scholarly journals Local Complexity of Delone Sets and Crystallinity

2002 ◽  
Vol 45 (4) ◽  
pp. 634-652 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Peter A. B. Pleasants
Keyword(s):  
Growth Rates ◽  
Linear Growth ◽  
Closed Ball ◽  
Covering Radius ◽  
Ideal Crystal ◽  
Minimum Radius ◽  
Local Complexity ◽  
Delone Set ◽  
Delone Sets

AbstractThis paper characterizes when a Delone set X in is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X, let NX(T) count the number of translation-inequivalent patches of radius T in X and let MX(T) be the minimum radius such that every closed ball of radius MX(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to X being an ideal crystal.Explicitly, for NX(T), if R is the covering radius of X then either NX(T) is bounded or NX(T) ≥ T/2R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions.For MX(T), either MX(T) is bounded or MX(T) ≥ T/3 for all T > 0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has MX(T) ≥ c(n)T for all T > 0, for a certain constant c(n) which depends on the dimension n of X and is > 1/3 when n > 1.

scholarly journals Equicontinuous Delone Dynamical Systems

2013 ◽  
Vol 65 (1) ◽  
pp. 149-170 ◽  
Author(s):  
Johannes Kellendonk ◽  
Daniel Lenz
Keyword(s):  
Dynamical Systems ◽  
The Other ◽  
Almost Periodic ◽  
Other Hand ◽  
Local Complexity ◽  
Dirac Comb ◽  
Delone Set ◽  
Delone Sets

AbstractWe characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with finite local complexity, the only equicontinuous systems are then shown to be the crystallographic ones. On the other hand, within the class without finite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems that are not crystallographic. Our results solve the problem posed by Lagarias as to whether a Delone set whose Dirac comb is strongly almost periodic must be crystallographic.

scholarly journals Number of bounded distance equivalence classes in hulls of repetitive Delone sets

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber ◽  
Lorenzo Sadun
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Bounded Distance ◽  
Local Complexity ◽  
Delone Set ◽  
Delone Sets ◽  
Uniformly Bounded

<p style='text-indent:20px;'>Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.</p>

scholarly journals A dichotomy for bounded displacement equivalence of Delone sets

2021 ◽  
pp. 1-18
Author(s):  
YOTAM SMILANSKY ◽  
YAAR SOLOMON
Keyword(s):  
Compact Space ◽  
Equivalence Classes ◽  
Natural Topology ◽  
Fell Topology ◽  
Local Complexity ◽  
Substitution Tilings ◽  
Local Topology ◽  
Delone Sets

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

scholarly journals On substitution tilings and Delone sets without finite local complexity

2019 ◽  
Vol 39 (6) ◽  
pp. 3149-3177 ◽  
Author(s):  
Jeong-Yup Lee ◽  
◽  
Boris Solomyak ◽  
◽  
Keyword(s):  
Local Complexity ◽  
Substitution Tilings ◽  
Delone Sets

COMPARATIVE STUDIES OF NORTH AMERICAN AND EUROPEAN CULTURES OF THE ROOT ROT FUNGUS, FOMES ANNOSUS (FR.) COOKE

1955 ◽  
Vol 33 (5) ◽  
pp. 416-428 ◽  
Author(s):  
D. E. Etheridge
Keyword(s):  
Growth Rate ◽  
North American ◽  
Growth Rates ◽  
Root Rot ◽  
Linear Growth ◽  
Morphological Characteristics ◽  
Dry Weight ◽  
Cellulolytic Activity ◽  
Ph Optimum ◽  
Two Cultures

Cultures of Fames annosus originating in Europe could not be distinguished from those originating in North America either by colony appearance, growth rate, pH optimum, or cellulolytic activity. Three growth rate types on 2.5% malt agar were recognized and these are ascribed to individual variation rather than to host or geographical influences. Successive subculturing produced variants that fell into three growth classes. Half of the isolates displayed spontaneous, but reversible, changes in growth rate and colony appearance during subculturing and this is discussed from the standpoint of genetical and environmental influences. Cultures displaying different morphological characteristics and linear-growth rates differed little metabolically; each had a similar pH optimum ranging from 4.6 to 5.5, and each proved capable of altering the initial acidity of the medium to a reaction which was more suitable for growth. Two cultures were characterized by double pH optima at 4.6 and 5.5. Cultures having different linear-growth rates produced about the same dry-weight of mycelium on a cellulose substrate in a semisynthetic nutrient solution. On the basis of a statistical analysis of cellulose utilization by representative isolates it was impossible to distinguish between North American and European cultures.

