Bernoulli Numbers, Wolstenholme's Theorem, and p^5 Variations of Lucas' Theorem

In this note we shall improve some congruences of D.F. Bailey [Two p^3 variations of Lucas' Theorem, JNT 35(1990), pp. 208-215] to higher prime power moduli, by studying the relation between irregular pairs of the form (p,p-3) and refined version of Wolstenholme's theorem.Comment: 7 pages. Final version accepted by J. of Number Theor

Marcus S. Zuber Collection

Zuber (January 10, 1912 - ?) was well known for his contributions to corn breeding. His development of tools and techniques and genetically improved populations or inbreds resulting from his research was released to hybrid corn breeders for utilization and improvement of hybrids grown by farmers. The collection is composed of scrapbooks documenting Marcus S. Zuber\u27s years of corn breeding research

Instantaneous Bethe-Salpeter Equation and Its Exact Solution

We present an approach to solve a Bethe-Salpeter (BS) equation exactly without any approximation if the kernel of the BS equation exactly is instantaneous, and take positronium as an example to illustrate the general features of the solutions. As a middle stage, a set of coupled and self-consistent integration equations for a few scalar functions can be equivalently derived from the BS equation always, which are solvable accurately. For positronium, precise corrections to those of the Schr\"odinger equation in order $v$ (relative velocity) in eigenfunctions, in order $v^2$ in eigenvalues, and the possible mixing, such as that between $S$ ($P$) and $D$ ($F$) components in $J^{PC}=1^{--}$ ($J^{PC}=2^{++}$) states as well, are determined quantitatively. Moreover, we also point out that there is a problematic step in the classical derivation which was proposed first by E.E. Salpeter. Finally, we emphasize that for the effective theories (such as NRQED and NRQCD etc) we should pay great attention on the corrections indicated by the exact solutions.Comment: 4 pages, replace for shortening the manuscrip

The convection-diffusion equation for a finite domain with time varying boundaries

A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature, and the resulting representation should be useful for problems of practical interest. Authors of previous works on problems of this type have eliminated the unknown exit concentration by assuming a continuous concentration at the outflow boundary. This yields a well-posed problem by forcing a homogeneous Neumann exit, widely known as the Danckwerts [1] condition. We provide a solution to the Neumann problem and use it to produce an estimate which demonstrates that the Danckwerts condition implies a zero concentration at the outflow boundary, even for a long flow domain and a large time.Comment: W. J. Golz and J. R. Dorroh. 2001. The Convection-diffusion equation for a finite domain with time varying boundaries. Applied Mathematics Letters 14 : 983-988 (received by AML September 2000; accepted by AML October 2000

Corn planting rates and row spacing in Missouri

File: Agron. 2William J. Murphy (Department of Agronomy, College of Agriculture)Rev. 12/71/15M, 9/75/5

Orthogonal polynomial method and odd vertices in matrix models

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.Comment: 13 pages, 3 Postscript figure

Pioneer Venus polarimetry and haze optical thickness

The Pioneer Venus mission provided us with high-resolution measurements at four wavelengths of the linear polarization of sunlight reflected by the Venus atmosphere. These measurements span the complete phase angle range and cover a period of more than a decade. A first analysis of these data by Kawabata et al. confirmed earlier suggestions of a haze layer above and partially mixed with the cloud layer. They found that the haze exhibits large spatial and temporal variations. The haze optical thickness at a wavelength of 365 nm was about 0.06 at low latitudes, but approximately 0.8 at latitudes from 55 deg poleward. Differences between morning and evening terminator have also been reported by the same authors. Using an existing cloud/haze model of Venus, we study the relationship between the haze optical thickness and the degree of linear polarization. Variations over the visible disk and phase angle dependence are investigated. For that purpose, exact multiple scattering computations are compared with Pioneer Venus measurements. To get an impression of the variations over the visible disk, we have first studied scans of the polarization parallel to the intensity equator. After investigating a small subset of the available data we have the following results. Adopting the haze particle characteristics given by Kawabata et al., we find a thickening of the haze at increasing latitudes. Further, we see a difference in haze optical thickness between the northern and southern hemispheres that is of the same order of magnitude as the longitudinal variation of haze thickness along a scan line. These effects are most pronounced at a wavelength of 935 nm. We must emphasize the tentative nature of the results, because there is still an enormous amount of data to be analyzed. We intend to combine further polarimetric research of Venus with constraints on the haze parameters imposed by physical and chemical processes in the atmosphere