Related papers: Affine Maps and Feuerbach'sTheorem

We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach's Fixed Point Theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not…

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by…

We establish estimates for the number of solutions of certain affine congruences. These estimates are then used to prove Manin's conjecture for a cubic surface split over Q and whose singularity type is D_4. This improves on a result of…

The von Neumann-Halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces. We propose acceleration schemes making use of two ideas:…

Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics,…

We prove that entire conformal curves $\mathbb{R}^n \rightarrow \mathbb{R}^m$ fall into two classes: either the curve is affine or the average energy in a ball is strictly increasing for large radii and diverges to infinity. This rigidity…

We give synthetic proofs of many new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangle $DEF$ of a point $P$ with respect to a given triangle $ABC$, as…

We prove a conjecture of B. Gr\"unbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on…

Let $X$ be a complex Banach space with $\dim X\geq3$ and $B(X)$ the algebra of all bounded linear operators on $X$. Suppose $\phi:B(X)\longrightarrow B(X)$ is a surjective map satisfying the following property: $Fix(AB)=Fix(\phi(A)\phi(B)),…

We establish a fixed-point theorem for the face maps that consist in deleting the $i$th entry of an ordered set. Furthermore, we show that there exists random finite sets of integers that are almost invariant under such deletions.…

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…

We consider a new type of mappings in metric spaces which can be characterized as mappings contracting perimeters of triangles. It is shown that such mappings are continuous. The fixed-point theorem for such mappings is proved and the…

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…

This paper explores the Bernstein problem of smooth maps $f:\mathbb{R}^4 \to \mathbb{R}^3$ whose graphs form coassociative submanifolds in $\mathbb{R}^7$. We establish a condition, expressed in terms of the second elementary symmetric…

We consider some constructions in hyperbolic geometry that are analogous to classical constructions in Euclidean geometry. We show that both Monge's theorem and the theorem on the concurrence of the common chords of three circles also hold…

We give an elementary proof that, for a closed manifold with an integral-integral affine structure, its total volume and number of integral points coincide. The proof uses rational Ehrhart theory and elementary Fourier analysis to estimate…

By affine arithmetic is meant the set of affine consequences of Peano arithmetic. This is a continuous theory which is studied in the framework of affine logic, a sublogic of continuous logic. Affine arithmetic is undecidable. Also, its…

In \cite{branko} Gr{\"u}nbaum asked if the set of all affine invariant points of a given convex body is equal to the set of all points invariant under every affine automorphism of the body. In \cite{ivan} we have proven the case of a body…

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme…

In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number…