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Swedish Mathematician Lennart Carleson Wins Abel
William Robinson writes "Sci Tech is reporting that Swedish mathematician Lennart Carleson has won the Abel Prize on Thursday for proving a 19th century theorem on harmonic analysis. His theorems have been helpful in creating iPod. Prof Carleson's major contributions have come in two fields - the first has subsequently been used in the components of sound systems and the second helps to predict how markets and weather systems respond to change. One of Carleson's many triumphs was settling a conjecture that had remained unsolved for over 150 years. He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."
My math prof used to shoot chalk pieces onto me for saying something like that!
That's when I decided that statistics is more my kinda speed.
We used to have a Bill of Rights. Now, with the rights gone, all we have left is the bill.
I thought I was tought that in school...
"He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."
Negligible? In engineering, maybe. In mathematics, never.
This guy Carleson helped create the iPod and his brother, Arthur, ran a radio station in Cincinatti.
What're the odds on that?
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Young people today. You tell them about a deep result in real analysis, and the only thing they're interested in is how it relates to their iPod. And get off my lawn.
Athletic Scholarships to universities make as much sense as academic scholarships to sports teams.
Wiki Article on the Breakthrough
"Sure there's porn and piracy on the Web but there's probably a downside too."
Well I hope his blonde haired blue eyed and bikini clad assistants got recognised to.
"I'm going to f***ing bury that guy, I have done it before, and I will do it again. I'm going to f***ing kill Google"
"He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."
Someone correct me if I am wrong but hasn't this been common knowledge?? I was under the impression this was taught in every sytems, signals analysis course?? It would have been interesting to determine the fourier transform of many blackbox circuits without running under this assumption. Or is there more to what he solved??
I think the invisible hand of the market has its middle finger extended
--A wise old fart named SC0RN
Since TFA doesn't mention anything about how his connection to iPods, I don't see how he helped the iPod any more than I did by not joining Apple and botching up the manufacturer's contract for producing crack-resistant LCD screens ... oh wait.
"...Swedish mathematician Lennart Carleson has won the Abel Prize on Thursday for proving a 19th century theorem on harmonic analysis. His theorems have been helpful in creating iPod."
What. The. Fuck?
I am very glad for him.
The result he proved is nice mathematics, but you don't need it for iPods or audio coding. First of all, for many engineering purposes, it only matters that it works, not that you can prove that it works theoretically. Secondly, audio coding is done over discretely sampled signals, and most of those theorems become simple linear algebra in that case.
No wireless. Less decimals than pi. Lame.
Yeah, it might be that the MPEG-4/AAC/H.264 algorithms are based in part on Fourier analysis, but I fail to understand how Carleson's theorems have been used in making the iPod. Cupertino is hardly knowledgable in the more esoteric realms of theoretical mathematics, and there is simply no need to incorporate such stuff in an mp4 player.
This is bad journalism, written by bad reporters who lack the most basic understanding of mathematics and engineering. He just thought it might be cool to clam in an iPod in the mess.
The iPod reference is completely misleading, as simple harmonic analysis is way bigger than just an iPod. It's merely talking about this guy proving that Fourier was basically right, validating harmonic analysis and expanding the horizons for signal processing. That's the biggie: signal processing, not the bloody iPod. The stupid article probably includes iPod just for the sake of hits.
Oh really? Search Wikipedia entries, the articles, all links - no mention of iPod except in those annoying side adverts. Why? Because it has nothing to do with it
Credit where credit is due, and none is due here.
If you want credit, how about: Shannon, Fourier and Huffman. Then there's all the folks involved in working out noise masking and all the oddities of human hearing that I don't have the names of.
I seriously need a "No iPod mentions whatsoever" checkbox for my slashdot profile to pull some more signal out of the slashdot article noise.
one with a connected graph. For instance the function equal to sin(1/x) for x != 0 and 0 for x = 0 does have a connected graph but is NOT continuous.
Wondering why i am doing so strange posts? I am trying to get a "+5,Flamebait" or "-1,Insightful" rating.
