scholarly journals Efficient Covering of Thin Convex Domains Using Congruent Discs

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3056
Author(s):  
Shai Gul ◽  
Reuven Cohen
Keyword(s):  
Hexagonal Lattice ◽  
Lattice Points ◽  
Thin Domains ◽  
Convex Domains ◽  
Asymptotically Optimal ◽  
Single Row

We present efficient strategies for covering classes of thin domains in the plane using unit discs. We start with efficient covering of narrow domains using a single row of covering discs. We then move to efficient covering of general rectangles by discs centered at the lattice points of an irregular hexagonal lattice. This optimization uses a lattice that leads to a covering using a small number of discs. We compare the bounds on the covering using the presented strategies to the bounds obtained from the standard honeycomb covering, which is asymptotically optimal for fat domains, and show the improvement for thin domains.

scholarly journals Some Inequalities Involving Perimeter and Torsional Rigidity

2020 ◽  
Author(s):  
Luca Briani ◽  
Giuseppe Buttazzo ◽  
Francesca Prinari
Keyword(s):  
Lebesgue Measure ◽  
Torsional Rigidity ◽  
Thin Domains ◽  
Convex Domains

Abstract We consider shape functionals of the form $$F_q(\Omega )=P(\Omega )T^q(\Omega )$$ F q ( Ω ) = P ( Ω ) T q ( Ω ) on the class of open sets of prescribed Lebesgue measure. Here $$q>0$$ q > 0 is fixed, $$P(\Omega )$$ P ( Ω ) denotes the perimeter of $$\Omega $$ Ω and $$T(\Omega )$$ T ( Ω ) is the torsional rigidity of $$\Omega $$ Ω . The minimization and maximization of $$F_q(\Omega )$$ F q ( Ω ) is considered on various classes of admissible domains $$\Omega $$ Ω : in the class $$\mathcal {A}_{all}$$ A all of all domains, in the class $$\mathcal {A}_{convex}$$ A convex of convex domains, and in the class $$\mathcal {A}_{thin}$$ A thin of thin domains.

Geometry of the gauss map and lattice points in convex domains

Mathematika ◽  
2001 ◽  
Vol 48 (1-2) ◽  
pp. 107-117 ◽  
Author(s):  
L. Brandolini ◽  
L. Colzani ◽  
A. Iosevich ◽  
A. Podkorytov ◽  
G. Travaglini
Keyword(s):  
Gauss Map ◽  
Lattice Points ◽  
Convex Domains

scholarly journals Random fluctuations of convex domains and lattice points

1999 ◽  
Vol 127 (10) ◽  
pp. 2981-2985
Author(s):  
Alex Iosevich ◽  
Kimberly K. J. Kinateder
Keyword(s):  
Lattice Points ◽  
Convex Domains ◽  
Random Fluctuations

Lattice Points in Planar Convex Domains

2004 ◽  
Vol 143 (2) ◽  
pp. 145-162 ◽  
Author(s):  
Ekkehard Kr�tzel
Keyword(s):  
Lattice Points ◽  
Convex Domains ◽  
Planar Convex

Crystal Structures of Long-Period Stacking-Ordered Phases in the Mg-TM-RE Ternary Systems

2013 ◽  
Vol 1516 ◽  
pp. 291-302 ◽  
Author(s):  
Kyosuke Kishida ◽  
Hideyuki Yokobayashi ◽  
Atsushi Inoue ◽  
Haruyuki Inui
Keyword(s):  
Crystal Structures ◽  
Hexagonal Lattice ◽  
Ternary Systems ◽  
Dark Field ◽  
Lattice Points ◽  
Atomic Arrangement ◽  
Lpso Phase ◽  
Long Period ◽  
The Difference ◽  
Image Position

ABSTRACTCrystal structures of long-period stacking-ordered (LPSO) phases in the Mg-TM (transition-metal)-RE(rare-earth) systems were investigated by atomic resolution high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM). The 18R-type LPSO phase is constructed by stacking 6-layer structural blocks, each of which contains four consecutive close-packed planes enriched with TM and RE atoms. Formation of the TM6RE8 clusters with the L12 type atomic arrangement is commonly observed in both Mg-Al-Gd and Mg-Zn-Y LPSO phases. The difference between the crystal structures of Mg-Al-Gd and Mg-Zn-Y LPSO phases can be interpreted as the difference in the in-plane ordering of the TM6RE8 clusters in the structural block. The Mg-Al-Gd LPSO phase exhibits a long-range in-plane ordering of Gd and Al, which can be described by the periodic arrangement of the Al6Gd8 clusters with the L12 type atomic arrangement on lattice points of a two-dimensional 2$\sqrt 3 $aMg × 2$\sqrt 3 $aMg primitive hexagonal lattice, although the LPSO phase in the Zn/Y-poor Mg-Zn-Y alloys exhibits a shortrange in-plane ordering of the Zn6Y8 clusters.

scholarly journals Lattice points in rotated convex domains

2015 ◽  
Vol 31 (2) ◽  
pp. 411-438 ◽  
Author(s):  
Jingwei Guo
Keyword(s):  
Lattice Points ◽  
Convex Domains

scholarly journals On relations between principal eigenvalue and torsional rigidity

2020 ◽  
pp. 2050093
Author(s):  
Michiel van den Berg ◽  
Giuseppe Buttazzo ◽  
Aldo Pratelli
Keyword(s):  
Optimization Problem ◽  
Torsional Rigidity ◽  
Principal Eigenvalue ◽  
Higher Dimensions ◽  
First Eigenvalue ◽  
Thin Domains ◽  
Convex Domains ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Dimension One

We consider the problem of minimizing or maximizing the quantity [Formula: see text] on the class of open sets of prescribed Lebesgue measure. Here [Formula: see text] is fixed, [Formula: see text] denotes the first eigenvalue of the Dirichlet Laplacian on [Formula: see text], while [Formula: see text] is the torsional rigidity of [Formula: see text]. The optimization problem above is considered in the class of all domains [Formula: see text], in the class of convex domains [Formula: see text], and in the class of thin domains. The full Blaschke–Santaló diagram for [Formula: see text] and [Formula: see text] is obtained in dimension one, while for higher dimensions we provide some bounds.

scholarly journals Stretching Convex Domains to Capture Many Lattice Points

2018 ◽  
Vol 2020 (10) ◽  
pp. 2918-2951 ◽  
Author(s):  
Nicholas F Marshall
Keyword(s):  
Fourier Analysis ◽  
Convex Domain ◽  
Large Volume ◽  
Lattice Points ◽  
Convex Domains ◽  
Coordinate Hyperplane ◽  
Analysis Techniques ◽  
Volume Limit ◽  
Gauss Circle Problem ◽  
Large Volume Limit

Abstract We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant 1. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the (d − 1)-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes and Freitas, van den Berg, Bucur and Gittins, Ariturk and Laugesen, van den Berg and Gittins, and Gittins and Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =\pi r^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.

scholarly journals Quantum Transport in a Crystal with Short-Range Interactions: The Boltzmann–Grad Limit

2021 ◽  
Vol 184 (2) ◽  
Author(s):  
Jory Griffin ◽  
Jens Marklof
Keyword(s):  
Quantum Transport ◽  
Quantum Chaos ◽  
Quantum Dynamics ◽  
Short Range ◽  
Lorentz Gas ◽  
Lattice Points ◽  
Linear Boltzmann Equation ◽  
Thin Domains ◽  
Grad Limit ◽  
Random Flight

AbstractWe study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann–Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry–Tabor conjecture in quantum chaos.

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