Posts tagged with continuous

A history of 20th-century mathematics in ~500 words, with example pictures:

The inheritance of Cantor’s Set Theory allowed the 20th century to create the domain of “Functional Analysis”.
handwriting as functional dataa function that's neither concave nor convexweather data as functional dataTerence Tao's map of function spaceGoldman Sachs share price as functional data
This comes about as an extension of the classical Differential and Integral Calculus in which one considers not merely a particular function (like the exponential function or a trigonometric function), but the operations and transformations which can be performed on all functions of a certain type.
What is sine? What is cosine?
the point of calculus
The creation of a “new” theory of integration, by Émile Borel and above all Henri Lebesgue, at the beginning of the 20th century, followed by the invention of normed spaces by Maurice Fréchet, Norbert Wiener and especially Stefan Banach, yielded new tools for construction and proof in mathematics.

The theory is seductive by its generality, its simplicity and its harmony, and it is capable of resolving difficult problems with elegance. The price to pay is that it usually makes use of non-constructive methods (the Hahn-Banach theorem, Baire’s theorem and its consequences), which enable one to prove the existence of a mathematical object, but without giving an effective construction.

[I]n 1955 … [Nancy] … was the golden age of French mathematics, where, in the orbit of Bourbaki and impelled above all by Henri Cartan, Laurent Schwartz and Jean-Pierre Serre, mathematicians attacked the most difficult problems of geometry, group theory and topology.
a knotsome groups, drawn by Nathan Carter
John Baez's story of the reflection group of a pentagon, drawn by me
reflection group of the square in LaTeX
New tools appeared: sheaf theory and homological algebra … which were admirable for their generality and flexibility.
homological algebra
The apples of the garden of the Hesperides were the famous conjectures stated by André Weil in 1954: these conjectures appeared as a combinatorial problem … of a discouraging generality….

The fascinating aspect of these conjectures is that they … fuse … opposites: “discrete” and “continuous”, or “finite” and “infinite”.

Methods invented in topology to keep track of invariants under the continuous deformation of geometric objects, must be employed to enumerate a finite number of configurations.

Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel. André Weil … was not … ignorant of these techniques …. But [he] was suspicious of “big machinery” ….

Homological algebra, conceived as a general tool reaching beyond all special cases, was invented by Cartan and Eilenberg (… in 1956). This book is a very precise exposition, but limited to the theory of modules over rings and the associated functors “Ext” and “Tor”. It was already a vast synthesis of known methods and results, but sheaves do not enter into this picture. Sheaves … were created together with their homology, but the homology theory is constructed in an ad hoc manner….

In the autumn of 1950, Eilenberg …undertook with Cartan to [axiomatise] sheaf homology; yet the construction … preserves its initial ad hoc character.

When Serre introduced sheaves into algebraic geometry, in 1953, the seemingly pathological nature of the “Zariski topology” forced him into some very indirect constructions.

………




homotopy

homotopy

http://upload.wikimedia.org/wikipedia/commons/7/7e/HomotopySmall.gif

image

image


hi-res




Mapping from
discrete domain: length × width →
discrete codomain: {A,B,Q,Q+} = stocking size.
Two things.
First, it’s a scale in the sense of Hadley Wickham’s ggplot: an association between logic and graphics.
Second, it depicts a well-known phase space.Just like certain pressure & temperature combinations make plumbumappear as a solid, liquid, or gas [for instance the point (3180℉, 1 atmosphere)] — so do certain height & weight combinations recommend a stocking of A, B, or Q.

Mapping from

  • discrete domain: length × width →
  • discrete codomain: {A,B,Q,Q+} = stocking size.

Two things.

  1. First, it’s a scale in the sense of Hadley Wickham’s ggplot: an association between logic and graphics.
  2. Second, it depicts a well-known phase space.

    Just like certain pressure & temperature combinations make plumbum

    appear as a solid, liquid, or gas [for instance the point (3180, 1 atmosphere)] — so do certain height & weight combinations recommend a stocking of A, B, or Q.




Computer scientists use filters, ≥ signs, intersections (sql), and other forms of what I would call “hard boundaries”.

