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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.

If the left and right travelling-wave coefficients are related bythen

For $x,\xi >0$, the Green function satisfyingas $x\to \infty$ is

Simple harmonic motion is motion in which the acceleration is proportional to the displacement from equilibrium and points back towards it:Here $x$ is measured from equilibrium, and $\omega$ is the constant angular frequency. For a point mass attached to an ideal spring, Hooke's law gives the restoring force $F=-kx$, with stiffness $k>0$, and Newton's second law yields $\omega =\sqrt{k/m}$. The minus sign matters: for $x>0$ the force is negative, and for $x<0$ it is positive. A restoring force need not oppose velocity; on the return towards equilibrium it increases speed.

The general solution isThe amplitude $A\ge 0$ and oscillation phase $\phi$ encode the initial conditions. We use $P$ for the oscillation period to distinguish it from kinetic energy $T$. For $A>0$, the oscillation period is independent of amplitude, the property of isochronous harmonic motion. At $A=0$ the body remains at equilibrium. ^[ref]

Choose $m=0.5\mathrm{kg}$, $k=2N/m$, and release the mass from rest at displacement $A=0.3\mathrm{m}$. Then $\omega =2{\mathrm{s}}^{-1}$, $P=\pi \mathrm{s}$, and $\phi =0$. Writing time $t$ in seconds,At $t=0$ the velocity vanishes, but the acceleration is $-1.2m/{\mathrm{s}}^2$: a turning point is not a state of rest that persists. At $t=\pi /4\mathrm{s}$, the body crosses equilibrium with velocity $-0.6m/s$ and zero acceleration. At $t=\pi /2\mathrm{s}$ it reaches the opposite turning point, $x=-0.3\mathrm{m}$.

Taking potential energy zero at equilibrium, the same solution givesAt each turning point all mechanical energy is potential energy; at each equilibrium crossing it is kinetic energy. Both repeat twice during one complete position cycle because squaring removes the sign. At $t=\pi /8\mathrm{s}$, for example, each contributes $0.045\mathrm{J}$.

The physical position trace covers a line segment, $-0.3\le x\le 0.3$ in metres. In the phase plane, the paired position and velocity instead trace the ellipseThis is precisely $m{v}^2/2+k{x}^2/2=E$, rewritten: the single conserved mechanical energy fixes the amplitude and the whole ellipse. It does not fix the starting oscillation phase. At $x=0$ both $v=-0.6m/s$ and $v=+0.6m/s$ lie on it; position alone cannot distinguish them. Replacing $v$ by momentum $p=mv$ gives the equivalent phase-space trajectory. The arrows show increasing time, clockwise in $(x,v)$ coordinates.

This closed ellipse assumes a time-independent spring and no damping or applied drive. Adding a resisting force $-bv$, $b>0$, gives $m\ddot{x}+b\dot{x}+kx=0$ and $dE/dt=-b{v}^2\le 0$. An underdamped damped harmonic oscillator spirals inward in the phase plane, crossing successively smaller energy contours. The ideal harmonic oscillator is also useful beyond springs: near a smooth potential energy minimum with $V^{\prime \prime }(0)>0$, its Taylor expansion begins with $V(0)+V^{\prime \prime }(0)x^2/2$. For sufficiently small displacements, this gives the same restoring force to leading order, explaining why small oscillations share one solvable pattern.

A trajectory approaching a nondegenerate unstable maximum of its potential energy has a logarithmically divergent period.

Every one-dimensional character kills the commutator subgroup and therefore factors through the abelianization. Conversely, every character of the abelianization pulls back to a one-dimensional character of the group.

Wind stress is the tangential traction exerted by the atmosphere on the ocean surface. Its curl injects vorticity into the ocean.