[QUOTE=thefreemann;52992470]Can anyone sort of “generalize” the Fourier Transform mathematically? As in, what does it mean to take a Fourier transform? Is it basically changing the units to their inverse counterpart? Like in my EE classes it’s of course used to go between the time and frequency domain, but I’ve also been introduced to it in my physics classes as a method between the normal spatial domain (meters) and k-space? (Inverse meters)
The wiki seems to only talk about it in terms of the EE application, obviously given it has the most application in that field.[/QUOTE]
[url=https://en.wikipedia.org/wiki/Pontryagin_duality]Pontryagin duality[/url] expresses the Fourier transform in an abstract form. I can't claim to understand it in much detail.
[QUOTE=JohnnyMo1;52993007][url=https://en.wikipedia.org/wiki/Pontryagin_duality]Pontryagin duality[/url] expresses the Fourier transform in an abstract form. I can't claim to understand it in much detail.[/QUOTE]
Jesus Christ what am I reading
The most "general" explanation of Fourier transforms that I've seen claim that they're analogous to matrix multiplication (and if you consider the Discrete Fourier transform, they're identical). A matrix transforms a vector from one set of basis vectors to another, and the continuous time Fourier Transform transforms a function from one set of basis functions to another (complex exponentials).
So basically what you're doing is expressing your function as a linear combination of complex exponentials. The function F(jw) determines the weighting of each complex exponential, where jw is the argument to the basis function (in this case, the frequency of the complex exponential). I feel like my ability to describe Fourier transforms decreases the more I use them so god knows if this makes any sense.
Other integral transforms use the same idea. Z-transforms express the function as a linear combination of z^t basis functions, where z is the argument. Laplace transforms are a generalization of Fourier transforms, expressing functions using exp(st) basis functions, where s is the argument. s is complex, and setting the real part to zero gives exp(jwt), which is exactly the Fourier transform basis function.
I have no idea where the interpretation from the physics classes came from. Outside of EE, I only see them used to solve differential equations, and rarely is an interpretation of the Fourier domain ever of use in those fields. JohnnyMo's answer is probably the most correct and abstract, but this is the extent to which I understand it and I have an unhealthy emotional attachment to the math that governs EE.
In other news, finished my first real analysis class and aside from the part where I got 37% on the final, I kind of wonder if I should have gone into math instead of EE. Just applied to grad school last week so changing majors is slightly out of the question. I've never enjoyed a class nearly as much as I did this one.
[QUOTE=JohnnyMo1;52993007][url=https://en.wikipedia.org/wiki/Pontryagin_duality]Pontryagin duality[/url] expresses the Fourier transform in an abstract form. I can't claim to understand it in much detail.[/QUOTE]
Would a slightly less difficult generalization be that of Hilbert spaces? I never saw it that formally but that's my intuition.
[QUOTE=JohnnyMo1;52993007][url=https://en.wikipedia.org/wiki/Pontryagin_duality]Pontryagin duality[/url] expresses the Fourier transform in an abstract form. I can't claim to understand it in much detail.[/QUOTE]
I am Fourier and I am confused by this.
p-adic metrics are the most annoying things ever, it's so hard to get a good intuition with them
[QUOTE=halofreak472;52994092]Jesus Christ what am I reading[/QUOTE]
[QUOTE=Fourier;53006813]I am Fourier and I am confused by this.[/QUOTE]
Fear the power of categories.
Analysts: "Finally, a perfect theorem with lots of epsilons and deltas and absolutely no elegant machinery anywhere."
Algebraists: "THOU FOOL! THIS IS JUST THE NATURALITY OF A PARTICULAR FUNCTOR."
Analysts: "Nooooooooo!"
In all seriousness, as I said, I don't know much about it, but I think it's cool how it unifies the Fourier transform and series and the discrete Fourier transform. They all seem closely related and there's a reason for it.
[QUOTE=Number-41;52994490]Would a slightly less difficult generalization be that of Hilbert spaces? I never saw it that formally but that's my intuition.[/QUOTE]
Yep. Well, I dunno about difficulty, but this subsumes that. Not quite sure how they're related when you generalize to distributions though.
