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Obtuse Lattice Bases
[article]

2020
*
arXiv
*
pre-print

We define a class of

arXiv:2009.00384v2
fatcat:b7gi53l2bjg47kzz66i673kcyy
*bases*called*obtuse**bases*and show that any*lattice*basis can be transformed to an*obtuse*basis. A shortest vector 𝐬 can be written as 𝐬=v_1𝐛_1+...+v_n𝐛_n where 𝐛_1,... ... This property of*obtuse**bases*makes the*lattice*enumeration algorithm for finding a shortest vector exponentially faster. ... In this paper, we present our work on a class of*lattice**bases*called*Obtuse**Bases*, introduced in [4] . ...##
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Faster Lattice Enumeration
[article]

2019
*
arXiv
*
pre-print

We define a class of

arXiv:1912.01781v1
fatcat:ocn2p37qm5bihkfsuwossx7rtq
*bases*called*obtuse**bases*and show that any*lattice*basis can be transformed to an*obtuse*basis in O(n^4) time. A shortest vector s can be written as v_1b_1+... ... Moreover, extreme pruning, the current fastest algorithm for*lattice*enumeration, can be run on an*obtuse*basis. ... New*lattice*reduction In this section, we define a special class of basis vectors called*obtuse**bases*and show that it is possible to transform any basis to an*obtuse*basis in polynomialtime. ...##
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A complete isometry classification of 3-dimensional lattices
[article]

2022
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arXiv
*
pre-print

A periodic

arXiv:2201.10543v1
fatcat:6iwgon7fajcuhppbn23h6z66xi
*lattice*in Euclidean 3-space is the infinite set of all integer linear combinations of basis vectors. Any*lattice*can be generated by infinitely many different*bases*. ... This paper completes a continuous classification of 3-dimensional*lattices*up to Euclidean isometry (or congruence) and similarity (with uniform scaling).The new homogeneous invariants are uniquely ordered ...*lattice*representations by their*bases*. ...##
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A complete and continuous map of the Lattice Isometry Space for all 3-dimensional lattices
[article]

2021
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arXiv
*
pre-print

The resulting space LISP consists of infinitely many isometry classes of

arXiv:2109.11538v1
fatcat:yuwrvimsajfmta4laggh5ve5xi
*lattices*. ... A periodic 3-dimensional*lattice*is an infinite set of all integer linear combinations of basis vectors in Euclidean 3-space. ... Any fixed*lattice*Λ ⊂ R 3 has infinitely many (super)*bases*but only a few*obtuse*superbases, maximum eight for rectangular Voronoi domains. ...##
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Communication-Efficient Search for an Approximate Closest Lattice Point
[article]

2018
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arXiv
*
pre-print

*Based*on available parameterizations of reduced

*bases*, we determine the error probability of the nearest plane algorithm for two dimensional

*lattices*analytically, and present a computational error estimation ... Assuming a triangular special basis for the

*lattice*, we develop communication-efficient protocols for computing the approximate

*lattice*point and determine the communication cost for

*lattices*of dimension ... We will introduce next two types of special

*bases*we will work closely in this paper: Minkowski-reduced basis and

*obtuse*superbase. ...

##
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6-Fold Quasiperiodic Tilings With Two Diamond Shapes
[article]

2015
*
arXiv
*
pre-print

They are distinguished by labeling one as an acute diamond with a

arXiv:1503.03018v1
fatcat:sfjaa7ezcrforovix3kv6q66sy
*base*angle of 60 degrees, the other one as an*obtuse*diamond with a*base*angle of 120 degrees. ... Similarly, three*obtuse*diamonds can be matched with 3-fold rotational symmetry to form a hexagon among other possibilities. ... They are distinguished by labeling one as an acute diamond with a*base*angle of 60 degrees, the other one as an*obtuse*diamond with a*base*angle of 120 degrees. ...##
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On Communication for Distributed Babai Point Computation
[article]

2020
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arXiv
*
pre-print

In such cases, a distributed algorithm for finding the approximate nearest

arXiv:2008.13075v1
fatcat:c2fythtagfhi3itb64smdpwogu
*lattice*point is sufficient for finding the nearest*lattice*point. ... Our investigations suggest that for uniform distributions, the error probability becomes large with the dimension of the*lattice*, for*lattices*with good packing densities. ... Special*Bases*: Minkowski and*Obtuse*Superbase A basis {v 1 , v 2 , ..., v n } of a*lattice*Λ ⊂ R n is said to be Minkowski-reduced if v j , j = 1, . . . , n, is such that v j ≤ v , for any v such that ...##
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On Extremal Finite Packings

2002
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Discrete & Computational Geometry
*

