On the ranked points of a Π10 set

1989 ◽  
Vol 54 (3) ◽  
pp. 975-991 ◽  
Author(s):  
Douglas Cenzer ◽  
Rick L. Smith
Keyword(s):  
Baire Space ◽  
High Rank ◽  
Joint Work ◽  
Cantor Space ◽  
Closed Set ◽  
Applied Logic ◽  
Rank One ◽  
Author Paper

AbstractThis paper continues joint work of the authors with P. Clote, R. Soare and S. Wainer (Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145–163). An element x of the Cantor space 2ω is said have rank α in the closed set P if x is in Dα(P)/Dα + 1(P), where Dα is the iterated Cantor-Bendixson derivative. The rank of x is defined to be the least α such that x has rank a in some set. The main result of the five-author paper is that for any recursive ordinal λ + n (where λ is a limit and n is finite), there is a point with rank λ + n which is Turing equivalent to O(λ + 2n) All ranked points constructed in that paper are singletons. We now construct a ranked point which is not a singleton. In the previous paper the points of high rank were also of high hyperarithmetic degree. We now construct points with arbitrarily high rank. We also show that every nonrecursive RE point is Turing equivalent to an RE point of rank one and that every nonrecursive point is Turing equivalent to a hyperimmune point of rank one. We relate Clote's notion of the height of a singleton in the Baire space with the notion of rank. Finally, we show that every hyperimmune point x is Turing equivalent to a point which is not ranked.

The weak König lemma and uniform continuity

2008 ◽  
Vol 73 (3) ◽  
pp. 933-939 ◽  
Author(s):  
Josef Berger
Keyword(s):  
Continuous Function ◽  
Uniform Continuity ◽  
Baire Space ◽  
Cantor Space

AbstractWe prove constructively that the weak König lemma and quantifier-free number–number choice imply that every pointwise continuous function from Cantor space into Baire space has a modulus of uniform continuity.

Taking the path computably traveled

2019 ◽  
Vol 29 (6) ◽  
pp. 969-973 ◽  
Author(s):  
Johanna N Y Franklin ◽  
Dan Turetsky
Keyword(s):  
Baire Space ◽  
Cantor Space

Abstract We define a real $A$ to be low for paths in Baire space (or Cantor space) if every $\varPi ^0_1$ class with an $A$-computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.

Comparing the Medvedev and Turing degrees of Π01 classes

2014 ◽  
Vol 25 (8) ◽  
pp. 1649-1668
Author(s):  
TAKAYUKI KIHARA
Keyword(s):  
Turing Degrees ◽  
Cantor Space ◽  
Closed Set ◽  
Medvedev Lattice ◽  
Π01 Classes

Every co-c.e. closed set (Π01 class) in Cantor space is represented by a co-c.e. tree. Our aim is to clarify the interaction between the Medvedev and Muchnik degrees of co-c.e. closed subsets of Cantor space and the Turing degrees of their co-c.e. representations. Among other results, we present the following theorems: if v and w are different c.e. degrees, then the collection of the Medvedev (Muchnik) degrees of all Π01 classes represented by v and the collection represented by w are also different; the ideals generated from such collections are also different; the collections of the Medvedev and Muchnik degrees of all Π01 classes represented by incomplete co-c.e. sets are upward dense; the collection of all Π01 classes represented by K-trivial sets is Medvedev-bounded by a single Π01 class represented by an incomplete co-c.e. set; and the Π01 classes have neither nontrivial infinite suprema nor infima in the Medvedev lattice.

scholarly journals Heine-Borel does not imply the Fan Theorem

1984 ◽  
Vol 49 (2) ◽  
pp. 514-519 ◽  
Author(s):  
Ieke Moerdijk
Keyword(s):  
Function Space ◽  
Real Line ◽  
General Framework ◽  
Geometric Theory ◽  
Continuous Functions ◽  
Unit Interval ◽  
Baire Space ◽  
Locally Compact ◽  
Cantor Space ◽  
Fan Theorem

This paper deals with locales and their spaces of points in intuitionistic analysis or, if you like, in (Grothendieck) toposes. One of the important aspects of the problem whether a certain locale has enough points is that it is directly related to the (constructive) completeness of a geometric theory. A useful exposition of this relationship may be found in [1], and we will assume that the reader is familiar with the general framework described in that paper.We will consider four formal spaces, or locales, namely formal Cantor space C, formal Baire space B, the formal real line R, and the formal function space RR being the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(C), pt(B), pt(R) and pt(RR). Classically, these locales all have enough points, of course, but constructively or in sheaves this may fail in each case. Let us recall some facts from [1]: the assertion that C has enough points is equivalent to the compactness of the space of points pt(C), and is traditionally known in intuitionistic analysis as the Fan Theorem (FT). Similarly, the assertion that B has enough points is equivalent to the principle of (monotone) Bar Induction (BI). The locale R has enough points iff its space of points pt(R) is locally compact, i.e. the unit interval pt[0, 1] ⊂ pt(R) is compact, which is of course known as the Heine-Borel Theorem (HB). The statement that RR has enough points, i.e. that there are “enough” continuous functions from R to itself, does not have a well-established name. We will refer to it (not very imaginatively, I admit) as the principle (EF) of Enough Functions.

