On a Problem of Erd\H{o}s, Herzog and Sch\"onheim

Let $p_1, p_2,..., p_n$ be distinct primes. In 1970, Erd\H os, Herzog and Sch\"{o}nheim proved that if $\cal D$ is a set of divisors of $N=p_1^{\alpha_1}...p_n^{\alpha_n}$, $\alpha_1\ge \alpha_2\ge...\ge \alpha_n$, no two members of the set being coprime and if no additional member may be included in $\cal D$ without contradicting this requirement then $|{\cal D}|\ge \alpha_n \prod_{i=1}^{n-1} (\alpha_i +1)$. They asked to determine all sets $\cal D$ such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research.Comment: 12 page

Jonathan H. v. Souderton Sch Dist

USDC for the Eastern District of Pennsylvani

Nuevos Haeterius del grupo incisus Schmidt (Col. Histeridae)

Al largo y en la falda Sur de nuestra cordillera central aparece una serie de formas del género Haeterius próximas al H. incisus Sch., pero perfectamente distinguibles entre sí; y bien que no me haya sido posible examinar el tipo o cotipos de esa especie, y aceptando como exacta la denominada así sobre un ejemplar de Madrid (Sanz) existente en las colecciones de nuestro Museo Nacional, consultada en tiempos a París y devuelta con ese nombre (haciendo la salvedad de que no concuerdan exactamente los caracteres asignados en los Bestimmungs Tabellen al H. incisus Sch., en algún pequeño detalle, con el ejemplar citado), precedo esta nota de los epígrafes que corresponden en la Monografía de Schmidt a la especie H. incisus, y doy a continuación, en mis cuadros, la característica exacta en cuanto a puntuación, pubescencia, etc., del ejemplar de Madrid (Sanz), de su variedad gili y de la nueva H. cruzi y de su variedad micropilosus, todas las cuatro pudiéndose incluir en la característica reseñada del H. incisus Sch. en sus caracteres generales, y sin embargo distintas entre sí, como se dijo al principio, dándose los dibujos de H. incisus, gili y cruzi para mayor claridad, hechos al microscopio por la señorita Simón.Peer reviewe

Global well-posedness of the short-pulse and sine-Gordon equations in energy space

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space $H^2$. Our analysis relies on local well-posedness results of Sch\"afer & Wayne, the correspondence of the short-pulse equation to the sine-Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and global well-posedness of the sine-Gordon equation in an appropriate function space.Comment: 17 pages, revised versio

Optimization of the industrial production of bacterial alpha amylase in Egypt. IV. Fermentor production and characterization of the enzyme of two strains of Bacillus subtilis and Bacillus amyloliquefaciens

Production of alpha amylase using amplified variants of Bacillus subtilis (strain SCH) and of Bacillus amyloliquefaciens (strain 267CH) was conducted in a bioreactor with multiprotein-mineral media. Thetime course of fermentation in a bioreactor revealed that the highest yield (about 8 x 104 U/ml within 60 h) by strain SCH was obtained by applying: 3.5% initial starch, 2% additional starch after 19 h, 3 vvmaeration and 300 rpm agitation. The highest yield (about 19 x 104 U/ml within 100 h) by strain 267CH was obtained by applying: 2.5% initial starch, 2% additional starch after 24 h, 3 vvm aeration, and 300 rpmagitation with the productivity after 60 h reaching only about 14 x 104 U/ml. Production occurred in both the logarithmic and postlogarithmic phases of growth. Maximum consumption of starch and proteinoccurred during the first day of incubation. The optical density peak coincided with enzyme production peak in case of strain SCH and preceded that of enzyme production in case of strain 267CH. The alphaamylase produced by the two strains was shown to be of the liquefying and not the saccharifying type. Both enzymes liquefied starch to a dextrose equivalent of about 15 - 17 at 95oC hence they areclassified among thermostable alpha amylases. They exhibited broad pH and temperature activity profiles. The optimum pH for activity was 4 - 7 for alpha amylase produced by strain SCH and 4 - 8 foralpha amylase produced by strain 267CH while the optimum temperatures for their activities were in the range 37 -75oC at 0.5% starch and in the range 85 - 95oC at 35% starch