Related papers: A Special Conic Associated with the Reuleaux Negat…
We study Poncelet's Theorem in finite projective coordinate planes over the field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of such conics we investigate whether we can find polygons with $n$ sides, which are…
We study six pedal-like curves associated with the ellipse which are area-invariant for pedal points lying on one of two shapes: (i) a circle concentric with the ellipse, or (ii) the ellipse boundary itself. Case (i) is a corollary to…
Roughly speaking, to any space $M$ with perfect obstruction theory we associate a space $N$ with symmetric perfect obstruction theory. It is a cone over $M$ given by the dual of the obstruction sheaf of $M$, and contains $M$ as its zero…
Using the duality of positive cones, we show that applying the polar transform from convex analysis to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants have nice…
An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb{T}^2$, which induces a Poincar\'e self-duality for $\mathbb{T}^2$, can be 'quantized' to give a…
By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…
The space of quadrilaterals with fixed side lengths is an elliptic curve. Darboux used this to prove a porism on foldings. In this article, the space of oriented quadrilaterals is studied on the base of biquadratic equations between their…
An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality…
In this paper, we investigate analytic and geometric properties of obstruction flatness of strongly pseudoconvex CR hypersurfaces of dimension $2n-1$. Our first two results concern local aspects. Theorem 3.2 asserts that any strongly…
We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. In order to deal with the lack of control of the reflection measure the proof uses some…
Pedal and contrapedal curves are important study objects of plane curves. As for a mixed-type Minkowski plane curve, since the definitions of the pedal and contrapedal curves at lightlike points can not always be given, the investigation of…
Two retractions Q and R on closed convex cones M and respectively N of a Banach space are called mutually polar if Q+R=I and QR=RQ=0. This note investigates the existence of a pair of mutually polar retractions for given cones M and N. It…
This paper proposes direct and inverse results for the Dirichlet and Dirichlet to Neumann problems for complex curves with nodal type singularities. As an application, we give a method to reconstruct the conformal structure of a compact…
We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a…
A unibike curve is a track that can be made by either a bicycle or a unicycle. More precisely, the end of a unit tangent vector at any point on a unibike curve lies on the curve (so the bike's front wheel always lies on the track made by…
Inspired by the recent results toward Birkhoff conjecture (a rigidity property of billiards in ellipses), we discuss two rigidity properties of conics. The first one concerns symmetries of an analog of polar duality associated with an oval,…
This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, $3$-connected and strongly self-dual. We study the question of when these conditions are sufficient. If $G$ is any such a graph with…
The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that…
The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane - affine or projective, when that is not the case, i.e. the conics have at least…
Let f be a rational self-map of P^2 which leaves invariant an elliptic curve C with strictly negative transverse Lyapunov exponent. We show that C is an attractor, i.e. it possesses a dense orbit and its basin is of strictly positive…