A closed surface of genus $g$, like a doughnut with $g$ holes, can be given many different shapes as a Riemann surface. The moduli space lists these shapes up to isomorphism, but surfaces with extra symmetries make it an orbifold rather than a manifold. Teichmüller space adds a marking, a way of naming the loops on the surface, which removes this problem. For genus $1$ a marked torus is $\C / (\Z + \tau \Z)$ with $\tau$ in the upper half-plane, and $T_1$ is the upper half-plane itself.

Tanh-sinh quadrature stretches the interval $(-1, 1)$ to the whole real line with a clever change of variable. After the change, the function to be integrated dies off extremely fast at both ends, even if the original one blew up at an endpoint. Then the simplest rule, adding equally spaced samples, gives an excellent answer.

A semidirect product builds a group from a group $N$ and a group $H$ that acts on $N$ by automorphisms. It generalises the direct product, and $N$ becomes a normal subgroup of the result.

Walk once around a closed path and add up all the turning you do, ignoring left and right. That sum is the total curvature. Walking once around a circle you turn by a full turn, so the total curvature is $2\pi$. A wiggly path makes you turn back and forth, and the total grows.

The Riemann mapping theorem gives a conformal map from the disc onto any proper simply connected region, but it says nothing about the edge. Carathéodory's theorem says that if the edge of the region is a simple closed curve, the map behaves well up to the edge. The circle is then mapped continuously and one-to-one onto the boundary curve.

Ordinary logic says whether a statement is true. Modal logic can also say whether it must be true, or could be true. The formula $\square p$ says "$p$ is necessary", and $\lozenge p$ says "$p$ is possible". For example, $\square p \to p$ says "what is necessary is true", which holds for necessity but fails if $\square$ means "it is believed that".

The true solution lives in an infinite-dimensional space of functions. The Galerkin method picks a few simple functions, such as polynomials or piecewise linear "hat" functions, and looks for the best combination of them. "Best" means that the equation holds when tested against each of the chosen functions. This gives as many equations as unknown coefficients, which a computer can solve.

Picture an infinite street of lamps, one at each integer, and a lamplighter walking along it. A state is a finite set of lit lamps and a position of the lamplighter. The two moves are: walk one step, or switch the lamp where you stand. Combining moves is the group operation. This simple group is a standard source of examples and counterexamples in geometric group theory.

Two polynomials are orthogonal for a weight if the integral of their product against the weight is zero. Start with $1, x, x^2, \dots$ and straighten them one at a time so that each is orthogonal to all earlier ones. The result is a list of polynomials, one of each degree, that act like perpendicular axes in a space of functions. For the plain weight on $[-1, 1]$ one gets the Legendre polynomials $1$, $x$, $\frac12(3x^2 - 1)$, and so on.

For numbers, $e^{a + b} = e^a e^b$. For Hermitian matrices that do not commute, the two sides differ, but their traces compare in one direction: the trace of $e^{A + B}$ is at most the trace of $e^A e^B$. This one-sided rule is often enough to bound quantities in physics and probability.

Whether a boundary point is good for the Dirichlet problem depends on how much of the outside of the set lies near that point. Wiener's criterion measures this outside part in rings around the point, using capacity, the electrical size of a set. If the rescaled sizes of the rings add up to infinity, the point is regular; otherwise it is irregular.

With an orthonormal wavelet, the same function is used to measure a signal and to rebuild it. A biorthogonal wavelet splits these two jobs between two different functions, an analysis wavelet and a synthesis wavelet. The two families fit together like a basis and its dual basis in linear algebra: each analysis function sees exactly one synthesis function and is blind to all the others. This extra freedom lets designers make both functions symmetric and short, which avoids visible distortions in compressed images.

The logarithm brings an exponent down as a factor. For example, $\ln(x^3) = 3 \ln x$.

A series of functions is an infinite sum of functions defined on a common set. Its convergence is the convergence of its partial sums, pointwise or uniformly. The node series of functions gives the precise definition.

The type of a second-order equation can change from point to point when its coefficients vary. In the Tricomi equation $u_{yy} + y \, u_{xx} = 0$, the upper half plane $y > 0$ is elliptic and the lower half plane $y < 0$ is hyperbolic. So in one region the equation behaves like the Laplace equation and in the other like the wave equation. For gas flow past an airplane wing, the flow is slower than sound in one region and faster in another, and this is the physical origin of such equations.

A positive operator behaves like a nonnegative number, so it should have a square root that is again nonnegative. For a diagonal operator with nonnegative entries $d_1, d_2, \ldots$, the square root is the diagonal operator with entries $\sqrt{d_1}, \sqrt{d_2}, \ldots$. An operator can have many square roots, for instance $I$ and $-I$ both square to $I$, but only one of them is positive.

The product of the two neighbours of a Fibonacci number differs from its square by exactly $1$. The sign alternates: $1 \cdot 3 - 2^2 = -1$, then $2 \cdot 5 - 3^2 = 1$, then $3 \cdot 8 - 5^2 = -1$.

There are two ways to measure how chaotic a system is. Topological entropy counts distinguishable orbits, while measure-theoretic entropy measures the unpredictability of typical orbits for a given statistics. The variational principle says that the first is the best possible value of the second.

An area-minimizing surface of any dimension, in a space of any dimension, can have singular points where it is not smooth. Almgren's theorem says that these singular points form a set that is small: two dimensions smaller than the surface itself. For two-dimensional surfaces, the singular points are isolated, as for complex curves crossing themselves.

A tensor $u = \sum_i x_i \otimes y_i$ gives a number $\sum_i f(x_i) g(y_i)$ for each pair of functionals $f$ and $g$. The injective norm is the largest such number when $f$ and $g$ have norm at most $1$. If $X$ and $Y$ are spaces of continuous functions on compact spaces $K$ and $L$, the tensor becomes the function $(s, t) \mapsto \sum_i x_i(s) y_i(t)$ on $K \times L$. Then the injective norm is just the maximum of its absolute value.