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Posts tagged with integrals

Double integrals ∫∫ƒ(x)dA are introduced as a “little teacher’s lie” in calculus. The “real story” requires “geometric algebra”, or “the logic of length-shape-volume relationships”. Keywords

  • multilinear algebra
  • Grassmann algebra / Grassmanian
  • exterior calculus
  • Élie Cartán’s differential-forms approach to tensors

These equivalence-classes of blobs explain how

  • volumes (ahem—oriented volumes!)
  • areas (ahem—oriented areas!)
  • arrows (vectors)
  • numbers (scalars)

"should" interface with each other. That is, Clifford algebra or Grassman algebra or "exterior algebra" or "geometrical algebra" encodes how physical quantities with these dimensionalities do interface with each other.

(First the volumes are abstracted from their original context—then they can be “attached” to something else.)

 

EDIT:user mrfractal points out that Clifford algebras can only have dimensions of 2,4,8,16,… https://en.wikipedia.org/wiki/Clifford_algebra#Basis_and_dimension Yes, that’s right. This post is not totally correct. I let it fly out of the queue without editing it and it may contain other inaccuracies. I was trying to throw out a bunch of relevant keywords that go along with these motivating pictures, and relate it to equivalence-classing, one of my favourite themes within this blog. The text here is disjointed, unedited, and perhaps wrong in other ways. Mostly just wanted to share the pictures; I’ll try to fix up the text some other time. Grazie.

(Source: arxiv.org)










Lebesgue’s approach to integration was summarized in a letter to Paul Montel. He writes:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
Siegmund-Schultze, Reinhard (2008), “Henri Lebesgue”, in Timothy Gowers, June Barrow-Green, Imre Leader, Princeton Companion to Mathematics

(Source: Wikipedia)




There is a century-old tree at the end of my street. Right before you get to the graveyard with its wrought-iron gates. That tree saw my grandmother play in the street when she was a little girl. It saw her ride the train to the college, carry groceries in a paper sack. The tree—I don’t know its name—it saw my da walk across town—from school to that house on Broad, when they used to live there. It can see my great-grandfather’s grave right now—it’s tall enough. He built this house in 1921. They say he was a drunk. The floor slants a little and the window frames aren’t square. He built the other houses on our block, too. Before he built them, it was just this house and greenhouses. The greenhouses were filled with roses. The whole neighbourhood used to smell like roses. At some point they used to call this Rose City, even though there’s a meatpacking factory only two kilometres away.

 

They also say he could multiply long numbers in his head, without any paper. Now this house is holding a different kind of “family”. I can’t even say it’s a modern one. More like a gathering of moneyless relations. Ambitious failures; I sometimes wonder what the house thinks of us. It’s certainly used to the self-help books: Latin; Linux; teach yourself guitar. The trains in this town used to carry passengers. They took my grandmother to the teacher’s college. My da must have walked past this graveyard a thousand times. No, more—maybe even ten thousand. I walk in the graveyard every day. The tree sees me. My favourite is when it’s snowy. Some of the graves announce strange names. A woman named Ruby. She would be 136 now. A man named Reason. Apparently the brothers who lie beneath the massive Romanesque columns at the highest point in the graveyard invented a transport that was used massively during the War. You can see most of the town standing among those columns. Past the roads there’s a small forest, beyond that farms.

I’m thinking about my path γ(t) versus the tree’s λ(t). Neither of us can be everywhere at once. We’ve stood at or around the same spot often enough. But every time I’ve gone “adventuring”, I haven’t seen what’s happening in λ(t). Is the small-town life “worse” than the jet-setter lifestyle? It depends what functional you convolve against γ(t). I don’t like repetitiveness, but maybe what the tree has seen isn’t so repetitive. Two World Wars. The rise of feminism. A time before plastic, a time before tarmac, a time when white supremacists would parade through the streets. My grandmother recognised someone’s shoes and shouted his surname; her mother covered her mouth. The tree saw her in most stages of life.

On we go, hurtling through spacetime. The speed of γ equals the speed of λ. From a galactic perspective the tree and I are whirling in almost the same place — regardless of whether I whisk from here on the earth to there on the earth by plane. I’m bound to the ground, ultimately. The tree just recognises that. People used to wear hats here. Everybody wore hats. Now it’s practically a ghost town except for pensioners and welfare recipients. The tree’s children can’t have blown too far.

Spinning in the same spot on 360° × [−90°, +90°] = ∂(S²×[0,1]). γ torques and twists about the sphere but its length is exactly the same. Does the tree wish λ had summited a mountain at some point? Perhaps, but it would be blown down up there, and the ground is tough and nutritionless anyway. It’s suited to this life.

It bears the snow. It puts up with the heat.

I go inside after a couple hours out of doors, of course. But the tree spends all night, every night facing the elements. Maybe it likes being strong. Digging. Growing big. Drinking in sunlight like an athlete at a water fountain.

I’m more like a tumbleweed, rootless, quick to change course. Hanging out for a bit and then rolling—without announcing a goodbye. Untethered. Free, yet constrained by the same holonomy constraining the tree. One path, and one path only. The same width as all the others.

γ isn’t so much more interesting than λ. My γ is filled with magazines, airports, computer screens. Parties where people say more or less the same things, always indicating the hope that their gradient’s pointing in the right direction.




Things that are easy in J but hard in other languages:

  • generating a bunch of crazy surfaces
  • computing Gaussian curvatures of areas and geodesic curvatures of (closed) boundary curves
  • checking the Gauß-Bonnet Theorem

(Source: jsoftware.com)










Given a time-series of one security’s price-train P[t], a low-frequency trader’s job (forgetting trading costs) is to find a step function S[t] to convolve against price changes P[t]

image

with the proviso that the other side to the trade exists.

S[t] represents the bet size long or short the security in question. The trader’s profit at any point in time τ is then given by the above definite integral.

 
  • I haven’t seen anyone talk this way about the problem, perhaps because I don’t read enough or because it’s not a useful idea. But … it was a cool thought, representing a >0 amount of cogitation.
  • This came to mind while reading a discussion of “Monkey Style Trading” on NuclearPhynance. My guess is that monkey style is a Brownian ratchet and as such should do no useful work.
  • If I were doing a paper investigating the public-welfare consequences of trading, this is how I’d think about the problem.

    Each hedge fund / central bank / significant player is reduced to a conditional response strategy, chosen from the set of all step functions uniformly less than a liquidity constraint. This endogenously coughs up the trading volume which really should be fed back into the conditional strategies.
  • Does this viewpoint lead to new risk metrics?
  • Should be mechanical to expand to multiple securities. Would anything interesting come from that?

I wouldn’t usually think that multiplication of functions has anything to do with trading. Maybe some theorems can do a bit of heavy lifting here; maybe not.

It at least feels like an antidote to two wrongful axiomatic habits. For economists who look for real value, logic, and Information Transmission, it says The market does whatever it wants, and the best response is a response to whatever that is. For financial engineering graduates who spent too long chanting the mantraμ dt + σ dBt" this is just another way of emphasising: you can’t control anything except your bet size.

UPDATE: Thanks to an anonymous commenter for a correction.