Related papers: The circular mirror and Ian Stewart's polynomials
In this short paper we show that with a small change of the common ruler and compass construction of the regular pentadecagon, we can produce more regular polygons
It is demonstrated how the software system polymake can be used for computations in toric geometry. More precisely, counter-examples to conjectures related to A-determinants and defect polytopes are constructed.
This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…
In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some…
An introduction and overview of constructive reverse mathematics.
We construct some matroids that have a circuit and a cocircuit with infinite intersection. This answers a question of Bruhn, Diestel, Kriesell, Pendavingh and Wollan. It further shows that the axiom system for matroids proposed by Dress…
We make two observations regarding the invertibility of Keller maps. i.e., polynomial maps for which the determinant of their Jacobian matrix is identically equal to 1. In our first result, we show that if P is a n-dimensional Keller map,…
We introduce a model to design reflectors that take into account the inverse square law for radiation. We prove existence of solutions, both in the near and far field cases, when the input and output energies are prescribed.
We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…
This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article…
We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational K\"ahler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type as well. We also…
This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with…
Algebraic deformations of modules over a ring are considered. The resulting theory closely resembles Gerstenhaber's deformation theory of associative algebras.
This is a survey of recently published results. We introduce and study a wide class algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many well-known graphs…
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries…
We derive certain constraints on the reflection matrix for reflection from a plane, nonmagnetic, optically anisotropic surface using a reciprocity theorem stated long ago by van de Hulst in the context of scattering of polarized light. The…
We introduce a geometric construction which relates to the pentagram map much in the way that a logarithmic spiral relates to a circle. After introducing the construction, we establish some basic geometric facts about it, and speculate on…
If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact.
We study the inverse problem in the theory of (standard) orthogonal polynomials involving two polynomials families $(P_n)_n$ and $(Q_n)_n$ which are connected by a linear algebraic structure such as $$P_n(x)+\sum_{i=1}^N…