# 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`

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The $\LaTeX$ code for \(\varsigma\) is `\varsigma`

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The $\LaTeX$ code for \(\Sigma\) is `\Sigma`

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### 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`

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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`

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The $\LaTeX$ code for \(\ds \sum_{\map \Phi j} a_j\) is `\ds \sum_{\map \Phi j} a_j`

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### Divisor Function

- $\map {\sigma_\alpha} n$

Let $\alpha \in \Z_{\ge 0}$ be a non-negative integer.

A **divisor function** is an arithmetic function of the form:

- $\ds \map {\sigma_\alpha} n = \sum_{m \mathop \divides n} m^\alpha$

where 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`

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### Divisor Count Function

- $\map {\sigma_0} n$

Let $n$ be an integer such that $n \ge 1$.

The **divisor count 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`

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### 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`

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### Standard Deviation

- $\sigma_X$

Let $X$ be a random variable.

Then the **standard deviation of $X$**, written $\sigma_X$ or $\sigma$, is defined as the principal square root of the variance of $X$:

- $\sigma_X := \sqrt {\var X}$

The $\LaTeX$ code for \(\sigma_X\) is `\sigma_X`

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### Surface Charge Density

- $\map \sigma {\mathbf r}$

Let $B$ be a body made out of an electrically conducting substance.

Let $B$ be under the influence of an electric field $\mathbf E$ under which a surface charge is induced on $B$.

Let $\delta S$ be an area element which is smaller than the scale used for a macroscopic electric field, but still large enough to contain many atoms on the surface of $B$.

Let $P$ be a point in the vicinity of $\delta S$ whose position vector is $\mathbf r$.

Let $\delta V$ be a volume element just thick enough to enclose the whole of the surface charge $\map \sigma {\mathbf r} \delta S$ associated with $\delta S$.

The **surface charge density** is the charge density of the macroscopic electric field on the surface $P$, defined as:

- $\ds \map \sigma {\mathbf r} = \dfrac 1 {\delta S} \int_{\delta V} \map {\rho_{\text {atomic} } } {\mathbf r'} \rd \tau'$

where:

- $\d \tau'$ is an infinitesimal volume element
- $\mathbf r'$ is the position vector of $\d \tau'$
- $\map {\rho_{\mathrm {atomic} } } {\mathbf r'}$ is the atomic charge density caused by the electric charges within the atoms that make up $B$.

The $\LaTeX$ code for \(\map \sigma {\mathbf r}\) is `\map \sigma {\mathbf r}`

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### 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:

### Countability

- $\sigma$

Used to denote the property of **countability**.

The $\LaTeX$ code for \(\sigma\) is `\sigma`

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## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**sigma**