Ordered Groups: A Case Study in Reverse Mathematics

1999 ◽  
Vol 5 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Reed Solomon
Keyword(s):  
Complex Analysis ◽  
Metric Spaces ◽  
Reverse Mathematics ◽  
Ordered Group ◽  
Ordered Groups ◽  
Countable Basis ◽  
Order Arithmetic ◽  
Quotient Groups

The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson (1985) gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups (including an equivalent of ACA0), groups and semigroup conditions which imply orderability (WKL0), the orderability of free groups (RCA0), Hölder's Theorem (RCA0), Mal'tsev's classification of the order types of countable ordered groups ( – CA0)

scholarly journals – CA0 and order types of countable ordered groups

2001 ◽  
Vol 66 (1) ◽  
pp. 192-206 ◽  
Author(s):  
Reed Solomon
Keyword(s):  
Group Theory ◽  
Related Problem ◽  
Reverse Mathematics ◽  
Order Type ◽  
Second Order ◽  
Ordered Group ◽  
Second Order Arithmetic ◽  
Ordered Groups ◽  
Countable Ordinal ◽  
Order Arithmetic

Reverse mathematics uses subsystems of second order arithmetic to determine which set existence axioms are required to prove particular theorems. Surprisingly, almost every theorem studied is either provable in RCA0 or equivalent over RCA0 to one of four other subsystems: WKL0, ACA0, ATR0 or – CA0. Of these subsystems, – CA0 has the fewest known equivalences. This article presents a new equivalence of – C0 which comes from ordered group theory.One of the fundamental problems about ordered groups is to classify all possible orders for various classes of orderable groups. In general, this problem is extremely difficult to solve. Mal'tsev [1949] solved a related problem by showing that the order type of a countable ordered group is ℤαℚε where ℤ is the order type of the integers, ℚ is the order type of the rationals, α is a countable ordinal, and ε is either 0 or 1. The goal of this article is to prove that this theorem is equivalent over RCA0 to – CA0.In Section 2, we give the basic definitions and notation for RCA0, ACA0 and CA0 as well as for ordered groups. For more information on reverse mathematics, see Friedman, Simpson, and Smith [1983] or Simpson [1999] and for ordered groups, see Kokorin and Kopytov [1974] or Fuchs [1963]. Our notation will follow these sources. In Section 3, we show that – CA0 suffices to prove Mal'tsev's Theorem and the reversal is done over RCA0 in Section 4.

scholarly journals Metric spaces related to Abelian groups

2021 ◽  
Vol 22 (1) ◽  
pp. 169
Author(s):  
Amir Veisi ◽  
Ali Delbaznasab
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Additive Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Ordered Groups ◽  
The Family ◽  
Nonempty Compact ◽  
Metric Topology ◽  
Induced Topology

<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl<sub>X</sub>A if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N<sub>d</sub>(B, g) and B ⊆ N<sub>d</sub>(A, g), where N<sub>d</sub>(A, g) = {x ∈ X : d(x, A) &lt; g}, d<sub>B</sub>(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d<sub>A</sub>(B), d<sub>B</sub>(A)}.</p>

REVERSE MATHEMATICS OF MF SPACES

2006 ◽  
Vol 06 (02) ◽  
pp. 203-232 ◽  
Author(s):  
CARL MUMMERT
Keyword(s):  
Metric Space ◽  
Closed Subset ◽  
Metric Spaces ◽  
Reverse Mathematics ◽  
General Topology ◽  
Second Order Arithmetic ◽  
Perfect Set ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF (P) denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF (P) | p ∈ F}. We define a countably based MF space to be a space of the form MF (P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text].

scholarly journals Generalized absolute values, ideals and homomorphisms in mixed lattice groups

Positivity ◽  
2020 ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Jani Jokela ◽  
Lassi Paunonen
Keyword(s):  
Basic Structure ◽  
Vector Spaces ◽  
Ordered Group ◽  
Lattice Vector ◽  
Absolute Value ◽  
Lattice Group ◽  
Riesz Spaces ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Quotient Groups

Abstract A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

scholarly journals Assessment of health/functioning of older adults who consume psychoactive substances

2018 ◽  
Vol 71 (3) ◽  
pp. 942-950
Author(s):  
Vania Dias Cruz ◽  
Silvana Sidney Costa Santos ◽  
Jamila Geri Tomaschewski-Barlem ◽  
Bárbara Tarouco da Silva ◽  
Celmira Lange ◽  
...  
Keyword(s):  
Older Adults ◽  
Qualitative Case Study ◽  
International Classification Of Functioning ◽  
Psychoactive Substances ◽  
Systematic Observation ◽  
Health Functioning ◽  
Final Consideration ◽  
Theory Of Complexity

