must then consider either the absolute value of the C" (p,pl) cross section or that of the AP7(p, 3prs) cross section (or both) to be in error. We have rather arbitrarily chosen to base our data on the 10.8-mb value for the AP'(p, 3prs) cross section at 420 Mev. Figure 1 shows that the cross section of the C" (p,pcs) C" reaction is a fairly insensitive function of the energy of the incident proton in the energy range studied here. Since similar results were found for the production of Na", Na", and F" from aluminum and for Be~f ormation from carbon, 6 it appears to be generally true that the probability of ejecting a small number of nucleons from a small nucleus remains substantially constant over a range of bombarding energies from a few hundred Mev to at least 3 Bev. This implies that the probability that the incident particle leaves behind a relatively small amount of energy ( & 100 Mev) in the ieitia/ interaction with the nucleus is relatively constant over the wide energy range studied. However within this energy range meson production increases very markedly with energy and becomes a probable process. If the nucleus is large these mesons would have a good chance of being reabsorbed in the nucleus in which they were produced. This would result in a shift of the maximum in the total energy deposition spectrum to higher values, and reactions in which only a small 'Hudis, Wolfgang, and Friedlander (unpublished). (Received June 28, 1954) It is pointed out that the usual principle of invariance under isotopic spin rotation is not consistant with the concept of localized fields. The possibility is explored of having invariance under local isotopic spin rotations. This leads to formulating a principle of isotopic gauge invariance and the existence of a b Geld which has the same relation to the isotopic spin that the electromagnetic Geld has to the electric charge. The b Geld satisGes nonlinear differential equations. The quanta of the b field are particles with spin unity, isotopic spin unity, and electric charge +e or zero.

This is a contribution to the theory of Hecke algebras. A class of algebras called generic pro-p Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-p Hecke algebras are given by pro-p-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define 'Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra hold true, equivalent to Bernstein's relations in the case of Iwahori-Hecke algebras.For a certain subclass called affine pro-p Hecke algebras, containing Iwahori-Hecke and pro-p-Iwahori Hecke algebras, an explicit canonical and integral basis of the center is constructed and finiteness results are proved about the center and the module-structure of the algebra over its center, recovering results of Bernstein-Zelevinsky-Lusztig and Vignéras. ZusammenfassungEs wird ein Beitrag zur Theorie der Hecke-Algebren geleistet. Speziell wird eine Klasse von Algebren eingeführt, die generischen pro-p Hecke-Algebren, welche die Klasse der generischen Hecke-Algebren erweitert durch Übergang von Coxetergruppen zu Erweiterungen solcher durch abelsche Gruppen. Beispiele sind gegeben durch pro-p-Iwahori Hecke-Algebren und Yokonuma-Hecke Algebren. Es wird der Begriff der Orientierung einer Coxetergruppe eingeführt und benutzt um sogenannte Bernsteinabbildungen definieren, welche eng verwandt sind mit der Bernsteinpräsentierung und dem Cherednik-Kozykel. Sodann wird gezeigt, dass zwischen den Bildern der Bernsteinabbildungen gewisse Relationen herrschen, welche sich im Spezialfall der Iwahori-Hecke Algebra auf die bekannten Bernsteinrelationen reduzieren.Ferner wird für die Unterklasse der affinen pro-p Hecke-Algebren, welche sowohl die Iwahori-Hecke als auch die pro-p-Iwahori Hecke-Algebren umfassen, eine kanonische und ganzzahlige Basis des Zentrums konstruiert und es werden Endlichkeitssätze über das Zentrum, aufgefasst als Algebra, und über die Hecke-Algebra selbst, aufgefasst als Modul über dem Zentrum, bewiesen. Dabei werden bereits bekannte Ergebnisse von Bernstein-Zelevinsky-Lusztig und Vignéras verallgemeinert.I would like to thank my advisor for his support, and I would like to thank my family, without which this thesis would not exist. Generic pro-p Hecke algebras and Bernstein maps 1.Basic definitions and some geometric terminologyWe recall some standard facts and terminology from the theory of Coxeter groups (cf. [Bou07, Ch. IV] or [Bro89, II]). Definition.A Coxeter group W = (W, S) consists of a group W and a set S ⊆ W of generators of order 2 satisfying the action condition. That is, there exists an actionon the set H × {±1}, wheresuch that a generator s ∈ S acts asThere are several other equivalent definitions of the notion of a Coxeter group (see [Bro89, II.4]). In particular, given a group W and a set S of generators of order 2, the action conditio...

The question of parity conservation in P decays and in hyperon and meson decays is examined. Possible experiments are suggested which might test parity conservation in these interactions.

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