Here ∆ is the Laplacian on a complete negatively curved (not bounded away from zero) noncompact manifold. The spectrum is the L^2 spectrum.
I am reading (in a paper) that the spectrum being contained in (-oo,-c] follows from the "spectral theorem." Here, if I'm not mistaken, ∆ is essentially self-adjoint. I don't see how the result follows...
They conclude in the paper that the spectrum equals (-oo,-c] by finding a sequence f*n* with <f*n,fn>=1 such that <∆fn,fn*>->-c. How does this conclude the result?
My guess: there is a general result that ∆ should have only a continuous spectrum, and that it must be connected. Then, by <∆f,f>=<Lf,f><=-c<f,f> for eigenfunction f, we get L≤-c. The sequence argument (I'm guessing) should show that -c is also in the spectrum.
Any help would be appreciated.
Thanks! Please let me know if this should go to r/math.
Let T be an Operator that maps a smooth function f (Domain [0,1]) to a smooth function T(f) with T(f) = h*f for some smooth function h. What is the continous spectrum of T? I already showed that the spectrum of T must be h([0,1]) but i don't know how to get to the continous spectrum.
Is functional analysis just insanely harder than other fields of math? I'm just about to do my first set of exams to get my masters in pure math. I have 3 exams in the next 2 weeks. I know I could have put in more work over the semester but at the same time I have done absolutely loads of work too.
Complex analysis and Galois theory just seem wayyyyy easier than functional analysis. I am not sure if its because the lecturer isn't as good. There's significantly less material online for this course and what is there is very unorganised and difficult to get through. I used to love functional analysis - i wrote a paper on the history of functional analysis for an undergraduate module and completed a thesis titled "Integral Categories in Functional Analysis" for undergrad! I was so excited to continue learning about it but this module has completely confused me!!
Anyway, yeah i don't know what I'm looking for in this post, just need to vent to my fellow mathematicians! Functional analysis has become a true love/hate topic for me, I just can't wrap my head around all the definitions and proof methods anymore. Maybe I will switch to algebra. 🥲
Do you guys have a favorite text? Or one that you think is important in someone's library.
Current text is course in functional analysis by Conway.
The more rudimentary the better. While I have the standard foundations in undergrad linear algebra the graduate level is a bit lacking.
I like the casual text of axlers Mira text, so if there is something like that as well all the better. This in contrast to an analysis text by j.yeh, which has a lot of info but doesn't read as nicely, if that makes sense.
Ordinarily, I’d put this in the Quick Questions thread, but, alas, it’s not a quick question, nor is it well-known. Let p and q be distinct primes, and let f be a continuous function from the p-adic integers to the q-adic rationals (a “(p,q)-adic function”, as I call it).
I am performing analysis (specifically, (p,q)-adic Fourier analysis) with such functions. This is a rarity, so much so that even Keith Conrad asked me on mathoverflow “what’s the point” (seeing as most uses of the p-adics require much more structure (ex: analyticity) than what can be found in (p,q)-adic functions—the zeroes of which, for example, can be any closed set of p-adic integers.
Well... they come up naturally in the study of the Collatz Conjecture and the dynamics of Collatz-type maps. For classically trained analysts such as myself, as weird as p-adic analysis is, (p,q)-adic Analysis is even weirder.
For example, for a (p,q)-adic function f, the following are equivalent:
• f is continuous
• f is integrable (with respect to the q-adic-valued Haar probability measure on the p-adics)
• f is representable by an absolutely convergent Fourier series
Other oddities include the fact that there is no analogue of the classical notion of two functions being equal “almost everywhere”; two (p,q)-adic functions have the same integral if and only if they are the same, everywhere.
In my work, I’ve found a way to extend the notion of integrability to include a class of discontinuous (p,q)-adic functions that arise out of interpolating certain well-behaved rational-valued functions on the non-negative rational integers to (p,q)-adic functions, for appropriately chosen values of p and q. The idea, in short, is to treat these discontinuous functions as measures (or, rather, as the (Radon-Nikodym?) derivatives of measures). In this way, one can show that these functions admit a unique Fourier transform, even though the usual integral transform formula fails to be convergent because the integrand, in this case, is not continuous.
