For an open identity neighbourhood $V$ in a compact topological group, a finite left covering net is a finite set $F$ such that the left translates $fV$, $f\in F$, cover the group. Compactness gives such finite sets; minimum cardinality gives a minimal one. Left translation preserves minimality because $G=FV$ implies $G=gFV$.

The hemisphere graph $u_0=\pm \sqrt{1-|\mathbf{u}{|}^2}$ of SU(2) as the three-sphere has induced metric $g_{ij}={\delta }_{ij}+u_i{u}_j/u_0^2$, whose determinant is $1/u_0^2$. Its volume density is therefore $d^{3}u/|u_0|$. Fixed left or right multiplication preserves the Euclidean norm on the four Pauli coefficients, so the round volume is Haar measure. Both hemispheres have volume ${\pi }^2$, giving total volume $2{\pi }^2$ and normalized density $d^{3}u/(2{\pi }^2|u_0|)$ on each hemisphere. The apparent equatorial divergence is an integrable coordinate singularity.

For a class function on SU(2) and normalized Haar measure, the displayed formula reduces a group integral to its conjugacy-class angle. The weight follows by identifying SU(2) with the unit sphere in four real dimensions: writing a unit quaternion as $\cos \theta +\mathbf{n}\sin \theta$ gives spherical volume proportional to ${\sin }^2\theta d\theta d\mathbf{n}$. Since ${\int }_0^{\pi }{\sin }^2\theta d\theta =\pi /2$, normalization gives $2/\pi$.

A Haar integral integrates a function against Haar measure. On a compact group, normalizing the measure to total mass one makes it an averaging operation. Translation invariance then implies that the average of a nontrivial continuous unitary character is zero: translating the integral multiplies it by a character value different from one.

For a compact group $G$, positive normalized linear functionals on $C(G)$ form a nonempty weak-star topology compact convex set. For finitely many translations, apply the Schauder-Tychonoff fixed-point theorem to the average of their induced affine dual maps. A fixed functional is stationary under this average. For each real continuous $f$, its orbit function $h(g)=\varphi (l_{g}f)$ is continuous on the compact subgroup generated by those translations and satisfies a finite averaging identity. At a maximum, every summand must attain that same maximum; iteration and density of the positive-word semigroup force $h$ to be constant. Thus the functional is fixed by each of the finitely many translations. Compactness and the finite intersection property now give a functional invariant under every translation. The Riesz-Markov-Kakutani representation theorem turns it into a normalized Haar measure, without assuming commutativity of the group.

For nonzero nonnegative $u,h\in C_c(G)$, let $(f:u)$ be the infimum of ${\sum }_j{a}_j$ over finite dominations $f\le {\sum }_j{a}_{j}u(x_j^{-1}\cdot )$ with $a_j>0$. Composition of covers gives $(f:u)\le (f:h)(h:u)$, so $I_u(f)$ is uniformly bounded while $I_u(h)=1$. It is invariant under left translation, homogeneous, monotone and subadditive. As the support of $u$ shrinks to the identity, its additivity defect tends to zero: use the continuous proportions $f_i/(f_1+f_2+\delta p)$, with a cutoff $p$ positive on both supports, whose oscillations are small on every translate of that small support. In a locally compact sigma-compact metrizable group, diagonal convergence on a countable dense family, followed by uniform approximation on fixed compact supports, gives a positive invariant linear functional on all $C_c(G)$. The Riesz representation on compactly supported continuous functions turns it into Haar measure. Local compactness is essential for the compactly supported cutoffs and small-support test functions.

On a $d$-dimensional Lie group, take the coframe dual to a basis of left-invariant vector fields. Its top wedge is nonvanishing and invariant under left translations, so its absolute density defines a left Haar measure. If the frame has coordinate matrix $A(x)$, the density is $|\det A(x){|}^{-1}d{x}^1\cdots d{x}^d$. This makes explicit computation of Haar measure a determinant calculation.