We show that in contrast to the classical infinite packing problem, even in the Euclidian plane, the solutions to several finite packing problems are non-

doi:10.1007/s00454-002-0747-6
fatcat:gvp2rskg3vfurishhrp4z25tl4
*lattice*packings if the number of translates is ... Since a linear transformation keeps information on neighborhood,*obtuseness*and looseness, we may think of as I. the integral*lattice*Z 2 , or II. a hexagonal*lattice*, e.g. with*base*a 1 = (1, 0) , a ... Otherwise we could find a non-*obtuse*vertex of P. ...##
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Low-Dimensional Lattices. VI. Voronoi Reduction of Three-Dimensional Lattices

1992
*
Proceedings of the Royal Society A
*

The aim of this paper is to describe how the Voronoi cell of a

doi:10.1098/rspa.1992.0004
fatcat:p6lztifutbgc5ciga5pduvvvy4
*lattice*changes as that*lattice*is continuously varied. ... classification of the three-dimensional*lattices*into five types. ... Let A be an arbitrary three-dimensional*lattice*, with*obtuse*superbase v0, v, V2, V3 and conorms Pij. ...##
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Realizing Corner States in Artificial Crystals Based on Topological Spin Textures
[article]

2019
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arXiv
*
pre-print

Interestingly, we observe corner states at either

arXiv:1910.03956v1
fatcat:nboxzpvy6ngvjlviiv3tdpvjnu
*obtuse*-angled or acute-angled corners, depending on whether the*lattice*boundary has an armchair or zigzag shape. ... the conventional chiral symmetry of bipartite*lattices*is absent. ... corners instead of*obtuse*-angled corners. ...##
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Elastic higher-order topological insulator with topologically protected corner states
[article]

2018
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arXiv
*
pre-print

Compared with the trivial corner modes, the topological ones, immunizing against defects, are robustly localized at the

arXiv:1811.04412v1
fatcat:mispvothszfpnbj6ecwcrlkxkm
*obtuse*-angled but not the acute-angled corners. ... Aiming as this issue, a new mechanism*based*on the modulation of the inter-and intra-cell couplings will be developed. ... phononic*lattice*with 9 cells along the y direction. ...##
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Emergent geometry and duality in the carbon nucleus
[article]

2022
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arXiv
*
pre-print

We find that the well-known but enigmatic Hoyle state is composed of a "bent-arm" or

arXiv:2202.13596v2
fatcat:gu2ms3wcfjeshapagqswd35mc4
*obtuse*triangular arrangement of alpha clusters. ... In this work, we provide the first model-independent tomographic scan of the three-dimensional geometry of the nuclear states of ^12C using the ab initio framework of nuclear*lattice*effective field theory ... or*obtuse*triangle arrangements of alpha clusters. ...##
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Visualization of the effect of additives on the nanostructures of individual bio-inspired calcite crystals

2019
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Chemical Science
*

The acute (À) and

doi:10.1039/c8sc03733g
pmid:30774916
pmcid:PMC6349071
fatcat:2zwvf25xbbh7tpha6255bmgs2y
*obtuse*(+) steps are indicated. The hillock was overgrown with pure "CaCO 3 " growth solution between additives to establish a common*base*. ... The ratio of terrace widths changes upon addition of lysine from 1*obtuse*: 1 acute to 1*obtuse*: 0.22 acute . ...##
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The shapes of Galois quartic fields
[article]

2019
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arXiv
*
pre-print

For C_4-quartic fields, each family is a one-dimensional space of tetragonal

arXiv:1908.03969v1
fatcat:r7mb76474jhozlbzqm2mhaflai
*lattices*and the shapes make up a discrete subset of points in these spaces. ... In the V_4 case, each family is a two-dimensional space of orthorhombic*lattices*and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity ... If we add a*lattice*point in the centre of every*base*, we get a*base*-centered orthorhombic*lattice*(oC). A special case occurs when b = √ 3a: a primitive hexagonal*lattice*(hP ). ...##
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Geometrical folding transitions of the triangular lattice in the face-centred cubic lattice

1997
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Nuclear Physics B
*

We study the folding of the regular two-dimensional triangular

doi:10.1016/s0550-3213(97)00198-3
fatcat:tlmyodylyrb6hakp252hv2nxh4
*lattice*embedded in the regular three-dimensional Face-Centred Cubic*lattice*, a discrete model for the crumpling of membranes. ... A second continuous transition separates this latter phase from a phase of complete folding of the*lattice*on top of a single triangle. ... When embedded in the FCC*lattice*, the links of the triangular*lattice*must belong to one of the three crystalline planes (200), (020) or (002) which cross the octahedra by their three square*bases*. ...
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