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Binomial Theorem

Basis for dualities?

How does the binomial theorem relate to the triangle of power? (The relationship between a, b, c in {$a=b^c$}?)

Examples

  • 2 x 2 square - conceptual factors - "and" and "or"
  • eightsome - logical square 2 x 2 vertices and 4 pairs (edges)
  • 2 x 2 multiplication
  • 3 x 3 factorial experiment - Norman Anderson - multinomial
  • girl + boy
  • genetics - dominant and recessive
  • truth tables
  • expand 1.9 to a power = 2 - .01
  • |A union B| + |A intersect B| = |A| + |B|
  • Russian roulette
  • interest rates - and taking the limit - formula for {$e^x$}
  • simplexes, cross polytopes, hypercubes, coordinate systems

Infinite Series. Dissecting Hypercubes with Pascal's Triangle.

3 Blue 1 Brown. Derivative formulas through geometry.

Product rule {$u\cdot v + \textrm{d}(u\cdot v) = (u + \textrm{d}u)(v + \textrm{d}v)$}

Proof of the binomial theorem

Sum of the row is {$2^n$}.

Inclusion-exclusion principle Counting nondivisible integers. Counting derangements. Number of surjective functions. Stirling numbers of the second kind. Rook polynomials. Main proof and algebraic proof. Combinatorial sieve methods.(Brun.)

The coefficient of {$q^r$} in {$\binom{n}{k}_q$} counts the number of inversions, for example, transforming 00011 to 01100 so that 1 is to the left of 0, here in 4 ways.

The coefficient of {$q^r$} in {$\binom{n+m}{m}_q$} counts the number of partitions of r with m or fewer parts each less than or equal to n.

{$\binom{n}{k}_q$} counts the number of k-dimensional vector subspaces of an n-dimensional vector space over {$F_q$} (a Grassmannian).

When k=1 and q=1 we have the Euler characteristic of the complex Grassmannian, and when q=-1, the Euler characteristic of the real Grassmannian.

The number of k-dimensional affine subspaces of {$F_{q^n}$} is {$q^{n-k}\binom{n}{k}_q$}. This allows an interpretation of the identity {$\binom{m}{r}_q=\binom{m-1}{r}_q + q^{m-r}\binom{m-1}{r-1}_q$}.

  • Grassmannian Key to Moore's proof of Bott periodicity.
  • God's Binomial Identity Expressing a number {$d=d_1*d_2*d_3$} as the sum of signed binomial coefficients {$d=\binom{d_1+d_2+d_3}{3}-\sum_{i<j}\binom{d_i+d_j}{3}+\sum_{i}\binom{d_i}{3}-\binom{0}{3}$}

Directed simplicial complexes

Finite field with one element

Grassmannians

Bottom-up and top-down duality

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Šis puslapis paskutinį kartą keistas February 03, 2023, at 05:51 PM