Symbols:Greek/Lambda
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 .
Triangle Function
- $\map \Lambda x$
The triangle function is the real function $\Lambda: \R \to \R$ defined as:
- $\forall x \in \R: \map \Lambda x := \begin {cases} 1 - \size x : & \size x \le 1 \\ 0 : & \size x > 1 \end {cases}$
where $\size x$ denotes the absolute value function.
The $\LaTeX$ code for \(\map \Lambda x\) is \map \Lambda x .
Linear Density
- $\lambda$
Used to denote the linear density of a given one-dimensional body:
- $\lambda = \dfrac m l$
where:
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 .