Symbols:Greek/Sigma
Sigma
The $18$th letter of the Greek alphabet.
- Minuscules: $\sigma$ and $\varsigma$
- Majuscule: $\Sigma$
The $\LaTeX$ code for \(\sigma\) is \sigma .
The $\LaTeX$ code for \(\varsigma\) is \varsigma .
The $\LaTeX$ code for \(\Sigma\) is \Sigma .
Event Space
- $\Sigma$
Let $\EE$ be an experiment whose probability space is $\struct {\Omega, \Sigma, \Pr}$.
The event space of $\EE$ is usually denoted $\Sigma$ (Greek capital sigma), and is the set of all outcomes of $\EE$ which are interesting.
By definition, $\struct {\Omega, \Sigma}$ is a measurable space.
Hence the event space $\Sigma$ is a sigma-algebra on $\Omega$.
Summation
Let $\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers.
Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
The composite is called the summation of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:
- $\ds \sum_{j \mathop = 1}^n a_j = \tuple {a_1 + a_2 + \cdots + a_n}$
The $\LaTeX$ code for \(\ds \sum_{j \mathop = 1}^n a_j\) is \ds \sum_{j \mathop = 1}^n a_j .
The $\LaTeX$ code for \(\ds \sum_{1 \mathop \le j \mathop \le n} a_j\) is \ds \sum_{1 \mathop \le j \mathop \le n} a_j .
The $\LaTeX$ code for \(\ds \sum_{\map \Phi j} a_j\) is \ds \sum_{\map \Phi j} a_j .
Divisor Function
- $\map {\sigma_\alpha} n$
The divisor function:
- $\ds \map {\sigma_\alpha} n = \sum_{m \mathop \divides n} m^\alpha$
(meaning the summation is taken over all $m \le n$ such that $m$ divides $n$).
The $\LaTeX$ code for \(\map {\sigma_\alpha} n\) is \map {\sigma_\alpha} n .
Divisor Counting Function
- $\map {\sigma_0} n$
Let $n$ be an integer such that $n \ge 1$.
The divisor counting function is defined on $n$ as being the total number of positive integer divisors of $n$.
It is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\sigma_0$ (the Greek letter sigma).
That is:
- $\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
The $\LaTeX$ code for \(\map {\sigma_0} n\) is \map {\sigma_0} n .
Divisor Sum Function
- $\map {\sigma_1} n$
Let $n$ be an integer such that $n \ge 1$.
The divisor sum function $\map {\sigma_1} n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$.
That is:
- $\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
The $\LaTeX$ code for \(\map {\sigma_1} n\) is \map {\sigma_1} n .
Surface Charge Density
- $\sigma$
Denotes the surface charge density of a given body:
- $\sigma = \dfrac q A$
where:
- $q$ is the body's electric charge;
- $A$ is the body's area.
Area Density
- $\sigma$
Sometimes used, although $\rho_A$ (Greek letter rho) is more common, to denote the area density of a given two-dimensional body:
- $\sigma = \dfrac m A$
where: