Related papers: Finite sets of $d$-planes in affine space

For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair…

A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no…

We study self-similar attractors in the space $\mathbb{R}^d$, i.e., self-similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix $M$ of the linear part of affine…

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or…

A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$…

We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully…

Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $\lambda$ of $G$, the $T$-fixed point subscheme $(\bar{Gr}_G^\lambda)^T$ of the Schubert variety…

Given a finite index subfactor, we show that the {\em affine morphisms at zero level} in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a *-algebra. This…

A residual design ${\cal{D}}_B$ with respect to a block $B$ of a given design $\cal{D}$ is defined to be linearly embeddable over $GF(p)$ if the $p$-ranks of the incidence matrices of ${\cal{D}}_B$ and $\cal{D}$ differ by one. A sufficient…

The aim of this paper is mainly, after some theoretical explanations, to provide a program on Maple for computing, whatever be d, the curvature of the planar d-web implicitely defined by a differential equation F(x,y,y')=0, F being…

We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced irreducible plane curve of degree d with…

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance $\epsilon>0$ and an $\epsilon$-irreducible algebraic affine plane curve $\mathcal C$ of…

This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe…

Lattice theoretical generalizations of some classical linear algebra results are formulated. A vector space is replaced by its subspace lattice and a linear map is replaced by the induced lattice map. This map is a complete join…

A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming…

For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the…

In differential geometry, the notation d^n f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher order are useful objects that can be interpreted in terms of…

For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of…

For the finite ordered sets $A, D$, write $A^D$ for the ordered set of isotone maps $D \to A$ with the pointwise order. It was proved in earlier work that the order structure of $A^A$ determines~$A$ up to isomorphism. In this note we extend…