scholarly journals The Structure of d.r.e. Degrees

2021 ◽  
Vol 27 (2) ◽  
pp. 218-219
Author(s):  
Yong Liu
Keyword(s):  
Boolean Algebra ◽  
Recursion Theory ◽  
Density Theorem ◽  
The Other ◽  
Maximal Degree ◽  
Joint Project ◽  
Final Segment ◽  
E Mail

AbstractThis dissertation is highly motivated by d.r.e. Nondensity Theorem, which is interesting in two perspectives. One is that it contrasts Sacks Density Theorem, and hence shows that the structures of r.e. degrees and d.r.e. degrees are different. The other is to investigate what other properties a maximal degree can have.In Chapter 1, we briefly review the backgrounds of Recursion Theory which motivate the topics of this dissertation.In Chapter 2, we introduce the notion of $(m,n)$ -cupping degree. It is closely related to the notion of maximal d.r.e. degree. In fact, a $(2,2)$ -cupping degree is maximal d.r.e. degree. We then prove that there exists an isolated $(2,\omega )$ -cupping degree by combining strategies for maximality and isolation with some efforts.Chapter 3 is part of a joint project with Steffen Lempp, Yiqun Liu, Keng Meng Ng, Cheng Peng, and Guohua Wu. In this chapter, we prove that any finite boolean algebra can be embedded into d.r.e. degrees as a final segment. We examine the proof of d.r.e. Nondensity Theorem and make developments to the technique to make it work for our theorem. The goal of the project is to see what lattice can be embedded into d.r.e. degrees as a final segment, as we observe that the technique has potential be developed further to produce other interesting results.Abstract prepared by Yong Liu.E-mail: [email protected]

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]

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.

Doctrine and Dialectic in Plato’s Dialogues

2018 ◽  
Author(s):  
Rachana Kamtekar
Keyword(s):  
Methodological Approach ◽  
The Other ◽  
Set Up

Chapter 1 lays out the methodological approach employed throughout the book, which is to pay attention to the dialectical dependence of what the main speaker in the dialogue says on the intellectual problem(s) set up in the dialogue both by himself and the other speakers. To illustrate, Chapter 1 describes Socrates’ use of the method of hypotheses from the Meno and Phaedo to answer questions that go beyond his claims to knowledge in the Republic.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

scholarly journals Virtual Racism Rears its Head: Uncovering Librarian Bias in E-mail Reference Services

2007 ◽  
Vol 2 (2) ◽  
pp. 97
Author(s):  
Wendy Furlan
Keyword(s):  
African American ◽  
Sample Size ◽  
Response Times ◽  
The United States ◽  
The Other ◽  
Level Of Service ◽  
Virtual Reference ◽  
Reference Services ◽  
User Groups ◽  
E Mail

