In orientation-preserving annular coordinates $(\theta ,r)$ with orientation $d\theta \wedge dr$, the twist is $(\theta ,r)\mapsto (\theta +2\pi f(r),r)$, where $f$ increases from zero to one. The map is the identity outside this annulus and near its boundary. The orientation of the ambient topological surface, rather than the orientation assigned to the curve, fixes the handedness.

If $\gamma \cdot u$ uses the ordered-tangent orientation convention for algebraic intersection number of curves on an oriented surface, a right-handed Dehn twist takes $u$ to $u+([\gamma ]\cdot u)[\gamma ]$ on first homology. In particular a separating-curve twist acts trivially there.

A degree of a map between oriented manifolds equal to one makes the pullback on degree-one cohomology injective by the Poincare duality pairing. Thus a degree-one map of closed oriented surfaces requires $2h\le 2g$. Conversely, collapse the extra handles in ${\Sigma }_h\#{\Sigma }_{g-h}$ onto a point. The retained oriented disc shows that this map has degree one.

For a smooth map between oriented unit spheres, integrate the pullback of the target area form. Localizing a normalized top-degree form near a regular value, and using Stokes theorem to discard exact-form differences, identifies this integral with the degree as a sum of local degrees.

For a smooth map $f:M\to N$ between oriented connected closed manifolds of the same dimension, a normalized volume form on the target gives $\mathrm{deg}f={\int }_M{f}^{\ast }\omega$. The formula agrees with the degree as a sum of local degrees. A change of normalized top-form adds an exact form by top-dimensional de Rham cohomology, whose pullback has zero integral by Stokes theorem.

For $n\ge 1$, topological degree classifies continuous maps $S^n\to S^n$ up to homotopy. It is not a complete invariant for maps of general manifolds. On the torus, the identity and the integer shear $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ both have degree one, but induce distinct homomorphisms on $H_1(T^2;\mathbb{Z})$ and cannot be homotopic. The positive-dimensional sphere hypothesis matters: the two constant maps of the sphere $S^0$ have the same degree but are not homotopic in its discrete target.

For $u_{tt}-\Delta u+u^3=0$ in three spatial dimensions, conserved positive wave energy $E$ controls $\Vert \nabla u{\Vert }_2$. The Sobolev inequality gives $\Vert u^2\nabla u{\Vert }_2\lesssim \Vert \nabla u{\Vert }_2^2\Vert D^{2}u{\Vert }_2\lesssim E\Vert D^{2}u{\Vert }_2$. Differentiated wave energy estimates and the Gronwall inequality then give $\Vert u(t){\Vert }_{{\dot{H}}^2}+\Vert u_t(t){\Vert }_{{\dot{H}}^1}\le CD{e}^{CEt}$, where $D$ bounds the corresponding initial Sobolev norms.

A bounded trajectory in the damped Binet equation with nonzero angular momentum asymptotically becomes circular. Energy loss is nonnegative in magnitude and vanishes only for zero radial velocity. The centrifugal effective potential energy excludes collision; bounded radius forces indefinitely increasing forward angular distance, so the reciprocal-radius oscillation decays. Escaping trajectories are excluded by the boundedness hypothesis.

For a regular cylindrical flux tube with $\mathbf{B}=(0,B_{\varphi }(s),B_z(s))$, vanishing field beyond radius $s_0$, and exterior pressure $p_e$, define $⟨h⟩=2s_0^{-2}{\int }_0^{s_0}sh(s)ds$ and $B_p^2=2{\mu }_0{p}_e$. Integrating cylindrical magnetostatic pressure balance against $s^2$ gives the displayed identity. The toroidal magnetic pressure and hoop magnetic tension cancel in this cross-sectional axial-field average. Nonnegative gas pressure gives $⟨B_z^2⟩\le B_p^2$, with strict inequality precisely when $⟨p⟩>0$.

In a finite group, a cyclic subgroup of order $m$ has exactly $\phi (m)$ generators, where $\phi$ is the Euler totient function. Each element generates exactly one cyclic subgroup, so dividing the number of elements of order $m$ by $\phi (m)$ counts these subgroups. Include the identity subgroup using $m=1$ and $\phi (1)=1$. In a symmetric group, element counts come from cycle type and order of a group element is the least common multiple of cycle lengths.

