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
- $\map {\sigma_\alpha} n$
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
.
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
.
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
.
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
.
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}
.
Area Mass Density
- $\sigma$
Sometimes used, although $\rho_A$ (Greek letter rho) is more common, to denote the area mass density of a given two-dimensional body:
- $\sigma = \dfrac m A$
where:
Stefan-Boltzmann Constant
- $\sigma$
The symbol for the Stefan-Boltzmann constant is $\sigma$.
Its $\LaTeX$ code is \sigma
.
Poisson's Ratio
- $\sigma$
The symbol for Poisson's ratio is $\sigma$.
Its $\LaTeX$ code is \sigma
.
Countability
- $\sigma$
Used to denote the property of countability.
The $\LaTeX$ code for \(\sigma\) is \sigma
.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): sigma