A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Fluorescence determination of steroidal dienones

Reaction of Δ^1,4-3-keto steroids with zinc and acid produces a characteristic fluorescence. Estrogenic, 3-keto and Δ⁴-3-keto steroids did not yield this fluorescence. Stabilization of the dienone fluorescence by extraction into butyl ether allowed quantitation of triamcinolone acetonide in liquid formulations in which this dienone was present at the 0.01% level. A mechanism for the reaction is proposed.     

An elementary proof of the prime number theorem with a remainder term

An elementary proof of the prime number theorem in the form $$\psi (x) - x = O(x exp\{ - (\log x)^{\tfrac{1}{7}} (log log x)^{ - 2} \} )$$ is given. The proof uses a generalization of Selberg's formula and a tauberian argument.     

High-resolution spectroscopy of the 16-μm bending fundamental of CF4

The 16-μm bending fundamentals (ν₄) of ¹²CF₄, ¹³CF₄, and ¹⁴CF₄ have been observed at Doppler-limited resolution using a tunable PbSnSe semiconductor diode laser. The tetrahedral splittings of the rotational manifolds have been observed in all three branches, and in particular the dense and partially overlapping transitions in the Q branches have been resolved and assigned. A least-squares fit of the Hamiltonian, including off-diagonal terms, yielded five scalar and three tensor spectroscopic constants for each of the three isotopes. From these constants the upper-state rotational constant B₄ and the Coriolis constant ζ₄ have been calculated, together with some of the other molecular constants. An absorption feature at about 0.18 cm^?1 to the red of the main Q branch of each isotopic species has been identified as the Q branch of (ν₂ + ν₄) ? ν₂, which is the transition that lases when CF₄ is pumped by a CO₂ laser at 9.4 μm (i.e., in ν₂ + ν₄).     

Coulomb blockade of small Pd clusters

Single-electron tunneling through Au substrate-alkanethiol-Pd cluster-tip junctions is investigated with scanning tunneling spectroscopy. The measured I(V) curves reveal several characteristic features of the Coulomb blockade, namely, the presence of a Coulomb gap and a Coulomb staircase. By using the orthodox theory of single-electron tunneling, the capacitances and resistances of the double junction system as well as the fractional charge are extracted from the experimental data.     

MO imagery techniques using arrays of large aperture telescopes

Simultaneous multiple-path looks at a moving object (MO) of low signal to noise ratio (SNR) is proposed using an array of telescopes without using either interferometric or adaptive optics techniques. Instead, proper Affine overlays, of all looks through nonisoplanatic regions, give a better SNR and a better turning view in the real time and image plane, and can be furthermore integrated temporally using Affine transformation to yield new templates of a MO. Thereby a scheme of open, self-taught and therefore smart pattern recognition of MO's becomes possible. The system performance of a multi-opto-sensor-Affine (MOSA) system is generally derived in terms of system parameters and field conditions. Geometric constructions of Affine transformations are shown to be either area-ration preserving among three control picture elements (pixels), which are proved to be sufficient using Euler's theorem of solid body motion, or the first order moment of area-ratio preserving, when more than three pixels are needed due to poor SNR.