math basics
Contents
3.9. math basics#
3.9.1. misc#
\(\left( \frac{n}{k} \right) < \left( \frac{ne}{k} \right)^k\)
Stirling’s formula: \( n! ~= (\frac{n}{e})^n \)
corollary: log(n!) = 0(n log n)
gives us a bound on sorting
\(\left( \frac{n}{e} \right)^n < n!\)
\((1-x)^N \leq e^{-Nx}\)
Poisson pmf approximates binomial when N large, p small
3.9.2. functions#
Gamma: \(\Gamma(n)=(n-1)!=\int_0^\infty x^{n-1}e^{-x}dx\)
Zeta: \(\zeta(x) = \sum_1^\infty \frac{1}{x^2} \)
Sigmoid (logistic): \(f(x) = \frac{1}{1+e^{-x}} = \frac{e^x}{e^x+1}\)
Softmax: \(f(x) = \frac{e^{x_i}}{\sum_i e^{x_i}}\)
spline: piecewise polynomial
3.9.3. stochastic processes#
Stochastic - random process evolving with time
Markov: \(P(X_t=x\|X_{t-1})=P(X_t=x\|X_{t-1}...X_1)\)
Martingale: \(E[X_t]=X_{t-1}\)
3.9.4. abstract algebra#
Group: set of elements endowed with operation satisfying 4 properties:
closed 2. identity 3. associative 4. inverses
Equivalence Relation;
reflexive 2. transitive 3. symmetric
3.9.5. 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 \(\mathcal{P}\)(A) = set of all subsets
Converse: \(q\to p\) (same as inverse: \(-p \to -q\))
\(p_1 \to p_2 \iff - p_1 \lor p_2 \)
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 \(\land\) -q, show contradiction
Proof by Contrapositive - show -q \(\to\) -p
3.9.6. identities#
\(e^{-2lnx}= \frac{1}{e^{2lnx}} = \frac{1}{e^{lnx}e^{lnx}} = \frac{1}{x^2}\)
\(\ln(xy) = \ln(x)+\ln(y)\)
\(\ln x * \ln y = \ln(x^{\ln y})\)
difference between log 10n and log 2n is always a constant (about 3.322)
\(\log_b (x) = \log_d (x) / \log_d (b)\)
partial fractions: \(\frac{3x+11}{(x-3)(x+2)} = \frac{A}{x-3} + \frac{B}{x+2}\)
\((ax+b)^k = \frac{A_1}{ax+b}+\frac{A_2}{(ax+b)^2}+...\)
\((ax^2+bx+c)^k = \frac{A_1x+B_1}{ax^2+bx+c}+...\)
\(\cos(a\pm b) = \cos(a)\cos(b)\mp \sin(a)\sin(b)\)
\(\sin(a \pm b) = \sin(a)\cos(b) \pm \sin(b)\cos(a)\)
3.9.7. imaginary numbers#
complex conjugate of z=x+iy is \(z^*\) = x - iy
Euler’s formula \(e^{i \theta} = \cos (\theta) + i \sin (\theta)\)
sometimes we write imaginary numbers in polar form: \(z = |z| e^{i \theta}\)
makes multiplication / division simpler
absolute value / modules of imaginary numbers: \(|a + ib| = \sqrt{a^2 + b^2}\)
3.9.8. 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