6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I

2012 ◽  
pp. 297-360
Keyword(s):  
Inductive Step ◽  
Fundamental Proposition ◽  
Hard Cases

The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II

2012 ◽  
Author(s):  
Spyros Alexakis
Keyword(s):  
Inductive Step ◽  
Tensor Fields ◽  
Minimum Rank ◽  
Fundamental Proposition ◽  
Hard Cases

This chapter takes up the proof of Lemma 4.24 in Case B; this completes the inductive step of the proof of the fundamental proposition 4.13. The chapter recalls that Lemma 4.24 applies when all tensor fields of minimum rank μ‎ in (4.3) have all μ‎ of their free indices being nonspecial. We recall the setting of Case B in Lemma 4.24: Let M > 0 stand for the maximum number of free indices that can belong to the same factor, among all tensor fields in (4.3). Then consider all μ‎-tensor fields in (4.3) that have at least one factor T₁ containing M free indices; let M' ≤ M be the maximum number of free indices that can belong to the same factor, other than T₁. The setting of Case B in Lemma 4.24 is when M < 2. The chapter recalls that in the setting of case B the claim of Lemma 4.24 coincides with the claim of Proposition 4.13. To derive Lemma 4.24, the authors use all the tools developed in the Chapter 6, most importantly the grand conclusion, but also the two separate equations that were added to derive the grand conclusion.

The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I

2012 ◽  
Author(s):  
Spyros Alexakis
Keyword(s):  
Inductive Assumption ◽  
Inductive Step ◽  
Local Equation ◽  
New Equation ◽  
Fundamental Proposition ◽  
Hard Cases

This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the assumption of Proposition 4.13. However, it finds that this new equation is very far from proving the claim of Lemma 4.24. It then has to return to the hypothesis of Proposition 4.13 and extract an entirely new equation from its conformal variation. It proceeds with a detailed study of this new equation (again using the inductive assumption of Proposition 4.13); the result is a second new local equation which again is very far from proving the claim of our lemma. Next, it formally manipulates this second new local equation and adds it to the first one, and observes certain miraculous cancellations, which yield new local equations that can be collectively called the grand conclusion. Lemma 4.24 in case A then immediately follows from the grand conclusion.

The Inductive Step of the Fundamental Proposition: The Simpler Cases

2012 ◽  
Author(s):  
Spyros Alexakis
Keyword(s):  
Inductive Assumption ◽  
Inductive Step ◽  
Appropriate Use ◽  
Fundamental Proposition

This chapter takes up the proof of Lemmas 4.16 and 4.19, which are easier than Lemma 4.24. The proof of these two lemmas relies on the study of the first conformal variation of the assumption of Proposition 4.13. This study involves many complicated calculations and also the appropriate use of the inductive assumption of Proposition 4.13.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

scholarly journals Hard cases and bad law

2011 ◽  
Vol 17 (9) ◽  
pp. 803-806
Author(s):  
T. Molloy ◽  
T. Graham
Keyword(s):  
Hard Cases

Religious Neutrality and the Government’s Freedom of Speech in the Jurisprudence of the us Supreme Court

2015 ◽  
Vol 10 (1) ◽  
pp. 1-23
Author(s):  
András Koltay
Keyword(s):  
Supreme Court ◽  
Freedom Of Speech ◽  
The State ◽  
Social Conditions ◽  
Religious Symbols ◽  
The Us ◽  
Us Supreme Court ◽  
The Government ◽  
Hard Cases

The issue of the use of religious symbols by the State, the Government, the Municipalities and Courts has emerged as a practical constitutional problem during the last quarter of a century. Contradictory examples of us Supreme Court jurisprudence prove that this issue is among the constitutional ‘hard cases’. The relatively recent appearance of the problem clearly indicates the ways in which American social conditions have changed and the transformation of us society’s attitude to religion.

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