On the interval from $-1$ to $1$, two functions are called orthogonal when the integral of their product is zero. Start with $1$, $x$, $x^2$, and so on, and adjust each one so that it is orthogonal to all the earlier ones. The results, scaled to equal $1$ at $x = 1$, are $P_0 = 1$, $P_1 = x$, $P_2 = \frac{3x^2 - 1}{2}$, and so on. They play the role on $[-1, 1]$ that sines and cosines play on a circle.

The function $f(a, b)$ is a sum over all integers $n$, where positive $n$ mostly use powers of $a$ and negative $n$ mostly use powers of $b$. Choosing $a$ and $b$ as powers of one variable $q$ gives famous series. For example, $f(q, q) = \sum_{n \in \Z} q^{n^2} = 1 + 2q + 2q^4 + 2q^9 + \cdots$, where each square $n^2 > 0$ appears twice, once for $n$ and once for $-n$. Its square counts the ways to write a number as a sum of two squares.

To check that a sequence spreads evenly around a circle, it is enough to test it against waves. For each frequency $h$, average the points $e^{2\pi i h x_n}$ on the unit circle. The sequence is evenly spread exactly when all these averages tend to zero.

Place $n$ points evenly on a circle and reflect the circle in the line through point $a$ and the centre. Point $b$ moves to point $2a - b$. The dihedral quandle records exactly this rule. These simple quandles already detect interesting knots, through the familiar game of colouring a knot diagram with three colours.

There are several ways to put brackets into a long product, and moving brackets to the right is a single step. The Tamari lattice records which bracketings can be reached from which by such steps. With four factors there are five bracketings, arranged like a pentagon.

The NAND gate outputs $0$ only when both inputs are $1$. Chips can be built from NAND gates alone, because NAND can imitate every other gate. For example, $\neg x$ is $x \uparrow x$, and $x \wedge y$ is $(x \uparrow y) \uparrow (x \uparrow y)$. An operation with this "one gate does everything" property is called a Sheffer function.

A singular complex space can be replaced by a smooth one that maps onto it and agrees with it away from the singular points. Over the singular points, the smooth space may have whole curves or surfaces instead of single points. So questions about singular spaces can often be moved to manifolds, where calculus is available.

The Riemann integral of a function on a closed bounded interval is the limit of its Riemann sums as the partitions get finer. For a nonnegative function, it is the area under the graph. The node Riemann integral gives the precise definition.

Put one ball inside a larger ball, both possibly deformed but with tame surfaces. The theorem says that the shell between them is always a standard thick spherical shell, like the peel of an orange. In the plane, this says that the ring between two nested loops is a deformed round ring.

Let $n \ge 2$. The sign is a homomorphism $\operatorname{sgn} : S_n \to \{1, -1\}$, where $\{1, -1\}$ is a group under multiplication. A permutation has sign $1$ when it is a product of an even number of transpositions. Thus the kernel of $\operatorname{sgn}$ is the set of even permutations. This is the alternating group $A_n$. Thus $A_n$ is a normal subgroup of $S_n$.

Among all ways to fill in a function with given boundary values, the harmonic one is the least wiggly. It has the smallest Dirichlet energy. This is why a soap film or a stretched membrane with a fixed rim takes the shape of a harmonic function, for small slopes.

A finitely generated abelian group is a free part $\mathbb{Z}^n$ plus a finite part. The finite part is a sum of cyclic groups. So a short list of integers describes the group completely.

Write a prime in decimal and read its digits as the coefficients of a polynomial. That polynomial cannot be factored over the integers. For example, the prime $1013$ gives $x^3 + x + 3$, which is irreducible.

A minor of a matrix is the determinant of a square submatrix. We get such a submatrix when we choose some rows and the same number of columns.

Take two wedge-shaped regions of $\C^n$ that meet only along a common real edge. Suppose a holomorphic function on each wedge has boundary values on the edge, and the two sets of boundary values agree. Then the two functions are pieces of a single holomorphic function on a full neighbourhood of the edge. The edge acts like a seam that disappears once the two sides match.

A colouring of a graph assigns colours to the vertices so that adjacent vertices get different colours. Colourings model scheduling and other conflict problems. The node graph colouring gives the precise definition.

Pointwise convergence of functions does not control where their minima go. Γ-convergence asks instead for two things. The limit $F$ is never larger than what the $F_n$ give near a point. And every value of $F$ can be approached by values of $F_n$ at nearby points. Think of a landscape with finer and finer ripples: the ripples have no pointwise limit, but the lowest points of the landscape settle down. For example, $F_n(x) = \sin(nx)$ on $\R$ has no pointwise limit, yet it Γ-converges to the constant $-1$, because near every point there are points where $\sin(nx) = -1$.

When you multiply two polynomials, the highest terms multiply to give the highest term of the product. For example, $(x^2 + 1)(x^3 - x)$ has degree $2 + 3 = 5$. This needs the product of the leading coefficients to be nonzero.

Brouwer's theorem is about a map that sends each point to one point. Kakutani's theorem allows each point to be sent to a whole convex set of points, and still finds a point that lies in its own image. This is what one needs when a choice is not unique, such as the best reply to an opponent's move in a game.

Suppose one only knows that a quantity $x$ lies between $1$ and $3$. Interval arithmetic forgets where a range came from, so it treats $x$ and a second copy of $x$ as unrelated, and $x - x$ seems to lie anywhere between $-2$ and $2$. Affine arithmetic writes $x = 2 + \varepsilon$ with an unknown $\varepsilon \in [-1, 1]$, and keeps the same $\varepsilon$ wherever $x$ appears. Then $x - x = 0$ exactly, and sums such as $x + 2x = 6 + 3\varepsilon$ also come out sharp.