Jeffrey M. Lee

Riemannian Foliations and Eigenvalue Comparison

Eigenvalue comparison theorems for the Laplacian on a Riemannian manifold generally give bounds for the first Dirichlet eigenvalue on balls in the manifold in terms of an eigenvalue arising from a geometrically or analytically simpler situation. Chengs eigenvalue comparison theory assumes bounds on the curvature of the manifold and then compares this eigenvalue to the eigenvalue of a ball in a constant curvature space form. In this paper we examine the basic Laplacian – the appropriate Laplacian on functions that are constant on the leaves of the foliation. The main theorems generalize Chengs eigenvalue comparison theorem and other eigenvalue comparison theorems to the category of Riemannian foliations by estimating the first Dirichlet eigenvalue for the basic Laplacian on a metric tubular neighborhood of a leaf closure. Several other facts about the the first eigenvalue of such foliated tubes as well as some needed facts about the tubes themselves are established. This comparison theory, like Chengs theorem, remains valid for large tubes that are not homotopic to the middle leaf closure and that may have irregular boundaries. We apply these results to obtain upper bounds for the eigenvalues of the basic Laplacian on a closed manifold in terms of curvature bounds and the transverse diameter of the foliation.

On the geometry of the smallest circle enclosing a finite set of points

A number of numerical codes have been written for the problem of finding the circle of smallest radius in the Euclidean plane that encloses a finite set P of points, but these do not give much insight into the geometry of this circle. We investigate geometric properties of the minimal circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing P is minimal if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We show that the center of the minimal circle is in the convex hull of P. We use this rigidity result and an analysis of the case of three points to find sharp estimates on the diameter of the minimal circle in terms of the diameter of P.

A note on canonical forms for matrix congruence

Abstract Canonical forms for matrix congruence for general matrices are exhibited as an easy consequence of results presented in a recent paper by R. C. Thompson on pencils of symmetric and skew symmetric matrices. We also show how to exhibit standard equivalence class representatives in the orbits of the semiorthogonal groups acting naturally on skew symmetric forms.

Nonparametric curve estimation on stiefel manifolds

The main result is a speed of a.s. uniform convergence for estimators of nonparametric regression Junctions on Stiefel manifolds. The Sticfcl manifold is not only of interest in its own right but also because it generalizes both the sphere and the orthogonal group. The main tool for the variance pari is a local fluctuation inequality for the compound empirical process indexed by simple subsets of the manifold that we call caps. For the bias part the symmetry of the Stiefel manifold is exploited. It turns out that this symmetry is sufficient to obtain the same overall rate as in a Euclidean space of the same dimension.

Observability of permutations, and stream ciphers

We study the observability of a permutation on a finite set by a complex-valued function. The analysis is done in terms of the spectral theory of the unitary operator on functions defined by the permutation. Any function f can be written uniquely as a sum of eigenfunctions of this operator; we call these eigenfunctions the eigencomponents of f. It is shown that a function observes the permutation if and only if its eigencomponents separate points and if and only if the function has no nontrivial symmetry that preserves the dynamics. Some more technical conditions are discussed. An application to the security of stream ciphers is discussed.

The maximal number of pairwise communicating stations under limitations of maximal and minimal communication distance

We consider the problem of determining the maximal number of stations that can maintain a total network of communication. We assume that there is a distance R which is that maximal distance that two stations can be separated and remain in contact. We also assume that there is a distance r which is them minimal separation that allows communication. This problem is intimately related to the problem of packing disks within a circle. The problem of finding the circle of smallest radius enclosing a finite set of points in the plane arises in a number of applications. Many numerical codes have been written for this problem. We provide a framework for investigating the geometric properties of this circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing a finite set of points P is the minimal circle if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We use this result to find a sharp estimate on the diameter of the minimal circle in terms of the diameter of P. We also show that the center of the minimal circle is contained in the convex hull of P.

Geometry Detected by a Finite Part of the Spectrum

It is well known that although the spectrum of a manifold does not, in general, determine the geometry of the manifold, various geometric invariants such as volume and dimension are in fact determined. One of the goals of finite spectral geometry is to study, in an exact way, how information about these invariants is revealed by only a finite part of the spectrum. One is reminded of the fact that geometric detail about an optically detected object cannot be determined to a greater extent than the wavelength of the impinging radiation allows. Shorter wavelengths potentially reveal greater detail. This situation is studied in scattering theory and differs in many ways from the case of trying to use a finite part of the Laplace spectrum to gain geometric information about a manifold, but the general notion that access to higher eigenvalues (in analogy to shorter wavelengths) should give more geometric information still holds. The question is: what geometry can we expect to be missed due to not having information about the high end of the spectrum? If one where trying to estimate the area of a (microscopic) planar region via some kind of pixel counting, then the estimate can only be expected to be reasonable if one knew, a priori, that the region did not include something like a haze of thin tentacles extending beyond some central region (or some other complexity near the boundary). But this is exactly where the analogy with hearing the volume via the Laplace spectrum breaks down since a long thin tenticle, being similar to a long string, may indeed have a major effect on the smaller eigenvalues of the region and so cannot on principle be expected to escape detection. The question is: what aspects of a manifold can be expected to interfere with the finite spectral detection of a given geometric invariant such as volume?