A review of: Shachaf, Pnina, and Sarah Horowitz. "Are Virtual Reference Services Color Blind?" Library & Information Science Research 28.4 (Sept. 2006): 501-20. Abstract Objective – To examine whether librarians provide equitable virtual reference services to diverse user groups. Design – Unobtrusive method of defined scenarios submitted via e-mail. Setting – Twenty-three Association of Research Libraries (ARL) member libraries from across the United States. All ARL member libraries were invited to participate, with the 23 acceptances providing 19% participation. Subjects – Anonymous librarians from the 23 participating libraries’ virtual e-mail reference services. Up to 6 librarians from each library may have been involved. Six fictitious personas were developed to represent particular ethnic or religious groups, whereby the ethnic or religious affiliation was only indicated by the name chosen for each user and the corresponding e-mail address. Names were selected from lists of names or baby names available online: Latoya Johnson (African-American), Rosa Manuz (Hispanic), Chang Su (Asian - Chinese), Mary Anderson (Caucasian/Christian), Ahmed Ibrahim (Muslim), and Moshe Cohen (Caucasian/Jewish). These personas were used to submit reference queries via e-mail to the virtual reference services taking part in the study. Methods – Five different types of reference queries were developed for use in this study. Three were based on prior published research as they were deemed to be answerable by the majority of libraries. They included a dissertation query, a sports team query, and a population query all designed to be tailored to the target institution. The other 2 queries were developed with participating institutions’ virtual reference guidelines in mind, and were thought to not be answered by the target institutions when submitted by unaffiliated users. They consisted of a subject query on a special collection topic that asked for copies of relevant articles to be sent out, and an article query requesting that a copy of a specific article be e-mailed to the patron. The study was conducted over a 6 week period beginning the second week of September, 2005. Each week, 1 fictitious persona was used to e-mail a reference query to the virtual reference service of each of the 23 participating institutions. Five of each type of query were sent by each persona. During September and October 2005, a total of 138 queries were sent. Each institution received a different query for each of the first 5 weeks, and in the sixth week they received a repeat of a previous request with details of title or years altered. All other text in every request sent was kept consistent. Each institution only received 1 request from each persona during the study. In order to eliminate any study bias caused by an informed decision regarding the order in which personas were used, they were randomly arranged (alphabetically by surname). Furthermore, to avoid suspicions from responding librarians, queries were e-mailed on different days of the week at different times. This created some limitations in interpretating response times as some queries were submitted on weekends. All queries were analysed by Nvivo software in order to identify attributes and patterns to aid qualitative analysis. Each transaction (a single query and any related responses) was classified according to 12 attributes and 59 categories based on various associations’ digital reference guidelines. Transactions were coded and then 10% re-coded by a different coder. This led to the clarification and refinement of the coding scheme, resulting in the number of categories used being reduced to 23. Coding was then performed in 3 iterations until 90% agreement between the 2 coders was reached. The final inter-coder reliability was 92%. The study did not support cross tabulation among user groups on most content categories due to the small sample size. Main results – Response times varied greatly between users. Moshe (Caucasian/Jewish) received an average turn-around of less than a day. At the other end of the spectrum, Ahmed’s (Muslim) responses took an average of 3.5 days. Both Ahmed and Latoya (African-American) sent queries which took over 18 days to receive a response. The length (number of words) of replies also indicated a differing level of service with Mary (Caucasian/Christian) and Moshe receiving far lengthier responses than the other 4 personas. Number of replies (including automatic replies) was examined in comparison with the number of replies which answered the question, and again indicated Mary and Moshe were receiving a better level of service. The way in which the user was addressed by the librarian was examined as another measure of service, i.e. first name, full name, honorific. This again mirrored the low level of service received by Ahmed. The professional endings used by librarians in their replies also reinforced the high quality of service received by Moshe across other categories. Results for Rosa (Latino) and Chang (Asian - Chinese) were average for most categories presented. Conclusion – In this study, a discriminatory pattern was clearly evident, with the African-American and Muslim users receiving poor levels of service from virtual reference librarians across all dimensions of quality evaluated. The Caucasian (Christian and Jewish) users also noticeably received the best level of service. It is noted, however, that the sample size of the study is not large enough for generalisations to be drawn and that future, more statistically significant studies are warranted. Many other questions are raised by the study for possible future research into racism exhibited by library staff and services.

The Wilding of Cape York Peninsula

2017 ◽  
Author(s):  
Timothy Neale
Keyword(s):  
Indigenous Peoples ◽  
The Other ◽  
Northern Australia ◽  
Partial Failure ◽  
Hand Part ◽  
Cape York ◽  
The One ◽  
Colonial Encounters

In Chapter 1, I argue that ‘wildness’ is a product settler attempts to understand and thereby spatially remake the Northern Australia since the first colonial encounters in the 17th Century. For European explorers, a region like Cape York Peninsula was a wilderness to be surveyed, and through the misadventures and conflicts of inland expeditions it came to be understood as ‘wretched’ country populated with ‘treacherous’ peoples. Surveying subsequent uses of ‘the wild’ in this region, this chapter shows that if, on the one hand, part of the settler project has been to discursively and materially dictate the shape and texture of the region through such forms of wildness – ‘wilderness,’ ‘wild time,’ ‘wild blacks’ and ‘wild whites’ – then, on the other, the contemporary ‘wilderness’ should be understood not only as a product of the resistance and resilience of its Indigenous peoples, but also as the partial failure of this project.

Conclusion

2019 ◽  
pp. 117-119
Author(s):  
Charles F. Manski
Keyword(s):  
Patient Care ◽  
Guideline Development ◽  
Medical Knowledge ◽  
The Other ◽  
Adherence To Guidelines ◽  
Decisions Under Uncertainty

This concluding chapter provides suggestions that encourage putting the themes of the book into practice. It returns to the question from Chapter 1: “Should clinicians adhere to guidelines or exercise judgment?” The chapter cautions against universal adherence to guidelines. Motivated by some of these considerations, the chapter considers separating two tasks of guideline development that have commonly been performed in conjunction. One task is to characterize medical knowledge. The other is to make recommendations for patient care. An alternative to having guidelines make care recommendations would be to enhance the ability of clinicians to make reasonable patient care decisions under uncertainty.

