A self-adjoint operator on a finite-dimensional inner product space is a linear map from the space to itself that equals its adjoint.

The norm of a Gaussian integer $a + b i$ is $a^2 + b^2$, the square of its absolute value. The definition of the norm gives its properties.

Two solutions of a Lipschitz equation that meet at one time are the same solution. So the graphs of different solutions never cross or touch.

A power such as $n^{10}$ grows fast, but an exponential such as $1.01^n$ grows faster in the end. For small $n$ the power can be much larger. For large $n$ the exponential always wins.

Fix the last element and decide which other elements share its group. For each choice, the elements left over can be split in any way, so you add a Bell number for each case.

A divergent series is a series whose partial sums do not have a finite limit. Summation methods give values to some divergent series. The node divergent series gives the precise definition.

An antichain in a partially ordered set is a subset in which no two different elements are comparable.

The mesh of a partition is the length of its longest subinterval. Riemann sums approach the integral as the mesh tends to zero. The node mesh of a partition gives the precise definition.

Start at a point and follow the permutation until you return. This gives one cycle; then start again at a point that you did not visit yet. When all points are visited, the permutation is the product of the cycles that you found.

A family of random variables is independent, or mutually independent, when events about different members of the family are always mutually independent. The node independence of random variables gives the precise definition.

A real sequence assigns a real number to each natural number. Real sequences are the first setting for limits, convergence and series in real analysis. The node real sequence gives the precise definition.

A median of a distribution, or of a data set, is a value that splits it into two halves. At least half of the probability is at or below the value, and at least half is at or above it. The node median gives the precise definition for random variables.

If you know the first toss, you know nothing about $C$, and the same for the second toss. But if you know both tosses, you know $C$ exactly. So the three events are not independent as a group.

A positive definite form is a bowl: it is zero at the origin and positive everywhere else. An indefinite form is a saddle: it goes up in some directions and down in others.

The second parameter of the normal distribution is exactly its variance. So $\sigma$ is the standard deviation, the typical distance of a value from the center.

The zero ring has only one element. All the axioms of a ring hold, but in a trivial way. It is the only ring in which $1$ and $0$ are the same element.

This topic collects the standard methods of mathematical proof, such as direct proof, proof by contradiction, proof by cases and induction. A proof node names its methods with the method edge, and each target of that edge is a concept in this topic. The topic lets a reader find all the proofs that use one method.

The parameter of a Poisson distribution is its average. If a shop gets on average $4$ customers per hour, a Poisson model of the number of customers in one hour uses $\lambda = 4$.

The sum of two subspaces is the set of all sums of a vector of the first subspace and a vector of the second subspace. It is the smallest subspace that contains both.

The Euclidean group collects all the ways to move space without changing distances. Two such moves done one after the other give again such a move, and each move can be undone. In the plane, three numbers fix a direct move: two for a shift and one for a turn.