On the distribution of the order over residue classes

The multplicative order of an integer g modulo a prime p, with p coprime to g, is defined to be the smallest positive integer k such that g^k is congruent to 1 modulo p. For fixed integers g and d the distribution of this order over residue classes mod d is considered as p runs over the primes. An overview is given of the most significant of my results on this problem obtained (mainly) in the three part series of papers `On the distribution of the order and index of g (modulo p) over residue classes' I-III (appeared in the Journal of Number Theory, also available from the ArXiv).Comment: 8 pages, 1 tabl

On orbital variety closures in sl(n). II. Descendants of a Richardson orbital variety

For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In sl(n) orbital varieties are described by Young tableaux. In this paper we consider so called Richardson orbital varieties in sl(n). A Richardson orbital variety is an orbital variety whose closure is a standard nilradical. We show that in sl(n) a Richardson orbital variety closure is a union of orbital varieties. We give a complete combinatorial description of such closures in terms of Young tableaux. This is the second paper in the series of three papers devoted to a combinatorial description of orbital variety closures in sl(n) in terms of Young tableaux.Comment: 27 pages, to appear in Journal of Algebr

Hamiltonians defined by biorthogonal sets

In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of $\mathcal{G}$-quasi basis and we show a series of conditions under which a definition of non self-adjoint Hamiltonian with purely point real spectra is still possible.Comment: in press in Journal of Physics

Fifth ACRL National Conference: A Brief Report: April 5-8, 1989 Cincinnati, OH

One of the prevailing themes of the Fifth ACRL National Conference held in Cincinnati, Ohio, was the state of the library materials budget. This theme manifested itself in papers on educational and general (E&G) expenditures, journal use studies, approval plans, and cooperative collection development. Although other collection management issues were featured at the conference, this report will focus on six papers and programs which dealt with the library materials budget.No embarg

A Group Theoretical Identification of Integrable Equations in the Li\'enard Type Equation x¨+f(x)x˙+g(x)=0\ddot{x}+f(x)\dot{x}+g(x) = 0 : Part II: Equations having Maximal Lie Point Symmetries

In this second of the set of two papers on Lie symmetry analysis of a class of Li\'enard type equation of the form $\ddot {x} + f(x)\dot {x} + g(x)= 0$, where over dot denotes differentiation with respect to time and $f(x)$ and $g(x)$ are smooth functions of their variables, we isolate the equations which possess maximal Lie point symmetries. It is well known that any second order nonlinear ordinary differential equation which admits eight parameter Lie point symmetries is linearizable to free particle equation through point transformation. As a consequence all the identified equations turn out to be linearizable. We also show that one can get maximal Lie point symmetries for the above Li\'enard equation only when $f_{xx} =0$ (subscript denotes differentiation). In addition, we discuss the linearising transformations and solutions for all the nonlinear equations identified in this paper.Comment: Accepted for publication in Journal of Mathematical Physic

Goodnight Family (MSS 148)

Finding aid only for Manuscripts Collection 148. Letters, legal papers, articles, and data on the Goodnight family, Franklin, Kentucky, and journal, 1868-1880 (102 p.) of Cumberland Presbyterian minister Thomas Mitchell Goodnight. Also includes letters from William Jennings Bryan, J. G. Carlisle, and William Goebel. A scan of Thomas Mitchell Goodknight\u27s journal (Folder 7) has been scanned and is attached as an Additional File ; click on the link at the bottom of this page