An H-space whose multiplication is homotopy associative and admits a homotopy inverse. These operations give $[X,Y{]}_{\ast }$ a group structure for every based CW complex $X$. A loop space is the basic example: concatenate paths, use reversal for inversion, and use reparametrization for homotopy associativity. A bare H-space should not automatically be treated as a group object.

Restricting the tubular-neighborhood collapse of an embedding back to its source is the normal bundle's zero section. The Thom isomorphism in complex cobordism therefore makes pullback after the Gysin map multiplication by the Euler class in a generalized cohomology theory. For an actual complex normal bundle this is its top Chern class in complex cobordism.

Close to the upward pole, the two Cartesian tilt components of bottom-heavy spherical-cell orientation dynamics obey the displayed Ornstein-Uhlenbeck process. Thus $d⟨|\mathbf{q}{|}^2⟩/dt=-2⟨|\mathbf{q}{|}^2⟩/B+4D_r$. The stationary spread is $2B{D}_r$, and late axisymmetric variance perturbations relax on $B/2$ in this weak-noise approximation. The zero-noise mean from deterministic alignment of an initially isotropic orientation distribution has the same exponential scale but a different transient prefactor. Reaching a tiny diffusion-supported mean-deficit floor $B{D}_r$ from an initially isotropic population takes order $B\log [(B{D}_r{)}^{-1}]$, with logarithmic corrections. Equality of limiting mean directions does not make the full transients identical, and the alignment time is not automatically the pure rotational-diffusion time $D_r^{-1}$.

The group delay is the frequency derivative of an unwrapped transfer wave phase, using the $e^{-i\omega t}$ convention. It predicts the leading delay of a narrow wave envelope when the transfer wave amplitude and the delay vary little across its band; it is not generally a signal-front arrival time.

For a fixed finite generating alphabet and integer $K$, use the elements of the word-metric ball $B_K(1)$ as states, together with a reject state and padding-status flags. Starting at the identity, a padded letter pair $(u,v)$ changes the state to $u^{-1}dv$ if it stays in the ball. A padding letter acts as the identity. Final state $a$ recognizes endpoint difference $a$ among pairs whose prefixes remain within $K$. On a regular synchronous combing, intersecting with the two word acceptors gives the full group multiplier automaton. It does not recognize unrestricted endpoint relations for all input words.

Every bounded linear operator $T:L^1(\mu )\to H$ into a Hilbert space is an absolutely summing operator, with ${\pi }_1(T)\le K_G\Vert T\Vert$. For simple functions on a common disjoint partition, apply the Grothendieck inequality to the vectors $T(1_A/\mu (A))$ and norming vectors of the images. The scalar sign or phase supremum is the weak 1-norm of the family. Approximation by simple functions proves the general result.

For three noncompact spatial coordinates of radius $r$ and a circle of circumference $L$, the positive Laplace Green function on the source slice is $G(r,0)={\sum }_{n\in \mathbb{Z}}[4{\pi }^2(r^2+n^2{L}^2){]}^{-1}=(4\pi Lr{)}^{-1}\coth (\pi r/L)$. At $r\ll L$ this is $1/(4{\pi }^2{r}^2)$; at $r\gg L$ it is $1/(4\pi Lr)$ with exponentially small corrections. The long-distance coefficient also follows by replacing the dense image sum with $L^{-1}{\int }_{-\infty }^{\infty }dy/(r^2+y^2)=\pi /(Lr)$. Electric and gravitational potentials inherit this crossover after multiplying by their signed source couplings.

For a mean-field gas of parallel infinite rods with total mass per unit length $M=Nm$ confined to radius $R_e$, put $q=GM\beta /2$, $\beta =m/(kT)$ and $F=1-q+q{R}^2/R_e^2$. The isothermal solution is $\psi =\psi (R_e)-2\log F/\beta$ with mass density $\rho =M(1-q)/(\pi R_e^2{F}^2)$ and enclosed line mass $M(<R)=M(R^2/R_e^2)/F$. Direct differentiation verifies the cylindrical Poisson equation for Newtonian gravity. Regularity requires $q<1$, or $T>T_{\min }$; equality is a singular core-concentration limit, not a regular equilibrium.

