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Ordinal addition is ordered concatenation and is associative but generally not commutative.

If a property of an ordinal follows whenever it holds for every smaller ordinal, then it holds for every ordinal.

Transfinite recursion defines a value at each ordinal from the function of all earlier values.

A nonzero ordinal is a limit ordinal when it is not a successor ordinal. Equivalently, it is the supremum of all smaller ordinals.

The order type of a well-order is the unique ordinal isomorphic to it.

A countable ordinal has a countable set as its underlying set.

These are the ordinals strictly below the first uncountable ordinal.

The successor of an ordinal $\alpha$ is $\alpha +1=\alpha \cup \{\alpha \}$.

Independent and identically distributed random variables with a continuous probability distribution realize each of their possible strict orderings with equal probability.

Conserved relaxational order-parameter dynamics has ${\partial }_t\varphi =M{\nabla }^2(\delta F/\delta \varphi )$ plus conserved thermal noise. Its relaxation rate vanishes as $q^2$ at small wavenumber.

Nonconserved relaxational order-parameter dynamics has ${\partial }_t\psi =-\Gamma \delta F/\delta \psi$ plus thermal noise. Its zero-wavenumber mode may decay.

An involution is a group element $s$ satisfying $s^2=1$. Some conventions require $s\ne 1$.

A mean-motion resonance has $j{n}_2-k{n}_1\simeq 0$ for small integers $j,k$. Its slow resonant angle combines mean longitudes with apsidal or nodal longitudes.

A helical extremal has constant radius and a polar angle linear in axial position.

The open disc with centre $z$ and radius $r>0$ is $\{w:|w-z|<r\}$.

In a metric space, the open ball of radius $r>0$ about $x$ is $\{y:d(x,y)<r\}$; it is the open-boundary convention for a ball.

This is the open-boundary case of an interval.

An open interval in the real line is a set $(a,b)=\{x:a<x<b\}$, allowing either endpoint to be infinite.

A bounded bijective linear map between Banach spaces has a bounded inverse. This follows because an open bijection has continuous inverse.

Order source symbols by decreasing probability and assign the $i$th symbol a binary word of length $\lfloor {\log }_{2}i\rfloor$. This minimizes expected length among one-to-one binary codes because the available shortest words are assigned to the most probable symbols.

If an interceptor knows that an additive one-time-pad ciphertext $C$ encrypts plaintext $P$ and wants it to decrypt as $P^{\prime }$, replacing it byworks without knowing the key, since $C^{\prime }=P^{\prime }+K$.

An encryption scheme has perfect secrecy when the ciphertext and plaintext are independent random variables, so observing the ciphertext does not change the plaintext's probability distribution.