[QUOTE=Number-41;52994490]Would a slightly less difficult generalization be that of Hilbert spaces? I never saw it that formally but that's my intuition.[/QUOTE]
Yeah, Fourier Transform usually work on multi-dimensional vectors (time series for example), but you can also do Fourier Transform on functions (you still do dot products, it's just integrals with sines/cosines/e^x-es in this case).
Number-41, were you guy dealing with tensor imaging. If so, how is it going?
Good, just had to spend months writing a grant proposal that got denied so I did absolutely fuck all in terms of research :v:
But I got my signal model working (the one that estimates a single white matter fiber diffusion MRI signal). I also saw a pretty cool [URL="https://arxiv.org/abs/1709.04281"]PhD defense[/URL] that deals with estimating multi-exponential signals (even with damping). It is able to combine the Prony method with the DFT. I wonder if it could be generalized to SH basis functions instead of ordinary complex exponentials.
Next up is cleaning up lots of code to make it pretty and bug-free, writing a paper and then Cramer-Rao Lower Bound stuff.
Well, even though it got denied you did lots of exercise and gained experience. So it's not complete waste.
The lower bound stuff is to calculate & prove how precise & accurate your method is?
I'm studying up for this moduli spaces course in spring. I'm hoping to read Mumford's Red Book plus brush up on rings and fields in the next month, probably amounting to some 400 pages of reading. Time to see just how much brain-work I can muster before I burn out. :v:
[QUOTE=JohnnyMo1;53011223]I'm studying up for this moduli spaces course in spring. I'm hoping to read Mumford's Red Book plus brush up on rings and fields in the next month, probably amounting to some 400 pages of reading. Time to see just how much brain-work I can muster before I burn out. :v:[/QUOTE]
If you will be in the flow all the time, you will probably do it just fine :).
[QUOTE=Fourier;53011349]If you will be in the flow all the time, you will probably do it just fine :).[/QUOTE]
Going alright so far, hoping to finish the rings chapter of Aluffi by the end of the day. Not working all the problems, of course, there's not enough time for that. Just one or two from each section.
[editline]28th December 2017[/editline]
I'll admit my background on commutative algebra and fields sucks, so even seeing the little bit of algebraic geometry sprinkled in so far (Krull dimension, and Spec of a ring) that have been sprinkled in so far is neat.
[QUOTE=Fourier;53011198]Well, even though it got denied you did lots of exercise and gained experience. So it's not complete waste.
The lower bound stuff is to calculate & prove how precise & accurate your method is?[/QUOTE]
We will use it to optimise how we measure our signal (optimal experimental design).
I've gotta make notes for ring theory because there are so many different classes of things that aren't named so you can easily remember what they mean.
"Integral domain." It's... like the integers? Sure, but how.
"Noetherian domain." Thanks Noether.
"Euclidean domain." STOP WITH THE NAMES.
At least there's "unique factorization domain" and "principal ideal domain," but "ideal" doesn't really bring to mind what it is either. Groups don't seem to have this problem.
"Group." "Group action." "Free group." "Subgroup." Pretty descriptive. Admittedly "normal subgroup" kinda sucks.
[QUOTE=JohnnyMo1;53023291]I've gotta make notes for ring theory because there are so many different classes of things that aren't named so you can easily remember what they mean.
"Integral domain." It's... like the integers? Sure, but how.
"Noetherian domain." Thanks Noether.
"Euclidean domain." STOP WITH THE NAMES.
At least there's "unique factorization domain" and "principal ideal domain," but "ideal" doesn't really bring to mind what it is either. Groups don't seem to have this problem.
"Group." "Group action." "Free group." "Subgroup." Pretty descriptive. Admittedly "normal subgroup" kinda sucks.[/QUOTE]
Just a case of having to remember them all, was an utter pain having to remember them all when I was first studying ring theory. Infact I made a diagram from semi-groups up to finite fields, all having one extra condition for them to "escalate" up one rank. I'm finishing up my paper on covering the basics of ring theory, ideal, local rings, radicals, and finishing on nullstellensatz if you're interested.