Abstract 13513: Serial CT Surveillance of Abdominal Aortic Aneurysm Diameter Demonstrates Predominantly Linear Growth (N-TA 3 CT)

Circulation ◽  
2020 ◽  
Vol 142 (Suppl_3) ◽  
Author(s):  
Sydney Olson ◽  
Marniker Wijesinha ◽  
Annalise Panthofer ◽  
William Blackwelder ◽  
Gilbert R Upchurch ◽  
...  
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Linear Growth ◽  
Growth Patterns ◽  
Aortic Aneurysms ◽  
Transverse Diameter ◽  
Invasive Treatment ◽  
Abdominal Aortic ◽  
Risk Of Rupture ◽  
Serial Ct

Objective: Small abdominal aortic aneurysms (AAAs) have a low risk of rupture. Intervention is indicated when diameters exceed established thresholds. This study assessed the growth rates and patterns of AAAs over 2 years as documented on serial CT scans from the Non-Invasive Treatment of AAA Clinical Trial. Methods: 254 patients, 35 females with baseline AAA maximum transverse diameter (MTD) between 3.5-4.5 cm and 219 males with baseline MTD 3.5-5.0 cm, were included in this study. Linear regressions and segmental growth rates were used to model growth rates and patterns. Results: The yearly growth rates of AAA MTDs had a median of 0.17 cm/yr and mean of 0.19 cm/yr ± 0.14 (Figure 1). 10% of AAA displayed minimal to no growth (< 0.05 cm/yr), 62% low growth (0.05-0.25 cm/yr), 28% high growth (> 0.25 cm/yr). Baseline AAA diameter accounted for only 5.4% of growth rate variance (P<0.001, R 2 0.05). Most AAAs displayed linear growth (70%); large variations in interval growth rates occurred infrequently (3% staccato growth, 4% exponential growth); a minority of subjects’ growth patterns were not clearly classifiable (11% indeterminate-not growing, 12% indeterminate-growing) (Figure 2). No patients with baseline MTD < 4.25 cm exceeded sex-specific repair thresholds (males 0 / 92, [95% CI, 0.00-0.06]; females 0 / 25 [95% CI, 0.00-0.25]) in the course of follow-up for as long as two years. Conclusions: The majority of small AAAs exhibit linear growth; large intra-patient growth rate variations were infrequently observed over 2 years. AAA < 4.25 cm can be followed with a CT scan in 2 years with little chance of exceeding interventional MTD thresholds of 5.5 cm for men.

Preliminary Observations on the Growth of Transplanted Peltigera Canina Under Semi-Natural Conditions

1997 ◽  
Vol 29 (1) ◽  
pp. 91-96 ◽  
Author(s):  
M. Webster ◽  
D. H. Brown
Keyword(s):  
Growth Rates ◽  
Linear Growth ◽  
Natural Conditions ◽  
Peltigera Canina

AbstractPeltigera canina thalli have been successfully transplanted onto soil in a garden and in flowerpots. Garden samples showed marked seasonality and achieved growth rates of 6·4 cm per year. Pot-grown samples showed variation in the growth of individual thallus lobes and established that, under different soil hydration regimes, permanently hydrated thalli could sustain considerable linear growth rates for at least 140 days.

Differential thermal analyses of solid-solid transitions in AgNO3 under hydrostatic pressures to 0.7 Gpa

1981 ◽  
Vol 34 (9) ◽  
pp. 1845 ◽  
Author(s):  
LH Cohen ◽  
Jr W Klement
Keyword(s):  
High Temperature ◽  
Growth Rates ◽  
Linear Growth ◽  
Thermal Analyses ◽  
Quantitative Description ◽  
Initial Slope ◽  
Transition Temperatures ◽  
Cooling Rates ◽  
Zero Rate ◽  
Show Temperature Dependence

Differential thermal analyses have located the transitions between low-temperature I and high-temperature II phases in AgNO3 powder and single crystals under hydrostatic pressures </~0.7 GPa, with heating/cooling rates in the range 0.1-1.5 K s-1. Isobaric transition temperatures plot linearly against heating/cooling rates and, extrapolated to zero rate, show hystereses between I → II and II → I transition temperatures which are comparable with the 'regions of indifference' of Bridgman's isothermal experiments. The present results suggest an initial slope of -0.090 μK Pa-1 and zero initial curvature for the I-II phase boundary. Greater hystereses are observed for the I-II transitions near intersection with the II-II' λ transition at </~0.7 GPa. An improved, quantitative description is achieved for the Kennedy-Schultz data on the linear growth rates of II,II? → I at 0.1 MPa. For II' (with ordered NO3- ions) → I the growth rates show temperature dependence markedly different than growth rates for II(with disordered NO3-) → 1 and I → II.

Linear growth rates of ice crystals grown from the vapor phase

1972 ◽  
Vol 12 (1) ◽  
pp. 21-31 ◽  
Author(s):  
D. Lamb ◽  
W.D. Scott
Keyword(s):  
Vapor Phase ◽  
Growth Rates ◽  
Linear Growth ◽  
Ice Crystals
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