Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

On strongly-non-reflexive groups

A strong negative answer is given to the old question of whether every dual group is reflexive. Using ◊_ω1 a groupA is constructed so thatA, A*, A**, andA*** are weakly ω₁-separable groups of cardinalityω ₁ andA* is not isomorphic toA***. Research partially supported by NSF Grant No. DMS-8400451. Research partially supported by NSERC Grant No. A8948.     

Measurement and analysis of the ν2 and ν4 infrared bands of 13CD4

A Fourier transform infrared spectrum of 91% ¹³CD₄ has been recorded between 885 and 1193 cm^?1 with a resolution of 0.04 cm^?1. The frequencies of 600 lines were measured with an accuracy of ±0.003 cm^?1. Of these, approximately 368 are assigned as allowed transitions in ν₄, 95 are forbidden ν₄ transitions, and 137 belong to ν₂; maximum upper state J values are 20 for ν₄ and 19 for ν₂. The ground state tensor constants D_t, H_4t, and H_6t were obtained by fitting them to four rotational transitions observed by Kreiner and Opferkuch in the infrared-microwave double-resonance spectrum. An interacting-band analysis of the $\text{ν}{}_2\text{ν}{}_4$ diad then yields 22 spectroscopic constants for these Coriolis-coupled fundamentals and fits the experimental frequencies with an rms deviation of 0.0055 cm^?1 for 432 unblended lines that were assigned unit weight. A ν₄P⁺(12) transition at 943.3812 cm^?1, nearly coincident with the 10P(22) emission of the ¹²C¹⁶O₂ laser, has been investigated by heterodyne spectroscopy and its detuning (?64 MHz) and absorption coefficient have been determined. Such coincidences may lead to the development of laser analytical techniques for ¹³CD₄, which is a useful nonradioactive atmospheric tracer. ¹³CD₄ transitions that are within 300 MHz of isotopic CO₂ laser lines are tabulated for this purpose and for use in double-resonance experiments.     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.     

A universal phase space for particles in Yang-Mills fields

Given a principal G-bundle P over M and a Hamiltonian G-space Q, one may construct the reduced symplectic manifold (T*P x Q)₀. When a connection on P is chosen, this manifold becomes a bundle over T*M with fibre Q. It is shown that this bundle is precisely the phase space constructed by Sternberg for a classical particle in a Yang-Mills field.Research partially supported by NSF Grant MCS 74-23180.A01.     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

Kinetic Limit for a System of Coagulating Planar Brownian Particles

We study a model of mass-bearing coagulating planar Brownian particles. The coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order |logε|. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.     

Up to 30 mW of broadly tunable CW green-to-orange light, based on sum-frequency mixing of Cr:forsterite and Nd:YVO4 lasers

Efficient generation of continuous-wave (CW) tunable light in the yellow region is reported. The method is based on sum-frequency mixing of a tunable Cr⁴⁺:forsterite laser with a Nd:YVO₄ laser. A periodically poled lithium niobate crystal was placed intra-cavity in a Nd:YVO₄ laser, and the Cr⁴⁺:forsterite laser was single-passed through the non-linear media. With this setup, it was possible to generate up to 3 mW of yellow light smoothly tunable from 573 to 587 nm. This is the highest output demonstrated to date for a tunable diode pumped solid-state CW laser in this wavelength region. The ways to improve the efficiency further are discussed.     

Existence of compatible families of proper regular conditional probabilities

Summary Let (, , ) be a perfect probability space with countably generated, and let IB be a family of sub--fields of . Under a countability condition on the family IB, I show that there exists a family {}_IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.Research supported in part by NSF PHY-78-23952NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81).