Class Numbers and Biquadratic Reciprocity

The research of the first author was supported by Natural Sciences and Engineering Research Council Canada Grant No. A-7233, while that of the second was supported by a Natural Sciences and Engineering Research Council Canada Undergraduate Summer Research Award.https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/class-numbers-and-biquadratic-reciprocity/00FECEE10C8CB10943F2ED8E7AAE2B8

REAL QUADRATIC NUMBER FIELDS WITH LARGE FUNDAMENTAL UNITS

There are many works on the determination or the estimation of the fundamental unit £ and the ideal class number h of real quadratic number fields F ([1], [3], [6] and [10], for example). The £'s which are treated in them have small orders of absolute value in comparison to their discriminants Z), that is

The Geometry of the Gibbs-Appell Equations and Gauss' Principle of Least Constraint

We present a generalisation of the Gibbs-Appell equations which is valid for general Lagrangians. The general form of the Gibbs-Appell equations is shown to be valid in the case when constraints and external forces are present. In the case when the Lagrangian is the kinetic energy with respect to a Riemannian metric, the Gibbs function is shown to be related to the kinetic energy on the tangent bundle of the configuration manifold with respect to the Sasaki metric. We also make a connection with the Gibbs-Appell equations and Gauss' principle of least constraint in the general case

A General Theory of Geodesics with Applications to Hyperbolic Geometry

In this thesis, the geometry of curved surfaces is studied using the methods of differential geometry. The introduction of manifolds assists in the study of classical two-dimensional surfaces. To study the geometry of a surface a metric, or way to measure, is needed. By changing the metric on a surface, a new geometric surface can be obtained. On any surface, curves called geodesics play the role of straight lines in Euclidean space. These curves minimize distance locally but not necessarily globally. The curvature of a surface at each point p affects the behavior of geodesics and the construction of geometric objects such as circles and triangles. These fundamental ideas of manifolds, geodesics, and curvature are developed and applied to classical surfaces in Euclidean space as well as models of non-Euclidean geometry, specifically, two-dimensional hyperbolic space

A variational approach to magnetoelastic buckling problems for systems of superconducting tori

A variational principle for the magnetoelastic stability problem of superconductors is constructed. Independently, a pair of integral equations is derived, from which the initial and the perturbed field can be computed. The integral equations are solved for in-plane buckling of a slender pair of concentric tori, and out-of-plane buckling of a slender pair of equal coaxial tori. By using the variational principle, it is shown that both cases can become unstable when the currents on the two tori are equally directed, and the pertinent buckling values are calculated. The thus obtained buckling values are compared with the results of an alternative, mathematically less rigorous, method. A good correspondence between the two methods is found (at least as long as the two tori are not too near)

Tau-Function Constructions of the Recurrence Coefficients of Orthogonal Polynomials

AbstractIn this paper we compute the recurrence coefficients of orthogonal polynomials using τ-function techniques. It is shown that for polynomials orthogonal with respect to positive weight functions on a noncompact interval, the recurrence coefficient can be expressed as the change in the chemical potential which, for sufficiently largeNis the second derivative of the free energy with respect toN, the particle number. We give three examples using this technique: Freud weights, Erdős weights, and weak exponential weights