24 Matching Annotations
1. May 2023
2. Local file Local file
1. 4.1-6 Zorn's lemma.

Useful for abstract vector spaces Banach Hahn.

2. 4.8-3 Lemma (Weak convergence).

weak convergences and its lemmas.

3. 4.6-2 Lemma (Canonical mapping).

The canonical mapping properly defined

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3. Apr 2023
4. Local file Local file
1. Lemma 2.13

The cute formula, generalized version.

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5. Local file Local file
1. 2.9-2 Lemma· (Zero vector)
2. 2.4-1 Lemma (Linear combinations)

Norm of the linear combinations of vectors are, larger than the sum of absolute value of all the scalar weight, by a strictly positive constant. Only for finite dimensional spaces.

$$\Vert \alpha_1 + \cdots + \alpha_n\Vert \ge c(|\alpha_1| + \cdots + |\alpha_n|)$$

3. 2.7-2 Lemma (Norm).

operator norm is actually a type of norm. Linear operators as homomorphism between the vector spaces are indeed a vector space itself. See here.

The properties of the operator norm basically interits the properties from the normed vector space. This is direct from the definition. However, these results are only for the bounded linear operators.

4. 2.6-11 Lemma (Inverse of product)
5. 2.5-4 F. Riesz's Lemma

Riesz's Lemma, preparing for a theorem about the norm ball in finite dimensional spaces.

6. 2.5-2 Lemma (Compactness)

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6. www.dl.behinehyab.com www.dl.behinehyab.com
1. Lemma 9.14

out of kilter algorithm

2. Lemma 9.13.

Out of Kilter Algorithm

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7. Mar 2023
8. www.dl.behinehyab.com www.dl.behinehyab.com
1. Lemma 7.12.

Upper bound for the distance label of the preflow push algorithm

2. Lemma 7.11.

dipath connection for all nodes in the graph.

3. Lemma 7.5

The shortest augmenting path matains a correct distance label for all the vertices during the iterations. By valid, we mean that there exists some way such that $$d(j) = d(i) + 1$$, where $$(i, j)\in A$$, for all $$i\in N$$.

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9. Feb 2023
10. Local file Local file
1. 5.1-5 Lemma (Continuity).

This is direct because the characterization contractive mapping resemble the Lipschitz Continuity in real analysis.

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11. Jul 2022
12. docdrop.org docdrop.org

This question regards the use of logic by Nagarjuna in his tetralemma and parallels in quantum mechanics.

Jay L. Garfield has some interesting and insightful observations about how Nagarjuna's logic works, and it relates to the different types of experiences where such statements could make sense.

2. it works on on logic essentially and shows look if you if you take this entity as existence you get into a contradiction for this and this reason 00:36:31 and it slowly demolishes all the possible foundations of our thinking not only objects but also causation itself also time itself also the self 00:36:44 itself and so on and so forth one one by one um showing that uh thinking that they are foundational they're they're they have intrinsic existence uh doesn't doesn't hold

Nagarjuna uses his tetralema to deconstruct logical arguments of thinking, existence of self, causation, time, intrinsic existence using logical arguments.j

For other viewpoints of Nagarjuna's Tetralemma, visit:

Judith Ragir discusses parallels between Dogen and Nagarjuna and employs Trungpha Rinpoche's Diamond Sliver diagram https://hyp.is/go?url=http%3A%2F%2Fwww.judithragir.org%2F2017%2F08%2Fdogen-nagarjunas-tetralemma-6%2F&group=world Graham Priest's paper on the Catuskoti / Tetralemma technique https://hyp.is/go?url=https%3A%2F%2Fbafybeifum5ioeus3y3hl4lqdwclgxpd6in4muleocuhsk3jev2rd7j3hpu.ipfs.dweb.link%2FThe-Logic-of-the-Catuskoti-by-G.-Priest.pdf&group=world

Professor Peter Adamson:of King's College London on his "History of Philosophy without any Gaps" podcast series https://hyp.is/go?url=https%3A%2F%2Fhistoryofphilosophy.net%2Fnagarjuna-tetralemma&group=world Here Peter interviews Nagarjuna expert, Jan Westerhoff of Oxford,in an insightful interview https://historyofphilosophy.net/nagarjuna-westerhoff

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1. THE LOGIC OF THE CATUSKOTI

Title: THE LOGIC OF THE CATUSKOTI Author: GRAHAM PRIEST Year: 2010

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14. historyofphilosophy.net historyofphilosophy.net
1. Jan Westerhoff on Nāgārjuna

Title: JAN WESTERHOFF ON NĀGĀRJUNA Author:Adamson, Peter & Negary, Jardin (???) Date: 23 July 2017

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15. historyofphilosophy.net historyofphilosophy.net
1. No Four Ways About It: Nāgārjuna’s Tetralemma

Title: 46. NO FOUR WAYS ABOUT IT: NĀGĀRJUNA’S TETRALEMMA Author:

insightful explanation of Nagarjuna's tetralemma!

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16. www.judithragir.org www.judithragir.org
1. Dogen will take all the 4 positions of Nagarjuna’s tetralemma (either, or, both, neither) and present them in one sentence or one paragraph. He might even debate the righteousness of one point of view and then combat it with the other point of view. He breaks up our attachment to our point of views and points his finger at total dynamic functioning.

Obviously this will appear confusing and frustrating to someone still trying to find "the right perspective" or "the right concept".

2. After bringing the opposites to our consciousness, we then want to decide; do they happen as “both” (at the top of the chart) or neither (which is at the bottom of the chart).  Is it the truth that everything is annihilated by the word “neither”?

Is it the truth that both form and emptiness coexist (Top of diagram)? Is that enough? No! THAT is still a position, that both exists. Is it then that neither exists? Is that enough? No! For even if neither exists, REALITY still exists!

The whole exercise gets us to rise above our thinking and look at the thinking process itself.

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17. Jul 2021
18. en.wikipedia.org en.wikipedia.org
1. The term 'lemma' comes from the practice in Greco-Roman antiquity of using the word to refer to the headwords of marginal glosses in scholia; for this reason, the Ancient Greek plural form is sometimes used, namely lemmata (Greek λῆμμα, pl. λήμματα).