Exceptional Values of Metric Density

Lebesgue\u27s density theorem states that at almost every point of a measurable set S in En, the metric density of S exists and is 1 and at almost every point of the complement of S, the density of S exists and is 0. This theorem was first proven for E1 by Lebesgue using his theory of integration. It was later proven by Denjoy [1], Lusin [2], and Sierpinski [3] for E1 without the use of integration. The theorem was first proven for En by de la Vallee Poussin

Influence of Grazing Frequency on Biomass Production Using Several Selected Tropical Grasses

To provide commercial growers with forage grasses that produce well throughout the year, there is a constant need for screening and testing new germplasm. Two rhodesgrasses (Chloris gayana cv. Rhods and Callide), four stargrasses (Cynodon nlemfuensis Vanderyst var. nlemfuensis cv. Florona, Zimbabwe, Okeechobee, and Rhodesian No. 2), one bermudagrass (C. dactylon var. dactylon cv. Jiggs), and one creeping signalgrass (Brachiaria humidicola CIAT 6369) were tested under a mob-grazing system. Dry biomass yield increased linearly as grazing frequency (GF) was delayed from 2 to 7 weeks. The cultivars, Florona, Zimbabwe and Okeechobee stargrasses and Jiggs bermudagrass, yielded best during the warm season regardless of GF. However, during the cool season Rhods rhodesgrass, Florona stargrass and Jiggs bermudagrass were generally most productive. These grasses were also the most persistent, averaging better than 97% ground cover after 3 years of grazing

Does the Second Caustic Ring of Dark Matter Cause the Monoceros Ring of Stars ?

Caustic rings of dark matter were predicted to exist in the plane of the Galaxy at radii $a_n \simeq 40 {\rm kpc}/n$ for $n = 1,2,3 ..$. The recently discovered Monoceros Ring of stars is located near the $n=2$ caustic, prompting us to consider a possible connection between these two objects. We identify two processes through which the Monoceros Ring of stars may have formed. One process is the migration of gas to an angular velocity minimum at the caustic leading to enhanced star formation there. The other is the adiabatic deformation of star orbits as the caustic slowly grows in mass and radius. The second process predicts an order 100% enhancement of the density of disk stars at the location of the caustic ring.Comment: 21 pages, 3 figure

Floating s- and p-type Gaussian Orbitals

The advantages of including a small number of p-type gaussian functions in a floating spherical gaussian orbital calculation are pointed out and illustrated by calculations on molecules which previously have proved to be troublesome. These include molecules such as F2 with multiple lone pairs and C2H2 with multiple bonds. A feature of the results is the excellent correlation between the orbital energies and those of a double zeta calculation reported by Snyder and Basch

Light scattering by an elongated particle: spheroid versus infinite cylinder

Using the method of separation of variables and a new approach to calculations of the prolate spheroidal wave functions, we study the optical properties of very elongated (cigar-like) spheroidal particles. A comparison of extinction efficiency factors of prolate spheroids and infinitely long circular cylinders is made. For the normal and oblique incidence of radiation, the efficiency factors for spheroids converge to some limiting values with an increasing aspect ratio a/b provided particles of the same thickness are considered. These values are close to, but do not coincide with the factors for infinite cylinders. The relative difference between factors for infinite cylinders and elongated spheroids (a/b \ga 5) usually does not exceed 20 % if the following approximate relation between the angle of incidence $\alpha (in degrees)$ and the particle refractive index m=n+ki takes the place: \alpha \ga 50 |m-1| + 5 where 1.2 \la n \la 2.0 and k \la 0.1. We show that the quasistatic approximation can be well used for very elongated optically soft spheroids of large sizes.Comment: 12 pages, 7 figures, Accepted by Measurement Science and Technology (special OPC issue