Posts tagged with manifolds













The tropical semiring is arithmetic piped through a log with base →.

Also if you or someone you know  is first encountering a squeeze theorem or other a≤x≤A type reasoning, remark 2.1 might be a relatively painless calisthenic to warm you/them up to a≤x≤A type arguments.

by Gregory Mikhalkin











[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …
A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …
We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory
(I rearranged their words liberally but the substance is theirs.)
in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.

[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …

A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …

We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory

(I rearranged their words liberally but the substance is theirs.)

in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.




It was the high zenith of autumn’s colour.

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We drove her car out to the countryside, to an orchard. Whatever the opposite of monocropping is, that’s how the owners had arranged things.

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The apple trees shared their slopey hillside with unproductive bushes, tall grasses, and ducks in a small pond in the land’s lazy bottom.

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Barefoot I felt the trimmed grass with my toes. A mother pulled her daughter away from the milkweeds—teeming with milkweed nymphs—because “They’re dangerous”.

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It was only walking along the uneven ground between orchard and forest that I realised that I almost never walk on surfaces that aren’t totally flat, level, hard, and constant.

 

In the Chauvet cave paintings of 32 millennia before sidewalks, the creator — rather than being hampered by the painting surface — used its unevenness to their advantage.

Photo: Horse paintings in  Chauvet Cave

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But today

  • sidewalks are completely flat in New York City; if you trip and hurt yourself because of their ill repair you can actually sue the City
  • art (not all art but a lot of painting or screen-media) is conceived on a flat surface
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  • houses are square; efficient industrial production of the straight and right-angle-based construction materials
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    and work plans
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    means it would be relatively expensive to build otherwise.
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  • yards are square
  • parks are square
  • city blocks are square
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  • (…except older cities which resemble a CW complex more than a grid)
    ComplexCity: Moscow
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In general relativity flat Euclidean spaces are deformed by massive or quick-spinning objects. 

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Due to an uneven distribution of mass inside the Earth, its gravity field is not uniform, as indicated by the lumps in this illustration.

Still image of world gravity map

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and in sheaf theory things can be different around different localities.

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The cave walls in Chauvet have been locally deformed even to the point that knobs protrude from them—and the 32,000-year-old artist utilised these as well.

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Maybe when Robert Ghrist gets his message to the civil engineers, we too will have a bump-tolerant—even bump-loving—future ahead of us.

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EDIT: Totally forgot about tattoos. 














M. C. Escher’s painting Ascending and Descending illustrates a non-conservative vector field, impossibly made…. In reality, the height above the ground is a scalar potential field [the scalar (single number attached to a point) being the height above the ground]. If one returns to the same horizontal place, one has gone up exactly as much as one goes down.


So that’s that picture related to
vector fields
vectors
physics
chain rule
product rule
calculus
scalar fields
gradients



Conservative vector fields obey the product rule:

(and the conservative scalar field is also the output of a derivative operation…just a different dimensionality)

M. C. Escher’s painting Ascending and Descending illustrates a non-conservative vector field, impossibly made…. In reality, the height above the ground is a scalar potential field [the scalar (single number attached to a point) being the height above the ground]. If one returns to the same horizontal place, one has gone up exactly as much as one goes down.

So that’s that picture related to

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Conservative vector fields obey the product rule:

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(and the conservative scalar field is also the output of a derivative operation…just a different dimensionality)


hi-res













horizontal composition of homotopies
within 𝒵:
red→yellow then orange→green
ends the same as
orange→green then red→yellow

horizontal composition of homotopies

  • within 𝒵:
  • red→yellow then orange→green
  • ends the same as
  • orange→green then red→yellow

(Source: youtube.com)


hi-res




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One of the more consequential kinds of extrapolation happens in the law.

In the case of Islamic law شريعة  the Hadith and the Qu’ran contain some examples of what’s right and wrong, but obviously don’t cover every case.

