Posts tagged with tensors










Without using a calculator or even pen and paper—or even doing the arithmetic in your head—¿which is the larger product? of:

  • 3.01 × 6.99
  • 2.99 × 7.01

You know 3 × 7  =  21. But what happens if you gently place your finger on the teeter-totter?

 

Although this is “just arithmetic” and so doesn’t require a learnèd vocabulary of higher mathematics, it touches on a few “higher maths” topics:

  • bilinearity being the way I would answer it “conceptually rather than computationally”
    Bilinear maps and dual spaces  Think of a function that takes two inputs and gives one output. The + operator is like that. 9+10=19 or, if you prefer to be computer-y about it, plus(9, 10) returns 19.  So is the relation “the degree to which X loves Y”. Takes as inputs two people and returns the degree to which the first loves the second. Not necessarily symmetrical! I.e. love(A→B) ≠ love(B→A). * It can get quite dramatic.    An operator could also take three or four inputs.  The vanilla Black-Scholes price of a call option asks for {the current price, desired exercise price, [European | American | Asian], date of expiry, volatility}.  That’s five inputs: three ℝ⁺ numbers, one option from a set isomorphic to {1,2,3} = ℕ₃, and one date.    A bilinear map takes two inputs, and it’s linear in both terms.  Meaning if you adjust one of the inputs, the final change to the output is only a linear difference.  Multiplication is a bilinear operation (think 3×17 versus 3×18). Vectorial dot multiplication is a bilinear operation. Vectorial cross multiplication is a bilinear operation but it returns a vector instead of a scalar. Matrix multiplication is a bilinear operation which returns another matrix. And tensor multiplication ⊗, too, is bilinear.  Above, Juan Marquez shows the different bilinear operators and their duals. The point is that it’s just symbol chasing.    * The distinct usage “I love sandwiches” would be considered a separate mathematical operator since it takes a different kind of input.
  • wiggly numbers
    image
  • ± ε
  • statistical robustness
  • tensor product of modules
    Tensor product of modules.png
  • isoperimetric inequality

I’ll leave the question/reasoning unanswered with a space for you to answer which is larger and why.

3±ε  ×   7∓ε    =    21±¿?

(Source: Wikipedia)





The Crown of Thorns is a woodworking technique using interlocking wooden sticks that are notched to intersect at right angles forming joints and self-supporting objects….
Large-scale crowns may use the principles of tensegrity structures, where the wooden sticks provide rigidity and separate cables in tension carry the forces that hold them together.

The Crown of Thorns is a woodworking technique using interlocking wooden sticks that are notched to intersect at right angles forming joints and self-supporting objects….

Large-scale crowns may use the principles of tensegrity structures, where the wooden sticks provide rigidity and separate cables in tension carry the forces that hold them together.


hi-res




In 4 dimensions, it takes 20 numbers to specify the curvature at each point. 10 of these numbers are captured by the “Ricci tensor”, while the remaining 10 are captured by the “Weyl tensor”.
John Baez

(Source: math.ucr.edu)













To efficiently travel in the solar system (like, using gravity as much as possible and fighting it as little as possible — cf, Dao/wuwei 無爲) you want to weave among these moving pullers like Tarzan brachiating among the vines.
via @vruba:


… I was delighted the other day when one Peter R. – no, that won’t do; let’s call him P. Richardson – linked to the interplanetary transport network. This is a fluid set of easy paths around the Solar System that has only a very abstract kind of existence but is of first importance if you want to, say, send something to Mars.
Orbital mechanics are way beyond my understanding, but they’re really cool. Twice now I’ve implemented the velocity Verlet integrator, without really getting why it works, to play. Actually, a friend of my family is a mathematician whose entire – quite successful – career is in geometric integration. He’s published a couple dozen papers on, basically, the plusses and minuses of various ways of simulating simple physical systems. It’s all just super neat. Did you know Buzz Aldrin figured out an important Earth-Mars orbit?

