1. CMB Online first
||Discrete space-time and Lorentz transformations|
Alfred Schild has established conditions
that Lorentz transformations map world-vectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. The problem
was dealt with in the context of tensor and spinor calculus.
Due to Schild's number-theoretic arguments, the subject is also
interesting when isolated from its physical background.
The paper of Schild is not easy to understand. Therefore we first
present a streamlined version of his proof which is based on
the use of null vectors. Then we present a purely algebraic proof
that is somewhat shorter. Both proofs rely on the properties
of Gaussian integers.
Keywords:Lorentz transformation, integer lattice, Gaussian integers
Categories:22E43, 20H99, 83A05
2. CMB 2011 (vol 56 pp. 161)
||An Extension of the Dirichlet Density for Sets of Gaussian Integers|
Several measures for the density of sets of integers have been proposed,
such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and
investigate some of its properties.
Keywords:Gaussian integers, Dirichlet density
Categories:11B05, 11M99, 11N99
3. CMB 2008 (vol 51 pp. 57)
||A Note on Integer Symmetric Matrices and Mahler's Measure |
We find a lower bound on the absolute value of the discriminant of
the minimal polynomial of an integral symmetric matrix and apply
this result to find a lower bound on Mahler's measure of related
polynomials and to disprove a conjecture of D. Estes and R. Guralnick.
Keywords:integer matrices, Lehmer's problem, Mahler's measure
4. CMB 2007 (vol 50 pp. 399)
||Expansions in Complex Bases |
Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely
studied in the last
forty years. They turned out to be relevant in
various domains of mathematics, such as the theory of finite
theory, fractals or dynamical systems.
Several results were extended by Dar\'oczy and K\'atai
in complex bases. We introduce an adaptation of the so-called greedy
algorithm to the complex case, and we
generalize one of their main theorems.
Keywords:non-integer bases, greedy expansions, beta-expansions
Categories:11A67, 11A63, 11B85