Replicable Rooms and Boundary Shrinkable Skyscrapers

2021 ◽  
pp. 385-390
Author(s):  
Stefan Behrens ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Solid Torus ◽  
Embedding Theorem ◽  
Vertical Boundary ◽  
Subsequent Chapter

‘Replicable Rooms and Boundary Shrinkable Skyscrapers’ points out precisely which properties of skyscrapers are required in the remainder of the proof of the disc embedding theorem. To achieve this, it introduces an abstraction of towers, known as buildings. The required properties for a generalized skyscraper include boundary shrinkability and replicability. The former allows the conclusion that the vertical boundary of a generalized skyscraper is a solid torus. Replicability ensures that any generalized skyscraper contains uncountably many other skyscrapers as subsets. Both of the above properties will be essential in the construction of the design in a subsequent chapter.

The Whitehead Decomposition

2021 ◽  
pp. 95-102
Author(s):  
Xiaoyi Cui ◽  
Boldizsár Kalmár ◽  
Patrick Orson ◽  
Nathan Sunukjian
Keyword(s):  
Euclidean Space ◽  
Real Line ◽  
Embedding Theorem ◽  
The Real ◽  
Subsequent Chapter ◽  
Simply Connected

‘The Whitehead Decomposition’ introduces this historically significant decomposition. Not only is the quotient of the 3-sphere by the Whitehead decomposition not homeomorphic to the 3-sphere, it is not even a manifold. In order to detect this curious fact, the notion of a noncompact space being simply connected at infinity is introduced. The chapter also describes the Whitehead manifold, which is a contractible 3-manifold not homeomorphic to Euclidean space. While the Whitehead decomposition does not shrink, its product with the real line does, as is proved in this chapter; in other words, the quotient of the 3-sphere by the Whitehead decomposition is a manifold factor. The proof of the disc embedding theorem utilizes Bing–Whitehead decompositions, which may be understood to be a mix between the Whitehead decomposition and the Bing decomposition from a previous chapter. In a subsequent chapter, precisely when Bing–Whitehead decompositions shrink is explained.

Key Facts about Skyscrapers and Decomposition Space Theory

2021 ◽  
pp. 395-398
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Solid Torus ◽  
Embedding Theorem ◽  
Space Theory ◽  
Decomposition Space

‘Key Facts about Skyscrapers and Decomposition Space Theory’ summarizes the input from Parts I and II needed for the remainder of the proof of the disc embedding theorem. Precise references to previous chapters are provided. This enables Part IV to be read independently from the previous parts, provided the reader is willing to accept the facts from Parts I and II summarized in this chapter. The listed facts include the shrinking of mixed, ramified Bing–Whitehead decompositions of the solid torus, the shrinking of null decompositions consisting of recursively starlike-equivalent sets, the ball to ball theorem, the skyscraper embedding theorem, and the collar adding lemma.

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

2021 ◽  
Vol 29 (6) ◽  
pp. 863-868
Author(s):  
Danila Shubin ◽  
◽  
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Solid Torus ◽  
Lens Space ◽  
Tubular Neighborhood ◽  
Lens Spaces ◽  
Ambient Manifold ◽  
Orientable Manifold ◽  
Smale Flows ◽  
Published Result

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

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