# Tag Info

5

The distribution is called the Dirac measure at $x$, often denoted by $\delta_x$. Thus, for every $A\subseteq\mathbb R$, $\delta_x(A)=1$ is $x\in A$ and $\delta_x(A)=0$ otherwise. This distribution has no PDF and its CDF is a Heaviside function, namely, $P(X\leqslant y)=0$ if $y\lt x$ and $P(X\leqslant y)=1$ if $y\geqslant x$.

3

The phrase "maximal clique" is usually used in terms of a subgraph of a given graph $G$. So a subgraph $H$ of a graph $G$ is a maximal clique in $G$ if $H$ is isomorphic to a complete graph and there is no vertex $v \in V(G)\backslash V(H)$ so that $v$ is adjacent to each vertex of $H$. In other words, a subgraph $H$ of a graph $G$ is a maximal clique in ...

1

Let $V$ be a complex vector space. Then the inner product will be a map: $$\langle \;\; , \;\; \rangle : V \times V \to \mathbb{C}$$ which is (1) conjugate symmetric in both arguments: $$\langle \mathbf{u} , \mathbf{v} \rangle = \overline{\langle \mathbf{v} , \mathbf{u} \rangle}$$ (2) linear in the ...

1

As I understand it, the statement that "a set $S$ freely generates the group $G$" means that the group generated by $S$ is free: in other words, there are no relations between elements of $S$. In your example, $a$ and $b$ would not freely generate the group because they satisfy the relations $a^3=1$ and $b^2=1$. (When I say "relation", I exclude the obvious ...

1

Think of the additive group of integers; it is free of rank 1. The integer 1 is its free generator. The set of integers, say 2 and 3, are generators but not free generators. Edit: Wikipedia is a great source but cannot replace a textbook. Consider reading Lyndon and Schupp "Combinatorial group theory" or Karras, Magnus, Solitar, same title. This should ...

1

If you have a sequence, like $a_k$, it's generating function is a (formal) expression of the form: $$A(z) = \sum_{k \ge 0} a_k z^k$$ (instead of $z^k$ sometimes other base functions are used). Note that this is purely formal, $A(z)$ doesn't have to define a function anywhere (a favorite of mine is $\sum_{n \ge 0} n! z^n$, which does have it's uses). You ...

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