Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

Ligand Substitution Dynamics

Substitution of a ligand in an inner sphere complex by an outside group is the most fundamental reaction in metal ion chemistr

Substitute Arguments in Constitutional Law

In this article, I argue that that substitution is crucial to our practice of constitutional law. Of course, if one wished, one could easily extend the domain of substitution beyond these boundaries. Substitute arguments are an important aspect of law more generally and, indeed, of life. I have nonetheless chosen to limit my discussion to constitutional substitution because, I believe, overt discussion of substitution in this particular area illuminates important aspects of our constitutional regime-–aspects that substitution itself regularly obscures. To put my central point directly, I hope to show that constitutional law amounts to one, giant substitute argument

Substitution Delone Sets

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems reformulate

Ligand Substitution Processes

From the preface: The subject of the mechanistic study of ligand substitution reactions is currently undergoing an exciting growth. New fast-reaction techniques have removed the upper limit on rates that can be measured, and extension to less familiar central metal atoms has begun in earnest. This might seem the wrong moment for review of the field. As yet, definitive treatment is possible only for those complexes involving monodentate ligands with cobalt(III) and platinurn(II). But, because information is so extensive for these systems, it is clear that they are functioning as models from which concepts and experiments are generated for application over the fast-growing range of the subject. We believe that this is an important moment to reopen debate on fundamentals so that concepts will be most felicitously formulated to aid growth of understanding. This monograph is centrally concerned with three aspects of those fundamentals. We have attempted to develop an approach to classification of ligand substitution reactions that is adapted to what seem to have emerged as the characteristic features of these reactions and is susceptible to operational tests. (We do recognize that any such scheme of ideas is necessarily obsolescent once it is formulated since new experiments will certainly follow immediately.) We have tried to evaluate the basis for making generalizations about ligand substitution processes and to formulate tests to show whether new reactions fall within familiar patterns. Finally, we have sought to base the models of ligand substitution processes in the language of molecular-orbital theory. We believe that MO theory is most useful, because it may be used to correlate rate data on complexes with the extensive information available from spectral and magnetic studies, yet differs from crystal-field theory in providing a natural place for consideration of the bonding electrons, which must be a principal determinant of reaction processes. To keep this essay within bounds, we assume familiarity with the elements of experimental kinetics, transition-state theory, and the simple molecular-orbital theory of complexes. Introductory physical chemistry, some familiarity with the study of reaction mechanisms, and mastery of one of the qualitative treatments of MO theory as applied to transition-metal complexes should provide sufficient background. Thus, we hope that this book will be useful to students, relatively early in their careers, who wish to explore this field. Our debts to very many workers will be obvious throughout. We want to record here our special personal debt to Professors Ralph G. Pearson and Fred Basolo and to Dr. Martin Tobe. We particularly thank Professor George S. Hammond for his interest and enthusiasm in this project. Professor Hammond carefully read and criticized the entire manuscript in the final drafts. We received many other valuable criticisms at various stages of this project from Professors R. D. Archer, F. Basolo, J. O. Edwards, J. Finholt, P. Haake, J. Halpern, A. Kropf, R. G. Pearson, S. I. Shupack, M. S. Silver, and C. Walling, and Dr. U. Belluco and Dr. L. Cattalini. We very much appreciate their help and probably should have followed their suggestions more closely. We warmly acknowledge expert assistance from Mrs. Madeline deFriesse, Miss Jan Denby, and Mrs. Diane Celeste in preparation of the manuscript. COOPER H. LANGFORD HARRY B. GRAY Amherst, Massachusetts New York, New York October 196