Special subsets of the generalized Cantor space and generalized Baire space

2020 ◽  
Vol 66 (4) ◽  
pp. 418-437
Author(s):  
Michał Korch ◽  
Tomasz Weiss
Keyword(s):  
Baire Space ◽  
Cantor Space

Basic Problems of Logic in Drug Investigations

1974 ◽  
Vol 13 (04) ◽  
pp. 248-253
Author(s):  
H. Röpke
Keyword(s):  
Animal Experiments ◽  
Point Of View ◽  
Applied Logic ◽  
Clinical Investigations

The basic problems of the theory of cognition when extrapolating the results of animal experiments to man and the limitations of inferences from clinical investigations of drugs are discussed from the point of view of applied logic.

Which human qualities can economic liberalism be based on?

2015 ◽  
pp. 5-24 ◽  
Author(s):  
V. Avtonomov
Keyword(s):  
Economic Policy ◽  
Human Values ◽  
High Rank ◽  
Economic Liberalism ◽  
Von Mises ◽  
Liberal Policy

The article asks which human qualities can serve as the base for a liberal economic policy. The author is looking for an answer to this question in the classical works of economic liberalism - in the books by F. Bastiat, L. von Mises, F. von Hayek, W. Eucken, and M. Friedman. The two main qualities can be summarized as follows: the relatively high rank of freedom among human values and rational utilitarian calculus. It is assumed that in those countries where the both human prerequisites are present the liberal policy (for instance, liberal reforms) can be self-supportive and may have good results. On the contrary, if the first prerequisite is missing and a liberal policy can be based only on rational calculus, the “paternalistic” liberalism prevails and chances for success are much smaller.

Coaction of Electrostatic and Hydrophobic Interactions: Dynamic Constraints on Disordered TrkA Juxtamembrane Domain

2019 ◽  
Author(s):  
Zichen Wang ◽  
Huaxun Fan ◽  
Xiao Hu ◽  
John Khamo ◽  
Jiajie Diao ◽  
...  
Keyword(s):  
Single Molecule ◽  
Protein Function ◽  
Hydrophobic Interactions ◽  
Transmembrane Helix ◽  
Point Mutations ◽  
Salt Bridge ◽  
Receptor Activation ◽  
Membrane Dynamics ◽  
Juxtamembrane Domain ◽  
Joint Work

<p>The receptor tyrosine kinase family transmits signals into cell via a single transmembrane helix and a flexible juxtamembrane domain (JMD). Membrane dynamics makes it challenging to study the structural mechanism of receptor activation experimentally. In this study, we employ all-atom molecular dynamics with Highly Mobile Membrane-Mimetic to capture membrane interactions with the JMD of tropomyosin receptor kinase A (TrkA). We find that PIP<sub>2 </sub>lipids engage in lasting binding to multiple basic residues and compete with salt bridge within the peptide. We discover three residues insertion into the membrane, and perturb it through computationally designed point mutations. Single-molecule experiments indicate the contribution from hydrophobic insertion is comparable to electrostatic binding, and in-cell experiments show that enhanced TrkA-JMD insertion promotes receptor ubiquitination. Our joint work points to a scenario where basic and hydrophobic residues on disordered domains interact with lipid headgroups and tails, respectively, to restrain flexibility and potentially modulate protein function.</p>

Methodology of studying the mechanical properties of composite materials (reinforced concrete)

2018 ◽  
Vol 84 (12) ◽  
pp. 61-67
Author(s):  
V. A. Eryshev
Keyword(s):  
Mechanical Properties ◽  
Reinforced Concrete ◽  
Experimental Studies ◽  
Analytical Relationship ◽  
Hardened Concrete ◽  
Deformation Model ◽  
Joint Work ◽  
Concrete Shrinkage ◽  
Axial Tension ◽  
Tension And Compression

The mechanical properties of a complex composite material formed by steel and hardened concrete, are studied. A technique of operative quality control of new credible concrete and reinforcement, both in laboratory and field conditions is developed for determination of the strength and strain characteristics of materials, as well as cohesion forces determining their joint operation under load. The design of the mobile unit is presented. The unit provides a possibility of changing the direction of loading and testing the reinforced element of the given shape both for tension and compression. Moreover, the nomenclature of testing equipment and the number of molds for manufacturing concrete samples substantially decrease. Using the values of forcing resulting in concrete cracking when the joint work of concrete and reinforcement is disrupted the values of the inherent stresses and strains attributed to the concrete shrinkage are determined. An analytical relationship between the forces and deformations of the reinforced concrete sample with central reinforcement is derived for axial tension and compression, with allowance for strains and stresses in the reinforcement and concrete resulted from concrete shrinkage. The results of experimental studies are presented, including tension diagrams and diagrams of developing axial deformations with an increase in the load under the central loading of the reinforced elements. A methodology of accounting for stresses and deformations resulted from concrete shrinkage is developed. The applicability of the derived analytical relationships between stresses and deformations on the material diagrams to calculations of the reinforced concrete structures in the framework of the deformation model is estimated.

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