Related papers: Integer points on spheres and their orthogonal gri…

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…

We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we…

The purpose of this paper is to give an effective construction for some induced structures on spheres or product of spheres of codimension 1, 2 or 3, respectively, in Euclidean space endowed with an almost product structure.

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally…

In this article, we introduce and study the concept of $\textit{spherical-vectors}$, which can be perceived as a natural extension of the arguments of complex numbers in the context of quaternions. We initially establish foundational…

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…

Suppose that a finite-dimensional cube is orthogonally projected onto a central section of itself by a subspace of one dimension less. Up to dimension $9$, at least one vertex is projected onto the section, but for dimension $10$ or larger,…

Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by…

In this paper we prove Schur's conjecture in $\mathbb R^d$, which states that any diameter graph $G$ in the Euclidean space $\mathbb R^d$ on $n$ vertices may have at most $n$ cliques of size $d$. We obtain an analogous statement for…

The program of understanding Shape Theory layer by layer topologically and geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$. Topological shape space graphs are far more complex here, whereas metric shape spaces…

We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either…

We prove that the discrete Hardy-Littlewood maximal function associated with Euclidean spheres with small radii has dimension-free estimates on $\ell^p(\mathbb{Z}^d)$ for $p\in[2,\infty).$ This implies an analogous result for the Euclidean…

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower…

We study distributions on a Euclidean Jordan algebra V with values in a finite dimensional representation space for the identity component G of the structure group of V and homogeneous equivariance condition. We show that such distributions…

The minimal spherical cap dispersion ${\rm disp}_{\mathcal{C}}(n,d)$ is the largest number $\varepsilon\in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere $\mathbb{S}^d$, there exists a spherical cap with…

In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We…

We construct stable vector bundles on the space of symmetric forms of degree d in n+1 variables which are equivariant for the action of SL_{n+1}(C), and admit an equivariant free resolution of length 2. For n=1, we obtain new examples of…

In this work, the classical Borsuk conjecture is discussed, which states that any set of diameter 1 in the Euclidean space $ {\mathbb R}^d $ can be divided into $ d+1 $ parts of smaller diameter. During the last two decades, many…

In this paper is studied the problem concerning the angle between two subspaces of arbitrary dimensions in Euclidean space $E_{n}$. It is proven that the angle between two subspaces is equal to the angle between their orthogonal subspaces.…