"You can use the metric to raise an index" is what I found in one or the other physics text book. Surely I missed some context each time, so I couldn't help to swear: 🤢 , I like the indexes where they are, what is the point of raising or lowering them.

I found the answer in Gravitation, by Misner, Thorne, Wheeler but felt the need to write it down in a way I understand it best: https://miamao.de/blog/2024-06/04.Raising_or_Lowering_an_Index.html

#physics #tensor #metric #vectorspace #dotproduct #raisinganindex #mtw

The Double Helix inside the NLP Transformer
https://arxiv.org/abs/2306.13817

We introduce a framework for analyzing types of information in a NLP Transformer. We distinguish four layers of information: positional, syntactic, semantic, and contextual.

We show that the distilled positional components of the embedding vectors follow the path of a helix, both on the encoder side & on the decoder side.
...

#NLP #MachineLearning #transformers #DependencyParsing #PartsOfSpeech #VectorSpace #informatics

I'm trying to educate myself in graduate level #mathematics and picked up this book by Isaacs on #GroupTheory characters. I managed to just so survive the first chapter which is a dense write up on #vectorspace #module theory. I'm able to read on, chapter 2 and 3 were still possible, but are there better entries into representations and characters?

Appreciate advice on:
1. easier but more specific exposition of central results?
2. advantages of developing everything in the A-module framework?
3. which chapters are key for subsequent developments?
4. which theorems are the ultimate highlights and which are 'just' key to the particular question at hand?
5. What's particularly worthwhile in the presentation approach taken in this book?

😂

https://youtu.be/SXPKYR417T8

#math #Monk #MrsMaisel #vectors #vectorspace #college #TonyShalhoub #1950s #vectorspace