Math Basics

# 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

# 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

# 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: $E[X_t]=X_{t-1}$

# 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 $\mathcal{P}$(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

# 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: $\frac{3x+11}{(x-3)(x+2)} = \frac{A}{x-3} + \frac{B}{x+2}$
• $(ax+b)^k = \frac{A_1}{ax+b}+\frac{A_2}{(ax+b)^2}+…$
• $(ax^2+bx+c)^k = \frac{A_1x+B_1}{ax^2+bx+c}+…$
• $\cos(a\pm b) = \cos(a)\cos(b)\mp \sin(a)\sin(b)$
• $\sin(a \pm b) = \sin(a)\cos(b) \pm \sin(b)\cos(a)$

# 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: $a + ib = \sqrt{a^2 + b^2}$

# 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