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For a bound noncollision Kepler orbit with gravitational parameter $\mu$, the canonical actions per unit mass are $\mathcal{L}=\sqrt{\mu a}$, $\mathcal{G}=\mathcal{L}\sqrt{1-e^2}$ and $\mathcal{H}=\mathcal{G}\cos i$. Their angles are mean anomaly $\ell$, argument of periapsis $\omega$ and longitude of ascending node $\Omega$. The Hamiltonian is $-{\mu }^2/(2{\mathcal{L}}^2)$, so only $\ell$ advances in the unperturbed problem. In the prograde spherical-action chart, $J_r=\mathcal{L}-\mathcal{G}$, $J_{\theta }=\mathcal{G}-\mathcal{H}$, $J_{\varphi }=\mathcal{H}$; the type-two canonical transformation is generated by $F_2=(\mathcal{L}-\mathcal{G})w_r+(\mathcal{G}-\mathcal{H})w_{\theta }+\mathcal{H}{w}_{\varphi }$. Retrograde spherical actions have $J_{\theta }=\mathcal{G}-|\mathcal{H}|$, so the absolute value cannot be dropped globally. Circular and coplanar limits are singular angle charts, not singular orbits.

A point source moving at angular speed $\Omega$ on a circle of radius $a$ has observer-direction Mach number $m=\Omega a\sin \iota /c_0$. In the far field its retarded phase satisfies $\Theta +\theta -\pi /2=\Omega (t-r/c_0)+m\sin \Theta$. For $m<1$ there is one retarded root, and a constant source strength gives ${\rho }^{\prime }=q\dot{\Theta }/(4\pi c_0^2\Omega r)$. For $m>1$, multiple retarded roots can occur; their contributions require $|1-m\cos \Theta |$ and a sum over roots. Coalescing roots produce a caustic in the ideal point-source model.

For a small body of size $l$ in flow of speed $u$, force of order ${\rho }_0{u}^2{l}^2$ varying on time $l/u$ produces an acoustic dipole density amplitude of the displayed order, where $M=u/c_0$ is the Mach number. Its sound power scales as ${\rho }_0{u}^6{l}^2/c_0^3$. The corresponding compact acoustic quadrupole Mach-number scaling has density amplitude of order $(l/r)M^4$ and power ${\rho }_0{u}^8{l}^2/c_0^5$.

For a closed operator $L$ on a Banach space $H$, a positive-time classical solution belongs to $C([0,T],H)\cap C^1((0,T],H)$, takes values in $D(L)$ for positive time, and has $Lu\in C((0,T],H)$. It satisfies the differential equation and the initial condition in $H$. Classical regularity also at time zero requires the corresponding regularity of the initial datum, in particular $u_0\in D(L)$. For an analytic semigroup and Hölder continuous forcing, the variation-of-constants formula gives such a solution: subtracting the forcing at the current time turns the singular estimate $\Vert LS(r)\Vert \le C/r$ into the integrable bound $C{r}^{\theta -1}$.

Absorption distance is a dimensionless path variable chosen so a non-evolving comoving absorber population with fixed proper cross-section has constant $d\mathcal{N}/dX$. It absorbs the geometrical factor in absorber incidence and comoving number density. A distribution per $dNdX$ and one per $dNdz$ therefore differ by $dX/dz$.

Let the locations $a_n$ be distinct and bounded, and let $c_n>0$ with ${\sum }_n{c}_n<\infty$. The Weierstrass M-test gives locally uniform convergence of $G$, so the uniform limit theorem makes it continuous. The reverse triangle inequality bounds each term's difference quotient by $c_n$. Consequently a uniformly small tail permits passing each one-sided difference-quotient limit through the series. At every point outside the locations, $G^{\prime }(x)={\sum }_n{c}_n\mathrm{sgn}(x-a_n)$; at $a_m$, the right derivative minus the left derivative is $2c_m>0$. Thus the function has precisely the prescribed cusp locations, even when they accumulate at another point where the function remains differentiable.

