Hyperhypersimple supersets in admissible recursion theory

1983 ◽  
Vol 48 (1) ◽  
pp. 185-192
Author(s):  
C. T. Chong
Keyword(s):  
Recursion Theory ◽  
Order Type ◽  
Complete Characterization ◽  
Point Of View ◽  
Extensive Literature ◽  
Careful Study ◽  
Finite Sets ◽  
Natural Class ◽  
Recursively Enumerable Set ◽  
Simple Sets

Let α be an admissible ordinal. An α-recursively enumerable set H is hyper-hypersimple (hh-simple) if its lattice of α-r.e. supersets forms a Boolean algebra. In [3], Chong and Lerman characterized the class ℋ() of hh-simple -r.e. sets as precisely those -r.e. sets whose complements are unbounded and of order type less than . Perhaps a nice example of such a set is {σ∣σ is not for any n < ω}. It follows that all hh-simple sets in are nonhyperregular and therefore of degree 0′. That ℋ() is a natural class to study can be seen from the role played by its ω-counterpart in the study of decision problems and automorphisms of ℰ*(ω), the lattice of ω-r.e. sets modulo finite sets (Soare [13] gives an extensive literature on these topics). In α-recursion theory the existence of hh-simple sets is not an all pervasive phenomenon, and there is as yet no complete characterization of the admissible ordinal α for which ℋ(α) is nonempty. While this situation is admittedly unsatisfactory, we feel that the lattice ℰ*(α) of α-r.e. sets modulo α*-finite sets for which ℋ(α) ≠ ∅ deserves a careful study. Indeed armed with some understanding over the last few years of the general theory of admissible ordinals, it is tempting to focus one's attention on some specific ordinals whose characteristics admit a more detailed analysis of the fine structure of sets and degrees. From this point of view, and ℋ() are natural objects of study since the former is a typical example of a non-Σ2-projectible, Σ2-inadmissible ordinal, while the latter is important for the investigations of automorphisms over ℰ*().

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

scholarly journals Encephalocele: know it to deal with it

2021 ◽  
Vol 52 (1) ◽  
Author(s):  
Nilkanth Laxman Pal ◽  
Ambika Sumeet Juwarkar ◽  
Sanjaya Viswamitra
Keyword(s):  
Prognostic Factors ◽  
Imaging Modality ◽  
Point Of View ◽  
Extensive Literature ◽  
Resonance Imaging ◽  
Imaging Protocol ◽  
Imaging Protocols ◽  
Associated Malformations ◽  
Associated Abnormalities

Abstract Background Encephaloceles refer to the herniation of intracranial contents through the defect in the dura and calvarium. This article aims to equip the radiologists with all the necessary information to approach a case of encephalocele in a systematic manner and report in an organized format. Thus reduce the reporting time without overlooking any detail. Main text An extensive literature review was performed searching through the standard textbooks and electronic databases to obtain thorough information on the various encephaloceles particularly from a radiological point of view. The author’s observations were also incorporated in the review. Among the available imaging modalities, magnetic resonance imaging is the most versatile and considered the imaging modality of choice because of superior anatomic resolution in the characterization of central nervous system malformation. This study provides the reporting radiologist with a methodical guide to approach a case of encephalocele discussing the imaging protocol, relevant anatomy, classification, associated abnormalities, and imaging prognostic factors. Conclusion Encephalocele has always been a subject of predicament for radiologists. Understanding the relevant terminologies, anatomy, imaging protocols, classification, associated malformations/anomalies, and imaging prognostic factors will help to methodically approach each case and provide a systematic and comprehensive report.

scholarly journals Linking the spare parts management with the total costs of ownership: An agenda for future research

2016 ◽  
Vol 9 (5) ◽  
pp. 991 ◽  
Author(s):  
Orlando Duran ◽  
Irene Roda ◽  
Marco Macchi
Keyword(s):  
Life Cycle ◽  
Conceptual Framework ◽  
Spare Parts ◽  
Point Of View ◽  
Management Decision ◽  
Future Research ◽  
Extensive Literature ◽  
Economic Point ◽  
Spare Parts Management ◽  
Physical Asset

Purpose: This manuscript explores the link between Spare Parts Management and Total Costs of Ownership or Life Cycle Costs (LCC).Design/methodology/approach: First, this work enumerates the different managerial decisions instances in spare parts management that are present during the life cycle of a physical asset. Second, we analyse how those decision instances could affect the TCO of a physical asset (from the economic point of view). Finally, we propose a conceptual framework for incorporating the spare parts management into a TCO model.Findings: The recent literature lacks discussions on the integration of spare parts management with the Total Costs of Ownership (TCO). Based in an extensive literature revision we can declare that the computation of costs related to spare parts management has been neglected by TCO models.Originality/value: The contribution of this paper is twofold. First, a literature review and identification of a series of spare parts management decision instances and its relationship with TCOs is presented in this paper. Second, a conceptual framework is suggested for linking those decisions instances to a total cost of ownership perspective. Some research questions and future research challenges are presented at the end of this work.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (3) ◽  
pp. 681-694
Author(s):  
Anne Leggett ◽  
Richard A. Shore
Keyword(s):  
Lattice Theory ◽  
Lattice Structure ◽  
Sufficient Conditions ◽  
Recursion Theory ◽  
Recursively Enumerable ◽  
Countable Ordinal ◽  
Congruence Relations ◽  
General Program ◽  
Simple Sets

