Related papers: Lines, Quadrics, and Cremona Transformations in Tw…
Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal G}_{k}$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak G}_{k}$ for the set of all pairs $(S,U)\in…
We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\geq$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point)…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
Zernike polynomials are widely used mathematical models of experimentally observed optical aberrations. Their useful mathematical properties, in particular their orthogonality, make them a ubiquitous basis set for solving various problems…
We discuss the local and global problems for the equivalence of geometric structures of an arbitrary order and, in later sections, attention is given to what really matters, namely the equivalence with respect to transformations belonging…
We give a list of possibilities for surfaces of general type with $p_g=0$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$ and $S/i$ is not rational. Some examples with $K^2=4,...,7$ are constructed as…
We obtain an alternate proof of an injectivity result by Beauville for a map from the moduli space of quartic del Pezzo surfaces to the set of conjugacy classes of certain subgroups of the Cremona group. This amounts to showing that a…
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix…
We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
Given a collection of points in the plane, classifying which subsets are collinear is a natural problem and is related to classical geometric constructions. We consider collections of points in a projective plane over a finite field such…
We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…
Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…
We classify smooth projective varieties of Picard rank 2 which has two structures of blow-up of projective space along smooth subvarieties of different dimensions. This gives a characterization of the so called quadro-cubic Cremona…
A five dimensional model containing both left-right and quark-lepton symmetries is constructed, with the gauge group broken by a combination of orbifold compactification and the Higgs mechanism. An analysis of the gauge and scalar sectors…
An efficient way to get implicit equations of conics on five points and quadrics on nine, using pencils of conics and quadrics, is revealed. Parallel axis right cones intersect on a conic. An example, to show how to place five coplanar…
We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in "Collinear Points in Permutations", 2005) and the well-known finite plane Kakeya problem are intimately…
Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering…