Related papers: A short proof for the open quadrant problem

Consider a finite collection of affine hyperplanes in $\mathbb R^d$. The hyperplanes dissect $\mathbb R^d$ into finitely many polyhedral chambers. For a point $x\in \mathbb R^d$ and a chamber $P$ the metric projection of $x$ onto $P$ is the…

This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove…

With the popularity of drone technologies, aerial photography has become prevalent in many daily scenarios such as environment monitoring, structure inspection, law enforcement etc. A central challenge in this domain is the efficient…

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…

In this paper, we consider polynomial correspondences $f (x, y)$ in $\mathbb{C}[x, y]$ of degree $d \ge 2$ in both the variables and obtain necessary and sufficient conditions in order that the equation $f (x, y) = 0$ can be expressed as…

We study geometric characterizations of unbounded integer polynomial optimization problems. While unboundedness along a ray fully characterizes unbounded integer linear and quadratic optimization problems, we show that this is not the case…

We present computational methods for constructing orthogonal/orthonormal polynomials over arbitrary polygonal domains in $\mathbb{R}^2$ using bivariate spline functions. Leveraging a mature MATLAB implementation which generates spline…

In this paper, the intersection of bivariate orthogonal polynomials on triangle patches will be investigated. The result is interesting by its own but also has important applications in the theory of a posteriori error estimation for finite…

We show that packing axis-aligned unit squares into a simple polygon $P$ is NP-hard, even when $P$ is an orthogonal and orthogonally convex polygon with half-integer coordinates. It has been known since the early 80s that packing unit…

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…

Let X be a minimal complex surface of general type such that its image via the canonical map is a surface; we denote by d the degree of the canonical map. In this expository work, first of all we recall the known possibilities for the…

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and…

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…

For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector…

Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by m endomorphism is a powerful and much-used tool. The pre-images for this map are…

The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in…

Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal…

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…

Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be drawn.…

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Latt\`{e}s map. In particular, this provides some…