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Graphs of difference operators for p-ary sequences
Abstract For a given prime p and positive integer n, we consider the graph Gn of the difference operator acting on p-ary sequences of length n. We suggest new proofs of some results of V.I. Arnold onExpand
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Symmetries of Monocoronal Tilings
TLDR
We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups. Expand
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The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices.
This paper reports on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. A complete list is obtained of 110 244 affine typesExpand
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On the origin of crystallinity: a lower bound for the regularity radius of Delone sets.
The mathematical conditions for the origin of long-range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that theExpand
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On $${\pi}$$π-Surfaces of Four-Dimensional Parallelohedra
We show that every four-dimensional parallelohedron P satisfies a recently found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi conjecture being true for P. Namely, we showExpand
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The Voronoi Conjecture for Parallelohedra with Simply Connected $$\delta $$δ-Surfaces
TLDR
We show that the Voronoi conjecture is true for parallelohedra with simply connected $$\delta $$δ-surfaces and give another condition on the homology group of the constructed surface. Expand
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Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary
This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence isExpand
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Another Ham Sandwich in the Plane
We show that every two nice measures in the plane can be partitioned into equal halves by translation of an angle from any k-fan when k is odd and in some cases when k is even. We also give someExpand
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On Helly number for crystals and cut-and-project sets
We prove existence of finite Helly numbers for crystals and for cut-and-project sets with convex windows; also we prove exact bound of $k+6$ for the Helly number of a crystal consisting of $k$ copiesExpand
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