On Σ1-Structural Differences Among Finite Levels of the Ershov Hierarchy

2006 ◽  
Vol 71 (4) ◽  
pp. 1223-1236 ◽  
Author(s):  
Yue Yang ◽  
Liang Yu
Keyword(s):  
Elementary Substructure ◽  
Recursively Enumerable ◽  
Ershov Hierarchy ◽  
Structural Differences

AbstractWe show that the structure of recursively enumerable degrees is not a Σ1-elementary substructure of , where (n > 1) is the structure of n-r.e. degrees in the Ershov hierarchy.

scholarly journals On Transfinite Levels of the Ershov Hierarchy

2021 ◽  
Vol 27 (2) ◽  
pp. 220-221
Author(s):  
Cheng Peng
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Elementary Substructure ◽  
Ordered Structures ◽  
Research Areas ◽  
Ershov Hierarchy ◽  
Sigma 1 ◽  
E Mail

AbstractIn this thesis, we study Turing degrees in the context of classical recursion theory. What we are interested in is the partially ordered structures $\mathcal {D}_{\alpha }$ for ordinals $\alpha <\omega ^2$ and $\mathcal {D}_{a}$ for notations $a\in \mathcal {O}$ with $|a|_{o}\geq \omega ^2$ .The dissertation is motivated by the $\Sigma _{1}$ -elementary substructure problem: Can one structure in the following structures $\mathcal {R}\subsetneqq \mathcal {D}_{2}\subsetneqq \dots \subsetneqq \mathcal {D}_{\omega }\subsetneqq \mathcal {D}_{\omega +1}\subsetneqq \dots \subsetneqq \mathcal {D(\leq \textbf {0}')}$ be a $\Sigma _{1}$ -elementary substructure of another? For finite levels of the Ershov hierarchy, Cai, Shore, and Slaman [Journal of Mathematical Logic, vol. 12 (2012), p. 1250005] showed that $\mathcal {D}_{n}\npreceq _{1}\mathcal {D}_{m}$ for any $n < m$ . We consider the problem for transfinite levels of the Ershov hierarchy and show that $\mathcal {D}_{\omega }\npreceq _{1}\mathcal {D}_{\omega +1}$ . The techniques in Chapters 2 and 3 are motivated by two remarkable theorems, Sacks Density Theorem and the d.r.e. Nondensity Theorem.In Chapter 1, we first briefly review the background of the research areas involved in this thesis, and then review some basic definitions and classical theorems. We also summarize our results in Chapter 2 to Chapter 4. In Chapter 2, we show that for any $\omega $ -r.e. set D and r.e. set B with $D<_{T}B$ , there is an $\omega +1$ -r.e. set A such that $D<_{T}A<_{T}B$ . In Chapter 3, we show that for some notation a with $|a|_{o}=\omega ^{2}$ , there is an incomplete $\omega +1$ -r.e. set A such that there are no a-r.e. sets U with $A<_{T}U<_{T}K$ . In Chapter 4, we generalize above results to higher levels (up to $\varepsilon _{0}$ ). We investigate Lachlan sets and minimal degrees on transfinite levels and show that for any notation a, there exists a $\Delta ^{0}_{2}$ -set A such that A is of minimal degree and $A\not \equiv _T U$ for all a-r.e. sets U.Abstract prepared by Cheng Peng.E-mail: [email protected]

Classifications of generalized index sets of open classes

1978 ◽  
Vol 43 (4) ◽  
pp. 694-714 ◽  
Author(s):  
Nancy Johnson
Keyword(s):  
Finite Collection ◽  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Ershov Hierarchy ◽  
Index Sets ◽  
Finite Set ◽  
The Difference ◽  
Original Classification

The Rice-Shapiro Theorem [4] says that the index set of a class of recursively enumerable (r.e.) sets is r.e. if and only if consists of all sets which extend an element of a canonically enumerable sequence of finite sets. If an index of a difference of r.e. (d.r.e.) sets is defined to be the pair of indices of the r.e. sets of which it is the difference, then the following generalization due to Hay [3] is obtained: The index set of a class of d.r.e. sets is d.r.e. if and only if is empty or consists of all sets which extend a single fixed finite set. In that paper Hay also classifies index sets of classes consisting of d.r.e. sets which extend one of a finite collection of finite sets. These sets turn out to be finite Boolean combinations of r.e. sets. The question then arises “What about the classification of the index set of a class consisting of d.r.e. sets which extend an element of a canonically enumerable sequence of finite sets?” The results in this paper come from an attempt to answer this question.Since classes of sets which are Boolean combinations of r.e. sets form a hierarchy (the finite Ershov hierarchy, see Ershov [1]) with the r.e. and d.r.e. sets respectively levels 1 and 2 of this hierarchy, we may define index sets of classes of level n sets. If is a class of level n sets which extend some element of a canonically enumerable sequence of finite sets and if we let co-, then we extend the original classification question to the classification of the index sets of the classes and co-.Now if the sequence of finite sets enumerates only finitely many sets or if only finitely many of the finite sets are minimal under inclusion, then it is a routine computation to verify that the index sets of and co- are in the finite Ershov hierarchy. Thus we are interested in the case in which infinitely many of the sequence of finite sets are minimal under inclusion. However if the infinite sequence is fairly simple, for instance{0}, {1}, {2}, … then the r.e. index set of co- is Σ20-complete as well as the index sets of and co- for all levels n > 2. Since the finite Ershov hierarchy does not exhaust ⊿20 there is a lot of “room” between these two extreme cases.

