Related papers: Curves in R^d intersecting every hyperplane at mos…

Consider a random geometric graph over a random point process in $\mathbb{R}^d$. Two points are connected by an edge if and only if their distance is bounded by a prescribed distance parameter. We show that projecting the graph onto a two…

A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is…

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as…

Rotation curves of interacting galaxies often show that velocities are either rising or falling in the direction of the companion galaxy. We seek to reproduce and analyse these features in the rotation curves of simulated equal-mass…

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…

We show that any graded digraph $D$ on $n$ vertices with maximum degree $\Delta$ has an oriented Ramsey number of at most $C^\Delta n$ for some absolute constant $C > 1$, improving upon a recent result of Fox, He, and Wigderson. In…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions…

A family of closed simple (i.e., Jordan) curves is $m$-intersecting if any pair of its curves have at most $m$ points of common intersection. We say that a pair of such curves touch if they intersect at a single point of common tangency. In…

Let $\mathcal{H}_q$ denote the Hermitian curve in $\mathbb{P}^2$ over $\mathbb{F}_{q^2}$ and $\mathcal{C}_d$ be an irreducible plane projective curve in $\mathbb{P}^2$ also defined over $\mathbb{F}_{q^2}$ of degree $d$. Can $\mathcal{H}_q$…

Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H…

We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a result, we can give an upperbound of the level $N$ by means of $d$. This generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a corollary of this…

Consider l lines in P^2 such that no three lines meet in a point. Let X(l) denote all points of intersections of these l lines. We describe all pairs (d,l) such that generic degree d curve in P^2 contains a X(l).

A point P on a smooth hypersurface X of degree d in an N-dimensional projective space is called a star point if and only if the intersection of X with the embedded tangent space T_P(X) is a cone with vertex P. This notion is a…

We introduce the Frenet theory of curves in dual space $\d^3$. After defining the curvature and the torsion of a curve, we classify all curves in dual plane with constant curvature. We also establish the fundamental theorem of existence in…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…

Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to…

A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if…

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation…

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show…