167 lines
7.3 KiB
TeX
167 lines
7.3 KiB
TeX
% Written by Anders Sjoqvist and Ulf Lundstrom, 2009
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% The main sources are: tinyKACTL, Beta and Wikipedia
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\subsection{Equations}
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\[ax^2+bx+c=0 \Rightarrow x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
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The extremum is given by $x = -b/2a$.
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\[\begin{aligned}ax+by=e\\cx+dy=f\end{aligned}
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\Rightarrow
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\begin{aligned}x=\dfrac{ed-bf}{ad-bc}\\y=\dfrac{af-ec}{ad-bc}\end{aligned}\]
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In general, given an equation $Ax = b$, the solution to a variable $x_i$ is given by
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\[x_i = \frac{\det A_i'}{\det A} \]
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where $A_i'$ is $A$ with the $i$'th column replaced by $b$.
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\subsection{Inequality}
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For positive $a$, $b$, $c$, $d$ with $\frac{a}{b} < \frac{c}{d}$, $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$ hold.
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Also, for positive $a$, $b$, $c$, $d$, $\min(\frac{a}{b}, \frac{c}{d}) \leqslant \frac{a+c}{b+d} \leqslant max(\frac{a}{b}, \frac{c}{d})$.
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e.g., for two sets of disjoint vertices $A$, $B$ of $G$,
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\begin{align*}
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\min \left( \frac{E(G[A])}{V(G[A])}, \frac{E(G[B])}{V(G[B])} \right) & \leqslant
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\frac{E(G[A\cup B])}{V(G[A\cup B])} = \frac{E(G[A]) + E(G[B])}{V(G[A]) + V(G[B])} \\&
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\leqslant \max \left( \frac{E(G[A])}{V(G[A])}, \frac{E(G[B])}{V(G[B])} \right)
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\end{align*}
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(therefore, the average degree of the disjoint set is larger when we divide them.)
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\subsection{Recurrences}
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If $a_n = c_1 a_{n-1} + \dots + c_k a_{n-k}$, and $r_1, \dots, r_k$ are distinct roots of $x^k - c_1 x^{k-1} - \dots - c_k$, there are $d_1, \dots, d_k$ s.t.
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\[a_n = d_1r_1^n + \dots + d_kr_k^n. \]
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Non-distinct roots $r$ become polynomial factors, e.g. $a_n = (d_1n + d_2)r^n$.
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% \subsection{Trigonometry}
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% \begin{align*}
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% \sin(v+w)&{}=\sin v\cos w+\cos v\sin w\\
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% \cos(v+w)&{}=\cos v\cos w-\sin v\sin w\\
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% \tan(v+w)&{}=\dfrac{\tan v+\tan w}{1-\tan v\tan w}\\
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% \sin v+\sin w&{}=2\sin\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
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% \cos v+\cos w&{}=2\cos\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
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% (V+W)\tan(v-w)/2&{}=(V-W)\tan(v+w)/2
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% \end{align*}
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% where $V, W$ are lengths of sides opposite angles $v, w$.
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% \begin{align*}
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% a\cos x+b\sin x&=r\cos(x-\phi)\\
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% a\sin x+b\cos x&=r\sin(x+\phi)
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% \end{align*}
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% where $r=\sqrt{a^2+b^2}, \phi=\operatorname{atan2}(b,a)$.
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\subsection{Geometry}
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\subsubsection{Triangles}
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Side lengths: $a,b,c$
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Semiperimeter: $p=\dfrac{a+b+c}{2}$
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Area: $A=\sqrt{p(p-a)(p-b)(p-c)}$
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Circumradius: $R=\dfrac{abc}{4A}$
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Inradius: $r=\dfrac{A}{p}$
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Length of the median (divides the triangle into two equal area triangles): $m_a=\tfrac{1}{2}\sqrt{2b^2+2c^2-a^2}$
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Length of the bisector (divides angles into two): $s_a=\sqrt{bc\left[1-\left(\dfrac{a}{b+c}\right)^2\right]}$
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Law of sines: $\dfrac{\sin\alpha}{a}=\dfrac{\sin\beta}{b}=\dfrac{\sin\gamma}{c}=\dfrac{1}{2R}$
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Law of cosines: $a^2=b^2+c^2-2bc\cos\alpha$
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Law of tangents: $\dfrac{a+b}{a-b}=\dfrac{\tan\dfrac{\alpha+\beta}{2}}{\tan\dfrac{\alpha-\beta}{2}}$
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\subsubsection{Quadrilaterals}
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With side lengths $a,b,c,d$, diagonals $e, f$, diagonals angle $\theta$, area $A$ and
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magic flux $F=b^2+d^2-a^2-c^2$:
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\[ 4A = 2ef \cdot \sin\theta = F\tan\theta = \sqrt{4e^2f^2-F^2} \]
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For cyclic quadrilaterals the sum of opposite angles is $180^\circ$,
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$ef = ac + bd$, and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$.
