On computable self-embeddings of computable linear orderings

AbstractWe solve a longstanding question of Rosenstein, and make progress toward solving a long-standing open problem in the area of computable linear orderings by showing that every computable η-like linear ordering without an infinite strongly η-like interval has a computable copy without nontrivial computable self-embedding.The precise characterization of those computable linear orderings which have computable copies without nontrivial computable self-embedding remains open.

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ALGEBRAIC LINEAR ORDERINGS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111008155 ◽

2011 ◽

Vol 22 (02) ◽

pp. 491-515 ◽

Cited By ~ 10

Author(s):

S. L. BLOOM ◽

Z. ÉSIK

Keyword(s):

Order Type ◽

Linear Ordering ◽

Type Result ◽

First Order ◽

Linear Orderings ◽

Context Free Language ◽

Categorical Algebra ◽

Context Free ◽

Free Language

An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

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Two further combinatorial theorems equivalent to the 1-consistency of peano arithmetic

Journal of Symbolic Logic ◽

10.2307/2273672 ◽

1983 ◽

Vol 48 (4) ◽

pp. 1090-1104 ◽

Cited By ~ 5

Author(s):

Peter Clote ◽

Kenneth Mcaloon

Keyword(s):

Peano Arithmetic ◽

Infinite Subset ◽

Linear Ordering ◽

Tree Property ◽

Preceding Theorem ◽

Linear Orderings ◽

Finite Set ◽

The One ◽

König’S Lemma

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.

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Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Journal of Geometric Analysis ◽

10.1007/s12220-021-00724-y ◽

2021 ◽

Author(s):

Bin Liu ◽

Jouni Rättyä ◽

Fanglei Wu

Keyword(s):

Bergman Space ◽

Open Problem ◽

Lebesgue Space ◽

Composition Operators ◽

Bergman Spaces ◽

Weighted Bergman Space ◽

Carleson Measures ◽

Doubling Weights ◽

The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

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Highly precise characterization of the hydration state upon thermal denaturation of human serum albumin using a 65 GHz dielectric sensor

Physical Chemistry Chemical Physics ◽

10.1039/d0cp02265a ◽

2020 ◽

Vol 22 (35) ◽

pp. 19468-19479 ◽

Cited By ~ 1

Author(s):

Keiichiro Shiraga ◽

Mako Urabe ◽

Takeshi Matsui ◽

Shojiro Kikuchi ◽

Yuichi Ogawa

Keyword(s):

Human Serum Albumin ◽

Serum Albumin ◽

Human Serum ◽

Thermal Denaturation ◽

Hydration Water ◽

Biological Functions ◽

Dielectric Sensor ◽

Hydration State ◽

Precise Characterization

The biological functions of proteins depend on harmonization with hydration water surrounding them.

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Distorted chiolite crystal structures of α-Na5M3F14 (M=Cr,Fe,Ga) studied by X-ray powder diffraction

Powder Diffraction ◽

10.1154/1.1556990 ◽

2003 ◽

Vol 18 (2) ◽

pp. 128-134 ◽

Cited By ~ 4

Author(s):

A. Le Bail ◽

A.-M. Mercier

Keyword(s):

Crystal Structures ◽

Powder Diffraction ◽

Room Temperature ◽

Rietveld Refinements ◽

Space Group ◽

X Ray ◽

Precise Characterization

The crystal structures of the chiolite-related room temperature phases α-Na5M3F14 (MIII=Cr,Fe,Ga) are determined. For all of them, the space group is P21/n, Z=2; a=10.5096(3) Å, b=7.2253(2) Å, c=7.2713(2) Å, β=90.6753(7)° (M=Cr); a=10.4342(7) Å, b=7.3418(6) Å, c=7.4023(6) Å, β=90.799(5)° (M=Fe), and a=10.4052(1) Å, b=7.2251(1) Å, c=7.2689(1), β=90.6640(4)° (M=Ga). Rietveld refinements produce final RF factors 0.036, 0.033, and 0.035, and RWP factors, 0.125, 0.116, and 0.096, for MIII=Cr, Fe, and Ga, respectively. The MF6 polyhedra in the defective isolated perovskite-like layers deviate very few from perfect octahedra. Subtle octahedra tiltings lead to the symmetry decrease from the P4/mnc space group adopted by the Na5Al3F14 chiolite aristotype to the P21/n space group adopted by the title series. Facile twinning precluded till now the precise characterization of these compounds.

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Simple and Precise Characterization of Differential Modal Group Delay arising in Few-Mode Fiber

Optics Letters ◽

10.1364/ol.423950 ◽

2021 ◽

Author(s):

Zhe Fu ◽

Junjie Zhang ◽

Ziheng Zhang ◽

Songnian Fu ◽

Yuwen Qin ◽

...

Keyword(s):

Group Delay ◽

Precise Characterization ◽

Modal Group

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Thin-Film Elemental Analyses for Precise Characterization of Minerals

ACS Symposium Series - Spectroscopic Characterization of Minerals and Their Surfaces ◽

10.1021/bk-1990-0415.ch002 ◽

1990 ◽

pp. 32-53 ◽

Cited By ~ 2

Author(s):

Ian D. R. Mackinnon

Keyword(s):

Thin Film ◽

Elemental Analyses ◽

Precise Characterization

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A Model of the Real Numbers

Canadian Mathematical Bulletin ◽

10.4153/cmb-1963-022-0 ◽

1963 ◽

Vol 6 (2) ◽

pp. 239-255

Author(s):

Stanton M. Trott

Keyword(s):

Irrational Number ◽

Additive Group ◽

Linear Ordering ◽

Real Numbers ◽

Ordered Group ◽

The Real ◽

Linear Orderings ◽

Positive Elements ◽

The Law ◽

Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

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