# Symbols:Greek/Lambda

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## Lambda

The $11$th letter of the Greek alphabet.

Minuscule: $\lambda$
Majuscule: $\Lambda$

The $\LaTeX$ code for $\lambda$ is \lambda .

The $\LaTeX$ code for $\Lambda$ is \Lambda .

## Von Mangoldt Function

$\map \Lambda n$

The von Mangoldt function $\Lambda: \N \to \R$ is defined as:

$\Lambda \left({n}\right) = \begin{cases} \ln \left({p}\right) & : \exists m \in \N, p \in \mathbb P: n = p^m \\ 0 & : \text{otherwise} \end{cases}$

where $\mathbb P$ is the set of all prime numbers.

The $\LaTeX$ code for $\map \Lambda n$ is \map \Lambda n .

## Linear Density

$\lambda$

Used to denote the linear density of a given one-dimensional body:

$\lambda = \dfrac m l$

where:

$m$ is the body's mass
$l$ is the body's length.

## Poisson Distribution

$\lambda$

Used to denote the parameter of a given Poisson distribution:

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ has the Poisson distribution with parameter $\lambda$ (where $\lambda > 0$) if and only if:

$\Img X = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$

It is written:

$X \sim \Poisson \lambda$

## Left Regular Representation

$\lambda_a$

Let $\struct {S, \circ}$ be an algebraic structure.

The mapping $\lambda_a: S \to S$ is defined as:

$\forall x \in S: \map {\lambda_a} x = a \circ x$

This is known as the left regular representation of $\struct {S, \circ}$ with respect to $a$.

The $\LaTeX$ code for $\map {\lambda_a} x$ is \map {\lambda_a} x .