[QUOTE]The visible waves are approximately:
10.5 8 6.4 4.8 3.6 2.6
If we convert these to ratios:
4.04 3.08 2.46 1.85 1.38 1 [/QUOTE]
How exactly were these numbers converted? I'm having a brain fart thinking about it.
[QUOTE=dingusnin;53023465]Just a case of having to remember them all, was an utter pain having to remember them all when I was first studying ring theory. Infact I made a diagram from semi-groups up to finite fields, all having one extra condition for them to "escalate" up one rank. I'm finishing up my paper on covering the basics of ring theory, ideal, local rings, radicals, and finishing on nullstellensatz if you're interested.[/QUOTE]
Sure, that sounds helpful. Now I remember you saying before you were going to do a project on radical ideals :v:
Bet you know all about annihilating radical right-ideals now, eh comrade?
[QUOTE=REMBER;53025234]How exactly were these numbers converted? I'm having a brain fart thinking about it.[/QUOTE]
Divide all by 2.6?
I'd like to apologize in advance if/when I come off as ignorant because I'm in a somewhat unique situation right now. I'd also like to apologize if some of this seems slightly incoherent as I'm writing this very early in the morning.
Basically I've garnered all of my computer science knowledge up to this point through practical use and very little mathematical involvement. Recently I've been trying to tackle more complex problems during programming challenges and realized I completely lack the necessary mathematical background to appropriately solve them.
More specifically I was basically able to get by in college by doing up to pre-cal (which covered trig, since some colleges separate their trig courses here), and statistics but that's really about as far as I got since I didn't major in CS. So now I'm at a point of needing beyond-calculus level mathematics and I'm completely in the dark on where to go.
The problem I was working on in question dealt with number theory, namely with partitions and calculating the number of them: p(n). I decided to take a look at the [URL="https://en.wikipedia.org/wiki/Partition_(number_theory)#Partition_function"]Wikipedia page[/URL] and immediately realized I understood little to none of this stuff so I went to go find a different source.
Unfortunately I haven't found a single source that was able to easily explain this stuff (for obvious reasons, it's number theory), which is when I realized that I'm at an impasse.
So here's where my question comes into play. I can understand Sigma and Pi notation and understand what this does (although I might be misreading the p(n)x^n part as well):
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/50f07d199eb8f2c86f45c8dcfabe3048acbb896c[/img]
But what I don't understand is how one expands it into this:
(1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) ....
I understand how the summation produces the first term in the series, but I don't understand how or why Pi produces the subsequent terms.
Furthermore I'm having trouble understanding how "1-x^k" in the denominator of the product, even though Euler's function, becomes this:
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/5496271e059880797222bf1962908415440e7252[/img]
And finally, I'm struggling to understand how I'm supposed to calculate a finite result when this is the finalized function, and also how I'm meant to determine how the signs alternate in this function:
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/d24f0330f5fab348bc60fca0cf8e1bffaa5548ea[/img] - how the signs alternate according to Wiki page
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/5a264a17130eb6abede12d5a6d85df3ce36fd18c[/img]
This is obviously way too much to cover in a single post, but I'm more hoping for some pointers as to where I could go to more easily ground myself so I can better understand what's actually going on (although some information and guidance on this topic would be helpful as well).
[QUOTE=JohnnyMo1;53025352]Sure, that sounds helpful. Now I remember you saying before you were going to do a project on radical ideals :v:
Bet you know all about annihilating radical right-ideals now, eh comrade?[/QUOTE]
I'll send it over once I've finished my section on spectrums, given a nice paragraph at the end of each chapter describing how things behave though homomorphisms (i.e. prime ideals in a localised ring being prime in the original ring), and a few examples etc.
I'm having huge difficulty getting a decent page count. It's stupidly low at the minute with only 20 pages (although I'm expecting that to go up to 32), but I mean there isn't that much to say about radical ideals themselves, only what they are used for.
I have a bit of an issue. Lately in my classes I feel like I'm not able to do well simply because my mathematical prowess is kind of poor.
