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Stephan Foldes

Publications and source records attributed to Stephan Foldes.

25 records · Page 2Linked to original sources
arXiv2010

Hyperboloid preservation implies the Lorentz and Poincaré groups without dilations

An analogue of the Alexandrov-Zeeman theorem, based on hyperboloid preservation, as opposed to light cone preservation, is provided. This characterizes exactly the Poincaré group, as opposed to the Poincaré group extended by dilations. The hyperbolic analogue, as opposed to the cone-based Alexandrov-Zeeman theorem, is also valid in the case of a single space dimension. An orthochronous version also holds, based on the preservation of forward hyperboloid shells.

math-ph↗
arXiv2009

The Cauchy-Schwarz inequality in Cayley graph and tournament structures on finite fields

The Cayley graph construction provides a natural grid structure on a finite vector space over a field of prime or prime square cardinality, where the characteristic is congruent to 3 modulo 4, in addition to the quadratic residue tournament structure on the prime subfield. Distance from the null vector in the grid graph defines a Manhattan norm. The Hermitian inner product on these spaces over finite fields behaves in some respects similarly to the real and complex case. An analogue of the Cauchy-Schwarz inequality is valid with respect to the Manhattan norm. With respect to the non-transitive order provided by the quadratic residue tournament, an analogue of the Cauchy-Schwarz inequality holds in arbitrarily large neighborhoods of the null vector, when the characteristic is an appropriate large prime.

math.CO↗
arXiv2008

The Lorentz group and its finite field analogues: local isomorphism and approximation

Finite Lorentz groups acting on 4-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real 4-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary accuracy to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields.

math-ph↗
arXiv2008

On McMillan's theorem about uniquely decipherable codes

Karush's proof of McMillan's theorem is recast as an argument involving polynomials with non-commuting indeterminates certain evaluations of which yield the Kraft sums of codes, proving a strengthened version of McMillan's theorem.

math.CO↗
arXiv2007

A half-space approach to order dimension

The aim of the present paper is to investigate the half-spaces in the convexity structure of all quasiorders on a given set and to use them in an alternative approach to classical order dimension. The main result states that linear orders can almost always be replaced by half-space quasiorders in the definition of the dimension of a partially ordered set.

math.CO↗
arXiv2006

Column-partitioned matrices over rings without invertible transversal submatrices

Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p \times p$ matrix $Q$ and a positive integer $m \leq p$ such that $QM = [QM_1, ..., QM_n]$ is in reduced echelon form and in all but at most $m-1$ blocks $QM_i$ the last $m$ entries of each column are either all zero or they include a non-zero non-unit.

math.CO↗