To efficiently travel in the solar system (like, using gravity as much as possible and fighting it as little as possible — cf, Dao/wuwei 無爲) you want to weave among these moving pullers like Tarzan brachiating among the vines.

via @vruba:

… I was delighted the other day when one Peter R. – no, that won’t do; let’s call him P. Richardson – linked to the interplanetary transport network. This is a fluid set of easy paths around the Solar System that has only a very abstract kind of existence but is of first importance if you want to, say, send something to Mars.

Orbital mechanics are way beyond my understanding, but they’re really cool. Twice now I’ve implemented the velocity Verlet integrator, without really getting why it works, to play. Actually, a friend of my family is a mathematician whose entire – quite successful – career is in geometric integration. He’s published a couple dozen papers on, basically, the plusses and minuses of various ways of simulating simple physical systems. It’s all just super neat. Did you know Buzz Aldrin figured out an important Earth-Mars orbit?


hi-res




J is hott. Some highlights from the Wikipedia article and J's homepage:

  • you can do a lot with just a few characters in J. Define a moving average in 8 characters, including spaces, for example.
  • Have you ever felt like whether it’s Java or C, Python or Ruby, all these languages are just the Same Old Thing?

J makes thinking in high-dimensional arrays easy.

  1. The sentence .i 7 8 means “Show me a 7×8 two-array” (ok, “matrix” but … matrices are verbs and arrays are nouns)
  2. The sentence .i 7 8 3 means “Show me a 7×8×3 three-array”.
  3. The sentence .i 7 8 3 4 13 2 66 means "Show me a 7×8×3×4×13×2×66 dimensional seven-array”.

I won’t reprint the long outputs but here’s a shorter one.

   i.4 5 3
 0  1  2
 3  4  5
 6  7  8
 9 10 11
12 13 14

15 16 17
18 19 20
21 22 23
24 25 26
27 28 29

30 31 32
33 34 35
36 37 38
39 40 41
42 43 44

45 46 47
48 49 50
51 52 53
54 55 56
57 58 59

And another for clarity:

   i.3 5 4
 0  1  2  3
 4  5  6  7
 8  9 10 11
12 13 14 15
16 17 18 19

20 21 22 23
24 25 26 27
28 29 30 31
32 33 34 35
36 37 38 39

40 41 42 43
44 45 46 47
48 49 50 51
52 53 54 55
56 57 58 59

This is reminiscent of using R's combn function to visualise higher-dimensional stuff, right?

I guess this is how computers think all the time! I wonder what they say about us when we’re not around.




It wasn’t Einstein, but the mathematician Hermann Weyl who first addressed the [distinction] [between gravitational and non-gravitational fields] in 1918 in the course of reconstructing Einstein’s theory on the preferred … basis of a “pure infinitesimal geometry”….

Holding that direct…comparisons of length or duration could be made at near-by points of spacetime, but not … “at a distance”, Weyl discovered additional terms in his expanded geometry that he … formally identified with the potentials of the electromagnetic field. From these, the electromagnetic field strengths can be immediately derived.
Maxwell's equations in differential form (reduces 20 to 4)Choosing an action integral to obtain both [sorts of] Maxwell equations as well as Einstein’s gravitational theory, Weyl could express electromagnetism as well as gravitation solely within the confines of a spacetime geometry. As no other interactions were definitely known to occur, Weyl proudly declared that the concepts of geometry and physics were the same.
Gauss' law for rmagnetism
Hence, everything in the physical world was a manifestation of spacetime geometry. (The) distinction between geometry and physics is an error, physics extends not at all beyond geometry: the world is a (3+1) dimensional metrical manifold, and all physical phenomena transpiring in it are only modes of expression of the metric field, …. (M)atter itself is dissolved in “metric” and is not something substantial that in addition exists “in” metric space. (1919, 115–16)

Riemannian

Ryckman, Thomas A., "Early Philosophical Interpretations of General Relativity", The Stanford Encyclopedia of Philosophy (Fall 2012 Edition), Edward N. Zalta (ed.), forthcoming URL = <http://plato.stanford.edu/archives/fall2012/entries/genrel-early/>.

via University of David