35 research outputs found
A Characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2
An integral quadratic polynomial (with positive definite quadratic part) is
called almost universal if it represents all but finitely many positive
integers. In this paper, we provide a characterization of almost universal
ternary quadratic polynomials with conductor 2
Almost universal ternary sums of polygonal numbers
For a natural number , generalized -gonal numbers are those numbers of
the form with . In this
paper we establish conditions on for which the ternary sum
is almost universal
Primitive prime divisors in zero orbits of polynomials
Let be a sequence of integers. A primitive prime
divisor of a term is a prime which divides but does not divide any
of the previous terms of the sequence. A zero orbit of a polynomial is a
sequence of integers where the -th term is the -th iterate of
at 0. We consider primitive prime divisors of zero orbits of polynomials. In
this note, we show that for integers and , where and , every iterate in the zero orbit of contains a primitive
prime whenever zero has an infinite orbit. If , then every iterate
after the first contains a primitive prime.Comment: 6 page
A characterization of almost universal ternary quadratic polynomials with odd prime power conductor
An integral quadratic polynomial (with positive definite quadratic part) is
called almost universal if it represents all but finitely many positive
integers. In this paper, we introduce the conductor of a quadratic polynomial,
and give an effective characterization of almost universal ternary quadratic
polynomials with odd prime power conductor
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Reflections on hyperbolic space
In school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research
30 Years of Synthetic Data
The idea to generate synthetic data as a tool for broadening access to
sensitive microdata has been proposed for the first time three decades ago.
While first applications of the idea emerged around the turn of the century,
the approach really gained momentum over the last ten years, stimulated at
least in parts by some recent developments in computer science. We consider the
upcoming 30th jubilee of Rubin's seminal paper on synthetic data (Rubin, 1993)
as an opportunity to look back at the historical developments, but also to
offer a review of the diverse approaches and methodological underpinnings
proposed over the years. We will also discuss the various strategies that have
been suggested to measure the utility and remaining risk of disclosure of the
generated data.Comment: 42 page
A Canonical Form for Positive Definite Matrices
We exhibit an explicit, deterministic algorithm for finding a canonical form
for a positive definite matrix under unimodular integral transformations. We
use characteristic sets of short vectors and partition-backtracking graph
software. The algorithm runs in a number of arithmetic operations that is
exponential in the dimension , but it is practical and more efficient than
canonical forms based on Minkowski reduction