Spectral Theory Puns

My professor suggested me "Spectra of Graphs" by Brouwer and Haemers. But I think the book assumes a lot and skips some steps that I eventually figure out but it is time-consuming for me.

For some more context, I've done a course on Graph Theory and a basic Linear Algebra course.

I have a BS in physics, so I have taken a course in linear algebra, plus two courses in QM, so I have some familiarity with abstract vector spaces/Hilbert spaces/linear operators etc.

Despite this I have been having trouble parsing some of the textbooks I've been perusing on the subject (all graduate level texts that assume just a bit more knowledge than I already have, and use a lot of foreign terminology).

I am trying to get a grasp of spectral theory for its applications in data science (like 90% of r/math, I currently work as a software developer), and I'm looking for a textbook/lecture notes that will hold my hand a little bit with some of the rigor (physics education was fast and loose).

Thanks

I've been studying color theory for 2 years now, but I'm a little confused about the RGB color model. I know that RGB is the objectively best color model (that uses 3 primaries). I learned that this was because you have three cones in your eyes, the S M and L cones, and each one was sensitive to a different wavelength, corresponding to the wavelengths for blue, green, and red respectively. I learned that because they corresponded to those wavelengths, they'd give us the brightest colors and largest gamut.

...However, I've recently come across the spectral sensitivity chart, which is... making my head hurt. (imagine chidi anagonye from the good place seeing the dot in the 'i'. That's my mood.) In case you aren't aware, this chart visually shows the wavelengths that each cone is receptive to. Here are some examples:

https://preview.redd.it/qyp9y5edg4r51.png?width=287&format=png&auto=webp&s=b5b545e797151234558944eee34926f48330944a

https://preview.redd.it/sp1f9hbfg4r51.png?width=1200&format=png&auto=webp&s=a1c83fde29d4bd4a0531188bad03deccc83c3cef

What confuses me is that yes, the S cone has the highest sensitivity for blue/violet, and yes, the M cone has the highest sensitivity for green. But the L cone's peak sensitivity is around yellow or yellow/green, not red. Every article I've read about spectral sensitivity confirms this observation.

This discovery is unraveling my knowledge of color theory. Why do people keep saying that the L cone sees red the best? Why do color theory articles perpetuate that lie? Couldn't people just say "blue and red are on the farthest ends of the visible color spectrum, and green is in the middle"? Since it appears that a cone's peak sensitivity doesn't matter to the RGB color model, why can't we use violet instead of blue, since violet is closer to the end of the visible light spectrum, therefore allowing a larger color range/gamut? If the cones' peak sensitivity doesn't really matter to make an ideal color model (otherwise we'd be using blue, green, and yellow-green as primaries), is there any point to using a color model with just 3 colors anymore, since the cones /apparently/ don't really matter? sdafhsdkjfhaskldfhsd? askdfhlasdkjfsdfasf????

TL;DR: Can someone please explain why RGB is the best color model without using the "each cone in your eye corresponds to red, green, or blue" argument? Can someone explain why we don't use purple instead of blue then?

The novel always told us that the reason Spectral Soul Demon was targeted by fate gu was because he tried to come back to life and take over the world but honestly there are many examples of venerable not fully dying or tampering with the future like giant sun saving up for a last attack or red lotus having a chess match (can't remember details)

Spectral soul demon was near max attainment level in every path meaning he was the closest to omniscience and finding out how to obtain eternal life so Fate Gu had to do something. This was the real more important reason that fate gu was against spectral soul demon. Otherwise why would it resort to using an otherworldly demon beyond its control to deal with the situation instead of just limiting lifespan gu and letting spectral soul demon rule for a little bit (who is an incomplete otherworldly demon and no threat to it).

Ok, so this theory has a massive hole which is if Spectral soul demon was really a venerable with the highest attainment in every path he would have unfathomable methods that could maybe hurt fate gu further or do something else but it still seems risky. I don't mean to be a hindsight Harald but helping an otherworld demon (the only thing you can't influence) seems like a huge oversight/misplay.

FY > spectral soul demon

I recently finished a course on spectral theory. However, other than some remarks on how it can be applied to the study of differential equations, we never really discussed any major applications of it. Does anyone here know of any nice applications of spectral theory?

I am a math/cs undergrad and I was working on an analysis of algorithms homework yesterday and stumbled upon the Kirchhoff's theorem while trying to find a general formula for the number of Spanning Trees in a graph. This sent me down a very interesting rabbit hole, and was wondering where I should start to really understand what is going on here. I've taken proof based LA, Discrete Math, and Real Analysis.

