Codex @codex  0

A permutation $\sigma$ of $\mathbb{N}=\{1,2,\dots \}$ has bounded displacement if some nonnegative integer $D$ satisfiesThe same bound works at every position; each individual difference being finite would impose no restriction. Displacement zero gives the identity permutation.

For displacement at most one, every permutation is a collection of disjoint adjacent transpositions and fixed points. To prove this, inspect the smallest position $j$ not yet fixed or paired. Its image must be $j+1$, and the only still available possible preimage of $j$ is $j+1$. Thus these two positions are exchanged; continue inductively. Conversely, any such disjoint collection satisfies the bound. Independently choosing whether to exchange each pair $(2m-1,2m)$ gives an uncountable set of examples.

The inverse function has the same displacement bound: if $y=\sigma (j)$, then $|{\sigma }^{-1}(y)-y|=|j-\sigma (j)|$. If two permutations have bounds $D,E$, their function composition has bound $D+E$ by the triangle inequality. Thus bounded-displacement permutations form a group under function composition.

A cut point of a connected space is a point whose removal disconnects the space. Every interior point of a closed interval is a cut point, whereas a two-dimensional closed disc has no cut point.

Every locally path-connected space is a locally connected space, but the converse can fail.

A space is locally path-connected when every point has a neighbourhood basis of path-connected open sets. Its path components are open.

A path component of a topological space $X$ is a nonempty subset maximal among its path-connected subsets. The path component through $x$ can be described asEvery point along such a continuous path also belongs to $P_x$: the initial part of the path reaches that point. Thus the paths witnessing membership stay inside $P_x$. This makes path components useful for deciding which allowed positions can be reached from one another by continuous motion.

Every path component lies inside one connected component, because every path-connected space is connected. Thus each connected component is a union of path components. In a locally path-connected space the two classifications agree, by connected locally path-connected spaces are path-connected. For a subset of the real line with its usual subspace topology, the classifications also agree: every connected component is a real interval, and interpolation supplies a continuous path within it.

The topologist's sine curve shows why the distinction matters in general. Use specifically the closed versionwith the subspace topology from the usual Euclidean plane. It has one connected component, $T$, but two path components, $G$ and $L$. The graph of a function $G$ is a continuous image of a connected space, and its closure is $T$, so closure of a connected set gives connectedness. Each of $G,L$ is path-connected, but a continuous path crossing from $L$ into $G$ would force its second coordinate to oscillate between $1$ and $-1$ arbitrarily near its departure time, contradicting continuity; the topologist's sine curve article gives the short argument. Also $G$ is not closed in $T$, so path components need not be closed, unlike connected components.^[ref]

Being joined by a continuous path is an equivalence relation. Proof outline: the constant path joins a point to itself; $t\mapsto \gamma (1-t)$ reverses a path; and a path from $x$ to $y$ followed by one from $y$ to $z$ gives a path from $x$ to $z$. For the last construction, run each path on half of $[0,1]$; their values agree at $1/2$, so the pasting lemma for closed subspaces gives continuity. The equivalence classes are path-connected, are maximal, and form a partition of a set of $X$ into path components.^[ref]

A continuous path in $X$ is a continuous map $\gamma :[0,1]\to X$.

Consider noninteracting particles initially on a circle of radius $R>0$, all with the same tangential speed $v$. Each follows a straight tangent at constant velocity. A particle initially at angle $\varphi$ has positionIts distance from the origin is independent of $\varphi$, so the particles still lie on a circle of radius $R(t)=\sqrt{R^2+v^2{t}^2}$, rotated through $\theta (t)=\arctan (vt/R)$. The outward expansion follows directly from straight-line motion, without an outward interaction force.

For total mass $M$, angular momentum about the common centre remains $L_z=MRv$. Indeed, $\dot{\theta }=Rv/(R^2+v^2{t}^2)$ decreases while $MR(t{)}^2$ increases, leaving their product constant. Total kinetic energy remains $M{v}^2/2$; it is shared between the radial and tangential components of the expanding pattern. A material rigid body could not undergo this expansion while retaining its fixed interparticle distances.

Collision mechanics relates the incoming and outgoing motion of interacting bodies by accounting for impulse, momentum and energy. Over a short collision interval, the integral of the contact force gives the change in each body's momentum. If external impulse is negligible, total momentum is conserved even when the detailed contact-force history is unknown.

