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pro vyhledávání: '"Garber, Alexey"'
The local angle property of the (order-$1$) Delaunay triangulations of a generic set in $\mathbb{R}^2$ asserts that the sum of two angles opposite a common edge is less than $\pi$. This paper extends this property to higher order and uses it to gener
Externí odkaz:
http://arxiv.org/abs/2310.18238
Lattices and periodic point sets are well known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces of lattices
Externí odkaz:
http://arxiv.org/abs/2310.07594
Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the interstice
Externí odkaz:
http://arxiv.org/abs/2306.11127
Autor:
Bajo, Esme, Davis, Robert, De Loera, Jesús A., Garber, Alexey, Mora, Sofía Garzón, Jochemko, Katharina, Yu, Josephine
We generalize R. P. Stanley's celebrated theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for we
Externí odkaz:
http://arxiv.org/abs/2303.09614
We study flips in hypertriangulations of planar points sets. Here a level-$k$ hypertriangulation of $n$ points in the planes is a subdivision induced by the projection of a $k$-hypersimplex, which is the convex hull of the barycenters of the $(k-1)$-
Externí odkaz:
http://arxiv.org/abs/2212.11380
In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting function i
Externí odkaz:
http://arxiv.org/abs/2211.02548
For any $\lambda>2$, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor $\lambda$. In particular, we obtain the first class of examples of substitutive systems with transcendental infla
Externí odkaz:
http://arxiv.org/abs/2208.01327
Autor:
Edelsbrunner, Herbert, Garber, Alexey, Ghafari, Mohadese, Heiss, Teresa, Saghafian, Morteza, Wintraecken, Mathijs
Publikováno v:
SIAM Journal on Discrete Mathematics 38.2 (2024): 1784-1807
For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which $\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $
Externí odkaz:
http://arxiv.org/abs/2204.01077
Publikováno v:
Discrete & Computational Geometry 72.1 (2024): 29-48
For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum an
Externí odkaz:
http://arxiv.org/abs/2204.01076
Publikováno v:
In Indagationes Mathematicae September 2024 35(5):890-913