Open Questions in Reverse Mathematics

2011 ◽  
Vol 17 (3) ◽  
pp. 431-454 ◽  
Author(s):  
Antonio Montalbán
Keyword(s):  
Reverse Mathematics ◽  
Background Information ◽  
Open Questions ◽  
Open Question

AbstractWe present a list of open questions in reverse mathematics, including some relevant background information for each question. We also mention some of the areas of reverse mathematics that are starting to be developed and where interesting open question may be found.

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Richard A. Shore
Keyword(s):  
Reverse Mathematics ◽  
The Other ◽  
Base System ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Points Of View ◽  
The One ◽  
Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

Avatars in E- and U-Learning

2016 ◽  
pp. 792-815
Author(s):  
Raymond Szmigiel ◽  
Doris Lee
Keyword(s):  
Collaborative Learning ◽  
Virtual Environments ◽  
Learning Environments ◽  
Learning Experience ◽  
Three Dimensional ◽  
Virtual Learning Environments ◽  
Background Information ◽  
Virtual Agents ◽  
Learning Context ◽  
Open Question

Avatars are virtual agents or characters that graphically represent users within virtual environments. Avatars can be implemented in three-dimensional (3-D) virtual environments for training purposes. While there are promising findings indicating that avatars can enhance the learning experience, conclusive and generalized evaluations cannot be made at this time. The effectiveness of these virtual agents in a learning context remains an open question. The purpose of this chapter is to present background information on the definitions and use of avatars in e-based, virtual learning environments and to address the applicability of avatars to ubiquitous learning (u-learning). This chapter examines the available empirical research on the effectiveness of avatars in facilitating social interactivity, motivation, and collaborative learning in 3-D environments. Finally, this chapter provides suggestions for future studies on the design of avatars in both e- and u-learning.

Open question argument

2018 ◽  
Author(s):  
Tristram McPherson
Keyword(s):  
Natural Sciences ◽  
Ethical Naturalism ◽  
Natural Properties ◽  
The Core ◽  
Open Questions ◽  
Open Question Argument ◽  
Naturalistic Account ◽  
Open Question

The open question argument is the heart of G.E. Moore’s case against ethical naturalism. Ethical naturalism is the view that goodness, rightness, etc. are natural properties; roughly, the sorts of properties that can be investigated by the natural sciences. Moore claims that, for any candidate naturalistic account of an ethical term according to which ‘good’ had the same meaning as some naturalistic term A, we might without confusion ask: ‘I see that this act is A, but is it good?’ Moore claimed that the existence of such open questions shows that ethical naturalism is mistaken. In the century since its introduction, the open question argument has faced a battery of objections. Despite these challenges, some contemporary philosophers claim that the core of Moore’s argument can be salvaged. The most influential defences link Moore’s argument to the difficulty that naturalistic ethical realists face in explaining the practical role of ethical concepts in deliberation.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

2019 ◽  
Vol 18 (09) ◽  
pp. 1950167 ◽  
Author(s):  
M. Chacron ◽  
T.-K. Lee
Keyword(s):  
Finite Order ◽  
Division Ring ◽  
Affirmative Answer ◽  
Major Result ◽  
Division Rings ◽  
Open Questions ◽  
Engel Condition ◽  
Open Question ◽  
Positive Integers

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

scholarly journals SEPARATING PRINCIPLES BELOW RAMSEY'S THEOREM FOR PAIRS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350007 ◽  
Author(s):  
MANUEL LERMAN ◽  
REED SOLOMON ◽  
HENRY TOWSNER
Keyword(s):  
Big Five ◽  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.

RADICAL RELATED TO SPECIAL ATOMS REVISITED

2014 ◽  
Vol 91 (2) ◽  
pp. 202-210
Author(s):  
HALINA FRANCE-JACKSON ◽  
SRI WAHYUNI ◽  
INDAH EMILIA WIJAYANTI
Keyword(s):  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Radical ◽  
Radical Theory ◽  
Factor Ring ◽  
Open Questions ◽  
Open Question ◽  
The Way

AbstractA semiprime ring $R$ is called a $\ast$-ring if the factor ring $R/I$ is in the prime radical for every nonzero ideal $I$ of $R$. A long-standing open question posed by Gardner asks whether the prime radical coincides with the upper radical $U(\ast _{k})$ generated by the essential cover of the class of all $\ast$-rings. This question is related to many other open questions in radical theory which makes studying properties of $U(\ast _{k})$ worthwhile. We show that $U(\ast _{k})$ is an N-radical and that it coincides with the prime radical if and only if it is complemented in the lattice $\mathbb{L}_{N}$ of all N-radicals. Along the way, we show how to establish left hereditariness and left strongness of important upper radicals and give a complete description of all the complemented elements in $\mathbb{L}_{N}$.

