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Register a new userOn the Number of Faces and Radii of Cells Induced by Gaussian Spherical Tessellations
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Added by
Anna Ma
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study a geometric property related to spherical hyperplane tessellations in $mathbb{R}^{d}$. We first consider a fixed $x$ on the Euclidean sphere and tessellations with $M gg d$ hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of $dlog(d)log(M)$ such that the radius of the cell containing $x$ induced by these hyperplanes is bounded above by, up to constants, $dlog(d)log(M)/M$. We extend this result to hold for all cells in the tessellation with high probability. Up to logarithmic terms, this upper bound matches the previously established lower bound of Goyal et al. (IEEE T. Inform. Theory 44(1):16-31, 1998).
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The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermass integrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $ntoinfty$ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.
The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $dgeq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $kin{1,ldots,d-1}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.
We present a theorem on the compatibility upon deployment of kirigami tessellations restricted on a spherical surface with patterned cuts forming freeform quadrilateral meshes. We show that the spherical kirigami tessellations have either one or two compatible states, i.e., there are at most two isolated strain-free configurations along the deployment path. The proof of the theorem is based on analyzing the number of roots of the compatibility condition, under which the kirigami pattern allows a piecewise isometric transformation between the undeployed and deployed configurations. As a degenerate case, the theorem further reveals that neutral equilibrium arises for planar quadrilateral kirigami tessellations if and only if the cuts form parallelogram voids. Our study provides new insights into the rational design of morphable structures based on Euclidean and non-Euclidean geometries.
Information-theoretic methods have proven to be a very powerful tool in communication complexity, in particular giving an elegant proof of the linear lower bound for the two-party disjointness function, and tight lower bounds on disjointness in the multi-party number-in-the-hand (NIH) model. In this paper, we study the applicability of information theoretic methods to the multi-party number-on-the-forehead model (NOF), where determining the complexity of disjointness remains an important open problem. There are two basic parts to the NIH disjointness lower bound: a direct sum theorem and a lower bound on the one-bit AND function using a beautiful connection between Hellinger distance and protocols revealed by Bar-Yossef, Jayram, Kumar and Sivakumar [BYJKS04]. Inspired by this connection, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party NOF protocols and provide a small toolbox that allows one to manipulate several Hellinger volume terms and lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. We then apply these new tools to obtain a lower bound on the informational complexity of the AND_k function in the NOF setting. Finally, we discuss the difficulties of proving a direct sum theorem for information cost in the NOF model.
For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area $R^2$. The mean number of components is known to be of order $R^2$ for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels $ell$, these random variables have fluctuations of order at least $R$, and hence variance of order at least $R^2$. In particular, this holds for excursion sets when $ell$ is in some neighbourhood of zero, and it holds for excursion/level sets when $ell$ is sufficiently large. We prove stronger fluctuation lower bounds of order $R^alpha$, $alpha in [1,2]$, in the case that the spectral density has a singularity at the origin. Finally, we show that the number of excursion/level sets for the Random Plane Wave at certain levels has fluctuations of order at least $R^{3/2}$, and hence variance of order at least~$R^3$. We expect that these bounds are of the correct order, at least for generic levels.
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