Posts tagged with math

(x²−y<²−1) • (x²−z²−1) •  (y²−z²−1)   =   0

(Source: imaginary.org)

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)

A jet can be thought of as the infinitesimal germ of a section of some bundle or of a map between spaces.

Jets are a coordinate-free version of … Taylor series.


Pictorial glossary

Bundle:

Sections:

Mapping between spaces:

• Mapping a circle onto a line
• Wrapping a line around a circle
• Associating a number to a Euclidean=flat subregion

• Thinking of knots as mappings of a loop 𝕊¹ into ambient space ℝ³.

• Thinking of homotopies as mappings of cubes into spaces.

• Associating photos to geolocations (Flickr, stochasticplanet)
• Mappings from manifold to manifold—for example the derivative (pushforward) at each point of the surface.
• Associating vectors (based force-arrows) to spots on the planet (2-sphere):

Extrinsic vs Intrinsic

In The Geometry and Topology of 3-ManifoldsThurston speaks of the “extrinsic” vs “intrinsic” view of {the 3-sphere − a knot}.

I like learning words for something that’s been kicking the walls of my head trying to get out. Sometimes I can look at the world in an objective way, and sometimes everything centres on me.

• Extrinsic I and you and they are cerebral plains-apes, doing all of the things one would expect such to do. Chasing after money, status, sex, it’s all pretty simple when you see things this way.
• Extrinsic your exotic mind-expanding holiday is just a status signal.
• Extrinsic the hobby or job you hang your identity on tells me that you’re bourgeois and therefore define yourself through your activities or achievements.
• Intrinsic reinterprets monkeys’ “social grooming” as an activity that actually feels like something.
• Intrinsic is being in love.
• Extrinsic is love as a neurochemical sequence—dopamine, adrenaline, vasopressin, norepinephrine, oxytocin—generated by chance mutations and selective reproduction in response to an evolutionary problem.
• Extrinsic is 5,000 applicants for 5 jobs = 1‰ chance of getting it.
• Intrinsic my application was rejected so I’m a failure.
• Extrinsic is the pale blue dot, spaceship Earth, a wet rock bearing eukaryotes.
• Extrinsic I’m one of 10000 options s/he has on this dating site.
• Intrinsic is me; my likes, my interests; the homunculus behind the camera, 我, my life. Extrinsic this display is again an outgrowth of my bourgeois background.
• Intrinsic is exotic Peshawar, a world away and uncomfortably hot.

Extrinsic is more primates with a different culture-function applied to them. Going through the same life-stages, wearing different customs. Instead of going to a pub they do some sober, Pakistani alternative—but it’s all the same, all human stuff just expressed differently.
• Extrinsic when Parisians kiss each other on the cheek it’s not romantic, it’s just like a handshake.
• Intrinsic it’s gross that my parents have sex.
• Intrinsic I hate this town I hate the football games I hate the fakeness I hate the parochial small-mindedness there’s something better in store for me I need to get out of here I will get out of here
• Extrinsic is a visitor look how excited these people are about their quaint sports and amusements! That’s so nice.
• Intrinsic is being engrossed in reading.
• Extrinsic is seeing someone sitting still looking at paper with glyphs on it.
• Intrinsic is noticing how Thomas Piketty’s lower teeth are not-at-all straight like George Clooney’s, and being taken back a bit. Extrinsic is knowing that Piketty earns his stature from books and papers (arrayed behind him) whereas Clooney earns based on his looks.
• Intrinsic is waking up in the morning driven by the will to understand pseudo-Riemannian metrics and thereby, the Universe.

Chanting “sheaf cohomology" like a shamanic totem, carving it into my forehead my topknot my yarmulke my niqab, showing the world what’s inside my head and wishing they love me for it.
• Extrinsic I’m yet another plains-ape distracted by ego, status-signalling, and a scientific religion, cultured by stimuli that practically guaranteed I would behave this way.
• Intrinsic my new friend is so cool! She is friends with someone famous!
• Extrinsic of course she leads with her connection to the desirable; it’s all part of her personal branding scheme.
• Intrinsic is the many self-serious songs about my chain, my ding-a-ling, whatever.
• extrinsic is a boy in love with his homeostatic skin flap.
• Intrinsic is watching an ad targeted at you, and just reacting.
• Extrinsic is watching an ad that makes no sense to you, and thinking about the ad on a higher level.
• Intrinsic is appetite; craving; addiction.
• Extrinsic is flavourful sustenance.
• Extrinsic is asthexia; cachenia; syncope; renal failure.
• Intrinsic is the only death that matters (mine).
• Intrinsic is write drunk.
• Extrinsic is revise sober.

Beware of the pursuit of the Superhuman: it leads to an indiscriminate contempt for the Human. —George Bernard Shaw, Man and Superman

A farmer’s Markov transition matrix.

