Differentiable structures of central Cantor sets

1997 ◽  
Vol 17 (5) ◽  
pp. 1027-1042 ◽  
Author(s):  
RODRIGO BAMÓN ◽  
CARLOS G. MOREIRA ◽  
SERGIO PLAZA ◽  
JAIME VERA
Keyword(s):  
Real Line ◽  
Fractal Dimensions ◽  
Complete Characterization ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Diffeomorphisms ◽  
The Real ◽  
Global And Local

Central Cantor sets form a class of symmetric Cantor sets of the real line. Here we give a complete characterization of the $C^{k + \alpha}$ regularity of these Cantor sets. We also give a classification of central Cantor sets up to global and local diffeomorphisms. Examples of central Cantor sets with special dynamical and measure-theoretical properties are also provided. Finally, we calculate the fractal dimensions of an arbitrary central Cantor set.

scholarly journals The Underlying Order Induced by Orthogonality and the Quantum Speed Limit

2021 ◽  
Vol 3 (3) ◽  
pp. 376-388
Author(s):  
Francisco J. Sevilla ◽  
Andrea Valdés-Hernández ◽  
Alan J. Barrios
Keyword(s):  
Energy Spectrum ◽  
Energy Distribution ◽  
Finite Time ◽  
Complete Characterization ◽  
Comprehensive Analysis ◽  
Orthogonality Condition ◽  
Speed Limit ◽  
Physical Quantities

We perform a comprehensive analysis of the set of parameters {ri} that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time τ, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between τ and the energy spectrum and allowing the classification of {ri} into families organized in a 2-simplex, δ2. Furthermore, the states determined by {ri} are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those ris in δ2 correspondent to states whose orthogonality time is limited by the Mandelstam–Tamm bound from those restricted by the Margolus–Levitin one. Our results offer a complete characterization of the physical quantities that become relevant in both the preparation and study of the dynamics of three-level states evolving towards orthogonality.

Kenoplumbomicrolite, (Pb,□)2Ta2O6[□,(OH),O], a new mineral from Ploskaya, Kola Peninsula, Russia

2018 ◽  
Vol 82 (5) ◽  
pp. 1049-1055 ◽  
Author(s):  
Daniel Atencio ◽  
Marcelo B. Andrade ◽  
Luca Bindi ◽  
Paola Bonazzi ◽  
Matteo Zoppi ◽  
...  
Keyword(s):  
Kola Peninsula ◽  
Northern Region ◽  
Complete Characterization ◽  
New Mineral ◽  
X Ray Diffraction ◽  
International Mineralogical Association ◽  
Cell Parameters ◽  
Mohs Hardness

ABSTRACTThis study presents a complete characterization of kenoplumbomicrolite, (Pb,□)2Ta2O6[□,(OH),O], occurring in an amazonite pegmatite from Ploskaya Mountain, Western Keivy Massif, Kola Peninsula, Murmanskaja Oblast, Northern Region, Russia.Kenoplumbomicrolite occurs in yellowish brown octahedral, cuboctahedral and massive crystals, up to 20 cm, has a white streak, a greasy lustre and is translucent. The Mohs hardness is ~6. Attempts to measure density (7.310–7.832 g/cm3) were affected by the ubiquitous presence of uraninite inclusions. Reflectance values were measured in air and immersed in oil. Kenoplumbocrolite is optically isotropic. The empirical formula is (Pb1.30□0.30Ca0.29Na0.08U0.03)Σ2.00(Ta0.82Nb0.62Si0.23Sn4+0.15Ti0.07Fe3+0.10Al0.01)Σ2.00O6[□0.52(OH)0.25O0.23]Σ1.00 (from the crystal used for the structural study) and (Pb1.33□0.66Mn0.01)Σ2.00(Ta0.87Nb0.72Sn4+0.18Fe3+0.11W0.08Ti0.04)Σ2.00O6[□0.80(OH)0.10O0.10]Σ1.00 (average including additional fragments). The mineral is cubic, space group Fd$\overline 3 $m. The unit-cell parameters refined from powder X-ray diffraction data are a = 10.575(2) Å and V = 1182.6(8) Å3, which are in accord with those obtained previously from a single crystal of a = 10.571(1) Å, V = 1181.3(2) Å3 and Z = 8. The mineral description and its name have been approved by the Commission on New Minerals, Nomenclature and Classification of the International Mineralogical Association (IMA2015-007a).

