For a straight rod $\mathbf{X}=s\mathbf{p}$, $-L<s<L$, leading slender-body theory uses $\mathbf{f}=C(I-\mathbf{pp}/2)[\Delta \mathbf{U}+s(\Delta \mathbf{\Omega }\times \mathbf{p}-E\mathbf{p})]$. Integration gives $\mathbf{F}=2CL(I-\mathbf{pp}/2)\Delta \mathbf{U}$ and $\mathbf{G}=(2C{L}^3/3)[(I-\mathbf{pp})\Delta \mathbf{\Omega }-\mathbf{p}\times E\mathbf{p}]$. Axial spin is outside this centreline approximation.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.

The standard low-energy $B=2$ Skyrmion in the ordinary Skyrme model has a toroidal baryon-density shape, with axial density symmetry $D_{\infty h}$. Its proper combined field symmetries have an $O(2)$-type stabilizer. The angular ansatz $R(z)=z^2$ in the rational map approximation for Skyrmions explains the axial relation $R(e^{i\theta }z)=e^{2i\theta }R(z)$: spatial rotation through $\theta$ is accompanied by isorotation through $2\theta$. The angular density vanishes at the two axial directions, yielding the toroidal hole. This describes the familiar minimal branch and a useful approximate field; it is not an assertion that every degree-two field is toroidal.

The familiar low-energy $B=3$ Skyrmion has tetrahedral baryon-density symmetry $T_d$. Proper rotations form the tetrahedral symmetry group $T\cong A_4$, acting on the field with compensating isorotations. An angular approximation is $R(z)=(i\sqrt{3}{z}^2-1)/(z^3-i\sqrt{3}z)$. Its Wronskian of a rational map is proportional to $z^4+2i\sqrt{3}{z}^2+1$. The four branch directions are the face-hole directions of a tetrahedral shell. A rational map and a variational radial profile approximate the actual field, and baryon-density symmetry must be distinguished from invariance of the pion field under an unaccompanied spatial rotation.

The familiar low-energy $B=4$ Skyrmion has a cubic baryon-density shell with six face-hole directions and full density symmetry $O_h$, the symmetry group of a cube. The proper subgroup is $O\cong S_4$, the rotational symmetry group of a cube; its field invariance includes compensating isorotations. A useful rational map approximation for Skyrmions has $R(z)=(z^4+2i\sqrt{3}{z}^2+1)/(z^4-2i\sqrt{3}{z}^2+1)$. The relation $R(iz)=1/R(z)$ pairs a spatial quarter-turn with a target half-turn. Its branch directions are $z=0,\infty ,\pm 1,\pm i$, corresponding to the six coordinate-axis directions. The map gives an approximate field, not an exact analytic energy-minimizing solution.

For a centered static Skyrmion $U_0$, the effective rotation-isorotation group is $G=SO(3{)}_{\mathrm{space}}\times SO(3{)}_{\mathrm{iso}}$. Its stabilizer subgroup $H$ consists of pairs obeying $A{U}_0(R^{-1}x)A^{†}=U_0(x)$. Thus distinct rigid orientations form the homogeneous space $G/H$, by the orbit-stabilizer theorem. The unit hedgehog has diagonal $SO(3)$ stabilizer and three orientation coordinates; the toroidal two-Skyrmion has a one-dimensional continuous stabilizer and five; the tetrahedral and cubic cases have discrete stabilizers and six. Adding three translations gives zero-mode orbit dimensions 6, 8, 9, 9. These are symmetry-generated low-energy coordinates, not a claim that non-Bogomolny multi-Skyrmion solutions possess an exact flat moduli space of arbitrary separations.

A charge-$A$ multi-Skyrmion provides an intrinsic field configuration for an atomic nucleus with mass number $A$. Its spatial and isospin collective coordinates must be quantized, with Finkelstein-Rubinstein constraints, before assigning physical nuclear states. A classical intrinsic shape is distinct from rotationally invariant observables in a spin-zero state.

The real vector space of traceless skew-Hermitian matrices of size two is the Lie algebra $\mathfrak{su}(2)$. For the basis $M_1=\frac{1}{2}\mathrm{diag}(i,-i)$, $M_2=\frac{1}{2}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, $M_3=\frac{1}{2}\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)$, cyclic commutators satisfy $[M_1,M_2]=M_3$. The map $a\mapsto \sum a_j{M}_j$ carries the cross product to the commutator. Every target is a single commutator: choose a unit vector perpendicular to its associated vector $c$ and use $u\times (c\times u)=c$.

The displayed singular value system applies when the Fourier multipliers $c_n$ are nonzero. The phase in the left singular vector is necessary for complex kernels. If a multiplier vanishes, its mode instead belongs to the corresponding null spaces and is excluded from the positive singular system.

Adiabatic elimination replaces a rapidly relaxing variable by its instantaneous stable quasistatic value. If $\dot{R}=-\Gamma (t)R+P(t)$ with $\Gamma >0$, write $R=R_0+\delta R$, $R_0=P/\Gamma$. Then $\dot{\delta R}+\Gamma \delta R=-{\dot{R}}_0$. After the transient, $\delta R\simeq -{\dot{R}}_0/\Gamma$ when coefficients vary slowly compared with ${\Gamma }^{-1}$. This gives both the leading reduction and its time-scale error; rapid relaxation alone does not justify an additional Taylor expansion in another parameter.

