Related papers: Nondegenerate spheres in four dimensions
We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex…
Symmetries are manifested in nature through degeneracies in the spectra of physical systems. In the case of heavy deformed nuclei, when described in the framework of the Interacting Boson Model, within which correlated proton (neutron)…
Degenerate dynamical systems are characterized by symplectic structures whose rank is not constant throughout phase space. Their phase spaces are divided into causally disconnected, nonoverlapping regions such that there are no classical…
New examples of noncommutative 4-spheres are introduced.
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…
An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of ${\rm Aut}(T_p M)$ for all $p\in M$, which preserves (pointwise) the affine metric $h$, the difference tensor $K$ and the affine shape…
Let $(S,H)$ be a general primitively polarized $K3$ surface of genus $\p$ and let $p_a(nH)$ be the arithmetic genus of $nH.$ We prove the existence in $|\mathcal O_S(nH)|$ of curves with a triple point and $A_k$-singularities. In…
Richmond and Richmond (Amer. Math. Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove…
A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower…
There are three parameter degeneracies inherent in the three--neutrino analysis of long--baseline neutrino experiments. We develop a systematic method for determining whether or not a set of measurements in neutrino oscillation appearance…
Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent…
For a $(d-1)$-dimensional simplicial complex $\Delta$ and $1\leq i\leq d$, let $f_{i-1}$ be the number of $(i-1)$-faces of $\Delta$ and $m_i$ be the number of missing $i$-faces of $\Delta$. In the nineties, Kalai asked for a…
We study "distance spheres": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is "too dense" and a set of small volume, we can decompose $[0,1]^d$…
We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for…
We study how the eightfold-degeneracy in the ($\theta_{13},\delta$) plane observed in $\gamma \sim 100$ $\beta$-beams can be reduced by periodically changing the ions in the storage ring. This ``ions cocktail'' allows to change the neutrino…
``Rubber'' coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by ``marble'' coated bodies. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the…
We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials $\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $ on a unit--sphere in $d$ dimensions. Sidelnikov's…
We solved numerically the implicit, trascendental equation that defines the eigenenergy surface of a degenerating isolated doublet of unbound states in the simple but illustrative case of the scattering of a beam of particles by a double…
The notion of synthetic dimensions has expanded the realm of topological physics to four dimensional (4D) space lately. In this work, non-Hermiticity is used as a synthetic parameter in PT-symmetric photonic crystals to study the…
A subset $S$ of the unit sphere $\mathbb{S}^2$ is called orthogonal-pair-free if and only if there do not exist two distinct points $u, v \in S$ at distance $\frac{\pi}{2}$ from each other. Witsenhausen \cite{witsenhausen} asked the…