In a nil ideal each element dies after enough self-multiplications. In a nilpotent ideal a single number $n$ works for all of them at once: any product of $n$ elements of the ideal, equal or not, is zero. In the ring of upper triangular $2 \times 2$ matrices, the matrices with zero diagonal form a nilpotent ideal. The product of any two of them is zero.

Sometimes the energy only stops decreasing on a set of states, not only at the equilibrium. LaSalle's principle says that the motion must still end at the equilibrium, if no motion can stay forever in that set.

A Kakeya set is a set that contains a full line pointing in every direction. Over the real numbers such sets can have measure zero, which is surprising. Over a finite field the theorem says the opposite: such a set must contain a fixed positive fraction, about $1/n!$, of all the points of the space.

The total variation of a list of numbers is the sum of the sizes of the jumps between neighbours: it measures how much the profile goes up and down. Wiggles near a shock increase this sum. A TVD scheme guarantees that the sum never grows, so wiggles cannot appear. For example, a step from $1$ to $0$ has total variation $1$, and a TVD scheme can smear the step but never turn it into an overshoot such as $1, 1.1, 0.4, 0$.

A holomorphic function solves the Cauchy–Riemann equations, which can be packed into one equation $\partial f / \partial \bar z = 0$. In $\R^n$ one packs the derivatives into one operator $\sum_j e_j \partial_{x_j}$ with anticommuting units $e_j$. A monogenic function is one sent to $0$ by this operator. For example, a constant function is monogenic, and so is $x \mapsto x / \abs{x}^n$ away from $0$, where $x = \sum_j x_j e_j$; this is the Cauchy kernel up to a constant.

Start with a proper ideal and make it larger, step by step, without reaching the whole ring. In an infinite ring this process may not stop by itself. Zorn's lemma guarantees that an ideal that you cannot enlarge exists.

A metric is a way to measure distances, and many spaces are given only by their open sets. The theorem says when distances can be put back in. It is enough that the space has a countable family of basic open sets and that points and closed sets can be kept apart by open sets. So a large class of spaces that look abstract can be treated as metric spaces.

A Bernoulli trial is a yes-or-no experiment. Many questions have this form: does the coin show heads, does the patient recover, does the part fail.

A nilpotent group is a group that has a central series of finite length. Equivalently, its lower central series reaches the trivial subgroup, or its upper central series reaches the whole group.

The Dirichlet distribution is a law on random probability vectors: lists of $K$ positive numbers that add up to $1$. It is used to model an unknown die with $K$ faces, before the die has been thrown. Large parameters concentrate the vector near the mean $(\alpha_1, \dots, \alpha_K)/\alpha_0$, and small parameters push it towards the corners of the simplex. For two categories it is the beta distribution.

The property says that a set of reals cannot be uncountable in a thin or scattered way. Either the set can be listed as a sequence, or it contains a whole copy of the Cantor set. For example, the interval $[0, 1]$ is uncountable and is itself perfect: it is closed and none of its points is isolated. The set of rational numbers is countable, so it has the property for the other reason. Descriptive set theory asks which definable sets have the property, and the answer depends on the axioms one adopts.

Different cohomology theories of the same variety give groups that look alike, with the same dimensions and similar structures. Grothendieck asked for a single object behind all of them, the motive of the variety, of which each cohomology is a "realization". Motives can be split into pieces even when the variety cannot. For example, the motive of the projective line is the sum of the motive of a point and the Lefschetz motive $\mathbb{L}$. The piece $\mathbb{L}$ accounts for the cohomology in degree $2$.

If each diagonal entry is larger in absolute value than all other entries of its row together, the matrix cannot send a nonzero vector to zero. The diagonal wins against the rest of the row, so it can never be cancelled.

Suppose a polynomial is at most $1$ in size on a set $E$. How large can it be at a point $z$ outside $E$, given its degree? The Siciak extremal function answers this: the best bound is $\abs{p(z)} \le e^{d \, V_E(z)}$ for polynomials of degree $d$. For the closed unit ball in $\C^n$, the answer is $V_E(z) = \log\abs{z}$ outside the ball. So a polynomial of degree $d$ bounded by $1$ on the ball is bounded by $\abs{z}^d$ outside it.

Feed a starting value into $f_n$, feed the result into $f_{n-1}$, and so on down to $f_1$. Doing this for larger and larger $n$ gives a sequence of numbers, and the infinite composition is its limit when it exists. For example, the continued fraction $\cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$ comes from composing the maps $w \mapsto \frac{1}{2 + w}$ and evaluating at $0$.

Midpoint convexity alone allows very irregular functions, but all of them are so wild that they cannot even be measured. Sierpiński's theorem says that any midpoint convex function that is measurable is a genuine convex function. In practice this means that every midpoint convex function one meets in analysis is convex.

Put a flat sheet of glass on a surface so that it touches at one point. Near that point the surface pulls away from the glass, faster in some directions and slower in others. The second fundamental form records, for each direction, how fast the surface pulls away. On a sphere it pulls away equally in all directions, and on a saddle it rises in some directions and falls in others.

From a point outside a circle, draw a tangent segment to the circle. The radical axis of two circles is where these tangent segments to the two circles have equal length. If two circles cross, the radical axis is the line through their two crossing points. For example, the unit circles centred at $(0, 0)$ and $(2, 0)$ touch at $(1, 0)$, and their radical axis is the vertical line $x = 1$.

The torsion subgroup of an abelian group is the subgroup of its elements of finite order. The quotient by the torsion subgroup is torsion-free.

A weighted-average method gives the right answer on every convergent sequence under three conditions. The weights stay bounded in total size, each fixed weight fades away, and the weights in each row add up to nearly $1$.