A finite spherical subvolume has radial-pressure virial surface term $4\pi R^3\rho (R){\sigma }_r^2$. The infinite singular isothermal sphere has divergent total mass and energies, so a finite isolated virial theorem cannot omit this term. Different orbital distributions with the same density can have different radial boundary pressures.

For the $r^{-2}$ singular isothermal sphere, the isotropic Spherical Jeans equation gives ${\sigma }_r^2={\sigma }_{\theta }^2={\sigma }_{\varphi }^2=v_0^2/2$, so $⟨v^2⟩=3v_0^2/2$. Randomly oriented circular orbits instead have ${\sigma }_r^2=0$, each transverse second moment $v_0^2/2$, and total mean square speed $v_0^2$.

A singular $n$-cochain with coefficients in an abelian group $A$ is a homomorphism from the singular chain group $C_n(X;\mathbb{Z})$ to $A$. It is specified by its values on all singular simplices, without a finite-support restriction. Its coboundary evaluates on the alternating sum of the faces. The resulting cochain complex computes singular cohomology.

For an absolutely continuous valuation distribution with density $f(v)>0$, its virtual valuation is $v-(1-F(v))/f(v)$. The virtual-surplus revenue identity converts expected incentive-compatible payments into expected allocations weighted by virtual valuations. A regular prior makes this quantity nondecreasing.

The static kink ${\varphi }_K(x)=4\arctan e^{x-a}$ joins adjacent scalar-field vacua $0$ and $2\pi$. It obeys ${\varphi }_K^{\prime }=2\mathrm{sech}(x-a)$ and has topological charge $Q=[\varphi (+\infty )-\varphi (-\infty )]/(2\pi )=1$. In the Sine-Gordon theory normalization with physical mass scale $m$ and coupling $\beta$, its classical mass is $8m/{\beta }^2$. A Lorentz boost gives ${\varphi }_K=4\arctan \exp [(x-vt-a)/\sqrt{1-v^2}]$. Spatial reflection gives an antikink.

For real parameters with ${\kappa }_i^2-b_i^2=1$, set $E_i=e^{{\kappa }_{i}x-b_{i}t+{\gamma }_i}$ and $a_{ij}=[({\kappa }_i-{\kappa }_j{)}^2-(b_i-b_j{)}^2]/[({\kappa }_i+{\kappa }_j{)}^2-(b_i+b_j{)}^2]$. Sum ${\prod }_i{E}_i^{{\mu }_i}{\prod }_{i<j}{a}_{ij}^{{\mu }_i{\mu }_j}$ over binary vectors of even parity for $f$ and odd parity for $g$. The Sine-Gordon equation solution is the continuous field $4\arg (f+ig)$. In the all-kink sector, ${\kappa }_i=\cosh {\theta }_i$, $b_i=\sinh {\theta }_i$ and $a_{ij}=-{\tanh }^2[({\theta }_i-{\theta }_j)/2]$. Treating these coefficients as positive would change the solution. Distinct rapidities give separated incoming and outgoing solitons.

With canonical field $\phi =\varphi /\beta$, the quartic interaction has coupling $-m^2{\beta }^2$. Cancelling the vacuum tadpole diagram requires the divergent mass counterterm $\delta m^2=(m^2{\beta }^2/4){\int }_{-\Lambda }^{\Lambda }dk/[2\pi \sqrt{k^2+1}]$. Its kink energy is $\Delta M_{\mathrm{ct}}=4\delta m^2/(m{\beta }^2)$ because $\int (1-\cos {\varphi }_K)dx=4$. This cancels the logarithmic ultraviolet divergence in the one-loop soliton mass correction. Finite parts depend on the renormalization condition.

For the angular field $u=\beta \varphi$ and coordinates $\xi =m(x+t)/2$, $\eta =m(x-t)/2$, let $w=(v+u)/2$ and $d=(v-u)/2$. The displayed relations imply $u_{\xi \eta }=\sin u$ and $v_{\xi \eta }=\sin v$. They define a Bäcklund transformation with the reciprocal parameter convention used in the Bianchi permutability for sine-Gordon Bäcklund transformations. Physical-field trigonometric arguments include the coupling $\beta$.

At renormalized coupling $\gamma =8\pi /n$, the kink-antikink transmission poles lie at $\vartheta =i\pi (1-k/n)$. The relativistic bound-state mass from a rapidity pole gives the displayed increasing breather masses. The kink mass is $M=mn/\pi$, and $k=n$ is an excluded threshold state. No breathers occur at $n=1$; scattering involving a physical second breather requires $n\ge 3$.

For a prize of value $w$, let $G(b)$ be the probability that a rival is absent or enters with effort at most $b$. When rivals act independently and have no atoms at positive efforts, effort $b>0$ wins with probability $G(b{)}^{n-1}$. If a player mixes over an interval of best responses, its payoff on that interval is constant, so $G(b)=((b+u)/w{)}^{1/(n-1)}$. If the absence probability is $q$ and the active effort support starts at zero, then $u=w{q}^{n-1}$.