Today I was embarrassed after I failed to realize what the Fundamental Theorem of Algebra was. I mean I've always known that an n-degree polynomial can have at most n roots but not necessarily by its name.
Granted was just me not knowing a definition, but I think the core issue is that my math fundamentals are weak, and when I try to do this higher level math I'm often stumbling in places which simply build off older concepts from past classes.
I've basically concluded that this is due to the following:
[B]1. [/B]My high school education was relatively poor compared to the rest of the nation (My state is like almost #50 in terms of public education rankings) and my math teachers rarely provided a rigorous challenge.
[B]2. [/B]But at the same time I was not very motivated in high school, it's unreasonable for me to blame it entirely on my teachers.
I mean, I got 5's on my AP tests but you don't have to be great at Calculus to do that, and AP classes are extraordinarily easy compared to a university equivalent.
It's really quite sad for me knowing that a weak foundation in math is what's preventing me from doing better. If you guys have any advice to give I'd greatly appreciate it. What should I do? Work through textbooks in every subject from Algebra 1 --> Multivariable Calculus? Buy an exercises book in a certain topic and work on it a little every day? I'm really at a loss.
Why is the Monty Hall problem always explained so poorly?
I only [I]just[/I] got the logic behind it because I finally saw the (incredibly simple) explanation behind it, instead of a vague "you have more information".
Anyone know what the total derivative of this function is? Wolfram is giving me a strange answer.
T(x,y)=cos^2(x)sin^2(y)
[QUOTE=WitheredGryphon;53037508]I'd like to apologize in advance if/when I come off as ignorant because I'm in a somewhat unique situation right now. I'd also like to apologize if some of this seems slightly incoherent as I'm writing this very early in the morning.
Basically I've garnered all of my computer science knowledge up to this point through practical use and very little mathematical involvement. Recently I've been trying to tackle more complex problems during programming challenges and realized I completely lack the necessary mathematical background to appropriately solve them.
More specifically I was basically able to get by in college by doing up to pre-cal (which covered trig, since some colleges separate their trig courses here), and statistics but that's really about as far as I got since I didn't major in CS. So now I'm at a point of needing beyond-calculus level mathematics and I'm completely in the dark on where to go.
The problem I was working on in question dealt with number theory, namely with partitions and calculating the number of them: p(n). I decided to take a look at the [URL="https://en.wikipedia.org/wiki/Partition_(number_theory)#Partition_function"]Wikipedia page[/URL] and immediately realized I understood little to none of this stuff so I went to go find a different source.
Unfortunately I haven't found a single source that was able to easily explain this stuff (for obvious reasons, it's number theory), which is when I realized that I'm at an impasse.
So here's where my question comes into play. I can understand Sigma and Pi notation and understand what this does (although I might be misreading the p(n)x^n part as well):
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/50f07d199eb8f2c86f45c8dcfabe3048acbb896c[/img]
But what I don't understand is how one expands it into this:
(1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) ....
I understand how the summation produces the first term in the series, but I don't understand how or why Pi produces the subsequent terms.
Furthermore I'm having trouble understanding how "1-x^k" in the denominator of the product, even though Euler's function, becomes this:
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/5496271e059880797222bf1962908415440e7252[/img]
And finally, I'm struggling to understand how I'm supposed to calculate a finite result when this is the finalized function, and also how I'm meant to determine how the signs alternate in this function:
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/d24f0330f5fab348bc60fca0cf8e1bffaa5548ea[/img] - how the signs alternate according to Wiki page
[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/5a264a17130eb6abede12d5a6d85df3ce36fd18c[/img]
This is obviously way too much to cover in a single post, but I'm more hoping for some pointers as to where I could go to more easily ground myself so I can better understand what's actually going on (although some information and guidance on this topic would be helpful as well).[/QUOTE]
To answer your first question: each term in the product on the right-hand side is reminiscent of the formula for the sum of a geometric series. (1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) .... is just the first few terms of the first three geometric series written out. I think [url=https://en.wikipedia.org/wiki/Geometric_series#Formula]this section[/url] of the geometric series page might clear up any confusion.
