assignments of the compounds derived were established by analysis of their NMR spectra (¹H, ¹³C, DEPT, COSY, NOESY, HETCOR C? H, and COLOC C? H). The chemoselectivity of the reactions of 1a and 1c with benzohydrazide was studied by conformational analysis with MM2 and semiempirical (AM1 and PM3) MO calculations.     

Forced lattice vibrations: Part II

This is the second part of a two-part series on forced lattice vibrations in which a semi-infinite lattice of one-dimensional particles {x_n}_n≧1, is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + h(yt) where a and γ are constants and h(·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling-wave solutions of the doubly infinite system exist in the cases γ > γ₁ and γ₁ > γ > γ₂ for general restoring forces F. In the case with Toda forces, F(x) = e^x, the authors prove that sufficiently ample families of traveling-wave solutions exist for all k, γ_k > γ > γ_k+1. By a general result proved in Part I, this implies that there exist time-periodic solutions of the driven system (i) with k-phase wave asymptotics in n of the type with k = 0 or 1 for general F and k arbitrary for F(x) = e^x (when k = 0, take γ₀ = ∞ and X₀ ≡ 0).