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Williams, Robert F. Expanding attractors. Publications Mathématiques de l'IHÉS, 43 (1974), p. 169-203

Ode to Nothing.

Oh, sweet emptiness.

A full room is often empty

The essence of nothing all perception

If one says there is nothing,

There is nothing.

Nothing could be what you see

When your eyes are closed

It could be what you feel

When you are alone.

The empty space

Filled like a black abyss

Nothing is infinite

It never ends

Trying to grasp the idea-

Rather impossible.

As one would drift into

Nothing

They feel alone

Empty.

It could be comforting know that

Although you are alone

And are lost in an expanse of nothing.

Once lost, never to be found

A shimmer of hope arises,

Your emotions return,

You feel joy,

You feel lost,

But alas

In that empty nothingness

Something appears, something-

You thought you had lost.

#Ode #Ode's #Poetry #I suck

SINGULAR POINTS

The prodigal sons of ordinary differential equations.

Yes, I will do anything to avoid doing practice exams right now. No, don't remind me I may regret this next Tuesday.

#math #ode's #pam dumbs down engineering #religion and science collide #not studying for ode's

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.

#ODE's #1-forms #differential forms #vector fields #differential geometry #differential equations #calculus #Euler characteristic #Rolle's Theorem #intermediate value theorem #Robert Ghrist #John D Cook #math #maths #mathematics #heroes

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.)

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.

Your left hand goes across and your right hand goes up/down.

where

There need to be as many abcdefg's as there are 1234567's or else the operation can't be done.

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.

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.)

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.

#2-D #ODE's #array #econometrics #inner product #linear algebra #linear operators #math #mathematics #maths #matrices #matrix multiplication #numbers #operators #systems of differential equations #tensor #vectors #angle

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 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.

#Carl Sagan #Dave Barry #Erwin Schrödinger #Laplacian #ODE's #The universe is a song singing itself.#differential equations #math #mathematics #maths #operators #science #reality #facts

Laplace Transform

The LaPlace Transform is the continuous version of a power series.

Think of a power series

as mapping a sequence of constants to a function.

Well, it does, after all.

Then turn the ∑ into a ∫. And turn the x^k into a exp ( ‒k ⨯ ln x ). Now you have the continuous version of the "spectrum" view that allows so many tortuous ODE's to be solved in a flash. I wonder what the economic value of that formula is?

In addition to solving some ODE's that occur in engineering applications, there is also wisdom to be had here. Thinking of functions as all being made up of the same components allows fair comparisons between them.

(If you really want to know what a power series is, read Roger Penrose's book.

To summarise: a lot of functions can be approximated by summing weighted powers of the input variable, as an equally valid alternative to applying the function itself. For example, adding input¹ + 1/2 ⨯ input² + 1/2/3 ⨯ input³ + 1/2/3/4 ⨯ input⁴ and so on, eventually approximates e^input.)