Infinite-dimensional 4-manifold of finite cohomological dimension under the continuum hypothesis

1999 ◽  
Vol 66 (5) ◽  
pp. 550-555
Author(s):  
A. V. Karasev
Keyword(s):  
Continuum Hypothesis ◽  
Cohomological Dimension ◽  
Infinite Dimensional ◽  
The Continuum

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

Flow and Solute Transport in Saturated Porous Media: 1. The Continuum Hypothesis

2008 ◽  
Vol 11 (4) ◽  
pp. 403-413 ◽  
Author(s):  
Amgad Salama ◽  
Paul J. Van Geel
Keyword(s):  
Porous Media ◽  
Solute Transport ◽  
Continuum Hypothesis ◽  
Saturated Porous Media ◽  
The Continuum

Pattern Theory

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Computational Linguistics ◽  
Parametric Representation ◽  
Bayes Estimation ◽  
Geometric Transformation ◽  
Central Component ◽  
Pattern Theory ◽  
Infinite Dimensional ◽  
Engineering Mathematics ◽  
Dimensional Shape ◽  
The Continuum

Pattern Theory provides a comprehensive and accessible overview of the modern challenges in signal, data, and pattern analysis in speech recognition, computational linguistics, image analysis and computer vision. Aimed at graduate students in biomedical engineering, mathematics, computer science, and electrical engineering with a good background in mathematics and probability, the text includes numerous exercises and an extensive bibliography. Additional resources including extended proofs, selected solutions and examples are available on a companion website. The book commences with a short overview of pattern theory and the basics of statistics and estimation theory. Chapters 3-6 discuss the role of representation of patterns via condition structure. Chapters 7 and 8 examine the second central component of pattern theory: groups of geometric transformation applied to the representation of geometric objects. Chapter 9 moves into probabilistic structures in the continuum, studying random processes and random fields indexed over subsets of Rn. Chapters 10 and 11 continue with transformations and patterns indexed over the continuum. Chapters 12-14 extend from the pure representations of shapes to the Bayes estimation of shapes and their parametric representation. Chapters 15 and 16 study the estimation of infinite dimensional shape in the newly emergent field of Computational Anatomy. Finally, Chapters 17 and 18 look at inference, exploring random sampling approaches for estimation of model order and parametric representing of shapes.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

Kreisel, the continuum hypothesis and second order set theory

1976 ◽  
Vol 5 (2) ◽  
Author(s):  
Thomas Weston
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Second Order ◽  
The Continuum ◽  
Order Set

The Operational Significance of the Continuum Hypothesis in the Theory of Water Movement Through Soils and Aquifers

1984 ◽  
Vol 20 (5) ◽  
pp. 521-530 ◽  
Author(s):  
Philippe Baveye ◽  
Garrison Sposito
Keyword(s):  
Water Movement ◽  
Continuum Hypothesis ◽  
The Continuum

Regular Conditional Expectations and the Continuum Hypothesis

1994 ◽  
pp. 85-93
Author(s):  
I. V. Evstigneev
Keyword(s):  
Continuum Hypothesis ◽  
Conditional Expectations ◽  
The Continuum

HAS THE CONTINUUM HYPOTHESIS BEEN SETTLED?

2006 ◽  
pp. 56-75
Author(s):  
MATTHEW FOREMAN
Keyword(s):  
Continuum Hypothesis ◽  
The Continuum
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