Playing around with the (p,q)-adic Hahn-Banach Theorem gives me an argument which would make sense in a non-Archimedean context, but which produces something very odd in the (p,q)-adic context: namely, I obtain a series formula for my (p,q)-adic function which converges to the correct value for every p-adic integer input, but with the caveat that, if the input is a non-negative rational integer, the series must be treated as converging in the usual topo
... keep reading on reddit ➡Edit. Forget about what in brackets
Edit2. I don't want to delete this post, because there's a lot of interesting things was said in the comments, but just so you know: I formulated my question really bad and a lot stuff just incorrect, so I'm not expecting you actually answering this, especially if you don't want to waste time on dumb and apparently common misconceptions
Yesterday I asked a question if you can say that the F chord in C major chord progression slightly leaning towards Lydian mode (assuming that melody above chord progression is still in C major and hits Bnat). That F chord creates a sense of Lydian in C major progression.
And answers were actually divided.
Some people said that progressions like I-V-vi-IV are very simple and traditional, they're purely tonal and have nothing to do with modes.
Well, I disagree. Functional harmony is not chords themselves it's just how you look at them.
I have an argument.
Let's take Dm-G-C-F. Very strong ii-V-I resolution and just IV at the end (I know C-F is kinda V-I, but F on the weak beat and I'm pretty sure it gives more momentum rather than resolving). Seems very tonal, everything is just in major, but what if we were playing this progression at a very slow tempo? 60 for example. Wouldn't the distance between chords change be so big, that connections of degrees to the tonic became kinda weak? Our ear would just "forget" that there was a Cmajor chord until we reach G.
Harmony is not written in our heads when we hear it - it's happening. If you play strong Dm-G-C resolution, but then 7 bars of F chord it doesn't matter - it will be Lydian even tho we didn't change the key at all. When we hit F chord under C major - harmony starts to slowly morph inLydian, and more time goes by the more obvious it became. That's why I thought that the title "I think there are no clear vivid borders between modes" makes sense
Key change makes modes sound more obvious and saturated. Some people said that to create Lydian sound in major progression you need borrowed chord - modal mixture. But that's not the strongest sense of Lydian either. Lydian modal mixture is just a brief key change to the closest, from the right side, key (Gmaj for C). If we modulate to F# it will create a more obvious Lydian, yet we do consider borrowed chords as a good example of modal harmony. https://youtu.be/YoX6OQOBAtE
I just don't like the fact that just because we traditionally analyzed simple di
... keep reading on reddit ➡I know Research exists but is extremely limited and is mostly just a news aggregator (unless there’s some hidden feature that I’m dumb and haven’t discovered). I mean something like Seeking Alpha, Yahoo Finance et al with statements data, projections, commentary, etc. I think this would be a nice add up to Plus and I would seriously consider paying for it
I finished a physics degree with a math minor 6 years ago, and finished grad school with an engineering phd a year ago. My math education went through first semester real analysis and complex analysis, but I never had to go beyond there when I switched to engineering. I’ve always been interested in functional analysis, so I’m looking for a book appropriate to my level of knowledge and experience.
Hey, I'm going to be reading Royden's book chapters 7,8,17,18, and so on as far as I can go from there. I'm taking a course in the subject next semester so figured I'd get a head-start on the reading.
If anyone wants to join, I'd be down for weekly meetings to talk about the text and solve problems from the end of the chapters.
https://preview.redd.it/ywwgcsu9ob981.png?width=1143&format=png&auto=webp&s=694672eff335e3c6b6055e3f502245f5a5ba5243
I'm currently reading this paper (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.297.8841&rep=rep1&type=pdf) on invariant sets for PDEs and am confused by an argument in Lemma 1. Hopefully you can access the paper, but if not I'll try to make my question self contained.
Essentially, you have a function U:Rn x R → Rm, U: (x,t) → U(x,t) that's a solution to a system of PDEs on the domain D x [0,T]. It is assumed that U(x,t) is unique and is also continuous everywhere in D x [0,T]. Importantly, up to time t*, the set {U(x,t):(x,t) ∈ D x [0,t*]} is contained inside a closed convex set Sp ⊂ Rm and there is a special point (x*,t*) with the property that U(x*,t*)∈∂Sp .
At one point in Lemma 1 of the paper, they consider the function p∙U(x,t) where p is the outward normal vector to the set Sp at the special point U(x*,t*). The paper argues that "since Sp is convex, the function p∙U(x,t) attains its maximum value in D x [0,t*] at (x*,t*)." "Therefore at (x*,t*):
p∙∂U(x,t)/∂t ≥ 0, p∙∂U(x,t)/∂xi=0 i=1,...,n, and p∙∂^2U(x,t)/∂xi∂xj is negative semidefinite.