For a weight-$k$ cusp form $f(\tau )={\sum }_{n\ge 1}{a}_n{e}^{2\pi in\tau }$ of level $N$, its L-function is the Dirichlet seriesinitially in a right half-plane of absolute convergence. With the determinant-normalized slash operator and $W_N=\left(\begin{array}{cc}0& -1\\ N& 0\end{array}\right)$, put $Jf=i^{k}f{|}_k{W}_N$. For example, on the space of cusp forms for ${\Gamma }_1(N)$ this is again a cusp form, and the level-normalized completion isExponential decay at the modular cusps zero and infinity makes this Mellin transform entire. Modular inversion gives the phase-normalized Fricke functional equation ${\Lambda }_N(f,s)={\Lambda }_N(Jf,k-s)$. Only when $f$ is an eigenvector of $J$ does this become a scalar functional equation relating the completion of $f$ to itself. At level one the completion reduces to $(2\pi {)}^{-s}\Gamma (s)L(f,s)$. A normalized Hecke eigenform also has an appropriate Euler product; at general level its bad-prime factors and character must be specified.

Three external fields can be joined to the three cubic vertex fields in six connected Wick contractions. There are three choices for the external field at the undifferentiated slot and two exchanges of the differentiated slots. Three cyclic terms therefore require an explicit factor of two. Ordered contractions with an earlier vertex use complex-conjugated mode functions.

The undifferentiated soft-leg permutation gives $B(q,k,k)\simeq (ϵ/2)P(q)P(k)$ as $q/k\to 0$. It retains a squeezed $q^{-3}$ enhancement, but corresponds to local amplitude $f_{\mathrm{NL}}^{\mathrm{effective}}=5ϵ/24$. Soft momentum placed only at derivative slots would miss this dominant term.

The comoving curvature perturbation is dimensionless. With spatial coordinates and time carrying inverse mass dimensions, the cubic interaction $\int dt{d}^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ needs a mass-squared coefficient. Its standard gravitational normalization is $M_{\mathrm{Pl}}^2$. Six De Sitter curvature mode functions supply $M_{\mathrm{Pl}}^{-6}$; this vertex then supplies $M_{\mathrm{Pl}}^2$, leaving the physical $M_{\mathrm{Pl}}^{-4}$ curvature primordial bispectrum. The Wick-pairing multiplicity for a zeta zeta-prime-squared vertex is independently six: three choices of undifferentiated external leg times two differentiated-leg assignments.

For a proper continuous event time and invertible cumulative hazard function $H$, the survivor function identity $F(t)=e^{-H(t)}$ gives $H(T)\sim \mathrm{Exp}(1)$.

For $H_{\mathrm{int}}=-\int d^{3}x{a}^3ϵ\mathcal{F}{\dot{\zeta }}^3/H$, $a=-1/(H\tau )$ and stipulated modes $u_k=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵk^3}$, six connected Wick contractions and ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{iK\tau }d\tau =2i/K^3$ yield $B=3H^4\mathcal{F}/(8{ϵ}^2{k}_1{k}_2{k}_3{K}^3)$. Here $⟨{\zeta }^3{⟩}_c=(2\pi {)}^3\delta (\sum {\mathbf{k}}_i)B$. Using unstarred vertex modes with the external-before-vertex prefactor incorrectly reverses the sign. Non-unit sound speed also changes the quadratic modes, so this result is conditional on the specified unit-speed modes.

For a fixed linear smoother, generalized cross-validation replaces the individual leverage corrections of Leave-one-out cross-validation by their average. It selects smoothing parameters by minimizing the displayed criterion. It approximates a predictive error criterion; it is not a universal exact estimate of future error and requires care when observation errors are dependent.

This cross-ratio convention is the value of $z_3$ under the Möbius transformation sending $z_1,z_2,z_4$ to $1,0,\infty$. It is one minus the convention $(z_1-z_3)(z_2-z_4)/[(z_1-z_2)(z_3-z_4)]$. Exchanging the first two points replaces $r$ by $1-r$. For four points in cyclic order on a circle, this convention gives $r<0$ and $1-r>1$.

Every Möbius transformation preserves the cross-ratio. The identity $M(u)-M(v)=(ad-bc)(u-v)/[(cu+d)(cv+d)]$ makes all extra factors cancel in the ratio; limiting conventions handle points at infinity.

For four vertices in cyclic order on a circle, the product of the diagonal lengths is the sum of the products of opposite side lengths. One proof uses the cross-ratio with second point mapped to zero: its negative sign gives $|r|+1=|1-r|$, which becomes the displayed length relation.