Related papers: The D_4 root system is not universally optimal
In the point set embeddability problem, we are given a plane graph $G$ with $n$ vertices and a point set $S$ with $n$ points. Now the goal is to answer the question whether there exists a straight-line drawing of $G$ such that each vertex…
In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert's Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric…
The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating…
We show that among antipodal $2d$-point configurations on the sphere $S^{d-1}$ in $\mathbb R^d$, the set of vertices of a regular cross-polytope inscribed in $S^{d-1}$ uniquely solves the best-covering problem (this is new for $d\geq 5$)…
It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally, for all…
We investigate the distribution of cells by dimension in cylindrical algebraic decompositions (CADs). We find that they follow a standard distribution which seems largely independent of the underlying problem or CAD algorithm used. Rather,…
Planar N=4 super Yang-Mills appears to be integrable. While this allows to find this theory's exact spectrum, integrability has hitherto been of no direct use for scattering amplitudes. To remedy this, we deform all scattering amplitudes by…
We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d at least about 2n/3, a very general complex hypersurface of degree d in P^{n+1} is not stably rational. The statement generalizes…
An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…
We present an analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric, focusing on highly symmetric configurations on the unit sphere $\mathbb S^2$. Three discrete uniform…
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…
We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree…
Tree level scattering amplitudes in N=4 super Yang-Mills theory are almost, but not exactly invariant under the free action of the N=4 superconformal algebra. What causes the non-invariance is the holomorphic anomaly at poles where external…
This paper considers the geometry of $E_8$ from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using…
For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of the three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is…
The quartic hypersurfaces in P^4 invariant under the standard representation of S_6 form a linear pencil. We prove that a general member of this pencil is not rational.
In this paper we discuss reflection groups and root systems, in particular non-crystallographic ones, and a Clifford algebra framework for both these concepts. A review of historical as well as more recent work on viral capsid symmetries…
We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In $\S$7 we apply our results to the even unimodular lattice $E_8$ and show how the 600-cell transforms $E_8$/2$E_8$, an 8-space over the field $\bf{F}$$_2$,…
In this paper, we propose an exact general algorithm for solving non-convex optimization problems, where the non-convexity arises due to the presence of an inverse S-shaped function. The proposed method involves iteratively approximating…
A set ${X}_{N}=\{x_1,\ldots,x_N\}$ of $N$ points on the unit sphere $\mathbb{S}^d,\,d\geq 2$ is a spherical $t$-design if the average of any polynomial of degree at most $t$ over the sphere is equal to the average value of the polynomial…