Related papers: Triangles on planar Jordan $C^1$-curves
In this article, we show that every Jordan {g, h}-derivation over T_n(C) is a {g, h}-derivation under an assumption, where C is a commutative ring with unity 1 not equal to 0. We give an example of a Jordan {g, h}-derivation over T_n(C)…
We study the number of solutions of the asymptotic Plateau problem in H^3. By using the analytical results in our previous paper, and some topological arguments, we show that there exists an open dense subset of C^3 Jordan curves in…
We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable…
In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non--zero. We will prove an analogous statement for complements of any algebraic curve in C^2.…
In this paper, a study of topological and algebraic properties of two families of functions from the unit interval $I$ into the plane $\mathbb{R}^2$ is performed. The first family is the collection of all Peano curves, that is, of those…
We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite…
We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely. We also prove the same result for…
We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jordan curve has inflection minimum zero. For tree-like shadows,…
We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree (3n+1)/2 \leq d \leq 2n-3…
We extend the group law of curves of degree three by chords and tangents to the Jacobi variety of plane curves of degree n>4 by replacing points by point groups and lines by algebraic curves. The curves are nonsingular or have simple…
We develop a connection between the inscribed square problem and the question of understanding relation avoiding paths in a complex vector space. Our main theorem is that a Jordan curve with no inscribed squares would have a seemingly…
For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a…
Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide…
We show that every possible value for the Clifford index and gonality of a curve of a given genus on a $K3$ surface occurs.
Let $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ denote the set of $C^1$ regular curves in the $2$-sphere $\mathbb{S}^2$ that start and end at given points with the corresponding Frenet frames $\boldsymbol{P}$ and…
In the article we study the simple unital communitative three-dimensional algebras over an algebraically closed field of characteristic not equal to 2. It is proved that every simple unital communitative three-dimensional algebra of…
We prove some restriction theorems for flat homogeneous surfaces of codimension greater than one.
The Heesch problem 'grades' polygons that fail to tile the plane in terms of the number of layers (or corollas) of copies of it that can be formed around a central unit. We study the different topology of ' walls', which we define to be…
We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be…
In this paper we consider Erd\"os-Mordell inequality and its extension in the plane of triangle to the Erd\"os-Mordell curve. Algebraic equation of this curve is derived, and using modern computer tools in mathematics, we verified one…