Related papers: VC dimension of ellipsoids

We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $d$th Veronese embedding of the projective $n$-space $\mathbb{P}^n$ have the expected…

We show analytically that the $[0,1]$, $[1,1]$ and $[2,1]$ Pad{\'e} approximants of the mean cluster number $S(p)$ for site and bond percolation on general $d$-dimensional lattices are upper bounds on this quantity in any Euclidean…

We study combinatorial complexity of certain classes of products of intervals in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the Vapnik-Chervonenkis…

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric…

We give a bound for the geometric dimension for the family of virtually cyclic groups in mapping class groups of a compact surface with punctures, possibly with nonempty boundary and negative Euler characteristic.

We propose a novel approach for probabilistic generative modeling of 3D shapes. Unlike most existing models that learn to deterministically translate a latent vector to a shape, our model, Point-Voxel Diffusion (PVD), is a unified,…

We adapt a recent theory for the random close packing of polydisperse spheres in three dimensions [R. S. Farr and R. D. Groot, J. Chem. Phys. {\bf 131} 244104 (2009)] in order to predict the Hausdorff dimension $d_{A}$ of the Apollonian…

The box counting dimension $d_C$ and the correlation dimension $d_G$ change with the number of numerically generated points forming the attractor. At a sufficiently large number of points the fractal dimension tends to a finite value. The…

A set of $N$ points is chosen randomly in a $D$-dimensional volume $V=a^D$, with periodic boundary conditions. For each point $i$, its distance $d_i$ is found to its nearest neighbour. Then, the maximal value is found, $d_{max}=max(d_i,…

In this paper we show that the diameter of a d-dimensional lattice polytope in [0,k]^n is at most (k - 1/2) d. This result implies that the diameter of a d-dimensional half-integral polytope is at most 3/2 d. We also show that for…

We present the multidimensional membership mixture (M3) models where every dimension of the membership represents an independent mixture model and each data point is generated from the selected mixture components jointly. This is helpful…

Higher-dimensional Dedekind sums are defined as a generalization of a recent 1-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the frequency distribution of marked lattice points leads to new…

A method for detecting voids in the galaxy distribution is presented. Using this method, we have identified 732 voids with a radius of the seed sphere Rseed > 4.0h^{-1}Mpc in a volume-limited sample of galaxies from the southern part of the…

Let $\mathcal{E}$ be an ellipsoid in $\mathbb{R}^n$. A. Gusakova and D. Zaporozhets conjectured that $\mathcal{E}$ is uniquely (up to rigid motions) determined by its intrinsic volumes. We prove this conjecture for $n = 3$.

We compute the $k$-width of a round $2$-sphere for $k=1,\ldots,8$ and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max critical varifold.

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to…

We prove that subsets of ${\Bbb R}^d$, $d \ge 4$ of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two…

The dimension of a linear space is the maximum positive integer $d$ such that any $d$ of its points generate a proper subspace. For a set $K$ of integers at least two, recall that a pairwise balanced design PBD$(v,K)$ is a linear space on…

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible…

Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected…