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Register a new userAn algorithm for the classification of smooth Fano polytopes
الخوارزمية لتصنيف المضلعات الفانو الناعمة
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Authors
Mikkel {O}bro
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نقدم الآن خوارزمية تنتج قائمة التصنيف للبوليتوبات الفانو الناعمة لأي د. المدخل للخوارزمية هو رقم واحد، أي الرقم الصحيح الموجب. تم استخدام الخوارزمية لتصنيف البوليتوبات الفانو الناعمة لد<=7. هناك 7622 صنف من البوليتوبات الفانو الناعمة للد 6 و 72256 صنف من البوليتوبات الفانو الناعمة للد 7.
We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes for d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.
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This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all entries on each line (i.e., one free index) is equal to 1; a plane-stochastic tensor is a nonnegative tensor in which the sum of all entries on each plane (i.e., two free indices) is equal to 1. In enumerating extreme points of the polytopes of line- and plane-stochastic tensors of order 3 and dimension $n$, we consider the approach by linear optimization and present new lower and upper bounds. We also study the coefficient matrices that define the polytopes.
We consider the problem of sampling from a density of the form $p(x) propto exp(-f(x)- g(x))$, where $f: mathbb{R}^d rightarrow mathbb{R}$ is a smooth and strongly convex function and $g: mathbb{R}^d rightarrow mathbb{R}$ is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis-Hastings framework, and prove that it mixes to within TV distance $varepsilon$ of the target density in at most $O(d log (d/varepsilon))$ iterations. This guarantee extends previous results on sampling from distributions with smooth log densities ($g = 0$) to the more general composite non-smooth case, with the same mixing time up to a multiple of the condition number. Our method is based on a novel proximal-based proposal distribution that can be efficiently computed for a large class of non-smooth functions $g$.
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Let $K subset R^d$ be a smooth convex set and let $P_la$ be a Poisson point process on $R^d$ of intensity $la$. The convex hull of $P_la cap K$ is a random convex polytope $K_la$. As $la to infty$, we show that the variance of the number of $k$-dimensional faces of $K_la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.
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