Posts tagged with ODE's

In harmonic analysis and PDE, one often wants to place a function `ƒ:ℝᵈ→ℂ` on some domain (let’s take a Euclidean space `ℝᵈ` for simplicity) in one or more function spaces in order to quantify its “size”….

[T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.

For function spaces `X` on Euclidean space, two such exponents are the regularity `s` of the space, and the integrability `p` of the space.

—Terence Tao

Hat tip: @AnalysisFact

hi-res

hi-res

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

## My sister isn’t “irrational”, her utility function just has large interaction terms.

What happens if, instead of doing a linear regression with sums of monomial terms, you do the complete opposite? Instead of regressing the phenomenon against $\large \inline \dpi{200} \bg_white x_t + y_t + z_t + \epsilon_t$ , you regressed the phenomenon against an explanation like $\dpi{200} \bg_white \sqrt[ \sum \text{powers} ]{{x_t}^{66} \; {y_t}^{13} \; {z_t}^{282} + {x_t}^3 \, {y_t}^9 \, {z_t}^7 + \ldots + {x_t}^{17} \, {z_t}^{1377} }$ ?

I first thought of this question several years ago whilst living with my sister. She’s a complex person. If I asked her how her day went, and wanted to predict her answer with an equation, I definitely couldn’t use linearly separable terms. That would mean that, if one aspect of her day went well and the other aspect went poorly, the two would even out. Not the case for her. One or two things could totally swing her day all-the-way-to-good or all-the-way-to-bad.

The pattern of her moods and emotional affect has nothing to do with irrationality or moodiness. She’s just an intricate person with a complex utility function.

If you don’t know my sister, you can pick up the point from this well-known stereotype about the difference between men and women:

"Men are simple, women are complex.” Think about a stereotypical teenage girl describing what made her upset. "It’s not any one thing, it’s everything.”

I.e., nonseparable interaction terms.

I wonder if there’s a mapping that sensibly inverts strongly-interdependent polynomials with monomials — interchanging interdependent equations with separable ones. If so, that could invert our notions of a parsimonious model.

Who says that a model that’s short to write in one particular space or parameterisation is the best one? or the simplest? Some things are better understood when you consider everything at once.

A beautiful depiction of a 1-form by Robert Ghrist. You never thought understanding a 1→1-dimensional ODE (or a 1-D vector field) would be so easy!

What his drawing makes obvious, is that images of Phase Space wear a totally different meaning than “up”, “down”, “left”, “right”. In this case up = more; down = less; left = before and right = after. So it’s unhelpful to think about derivative = slope.

BTW, the reason that ƒ must have an odd number of fixed points, follows from the “dissipative” assumption (“infinity repels”). If ƒ (−∞)→+, then the red line enters from the top-left. And if ƒ (+∞)→−∞, then the red line exits toward the bottom-right. So no matter how many wiggles, it must cross an odd number of times. (Rolle’s Thm / intermediate value theorem from undergrad calculus / analysis)

Found this via John D Cook.

(Source: math.upenn.edu)

## How to multiply matrices

This is for my homies in maths class.

Mathematical matrices are blocks of numbers, arrayed in 2-D. (Higher-dimensional array-verbs are called tensors.)

1. Left “times” right equals target. Each entry in the target is the result of a series of +'s and ×'s along the red and blue. A long sum of pairwise products.

2. Your left hand goes across and your right hand goes up/down.
where
.
3. There need to be as many `abcdefg`'s as there are `1234567`'s or else the operation can't be done.
4. Also you can tell how big the output matrix will be. There can be three blue rows so the output has three rows. There can be four red columns so the output has four columns.
5. This is the “inner product” because multiplying vector-shaped blocks (tall blocks) like Aᵀ•B results in an equal or smaller sized output.

(There is also an “outer product" which is a different way of combining the info from the two matrices. That gives you an equal or larger shaped result when you multiply vector/list-shaped tall blocks A∧B.)
6. Try playing around with this one or that one.

Matrix multiplication is the simplest example of a linear operator, the broad class of which explains quantum mechanics and ODE’s. You can also apply different matrices at different points as in a vector field — on a flat surface or a curvy, holey surface.

### Proof that differential equations are real.

The shapes the salt is taking at different pitches are combinations of eigenfunctions of the Laplace operator.

(The Laplace operator $\dpi{200} \bg_white \large \bigtriangleup f = \sum_{i=1}^n {\partial ^2 f \over \partial {x_i}^2 }$ tells you the flux density of the gradient flow of a many-to-one function ƒ. As eigenvectors summarise a matrix operator, so do eigenfunctions summarise this differential operator.)

Remember that sound is compression waves — air vibrating back and forth — so that pressure can push the salt (or is it sand?) around just like wind blows sand in the desert.

Notice the similarity to solutions of Schrödinger PDE’s from the hydrogen atom.

When the universe sings itself, the probability waves of energy hit each other and form material shapes in the same way as the sand/salt in the video is doing. Except in 3-D, not 2-D. Everything is, like, waves, man.

To quote Dave Barry: I am not making this up. Science fact, not science fiction.