Related papers: Quantization Bounds on Grassmann Manifolds of Arbi…
In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By `universal' we mean without curvature…
The volume of a k-dimensional foliation $\mathcal{F}$ in a Riemannian manifold $M^{n}$ is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a…
We derive simplified sphere-packing and Gilbert--Varshamov bounds for codes in the sum-rank metric, which can be computed more efficiently than previous ones. They give rise to asymptotic bounds that cover the asymptotic setting that has…
This paper introduces a new quantization scheme for real and complex Grassmannian sources. The proposed approach relies on a structured codebook based on a geometric construction of a collection of bent grids defined from an initial mesh on…
We derive quantitative volume constraints for sampling measures $\mu_t$ on the unit sphere $\mathbb{S}^d$ that satisfy Marcinkiewicz-Zygmund inequalities of order $t$. Using precise localization estimates for Jacobi polynomials, we obtain…
The capacity of a deterministic multiple-input multiple-output (MIMO) channel under the peak and average power constraints is investigated. For the identity channel matrix, the approach of Shamai et al. is generalized to the higher…
We present a novel approach to Riemannian interpolation on the Grassmann manifold. Instead of relying on the Riemannian normal coordinates, i.e. the Riemannian exponential and logarithm maps, we approach the interpolation problem with an…
Given metric quotients $S$ and $S_n$, $n \in \mathbb{N}$, of a metric space $X$, sufficient conditions are provided on the data defining them guaranteeing that $S$ is the Gromov-Hausdorff limit of $S_n$. These conditions are recognized…
In Gel'fand's inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold $M$ with boundary from the knowledge of the boundary $\partial M,$ the Neumann eigenvalues $\lambda_j$ and…
A multi-input multi-output (MIMO) Gaussian channel with two transmit antennas and two receive antennas is studied that is subject to an input peak-power constraint. The capacity and the capacity-achieving input distribution are unknown in…
This paper provides new bounds on the size of spheres in any coordinate-additive metric with a particular focus on improving existing bounds in the sum-rank metric. We derive improved upper and lower bounds based on the entropy of a…
In this work, novel upper and lower bounds for the capacity of channels with arbitrary constraints on the support of the channel input symbols are derived. As an immediate practical application, the case of multiple-input multiple-output…
A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two…
We study the high-dimensional limit of (projective) Stiefel and Grassmann manifolds as metric measure spaces in Gromov's topology. The limits are either the infinite-dimensional Gaussian space or its quotient by an mm-isomorphic group…
We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the…
The polymer representations, which are partially motivated by loop quantum gravity, have been suggested as alternative schemes to quantize the matter fields. Here we apply a version of the polymer representations to the free electromagnetic…
The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to…
We consider the inverse problem for the wave equation on a compact Riemannian manifold or on a bounded domain of $\R^n$, and generalize the concept of {\em domain of influence}. We present an efficient minimization algorithm to compute the…
Generalized spatial modulation (GSM) uses $N$ antenna elements but fewer radio frequency (RF) chains ($R$) at the transmitter. Spatial modulation and spatial multiplexing are special cases of GSM with $R=1$ and $R=N$, respectively. In GSM,…
Through the analyses of volume-forms in differentiable manifolds, it is shown that the usual way of defining minimal action principles for field theory on curved space-times is not appropriate on non-riemannian manifolds. An alternative…