This leaves it up to jurist philosophers to figure out what’s G-d’s underlying message, from a sparse sample of data. If this sounds to you like Nyquist-Shannon sampling, you and I are on the same wavelength! (ha, ha)

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File:CriticalFrequencyAliasing.svg

File:AliasedSpectrum.png
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Of course the geometry of all moral quandaries is much more interesting than a regular lattice like the idealised sampling theorem.

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Lattice of beverages
Revised lattice



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imagering lattice
ring lattice

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Escher's grid







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Evenly-spaced samples mapping from a straight line to scalars could be figured out by these two famous geniuses, but the effort of interpreting the law has taken armies of (good to) great minds over centuries.

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A_2 lattice Voronoi and Delaunay cells
PCA of British MPs in the space of rollcalls

 

The example from this episode of In Our Time is the prohibition on grape wine:

  • What about date wine?
  • What about other grape products?
  • What about other alcoholic beverages?
  • What about coffee?
  • What about intoxicants that are not in liquid form?

The jurists face the big p, small N problem—many features to explain, less data than desirable to draw on.

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splines_2d_linear
splines_2d_cardinal
splines_2d_catmullrom
splines_1d_step

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Clearly the reason why cases A, B, or D are argued to connect to the known parameter from the Hadith matters quite a lot. Just like in common-law legal figuring, and just like the basis matters in functional data analysis. (Fits nicely how the “basis for your reasoning” and “basis of a function space” coincide in the same word!) .

Think about just two famous functional bases:

  1. Polynomials (think Taylor series),
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    and
  2. Sinusoidals (think Fourier series).
    Fourier Series
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Even polynomials look like a ‿or   ͡ ; odd polynomials look at wide range like a \ or / (you know how looks: a small kink in the centre ՜𝀱 but in broad distances like /), and sinusoidal functions look like ∿〜〜〜〜〜〜∿∿∿∿〰〰〰〰〰〰〰〰〰〰〰𝀨𝀨𝀨𝀨𝀨.

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So imagine I have observations for a few nearby points—say three near the origin. Maybe I could fit a /, or a 𝀱, a , or a ‿.

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All three might fit locally—so we could agree that

  • if grape wine is prohibited
  • and date wine is prohibited
  • and half-grape-half-date-wine is prohibited,
  • then it follows that so should be two-thirds-grape-one-third-date-wine prohibited—
  • but, we mightn’t agree whether rice wine, or beer, or qat, or all grape products, or fermented grape products that aren’t intoxicating, or grape trees, or trees that look like grape trees, and so on.

The basis-function story also matches how a seemingly unrelated datum (or argument) far away in the connected space could impinge on something close to your own concerns.

If I newly interpret some far-away datum and thereby prove that the basis functions are not  but 𝀨𝀱/, then that changes the basis function (changes the method of extrapolation) near where you are as well. Just so a change in hermeneutic reasoning or justification strategy could sweep through changes throughout the connected space of legal or moral quandaries.

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This has to be one of the oldest uses of logic and consistency—a bunch of people trying to puzzle out what a sacred text means, how its lessons should be applied to new questions, and applying lots of brainpower to “small data”. Of course disputes need to have rules of order and points of view need to be internally consistent, if the situation is a lot of fallible people trying to consensually interpret infallible source data. Yet hermeneutics predates Frege by millennia—so maybe Russell was wrong to say we presently owe our logical debt to him.

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In the law I could replace the mathematician’s “Let” or “Suppose” or “Consider”, with various legalistic reasons for taking the law at face value. Either it is Scripture and therefore infallible, or it has been agreed by some other process such as parliamentary, and isn’t to be questioned during this phase of the discussion. To me this sounds exactly like the hypothetico-deductive method that’s usually attributed to scientific logic. According to Einstein, the hypothetico-deductive method was Euclid’s “killer app” that opened the door to eventual mathematical and technological progress. If jurisprudence shares this feature and the two are analogous like I am suggesting, that’s another blow against the popular science/religion divide, wherein the former earns all of the logic, technology, and progress, and the latter gets superstition and Dark Ages.

(Source: BBC)