But this isn't the only way to view the right-hand side. You can also view the denominator as Euler's formula, in which case the whole thing expands to 1 / (1 - x - x^2 + x^5 + ... ). Whereas the first way of expanding the right-hand side (using geometric series) kept the infinite product, this expansion uses the pentagonal number theorem to relate the original infinite product (of the denominator) to the series (1 - x - x^2 + x^5 + ...).
This second way of expanding the right-hand side has the nice result of revealing a recurrence in the partition function, meaning you can calculate p(n) as a recursive function. First, notice that p(n) = p(n - 1) + p(n - 2) + p(n - 5) + ... always evaluates to a finite series. While this doesn't seem to be the case at first glance, remember that p(n) is 0 for negative integers. Now, focus on the numbers being subtracted from n in each of the terms. These are the generalized pentagonal numbers, given by expanding and alternating m in m(3m - 1)/2. That is, if f(m) = m(3m − 1)/2, then f(1) = 1, f(-1) = 2, f(2) = 5, f(-2) = 7, f(3) = 12, etc. Using the same value of m you used to compute a term, you simply compute the coefficient (-1)^(|m| - 1) to find the term's sign.
From here, you can use p(0) = 1 to find any other p(n) (though it may take an unreasonably long time to do so).
[QUOTE=thefreemann;53076612]I have a bit of an issue. Lately in my classes I feel like I'm not able to do well simply because my mathematical prowess is kind of poor.
Today I was embarrassed after I failed to realize what the Fundamental Theorem of Algebra was. I mean I've always known that an n-degree polynomial can have at most n roots but not necessarily by its name.
Granted was just me not knowing a definition, but I think the core issue is that my math fundamentals are weak, and when I try to do this higher level math I'm often stumbling in places which simply build off older concepts from past classes.
I've basically concluded that this is due to the following:
[B]1. [/B]My high school education was relatively poor compared to the rest of the nation (My state is like almost #50 in terms of public education rankings) and my math teachers rarely provided a rigorous challenge.
[B]2. [/B]But at the same time I was not very motivated in high school, it's unreasonable for me to blame it entirely on my teachers.
I mean, I got 5's on my AP tests but you don't have to be great at Calculus to do that, and AP classes are extraordinarily easy compared to a university equivalent.
It's really quite sad for me knowing that a weak foundation in math is what's preventing me from doing better. If you guys have any advice to give I'd greatly appreciate it. What should I do? Work through textbooks in every subject from Algebra 1 --> Multivariable Calculus? Buy an exercises book in a certain topic and work on it a little every day? I'm really at a loss.[/QUOTE]
Instead of buying anything, why not check out all the free sites online dedicated to these subjects? There's Khan Academy and Lemma for exercises in everything from arithmetic to linear algebra, and places like Coursera and EdX for more advanced courses. I don't know where you're at exactly (or what your goals are), but if it's just fluency in manipulation you're looking for, try getting through [url=https://www.lem.ma/content/AqolhXSAxkGDIdSYQJ3q-A?book_id=CwXXPubAmVbSjFn6FOWZfw]these[/url]. If you're looking to better understand proofs, check out Coursera's [url=https://www.coursera.org/learn/mathematical-thinking]Introduction to Mathematical Thinking[/url].
You might be different, but I always learn best (and am most engaged) when I'm combining easier studies with harder ones. Maybe try working through some exercises and videos online in simpler subjects while reading through a book on something that's challenging you. Take your time and see how far you can get.
[QUOTE=download;53079115]Anyone know what the total derivative of this function is? Wolfram is giving me a strange answer.
T(x,y)=cos^2(x)sin^2(y)[/QUOTE]
I'm assuming that x and y are both dependent on some unspecified variable t. If so, then the total derivative is 2cos^2(x)sin(2y)dy/dt - sin(2x)sin^2(y)dx/dt.
Statistician here, moving into biostats, do I belong here? Been recently working with taxonomists and behavior ecologists, and negative binomial regression models are cancer.
[QUOTE=thefreemann;53076612]I have a bit of an issue. Lately in my classes I feel like I'm not able to do well simply because my mathematical prowess is kind of poor.
