Useful mathematical symbols

symbol tex digraph how it reads
<: is a subtype of
\vDash <bar> = entails
\vdash <bar> - infers
\to -> is mapped to maps sets to sets
\mapsto <bar> > is mapped to maps elements to elements

Euler’s Characteristic

The second most beautiful equation and its surprising applications - YouTube

Not every shape has an Euler Characteristic that equals 2.

For example, donut or torus.

This is a specific case where it equals 2
\(V - E + F = 2\)

For any convex polyhedron the vertices, minus edges, plus faces will always equal 2.

It also holds on curved surfaces (e.g. the longitude and latitude lines on a sphere).

The Euler characteristic remains the same under homeomorphisms (topological stretching).

Sphere packing

  • square contains \(2^2 = 4\) circles, one for each quadrant
  • cube contains \(2^3 = 8\) spheres, one for each octant
  • 9D-hypercube contains \(2^9 = 512\) 9D hyperspheres, one for each division

A 9D-cube is able to fit one hypersphere between and touching all 512 which tangents on all the edges of the hypercube too.

As the number of dimensions goes up, the distance between opposing faces of the cube stay the same while the diagonal distance between opposite corners gets longer and longer

Bayes theorem

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                   P(Netflix|chill) P(chill)
P(chill|Netflix) = -------------------------
                           P(Netflix)
Posterior probability
P(Netflix)
Likelihood
P(Netflix|chill)
Prior probability
P(chill)

5-Sided Square

5-Sided Square - Numberphile - YouTube

3-sided squares are possible if you draw right angles on a sphere.

A sphere has constant Gaussian curvature.

You can make a pseudosphere.

A pseudosphere has constant negative curvature.

Hyperbolic secants and co-secants.

Hyperbolic space is amazing.

A sphere can create a 3-sided square

A pseudosphere can create a 5-sided square.

If spacetime is curved, does this mean we can use curvature to create atomic structures that could not exist in regular, flat space?

The universe appears to have some kind of negative curvature.

This means that at a large scale things appear to fit inside it which could not otherwise fit if there was not equal curvature.

Glossary

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Particle filters
Sequential Monte Carlo method
SMC method
    [set of MC algorithms]

    Used to solve filtering problems arising
    in signal processing and Bayesian
    statistical inference.

Pascal’s Triangle

Generate it with code

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#+BEGIN_SRC sh -n :async :results verbatim drawer
  pascals-triangle
#+END_SRC
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#!/bin/bash
export TTY

is_tty() {
    # If stout is a tty
    [[ -t 1 ]]
}

pager() {
    if is_tty; then
        vs $@
    else
        cat
    fi
}

runhaskell $HOME/scripts/pascals-triangle.hs 10 | tabulate | tr '\t' ' ' | pager
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#!/home/shane/scripts/runhaskell

import Data.List (transpose)
import System.Environment (getArgs)
import Text.Printf (printf)

-- Pascal's triangle.
pascal :: [[Int]]
pascal = iterate (\row -> 1 : zipWith (+) row (tail row) ++ [1]) [1]

-- The n by n Pascal lower triangular matrix.
pascLow :: Int -> [[Int]]
pascLow n = zipWith (\row i -> row ++ replicate (n-i) 0) (take n pascal) [1..]

-- The n by n Pascal upper triangular matrix.
pascUp :: Int -> [[Int]]
pascUp = transpose . pascLow

-- The n by n Pascal symmetric matrix.
pascSym :: Int -> [[Int]]
pascSym n = take n . map (take n) . transpose $ pascal

-- Format and print a matrix.
printMat :: String -> [[Int]] -> IO ()
printMat title mat = do
  putStrLn $ title ++ "\n"
  -- mapM_ (putStrLn . concatMap (printf " %2d")) mat
  mapM_ (putStrLn . concatMap (printf "\t%2d")) mat
  putStrLn "\n"

main :: IO ()
main = do
  ns <- fmap (map read) getArgs
  case ns of
    [n] -> do printMat "Lower triangular" $ pascLow n
              printMat "Upper triangular" $ pascUp  n
              printMat "Symmetric"        $ pascSym n
    _   -> error "Usage: pascmat <number>"

Output

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Lower triangular

                  1  0  0  0   0   0    0    0     0     0
                  1  1  0  0   0   0    0    0     0     0
                  1  2  1  0   0   0    0    0     0     0
                  1  3  3  1   0   0    0    0     0     0
                  1  4  6  4   1   0    0    0     0     0
                  1  5 10 10   5   1    0    0     0     0
                  1  6 15 20  15   6    1    0     0     0
                  1  7 21 35  35  21    7    1     0     0
                  1  8 28 56  70  56   28    8     1     0
                  1  9 36 84  126 126  84   36     9     1


