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A locally compact group is a topological group whose underlying space is a locally compact space. In Haar measure and Fourier transform theory one normally also assumes it is a Hausdorff space. Local compactness supplies compact neighbourhoods, not just compact sets which algebraically generate the group.

A topological group is compactly generated in the algebraic sense when a compact subset generates its underlying group. Adjoining its inverse and the identity gives a compact symmetric generating set $K$, and $G={\bigcup }_{n\ge 1}{K}^n$ is a sigma-compact space. Some conventions restrict the term to locally compact groups or require a compact generating neighbourhood; those stronger conventions supply local compactness. Compact generation by an arbitrary compact subset alone does not.

An open subgroup is also closed, since its complement is the union of its other open cosets. Conversely a closed subgroup need not be open, as $\mathbb{Z}\subset \mathbb{R}$ shows.

A closed subgroup is a subgroup which is a closed set in the ambient topology. It is a topological group in its subspace topology, but need not be an open subgroup.

The additive subgroup $\mathbb{Z}\subset \mathbb{R}$ of integers is a closed subgroup: every noninteger lies in an open interval $(n,n+1)$ avoiding $\mathbb{Z}$. It is also a discrete subgroup, since $(n-1/2,n+1/2)\cap \mathbb{Z}=\{n\}$. Nevertheless it is not an open subgroup of $(\mathbb{R},+)$: every open interval containing an integer contains nonintegers too. Thus having a discrete topology internally says nothing about being an open set in the ambient topology.

This is a distance function satisfying the displayed identity, rather than a Riemannian metric tensor. When it induces the group's given topology, it is a compatible left-invariant group metric. Left translations are isometries.

Right translations are isometries for a right-invariant group metric. Given a compatible left-invariant group metric $d$, the formula $d_R(x,y)=d(x^{-1},y^{-1})$ defines a compatible right-invariant group metric, since inversion is a homeomorphism and $(xa{)}^{-1}=a^{-1}{x}^{-1}$. Compatible invariant metrics need not be Lipschitz equivalent; on the additive group $\mathbb{R}$, both $|x-y|$ and $\sqrt{|x-y|}$ are compatible invariant distances.

A continuous unitary character is a continuous group homomorphism from a topological group to the unit circle with multiplication. For abelian groups, multiplication of continuous unitary characters is pointwise. Such continuous unitary characters must be distinguished from the traces of higher-dimensional group representations, called characters of a representation, and from characters of an algebra.

For every integer $n$, the map ${\chi }_n(z)=z^n$ is a continuous unitary character of the circle group: ${\chi }_n(zw)={\chi }_n(z){\chi }_n(w)$ and $|{\chi }_n(z)|=1$. By contrast, the two-dimensional group representation $\rho (z)=\mathrm{diag}(z,z^{-1})$ has character of a representation $\mathrm{tr}\rho (e^{i\theta })=2\cos \theta$. Its value at the identity element is $2$, so this trace is not a continuous unitary character.

A locally profinite group is a Hausdorff, locally compact, totally disconnected topological group. It has a neighbourhood basis at the identity consisting of compact-open subgroups. In smooth representation theory this is also called an $\ell$-group; that terminology is unrelated to a lattice-ordered group.

For a continuous map on a compact metric space, the Bowen metric compares the first $n$ points of two forward orbits. Its radius-$ϵ$ ball is $B_n(x,ϵ)=\{y:d_n(x,y)<ϵ\}$. If $s_n(ϵ)$ is the largest cardinality of an $ϵ$-separated set in this metric, then topological entropy is ${\lim }_{ϵ\downarrow 0}{\mathrm{limsup}}_{n\to \infty }{n}^{-1}\log s_n(ϵ)$. The uniform continuity of the identity between two compatible metrics gives the comparison of separated-set counts needed for independence of the original metric.

For a two-dimensional hyperbolic toral automorphism, the stable and unstable eigenvalues obey $|{\lambda }_s|<1<|{\lambda }_u|$. In a sufficiently small Bowen ball, lifting to the stable and unstable coordinates gives widths proportional to $ϵ$ and $ϵ|{\lambda }_u{|}^{-(n-1)}$. Its normalized Haar measure is therefore proportional to ${ϵ}^2|{\lambda }_u{|}^{-(n-1)}$, uniformly in the centre. Maximal separated sets cover by radius-$ϵ$ Bowen balls and pack disjoint radius-$ϵ/2$ Bowen balls, so their exponential growth rate is $\log |{\lambda }_u|$. Smallness of $ϵ$ must exclude nonzero lattice jumps between successive lifted iterates; this is essential to the ball-volume argument.

Use half-width $b$, upward speed $w$, constant inertial reference density $R$, and positive buoyant reduced gravity $g^{\prime }$. Half-section fluxes are $Q=Rbw$, $M=Rb{w}^2$ and $F=R{g}^{\prime }bw$. With two-sided entrainment $u_e=\alpha w$, the Boussinesq balances areFull-width fluxes are twice these definitions. The shape coefficients differ from triangular transverse profiles, so their unsteady balances cannot be used unchanged here.

