102.22 Proof without Words: sums of sums of factorials is one less than a factorial: inductive step

2018 ◽  
Vol 102 (554) ◽  
pp. 309-310
Author(s):  
Günhan Caglayan
Keyword(s):  
Inductive Step

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.

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 basic inductive step

1983 ◽  
pp. 68-86
Author(s):  
Colin J. Bushnell ◽  
Albrecht Fröhlich
Keyword(s):  
Inductive Step

The Intertwining Lemma

2019 ◽  
pp. 199-210
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Single Point ◽  
Inductive Step ◽  
Induction Step ◽  
Definition Of

This chapter gives a proof of the Intertwining Lemma. Section 20.2 lists out the formulas for all the maps involved. Section 20.3 recalls the definition of Z* and proves Statement 3 of the Intertwining Lemma. Section 20.4 proves statements 1 and 2 of the Intertwining Lemma for a single point. Section 20.5 decomposes Z* into two smaller pieces as a prelude to giving the inductive step in the proof. Section 20.6 proves the following induction step: If the Intertwining Lemma is true for g ɛ GA then it is also true for g + dTA (0, 1). Section 20.7 explains what needs to be done to finish the proof of the Intertwining Lemma. Section 20.8 proves the Intertwining Theorem for points in Π‎A corresponding to the points gn = (n + 1/2)(1 + A, 1 − A) for n = 0, 1, 2, ... which all belong to GA. This result combines with the induction step to finish the proof, as explained in Section 20.7.

scholarly journals What the argument from evil should, but cannot, be

2018 ◽  
Vol 56 (3) ◽  
pp. 333-348
Author(s):  
JAMES HENRY COLLIN
Keyword(s):  
A Priori ◽  
Base Rate ◽  
Inductive Step ◽  
Base Rate Fallacy ◽  
Inductive Inferences ◽  
Argument From Evil

AbstractMichael Tooley has developed a sophisticated evidential version of the argument from evil that aims to circumvent sceptical theist responses. Evidential arguments from evil depend on the plausibility of inductive inferences from premises about our inability to see morally sufficient reasons for God to permit evils to conclusions about there being no morally sufficient reasons for God to permit evils. Tooley's defence of this inductive step depends on the idea that the existence of unknown rightmaking properties is no more likely, a priori, than the existence of unknown wrongmaking properties. I argue that Tooley's argument begs the question against the theist, and, in doing so, commits an analogue of the base rate fallacy. I conclude with some reflections on what a successful argument from evil would have to establish.

scholarly journals Regeneration of Plants from Peach Embryo Cells Infected with a Shooty Mutant Strain of Agrobacterium

1991 ◽  
Vol 116 (6) ◽  
pp. 1092-1097 ◽  
Author(s):  
Ann C. Smigocki ◽  
Freddi A. Hammerschlag
Keyword(s):  
Mutant Strain ◽  
Prunus Persica ◽  
Zeatin Riboside ◽  
Inductive Step ◽  
Chain Reaction ◽  
Regenerated Plants ◽  
Embryo Cells ◽  
Total Plant ◽  
Polymerase Chain ◽  
Callus Tissues

Immature `Redhaven' peach [Prunus persica (L.) Batsch] embryos were infected with a shooty mutant strain of Agrobacterium tumefaciens, tms328::Tn5, which carries an octopine-type Ti plasmid with a functional cytokinin gene and a mutated auxin gene. Shoots were regenerated from embryo-derived callus that was initiated on MS medium lacking phytohormones. Shoots exhibited increased frequency of branching and were more difficult to root than the noninfected. Transcripts of the tms328::Tn5-cytokinin gene were detected using northern analyses of total plant RNA. Polymerase chain reaction of genomic DNA and cDNA resulted in amplification of DNA fragments specific for the cytokinin gene, as determined by restriction enzyme and Southern analyses. The concentrations of the cytokinins zeatin and zeatin riboside in the leaves of regenerated plants were on the average 51-fold higher than in leaves taken from nontransformed plants. None of the shoots or callus tissues were postive for octopine. The expression of the T-DNA encoded cytokinin gene promotes growth of peach cells in the absence of phytohormones, thus serving as a marker for transformation. In addition, this gene appears to promote morphogenesis without an auxin inductive step.

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