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

Study of Phase Transformation in Alloys of the Al-Fe System

2012 ◽  
Vol 326-328 ◽  
pp. 573-577 ◽  
Author(s):  
Magdalena Jabłońska ◽  
Marta Mikuśkiewicz ◽  
Anna Śmiglewicz ◽  
E. Bernstock-Kopaczyńska
Keyword(s):  
Thermal Analysis ◽  
Solid Solution ◽  
Phase Transformation ◽  
Phase Transformations ◽  
Order Type ◽  
Point Of View ◽  
Gravity Casting ◽  
Mechanical And Physical Properties ◽  
Temperatures Of Phase Transformations ◽  
Alloys Of The Al

In this paper, phase transformation in alloys of the Al-Fe and Al-Fe-Cr systems was investigated. The alloys were prepared by melting and gravity casting. The studies of phase transformation were carried out on samples after casting and annealing, using the differential thermal analysis (DTA) and dilatometric method. The knowledge on the phase transformations in these alloys including the information about order-disorder transition is very important from the point of view of obtained mechanical and physical properties of alloys of the Al-Fe system. These results are an important contribution in development of knowledge on iron aluminides. In the article, temperatures of phase transformations connected with a change in order type and transition into disordered solid solution were defined. Conformity of the recorded DTA results and dilatometric analysis was achieved.

The Digital Transformation of the Product Management Process: Conception of Digital Twin Impacts for the Different Stages

2019 ◽  
Vol 5 (2) ◽  
pp. 74-86 ◽  
Author(s):  
Günter Hofbauer ◽  
Anita Sangl ◽  
Sonja Engelhardt
Keyword(s):  
Digital Transformation ◽  
Point Of View ◽  
Product Management ◽  
Extensive Literature ◽  
Management Process ◽  
Digital Twin ◽  
Digital Twins ◽  
Analytical Point ◽  
Potential Benefits ◽  
Process Conception

In this paper the potential impact of digital transformation in general and digital twin applications on the Product Management Process (PMP) in particular will be investigated. The object is to figure out the potential benefits of digital twin utilization. Methodologically, a conceptual and analytical approach are applied in using statistical data, surveys, latest literature and logical conclusions. Thus, the approach is twofold: from the conceptual point of view the different types of digital twins are introduced; from the analytical point of view different applications in different stages of the PMP will be subject to investigation. So, this paper can be categorized as conceptual paper based on an extensive literature research. The findings of research revealed that the digital transformation can offer various benefits for the PMP. It can be summarized that in all phases of the PMP potential benefits can be identified. The most important is saving time and money, avoiding waste of physical resources and simultaneously raising quality, reliability and competitive advantage. Regarding the originality value it can be stated that this is the first comprehensive examination of the entire PMP with regard to digital transformation in order to identify sources of potential benefits.

The Rebellion of the Earls, 1569

1906 ◽  
Vol 20 ◽  
pp. 171-203 ◽  
Author(s):  
R. R. Reid
Keyword(s):  
British Museum ◽  
Point Of View ◽  
The Other ◽  
Careful Study ◽  
Modern History ◽  
The Public ◽  
Local Point ◽  
Local Sources ◽  
Public Record ◽  
Parish Registers

A popular movement like the Rebellion of the Earls can always be treated from two distinct standpoints, the national and the local. Hitherto, the Rebellion has always been treated from the national standpoint, with the result that, so far as I am aware, there is no book dealing with the Rebellion alone. All accounts of it must be sought in general histories such as those named below. I would specially mention the chapter in the ‘Cambridge Modern History’ in which Mr. Law has anticipated all the conclusions which I have been able to draw from my own examination of the sources. The local point of view, on the other hand, has been almost wholly ignored, and affords more opportunity for investigation; to it, therefore, I have confined myself. I cannot pretend that the essay is exhaustive, as circumstances have prevented me from investigating the local sources, such as Corporation and Town Records, Parish Registers and the like. Nevertheless, this contribution may not be wholly without value, since it is based on a careful study of the material preserved at the Public Record Office and in the British Museum.

Effectively nowhere simple sets

1984 ◽  
Vol 49 (1) ◽  
pp. 129-136 ◽  
Author(s):  
D. Miller ◽  
J. B. Remmel
Keyword(s):  
Simple Proof ◽  
Recursive Function ◽  
Finite Sets ◽  
Definition Of ◽  
Simple Sets

An r.e. set A is nowhere simple if for every r.e. set We such that We − A is infinite, there is an infinite r.e. set W such that W ⊆ We − A. The definition of nowhere simple sets is due to R. Shore in [4]. In [4], Shore studied various properties of nowhere simple sets and showed that they could be used to give an elegant and simple proof of the fact that every nontrivial class of r.e. sets C closed under recursive isomorphisms is an automorphism base for , the lattice of r.e. sets modulo finite sets, (that is, an automorphism α of is completely determined by its action on C; see Theorem 8 of [4]). Shore also defined the notion of effectively nowhere simple sets.Definition. An r.e. set A is effectively nowhere simple if there is a recursive function f such that for every i, Wf(i) ⊆ Wi − A and Wf(i) is infinite iff Wi − A is infinite. f is called a witness function for A.Other than to produce examples of effectively nowhere simple sets and nowhere simple sets that are not effectively nowhere simple, Shore did not concern himself with the properties of effectively nowhere simple sets since he felt that effectively nowhere simple sets were unlikely to be lattice invariant in either E, the lattice of r.e. sets, or in .

Sign in / Sign up

Export Citation Format

Share Document