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

Quadratic-monomial generated domains from mixed signed, directed graphs

2019 ◽  
Vol 29 (02) ◽  
pp. 279-308
Author(s):  
Michael A. Burr ◽  
Drew J. Lipman
Keyword(s):  
Directed Graphs ◽  
Complete Characterization ◽  
Combinatorial Structure ◽  
Signed Directed Graph ◽  
Odd Cycle ◽  
Combinatorial Classification ◽  
Special Case ◽  
Text Condition

Determining whether an arbitrary subring [Formula: see text] of [Formula: see text] is a normal or Cohen-Macaulay domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. We provide a complete characterization of the normality, normalizations, and Serre’s [Formula: see text] condition for quadratic-monomial generated domains. For a quadratic-monomial generated domain [Formula: see text], we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph [Formula: see text], i.e. a graph with signed edges and directed edges. We classify the normality and the normalizations of such rings in terms of a generalization of the combinatorial odd cycle condition on [Formula: see text]. We also generalize and simplify a combinatorial classification of Serre’s [Formula: see text] condition for such rings and construct non-Cohen–Macaulay rings.

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

Lebesque measure zero subsets of the real line and an infinite game

1996 ◽  
Vol 61 (1) ◽  
pp. 246-249 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
Measure Zero ◽  
Cardinal Number ◽  
Minimal Cardinality ◽  
Combinatorial Characterization ◽  
The Real ◽  
Lebesque Measure ◽  
The Ideal

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

FIRST BETTI NUMBERS OF ABELIAN COVERINGS OF THE COMPLEX PROJECTIVE PLANE BRANCHED OVER LINE CONFIGURATIONS

2000 ◽  
Vol 09 (02) ◽  
pp. 271-284 ◽  
Author(s):  
IKUO TAYAMA
Keyword(s):  
Projective Plane ◽  
Real Line ◽  
General Position ◽  
Betti Numbers ◽  
Complex Projective Plane ◽  
The Real ◽  
Complex Projective

We prove the following results concerning the first Betti numbers of abelian coverings of CP2 branched over line configurations: (1) An estimate of the first Betti numbers. (2) A characterization of central and general position line configurations in terms of the first Betti numbers of abelian coverings. (3) The first Betti numbers of the abelian coverings of the real line configurations up to 7 components.

Subgradient Criteria for Monotonicity, The Lipschitz Condition, and Convexity

1993 ◽  
Vol 45 (6) ◽  
pp. 1167-1183 ◽  
Author(s):  
F. H. Clarke ◽  
R. J. Stern ◽  
P. R. Wolenski
Keyword(s):  
Hilbert Space ◽  
Lipschitz Condition ◽  
Real Line ◽  
Nonsmooth Analysis ◽  
Real Hilbert Space ◽  
Lower Semicontinuous ◽  
Multidimensional Case ◽  
The Real

AbstractLet ƒ H → (—∞,∞] be lower semicontinuous, where H is a real Hilbert space. An approach based upon nonsmooth analysis and optimization is used in order to characterize monotonicity of ƒ with respect to a cone, as well as Lipschitz behavior and constancy. The results, which involve hypotheses on the proximal subgradient ∂ πƒ, specialize on the real line to yield classical characterizations of these properties in terms of the Dini derivate. They also give new extensions of these results to the multidimensional case. A new proof of a known characterization of convexity in terms of proximal subgradient monotonicity is also given.

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