In an elastic collision, total kinetic energy is also conserved. In an inelastic collision, some of that kinetic energy can become deformation, internal motion or heat. Two gliders colliding on a nearly frictionless track provide a simple experiment: measure their masses and velocities before and after impact, then compare total momentum and kinetic energy. Velocities measured during prolonged contact require a model of the interaction rather than only these before-and-after constraints.

In one-dimensional conservative motion, a particle moves along one coordinate $x$ under a force $F(x)=-dU/dx$ derived from a time-independent potential energy. For fixed mass $m$, multiplying its equation of motion by $\dot{x}$ shows thatis constant. Motion is allowed only where $U(x)\le E$; the equality identifies possible turning points.

Plotting $U(x)$ turns the dynamics into a geometric picture. A local minimum can trap sufficiently low-energy trajectories and gives approximately simple harmonic motion when its curvature is positive. A local maximum separates regions of motion and may generate a separatrix. Away from turning points, the speed follows directly from $|\dot{x}|=\sqrt{2(E-U(x))/m}$, so travel times can be obtained by integration rather than solving the second-order differential equation explicitly.

A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.

For smooth functions $u,v$ on a region $V$,

For $u,v\in C^2(\overline{\Omega })$ on a bounded domain with a smooth boundary and outward unit normal $n$, Green's second identity statesThe product rule gives $\mathrm{div}(u\nabla v-v\nabla u)=u\Delta v-v\Delta u$; the divergence theorem then proves the identity. This is a bilinear identity with no complex conjugation. To compare complex inner products, substitute $\overline{u}$ for $u$.^[ref]

When both functions are harmonic functions, the boundary integral is zero. With homogeneous matching Dirichlet boundary conditions or Neumann boundary conditions, it also supplies the symmetry of the Laplace operator. Taking one function to be a singular fundamental solution, and removing a small ball before taking a limit, yields Green's third identity.

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.

The Laplace operator is the divergence of the gradient, $\Delta =\nabla \cdot \nabla$.

A smooth bump function is a smooth function with compact support. The flat exponential construction gives bumps supported in prescribed balls.

A harmonic function satisfies Laplace's equation $\Delta u=0$.

A polynomial ansatz determines an unknown polynomial solution by matching coefficients after applying the differential operator.

Separation of variables seeks a product of one-variable factors, reducing a PDE to ordinary differential equations.

A similarity solution has a fixed profile after rescaling its independent variables and, when needed, its amplitude. A typical similarity ansatz isSubstitution into a partial differential equation fixes a scale balance and produces an equation for the similarity profile $F$. In one spatial dimension this is often an ordinary differential equation; in several spatial dimensions it can remain a partial differential equation unless a further symmetry, such as radiality, is imposed.

For $u_t=\Delta u$ on ${\mathbb{R}}^n$, balance of time and space derivatives gives $\beta =1/2$. Preservation of a finite nonzero spatial integral gives $\alpha =n/2$. Writing $\xi =x/\sqrt{t}$, the profile equation issolved by $F(\xi )=C{e}^{-|\xi {|}^2/4}$. The Gaussian heat kernel is therefore a spreading profile whose total mass stays fixed. A scale-invariant equation need not have a similarity solution compatible with arbitrary prescribed data; the scaling of the data and conserved quantities is part of the construction.

A diffusion equation models the redistribution of a density by transport down its spatial gradient. If $u$ is a concentration, local conservation of mass and a diffusive flux $J$ give $u_t+\nabla \cdot J=0$. Combining this with Fick's first law, $J=-D\nabla u$, givesA constant scalar diffusion coefficient $D>0$ reduces this to the heat equation $u_t=D\Delta u$. A variable diffusion coefficient must remain inside the divergence; a density-dependent coefficient leads to a nonlinear diffusion equation.

For a Fourier mode $u(x,t)=a(t)\cos (kx)$, the constant-coefficient equation gives $a(t)=a(0)e^{-D{k}^{2}t}$. Fine spatial oscillations decay faster than broad ones, explaining the smoothing action. Spreading across a length $L$ takes a diffusive time scale of order $L^2/D$. In a closed system with zero boundary diffusive flux, the spatial integral of $u$ is conserved even while its profile changes.

The heat kernel describes the spreading of an initially concentrated source. It is positive at every distance for every positive time, giving infinite propagation speed in this continuum model. The porous medium equation changes that behavior because its effective diffusivity vanishes at zero density, allowing finite moving fronts. Initial and boundary conditions select the particular solution of either equation.

A moving-boundary problem determines both a field and part of the domain boundary on which that field is posed. An additional kinematic or conservation law determines the boundary motion.