Posts tagged with wikipedia

creator: Alberto Sevoso

  • This really happens, all the time:
  • Diffusion
    \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \, \nabla\phi(\mathbf{r},t) \big],
    \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \sum_{i=1}^3\sum_{j=1}^3 \frac{\partial}{\partial x_i}\left[D_{ij}(\phi,\mathbf{r})\frac{\partial \phi(\mathbf{r},t)}{\partial x_j}\right]
     \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla\cdot \left[D(\phi,\mathbf{r})\right] \nabla \phi(\mathbf{r},t) + {\rm tr} \Big[ D(\phi,\mathbf{r})\big(\nabla\nabla^T \phi(\mathbf{r},t)\big)\Big]
  • Schooly mathematics is all about rigid, blocky shapes. But since people realised that the infinite limit of a curve is straight, that dynamics are just another dimension (time), then all tame ploop-ploppulous and fandangulous shapes become fair game.
  • "Later" mathematics takes those simple forms—triangles, squares, circles—to the limit and comes up with these kinds of shapes.
  • A few of them have non-trivial topologyknotting within themselves or linking with other hoops of different colour.
  • Those are some nice 3-manifolds.

via chels

(Source: behance.net)

Just playing with z² / z² + 2z + 2


on WolframAlpha. That’s Wikipedia’s example of a function with two poles (= two singularities = two infinities). Notice how “boring” line-only pictures are compared to the the 3-D ℂ→>ℝ picture of the mapping (the one with the poles=holes). That’s why mathematicians say ℂ uncovers more of “what’s really going on”.

As opposed to normal differentiability, ℂ-differentiability of a function implies:

  • infinite descent into derivatives is possible (no chain of C¹ ⊂ C² ⊂ C³ ... Cω like usual)

  • nice Green’s-theorem type shortcuts make many, many ways of doing something equivalent. (So you can take a complicated real-world situation and validly do easy computations to understand it, because a squibbledy path computes the same as a straight path.)

Pretty interesting to just change things around and see how the parts work.

  • The roots of the denominator are 1+i and 1−i (of course the conjugate of a root is always a root since i and −i are indistinguishable)
  • you can see how the denominator twists
  • a fraction in ℂ space maps lines to circles, because lines and circles are turned inside out (they are just flips of each other: see also projective geometry)
  • if you change the z^2/ to a z/ or a 1/ you can see that.
  • then the Wikipedia picture shows the poles (infinities) 

Complex ℂ→ℂ maps can be split into four parts: the input “real”⊎”imaginary”, and the output “real"⊎"imaginary”. Of course splitting them up like that hides the holistic truth of what’s going on, which comes from the perspective of a “twisted” plane where the elements z are mod z • exp(i • arg z).

a conformal map (angle-preserving map)

ℂ→ℂ mappings mess with my head…and I like it.

When you see the context in which something was written and you know who the author was beyond just a name, you learn so much more than when you find the same text placed in the anonymous, faux-authoritative, anti-contextual brew of the Wikipedia.

The question isn’t just one of authentication and accountability, though those are important, but something more subtle. A voice should be sensed as a whole.
Jaron Lanier

(Source: edge.org)

Are mathematicians deliberately obscure? Or is it really so hard for them to write prose?

Check out this description of σ-algebras from Wik***dia.

In mathematics, a σ-algebra (also sigma-algebraσ-fieldsigma-field) is a technical concept for a collection of sets satisfying certain properties.[1] The main use of σ-algebras is in the definition of measures; specifically, a σ-algebra is the collection of sets over which a measure is defined.

No sh_t? It’s technical? And it satisfies properties. You don’t say.

The kernel of that paragraph is just one sentence.

In mathematics, a σ-algebra is a measurable collection of sets.

I changed the W*****dia page at 8:40pm on 3 Mar ‘11. Let’s see if I get in trouble. (I bet if I do it will be for “not being rigorous” or “original research”.)

Yes this is a specialist topic, but that doesn’t require gobbledegook. A σ-algebra is measurable like , but is not ℝ. Why can’t we just use normal words?

UPDATE: It hurts to be this right. My changes were reverted about an hour after I put them up. Am I wrong here?

I’m reminded of a story Doug Hofstadter told us about a friend of his who submitted an article in clear, everyday language to an academic journal. According to DH, the journal’s editors rejected the piece, saying it was too unprofessional. They confused jargon with sophistication, bombast with wisdom.

I don’t know the friend’s name or the journal’s name, and I half-wonder if I am just being a pr$ck about this Wikipedia article. But no, think about how people react to the word “maths”. This has got to be the reason—this and boring maths classes. Mathematicians literally refuse to write simply.

UPDATE 2: Another offender is the article on compact topological spaces. I’m actually removing some text from the garbled lede when I say:

In mathematics, specifically general topology, a compact topology is a topological space whose topology has the compactness property.

I think I’ve found a new candidate for worst sentences in the English language. Does anyone have George Orwell’s e-mail address?