Lebesgue's Remarkable Result

Remarkable cardinals

To a large extent, the scientific work of both Peter Koepke and Philip Welch has always been inspired by Ronald Jensen’s fine structure theory and his Covering Lemma for L, the constructible universe. Cf., e.g., their joint papers [5] and [6]. In this paper we aim to play with the theme that large cardinals compatible with “V=L,” specifically: remarkable cardinals, allow us to create situations...

Remarkable Asymmetric Random Walks

There exists an asymmetric probability measure P on the real line with P n (]0; 1)+ (1=2)P n (f0g) = 1=2 for every n 2 N. P can be chosen absolutely continuous and P can be chosen to be concentrated on the integers. In both cases, P can be chosen to have moments of every order, but P can not be determined by its moments. 1 Questions and results Let P be a probability measure on the real line R,...

Technology Critique: ReMarkable Texts

ReMarkable Texts (RMT) is an “electronic student notebook” currently under development by Brown Computer Science Department's Computer Graphics Group. Running on a TabletPC, the system aims to replace and augment the standard paper notebook with an electronic device. RMT offers a variety of tools for manipulating and annotating documents, including simple note-taking with the Tablet's stylus, s...

A Remarkable Monster

Some Remarkable Combinatorial Matrices

In this paper we describe, in combinatorial terms, some matrices which arise as Laplacians connected to the three-dimensional Heisenberg Lie algebra. We pose the problem of finding the eigenvalues and eigenvectors of these matrices. We state a number of conjectures including the conjecture that all eigenvalues of these matrices are non-negative integers. We determine the eigenvalues and eigenve...