For those not in the know, continuity at a point p for a function f means If the function is continous for all x in the interval [a,b] the function itself is said to be continuous about that interval. This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.
Don't you just hate it when people reply to your signature?
...was published in the Swedish maths journal Acta Mathematica and is calles On the convergence and growth of partial sums of Fourier series.
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
You have to love the new vernacular; where everything is defined in its relation to pop culture and the standard unit of size is the football field.
Rigor is very well for the rigorous mathematician. For the rest of us, and particularly for the purposes of talking to the layman (i.e. slashdot), it is a useless pedantry.
Anyone you might care to name who understands mathematics well enough to be able to understand a distinction of rigor most does not need anyone to tell the difference between what is "propper" and what is easy to say.
When things get complex, multiply by the complex conjugate.
Let's try this: Replace iPod with Windows...
His theorems have been helpful in creating Windows.
iPod is a proper noun, just like Windows, it works, and it's how Apple refers to their product. I'm not saying I agree with it, but don't yell about a grammar mistake that isn't there.
The right to protest the State is more sacred than the State.
It was just a calculated win on his part. Pro-math has become so phony.
His theorems have been helpful in creating iPod.
... so math is good for something after all!!!
Wow
...IMHO.
Why not mention signal processing, that makes it possible to filter out unimportant data from sound so that iPods (and it's likes) can store more music and MP3 and Vorbis files (and their likes) doesn't can be as small as they are. See there, iPod is still mentioned, but not in a way that makes you think there was something special (mabye mechanically) about the construction of an iPod that the world never before had seen.
When seeing the iPod referenche, I at first thought there mabye was something cool with it, related to this maths work, that I didn't know about. It wasn't. Just a boring sound player.
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
Andy
Hoping that he hasn't misapplied some math that he hasn't used since his degree more than 35 years ago.
This is interesting, please give us more!
In all topological spaces you have a sense of convergence of a sequenceI must admit I didn't know of any way to speak of convergence without the notion of a metric. How is that possible?
For even more general topological spaces you need the concept of a netMore general than what? And do you mean we need the "net" to replace the sequence? If you say so I'll believe you. However, one must still be able to define a sequence (a function from "the set of all natural numbers" to "the topological space in question"), since it doesn't really require much of the space, right?
(We need more maths on Slashdot!)
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
Saying that this has "been helpful in creating iPod" is at least weird.
m
It's like saying Einstein's special theory of relativity helped to invent the automobile. After all it deals with motion.
For practical purposes lot's of convergence theorems for Fourier series had been known before this one and those would be more than enough to show that in practice things would work.
Take for instance
http://en.wikipedia.org/wiki/Riesz-Fischer_theore
from 1907.
And before that even others, though this one is quite nice.
Second
saying that a continuous function is "(one with a connected graph)" is not very accurate but can be understandable given the expected audience, using "negligible points" is not understandable since using the right term "almost everywhere" would be more understandable for the layman and since the "negligible points" gives the idea that the points are somewhat special (at least to me) when in reality it should mean that when you calculate the measure of these points it's zero.
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Here is a link to the site with the announcement of the winner.
I believe this relates to shannon's theorem as used in audio. This states that a continuous waveform may be reconstructed completely from samples taken at greater than twice the highest component frequency of the waveform (Nyquist rate) -- and the waveform can be analyzed for frequency content via fourier analysis. This is EXTREMEMLY important in digital audio -- because that's how it works and how we reconstruct an analog wveform from 1's and 0's.e s/Winter2000/SamplingTheory.html
Admittedly, throwing the ipod reference in was a troll, but that's how digital audio works ladies and gents -- and that's how your ipod works too.
http://graphics.cs.ucdavis.edu/~okreylos/PhDStudi
Engineer 1: Who wants to listen to ... oh, never mind. What's the problem?
Time passes. Testing occurs.
Engineer 1: It turns out that the keyboard her band uses isn't encodable because the particular waveform it produces yields a Fourier series that doesn't converge. Net result -- the keyboard makes the codec explode. Too bad we didn't know that some fourier series don't converge to the functions they represent.