Behaviour of Fluids (Liquids and Gases, Flow and Pressure)

2011 ◽  
Author(s):  
Patrick Magee ◽  
Mark Tooley
Keyword(s):  
Laminar Flow ◽  
Average Velocity ◽  
Fluid Transport ◽  
Minimum Energy ◽  
Flow Characteristics ◽  
The Other ◽  
Transport Mode ◽  
Density Flow ◽  
Fluid Layers

A fluid can be either a liquid or a gas. Fluids exhibit different flow behaviours depending on their physical properties, in particular viscosity and density. Flow characteristics also depend on the geometry of the pipes or channels through which they flow, and on the driving pressure regimes. These principles can be applied to any fluid, and the complexity of the analysis depends on the flow regimes described in this section [Massey 1970]. Fluid flow is generally described as laminar or turbulent. Laminar flow, demonstrated by Osborne Reynolds in 1867, is flow in which laminae or layers of fluid run parallel to each other. In a circular pipe, such as a blood vessel or a bronchus, velocity within the layers nearest the wall of the pipe is least; in the layer immediately adjacent to the wall it is probably actually zero. In fully developed laminar flow, the velocity profile across the pipe is parabolic, as shown in Figure 7.1, and as discussed in Chapter 1. Peak velocity of the fluid occurs in the mid line of the pipe, and is twice the average velocity across the pipe at equilibrium, and layers equidistant from the wall have equal velocity. The importance of laminar flow is that there is minimum energy loss in the flow, i.e. it is an efficient transport mode. This is in contrast to turbulent flow, where eddies and vortices (flow in directions other than the predominant one) mean that energy in fluid transport is wasted in production of heat, additional friction and noise. The result is that the pressure drop required to drive a given flow from one end of the pipe to the other is greater in turbulent than in laminar flow. The shear stress τ, which is the mechanical stress between layers of fluid and between the fluid and the tube wall, is proportional to the velocity gradient across the tube (dv/dr) of the fluid layers. The constant of proportionality between these two variables is the dynamic viscosity, η.

Aesthetic trials: taking care of oneself, of the other and of the environment in which we live Experimentações estéticas: o cuidado de si, do outro e do ambiente em que vivemos

2017 ◽  
Vol 9 (4) ◽  
pp. 946
Author(s):  
Cláudio Tarouco De Azevedo ◽  
Augusto Luis Medeiros Amaral
Keyword(s):  
The Other ◽  
E Mail ◽  
Taking Care

Objetivo: Apresentar a pesquisa-intervenção desenvolvida pelos autores em forma de atividades artístico-pedagógicas, denominadas experimentações estéticas, analisando as mesmas e seus efeitos enquanto conjunto prático-teórico voltado ao cuidado de si, do outro e do ambiente. Métodos: Construído a partir da experiência dos autores com práticas de pesquisa inspiradas nos estudos sobre As Três Ecologias e na metodologia Sociopoética, sendo utilizada na análise a escrita presencial dos participantes durante as oficinas e as respostas dos mesmos às questões enviadas por e-mail. Resultados: Apontam para a superação dos medos internos e a descoberta das potencialidades pessoais para enfrentar desafios. A sensibilidade do olhar e o envolvimento com o outro como elemento fundamental no exercício de confiança das relações cotidianas. Conclusão: O processo de experimentação indica que existe uma potência do encontro, dos intercâmbios e da sensibilidade quando privilegiamos um conhecimento que se apreende com o corpo em movimento, relacionando-se com o outro e o mundo. Descritores: Experimentações estéticas, corpo, intervenção socioambiental

scholarly journals Indecomposable orthogonal invariants of several matrices over a field of positive characteristic

2020 ◽  
pp. 1-11
Author(s):  
Artem Lopatin
Keyword(s):  
Characteristic Polynomial ◽  
Orthogonal Group ◽  
General Linear Group ◽  
Positive Characteristic ◽  
The Other ◽  
Symmetric Matrices ◽  
Maximal Degree ◽  
Infinite Field ◽  
The Given ◽  
Field Of Positive Characteristic

We consider the algebra of invariants of [Formula: see text]-tuples of [Formula: see text] matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic [Formula: see text] different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic [Formula: see text] matrices. We establish that in case [Formula: see text] the maximal degree of indecomposable invariants tends to infinity as [Formula: see text] tends to infinity. In other words, there does not exist a constant [Formula: see text] such that it only depends on [Formula: see text] and the considered algebra of invariants is generated by elements of degree less than [Formula: see text] for any [Formula: see text]. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of [Formula: see text] the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

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