Quantcast

Posts tagged with topology

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.

Michael BächtoldDavid CorfieldUrs Schreiber

 

Pictorial glossary

Bundle:

Vector Bundle Construction
image
image

Sections:
\begin{figure}\begin{center}  \small\psfrag{PictureZeroSection} [l]{Zero sectio...  ...g{figure=figures/fibration_2.eps, width=0.4\textwidth}\end{center}\end{figure}
imagepicture that illustrates the research done

Mapping between spaces:




Statisticians are crystal clear on human variation. They know that not everyone is the same. When they speak about groups in general terms, they know that they are reducing N-dimensional reality to a 1-dimensional single parameter.

http://media.tumblr.com/tumblr_m0imzi2RqU1qbdydv.png

Nevertheless, statisticians permit, in their regression models, variables that only take on one value, such as {0,1} for male/female or {a,b,c,d} for married/never-married/divorced/widowed.
https://upload.wikimedia.org/wikipedia/en/thumb/6/6c/2_point_sheaf_sections.svg/1000px-2_point_sheaf_sections.svg.png
No one doing this believes that all such people are the same. And anyone who’s done the least bit of data cleaning knows that there will be NA's, wrongly coded cases, mistaken observations, ill-defined measures, and aberrances of other kinds. It can still be convenient to use binary or n-ary dummies to speak simply. Maybe the marriages of some people coded as currently married are on the rocks, and therefore they are more like divorced—or like a new category of people in the midst of watching their lives fall apart. Yes, we know. But what are you going to do—ask respondents to rate their marriage on a scale of one to ten? That would introduce false precision and model error, and might put respondents in such a strange mood that they answer other questions strangely. Better to just live with being wrong. Any statistician who uses the cut function in R knows that the variable didn’t become basketed←continuous in reality. But a facet_wrap plot is easier to interpret than a 3D wireframe or cloud-points plot.
http://www.cerebralmastication.com/wp-content/uploads/2009/09/wireframe.png
http://i.stack.imgur.com/SdfTq.png
http://www.wekaleamstudios.co.uk/wp-content/uploads/2010/05/surface-base.jpeg
http://docs.ggplot2.org/0.9.3.1/facet_wrap-22.png
http://i.stack.imgur.com/r272c.png
http://docs.ggplot2.org/0.9.3.1/facet_wrap-14.png





To the precise mind, there’s a world of difference between saying

  • "the mean height of men > the mean height of women", and saying
  • "men are taller than women".

image

Of course one can interpret the second statement to be just a vaguer, simpler inflection of the first. But some people understand  statements like the second to mean “each man is taller than each woman”. Or, perniciously, they take “Blacks have lower IQ than Whites” to mean “every Black is mentally inferior to every White.”

http://upload.wikimedia.org/wikipedia/commons/f/fa/Miles_Davis_by_Palumbo.jpg

http://www.timecube.com/TheWisestHuman_newimg_GeneRayCube.jpg

I want to live somewhere between pedantry and ignorance. We can give each other a break on the precision as long as the precise idea behind the words is mutually understood.

 

Dummyisation is different to stereotyping because:

  • stereotypes deny variability in the group being discussed
  • dummyisation acknowledges that it’s incorrect, before even starting
  • stereotyping relies on familiar categories or groupings like skin colour
  • dummyisation can be applied to any partitioning of a set, like based on height or even grouped at random
    VoronoiDiagramPlots
    http://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Coloured_Voronoi_3D_slice.svg/1000px-Coloured_Voronoi_3D_slice.svg.png
    http://xlr8r.info/mPower/img/boundedVoronoi.jpg

It’s the world of difference between taking on a hypotheticals for the purpose of reaching a valid conclusion, and bludgeoning someone who doesn’t accept your version of the facts.

So this is a word I want to coin (unless a better one already exists—does it?):

  • dummyisation is assigning one value to a group or region
  • for convenience of the present discussion,
  • recognising fully that other groupings are possible
  • and that, in reality, not everyone from the group is alike.
  • Instead, we apply some ∞→1 function or operator on the truly variable, unknown, and variform distribution or manifold of reality, and talk about the results of that function.
  • We do this knowing it’s technically wrong, as a (hopefully productive) way of mulling over the facts from different viewpoints.

In other words, dummyisation is purposely doing something wrong for the sake of discussion.













1. The monad, of which we will speak here, is nothing else than a simple substance, which goes to make up compounds; by simple, we mean without parts.

2. There must be simple substances because there are compound substances; for the compound is nothing else than a collection or aggregatum of simple substances.

Gottfried W. Leibniz, Monadology

 

Crazy how a “father” of calculus was so illogical in his seminal work of 1714.

  • The existence of compound things does not imply the existence of partless atoms.
  • He asserts, doesn’t prove, that a compound is “nothing more than" a collection of simple substances. (atoms)







I’ve collected a few tidbits about non-wellfoundedness on isomorphismes:

  • the opposite of the idea of “indivisible atoms" at the "bottom" of everything
  • turtles all the way down
  • (infinite regress is OK)
  • a > b > c > a
  • (so the two options I can think of for non-wellfounded sets are either an infinite straight line or a circle—which biject by stereographic projection)

as well as examples of irreducible things:

  • if you take away one Borromean ring
    http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Borromean_Rings_Illusion.png/774px-Borromean_Rings_Illusion.png
    then the whole is no longer interlinked
  • Twisted products in K-theory are different to straight products.
    https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/M%C3%B6biusStripAsSquare.svg/1000px-M%C3%B6biusStripAsSquare.svg.png
    A Möbius band is different to a wedding band.
    image

    Yet 100% of the difference is in how two 1-D lines are put together. The parts in the recipe are the same, it’s the way they’re combined (twisted or straight product) that makes the difference.

So Leibnitz’s assertions are not only unsupported, but wrong. (Markov, causality, St Anselm’s argument, conservation of mass, etc. in Monadology 4, 5, 22, 44, 45.)

tl,dr: Leibniz, like Spinoza, uses the word “therefore” to mean “and here’s another thing I’m assuming”.





[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.




the sine of the reciprocal of [some angle between −1/π and 1/π]

at increasing resolution

s <- function(x) sin( 1/x )
    plot( s, xlim=c(-1/pi, 1/pi), col=rgb(0,0,0,.7), type = "l", ylab="output", xlab="input", main="compose [multiplicative inverse] with [vertical rect of a circle]" )

(Source: amzn.to)










Why is the boundary of a boundary always null? What is a coboundary?

Beware, some of the + signs in Ghrist’s Elementary Applied Topology are formal sums, which is pasting things together
http://upload.wikimedia.org/wikipedia/en/e/e8/F2_Cayley_Graph.png
rather than umming in the usual dimension-reducing sense.










We often speak of an object being composed of various other objects. We say that the deck is composed of the cards, that a road is [composed of asphalt or concrete], that a house is composed of its walls, ceilings, floors, doors, etc.

Suppose we have some material objects. Here is a philosophical question: what conditions must obtain for those objects to compose something?

If something is made of atomless gunk then it divides forever into smaller and smaller parts—it is infinitely divisible. However, a line segment is infinitely divisible, and yet has atomic parts: the points. A hunk of gunk does not even have atomic parts ‘at infinity’; all parts of such an object have proper parts.













Another interesting space.