• At every step of the life cycle, the farmer (well, the plant, really…) contends with new natural enemies: wind, birds, mice, disease, varmints, frost, deer, ….
• This is a not-really-Markov transition matrix.
• Some states (such as “into a deer’s mouth”) that a plant could come to occupy aren’t shown.
• In the final row I made up some number for volume change.
• I guess the bottom right is the difference between a perennial and an annual, right? Like if that 1 were a 0 the plant would be an annual.

pics from Wikipedia or Purdue

Vector Duals and Musical Isomorphism in 1-D

The dual V* of a vector space V  over ℝ matches lists of reals to linear functionals.

What’s the simplest way to say this? Talk about a number like “5”. Initially I think of it as 5 stones ⬤⬤⬤⬤⬤. But I could also imagine a line through the origin with a slope of 5, representing the verb quintuple.

pictures of lines through the origin with various slopes

Seen as a function ƒ₅=quintuple, the-line-through-the-origin-with-a-slope-of-5, is ƒ₅(x)=5•x. That ƒ₅ does things like

• ƒ₅(■■■)=5+5+5 and
• ƒ₅(■■■■■■)=5+5+5+5+5+5.

Counting in the dual space ƒ₀,ƒ₁,ƒ₂,… would look like _ / ∕ ...|. Increasing slope from _ to ⁄ to | instead of increasing number from 0 to 1 to ∞. Or I could say id, double, triple, quadruple, quintuple, ….

(Why did I jump so suddenly 0,1,… from _ flat to ⁄  45°? This just proves that half of the ℝ⁺ are stuffed between [0,1) and the other half are between (1,∞).

To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T

hen you’d be counting id, half, third, fourth, fifth, sixth, seventh… Infinity in a teacup.)



These two things—the five rocks ⬤⬤⬤⬤⬤ and the function ƒ₅—aren’t even the same kind of thing. One is nouns and one is a verb.

But still, for any real number that I “counted” I could match up a function, just like I did with

• "5 that I counted" and
• "function ƒ₅ = quintuple

So these two qualitatively different things are in bijection. (One can hope for insights by viewing things through one lens or the other, noun or verb version.)

This one-dimensional story can be upgraded to a multi-dimensional one where

• lists of reals (3.1, √2, −2.1852, ..., 6)

match to

• many-to-one functions ƒ( list ) = 3.1•list[first] + √2•list[second] − 2.1852•list[third] + ... + 6•list[Nth].

Translating between the noun and verb viewpoints is then called musical isomorphism, represented with ♭ and ♯ symbols. Raising and lowering indices in a tensor is ♯ and ♭.

my illustration of the first isomorphism theorem, which says you can replace an arrow ƒ:X→Y by a sequence of arrows surjection ∘ bijection ∘ injection.

(Source: tjsullivan.org.uk)

Rank-Nullity Theorem

The rank-nullity theorem in linear algebra says that dimensions either get

• thrown in the trash
• or show up

after the mapping.

By “the trash” I mean the origin—that black hole of linear algebra, the /dev/null, the ultimate crisscross paper shredder, the ashpile, the wormhole to void and cancelled oblivion; that country from whose bourn no traveller ever returns.

The way I think about rank-nullity is this. I start out with all my dimensions lined up—separated, independent, not touching each other, not mixing with each other. ||||||||||||| like columns in an Excel table. I can think of the dimensions as separable, countable entities like this whenever it’s possible to rejigger the basis to make the dimensions linearly independent.

I prefer to always think about the linear stuff in its preferably jiggered state and treat how to do that as a separate issue.

So you’ve got your 172 row × 81 column matrix mapping 172→ separate dimensions into →81 dimensions. I’ll also forget about the fact that some of the resultant →81 dimensions might end up as linear combinations of the input dimensions. Just pretend that each input dimension is getting its own linear λ stretch. Now linear just means multiplication.

Linear stretches λ affect the entire dimension the same. They turn a list like [1 2 3 4 5] into [3 6 9 12 15] (λ=3). It couldn’t be into [10 20 30 − 42856712 50] (λ=10 except not everywhere the same stretch=multiplication).

Also remember – everything has to stay centred on 0. (That’s why you always know there will be a zero subspace.) This is linear, not affine. Things stay in place and basically just stretch (or rotate).

So if my entire 18th input dimension [… −2 −1 0 1 2 3 4 5 …] has to get transformed the same, to [… −2λ −λ 0 λ 2λ 3λ 4λ 5λ …], then linearity has simplified this large thing full of possibility and data, into something so simple I can basically treat it as a stick |.

If that’s the case—if I can’t put dimensions together but just have to λ stretch them or nothing, and if what happens to an element of the dimension happens to everybody in that dimension exactly equal—then of course I can’t stick all the 172→ input dimensions into the →81 dimension output space. 172−81 of them have to go in the trash. (effectively, λ=0 on those inputs)

So then the rank-nullity theorem, at least in the linear context, has turned the huge concept of dimension (try to picture 11-D space again would you mind?) into something as simple as counting to 11 |||||||||||.

Define the derivative to be the thing that makes the fundamental theorem of calculus work.