ABSTRACT Objective: To assess the health/functioning of the older adult who consumes psychoactive substances through the International Classification of Functioning, Disability and Health, considering the theory of complexity. Method: Qualitative case study, with 11 older adults, held between December 2015 and February 2016 in the state of Rio Grande do Sul, using interviews, documents and non-systematic observation. It was approved by the ethics committee. The analysis followed the propositions of the case study, using the complexity of Morin as theoretical basis. Results: We identified older adults who consider themselves healthy and show alterations - the alterations can be exacerbated by the use of psychoactive substances - of health/functioning expected according to the natural course of aging such as: systemic arterial hypertension; depressive symptoms; dizziness; tinnitus; harmed sleep/rest; and inadequate food and water consumption. Final consideration: The assessment of health/functioning of older adults who use psychoactive substances, guided by complex thinking, exceeds the accuracy limits to risk the understanding of the phenomena in its complexity.

scholarly journals Sparse Autoencoder-based Multi-head Deep Neural Networks for Machinery Fault Diagnostics with Detection of Novelties

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Zhe Yang ◽  
Dejan Gjorgjevikj ◽  
Jianyu Long ◽  
Yanyang Zi ◽  
Shaohui Zhang ◽  
...  
Keyword(s):  
Fault Diagnosis ◽  
Supervised Classification ◽  
Deep Neural Networks ◽  
Activation Function ◽  
Reconstruction Error ◽  
Fine Tuning ◽  
Fault Diagnostics ◽  
Sparse Autoencoder

AbstractSupervised fault diagnosis typically assumes that all the types of machinery failures are known. However, in practice unknown types of defect, i.e., novelties, may occur, whose detection is a challenging task. In this paper, a novel fault diagnostic method is developed for both diagnostics and detection of novelties. To this end, a sparse autoencoder-based multi-head Deep Neural Network (DNN) is presented to jointly learn a shared encoding representation for both unsupervised reconstruction and supervised classification of the monitoring data. The detection of novelties is based on the reconstruction error. Moreover, the computational burden is reduced by directly training the multi-head DNN with rectified linear unit activation function, instead of performing the pre-training and fine-tuning phases required for classical DNNs. The addressed method is applied to a benchmark bearing case study and to experimental data acquired from a delta 3D printer. The results show that its performance is satisfactory both in detection of novelties and fault diagnosis, outperforming other state-of-the-art methods. This research proposes a novel fault diagnostics method which can not only diagnose the known type of defect, but also detect unknown types of defects.

scholarly journals Neither a Novel Tau Proteinopathy nor an Expansion of a Phenotype: Reappraising Clinicopathology-Based Nosology

2021 ◽  
Vol 22 (14) ◽  
pp. 7292
Author(s):  
Luca Marsili ◽  
Jennifer Sharma ◽  
Alberto J. Espay ◽  
Alice Migazzi ◽  
Elhusseini Abdelghany ◽  
...  
Keyword(s):  
Spinocerebellar Ataxia Type ◽  
Niemann Pick Disease ◽  
Whole Exome ◽  
Heterozygous Variant ◽  
History Of ◽  
Ataxia Syndrome ◽  
Niemann Pick

The gold standard for classification of neurodegenerative diseases is postmortem histopathology; however, the diagnostic odyssey of this case challenges such a clinicopathologic model. We evaluated a 60-year-old woman with a 7-year history of a progressive dystonia–ataxia syndrome with supranuclear gaze palsy, suspected to represent Niemann–Pick disease Type C. Postmortem evaluation unexpectedly demonstrated neurodegeneration with 4-repeat tau deposition in a distribution diagnostic of progressive supranuclear palsy (PSP). Whole-exome sequencing revealed a new heterozygous variant in TGM6, associated with spinocerebellar ataxia type 35 (SCA35). This novel TGM6 variant reduced transglutaminase activity in vitro, suggesting it was pathogenic. This case could be interpreted as expanding: (1) the PSP phenotype to include a spinocerebellar variant; (2) SCA35 as a tau proteinopathy; or (3) TGM6 as a novel genetic variant underlying a SCA35 phenotype with PSP pathology. None of these interpretations seem adequate. We instead hypothesize that impairment in the crosslinking of tau by the TGM6-encoded transglutaminase enzyme may compromise tau functionally and structurally, leading to its aggregation in a pattern currently classified as PSP. The lessons from this case study encourage a reassessment of our clinicopathology-based nosology.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

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