Posts tagged with Fourier transform

"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.

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.

$\large \dpi{200} \bg_white \alpha \cdot |1 \rangle \ + \ \beta \cdot |2 \rangle \ + \ \gamma \cdot |3 \rangle \ + \ \cdots$

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

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.)

## Polynomials as Sequences

One reason polynomials are interesting is that you can use them to encode sequences.

$\large \dpi{150} \bg_white \begin{matrix} 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + \cdots \\ \\ 5 + 2 \, x + 13 \, x^2 + 87.1 \, x^3 - x^4 + x^5 - 3 x^6 \cdots \\ \\ (\ 5, \ \ 2, \ \ 13, \ \; 87.1, \ -1, \ \ 1, \ -3, \ \ldots\ ) \\ \\ a + b \, x + c \, x^2 + d \, x^3 + e \, x^4 + f \, x^5 + g \, x^6 + \cdots \\ \\ \sum_0^N \mathrm{const}_i \cdot x^i \end{matrix}$

In fact some of the theory of abstract algebra (the theory of rings) deals specifically with how your perspective changes when you erase all of the x^297 and x^16 terms and think instead about a sequence of numbers, which actually doesn’t represent a sequence at all but one single functional.

$\large \dpi{150} \bg_white \begin{matrix} 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + \cdots \\ \\ (\ 1, \ \ 1, \ \ 1, \ \ 1, \ \ 1, \ \ 1, \ \ 1, \ \ldots\ ) \\ \\ 5 + 2 \, x + 13 \, x^2 + 87.1 \, x^3 - x^4 + x^5 - 3 x^6 \cdots \\ \\ (\ 5, \ \ 2, \ \ 13, \ \; 87.1, \ -1, \ \ 1, \ -3, \ \ldots\ ) \\ \\ a + b \, x + c \, x^2 + d \, x^3 + e \, x^4 + f \, x^5 + g \, x^6 + \cdots \\ \\ (\ a, \ \ b, \ \ c, \ \ d, \ \ e, \ \ f, \ \ g, \ \ldots\ ) \\ \\ \sum_0^N \mathrm{const}_i \cdot x^i \\ \\ (\ \mathrm{const}_0, \ \ \mathrm{const}_1, \ \ \mathrm{const}_2, \ \ \mathrm{const}_3, \ \ \mathrm{const}_4, \ \ \mathrm{const}_5, \ \ \mathrm{const}_6, \ \ldots\ ) \end{matrix}$

When you put that together with observations about polynomials

• Every sequence is a functional. (OK, can be made into a functional / corresponds to a functional)

• So plain-old sequences like 2, 8, 197, 1780, … actually represent curvy, warped things.

• Sequences of infinite length are just as admissible as sequences that finish.
(After all, you see infinite series all the time in maths: Laurent series, Taylor series, Fourier series, convergent series for pi, and on and on.)
• Any questions about analyticity, meromorphicity, convergence-of-series, etc, and any tools used to answer them, now apply to plain old sequences-of-numbers.
• Remember Taylor polynomials? There’s a calculus connection here.
• Derivatives and integrals can be performed on any sequence of plain-old-numbers. They correspond (modulo k!) to a left-shift and right-shift of the sequence.
• You can take the Fourier transform of a sequence of numbers.

• How about integer sequences from the OEIS? What do those functions look like? How about once they’re Taylored down? (each term divided by k!.)

• Sequences are lists. Sequences are polynomials. Vectors are lists. Ergo—polynomials are vectors?!
• Yes, they are, and due to Taylor’s theorem sequences-as-vectors constitute a basis for all smooth ℝ→ℝ functionals.
• The first question of algebraic geometry arises from this viewpoint as well. A sequence of "numbers" instantiates a polynomial, which has “zeroes”. (The places where the weighted x^1192 terms sum to 0.)

So middle-school algebra instantiates a natural mapping from one sequence to another. For example (1, 1−2−1, 1 (−1, 1−φ, 1, φ). Look, I don’t make the rules. That correspondence just is there, because of logic.

Instead of thinking sequence → polynomial → curve on a graph → places where the curve passes through a horizontal line, you can think sequence → sequence. How are sequences→ connected to →sequences? Here’s an example sequence (0.0, 1.1, 2.2, 3.3, 4.4, 0, 0, 7.7) to start playing with on WolframAlpha. Try to understand how the roots dance around when you change sequence.

• Looking at sequences as polynomials explains the partition function (how many ways can you split up 7?) As explained here.
• Also, general combinatorics http://en.wikipedia.org/wiki/Enumerative_combinatorics problems besides the partition example, are often answered by a polynomial-as-sequence.
• Did I mention that combinatorics are the basis for Algorithms that make computers run faster?
• Did I mention that Algorithms class is one of the two fundae that set hunky Computer Scientists above the rest of us dipsh_t programmers?
• There is a connection to knots as well.
• Which means that group theory / braid theory / knot theory can be used to simplify any problem that reduces to “some polynomial”.
• Which means that, if complicated systems of particles, financial patterns, whatever, can be reduced to a polynomial, then I can use a much simpler (and more visual) way of reasoning about the complicated thing.
• I think this stuff also relates to Gödel numbers, which encode mathematical proofs.
• You can encode all of the outputs of a ℕ→ℕ function as a sequence. Which means you may be able to factor a sequence into the product of other sequences. In other words, maybe you can multiply simple sequences together to get the complicated sequence—or function—you’re looking for.

This is an example of when the kind of language mathematics is, is quite nice. Every author’s sprawling thoughts coming from here and going to there while taking a detour to la-la land, are condensed by uniformity of notation. Then by force of reasoning, analogies are held fast, concrete is poured over them, and eventually you can walk across the bridge to Tarabithia. Try nailing down parallels between Marx & Engels, it’s much harder.

All of these connections give one an archaeological feeling, like … What exactly am I unearthing here?

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)

This is the reason for aliasing in music samples, Moire vibration in digital images, and is a useful formula for interpolation of data.

hi-res

## 1D, 2D, circle, back

I don’t know how one gets these ideas.  The Box-Muller algorithm—which quickly fashions a normal distribution out of a few dice rolls—comes from the following unintuitive chain of logic.

1. Take a problem in a simple domain (1-D).
2. Put it into a more complicated domain (2-D).
3. Change from grid coordinates to Navy coordinates (rect → polar).
4. Do the problem there, then push it back.

$\large \dpi{175} \bg_white \begin{matrix} \{ \text{length} \} \to \{ \text{square area} \} &\to \{ \text{area of part of a ring} \}\\ & \qquad \qquad \qquad \downarrow \\ & \qquad \qquad \text{solve the problem} \\ & \qquad \qquad \qquad \downarrow \\ \{ \text{length} \} \leftarrow \{ \text{square area} \} &\leftarrow \{ \text{area of part of a ring} \} \end{matrix}$

It’s normal to move problems to other domains to solve them: f⁻¹( g( f(●))).  Some might say that’s the whole point of math.  But this is so unintuitive, to make something more convoluted before making it simpler.

Using the Fourier transform, for example, you take a wave (like music) and turn it into a frequency (like notes).  Well both music and notes are understandable domains — sometimes you want one, sometimes you want the other, so you just transition back and forth between the domains as suits.  You use the equalizer to adjust things in the second domain, and then you go back to the music domain to listen to the transformed song.

Not this time, though.  You go into a seemingly unrelated domain, solve the problem there, and then come back.  Very weird.  Or as some people like to say, “clever”.

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