Spontaneous Generation of Modular Invariants

It is possible to compute and its modular equations with no perception of its related classical group structure except at . We start by taking, for prime, an unknown ``-Newtonian' polynomial equation with arbitrary coefficients (based only on Newton's polygon requirements at for and ). We then ask which choice of coefficients of leads to some consistent Laurent series solution , (where . It is conjectured that if the same Laurent series works for -Newtonian polynomials of two or more primes , then there is only a bounded number of choices for the Laurent series (to within an additive constant). These choices are essentially from the set of ``replicable functions,' which include more classical modular invariants, particularly . A demonstration for orders and is done by computation. More remarkably, if the same series works for the -Newtonian polygons of 15 special ``Fricke-Monster' values of , then is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise ``spontaneously.'

     

Cyclic Artinian Modules without a Composition Series

Let R be a ring (always understood to be associative with aunit element 1). It is well known that an R-module is Noetherianif and only if all its submodules are finitely generated andthat it has a finite composition series if and only if it isNoetherian and Artinian. This raises the question whether everyfinitely generated Artinian module is Noetherian; here it isenough to consider cyclic Artinian modules, by an inductionon the length. This question has been answered (negatively)by Brian Hartley [5], who gives a construction of an Artinianuniserial module of uncountable composition-length over thegroup algebra of a free group of countable rank. If we are justinterested in finding cyclic modules that are Artinian but notNoetherian, there is a very simple construction based on thefact that over a free algebra every countably generated Artinianmodule can be embedded in a cyclic module which is again Artinian.This is described in 2 below.     

Nachweis und Bestimmung einiger Konservierungsmittel in Nahrungsmitteln und Konserven Formaldehyd

Ohne Zusammenfassung     

Eine Bemerkung über die multiplikative Gruppe eines Körpers

Ohne ZusammenfassungHerrnReinhold Baer zum 60. Geburtstag gewidmetThis research was supported by the United States Air Force under Contract No. SAR/G/AF/ OSR/61-29.     

The effect of a carbon-carbon double bond on electron beam-generated plasma decomposition of trichloroethylene and 1,1,1-trichloroethane

The effect of a carbon-carbon double bond on the energy required for decomposition in an electron beam-generated plasma reactor is studied by comparing the decomposition of trichloroethylene and 1,1,1-trichloroethane. A reaction mechanism for TCE decomposition based on a chlorine radical chain reaction is presented which accounts for the formation of all of the experimentally observed reaction products. TCE decomposition is autocatalyzed by reaction products, whereas TCA decomposition is inhibited. The rate expression for the decomposition of TCE in the reactor is determined to be r=−[T](15.07[T₀]^−0.40+0.006{[T₀]−[T]}), where [T] and [T₀] are both in ppm, and r is in ppm Mrad⁻¹. The energy expense ɛ for TCE decomposition is determined as a function of inlet concentration. For 99% decomposition of 100 ppm TCE in air, ɛ=28 eV/molecule, and ɛ=2.5 eV/molecule at 3000 ppm. This is only 2.5–5% of the amount of energy required to decompose a similar amount of TCA as reported by the authors in a previous study. By comparing the energy requirements for TCE decomposition to those for TCA decomposition, the TCE reaction chain length is determined to increase from approximately 20 at 100 ppm initial TCE concentration, to 40 at 3000 ppm. This work was supported by the Contaminant Plume Containment and Remediation Focus Area, Office of Environmental Management, U.S. Department of Energy.