A laboratory gravity gradiometry experiment used two vertically separated clouds of chemical atoms in vacuum, launched into free fall. Shared laser beams applied three pulses ($\pi /2,\pi ,\pi /2$), splitting and recombining matter waves. Balanced photodiodes measured probe-light absorption to infer internal-state populations; scanning the last pulse's wave phase produced an interference fringe.^[ref]

The difference of fringe wave phases obeys $\delta \varphi ={\mathbf{k}}_{\mathrm{eff}}\cdot \Delta \mathbf{g}{\tau }^2$, with effective wave vector ${\mathbf{k}}_{\mathrm{eff}}$ and pulse spacing $\tau$: shared reference motion cancels, leaving differential gravitational acceleration. Eight lead bricks, each approximately $12.5\mathrm{kg}$, were placed symmetrically around the lower chamber, about $0.2\mathrm{m}$ from the fountain apex. The measured signal was $(8.1\pm 2.1)\times 10^{-9}g$, versus a calculated $8.2\times 10^{-9}g$, where $g$ denotes terrestrial gravitational acceleration.^[ref]

Controls deliberately shook the shared reference reflector and compared actively isolated and unisolated operation. Tilting tests exposed sensitivity to Coriolis acceleration and centrifugal acceleration. Thus shared translation is rejectable, while rotation and finite measurement uncertainty remain limitations. The brick signal detects spatially varying gravity; it does not establish a precision value of the gravitational constant.^[ref]

In uniformly rotating stellar hydrostatic equilibrium, composition-uniform thermodynamic quantities are functions of the total potential. Radiative conductivity gives $\mathbf{F}=f(\varphi )\nabla \varphi$, with $f=-\chi dT/d\varphi$. On one surface, $f$ is constant, so the outward flux is $F_n=fg$. The Stefan–Boltzmann law therefore gives $T_e\propto g^{1/4}$. This is a surface-flux law: the material temperature constant on an interior equipotential is not the latitude-dependent emergent effective temperature.

The shallow water equations at a dry bed, with negligible ambient mass density, end in a vanishing-depth edge, rather than a finite-depth gravity current head. In a prismatic triangular channel the Riemann invariant $u+4\sqrt{g^{\prime }h/2}$ gives the speed $2\sqrt{2g^{\prime }{h}_0}$ after release of a resting semi-infinite layer of depth $h_0$. In a rectangular channel the corresponding speed is $2\sqrt{g^{\prime }{h}_0}$; the channel geometry matters.

In an ideal Boussinesq gravity current advancing beneath a much deeper ambient, the local head Froude number is $u_f/\sqrt{g^{\prime }{h}_f}=\sqrt{2}$. The ambient must be displaced around the finite-depth head. This differs from the Saint-Venant dry-front condition, which neglects ambient inertia. The coefficient belongs to the ideal deep-ambient head model; confinement, mixing and drag can change it.

Let a bipartite graph have $n$ vertices in each class and at least $\rho n^2$ edges. There is a subset $B$ of one class with $|B|\ge \rho n/4$ such that every pair of its vertices has at least ${\rho }^5{n}^3/4096$ four-edge walks between them. Declare a pair of left vertices bad if it has fewer than $\tau n$ common neighbours, with $\tau ={\rho }^2/32$. For a random right vertex $y$, put $U=N(y)$ and let $b(U)$ count ordered bad pairs inside $U$. Then $\mathbb{E}|U|\ge \rho n$ and $\mathbb{E}b(U)\le \tau n^2$. Some $y$ has $|U|-16b(U)/(\rho n)\ge \rho n/2$, so $|U|\ge \rho n/2$ and $b(U)\le |U{|}^2/8$. Remove vertices with more than $|U|/4$ bad partners. At least half remain. For any two remaining vertices, at least $|U|/2$ middle vertices are good partners of both. Each supplies at least $(\tau n{)}^2$ walks, giving the bound.