scholarly journals PROFIL GURU, PEMAHAMAN KOOPERATIF NHT, DAN KEMAMPUAN BERPIKIR TINGKAT TINGGI DI SMP KABUPATEN MINAHASA UTARA

el–Hayah ◽  
2012 ◽  
Vol 1 (4) ◽  
Author(s):  
Femmy Kawuwung
Keyword(s):  
Science Teacher ◽  
Junior High ◽  
Higher Order Thinking ◽  
Thinking Skills ◽  
Higher Order ◽  
Higher Order Thinking Skills ◽  
Open Questions ◽  
Study Population ◽  
Open Question ◽  
General Profile

The research aims to uncover a general profile of teachers, understanding cooperative NHT, and higher-order thinking skills in middle school biology teacher North Minahasa regency. Research methods: survey with a research instrument is a questionnaire consisting of questions developed is an open question and a combination of semi-closed and open questions that have been validated by experts of learning. The study population is a junior high science teacher in North Minahasa Biology. The questionnaire that was circulated questionnaires amounted to 40 and who entered and analyzed totaled 31. The study began in July and August 2010. The results are: Understanding the Biology of junior high science teacher in North Minahasa regency NHT 6.45% towards learning, and teachers' understanding of higher-order thinking skills 12.90%. Teachers' understanding of the NHT and the results of cooperative learning is still low.<br /><br />Keywords: Profile of Teachers, Science-Biology, NHT, High Thinking, Junior North Minahasa.<br /><br />

scholarly journals Normal and Jones surfaces of knots

2018 ◽  
Vol 27 (06) ◽  
pp. 1850039 ◽  
Author(s):  
Efstratia Kalfagianni ◽  
Christine Ruey Shan Lee
Keyword(s):  
Experimental Evidence ◽  
Jones Polynomial ◽  
Normal Surface ◽  
Colored Jones Polynomial ◽  
Open Questions ◽  
Strong Slope ◽  
Open Question

We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open questions. We establish a relation between the Jones period of a knot and the number of sheets of the surfaces that satisfy the Strong Slope Conjecture (Jones surfaces). We also present numerical and experimental evidence supporting a stronger such relation which we state as an open question.

scholarly journals OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS

2016 ◽  
Vol 22 (2) ◽  
pp. 151-169 ◽  
Author(s):  
LUDOVIC PATEY
Keyword(s):  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
The Past ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Ramsey Type ◽  
Infinite Set

AbstractRamsey’s theorem states that for any coloring of then-element subsets of ℕ with finitely many colors, there is an infinite setHsuch that alln-element subsets ofHhave the same color. The strength of consequences of Ramsey’s theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey’s theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey’s theorem.

scholarly journals Living with ankylosing spondylitis: an open response survey exploring physical activity experiences

2019 ◽  
Vol 3 (2) ◽  
Author(s):  
Peter C Rouse ◽  
Martyn Standage ◽  
Raj Sengupta
Keyword(s):  
Physical Activity ◽  
Online Survey ◽  
Physical Activities ◽  
Activity Levels ◽  
Physically Active ◽  
Self Monitoring ◽  
Diverse Range ◽  
Open Questions ◽  
Specific Factors ◽  
Open Question

Abstract Objective The aim was to gather in-depth, rich accounts of physical activity experiences of people living with AS, to include symptom management, consequences for symptoms, factors that encourage and disrupt physical activity, and motivations that underpin participation in physical activity. Methods Participants (n = 149; 60% female) completed a Bristol Online Survey that consisted of open questions to capture rich qualitative data. In total, 96% of participants self-reported having AS (1% other arthritis; 3% missing), and 51% had this diagnosis for >20 years. A content analysis was conducted to identify the key themes/factors from within the open question responses. A frequency analysis was used to ascertain the most commonly identified themes and factors. Results Fifty different physical activities were participated in over the previous month. Physical activity can improve and worsen arthritis symptoms, and fluctuations in participation exist even in the most active. Pain and fatigue were the two most frequently identified factors that stopped people with AS from being physically active. Participants reported more autonomously driven motivations than controlled motivations for participating in physical activity. Conclusion People with AS can and do participate in a diverse range of physical activities, but fluctuations in activity levels occur owing to disease- and non-disease-specific factors. Individually tailored plans and self-monitoring are important to optimize levels of physical activity and maximize benefits for people living with AS. Multiple reasons why AS patients participate in physical activity were revealed that included both adaptive (i.e. autonomous) and maladaptive (i.e. controlled) forms of motivation.

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