Suppose a real function $g$ is differentiable at $a$, with $g(a)=0$ and $g^{\prime }(a)\ne 0$. The derivative definition gives $g(a+h)=g^{\prime }(a)h+o(|h|)$. The reverse triangle inequality then impliesThus $|g|$ has unequal one-sided derivatives, $|g^{\prime }(a)|$ on the right and $-|g^{\prime }(a)|$ on the left. More generally, multiplication by a function $w$ continuous at $a$ with $w(a)\ne 0$ gives right and left slopes $w(a)|g^{\prime }(a)|$ and $-w(a)|g^{\prime }(a)|$. This elementary expansion explains how a smooth oscillation can acquire infinitely many corners under the absolute value function, even after multiplication by a factor that makes it differentiable at an accumulation point.

This loss assigns different positive costs to underestimation and overestimation of a scalar parameter. Here $\gamma$ penalizes estimates below the parameter and $\delta$ estimates above it. Equal costs give a multiple of absolute-error loss.

Let $D_8$ denote the dihedral group of group order eight acting on a square. Here the subscript specifies group order; the same group is called $D_4$ in the convention that counts vertices of a polygon, as in the group action example. Let $r$ be a counterclockwise quarter-turn rotation and $s$ a reflection in a diagonal. ThenProducts act rightmost first. The group commutator of the generators of a group isIt is a half-turn rotation, so forcing $r$ and $s$ to commute must identify a half-turn with $e$.

More concretely, write each element uniquely as $r^i{s}^j$, with $0\le i<4$ and $j\in \{0,1\}$. The functionis a surjective group homomorphism. Indeed, $srs=r^{-1}$ giveswhere the exponents are reduced modulo four and two. Modulo two, $-k$ and $k$ agree, so $\varphi$ adds the two coordinates separately. Its group kernel is $N=\{e,r^2\}$. Since the target is an abelian group, $D_8^{\prime }\subseteq N$; since $[r,s]=r^2$, also $N\subseteq D_8^{\prime }$. ThusThe four cosets are $\{e,r^2\}$, $\{r,r^3\}$, $\{s,r^{2}s\}$, and $\{rs,r^{3}s\}$. Each coset forgets the difference of a half-turn. The quotient group still distinguishes rotations from reflections, and even from odd numbers of quarter-turns; the two surviving choices are independent. This is the Klein four-group, rather than the cyclic group $C_4$, because each nonidentity element squares to the identity element. For the general dihedral group calculation, see Conrad, Dihedral groups I, theorem A.1; that source uses the vertex-count convention and $xy{x}^{-1}{y}^{-1}$ for individual group commutators.

If a finitely generated group $S$ has exactly one subgroup of each finite index, every such subgroup is normal. The quotients of prime order give surjections of its finitely generated abelianization onto $C_p$ for all primes, forcing positive free rank. A free rank of at least two gives two distinct index-$p$ subgroups. Positive finite torsion together with the free rank-one factor also gives two index-$p$ subgroups for a suitable prime. Hence $S_{\mathrm{ab}}\cong \mathbb{Z}$.

// Target: algebra.bigb

For every abelian group $A$ and group homomorphism $f:G\to A$, there is a unique group homomorphism $\bar{f}:G^{\mathrm{ab}}\to A$ such thatThis is a universal property. To see why it holds, compute $f([x,y])=[f(x),f(y)]=e_A$, using commutativity in $A$. Hence $G^{\prime }\subseteq \ker f$, and $f$ is constant on each coset of $G^{\prime }$. The displayed formula is therefore well-defined and preserves multiplication. It is unique because $q$ is surjective. This is an instance of factoring a group homomorphism through a quotient group; see Conrad, Universal mapping properties, theorem 3.4.

In particular, every abelian quotient $G/N$ comes from a surjective group homomorphism $G^{\mathrm{ab}}\to G/N$, $g{G}^{\prime }\mapsto gN$. For $S_3$, any group homomorphism to an abelian group is determined by the common image $a$ of its three transpositions, with $a^2=e_A$. Every even permutation must map to $e_A$. Thus sign of a permutation captures all possible information of this kind, including maps with infinite abelian groups as their targets.

