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Obtuse Lattice Bases [article]

Kanav Gupta, Mithilesh Kumar, Håvard Raddum
2020 arXiv   pre-print
We define a class of 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] .  ... 
arXiv:2009.00384v2 fatcat:b7gi53l2bjg47kzz66i673kcyy

Faster Lattice Enumeration [article]

Mithilesh Kumar
2019 arXiv   pre-print
We define a class of 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.  ... 
arXiv:1912.01781v1 fatcat:ocn2p37qm5bihkfsuwossx7rtq

A complete isometry classification of 3-dimensional lattices [article]

Vitaliy Kurlin
2022 arXiv   pre-print
A periodic 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.  ... 
arXiv:2201.10543v1 fatcat:6iwgon7fajcuhppbn23h6z66xi

A complete and continuous map of the Lattice Isometry Space for all 3-dimensional lattices [article]

Matthew Bright, Andrew I Cooper, Vitaliy Kurlin
2021 arXiv   pre-print
The resulting space LISP consists of infinitely many isometry classes of 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.  ... 
arXiv:2109.11538v1 fatcat:yuwrvimsajfmta4laggh5ve5xi

Communication-Efficient Search for an Approximate Closest Lattice Point [article]

Maiara F. Bollauf, Vinay A. Vaishampayan, Sueli I. R. Costa
2018 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.  ... 
arXiv:1801.09796v1 fatcat:n32iegnaebacdagigkavpiw6la

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

Theo P. Schaad
2015 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1503.03018v1 fatcat:sfjaa7ezcrforovix3kv6q66sy

On Communication for Distributed Babai Point Computation [article]

Maiara F. Bollauf, Vinay A. Vaishampayan, Sueli I. R. Costa
2020 arXiv   pre-print
In such cases, a distributed algorithm for finding the approximate nearest 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  ... 
arXiv:2008.13075v1 fatcat:c2fythtagfhi3itb64smdpwogu

On Extremal Finite Packings

Schürmann
2002 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-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.  ... 
doi:10.1007/s00454-002-0747-6 fatcat:gvp2rskg3vfurishhrp4z25tl4

Low-Dimensional Lattices. VI. Voronoi Reduction of Three-Dimensional Lattices

J. H. Conway, N. J. A. Sloane
1992 Proceedings of the Royal Society A  
The aim of this paper is to describe how the Voronoi cell of a 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.  ... 
doi:10.1098/rspa.1992.0004 fatcat:p6lztifutbgc5ciga5pduvvvy4

Realizing Corner States in Artificial Crystals Based on Topological Spin Textures [article]

Z.-X. Li, Yunshan Cao, X. R. Wang, Peng Yan
2019 arXiv   pre-print
Interestingly, we observe corner states at either 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.  ... 
arXiv:1910.03956v1 fatcat:nboxzpvy6ngvjlviiv3tdpvjnu

Elastic higher-order topological insulator with topologically protected corner states [article]

Haiyan Fan, Baizhan Xia, Liang Tong, Shengjie Zheng, Dejie Yu
2018 arXiv   pre-print
Compared with the trivial corner modes, the topological ones, immunizing against defects, are robustly localized at the 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.  ... 
arXiv:1811.04412v1 fatcat:mispvothszfpnbj6ecwcrlkxkm

Emergent geometry and duality in the carbon nucleus [article]

Shihang Shen, Timo A. Lähde, Dean Lee, Ulf-G. Meißner
2022 arXiv   pre-print
We find that the well-known but enigmatic Hoyle state is composed of a "bent-arm" or 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.  ... 
arXiv:2202.13596v2 fatcat:gu2ms3wcfjeshapagqswd35mc4

Visualization of the effect of additives on the nanostructures of individual bio-inspired calcite crystals

Johannes Ihli, Jesse N. Clark, Nasima Kanwal, Yi-Yeoun Kim, Mark A. Holden, Ross J. Harder, Chiu C. Tang, Sharon E. Ashbrook, Ian K. Robinson, Fiona C. Meldrum
2019 Chemical Science  
The acute (À) and 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 .  ... 
doi:10.1039/c8sc03733g pmid:30774916 pmcid:PMC6349071 fatcat:2zwvf25xbbh7tpha6255bmgs2y

The shapes of Galois quartic fields [article]

Piper H, Robert Harron
2019 arXiv   pre-print
For C_4-quartic fields, each family is a one-dimensional space of tetragonal 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 ).  ... 
arXiv:1908.03969v1 fatcat:r7mb76474jhozlbzqm2mhaflai

Geometrical folding transitions of the triangular lattice in the face-centred cubic lattice

M. Bowick, O. Golinelli, E. Guitter, S. Mori
1997 Nuclear Physics B  
We study the folding of the regular two-dimensional triangular 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.  ... 
doi:10.1016/s0550-3213(97)00198-3 fatcat:tlmyodylyrb6hakp252hv2nxh4
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