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Math Basics view markdown


misc

  • (nk)<(nek)k
  • Stirling’s formula: n! =(ne)n
    • corollary: log(n!) = 0(n log n)
    • gives us a bound on sorting
    • (ne)n<n!
  • (1x)NeNx
  • Poisson pmf approximates binomial when N large, p small

functions

  • Gamma: Γ(n)=(n1)!=0xn1exdx
  • Zeta: ζ(x)=11x2
  • Sigmoid (logistic): f(x)=11+ex=exex+1
  • Softmax: f(x)=exiiexi
  • spline: piecewise polynomial

stochastic processes

  • Stochastic - random process evolving with time
  • Markov: P(Xt=x|Xt1)=P(Xt=x|Xt1X1)
  • Martingale: E[Xt]=Xt1

abstract algebra

  • Group: set of elements endowed with operation satisfying 4 properties:
  1. closed 2. identity 3. associative 4. inverses
  • Equivalence Relation;
  1. reflexive 2. transitive 3. symmetric

discrete math

  • Goldbach’s strong conjecture: Every even integer greater than 2 can be expressed as the sum of two primes (He considered one a prime).
  • Goldbach’s weak conjecture: All odd numbers greater than 7 are the sum of three primes.
  • Set - An unordered collection of items without replication
  • Proper subset - subset with cardinality less than the set
    • A U A = A Idempotent law
  • Disjoint: A and B = empty set
  • Partition: mutually disjoint, union fills space
  • powerset P(A) = set of all subsets
  • Converse: qp (same as inverse: pq)
  • p1p2p1p2
  • The greatest common divisor of two integers a and b is the largest integer d such that d | a and d | b
  • Proof Techniques
    • Proof by Induction
    • Direct Proof
    • Proof by Contradiction - assume p -q, show contradiction
    • Proof by Contrapositive - show -q -p

identities

  • e2lnx=1e2lnx=1elnxelnx=1x2
  • ln(xy)=ln(x)+ln(y)
  • lnxlny=ln(xlny)
    • difference between log 10n and log 2n is always a constant (about 3.322)
  • logb(x)=logd(x)/logd(b)
  • partial fractions: 3x+11(x3)(x+2)=Ax3+Bx+2
  • (ax+b)k=A1ax+b+A2(ax+b)2+
  • (ax2+bx+c)k=A1x+B1ax2+bx+c+
  • cos(a±b)=cos(a)cos(b)sin(a)sin(b)
  • sin(a±b)=sin(a)cos(b)±sin(b)cos(a)

imaginary numbers

  • complex conjugate of z=x+iy is z = x - iy
  • Euler’s formula eiθ=cos(θ)+isin(θ)
  • sometimes we write imaginary numbers in polar form: $z = z e^{i \theta}$
    • makes multiplication / division simpler
  • absolute value / modules of imaginary numbers: $ a + ib = \sqrt{a^2 + b^2}$

spaces

  • hilbert space - requires an inner product (useful in analyzing kernels) - more general than an inner product space
    • reproducing kernel hilbert space with extra property