A free massive $p$-form obeys the displayed equation, with the divergence constraint following by applying the codifferential. For $m>0$ and $0\le p\le D-1$, a rest frame leaves an antisymmetric tensor on $D-1$ spatial directions, giving $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$ physical polarizations. The mass term removes the ordinary gauge invariance, unless a Stueckelberg mechanism restores a redundant description.

// Target: quantum-field-theory.bigb

An Abelian Higgs vortex has quantized magnetic flux and a Higgs field whose phase winds at spatial infinity. Its vortex number is $N=(2\pi {)}^{-1}\int F$ under the convention $D=d-iA$.

This planar Abelian Higgs model replaces the usual magnetic energy by $B^2/h$. Use $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$. The magnetic term is singular at a zero of the Higgs field, so the admissible class must have finite energy; formal field equations are first derived where $h>0$. Its positive-flux Bogomolny equations are $(D_1+i{D}_2)\Phi =0$ and $B=h(1-h)/2$. On those solutions $B/h=(1-h)/2$ extends regularly across zeros.

In two dimensions, scale ${\Phi }_{\lambda }(x)=\Phi (\lambda x)$ and $A_{j,\lambda }(x)=\lambda A_j(\lambda x)$. The magnetic field has scaling weight two, the gauge covariant derivative of the Higgs field has weight one, and the potential has weight zero. Their integrated energies therefore have the powers shown. Stationarity gives $E_B=E_P$, a Derrick virial identity. This balance allows a planar Abelian Higgs vortex to have finite size even though a scalar-gradient-plus-nonnegative-potential model has no comparable magnetic term.

For $|D\Phi {|}^2-\lambda (|\Phi {|}^2-{\eta }^2{)}^2/4$, write $\Phi =(\eta +h/\sqrt{2})e^{i\theta }$ and absorb $\partial \theta /e$ into the vector field. The quadratic Lagrangian has a real scalar of mass $m_h=\sqrt{\lambda }\eta$ and a vector of mass $m_A=\sqrt{2}e\eta$. The two scalar plus two massless-vector polarizations in the symmetric phase become one radial scalar plus three massive-vector polarizations. The removed phase supplies the longitudinal mode; local vortices retain global winding and quantized flux.

// Target: cosmology.bigb

An abelian group of order $p^2$ is cyclic if it has an element of order $p^2$. Otherwise Lagrange's theorem makes every nonidentity element have order $p$. Choose $a\ne 1$ and $b\notin ⟨a⟩$. The group homomorphism $(i,j)\mapsto a^i{b}^j$ from $C_p\times C_p$ is injective: $b^j\in ⟨a⟩$ with $j\ne 0$ would imply $b\in ⟨a⟩$, using the inverse of $j$ modulo $p$. Equal finite orders then make this an isomorphism.

The nonzero complex numbers form an abelian group under multiplication, with identity one and inverse $z^{-1}$. As a topological group it is isomorphic to $\mathbb{R}\times S^1$ by logarithmic modulus and phase. It appears naturally as a Möbius pointwise stabilizer of two points.

A degree-zero complex analytic divisor $E$ on a compact Riemann surface is principal exactly when its Abel-Jacobi map of a compact Riemann surface is zero. Normalize a differential of the third kind with residues equal to the coefficients of $E$. The Riemann bilinear relations for a compact surface express its $b$-periods as $2\pi i$ times the Abel integrals. If those integrals are in the period lattice, subtract a suitable integral multiple of the normalized holomorphic differential forms; all periods then lie in $2\pi i\mathbb{Z}$, so exponentiation gives the required meromorphic function. Conversely, the logarithmic differential of a meromorphic function has integer residues and periods in $2\pi i\mathbb{Z}$, giving a zero Abel class by the same calculation.

For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.