Also what exactly is the significance of an eigenvalue in relation to a discrete graph?

I'm currently working through "Principles of Harmonic Analysis" by Deitmar and Echterhoff and they're also doing a little bit about spectral theory in a Banach-Algebra. They start by considering a unital Banach-Algebra and defining the spectrum as usual: 𝜆 is in the spectrum of a if and only if 𝜆 - a is not invertible. Of course in a non-unital Banach-Algebra this definition makes no sense because there is no unit element so "invertible" is meaningless. What they do is that they start with a non-unital Banach-Algebra A and then they adjoin a unit element as follows: Let B = A x ℂ with the usual vector space structure and define the multiplication by (a,𝜆)(b,𝜇) = (ab + 𝜆b + 𝜇a, 𝜆𝜇). The norm on B is given by ‖(a,𝜆)‖ = ‖a‖ + |𝜆|. One can prove that B is a unital Banach-Algebra and that the map a ↦ (a,0) isometrically embeds A into B. Now we can define the spectrum of a in A as the spectrum of (a,0) in B.

Now my question: Why is that the "right way" to define the spectrum in a non-unital Banach-Algebra? I can see that it works but it seems kind of arbitrary. B is not uniquely defined by the property that A isometrically embeds into it, right? We could use any other embedding of A into some unital Banach-Algebra C and define the spectrum with respect to C. So why exactly do we pick the unital Banach-Algebra B as constructed above?

Another question: We can do the very same construction if A is already unital itself. We then get a unital Banach-Algebra B such that A embeds into it but in general B will be a different unital Banach-Algebra than A. Can we at least guarantee that the definition of the spectrum coincides? So is the spectrum of a in A with respect to the unital Banach-Algebra A the same as the spectrum of (a,0) in B with respect to the unital Banach-Algebra B?

I’m kind of screwed with regards to a pretty basic undergraduate-level research paper I have to write on spectral graph theory and it’s applications. Anybody here have any recommendations for good introduction sources, surveys of the field and such?

I just completed an honours level Functional Analysis course and there was no motivation for the section on spectral theory.

I had a lot of trouble with this section because I didn't really understand the point of it, aside from maybe the definition of the continuous functional calculus.

My issue is that looking at an operator T on some space and asking which scalars t we have such that T-tI is not invertible seems like a very strange thing to ask. I realise that in some sense this is a generalisation of eigenvalues but the eigenvalues are just the point spectrum, so what's the point of the rest of the spectrum?

Keep in mind I'm more algebraically inclined than analytic.

So yeah why is spectral theory useful?

What are your opinions on this book? Is it worth reading, or is there some other useful book about spectral graph theory?

And join the Activity Club. Suzie will get all depressed and junk because everyone has left her for them. She ends up getting possessed by an evil spirit and takes her revenge, blowing up Max's house.

...and quite possibly can't control people with strong wills.

If he could control a spectral, he very well could have just jumped onto Isaac, Isabel, Max or Ed at any point, off to go start more drama in an attempt to get into the teacher's lounge.

And he very well could have just jumped into Johnny- no reason to use an innocent person, now, and also gets revenge on Johnny for Jeff, putting Jonny in detention and possibly sending him to juvy.

However, so far Hijack has only been capable of possessing a dog and a definitely not spectral, timid, orange haired nerd, plus some unknown person.

I've been meaning to learn more about spectral graph theory for some time. I'd prefer something along the lines of a free pdf, but if there're any really spectacular books out there any of you'd recommend, I love me a good textbook!

I was wondering, a differential equation of the form: [; -\frac{d^2 y}{dx^2} + V(x) y = E y ;] with V(x) as some quadratic polynomial, the eigenvalues are equally spaced. That is, [; E_n = a n + b ;].

My question is, does there exist another non-trivial V(x) which results in all equally space eigenvalues?

Note: One possible solution which I will consider trivial is V(x) = infinity for x < 0, and x^2 for x > 0.

This is relevant in quantum optics and quantum field theory, where each successive eigenvalue represents another photon in a particular mode, which only really makes sense if the eigenvalues are equally spaced. We assume the underlying potential is harmonic, but I was wondering if there is another possibility.

Any advice on the prerequisites leading up to spectral graph theory and a good textbook for self study?