  • Grep either finds what you’re looking for, or it doesn’t. 
  • The condition inside your while(){ loop either trips true and the interior code runs, or it trips false and it’s skipped.
  • You either follow someone on twitter, or you don’t.
  • You either crawled a webpage, or you didn’t.
  • In exploring a code tree or other graph, you either look at the node, or you don’t.
  • Two people either are Facebook friends, or they aren’t.
  • The tweet either included a word from this list, or it didn’t.
 

But, one needn’t be so conceptually constrained. Thinking in a fuzzy logic sense, it’s possible to create a “soft” boundary.

To use a classic example from Bart Kosko's book, although the American legal system imposes a “hard boundary” on adulthood (OK, a series of hard boundaries—16, 18, 21, 25), one really passes into adulthood gradually over time. (Unless you have your first kid at 16, in which case you grow up real quick. But talking about the upper-middle-class college-enrolled set here: most of them grow up slowly.)

That’s nice in a philosophical, contemplative way. But can we use the soft-boundary concept for anything useful? I think so.

For example, in this neo4j video (minute 5)  Marko Rodriguez gives us the following line of Gremlin code:

g.v(1).outE.filter{it.label=='knows' & since > 2006}.count()

We could either be naïve about this and treat 2006 as a hard boundary, or make it a variable and perform sensitivity analysis. In fact, any time we see a number we could turn it into a parameter — ending with a hull of list. We could poke about in that parameter space and by doing so get a better idea of the shape of things than setting a naïve tripwire.

Is there a design pattern for this?

Notice also his gremlins can “be” on multiple nodes at once. That’s certainly not a binary data structure to the codomain. Other non-binary aspects to his graphs:

  • different words (“coloured edges” in graph parlance) like “speaks”, “has worked with”, “had a child with” — all of the richness and drama of Quine’s ontology of language wrought in the connectome of the graph
  • the network structure itself
  • and of course edge weights
 

Here’s an example from Unix for Poets:

cat bible | grep Abel | uniq -c

So-called “bright lines” appear also in the law (married vs not), statistical regression (dummy/indicator variables), and tax brackets (under $15,000.00 or ≥ $15,000.01).

They’re frustrating because they’re discontinuous. (Actually tax brackets are not but the first derivative is discontinuous.) 

Imagine the following (non-existent, stupid) tax system:

  • If you make under £30,000/year you pay no tax.
  • If you make ≥ £30,000.01/year you pay 50% tax on every dollar you made (all the way down to £0.01).

It’s frustrating because it’s discontinuous. I might not go as far as to say that continuity, smoothness, holomorphicity, analyticity and so on are “natural to the human mind” — if in fact we can just take a monolithic view on “the” mind — but continuity and smoothness certainly seem—to me and to other mathematical writers I’m thinking of—like they’re more fair, just, or sensible.

 

Imagine you’re trying to catch an email spammer, and you’ve determined that the character ! is a good trigger for spams. You could either

  1. set a hard boundary: more than 3 !’s, flagged for spam; ≤ 3 !’s, not flagged
  2. or you could count the number of !’s in the text

The latter approach is more flexible:

  • you can change the parameter 3 to something else
  • you can pass the count through a function (like a sigmoid, monotone convex or monotone concave function, or the cumulative-prospect-theory function)
     
  • As in minute 14 of this d3.js video you can add (something like a) “blending” parameter
  • you can set a known algorithm (like logistic regression) to find the optimal parameter value for you
  • you can combine the ! count with other variables (like counts of the word herbal or counts of the forenames of people in the mail user’s address book)
  • you can combine the ! count with other variables and use a known algorithm (like a backprop net) to set all the optimal values for you
  • maybe you can find a way to half-instantiate your desired response when the count is “at half mast” or “in a middling range”.

