The convergence of the Fourier integral of a unimodal distribution

We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine's criterion for unimodal distributions.

     

On Two-Dimensional Sonic-Subsonic Flow

A compensated compactness framework is established for sonic-subsonic approximate solutions to the two-dimensional Euler equations for steady irrotational flows that may contain stagnation points. Only crude estimates are required for establishing compactness. It follows that the set of subsonic irrotational solutions to the Euler equations is compact; thus flows with sonic points over an obstacle, such as an airfoil, may be realized as limits of sequences of strictly subsonic flows. Furthermore, sonic-subsonic flows may be constructed from approximate solutions. The compactness framework is then extended to self-similar solutions of the Euler equations for unsteady irrotational flows.     

The effect of detuning upon resonance fluorescence

The excitation spectrum of a two-level atom interacting with a strong laser field is considered as a function of the detuning. The spectral function of the system is calculated by making use of the Green's function formalism and includes a physical process which is neglected in many earlier treatments of the one photon problem. It is found that for zero and small detunings, the spectral function describes the well known three peak spectrum which evolves into a four line spectrum as the detuning is increased. Results of numerical calculations for a wide range of detuning are presented and analytic expressions are derived to describe the small and large detuning limits.     

Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.     

Solid state polymerization of poly(lactic acid): Some fundamental parameters

Poly(lactic acid) (PLA) was submitted to solid state polymerization (SSP) in a fixed bed reactor under nitrogen flow, so as to examine technique efficiency for increasing the molecular weight and hence permitting the reduction of the melt polymerization residence times. In order to use a suitable starting material, SSP prepolymers of low and medium molecular weight were first prepared through solid state hydrolysis of commercial PLA grade under acidic and alkaline conditions. During these degradation runs, hydrolysis involved the random scission of ester groups in the polymer backbone, while the relevant kinetics and the resulting thermal properties were also examined. In a subsequent step, the prepolymers obtained were subjected to SSP at three temperatures, approximately 2.5–25.0 °C below their melting point. The process achieved an increase of up to 1.7 times the initial molecular weight, however, with different trends depending on the prepolymer characteristics, reaction temperature and time, as well as the pH of the hydrolysis medium. In addition to molecular weight build up, the effect of the SSP process on end product thermal properties was also investigated.     

A note on minimum distance estimates

It is shown that minimum distance estimates enjoy the invariance property of maximum likelihood estimates. Supported under Grant No. 18-2773 of the American University of Beirut.