For plume area $A$ and reduced gravity $g^{\prime }$, the principal matrix in variables $(A,w,g^{\prime })$ is $\left(\begin{array}{ccc}w& A& 0\\ 0& w& 0\\ 0& 0& w\end{array}\right)$. At $A>0$ it has a triple real speed $w$ but only two independent eigenvectors. Thus the model is weakly hyperbolic, not strongly or strictly hyperbolic. Real speeds alone do not provide a complete set of characteristic fields or a uniform high-frequency estimate. Shape factors representing nonuniform radial profiles can alter this degeneracy.

With zero source volume flux and zero buoyancy flux but positive source momentum flux, the axisymmetric top-hat plume model reduces to a turbulent round jet. For full-cross-section kinematic fluxes, put $c=2{\alpha }_e\sqrt{\pi }$. Entrainment gives $Q^{\prime }=c\sqrt{M}$, while momentum conservation gives $M=M_0$. Hence $Q=c\sqrt{M_0}z$, speed $w=\sqrt{M_0}/(cz)$ and radius $r=cz/\sqrt{\pi }$. A finite source flux replaces $z$ by $z+Q_0/(c\sqrt{M_0})$. The ideal zero-flux source has singular speed and is a far-field model, not a finite nozzle.

Source volume flux is negligible only when entrained volume greatly exceeds it. In the jet limit of the top-hat plume model, this requires $z\gg Q_0/(c\sqrt{M_0})$. In a pure plume with $Q=C_Q{z}^{5/3}$, it requires $z\gg (Q_0/C_Q{)}^{3/5}$, where $C_Q=(3c/5)(9c{F}_0/20{)}^{1/3}$ for full-cross-section fluxes and $c=2{\alpha }_e\sqrt{\pi }$. These scales differ from the jet length which measures imposed momentum relative to buoyancy. Neglecting source volume does not automatically permit neglecting source momentum.

In an impermeable cylinder, an axisymmetric turbulent plume with negligible source volume flux requires a downward return flow. With uniform upward plume velocity $w$ and ambient velocity $v$, volume conservation gives $b^{2}w+(R^2-b^2)v=0$. An entrainment coefficient $\alpha$ based on the relative speed gives $Q^{\prime }=2\alpha b(w-v)$. A specified pressure closure is essential for momentum conservation. If the departure from ambient hydrostatic pressure and wall force are neglected in the whole-section integral, the total specific momentum flux $J=b^2{w}^2+(R^2-b^2)v^2$ obeys $J^{\prime }=b^2{g}^{\prime }$. Homogeneous unmodified ambient and no buoyancy loss give $b^{2}w{g}^{\prime }=B_s$. These are leading integral-model closures; restoring the ambient dynamic pressure can change the singularity criterion.

A form of tomography that reconstructs an internal radioactive source from directional counts of individually emitted photons. Known spatial photon absorption gives the exponential path weight in an attenuated Radon transform. Full directed data and a suitable reconstruction formula recover the source intensity rather than a transmitted exterior-beam attenuation map.

Transmission measurements determine line integrals of an absorption coefficient after taking the logarithm of incident intensity divided by transmitted intensity. Under the ideal straight-ray model, these measurements are a Radon transform; filtered backprojection reconstructs a planar slice. Angular coverage, calibration and noise control matter because differentiating measured projections amplifies high frequencies. In contrast, single-photon emission computed tomography reconstructs an emitting source through a possibly attenuating medium.

A light clock measures time by counting round trips of a light pulse between mirrors. In its rest frame, mirrors separated by $h$ give a tick duration $2h/c$, where $c$ is the speed of light. Comparing this round trip in another inertial frame provides the geometric time dilation argument above.

Amortized analysis bounds the total cost of every permitted sequence of operations from a specified initial state. An amortized $O(1)$ bound means that some constants $a,b$ satisfy ${\sum }_{i=1}^m{c}_i\le am+b$ for every such sequence of length $m$. The cost model and allowed operations are part of the claim; Big O notation alone does not specify them.

This distinction, and the credit and potential method viewpoints, are developed in R. Endre Tarjan's 1985 paper Amortized Computational Complexity, sections 1–2, pp. 306–308. The dynamic array doubling example below shows how an expensive operation can pay for itself through the cheap operations that precede it.

An algorithm has pseudo-polynomial time when its running time is bounded by a polynomial in the numerical magnitudes of input quantities, rather than necessarily in their encoded lengths. The dynamic programming table for the 0-1 knapsack problem uses $O(nW)$ arithmetic operations for $n$ items and capacity $W$.

A binary numeral system representation needs only $O(\log (W+1))$ bits for that capacity, so the table bound can be exponential in the number of capacity bits. For example, capacities below $2^b$ fit in $b$ bits, while a table with one entry for every capacity can have nearly $2^b$ columns. The numerical-value bound does not claim that every instance requires exponential work.