Engineer 2: yeah, well, we'll just have to put an advisory on those iPods -- "Warning! The music of Britney Spears is not suitable for listening through this device."
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One use of the Fourier transform is from the time domain to the frequency domain, and vice versa. So for example, in writing a function for EQ on an mp3 player you might use a time function that you wrote in the frequency domain, thus utilizing the Fourier transform to do so. I'm not saying iPods run FFT (Fast Fourier Transforms) real time, but this guy's work really handed down a lot of usefull tools to digital audio: playback, recording, and acoustic testing and research.
"I understand my tests are popular reading in the teacher's lounge." -Calvin to Hobbes
WTF is going on here? I took the class in Fourier series and Fourier analysis and thought that the Fourier series approximated any continuous function at all points! Now I'm seeing an article talking about "negligible points"! Could a mathematician explain how we get away using the approximation so successfully in engineering applications? Or maybe this is why some things fail but we don't know why?
P.S. I think you meant Microsoft Windows.
No no, grandparent was entirely correct.
- Windows is a piece of software, like iTunes or Photoshop. No definite article required.
- iPod is a class of discrete objects, Like computer, Walkman or automobile. Try saying "wheels have been useful in creating automobile"... does that sound OK to you?
You're right that it isn't a mistake (it's intentional of course), but it is still a grammatical error.You need some kind of smoothness (piecewise continuous differentiability works, for example) in order to prove pointwise convergence.
Is he Swedish or is he a Norwegian living in Sweden? I believe the submission is in error on a non-negligible point!
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It's worth mentioning that Carleson was on the faculty at UCLA, usually spending at least the winter quarter there (it doesn't take a genius to prefer Los Angeles in February to Uppsala in February.) I think all of the graduate students he advised were in Sweden though, which seems to be the case from the math genealogy site: http://www.genealogy.ams.org/html/id.phtml?id=1978 1 He did at least intermittently teach the first-year graduate analysis course at UCLA, and made those students suffer (and learn...)
It's psychosomatic. You need a lobotomy. I'll get a saw.
There is a minor difference between what you said and what the article text said, although no one except math PhDs would be likely to care. (For the record: your statement is correct, but the article text is not.)
A path connected graph is not the same thing as a connected graph. There exist examples of graphs which are connected but not path connected. The article text claims that a continuous function is one whose graph is connected. This statement is wrong. The correct statement is what you said, that a function is continuous if and only if the graph is path connected.
The topologists' sine curve is an example of a discontinuous function which has a connected graph.
My Engineering professors all beat this guy by years in proving that Fourier Analyisis works. And they used to routinely prove it and use it on the blackboard during my Electrical Engineering course. In fact, superposition of functions is the basis for many mathematical frameworks, including quantum mechanics.
The basis for my project at http://sourceforge.net/projects/buzz-like is Fourier Analysis.
It's OK Bender, there's no such thing as 2.
wtf?
by definition you can't take the measure of an unmeasurable set.
measure is defined on measurable sets. the word 'measurable' gives it away a bit.
nowhere in your link to the banach-tarski paradox does it say anything about taking the measure of the unmeasurable sets used in the construction.
my password really is 'stinkypants'
you've just defined an outer measure, a very common way to construct a measure. if you have an outer measure defined on a whole bunch of sets, you can let the measure be the same as that on the measurable sets, which like you say, can be those that satisfy a Caratheodory condition.
it's similar to a measure, i'll grant you that. but AFAIK the term 'measure' is specifically defined to be countably additive etc.
i wasn't aware that the term measure is used anywhere in the literature to mean an outer measure or equivalent. i could be wrong though in which case i apologise, there are too many different mathematical concepts, and not enough words for all definitions to be consistent!
it is kind of tangential, you are right. unmeasurable sets behave weirdly but you don;t actually need the 'measure' or outer measure of the sets in the B-T paradox. there's enough weirdness going on w/o worrying about it.
like me, you seem to have learnt measure theory a slightly cooler and nonstandard way? i think it's usual to define measure on Borel sets and build it up? but the reverse approach that you described works as well and is easier to get your head around, imho..
my password really is 'stinkypants'