Origins of Dimension Theory

DIMENSION THEORY OF LINEAR SOLENOIDS

Fractals ◽

10.1142/s0218348x07003423 ◽

2007 ◽

Vol 15 (01) ◽

pp. 63-72 ◽

Cited By ~ 1

Author(s):

JÖRG NEUNHÄUSERER

Keyword(s):

Dimension Theory ◽

Ergodic Measures ◽

Full Dimension ◽

Fractal Attractor

We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question of whether there exists a measure of full dimension.

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Model development to predict vibration response of roller bearings considering the material and thermal parameters using dimension theory

10.1063/5.0019375 ◽

2020 ◽

Author(s):

Surajkumar G. Kumbhar ◽

P. Edwin Sudhagar

Keyword(s):

Model Development ◽

Vibration Response ◽

Thermal Parameters ◽

Dimension Theory ◽

Roller Bearings

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L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002748 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 1169-1188 ◽

Cited By ~ 9

Author(s):

ROMAN SAUER

Keyword(s):

Probability Space ◽

Betti Numbers ◽

Free Action ◽

Countable Group ◽

Equivalence Relations ◽

Dimension Theory ◽

Discrete Groups ◽

Type Ii ◽

Orbit Equivalent ◽

Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

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Stone-čech Compactification and Dimension Theory for Regular σ-Frames

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-39.1.1 ◽

1989 ◽

Vol s2-39 (1) ◽

pp. 1-8 ◽

Cited By ~ 28

Author(s):

Bernhard Banaschewski ◽

Christopher Gilmour

Keyword(s):

Dimension Theory

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TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

Journal of Symbolic Logic ◽

10.1017/jsl.2016.45 ◽

2017 ◽

Vol 82 (1) ◽

pp. 347-358 ◽

Cited By ~ 4

Author(s):

PABLO CUBIDES KOVACSICS ◽

LUCK DARNIÈRE ◽

EVA LEENKNEGT

Keyword(s):

Algebraic Geometry ◽

Open Subset ◽

Cell Decomposition ◽

Real Algebraic Geometry ◽

Dimension Theory ◽

Definable Set ◽

Image Position ◽

Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

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