Upper triangular

                  1  1  1  1   1   1    1    1     1     1
                  0  1  2  3   4   5    6    7     8     9
                  0  0  1  3   6  10   15   21    28    36
                  0  0  0  1   4  10   20   35    56    84
                  0  0  0  0   1   5   15   35    70    126
                  0  0  0  0   0   1    6   21    56    126
                  0  0  0  0   0   0    1    7    28    84
                  0  0  0  0   0   0    0    1     8    36
                  0  0  0  0   0   0    0    0     1     9
                  0  0  0  0   0   0    0    0     0     1


Symmetric

                  1  1  1  1   1   1    1    1     1     1
                  1  2  3  4   5   6    7    8     9    10
                  1  3  6 10  15  21   28   36    45    55
                  1  4 10 20  35  56   84   120   165   220
                  1  5 15 35  70  126  210  330   495   715
                  1  6 21 56  126 252  462  792   1287  2002
                  1  7 28 84  210 462  924  1716  3003  5005
                  1  8 36 120 330 792  1716 3432  6435  11440
                  1  9 45 165 495 1287 3003 6435  12870 24310
                  1 10 55 220 715 2002 5005 11440 24310 48620

Explanation

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Lower triangular
To get the fibonacci sequence, sum the diagonals.

  1 1 2 3 etc.
 / / / /
1 / / /
 / 1 /
1 / /
 / 2
1 /  1
 /
1  3  3  1
1  4  6  4   1
1  5 10 10   5   1
1  6 15 20  15   6    1
1  7 21 35  35  21    7    1
1  8 28 56  70  56   28    8     1
1  9 36 84  126 126  84   36     9     1


Upper triangular
To get the fibonacci sequence, sum the diagonals as above.

1  1  1  1   1   1    1    1     1     1
   1  2  3   4   5    6    7     8     9
      1  3   6  10   15   21    28    36
         1   4  10   20   35    56    84
             1   5   15   35    70    126
                 1    6   21    56    126
                      1    7    28    84
                           1     8    36
                                 1     9
                                       1


Symmetric
This is the traditional pascal's triangle.
To get the fibonaccy sequence, translate across the grid in an L shape. 1 right, 2 up.

1  1  1  1   1   1    1    1     1     1
1  2  3  4   5   6    7    8     9    10
1  3  6 10  15  21   28   36    45    55
1  4 10 20  35  56   84   120   165   220
1  5 15 35  70  126  210  330   495   715
1  6 21 56  126 252  462  792   1287  2002
1  7 28 84  210 462  924  1716  3003  5005
1  8 36 120 330 792  1716 3432  6435  11440
1  9 45 165 495 1287 3003 6435  12870 24310
1 10 55 220 715 2002 5005 11440 24310 48620

Bayes’ Rule

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\begin{equation}
\underbrace{p(\mathbf{z} \mid
\mathbf{x})}_{\text{Posterior}} =
\underbrace{p(\mathbf{z})}_{\text{Prior}}
\times \frac{\overbrace{p(\mathbf{x} \mid \mat
hbf{z})}^{\text{Likelihood}}}{\underbrace{\int
p(\mathbf{x} \mid \mathbf{z}) \, p(\mathbf{z})
\, \mathrm{d}\mathbf{z}}_{\text{Marginal
Likelihood}}} \enspace ,
\end{equation}
where $\mathbf{z}$ denotes latent parameters we want to infer and $\mathbf{x}$ denotes data.

\begin{equation} \underbrace{p(\mathbf{z} \mid \mathbf{x})}_{\text{Posterior}} = \underbrace{p(\mathbf{z})}_{\text{Prior}} \times \frac{\overbrace{p(\mathbf{x} \mid \mathbf{z})}^{\text{Likelihood}}}{\underbrace{\int p(\mathbf{x} \mid \mathbf{z}) , p(\mathbf{z}) , \mathrm{d}\mathbf{z}}_{\text{Marginal Likelihood}}} \enspace , \end{equation}

where \(\mathbf{z}\) denotes latent parameters we want to infer and \(\mathbf{x}\) denotes data.** Euclid

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\Procedure{Euclid}{$a,b$}
\State $r\gets a\bmod b$
\While{$r\not=0$}
\State $a\gets b$
\State $b\gets r$
\State $r\gets a\bmod b$
\EndWhile
\State \textbf{return} $b$
\EndProcedure

Appendix

This person talks about math related to statistics and computing.