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.

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 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.

Fifth Edition: the New Standard

2000 ◽  
Vol 5 (6) ◽  
pp. 1-7
Author(s):  
Christopher R. Brigham ◽  
James B. Talmage ◽  
Leon H. Ensalada
Keyword(s):  
Musculoskeletal System ◽  
Medical Information ◽  
Objective Assessment ◽  
Scientific Data ◽  
Practical Application ◽  
Fourth Edition ◽  
Appropriate Use ◽  
Medical Evaluation ◽  
Single Set

Abstract The AMA Guides to the Evaluation of Permanent Impairment (AMA Guides), Fifth Edition, is available and includes numerous changes that will affect both evaluators who and systems that use the AMA Guides. The Fifth Edition is nearly twice the size of its predecessor (613 pages vs 339 pages) and contains three additional chapters (the musculoskeletal system now is split into three chapters and the cardiovascular system into two). Table 1 shows how chapters in the Fifth Edition were reorganized from the Fourth Edition. In addition, each of the chapters is presented in a consistent format, as shown in Table 2. This article and subsequent issues of The Guides Newsletter will examine these changes, and the present discussion focuses on major revisions, particularly those in the first two chapters. (See Table 3 for a summary of the revisions to the musculoskeletal and pain chapters.) Chapter 1, Philosophy, Purpose, and Appropriate Use of the AMA Guides, emphasizes objective assessment necessitating a medical evaluation. Most impairment percentages in the Fifth Edition are unchanged from the Fourth because the majority of ratings currently are accepted, there is limited scientific data to support changes, and ratings should not be changed arbitrarily. Chapter 2, Practical Application of the AMA Guides, describes how to use the AMA Guides for consistent and reliable acquisition, analysis, communication, and utilization of medical information through a single set of standards.

Regime of Fiscal Incorporation (RIF): A Tool That Gives Competitiveness to Start-Ups in Mexico

2018 ◽  
Vol 9 (12) ◽  
pp. 1042-1046
Author(s):  
Tadeo Armando Barrón López ◽  
Keyword(s):  
Latin America ◽  
Income Tax ◽  
Appropriate Use ◽  
Old Regime ◽  
Advantages And Disadvantages ◽  
Start Ups

The following text will show the different tax forms for a newly created company to become competitive, analyze the subsidies they have in a federal tax (Income Tax), compare the tax incorporation regime (RIF) with The old regime of small taxpayers (REPECO), analyzes the advantages and disadvantages of the appropriate use of RIF for start-ups, and finally, the tax incorporation regime is compared with similar ones in Latin America, reflecting on tax contributions Which each government has to raise so that its governments are efficient and effective within a country.

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.

Sign in / Sign up

Export Citation Format

Share Document