On an embedding property of generalized Carter subgroups

If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with E-subgroup, E, and F-subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation E.

Our main results are restricted to formations, E, such that E = {G|G/F(G) ϵT}, where T is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If T consists only of the identity, then E=N, the class of nilpotent groups, and for any solvable group, G, the N-subgroups of G are the Carter subgroups of G.

We give a characterization of strong containment which depends only on the formations E, and F. From this characterization, we prove:

If T is a non-empty formation of solvable groups, E = {G|G/F(G) ϵT}, and E is strongly contained in F, then

(1) there is a formation V such that F = {G|G/F(G) ϵV}.

(2) If for each prime p, we assume that T does not contain the class, S_p’, of all solvable p’-groups, then either E = F, or F contains all solvable groups.

This solves the problem for the Carter subgroups.

We prove the following result to show that the hypothesis of (2) is not redundant:

If R = {G|G/F(G) ϵS_r’}, then there are infinitely many formations which strongly contain R.

Descriptive set theory and the ergodic theory of countable groups

The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Abért and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C^*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.

Travel time tomographic studies of seismic structures around subducted lithospheric slabs

The nature of the subducted lithospheric slab is investigated seismologically by tomographic inversions of ISC residual travel times. The slab, in which nearly all deep earthquakes occur, is fast in the seismic images because it is much cooler than the ambient mantle. High resolution three-dimensional P and S wave models in the NW Pacific are obtained using regional data, while inversion for the SW Pacific slabs includes teleseismic arrivals. Resolution and noise estimations show the models are generally well-resolved.

The slab anomalies in these models, as inferred from the seismicity, are generally coherent in the upper mantle and become contorted and decrease in amplitude with depth. Fast slabs are surrounded by slow regions shallower than 350 km depth. Slab fingering, including segmentation and spreading, is indicated near the bottom of the upper mantle. The fast anomalies associated with the Japan, Izu-Bonin, Mariana and Kermadec subduction zones tend to flatten to sub-horizontal at depth, while downward spreading may occur under parts of the Mariana and Kuril arcs. The Tonga slab appears to end around 550 km depth, but is underlain by a fast band at 750-1000 km depths.

The NW Pacific model combined with the Clayton-Comer mantle model predicts many observed residual sphere patterns. The predictions indicate that the near-source anomalies affect the residual spheres less than the teleseismic contributions. The teleseismic contributions may be removed either by using a mantle model, or using teleseismic station averages of residuals from only regional events. The slab-like fast bands in the corrected residual spheres are are consistent with seismicity trends under the Mariana Tzu-Bonin and Japan trenches, but are inconsistent for the Kuril events.

The comparison of the tomographic models with earthquake focal mechanisms shows that deep compression axes and fast velocity slab anomalies are in consistent alignment, even when the slab is contorted or flattened. Abnormal stress patterns are seen at major junctions of the arcs. The depth boundary between tension and compression in the central parts of these arcs appears to depend on the dip and topology of the slab.

Sterochemically modified polyamides for recognition in the minor groove of DNA

The design of synthetic molecules that recognize specific sequences of DNA is an ongoing challenge in molecular medicine. Cell-permeable small molecules targeting predetermined DNA sequences offer a potential approach for offsetting the abnormal effects of misregulated gene-expression. Over the past twenty years, Professor Peter B. Dervan has developed a set of pairing rules for the rational design of minor groove binding polyamides containing pyrrole (Py), imidazole (Im), and hydroxypyrrole (Hp). Polyamides have illustrated the capability to permeate cells and inhibit transcription of specific genes in vivo. This provides impetus to identify structural elements that expand the repetoire of polyamide motifs with recognition properties comparable to naturally occurring DNA binding proteins. Through the introduction of chiral amino acids, we have developed chiral polyamides with stereochemically regulated binding characteristics. In addition, chiral substituents have facilitated the development of new polyamide motifs that broaden binding site sizes targetable by this class of ligands.

Class two p groups as fixed point free automorphism groups

Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If p^c ≠ r^d + 1 for any c = 1, 2 and any prime r where r^2d+1 divides |G| and if C_G(A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|.

The theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A₁, a subgroup of A, where A₁ centralizes D(R), then all irreducible characters of A₁R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed.

Normal structures and automorphism groups of t-designs

Combinatorial configurations known as t-designs are studied. These are pairs ˂B, ∏˃, where each element of B is a k-subset of ∏, and each t-design occurs in exactly λ elements of B, for some fixed integers k and λ. A theory of internal structure of t-designs is developed, and it is shown that any t-design can be decomposed in a natural fashion into a sequence of “simple” subdesigns. The theory is quite similar to the analysis of a group with respect to its normal subgroups, quotient groups, and homomorphisms. The analogous concepts of normal subdesigns, quotient designs, and design homomorphisms are all defined and used.

This structure theory is then applied to the class of t-designs whose automorphism groups are transitive on sets of t points. It is shown that if G is a permutation group transitive on sets of t letters and ф is any set of letters, then images of ф under G form a t-design whose parameters may be calculated from the group G. Such groups are discussed, especially for the case t = 2, and the normal structure of such designs is considered. Theorem 2.2.12 gives necessary and sufficient conditions for a t-design to be simple, purely in terms of the automorphism group of the design. Some constructions are given.

Finally, 2-designs with k = 3 and λ = 2 are considered in detail. These designs are first considered in general, with examples illustrating some of the configurations which can arise. Then an attempt is made to classify all such designs with an automorphism group transitive on pairs of points. Many cases are eliminated of reduced to combinations of Steiner triple systems. In the remaining cases, the simple designs are determined to consist of one infinite class and one exceptional case.

Nuclear magnetic resonance studies of protein conformation: I. Ribonuclease S. II. Hemoglobin

Fluorine nuclear magnetic resonance techniques have been used to study conformational processes in two proteins labeled specifically in strategic regions with covalently attached fluorinated molecules. In ribonuclease S, the ϵ-amino groups of lysines 1 and 7 were trifluoroacetylated without diminishing enzymatic activity. As inhibitors bound to the enzyme, changes in orientation of the peptide segment containing the trifluoroacetyl groups were detected in the nuclear magnetic resonance spectrum. pH Titration of one of the histidines in the active site produced a reversal of the conformational process.

Hemoglobin was trifluoroacetonylated at the reactive cysteine 93 of each β chain. The nuclear magnetic resonance spectrum of the fluorine moiety reflected changes in the equilibrium position of the β chain carboxy terminus upon binding of heme ligands and allosteric effectors. The chemical shift positions observed in deoxy- and methemoglobin were pH dependent, undergoing an abnormally steep apparent titration which was not observed in hemoglobin from which histidine β 146 had been removed enzymatically. The abnormal sharpness of these pH dependent processes is probably due to interactions between several ionizing groups.

The carbon monoxide binding process was studied by concurrent observation of the visible and nuclear magnetic resonance spectra of trifluoroacetonylated hemoglobin at fractional ligand saturations throughout the range 0-1.0. Comparison of the ligand binding process observed in these two ways yields evidence for a specific order of ligand binding. The sequence of events is sensitive to the pH and organic phosphate concentration of the medium, demonstrating the delicately balanced control system produced by interactions between the hemoglobin subunits and the effectors.