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% \subsubsection{Spherical coordinates}
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% \begin{center}
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% \includegraphics[width=25mm]{source/Math/sphericalCoordinates.pdf}
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% \end{center}
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% \[\begin{array}{cc}
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% x = r\sin\theta\cos\phi & r = \sqrt{x^2+y^2+z^2}\\
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% y = r\sin\theta\sin\phi & \theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2})\\
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% z = r\cos\theta & \phi = \textrm{atan2}(y,x)
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% \end{array}\]
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\subsection{Derivatives/Integrals}
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\begin{align*}
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\dfrac{d}{dx}\arcsin x = \dfrac{1}{\sqrt{1-x^2}} &&& \dfrac{d}{dx}\arccos x = -\dfrac{1}{\sqrt{1-x^2}} \\
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\dfrac{d}{dx}\tan x = 1+\tan^2 x &&& \dfrac{d}{dx}\arctan x = \dfrac{1}{1+x^2} \\
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\int\tan ax = -\dfrac{\ln|\cos ax|}{a} &&& \int x\sin ax = \dfrac{\sin ax-ax \cos ax}{a^2} \\
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\int e^{-x^2} = \frac{\sqrt \pi}{2} \text{erf}(x) &&& \int xe^{ax}dx = \frac{e^{ax}}{a^2}(ax-1)
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\end{align*}
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Integration by parts:
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\[\int_a^bf(x)g(x)dx = [F(x)g(x)]_a^b-\int_a^bF(x)g'(x)dx\]
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\subsection{Sums}
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\[ c^a + c^{a+1} + \dots + c^{b} = \frac{c^{b+1} - c^a}{c-1}, c \neq 1 \]
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\begin{align*}
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1 + 2 + 3 + \dots + n &= \frac{n(n+1)}{2} \\
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1^2 + 2^2 + 3^2 + \dots + n^2 &= \frac{n(2n+1)(n+1)}{6} \\
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1^3 + 2^3 + 3^3 + \dots + n^3 &= \frac{n^2(n+1)^2}{4} \\
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1^4 + 2^4 + 3^4 + \dots + n^4 &= \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30} \\
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\end{align*}
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\subsection{Series}
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$$e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots,\,(-\infty<x<\infty)$$
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$$\ln(1+x) = x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots,\,(-1<x\leq1)$$
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$$\sqrt{1+x} = 1+\frac{x}{2}-\frac{x^2}{8}+\frac{2x^3}{32}-\frac{5x^4}{128}+\dots,\,(-1\leq x\leq1)$$
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$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots,\,(-\infty<x<\infty)$$
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$$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots,\,(-\infty<x<\infty)$$
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\subsection{Probability theory}
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Let $X$ be a discrete random variable with probability $p_X(x)$ of assuming the value $x$. It will then have an expected value (mean) $\mu=\mathbb{E}(X)=\sum_xxp_X(x)$ and variance $\sigma^2=V(X)=\mathbb{E}(X^2)-(\mathbb{E}(X))^2=\sum_x(x-\mathbb{E}(X))^2p_X(x)$ where $\sigma$ is the standard deviation. If $X$ is instead continuous it will have a probability density function $f_X(x)$ and the sums above will instead be integrals with $p_X(x)$ replaced by $f_X(x)$.
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Expectation is linear:
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\[\mathbb{E}(aX+bY) = a\mathbb{E}(X)+b\mathbb{E}(Y)\]
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For independent $X$ and $Y$, \[V(aX+bY) = a^2V(X)+b^2V(Y).\]
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\subsubsection{Binomial distribution}
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The number of successes in $n$ independent yes/no experiments, each which yields success with probability $p$ is $\textrm{Bin}(n,p),\,n=1,2,\dots,\, 0\leq p\leq1$.
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\[p(k)=\binom{n}{k}p^k(1-p)^{n-k}\]
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\[\mu = np,\,\sigma^2=np(1-p)\]
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$\textrm{Bin}(n,p)$ is approximately $\textrm{Po}(np)$ for small $p$.
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\subsubsection{First success distribution}
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The number of trials needed to get the first success in independent yes/no experiments, each which yields success with probability $p$ is $\textrm{Fs}(p),\,0\leq p\leq1$.
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\[p(k)=p(1-p)^{k-1},\,k=1,2,\dots\]
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\[\mu = \frac1p,\,\sigma^2=\frac{1-p}{p^2}\]
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\subsubsection{Poisson distribution}
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The number of events occurring in a fixed period of time $t$ if these events occur with a known average rate $\kappa$ and independently of the time since the last event is $\textrm{Po}(\lambda),\,\lambda=t\kappa$.
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\[p(k)=e^{-\lambda}\frac{\lambda^k}{k!}, k=0,1,2,\dots\]
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\[\mu=\lambda,\,\sigma^2=\lambda\]
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% \subsubsection{Uniform distribution}
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% If the probability density function is constant between $a$ and $b$ and 0 elsewhere it is $\textrm{U}(a,b),\,a<b$.
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% \[f(x) = \left\{
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% \begin{array}{cl}
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% \frac{1}{b-a} & a<x<b\\
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% 0 & \textrm{otherwise}
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% \end{array}\right.\]
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% \[\mu=\frac{a+b}{2},\,\sigma^2=\frac{(b-a)^2}{12}\]
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\subsubsection{Exponential distribution}
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The time between events in a Poisson process is $\textrm{Exp}(\lambda),\,\lambda>0$.
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\[f(x) = \left\{
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\begin{array}{cl}
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\lambda e^{-\lambda x} & x\geq0\\
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0 & x<0
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\end{array}\right.\]
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\[\mu=\frac{1}{\lambda},\,\sigma^2=\frac{1}{\lambda^2}\]
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% \subsubsection{Normal distribution}
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% Most real random values with mean $\mu$ and variance $\sigma^2$ are well described by $\mathcal{N}(\mu,\sigma^2),\,\sigma>0$.
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% \[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
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% If $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$ then
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% \[ aX_1 + bX_2 + c \sim \mathcal{N}(\mu_1+\mu_2+c,a^2\sigma_1^2+b^2\sigma_2^2) \] |