Broadly speaking, I understand where the argument comes from. Namely, the dot product is a linear (convex and concave) function and hence a maximum is going to be obtained on the boundary of a convex domain. Moreover, the three derivative conditions come from necessary conditions for optimality and from properties of the Hessian matrix at a maximum. Where I'm having issues though is that the function is nested. So while the range of U is a subset of the convex set Sp, it need not be convex itself, and hence those optimality conditions need not apply. Also, supposing those conditions did hold, if the x* where the maximum occurs is on the boundary of D, directional derivatives are not going to be defined in every dimension. So the second condition of p∙∂U(x,t)/∂xi=0 could fail as well.
Sorry for the long post, but any clarification that you can offer would be great. I get the impression I'm missing something obvious because the paper goes over these points so quickly, but I can't seem to find something that clears things up.
Hi, I do not understand the part where he mentions" between two consecutive zeroes"
https://ibb.co/Vj7z1g1
This post is to the functional analysts out there! I have been searching online to find a “pathway in functional analysis” which I (and many other people interested in the area) could use but I really don’t think this was asked before. I am a sophomore and so far my background and the path I followed is:
- Banach and Hilbert spaces with the big theorems like Hahn-Banach, open mapping, Riesz Representation
- Spectral theory in Banach algebras
What do you think is the best pathway to learn functional analysis? I think I like it because it is the intersection of topology, algebra and analysis. Why do you like it, or not?
Lastly, I think I will be doing research in functional analysis in the future. That seems to be my goal right now. I know functional analysis is widely used in PDEs. What are some non-PDE areas in math that use/are related to functional analysis? If your research is in functional analysis, what exactly do you work on?
(CSE: SPR) (OTCQB: SRUTF) (FSE: 38G)
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They’re trading at CAD$0.045 with a market cap of CAD$16.21 million. They’re pre profit but they’re getting close to commercializing their products and have virtually no debt.
Sproutly is also looking to expand into Europe and Israel and are keeping a close eye on regulatory authorities and potential strategic partnerships in Europe, especially good timing with Germany’s new government announcing this week that they’re going to legalize weed.
Most recently they signed a LOI with Canadian licensed Kingston Cannabis to launch their cannabis infused beverages.
They already have a partnership with Halo Collective in the US for a hemp beverage, launched during the pandemic, it’s one of the top selling hemp beverages in the US.
They believe that gap they can fill in the cannabis market is the lack of proper, scientifically backed products, focusing on water soluble oils, a process they have perfected and is a significant barrier for many other companies. A focus on the whole plant is where they differentiate themselves from the market, they don’t want to focus on THC or CBD isolate, instead of focusing on making these compounds as high as possible in the plant they take a more natural approach, using a combination of different compounds in naturally occurring cannabis plants.
I think their approach to what’s still a relatively new market is good, I think it’s good for first time users to buy from a company focused not he science behind the products. I also think they have huge potential in the functional cannabis product market, by analyzing the different compounds and using them in combination they can differentiate themselves from companies just selling CBD.
Not financial advice, always do your own research.
Hi, guys. I'm currently in my final year of my undergrad and for the next semester, I have to choose a dissertation topic. This is the first time I will be doing anything remotely close to this, so I am pretty nervous.
I have always been inclined towards topics like metric spaces, group theory but I have also enjoyed my linear algebra course a lot and I really want to work under that professor. He told me that the topic would be Hilbert spaces which are a great mixture of linear algebra and analysis and I was stoked. But I googled it later and it said that it had heavy applications in thermodynamics and partial differential equations and so on. I would love to study this, but I also don't want to go into applications too much and I want to instead veer more into the algebra and analysis part of it. He told me that I should choose my dissertation topic properly and not make it something wildly different from my future interests and when I asked him if it was application based or more theoretical, he just said that if applications come into the purview, we have to deal with them as they do. He taught us mechanics too and while I enjoyed it, I clearly know that I would love to go into the theoretical side more.
So, I just want to know what are my prospects after functional analysis? Like where does it go into next? Like how our real analysis course lead to metric spaces and then topology and so on.
Any advice is appreciated ;-;
Sorry if this comes off as an incredibly dumb question, I'm just very anxious about not making the wrong choice.
Tldr; What are the future branches of functional analysis?
I finished a physics degree with a math minor 6 years ago, and finished grad school with an engineering phd a year ago. My math education went through first semester real analysis and complex analysis, but I never had to go beyond there when I switched to engineering. I’ve always been interested in functional analysis, so I’m looking for a book appropriate to my level of knowledge and experience.
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