Back to catching spammers, I drew up an idea for tumblr to catch its spammers a while ago. I noticed a few telltale markers of spam accounts:

  • quick liking in succession
  • squatting on a hashtag
  • high number of likes
  • no / low content in the title
  • at first the spammrs were not reblogging stuff (now they not only reblog but post fakey “original” looking text posts … that’s counter-evolution for ya) so they usually had no posts on their blog page
  • exist ads on the sidebar

They opted for social proof (let people “block” spammy likers from their dashboard and flag them as suspected spammers), which seems to have worked out very well. So I’m not saying “soft boundaries are always better” or something — just that if a “hard boundary” is preventing you from thinking about a problem like you want to, you can get around it pretty easily!

 

I think computer scientists do use soft boundaries, although they might not draw the same analogy to the “crisp” > sign as I am.

  • tag clouds don’t just count words — they increase the display size of the word depending how large the count is (maybe the sqrt of the count?). That tag clouds count different words rather could also be construed as a “coloured” codomain.
  • you don’t just return a webpage or not return a webpage in your crawler. You might get a 404, or you might get a 302. Or you might get a 200, 500, 303, 504, and so on. Additionally the page might be in HTML, JSON, or might simply flip a switch (“turn on my  remote TV recording device”).

Business people (I’ve found) think naturally in terms of soft boundaries as well. If your client / boss is using the word “score” you can mimic that directly with what I’m calling a “soft boundary”.

All you’ve got to do is make up a functional that “measures stuff” any way you want, and slide your > sign along the resulting smooth scale.




  • "Adults have to deal with moral grey areas"
  • "I’m not liberal or conservative, I guess I’m somewhere in the middle"
  • "It may be helpful to think of data science and business intelligence as being on two ends of the same spectrum” (source)
  • "On a sliding scale from 1 to 10, how happy are you with life?"
  • "[S]cientific bias…is a model for separating plausible hypotheses from their opposite.” (source)
  • Please rate your attitude toward the following statements from “strongly agree” to “strongly disagree”.
  • How did you like that book, movie, play, album? Please answer anywhere between ★ and ★★★★★.
  • "The truth lies somewhere in between"

image

People talk about “grey areas” as if [0,1] is so much more sophisticated than {0,1}. I find such rhetoric limiting. After all, the convex combinations of black and white are totally ordered, completely linear, and only one-dimensional! A painting in B&W couldn’t display much variation. (Not that it couldn’t be interesting.) We deal everyday with things more complicated than “a grey area” because the world is 3-D and colour is Lab (3-D nonlinear). Add in texture and smell and you’ve increased the psychological dimensionality manyfold.

image

The metaphor is insufficiently rich. Adult situations don’t fall on a straight line. Political viewpoints don’t sit neatly next to each other in 1-D. Moral ambiguity is certainly more colourful and convoluted than the path from #000000 to #FFFFFF.

Me, I’m more interested in 2.7-dimensional hornspheres, quartz crystal spires, hot-air balloons with a row of golden rings piercing the spine, and quasi-polar negatively bent inside-out torii-cum-logcabins. Or even just something as “pedestrian” as a mountaintop pine forest, which is already much more intricate than, cough cough, the unit interval [0,1].

image

So—back to my original point—I think moral ambiguity resembles a cell complex more than a line segment. Real situations—the layered tragedies, ironies, comedies, and lengthy mediocrities that desirous, egocentric humans instinctively generate—have a much more interesting shape than “the span between 0 and 1.”

image

I guess I shouldn’t be so critical. The people using the grey-area metaphor probably don’t avail themselves of the whimsical thought-gardens in which more exciting shapes live. Sorry there, I was just feeling constricted.

image

I hope you’ve enjoyed these drawings by Robert Ghrist from his (free) notes on homotopy.




Here is a really broad question.  What’s the lay of the land regarding maps of continuous surfaces to discrete ones?  I’m thinking here of credit scores. Someone’s credit score is continuous, one-dimensional — and derives from a multitude of measures that are both continuous and discrete.  As a lender, you have to decide at some point, whether or not to lend to this person — a yes/no proposition.  Granted, you can charge different interest rates.  So maybe it’s not a continuous-to-discrete problem?  Nevertheless it has that flavour.

Logistic maps come to mind.  As does the expansion of a point into several branches.  Maybe someone can lay this out better….  Takers?