Today I was embarrassed after I failed to realize what the Fundamental Theorem of Algebra was. I mean I've always known that an n-degree polynomial can have at most n roots but not necessarily by its name.
-snipp-[/QUOTE]
I can empathize with this. Currently at my sophomore year in a heavy-duty engineering college and although I haven't been really shafted from my math prowess, I can feel the pressure occasionally. I'm not in a super math-heavy major either, but junior year is when things really kick into gear, so.
Having a solid math foundation and math 'prowess' (honestly probably the best word to describe the general sense of your familiarity with math) really helps such a huge amount with engineering problems. Being able to think logically, mathematically, etc can make approaching certain problems super easy.
In my two years here I've already improved quite a bit. If I stopped thinking and simply started doing and really hammering away at online resources I think I'd be able to make progress as well, so I should probably do that. Of course, the nature of time and school and busyness makes this not so simple.
Part of the thing that gets me, though, is being unaware of what is [I]good[/I] to know. Should I focus on the underlying logic/interaction/etc? I may need to spend some time defining my goals clearly.
[QUOTE=Repulsion;53136577]I can empathize with this. Currently at my sophomore year in a heavy-duty engineering college and although I haven't been really shafted from my math prowess, I can feel the pressure occasionally. I'm not in a super math-heavy major either, but junior year is when things really kick into gear, so.
Having a solid math foundation and math 'prowess' (honestly probably the best word to describe the general sense of your familiarity with math) really helps such a huge amount with engineering problems. Being able to think logically, mathematically, etc can make approaching certain problems super easy.
In my two years here I've already improved quite a bit. If I stopped thinking and simply started doing and really hammering away at online resources I think I'd be able to make progress as well, so I should probably do that. Of course, the nature of time and school and busyness makes this not so simple.
Part of the thing that gets me, though, is being unaware of what is [I]good[/I] to know. Should I focus on the underlying logic/interaction/etc? I may need to spend some time defining my goals clearly.[/QUOTE]
Lately I think I've been getting better with regards to my post. I think for me, one of the hardest things was forming connections between different concepts. However, I think that part of the process has been improving recently.
Last week we saw how the orthogonality concept in Fourier Series arises from its linear algebra interpretation, and taking linear algebra myself right now, I was able to follow along really easily, and that lecture by far has been one of my favorites.
For me, I think another thing was realizing that the problems we're given usually have some underlying pattern/theme to them. It's kind of obvious in retrospect, but reframing the problem with the attitude that "hey I learned everything about these problems, now all I have to do is apply that knowledge" really helps with some of the intimidation and frustration that comes with solving them. To put it another way, I think my confidence in my own ability was one of the things holding me back from doing better.
Just wrote a Python function for numerical analysis homework. It adaptively partitions an interval for piecewise linear interpolation subject to some tolerances. I thought I noticed a way to improve the speed, so I added 2 lines of code and changed a couple of characters in a third.
First version: >300,000 loop iterations
Second version: a bit over 1,000 loop iterations
That is a very satisfying optimization.
I'm taking a course on differential geometry, and while most of the time I feel like I'm understanding what's going on, the assignments have been owning me big-time
[img]https://i.imgur.com/Jt2CPY1.png[/img]
Where a surface normal [B]n[/B] is given by
[img]https://i.imgur.com/p098m6u.png[/img]
This (part a) seems pretty straightforward, and it would be, if the algebra didn't make it a complete disaster. For example, for sigma(0, 0), sigma_u = [B]b_01[/B] - [B]b_00[/B] and sigma_v = [B]b_10[/B] - [B]b_00[/B]. The cross product of these is absolutely disgusting, much less its norm, and it makes me think I'm going about this entirely wrong. I feel like I must be missing something pretty obvious but I've been kinda stuck just thinking about it for a liiiiittle bit too long and if anyone has any insight I would actually love you so much
I've taken differential topology and I'm in a differential geometry reading group now and I have no idea what any of that means. Are you using a textbook?
Sorry, you need to Log In to post a reply to this thread.