High Resolution Observations of Ion-Milling Induced Transformation in Synthetic Hollandses

1981 ◽  
Vol 39 ◽  
pp. 114-115 ◽  
Author(s):  
T. J. Headley
Keyword(s):  
High Resolution ◽  
Radioactive Waste Disposal ◽  
Ion Milling ◽  
Minor Elements ◽  
Waste Forms ◽  
Carbon Substrates ◽  
Structural Differences ◽  
Oxide Phases ◽  
Argon Ions ◽  
High Resolution Microscopy

Oxide phases having the hollandite structure have been identified in multiphase ceramic waste forms being developed for radioactive waste disposal. High resolution studies of phases in the waste forms described in Ref. [2] were initiated to examine them for fine scale structural differences compared to natural mineral analogs. Two hollandites were studied: a (Ba,Cs,K)-titan-ate with minor elements in solution that is produced in the waste forms, and a synthesized BaAl2Ti6O16 phase containing ∼ 4.7 wt% Cs2O. Both materials were consolidated by hot pressing at temperatures above 1100°C. Samples for high resolution microscopy were prepared both by ion-milling (7kV argon ions) and by crushing and dispersing the fragments on holey carbon substrates. The high resolution studies were performed in a JEM 200CX/SEG operating at 200kV.

Weakly precomplete equivalence relations in the Ershov hierarchy

2019 ◽  
Vol 58 (3) ◽  
pp. 297-319
Author(s):  
N. A. Bazhenov ◽  
B. S. Kalmurzaev
Keyword(s):  
Equivalence Relations ◽  
Ershov Hierarchy

CHEMISTRY OF GONADOTROPHINS IN RELATION TO THEIR ANTIGENIC PROPERTIES

1969 ◽  
Vol 62 (1_Suppl) ◽  
pp. S13-S30 ◽  
Author(s):  
W. R. Butt
Keyword(s):  
Biological Activity ◽  
Guinea Pigs ◽  
Complement Fixation ◽  
Neuraminic Acid ◽  
Fixation Technique ◽  
Immunological Activity ◽  
Antigenic Properties ◽  
Specific Antisera ◽  
Structural Differences

ABSTRACT Several chemical differences between FSH, LH and HCG have been reported: thus LH and HCG are richer in proline than FSH and FSH and HCG contain more N-acetyl neuraminic acid than LH. Sub-units of LH are formed by treatment with urea, guanidine or acid. HCG also may contain two sub-units. The sub-units from LH are biologically inert but retain their immunological activity: biological activity is restored when the sub-units are incubated together. There is much evidence from chemical and enzymic reactions that antigenic groups are distinct from those parts of the molecule essential for biological activity. N-acetyl neuraminic acid and probably other carbohydrates in FSH and HCG are not involved in immunological activity but are necessary for biological activity. Histidine, methionine and possibly cysteine appear to be essential for biological but not immunological activity of FSH, while tryptophan and possibly tyrosine are not essential for either. A few highly specific antisera to gonadotrophins have been prepared in rabbits and guinea pigs to crude antigens: there is no evidence that purified antigens are more likely to produce specific antisera. Differences in the immunological reactivities of urinary compared with pituitary gonadotrophins have been observed both by radioimmunoassay and by the complement fixation technique. The latter may be particularly useful for detecting structural differences in the hormones.

Structural Differences between the Nanocrystalline Soft Magnetic Fe73.5Si13.5B9Nb3Cu1and Fe86Zr7B6Cu1Alloys

2002 ◽  
Vol 93 (1) ◽  
pp. 21-27
Author(s):  
Á. Cziráki ◽  
I. Gerocs ◽  
L. K. Varga ◽  
I. Bakonyi ◽  
U. Falke ◽  
...  
Keyword(s):  
Soft Magnetic ◽  
Structural Differences

Different Morphology of Neuritic Plaques in the Archicortex of Alzheimer’s Disease with Comorbid Synucleinopathy: A Pilot Study

2020 ◽  
Vol 17 ◽  
Author(s):  
Nikol Jankovska ◽  
Tomas Olejar ◽  
Jaromir Kukal ◽  
Radoslav Matej
Keyword(s):  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
Pilot Study ◽  
Time Course ◽  
Clinical Course ◽  
Lewy Bodies ◽  
Dystrophic Neurites ◽  
Neuritic Plaques ◽  
Structural Differences ◽  
Lewy Body Variant

Background: Bulbous neuritic changes in neuritic plaques have already been described, and their possible effect on the clinical course of the disease has been discussed. OBJECTIVE: In our study, we focused on the location and density of these structures in patients with only Alzheimer’s disease (AD) and patients with AD in comorbidity with synucleinopathies. Methods: Utilizing immunohistochemistry and confocal microscopy, we evaluated differences of neocortical and archicortical neuritic plaques and the frequency of bulbous changes in the archicortex of 14 subjects with Alzheimer’s disease (AD), 10 subjects with the Lewy body variant of Alzheimer's disease (AD/DLB), and 4 subjects with Alzheimer's disease with amygdala Lewy bodies (AD/ALB). Also, the progression and density of neuritic changes over the time course of the disease were evaluated. Results: We found structural differences in bulbous dystrophic neurites more often in AD/DLB and AD/ALB than in pure AD cases. The bulbous neuritic changes were more prominent in the initial and progressive phases and were reduced in cases with a long clinical course. Conclusion: Our results indicate that there is a prominent difference in the shape and composition of neocortical and archicortical neuritic plaques and, moreover, that bulbous neuritic changes can be observed at a higher rate in AD/DLB and AD/ALB subjects compared to pure AD subjects. This observation probably reflects that these subacute changes are more easily seen in the faster clinical course of AD patients with comorbidities.

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