Posts tagged with fourier analysis

Mirror symmetry is an example of a duality, which occurs when two seemingly different systems are isomorphic in a non-trivial way. The non-triviality of mirror symmetry involves quantum corrections. It’s like the Fourier transform, where “local” in one domain translates to “global”—something requiring information from over the whole space—in the other domain.


a Fourier spike
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Under a local/global isomorphism, complicated quantities get mapped to simple ones in the dual domain. For this reason the discovery of duality symmetries has revolutionized our understanding of quantum theories and string theory.

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summer school on mirror symmetry (liberally edited)

 

Thinking about local-global dualities gave me another idea about my model-sketch of knowledge, ignorance & expectation.

  • Under physical limitations, at a fixed energy level, Fourier duality causes a complementary tradeoff between frequency and time domains—not both can be specific. Same with position & momentum, again at a fixed energy level.
  • Under human limitations, at a fixed commitment of effort|time|concentration, you can either dive deep into a few areas of knowledge|skill, or swim broadly over many areas of knowledge|skill.

If I could come up with a specific implementation of that duality it would impose a boundary constraint on that model-sketch. Which would be great as optimal time|effort|concentration|energy could be computed from other parts of decision theory.




Dave Rusin

(Source: math.niu.edu)




http://ecx.images-amazon.com/images/I/51ohaNaIEGL.jpg
from C. H. Edwards, Jr.
come some examples of linear (vector) spaces less familiar than span{(1,0,0,0,0), (0,1,0,0,0), ..., (0,0,0,0,1)}.

  • The infinite set of functions {1, cos θ, sin θ, ..., cos n • θ, sin n • θ, ...} is orthogonal in 𝓒[−π,+π]. This is the basis for Fourier series.
    http://www.math.harvard.edu/archive/21b_fall_03/fourier/approximation.gif
  • Let 𝓟 denote the vector space of polynomials, with inner product (multiplication) of p,q ∈ 𝓟 given by ∫_1¹ p(x) • q(x) • dx. Applying Gram-Schmidt orthogonalisation gets us within constant factors of the Legendre polynomials 1, x, x²−⅓, x³−⅗x, x⁴−6/7 x²+9/5, ...
    http://www.math.kth.se/math/GRU/2008.2009/SF1625/CMIEL/taylor%20Files/appviewer_010.gif
  • (and, from M A Al-Gwaiz)
    The set of all infinitely-smooth complex-valued functions that map to zero outside a finite interval (i.e., have compact support). These tempered distributions lead to generalised distributions (hyperfunctions) and imprecision on purpose.




"a mixing console to your personality”

You may not be this bold or ferocious in your day-to-day life, but on stage you amplify these things in you that already exist.

File:Dirichlet.png

I could talk about this in equation form: imagine the personality is a vector (list) and some of these aspects are in some way independent or separable to each other.

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where |1⟩, |2⟩, |3⟩ are projections of the whole personality down to one “aspect”.

File:KleinInvariantJ.jpg

Then St Vincent’s idea is simply to lower and raise some of the α, β, γ, δ …sliders”. So like when doves cry inside a convex hull, it’s just linear combinations of pre-existing stuff, rather than the generation of “truly new” (orthogonal) things. (Properly in maths one needs multiple distinct examples to do linear combinations and create a span. I wonder if she would agree that “projecting" (isolating) the "elements" of her personality is a step requiring work in finding out what the aspects of the personality are, to amplify or mute them.)

a well-known stereotype: women are complex, men are simple

http://ars.els-cdn.com/content/image/1-s2.0-S0272771412001072-gr3.jpg

http://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Fourier_Series.svg/500px-Fourier_Series.svg.png
http://www.learner.org/courses/mathilluminated/images/units/10/1873.png
http://music.columbia.edu/cmc/musicandcomputers/images/chapter3/sumofsines.jpg




Standing waves in 2-D via dhiyamuhammad.
Pretty amazing that if you simply add together oscillations = vibrations = waves = harmonics and constrain them within a box, that all these shapes emerge. (See this video for such waves being constructed in real life). By the way, mathematicians sometimes refer to these as “square drumhead” problems because a drumhead is a real-life 2-D surface that vibrates in exactly these kinds of ways to produce the sounds we associate with various drums.
In the link Muhammad points to—Harmonic Resonance Theory—the mathematics of standing waves are applied to the problem of Gestalt in psychology of sense experience.

Standing waves in 2-D via dhiyamuhammad.

Pretty amazing that if you simply add together oscillations = vibrations = waves = harmonics and constrain them within a box, that all these shapes emerge. (See this video for such waves being constructed in real life). By the way, mathematicians sometimes refer to these as “square drumhead” problems because a drumhead is a real-life 2-D surface that vibrates in exactly these kinds of ways to produce the sounds we associate with various drums.

In the link Muhammad points to—Harmonic Resonance Theory—the mathematics of standing waves are applied to the problem of Gestalt in psychology of sense experience.


hi-res




Me and my friends been too busy bathing off the southern coast of St Barts for the past two weeks, contemplating the Fourier transforms of spider monkeys. Changed our whole perspective on sh%t.

Hansel, in the movie Zoolander


(Source: imdb.com)




File this under fourier analysis + linear algebra = bad#ss.

Fourier transform of toes

On the right you’re seeing the configuration space of the toes as opposed to physical space of the toes.

Ponies

Take a 3-D mesh wireframe stallion and do the Fourier transform.

Now you have a summary of the position, so you can move hoof-leg-and-shoulder by just moving 1 point in the transformed space.

In other words the DFT takes you into the configuration space of the horsie. Inverse DFT takes a leg-and-hoof configuration and gives you back a wireframe horsie.

clustering

The discrete Fourier transform also helps sort out the clustering problem:

smoothing

From the slides, I don’t get what the connection is to (anti-fractal) smoothing. But…seahorses and seagulls:

PDF SLIDES via Artemy Kolchinsky

If you thought linear regression was a hammer for every nail … wait until you play around with the Fourier transform!




wavelets and multiresolution