Posts tagged with terence tao

Transgressing boundaries, smashing binaries, and queering categories are important goals within certain schools of thought.

Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.

In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the `[0,1]` of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.

By contrast `ℝ→ℝ` functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.

I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.

But in the case I drew above, ƒ′′≹0. In fact “most” 𝒞² functions on that same interval wouldn’t fully fit into either “concave" or "convex”.

So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into

• one type,
• the other,
• or “something in between”.

Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".

hi-res

In harmonic analysis and PDE, one often wants to place a function `ƒ:ℝᵈ→ℂ` on some domain (let’s take a Euclidean space `ℝᵈ` for simplicity) in one or more function spaces in order to quantify its “size”….

[T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.

For function spaces `X` on Euclidean space, two such exponents are the regularity `s` of the space, and the integrability `p` of the space.

—Terence Tao

Hat tip: @AnalysisFact

hi-res

[T]he point of introducing L^p spaces in the first place is … to exploit … Banach space. For instance, if one has |ƒ − g| = 0, one would like to conclude that ƒ = g. But because of the equivalence class in the way, one can only conclude that ƒ is equal to g almost everywhere.

The Lebesgue philosophy is analogous to the “noise-tolerant” philosophy in modern signal progressing. If one is receiving a signal (e.g. a television signal) from a noisy source (e.g. a television station in the presence of electrical interference), then any individual component of that signal (e.g. a pixel of the television image) may be corrupted. But as long as the total number of corrupted data points is negligible, one can still get a good enough idea of the image to do things like distinguish foreground from background, compute the area of an object, or the mean intensity, etc.

Terence Tao

If you’re thinking about points in Euclidean space, then yes — if the distance between them is nil, they are in the exact same spot and therefore the same point.

But abstract mathematics opens up more possibilities.

• Like TV signals. Like 2-D images or 2-D × time video clips.
• Like crime patterns, dinosaur paw prints, neuronal spike-trains, forged signatures, songs (1-D × time), trajectories, landscapes.
• Like, any completenormedvector space. (= it’s thick + distance exists + addition exists + everything’s included = it’s a Banach space)

(Source: terrytao.wordpress.com)